Direct interpretation of magnetotelluric sounding data based on the frequency-normalized impedance...

Direct interpretation of magnetotelluric soundingdata based on the frequency-normalizedimpedance function1

Ahmet T. Basokur,2 Cemal Kaya3 and Emin U. Ulugergerli 2

Abstract

An important aspect of any non-linear inversion method is the generation of a suitableor good initial model as this controls the rate of convergence and accuracy of the result.To overcome the problem, a numerical method is presented for direct interpretation ofmagnetotelluric sounding data based on the frequency-normalized impedance (FNI)function. The expressions used to calculate the parameters are developed, first for atwo-layer case under the assumption that deeper layers do not contribute to the earlypart of the FNI curve, and they are then generalized for an n-layer situation. Theparameters of the first layer are computed by using successive sample values and thefinal estimate is obtained from the arithmetic mean of selected values by excludingunacceptable results in the logarithmic space. The top layer is then removed using areduction equation. The repetition of the procedure on successive branches of the FNIfunction gives successive layer parameters, the resistivity of the substratum beingobtained at the final step, when the reduction equation becomes equal to the squareroot of that resistivity.

The proposed method can be used as a complementary method for iterativeinversion as it creates an initial guess which is close to the optimal solution. Thesolution produced by the direct interpretation may also be modified by the interpreterto incorporate prior geological information before being input to iterativeinterpretation schemes.

Introduction

The interpretation of electrical and electromagnetic sounding data is a non-linearinverse problem. Iterative least-squares inversion methods are traditionally used tointerpret various types of geophysical data and there are several examples of theinterpretation of electrical and electromagnetic data using Levenberg–Marquardt-typeinversion techniques (e.g. Inman 1975; Johansen 1975; Jupp and Vozoff 1975; Ilkisik

q 1997 European Association of Geoscientists & Engineers 21

Geophysical Prospecting, 1997, 45, 21–37

1 Received May 1994, revision accepted February 1995.2 Ankara Universitesi, Fen Fak., Jeofizik Muh.Bolumu, Besevler, 06100 Ankara, Turkey.3 MTA Genel Mudurlugu, Jeofizik Etudleri Dairesi, Eskisehir Yolu, 06520 Ankara, Turkey.

and Jones 1984; Pedersen and Rasmussen 1989; Meju and Hutton 1992). The iterativeleast-squares non-linear inversion method requires a guess supplied by the interpreterand this model is updated at each iteration step to improve the fit between the measuredand model responses. However, the solution and the speed of convergence depend onthe selected model and, as is well known, a poor initial model may lead the inversionprocess astray or cause slow convergence. It is easy to explain why this is so. The non-linear problem is generally linearized by expanding the calculated function in a Taylorseries about an initial guess in the parameter space and neglecting the second and thehigher order terms. Since this approximation will be valid for initial guesses close to thetrue parameters, the correction step for the model parameters at intermediate iterationswill be valid in the neighbourhood of the solution. For these reasons, the number ofiterations required to minimize the differences between the model responses andmeasured data depends on the goodness of the initial guess and the noisecontamination of the data which sometimes leads to no convergence. Some suggestionshave been made to overcome the problem of the need for a reasonable guess innon-linear inversion. For example, Sen, Bhattacharya and Stoffa (1993) proposed anon-linear global optimization method referred to as simulated annealing (SA).However, SA requires a large amount of computation time for thousands of iterations,which is feasible only on powerful computers. The typical geoelectrical interpreter usesa one-dimensional (1D) interpretation scheme that can be implemented on a portablefield microcomputer, then two-dimensional (2D) or three-dimensional (3D)interpretations are usually carried out on powerful computers using the 1D solutions.

In any of the optimization techniques, the cost of a poor initial guess is a substantialincrease in the number of iterations. A reasonable guess can be obtained from directmethods that estimate a conductivity profile from the data by some algebraicoperations. Well-known examples which produce a continuous resistivity–depthdistribution are the Bostick transformation (Bostick 1977) and the r*–z*transformation (Schmucker 1970, 1987). However, the output of these methodscannot be used directly to initialize a 1D inversion scheme, because the interpreter hasto make decision about the number of layers and the initial values of their resistivitiesand thicknesses by visual inspection of the resistivity–depth diagram.

A direct method for a horizontally stratified earth model consisting of homogeneousand isotropic layers was suggested by Patella (1976). In this method, the basicprinciples of Koefoed’s (1970) method of direct interpretation for direct current (DC)resistivity soundings have been adapted for the quantitative analysis of MT data. Theresistivity and thickness of the first layer are obtained by fitting the early parts of theapparent resistivity and phase curves with previously calculated two-layer theoreticalcurves. After this fitting, the first layer is removed by a reduction equation and the sameprocedure is repeated until all layer parameters are derived. Patella’s (1976) method ofdirect interpretation suffers from the restrictions of any curve-matching proceduresand it is not suitable for machine calculations.

