GtOPHYSICS VOL 53 NO 51MAY 1918) 169170618 FIGS

Transient electromagnetic responses ot high-contrast prisms in a layered earth

Gregory A Newman and Gerald W Hohmann]

ABSTRA(

An integral-equation solution for transient electroshymagnetic (TEM) scattering by prisms in layered halfshy

spaces is formulated to provide meaningful results when the prisms are in highly resistive layers A prism is reshyplaced with an unk nown scattering current which is approximated with pulse and divergence-free basis funcshytions in the frequency domain Divergence-free basis functions model eddy currents that exist in confined bodies in a very resistive host and hence simulate the inductive responses of the prisms A Galerkin solution

for the scattering current is obtained where the domishynant charge operator is eliminated from part of the solution by integrating the tensor Greens function around rectangular paths After the scattering current is determined the electric and magnetic fields scattered by the prisms are calculated and the corresponding TEM responses arc obtained by inverse Fourier transformashytion The resulting solution provides meaningful results over a wide range of resistivities in layered hosts includshy

ing the case of free space The masking effect of a conductive overburden delays

and suppresses the three-dimensional TEM response of

a conductor The overburden response must be removed

for the conductors response to be fully interpretable An interpretation of the conductor with free-space models is a poor approximation when the basement rock is conductive Instead of an exponential decay at late times the conductors response decays in an inverse power relationship When the basement resistivity is inshycreased the conductor exhibits an exponential decay at late times For a thin dike the time constant estimated from this decay is identical to that for a thin plate in free space However the response of the dike buried beneath the overburden is larger than the response of the dike in free space This increase in the response of the dike will bias modeling it in free space with thin plates

We have used the solution to gain insight regarding the lateral resolution of two vertical conductors for the fixed-loop and central-loop survey configurations The results suggest that resolution of multiple conductors is very poor in a fixed-loop survey but in a central-loop survey the resolution is much better provided the data are- interpreted at early times At later times multiple conductors may not be resolvable and interpretational ambiguities could arise

INTRODUCT10N

High-quality transient electromagnetic (TEM) data arc colshylected with fast accurate and reliable instrumentation but interpretation techniques are still limited In recent years sigshynificant progress has been made in interpretation of data with the development of three-dimensional (3-0) forward-modeling capabilities 3-D modeling has provided new insight into the behavior of TEM fields such that new and old interpretation techniques are now being investigated and verified (cf Newman et al 1987 Eaton and Hohmann 1987)

Existing 3-D solutions include those for a thin-plate conshyductor in free space (Annan 1974) and an asymptotic solution for a sphere in a layered host (Lee 1981) SanFilipo and Hohshymann (1985) published a numerical solution for a prism in a conductive half-space Their solution uses the method of inteshygral equations and is formulated directly in the time domain San Filipo and Hohmanns solution is also valid when the half-space is very resistive including the case of free space Frequency-domain integral-equation solutions have also had very good success in 3-D TEM modeling Transformation from the frequency domain to the time domain is done by

Manuscript received by the Editor April 20 987 revised manuscript received September 9 1987 Formerly Department of Geology and Geophysics University of Utah presently Department of Geophysics and Meteorology University of Cologne Albertus-Magnus-Platz 5000Cologne 41 West Germany Dcpartmtnt of Geology and Geophysics University of Utah Salt Lake City UT amp4112-1183 ( 1988 Society of Exploration Geophysicists All rights reserved

691

692 Newman and Hohmann

inverse Fourier transformation Two important solutions of this type are the Wcidelt (1981) thin-plate solution and the Newman et pound11 (198b) solution for 3-D bodies in a layered half-space Weidelts solution allows for an overburden layer

and is valid when the resistivity of the basal half-space is very large The Newman et al solution will allow for arbitrary 3-D structures but is only valid for moderate resistivity contrasts between the bodies and their host layers

The intent of this paper is to extend the capabilities of 3-D forward modeling We first develop a 3-D frequency-domain solution for multiple prisms in a layered host OUf solution is an improvement over the Newman et al (1986) solution beshycause it will produce good results when prisms are in highly resistive layers 3-D transient responses will then be obtained by inverse Fourier transformation Next we use the solution to study the effect of conductive overburden on 3-D TEM reshysponses for the fixed-loop configuration We conclude the paper with a limited study of the TEM response of multiple

conductive prisms In this investigation we discuss the derecshyrability and resolution of two conductors based on the design

of field surveys

THEORY

Integral-equation formulation

Consider the 3-D earth in Figure I Several 3-D conducshytivity inhomogeneities are imbedded in an n-layered earth and the conductivity in the earth is represented by the function CT(r) The space and frequency dependence of EM fields in a 3-D earth obeys Maxwells equations

v x E(r) = itl)lo Hlr) - M p (r] (1)

and

v x Hr) = a(r)E(rl + Jp(r) (2)

The source vectors in equations (1) and (2) Jp(r) and Mp(r)

are current densities for the impressed electric and magnetic sources such as large loops grounded wires magnetic dipoles

or electric dipoles Because the earth is a conductor (conductivitygt 10- 5 Srrn) EM fields in the earth are strongly attenuated Any significant depth of penetration requires freshyquencies below to 000 Hz therefore displacement currents are ignored in equation (2) We also set ~r) - 1-40 (4n x 10- 7

Hrrn] in equation (1) because the effects of magnetic pershymeability changes are usually very small compared with those of conductivity changes otr)

The corresponding layered half-space fields which would apply in the absence of any 3-D conductivity inhomogeneities satisfy the equations

v x E(r) = -iooloHp(J) - M(r) (3)

and

V)( Hp(r) = Clp(r)Ep(r) + Jp(r) (4)

where o p(r) is the conductivity of a layered earth The solution for the layered-earth fields in equations (3) and (4) is usually in the form of integrals that are evaluated numerically

The equations for the fields scattered by 3-D bodies are obtained by subtracting the layered half-space fields in equashy

lions Ol and (4) from the total fields in equations (1) and (2) Thus we have

v x Iltslr) - -i(l)~Lo H(r) (5)

and

v x H (r) = (Jp In IT) + J (f) (6)

where in equation (61 an equivalent source has replaced all 3-D inhomogeneities in the earth This equivalent source radishyates in the presence of the layered half-space and is defined by

J[f) = lotr) - Op(rl] E(r) (7)

which is nonzero only In the 3-D inhomogeneities For the moment consider all 3-D inhomogeneities in

Figure I to be confined to layer j The electric-field Helmholtz equation derived from equations (5) and (6) is

V2E(T) + kJ EJr) = -l-i(l)JloJ(r) + lcr j V[V J(r)]l (8)

where 1) is the layered half-space wavenumber in layer j and is

defined by -i(J)~() cr 2 The solution to equation (8) is given

by the integral equation

E (r) = 1 GJ-middot(r r)J(r) di (9)

where lt(r r) is the electric-field tensor Greens function which relates the electric field at r in layer 1 to a current

element at r in layer j including the case where J = j In

pj Jp M do 77

CTI d

cs2

G ~

CTt G0

CT-1 a

ltTn

1-1G 1 3-D bodies in a layered host J p

and M p are impressed electric and magnetic current densities of the sources

693 3-D Model ing of TEM Sceltering

general when layer boundaries intersect the 3-D bodies or when the bodies are in different layers the subscript j refers to any layer containing the bodies

An integral equation for the scattered magnetic field is derived from equation 19) with the use of equation (5) its solution is expressed a

H(rl = 1 (1I(r r)J (r) ell (10)

where Glltr r) is magnetic-field tensor Greens function The specification of the electric and magnetic tensor Greens funcshytions is given in Wannamaker et al (19g4)

NUMlRICAL SOLCTION

Freqsency-dornain solution

Electric and magnetic fields can be computed anywhere in the earth from equations (9) and (lO) if the scattering current J is known An integral equation for the scattering current can be obtained by adding the layered half-space electric field in layerj to both sides of equation (9)

J(r-)1a(r-)-=-E (T)-+ f ~Er r)J(r)dvp crn

where Aa(rl = orr) - (Jj Equation (II) is a Fredholm integral equation of the second kind Van Blade] (1961) has shown that equation (III is also valid within any inhomogeneity beshycause the integral equation has a principal value Equation (II) is solved numerically and a matrix solution for the scatshytering current is constructed using the method of moments (Harrington 19(8) The scattering current can be approxishymated with subdomain or full-domain basis functions where a suitable weighting function IS defined However if the basis function is not carefully selected the numerical solution will fail as the host conductivity Jj becomes small Simply using a

higher order basis function will not solve the problem (cf Lajoie and West 1976 Petrick 1984 Hanneson and West 1984 SanFilipo and Hohmann 1985 and Newman et at ]986)

In order to explain and avoid these numerical difficulties let us write equation (Ill in operator notation where the tensor Greens function is replaced by the scalar whole-space Greens function The same numerical problem exists in equation (11) for a tensor or scalar Greens function and switching to a scalar Greens function simplifies the following analysis Thus

) + JLllJS(r) + L1J(r (r 1cr(r ) = Ep(r) (12)s

with the current and charge operators defined as

i = iroflo J de G(r r) (13)

and

L bull = -10) V i dv G(r rIV (14)

and where G(r r) is a scalar Greens function which for a whole space is given by

C(r f) = 114ft exp (-- ikj Ir - r 1) r - r Imiddot (15)

The 1 operator is associated with the current source J~(r)

while L is associated with electric charge since it involves V J~(r) To show that V bull J(r is related to charge apply the divergence operator to both sides of equation (6)

v J(rl = -OJ V Es(r) (16)

where C1 p (r ) = u j bull Then from Gausss law we have

v E(r) = OlE (17)

where ~ is the charge density and e is the dielectric constant or the medium which is assumed to he constant and usually equal to the dielectric constant of free space 136rr x 10- 9

Frm The charges occur at the boundaries of the inhomogeshyneities because the normal component of E is discontinuous there

For 3-D inhomogeneities in a layered half-space additional terms must be added to equations (13) and (14) to account for image currents and charges in the other layers Equation (11)

is really the symbolic representation of equations (12) (t 3) and (14) plus additional layered half-space terms The numerishy

cal difficulty associated with solving equation (12) for J is dues

to the disparity in the sizes of the current and charge opershyators Note the different magnitudes of the leading terms in equations (13) and (14)

(lO)(OJllo)~ (IOou)(J j ~ (10)5

for a frequency of 1000 Hz and a layer conductivity of 0001 Srrn this disparity becomes larger for smaller background conductivities Our comparison of the operators does not inshydude the effect of the gradient and divergence operators in equation (14) and therefore is not rigorous However in numerical-modeling experiments Lajoie and West (1976) conshysidered the gradient and divergence operators in equation (14)

and demonstrated the disparity in the two operators when the host medium becomes resistive

The scattering current in equation (12) can be written as a sum or curl-free and divergence-free parts where J = J~ + Js

and where V bull J~ = 0 and V x J~ = O Note that J~ is in the

null space of L because LJ~ = O Since L is the dominant operator in equation (12) a poorly constructed numerical

solution will give no information about the component of J that is divergence-free This is the component which domishynates the EM response of a body in a resistive earth

The disparity in the sizes of the two operators can be reshymoved by solving for curl-free and divergence-free scattering

currents Lajoie and West (1976) first obtained a solution valid in the limit of free space by solving for these two types of current in a thin 3-D plate SanFilipo and Hohmann (1985) recently published a solution for a prism based on a similar approach Their direct time-domain solution incorporates a specialized subset of basis functions that represent divergenceshyfree scattering current The solution produces good results for large conducti vity contrasts but the model is limited to a single prism in a half-space The frequency-domain solution we present here is based on SanFilipo and Hohmanns solushytion and is valid for multiple prisms in a layered half-space

Because they are the simplest geometries to model we start by assuming that the inhomogeneities in Figure 1 are prisms with constant conductivities The prisms are divided into N cells and ~1 closed current tubes as shown in Figure 2 The tubes making up a prism are concentric with respect to the

694 Newman and Hohmann

prism center and are constructed out of the cells making up that prism (Figure 2) The cells that make up a prism can also be rectangular which is an efficient shape because the scattershying current in an elongate prism varies more rapidly over its short dimension than over its long dimension

The scattering current within the prisms is approximated with a linear combination of pulse basis functions and divergence-free basis functions where

N M

J (r) ~ r J i Pi (r) + I c V (r) (18) ~ I i=l

The pulse and the divergence-free basis functions are defined by

I r in the ith cell (19)P (r) = O otherwise

and

Uj(r) = Qi(f)Ui(f) r in the ith tube (20)O otherwise

where Qj(r) is the unit vector in the direction of current flow in the tube according to the right-hand rule and Q j is the (variable) cross-sectional area of the tube The pulse basis function in equation (19) requires a component of the current density to be constant within a cell while the divergence-free basis function in equation (20) requires a component of the

current flowing within the tube to be constant

rUy U z

Uj ltd

(0) (b)

We note that the solution of Newman et al (1986) uses only pulse basis functions for tracking the scattering current While the pulse basis functions produce good results when the conshyductivity contrast between the bodies and their host layers is low they rail badly at higher contrasts The solution of Newman et al (1986) is reliable for contrasts below 200 to l where galvanic effects are important in determining the EM response of a 3-D body Galvanic responses are caused by charges on the surface of a body (Kaufman ]985) current is channeled into a conductive body or deflected away from a resistive body When the host is a perfect insulator no current elm be gathered into it body and its EM response is caused only by induction Inductive current flow is characterized by currents flowing in closed loops inside a body These eddy currents are caused by a magnetic field changing over time Thus the addition or the divergence-free (current-tube) basis function in equation (18) is essential for a valid numerical solution in the case of a high conductivity contrast

Because the current tubes of any prism are concentric with its center as shown in Figure 2 equation (18) docs not allow for completely general eddy-current patterns Although more general eddy-current patterns are possible they are compushyrationally prohibitive because the solution would then demand excessive amounts of computer time and memory Thus equation (18) will not be a good approximation to the scattering current for all types of source-prism geometries at high contrast For example the eddy-current patterns induced in a large horizontal plate are not always expected to be conshy

(c)

FIG 2 The prism is divided into 32 rectangular cells and 12 tubes The tubes are outlined by the heavy lines and the cells by the light lines The vector Uj(r) is the direction the current flows in the ith tube following the dotted path The magnitude of the current in the tube is Ct (a) shows four tubes normal to Ux _ (b) and (c) show four tubes normal to u

yand u respectively

695 3-0 Modeling of TEM Scattering

centric with the center of the plate A magnetic dipole placed air the axis of the plate will induce current vortices that are not concentric

The numerical solution for J is obtained by substituting equation (I~) into equation (12) to obtain

N

E(T) = L J j Pi (l)tOj(r) + (ttl + LI)J Pdr) i~ 1

Af

+ L Ci u (r)1Oj (r) I- Le Lj(r) (21) i ~ 1

where Au (r) is the difference net ween the conductivity of the prism occupied by the ith cell or the ith tube and the conducshytivity of its layered host The LJ i Pi(r) term involves the divergence of J and requires careful evaluation because J Pi(r) is discontinuous from cell to cell This term is evalushyatcd using an integro-diffcrence technique described by Hohshy

mann 11983) A Galerkin solution using weighting functions that are the same as the basis functions yields the following coupled equations for 311 + M

Munk nowns JN I and C 1

(Pk bull Ep ) = [ laquor J P)1Cfi+ (Pk LaJi Pi) + crLJ Piraquo

i gt 1

u

+ L (Pk bull cU i)1Oi + (Pk rcUraquo) i~ I

k = l 2 N (22)

and

v (Uk [1)= L(UkmiddotJiP)AOi+ltUkLJjP)1

i gt I

H

+ L(U k ClJ)Licr i + (Uk LnCU) i 1

k = 1 2 AI (23

where the inner product of the two functionsj a is defined by

ltI flgt = f(rgr) dr (24)JIf both f and g are vectors a dot multiplication is assumed in equation (24)

Because U is divergence-free the dominant L operator does not appear in the second sums of equations (21) (22) and (23) Furthermore the L operator does not appear in the first sum of equation (23) because the inner product of U and L J i Pi is zero by Stokes theorem we are integrating the gradient of a scalar around a closed path The coupled equashytions (22) and (23) result in a stable solution for the scattering current for a host of any conductivity

A concise (3N + M) x (3N + M) matrix equation is written from equations (22) and (23) for the unknown coefficients J j

and C i where

[ I n ~PlJ [J] = [ltP Ep ) ] (25) Ivp tSuu c (V Ellgt

The (3N x 3N) Kpp submatrix represents the coupling coefshyficients between the pulse functions The Kpu and the fSuP

submatrices represent the coupling coefficients between the tube and pulse functions Their sizes are (3N x M) and (M x 31) respectively Finally the (M x M) KlJll submatrix represents the coupling coefficients between the tube functions The impedance matrix in equation (25) is full and the compushytation time required to build and factor the matrix can be substantial Tripp and Hohmann (1984) have shown that a significant reduction in matrix formulation and factoring time can be achieved for a single prism with two vertical planes of symmetry by a similarity transformation The impedance matrix is block diagonalized into four (3N x 4) 4(3N x M)4 matrices and the source vector is transformed into four new source vectors However in general for multiple prisms this similarity transformation cannot be used and equation (251 must be solved directly

Once the coefficients J and C are determined the scattershying current is given by equation (18) The electric and magshynetic fields outside the prisms are then obtained by addition of the layered half-space fields with the discrete versions of equashytions (9) and (0)

N

E(r) = E(r) + L rE(r r n ) bull J(rn ) (26) n ~ 1

and

Ii

Htr) = Hp(r) + L rH(r r) J(r) (27) =1

with the tensor Greens functions for a prism of current in cell n given by

rE(r rJ = f (E(r r) do (28) Ill

and

[H(r r) = r (i(r r) dv (29)

J Time-domain solution

Transformation of frequency-domain responses to the time domain is accomplished by inverse Fourier transformation The electric-field and magnetic-field responses in the time domain for a step current are computed using the cosine transforms

e(t) = -21n 1 1m [E(W)Jco cos wt dtraquo (30)

and

h(t) = rot dsraquo (31)-2rr LX 1m [H((l))]m cos

and for the time derivative of the magnetic field by using the sine transform

iih(l)ilt = 2re fX 1m [ H(W)] sin rot am (32)

696 Newman and Hohmann

where 1m [H(m)] and Im [E(I))] are the imaginary parts of the magnetic and electric fields in the frequency domain The inteshygrations in equations (30)-(32) are carried out using the disshycrete convolution technique described by Newman et al (19lS6) At late times the time derivative of the magnetic field is not calculated directly from equation (32) because of numerishycal noise Rather we numerically differentiate the magneticshyfield transient since equation 131) has a smaller dynamic range from early to late times and is more accurate than equation (32) The magnetic-field transient is replaced with a cubic spline and the spline is then differentiated to obtain (~h(t)ilt

However evaluating th(t)tt in this way is still inaccurate at

the latest times because of numerical noise in the magnetic field after it has decayed six orders in magnitude from early to late times Equations (30) (31) and (32) usually require 20 to 40 3-D lrequency-domain evaluations at five to eight points per decade for an accurate solution We note that the calculashytion of 3-D transient responses requires substantial amounts of computer time because each frequency-domain evaluation involves solving equations (25) and (26) for the electric field

and equations (25) and (27) for the magnetic field

Checks Oil numerical solution

Essential to the development of any numerical solution is its verilication Our solution can be checked for internal consist shyency by verifying reciprocity by comparing decay rates with

the late-time decay rates of thin plates and spheres in free space and by carrying out convergence tests based on the level of cell discretization However the best check is to comshypare the solution with other numerical solutions andor scaleshymodel results Because we arc interested in computing 3-D TEM responses we present checks in the time domain only

I t is necessary to design models with a minimum number of cells because computation time increases rapidly with more cells Therefore we have carefully discretized all the models in this paper with as few cells as possible based on convergence studies For more information on model discretization refer to Newman et a1 (1986)

figures 3 and 4 show that the current-tube basis functions in equation (I8) are required to obtain the correct late-time decay rate of a thin plate in free space We model a thin plate by letting the prism in Figure 2 be one cell wide and thereby restrict the eddy currents to flow normal to one coordinate direction Our solution using both current-tube and pulse functions (solid and dashed lines in Figures 3 and 4) is comshypared with a solution using only pulse functions (crosses) for a 40 Sm prism (cr) 20 m thick (r) in a 10 000 nmiddot m half-space Both solutions are also compared with a solution using only current tubes (triangles) The top of the prism in Figure 3 is at a depth of 100 m (DR) and its depth extent (DE) and strike length (Sl) are gOand gOO m respectively The prisms vertical magnetic-field response [or a step shutoff in the transmitter is plotted at the surface for 21 receiver positions where the

time X_--)(---X---IC-~-X-_-X lt Complete solution07msx_x-x-x-x-x-x-x -L-x

x l Solution with pulse- Functions only x x

_x_x-x-x-x-x- _IC--x---lC---lC---x-- - __ x

bull05 liS lC x I x---)(10-6 ~ _~A~ lC ___~li_A-A-A~ raquo (-pound-_f-~~=~-A

_A 6~ ~~ l ~A-f-A Solution with tube 1ms --t-A- ~ i x x )( 6 functions only1 A x X x x x ~ l x )( IC ol)( x

Ii X _A-A-A _Igt---6---A~~6__A x A_A lt

-~ )( A -A_-II shy

A-A- ~ 10-7 1- 10ms A II ~~

-- r _A __-A__ ~_A_A-~AJIIIl A- lt6 11_

lJ -A_bull-V- I ~b ~10ms AA A_-A

MODEl

10-8 I shy

surFac umiddot40Sm

10000n-1ll Otmiddot 20m

Slmiddot 800m Near edge of loop DEmiddot80m

D8middot100m

I I I I l I I L I I I I I I I I I I I

-200 -160 -120 -80 -40 0 40 80 120 160 200

DISTANCE FROM CONDUCTOR CENTER (m)

FIG 3 Comparison of the vertical component of the scattered magnetic field calculated with different basis functions The model is a vertical thin prism Solid and dashed curves show the solution based on pulse and tube functions Crosses show the solution using only the pulse functions at 001 005 and 01 ms and triangles show the solution using only the tube functions at 01 1 10 and 20 IDS

697

1

3-D Modeling of TEM Scattering

- i bull ~ bull toSotio bull luhCbullbullpl f tlbullbull only unc10

~ __~___ -

_SollIon 1I1t pul

~ 10 [ bullbull

10- ~ ~ EUltotd tlbullbull oltatod tiM co~ant T 9 ItftIQ~ 08)( Ae

10 I-shy

I I

transmitter is a 160 x 160 m loop with its near wire located at - 250 m Results for six time channels are shown in Figure 3

but results are not plotted before 01 ms for the current-tube solution or after 01 ms for the pulse function solution

In the two early-time channels 001 and 003 rns the comshyplete solution compares well with the solution using only pulse functions This early time range is where the galvanic respolHe is important ~ diffusing currents in the half-space are concentrated around and channeled through the prism Howshyever beyond 003 ms the solution using only pulse functions decays faster than the complete solution (Figure 4) The lateshytime decay of the pulse solution is an exponential with a time constant of OO~ ms The empirical formula for the time conshystant of a thin plate given in Lamontagne (1975) is

r = (110 otL) 10 (33)

where L is the smaller dimension of the plate and crt is its conductivity-thickness product The time constant calculated from equation (33) for the prism in Figure 3 is 8 ms

The complete solution agrees with the current-tube solution at later times (Figure 3) and its late-time decay rate is exshy

01 10 100 ponential from Figure 4 we estimate a time constant of 8 ms

tim) This time constant agrees with that calculated from equation (33) and thus the TEM response of the prism decays as that

nO 4 Scattered vertical-field decay of the three solutions at for a thin plate in free space at later times Note that the x = -80 m in Figure 3 solution using only the current tubes appears to be constant at

VERTICAL HORIZONTAL100

Son Fllrpo end Hohmonrls solutlon( A)

A nnoll egflnmodfl solution (l( )

10 0335

MODEL

lOOOm

~oo- free space

llltegrol equation solution (-)

00 0 0 0- 0 - 0 - 0 0

0 bull amp132ms l amp po oOAJo-oo ~ ~ tE bull p

gt 33ms I

0 shyc iI ~ r _~o-ltl--o-

I gXcglshy~

0 -s

bull I

f 6 7718S II llt X

I l( lCbull pKJImiddotoIlo -ilrO-oo

01 lC I ~ 1 r I

o 50 100 150 200 250 o 50 100 150 200 250

DISTANCE FROM PLATE (II) DISTANCE fROM PLATE (m)

FIG 5 Comparison of the integral-equation solution (solid and dashed curves) for a plate in free space with the eigenmode solution of Annan (1974) (crosses) and the integral-equation solution of SanFilipo and Hohmann (triangles) Vertical and horizontal components of ob(t)ot are compared

692 Newman and Hohmann

inverse Fourier transformation Two important solutions of this type are the Wcidelt (1981) thin-plate solution and the Newman et pound11 (198b) solution for 3-D bodies in a layered half-space Weidelts solution allows for an overburden layer

and is valid when the resistivity of the basal half-space is very large The Newman et al solution will allow for arbitrary 3-D structures but is only valid for moderate resistivity contrasts between the bodies and their host layers

The intent of this paper is to extend the capabilities of 3-D forward modeling We first develop a 3-D frequency-domain solution for multiple prisms in a layered host OUf solution is an improvement over the Newman et al (1986) solution beshycause it will produce good results when prisms are in highly resistive layers 3-D transient responses will then be obtained by inverse Fourier transformation Next we use the solution to study the effect of conductive overburden on 3-D TEM reshysponses for the fixed-loop configuration We conclude the paper with a limited study of the TEM response of multiple

conductive prisms In this investigation we discuss the derecshyrability and resolution of two conductors based on the design

of field surveys

THEORY

Integral-equation formulation

Consider the 3-D earth in Figure I Several 3-D conducshytivity inhomogeneities are imbedded in an n-layered earth and the conductivity in the earth is represented by the function CT(r) The space and frequency dependence of EM fields in a 3-D earth obeys Maxwells equations

v x E(r) = itl)lo Hlr) - M p (r] (1)

and

v x Hr) = a(r)E(rl + Jp(r) (2)

The source vectors in equations (1) and (2) Jp(r) and Mp(r)

are current densities for the impressed electric and magnetic sources such as large loops grounded wires magnetic dipoles

or electric dipoles Because the earth is a conductor (conductivitygt 10- 5 Srrn) EM fields in the earth are strongly attenuated Any significant depth of penetration requires freshyquencies below to 000 Hz therefore displacement currents are ignored in equation (2) We also set ~r) - 1-40 (4n x 10- 7

Hrrn] in equation (1) because the effects of magnetic pershymeability changes are usually very small compared with those of conductivity changes otr)

The corresponding layered half-space fields which would apply in the absence of any 3-D conductivity inhomogeneities satisfy the equations

v x E(r) = -iooloHp(J) - M(r) (3)

and

V)( Hp(r) = Clp(r)Ep(r) + Jp(r) (4)

where o p(r) is the conductivity of a layered earth The solution for the layered-earth fields in equations (3) and (4) is usually in the form of integrals that are evaluated numerically

The equations for the fields scattered by 3-D bodies are obtained by subtracting the layered half-space fields in equashy

lions Ol and (4) from the total fields in equations (1) and (2) Thus we have

v x Iltslr) - -i(l)~Lo H(r) (5)

and

v x H (r) = (Jp In IT) + J (f) (6)

where in equation (61 an equivalent source has replaced all 3-D inhomogeneities in the earth This equivalent source radishyates in the presence of the layered half-space and is defined by

J[f) = lotr) - Op(rl] E(r) (7)

which is nonzero only In the 3-D inhomogeneities For the moment consider all 3-D inhomogeneities in

Figure I to be confined to layer j The electric-field Helmholtz equation derived from equations (5) and (6) is

V2E(T) + kJ EJr) = -l-i(l)JloJ(r) + lcr j V[V J(r)]l (8)

where 1) is the layered half-space wavenumber in layer j and is

defined by -i(J)~() cr 2 The solution to equation (8) is given

by the integral equation

E (r) = 1 GJ-middot(r r)J(r) di (9)

where lt(r r) is the electric-field tensor Greens function which relates the electric field at r in layer 1 to a current

element at r in layer j including the case where J = j In

pj Jp M do 77

CTI d

cs2

G ~

CTt G0

CT-1 a

ltTn

1-1G 1 3-D bodies in a layered host J p

and M p are impressed electric and magnetic current densities of the sources

693 3-D Model ing of TEM Sceltering

general when layer boundaries intersect the 3-D bodies or when the bodies are in different layers the subscript j refers to any layer containing the bodies

An integral equation for the scattered magnetic field is derived from equation 19) with the use of equation (5) its solution is expressed a

H(rl = 1 (1I(r r)J (r) ell (10)

where Glltr r) is magnetic-field tensor Greens function The specification of the electric and magnetic tensor Greens funcshytions is given in Wannamaker et al (19g4)

NUMlRICAL SOLCTION

Freqsency-dornain solution

Electric and magnetic fields can be computed anywhere in the earth from equations (9) and (lO) if the scattering current J is known An integral equation for the scattering current can be obtained by adding the layered half-space electric field in layerj to both sides of equation (9)

J(r-)1a(r-)-=-E (T)-+ f ~Er r)J(r)dvp crn

where Aa(rl = orr) - (Jj Equation (II) is a Fredholm integral equation of the second kind Van Blade] (1961) has shown that equation (III is also valid within any inhomogeneity beshycause the integral equation has a principal value Equation (II) is solved numerically and a matrix solution for the scatshytering current is constructed using the method of moments (Harrington 19(8) The scattering current can be approxishymated with subdomain or full-domain basis functions where a suitable weighting function IS defined However if the basis function is not carefully selected the numerical solution will fail as the host conductivity Jj becomes small Simply using a

higher order basis function will not solve the problem (cf Lajoie and West 1976 Petrick 1984 Hanneson and West 1984 SanFilipo and Hohmann 1985 and Newman et at ]986)

In order to explain and avoid these numerical difficulties let us write equation (Ill in operator notation where the tensor Greens function is replaced by the scalar whole-space Greens function The same numerical problem exists in equation (11) for a tensor or scalar Greens function and switching to a scalar Greens function simplifies the following analysis Thus

