Turkish Journal of Earth Sciences

http://journals.tubitak.gov.tr/earth/

Research Article

Turkish J Earth Sci

(2018) 27: 152-166

© TÜBİTAK

doi:10.3906/yer-1702-6

Resistivity inversion of transversely isotropic media

1,

2

Ertan PEKŞEN *, Türker YAS

Department of Geophysical Engineering, Faculty of Engineering, Kocaeli University, Kocaeli, Turkey

2

Department of Marine and Environmental Researches, General Directorate of Mineral Research & Exploration, Ankara, Turkey

1

Received: 08.02.2017

Accepted/Published Online: 27.12.2017

Final Version: 19.03.2018

Abstract: In this paper, we have suggested a finite difference algorithm for resistivity forward and inverse modeling with electrically

anisotropic media in applied geophysics. Electrical anisotropy affects the surface measurement in a fashion that may make interpretation

erroneous. This means that the resistivity section obtained by an inversion method that does not incorporate electrical anisotropy gives

the wrong subsurface structure. We used a classical multielectrode dataset of a profile for estimating not only horizontal but vertical

resistivity as well. Thus, the electrical anisotropy can be calculated. Finally, the finite difference mesh can be corrected by using the

estimated anisotropic coefficient. The result of the developed algorithm was verified with a 2-layered analytic solution. Furthermore, the

present method was also tested on a field dataset.

Key words: Electrical resistance tomography, electrical anisotropy, 2D resistivity modeling, 2D resistivity inversion, direct resistivity

method, transversely isotropic media

1. Introduction

Electrical conductivity of a geological formation may

vary in different directions depending on the formation’s

properties. This type of directional property of conductivity

gives rise to electrical anisotropy (Maillet, 1947). Ignoring

the electrical anisotropy of a geological formation or

assuming that the formation is isotropic may yield to a

misleading interpretation (Nguyen et al., 2007; Wiese et

al., 2009; Greenhalgh et al., 2010).

Electrical anisotropy detection in the field can be

determined by using a square array, since the square array

is more sensitive to electrical anisotropy in a geological

formation than the most commonly used Schlumberger

and Wenner arrays (Habberjam, 1972, 1975; Matias, 2002;

Yeboah-Forson and Whitman, 2014; Yeboah-Forson et

al., 2014). The second approach for determining electrical

anisotropy in the field based on the surface measurement

is an azimuthal resistivity measurement (Busby, 2000).

However, the square array requires more fieldwork than

the more commonly used electrode arrays to get a strike

direction and to determine electrical anisotropy of the

medium. If we need to estimate what the strike direction

is in an electrically anisotropic formation, we can apply the

square array (Yeboah-Forson and Whitman, 2014; YeboahForson et al., 2014). Otherwise, we often prefer the most

commonly used collinear arrays rather than a square one.

*Correspondence: ertanpeksen@kocaeli.edu.tr

152

In some situations, such as one we came across in an

archaeological site in Bathonea in Turkey, there was a shift

in depth estimation due to possible electrical anisotropy. To

show this misleading interpretation as an example, one can

compare inversion results obtained using the same dataset

with and without electrical anisotropy assumptions. To do

this, we compared both results. The datasets used in this

study were collected at an archaeological site in Turkey in

2013, during the field season. The dataset was then inverted

with the assumption of the earth as a 2-dimensional (2D)

isotropic and anisotropic model, respectively. Comparison

of the excavated site and inverted resistivity sections

suggests that electrical anisotropy affects the depth

information, which is directly related to the electrical

anisotropy coefficient as expected.

In this study, we considered the 2D direct current (DC)

method in anisotropic media. This method is also known

as electrical resistance tomography (ERT) or imaging

(ERI). The method used in the present study can also

be called 2.5D due to an infinite strike direction normal

to the resistivity profile. Here, we developed a 2D finitedifference (FD) code for forward and inverse modeling.

The 2D code validation was tested against an analytical

solution for the 1-dimensional DC method in a layered

anisotropic medium introduced by Pekşen et al. (2014).