The automatic inversion scheme proposed by Fischer et al. (1981) needs anassumption about the resistivity of the top layer. The thickness of the first layer and the

22 A.T. Basokur, C. Kaya and E.U. Ulugergerli

q 1997 European Association of Geoscientists & Engineers, Geophysical Prospecting, 45, 21–37

resistivity of the second layer are derived from the high-frequency part of the MT data.The parameters of the deeper layers are obtained by switching the operation to lowerfrequencies. An initial guess about the number of layers is not required and the methodyields a multilayer model in which the resistivities of some neighbouring layers are closeto each other. The multilayer model has to be reduced to fewer layers to satisfy knowngeological information.

We therefore developed a simple method for calculating a reasonable guess for 1Dinterpretations that exploits our observations. The direct method proposed here usesthe concept of ‘reduction to a lower boundary plane’ suggested by Koefoed (1970) forDC soundings. After the calculation of the resistivity and the thickness of an individuallayer, the contribution of that layer to the frequency-normalized impedance data isremoved by the reduction equation. The repetition of the procedure computes all layerparameters giving an equivalent solution (one of the many possible solutions) withoutany prior knowledge of the subsurface structure. The layer parameters determined bythis direct interpretation method can then be used directly as input to a damped least-squares iterative interpretation software to improve the interpretation. A similarmethod has been successfully applied to DC resistivity sounding curves (Basokur1984, 1990, 1993).

Approach and assumptions

In the magnetotelluric sounding method, the impedance Z was defined by Cagniard(1953 ) as the ratio of the electric field to the magnetic field, i.e.

Z ¼EH

: ð1Þ

The frequency-normalized impedance function can be defined as (Basokur 1994)

Y ¼Z

ðiqmÞ1=2 ¼1

ðiqmÞ1=2

Ex

Hy; ð2Þ

where q is the angular frequency, m is the magnetic permeability of free space andY is known as the frequency-normalized impedance (FNI) function. The forwardsolution of the FNI function for the horizontally stratified earth model is given in theAppendix.

The real part of the FNI function is equal to the square root of the resistivity of thetop layer and of the basement for high- and low-frequency limits, respectively. Atintermediate frequencies, the real part shows an increase if the underlying layer has amuch greater resistivity and vice versa. The general shape of the real part curveresembles the corresponding apparent resistivity curve but the magnitudes of theminimum and maximum of the real part lie within the range of the square roots of thetrue resistivities of the subsurface layers. The real part increases or decreasescontinuously when the resistivities of successive layers are respectively increasing ordecreasing. However, the slope of the real part depends on the ratio of the resistivities

Interpretation of MT data using FNI function 23


of the consecutive layers and the transition from one layer to the next can easily berecognized.

The imaginary part is zero for the high- and low-frequency limits. At intermediatefrequencies, the imaginary part describes a semicircular-shaped arc for each boundaryplane between two consecutive layers. The sign of the imaginary part distinguishesbetween the descending and ascending branches of the real part. The term ‘branch’ asused here refers to a part of the FNI curve that represents the transition from one layerto the next. For example, a two-layer curve has one branch, a three-layer curve has twobranches, and so on. The FNI curve can easily be subdivided into its branches byexamining the behaviour of the real and imaginary parts. The interpreter decides atwhich discrete values the branch starts and ends and these key numbers are fed into thecomputer. In this way, the interpreter also makes a decision about the number of layers.

The general strategy applied in the proposed method is based on the assumption thatthe contribution of the deeper layer is neglected for a given branch. The procedure isfirst applied to the first branch at the highest frequencies and is switched to lowerfrequencies.