) + JLllJS(r) + L1J(r (r 1cr(r ) = Ep(r) (12)s

with the current and charge operators defined as

i = iroflo J de G(r r) (13)

and

L bull = -10) V i dv G(r rIV (14)

and where G(r r) is a scalar Greens function which for a whole space is given by

C(r f) = 114ft exp (-- ikj Ir - r 1) r - r Imiddot (15)

The 1 operator is associated with the current source J~(r)

while L is associated with electric charge since it involves V J~(r) To show that V bull J(r is related to charge apply the divergence operator to both sides of equation (6)

v J(rl = -OJ V Es(r) (16)

where C1 p (r ) = u j bull Then from Gausss law we have

v E(r) = OlE (17)

where ~ is the charge density and e is the dielectric constant or the medium which is assumed to he constant and usually equal to the dielectric constant of free space 136rr x 10- 9

Frm The charges occur at the boundaries of the inhomogeshyneities because the normal component of E is discontinuous there

For 3-D inhomogeneities in a layered half-space additional terms must be added to equations (13) and (14) to account for image currents and charges in the other layers Equation (11)

is really the symbolic representation of equations (12) (t 3) and (14) plus additional layered half-space terms The numerishy

cal difficulty associated with solving equation (12) for J is dues

to the disparity in the sizes of the current and charge opershyators Note the different magnitudes of the leading terms in equations (13) and (14)

(lO)(OJllo)~ (IOou)(J j ~ (10)5

for a frequency of 1000 Hz and a layer conductivity of 0001 Srrn this disparity becomes larger for smaller background conductivities Our comparison of the operators does not inshydude the effect of the gradient and divergence operators in equation (14) and therefore is not rigorous However in numerical-modeling experiments Lajoie and West (1976) conshysidered the gradient and divergence operators in equation (14)

and demonstrated the disparity in the two operators when the host medium becomes resistive

The scattering current in equation (12) can be written as a sum or curl-free and divergence-free parts where J = J~ + Js

and where V bull J~ = 0 and V x J~ = O Note that J~ is in the

null space of L because LJ~ = O Since L is the dominant operator in equation (12) a poorly constructed numerical

solution will give no information about the component of J that is divergence-free This is the component which domishynates the EM response of a body in a resistive earth

The disparity in the sizes of the two operators can be reshymoved by solving for curl-free and divergence-free scattering

currents Lajoie and West (1976) first obtained a solution valid in the limit of free space by solving for these two types of current in a thin 3-D plate SanFilipo and Hohmann (1985) recently published a solution for a prism based on a similar approach Their direct time-domain solution incorporates a specialized subset of basis functions that represent divergenceshyfree scattering current The solution produces good results for large conducti vity contrasts but the model is limited to a single prism in a half-space The frequency-domain solution we present here is based on SanFilipo and Hohmanns solushytion and is valid for multiple prisms in a layered half-space

Because they are the simplest geometries to model we start by assuming that the inhomogeneities in Figure 1 are prisms with constant conductivities The prisms are divided into N cells and ~1 closed current tubes as shown in Figure 2 The tubes making up a prism are concentric with respect to the

694 Newman and Hohmann

prism center and are constructed out of the cells making up that prism (Figure 2) The cells that make up a prism can also be rectangular which is an efficient shape because the scattershying current in an elongate prism varies more rapidly over its short dimension than over its long dimension

The scattering current within the prisms is approximated with a linear combination of pulse basis functions and divergence-free basis functions where

N M

J (r) ~ r J i Pi (r) + I c V (r) (18) ~ I i=l

The pulse and the divergence-free basis functions are defined by

I r in the ith cell (19)P (r) = O otherwise

and

Uj(r) = Qi(f)Ui(f) r in the ith tube (20)O otherwise

where Qj(r) is the unit vector in the direction of current flow in the tube according to the right-hand rule and Q j is the (variable) cross-sectional area of the tube The pulse basis function in equation (19) requires a component of the current density to be constant within a cell while the divergence-free basis function in equation (20) requires a component of the

current flowing within the tube to be constant

rUy U z

Uj ltd

(0) (b)

We note that the solution of Newman et al (1986) uses only pulse basis functions for tracking the scattering current While the pulse basis functions produce good results when the conshyductivity contrast between the bodies and their host layers is low they rail badly at higher contrasts The solution of Newman et al (1986) is reliable for contrasts below 200 to l where galvanic effects are important in determining the EM response of a 3-D body Galvanic responses are caused by charges on the surface of a body (Kaufman ]985) current is channeled into a conductive body or deflected away from a resistive body When the host is a perfect insulator no current elm be gathered into it body and its EM response is caused only by induction Inductive current flow is characterized by currents flowing in closed loops inside a body These eddy currents are caused by a magnetic field changing over time Thus the addition or the divergence-free (current-tube) basis function in equation (18) is essential for a valid numerical solution in the case of a high conductivity contrast

Because the current tubes of any prism are concentric with its center as shown in Figure 2 equation (18) docs not allow for completely general eddy-current patterns Although more general eddy-current patterns are possible they are compushyrationally prohibitive because the solution would then demand excessive amounts of computer time and memory Thus equation (18) will not be a good approximation to the scattering current for all types of source-prism geometries at high contrast For example the eddy-current patterns induced in a large horizontal plate are not always expected to be conshy

(c)

FIG 2 The prism is divided into 32 rectangular cells and 12 tubes The tubes are outlined by the heavy lines and the cells by the light lines The vector Uj(r) is the direction the current flows in the ith tube following the dotted path The magnitude of the current in the tube is Ct (a) shows four tubes normal to Ux _ (b) and (c) show four tubes normal to u

yand u respectively

695 3-0 Modeling of TEM Scattering

centric with the center of the plate A magnetic dipole placed air the axis of the plate will induce current vortices that are not concentric

The numerical solution for J is obtained by substituting equation (I~) into equation (12) to obtain

N

E(T) = L J j Pi (l)tOj(r) + (ttl + LI)J Pdr) i~ 1

Af

+ L Ci u (r)1Oj (r) I- Le Lj(r) (21) i ~ 1

where Au (r) is the difference net ween the conductivity of the prism occupied by the ith cell or the ith tube and the conducshytivity of its layered host The LJ i Pi(r) term involves the divergence of J and requires careful evaluation because J Pi(r) is discontinuous from cell to cell This term is evalushyatcd using an integro-diffcrence technique described by Hohshy

mann 11983) A Galerkin solution using weighting functions that are the same as the basis functions yields the following coupled equations for 311 + M

Munk nowns JN I and C 1

(Pk bull Ep ) = [ laquor J P)1Cfi+ (Pk LaJi Pi) + crLJ Piraquo

i gt 1

u

+ L (Pk bull cU i)1Oi + (Pk rcUraquo) i~ I

k = l 2 N (22)

and

v (Uk [1)= L(UkmiddotJiP)AOi+ltUkLJjP)1

i gt I

H

+ L(U k ClJ)Licr i + (Uk LnCU) i 1

k = 1 2 AI (23

where the inner product of the two functionsj a is defined by

ltI flgt = f(rgr) dr (24)JIf both f and g are vectors a dot multiplication is assumed in equation (24)

Because U is divergence-free the dominant L operator does not appear in the second sums of equations (21) (22) and (23) Furthermore the L operator does not appear in the first sum of equation (23) because the inner product of U and L J i Pi is zero by Stokes theorem we are integrating the gradient of a scalar around a closed path The coupled equashytions (22) and (23) result in a stable solution for the scattering current for a host of any conductivity

A concise (3N + M) x (3N + M) matrix equation is written from equations (22) and (23) for the unknown coefficients J j

and C i where

[ I n ~PlJ [J] = [ltP Ep ) ] (25) Ivp tSuu c (V Ellgt

The (3N x 3N) Kpp submatrix represents the coupling coefshyficients between the pulse functions The Kpu and the fSuP

submatrices represent the coupling coefficients between the tube and pulse functions Their sizes are (3N x M) and (M x 31) respectively Finally the (M x M) KlJll submatrix represents the coupling coefficients between the tube functions The impedance matrix in equation (25) is full and the compushytation time required to build and factor the matrix can be substantial Tripp and Hohmann (1984) have shown that a significant reduction in matrix formulation and factoring time can be achieved for a single prism with two vertical planes of symmetry by a similarity transformation The impedance matrix is block diagonalized into four (3N x 4) 4(3N x M)4 matrices and the source vector is transformed into four new source vectors However in general for multiple prisms this similarity transformation cannot be used and equation (251 must be solved directly

Once the coefficients J and C are determined the scattershying current is given by equation (18) The electric and magshynetic fields outside the prisms are then obtained by addition of the layered half-space fields with the discrete versions of equashytions (9) and (0)

N

E(r) = E(r) + L rE(r r n ) bull J(rn ) (26) n ~ 1

and

Ii

Htr) = Hp(r) + L rH(r r) J(r) (27) =1

with the tensor Greens functions for a prism of current in cell n given by

rE(r rJ = f (E(r r) do (28) Ill

and

[H(r r) = r (i(r r) dv (29)

J Time-domain solution

Transformation of frequency-domain responses to the time domain is accomplished by inverse Fourier transformation The electric-field and magnetic-field responses in the time domain for a step current are computed using the cosine transforms

e(t) = -21n 1 1m [E(W)Jco cos wt dtraquo (30)

and

h(t) = rot dsraquo (31)-2rr LX 1m [H((l))]m cos

and for the time derivative of the magnetic field by using the sine transform

iih(l)ilt = 2re fX 1m [ H(W)] sin rot am (32)

696 Newman and Hohmann

where 1m [H(m)] and Im [E(I))] are the imaginary parts of the magnetic and electric fields in the frequency domain The inteshygrations in equations (30)-(32) are carried out using the disshycrete convolution technique described by Newman et al (19lS6) At late times the time derivative of the magnetic field is not calculated directly from equation (32) because of numerishycal noise Rather we numerically differentiate the magneticshyfield transient since equation 131) has a smaller dynamic range from early to late times and is more accurate than equation (32) The magnetic-field transient is replaced with a cubic spline and the spline is then differentiated to obtain (~h(t)ilt

However evaluating th(t)tt in this way is still inaccurate at

the latest times because of numerical noise in the magnetic field after it has decayed six orders in magnitude from early to late times Equations (30) (31) and (32) usually require 20 to 40 3-D lrequency-domain evaluations at five to eight points per decade for an accurate solution We note that the calculashytion of 3-D transient responses requires substantial amounts of computer time because each frequency-domain evaluation involves solving equations (25) and (26) for the electric field

and equations (25) and (27) for the magnetic field

Checks Oil numerical solution

Essential to the development of any numerical solution is its verilication Our solution can be checked for internal consist shyency by verifying reciprocity by comparing decay rates with

the late-time decay rates of thin plates and spheres in free space and by carrying out convergence tests based on the level of cell discretization However the best check is to comshypare the solution with other numerical solutions andor scaleshymodel results Because we arc interested in computing 3-D TEM responses we present checks in the time domain only

I t is necessary to design models with a minimum number of cells because computation time increases rapidly with more cells Therefore we have carefully discretized all the models in this paper with as few cells as possible based on convergence studies For more information on model discretization refer to Newman et a1 (1986)

figures 3 and 4 show that the current-tube basis functions in equation (I8) are required to obtain the correct late-time decay rate of a thin plate in free space We model a thin plate by letting the prism in Figure 2 be one cell wide and thereby restrict the eddy currents to flow normal to one coordinate direction Our solution using both current-tube and pulse functions (solid and dashed lines in Figures 3 and 4) is comshypared with a solution using only pulse functions (crosses) for a 40 Sm prism (cr) 20 m thick (r) in a 10 000 nmiddot m half-space Both solutions are also compared with a solution using only current tubes (triangles) The top of the prism in Figure 3 is at a depth of 100 m (DR) and its depth extent (DE) and strike length (Sl) are gOand gOO m respectively The prisms vertical magnetic-field response [or a step shutoff in the transmitter is plotted at the surface for 21 receiver positions where the

time X_--)(---X---IC-~-X-_-X lt Complete solution07msx_x-x-x-x-x-x-x -L-x

x l Solution with pulse- Functions only x x

_x_x-x-x-x-x- _IC--x---lC---lC---x-- - __ x

bull05 liS lC x I x---)(10-6 ~ _~A~ lC ___~li_A-A-A~ raquo (-pound-_f-~~=~-A

_A 6~ ~~ l ~A-f-A Solution with tube 1ms --t-A- ~ i x x )( 6 functions only1 A x X x x x ~ l x )( IC ol)( x

Ii X _A-A-A _Igt---6---A~~6__A x A_A lt

-~ )( A -A_-II shy

A-A- ~ 10-7 1- 10ms A II ~~

-- r _A __-A__ ~_A_A-~AJIIIl A- lt6 11_

lJ -A_bull-V- I ~b ~10ms AA A_-A

MODEl

10-8 I shy

surFac umiddot40Sm

10000n-1ll Otmiddot 20m

Slmiddot 800m Near edge of loop DEmiddot80m

D8middot100m

I I I I l I I L I I I I I I I I I I I

-200 -160 -120 -80 -40 0 40 80 120 160 200

DISTANCE FROM CONDUCTOR CENTER (m)

FIG 3 Comparison of the vertical component of the scattered magnetic field calculated with different basis functions The model is a vertical thin prism Solid and dashed curves show the solution based on pulse and tube functions Crosses show the solution using only the pulse functions at 001 005 and 01 ms and triangles show the solution using only the tube functions at 01 1 10 and 20 IDS

697

1

3-D Modeling of TEM Scattering

- i bull ~ bull toSotio bull luhCbullbullpl f tlbullbull only unc10

~ __~___ -

_SollIon 1I1t pul

~ 10 [ bullbull

10- ~ ~ EUltotd tlbullbull oltatod tiM co~ant T 9 ItftIQ~ 08)( Ae

10 I-shy

I I

transmitter is a 160 x 160 m loop with its near wire located at - 250 m Results for six time channels are shown in Figure 3

but results are not plotted before 01 ms for the current-tube solution or after 01 ms for the pulse function solution

In the two early-time channels 001 and 003 rns the comshyplete solution compares well with the solution using only pulse functions This early time range is where the galvanic respolHe is important ~ diffusing currents in the half-space are concentrated around and channeled through the prism Howshyever beyond 003 ms the solution using only pulse functions decays faster than the complete solution (Figure 4) The lateshytime decay of the pulse solution is an exponential with a time constant of OO~ ms The empirical formula for the time conshystant of a thin plate given in Lamontagne (1975) is

r = (110 otL) 10 (33)

where L is the smaller dimension of the plate and crt is its conductivity-thickness product The time constant calculated from equation (33) for the prism in Figure 3 is 8 ms

The complete solution agrees with the current-tube solution at later times (Figure 3) and its late-time decay rate is exshy

01 10 100 ponential from Figure 4 we estimate a time constant of 8 ms

tim) This time constant agrees with that calculated from equation (33) and thus the TEM response of the prism decays as that

nO 4 Scattered vertical-field decay of the three solutions at for a thin plate in free space at later times Note that the x = -80 m in Figure 3 solution using only the current tubes appears to be constant at

VERTICAL HORIZONTAL100

Son Fllrpo end Hohmonrls solutlon( A)

A nnoll egflnmodfl solution (l( )

10 0335

MODEL

lOOOm

~oo- free space

llltegrol equation solution (-)

00 0 0 0- 0 - 0 - 0 0

0 bull amp132ms l amp po oOAJo-oo ~ ~ tE bull p

gt 33ms I

0 shyc iI ~ r _~o-ltl--o-

I gXcglshy~

0 -s

bull I

f 6 7718S II llt X

I l( lCbull pKJImiddotoIlo -ilrO-oo

01 lC I ~ 1 r I

o 50 100 150 200 250 o 50 100 150 200 250

DISTANCE FROM PLATE (II) DISTANCE fROM PLATE (m)

FIG 5 Comparison of the integral-equation solution (solid and dashed curves) for a plate in free space with the eigenmode solution of Annan (1974) (crosses) and the integral-equation solution of SanFilipo and Hohmann (triangles) Vertical and horizontal components of ob(t)ot are compared

693 3-D Model ing of TEM Sceltering

general when layer boundaries intersect the 3-D bodies or when the bodies are in different layers the subscript j refers to any layer containing the bodies

An integral equation for the scattered magnetic field is derived from equation 19) with the use of equation (5) its solution is expressed a

H(rl = 1 (1I(r r)J (r) ell (10)

where Glltr r) is magnetic-field tensor Greens function The specification of the electric and magnetic tensor Greens funcshytions is given in Wannamaker et al (19g4)

NUMlRICAL SOLCTION

Freqsency-dornain solution

Electric and magnetic fields can be computed anywhere in the earth from equations (9) and (lO) if the scattering current J is known An integral equation for the scattering current can be obtained by adding the layered half-space electric field in layerj to both sides of equation (9)

J(r-)1a(r-)-=-E (T)-+ f ~Er r)J(r)dvp crn

where Aa(rl = orr) - (Jj Equation (II) is a Fredholm integral equation of the second kind Van Blade] (1961) has shown that equation (III is also valid within any inhomogeneity beshycause the integral equation has a principal value Equation (II) is solved numerically and a matrix solution for the scatshytering current is constructed using the method of moments (Harrington 19(8) The scattering current can be approxishymated with subdomain or full-domain basis functions where a suitable weighting function IS defined However if the basis function is not carefully selected the numerical solution will fail as the host conductivity Jj becomes small Simply using a

higher order basis function will not solve the problem (cf Lajoie and West 1976 Petrick 1984 Hanneson and West 1984 SanFilipo and Hohmann 1985 and Newman et at ]986)

In order to explain and avoid these numerical difficulties let us write equation (Ill in operator notation where the tensor Greens function is replaced by the scalar whole-space Greens function The same numerical problem exists in equation (11) for a tensor or scalar Greens function and switching to a scalar Greens function simplifies the following analysis Thus

) + JLllJS(r) + L1J(r (r 1cr(r ) = Ep(r) (12)s

with the current and charge operators defined as

i = iroflo J de G(r r) (13)

and

L bull = -10) V i dv G(r rIV (14)

and where G(r r) is a scalar Greens function which for a whole space is given by

C(r f) = 114ft exp (-- ikj Ir - r 1) r - r Imiddot (15)

The 1 operator is associated with the current source J~(r)

while L is associated with electric charge since it involves V J~(r) To show that V bull J(r is related to charge apply the divergence operator to both sides of equation (6)

v J(rl = -OJ V Es(r) (16)

where C1 p (r ) = u j bull Then from Gausss law we have

v E(r) = OlE (17)

where ~ is the charge density and e is the dielectric constant or the medium which is assumed to he constant and usually equal to the dielectric constant of free space 136rr x 10- 9

Frm The charges occur at the boundaries of the inhomogeshyneities because the normal component of E is discontinuous there

For 3-D inhomogeneities in a layered half-space additional terms must be added to equations (13) and (14) to account for image currents and charges in the other layers Equation (11)

is really the symbolic representation of equations (12) (t 3) and (14) plus additional layered half-space terms The numerishy

cal difficulty associated with solving equation (12) for J is dues

to the disparity in the sizes of the current and charge opershyators Note the different magnitudes of the leading terms in equations (13) and (14)

(lO)(OJllo)~ (IOou)(J j ~ (10)5

for a frequency of 1000 Hz and a layer conductivity of 0001 Srrn this disparity becomes larger for smaller background conductivities Our comparison of the operators does not inshydude the effect of the gradient and divergence operators in equation (14) and therefore is not rigorous However in numerical-modeling experiments Lajoie and West (1976) conshysidered the gradient and divergence operators in equation (14)

and demonstrated the disparity in the two operators when the host medium becomes resistive

The scattering current in equation (12) can be written as a sum or curl-free and divergence-free parts where J = J~ + Js

and where V bull J~ = 0 and V x J~ = O Note that J~ is in the

null space of L because LJ~ = O Since L is the dominant operator in equation (12) a poorly constructed numerical

solution will give no information about the component of J that is divergence-free This is the component which domishynates the EM response of a body in a resistive earth

The disparity in the sizes of the two operators can be reshymoved by solving for curl-free and divergence-free scattering

currents Lajoie and West (1976) first obtained a solution valid in the limit of free space by solving for these two types of current in a thin 3-D plate SanFilipo and Hohmann (1985) recently published a solution for a prism based on a similar approach Their direct time-domain solution incorporates a specialized subset of basis functions that represent divergenceshyfree scattering current The solution produces good results for large conducti vity contrasts but the model is limited to a single prism in a half-space The frequency-domain solution we present here is based on SanFilipo and Hohmanns solushytion and is valid for multiple prisms in a layered half-space

Because they are the simplest geometries to model we start by assuming that the inhomogeneities in Figure 1 are prisms with constant conductivities The prisms are divided into N cells and ~1 closed current tubes as shown in Figure 2 The tubes making up a prism are concentric with respect to the

694 Newman and Hohmann

prism center and are constructed out of the cells making up that prism (Figure 2) The cells that make up a prism can also be rectangular which is an efficient shape because the scattershying current in an elongate prism varies more rapidly over its short dimension than over its long dimension

The scattering current within the prisms is approximated with a linear combination of pulse basis functions and divergence-free basis functions where

N M

J (r) ~ r J i Pi (r) + I c V (r) (18) ~ I i=l

The pulse and the divergence-free basis functions are defined by

I r in the ith cell (19)P (r) = O otherwise

and

Uj(r) = Qi(f)Ui(f) r in the ith tube (20)O otherwise

where Qj(r) is the unit vector in the direction of current flow in the tube according to the right-hand rule and Q j is the (variable) cross-sectional area of the tube The pulse basis function in equation (19) requires a component of the current density to be constant within a cell while the divergence-free basis function in equation (20) requires a component of the

current flowing within the tube to be constant

rUy U z

Uj ltd

(0) (b)

We note that the solution of Newman et al (1986) uses only pulse basis functions for tracking the scattering current While the pulse basis functions produce good results when the conshyductivity contrast between the bodies and their host layers is low they rail badly at higher contrasts The solution of Newman et al (1986) is reliable for contrasts below 200 to l where galvanic effects are important in determining the EM response of a 3-D body Galvanic responses are caused by charges on the surface of a body (Kaufman ]985) current is channeled into a conductive body or deflected away from a resistive body When the host is a perfect insulator no current elm be gathered into it body and its EM response is caused only by induction Inductive current flow is characterized by currents flowing in closed loops inside a body These eddy currents are caused by a magnetic field changing over time Thus the addition or the divergence-free (current-tube) basis function in equation (18) is essential for a valid numerical solution in the case of a high conductivity contrast

Because the current tubes of any prism are concentric with its center as shown in Figure 2 equation (18) docs not allow for completely general eddy-current patterns Although more general eddy-current patterns are possible they are compushyrationally prohibitive because the solution would then demand excessive amounts of computer time and memory Thus equation (18) will not be a good approximation to the scattering current for all types of source-prism geometries at high contrast For example the eddy-current patterns induced in a large horizontal plate are not always expected to be conshy

(c)

FIG 2 The prism is divided into 32 rectangular cells and 12 tubes The tubes are outlined by the heavy lines and the cells by the light lines The vector Uj(r) is the direction the current flows in the ith tube following the dotted path The magnitude of the current in the tube is Ct (a) shows four tubes normal to Ux _ (b) and (c) show four tubes normal to u

yand u respectively

695 3-0 Modeling of TEM Scattering

centric with the center of the plate A magnetic dipole placed air the axis of the plate will induce current vortices that are not concentric

The numerical solution for J is obtained by substituting equation (I~) into equation (12) to obtain

N

E(T) = L J j Pi (l)tOj(r) + (ttl + LI)J Pdr) i~ 1

Af

+ L Ci u (r)1Oj (r) I- Le Lj(r) (21) i ~ 1

where Au (r) is the difference net ween the conductivity of the prism occupied by the ith cell or the ith tube and the conducshytivity of its layered host The LJ i Pi(r) term involves the divergence of J and requires careful evaluation because J Pi(r) is discontinuous from cell to cell This term is evalushyatcd using an integro-diffcrence technique described by Hohshy

mann 11983) A Galerkin solution using weighting functions that are the same as the basis functions yields the following coupled equations for 311 + M

Munk nowns JN I and C 1

(Pk bull Ep ) = [ laquor J P)1Cfi+ (Pk LaJi Pi) + crLJ Piraquo

i gt 1

u

+ L (Pk bull cU i)1Oi + (Pk rcUraquo) i~ I

k = l 2 N (22)

and

v (Uk [1)= L(UkmiddotJiP)AOi+ltUkLJjP)1

i gt I

H

+ L(U k ClJ)Licr i + (Uk LnCU) i 1

k = 1 2 AI (23

where the inner product of the two functionsj a is defined by

ltI flgt = f(rgr) dr (24)JIf both f and g are vectors a dot multiplication is assumed in equation (24)

Because U is divergence-free the dominant L operator does not appear in the second sums of equations (21) (22) and (23) Furthermore the L operator does not appear in the first sum of equation (23) because the inner product of U and L J i Pi is zero by Stokes theorem we are integrating the gradient of a scalar around a closed path The coupled equashytions (22) and (23) result in a stable solution for the scattering current for a host of any conductivity

A concise (3N + M) x (3N + M) matrix equation is written from equations (22) and (23) for the unknown coefficients J j

and C i where

[ I n ~PlJ [J] = [ltP Ep ) ] (25) Ivp tSuu c (V Ellgt

The (3N x 3N) Kpp submatrix represents the coupling coefshyficients between the pulse functions The Kpu and the fSuP

submatrices represent the coupling coefficients between the tube and pulse functions Their sizes are (3N x M) and (M x 31) respectively Finally the (M x M) KlJll submatrix represents the coupling coefficients between the tube functions The impedance matrix in equation (25) is full and the compushytation time required to build and factor the matrix can be substantial Tripp and Hohmann (1984) have shown that a significant reduction in matrix formulation and factoring time can be achieved for a single prism with two vertical planes of symmetry by a similarity transformation The impedance matrix is block diagonalized into four (3N x 4) 4(3N x M)4 matrices and the source vector is transformed into four new source vectors However in general for multiple prisms this similarity transformation cannot be used and equation (251 must be solved directly

Once the coefficients J and C are determined the scattershying current is given by equation (18) The electric and magshynetic fields outside the prisms are then obtained by addition of the layered half-space fields with the discrete versions of equashytions (9) and (0)

N

E(r) = E(r) + L rE(r r n ) bull J(rn ) (26) n ~ 1

and

Ii

Htr) = Hp(r) + L rH(r r) J(r) (27) =1

with the tensor Greens functions for a prism of current in cell n given by

rE(r rJ = f (E(r r) do (28) Ill

and

[H(r r) = r (i(r r) dv (29)

J Time-domain solution

Transformation of frequency-domain responses to the time domain is accomplished by inverse Fourier transformation The electric-field and magnetic-field responses in the time domain for a step current are computed using the cosine transforms

e(t) = -21n 1 1m [E(W)Jco cos wt dtraquo (30)

and

h(t) = rot dsraquo (31)-2rr LX 1m [H((l))]m cos

and for the time derivative of the magnetic field by using the sine transform

iih(l)ilt = 2re fX 1m [ H(W)] sin rot am (32)

696 Newman and Hohmann

where 1m [H(m)] and Im [E(I))] are the imaginary parts of the magnetic and electric fields in the frequency domain The inteshygrations in equations (30)-(32) are carried out using the disshycrete convolution technique described by Newman et al (19lS6) At late times the time derivative of the magnetic field is not calculated directly from equation (32) because of numerishycal noise Rather we numerically differentiate the magneticshyfield transient since equation 131) has a smaller dynamic range from early to late times and is more accurate than equation (32) The magnetic-field transient is replaced with a cubic spline and the spline is then differentiated to obtain (~h(t)ilt

However evaluating th(t)tt in this way is still inaccurate at

the latest times because of numerical noise in the magnetic field after it has decayed six orders in magnitude from early to late times Equations (30) (31) and (32) usually require 20 to 40 3-D lrequency-domain evaluations at five to eight points per decade for an accurate solution We note that the calculashytion of 3-D transient responses requires substantial amounts of computer time because each frequency-domain evaluation involves solving equations (25) and (26) for the electric field

and equations (25) and (27) for the magnetic field

Checks Oil numerical solution

Essential to the development of any numerical solution is its verilication Our solution can be checked for internal consist shyency by verifying reciprocity by comparing decay rates with

the late-time decay rates of thin plates and spheres in free space and by carrying out convergence tests based on the level of cell discretization However the best check is to comshypare the solution with other numerical solutions andor scaleshymodel results Because we arc interested in computing 3-D TEM responses we present checks in the time domain only

I t is necessary to design models with a minimum number of cells because computation time increases rapidly with more cells Therefore we have carefully discretized all the models in this paper with as few cells as possible based on convergence studies For more information on model discretization refer to Newman et a1 (1986)

figures 3 and 4 show that the current-tube basis functions in equation (I8) are required to obtain the correct late-time decay rate of a thin plate in free space We model a thin plate by letting the prism in Figure 2 be one cell wide and thereby restrict the eddy currents to flow normal to one coordinate direction Our solution using both current-tube and pulse functions (solid and dashed lines in Figures 3 and 4) is comshypared with a solution using only pulse functions (crosses) for a 40 Sm prism (cr) 20 m thick (r) in a 10 000 nmiddot m half-space Both solutions are also compared with a solution using only current tubes (triangles) The top of the prism in Figure 3 is at a depth of 100 m (DR) and its depth extent (DE) and strike length (Sl) are gOand gOO m respectively The prisms vertical magnetic-field response [or a step shutoff in the transmitter is plotted at the surface for 21 receiver positions where the

time X_--)(---X---IC-~-X-_-X lt Complete solution07msx_x-x-x-x-x-x-x -L-x

x l Solution with pulse- Functions only x x

_x_x-x-x-x-x- _IC--x---lC---lC---x-- - __ x

bull05 liS lC x I x---)(10-6 ~ _~A~ lC ___~li_A-A-A~ raquo (-pound-_f-~~=~-A

_A 6~ ~~ l ~A-f-A Solution with tube 1ms --t-A- ~ i x x )( 6 functions only1 A x X x x x ~ l x )( IC ol)( x

