Electromagnetic Methods - University of Nevada,...

Chapter 7

Electromagnetic Methods

7.1. INTRODUCTION AND HISTORICAL BACKGROU N D

ploration (Dolan, 1970). Since the early 1970s, there has been a dramatic increase in the development of

With the exception of magnetics, the electromagnetic (EM) prospecting technique is the most commonly used in mineral exploration. In its usual form the equipment is not suitable for oil search, because it responds best to good electrical conductors at shallow depth. Nor has it been much employed in civil engineering work, although it is used occasionally to locate buried pipe and cable, for the detection of land mines, and for mapping surficial areas infil- trated by contaminants.

As the name implies, the method involves the propagation of continuous-wave or transient electromagnetic fields in and over the earth. There is a close analogy between the transmitter, receiver, and buried conductor in the EM field situation, and a trio of electric circuits coupled by electromagnetic induction. In a few EM ground systems the source energy may be introduced into the ground by direct contact, although generally inductive coupling is used; invari- ably the detector receives its signal by induction.

The EM ground method was developed during the 1920s in Scandinavia, the United States, and Canada, regions where the detection of conductive base-metal deposits was facilitated by their large contrast with the resistive host rock and generally thin overburden. The airborne version was introduced some 30 years later.

Until the early 1960s, practically all EM equipment transmitted and received continuously on one frequency at a time. Such a continuous wave system is said to be operating in the frequency domain (FEM or FDEM). Although several attempts were made, dating back to the 1930s, to transmit transient pulses and detect the ground response during off-time (Statham, 1936; Hawley, 1938), the first successful applications of this type did not appear until 1962. These were the airborne Input (Barringer, 1962), the MPPO-1 ground transient system in the USSR, and the EMP pulse ground equipment of Newmont Ex-

such time-domain systems (TEM or TDEM). Almost all EM field sets include a portable power

source. However, limited use has also been made of radio transmission stations in the frequency range 100 kHz to 10 MHz and particularly in the very low frequency range (VLF), 15 to 25 kHz. One other field method that can be included with EM, AF- MAG, makes use of atmospheric energy resulting from worldwide thunderstorm activity (56.2.1).

A great advantage of the inductive coupling is that it permits the use of EM systems in aircraft. Airborne EM, usually in combination with other aeromagnetic methods, has been widely applied in mineral exploration reconnaissance and recently in detailed surveys.

7.2. ELECTROMAGNETIC THEORY

7.2.1. Vector and Scalar Potentials

The propagation and attenuation of electromagnetic waves were discussed in Sections 6.2.2 to 6.2.5 in connection with the magnetotelluric method. Al- though the frequencies employed in EM prospecting are somewhat higher than in most MT work, the general theory, limiting assumptions (negligible displacement current and spatial phase shift), and boundary conditions are identical in the two methods.

In general potential theory it is usually easier to solve problems by starting with the potential and obtaining the field vectors by appropriate differenti- ation. The same rule applies in electromagnetics, where it is convenient to introduce certain potentials from which both electric and magnetic field vectors may be derived. We define the oector magnetic potential A in terms of the magnetic induction

344 Electromagnetic methods

Still another potential, the Hertz vector potential II, is available to define the EM field. It is defined in terms of A and $ thus:

Then from Equation (6.3) we have

v x (E + aA/a t ) = o The vector inside the bracket is irrotational [see Eq. (A.31)] and can be expressed as the gradient of a scalar potential $:

E + aA/at = -v+

E = -( aA/at + v+) or

( 7 4

The term - dA/d t is the induced emf portion of E while -v$ is the potential gradient due to current flow in a resistive medium.

Using Equations (7.1) and (7.2) along with Equa- tions (6.4) and (6.9), we have

v x v x A + p.{ a2A/at2 + V( a+ /a t ) } = p~ (7.3a)

[If external sources are present, J must be replaced by J + &-see Eq. (6.11b).] The electric field E in Equation (7.2) produces a current in a conducting medium such that (see Eq. (6.9))

J = u~ = -u( aA/at + v+) Substituting for J in Equation (7.3a) and using Equation (A.25), we obtain

V(V . A) - V ~ A + pE{ a2A/at2 + V( a + / a t ) } + aA/a t + v+) = o (7.3b)

Equation (6.6b) states that v - D = Q, Q being the charge density. Using this to eliminate E in Equation (7.2) gives

v2$ + v . ( a A / a t ) = -elE (7.4)

We now make use of the Lorentz condition (Ward, 1967):

v . A = - ( p e a+/a t + pu+) (7.5)

(this forces both A and $ to satisfy the same wave equation as H and E). Substituting Equation (7.5) into Equations (7.3b) and (7.4) we get

V ~ A - pea2A/a t2 - puaA/at = o (7.6)

v2+ - pEa2+/at2 - pua$ /a t = -elE (7.7)

Thus we have a pair of potentials, one vector and the other scalar, that in homogeneous media satisfy the same wave equation as the fields.

} (7.8) A = a n / a t + ~ U I I

+ = -v a l l

Following manipulation similar to that for A and $, we arrive at the expression (Ward, 1967)

v2n - pE a 2 n / a t 2 - p a a n / a t = K (7.9)

where K = - P / E , or -M, or zero. P, M are the respective electric and magnetic-dipole moments per unit volume. The value of K depends on the conditions of the system.

Although these potentials do not have the physi- cal significance of the scalar potential in gravity or electrostatics, they provide convenient mathematical tools for determining various EM fields, as we shall see in the following sections. The similarity in form between Equations (7.6), ( 7 3 , (7.9), and (6.13) is worth noting.

7.2.2. Description of EM Fields; Biot - Savart Law

(a) General. The primary or soufce fields used in EM prospecting are normally generated by passing alternating or pulsed current through long wires or coils. For simple geometric configurations such as these the resultant fields can be calculated exactly for points in the surrounding region, although this is not generally possible. With FDEM one must measure the disturbing field in the presence of the original primary field, hence it is often necessary to calculate the latter at the receiver in order to eliminate, or at least reduce, its effect. With TDEM the primary field is eliminated by source cutoff. In both cases the secondary fields of interest are small in comparison with the primary fields.

(b) Biot- Savart law; magnetic field of a long wire. Originally stated for static magnetic fields, this law is valid also for low-frequency ac, provided the linear distances involved are much less than the wavelength (s.2.3). From Equation (3.4) and Figure 7.1 we have

AH = (I d l ) x r,/4.nr2

In scalar form this becomes

AH = I p { sec2 $ sin( 7r/2 - +) } d $ / 4 d Sec2 9 = ZCOS $ d$/4~p (7.10a)

Electromagnetic theory 34s

dH (out of paper)

dl

Figure 7.1. Illustrating the Biot-Savart law.