Two-layer case

Determination of the resistivity of the first layer

For a two-layer case, equation (A1) gives

Y ¼ P1 tanhut1

P1þ arctanh

P2

P1

� ��

: ð3Þ

With discrete values Y ðu1Þ, Y ðu2Þ; . . . ;Y ðujÞ of abscissa points u1, u2; . . . ; uj , thefollowing equations can be developed from (3):

ujt1P1

¼ arctanhY ðujÞ

P1

� �

¹ arctanhP2

P1

� �

; ð4Þ

uj þ1t1P1

¼ arctanhY ðuj þ1Þ

P1

� �

¹ arctanhP2

P1

� �

; ð5Þ

uj þ2t1P1


P1

� �

¹ arctanhP2

P1

� �

: ð6Þ

P2 can be eliminated by subtracting (5) from (4) and also (6) from (5). The resultingequations are

ðuj ¹ uj þ1Þt1P1

¼ arctanhY ðujÞ

P1

� �

¹ arctanhY ðuj þ1Þ

P1

� �

; ð7Þ

and

ðuj þ1 ¹ uj þ2Þt1P1


P1

� �


P1

� �

: ð8Þ



Eliminating t1 by dividing (7) by (8), we obtain

arctanhY ðujÞ

P1

� �

¹ ð1 þ vÞ arctanhY ðuj þ1Þ

P1

� �

þ v arctanhY ðuj þ2Þ

P1

� �

¼ 0; ð9Þ

where

v ¼ ðuj ¹ uj þ1Þ=ðuj þ1 ¹ uj þ2Þ: ð10Þ

The resistivity of the first layer cannot be directly solved from (9). Note that if

uj þ1 ¼ ðuj þ uj þ2Þ=2; ð11Þ

v becomes equal to unity, with the relationship between the corresponding frequencies:

fj þ1 ¼ ð fj þ 2��

fj fj þ2

q

þ fj þ2Þ=4; ð12Þ

which with (9) leads to

ðP1 þ Y ðujÞÞ

ðP1 ¹ Y ðujÞÞ

ðP1 ¹ Y ðuj þ1ÞÞ

ðP1 þ Y ðuj þ1ÞÞ

ðP1 þ Y ðuj þ2ÞÞ

ðP1 ¹ Y ðuj þ2ÞÞ¼ 1; ð13Þ

since

arctanhðxÞ ¼ 0:5 lnðð1 þ xÞ=ð1 ¹ xÞÞ: ð14Þ

The expression for the resistivity of the first layer is now determined from (13), giving

r1 ¼Y 2ðuj þ1Þð2Y ðujÞY ðuj þ2Þ=Y ðuj þ1Þ ¹ Y ðujÞ ¹ Y ðuj þ2ÞÞ

Y ðujÞ ¹ 2Y ðuj þ1Þ þ Y ðuj þ2Þ: ð15Þ

Thus, the resistivity of the first layer can be calculated from three successive discretevalues by using (15). The calculation of r1 can be repeated by shifting the operation toa new set of three successive values. The arithmetic mean of the closest values selectedfrom among all calculated values of r1 will give an estimate of the resistivity of the firstlayer. This calculation is carried out in logarithmic space.

Determination of the thickness of the first layer

Once the resistivity is determined, the thickness of the first layer can then be foundfrom (7) as

t1 ¼ wjP1 arctanhY ðujÞ

P1

� �


P1

� ��

; ð16Þ

where

wj ¼ 1=ðuj ¹ uj þ1Þ;

or

t1 ¼ 0:5wjP1 lnfðP1 þ Y ðujÞÞðP1 ¹ Y ðuj þ1ÞÞ=ðP1 ¹ Y ðujÞÞ=ðP1 þ Y ðuj þ1ÞÞg: ð17Þ



Thus, t1 can be calculated from each of the successive pairs of the FNI function. Byexcluding unacceptable results, the final estimation of the thickness can be computedfrom the arithmetic mean of the selected values in logarithmic space.

Determination of the resistivity of the electrical basement

The resistivity r2 of the second layer can be found from (3) for each discrete value ofthe FNI function since r1 and t1 have been determined in the previous steps. It issimply

r2 ¼Y ðuÞ ¹ P1 tanhðut1=P1Þ

1 ¹ ðY ðuÞ=P1Þ tanhðut1=P1Þ

� �2

: ð18Þ

n-layer case

Determination of the resistivity and thickness of the first layer

If we assume that the early part of the FNI curve does not contain information ondeeper layers, then the resistivity of the first layer can be calculated from threesuccessive discrete values by using (15). The procedure is terminated at the firstdiscrete value of the second branch where the FNI curve contains information on thethird layer. The thickness of the first layer is obtained from the sample values used forcomputation of r1 by the procedure described for the two-layer case.