Ii X _A-A-A _Igt---6---A~~6__A x A_A lt

-~ )( A -A_-II shy

A-A- ~ 10-7 1- 10ms A II ~~

-- r _A __-A__ ~_A_A-~AJIIIl A- lt6 11_

lJ -A_bull-V- I ~b ~10ms AA A_-A

MODEl

10-8 I shy

surFac umiddot40Sm

10000n-1ll Otmiddot 20m

Slmiddot 800m Near edge of loop DEmiddot80m

D8middot100m

I I I I l I I L I I I I I I I I I I I

-200 -160 -120 -80 -40 0 40 80 120 160 200

DISTANCE FROM CONDUCTOR CENTER (m)

FIG 3 Comparison of the vertical component of the scattered magnetic field calculated with different basis functions The model is a vertical thin prism Solid and dashed curves show the solution based on pulse and tube functions Crosses show the solution using only the pulse functions at 001 005 and 01 ms and triangles show the solution using only the tube functions at 01 1 10 and 20 IDS

697

1

3-D Modeling of TEM Scattering

- i bull ~ bull toSotio bull luhCbullbullpl f tlbullbull only unc10

~ __~___ -

_SollIon 1I1t pul

~ 10 [ bullbull

10- ~ ~ EUltotd tlbullbull oltatod tiM co~ant T 9 ItftIQ~ 08)( Ae

10 I-shy

I I

transmitter is a 160 x 160 m loop with its near wire located at - 250 m Results for six time channels are shown in Figure 3

but results are not plotted before 01 ms for the current-tube solution or after 01 ms for the pulse function solution

In the two early-time channels 001 and 003 rns the comshyplete solution compares well with the solution using only pulse functions This early time range is where the galvanic respolHe is important ~ diffusing currents in the half-space are concentrated around and channeled through the prism Howshyever beyond 003 ms the solution using only pulse functions decays faster than the complete solution (Figure 4) The lateshytime decay of the pulse solution is an exponential with a time constant of OO~ ms The empirical formula for the time conshystant of a thin plate given in Lamontagne (1975) is

r = (110 otL) 10 (33)

where L is the smaller dimension of the plate and crt is its conductivity-thickness product The time constant calculated from equation (33) for the prism in Figure 3 is 8 ms

The complete solution agrees with the current-tube solution at later times (Figure 3) and its late-time decay rate is exshy

01 10 100 ponential from Figure 4 we estimate a time constant of 8 ms

tim) This time constant agrees with that calculated from equation (33) and thus the TEM response of the prism decays as that

nO 4 Scattered vertical-field decay of the three solutions at for a thin plate in free space at later times Note that the x = -80 m in Figure 3 solution using only the current tubes appears to be constant at

VERTICAL HORIZONTAL100

Son Fllrpo end Hohmonrls solutlon( A)

A nnoll egflnmodfl solution (l( )

10 0335

MODEL

lOOOm

~oo- free space

llltegrol equation solution (-)

00 0 0 0- 0 - 0 - 0 0

0 bull amp132ms l amp po oOAJo-oo ~ ~ tE bull p

gt 33ms I

0 shyc iI ~ r _~o-ltl--o-

I gXcglshy~

0 -s

bull I

f 6 7718S II llt X

I l( lCbull pKJImiddotoIlo -ilrO-oo

01 lC I ~ 1 r I

o 50 100 150 200 250 o 50 100 150 200 250

DISTANCE FROM PLATE (II) DISTANCE fROM PLATE (m)

FIG 5 Comparison of the integral-equation solution (solid and dashed curves) for a plate in free space with the eigenmode solution of Annan (1974) (crosses) and the integral-equation solution of SanFilipo and Hohmann (triangles) Vertical and horizontal components of ob(t)ot are compared

694 Newman and Hohmann

prism center and are constructed out of the cells making up that prism (Figure 2) The cells that make up a prism can also be rectangular which is an efficient shape because the scattershying current in an elongate prism varies more rapidly over its short dimension than over its long dimension

The scattering current within the prisms is approximated with a linear combination of pulse basis functions and divergence-free basis functions where

N M

J (r) ~ r J i Pi (r) + I c V (r) (18) ~ I i=l

The pulse and the divergence-free basis functions are defined by

I r in the ith cell (19)P (r) = O otherwise

and

Uj(r) = Qi(f)Ui(f) r in the ith tube (20)O otherwise

where Qj(r) is the unit vector in the direction of current flow in the tube according to the right-hand rule and Q j is the (variable) cross-sectional area of the tube The pulse basis function in equation (19) requires a component of the current density to be constant within a cell while the divergence-free basis function in equation (20) requires a component of the

current flowing within the tube to be constant

rUy U z

Uj ltd

(0) (b)

We note that the solution of Newman et al (1986) uses only pulse basis functions for tracking the scattering current While the pulse basis functions produce good results when the conshyductivity contrast between the bodies and their host layers is low they rail badly at higher contrasts The solution of Newman et al (1986) is reliable for contrasts below 200 to l where galvanic effects are important in determining the EM response of a 3-D body Galvanic responses are caused by charges on the surface of a body (Kaufman ]985) current is channeled into a conductive body or deflected away from a resistive body When the host is a perfect insulator no current elm be gathered into it body and its EM response is caused only by induction Inductive current flow is characterized by currents flowing in closed loops inside a body These eddy currents are caused by a magnetic field changing over time Thus the addition or the divergence-free (current-tube) basis function in equation (18) is essential for a valid numerical solution in the case of a high conductivity contrast

Because the current tubes of any prism are concentric with its center as shown in Figure 2 equation (18) docs not allow for completely general eddy-current patterns Although more general eddy-current patterns are possible they are compushyrationally prohibitive because the solution would then demand excessive amounts of computer time and memory Thus equation (18) will not be a good approximation to the scattering current for all types of source-prism geometries at high contrast For example the eddy-current patterns induced in a large horizontal plate are not always expected to be conshy

(c)

FIG 2 The prism is divided into 32 rectangular cells and 12 tubes The tubes are outlined by the heavy lines and the cells by the light lines The vector Uj(r) is the direction the current flows in the ith tube following the dotted path The magnitude of the current in the tube is Ct (a) shows four tubes normal to Ux _ (b) and (c) show four tubes normal to u

yand u respectively

695 3-0 Modeling of TEM Scattering

centric with the center of the plate A magnetic dipole placed air the axis of the plate will induce current vortices that are not concentric

The numerical solution for J is obtained by substituting equation (I~) into equation (12) to obtain

N

E(T) = L J j Pi (l)tOj(r) + (ttl + LI)J Pdr) i~ 1

Af

+ L Ci u (r)1Oj (r) I- Le Lj(r) (21) i ~ 1

where Au (r) is the difference net ween the conductivity of the prism occupied by the ith cell or the ith tube and the conducshytivity of its layered host The LJ i Pi(r) term involves the divergence of J and requires careful evaluation because J Pi(r) is discontinuous from cell to cell This term is evalushyatcd using an integro-diffcrence technique described by Hohshy

mann 11983) A Galerkin solution using weighting functions that are the same as the basis functions yields the following coupled equations for 311 + M

Munk nowns JN I and C 1

(Pk bull Ep ) = [ laquor J P)1Cfi+ (Pk LaJi Pi) + crLJ Piraquo

i gt 1

u

+ L (Pk bull cU i)1Oi + (Pk rcUraquo) i~ I

k = l 2 N (22)

and

v (Uk [1)= L(UkmiddotJiP)AOi+ltUkLJjP)1

i gt I

H

+ L(U k ClJ)Licr i + (Uk LnCU) i 1

k = 1 2 AI (23

where the inner product of the two functionsj a is defined by

ltI flgt = f(rgr) dr (24)JIf both f and g are vectors a dot multiplication is assumed in equation (24)

Because U is divergence-free the dominant L operator does not appear in the second sums of equations (21) (22) and (23) Furthermore the L operator does not appear in the first sum of equation (23) because the inner product of U and L J i Pi is zero by Stokes theorem we are integrating the gradient of a scalar around a closed path The coupled equashytions (22) and (23) result in a stable solution for the scattering current for a host of any conductivity

A concise (3N + M) x (3N + M) matrix equation is written from equations (22) and (23) for the unknown coefficients J j

and C i where

[ I n ~PlJ [J] = [ltP Ep ) ] (25) Ivp tSuu c (V Ellgt

The (3N x 3N) Kpp submatrix represents the coupling coefshyficients between the pulse functions The Kpu and the fSuP

submatrices represent the coupling coefficients between the tube and pulse functions Their sizes are (3N x M) and (M x 31) respectively Finally the (M x M) KlJll submatrix represents the coupling coefficients between the tube functions The impedance matrix in equation (25) is full and the compushytation time required to build and factor the matrix can be substantial Tripp and Hohmann (1984) have shown that a significant reduction in matrix formulation and factoring time can be achieved for a single prism with two vertical planes of symmetry by a similarity transformation The impedance matrix is block diagonalized into four (3N x 4) 4(3N x M)4 matrices and the source vector is transformed into four new source vectors However in general for multiple prisms this similarity transformation cannot be used and equation (251 must be solved directly

Once the coefficients J and C are determined the scattershying current is given by equation (18) The electric and magshynetic fields outside the prisms are then obtained by addition of the layered half-space fields with the discrete versions of equashytions (9) and (0)

N

E(r) = E(r) + L rE(r r n ) bull J(rn ) (26) n ~ 1

and

Ii

Htr) = Hp(r) + L rH(r r) J(r) (27) =1

with the tensor Greens functions for a prism of current in cell n given by

rE(r rJ = f (E(r r) do (28) Ill

and

[H(r r) = r (i(r r) dv (29)

J Time-domain solution

Transformation of frequency-domain responses to the time domain is accomplished by inverse Fourier transformation The electric-field and magnetic-field responses in the time domain for a step current are computed using the cosine transforms

e(t) = -21n 1 1m [E(W)Jco cos wt dtraquo (30)

and

h(t) = rot dsraquo (31)-2rr LX 1m [H((l))]m cos

and for the time derivative of the magnetic field by using the sine transform

iih(l)ilt = 2re fX 1m [ H(W)] sin rot am (32)

696 Newman and Hohmann

where 1m [H(m)] and Im [E(I))] are the imaginary parts of the magnetic and electric fields in the frequency domain The inteshygrations in equations (30)-(32) are carried out using the disshycrete convolution technique described by Newman et al (19lS6) At late times the time derivative of the magnetic field is not calculated directly from equation (32) because of numerishycal noise Rather we numerically differentiate the magneticshyfield transient since equation 131) has a smaller dynamic range from early to late times and is more accurate than equation (32) The magnetic-field transient is replaced with a cubic spline and the spline is then differentiated to obtain (~h(t)ilt

However evaluating th(t)tt in this way is still inaccurate at

the latest times because of numerical noise in the magnetic field after it has decayed six orders in magnitude from early to late times Equations (30) (31) and (32) usually require 20 to 40 3-D lrequency-domain evaluations at five to eight points per decade for an accurate solution We note that the calculashytion of 3-D transient responses requires substantial amounts of computer time because each frequency-domain evaluation involves solving equations (25) and (26) for the electric field

and equations (25) and (27) for the magnetic field

Checks Oil numerical solution

Essential to the development of any numerical solution is its verilication Our solution can be checked for internal consist shyency by verifying reciprocity by comparing decay rates with

the late-time decay rates of thin plates and spheres in free space and by carrying out convergence tests based on the level of cell discretization However the best check is to comshypare the solution with other numerical solutions andor scaleshymodel results Because we arc interested in computing 3-D TEM responses we present checks in the time domain only

I t is necessary to design models with a minimum number of cells because computation time increases rapidly with more cells Therefore we have carefully discretized all the models in this paper with as few cells as possible based on convergence studies For more information on model discretization refer to Newman et a1 (1986)

figures 3 and 4 show that the current-tube basis functions in equation (I8) are required to obtain the correct late-time decay rate of a thin plate in free space We model a thin plate by letting the prism in Figure 2 be one cell wide and thereby restrict the eddy currents to flow normal to one coordinate direction Our solution using both current-tube and pulse functions (solid and dashed lines in Figures 3 and 4) is comshypared with a solution using only pulse functions (crosses) for a 40 Sm prism (cr) 20 m thick (r) in a 10 000 nmiddot m half-space Both solutions are also compared with a solution using only current tubes (triangles) The top of the prism in Figure 3 is at a depth of 100 m (DR) and its depth extent (DE) and strike length (Sl) are gOand gOO m respectively The prisms vertical magnetic-field response [or a step shutoff in the transmitter is plotted at the surface for 21 receiver positions where the

time X_--)(---X---IC-~-X-_-X lt Complete solution07msx_x-x-x-x-x-x-x -L-x

x l Solution with pulse- Functions only x x

_x_x-x-x-x-x- _IC--x---lC---lC---x-- - __ x

bull05 liS lC x I x---)(10-6 ~ _~A~ lC ___~li_A-A-A~ raquo (-pound-_f-~~=~-A

_A 6~ ~~ l ~A-f-A Solution with tube 1ms --t-A- ~ i x x )( 6 functions only1 A x X x x x ~ l x )( IC ol)( x

Ii X _A-A-A _Igt---6---A~~6__A x A_A lt

-~ )( A -A_-II shy

A-A- ~ 10-7 1- 10ms A II ~~

-- r _A __-A__ ~_A_A-~AJIIIl A- lt6 11_

lJ -A_bull-V- I ~b ~10ms AA A_-A

MODEl

10-8 I shy

surFac umiddot40Sm

10000n-1ll Otmiddot 20m

Slmiddot 800m Near edge of loop DEmiddot80m

D8middot100m

I I I I l I I L I I I I I I I I I I I

-200 -160 -120 -80 -40 0 40 80 120 160 200

DISTANCE FROM CONDUCTOR CENTER (m)

FIG 3 Comparison of the vertical component of the scattered magnetic field calculated with different basis functions The model is a vertical thin prism Solid and dashed curves show the solution based on pulse and tube functions Crosses show the solution using only the pulse functions at 001 005 and 01 ms and triangles show the solution using only the tube functions at 01 1 10 and 20 IDS

697

1

3-D Modeling of TEM Scattering

- i bull ~ bull toSotio bull luhCbullbullpl f tlbullbull only unc10

~ __~___ -

_SollIon 1I1t pul

~ 10 [ bullbull

10- ~ ~ EUltotd tlbullbull oltatod tiM co~ant T 9 ItftIQ~ 08)( Ae

10 I-shy

I I

transmitter is a 160 x 160 m loop with its near wire located at - 250 m Results for six time channels are shown in Figure 3

but results are not plotted before 01 ms for the current-tube solution or after 01 ms for the pulse function solution

In the two early-time channels 001 and 003 rns the comshyplete solution compares well with the solution using only pulse functions This early time range is where the galvanic respolHe is important ~ diffusing currents in the half-space are concentrated around and channeled through the prism Howshyever beyond 003 ms the solution using only pulse functions decays faster than the complete solution (Figure 4) The lateshytime decay of the pulse solution is an exponential with a time constant of OO~ ms The empirical formula for the time conshystant of a thin plate given in Lamontagne (1975) is

r = (110 otL) 10 (33)

where L is the smaller dimension of the plate and crt is its conductivity-thickness product The time constant calculated from equation (33) for the prism in Figure 3 is 8 ms

The complete solution agrees with the current-tube solution at later times (Figure 3) and its late-time decay rate is exshy

01 10 100 ponential from Figure 4 we estimate a time constant of 8 ms

tim) This time constant agrees with that calculated from equation (33) and thus the TEM response of the prism decays as that

nO 4 Scattered vertical-field decay of the three solutions at for a thin plate in free space at later times Note that the x = -80 m in Figure 3 solution using only the current tubes appears to be constant at

VERTICAL HORIZONTAL100

Son Fllrpo end Hohmonrls solutlon( A)

A nnoll egflnmodfl solution (l( )

10 0335

MODEL

lOOOm

~oo- free space

llltegrol equation solution (-)

00 0 0 0- 0 - 0 - 0 0

0 bull amp132ms l amp po oOAJo-oo ~ ~ tE bull p

gt 33ms I

0 shyc iI ~ r _~o-ltl--o-

I gXcglshy~

0 -s

bull I

f 6 7718S II llt X

I l( lCbull pKJImiddotoIlo -ilrO-oo

01 lC I ~ 1 r I

o 50 100 150 200 250 o 50 100 150 200 250

DISTANCE FROM PLATE (II) DISTANCE fROM PLATE (m)

FIG 5 Comparison of the integral-equation solution (solid and dashed curves) for a plate in free space with the eigenmode solution of Annan (1974) (crosses) and the integral-equation solution of SanFilipo and Hohmann (triangles) Vertical and horizontal components of ob(t)ot are compared

695 3-0 Modeling of TEM Scattering

centric with the center of the plate A magnetic dipole placed air the axis of the plate will induce current vortices that are not concentric

The numerical solution for J is obtained by substituting equation (I~) into equation (12) to obtain

N

E(T) = L J j Pi (l)tOj(r) + (ttl + LI)J Pdr) i~ 1

Af

+ L Ci u (r)1Oj (r) I- Le Lj(r) (21) i ~ 1

where Au (r) is the difference net ween the conductivity of the prism occupied by the ith cell or the ith tube and the conducshytivity of its layered host The LJ i Pi(r) term involves the divergence of J and requires careful evaluation because J Pi(r) is discontinuous from cell to cell This term is evalushyatcd using an integro-diffcrence technique described by Hohshy

mann 11983) A Galerkin solution using weighting functions that are the same as the basis functions yields the following coupled equations for 311 + M

Munk nowns JN I and C 1

(Pk bull Ep ) = [ laquor J P)1Cfi+ (Pk LaJi Pi) + crLJ Piraquo

i gt 1

u

+ L (Pk bull cU i)1Oi + (Pk rcUraquo) i~ I

k = l 2 N (22)

and

v (Uk [1)= L(UkmiddotJiP)AOi+ltUkLJjP)1

i gt I

H

+ L(U k ClJ)Licr i + (Uk LnCU) i 1

k = 1 2 AI (23

where the inner product of the two functionsj a is defined by

ltI flgt = f(rgr) dr (24)JIf both f and g are vectors a dot multiplication is assumed in equation (24)

Because U is divergence-free the dominant L operator does not appear in the second sums of equations (21) (22) and (23) Furthermore the L operator does not appear in the first sum of equation (23) because the inner product of U and L J i Pi is zero by Stokes theorem we are integrating the gradient of a scalar around a closed path The coupled equashytions (22) and (23) result in a stable solution for the scattering current for a host of any conductivity

A concise (3N + M) x (3N + M) matrix equation is written from equations (22) and (23) for the unknown coefficients J j

and C i where

[ I n ~PlJ [J] = [ltP Ep ) ] (25) Ivp tSuu c (V Ellgt

The (3N x 3N) Kpp submatrix represents the coupling coefshyficients between the pulse functions The Kpu and the fSuP

submatrices represent the coupling coefficients between the tube and pulse functions Their sizes are (3N x M) and (M x 31) respectively Finally the (M x M) KlJll submatrix represents the coupling coefficients between the tube functions The impedance matrix in equation (25) is full and the compushytation time required to build and factor the matrix can be substantial Tripp and Hohmann (1984) have shown that a significant reduction in matrix formulation and factoring time can be achieved for a single prism with two vertical planes of symmetry by a similarity transformation The impedance matrix is block diagonalized into four (3N x 4) 4(3N x M)4 matrices and the source vector is transformed into four new source vectors However in general for multiple prisms this similarity transformation cannot be used and equation (251 must be solved directly

Once the coefficients J and C are determined the scattershying current is given by equation (18) The electric and magshynetic fields outside the prisms are then obtained by addition of the layered half-space fields with the discrete versions of equashytions (9) and (0)

N

E(r) = E(r) + L rE(r r n ) bull J(rn ) (26) n ~ 1

and

Ii

Htr) = Hp(r) + L rH(r r) J(r) (27) =1

with the tensor Greens functions for a prism of current in cell n given by

rE(r rJ = f (E(r r) do (28) Ill

and

[H(r r) = r (i(r r) dv (29)

J Time-domain solution

Transformation of frequency-domain responses to the time domain is accomplished by inverse Fourier transformation The electric-field and magnetic-field responses in the time domain for a step current are computed using the cosine transforms

e(t) = -21n 1 1m [E(W)Jco cos wt dtraquo (30)

and

h(t) = rot dsraquo (31)-2rr LX 1m [H((l))]m cos

and for the time derivative of the magnetic field by using the sine transform

iih(l)ilt = 2re fX 1m [ H(W)] sin rot am (32)

696 Newman and Hohmann

where 1m [H(m)] and Im [E(I))] are the imaginary parts of the magnetic and electric fields in the frequency domain The inteshygrations in equations (30)-(32) are carried out using the disshycrete convolution technique described by Newman et al (19lS6) At late times the time derivative of the magnetic field is not calculated directly from equation (32) because of numerishycal noise Rather we numerically differentiate the magneticshyfield transient since equation 131) has a smaller dynamic range from early to late times and is more accurate than equation (32) The magnetic-field transient is replaced with a cubic spline and the spline is then differentiated to obtain (~h(t)ilt

However evaluating th(t)tt in this way is still inaccurate at

the latest times because of numerical noise in the magnetic field after it has decayed six orders in magnitude from early to late times Equations (30) (31) and (32) usually require 20 to 40 3-D lrequency-domain evaluations at five to eight points per decade for an accurate solution We note that the calculashytion of 3-D transient responses requires substantial amounts of computer time because each frequency-domain evaluation involves solving equations (25) and (26) for the electric field

and equations (25) and (27) for the magnetic field

Checks Oil numerical solution

Essential to the development of any numerical solution is its verilication Our solution can be checked for internal consist shyency by verifying reciprocity by comparing decay rates with

the late-time decay rates of thin plates and spheres in free space and by carrying out convergence tests based on the level of cell discretization However the best check is to comshypare the solution with other numerical solutions andor scaleshymodel results Because we arc interested in computing 3-D TEM responses we present checks in the time domain only

I t is necessary to design models with a minimum number of cells because computation time increases rapidly with more cells Therefore we have carefully discretized all the models in this paper with as few cells as possible based on convergence studies For more information on model discretization refer to Newman et a1 (1986)

figures 3 and 4 show that the current-tube basis functions in equation (I8) are required to obtain the correct late-time decay rate of a thin plate in free space We model a thin plate by letting the prism in Figure 2 be one cell wide and thereby restrict the eddy currents to flow normal to one coordinate direction Our solution using both current-tube and pulse functions (solid and dashed lines in Figures 3 and 4) is comshypared with a solution using only pulse functions (crosses) for a 40 Sm prism (cr) 20 m thick (r) in a 10 000 nmiddot m half-space Both solutions are also compared with a solution using only current tubes (triangles) The top of the prism in Figure 3 is at a depth of 100 m (DR) and its depth extent (DE) and strike length (Sl) are gOand gOO m respectively The prisms vertical magnetic-field response [or a step shutoff in the transmitter is plotted at the surface for 21 receiver positions where the

time X_--)(---X---IC-~-X-_-X lt Complete solution07msx_x-x-x-x-x-x-x -L-x

x l Solution with pulse- Functions only x x

_x_x-x-x-x-x- _IC--x---lC---lC---x-- - __ x

bull05 liS lC x I x---)(10-6 ~ _~A~ lC ___~li_A-A-A~ raquo (-pound-_f-~~=~-A

_A 6~ ~~ l ~A-f-A Solution with tube 1ms --t-A- ~ i x x )( 6 functions only1 A x X x x x ~ l x )( IC ol)( x

Ii X _A-A-A _Igt---6---A~~6__A x A_A lt

-~ )( A -A_-II shy

A-A- ~ 10-7 1- 10ms A II ~~

-- r _A __-A__ ~_A_A-~AJIIIl A- lt6 11_

lJ -A_bull-V- I ~b ~10ms AA A_-A

MODEl

10-8 I shy

surFac umiddot40Sm

10000n-1ll Otmiddot 20m

Slmiddot 800m Near edge of loop DEmiddot80m

D8middot100m

I I I I l I I L I I I I I I I I I I I

-200 -160 -120 -80 -40 0 40 80 120 160 200

DISTANCE FROM CONDUCTOR CENTER (m)

FIG 3 Comparison of the vertical component of the scattered magnetic field calculated with different basis functions The model is a vertical thin prism Solid and dashed curves show the solution based on pulse and tube functions Crosses show the solution using only the pulse functions at 001 005 and 01 ms and triangles show the solution using only the tube functions at 01 1 10 and 20 IDS

697

1

3-D Modeling of TEM Scattering

- i bull ~ bull toSotio bull luhCbullbullpl f tlbullbull only unc10

~ __~___ -

_SollIon 1I1t pul

~ 10 [ bullbull

10- ~ ~ EUltotd tlbullbull oltatod tiM co~ant T 9 ItftIQ~ 08)( Ae

10 I-shy

I I

transmitter is a 160 x 160 m loop with its near wire located at - 250 m Results for six time channels are shown in Figure 3

but results are not plotted before 01 ms for the current-tube solution or after 01 ms for the pulse function solution

In the two early-time channels 001 and 003 rns the comshyplete solution compares well with the solution using only pulse functions This early time range is where the galvanic respolHe is important ~ diffusing currents in the half-space are concentrated around and channeled through the prism Howshyever beyond 003 ms the solution using only pulse functions decays faster than the complete solution (Figure 4) The lateshytime decay of the pulse solution is an exponential with a time constant of OO~ ms The empirical formula for the time conshystant of a thin plate given in Lamontagne (1975) is

r = (110 otL) 10 (33)

where L is the smaller dimension of the plate and crt is its conductivity-thickness product The time constant calculated from equation (33) for the prism in Figure 3 is 8 ms

The complete solution agrees with the current-tube solution at later times (Figure 3) and its late-time decay rate is exshy

01 10 100 ponential from Figure 4 we estimate a time constant of 8 ms

tim) This time constant agrees with that calculated from equation (33) and thus the TEM response of the prism decays as that

nO 4 Scattered vertical-field decay of the three solutions at for a thin plate in free space at later times Note that the x = -80 m in Figure 3 solution using only the current tubes appears to be constant at

VERTICAL HORIZONTAL100

Son Fllrpo end Hohmonrls solutlon( A)

A nnoll egflnmodfl solution (l( )

10 0335

MODEL

lOOOm

~oo- free space

llltegrol equation solution (-)

00 0 0 0- 0 - 0 - 0 0

0 bull amp132ms l amp po oOAJo-oo ~ ~ tE bull p

gt 33ms I

0 shyc iI ~ r _~o-ltl--o-

I gXcglshy~

0 -s

bull I

f 6 7718S II llt X

I l( lCbull pKJImiddotoIlo -ilrO-oo

01 lC I ~ 1 r I

o 50 100 150 200 250 o 50 100 150 200 250

DISTANCE FROM PLATE (II) DISTANCE fROM PLATE (m)

FIG 5 Comparison of the integral-equation solution (solid and dashed curves) for a plate in free space with the eigenmode solution of Annan (1974) (crosses) and the integral-equation solution of SanFilipo and Hohmann (triangles) Vertical and horizontal components of ob(t)ot are compared

696 Newman and Hohmann

where 1m [H(m)] and Im [E(I))] are the imaginary parts of the magnetic and electric fields in the frequency domain The inteshygrations in equations (30)-(32) are carried out using the disshycrete convolution technique described by Newman et al (19lS6) At late times the time derivative of the magnetic field is not calculated directly from equation (32) because of numerishycal noise Rather we numerically differentiate the magneticshyfield transient since equation 131) has a smaller dynamic range from early to late times and is more accurate than equation (32) The magnetic-field transient is replaced with a cubic spline and the spline is then differentiated to obtain (~h(t)ilt

However evaluating th(t)tt in this way is still inaccurate at

the latest times because of numerical noise in the magnetic field after it has decayed six orders in magnitude from early to late times Equations (30) (31) and (32) usually require 20 to 40 3-D lrequency-domain evaluations at five to eight points per decade for an accurate solution We note that the calculashytion of 3-D transient responses requires substantial amounts of computer time because each frequency-domain evaluation involves solving equations (25) and (26) for the electric field

and equations (25) and (27) for the magnetic field

Checks Oil numerical solution

Essential to the development of any numerical solution is its verilication Our solution can be checked for internal consist shyency by verifying reciprocity by comparing decay rates with

the late-time decay rates of thin plates and spheres in free space and by carrying out convergence tests based on the level of cell discretization However the best check is to comshypare the solution with other numerical solutions andor scaleshymodel results Because we arc interested in computing 3-D TEM responses we present checks in the time domain only

I t is necessary to design models with a minimum number of cells because computation time increases rapidly with more cells Therefore we have carefully discretized all the models in this paper with as few cells as possible based on convergence studies For more information on model discretization refer to Newman et a1 (1986)

figures 3 and 4 show that the current-tube basis functions in equation (I8) are required to obtain the correct late-time decay rate of a thin plate in free space We model a thin plate by letting the prism in Figure 2 be one cell wide and thereby restrict the eddy currents to flow normal to one coordinate direction Our solution using both current-tube and pulse functions (solid and dashed lines in Figures 3 and 4) is comshypared with a solution using only pulse functions (crosses) for a 40 Sm prism (cr) 20 m thick (r) in a 10 000 nmiddot m half-space Both solutions are also compared with a solution using only current tubes (triangles) The top of the prism in Figure 3 is at a depth of 100 m (DR) and its depth extent (DE) and strike length (Sl) are gOand gOO m respectively The prisms vertical magnetic-field response [or a step shutoff in the transmitter is plotted at the surface for 21 receiver positions where the

time X_--)(---X---IC-~-X-_-X lt Complete solution07msx_x-x-x-x-x-x-x -L-x

x l Solution with pulse- Functions only x x

_x_x-x-x-x-x- _IC--x---lC---lC---x-- - __ x

bull05 liS lC x I x---)(10-6 ~ _~A~ lC ___~li_A-A-A~ raquo (-pound-_f-~~=~-A

_A 6~ ~~ l ~A-f-A Solution with tube 1ms --t-A- ~ i x x )( 6 functions only1 A x X x x x ~ l x )( IC ol)( x

