An alternative algorithm for one-dimensionalmagnetotelluric response calculation

Hendra Grandis *

Jurusan Geo®sika dan Meteorologi, Institut Teknologi Bandung (ITB), Jalan Ganesha 10, Bandung 40132, Indonesia

Received 16 April 1998; revised 18 June 1998

Abstract

The magnetotelluric response of a given conductivity model is obtained from the solution of Maxwell's di�erential

equations. In the case of a one-dimensional model, the response function is usually expressed as a recursive formulaconnecting the impedance at the surface of two successive layers. The present paper describes an alternativealgorithm for computing the 1D-magnetotelluric response function using a recursive formula for electromagnetic®elds at the top of two consecutive layers. Special attention is paid to the conception of the algorithm in the form

of matrix multiplication, which o�ers advantages in its application. An example is presented to illustrate thepotential use of the proposed scheme. # 1999 Elsevier Science Ltd. All rights reserved.

Code available at http://www.iang.org/CGEditor/index.htm

Keywords: Electromagnetic method; Magnetotelluric sounding; 1D forward modelling

1. Introduction

The magnetotelluric (MT) method is a frequency-

domain electromagnetic sounding technique that uses

naturally existing electromagnetic ®elds as the signal

source. These primary ®elds induce secondary electric

and magnetic ®elds in the conductive earth. The transi-

ent variation of the electromagnetic ®elds recorded at

the surface of the Earth is therefore diagnostic of the

subsurface electrical conductivity (or resistivity) distri-

bution. The magnetotelluric method has a wide range

of application, from shallow investigations (geotech-

nics, groundwater and environment) to moderate and

deep exploration of natural resources (mineral,

geothermal and petroleum).

Estimating magnetotelluric response for a given

earth model (forward modelling) is of central import-

ance in the interpretation of magnetotelluric data.

Despite considerable progress in multi-dimensional

modelling (2D and 3D) over the last several years, 1D

modelling is still routinely employed, at least during

®eld work and at the preliminary stage of interpret-

ation. A 1D model that consists of a succession of

homogeneous horizontal layers represents the subsur-

face resistivity that varies only with depth. In most

cases, collating 1D models obtained at stations in a

pro®le can reasonably give some idea about the overall

resistivity structure of the area investigated.

To obtain the magnetotelluric response function

from solution of Maxwell's equations is straightfor-

ward when a 1D or layered medium is considered.

This results in a recursive formula that relates the com-

plex impedance at the top of two successive layers.

The formula, as can be found in most textbooks on

geophysical exploration methods (see for example

Kau�man and Keller, 1981; Rokityansky, 1982), con-

tains hyperbolic trigonometric functions. The hyper-

bolic (co)tangent of complex quantities is transformed

into its exponential form with only negative argument

and terms involving quotients are forced to have non-

zero denominator. Such a modi®cation leads to a

Computers & Geosciences 25 (1999) 119±125

0098-3004/99/$ - see front matter # 1999 Elsevier Science Ltd. All rights reserved.

PII: S0098-3004(98 )00110-1

PERGAMON

* E-mail: grandis@geoph.itb.ac.id.

straightforward and stable algorithm (Pedersen andHermance, 1986).

This paper describes an alternative scheme for com-puting the complex impedance as a function of 1Dmodel parameters (resistivity and thickness of layers).

The recursive formula in the form of matrix multipli-cation is de®ned for electric and magnetic ®elds inde-pendently, instead of impedance, at the surface of two

successive layers. Unlike the formula presented byWard and Hohmann (1988), the exponentials in thepresent algorithm are preserved and restricted to have

only negative argument in order to ensure the stabilityof the algorithm. Although the proposed scheme in itsoriginal form is not essentially di�erent from the classi-cal algorithm, the matrix multiplication method can be

optimized for particular purposes; for example, whencomputing the response of a 1D model that consists ofa large number of layers and all possible values of one

parameter (the resistivity) are used in turn.