Integrating over a straight wire, the result is

H = ( I/4rp) (sin ‘pz - sin 9,) (7.10b)

If the wire is extremely long (infinite length), & =

H = I / 2 r p (7.10~)

r / 2 , +, = - ~ / 2 , and

(c) Vector potential of a current element. To get the potential A for a current element Idl, we simplify Equation (7.3a) by neglecting a l l currents (in- cluding displacement currents) except those induced in the conductor. Thus, we set pe aZA/atZ = 0 (this eliminates displacement currents-see discussion of eqs. (6.13b) and (6.14)), and we also set + = 0. This reduces eq. (7.3a) to

v X V x A = p J

But v - A = 0 from Equation (7.5) when + = 0, hence Equaton (A.25) gives

v 2 A = -pJ

This equation resembles Poisson’s equation in gravity and magnetics. It has the following solution for a current element I dl

A=(p/4r) /Jdo/r = (p/4r)#Idl/r (7.11) V

where the line integral (5A.3.4) is over a closed path encircling the element df or its prolongation, A is the potential at P ( x , y, z), and r is the distance from P to dl.

7.2.3. Fields in the Frequency Domain

(a) General. In this section we consider propagation of alternating fields from several common trans-

mitter sources in free space and over a conductive half-space. The assumption of free space is generally valid because the host rocks surrounding the conductor often are highly resistive.

Ward and Hohmann (1988) give an excellent theoretical treatment of electromagnetic fields in EM prospecting. The discussion in Section 7.2.3 is based largely on their paper.

(b) Long straight horizontal wire. We take the y axis along the wire. The field is now given by Equa- tion (7.10~) where 2 = xz + zz, and H is a vector perpendicular to p at the point P ( x , y , z ) in the direction given by the righthand rule (Fig. 7.1). The x and z components are

H, = I z /2 rp2 H, = I x /2 rpz (7.10d)

There is also an electric component (Ward and Hohmann, 1988) given by

Ey = - jwplKo( j k p ) / 2 r (7.1Oe)

where KO( j k p ) is the modified Bessel function of the second kind of order zero and k is the propagation constant given by

k Z = oZp& - j o p u ;

in free space u = 0 and

k Z = w ~ ~ ~ E ~ = oZ/c2 = ( Z r / X ) ’ ,

c being the velocity of light in free space. When kp is small, KO( j k p ) = -In( j k p / 2 ) (Abramowitz and Stegun, 1964: 375) and Equation (7.10e) reduces to

Ey = j o p l In( j k p / 2 ) / 2 r (7.10f)

while Equation (7.1Od) is unchanged. When the long-wire source lies on the ground (the

usual case), Equations (7.10d, e, f ) are modified because the propagation constant becomes k Z = -&pu (Ward and Hohmann, 1988). For ( k p ) small, the components become

H, = j k I /2 ,12r , H, = - ( I/rx)ln( j k x / 2 ) E,, = 0

When the current is alternating, I in Equations (7.10) should be replaced with Ie-J,‘ but in practice the exponential factor is understood and not written out (however it must be taken into account when time derivatives are involved). Also, we assume that the FDEM fields are H(o), E ( w ) in contrast to the


A D

Figure 7.2. Geometrical parameters involved in calculating the magnetic field of a rectangular loop. (a) Calculating the field at an internal point; (b) Calculating the field at an external point.

TDEM fields H( t), E( t) to be considered later (57.2.6).

(c) Large rectangle. The geometry is shown in Fig- ure 7.2. Inside the rectangle the field (which is normal to the paper) at P , produced by the current in segment A E , is from Equation (7.10b) with +1 = 0,

H( A E ) = z sin+,/4s( E P ) = I ( ~ ~ ) ’ / 4 s r , d ~

where d, is the area of the rectangle with diagonal r,. Adding the field H( AH),

H( A E ) + H( A H )

= ( z / 4 . n r 1 d 1 ) { ( H P ) ’ + ( EP)’ }

= ( Z/477) ( r 1 / 4 )

Adding the fields of the four subrectangles, we get the total field

4

H = (Z/4 .n) ( r n / d n ) (7.12a) n-1

When P is outside the rectangle, we find

H’ = (Z/477)( r { / d { + r i / d {

- r { / d { - r i / d / ) (7.12b)

Within the loop the field varies about 40% over a rectangle concentric with ABCD and one-quarter the area, being a minimum at the center. Outside the loop the field will be mainly determined by the near side of the rectangle, provided the distance from it is

considerably less than the distance to the far side. It is important to note that the field in the neighborhood of the long wire, or outside a large rectangle and close to one side, falls off inversely as the distance, a relatively slow decrease in intensity.

The preceding analysis is for a loop in free space. The case of a rectangle lying on a homogeneous half-space has not been solved up to the present (see also 57.2.6d).

Both the long wire with and without the ends grounded and the large rectangle or square have been extensively used for generating EM primary fields in FD systems such as Turam, and also for pulse transmission (UTEM, Newmont EMP, SIROTEM, etc.). Dimensions are generally several hundred meters.

(d) Small horizontal circular coil (vertical magnetic dipole). By using the vector potential we can calculate the field at a point in the neighborhood of the loop, not necessarily in its plane or on the axis. The cylindrical coordinate system is shown in Figure 7.3. The cross section of the winding on the loop is assumed to be very small with respect to its radius. Because the current is confined to a circular path there is only one component of A, A,. Then dv = a d+ dp dz, J+ = I / d p dz, and Equation (7.11) simplifies to

Electromagnetic theory

,I 347

0; inserting the time factor exp( -jot), we have

E, = - a ~ , / a t

= j ~ p I 8 d p e - ~ ‘ “ / 4 r ( p’ + z ’ ) ~ ’ ~ (7.14d)

However, its effect is small compared to the magnetic fields.

All three components may also be expressed in spherical coordinates as H,, He, and E, [the last being identical to E, in Eq. (7.14d)J Because H,? + H ~ = H ~ + H ~ a n d z = r c o s B , p=rsinB,weget

H, = 2 Hz/( 2 cos B - sin B tan 0 ) = 2 H,/3 sin B

H, = H~/( 2 cos B cot B - sin 0 ) = H,/3 cos B

Figure 7.3. Field of a small circular coil at a point outside the plane of the loop.

If we assume that 3 + z’ w a’ and expand the denominator, we obtain

a’ up cos + * a * } d+ x(1 - 2( p’ + z’) + (p’ + z’)

PZa’P = 4( p’ + z’)3’2 (7.13)

Now we can get the magnetic field using Equation (A.36) (see 9A.4)

Ia ’ 5/2 { 3Pzip + (2z2 - P’)iz} - -

4( p’ + z’) (7.14a)

where i, and i, are unit vectors along the p and z axes. Thus the magnetic field has two components: one in the p direction and the other in the 2 direction:

H, = 316~dpa/4r( p’ + z ’ ) ’ ~ (7.14b)

H, = 1 6 d ( 2 z 2 - $)/4r( $ + z’)”’ (7.14~)

where a d = = ,a2, the area of the loop. There is also an electric field component E,,

obtained from Equation (7.2) with the potential + =

Thus,

H, = I 6 d c o s B/2rr3 (7.14e)

H, = 16dsinB/4rr3 (7.14f)

E+ = jop16dsinB/4rr2 (7.14g)

There are three limiting cases of interest in regard to the H fields:

(i) p = 0, so B = 0 and the field is axial:

H, = H, = 16.4/2~~3

(ii) z = 0 , so 0 = r/2 and the loop and field are coplanar:

(7.14h)

H, = -He = -I6d/4r? (7.14i)

These are approximations because we took (p’ + z’) >> a’; the error, however, is less than 3% if either z or p is larger than 7a. Replacing the latter in the original expression for A, and setting p = z = 0, we find the field at the coil center to be

(iii) H, = I/2a H, = 0 (7.14j)

Clearly the small coil is equivalent to an oscillat- ing magnetic dipole lying in the 2 axis at the origin with magnetic moment given by m = I S d e - J a ‘ . Furthermore its shape is not significant, provided a < 0.14~ (or 0.142); for a square loop = ISd4r? . It is also of interest that the z components in Equations (7.14h) and (7.14i) resemble the Gauss A and B positions ($3.2.3) with m in Equa- tion (3.15~) replaced with I6d/4r .