Determination of the resistivity and thickness of the subsequent layer

Once r1 and t1 are determined, the contribution of the first layer to the FNI curve canbe removed. The reduction equation is obtained by rearranging (A4) as

Y2ðuÞ ¼Y1ðuÞ ¹ P1 tanhðt1u=P1Þ

1 ¹ ðY1ðuÞ=P1Þ tanhðt1u=P1Þ: ð19Þ

It is assumed that the measurements are carried out on the top of the second layer.Thus, replacing Y ðuÞ and r1 by Y2ðuÞ and r2 in (15), we obtain

r2 ¼Y 2

2 ðuj þ1Þð2Y2ðujÞY2ðuj þ1Þ=Y2ðuj þ1Þ ¹ Y2ðuj þ2ÞÞ

Y2ðujÞ ¹ 2Y2ðuj þ1Þ þ Y2ðuj þ2Þ: ð20Þ

Similarly, an equation to determine t2 can be derived from (16) as

t2 ¼ wjP2 arctanhY2ðujÞ

P2

� �

¹ arctanhY2ðuj þ1Þ

P2

� ��

; ð21Þ

where

wj ¼ 1=ðuj ¹ uj þ1Þ:



r2 and t2 are calculated from the discrete values of the reduced FNI curve whichcorresponds to the second branch. The reason for disregarding the first branch is that asmall amount of noise contamination and/or errors made in the determination of theresistivity and the thickness of the top layer make (19) numerically unstable. This isbecause the sample values of the real part of the FNI curve are close to P1 in this rangeof abscissa values. The other values of the reduced FNI curve lying on the right-handside of the discrete value where the second branch starts can be safely used for thecalculation of r2 and t2, and for the next steps of the computations.

Note that (20) and (21) suggest that the method can be easily generalized. In the k thstep of the method, it is assumed that the layer parameters, r1, r2; . . . ; rk¹1 and t1,t2; . . . ; tk¹1, have been determined, and the top layers corresponding to thoseparameters can be removed by the reduction equation,

YkðuÞ ¼Yk¹1ðuÞ ¹ Pk ¹1 tanhðutk ¹1=Pk ¹1Þ

1 ¹ ðYk¹1ðuÞ=Pk¹1Þ tanhðutk¹1=Pk¹1Þ: ð22Þ

Thus rk and tk can be calculated from the discrete values of YkðuÞ instead of r1, t1 andY ðuÞ in (15) and (16) using the formulae

rk ¼Y 2

k ðuj þ1Þð2YkðujÞYkðuj þ2Þ=Ykðuj þ1Þ ¹ YkðujÞ ¹ Ykðuj þ2ÞÞ

YkðujÞ ¹ 2Ykðuj þ1Þ þ Ykðuj þ2Þð23Þ

and

tk ¼ wjPkðarctanhðYkðujÞ=PkÞ ¹ arctanhðYkðuj þ1Þ=PkÞÞ; ð24Þ

where

wj ¼ 1=ðuj ¹ uj þ1Þ:

Once rk and tk are computed (as for r1 and t1) then Ykþ1ðuÞ and the correspondinglayer parameters can be evaluated. This procedure is repeated until rn¹1 and tn¹1 areobtained. The values of the resistivity and thickness of one layer are calculated from thecorresponding branch. The only input to the algorithm are the key numbers thatinstruct the computer program at which discrete values the branch starts and ends.

At the final step of the procedure, the reduction equation becomes equal to thesquare root of the resistivity of the last layer. This can be deduced from (22) as

rn ¼Yn¹1ðujÞ ¹ Pn¹1 tanhðujtn¹1=Pn¹1Þ

1 ¹ ðYn¹1ðujÞ=Pn¹1Þ tanhðujtn¹1=Pn¹1Þ

� �2

: ð25Þ

Thus, rn can be calculated directly from the values of the reduced FNI function on thelast branch.

The proposed method of direct interpretation gives an equivalent solution withoutan initial guess. The algorithm can recover the true parameters from noise-free data(i.e. synthetic data calculated for the given set of parameters). The success of the directmethod is greatly affected by the noise contamination of measured data. This isbecause a small error made in the determination of the resistivity and thickness of a



layer at one step leads to the magnification of noise by the reduction equation (22) inthe next steps. For this reason, we have developed a method for checking the resistivityof the layers, using the apparent resistivity values defined by Basokur (1994). Despitethis control method, the fit between the measured data and the theoretical curvecomputed from the layer parameters solved by the method described is not sufficientlygood in most cases. However, the algorithm described for direct interpretation can beincluded in a standard 1D inversion program as a complementary method and can becalled as a subroutine by the interpreter to create an initial guess (e.g. Basokur 1993).Then the adjustment of the layer parameters in order to minimize the differencesbetween measured and theoretical data can be carried out using a damped least-squaresalgorithm for an over-determined system (e.g. Inman 1975; Jupp and Vozoff 1975;Ilkisik and Jones 1984; Pedersen and Hermance 1986; Pedersen and Rasmussen 1989;Meju and Hutton 1992; Meju 1992). Our inversion is carried out in the conventionalway by iteratively adjusting the layer parameters as described in the papers mentioned,but using the FNI formulations.