2. Maxwell's equations

In the following we summarize, for completeness,

the formulation of a 1D magnetotelluric response func-tion from Maxwell's equations which de®ne the funda-mental relationship between electromagnetic ®eld

vectors. Assuming harmonic time dependence of theelectromagnetic ®elds as exp(+iot), two of Maxwell'sequations in a generalized di�erential form are(Kau�man and Keller, 1981; Pedersen and Hermance,

1986)

r � E � ÿiomH, �1�

r �H � sE� ioeE, �2�where E and H are the electric and magnetic ®elds, sis the conductivity, m is the magnetic permeability, e isthe dielectric permittivity and o= 2p/T is the angularfrequency for a period T, all in SI units. When conduc-

tion currents dominate over displacement currents asin most geoelectromagnetic studies, the second term ofthe right-hand side of Eq. (2) can be neglected. In all

subsequent equations we also considerm = m0= 4p � 10ÿ7 H/m.In a 1D medium, the electromagnetic ®elds and con-

ductivity do not vary in the horizontal directions, xÃ

and yÃ. Then, the x-component of the curl of Eq. (1)can be expressed as

@ 2Ex

@z2� iom0sEx, �3�

whereas the y-component of Eq. (1) is

@Ex

@z� iom0Hy: �4�

Eqs. (3) and (4) and a similar pair of equations thathold for Ey and Hx, along with appropriate boundaryconditions, are the governing equations for 1D electro-

magnetic induction. For the rest of the paper, only thepair, Ex and Hy, is considered and their subscripts aredropped for clarity.

3. One-dimensional response function

An elementary solution to Eq. (3) is given by

E � A exp�ÿkz� � B exp��kz�, �5�where terms in exp(ÿkz) and exp(+kz) correspond toattenuation of the electric ®eld in the +z and ÿz direc-

tions, respectively (z being depth, that is, positivedownwards), and k= (iom0/r)

1/2 is the wave number.A and B are constants to be determined from bound-

ary conditions. From Eq. (4), we obtain the corre-sponding (orthogonal) magnetic ®eld as

H � k

iom0�A exp�ÿkz� ÿ B exp��kz��: �6�

The magnetotelluric response is the complex impe-dance de®ned as the ratio of the horizontal electric®eld to the orthogonal horizontal magnetic ®eld (i.e.E/H). For a homogeneous half-space of constant resis-

tivity, the coe�cient B is zero since we consider onlyexternal sources of the ®elds such that the electromag-netic ®elds must vanish at in®nite depth. This yields

(at the surface of the homogeneous medium)

ZI � E

H� �������������

iom0rp �7�

where ZI in Eq. (7) stands for intrinsic impedance, i.e.the impedance at the surface of a homogeneous half-space, which is a function of resistivity and frequency

Fig. 1. One-dimensional earth model composed of N homo-

geneous horizontal layers, bottom layer being homogeneous

half-space extending to in®nity.

H. Grandis / Computers & Geosciences 25 (1999) 119±125120

(or period). The resistivity of the half-space and thephase of the impedance are related to the intrinsic

impedance as follows:

r � 1

om0jZIj2, �8�

f � tanÿ1�Im ZI

Re ZI

�� 458: �9�

For a more general 1D situation (such as a layered

earth), the impedance is no longer intrinsic impedanceso that the resistivity in Eq. (8) becomes apparent res-istivity and the phase in Eq. (9) becomes a function of

period.We now consider a 1D earth model consisting of N

horizontal layers with constant resistivity (Fig. 1).Following Ward and Hohmann (1988), we de®ne a

recursive relationship that holds for electric and mag-netic ®elds separately. From Eqs. (5) and (6), the elec-tric and magnetic ®elds in the jth layer at depth z1 are

given by

Ej�z1� � Aj exp�ÿkjz1� � Bj exp��kjz1�, �10�

Hj�z1� � kjiom0

�Aj exp�ÿkjz1� ÿ Bj exp��kjz1��: �11�

Similar relations hold for the ®elds at depth z2 in thesame jth layer from which the coe�cients Aj and Bj

are obtained

Aj � 1

2�Ej�z2� � ZI,jHj�z2��exp��kjz2�, �12�

Bj � 1

2�Ej�z2� ÿ ZI,jHj�z2��exp�ÿkjz2�; �13�

where the intrinsic impedance of the jth layer ZI,j has

been used to substitute iom0/kj.Substituting Eqs. (12) and (13) into Eqs. (10) and

(11) yields

Ej�z1� � 1

2�Ej�z2� � ZI,jHj�z2��exp��kj�z2 ÿ z1��

� 1

2�Ej�z2� ÿ ZI,jHj�z2��exp�ÿkj�z2 ÿ z1��, �14�

Hj�z1� � 1

2�Z ÿ1I,j Ej�z2� �Hj�z2��exp��kj�z2 ÿ z1��

ÿ 1

2�Z ÿ1I,j Ej�z2� ÿHj�z2��exp�ÿkj�z2

ÿ z1��: �15�

Rearranging terms, we obtain a matrix equation equiv-alent to Eqs. (14) and (15)

�Ej�z1�Hj�z1�

���a ZI,jbZ ÿ1I,j b a

��Ej�z2�Hj�z2�

�, �16�

where a=12(exp(+kj(z2ÿz1)) + exp(ÿkj(z2ÿz1))),

b=12(exp(+kj(z2ÿz1))ÿ exp(ÿkj(z2ÿz1))).