The electrical anisotropy coefficient of a geological

formation can be estimated by using an inversion based

PEKŞEN and YAS / Turkish J Earth Sci

on the least-squares method. For this purpose, electrical

data as mean apparent resistivity values collected using

a multielectrode system along profiles on the surface

can be used. We then convert the data for estimating

vertical and horizontal resistivity of geological formations

in the corresponding field, based on surface electrical

measurements. In general, it is impossible to distinguish

horizontal and vertical resistivity values from a single

surface measurement without constraint, due to the

principle of equivalence. To overcome this difficulty, we

assume that the vertical resistivity value is greater than or

at least equal to the horizontal resistivity value.

Anisotropic inversion is not new in resistivity methods:

Pain et al. (2003), Herwanger et al. (2004), LaBrecque

et al. (2004), Kim et al. (2006), and Wiese et al. (2015)

have studied the inversion method within electrically

anisotropic media for different types of applications.

LaBrecque et al. (2004) studied 3D resistivity inversion

using surface DC data. In our inversion process, we solve

the vertical and horizontal resistivity values of model mesh

iteratively. After we estimate the anisotropy coefficient, the

last step of our algorithm modifies the mesh for the depth

correction. The proposed method was tested on synthetic

models and the field dataset successfully.

2. 2D DC FD modeling

2.1. Forward modeling in anisotropic media

Electrical conductivities of formations can vary in different

directions. In such formation, Ohm’s law can be written as:

(1)

where is the current density (A/m2), is the electric field

(V/m), and is the conductivity tensor (S/m), which has

the following form:

(2)

The conductivity tensor is a positive-definite and

symmetric matrix with respect to the main diagonal in

the Cartesian coordinate (Marti, 2014). In this paper, we

used the right-handed coordinate system that is positive

downward. In the anisotropic earth model, electrical

current is not parallel to the electric field. However, if the

medium is isotropic, the conductivity is a scalar. In an

isotropic model, current flows in the same direction as

the applied electric field. In sedimentary sequences, the

conductivity can be assumed to be equal in the x and y

directions, while it is different in the z direction. In our

assumption, the formation is generally isotropic in the

horizontal plane (x–y). However, it is anisotropic in the

vertical plane (x–z). This kind of anisotropic model is

known as transverse isotropy in the vertical direction

(Figure 1). The conductivity tensor given by Eq. (2) can

be rotated through the Euler angles, which makes the

principal axes and the recording frame coincide with

each other. This diagonalization can be applied using the

following:

(3)

where R is the Euler matrix. The superscript T stands for

a transposition of the Euler matrix. Note that the Euler

matrix is an orthonormal matrix. Thus, the inverse of the

Euler matrix is equal to its transposition, which is RT=R-1.

Based on our assumption, the conductivity tensor can

be written as:

(4)

where the prime denotes the principal axis values. The

conductivity tensor can be easily transferred from the

primed to the unprimed or the unprimed to the primed

coordinate system through the Euler matrix without any

error with respect to the mathematical point of view, since

the Euler matrix is orthonormal. To perform this rotation,

Eq. (3) can be used. However, this requires a priori

information about the Euler angles, such as strike and dip

directions. In this study, we assume that the Euler angles

are known. More specifically, the strike and dip angles are

zeros, which means that the primed and the unprimed

coordinate systems coincide.

For anisotropic media, the forward response can be

obtained by a numerical solution of the following equation:

(5)

where v(x,y,z) is the potential field distribution in a 3D

domain, I is a point source with its location indicated by

subindex s,

is the conductivity tensor, and δ is the

Dirac delta function.

To develop the forward response of the isotropic

model, we used the theory introduced by Dey and

Morrison (1979). They solved Poisson’s equation in a 2D

arbitrarily shaped model by the FD method, assuming the

earth to be isotropic while employing a point source and

mixed boundary condition. In this paper, we follow the

same methodology for the forward modeling.

To solve Poisson’s equation using the FD method, we

need to discretize the corresponding domain in the x and

z directions (Figure 2). This discretization determines the

model size as well. In general, a finer mesh design gives

more accurate results. We cannot continuously increase

the number of cells in the x and z directions, since the

computational time increases (Pidlisecky and Knight,

2008). A typical value of the matrix dimension is 100 × 20

in our cases. This coarse mesh design can also be divided

into 2, 4, 8, etc. Our experience suggests that using 4

divisions of a single cell gives much better results, and it is

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PEKŞEN and YAS / Turkish J Earth Sci

Transverse isotropy in the vertical direction

σh

electrodes

0

z(m)

σh

-5

-10

40

σv

20

0

y(m)

-20

-40

-40

-30

-20

-10

10

0

x(m)

20

30

40

Figure 1. Sketch shows transversely isotropic medium. Red and blue colors represent various formations

of the corresponding earth models. Horizontal conductivities are in the x and y directions (σh). Vertical

conductivity is in the vertical direction (σv). Arrows show the location of potential and current electrodes

along the measuring profile.