The dipole field intensity, then, falls off with the inverse cube of the distance; compared to the long- wire systems, the intensity decreases much faster. However, it should be kept in mind that both are approximations. The straight wire is assumed to be considerably longer than the perpendicular distance to the station at which the field is measured, whereas


the closed-loop dimensions are considerably smaller than its distance from the field station.

If the source loop, instead of being in free space, is lying on homogeneous ground, the field components in Equations (7.14b, c, d) become (Ward and Hohmann, 1988:211-2) in cyclindrical coordinates

Hp = - ( I S d k 2 / 4 ~ p ) { I,( fjkp) K,( f jkp)

-12( f j k p ) G( i j k p ) } ( 7 . W 4 81 t H, = ( ISd/27rk2p5) { 9 - (9 + 9jkp

J'

-4k*$ - j k 3 $ ) e- jkp} (7.141)

E+ = - ( z ~ & / 2 ~ ~ ~ 4 ) { 3 - (3 + 3 jkp r

Figure 7.4. Field of a short vertical wire at a distant point. - k2$) e-JkP} (7.14m)

where the propagation constant (57.2.3b) k = ( w ' p ~ - jwpu)' / ' = ( - j w p ~ ) ' / ~ over homogeneous ground, I , ( f j k p ) and K,(fjkp) are modified Bessel functions of the first and second kinds of order n (Pipes and Harvill, 1970). For ( k p ) = 0, the above equations become

Hp = -( ISdk2/167rp) (7.1.h)

H, = -( I6d/47r$) (7.140)

E+ = ( IS&k2/47ru$) (7.14P)

Using reasonable values such as p = 100 m, u = 0.01 S/m, we find that the magnitudes of all three components in Equations (7.14~1, 0, p) are smaller than those in free space by a factor of about The vertical magnetic dipole is widely used in ground systems.

(e) Horizontal magnetic dipole. This type of source is commonly used in vertical-loop fixed- transmitter and broadside ground units (57.4.2b, c) and in airborne systems. The results for propagation in free space are easily obtained. We take the coil in the XI plane with its center at the origin; the expressions for the components are then obtained by replacing z in Equations (7.14b, c, d) with y.

When the coil is on homogeneous ground, the components are (Ward and Hohmann, 1988)

H, = 31 Sdxy/47r$ (7.146)

Hy = 1 6 d ( 2 y 2 - x2)/47rp5 (7.14r)

H, = -jwpOaI S d y / l 6 ~ $ (7.14s)

where $ = x 2 + zz. In all the preceding magnetic field sources we

have also assumed only one turn of wire. If there are

n turns, the calculated fields are increased by a factor of n in all cases.

(f) Vertical straight wire (vertical electric dipole). One other artificial source of EM waves that has been employed in prospecting should be considered here. This is the high power VLF (very-low- frequency) transmission in the range 15 to 25 ~Hz, which is normally used for air and marine naviga- tion. There are, of course, many RF stations available as well, but the frequencies are considerably higher (500 kHz and up) and the power lower than the VLF sources, so that the range and depth of penetration are limited.

The VLF antenna is effectively a grounded vertical wire, several hundred meters high. Consequently it is much shorter than a transmission wavelength, which, for a frequency of 20 kHz, is 15 km. Whereas the small loop is equivalent to a magnetic dipole, the short wire behaves as an electric dipole.

There are several possible modes of radiation from this type of antenna, but in the low-frequency range and at distances considerably greater than a wavelength, the propagation is a combination of ground wave and sky wave. The former travels over the earth's surface, whereas the sky wave is refracted and reflected by the ionized layers in the upper atmosphere (50 km and higher) to return to ground. At large distances from the antenna the VLF waves appear to be propagated in the space between the spherical reflecting shell formed by the earth surface and the lower ionosphere. The attenuation is com- paratively small in both surfaces.

From Figure 7.4 it is apparent that the electric dipole is quite like the magnetic dipole if we inter- change the E and H components of the wave, that is, there are two components of electric field, E, and E, in spherical coordinates, and one magnetic com-


ponent H+ in the azimuth. The vector potential of a current element 161 can be found from Equation (7.11); in spherical coordinates,

A = pZ6le-J"'*iZ/4~r

= p16[e-ja'*(cost)i, - sinBie)/4rr

349

where the current flow in the dipole length Sf is along the z axis, t * = t - r/c = t - r(p&)'/', and c is the velocity of light = 3 x lo8 m/s. The scalar portion, +', composed of the scalar potential plus an extra term caused by phase difference in the time- varying potential, becomes

Using Equations (7.1), (7.2), and (A.38), we find

(7.15a) I E = -v+! - aA/at = - ( a+'/ar + aA,/at)i, - ( a+!/rae + aA,/at)i,

= [ a( rAe)/ar - a~,/afl]i+/pr H = (V X A) /p

These expressions produce one magnetic and two electric components:

H+ = I61 sin &-jot*( jw/cr + l/r') /4r' (7.15b)

E, = I61 cos Be-Jw'

x ( l/cr2 - j/w3) /2m (7.15~)

E, = 161 sin Be-Ja'*

X( jw/c2r + l/cr2 - j/wr3)/4m (7.15d)

The three components may be converted to cyh- drical coordinates as in Section 7.2.3d. For example, using Equation (A.36) and neglecting terms in l/r2, l /r3 (note that r' = pz + z' in Fig. 7.3),

Ep = 3jopZSlzpe-jo'*/4.n( p' + 2')"' (7.15e)

E, = 0 (7.15f)

H+ = j w ( pe),'/'Z6f pe-ja"/4a( p' + z') (7.158)

These formulas are more complete and more complex than those for the magnetic dipole, and they contain terms in l/r, l/r2 as well as 11r3. Clearly the VLF system is designed for long-distance transmission; the so-called radiation jieldr or far jieldr, varying as l /r , are significant when the induction and static jielak (near jieldr), varying as l / r2 and l/r3, have become neghgible. Consequently, we

consider only the first term inside the brackets in Equations (7.15b, d) and assume that E, = E, = 0 at great distance.

The preceding development may be carried out for Equations (7.14e, f, g) (Stratton, 1941; Smythe, 1950). Apart from the dipole moments and a phase shift of +r /2 (indicated by * j ; see §A.7), for all terms, Equations (7.14g) and (7.15b) are identical, and Equations (7.15~) and (7.15d) are the same as Equations (7.14e) and (7.140 if the latter are multi- plied by (l/we) and H and E interchanged. How- ever, because the magnetic dipole is very rarely used for transmission beyond a few kilometers, only the near-field term is significant.