Practical strategy for very noisy data

The method described is not directly applicable to very noisy data due to the noiseamplification by the reduction equation at each reduction step. The method needs asmooth curve since the expressions (20) and (21) yield meaningless results if scattereddata are used. To overcome this problem, we adapt a DC electrical method used fordifferent purposes (Santini and Zambrano 1981, 1983; Niwas and Israil 1986).

This method involves the approximation of the measured data by a linearcombination of simple fitting functions

Y ¬ð fjÞ ¼X

m

i ¼1

bigð fj ;eiÞ; j ¼ 1; 2; . . . ; n; i ¼ 1; 2; . . . ;m; ð26Þ

where m and n are the number of fitting functions and the number of sample values ofthe FNI function, respectively. The coefficients ei determine the position of eachfunction along the horizontal frequency axis and are fixed depending on the frequencyrange and the number of fitting functions. Y ¬ð f Þ is an approximation of the measuredY ð f Þ function and is determined to within a desired precision. The unknowncoefficients bi (referred to as decomposition coefficients) are determined by solving asystem of linear equations with the help of the least-squares method, i.e. we require that

X

n

j ¼1

½Y ¬ð fjÞ ¹ Y ð fjÞÿ2 ¼ a minimum; ð27Þ

where Y ð fjÞ are the measured data. Substituting (26) in (27), we obtain

X

n

j ¼1

X

m

i ¼1

bigð fj ; eiÞ ¹ Y ð fjÞ

" #2

¼ a minimum: ð28Þ



The error energy is minimized if its partial derivatives with respect to the coefficients bi

equal zero. In this way, we obtain a symmetric normal linear system of m equations form unknowns which can be solved by the algorithm of Levinson (1947). The solutioncan be written in matrix notation as

B ¼ ðGTGÞ¹1GTY; ð29Þ

where B is the m × 1 matrix whose elements are the unknown coefficients bi , G is then × m matrix whose elements consist of the fitting functions and Y is the n × 1 matrixwhich represents the measured data.

Once the coefficients bi are determined, then the approximated values of Y ¬ð f Þ canbe constructed by the linear combination of the fitting functions using the knowncoefficients bi at any desired abscissa points. Thus

Y ¬ð fjÞ ¼X

m

i ¼1

bigð fj ; eiÞ; ð30Þ

or in matrix notation we can write

Y ¬ ¼ GB; ð31Þ

or substituting (29) in (31), we can construct the approximated function as

Y ¬ ¼ GðGTGÞ¹1GTY: ð32Þ

The procedure described differs from the conventional smoothing operation whichseeks a function passing through a given series of measurement points with the smallestpossible error. Here, we choose a fitting function which simulates the behaviour of theFNI function. By allowing a variable misfit, the interpreter can freely decrease orincrease the number of fitting functions in an interactive way in order to find anacceptable approximated curve from among many possible solutions. The procedure isequivalent to low-pass filtering in linear filter theory. However, the proposed methodhas an advantage in that the number of fitting functions can be decreased to find asmoother curve than the previous one or it can be increased to obtain a curve closer tothe sample values of the measured FNI function.

The solution of the differential equation for the electric field can be expressed as thesum of the exponential functions corresponding to the downgoing and upgoing waves(Levy, Oldenberg and Wang 1988). Therefore, for the decomposition of the FNIfunction, we chose the following fitting functions, which are similar to the functionsused by Levy et al. (1988) and Lee, McMechan and Aiken (1987) for the constructionof a pseudo-impulse response to obtain the seismic equivalent of MT data,

gð fj ; eiÞ ¼ exp ¹��

ifj =ei

q� �

: ð33Þ

By separating (33) into real and imaginary parts, we obtain

grð fj ; eiÞ ¼ exp ¹��

fj =2ei

q� �

cos��

fj =2ei

q� �

ð34Þ



and

giðfj ; eiÞ ¼ ¹ exp ¹��

fj =2ei

q� �

sin��

fj =2ei

q� �

: ð35Þ

The fitting functions gr and gi are used for the computation of the real and theimaginary parts of the FNI function. Thus the coefficients bi simulate pseudo-reflection coefficients. In order to carry out the numerical calculations, we set em ¼ fm,which is the lowest frequency in the measurement sequence, then the other ei

coefficients are calculated by distributing the fitting functions along the logarithmichorizontal axis by providing an equal number of fitting functions per decade. Thedecomposition of the FNI function, i.e. the calculation of the coefficients bi can becarried out separately for the real and imaginary parts, also different numbers ofcorresponding fitting functions can be used. If data are noisy, the following simpleexponential function is successful for the approximation of both the real and imaginaryparts of the measured FNI function:

gð fj ; eiÞ ¼ exp ¹��

fj =2ei

q� �

: ð36Þ

After the calculation of the coefficients bi by the approximation of the measured datausing a certain number of fitting functions interactively, we then reconstruct the FNIfunction at new abscissa values that give equally spaced data along the horizontal axiswith four sample points per decade. In other words, the approximated data areresampled. We also calculate sample values whose abscissa values satisfy the criterion(12) and permit the use of the expression (23) that calculates the resistivity of thelayers. The algorithm also permits extrapolation by reconstructing the FNI function atabscissa points beyond the frequency range used for the measurements. This is anoptional facility at the disposal of the interpreter.

Real data examples

Since the proposed method of direct interpretation can recover the true parametersfrom noise-free data, we shall not give an example which uses a theoretical data set.Figure 1 shows the real and imaginary parts of the FNI function converted from theapparent resistivity and phase data measured in the central part of the Danish Basin aspart of a study for the exploration of regional conductivity structure in SouthScandinavia (Thomsen et al. 1989). The real and imaginary parts of the FNI functionhave been transformed to their approximations by using 14 exponential fittingfunctions (solid curves in Fig. 1). These approximations permit resampling of the dataat abscissa values that satisfy criterion (12). The layer parameters are solved directly,using the equations given in the previous sections without any initial guess. Thedecision about the number of subsurface layers is made by the interpreter inputting thekey numbers indicating the beginning and end points of each branch of the FNIfunction. This example is interpreted assuming that the subsurface consists of six



layers. The measured and model FNI curves and layer parameters solved by directinterpretation are shown in Fig. 2.

Using the solved layer parameters as an initial guess, and proceeding with theiterative interpretation, a new set of layer parameters, shown in Fig. 3, is obtained. Themodel response and the measured data are also shown in Fig. 3 in the form of the FNIfunction. Figure 4 shows a conventional plot of the measured and theoretical data in theform of apparent resistivity and phase curves for the same model.

The data of the second example (Fig. 5) were measured at the same site in a boreholeconstructed by the Turkish Petroleum Company in the central part of the ThraceSedimentary Basin located on the north side of the Sea of Marmara, Turkey. Theborehole intersected the Pliocene–Oligocene boundary at a depth of 700 m and itended at a depth of 2205 m within Oligocene units. The Pliocene is composed ofconglomerates, siltstones and clay. The Oligocene consists of three differentdepositional sequences which are mainly interbedded shales, sandstones andmudstones. The boundaries of the Oligocene formations were intersected at depthsof 1720 and 2177 m, respectively.

A remote-reference MT station provided a high-quality data set except for a fewdiscrete values at high frequencies (Fig. 5). The imaginary part of the FNI data startstaking negative values at high frequencies and helps to distinguish the first branch. Atintermediate frequencies, the sign of the imaginary part changes to positive and twosemicircular-shaped arcs indicate a descending type of resistivity sequence. Theminimum between two arcs separates two branches. The negative discrete values ofthe imaginary part at low frequencies correspond to a highly resistive bottom layer. The



Figure 1. The real part (asterisks) and imaginary part (plus signs and rectangles) of the FNIcurve versus decreasing frequency measured in the Danish Basin. The solid curve is theapproximated FNI curve calculated using 14 exponential fitting functions.



Figure 2. The measured data and model FNI curve (solid curve). The layer parameters havebeen solved from the direct interpretation without any initial guess. (∧) shows the starting andending points of each branch. The computed layer parameters (resistivities (Q m) and depths(m)) are: r1 ¼ 7:5, r2 ¼ 0:31, r3 ¼ 0:28, r4 ¼ 24, r5 ¼ 145, r6 ¼ 49 and d1 ¼ 575, d2 ¼ 1129,d3 ¼ 1256, d4 ¼ 23938, d5 ¼ 46223.

Figure 3. The measured data and model FNI curve after proceeding with iterative inversion.The final interpreted model parameters (resistivities (Q m) and depths (m)) are: r1 ¼ 9:1,r2 ¼ 0:71, r3 ¼ 2:25, r4 ¼ 26, r5 ¼ 12452, r6 ¼ 34 and d1 ¼ 333, d2 ¼ 2103, d3 ¼ 3072,d4 ¼ 20157, d5 ¼ 118597.

ascending and descending branches of the real part support the decision made on thebehaviour of the imaginary part. Thus, the total FNI curve can be subdivided into fourbranches and consequently the geoelectric section is assigned to a five-layer case.