We assume that z1 and z2 coincide with the depth tothe top and bottom of the jth layer, respectively, anddenote hj=z2ÿz1 the thickness of the jth layer. Theboundary conditions demand that tangential ®elds are

continuous across the layer interface so that the ®eldsat the bottom of the jth layer are equal to the ®elds atthe top of the ( j + 1)st layer, i.e. Fj(z2) = Fj+1(z2)

where F= E or H. If we de®ne the ®elds systemati-cally at the upper surface of a layer, then the ®elds canbe identi®ed simply by the layer's index such that

Fj(z1) = Fj and Fj+1(z2) = Fj+1. Then Eq. (16)becomes� ~Ej

~Hj

���

1� exp�ÿ2kjhj � ZI,j�1ÿ exp�ÿ2kjhj ��Z ÿ1I,j �1ÿ exp�ÿ2kjhj �� 1� exp�ÿ2kjhj �

��Ej�1Hj�1

�,� ~Ej

~Hj

�� Tj

�Ej�1Hj�1

��17�

where EÄj=2 exp(ÿkjhj)Ej and HÄ j=2 exp(ÿkjhj)Hj. InEq. (17), only exponential terms with a negative argu-

ment are retained after multiplying both sides ofEq. (16) by 2 exp(ÿkjhj).Eq. (17) represents a recursive formula connecting

the electromagnetic ®elds at the upper surfaces of two

consecutive layers. At a given period, the transfermatrix T is de®ned by the parameters (thickness andresistivity) of the jth layer. For an earth model consist-

ing of N homogeneous layers, the surface electromag-netic ®elds are related to the ®elds at the Nth layer bythe product of a succession of Tj's, i.e.�

~E1~H1

��YNÿ1j�1

Tj

�EN

HN

��18�

where, in this case, EÄ1=E1 and HÄ 1=H1 if N is oddand EÄ1=2 exp(ÿk1h1)E1 and HÄ 1=2 exp(ÿk1h1)H1 if Nis even. The product of Nÿ 1 transfer matrices is a

(2 � 2) matrix S= [skl]; k, l = 1, 2 so that the surfaceimpedance (impedance at the top of the ®rst layer) canbe written as

Z1 � E1

H1�

~E1

~H1

,

Z1 � s11EN � s12HN

s21EN � s22HN� s11ZI,N � s12

s21ZI,N � s22,

�19�

where, in Eq. (19), the impedance of the homogeneoushalf-space, ZI,N replaces the electromagnetic ®elds at

H. Grandis / Computers & Geosciences 25 (1999) 119±125 121

the surface of the last layer. In this case, the matrixmultiplication can be performed in any order, fromtop to bottom or vice versa, as long as we obtain the

total transfer matrix S.

4. Application

The algorithm coded in FORTRAN 77 is given inAppendix A as a subroutine that calls another subrou-

tine for (2 � 2) matrix multiplication. The input vari-ables transferred within the subroutine are: the numberof layers, the layer parameters (resistivity and thick-

ness), the number of periods and the periods at whichthe complex impedances (the output) are computed, allin SI units. A typical main program for computing the1D magnetotelluric response function (forward model-

ling) ®rst de®nes the input variables before calling thesubroutine. The apparent resistivity and phase are thencalculated using the complex impedances transferred

from the subroutine. For normal purposes, we do notclaim that the e�ectiveness of the proposed scheme issuperior to the classical method (e.g. Pedersen and

Hermance, 1986). A simple test suggests that the ex-ecution time of the two algorithms in computing theresponse function of a 1D model composed of 60layers at 30 frequencies or periods 10,000 times is vir-

tually the same (8 min 54 s for the classical methodand 8 min 56 s for the present algorithm using a PCwith a Pentium1 133 MHz processor). However, the

matrix multiplication method has signi®cant advan-tages when used in certain circumstances as discussedin the following.