X

Z

i=1

ΔZ1

ΔX1

i=2

ΔX2

ΔXN-1

i=N

j=1

j=2

ΔZM-1

j=M

Figure 2. The rectangular grid displays the discretization of the finite difference method.

suitable for us with respect to accuracy and computational

time. Thus, a cell was divided into 2 or 4 subcells to get a

finer grid in the 2D conductivity domain in this study.

Once the mesh design was completed, we assigned each

intersection of the discretized domain with a conductivity

σn(x,z) value (Figure 3a). Therefore, a capacitance matrix

can be set up properly by employing a proper boundary

condition such as Neumann, Dirichlet, or mixed. Using

matrix notation, a discrete system of equation can be

written as:

Cv=s

(6)

where C is the capacitance matrix, v is a vector with

unknown potential values in the modeling domain,

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and s is a point source vector. The capacitance matrix

depends on the model geometry and physical properties

of the domain. Eq. (5) can be solved by the Cholesky

decomposition method (LU), or it may be solved with

other methods, such as QR factorization (Haber and

Oldenburg, 2000; Candansayar, 2008). For each source

position, Eq. (5) must be solved.

Even though we assume that a geological subsurface

model is 2D, the distribution of potential is in the 3D

domain. Domain transformation from 3D to 2D can be

achieved by integration in the Fourier domain (Dey and

Morrison, 1979; Pidlisecky and Knight, 2008; Xu et al.,

2000). Finally, from this 2D model’s potential values,

PEKŞEN and YAS / Turkish J Earth Sci

ΔXi-1

ΔXi

ΔXi-1

i,j-1

ΔZj-1

ΔZj

i-1,j

i,j

σi-1,j

i,j-1

σi,j-1

σi-1,j-1

ΔXi

i+1,j

σi,j

λΔZj-1

λΔZj

i-1,j

v

i-1,j-1

σ

i,j

v

σi,j-1

h

s i-1,j

i+1,j

σi,jh

v

i,j

σ

σvi-1,j

i,j+1

i,j+1

(a)

h

σi,j-1

h

σi-1,j-1

(b)

Figure 3. (a) A rectangular grid discretization of a cell, assuming the earth to be electrically isotropic. (b) A rectangular grid discretization

of a cell, assuming the earth to be electrically anisotropic.

one can calculate apparent resistivity for any electrode

configuration. Apparent resistivity calculations can be

classically achieved by multiplication of a geometric

factor, with potential difference normalized by injected

currents (Dey and Morrison, 1979). Furthermore, we

also normalize these apparent resistivity values by a

homogeneous response in the model introduced by

Marescot et al. (2006) as:

,

(7)

where Δv is the potential differences measured in the field,

ΔvH is the potential differences in a homogeneous model,

and ρH is the resistivity value of a homogeneous model,

e.g., 1 ohm-m (Marescot et al., 2006).

In our anisotropic model, we assume that the earth

consists of arbitrary shapes with electrical anisotropy.

Figure 3b shows a cell in an anisotropic medium. Based

on our assumption that the horizontal conductivity values

are in the x direction and the vertical conductivity values

are in the z direction, the vertical conductivity values

are always smaller than the horizontal value in each cell

(Maillet, 1947). The geometric mean of the conductivity

values is given as (Maillet, 1947):

,

(8)

where σh(x,z) and σv(x,z) are conductivities in the horizontal

and vertical directions, respectively. The subindices h and

v stand for horizontal and vertical directions. Note that

. From these 2 conductivity values, one

can calculate the electrical anisotropy coefficient of each

cell of the medium. Thus, the anisotropy coefficient can

be calculated with the following equation (Maillet, 1947;

Grant and West, 1965):

(9)

Laboratory measurements show

in most

sedimentary rocks. The range of the coefficient of the

anisotropy value is between 1.0 and 7.5 (Carmichael,

1989; Negi and Saraf, 1989). Without electrical anisotropy

assumption, interpretation may be erroneous (Nguyen

et al., 2007). Electrical anisotropy should be considered

even when the anisotropy coefficient is

(Wiese et

al., 2009). Electrical anisotropy can even be detected in

alluvium (Greenhalgh et al., 2010).