The FNI curve has been resampled using the fitting functions (34) and (35). After adecision about the number of layers, the parameters of the layered earth model havebeen solved using the method of direct interpretation. The theoretical curve computedfrom the solved layer parameters and the measured FNI data are shown in Fig. 5.Accepting these parameters as the first approximation to the model, we arrive the finalinterpreted geoelectric section (Fig. 6) by proceeding with the conventional dampedleast-squares inversion. The interpreted depths of 659 m and 2213 m are wellcorrelated with the formation boundaries. However, the boundary known from theborehole at a depth of 1710 m which subdivides the first and the second depositionalsequences of the Oligocene could not be resolved. The interpreted fourth boundary



Figure 4. Apparent resistivity and phase-p/4 display of the magnetotelluric data given in Fig. 3.



Figure 5. The measured data and model FNI curve (solid curve). The layer parameters havebeen solved from the direct interpretation without any initial guess. (∧ ) shows the starting andending points of each branch. The computed layer parameters (resistivities (Q m) and depths(m)) are: r1 ¼ 6:5, r2 ¼ 18, r3 ¼ 4:64, r4 ¼ 3:75, r5 ¼ 94 and d1 ¼ 188, d2 ¼ 682, d3 ¼ 2611,d4 ¼ 3639.

Figure 6. The measured data and model FNI curve after proceeding with iterative inversion.The final interpreted model parameters (resistivities (Q m) and depths (m)) are: r1 ¼ 4:03,r2 ¼ 11, r3 ¼ 5:07, r4 ¼ 2:97, r5 ¼ 74 and d1 ¼ 46, d2 ¼ 659, d3 ¼ 2213, d4 ¼ 3183.

plane at a depth of 3183 m overlies a high-resistivity layer and it might correspond tothe top of the limestone intercalated unit of the Upper Eocene which was observed inother boreholes in the region.

Conclusion

The direct inversion method described here produces an acceptable solution withoutany prior knowledge about the subsurface layering. The decision about the number oflayers is made by the interpreter by dividing the FNI curve into branches. Thealgorithm is fast and the interpreter obtains information about the possible subsurfacestructure and can use the model to initialize any conventional iterative inversionscheme. Suggestions have been made for implementing such a combined direct anditerative scheme. In this way, the interpreter can test many possible solutions beforeproceeding with the inversion process by adding or excluding extra layers to thegeoelectric section.

The success of the proposed method is directly related to the noise contamination ofthe data. However, the method usually produces a good initial guess so that the numberof iterations is substantially reduced and convergence of the inversion is guaranteed.The method described is used to obtain a first approximation to the model and it is notintended to replace the subsequent conventional methods of data interpretation. Thenon-uniqueness and equivalence effects should be examined in view of the final resultsobtained by the damped least-squares or any other inversion method.

Acknowledgements

We are grateful to Dr. M. Meju of Leicester University for critically reading themanuscript and we thank Dr. T.M. Rasmussen of Uppsala University for providing uswith the MT data set measured in the Danish Basin. Constructive comments by theassociate editor P. Valla and an anonymous referee were helpful in improving the paper.This work was supported by the Scientific and Technical Research Council of Turkey(TUBITAK) under grant no. YBAG-102.

Appendix

For a horizontally stratified earth model, consisting of n homogeneous and isotropiclayers of resistivities r1, r2; . . . ; rn and thicknesses t1, t2; . . . ; tn¹1, the FNI function canbe derived following the conventional method described in many text books (e.g.Kaufman and Keller 1981), i.e.

Y ¼ P1 tanhut1

P1þ arctanh

P1

P2tanh

ut2

P2þ . . . þ arctanh

Pn

Pn¹1

� ��

; ðA1Þ

where

u ¼ ðiqmÞ1=2 ¼ ðmpf Þð1 þ iÞ ðA2Þ



and

P ¼ r1=2:

The above expression can be written as a recurrence formula,

Yk ¼ Pk tanhutk

Pkþ arctanh

Ykþ1

Pk

� ��

ðA3Þ

or

Yk ¼

Ykþ1 þ Pk tanhutk

Pk

� �

1 þYkþ1

Pk

� �

tanhutk

Pk

� � ; ðA4Þ

for

k ¼ n ¹ 1; . . . ; 2; 1:

Starting from the electrical basement for which

Yn ¼ Pn;

the recursive application of (A3) gives the final Y function as

Y ¼ Y1:

References

Basokur A.T. 1984. A numerical direct interpretation of resistivity sounding using the Pekerismodel. Geophysical Prospecting 32, 1131–1146.