Because the amplitude of VLF fields decreases only as l / r and the station output power is large (100 to 1,000 kw), it is possible to detect these sources over continental distances, occasionally nearly half way around the world.

At great distances from the source, E, is neghgible and E would appear to be nearly vertical; at the ground-air boundary, however, there is a considerable horizontal component in the direction of propagation. The magnetic field lines are horizontal circles concentric about the antenna; at distances of several hundred kilometers this field is practically uniform over, say, a few square kilometers, and is at right angles to the station direction.

Figure 7.5 compares the magnitude of primary fields from the various sources described in this section: VLF transmitter, long wire, square loop, and circular coil. These were calculated from Equa- tions (7.10b), (7.14), and a modification of Equation (7.15b), following, that allows for attenuation:

I61 2hr

(7.15h) H - e-l .SX10-3r/X

where A , r, and 61 are in kilometers. The curves are drawn for the following parameters:

1. VLF transmitter power = lo6 W, frequency = 20 kHz, antenna current = 5,000 A, and height = 300 ft (100 m).

2. Long-wire power = 1,000 W, current = 3 A, and length = 4,000 ft (1,200 m).

3. Square-loop power = 300 W, current = 3 A, section area = 36 ft', (3.3 rd), and turns of wire = 100.

4. Circular-loop power = 5 W, current = 100 mA, diameter = 3 ft (1 m), and turns of wire = 1,000.

These values correspond roughly to (1) VLF, (2) Turam, (3) vertical-loop dip-angle, and (4) horizontal-loop systems used in EM field work.


L

10-2 - Magnetic 1

field (ampere-turns

per meter) 10-3-

lo-' - -

10-5 - - -

. , I I I

I I I I I I I l l 1 I I l l 10 102 I 03 l(r 105 106 (Ti) 107

Distance from source

Figure 7.5. Comparison of magnetic fields produced by various sources.

7.2.4. Combination of FD Fields

(a) General. So far we have described the propagation, attenuation, and generation of alternating magnetic fields. We have seen that such fields can be initiated by various current configurations and atten- uated more or less depending on their frequency and the conductivity (and permeability) of the medium through which they travel.

In Section 6.2.3 it was also noted that the EM field was shifted in phase on encountering a relatively good conductor. In fact, this conductor becomes the source of a secondary field, which differs in phase from the primary field, while having the same frequency. Hence a suitable detector in the vicinity will be energized by both the primary and secondary fields simultaneously. The existence of the secondary field, indicating the presence of a subsurface conductor, may be established with respect to the primary field by a change of amplitude and/or phase in the normal detector signal. Some EM systems measure both quantities; some respond to one or the other.

(b) Amplitude and phase relations. The character of the secondary magnetic field is best illustrated by a consideration of the coupling between ac circuits. We assume a trio of coils having inductance and resistance and negligible capacitance; the first is the primary source, the second is equivalent to the con-

Figure 7.6. Electric-circuit analogy for EM system.

ductor, and the third is the detector (Fig. 7.6). The primary EM field at a point near the conductor (coil 2), resulting from a current ip flowing in the first coil, is given by

Hp = Kip = KIP sin w t

where K depends on the geometry of the system, the area and number of turns of the primary coil, and attenuation of the wave.

As a result of this field, coil 2 has an emf induced in it that lags behind the primary field by a / 2 ; thus,

dip

dt e, = -M- = -woMZpcosut

= UMZP sin( wt - a / 2 )

= - jwMHp/K

Electromagnetic theory 351

, ./

Figure 7.7. Vector diagram showing phase shift between H, and Hp.

(5A.7) where M = M , = mutual inductance (see 57.2.5) between coils 1 and 2. Then the current flowing in coil 2 will be

The phase difference between primary and secondary fields is

is = eJz, = e,/( r, + j o L , ) ep - e, = - + tan-'- = - + + (7.17) i; "") 'S (T 1 where z, = ( r , + jwL, ) is the effective impedance of the conductor of resistance r,, and inductance L,. The secondary field near the detector (coil 3) as a result of this current will be

where tan+ = wL,/r,. The lag in phase of 7r/2 is due to the inductive coupling between coils 1 and 2, whereas the additional phase lag + is determined by

- K jwMHp H, = K i , =

K( r, + j%)

- - K M H ~ ( jar , + JL,) -

K ( r,' + o'L:)

- K'MHp( Q2 + j Q ) - - (7.16a)

KL,(1 + Q')

where K' is a constant similar to K and Q = oL,/r, is the jigure of merit.

The primary field at the detector coil will be

Hd = K"ip = K"ZP sinat = K"Hp/K

where K" is similar to K and K'.

detector is Thus the relative magnitude of the fields at the

K'i, 151 = Kllip

Q4 =- :":( (1 + Q')2 -k (1

K M 1

K"Ls (1 + l/Q2) 1/2 =- (7.16b)

Because the ratio K'M/K''L, is generally very small, the ratio H,/H,' is small, regardless of the value of Q.

the properties of the conductor as an electrical circuit. That is,

H, = K'I, sin{ w t - ( ~ / 2 + +))

= - K'Z, COS( ~t - +)

The phase shift is most clearly illustrated by the vector diagram in Figure 7.7. (In this diagram the magnitude of H, with respect to Hp is greatly exag- gerated.) The resultant of Hp and H, is H,.

From this diagram and Equation (7.17), it can be seen that when we have a very good conductor, Q = oL,/r, 4 00 and + 4 n/2. In this case, the phase of the secondary field is practically 180" (T) behind the primary field. For a very poor conductor wL,/r, + 0 and + + 0; the secondary field lags lr/2 behind the primary. Generally H, is somewhere between ~ / 2 and T (90" and 180") out of phase with

The component of H, 180° out of phase with Hp is H, sin+, whereas the component 90" out of phase is H,cos+. In EM parlance, the 180" out-of-phase fraction of H, is called the real or in-phase component. The 90" out-of-phase fraction of H, is called the imaginary, out-phase, or quadrature component. These terms originated in ac circuit theory and, in fact, there is nothing imaginary about the quadrature component. From Figure 7.7 we get the important relation [see Eiq. (A.47a)l

HP.

t an+= [ 9 m { H s } / % { H , } ] (7.18)


Consequently the superposition of fields produces a single field that is elliptically polarized, the vector being finite at all times [although Hp and H, become zero at or = na and (2n + l)a/2, respec- tively] and rotating in space with continuous amplitude change. Its extremity sweeps out an ellipse.

This ellipse may lie in any space plane, although the plane will normally be only slightly tilted off horizontal or vertical. This is because the major axis of the ellipse is determined by Hp - because it is usually much larger than H, - and the primary field is normally either horizontal or vertical in EM systems.

There are two special cases of Equation (7.19) of considerable importance. Figure 7.7 is again helpful in visualizing these situations.