Based on our assumption given above, the forward

response of an isotropic earth model can be extended to

an anisotropic earth model by replacing the conductivity

tensor with the geometric mean of the conductivity values.

Thus, we have:

(10)

where v(x,y,z) is the potential field distribution in the 3D

domain, I is a point source with its location indicated by a

subindex s, σn(x,y) is the geometric mean of conductivity

values calculated by Eq. (8), and δ is the Dirac delta function.

The FD method requires a mesh designed by dividing

rectangular cells in the region as mentioned previously.

The corresponding 2D region consists of 2 dimensions

(assuming that the conductivity does not change along the

y direction in a Cartesian coordinate system). This region

can be divided into Nx by Mz cells (Figure 2). Here, Nx is the

number of cells in the x direction with step size Δx. Similarly,

Mz is the number of cells in the z direction with step size

Δz. In cases where electrical anisotropy exists, the vertical

step size is divided by the electrical anisotropy coefficient

with Δz / λ. However, we do not know initially what the

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anisotropy coefficients are. Thus, we need to determine

the anisotropy coefficient so that the mesh design can then

be corrected. Our inversion algorithm iteratively finds

not only horizontal but also vertical resistivity values. At

the end of the inversion process, we also need to calculate

electrical anisotropy coefficients by using Eq. (9) so that

the electrical anisotropy correction can be employed. Note

that any type of FD discretization requires mesh design as

mentioned above. In our model, the mesh does not change

during iteration as usual. The pseudosection is represented

by using median depth of investigation (Edwards, 1977;

Loke, 2016). This is a crucial point: if electrical anisotropy

exists in a geological formation, interpretation using 2D

inversion code may be erroneous without considering

anisotropy.

We developed a new inversion code by using MATLAB.

The code can be used for inverting resistivity sections of

2D isotropic and anisotropic earth models. Note that here

we assume that the measurement profile and layers of the

formation are parallel to each other. The newly developed

forward code was validated against 1D analytical solutions

on anisotropic examples (Pekşen et al., 2014). Comparison

of the results for an analytic solution for a 2-layered earth

model and a 2D DC FD anisotropic model are illustrated

in Figure 4. The results are very similar. The purpose of

numerical comparison with analytic results is to be a

useful control for numerical modeling (Greenhalgh et al.,

2009a.; Yan et al., 2016).

2.2. Inversion

Estimation of subsurface conductivity distribution

from measurement requires inversion. There are many

algorithms and methods available in the literature for this

purpose (e.g., Pain et al., 2003; Herwanger et al., 2004;

LaBrecque et al., 2004; Kim et al., 2006; Greenhalgh et al.,

2009b; Wiese et al., 2015). The forward 2D DC FD method

can be formulated as the following matrix notation:

d=G(m)

(11)

where G is a nonlinear forward operator, m is a

parameter vector, and d is an observation vector (Meju,

1994). Various approximations can be followed in order

to estimate model parameters from observations. In

this study, we applied the singular value decomposition

(SVD) method for solving Eq. (11). The sensitivity matrix

of the system is calculated for each iteration to solve Eq.

(11). The partial derivative of the sensitivity matrix was

calculated by using the method proposed by Tripp et

al. (1984). The sensitivity matrix of the model consists

of the partial derivative of the model parameters; the

dimensions of the matrix are N by M. N is the number

of measurements and M is the number of parameters.

Depending on the condition (N > M, N < M, or N=M),

we may meet some difficulties while inverting the matrix.

Problems based on the number of parameters and

observation points can be categorized as follows: when N

> M, it is called an overdetermined problem; when N=M,

the problem is known as evenly determined; finally, when

Figure 4. Comparison of the results of an analytic solution for a 2-layered earth model and a 2D DC FD anisotropic model.

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N < M, it is called an underdetermined one (Menke, 1989).

This problem is ill-posed from the mathematical point

of view; the solution may not be unique. These kinds of

inversion problems can be coped with by using different

algorithms, such as a damped-least squares method (Meju,

1994), Tikhonov regularization (Tikhonov and Arsenin,

1977; Zhdanov, 2002), or the Occam method (DeGrootHedlin and Constable, 1990; Aster et al., 2005).