Basokur A.T. 1990. Microcomputer program for the direct interpretation of resistivity soundingdata. Computer and Geosciences 16, 587–601.

Basokur A.T. 1993. Interpretation of vertical electrical sounding data by combining direct anditerative methods. 2nd congress of the Hellenic Geophysical Union, Florina, Greece,Expanded Abstracts 62–72.

Basokur A.T. 1994. Definitions of apparent resistivity for the presentation of magnetotelluricsounding data. Geophysical Prospecting 42, 141–149.

Bostick F.X. 1977. A simple almost exact method of MT analysis. Workshop onElectrical Methods in Geothermal Exploration, U.S. Geological Survey, Contract No.14080001-8-359.

Cagniard L. 1953. Basic theory of the magnetotelluric method of geophysical prospecting.Geophysics 18, 605–653.

Fischer G., Schnegg P. A., Peguiron M. and Le Quang B. V. 1981. An analytic one-dimensionalmagnetotelluric inversion scheme. Geophysical Journal of the Royal Astronomical Society 67,257–278.

Ilkisik O.M. and Jones A.G. 1984. Statistical evaluation of MTand AMT methods applied to abasalt-covered area in southeastern Anatolia, Turkey. Geophysical Prospecting 32, 706–724.

Inmann J.R. 1975. Resistivity inversion with ridge regression. Geophysics 40, 798–817.



Johansen H.K. 1975. A man/computer interpretation system for resistivity sounding over ahorizontally stratified earth. Geophysical Prospecting 25, 667–691.

Jupp D.L.B. and Vozoff K. 1975. Stable iterative methods for the inversion of geophysical data.Geophysical Journal of the Royal Astronomical Society 42, 957–976.

Kaufman A.A. and Keller G.V. 1981. The Magnetotelluric Sounding Method. Elsevier SciencePublishing Society.

Koefoed O. 1970. A fast method for determining the layer distribution from the raised kernelfunction. Geophysical Prospecting 18, 564–570.

Lee S., McMechan G.A. and Aiken C.L.V. 1987. Phase-field imaging: The electromagneticequivalent of seismic migration. Geophysics 52, 678–693.

Levinson N. 1947. The Wiener RMS (root mean square) error criterion in filter design andprediction. Journal of Mathematics and Physics 25, 261–278.

Levy S., Oldenberg D. and Wang J. 1988. Subsurface imaging using magnetotelluric data.Geophysics 53, 104–117.

Meju M.A. 1992. An effective ridge regression procedure for resistivity data inversion.Computers and Geosciences 18, 99–118.

Meju M.A. and Hutton V.R.S. 1992. Iterative most-squares inversion: application tomagnetotelluric data. Geophysical Journal International 108, 758–766.

Niwas S. and Israil M.1986. Computation of apparent resistivities using an exponentialapproximation of kernel functions. Geophysics 51, 1594–1602.

Patella D. 1976. Interpretation of magnetotelluric resistivity and phase soundings over horizontallayers. Geophysics 41, 96–105.

Pedersen J. and Hermance J.F. 1986. Least-squares inversion of one-dimensional magnetotelluricdata: an assessment of procedures employed by Brown University. Surveys in Geophysics 8,187–231.

Pedersen L.B. and Rasmussen T.M. 1989. Inversion of magnetotelluric data. a non-linear least-squares approach. Geophysical Prospecting 37, 669–695.

Santini R. and Zambrano R.1981. A numerical method of calculating the kernel function fromSchlumberger apparent resistivity data. Geophysical Prospecting 29, 108–127.

Santini R. and Zambrano R. 1983. A generalized method to calculate standard curves forgeoelectrical soundings. Geoexploration 21, 93–103.

Schmucker U. 1970. Anomalies of Geomagnetic Variations in the Southwestern United States.Scripps Institute of Oceanography, Bulletin 13, University of California Press.

Schmucker U. 1987. Substitute conductors for electromagnetic response estimates. Pure andApplied Geophysics 125, 341–367.

Sen M.K., Bhattacharya B.B. and Stoffa P.L. 1993. Nonlinear inversion of resistivity soundingdata. Geophysics 58, 496–508.

Thomsen I.D., Jacobsen B.H., Rasmussen T.M. and Balling N. 1989, Magnetotelluricinvestigations in Jutland-Preliminary results from the Danish Basin. In: Basin Workshop,Aarhus 1989 (eds N. Balling, O.B. Nielsen, J.A. Korstgard and S.B. Nielsen), pp. 25–30.Proceedings of Basin Workshop held at Department of Earth Sciences, University of Aarhus.



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