Ii X _A-A-A _Igt---6---A~~6__A x A_A lt

-~ )( A -A_-II shy

A-A- ~ 10-7 1- 10ms A II ~~

-- r _A __-A__ ~_A_A-~AJIIIl A- lt6 11_

lJ -A_bull-V- I ~b ~10ms AA A_-A

MODEl

10-8 I shy

surFac umiddot40Sm

10000n-1ll Otmiddot 20m

Slmiddot 800m Near edge of loop DEmiddot80m

D8middot100m

I I I I l I I L I I I I I I I I I I I

-200 -160 -120 -80 -40 0 40 80 120 160 200

DISTANCE FROM CONDUCTOR CENTER (m)

FIG 3 Comparison of the vertical component of the scattered magnetic field calculated with different basis functions The model is a vertical thin prism Solid and dashed curves show the solution based on pulse and tube functions Crosses show the solution using only the pulse functions at 001 005 and 01 ms and triangles show the solution using only the tube functions at 01 1 10 and 20 IDS

697

1

3-D Modeling of TEM Scattering

- i bull ~ bull toSotio bull luhCbullbullpl f tlbullbull only unc10

~ __~___ -

_SollIon 1I1t pul

~ 10 [ bullbull

10- ~ ~ EUltotd tlbullbull oltatod tiM co~ant T 9 ItftIQ~ 08)( Ae

10 I-shy

I I

transmitter is a 160 x 160 m loop with its near wire located at - 250 m Results for six time channels are shown in Figure 3

but results are not plotted before 01 ms for the current-tube solution or after 01 ms for the pulse function solution

In the two early-time channels 001 and 003 rns the comshyplete solution compares well with the solution using only pulse functions This early time range is where the galvanic respolHe is important ~ diffusing currents in the half-space are concentrated around and channeled through the prism Howshyever beyond 003 ms the solution using only pulse functions decays faster than the complete solution (Figure 4) The lateshytime decay of the pulse solution is an exponential with a time constant of OO~ ms The empirical formula for the time conshystant of a thin plate given in Lamontagne (1975) is

r = (110 otL) 10 (33)

where L is the smaller dimension of the plate and crt is its conductivity-thickness product The time constant calculated from equation (33) for the prism in Figure 3 is 8 ms

The complete solution agrees with the current-tube solution at later times (Figure 3) and its late-time decay rate is exshy

01 10 100 ponential from Figure 4 we estimate a time constant of 8 ms

tim) This time constant agrees with that calculated from equation (33) and thus the TEM response of the prism decays as that

nO 4 Scattered vertical-field decay of the three solutions at for a thin plate in free space at later times Note that the x = -80 m in Figure 3 solution using only the current tubes appears to be constant at

VERTICAL HORIZONTAL100

Son Fllrpo end Hohmonrls solutlon( A)

A nnoll egflnmodfl solution (l( )

10 0335

MODEL

lOOOm

~oo- free space

llltegrol equation solution (-)

00 0 0 0- 0 - 0 - 0 0

0 bull amp132ms l amp po oOAJo-oo ~ ~ tE bull p

gt 33ms I

0 shyc iI ~ r _~o-ltl--o-

I gXcglshy~

0 -s

bull I

f 6 7718S II llt X

I l( lCbull pKJImiddotoIlo -ilrO-oo

01 lC I ~ 1 r I

o 50 100 150 200 250 o 50 100 150 200 250

DISTANCE FROM PLATE (II) DISTANCE fROM PLATE (m)

FIG 5 Comparison of the integral-equation solution (solid and dashed curves) for a plate in free space with the eigenmode solution of Annan (1974) (crosses) and the integral-equation solution of SanFilipo and Hohmann (triangles) Vertical and horizontal components of ob(t)ot are compared

697

1

3-D Modeling of TEM Scattering

- i bull ~ bull toSotio bull luhCbullbullpl f tlbullbull only unc10

~ __~___ -

_SollIon 1I1t pul

~ 10 [ bullbull

10- ~ ~ EUltotd tlbullbull oltatod tiM co~ant T 9 ItftIQ~ 08)( Ae

10 I-shy

I I

transmitter is a 160 x 160 m loop with its near wire located at - 250 m Results for six time channels are shown in Figure 3

but results are not plotted before 01 ms for the current-tube solution or after 01 ms for the pulse function solution

In the two early-time channels 001 and 003 rns the comshyplete solution compares well with the solution using only pulse functions This early time range is where the galvanic respolHe is important ~ diffusing currents in the half-space are concentrated around and channeled through the prism Howshyever beyond 003 ms the solution using only pulse functions decays faster than the complete solution (Figure 4) The lateshytime decay of the pulse solution is an exponential with a time constant of OO~ ms The empirical formula for the time conshystant of a thin plate given in Lamontagne (1975) is

r = (110 otL) 10 (33)

where L is the smaller dimension of the plate and crt is its conductivity-thickness product The time constant calculated from equation (33) for the prism in Figure 3 is 8 ms

The complete solution agrees with the current-tube solution at later times (Figure 3) and its late-time decay rate is exshy

01 10 100 ponential from Figure 4 we estimate a time constant of 8 ms

tim) This time constant agrees with that calculated from equation (33) and thus the TEM response of the prism decays as that

nO 4 Scattered vertical-field decay of the three solutions at for a thin plate in free space at later times Note that the x = -80 m in Figure 3 solution using only the current tubes appears to be constant at

VERTICAL HORIZONTAL100

Son Fllrpo end Hohmonrls solutlon( A)

A nnoll egflnmodfl solution (l( )

10 0335

MODEL

lOOOm

~oo- free space

llltegrol equation solution (-)

00 0 0 0- 0 - 0 - 0 0

0 bull amp132ms l amp po oOAJo-oo ~ ~ tE bull p

gt 33ms I

0 shyc iI ~ r _~o-ltl--o-

I gXcglshy~

0 -s

bull I

f 6 7718S II llt X

I l( lCbull pKJImiddotoIlo -ilrO-oo

01 lC I ~ 1 r I

o 50 100 150 200 250 o 50 100 150 200 250

DISTANCE FROM PLATE (II) DISTANCE fROM PLATE (m)

FIG 5 Comparison of the integral-equation solution (solid and dashed curves) for a plate in free space with the eigenmode solution of Annan (1974) (crosses) and the integral-equation solution of SanFilipo and Hohmann (triangles) Vertical and horizontal components of ob(t)ot are compared

698 Newman and Hol1mann

early times (Figure 4) This is the inductive limit of the magshynetic field of a conductor in free space (McCracken et al 1986) At the time of shutoff currents flow in a conductor to preserve the magnetic field of the transmitter that was linking

the conductor Internal checks on our solution show its consistency but are

not a guarantee of its accuracy More convincing checks are comparisons with other numerical solutions In Figure 5 we

compare model results from our solution (solid and dashed Jines) with those of SanFilipo and Hohmann (1985) (triangles) and Annan (1974) (crosses) The model is a thin vertical plate

in frcc space and the profiles are for both vertical and horishyzontal components of ab(t)iJt where b(t) = oh(t) In these results t1b(r)(i( is equivalent to the voltage induced in a small

coil provided the voltage is divided by the product of the

coils area and the number of turns it contains The 50 S plate is 600 m long 300 m in depth extent buried 30 m and is excited by a 600 m x 300 m loop SanFilipo and Hohmanns solution and Annans solution are based on a 1 A current ramp which is terminated linearly over 0165 IDS and where

the measurement limes are referred to the bottom of the ramp Our solution is for a I A current which is shut off in a step but is corrected for the ramp by an integration technique described by Fitterman and Anderson (l987) This correction is given by

~~(t) = lito Jlaquor) dr (34) to

where v and ~~ are the voltages measured for the ramp and step excitation respectively to is the ramp length and time

zero starts at the bottom of the ramp Equation (34) is evalushyated with an adaptive Gaussian quadrature procedure

Our solution is about ]5 percent larger than SanFiIipo and Hohmanns and Annans solutions in the rwc early-timeshychannels (033 and 132 rns) From 330 ms onward our solushytion agrees with that of SanFilipo and Hohmann The lateshytime decay of our solution is an exponential with a time conshystant of 19 rns which agrees with that calculated for the plate The time constant estimated from Annans solution is 21 ms which is 10 percent larger than ours This 10percent difference in the time constants results in a 25 percent difference in the field values between our solution and that of Annan at 6 ms

For another check at high contrast our solution is comshy

pared with the solution of SanFilipo and Hohmann for a cube in free space Figure 6 shows a check on the vertical fields (voltage normalized by a I m 2 single-turn coil) from I to 100 ms The transmitter position and the receivers are the same as in Figure 3 The cube is 60 m on a side and has a conductivity of 200 Sjm while its depth below the plane of the transmitter and receivers is 60 m Our solution is shown by the solid and dashed lines in Figure 6 and SanFilipo and Hohmanns solushytion is shown by the crosses Both solutions are based on a 01

ms linear ramp shutoff where the measurement times arc reshyferred to the bottom of the ramp From 1 ms to 30 ms the solutions compare favorably However beyond 30 ms SanFilshyipo and Hohmanns solution begins to decay Iaster than ours

10- 10

NshyE 10- 11

gt

---gt

10 12shy

11----11_ A ~ San Filipa anda

-_ middotA HohmannJ Solution

middotft - y- Our Solution wmiddot O - lI

~

0 ~

11

111 --0 A 0

IIbull Ti- 0~

A

0 A_ 0 10-r ~ raquo ---ltgt- 0

II ~

0 ~

- II bull I e

~f Iamp ~ 0- __ 0_

0 11 t e o II

30 ~

o middot0 -~O ~ bull bull i _ ltgt

1 11

o I

o 070

0o 0-

I

middotlI 0100

I

I I I Ibull I

-200 -180 -160 -40 -120 -00 -eo -60 -40 -20 0 20middot40 60 80 100 120 140 160 180 200

DISTANCE (m)

FIG 6 Comparison of the integral-equation solution (solid and dashed lines) for a cube in free space with the solution of Sanfilipo and Hohmann (crossesI The vertical component of 8b(t)Dt (voltage) is compared

699 3middot0 Modeling of TEM Scattering

Estimated time constants arc 31 rns for our solution and 26 ms for Sanl-ilipo and Hohmanns Both of these time conshy

slants are still below that for a sphere of equal volume 35 IDS

The time constant of a sphere is given by

r - (plloR)rr 2 (35)

where R is the radius of the sphere (Spies 1980)

Our solution can be checked easily at low contrast by comshyparing it with the Newman er al (1986) solution We used a model similar to that in Figure 3 except that the conductivity

of the prism and the resistivity of the half-space were I Srrn

and 100 Q m respectively Our solution for the vertical magshy

netic lield scattered by the prism agreed with the pulseshyfunction solution of Newman el al at all times ranging from 01 [0 30 rns We found that the late-time decay rate of the

magnetic field for both solutions was t A power-law

decay indicates that the prisms response is influenced by the conductive half-space The t 25 decay may not be entirely

due [0 a galvanic response because the inductive vortex reshy

sponse of the prism is also influenced by the half-space Kaufshy

man and Keller (19~5) showed that the vortex and galvanic responses for the magnetic field of a sphere in a conductive half-space both decay as t 15 at late times

One problem with our solution is its failure 10 give the

correct decay rates and time constants for square and nearly square plates in free space Figure 7 shows a plot of estimated

time constant versus plate length When the plate length is 800 m it is the same model as in Figure 3 The estimated and

calculated time constants differ by 26 percent for a square plate We believe that the time constant calculated for the sq uare plate [equation (33)] is the correct one because it

agrees with the time constant calculated for a thin circular

disk of equal area thickness and conductivity based on a formula given by Kaufman (197~) Therefore our solution

underestimates the time constant Sanfilipo and Hohmanns (19~1~) solution also shows this error When the plate in Figure

7 is three times as long as its depth extent the estimated and calculated time constants vary by 7 percent Agreement in the time constants is even better for larger plate lengths We find

that the error in the time constants is sensitive to the area of

the plate as well For example the plate in Figure 5 is 600 m in length and 300 m in depth extent and the time constant cstirnarcd for this plate agreed with the time constant calcushy

lated lrorn equation (33) However for a plate with a smaller

area in figure 7 (L = 160 rn] but with the same length-toshy

depth ratio as the plate in Figure 5 there is an 11 percent

diflcrence in the time constants

HIE EFFECT OF CONDUCTIVE OVERBlRDEN ON 3-D TEM R[SPONSES

In many places orebodies are buried beneath a weathered layer that is conductive This conductive overburden is a

source of geologic noise that masks the response of the target (Nabighian 1987) and removing the overburden response is often required before TEM data can be interpreted The intershy

pretation of data for mining applications is usually based 011

free-space models which are used to obtain information on the size position and conductance of the target When the

data are influenced by conductive host rock the usual practice

is to assume that the host and target responses are electroshymagnetically dccouplcd The response of the host is then subshytracted from the data Once the host response is removed the

data are interpreted with free-space models In the case of a conductive half-space SanFilipo et al (1985) showed that this

interpretation practice is not valid when the host is modershyately conductive because the host influences the TEM reshy

sponse of the target I n addition there is always the problem

of estimating the host response accurately In many surveys estimating the host response is not easy

One way to illustrate the influence of conductive overburshy

den on the detectability of the target is to compare responses

of a target in a highly resistive half-space with and without

overburden In Figure 8 we consider a 25 m thick 10 Srn prism embedded in a 10 000 nmiddot m half-space Its depth of

burial depth extent and strike length arc 100 100 and 800 m respectively The prism is centered at x = 0 m in the profile the transmitter loop is 500 m on a side with its near wire

10

TillCIt

_-4---------shye---- ---shy shy

TI -ltwiltlated r I

6

80 s~

Fro 7 A plot of the estimated time constant of a thin plate as a function of the strike length of the plate The depth extent and the conductance of the vertical plate are amp0 m and 800 S The correct time constant for the plate is 8 rns

wshy1 ~ -__df-------=~ -=-----~~ 1_-------~ Iv~ -

1 shyI)-It

~ ------ -cbullbull -~ ~

~ IQ

IS z-gt- _ ~ ILI bullbull - -

tr

IOmiddotIJ

sobullbull ~ rgt elf n 1)

doI~P I D IC Ibull

~ ~ ~ ~ ~ 0 ~ ~ ~ bull 00

DISTANCE FIlOII CCNOUCTQR CEITER (II

FIG K Profile of the vertical component of cb(t)iot (voltage) calculated for a prism in a 10000 nmiddot m half-space

700 Newman and Hohmann

located at x = -250 m Profiles in Figure 8 clearly show the position of the prism from early to late times

The total half-space and scattered decay curves at x =

- gO m (Figure 9) show that the response of the 10 000 nmiddot m half-space is not important in the interpretation or the prism from 1 to 20 ms The scattered response is the difference beshytween the total and layered half-space (I-Dl responses The late-time decay rate of the prism in Figure 9 is that of a thin plate in free space with an estimated time constant of 33 ms

Figure 10 shows the profound effect of conductive overburshyden on target detectability when the prism is buried beneath a 50 m thick 10 nmiddot rn overburden (compare Figure 8 with Figure 10) Conductive overburden limits the time range in which the prism is detectable the prism is detectable in only three of the time channels shown 10 15 and 20 ms This suppression or delay of the prism response at early times has been called overburden blanking by Lamontagne (1975)

The decay curves in Figure 1 (x = -SO m) clearly show that the response of the overburden is significant and always larger than the scattered response of the 3-D body The scattered response of the prism in Figure 11 decays exponentially at late times and the estimated time constant is 33 rns which is identical to that of the prism in the 10 000 Q m half-space without overburden Thus if the overburden response is subshytracted from the profile in Figure 10 the correct time constant of the prism will be obtained

However note that superposition of the free-space response of the prism and the overburden response is a first-order apshyproximation to the actual response of the prism with overburshydC1 (Nabighian (987) Figure 12 shows that the scattered response of the prism with overburden is shifted and has a larger amplitude than that of the prism in the 10 000 nmiddot m half-space 1 late times This amplitude shift is uniform across the profile at late times Thus it appears that an accurate estimate of the depth of the prism under the overburden can be obtained with half-width or peak-to-peak free-space rules for thin plates (cf Gallagher et aI 1985) However these rules require an estimate of the size of the plate before they can be applied Moreover the increase in the amplitude of the prism response affects free-space modeling of plate size (cf Gallashygher et al 1985) estimates of the depth of the plate and its conductance are also affected Note that the product of the prisms conductance and its depth extent is known since the time constant estimated [rom the data is that of a plate [equashytion (33)] Therefore the conductance and depth extent are constrained in an interpretation

Before 5 ms there is a fundamental difference between the responses of the prism in Figure 12 The voltage sounding with the overburden has a sign reversal between 2 and 3 ms caused by a buildup in the magnetic field of the prism at early times The magnetic field builds up because the electric and magnetic fields of the source must first penetrate the conducshytive media and subsequently build up at the prism If the prism is in free space there is no buildup in its magnetic field the magnetic field decays from an initial maximum at the time the current is shut ofTin the transmitter This maximum called the inductive limit is the initial condition required by Farashydays law for preserving the magnetic field of the transmitter that was linking the prism before shutoff In the case of the prism in the 10 000 Q m half-space a sign reversal in the voltage does occur but at extremely early times and is not

shown in Figures 9 and 12 Unike free space a 10 000 nmiddot m half-space still has finite resistivity

Figure 13 shows the response of the prism with overburden for a basement resistivity of 1000 Q m As in Figure 10 the response of the prism is suppressed and not detectable until to ms At 30 ms the prism is no longer detectable Thus the luyer-ed-host response must be- removed to interpret the proshyfile If the basement resistivity is decreased to 100 nmiddot m (not shown) there is no significant increase in the detectability of the prism due to current channeling

Figure 14 compares the scattered responses of the prism with overburden for 10000 1000 and tOO ( m basement resistivities elt x = -RO m Peak responses in the decay curves occur at early time before the curves change sign These peaks are interpreted as due to current channeling and correspond to the fastest rate at which host currents build up at the prism The largest peak occurs for the most conductive basement 100 nmiddot m However the peak response in the 100 Q m baseshyment docs not significantly increase the detectability of the prism there is an increase in the layered-host response as the basement resistivity is decreased Since host currents take longer to penetrate a more resistive basement and build up at the prism the decay curve of the 1000 nm basement changes sign earlier than does the decay curve of the 10 000 nmiddot m basement However this is not the case ill the 100 nmiddot III baseshyment where the decay curve changes sign later than does the decay curve of the 1000 nmiddot m basement Even though currents penetrate the basement earlier when the basement conducshytivity is increased they also require longer to build to a maxishymum at the prism if the basement is very conductive

Starting from 10 ms the responses and time constants of the prism arc identical for the 10 000 and 1000 nmiddot m basements However the responses and time constants are nut the same for the 100 U m basement The decay rate of the prism in the 100 nmiddot m basement is not exponential Rather its decay is an inverse power law that falls off as t -4 for the latest times shown in Figure 14 This decay is consistent with the late-time response of a confined conductor in a conductive medium where the late-time response can be represented by an inverse power law ie O( Ir) which is characteristic of the host roek (Nabighian 1997) Free-space interpretations of the prism in the Inon and I() 000 nmiddot m basements are possible at later times because their responses decay exponentially However as emphasized above free-space interpretations of the prism under overburden are biased since the response of the prism is influenced by its overburden at all times

Sometimes an orebody is in contact with the overburden in Figure 15 we show the response of the prism for such a case When the prism is in the 1000 Q m basement its detectability decreases compared to the dctectability of the prism that is detached from the overburden (compare Figure 13 with Figure 15) This decrease in dctectability results from the inshycrease in the overburden response which is caused by the increased thickness of the overburden When the basement resistivity is changed to 100 nmiddot m (not shown) we still do not sec an increase in the dctectability of the prism One might ha C suspected there would be an increase in delectability because of vertical current channeling where current is pulled down through the overburden and channeled into the prism Newman et al (l986) have shown that for a 1 Sm prism located closer to the surface (40 m) and in contact with the 10

701 3-D Modeling of TEM Scattering

I

Y~ Il-D

l I

I I I I I I I I I

IO8~ 1081- I I

scattJ~

6

II

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0--0d

10-I 10-l Nla

~

10-10 10-6 ~

10 -II0-11

10 12 bull

1 m10 100

t (ms) ttrnl

FIG 9 Decay curves of the total (triangles) scattered (circles) FI(i 11 Decay curves of the total (triangles) scattered (circles) and half-space (solid dots) vertical field cb(f)t at x = - 80 rn and layered half-space (solid dots) vertical field (~b(t)(ll at x = for the model of Figure 8 -~o m for the model of Figure 10

-------- -

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1 r

IO-l

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la-I

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gt 10-1

-~

I 10 ~ gt ~

IO-I~ rshy 75

20

MODELIO-~ ---------1urQCr

DISTANCE FRON CONDUCTOR CENTER (m) Fro to Profile of the vertical component of (1b(t)il( (voltage) calculated for a prism in a 10000 n m half-space with a

50 m thick 10 Q m overburden

I 50~------- --- Q~i~~ 10000(1middot StSOO

or- lJO DBmiddot 100~

30

Near ~r1 of loop

-200 -160 -120 -80 -40 Q 40 80 120 160 20~

I

--------------

--

702 Newman and Hohmann

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oloooonmiddot

10-12 10-12 1 I 10 100 I I 10

t(IU) tlIlIS)

FI( 14 Comparison of the scattered vertical field decayFIG 12 The scattered vertical field tb(r)rL at 1C= -80- m cbtV(~1 for variable basement resistivity with overburden at

with and without a 50 m thick to Omiddot In overburden = - 80 m The Ionmiddot 111 overburden is 50 111 thick The basement resistivity varies from to 000 to 100 Q m

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e

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MO)EL

10 11

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SLmiddot800 DEmiddot100 DBmiddot100m0 cJg 0 f leop

-200 -160 -120 middotao -40 a 40 BO 120 160 200

DISTANCE fROM CONDUCTOR CENTER (m)

FilL 3 Profile of the vertical component or tb(l)(l (voltage) calculated for a prism in a tooO nmiddot m half-space with a 50 m thick 10 nmiddot m overburden

100

-----------

703 3-D Modeling of TEM Scattering

n m overburden there is a pronounced enhancement in the prisms response However our findings show when the overshyburden is 100 III thick there wilt be no dramatic increase in the delectability of the prism even when it is in contact with the overburden

R[SOLLITION OF TWO CONDUCTIVE PRISMS WITH TFM SURVEYS

An important problem in mineral exploration for massive sulfides is the resolution and possible interpretational ambishy

guities of multiple conductors Very few theoretical modelshystudies have been published on the FM response of multiple conductors 1110st studies have been based all scale models

Ogilvy (J 983) published scale-model results on the horizontal resolution for the TEM coincident-loop configuration of two identical thin-sheet conductors that were vertically oriented in free space His findings show that when the separation beshytween the conductors is less than one-quarter of the loops diameter individual conductors cannot be resolved Gupta et aJ 19871 recently conducted scale-model experiments on the resolution of two vertical half-planes for the horizontal loopshyloop system Their results show that closely spaced scatterers arc resolved only when the separation between the half-planes is greater than the transmitter-receiver separation

Numerical models can also contribute interpretational inshy~ight into the lateral resolution and response of multiple conshyductors Obviously many multiple targets for different host layering and survey configurations can be modeled with our solution However our study is intended only as a preliminary

survey so we study the TEM response of two prisms in a resistive half-space for the fixed-loop and central-loop configushyrations We also discuss the resolution uf both conductors paying particular attention to survey design

In Figure 16 we show the vertical response (total voltage) of two prisms in a 1000 Q m half-space for a fixed-loop configushyration The profiles arc at seven different times ranging from 05 to 30 ms The prisms are identical in size depth and conductivity and are the same as the prism in Figure 8 One prism is positioned at x = 0 m and the other is at x = 80 m The fixed-loop transmitter is again 500 m on a side with its neurwirehxnrtedar x = -250m

At early times (05 and 1 ms] and at late times (which inshy

elude 15 20 ami 30 rns) the response of the tOOO n 111 halfshyspace is visible in figure 16 The half-space response is imporshytant at late times because it has a slower decay than the response of the two prisms If the response of the half-space is not considered in an interpretation the migrating crossover at intermediate and (ate times in Figure (6 will appear to indishycate a dipping conductor However once the half-space reshysponse is subtracted from the profile a vertical-plate conducshytor is interpreted at x = 30 m with a time constant of 37 ms This time constant is the same as that observed when the prisms arc together in free space The fixed-loop configuration clearly fails to distinguish the individual prisms even when the half-space response is removed Spies and Parker (1984) have demonstrated with scale-model experiments similar limishy

tations of large fixed-loop surveys The mutual interaction or the TEM fields of the two prisms

can he investigated by comparing responses with and without

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IO- T

3 shy

10

N- 9

I 1OE

gtshy10

15bullbull

s bullbull ----_~

--

J

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Win_

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MO_Qf~

10middot1

iocon-Nealt d~e 01 loop

I

middot200 -160 -120 middot80 -40 40 80 120 160 200

DISTANCE FROM CONDUCTOR CENTER ltII)

FIG 15 Profile of the vertical component of tb(lliltt (voltage) calculated for a prism in a tooo Q m half-space in contact with a 100 m thick 10 n m overburden

704 Newman and Hohmann

the interaction In Figure 17 the scattered response of each

prism in its own half-space is superimposed and compared

with the scattered response which includes the mutual interacshytion Over the time range of 05 to 5 IDS the superimposed response in the vertical component is larger (except near sign reversals) than the scattered response observed when the fields of the two prisms interact However at later times (greater than 5 ms) this pattern reverses Both responses in Figure 17 decay exponentially at late times and we estimate a time conshystant of 33 ms for the superimposed response which is 11 percent smaller than the time constant estimated from the

response with the mutual interaction 37 ms The II percent difference in the time constants reflects the 66 percent disparshyity between the two responses at _~O ms as shown in Figure Il

As suggested by Spies and Parker (1(84) a survey configushy

ration where the source moves along the profile can improve the lateral resolution or the two prisms For a central-loop configuration with loops 25 m OIl a side both prisms are resolved at I 3 and 5 ms as shown in Figure 18 The two

minima ill the profilesindicate the locations of both prisms at these times At later times interpretational ambiguities could arise because the two separate minima are replaced by a single minimum located half way between the two prisms This single minimum could be interpreted as an isolated conductor Overall the shapes of the profiles in Figure 18 are similar to those studied by Ogilvy (1983) for the coincident-loop configushy

ration Note that after I ms the half-space has little influence on the response of the prisms in Figure 18 When the halfshyspace response is removed the response of the prisms is the same as that observed in free space

till 5NS shy

10middot

Jshy

10- 9

3

til 5E-==shy O-ID

shy-==

15

10-11

10

3010-11

NOf edge of loop

The time constant estimated from the scattered response of the two prisms at late time is dependent upon the type of survey Unlike the case for a fixed-loop configuration the time constant varies with its position along the profile for the central-loop configuration When central-loop stations are loshycated away from the two prisms the estimated time constant is the same as in the fixed-loop case 37 rns However when a central-loop station approaches both prisms the time conshystant decreases and its smallest value is 29 ms at x = 40 m The time constant decreases because the transmitter induces eddy currents in the prisms that flow in opposite directions Since these eddy currents oppose each other the magnetic

field of (he eddy currents in one prism enhances the decay of the eddy currents in the other prism Consequently a time constant is produced which is smaller than that for each prism in free space 33 ms However the eddy currents also produce a larger time constant when the transmitter is away from the prisms Here the eddy currents in both prisms flow in the same direction and they tend to sustain each other with their magshynetic fields The variation in the time constant indicates the presence of multiple conductors since the time constant of a single conductor does not vary along the profile Finally the variation in the time constant also explains the peak and trough in the profiles between the prisms at early and late times In a resistive host the response of a poor conductor is larger than that of a good conductor at early time (Dyck and West 19~4) and smaller at later time

An important conclusion from our study is the inability of Iurge fixed-loop surveys to resolve multiple conductors Thereshyfore when an anomaly is located with this type of survey we

~-------

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I

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shy

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I

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1 ~(fot

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-200 -160 -120 -80 -40 0 40 80 126 ISO 200 2~0 280

DISfANCE (m) Fit 16 Profile of the vertical component of (1b(l)tE (voltage) calculated for two prisms in a WOO Omiddot m half-space for a

fixed-loop configuration One prism is located at x = am and the other prism is at x = 80 m

705 3-D Modeling of TEM Scattering

MODEL

bullbull II III 41 a 0

------- shy0

IG-I 0 I

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i I ----__

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N- 10- 10

II laquo ~ - 10I

~ =shy

10middot1

xmiddotODt xaoIO-I~ fiJrfCJ(~

---- shy U 105 ~ JS - iocoa SLmiddot800

0[100 n8middot100

I

-lOO -160 -120 -80 -40 40 80 120 160 OO l40 l80

DISTANCE (m)

FIG 17 Comparison of profiles for the scattered vertical component of (Ib(t)Dt (voltage) calculated by superimposing the scattered response of each prism in its own half-space (circles) with the response which includes mutual interaction between the prisms (curves)

1010

1=gt shy 10- 11 --gt

10 -1pound

t~ --------__

lmJ

3mJ

5ms

MODEL

XmiddotOIll xmiddot80m-Ll surface

o O middot0Sm t bull 25m

1000nm SLmiddot800m DElOOm DBlOOm

25 m x 25m central loop~

-200 -160 -120 -ao -40 120 160 200 240

FI( r8 Profile of the vertical component of (1b(t)(~[ (voltage] calculated for two prisms in a LOOO Q m half-space for a central-loop configuration The loop dimensions are 25 m by 25 m

706 Newman and Hohmann

recommend (if possible) further work with a moving-source survey because it is more likely to eliminate interpretational ambiguities of multiple conductors and can provide precise information for locating drill holes

CONCLUDING REMARKS

We have presented a 3-D integral-equation solution for the TEM responses of prisms in a layered half-space In summary our solution is based on a Galerkin solution for the scattering current where the weighting functions are the basis functions It is necessary that the solution include a set nf basis and weight functions that are divergence-free if model results are to be accurate at large background resistivities The

divergence-free basis functions are used to model eddy curshyrents which are closed in the prisms and hence simulate the inductive responses of the prisms in a very resistive earth More importantly divergence-free weight functions eliminate the charge operator from part of the solution by integrating it over rectangular paths Thus the solution satisfies the freeshyspace boundary condition for no current leaks and the reshysulting solution is stable for a host with any resistivity