In this study, we used the SVD method. The details of

the SVD method can be found in the work of Golub and

Loan (1996). The problem can be solved iteratively with

the SVD algorithm given by Meju (1994) in the following

matrix notation:

(12)

,

(13)

where Δp is a vector consisting of model parameters, V and

U are orthonormal matrices, λi is the ith eigenvalue of the

system, and ε is a constant. m stands for a parameter vector

and k denotes an iteration number. To clarify the inversion

steps we have used here, flowcharts of the 2 algorithms for

isotropic and anisotropic media are shown in Figures 5 and

6. We have followed these inversion steps. As initial values,

we use the geometric mean of the apparent resistivities of a

profile. Since the initial value is a single apparent resistivity

value, it is a sort of homogeneous model for isotropic cases.

As for the anisotropic model, the initial value we begin

with for an electrical anisotropy coefficient is 1.1. This is a

kind of threshold value for inversion with the anisotropic

model. It can be between 1.1 and 3. Without a small

perturbation of anisotropy, the method suffers in finding

the correct model. Using a classical inversion algorithm,

the horizontal and vertical resistivity values cannot be

distinguished from surface resistivity measurement, due to

the principle of equivalence. This can also be seen in Eq. (8).

To overcome this problem, we use the constraints

during iterations. We also apply a Laplacian

and

filter operator to the data as a smoothing factor (Sasaki,

1989). Note that the problem is an underdetermined one,

since the number of parameters is smaller than the number

of observations. In spite of these difficulties, we have found

the subsurface resistivity distribution of the isotropic and

anisotropic models. Once the inversion algorithm reaches a

predetermined RMS threshold value (e.g., 3%), the process

ends. For an anisotropic model, we further calculate the

electrical anisotropy coefficient, and then we finalize the

process by incorporating the vertical shift due to electrical

anisotropy into the mesh discretization by Δz/λ.

3. Micro- and macroanisotropy

In general, electrical anisotropy can be classified as macro

or micro. For geological studies, macroanisotropy exists in

homogeneous stratified rocks. The presence of successive

parallel and different conductivities with long linear

structures causes macroanisotropy. The layered internal

structure of shale results in microanisotropy (Negi and

Saraf, 1989). Electrical anisotropy is a scaling problem.

An anisotropic layer can equally be represented by a

stack of isotropic layers (Maillet, 1947). Furthermore, a

number of isotropic layers can cause pseudoanisotropy.

Elongated small particles in the sedimentation such as

clay or sand prefer certain directions, which also gives

rise to electrical anisotropy (Kunz and Moran, 1958;

Rey and Jongmans, 2007).

4. Numerical example

Figure 7 shows a synthetic 2D model. The background

of the model is anisotropic. The anisotropy coefficient is

2 with

. The resistivity of the buried

body is 200 ohm-m in the horizontal and vertical

directions. Therefore, it is an isotropic block. The depth

of the upper side of the block is 1.5 m. The inverted

resistivity sections with isotropic and anisotropic

models are given in Figures 8a and 8b.

The white color shows the exact position of the block

in the middle of Figures 8a and 8b. The vertical shift

can be seen with and without incorporating electrical

anisotropy when comparing both results. Based on

Figures 8a and 8b, one can state that inversion without

electrical anisotropy can make interpretation erroneous

regarding the depth of the anomalous body’s location.

Inversion results of the isotropic and anisotropic

assumptions give some differences in the vertical body

position. These differences should be related to the

anisotropy coefficient of the medium. Thus, we can say

that if any geological formation has electrical anisotropy

properties, we should use the 2D inversion process with

incorporated electrical anisotropy.

The second synthetic model consists of 2 electrically

anisotropic blocks embedded in a model. The exact

model is shown in Figure 9. The anisotropy coefficients

of the blocks are 2 and 2.5, respectively. We assume that

the background of the model is isotropic in this synthetic

example. Figures 10a and 10b show the inverted results

with and without anisotropy assumptions. It can be

seen from Figures 10a–10c that there is no significant

vertical shift in the depth, since the background of the

model is isotropic. Our inversion algorithm successfully

found the upper block close to the surface; however, the

lower block was not located precisely. Comparison of

both inverted results with and without incorporating

anisotropy shows a relatively small enhancement of the

result with anisotropy (see Figures 10a and 10b).

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Figure 5. The flow chart of inversion steps, assuming the earth to be electrically isotropic.