(i) + = n/2. Equation (7.19) then reduces to

(c) Elliptic polarization. The detector in an EM field system, generally a small coil with many turns of fine wire, measures the secondary field produced by a subsurface conductor, in the presence of the primary field. Consequently, the detected signal is a combination of the primary and one or more secondary fields. In general the combination is a magnetic field that is elliptically polarized. From the previous section we can write

Hp = A s i n w t and H, = Bcos(wr - +)

where A and B are functions of the geometry of the transmitter, conductor and detector. Because

cos(ot - +) = cosotcos+ + sinwtsin+

= (1 - H;/Az )~” cos + + Hp sin+/A

= H,/B

we get

H,’ 2HpH,sin+ - H i + - - = cosz + A’ B Z A B

which is a straight line through the origin of the coordinates, having a slope + B / A . This case corre- sponds to a very good conductor, because

+ = tan-’ oL,/r, = n/2 that is,

or, because tan+ = w , 2 Hp H, sin +

- = 1 H i + H,” A~ C O S ~ + B~ C O S ~ + A B cos2 +

(7.19)

This equation is of the form

Lzz - ~ M X Z + Nx’ = 1

which is the equation of an ellipse. We have made two simplifying assumptions in

obtaining the equation. The first is that Hp and H, are orthogonal in space, which is not generally true. However, if the angle between Hp and H, is a # n/2, these vectors may be resolved in two orthogonal components, say

H, = Hp + H s ~ s a and H, = H, sina

in which case the expression for H, and H, is more complicated and has constant terms, but is still the equation of an ellipse.

The second assumption was that H, is due to the current in only one conductor. This is not necessarily the case; however, a combination of vectors of dif- ferent amplitudes, directions, and phases, resulting from currents in several conductors, can be resolved into a single resultant for H,.

r , = o

The ellipse of polarization has collapsed into a

(ii) + = 0. The ellipse equation simplifies to straight line.

H,’ H,’ - + - = I A2 B Z

which signifies a poor conductor because r, wL, when + = 0. In the unlikely event that A = B as well, the combination of Hp and H, results in circular polarization.

Obviously a detector coil can always be oriented so that it lies in the plane of polarization, when a true null signal would be obtained. Some of the early EM methods were based on this fact; the dip and azimuth of the polarization ellipse and its major and minor axes were measured.

On the other hand, if the detector coil is rotated about its vertical or horizontal diameter, it will not always be possible to find a perfect null position, because the plane of the coil will not, in general, coincide with the plane of the ellipse. There will, however, be a minimum signal at one coil orientation.


7.2.5. Mutual Inductance

(a) General theory. It was noted in the discussion of phase shift that the inductive coupling between electrical circuits is proportional to the coefficient of mutual inductance M. This parameter can be used to some effect in determining the signal amplitude at the receiver due to both the transmitter and conductor. It has already been employed to estimate the current induced in the conductor as the result of the primary field.

If we can simulate the transmitter and receiver coils and the conductor by simple electric circuits, it may be possible to calculate the mutual inductances that couple them. Consider the coil system illustrated in Figure 7.6, in which the mutual inductances between t randt te r and conductor, conductor and receiver, and transmitter and receiver are, respec- tively, MTc, M c - , and MTR.

It was shown previously (57.2.4b) that the current i , induced in the conductor by the transmitter is related to the transmitter current by the expression

where Q = oL,/r, was a figure of merit for the conductor circuit.

Current is, in turn, will induce an emf in the receiver coil given by

ter loop inductance, and L R is the receiver loop inductance. Then we can write Equation (7.20a) in the form

Q2 +iQ kcRkTc ( + Q 2 ) (7.20b)

eCR - = - - ~

~ T R kTR

Although this eliminates L, in the first part of the expression, it does not simplify the first ratio much because the k values, like the M values, involve complicated geometry of the system. Equation (7.20b) does indicate, however, that this ratio, some- times called the coupling parameter, is usually a very small quantity because k , will tend to be much larger than the two coefficients in the numerator. That is to say, the transmitter and receiver are coupled through air, which means the attenuation is practically zero. In some FDEM field layouts, the source and detector coils are purposely oriented to reduce the direct coupling, whereas in others the decoupling is accomplished by electrical means.

In any case, the mutual inductances between the various components within the range of the magnetic field are a controlling factor in EM systems. Conse- quently it would be useful to determine M for simple geometrical configurations that simulate field situations, as an aid in interpretation.

The mutual inductance M12 between two circuits 1 and 2 is defined as the total flux Q12 through circuit 1, produced by unit current in circuit 2. Thus,

At the same time the primary field induces an emf in A2 4 the receiver

eTR = -jldMTRip where S, is any surface terminating on circuit C,, and the last result came from Stokes' theorem [Fq. (A.29c)l. From Equation (7.11) we have Because the secondary Or anomalous field is mea-

sured in the presence of the primary field, we have Pi2 dC2

A , = -$- _ - 4s r

This gives the general Neumann formuIa for mu- ~ T R

Mutual inductance mav also be written in terms tual inductance:

A2 dl; dl; - dC, M12 = #- = !#$-

12 4 s r

cos e = 'pp- 4 s r dl; dC, (7.21a)

where dd,, dd2 are elements of length in circuits 1 and 2, r is the distance between dd, and d f 2 , and 6 is the angle between dd, and dd,.

where k , is the coupling coefficient between conductor and receiver and so forth, LT is the transmit-


i -------- I

L r - I

Figure 7.8. Calculating mutual inductance of various geometrical figures. (a) Two parallel straight lines. (b) Coaxial circles. (c) Coplanar circles. (d) Circle and long straight wire.

In many cases it is necessary to integrate numeri- cally to get a particular answer from this general formula. However, several simple circuits can be worked out exactly. These are illustrated in Figure 7.8 and the formulas are given in the following text. Units are microhenrys and meters.

(i) Two parallel wires of equal length. Let the length be t and the distance apart s. Then,

When e/s > 1, this can be modified to give

More complicated forms of straight conductors may be found in Grover (1962).

(ii) Two coaxial circles. M is a function of a, b, and s, the radii and separation, which varies with a parameter k , given by

k Z = [ ( a - b ) 2 + s 2 ] / [ ( a + b ) Z + s 2 ]

which is found in tables (Grover, 1962). When s 2 10a or lob,

M = 0.2ra2abz/s3 = 0.2dL@/s3 = 2a2b2/s3 (7.21~)

d, 9 being areas of the circles.

(iii) Two coplanar circles. When r =*- a, b,

M = -0.1na2.nb2/r3 = -0.1dL@/r3 = -azb2/r3 (7.21d)

Again, more complicated arrangements of the circles are dealt with in Grover.

(iv) Circle and long straight wire. Let a long straight wire intersect at right angles the extension of a diameter of the circle of radius a. If the plane of the circle makes an acute angle a with the plane through the diameter and the long wire, and if s is the distance between the wire and the center of the circle, then

1/2 M = 4.n x lo-’{ s sec a - ( s z sec2 a - a’) } (7.21e)

and when s/a 2 5, this is simplified to

2a x 10-3a2 0.002d M = =- (7.21f)

s sec a s sec a

(b) Selfinductance. Values of L for the straight wire and circle are given by

L, = 0.2e[ln(2e/p) - 0.751 (7.22a)

L, = 4.n X 10-3a[ln(8a/p) - 21 (7.22b)

where p is the wire radius, and C, a are as shown in Figure 7.8. Self-inductance for several regular figures may be written

L = 0.2Ie[ln(41,/p) - X ] (7.22~)

where I, is the perimeter of the figure and x is a constant related to the shape; generally x < 3.