We used current-tube functions as the divergence-free basis functions because they are simple and acceptable for our comshyputational limitations The current-tube functions are crude and restrict the geometry of the 3-D bodies to prisms Howshyever they are effective and allow modeling prisms of moderate geometric extent An exception where our solution does not perform well is for square and nearly square plates at a high resistivity contrast In this case the time constants of the plates are underestimated We believe the accuracy and genershyality of our solution will improve with a more sophisticated divergence-free basis function

The usefulness of OUT solution has been demonstrated with TEM model studies We investigated 3-D responses under the influence of conductive overburden and made an evaluation of the resolution of two conductors based on different survey configurations However our solution is more versatile than we have illustrated since it is valid for any number of layers and prisms and is designed for many different types oC source and receiver configurations Important exploration problems where the present solution can be applied include 3-D airshyborne TFM model studies and evaluations of the effect of a conductive host on 3-D TEM borehole responses Finally the solution is not restricted to TEM applications Because its basic formulation is in the frequency domain it can be applied to frequency-domain exploration problems as well

ACK-NO-WLEDGMpound~TS

This paper is dedicated to the memory of Frank Frischshyknecht We are grateful to him and the US Geological Survey for making the Surveys VAX IInO computer available for much of this work We are also indebted to Walter Anderson for his help in installing and running the programs Financial support was provided by Amoco Production Co ARCO Oil and Gas Co Chevron Resources Company CRA Exploration Pty Ltd Standard Oil Production Co and Unocal Corp in conjunction with the University of Utah EM Modeling Reshysearch Project

REFERENCES

Annan A P 1974 The equivalent source method for electromagnetic scattering analysis and its geophysical applications PhD thesis Memorial lJniv of Newfoundland

Dyck A V and West G F 1984 The role of simple computer models in interpretations of wide-band drill-hole electromagnetic surveys in mineral exploration Geophysics 49 957-980

Eaton P A and Hohmann G W 1987 Approximate inversion for lransient electromagnetic soundings Phys of the Earth and Plan Int in press

Fitlcrman D Y and Anderson W l 1987 Effect of transmitter turn-off on transient soundings Geoexplor 2413-146

Gallaghtr P R Ward S H and Hohmann G W 19R5 A model study of a thin plate in free space for the EM37 transient electroshymagnetic system (ieophysies 50 ()())-lDlq

Gupta O P Kakirde S T and Negi J G bull 1987 Scale model experiments In study low frequency electromagnetic resolution of multiple conductors Trans Geoscience and Remote Sensing Inst of Electr Electron Eng in press

Hanneson J E and West G F 1984 The horizontal-loop electroshymagnetic response of a thin plate in a conductive earth Part _shyComputat ional method Geophysics 49411-420

Harrington R F 1968 Field computation by moment methods The MacMillan Pub Co

Hohmann G W 19lB Three-dimensional EM modeling Geophys Surv 6 27 53

Kaufman A A 1978 Frequency and transient responses of EM fields created by currents in confined conductors Geophysics 43 1002-1010

19~5 Distribution of alternating electrical charges in a conshyductive medium Geophys Prosp 33171-184

Kaufman A A and Keller G V 1985 Inductive mining prospectshying Part I Theory ElsevierScience Pub Co

Lajoie J J and West G r 1976 The electromagnetic response of a conductive inhomogeneity in a layered earth Geophysics 4l 1133-1156

Lamontagne Y 1975 Applications of wideband time-domain elecshytrornagnetic measurements in mineral exploration PhD thesis UnivofToronto

Lee T S 19~ I Transient electromagnetic response of a sphere in a layered medium Pure Appl Geophys 119309-338shy

MLCracken K G Oristaglio 1 L and Hohmann U W 19~6 A comparison of electromagnetic exploration systems Geophysics SI XIO XIX

Nabighian M N 1987 Inductive time-domain electromagnetic methods EM mining volume Soc Explor Geophys in press

Newman G A Anderson W L and Hohmann G W 1987 Intershyprerarion of transrent electromagnetic soundings over threeshydimensional structures for the central-loop configuration Geophys J Roy Astr Socbull 89 l~9-(14

Newman G A Hohmann G W and Anderson W L 19~6 Transhysient electromagnetic response or a three-dimensional body in a layered earth Geophysics 51 160~- [627

Ogd~y K D 1910 A model study of the transient electromagnetic coincident loop technique Geoexplor 21 231-264

Petrick W R 11)84A fully internal hybrid technique for calculating electromagnetic SCattering from three-dimensional bodies in the earth PhD thesis Univ of Utah

Sanfilipo W A and Hohmann G W 1985 Integral equation solushyion for [he transient electromagnetic response of a threeshydimensional body in a conductive half-space Geophysics 50 798shy~Ol

S1I1hlipo W A I-aton P A and Hohmann G W 1985 The effect raquof a conductive half-space on the transient electromagnetic reshysponse of a three-dimensional body Geophysics 50 1144-1162

Spies B R 1980 TE~ in Australian conditions Field examples and model studies PhD thesis Macquarie Univ

Spies B R and Parker P 0- l9H4~ Limirarions of large-loop transhysient surveys in conductive terrains Geophysics 49 902 -912

Tripp A C ami Hohmann G W 1984 Block diagonalization of the electromagnetic impedance matrix of a symmetric body using group theory Inst of Elect and Electron Eng Trans Geosci and Remote Sensing GE-22 6269

Van Bladcl J 1961 SIJOlC remarks un Greens dyadic for infinite SPltlIC Trans on Antennas and Propagation Insc Electr Electron Eng 9 563middot 566

Wannamaker P E Hohmann G W and Sanfilipo W A 1984 Electromagnetic modeling of three-dimensional bodies in layered cart h- using integral equations Geophysics 49 60-77

Wcidclt P 198 l Dipole induction in a thin plate with host medium arid ovcrburdcn Research Project NTS 10 no R9727 Federal Inst ior Earth Sciences and Ravv Materials Hannover West Germany

699 3middot0 Modeling of TEM Scattering

Estimated time constants arc 31 rns for our solution and 26 ms for Sanl-ilipo and Hohmanns Both of these time conshy

slants are still below that for a sphere of equal volume 35 IDS

The time constant of a sphere is given by

r - (plloR)rr 2 (35)

where R is the radius of the sphere (Spies 1980)

Our solution can be checked easily at low contrast by comshyparing it with the Newman er al (1986) solution We used a model similar to that in Figure 3 except that the conductivity

of the prism and the resistivity of the half-space were I Srrn

and 100 Q m respectively Our solution for the vertical magshy

netic lield scattered by the prism agreed with the pulseshyfunction solution of Newman el al at all times ranging from 01 [0 30 rns We found that the late-time decay rate of the

magnetic field for both solutions was t A power-law

decay indicates that the prisms response is influenced by the conductive half-space The t 25 decay may not be entirely

due [0 a galvanic response because the inductive vortex reshy

sponse of the prism is also influenced by the half-space Kaufshy

man and Keller (19~5) showed that the vortex and galvanic responses for the magnetic field of a sphere in a conductive half-space both decay as t 15 at late times

One problem with our solution is its failure 10 give the

correct decay rates and time constants for square and nearly square plates in free space Figure 7 shows a plot of estimated

time constant versus plate length When the plate length is 800 m it is the same model as in Figure 3 The estimated and

calculated time constants differ by 26 percent for a square plate We believe that the time constant calculated for the sq uare plate [equation (33)] is the correct one because it

agrees with the time constant calculated for a thin circular

disk of equal area thickness and conductivity based on a formula given by Kaufman (197~) Therefore our solution

underestimates the time constant Sanfilipo and Hohmanns (19~1~) solution also shows this error When the plate in Figure

7 is three times as long as its depth extent the estimated and calculated time constants vary by 7 percent Agreement in the time constants is even better for larger plate lengths We find

that the error in the time constants is sensitive to the area of

the plate as well For example the plate in Figure 5 is 600 m in length and 300 m in depth extent and the time constant cstirnarcd for this plate agreed with the time constant calcushy

lated lrorn equation (33) However for a plate with a smaller

area in figure 7 (L = 160 rn] but with the same length-toshy

depth ratio as the plate in Figure 5 there is an 11 percent

diflcrence in the time constants

HIE EFFECT OF CONDUCTIVE OVERBlRDEN ON 3-D TEM R[SPONSES

In many places orebodies are buried beneath a weathered layer that is conductive This conductive overburden is a

source of geologic noise that masks the response of the target (Nabighian 1987) and removing the overburden response is often required before TEM data can be interpreted The intershy

pretation of data for mining applications is usually based 011

free-space models which are used to obtain information on the size position and conductance of the target When the

data are influenced by conductive host rock the usual practice

is to assume that the host and target responses are electroshymagnetically dccouplcd The response of the host is then subshytracted from the data Once the host response is removed the

data are interpreted with free-space models In the case of a conductive half-space SanFilipo et al (1985) showed that this

interpretation practice is not valid when the host is modershyately conductive because the host influences the TEM reshy

sponse of the target I n addition there is always the problem

of estimating the host response accurately In many surveys estimating the host response is not easy

One way to illustrate the influence of conductive overburshy

den on the detectability of the target is to compare responses

of a target in a highly resistive half-space with and without

overburden In Figure 8 we consider a 25 m thick 10 Srn prism embedded in a 10 000 nmiddot m half-space Its depth of

burial depth extent and strike length arc 100 100 and 800 m respectively The prism is centered at x = 0 m in the profile the transmitter loop is 500 m on a side with its near wire

10

TillCIt

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TI -ltwiltlated r I

6

80 s~

Fro 7 A plot of the estimated time constant of a thin plate as a function of the strike length of the plate The depth extent and the conductance of the vertical plate are amp0 m and 800 S The correct time constant for the plate is 8 rns

wshy1 ~ -__df-------=~ -=-----~~ 1_-------~ Iv~ -

1 shyI)-It

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~ IQ

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DISTANCE FIlOII CCNOUCTQR CEITER (II

FIG K Profile of the vertical component of cb(t)iot (voltage) calculated for a prism in a 10000 nmiddot m half-space

700 Newman and Hohmann

located at x = -250 m Profiles in Figure 8 clearly show the position of the prism from early to late times

The total half-space and scattered decay curves at x =

- gO m (Figure 9) show that the response of the 10 000 nmiddot m half-space is not important in the interpretation or the prism from 1 to 20 ms The scattered response is the difference beshytween the total and layered half-space (I-Dl responses The late-time decay rate of the prism in Figure 9 is that of a thin plate in free space with an estimated time constant of 33 ms

Figure 10 shows the profound effect of conductive overburshyden on target detectability when the prism is buried beneath a 50 m thick 10 nmiddot rn overburden (compare Figure 8 with Figure 10) Conductive overburden limits the time range in which the prism is detectable the prism is detectable in only three of the time channels shown 10 15 and 20 ms This suppression or delay of the prism response at early times has been called overburden blanking by Lamontagne (1975)

The decay curves in Figure 1 (x = -SO m) clearly show that the response of the overburden is significant and always larger than the scattered response of the 3-D body The scattered response of the prism in Figure 11 decays exponentially at late times and the estimated time constant is 33 rns which is identical to that of the prism in the 10 000 Q m half-space without overburden Thus if the overburden response is subshytracted from the profile in Figure 10 the correct time constant of the prism will be obtained

However note that superposition of the free-space response of the prism and the overburden response is a first-order apshyproximation to the actual response of the prism with overburshydC1 (Nabighian (987) Figure 12 shows that the scattered response of the prism with overburden is shifted and has a larger amplitude than that of the prism in the 10 000 nmiddot m half-space 1 late times This amplitude shift is uniform across the profile at late times Thus it appears that an accurate estimate of the depth of the prism under the overburden can be obtained with half-width or peak-to-peak free-space rules for thin plates (cf Gallagher et aI 1985) However these rules require an estimate of the size of the plate before they can be applied Moreover the increase in the amplitude of the prism response affects free-space modeling of plate size (cf Gallashygher et al 1985) estimates of the depth of the plate and its conductance are also affected Note that the product of the prisms conductance and its depth extent is known since the time constant estimated [rom the data is that of a plate [equashytion (33)] Therefore the conductance and depth extent are constrained in an interpretation

Before 5 ms there is a fundamental difference between the responses of the prism in Figure 12 The voltage sounding with the overburden has a sign reversal between 2 and 3 ms caused by a buildup in the magnetic field of the prism at early times The magnetic field builds up because the electric and magnetic fields of the source must first penetrate the conducshytive media and subsequently build up at the prism If the prism is in free space there is no buildup in its magnetic field the magnetic field decays from an initial maximum at the time the current is shut ofTin the transmitter This maximum called the inductive limit is the initial condition required by Farashydays law for preserving the magnetic field of the transmitter that was linking the prism before shutoff In the case of the prism in the 10 000 Q m half-space a sign reversal in the voltage does occur but at extremely early times and is not

shown in Figures 9 and 12 Unike free space a 10 000 nmiddot m half-space still has finite resistivity

Figure 13 shows the response of the prism with overburden for a basement resistivity of 1000 Q m As in Figure 10 the response of the prism is suppressed and not detectable until to ms At 30 ms the prism is no longer detectable Thus the luyer-ed-host response must be- removed to interpret the proshyfile If the basement resistivity is decreased to 100 nmiddot m (not shown) there is no significant increase in the detectability of the prism due to current channeling

Figure 14 compares the scattered responses of the prism with overburden for 10000 1000 and tOO ( m basement resistivities elt x = -RO m Peak responses in the decay curves occur at early time before the curves change sign These peaks are interpreted as due to current channeling and correspond to the fastest rate at which host currents build up at the prism The largest peak occurs for the most conductive basement 100 nmiddot m However the peak response in the 100 Q m baseshyment docs not significantly increase the detectability of the prism there is an increase in the layered-host response as the basement resistivity is decreased Since host currents take longer to penetrate a more resistive basement and build up at the prism the decay curve of the 1000 nm basement changes sign earlier than does the decay curve of the 10 000 nmiddot m basement However this is not the case ill the 100 nmiddot III baseshyment where the decay curve changes sign later than does the decay curve of the 1000 nmiddot m basement Even though currents penetrate the basement earlier when the basement conducshytivity is increased they also require longer to build to a maxishymum at the prism if the basement is very conductive

Starting from 10 ms the responses and time constants of the prism arc identical for the 10 000 and 1000 nmiddot m basements However the responses and time constants are nut the same for the 100 U m basement The decay rate of the prism in the 100 nmiddot m basement is not exponential Rather its decay is an inverse power law that falls off as t -4 for the latest times shown in Figure 14 This decay is consistent with the late-time response of a confined conductor in a conductive medium where the late-time response can be represented by an inverse power law ie O( Ir) which is characteristic of the host roek (Nabighian 1997) Free-space interpretations of the prism in the Inon and I() 000 nmiddot m basements are possible at later times because their responses decay exponentially However as emphasized above free-space interpretations of the prism under overburden are biased since the response of the prism is influenced by its overburden at all times

Sometimes an orebody is in contact with the overburden in Figure 15 we show the response of the prism for such a case When the prism is in the 1000 Q m basement its detectability decreases compared to the dctectability of the prism that is detached from the overburden (compare Figure 13 with Figure 15) This decrease in dctectability results from the inshycrease in the overburden response which is caused by the increased thickness of the overburden When the basement resistivity is changed to 100 nmiddot m (not shown) we still do not sec an increase in the dctectability of the prism One might ha C suspected there would be an increase in delectability because of vertical current channeling where current is pulled down through the overburden and channeled into the prism Newman et al (l986) have shown that for a 1 Sm prism located closer to the surface (40 m) and in contact with the 10

701 3-D Modeling of TEM Scattering

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FIG 9 Decay curves of the total (triangles) scattered (circles) FI(i 11 Decay curves of the total (triangles) scattered (circles) and half-space (solid dots) vertical field cb(f)t at x = - 80 rn and layered half-space (solid dots) vertical field (~b(t)(ll at x = for the model of Figure 8 -~o m for the model of Figure 10

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50 m thick 10 Q m overburden

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--------------

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702 Newman and Hohmann

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FI( 14 Comparison of the scattered vertical field decayFIG 12 The scattered vertical field tb(r)rL at 1C= -80- m cbtV(~1 for variable basement resistivity with overburden at

with and without a 50 m thick to Omiddot In overburden = - 80 m The Ionmiddot 111 overburden is 50 111 thick The basement resistivity varies from to 000 to 100 Q m

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DISTANCE fROM CONDUCTOR CENTER (m)

FilL 3 Profile of the vertical component or tb(l)(l (voltage) calculated for a prism in a tooO nmiddot m half-space with a 50 m thick 10 nmiddot m overburden

100

-----------

703 3-D Modeling of TEM Scattering

n m overburden there is a pronounced enhancement in the prisms response However our findings show when the overshyburden is 100 III thick there wilt be no dramatic increase in the delectability of the prism even when it is in contact with the overburden

R[SOLLITION OF TWO CONDUCTIVE PRISMS WITH TFM SURVEYS

An important problem in mineral exploration for massive sulfides is the resolution and possible interpretational ambishy

guities of multiple conductors Very few theoretical modelshystudies have been published on the FM response of multiple conductors 1110st studies have been based all scale models

Ogilvy (J 983) published scale-model results on the horizontal resolution for the TEM coincident-loop configuration of two identical thin-sheet conductors that were vertically oriented in free space His findings show that when the separation beshytween the conductors is less than one-quarter of the loops diameter individual conductors cannot be resolved Gupta et aJ 19871 recently conducted scale-model experiments on the resolution of two vertical half-planes for the horizontal loopshyloop system Their results show that closely spaced scatterers arc resolved only when the separation between the half-planes is greater than the transmitter-receiver separation

Numerical models can also contribute interpretational inshy~ight into the lateral resolution and response of multiple conshyductors Obviously many multiple targets for different host layering and survey configurations can be modeled with our solution However our study is intended only as a preliminary

survey so we study the TEM response of two prisms in a resistive half-space for the fixed-loop and central-loop configushyrations We also discuss the resolution uf both conductors paying particular attention to survey design

In Figure 16 we show the vertical response (total voltage) of two prisms in a 1000 Q m half-space for a fixed-loop configushyration The profiles arc at seven different times ranging from 05 to 30 ms The prisms are identical in size depth and conductivity and are the same as the prism in Figure 8 One prism is positioned at x = 0 m and the other is at x = 80 m The fixed-loop transmitter is again 500 m on a side with its neurwirehxnrtedar x = -250m

At early times (05 and 1 ms] and at late times (which inshy

elude 15 20 ami 30 rns) the response of the tOOO n 111 halfshyspace is visible in figure 16 The half-space response is imporshytant at late times because it has a slower decay than the response of the two prisms If the response of the half-space is not considered in an interpretation the migrating crossover at intermediate and (ate times in Figure (6 will appear to indishycate a dipping conductor However once the half-space reshysponse is subtracted from the profile a vertical-plate conducshytor is interpreted at x = 30 m with a time constant of 37 ms This time constant is the same as that observed when the prisms arc together in free space The fixed-loop configuration clearly fails to distinguish the individual prisms even when the half-space response is removed Spies and Parker (1984) have demonstrated with scale-model experiments similar limishy

tations of large fixed-loop surveys The mutual interaction or the TEM fields of the two prisms

can he investigated by comparing responses with and without

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DISTANCE FROM CONDUCTOR CENTER ltII)

FIG 15 Profile of the vertical component of tb(lliltt (voltage) calculated for a prism in a tooo Q m half-space in contact with a 100 m thick 10 n m overburden

704 Newman and Hohmann

the interaction In Figure 17 the scattered response of each

prism in its own half-space is superimposed and compared

with the scattered response which includes the mutual interacshytion Over the time range of 05 to 5 IDS the superimposed response in the vertical component is larger (except near sign reversals) than the scattered response observed when the fields of the two prisms interact However at later times (greater than 5 ms) this pattern reverses Both responses in Figure 17 decay exponentially at late times and we estimate a time conshystant of 33 ms for the superimposed response which is 11 percent smaller than the time constant estimated from the

response with the mutual interaction 37 ms The II percent difference in the time constants reflects the 66 percent disparshyity between the two responses at _~O ms as shown in Figure Il

As suggested by Spies and Parker (1(84) a survey configushy

ration where the source moves along the profile can improve the lateral resolution or the two prisms For a central-loop configuration with loops 25 m OIl a side both prisms are resolved at I 3 and 5 ms as shown in Figure 18 The two

minima ill the profilesindicate the locations of both prisms at these times At later times interpretational ambiguities could arise because the two separate minima are replaced by a single minimum located half way between the two prisms This single minimum could be interpreted as an isolated conductor Overall the shapes of the profiles in Figure 18 are similar to those studied by Ogilvy (1983) for the coincident-loop configushy

ration Note that after I ms the half-space has little influence on the response of the prisms in Figure 18 When the halfshyspace response is removed the response of the prisms is the same as that observed in free space

till 5NS shy

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The time constant estimated from the scattered response of the two prisms at late time is dependent upon the type of survey Unlike the case for a fixed-loop configuration the time constant varies with its position along the profile for the central-loop configuration When central-loop stations are loshycated away from the two prisms the estimated time constant is the same as in the fixed-loop case 37 rns However when a central-loop station approaches both prisms the time conshystant decreases and its smallest value is 29 ms at x = 40 m The time constant decreases because the transmitter induces eddy currents in the prisms that flow in opposite directions Since these eddy currents oppose each other the magnetic

field of (he eddy currents in one prism enhances the decay of the eddy currents in the other prism Consequently a time constant is produced which is smaller than that for each prism in free space 33 ms However the eddy currents also produce a larger time constant when the transmitter is away from the prisms Here the eddy currents in both prisms flow in the same direction and they tend to sustain each other with their magshynetic fields The variation in the time constant indicates the presence of multiple conductors since the time constant of a single conductor does not vary along the profile Finally the variation in the time constant also explains the peak and trough in the profiles between the prisms at early and late times In a resistive host the response of a poor conductor is larger than that of a good conductor at early time (Dyck and West 19~4) and smaller at later time

An important conclusion from our study is the inability of Iurge fixed-loop surveys to resolve multiple conductors Thereshyfore when an anomaly is located with this type of survey we

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DISfANCE (m) Fit 16 Profile of the vertical component of (1b(l)tE (voltage) calculated for two prisms in a WOO Omiddot m half-space for a

fixed-loop configuration One prism is located at x = am and the other prism is at x = 80 m

705 3-D Modeling of TEM Scattering

MODEL

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-lOO -160 -120 -80 -40 40 80 120 160 OO l40 l80

DISTANCE (m)

FIG 17 Comparison of profiles for the scattered vertical component of (Ib(t)Dt (voltage) calculated by superimposing the scattered response of each prism in its own half-space (circles) with the response which includes mutual interaction between the prisms (curves)

1010

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-200 -160 -120 -ao -40 120 160 200 240

FI( r8 Profile of the vertical component of (1b(t)(~[ (voltage] calculated for two prisms in a LOOO Q m half-space for a central-loop configuration The loop dimensions are 25 m by 25 m

706 Newman and Hohmann

recommend (if possible) further work with a moving-source survey because it is more likely to eliminate interpretational ambiguities of multiple conductors and can provide precise information for locating drill holes

CONCLUDING REMARKS

We have presented a 3-D integral-equation solution for the TEM responses of prisms in a layered half-space In summary our solution is based on a Galerkin solution for the scattering current where the weighting functions are the basis functions It is necessary that the solution include a set nf basis and weight functions that are divergence-free if model results are to be accurate at large background resistivities The

divergence-free basis functions are used to model eddy curshyrents which are closed in the prisms and hence simulate the inductive responses of the prisms in a very resistive earth More importantly divergence-free weight functions eliminate the charge operator from part of the solution by integrating it over rectangular paths Thus the solution satisfies the freeshyspace boundary condition for no current leaks and the reshysulting solution is stable for a host with any resistivity

We used current-tube functions as the divergence-free basis functions because they are simple and acceptable for our comshyputational limitations The current-tube functions are crude and restrict the geometry of the 3-D bodies to prisms Howshyever they are effective and allow modeling prisms of moderate geometric extent An exception where our solution does not perform well is for square and nearly square plates at a high resistivity contrast In this case the time constants of the plates are underestimated We believe the accuracy and genershyality of our solution will improve with a more sophisticated divergence-free basis function

The usefulness of OUT solution has been demonstrated with TEM model studies We investigated 3-D responses under the influence of conductive overburden and made an evaluation of the resolution of two conductors based on different survey configurations However our solution is more versatile than we have illustrated since it is valid for any number of layers and prisms and is designed for many different types oC source and receiver configurations Important exploration problems where the present solution can be applied include 3-D airshyborne TFM model studies and evaluations of the effect of a conductive host on 3-D TEM borehole responses Finally the solution is not restricted to TEM applications Because its basic formulation is in the frequency domain it can be applied to frequency-domain exploration problems as well

ACK-NO-WLEDGMpound~TS

This paper is dedicated to the memory of Frank Frischshyknecht We are grateful to him and the US Geological Survey for making the Surveys VAX IInO computer available for much of this work We are also indebted to Walter Anderson for his help in installing and running the programs Financial support was provided by Amoco Production Co ARCO Oil and Gas Co Chevron Resources Company CRA Exploration Pty Ltd Standard Oil Production Co and Unocal Corp in conjunction with the University of Utah EM Modeling Reshysearch Project

REFERENCES

Annan A P 1974 The equivalent source method for electromagnetic scattering analysis and its geophysical applications PhD thesis Memorial lJniv of Newfoundland

Dyck A V and West G F 1984 The role of simple computer models in interpretations of wide-band drill-hole electromagnetic surveys in mineral exploration Geophysics 49 957-980

Eaton P A and Hohmann G W 1987 Approximate inversion for lransient electromagnetic soundings Phys of the Earth and Plan Int in press

Fitlcrman D Y and Anderson W l 1987 Effect of transmitter turn-off on transient soundings Geoexplor 2413-146

Gallaghtr P R Ward S H and Hohmann G W 19R5 A model study of a thin plate in free space for the EM37 transient electroshymagnetic system (ieophysies 50 ()())-lDlq

Gupta O P Kakirde S T and Negi J G bull 1987 Scale model experiments In study low frequency electromagnetic resolution of multiple conductors Trans Geoscience and Remote Sensing Inst of Electr Electron Eng in press

Hanneson J E and West G F 1984 The horizontal-loop electroshymagnetic response of a thin plate in a conductive earth Part _shyComputat ional method Geophysics 49411-420

Harrington R F 1968 Field computation by moment methods The MacMillan Pub Co

Hohmann G W 19lB Three-dimensional EM modeling Geophys Surv 6 27 53

Kaufman A A 1978 Frequency and transient responses of EM fields created by currents in confined conductors Geophysics 43 1002-1010

19~5 Distribution of alternating electrical charges in a conshyductive medium Geophys Prosp 33171-184

Kaufman A A and Keller G V 1985 Inductive mining prospectshying Part I Theory ElsevierScience Pub Co

Lajoie J J and West G r 1976 The electromagnetic response of a conductive inhomogeneity in a layered earth Geophysics 4l 1133-1156

Lamontagne Y 1975 Applications of wideband time-domain elecshytrornagnetic measurements in mineral exploration PhD thesis UnivofToronto

Lee T S 19~ I Transient electromagnetic response of a sphere in a layered medium Pure Appl Geophys 119309-338shy

MLCracken K G Oristaglio 1 L and Hohmann U W 19~6 A comparison of electromagnetic exploration systems Geophysics SI XIO XIX

Nabighian M N 1987 Inductive time-domain electromagnetic methods EM mining volume Soc Explor Geophys in press

Newman G A Anderson W L and Hohmann G W 1987 Intershyprerarion of transrent electromagnetic soundings over threeshydimensional structures for the central-loop configuration Geophys J Roy Astr Socbull 89 l~9-(14

Newman G A Hohmann G W and Anderson W L 19~6 Transhysient electromagnetic response or a three-dimensional body in a layered earth Geophysics 51 160~- [627

Ogd~y K D 1910 A model study of the transient electromagnetic coincident loop technique Geoexplor 21 231-264

Petrick W R 11)84A fully internal hybrid technique for calculating electromagnetic SCattering from three-dimensional bodies in the earth PhD thesis Univ of Utah

Sanfilipo W A and Hohmann G W 1985 Integral equation solushyion for [he transient electromagnetic response of a threeshydimensional body in a conductive half-space Geophysics 50 798shy~Ol

S1I1hlipo W A I-aton P A and Hohmann G W 1985 The effect raquof a conductive half-space on the transient electromagnetic reshysponse of a three-dimensional body Geophysics 50 1144-1162

Spies B R 1980 TE~ in Australian conditions Field examples and model studies PhD thesis Macquarie Univ

Spies B R and Parker P 0- l9H4~ Limirarions of large-loop transhysient surveys in conductive terrains Geophysics 49 902 -912

Tripp A C ami Hohmann G W 1984 Block diagonalization of the electromagnetic impedance matrix of a symmetric body using group theory Inst of Elect and Electron Eng Trans Geosci and Remote Sensing GE-22 6269

Van Bladcl J 1961 SIJOlC remarks un Greens dyadic for infinite SPltlIC Trans on Antennas and Propagation Insc Electr Electron Eng 9 563middot 566

Wannamaker P E Hohmann G W and Sanfilipo W A 1984 Electromagnetic modeling of three-dimensional bodies in layered cart h- using integral equations Geophysics 49 60-77

Wcidclt P 198 l Dipole induction in a thin plate with host medium arid ovcrburdcn Research Project NTS 10 no R9727 Federal Inst ior Earth Sciences and Ravv Materials Hannover West Germany

700 Newman and Hohmann

located at x = -250 m Profiles in Figure 8 clearly show the position of the prism from early to late times

The total half-space and scattered decay curves at x =

- gO m (Figure 9) show that the response of the 10 000 nmiddot m half-space is not important in the interpretation or the prism from 1 to 20 ms The scattered response is the difference beshytween the total and layered half-space (I-Dl responses The late-time decay rate of the prism in Figure 9 is that of a thin plate in free space with an estimated time constant of 33 ms