5. Field example

Bathonea is a prehistoric city in İstanbul. Figure 11 shows

the location of the Küçükçekmece Lake district in İstanbul.

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Many archaeological objects and structures in Bathonea

have been found and identified, such as harbor structures,

roads, buildings, and smaller artifacts. The periods of these

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Figure 6. The flow chart of inversion steps, assuming the earth to be electrically anisotropic.

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0

0

10

20

30

Depth (m)

ρh=200Ω.m

2

λ =1

3

5

6

50

1.5 m

1

4

40

ρh=2.5Ω.m

ρv=10Ω.m

λ =2

ρvv =200Ω.m

2m

5m

7

Figure 7. A synthetic 2D resistivity model is shown. There is a rectangular body (electrically isotropic) in the

middle of the model. The background of the resistivity model is assumed to be electrically anisotropic.

Figure 8. (a) Estimated 2D model obtained from 2D inversion; the model is assumed to be electrically isotropic. The

white rectangular shape in the inverted resistivity section shows the exact location of the body. (b) Estimated 2D model

obtained from 2D inversion assuming the model to be electrically anisotropic. The white rectangular shape in the inverted

resistivity section shows the exact location of the body.

findings are Hellenistic, Roman, and Byzantine. Excavations

and research in the ancient city continue to the present day

(Aydıngün, 2007). More information about and photos of

Bathonea can be found at http://www.bathonea.org.

Here, we show 3 ERT profiles measured with 30

electrodes. Each profile was collected with a multielectrode

instrument. The profile length was 58 m, with 2-m electrode

spacing. As an array, we used the Wenner alpha electrode

configuration. Apparent resistivity pseudosections are

displayed in Figures 12a–12c for each profile. Each profile

was inverted by using the presented inversion methods

with and without electrical anisotropy assumptions.

Inverted resistivity sections are given in Figures 13a–13c

without electrical anisotropy assumption. Figures 14a–14c

shows the inverted resistivity sections for each profile

160

incorporating electrical anisotropy. The vertical shift can

be seen when we compare Figures 13a–13c and 14a–14c.

Figure 15 shows the field before and after excavations.

At the beginning, we interpreted the resistivity section

without considering electrical anisotropy. We found some

anomalies that were related to structures such as ancient

remnants, possible ancient water channels, and some

remains of ancient walls. However, after excavation, we

realized that there was a difference between the interpreted

depth based on 2D inversion without electrical anisotropy

and the archaeological findings. The application of the 2D

inversion method with electrical anisotropy gave much

better results in this field study.

We estimated the location of the archaeological wall

based on our inversion results. The archaeologist found

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0

10

0

20

Depth (m)

1

1.4m

2

3

6

40

50

ρ =100Ω.m

2.9m

h

ρϖv=625Ω.m λ =2.5

5.5m

4

5

30

1.1m

ρ =100Ω.m

h

ρv =100Ω.m

λ =1

ρ =10Ω.m

h

ρϖv=40Ω.m

2.5m

λ =2.0

5.5m

7

Figure 9. A synthetic 2D resistivity model is shown. There are 2 rectangular bodies with electrical

anisotropy embedded in the model. The background of the resistivity model is assumed to be

electrically isotropic.

Isotropic Section

Figure 10. (a) Estimated 2D model obtained from 2D inversion; the model is assumed to be electrically isotropic. The white rectangular

shapes in the inverted resistivity section show the exact location of the bodies. (b) Estimated 2D model obtained from 2D inversion

assuming that the model is electrically anisotropic. The white rectangular shapes in the inverted resistivity section show the exact

location of the body. (c) The exact model.

the wall and some other archaeological objects as we

expected. The depth of the wall was approximately between

0.75 m and 2.00 m. Before any excavations, it is vitally

important to know the exact location of any buried objects

in an archaeological area with respect to the depth. Our

numerical and field experiences suggest that if electrical

anisotropy exists in an area, we should always expect that

the depth of layer boundaries or buried objects appearing

on an inverted resistivity section is closer to the earth’s

surface than what the actual depth is.

6. Conclusions

In this paper, we have developed and investigated an

inversion method in electrically anisotropic media. In

general, we do not expect electrical anisotropy to have

an effect on the data acquired in an archaeological area.

However, the background formation of archaeological

objects can be electrically anisotropic due to sedimentation.