In all these formulas the values of M are increased by N,N2 and L by NZ if these are the number of turns on the various circuits. Although M and L are easily determined for the regular figures of +e transmitter and receiver antennae, it is a difficult problem to simulate the conductor. Because we normally do not know the current-flow cross section, it is necessary to oversimplify the equivalent circuit.

(c) Numerical example. To illustrate the magnitude of the coupling factor, ( McRMTc)/( MTRLs), in Equation (7.20a), consider the case of a horizontal- loop system (97.4.3~) straddling a long, thin vertical conducting sheet that outcrops at the surface as in Figure 7.9. We assume that the transmitter and receiver coils are identical (cross section = A), the distance I between their centers being less than e, the length of the sheet, and that the ratio !/p = 2,500. The principal effect of the sheet is due to its upper edge; hence to a first approximation we can consider the sheet as a horizontal wire of length L‘ at the surface. Using Equations (7.21d, f), and (7.22a), we get for the coupling factor in Equation (7.20a)

Thus the signal response has been reduced by the factor lo4 or more.

(d) Conductor response. Returning to Equation (7.20b), the second factor on the right-hand side depends only on the conductor and the frequency (because Q = uLs/rs). In this situation Q is known as the response parameter of the conductor, whereas the complex ratio (Q’ + j Q ) / ( l + Q’) is called the response function or induction number.

Plotting the response function against Q, we get two curves for the real and imaginary parts of the function:

A = Q’/( 1 + Q’)

and

B = Q/(l + Q’)

Figure 7.9. Coupling factor of horizontal-loop system over a long vertical sheet conductor.

When Q is very small, both real and imaginary parts of the function are very small. The ratio of secondary to primary response in the receiver will be

This is the case of a poor conductor. As Q increases, the imaginary part increases at a faster rate at first and its magnitude is larger than the real fraction until Q = 1, when they are both equal to 0.5. Beyond this point the imaginary part decreases until, at large values of Q, it is again zero. Mean- while, the value of A increases to an upper limit of unity when Q is large, or

~ C R ~ C R ~ T C

IGI = k, which is the maximum value for a very good conductor.

When the value of Q is quite small the phase angle of this function is r/2; when Q = 1, it is 3s/4 after which it increases to w for a very good conductor, that is, the secondary signal is opposed to the primary.

In the range 0 I Q I 1 the imaginary or quadrature component is larger than the real component, whereas from 1 I Q I 00 the reverse is true. Thus the ratio of in-phase to quadrature components is somewhat diagnostic of the conductor. It is also clear that if one measures only the imaginary component in an EM system, a very good conductor will give a very poor response.

7.2.6. Fields in the Time Domain

where A and B are real. This plot is shown in Figure 7.10.

This example, of course, is oversimplified because it represents a wire loop. However, more realistic models of a sphere, thin sheet, or half-space are quite similar.

(a) General. The time-domain EM transmitter gen- erates repetitive signals of a transient or pulse character in various forms: half sine-wave, step, ramp functions, and so forth. All of these have a broad frequency spectrum that results in a secondary field time-decay curve beginning immediately following


our function, pl

n.6 7 Q2 + iQ Response function =

t Response parameter, Q 1 I l l I 1 1 1 1 1 1 I I 1 1 1

lo-’ 0.2 0.3 0.404 0.8 I 2 3 5 7 10 20 30 50 70 100

Fkure 7.70. ResDonse function of a conductor in an ac field. (After Grant and West, 7 k5).

the transmitter cutoff and in which the frequency decreases with time.

Material for Section 7.2.6 is taken, to a great extent, from Ward and Hohmann (1988) and from the classic papers of Wait (1951% c; 1960a; ,1971). It is interesting that the theoretical work was done so long before the serious development of TDEM.

In Section 7.2.3 we denoted FDEM fields by H and E; in this section we use h and e for TDEM fields.

(b) Method of TO analysis. Development in the time domain analogous to that in Section 7.2.3 for the frequency domain is best accomplished by using Laplace transforms (5A.12) which are well suited to discussion of transient fields.

Taking Laplace transforms of Equations (6.3) and (6.4), and using Equation (A.78a) with n = 1, we obtain

v X e(s) = -psh(s) (7.23a) v X h( s) = ( u + Es)e( s), (7.23b)

where e(s) - e(t), and so forth. Generally, potential functions such as those discussed in Section 7.2.1 are used. If we use the Hertz vector potential, we write for Equation (7.9) (Wait, 1951a),

{ V’ - ( perz + pus)} a(.) = -j(s)/( u + Es)

(7.24a)

Equations (7.2) and (7.8) give e(s) whereas h(s) is given by Equations (7.1) and (7.8); the results are

e(s) =V{V -a(s)} - ( p u s + p ~ ~ ~ ) a ( s ) (7.25a)

h(s) = ( u + E s ) ~ X n ( s ) (7.25b)

From here on we shall neglect displacement currents because we are considering propagation in a conducting medium; this eliminates terms in E in all the equations above, producing a diffusion equation as discussed in Section 6.2.3. Then a solution for Equation (7.24a) is (Stratton, 1941, Section 8.3)

a(.) = (1/4au)/ j(s)[exp{ - r (p~s )~”} / r ] du V

(7.24b)

where the integration is performed over a volume V containing the source of current, whereas a(s) is determined at a point at a distance r outside it. The source is a step function given by Z(t) = Zu(t) [Equation (A.79)]. From Equation (A.80) we have

Having found a(s) from Equation (7.24b), we find e(s), h(s) from Equations (7.25), then transform back to the time domain to get e(t) and h(t) for a

A simplifed version of the above approach is given by Ward and Hohmann (1988); they consider the FD field relations as Fourier transforms, then translate these into Laplace transforms of the TD response to a step input. This method will be discussed in Section 7.2.6e.

Zu(t) - z/s.

step input.

(c) Long straight horizontal wire. Ward and Hohmann (1988) give relations for the TDEM field for this source in free space. When the wire is along the y axis,

e,, = -poIee-e2p2/4at (7.26a)

h = Ie-e2p2(zi - x k ) / 2 ~ 8 (7.26b)

ah/at = z02e-ez~(zi - xk)/2at (7.26~)


where 0 = (pou/4t)lfl, 2 = x2 + z2 , and t is the decay time of the transient after current shutoff. [The time derivative of h is given here because it is fre- quently the measured parameter in TD surveys.] The x and z components of h and ah/& are easily found from Equations (7.26b, c). Strictly speaking, u = 0 so that the exponentials in Equations (7.26) are unity.

When the wire lies on homogeneous ground, the formulas are modified and we have

357

e Y = I( 1 - e-@zxz)/aux2 (7.27a)

h, = I { ezX2F(ex) - e x } / a 3 / 2 x h, = - I { 1 + eZx2( e - @ X 2 - 1)}/2ax

(7.2%)

(7.27~)

ah,/& = z{ (1 + i/eZx2) F( ex) - l iex} /a3/2Xt

+ I)! /2mt

(7.27d)

) (7.27e)

where F(8x) is Dawson's integral (Abramowitz and Stegun, 1964). Retaining only the first term in the series expansions of Equations (7.27d, e), the expressions for the time derivatives become

TD fields for the long straight insulated wire lying on a conductive half-space have been discussed in several reports (Wait, 1971; Kauahikaua, 1978; Oristagho, 1982). The second paper quoted discusses the effect of grounding the ends of the wire; obviously this introduces galvanic currents into the half- space. According to Kauahikaua, this affects various field components: e, and h, are modified, h,, ex, and e, components are introduced by the grounded ends, while only the h, field is entirely due to the insulated length.