The fact that the transfer matrix characterizes eachlayer independently and the calculation of the totaltransfer matrix S can be performed in any order o�ers

a ¯exibility in the application of the algorithm.Consider a situation in which we have to compute theresponse function of a 1D model consisting of N

layers. The parameter (for example, the resistivity) ofthe kth layer is changed by assigning M possible (apriori) values in turn. In this case, Eq. (18) can be

rewritten as�~E1~H1

��Ykÿ1j�1

Tj � Tk �YNÿ1

j�k�1Tj

�EN

HN

�

� S� � Tk � Sÿ�EN

HN

�: �20�

where S+ and Sÿ are constants corresponding to thetransfer matrices of the unchanged layers above andbelow the kth layer. With careful bookkeeping in theprogramming, the problem is then reduced to M multi-

plications of three matrices (or two matrices if k= 1or k = N) instead of multiplying Nÿ 1 matrices Mtimes using the original scheme. The use of the classi-

cal algorithm would also be very time-consuming sincethe recursive formula relates the complex impedance atthe top of two successive layers, i.e. the computation

always begins from the homogeneous half-space (bot-tom layer) and proceeds upwards for all intermediatelayers, although only one of them is changed.The above particular situation is encountered in the

resolution of the 1D magnetotelluric inverse problemusing a stochastic method proposed by Grandis (1994).The 1D model is composed by a large number of `thin'

layers whose thicknesses are kept ®xed. The resistivityof each layer is determined by assigning M possible apriori resistivity values sequentially (see Grandis, 1994;

Grandis et al., 1994 for more details). For large N andM, the matrix multiplication method has proven to bevery advantageous. Although we have not done any

Fig. 2. Resistivity vs depth function obtained from 1D magnetotelluric inversion using stochastic algorithm (thick line) and syn-

thetic model (dash). Grey scales represent computed uncertainties of inverse model.

H. Grandis / Computers & Geosciences 25 (1999) 119±125122

systematic tests, preliminary results show that for typi-cal numbers of N = 60, M = 20 and 30 frequencies or

periods, and for 200 complete sweeps over all layers(iterations), the execution time is approximately 2 husing the original algorithm or classical method, and 15

min when the optimized form of the present algorithmis employed (using a PC with the same speci®cation asbefore).

Fig. 2 shows the inversion result of magnetotelluricsynthetic data using the stochastic algorithm incorporat-ing the optimized 1D forward modelling scheme dis-

cussed above. A 5-layer synthetic model havingresistivities 250, 25, 100, 10 and 25 O m and layerboundaries at 0.6, 2.0, 6.0 and 10.0 km depth was usedto generate the synthetic data. The same model was also

used by Dosso and Oldenburg (1991) and others todemonstrate their inversion techniques. A 10%Gaussian noise was added to the synthetic data in the

manner suggested by Constable (1991). The depth rangefrom 100 to 30,000 m is discretized into 60 layers having®xed boundaries at a constant interval in a logarithmic

scale. Therefore the layer thicknesses increase withdepth representing the resolution of the magnetotelluricmethod that generally decreases with depth. The resis-

tivity variation with depth obtained from the inversionis presented as a curve with grey scales representing itscomputed uncertainties. The discussion on the inversionresult is beyond the scope of this paper as the example

serves only for illustrating the use of the proposed al-gorithm.

5. Conclusion

We have presented an alternative algorithm for 1Dmagnetotelluric forward modelling that can be directlyimplemented as a numerical routine. In this algorithm,

called the matrix multiplication method, the exponen-tials involved are forced to have only a negative argu-

ment in order to prevent instability. The fact that thetransfer matrix characterizes each layer independentlyand the calculation of the total transfer matrix can be

performed in any order is emphasized and is exploitedto optimize the algorithm.

Although the example given in this paper is of a par-

ticular case, the matrix multiplication algorithm canalso be incorporated into other inversion methods of asimilar type (1D model with a large number of layers)

but using a di�erent approach of exploring the modelspace. The simulated annealing technique proposed byDosso and Oldenburg (1991) and the linearized inver-sion method of Constable et al. (1987) and Oldenburg

and Ellis (1991) are a few examples. The matrix multi-plication algorithm will likely speed up the linearizedinversion method especially when ®nite-di�erence ap-

proximation is used to calculate the partial derivatives.In recent years, research in techniques to resolve

strongly non-linear inversion problems has been directed

toward the use of global search methods, in place of alocal or linearized approach. Despite the advent ofpowerful computing resources, an e�cient forward

modelling algorithm is essential in such inversionmethods because the response of a large number ofmodels has to be estimated. It is in this perspective thatthis alternative algorithm of 1D magnetotelluric forward

modelling will gain particular interest.

Acknowledgements

The author wishes to thank Tom C. Kerr, SriWidiyantoro and the reviewers for their valuable sugges-tions which helped to improve the manuscript.

H. Grandis / Computers & Geosciences 25 (1999) 119±125 123

Appendix A

H. Grandis / Computers & Geosciences 25 (1999) 119±125124

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