Some small elongated particles prefer a certain direction,

which is a well-known fact.

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Küçükçekmece

Lake

Küçükçekmece

Lake

Profile-2

Profile-3

Profile-1

Figure 11. Location of the study area in Küçükçekmece district in Turkey. A circle in the upper left corner of the figure indicates the

study area in Turkey. The aerial photo in the background shows Küçükçekmece Lake (Image Landsat Data SIO, NOAA, US Navy, NGA,

GEBCO © 2016 Başarsoft, Google Earth). Electrical resistivity tomography (ERT) profiles are illustrated on the right of the figure as

profiles 1–3 (http://www.google.com/maps).

Figure 12. Pseudosections of profile-1 (a), profile-2 (b), and profile-3 (c).

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Figure 13. Estimated 2D model obtained from 2D inversion assuming electrically isotropic earth for profile-1 (a), profile-2 (b), and

profile-3 (c). The lines with white in the sections indicate that the areas were excavated by archaeologists.

Figure 14. Estimated 2D model obtained from 2D inversion assuming electrically anisotropic earth for profile-1 (a), profile-2 (b), and

profile-3 (c). The lines with white in the sections indicate that the areas were excavated by archaeologists.

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Lake

Lake

Profile-2

Profile-3

Profile-1

Lake

Figure 15. Photo of the ERT profiles in the field before and after excavations next to Küçükçekmece Lake. There are some archaeological

findings shown: wall and channels (to see more pictures: http://www.bathonea.org).

The study shows that the depth estimated based on

inversion without electrical anisotropy can be erroneous.

Thus, interpreting the data with consideration of electrical

anisotropy gives much better results regarding the actual

depth of archaeological finds.

When considering geological formation complexity, it

is difficult to say that a formation is electrically isotropic.

To enhance the interpretation of an ERT profile, we may

incorporate the electrical anisotropy effect to the data.

We show that it is possible to estimate the anisotropy

coefficient of any cells in the FD mesh. The mesh can

then be corrected by using the anisotropy coefficient.

This can be done after inversion is completed. After the

vertical mesh step size is corrected by dividing all of them

by the electrical anisotropy coefficient, we finally get the

corrected mesh. From our synthetic examples and field

164

study, we can say that the proposed method can be useful

for interpretation of ERT data. This approximation was

successfully tested on synthetic and field datasets. The

method we used here requires running inversion codes

twice. If one finds the two sets of results to be very close

to each other, one may say that the medium is isotropic.

Otherwise, the medium may be electrically anisotropic.

Acknowledgments

We thank Prof Dr Şerif Barış for permission to use his

data. We also thank Associate Prof Dr Şengül G Aydıngün

for providing us with archeological information from

Bathonea and for her support during the fieldwork. We

appreciate the efforts of Demirkan Baylar, Erdinç Duman,

and Güngör Doğan during the fieldwork.

PEKŞEN and YAS / Turkish J Earth Sci

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http://journals.tubitak.gov.tr/earth/

Research Article

Turkish J Earth Sci

(2018) 27: 152-166

© TÜBİTAK

doi:10.3906/yer-1702-6

Resistivity inversion of transversely isotropic media

1,

2

Ertan PEKŞEN *, Türker YAS

Department of Geophysical Engineering, Faculty of Engineering, Kocaeli University, Kocaeli, Turkey

2

Department of Marine and Environmental Researches, General Directorate of Mineral Research & Exploration, Ankara, Turkey

1

Received: 08.02.2017

Accepted/Published Online: 27.12.2017

Final Version: 19.03.2018

Abstract: In this paper, we have suggested a finite difference algorithm for resistivity forward and inverse modeling with electrically

anisotropic media in applied geophysics. Electrical anisotropy affects the surface measurement in a fashion that may make interpretation

erroneous. This means that the resistivity section obtained by an inversion method that does not incorporate electrical anisotropy gives

the wrong subsurface structure. We used a classical multielectrode dataset of a profile for estimating not only horizontal but vertical

resistivity as well. Thus, the electrical anisotropy can be calculated. Finally, the finite difference mesh can be corrected by using the

estimated anisotropic coefficient. The result of the developed algorithm was verified with a 2-layered analytic solution. Furthermore, the

present method was also tested on a field dataset.