Figure 10 shows the profound effect of conductive overburshyden on target detectability when the prism is buried beneath a 50 m thick 10 nmiddot rn overburden (compare Figure 8 with Figure 10) Conductive overburden limits the time range in which the prism is detectable the prism is detectable in only three of the time channels shown 10 15 and 20 ms This suppression or delay of the prism response at early times has been called overburden blanking by Lamontagne (1975)

The decay curves in Figure 1 (x = -SO m) clearly show that the response of the overburden is significant and always larger than the scattered response of the 3-D body The scattered response of the prism in Figure 11 decays exponentially at late times and the estimated time constant is 33 rns which is identical to that of the prism in the 10 000 Q m half-space without overburden Thus if the overburden response is subshytracted from the profile in Figure 10 the correct time constant of the prism will be obtained

However note that superposition of the free-space response of the prism and the overburden response is a first-order apshyproximation to the actual response of the prism with overburshydC1 (Nabighian (987) Figure 12 shows that the scattered response of the prism with overburden is shifted and has a larger amplitude than that of the prism in the 10 000 nmiddot m half-space 1 late times This amplitude shift is uniform across the profile at late times Thus it appears that an accurate estimate of the depth of the prism under the overburden can be obtained with half-width or peak-to-peak free-space rules for thin plates (cf Gallagher et aI 1985) However these rules require an estimate of the size of the plate before they can be applied Moreover the increase in the amplitude of the prism response affects free-space modeling of plate size (cf Gallashygher et al 1985) estimates of the depth of the plate and its conductance are also affected Note that the product of the prisms conductance and its depth extent is known since the time constant estimated [rom the data is that of a plate [equashytion (33)] Therefore the conductance and depth extent are constrained in an interpretation

Before 5 ms there is a fundamental difference between the responses of the prism in Figure 12 The voltage sounding with the overburden has a sign reversal between 2 and 3 ms caused by a buildup in the magnetic field of the prism at early times The magnetic field builds up because the electric and magnetic fields of the source must first penetrate the conducshytive media and subsequently build up at the prism If the prism is in free space there is no buildup in its magnetic field the magnetic field decays from an initial maximum at the time the current is shut ofTin the transmitter This maximum called the inductive limit is the initial condition required by Farashydays law for preserving the magnetic field of the transmitter that was linking the prism before shutoff In the case of the prism in the 10 000 Q m half-space a sign reversal in the voltage does occur but at extremely early times and is not

shown in Figures 9 and 12 Unike free space a 10 000 nmiddot m half-space still has finite resistivity

Figure 13 shows the response of the prism with overburden for a basement resistivity of 1000 Q m As in Figure 10 the response of the prism is suppressed and not detectable until to ms At 30 ms the prism is no longer detectable Thus the luyer-ed-host response must be- removed to interpret the proshyfile If the basement resistivity is decreased to 100 nmiddot m (not shown) there is no significant increase in the detectability of the prism due to current channeling

Figure 14 compares the scattered responses of the prism with overburden for 10000 1000 and tOO ( m basement resistivities elt x = -RO m Peak responses in the decay curves occur at early time before the curves change sign These peaks are interpreted as due to current channeling and correspond to the fastest rate at which host currents build up at the prism The largest peak occurs for the most conductive basement 100 nmiddot m However the peak response in the 100 Q m baseshyment docs not significantly increase the detectability of the prism there is an increase in the layered-host response as the basement resistivity is decreased Since host currents take longer to penetrate a more resistive basement and build up at the prism the decay curve of the 1000 nm basement changes sign earlier than does the decay curve of the 10 000 nmiddot m basement However this is not the case ill the 100 nmiddot III baseshyment where the decay curve changes sign later than does the decay curve of the 1000 nmiddot m basement Even though currents penetrate the basement earlier when the basement conducshytivity is increased they also require longer to build to a maxishymum at the prism if the basement is very conductive

Starting from 10 ms the responses and time constants of the prism arc identical for the 10 000 and 1000 nmiddot m basements However the responses and time constants are nut the same for the 100 U m basement The decay rate of the prism in the 100 nmiddot m basement is not exponential Rather its decay is an inverse power law that falls off as t -4 for the latest times shown in Figure 14 This decay is consistent with the late-time response of a confined conductor in a conductive medium where the late-time response can be represented by an inverse power law ie O( Ir) which is characteristic of the host roek (Nabighian 1997) Free-space interpretations of the prism in the Inon and I() 000 nmiddot m basements are possible at later times because their responses decay exponentially However as emphasized above free-space interpretations of the prism under overburden are biased since the response of the prism is influenced by its overburden at all times

Sometimes an orebody is in contact with the overburden in Figure 15 we show the response of the prism for such a case When the prism is in the 1000 Q m basement its detectability decreases compared to the dctectability of the prism that is detached from the overburden (compare Figure 13 with Figure 15) This decrease in dctectability results from the inshycrease in the overburden response which is caused by the increased thickness of the overburden When the basement resistivity is changed to 100 nmiddot m (not shown) we still do not sec an increase in the dctectability of the prism One might ha C suspected there would be an increase in delectability because of vertical current channeling where current is pulled down through the overburden and channeled into the prism Newman et al (l986) have shown that for a 1 Sm prism located closer to the surface (40 m) and in contact with the 10

701 3-D Modeling of TEM Scattering

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50 m thick 10 Q m overburden

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702 Newman and Hohmann

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FI( 14 Comparison of the scattered vertical field decayFIG 12 The scattered vertical field tb(r)rL at 1C= -80- m cbtV(~1 for variable basement resistivity with overburden at

with and without a 50 m thick to Omiddot In overburden = - 80 m The Ionmiddot 111 overburden is 50 111 thick The basement resistivity varies from to 000 to 100 Q m

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-200 -160 -120 middotao -40 a 40 BO 120 160 200

DISTANCE fROM CONDUCTOR CENTER (m)

FilL 3 Profile of the vertical component or tb(l)(l (voltage) calculated for a prism in a tooO nmiddot m half-space with a 50 m thick 10 nmiddot m overburden

100

-----------

703 3-D Modeling of TEM Scattering

n m overburden there is a pronounced enhancement in the prisms response However our findings show when the overshyburden is 100 III thick there wilt be no dramatic increase in the delectability of the prism even when it is in contact with the overburden

R[SOLLITION OF TWO CONDUCTIVE PRISMS WITH TFM SURVEYS

An important problem in mineral exploration for massive sulfides is the resolution and possible interpretational ambishy

guities of multiple conductors Very few theoretical modelshystudies have been published on the FM response of multiple conductors 1110st studies have been based all scale models

Ogilvy (J 983) published scale-model results on the horizontal resolution for the TEM coincident-loop configuration of two identical thin-sheet conductors that were vertically oriented in free space His findings show that when the separation beshytween the conductors is less than one-quarter of the loops diameter individual conductors cannot be resolved Gupta et aJ 19871 recently conducted scale-model experiments on the resolution of two vertical half-planes for the horizontal loopshyloop system Their results show that closely spaced scatterers arc resolved only when the separation between the half-planes is greater than the transmitter-receiver separation

Numerical models can also contribute interpretational inshy~ight into the lateral resolution and response of multiple conshyductors Obviously many multiple targets for different host layering and survey configurations can be modeled with our solution However our study is intended only as a preliminary

survey so we study the TEM response of two prisms in a resistive half-space for the fixed-loop and central-loop configushyrations We also discuss the resolution uf both conductors paying particular attention to survey design

In Figure 16 we show the vertical response (total voltage) of two prisms in a 1000 Q m half-space for a fixed-loop configushyration The profiles arc at seven different times ranging from 05 to 30 ms The prisms are identical in size depth and conductivity and are the same as the prism in Figure 8 One prism is positioned at x = 0 m and the other is at x = 80 m The fixed-loop transmitter is again 500 m on a side with its neurwirehxnrtedar x = -250m

At early times (05 and 1 ms] and at late times (which inshy

elude 15 20 ami 30 rns) the response of the tOOO n 111 halfshyspace is visible in figure 16 The half-space response is imporshytant at late times because it has a slower decay than the response of the two prisms If the response of the half-space is not considered in an interpretation the migrating crossover at intermediate and (ate times in Figure (6 will appear to indishycate a dipping conductor However once the half-space reshysponse is subtracted from the profile a vertical-plate conducshytor is interpreted at x = 30 m with a time constant of 37 ms This time constant is the same as that observed when the prisms arc together in free space The fixed-loop configuration clearly fails to distinguish the individual prisms even when the half-space response is removed Spies and Parker (1984) have demonstrated with scale-model experiments similar limishy

tations of large fixed-loop surveys The mutual interaction or the TEM fields of the two prisms

can he investigated by comparing responses with and without

---- -

------1 gt --~~

IO- T

3 shy

10

N- 9

I 1OE

gtshy10

15bullbull

s bullbull ----_~

--

J

-gtshy2O

Win_

30n

MO_Qf~

10middot1

iocon-Nealt d~e 01 loop

I

middot200 -160 -120 middot80 -40 40 80 120 160 200

DISTANCE FROM CONDUCTOR CENTER ltII)

FIG 15 Profile of the vertical component of tb(lliltt (voltage) calculated for a prism in a tooo Q m half-space in contact with a 100 m thick 10 n m overburden

704 Newman and Hohmann

the interaction In Figure 17 the scattered response of each

prism in its own half-space is superimposed and compared

with the scattered response which includes the mutual interacshytion Over the time range of 05 to 5 IDS the superimposed response in the vertical component is larger (except near sign reversals) than the scattered response observed when the fields of the two prisms interact However at later times (greater than 5 ms) this pattern reverses Both responses in Figure 17 decay exponentially at late times and we estimate a time conshystant of 33 ms for the superimposed response which is 11 percent smaller than the time constant estimated from the

response with the mutual interaction 37 ms The II percent difference in the time constants reflects the 66 percent disparshyity between the two responses at _~O ms as shown in Figure Il

As suggested by Spies and Parker (1(84) a survey configushy

ration where the source moves along the profile can improve the lateral resolution or the two prisms For a central-loop configuration with loops 25 m OIl a side both prisms are resolved at I 3 and 5 ms as shown in Figure 18 The two

minima ill the profilesindicate the locations of both prisms at these times At later times interpretational ambiguities could arise because the two separate minima are replaced by a single minimum located half way between the two prisms This single minimum could be interpreted as an isolated conductor Overall the shapes of the profiles in Figure 18 are similar to those studied by Ogilvy (1983) for the coincident-loop configushy

ration Note that after I ms the half-space has little influence on the response of the prisms in Figure 18 When the halfshyspace response is removed the response of the prisms is the same as that observed in free space

till 5NS shy

10middot

Jshy

10- 9

3

til 5E-==shy O-ID

shy-==

15

10-11

10

3010-11

NOf edge of loop

The time constant estimated from the scattered response of the two prisms at late time is dependent upon the type of survey Unlike the case for a fixed-loop configuration the time constant varies with its position along the profile for the central-loop configuration When central-loop stations are loshycated away from the two prisms the estimated time constant is the same as in the fixed-loop case 37 rns However when a central-loop station approaches both prisms the time conshystant decreases and its smallest value is 29 ms at x = 40 m The time constant decreases because the transmitter induces eddy currents in the prisms that flow in opposite directions Since these eddy currents oppose each other the magnetic

field of (he eddy currents in one prism enhances the decay of the eddy currents in the other prism Consequently a time constant is produced which is smaller than that for each prism in free space 33 ms However the eddy currents also produce a larger time constant when the transmitter is away from the prisms Here the eddy currents in both prisms flow in the same direction and they tend to sustain each other with their magshynetic fields The variation in the time constant indicates the presence of multiple conductors since the time constant of a single conductor does not vary along the profile Finally the variation in the time constant also explains the peak and trough in the profiles between the prisms at early and late times In a resistive host the response of a poor conductor is larger than that of a good conductor at early time (Dyck and West 19~4) and smaller at later time

An important conclusion from our study is the inability of Iurge fixed-loop surveys to resolve multiple conductors Thereshyfore when an anomaly is located with this type of survey we

~-------

---shy

lt I I

I

-ltr gt

-shyI lt f

I I I I I

shy

1I

1 V ----_ II

I

I I

I

MODEL -----shy~middotO x-80

1 ~(fot

l]lCS tmiddot 25 SLmiddot 800IOOOSl Of- 100m DBmiddot 100

-200 -160 -120 -80 -40 0 40 80 126 ISO 200 2~0 280

DISfANCE (m) Fit 16 Profile of the vertical component of (1b(l)tE (voltage) calculated for two prisms in a WOO Omiddot m half-space for a

fixed-loop configuration One prism is located at x = am and the other prism is at x = 80 m

705 3-D Modeling of TEM Scattering

MODEL

bullbull II III 41 a 0

------- shy0

IG-I 0 I

J bull __- _J -_ I s

i I ----__

lomiddotg

( --~~ l~ ~_-

e I 5 I I _------- _gt

N- 10- 10

II laquo ~ - 10I

~ =shy

10middot1

xmiddotODt xaoIO-I~ fiJrfCJ(~

---- shy U 105 ~ JS - iocoa SLmiddot800

0[100 n8middot100

I

-lOO -160 -120 -80 -40 40 80 120 160 OO l40 l80

DISTANCE (m)

FIG 17 Comparison of profiles for the scattered vertical component of (Ib(t)Dt (voltage) calculated by superimposing the scattered response of each prism in its own half-space (circles) with the response which includes mutual interaction between the prisms (curves)

1010

1=gt shy 10- 11 --gt

10 -1pound

t~ --------__

lmJ

3mJ

5ms

MODEL

XmiddotOIll xmiddot80m-Ll surface

o O middot0Sm t bull 25m

1000nm SLmiddot800m DElOOm DBlOOm

25 m x 25m central loop~

-200 -160 -120 -ao -40 120 160 200 240

FI( r8 Profile of the vertical component of (1b(t)(~[ (voltage] calculated for two prisms in a LOOO Q m half-space for a central-loop configuration The loop dimensions are 25 m by 25 m

706 Newman and Hohmann

recommend (if possible) further work with a moving-source survey because it is more likely to eliminate interpretational ambiguities of multiple conductors and can provide precise information for locating drill holes

CONCLUDING REMARKS

We have presented a 3-D integral-equation solution for the TEM responses of prisms in a layered half-space In summary our solution is based on a Galerkin solution for the scattering current where the weighting functions are the basis functions It is necessary that the solution include a set nf basis and weight functions that are divergence-free if model results are to be accurate at large background resistivities The

divergence-free basis functions are used to model eddy curshyrents which are closed in the prisms and hence simulate the inductive responses of the prisms in a very resistive earth More importantly divergence-free weight functions eliminate the charge operator from part of the solution by integrating it over rectangular paths Thus the solution satisfies the freeshyspace boundary condition for no current leaks and the reshysulting solution is stable for a host with any resistivity

We used current-tube functions as the divergence-free basis functions because they are simple and acceptable for our comshyputational limitations The current-tube functions are crude and restrict the geometry of the 3-D bodies to prisms Howshyever they are effective and allow modeling prisms of moderate geometric extent An exception where our solution does not perform well is for square and nearly square plates at a high resistivity contrast In this case the time constants of the plates are underestimated We believe the accuracy and genershyality of our solution will improve with a more sophisticated divergence-free basis function

The usefulness of OUT solution has been demonstrated with TEM model studies We investigated 3-D responses under the influence of conductive overburden and made an evaluation of the resolution of two conductors based on different survey configurations However our solution is more versatile than we have illustrated since it is valid for any number of layers and prisms and is designed for many different types oC source and receiver configurations Important exploration problems where the present solution can be applied include 3-D airshyborne TFM model studies and evaluations of the effect of a conductive host on 3-D TEM borehole responses Finally the solution is not restricted to TEM applications Because its basic formulation is in the frequency domain it can be applied to frequency-domain exploration problems as well

ACK-NO-WLEDGMpound~TS

This paper is dedicated to the memory of Frank Frischshyknecht We are grateful to him and the US Geological Survey for making the Surveys VAX IInO computer available for much of this work We are also indebted to Walter Anderson for his help in installing and running the programs Financial support was provided by Amoco Production Co ARCO Oil and Gas Co Chevron Resources Company CRA Exploration Pty Ltd Standard Oil Production Co and Unocal Corp in conjunction with the University of Utah EM Modeling Reshysearch Project

REFERENCES

Annan A P 1974 The equivalent source method for electromagnetic scattering analysis and its geophysical applications PhD thesis Memorial lJniv of Newfoundland

Dyck A V and West G F 1984 The role of simple computer models in interpretations of wide-band drill-hole electromagnetic surveys in mineral exploration Geophysics 49 957-980

Eaton P A and Hohmann G W 1987 Approximate inversion for lransient electromagnetic soundings Phys of the Earth and Plan Int in press

Fitlcrman D Y and Anderson W l 1987 Effect of transmitter turn-off on transient soundings Geoexplor 2413-146

Gallaghtr P R Ward S H and Hohmann G W 19R5 A model study of a thin plate in free space for the EM37 transient electroshymagnetic system (ieophysies 50 ()())-lDlq

Gupta O P Kakirde S T and Negi J G bull 1987 Scale model experiments In study low frequency electromagnetic resolution of multiple conductors Trans Geoscience and Remote Sensing Inst of Electr Electron Eng in press

Hanneson J E and West G F 1984 The horizontal-loop electroshymagnetic response of a thin plate in a conductive earth Part _shyComputat ional method Geophysics 49411-420

Harrington R F 1968 Field computation by moment methods The MacMillan Pub Co

Hohmann G W 19lB Three-dimensional EM modeling Geophys Surv 6 27 53

Kaufman A A 1978 Frequency and transient responses of EM fields created by currents in confined conductors Geophysics 43 1002-1010

19~5 Distribution of alternating electrical charges in a conshyductive medium Geophys Prosp 33171-184

Kaufman A A and Keller G V 1985 Inductive mining prospectshying Part I Theory ElsevierScience Pub Co

Lajoie J J and West G r 1976 The electromagnetic response of a conductive inhomogeneity in a layered earth Geophysics 4l 1133-1156

Lamontagne Y 1975 Applications of wideband time-domain elecshytrornagnetic measurements in mineral exploration PhD thesis UnivofToronto

Lee T S 19~ I Transient electromagnetic response of a sphere in a layered medium Pure Appl Geophys 119309-338shy

MLCracken K G Oristaglio 1 L and Hohmann U W 19~6 A comparison of electromagnetic exploration systems Geophysics SI XIO XIX

Nabighian M N 1987 Inductive time-domain electromagnetic methods EM mining volume Soc Explor Geophys in press

Newman G A Anderson W L and Hohmann G W 1987 Intershyprerarion of transrent electromagnetic soundings over threeshydimensional structures for the central-loop configuration Geophys J Roy Astr Socbull 89 l~9-(14

Newman G A Hohmann G W and Anderson W L 19~6 Transhysient electromagnetic response or a three-dimensional body in a layered earth Geophysics 51 160~- [627

Ogd~y K D 1910 A model study of the transient electromagnetic coincident loop technique Geoexplor 21 231-264

Petrick W R 11)84A fully internal hybrid technique for calculating electromagnetic SCattering from three-dimensional bodies in the earth PhD thesis Univ of Utah

Sanfilipo W A and Hohmann G W 1985 Integral equation solushyion for [he transient electromagnetic response of a threeshydimensional body in a conductive half-space Geophysics 50 798shy~Ol

S1I1hlipo W A I-aton P A and Hohmann G W 1985 The effect raquof a conductive half-space on the transient electromagnetic reshysponse of a three-dimensional body Geophysics 50 1144-1162

Spies B R 1980 TE~ in Australian conditions Field examples and model studies PhD thesis Macquarie Univ

Spies B R and Parker P 0- l9H4~ Limirarions of large-loop transhysient surveys in conductive terrains Geophysics 49 902 -912

Tripp A C ami Hohmann G W 1984 Block diagonalization of the electromagnetic impedance matrix of a symmetric body using group theory Inst of Elect and Electron Eng Trans Geosci and Remote Sensing GE-22 6269

Van Bladcl J 1961 SIJOlC remarks un Greens dyadic for infinite SPltlIC Trans on Antennas and Propagation Insc Electr Electron Eng 9 563middot 566

Wannamaker P E Hohmann G W and Sanfilipo W A 1984 Electromagnetic modeling of three-dimensional bodies in layered cart h- using integral equations Geophysics 49 60-77

Wcidclt P 198 l Dipole induction in a thin plate with host medium arid ovcrburdcn Research Project NTS 10 no R9727 Federal Inst ior Earth Sciences and Ravv Materials Hannover West Germany

701 3-D Modeling of TEM Scattering

I

Y~ Il-D

l I

I I I I I I I I I

IO8~ 1081- I I

scattJ~

6

II

totol--b I I

0--0d

10-I 10-l Nla

~

10-10 10-6 ~

10 -II0-11

10 12 bull

1 m10 100

t (ms) ttrnl

FIG 9 Decay curves of the total (triangles) scattered (circles) FI(i 11 Decay curves of the total (triangles) scattered (circles) and half-space (solid dots) vertical field cb(f)t at x = - 80 rn and layered half-space (solid dots) vertical field (~b(t)(ll at x = for the model of Figure 8 -~o m for the model of Figure 10

-------- -

tiM

1 r

IO-l

3

la-I

5

ltIshy__

E

gt 10-1

-~

I 10 ~ gt ~

IO-I~ rshy 75

20

MODELIO-~ ---------1urQCr

DISTANCE FRON CONDUCTOR CENTER (m) Fro to Profile of the vertical component of (1b(t)il( (voltage) calculated for a prism in a 10000 n m half-space with a

50 m thick 10 Q m overburden

I 50~------- --- Q~i~~ 10000(1middot StSOO

or- lJO DBmiddot 100~

30

Near ~r1 of loop

-200 -160 -120 -80 -40 Q 40 80 120 160 20~

I

--------------

--

702 Newman and Hohmann

0-0 b

I

~1 I

~ 10-al- 1 10- 6 ~ I

II

I I I llI

0 Iwrth vbud

b cr-( ~O

cf J I I

_- ~ Ithout avhurd I1O-9~ W1f d

t

I shy~

~ I ~~ b ~

~

~

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10oeon

Qlt~ ~ MODEL (0)IOll~ 1011 MODELloalI 50 lOAmiddot 50

100001middot C] OOllmiddot 0Dcodllclor A oooa eohelot

oloooonmiddot

10-12 10-12 1 I 10 100 I I 10

t(IU) tlIlIS)

FI( 14 Comparison of the scattered vertical field decayFIG 12 The scattered vertical field tb(r)rL at 1C= -80- m cbtV(~1 for variable basement resistivity with overburden at

with and without a 50 m thick to Omiddot In overburden = - 80 m The Ionmiddot 111 overburden is 50 111 thick The basement resistivity varies from to 000 to 100 Q m

~------

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Iw1 I

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3

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5 ------~_

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20m - shy

MO)EL

10 11

30 --------------------- shy JurlbC

50 1~8Cn 0~ i~

SLmiddot800 DEmiddot100 DBmiddot100m0 cJg 0 f leop

-200 -160 -120 middotao -40 a 40 BO 120 160 200

DISTANCE fROM CONDUCTOR CENTER (m)

FilL 3 Profile of the vertical component or tb(l)(l (voltage) calculated for a prism in a tooO nmiddot m half-space with a 50 m thick 10 nmiddot m overburden

100

-----------

703 3-D Modeling of TEM Scattering

n m overburden there is a pronounced enhancement in the prisms response However our findings show when the overshyburden is 100 III thick there wilt be no dramatic increase in the delectability of the prism even when it is in contact with the overburden

R[SOLLITION OF TWO CONDUCTIVE PRISMS WITH TFM SURVEYS

An important problem in mineral exploration for massive sulfides is the resolution and possible interpretational ambishy

guities of multiple conductors Very few theoretical modelshystudies have been published on the FM response of multiple conductors 1110st studies have been based all scale models

Ogilvy (J 983) published scale-model results on the horizontal resolution for the TEM coincident-loop configuration of two identical thin-sheet conductors that were vertically oriented in free space His findings show that when the separation beshytween the conductors is less than one-quarter of the loops diameter individual conductors cannot be resolved Gupta et aJ 19871 recently conducted scale-model experiments on the resolution of two vertical half-planes for the horizontal loopshyloop system Their results show that closely spaced scatterers arc resolved only when the separation between the half-planes is greater than the transmitter-receiver separation

Numerical models can also contribute interpretational inshy~ight into the lateral resolution and response of multiple conshyductors Obviously many multiple targets for different host layering and survey configurations can be modeled with our solution However our study is intended only as a preliminary

survey so we study the TEM response of two prisms in a resistive half-space for the fixed-loop and central-loop configushyrations We also discuss the resolution uf both conductors paying particular attention to survey design

In Figure 16 we show the vertical response (total voltage) of two prisms in a 1000 Q m half-space for a fixed-loop configushyration The profiles arc at seven different times ranging from 05 to 30 ms The prisms are identical in size depth and conductivity and are the same as the prism in Figure 8 One prism is positioned at x = 0 m and the other is at x = 80 m The fixed-loop transmitter is again 500 m on a side with its neurwirehxnrtedar x = -250m

At early times (05 and 1 ms] and at late times (which inshy

elude 15 20 ami 30 rns) the response of the tOOO n 111 halfshyspace is visible in figure 16 The half-space response is imporshytant at late times because it has a slower decay than the response of the two prisms If the response of the half-space is not considered in an interpretation the migrating crossover at intermediate and (ate times in Figure (6 will appear to indishycate a dipping conductor However once the half-space reshysponse is subtracted from the profile a vertical-plate conducshytor is interpreted at x = 30 m with a time constant of 37 ms This time constant is the same as that observed when the prisms arc together in free space The fixed-loop configuration clearly fails to distinguish the individual prisms even when the half-space response is removed Spies and Parker (1984) have demonstrated with scale-model experiments similar limishy

tations of large fixed-loop surveys The mutual interaction or the TEM fields of the two prisms

can he investigated by comparing responses with and without

---- -

------1 gt --~~

IO- T

3 shy

10

N- 9

I 1OE

gtshy10

15bullbull

s bullbull ----_~

--

J

-gtshy2O

Win_

30n

MO_Qf~

10middot1

iocon-Nealt d~e 01 loop

I

middot200 -160 -120 middot80 -40 40 80 120 160 200

DISTANCE FROM CONDUCTOR CENTER ltII)

FIG 15 Profile of the vertical component of tb(lliltt (voltage) calculated for a prism in a tooo Q m half-space in contact with a 100 m thick 10 n m overburden

704 Newman and Hohmann

the interaction In Figure 17 the scattered response of each

prism in its own half-space is superimposed and compared

with the scattered response which includes the mutual interacshytion Over the time range of 05 to 5 IDS the superimposed response in the vertical component is larger (except near sign reversals) than the scattered response observed when the fields of the two prisms interact However at later times (greater than 5 ms) this pattern reverses Both responses in Figure 17 decay exponentially at late times and we estimate a time conshystant of 33 ms for the superimposed response which is 11 percent smaller than the time constant estimated from the

response with the mutual interaction 37 ms The II percent difference in the time constants reflects the 66 percent disparshyity between the two responses at _~O ms as shown in Figure Il

As suggested by Spies and Parker (1(84) a survey configushy

ration where the source moves along the profile can improve the lateral resolution or the two prisms For a central-loop configuration with loops 25 m OIl a side both prisms are resolved at I 3 and 5 ms as shown in Figure 18 The two

minima ill the profilesindicate the locations of both prisms at these times At later times interpretational ambiguities could arise because the two separate minima are replaced by a single minimum located half way between the two prisms This single minimum could be interpreted as an isolated conductor Overall the shapes of the profiles in Figure 18 are similar to those studied by Ogilvy (1983) for the coincident-loop configushy

ration Note that after I ms the half-space has little influence on the response of the prisms in Figure 18 When the halfshyspace response is removed the response of the prisms is the same as that observed in free space

till 5NS shy

10middot

Jshy

10- 9

3

til 5E-==shy O-ID

shy-==

15

10-11

10

3010-11

NOf edge of loop

The time constant estimated from the scattered response of the two prisms at late time is dependent upon the type of survey Unlike the case for a fixed-loop configuration the time constant varies with its position along the profile for the central-loop configuration When central-loop stations are loshycated away from the two prisms the estimated time constant is the same as in the fixed-loop case 37 rns However when a central-loop station approaches both prisms the time conshystant decreases and its smallest value is 29 ms at x = 40 m The time constant decreases because the transmitter induces eddy currents in the prisms that flow in opposite directions Since these eddy currents oppose each other the magnetic

field of (he eddy currents in one prism enhances the decay of the eddy currents in the other prism Consequently a time constant is produced which is smaller than that for each prism in free space 33 ms However the eddy currents also produce a larger time constant when the transmitter is away from the prisms Here the eddy currents in both prisms flow in the same direction and they tend to sustain each other with their magshynetic fields The variation in the time constant indicates the presence of multiple conductors since the time constant of a single conductor does not vary along the profile Finally the variation in the time constant also explains the peak and trough in the profiles between the prisms at early and late times In a resistive host the response of a poor conductor is larger than that of a good conductor at early time (Dyck and West 19~4) and smaller at later time

An important conclusion from our study is the inability of Iurge fixed-loop surveys to resolve multiple conductors Thereshyfore when an anomaly is located with this type of survey we

~-------

---shy

lt I I

I

-ltr gt

-shyI lt f

I I I I I

shy

1I

1 V ----_ II

I

I I

I

MODEL -----shy~middotO x-80

1 ~(fot

l]lCS tmiddot 25 SLmiddot 800IOOOSl Of- 100m DBmiddot 100

-200 -160 -120 -80 -40 0 40 80 126 ISO 200 2~0 280

DISfANCE (m) Fit 16 Profile of the vertical component of (1b(l)tE (voltage) calculated for two prisms in a WOO Omiddot m half-space for a

fixed-loop configuration One prism is located at x = am and the other prism is at x = 80 m

705 3-D Modeling of TEM Scattering

MODEL

bullbull II III 41 a 0

------- shy0

IG-I 0 I

J bull __- _J -_ I s

i I ----__

lomiddotg

( --~~ l~ ~_-

e I 5 I I _------- _gt

N- 10- 10

II laquo ~ - 10I

~ =shy

10middot1

xmiddotODt xaoIO-I~ fiJrfCJ(~

---- shy U 105 ~ JS - iocoa SLmiddot800

0[100 n8middot100

I

-lOO -160 -120 -80 -40 40 80 120 160 OO l40 l80

DISTANCE (m)