(d) Large rectangle. This geometry, much used in TDEM systems as well as in Turam, is discussed in Section 7.2.3~. Fundamentally it is similar to the magnetic dipole in Section 7.2.3d, apart from the dimensions of the loop relative to its distance from the detector. Nevertheless the equations are so difficult to solve that no analytical solution is currently available for the general case of a Turam-type loop on homogeneous ground. However, a solution is available for the transient response when the receiver

is at the center of a circular transmitter loop of radius u similar to that used in the Sirotem coinci- dent-loop equipment (97.4.4.d) and in EM depth sounding (97.7.6). We have for the field at the center (Ward and Hohmann, 1988)

h, = (1124 { 3e-@*9peu

+ (1 - 3/2e2u2) erf( Q u ) } (7.28a)

ah,/& = - ( z/p0uu3) 3 erf( e u ) i

where erf(8a) is the error function (9A.12.3).

these expressions become At late times of the transient response, for u = 0,

Ih a well-known paper, Nabighnn (1979) described the downward and outward diffusion from the dipole of a transient EM field through a conducting half-space, the velocity and amplitude decreasing with time very much like a smoke ring in air.

Oristagho (1982), using two parallel line sources, illustrated a similar effect in which the two current patterns move away from each line at an angle of about 25". Three sections drawn for current densities at successive times in the ratio 1 : 2 : 4 are displayed in Figure 7.11. Nabighlan and Oristaglio (1984) noted that at late decay times the response fell off as t-2.5 in both systems, provided the rectangular source was of reasonable size.

The velocity V, radius R, and diffusion distance d (the TD equivalent of the skin depth-see 96.2.3) of the current whorls are 'of considerable significance. They are given by the relations

The diffusion distance locates the center of a whorl in terms of the decay time and conductivity. At late times the eddy currents reach large distances from the source; for a fixed time interval, the increase in d will be less, the larger u is, hence the whorls take longer to travel a given distance, that is, they remain longer in high conductivity beds.


0 20 4?m I .

Double Line Source

Contour ~ e v e i r ( 1 0 - ~ rmps/m2)

B = 3.0 C = 2.0 0 = 1.0 E = 0.0

H = -3.0 G = -2.0 F = -1.0

( 0 )

Figure 7.11. Plots of subsurface current density. Current flow 1 A, into page at 0, out at 0. Current shut-off time t = 0. Contour intervals A/m2 in (c). (After Oristaglio, 1982). (a) Plot for t /ap = Id m2.

A/m2 in (a) ahd (b),

(e) Small horizontal coil (vertical magnetic dipole). There is no airborne TD source of this type in current use. As for ground systems, although Input uses a horizontal-loop transmitter, the shape is roughly triangular and the loop is considerably larger than the conventional small horizontal loop. How- ever, PEM and SIROTEM Slingram-type ground equipment use relatively small vertical magnetic dipoles (see 57.7.4b and Table 7.3).

The TD field response in free space for this source can be written (Ward and Hohmann, 1988)

h, = ( 3Z6dpz/4ar5) (1 + 403r3/3p) (7.32a)

h, = ( z6d/4ar5) (21’ - $) x { 1 - 4e3r3$/p(2z2 - $) } (7.3%)

e+ = ( jwpoz6dp/4ar3)

x { 1 - erf( Or) + 2Or~?-~’~*/p} (7.32~)

where r2 = $ + z2 = x2 + y 2 + z2.

Over homogeneous ground the TD components can be obtained l‘rom Equations (7.14k, 1, m). For a homogeneous conducting medium, k = (-jwpu)’/’ where p and u are constant. Thus, at a fixed point (that is, p also constant), the right hand sides of Equations (7.14k, 1, m) are functions of jo, that is, they are Fourier transforms of the field responses. Assuming the system is linear (sA.13) we see from the discussion of Equation (A.87a) that these expressions are transforms of the unit impulse responses. Moreover, for the functions involved in these equations, we can set the covergence factor u in Equation (A.75b) equal to zero, that is, s = j w . When we replace jo with s, the Fourier transforms become Laplace transforms (5A.12).

Writing (Y = (pt~)’/~p, we have

jkp = j ( -jwpa)’/’p = (jo)’/’a = &I2;

k2$ = jk3p3 = - ( ~ 3 ~ 3 / 2

Substituting these relations in Equations (7.14k, 1,


0 20 40m W

Contour Levels ( 1 0 - ~ amps/m*)

B = l . l C = l . O D = 0 . 9 E-0 .8 F = O . 7 G = O . 6 H = 0 . 5

X = -1.1 W * -1.0 V = -0.9 U = -0.8 T = -0.7 S = -0.6 R = -0.5

I = 0.4 J = 0.3 K = 0.2 1 - 0.1

a = -0.4 P - -0.3 o = -0.2 N = -0.1

( h )

Figure 7.11. (Continued) (b) t/ap = 2 X Id m2.

m), we get the Laplace transforms of the unit impulse responses. Dividing by s gives us the transforms of the unit step responses (Sheriff and Geldart, 1983:174-5) in the form

The last two are readily transformed to the time domain using Equations (A.84b) and (A.85a, b, c, d). Transformation of the first expression is more dif-

ficult; details are given in Ward and Hohmann (1988:216). The final results are

+erfc(Op) - (48p + 9/8p)e-82d/p} (7.32e)

-(28p/p)(3 + 202p2) e-82d} (7.32f)

where erfc(8p) = 1 - erf(8p) (5A.12). To get the time derivatives of the field responses,

we multiply the Laplace transforms by s (see Eq. (A.78a) with n = 1, g(O+) = 0), then transform to the time domain. The results for the magnetic field


Contour Levets ( mnps/m2)

B = 4.0 C = 3.0 D = 2.0 E = 1.0

J = -4.0 I = -3.0 H = - 2 0 G = -1.0

(c)

Figure 7.11. (Continued) (c) t/up = 4 x Id m2

are (Ward and Hohmann, 1988)

ah,/at = ( z ~ d e ~ / 2 ~ , , t ) e-e2$/2

x {.( 1 + 6'8) z,(iez8) -(2 + ezpz + 4~e2pz)zl(~ezpz)} (7.32g)

ah,/at = -( 1~&/2rpu#) {9er f (~p)

-(2ep/p)(9 + SOz$ + 4e4p4) e-'ld} (7.3211)

At late time, Equations (7.32d, e, g, h) become

(7.32i) i i

h, = - ( ZSdppz~Z/128~t2)

h, = ( ~ t ~ ~ a ~ / 3 0 ) ( p u / r r ) ~ / ~

(7.32j) ah,/at = ( ZSdppZuz/64~t3)

ah,/at = -( zSd/20t)(pu/af)3'z

Figure 7.12 illustrates theoretical responses for h, and h, and their time derivatives for the vertical magnetic dipole. The decay amplitude is the value at a point 100 m from the transmitter over a 100 i2m homogeneous earth. Thus the curves show maximum values of h and a h / a t at the surface for the circular currents or "smoke rings" (Nabighlan, 1979) as they diffuse during the decay cycle (Fig. 7.11). Note also that the currents change sign on 3 of the 4 plots, following a sharp cusp or crossover.

In Figure 7.12a the h, curve is positive through- out with a peak of - 10-7A/m at about 10 ps after current cutoff. At later time h, decays at t -2 , the curve being at an angle of about 63" to the horizontal. The ah,/at curve changes sign abruptly at - 10 ps and subsequently falls off as t-3 at 72". From Figure 7.12b it is seen that the early-time maximum of h, (lop7 pA/m) persists to -10 ps where it changes sign and decays as t-3/2 at an angle of 56"; ah , /a t also is constant (-10 mA/m.s) through early time to -20 ps after which it decays as r5I2 at an angle of 68" (Oristagho, 1982).

EM equipment 361

n a L \ < -- Y - la

wA'ms) \

1 I 1 1 1 \I 1

Figure 7.12. TD transient decay curves at a point 100 m from a vertical magnetic dipole on 100 Clm homogeneous earth. (From Ward and Hohmann, 1988.) (a) h, and ah,/&. (b) h, and ah,/&.

(f) Horizontal magnetic dipole; vertical electric dipole. Although Wait (1951a) has discussed the TD version of the electric dipole on homogeneous ground, neither of these sources has been developed for either airborne or ground TD surveys.

VLF and AFMAG, make use of remote power sources and consequently do not need a transmitter.

7.3.2. Power Sources

7.3. EM EQUIPMENT

7.3.1. General

The measuring equipment for EM systems includes a local ac power source operating at one or several recurrence frequencies, transmitter and receiver coils (which may be one and the same in some TD sets), receiver amplifier tuned to the transmitter frequency (FD units) or wide-band (- 40 kHz in TD receivers), and an indicator, such as headphones, meter, digital readout, or recorder. Some FD field sets require in addition an ac potentiometer (or phase and amplitude compensator) for comparison of primary and secondary field signals. There is very little difference in these components whether they are used for ground or airborne sets, except that the latter are more elaborate and bulky. Two field methods that properly come under the heading of EM, namely

Formerly the power supply for EM transmitters was normally either a gas-driven alternator or a small, light battery-powered oscillator with a power amplifier having a low impedance output. The choice between these depended on the type of field set, that is, whether the transmitter was only semiportable or continually moving. In the long-wire, the large horizontal-loop, and the vertical-loop fuced-transmitter systems the larger power source would be used, whereas for the various completely mobile transmitters, such as horizontal-loop and vertical-loop broadside, the small unit was necessary.

Actual output from the semiportable power supplies varies between 250 and 2,500 W. The moving sources range from 1 to 10 W. Weight of the equipment, of course, increases with the power output and may be anywhere from 2 to 100 kg.

In recent EM equipment power supplies have been modified to reduce weight and size. For portable units the transmitter coil is an integral part of the

n

\ <

c

E

Y N


oscillator, eliminating the low-efficiency power amplifier. High-power sets employ a type of “flip-flop” switching unit to feed the coil directly for the same purpose.

The transmitter output, sinusoidal in FD equipment, is in the lower audio range, 100 to 5,000 Hz. With TDEM the periodic transient frequencies are low (3 to 300 Hz), consisting of half-sine, square- wave, or ramp-type (sawtooth) pulses, all of alternating polarity. The off time may be equal to or several times longer than the on time. One TDEM unit, the UTEM system (West, Macnae, and Lamontagne, 1984), uses a triangular waveform with zero off time. Formerly some FD power sources were dual frequency, one low and one relatively high, for example, 875 and 2,200, 1,OOO and 5,000, and 400 and 2,200 Hz. Recent ground FDEM equipment has a range from 111 to 3,555 Hz in steps of 2 (approximately). Frequency range in one operating airborne system is 900 to 35,000 Hz. The advantage in using two or more frequencies lies in discriminating between shallow and deep conductors and/or an indi- cation of conductivity of the anomaly and the surrounding medium.

7.3.3. Transmitter loops

In order to generate the desired electromagnetic field, the output of the power source must energize the ground by passing a current through some wire system. In the semiportable field sets this is done by coupling power into a long straight wire grounded at each end, a large (usually a single turn) rectangle or square laid out on the ground, or a relatively large vertical loop supported on a tripod or hung from a tree. In the first two arrangements, the dimensions are 0.5 km or more. The vertical loop, which may be single- or multiple-turn winding, may be triangular, square, or circular, and of necessity has an area of only a few square meters. Some means must be provided for orienting this coil in any desired azimuth.

The completely mobile FDEM transmitters often employ multiple-turn coils (100 turns or more) wound on insulating frames of 1 m diameter or less. Some- times the coil may be merely a single turn of heavy conductor; in this case, a matching transformer may be necessary in the source output. An alternative to this type of aircore coil, in which the wire is wound as a solenoid on a ferrite or other high-permeability core, is now used in many EM systems. If the winding is distributed properly, it is possible to generate a field equivalent to a coil wound on a much larger frame, because the value of H increases with core permeability to compensate for the small

area enclosed. Because the value of p for certain ferrite is about 1,000, the cross section can be greatly reduced.

Many FD transmitter systems have a capacitance in series with the coil, the value being chosen to resonate approximately with the coil inductance at the source frequency. Because the coil has low resistance, this permits maximum current (within the limits of the power supply) to flow in the coil.

7.3.4. Receiver Coils

In FD systems, these are generally small enough to be entirely portable, that is, 3 ft (1 m) diameter or less, and have many turns of fine wire. Smaller coils with high-permeability cores are also used. The coil may be shunted by an appropriate capacitance to give parallel resonance at the source frequency. The resulting high impedance across the amplifier input acts as a bandpass filter to enhance the signal-to-noise ratio.

Receiver loops, and usually the portable transmitter loops as well, are electrostatically shielded to eliminate capacitive coupling between coil and ground and between coil and operator. The shielding is obtained by conductive paint or strips of conductive foil over the winding; the shield, which is split around the perimeter to permit emission and recep- tion of the signal field, is then connected to the main circuit ground.

Generally the receiver coil must be oriented in a certain direction relative to the detected field. In some ground systems this merely means that the loop is maintained approximately horizontal or vertical. In others it is necessary to measure inclination and/or azimuth at each station. In airborne EM systems it is often a difficult problem to maintain a fixed orientation between transmitter and receiver coils.

In some TDEM systems the same coil may be employed both for transmitting and receiving (obviously this is not possible with continuous transmission). In such cases the loop size may vary from a few meters to several hundred; hence the coil is not completely portable.

7.3.5. Receiver Amplifiers

Amplifiers are of fairly standard design. The overall voltage gain is usually between lo4 and lo5. One or more narrow bandpass filters, tuned to the source frequency or frequencies, are incorporated in the FD amplifier. A network of this type, called the Twin-T, which uses only resistors and capacitors, is illus-

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