FIG 17 Comparison of profiles for the scattered vertical component of (Ib(t)Dt (voltage) calculated by superimposing the scattered response of each prism in its own half-space (circles) with the response which includes mutual interaction between the prisms (curves)

1010

1=gt shy 10- 11 --gt

10 -1pound

t~ --------__

lmJ

3mJ

5ms

MODEL

XmiddotOIll xmiddot80m-Ll surface

o O middot0Sm t bull 25m

1000nm SLmiddot800m DElOOm DBlOOm

25 m x 25m central loop~

-200 -160 -120 -ao -40 120 160 200 240

FI( r8 Profile of the vertical component of (1b(t)(~[ (voltage] calculated for two prisms in a LOOO Q m half-space for a central-loop configuration The loop dimensions are 25 m by 25 m

706 Newman and Hohmann

recommend (if possible) further work with a moving-source survey because it is more likely to eliminate interpretational ambiguities of multiple conductors and can provide precise information for locating drill holes

CONCLUDING REMARKS

We have presented a 3-D integral-equation solution for the TEM responses of prisms in a layered half-space In summary our solution is based on a Galerkin solution for the scattering current where the weighting functions are the basis functions It is necessary that the solution include a set nf basis and weight functions that are divergence-free if model results are to be accurate at large background resistivities The

divergence-free basis functions are used to model eddy curshyrents which are closed in the prisms and hence simulate the inductive responses of the prisms in a very resistive earth More importantly divergence-free weight functions eliminate the charge operator from part of the solution by integrating it over rectangular paths Thus the solution satisfies the freeshyspace boundary condition for no current leaks and the reshysulting solution is stable for a host with any resistivity

We used current-tube functions as the divergence-free basis functions because they are simple and acceptable for our comshyputational limitations The current-tube functions are crude and restrict the geometry of the 3-D bodies to prisms Howshyever they are effective and allow modeling prisms of moderate geometric extent An exception where our solution does not perform well is for square and nearly square plates at a high resistivity contrast In this case the time constants of the plates are underestimated We believe the accuracy and genershyality of our solution will improve with a more sophisticated divergence-free basis function

The usefulness of OUT solution has been demonstrated with TEM model studies We investigated 3-D responses under the influence of conductive overburden and made an evaluation of the resolution of two conductors based on different survey configurations However our solution is more versatile than we have illustrated since it is valid for any number of layers and prisms and is designed for many different types oC source and receiver configurations Important exploration problems where the present solution can be applied include 3-D airshyborne TFM model studies and evaluations of the effect of a conductive host on 3-D TEM borehole responses Finally the solution is not restricted to TEM applications Because its basic formulation is in the frequency domain it can be applied to frequency-domain exploration problems as well

ACK-NO-WLEDGMpound~TS

This paper is dedicated to the memory of Frank Frischshyknecht We are grateful to him and the US Geological Survey for making the Surveys VAX IInO computer available for much of this work We are also indebted to Walter Anderson for his help in installing and running the programs Financial support was provided by Amoco Production Co ARCO Oil and Gas Co Chevron Resources Company CRA Exploration Pty Ltd Standard Oil Production Co and Unocal Corp in conjunction with the University of Utah EM Modeling Reshysearch Project

REFERENCES

Annan A P 1974 The equivalent source method for electromagnetic scattering analysis and its geophysical applications PhD thesis Memorial lJniv of Newfoundland

Dyck A V and West G F 1984 The role of simple computer models in interpretations of wide-band drill-hole electromagnetic surveys in mineral exploration Geophysics 49 957-980

Eaton P A and Hohmann G W 1987 Approximate inversion for lransient electromagnetic soundings Phys of the Earth and Plan Int in press

Fitlcrman D Y and Anderson W l 1987 Effect of transmitter turn-off on transient soundings Geoexplor 2413-146

Gallaghtr P R Ward S H and Hohmann G W 19R5 A model study of a thin plate in free space for the EM37 transient electroshymagnetic system (ieophysies 50 ()())-lDlq

Gupta O P Kakirde S T and Negi J G bull 1987 Scale model experiments In study low frequency electromagnetic resolution of multiple conductors Trans Geoscience and Remote Sensing Inst of Electr Electron Eng in press

Hanneson J E and West G F 1984 The horizontal-loop electroshymagnetic response of a thin plate in a conductive earth Part _shyComputat ional method Geophysics 49411-420

Harrington R F 1968 Field computation by moment methods The MacMillan Pub Co

Hohmann G W 19lB Three-dimensional EM modeling Geophys Surv 6 27 53

Kaufman A A 1978 Frequency and transient responses of EM fields created by currents in confined conductors Geophysics 43 1002-1010

19~5 Distribution of alternating electrical charges in a conshyductive medium Geophys Prosp 33171-184

Kaufman A A and Keller G V 1985 Inductive mining prospectshying Part I Theory ElsevierScience Pub Co

Lajoie J J and West G r 1976 The electromagnetic response of a conductive inhomogeneity in a layered earth Geophysics 4l 1133-1156

Lamontagne Y 1975 Applications of wideband time-domain elecshytrornagnetic measurements in mineral exploration PhD thesis UnivofToronto

Lee T S 19~ I Transient electromagnetic response of a sphere in a layered medium Pure Appl Geophys 119309-338shy

MLCracken K G Oristaglio 1 L and Hohmann U W 19~6 A comparison of electromagnetic exploration systems Geophysics SI XIO XIX

Nabighian M N 1987 Inductive time-domain electromagnetic methods EM mining volume Soc Explor Geophys in press

Newman G A Anderson W L and Hohmann G W 1987 Intershyprerarion of transrent electromagnetic soundings over threeshydimensional structures for the central-loop configuration Geophys J Roy Astr Socbull 89 l~9-(14

Newman G A Hohmann G W and Anderson W L 19~6 Transhysient electromagnetic response or a three-dimensional body in a layered earth Geophysics 51 160~- [627

Ogd~y K D 1910 A model study of the transient electromagnetic coincident loop technique Geoexplor 21 231-264

Petrick W R 11)84A fully internal hybrid technique for calculating electromagnetic SCattering from three-dimensional bodies in the earth PhD thesis Univ of Utah

Sanfilipo W A and Hohmann G W 1985 Integral equation solushyion for [he transient electromagnetic response of a threeshydimensional body in a conductive half-space Geophysics 50 798shy~Ol

S1I1hlipo W A I-aton P A and Hohmann G W 1985 The effect raquof a conductive half-space on the transient electromagnetic reshysponse of a three-dimensional body Geophysics 50 1144-1162

Spies B R 1980 TE~ in Australian conditions Field examples and model studies PhD thesis Macquarie Univ

Spies B R and Parker P 0- l9H4~ Limirarions of large-loop transhysient surveys in conductive terrains Geophysics 49 902 -912

Tripp A C ami Hohmann G W 1984 Block diagonalization of the electromagnetic impedance matrix of a symmetric body using group theory Inst of Elect and Electron Eng Trans Geosci and Remote Sensing GE-22 6269

Van Bladcl J 1961 SIJOlC remarks un Greens dyadic for infinite SPltlIC Trans on Antennas and Propagation Insc Electr Electron Eng 9 563middot 566

Wannamaker P E Hohmann G W and Sanfilipo W A 1984 Electromagnetic modeling of three-dimensional bodies in layered cart h- using integral equations Geophysics 49 60-77

Wcidclt P 198 l Dipole induction in a thin plate with host medium arid ovcrburdcn Research Project NTS 10 no R9727 Federal Inst ior Earth Sciences and Ravv Materials Hannover West Germany

--------------

--

702 Newman and Hohmann

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with and without a 50 m thick to Omiddot In overburden = - 80 m The Ionmiddot 111 overburden is 50 111 thick The basement resistivity varies from to 000 to 100 Q m

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FilL 3 Profile of the vertical component or tb(l)(l (voltage) calculated for a prism in a tooO nmiddot m half-space with a 50 m thick 10 nmiddot m overburden

100

-----------

703 3-D Modeling of TEM Scattering

n m overburden there is a pronounced enhancement in the prisms response However our findings show when the overshyburden is 100 III thick there wilt be no dramatic increase in the delectability of the prism even when it is in contact with the overburden

R[SOLLITION OF TWO CONDUCTIVE PRISMS WITH TFM SURVEYS

An important problem in mineral exploration for massive sulfides is the resolution and possible interpretational ambishy

guities of multiple conductors Very few theoretical modelshystudies have been published on the FM response of multiple conductors 1110st studies have been based all scale models

Ogilvy (J 983) published scale-model results on the horizontal resolution for the TEM coincident-loop configuration of two identical thin-sheet conductors that were vertically oriented in free space His findings show that when the separation beshytween the conductors is less than one-quarter of the loops diameter individual conductors cannot be resolved Gupta et aJ 19871 recently conducted scale-model experiments on the resolution of two vertical half-planes for the horizontal loopshyloop system Their results show that closely spaced scatterers arc resolved only when the separation between the half-planes is greater than the transmitter-receiver separation

Numerical models can also contribute interpretational inshy~ight into the lateral resolution and response of multiple conshyductors Obviously many multiple targets for different host layering and survey configurations can be modeled with our solution However our study is intended only as a preliminary

survey so we study the TEM response of two prisms in a resistive half-space for the fixed-loop and central-loop configushyrations We also discuss the resolution uf both conductors paying particular attention to survey design

In Figure 16 we show the vertical response (total voltage) of two prisms in a 1000 Q m half-space for a fixed-loop configushyration The profiles arc at seven different times ranging from 05 to 30 ms The prisms are identical in size depth and conductivity and are the same as the prism in Figure 8 One prism is positioned at x = 0 m and the other is at x = 80 m The fixed-loop transmitter is again 500 m on a side with its neurwirehxnrtedar x = -250m

At early times (05 and 1 ms] and at late times (which inshy

elude 15 20 ami 30 rns) the response of the tOOO n 111 halfshyspace is visible in figure 16 The half-space response is imporshytant at late times because it has a slower decay than the response of the two prisms If the response of the half-space is not considered in an interpretation the migrating crossover at intermediate and (ate times in Figure (6 will appear to indishycate a dipping conductor However once the half-space reshysponse is subtracted from the profile a vertical-plate conducshytor is interpreted at x = 30 m with a time constant of 37 ms This time constant is the same as that observed when the prisms arc together in free space The fixed-loop configuration clearly fails to distinguish the individual prisms even when the half-space response is removed Spies and Parker (1984) have demonstrated with scale-model experiments similar limishy

tations of large fixed-loop surveys The mutual interaction or the TEM fields of the two prisms

can he investigated by comparing responses with and without

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I

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DISTANCE FROM CONDUCTOR CENTER ltII)

FIG 15 Profile of the vertical component of tb(lliltt (voltage) calculated for a prism in a tooo Q m half-space in contact with a 100 m thick 10 n m overburden

704 Newman and Hohmann

the interaction In Figure 17 the scattered response of each

prism in its own half-space is superimposed and compared

with the scattered response which includes the mutual interacshytion Over the time range of 05 to 5 IDS the superimposed response in the vertical component is larger (except near sign reversals) than the scattered response observed when the fields of the two prisms interact However at later times (greater than 5 ms) this pattern reverses Both responses in Figure 17 decay exponentially at late times and we estimate a time conshystant of 33 ms for the superimposed response which is 11 percent smaller than the time constant estimated from the

response with the mutual interaction 37 ms The II percent difference in the time constants reflects the 66 percent disparshyity between the two responses at _~O ms as shown in Figure Il

As suggested by Spies and Parker (1(84) a survey configushy

ration where the source moves along the profile can improve the lateral resolution or the two prisms For a central-loop configuration with loops 25 m OIl a side both prisms are resolved at I 3 and 5 ms as shown in Figure 18 The two

minima ill the profilesindicate the locations of both prisms at these times At later times interpretational ambiguities could arise because the two separate minima are replaced by a single minimum located half way between the two prisms This single minimum could be interpreted as an isolated conductor Overall the shapes of the profiles in Figure 18 are similar to those studied by Ogilvy (1983) for the coincident-loop configushy

ration Note that after I ms the half-space has little influence on the response of the prisms in Figure 18 When the halfshyspace response is removed the response of the prisms is the same as that observed in free space

till 5NS shy

10middot

Jshy

10- 9

3

til 5E-==shy O-ID

shy-==

15

10-11

10

3010-11

NOf edge of loop

The time constant estimated from the scattered response of the two prisms at late time is dependent upon the type of survey Unlike the case for a fixed-loop configuration the time constant varies with its position along the profile for the central-loop configuration When central-loop stations are loshycated away from the two prisms the estimated time constant is the same as in the fixed-loop case 37 rns However when a central-loop station approaches both prisms the time conshystant decreases and its smallest value is 29 ms at x = 40 m The time constant decreases because the transmitter induces eddy currents in the prisms that flow in opposite directions Since these eddy currents oppose each other the magnetic

field of (he eddy currents in one prism enhances the decay of the eddy currents in the other prism Consequently a time constant is produced which is smaller than that for each prism in free space 33 ms However the eddy currents also produce a larger time constant when the transmitter is away from the prisms Here the eddy currents in both prisms flow in the same direction and they tend to sustain each other with their magshynetic fields The variation in the time constant indicates the presence of multiple conductors since the time constant of a single conductor does not vary along the profile Finally the variation in the time constant also explains the peak and trough in the profiles between the prisms at early and late times In a resistive host the response of a poor conductor is larger than that of a good conductor at early time (Dyck and West 19~4) and smaller at later time

An important conclusion from our study is the inability of Iurge fixed-loop surveys to resolve multiple conductors Thereshyfore when an anomaly is located with this type of survey we

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I

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I

I I

I

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1 ~(fot

l]lCS tmiddot 25 SLmiddot 800IOOOSl Of- 100m DBmiddot 100

-200 -160 -120 -80 -40 0 40 80 126 ISO 200 2~0 280

DISfANCE (m) Fit 16 Profile of the vertical component of (1b(l)tE (voltage) calculated for two prisms in a WOO Omiddot m half-space for a

fixed-loop configuration One prism is located at x = am and the other prism is at x = 80 m

705 3-D Modeling of TEM Scattering

MODEL

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I

-lOO -160 -120 -80 -40 40 80 120 160 OO l40 l80

DISTANCE (m)

FIG 17 Comparison of profiles for the scattered vertical component of (Ib(t)Dt (voltage) calculated by superimposing the scattered response of each prism in its own half-space (circles) with the response which includes mutual interaction between the prisms (curves)

1010

1=gt shy 10- 11 --gt

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t~ --------__

lmJ

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MODEL

XmiddotOIll xmiddot80m-Ll surface

o O middot0Sm t bull 25m

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-200 -160 -120 -ao -40 120 160 200 240

FI( r8 Profile of the vertical component of (1b(t)(~[ (voltage] calculated for two prisms in a LOOO Q m half-space for a central-loop configuration The loop dimensions are 25 m by 25 m

706 Newman and Hohmann

recommend (if possible) further work with a moving-source survey because it is more likely to eliminate interpretational ambiguities of multiple conductors and can provide precise information for locating drill holes

CONCLUDING REMARKS

We have presented a 3-D integral-equation solution for the TEM responses of prisms in a layered half-space In summary our solution is based on a Galerkin solution for the scattering current where the weighting functions are the basis functions It is necessary that the solution include a set nf basis and weight functions that are divergence-free if model results are to be accurate at large background resistivities The

divergence-free basis functions are used to model eddy curshyrents which are closed in the prisms and hence simulate the inductive responses of the prisms in a very resistive earth More importantly divergence-free weight functions eliminate the charge operator from part of the solution by integrating it over rectangular paths Thus the solution satisfies the freeshyspace boundary condition for no current leaks and the reshysulting solution is stable for a host with any resistivity

We used current-tube functions as the divergence-free basis functions because they are simple and acceptable for our comshyputational limitations The current-tube functions are crude and restrict the geometry of the 3-D bodies to prisms Howshyever they are effective and allow modeling prisms of moderate geometric extent An exception where our solution does not perform well is for square and nearly square plates at a high resistivity contrast In this case the time constants of the plates are underestimated We believe the accuracy and genershyality of our solution will improve with a more sophisticated divergence-free basis function

The usefulness of OUT solution has been demonstrated with TEM model studies We investigated 3-D responses under the influence of conductive overburden and made an evaluation of the resolution of two conductors based on different survey configurations However our solution is more versatile than we have illustrated since it is valid for any number of layers and prisms and is designed for many different types oC source and receiver configurations Important exploration problems where the present solution can be applied include 3-D airshyborne TFM model studies and evaluations of the effect of a conductive host on 3-D TEM borehole responses Finally the solution is not restricted to TEM applications Because its basic formulation is in the frequency domain it can be applied to frequency-domain exploration problems as well

ACK-NO-WLEDGMpound~TS

This paper is dedicated to the memory of Frank Frischshyknecht We are grateful to him and the US Geological Survey for making the Surveys VAX IInO computer available for much of this work We are also indebted to Walter Anderson for his help in installing and running the programs Financial support was provided by Amoco Production Co ARCO Oil and Gas Co Chevron Resources Company CRA Exploration Pty Ltd Standard Oil Production Co and Unocal Corp in conjunction with the University of Utah EM Modeling Reshysearch Project

REFERENCES

Annan A P 1974 The equivalent source method for electromagnetic scattering analysis and its geophysical applications PhD thesis Memorial lJniv of Newfoundland

Dyck A V and West G F 1984 The role of simple computer models in interpretations of wide-band drill-hole electromagnetic surveys in mineral exploration Geophysics 49 957-980

Eaton P A and Hohmann G W 1987 Approximate inversion for lransient electromagnetic soundings Phys of the Earth and Plan Int in press

Fitlcrman D Y and Anderson W l 1987 Effect of transmitter turn-off on transient soundings Geoexplor 2413-146

Gallaghtr P R Ward S H and Hohmann G W 19R5 A model study of a thin plate in free space for the EM37 transient electroshymagnetic system (ieophysies 50 ()())-lDlq

Gupta O P Kakirde S T and Negi J G bull 1987 Scale model experiments In study low frequency electromagnetic resolution of multiple conductors Trans Geoscience and Remote Sensing Inst of Electr Electron Eng in press

Hanneson J E and West G F 1984 The horizontal-loop electroshymagnetic response of a thin plate in a conductive earth Part _shyComputat ional method Geophysics 49411-420

Harrington R F 1968 Field computation by moment methods The MacMillan Pub Co

Hohmann G W 19lB Three-dimensional EM modeling Geophys Surv 6 27 53

Kaufman A A 1978 Frequency and transient responses of EM fields created by currents in confined conductors Geophysics 43 1002-1010

19~5 Distribution of alternating electrical charges in a conshyductive medium Geophys Prosp 33171-184

Kaufman A A and Keller G V 1985 Inductive mining prospectshying Part I Theory ElsevierScience Pub Co

Lajoie J J and West G r 1976 The electromagnetic response of a conductive inhomogeneity in a layered earth Geophysics 4l 1133-1156

Lamontagne Y 1975 Applications of wideband time-domain elecshytrornagnetic measurements in mineral exploration PhD thesis UnivofToronto

Lee T S 19~ I Transient electromagnetic response of a sphere in a layered medium Pure Appl Geophys 119309-338shy

MLCracken K G Oristaglio 1 L and Hohmann U W 19~6 A comparison of electromagnetic exploration systems Geophysics SI XIO XIX

Nabighian M N 1987 Inductive time-domain electromagnetic methods EM mining volume Soc Explor Geophys in press

Newman G A Anderson W L and Hohmann G W 1987 Intershyprerarion of transrent electromagnetic soundings over threeshydimensional structures for the central-loop configuration Geophys J Roy Astr Socbull 89 l~9-(14

Newman G A Hohmann G W and Anderson W L 19~6 Transhysient electromagnetic response or a three-dimensional body in a layered earth Geophysics 51 160~- [627

Ogd~y K D 1910 A model study of the transient electromagnetic coincident loop technique Geoexplor 21 231-264

Petrick W R 11)84A fully internal hybrid technique for calculating electromagnetic SCattering from three-dimensional bodies in the earth PhD thesis Univ of Utah

Sanfilipo W A and Hohmann G W 1985 Integral equation solushyion for [he transient electromagnetic response of a threeshydimensional body in a conductive half-space Geophysics 50 798shy~Ol

S1I1hlipo W A I-aton P A and Hohmann G W 1985 The effect raquof a conductive half-space on the transient electromagnetic reshysponse of a three-dimensional body Geophysics 50 1144-1162

Spies B R 1980 TE~ in Australian conditions Field examples and model studies PhD thesis Macquarie Univ

Spies B R and Parker P 0- l9H4~ Limirarions of large-loop transhysient surveys in conductive terrains Geophysics 49 902 -912

Tripp A C ami Hohmann G W 1984 Block diagonalization of the electromagnetic impedance matrix of a symmetric body using group theory Inst of Elect and Electron Eng Trans Geosci and Remote Sensing GE-22 6269

Van Bladcl J 1961 SIJOlC remarks un Greens dyadic for infinite SPltlIC Trans on Antennas and Propagation Insc Electr Electron Eng 9 563middot 566

Wannamaker P E Hohmann G W and Sanfilipo W A 1984 Electromagnetic modeling of three-dimensional bodies in layered cart h- using integral equations Geophysics 49 60-77

Wcidclt P 198 l Dipole induction in a thin plate with host medium arid ovcrburdcn Research Project NTS 10 no R9727 Federal Inst ior Earth Sciences and Ravv Materials Hannover West Germany

-----------

703 3-D Modeling of TEM Scattering

n m overburden there is a pronounced enhancement in the prisms response However our findings show when the overshyburden is 100 III thick there wilt be no dramatic increase in the delectability of the prism even when it is in contact with the overburden

R[SOLLITION OF TWO CONDUCTIVE PRISMS WITH TFM SURVEYS

An important problem in mineral exploration for massive sulfides is the resolution and possible interpretational ambishy

guities of multiple conductors Very few theoretical modelshystudies have been published on the FM response of multiple conductors 1110st studies have been based all scale models

Ogilvy (J 983) published scale-model results on the horizontal resolution for the TEM coincident-loop configuration of two identical thin-sheet conductors that were vertically oriented in free space His findings show that when the separation beshytween the conductors is less than one-quarter of the loops diameter individual conductors cannot be resolved Gupta et aJ 19871 recently conducted scale-model experiments on the resolution of two vertical half-planes for the horizontal loopshyloop system Their results show that closely spaced scatterers arc resolved only when the separation between the half-planes is greater than the transmitter-receiver separation

Numerical models can also contribute interpretational inshy~ight into the lateral resolution and response of multiple conshyductors Obviously many multiple targets for different host layering and survey configurations can be modeled with our solution However our study is intended only as a preliminary

survey so we study the TEM response of two prisms in a resistive half-space for the fixed-loop and central-loop configushyrations We also discuss the resolution uf both conductors paying particular attention to survey design

In Figure 16 we show the vertical response (total voltage) of two prisms in a 1000 Q m half-space for a fixed-loop configushyration The profiles arc at seven different times ranging from 05 to 30 ms The prisms are identical in size depth and conductivity and are the same as the prism in Figure 8 One prism is positioned at x = 0 m and the other is at x = 80 m The fixed-loop transmitter is again 500 m on a side with its neurwirehxnrtedar x = -250m

At early times (05 and 1 ms] and at late times (which inshy

elude 15 20 ami 30 rns) the response of the tOOO n 111 halfshyspace is visible in figure 16 The half-space response is imporshytant at late times because it has a slower decay than the response of the two prisms If the response of the half-space is not considered in an interpretation the migrating crossover at intermediate and (ate times in Figure (6 will appear to indishycate a dipping conductor However once the half-space reshysponse is subtracted from the profile a vertical-plate conducshytor is interpreted at x = 30 m with a time constant of 37 ms This time constant is the same as that observed when the prisms arc together in free space The fixed-loop configuration clearly fails to distinguish the individual prisms even when the half-space response is removed Spies and Parker (1984) have demonstrated with scale-model experiments similar limishy

tations of large fixed-loop surveys The mutual interaction or the TEM fields of the two prisms

can he investigated by comparing responses with and without

---- -

------1 gt --~~

IO- T

3 shy

10

N- 9

I 1OE

gtshy10

15bullbull

s bullbull ----_~

--

J

-gtshy2O

Win_

30n

MO_Qf~

10middot1

iocon-Nealt d~e 01 loop

I

middot200 -160 -120 middot80 -40 40 80 120 160 200

DISTANCE FROM CONDUCTOR CENTER ltII)

FIG 15 Profile of the vertical component of tb(lliltt (voltage) calculated for a prism in a tooo Q m half-space in contact with a 100 m thick 10 n m overburden

704 Newman and Hohmann

the interaction In Figure 17 the scattered response of each

prism in its own half-space is superimposed and compared

with the scattered response which includes the mutual interacshytion Over the time range of 05 to 5 IDS the superimposed response in the vertical component is larger (except near sign reversals) than the scattered response observed when the fields of the two prisms interact However at later times (greater than 5 ms) this pattern reverses Both responses in Figure 17 decay exponentially at late times and we estimate a time conshystant of 33 ms for the superimposed response which is 11 percent smaller than the time constant estimated from the

response with the mutual interaction 37 ms The II percent difference in the time constants reflects the 66 percent disparshyity between the two responses at _~O ms as shown in Figure Il

As suggested by Spies and Parker (1(84) a survey configushy

ration where the source moves along the profile can improve the lateral resolution or the two prisms For a central-loop configuration with loops 25 m OIl a side both prisms are resolved at I 3 and 5 ms as shown in Figure 18 The two

minima ill the profilesindicate the locations of both prisms at these times At later times interpretational ambiguities could arise because the two separate minima are replaced by a single minimum located half way between the two prisms This single minimum could be interpreted as an isolated conductor Overall the shapes of the profiles in Figure 18 are similar to those studied by Ogilvy (1983) for the coincident-loop configushy

ration Note that after I ms the half-space has little influence on the response of the prisms in Figure 18 When the halfshyspace response is removed the response of the prisms is the same as that observed in free space

till 5NS shy

10middot

Jshy

10- 9

3

til 5E-==shy O-ID

shy-==

15

10-11

10

3010-11

NOf edge of loop

The time constant estimated from the scattered response of the two prisms at late time is dependent upon the type of survey Unlike the case for a fixed-loop configuration the time constant varies with its position along the profile for the central-loop configuration When central-loop stations are loshycated away from the two prisms the estimated time constant is the same as in the fixed-loop case 37 rns However when a central-loop station approaches both prisms the time conshystant decreases and its smallest value is 29 ms at x = 40 m The time constant decreases because the transmitter induces eddy currents in the prisms that flow in opposite directions Since these eddy currents oppose each other the magnetic

field of (he eddy currents in one prism enhances the decay of the eddy currents in the other prism Consequently a time constant is produced which is smaller than that for each prism in free space 33 ms However the eddy currents also produce a larger time constant when the transmitter is away from the prisms Here the eddy currents in both prisms flow in the same direction and they tend to sustain each other with their magshynetic fields The variation in the time constant indicates the presence of multiple conductors since the time constant of a single conductor does not vary along the profile Finally the variation in the time constant also explains the peak and trough in the profiles between the prisms at early and late times In a resistive host the response of a poor conductor is larger than that of a good conductor at early time (Dyck and West 19~4) and smaller at later time

An important conclusion from our study is the inability of Iurge fixed-loop surveys to resolve multiple conductors Thereshyfore when an anomaly is located with this type of survey we

~-------

---shy

lt I I

I

-ltr gt

-shyI lt f

I I I I I

shy

1I

1 V ----_ II

I

I I

I

MODEL -----shy~middotO x-80

1 ~(fot

l]lCS tmiddot 25 SLmiddot 800IOOOSl Of- 100m DBmiddot 100

-200 -160 -120 -80 -40 0 40 80 126 ISO 200 2~0 280

DISfANCE (m) Fit 16 Profile of the vertical component of (1b(l)tE (voltage) calculated for two prisms in a WOO Omiddot m half-space for a

fixed-loop configuration One prism is located at x = am and the other prism is at x = 80 m

705 3-D Modeling of TEM Scattering

MODEL

bullbull II III 41 a 0

------- shy0

IG-I 0 I

J bull __- _J -_ I s

i I ----__

lomiddotg

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~ =shy

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---- shy U 105 ~ JS - iocoa SLmiddot800

0[100 n8middot100

I

-lOO -160 -120 -80 -40 40 80 120 160 OO l40 l80

DISTANCE (m)

FIG 17 Comparison of profiles for the scattered vertical component of (Ib(t)Dt (voltage) calculated by superimposing the scattered response of each prism in its own half-space (circles) with the response which includes mutual interaction between the prisms (curves)

1010

1=gt shy 10- 11 --gt

10 -1pound

t~ --------__

lmJ

3mJ

5ms

MODEL

XmiddotOIll xmiddot80m-Ll surface

o O middot0Sm t bull 25m

1000nm SLmiddot800m DElOOm DBlOOm

25 m x 25m central loop~

-200 -160 -120 -ao -40 120 160 200 240

FI( r8 Profile of the vertical component of (1b(t)(~[ (voltage] calculated for two prisms in a LOOO Q m half-space for a central-loop configuration The loop dimensions are 25 m by 25 m

706 Newman and Hohmann

recommend (if possible) further work with a moving-source survey because it is more likely to eliminate interpretational ambiguities of multiple conductors and can provide precise information for locating drill holes

CONCLUDING REMARKS

We have presented a 3-D integral-equation solution for the TEM responses of prisms in a layered half-space In summary our solution is based on a Galerkin solution for the scattering current where the weighting functions are the basis functions It is necessary that the solution include a set nf basis and weight functions that are divergence-free if model results are to be accurate at large background resistivities The

divergence-free basis functions are used to model eddy curshyrents which are closed in the prisms and hence simulate the inductive responses of the prisms in a very resistive earth More importantly divergence-free weight functions eliminate the charge operator from part of the solution by integrating it over rectangular paths Thus the solution satisfies the freeshyspace boundary condition for no current leaks and the reshysulting solution is stable for a host with any resistivity

We used current-tube functions as the divergence-free basis functions because they are simple and acceptable for our comshyputational limitations The current-tube functions are crude and restrict the geometry of the 3-D bodies to prisms Howshyever they are effective and allow modeling prisms of moderate geometric extent An exception where our solution does not perform well is for square and nearly square plates at a high resistivity contrast In this case the time constants of the plates are underestimated We believe the accuracy and genershyality of our solution will improve with a more sophisticated divergence-free basis function

The usefulness of OUT solution has been demonstrated with TEM model studies We investigated 3-D responses under the influence of conductive overburden and made an evaluation of the resolution of two conductors based on different survey configurations However our solution is more versatile than we have illustrated since it is valid for any number of layers and prisms and is designed for many different types oC source and receiver configurations Important exploration problems where the present solution can be applied include 3-D airshyborne TFM model studies and evaluations of the effect of a conductive host on 3-D TEM borehole responses Finally the solution is not restricted to TEM applications Because its basic formulation is in the frequency domain it can be applied to frequency-domain exploration problems as well

ACK-NO-WLEDGMpound~TS

This paper is dedicated to the memory of Frank Frischshyknecht We are grateful to him and the US Geological Survey for making the Surveys VAX IInO computer available for much of this work We are also indebted to Walter Anderson for his help in installing and running the programs Financial support was provided by Amoco Production Co ARCO Oil and Gas Co Chevron Resources Company CRA Exploration Pty Ltd Standard Oil Production Co and Unocal Corp in conjunction with the University of Utah EM Modeling Reshysearch Project

REFERENCES

Annan A P 1974 The equivalent source method for electromagnetic scattering analysis and its geophysical applications PhD thesis Memorial lJniv of Newfoundland

Dyck A V and West G F 1984 The role of simple computer models in interpretations of wide-band drill-hole electromagnetic surveys in mineral exploration Geophysics 49 957-980

Eaton P A and Hohmann G W 1987 Approximate inversion for lransient electromagnetic soundings Phys of the Earth and Plan Int in press

Fitlcrman D Y and Anderson W l 1987 Effect of transmitter turn-off on transient soundings Geoexplor 2413-146

Gallaghtr P R Ward S H and Hohmann G W 19R5 A model study of a thin plate in free space for the EM37 transient electroshymagnetic system (ieophysies 50 ()())-lDlq

Gupta O P Kakirde S T and Negi J G bull 1987 Scale model experiments In study low frequency electromagnetic resolution of multiple conductors Trans Geoscience and Remote Sensing Inst of Electr Electron Eng in press

Hanneson J E and West G F 1984 The horizontal-loop electroshymagnetic response of a thin plate in a conductive earth Part _shyComputat ional method Geophysics 49411-420

Harrington R F 1968 Field computation by moment methods The MacMillan Pub Co

Hohmann G W 19lB Three-dimensional EM modeling Geophys Surv 6 27 53

Kaufman A A 1978 Frequency and transient responses of EM fields created by currents in confined conductors Geophysics 43 1002-1010

19~5 Distribution of alternating electrical charges in a conshyductive medium Geophys Prosp 33171-184

Kaufman A A and Keller G V 1985 Inductive mining prospectshying Part I Theory ElsevierScience Pub Co

Lajoie J J and West G r 1976 The electromagnetic response of a conductive inhomogeneity in a layered earth Geophysics 4l 1133-1156

Lamontagne Y 1975 Applications of wideband time-domain elecshytrornagnetic measurements in mineral exploration PhD thesis UnivofToronto

Lee T S 19~ I Transient electromagnetic response of a sphere in a layered medium Pure Appl Geophys 119309-338shy

MLCracken K G Oristaglio 1 L and Hohmann U W 19~6 A comparison of electromagnetic exploration systems Geophysics SI XIO XIX

Nabighian M N 1987 Inductive time-domain electromagnetic methods EM mining volume Soc Explor Geophys in press

Newman G A Anderson W L and Hohmann G W 1987 Intershyprerarion of transrent electromagnetic soundings over threeshydimensional structures for the central-loop configuration Geophys J Roy Astr Socbull 89 l~9-(14

Newman G A Hohmann G W and Anderson W L 19~6 Transhysient electromagnetic response or a three-dimensional body in a layered earth Geophysics 51 160~- [627

Ogd~y K D 1910 A model study of the transient electromagnetic coincident loop technique Geoexplor 21 231-264

Petrick W R 11)84A fully internal hybrid technique for calculating electromagnetic SCattering from three-dimensional bodies in the earth PhD thesis Univ of Utah

Sanfilipo W A and Hohmann G W 1985 Integral equation solushyion for [he transient electromagnetic response of a threeshydimensional body in a conductive half-space Geophysics 50 798shy~Ol

S1I1hlipo W A I-aton P A and Hohmann G W 1985 The effect raquof a conductive half-space on the transient electromagnetic reshysponse of a three-dimensional body Geophysics 50 1144-1162

Spies B R 1980 TE~ in Australian conditions Field examples and model studies PhD thesis Macquarie Univ

Spies B R and Parker P 0- l9H4~ Limirarions of large-loop transhysient surveys in conductive terrains Geophysics 49 902 -912

Tripp A C ami Hohmann G W 1984 Block diagonalization of the electromagnetic impedance matrix of a symmetric body using group theory Inst of Elect and Electron Eng Trans Geosci and Remote Sensing GE-22 6269

Van Bladcl J 1961 SIJOlC remarks un Greens dyadic for infinite SPltlIC Trans on Antennas and Propagation Insc Electr Electron Eng 9 563middot 566

Wannamaker P E Hohmann G W and Sanfilipo W A 1984 Electromagnetic modeling of three-dimensional bodies in layered cart h- using integral equations Geophysics 49 60-77

Wcidclt P 198 l Dipole induction in a thin plate with host medium arid ovcrburdcn Research Project NTS 10 no R9727 Federal Inst ior Earth Sciences and Ravv Materials Hannover West Germany

704 Newman and Hohmann

the interaction In Figure 17 the scattered response of each

prism in its own half-space is superimposed and compared

with the scattered response which includes the mutual interacshytion Over the time range of 05 to 5 IDS the superimposed response in the vertical component is larger (except near sign reversals) than the scattered response observed when the fields of the two prisms interact However at later times (greater than 5 ms) this pattern reverses Both responses in Figure 17 decay exponentially at late times and we estimate a time conshystant of 33 ms for the superimposed response which is 11 percent smaller than the time constant estimated from the

response with the mutual interaction 37 ms The II percent difference in the time constants reflects the 66 percent disparshyity between the two responses at _~O ms as shown in Figure Il

As suggested by Spies and Parker (1(84) a survey configushy

ration where the source moves along the profile can improve the lateral resolution or the two prisms For a central-loop configuration with loops 25 m OIl a side both prisms are resolved at I 3 and 5 ms as shown in Figure 18 The two

minima ill the profilesindicate the locations of both prisms at these times At later times interpretational ambiguities could arise because the two separate minima are replaced by a single minimum located half way between the two prisms This single minimum could be interpreted as an isolated conductor Overall the shapes of the profiles in Figure 18 are similar to those studied by Ogilvy (1983) for the coincident-loop configushy

ration Note that after I ms the half-space has little influence on the response of the prisms in Figure 18 When the halfshyspace response is removed the response of the prisms is the same as that observed in free space

till 5NS shy

10middot

Jshy

10- 9

3

til 5E-==shy O-ID

shy-==

15

10-11

10

3010-11

NOf edge of loop

The time constant estimated from the scattered response of the two prisms at late time is dependent upon the type of survey Unlike the case for a fixed-loop configuration the time constant varies with its position along the profile for the central-loop configuration When central-loop stations are loshycated away from the two prisms the estimated time constant is the same as in the fixed-loop case 37 rns However when a central-loop station approaches both prisms the time conshystant decreases and its smallest value is 29 ms at x = 40 m The time constant decreases because the transmitter induces eddy currents in the prisms that flow in opposite directions Since these eddy currents oppose each other the magnetic

field of (he eddy currents in one prism enhances the decay of the eddy currents in the other prism Consequently a time constant is produced which is smaller than that for each prism in free space 33 ms However the eddy currents also produce a larger time constant when the transmitter is away from the prisms Here the eddy currents in both prisms flow in the same direction and they tend to sustain each other with their magshynetic fields The variation in the time constant indicates the presence of multiple conductors since the time constant of a single conductor does not vary along the profile Finally the variation in the time constant also explains the peak and trough in the profiles between the prisms at early and late times In a resistive host the response of a poor conductor is larger than that of a good conductor at early time (Dyck and West 19~4) and smaller at later time

An important conclusion from our study is the inability of Iurge fixed-loop surveys to resolve multiple conductors Thereshyfore when an anomaly is located with this type of survey we

~-------

---shy

lt I I

I

-ltr gt

-shyI lt f

I I I I I

shy

1I

1 V ----_ II

I

I I

I

MODEL -----shy~middotO x-80

1 ~(fot

l]lCS tmiddot 25 SLmiddot 800IOOOSl Of- 100m DBmiddot 100

-200 -160 -120 -80 -40 0 40 80 126 ISO 200 2~0 280

DISfANCE (m) Fit 16 Profile of the vertical component of (1b(l)tE (voltage) calculated for two prisms in a WOO Omiddot m half-space for a

fixed-loop configuration One prism is located at x = am and the other prism is at x = 80 m

705 3-D Modeling of TEM Scattering

MODEL

bullbull II III 41 a 0

------- shy0

IG-I 0 I

J bull __- _J -_ I s

i I ----__

lomiddotg

( --~~ l~ ~_-

e I 5 I I _------- _gt

N- 10- 10

II laquo ~ - 10I

~ =shy

10middot1

xmiddotODt xaoIO-I~ fiJrfCJ(~

---- shy U 105 ~ JS - iocoa SLmiddot800

0[100 n8middot100

I

-lOO -160 -120 -80 -40 40 80 120 160 OO l40 l80

DISTANCE (m)

FIG 17 Comparison of profiles for the scattered vertical component of (Ib(t)Dt (voltage) calculated by superimposing the scattered response of each prism in its own half-space (circles) with the response which includes mutual interaction between the prisms (curves)

1010

1=gt shy 10- 11 --gt

10 -1pound

t~ --------__

lmJ

3mJ

5ms

MODEL

XmiddotOIll xmiddot80m-Ll surface

o O middot0Sm t bull 25m

1000nm SLmiddot800m DElOOm DBlOOm

25 m x 25m central loop~

-200 -160 -120 -ao -40 120 160 200 240

FI( r8 Profile of the vertical component of (1b(t)(~[ (voltage] calculated for two prisms in a LOOO Q m half-space for a central-loop configuration The loop dimensions are 25 m by 25 m

706 Newman and Hohmann

recommend (if possible) further work with a moving-source survey because it is more likely to eliminate interpretational ambiguities of multiple conductors and can provide precise information for locating drill holes

CONCLUDING REMARKS

We have presented a 3-D integral-equation solution for the TEM responses of prisms in a layered half-space In summary our solution is based on a Galerkin solution for the scattering current where the weighting functions are the basis functions It is necessary that the solution include a set nf basis and weight functions that are divergence-free if model results are to be accurate at large background resistivities The

divergence-free basis functions are used to model eddy curshyrents which are closed in the prisms and hence simulate the inductive responses of the prisms in a very resistive earth More importantly divergence-free weight functions eliminate the charge operator from part of the solution by integrating it over rectangular paths Thus the solution satisfies the freeshyspace boundary condition for no current leaks and the reshysulting solution is stable for a host with any resistivity

We used current-tube functions as the divergence-free basis functions because they are simple and acceptable for our comshyputational limitations The current-tube functions are crude and restrict the geometry of the 3-D bodies to prisms Howshyever they are effective and allow modeling prisms of moderate geometric extent An exception where our solution does not perform well is for square and nearly square plates at a high resistivity contrast In this case the time constants of the plates are underestimated We believe the accuracy and genershyality of our solution will improve with a more sophisticated divergence-free basis function

The usefulness of OUT solution has been demonstrated with TEM model studies We investigated 3-D responses under the influence of conductive overburden and made an evaluation of the resolution of two conductors based on different survey configurations However our solution is more versatile than we have illustrated since it is valid for any number of layers and prisms and is designed for many different types oC source and receiver configurations Important exploration problems where the present solution can be applied include 3-D airshyborne TFM model studies and evaluations of the effect of a conductive host on 3-D TEM borehole responses Finally the solution is not restricted to TEM applications Because its basic formulation is in the frequency domain it can be applied to frequency-domain exploration problems as well

ACK-NO-WLEDGMpound~TS

This paper is dedicated to the memory of Frank Frischshyknecht We are grateful to him and the US Geological Survey for making the Surveys VAX IInO computer available for much of this work We are also indebted to Walter Anderson for his help in installing and running the programs Financial support was provided by Amoco Production Co ARCO Oil and Gas Co Chevron Resources Company CRA Exploration Pty Ltd Standard Oil Production Co and Unocal Corp in conjunction with the University of Utah EM Modeling Reshysearch Project

REFERENCES

Annan A P 1974 The equivalent source method for electromagnetic scattering analysis and its geophysical applications PhD thesis Memorial lJniv of Newfoundland

Dyck A V and West G F 1984 The role of simple computer models in interpretations of wide-band drill-hole electromagnetic surveys in mineral exploration Geophysics 49 957-980

Eaton P A and Hohmann G W 1987 Approximate inversion for lransient electromagnetic soundings Phys of the Earth and Plan Int in press

Fitlcrman D Y and Anderson W l 1987 Effect of transmitter turn-off on transient soundings Geoexplor 2413-146

Gallaghtr P R Ward S H and Hohmann G W 19R5 A model study of a thin plate in free space for the EM37 transient electroshymagnetic system (ieophysies 50 ()())-lDlq

Gupta O P Kakirde S T and Negi J G bull 1987 Scale model experiments In study low frequency electromagnetic resolution of multiple conductors Trans Geoscience and Remote Sensing Inst of Electr Electron Eng in press

Hanneson J E and West G F 1984 The horizontal-loop electroshymagnetic response of a thin plate in a conductive earth Part _shyComputat ional method Geophysics 49411-420

Harrington R F 1968 Field computation by moment methods The MacMillan Pub Co

Hohmann G W 19lB Three-dimensional EM modeling Geophys Surv 6 27 53

Kaufman A A 1978 Frequency and transient responses of EM fields created by currents in confined conductors Geophysics 43 1002-1010

19~5 Distribution of alternating electrical charges in a conshyductive medium Geophys Prosp 33171-184

Kaufman A A and Keller G V 1985 Inductive mining prospectshying Part I Theory ElsevierScience Pub Co

Lajoie J J and West G r 1976 The electromagnetic response of a conductive inhomogeneity in a layered earth Geophysics 4l 1133-1156

Lamontagne Y 1975 Applications of wideband time-domain elecshytrornagnetic measurements in mineral exploration PhD thesis UnivofToronto

Lee T S 19~ I Transient electromagnetic response of a sphere in a layered medium Pure Appl Geophys 119309-338shy

MLCracken K G Oristaglio 1 L and Hohmann U W 19~6 A comparison of electromagnetic exploration systems Geophysics SI XIO XIX

Nabighian M N 1987 Inductive time-domain electromagnetic methods EM mining volume Soc Explor Geophys in press

Newman G A Anderson W L and Hohmann G W 1987 Intershyprerarion of transrent electromagnetic soundings over threeshydimensional structures for the central-loop configuration Geophys J Roy Astr Socbull 89 l~9-(14

Newman G A Hohmann G W and Anderson W L 19~6 Transhysient electromagnetic response or a three-dimensional body in a layered earth Geophysics 51 160~- [627

Ogd~y K D 1910 A model study of the transient electromagnetic coincident loop technique Geoexplor 21 231-264

Petrick W R 11)84A fully internal hybrid technique for calculating electromagnetic SCattering from three-dimensional bodies in the earth PhD thesis Univ of Utah

Sanfilipo W A and Hohmann G W 1985 Integral equation solushyion for [he transient electromagnetic response of a threeshydimensional body in a conductive half-space Geophysics 50 798shy~Ol

S1I1hlipo W A I-aton P A and Hohmann G W 1985 The effect raquof a conductive half-space on the transient electromagnetic reshysponse of a three-dimensional body Geophysics 50 1144-1162

Spies B R 1980 TE~ in Australian conditions Field examples and model studies PhD thesis Macquarie Univ

Spies B R and Parker P 0- l9H4~ Limirarions of large-loop transhysient surveys in conductive terrains Geophysics 49 902 -912

Tripp A C ami Hohmann G W 1984 Block diagonalization of the electromagnetic impedance matrix of a symmetric body using group theory Inst of Elect and Electron Eng Trans Geosci and Remote Sensing GE-22 6269

Van Bladcl J 1961 SIJOlC remarks un Greens dyadic for infinite SPltlIC Trans on Antennas and Propagation Insc Electr Electron Eng 9 563middot 566

Wannamaker P E Hohmann G W and Sanfilipo W A 1984 Electromagnetic modeling of three-dimensional bodies in layered cart h- using integral equations Geophysics 49 60-77

Wcidclt P 198 l Dipole induction in a thin plate with host medium arid ovcrburdcn Research Project NTS 10 no R9727 Federal Inst ior Earth Sciences and Ravv Materials Hannover West Germany

705 3-D Modeling of TEM Scattering

MODEL

bullbull II III 41 a 0

------- shy0

IG-I 0 I

J bull __- _J -_ I s

i I ----__

lomiddotg

( --~~ l~ ~_-

e I 5 I I _------- _gt

N- 10- 10

II laquo ~ - 10I

~ =shy

10middot1

xmiddotODt xaoIO-I~ fiJrfCJ(~

---- shy U 105 ~ JS - iocoa SLmiddot800

0[100 n8middot100

I

-lOO -160 -120 -80 -40 40 80 120 160 OO l40 l80

DISTANCE (m)

FIG 17 Comparison of profiles for the scattered vertical component of (Ib(t)Dt (voltage) calculated by superimposing the scattered response of each prism in its own half-space (circles) with the response which includes mutual interaction between the prisms (curves)

1010

1=gt shy 10- 11 --gt

10 -1pound

t~ --------__

lmJ

3mJ

5ms

MODEL

XmiddotOIll xmiddot80m-Ll surface

o O middot0Sm t bull 25m

1000nm SLmiddot800m DElOOm DBlOOm

25 m x 25m central loop~

-200 -160 -120 -ao -40 120 160 200 240

FI( r8 Profile of the vertical component of (1b(t)(~[ (voltage] calculated for two prisms in a LOOO Q m half-space for a central-loop configuration The loop dimensions are 25 m by 25 m

706 Newman and Hohmann

recommend (if possible) further work with a moving-source survey because it is more likely to eliminate interpretational ambiguities of multiple conductors and can provide precise information for locating drill holes

CONCLUDING REMARKS

We have presented a 3-D integral-equation solution for the TEM responses of prisms in a layered half-space In summary our solution is based on a Galerkin solution for the scattering current where the weighting functions are the basis functions It is necessary that the solution include a set nf basis and weight functions that are divergence-free if model results are to be accurate at large background resistivities The

divergence-free basis functions are used to model eddy curshyrents which are closed in the prisms and hence simulate the inductive responses of the prisms in a very resistive earth More importantly divergence-free weight functions eliminate the charge operator from part of the solution by integrating it over rectangular paths Thus the solution satisfies the freeshyspace boundary condition for no current leaks and the reshysulting solution is stable for a host with any resistivity

We used current-tube functions as the divergence-free basis functions because they are simple and acceptable for our comshyputational limitations The current-tube functions are crude and restrict the geometry of the 3-D bodies to prisms Howshyever they are effective and allow modeling prisms of moderate geometric extent An exception where our solution does not perform well is for square and nearly square plates at a high resistivity contrast In this case the time constants of the plates are underestimated We believe the accuracy and genershyality of our solution will improve with a more sophisticated divergence-free basis function

The usefulness of OUT solution has been demonstrated with TEM model studies We investigated 3-D responses under the influence of conductive overburden and made an evaluation of the resolution of two conductors based on different survey configurations However our solution is more versatile than we have illustrated since it is valid for any number of layers and prisms and is designed for many different types oC source and receiver configurations Important exploration problems where the present solution can be applied include 3-D airshyborne TFM model studies and evaluations of the effect of a conductive host on 3-D TEM borehole responses Finally the solution is not restricted to TEM applications Because its basic formulation is in the frequency domain it can be applied to frequency-domain exploration problems as well

ACK-NO-WLEDGMpound~TS

This paper is dedicated to the memory of Frank Frischshyknecht We are grateful to him and the US Geological Survey for making the Surveys VAX IInO computer available for much of this work We are also indebted to Walter Anderson for his help in installing and running the programs Financial support was provided by Amoco Production Co ARCO Oil and Gas Co Chevron Resources Company CRA Exploration Pty Ltd Standard Oil Production Co and Unocal Corp in conjunction with the University of Utah EM Modeling Reshysearch Project

REFERENCES

Annan A P 1974 The equivalent source method for electromagnetic scattering analysis and its geophysical applications PhD thesis Memorial lJniv of Newfoundland

Dyck A V and West G F 1984 The role of simple computer models in interpretations of wide-band drill-hole electromagnetic surveys in mineral exploration Geophysics 49 957-980

Eaton P A and Hohmann G W 1987 Approximate inversion for lransient electromagnetic soundings Phys of the Earth and Plan Int in press

Fitlcrman D Y and Anderson W l 1987 Effect of transmitter turn-off on transient soundings Geoexplor 2413-146

Gallaghtr P R Ward S H and Hohmann G W 19R5 A model study of a thin plate in free space for the EM37 transient electroshymagnetic system (ieophysies 50 ()())-lDlq

Gupta O P Kakirde S T and Negi J G bull 1987 Scale model experiments In study low frequency electromagnetic resolution of multiple conductors Trans Geoscience and Remote Sensing Inst of Electr Electron Eng in press

Hanneson J E and West G F 1984 The horizontal-loop electroshymagnetic response of a thin plate in a conductive earth Part _shyComputat ional method Geophysics 49411-420

Harrington R F 1968 Field computation by moment methods The MacMillan Pub Co

Hohmann G W 19lB Three-dimensional EM modeling Geophys Surv 6 27 53

Kaufman A A 1978 Frequency and transient responses of EM fields created by currents in confined conductors Geophysics 43 1002-1010

19~5 Distribution of alternating electrical charges in a conshyductive medium Geophys Prosp 33171-184

Kaufman A A and Keller G V 1985 Inductive mining prospectshying Part I Theory ElsevierScience Pub Co

Lajoie J J and West G r 1976 The electromagnetic response of a conductive inhomogeneity in a layered earth Geophysics 4l 1133-1156

Lamontagne Y 1975 Applications of wideband time-domain elecshytrornagnetic measurements in mineral exploration PhD thesis UnivofToronto

Lee T S 19~ I Transient electromagnetic response of a sphere in a layered medium Pure Appl Geophys 119309-338shy

MLCracken K G Oristaglio 1 L and Hohmann U W 19~6 A comparison of electromagnetic exploration systems Geophysics SI XIO XIX

Nabighian M N 1987 Inductive time-domain electromagnetic methods EM mining volume Soc Explor Geophys in press

Newman G A Anderson W L and Hohmann G W 1987 Intershyprerarion of transrent electromagnetic soundings over threeshydimensional structures for the central-loop configuration Geophys J Roy Astr Socbull 89 l~9-(14

Newman G A Hohmann G W and Anderson W L 19~6 Transhysient electromagnetic response or a three-dimensional body in a layered earth Geophysics 51 160~- [627

Ogd~y K D 1910 A model study of the transient electromagnetic coincident loop technique Geoexplor 21 231-264

Petrick W R 11)84A fully internal hybrid technique for calculating electromagnetic SCattering from three-dimensional bodies in the earth PhD thesis Univ of Utah

Sanfilipo W A and Hohmann G W 1985 Integral equation solushyion for [he transient electromagnetic response of a threeshydimensional body in a conductive half-space Geophysics 50 798shy~Ol

S1I1hlipo W A I-aton P A and Hohmann G W 1985 The effect raquof a conductive half-space on the transient electromagnetic reshysponse of a three-dimensional body Geophysics 50 1144-1162

Spies B R 1980 TE~ in Australian conditions Field examples and model studies PhD thesis Macquarie Univ

Spies B R and Parker P 0- l9H4~ Limirarions of large-loop transhysient surveys in conductive terrains Geophysics 49 902 -912

Tripp A C ami Hohmann G W 1984 Block diagonalization of the electromagnetic impedance matrix of a symmetric body using group theory Inst of Elect and Electron Eng Trans Geosci and Remote Sensing GE-22 6269

Van Bladcl J 1961 SIJOlC remarks un Greens dyadic for infinite SPltlIC Trans on Antennas and Propagation Insc Electr Electron Eng 9 563middot 566

Wannamaker P E Hohmann G W and Sanfilipo W A 1984 Electromagnetic modeling of three-dimensional bodies in layered cart h- using integral equations Geophysics 49 60-77

Wcidclt P 198 l Dipole induction in a thin plate with host medium arid ovcrburdcn Research Project NTS 10 no R9727 Federal Inst ior Earth Sciences and Ravv Materials Hannover West Germany

706 Newman and Hohmann

recommend (if possible) further work with a moving-source survey because it is more likely to eliminate interpretational ambiguities of multiple conductors and can provide precise information for locating drill holes

CONCLUDING REMARKS

We have presented a 3-D integral-equation solution for the TEM responses of prisms in a layered half-space In summary our solution is based on a Galerkin solution for the scattering current where the weighting functions are the basis functions It is necessary that the solution include a set nf basis and weight functions that are divergence-free if model results are to be accurate at large background resistivities The

divergence-free basis functions are used to model eddy curshyrents which are closed in the prisms and hence simulate the inductive responses of the prisms in a very resistive earth More importantly divergence-free weight functions eliminate the charge operator from part of the solution by integrating it over rectangular paths Thus the solution satisfies the freeshyspace boundary condition for no current leaks and the reshysulting solution is stable for a host with any resistivity

We used current-tube functions as the divergence-free basis functions because they are simple and acceptable for our comshyputational limitations The current-tube functions are crude and restrict the geometry of the 3-D bodies to prisms Howshyever they are effective and allow modeling prisms of moderate geometric extent An exception where our solution does not perform well is for square and nearly square plates at a high resistivity contrast In this case the time constants of the plates are underestimated We believe the accuracy and genershyality of our solution will improve with a more sophisticated divergence-free basis function

The usefulness of OUT solution has been demonstrated with TEM model studies We investigated 3-D responses under the influence of conductive overburden and made an evaluation of the resolution of two conductors based on different survey configurations However our solution is more versatile than we have illustrated since it is valid for any number of layers and prisms and is designed for many different types oC source and receiver configurations Important exploration problems where the present solution can be applied include 3-D airshyborne TFM model studies and evaluations of the effect of a conductive host on 3-D TEM borehole responses Finally the solution is not restricted to TEM applications Because its basic formulation is in the frequency domain it can be applied to frequency-domain exploration problems as well

ACK-NO-WLEDGMpound~TS

This paper is dedicated to the memory of Frank Frischshyknecht We are grateful to him and the US Geological Survey for making the Surveys VAX IInO computer available for much of this work We are also indebted to Walter Anderson for his help in installing and running the programs Financial support was provided by Amoco Production Co ARCO Oil and Gas Co Chevron Resources Company CRA Exploration Pty Ltd Standard Oil Production Co and Unocal Corp in conjunction with the University of Utah EM Modeling Reshysearch Project

REFERENCES

Annan A P 1974 The equivalent source method for electromagnetic scattering analysis and its geophysical applications PhD thesis Memorial lJniv of Newfoundland

Dyck A V and West G F 1984 The role of simple computer models in interpretations of wide-band drill-hole electromagnetic surveys in mineral exploration Geophysics 49 957-980

Eaton P A and Hohmann G W 1987 Approximate inversion for lransient electromagnetic soundings Phys of the Earth and Plan Int in press

Fitlcrman D Y and Anderson W l 1987 Effect of transmitter turn-off on transient soundings Geoexplor 2413-146

Gallaghtr P R Ward S H and Hohmann G W 19R5 A model study of a thin plate in free space for the EM37 transient electroshymagnetic system (ieophysies 50 ()())-lDlq

Gupta O P Kakirde S T and Negi J G bull 1987 Scale model experiments In study low frequency electromagnetic resolution of multiple conductors Trans Geoscience and Remote Sensing Inst of Electr Electron Eng in press

Hanneson J E and West G F 1984 The horizontal-loop electroshymagnetic response of a thin plate in a conductive earth Part _shyComputat ional method Geophysics 49411-420

Harrington R F 1968 Field computation by moment methods The MacMillan Pub Co

Hohmann G W 19lB Three-dimensional EM modeling Geophys Surv 6 27 53

Kaufman A A 1978 Frequency and transient responses of EM fields created by currents in confined conductors Geophysics 43 1002-1010

19~5 Distribution of alternating electrical charges in a conshyductive medium Geophys Prosp 33171-184

Kaufman A A and Keller G V 1985 Inductive mining prospectshying Part I Theory ElsevierScience Pub Co

Lajoie J J and West G r 1976 The electromagnetic response of a conductive inhomogeneity in a layered earth Geophysics 4l 1133-1156

Lamontagne Y 1975 Applications of wideband time-domain elecshytrornagnetic measurements in mineral exploration PhD thesis UnivofToronto

Lee T S 19~ I Transient electromagnetic response of a sphere in a layered medium Pure Appl Geophys 119309-338shy

MLCracken K G Oristaglio 1 L and Hohmann U W 19~6 A comparison of electromagnetic exploration systems Geophysics SI XIO XIX

Nabighian M N 1987 Inductive time-domain electromagnetic methods EM mining volume Soc Explor Geophys in press

Newman G A Anderson W L and Hohmann G W 1987 Intershyprerarion of transrent electromagnetic soundings over threeshydimensional structures for the central-loop configuration Geophys J Roy Astr Socbull 89 l~9-(14

Newman G A Hohmann G W and Anderson W L 19~6 Transhysient electromagnetic response or a three-dimensional body in a layered earth Geophysics 51 160~- [627

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