Key words: Electrical resistance tomography, electrical anisotropy, 2D resistivity modeling, 2D resistivity inversion, direct resistivity

method, transversely isotropic media

1. Introduction

Electrical conductivity of a geological formation may

vary in different directions depending on the formation’s

properties. This type of directional property of conductivity

gives rise to electrical anisotropy (Maillet, 1947). Ignoring

the electrical anisotropy of a geological formation or

assuming that the formation is isotropic may yield to a

misleading interpretation (Nguyen et al., 2007; Wiese et

al., 2009; Greenhalgh et al., 2010).

Electrical anisotropy detection in the field can be

determined by using a square array, since the square array

is more sensitive to electrical anisotropy in a geological

formation than the most commonly used Schlumberger

and Wenner arrays (Habberjam, 1972, 1975; Matias, 2002;

Yeboah-Forson and Whitman, 2014; Yeboah-Forson et

al., 2014). The second approach for determining electrical

anisotropy in the field based on the surface measurement

is an azimuthal resistivity measurement (Busby, 2000).

However, the square array requires more fieldwork than

the more commonly used electrode arrays to get a strike

direction and to determine electrical anisotropy of the

medium. If we need to estimate what the strike direction

is in an electrically anisotropic formation, we can apply the

square array (Yeboah-Forson and Whitman, 2014; YeboahForson et al., 2014). Otherwise, we often prefer the most

commonly used collinear arrays rather than a square one.

*Correspondence: ertanpeksen@kocaeli.edu.tr

152

In some situations, such as one we came across in an

archaeological site in Bathonea in Turkey, there was a shift

in depth estimation due to possible electrical anisotropy. To

show this misleading interpretation as an example, one can

compare inversion results obtained using the same dataset

with and without electrical anisotropy assumptions. To do

this, we compared both results. The datasets used in this

study were collected at an archaeological site in Turkey in

2013, during the field season. The dataset was then inverted

with the assumption of the earth as a 2-dimensional (2D)

isotropic and anisotropic model, respectively. Comparison

of the excavated site and inverted resistivity sections

suggests that electrical anisotropy affects the depth

information, which is directly related to the electrical

anisotropy coefficient as expected.

In this study, we considered the 2D direct current (DC)

method in anisotropic media. This method is also known

as electrical resistance tomography (ERT) or imaging

(ERI). The method used in the present study can also

be called 2.5D due to an infinite strike direction normal

to the resistivity profile. Here, we developed a 2D finitedifference (FD) code for forward and inverse modeling.

The 2D code validation was tested against an analytical

solution for the 1-dimensional DC method in a layered

anisotropic medium introduced by Pekşen et al. (2014).

The electrical anisotropy coefficient of a geological

formation can be estimated by using an inversion based

PEKŞEN and YAS / Turkish J Earth Sci

on the least-squares method. For this purpose, electrical

data as mean apparent resistivity values collected using

a multielectrode system along profiles on the surface

can be used. We then convert the data for estimating

vertical and horizontal resistivity of geological formations

in the corresponding field, based on surface electrical

measurements. In general, it is impossible to distinguish

horizontal and vertical resistivity values from a single

surface measurement without constraint, due to the

principle of equivalence. To overcome this difficulty, we

assume that the vertical resistivity value is greater than or

at least equal to the horizontal resistivity value.

Anisotropic inversion is not new in resistivity methods:

Pain et al. (2003), Herwanger et al. (2004), LaBrecque

et al. (2004), Kim et al. (2006), and Wiese et al. (2015)

have studied the inversion method within electrically

anisotropic media for different types of applications.

LaBrecque et al. (2004) studied 3D resistivity inversion

using surface DC data. In our inversion process, we solve

the vertical and horizontal resistivity values of model mesh

iteratively. After we estimate the anisotropy coefficient, the

last step of our algorithm modifies the mesh for the depth

correction. The proposed method was tested on synthetic

models and the field dataset successfully.

2. 2D DC FD modeling

2.1. Forward modeling in anisotropic media

Electrical conductivities of formations can vary in different

directions. In such formation, Ohm’s law can be written as:

(1)

where is the current density (A/m2), is the electric field

(V/m), and is the conductivity tensor (S/m), which has

the following form:

(2)

The conductivity tensor is a positive-definite and

symmetric matrix with respect to the main diagonal in

the Cartesian coordinate (Marti, 2014). In this paper, we

used the right-handed coordinate system that is positive

downward. In the anisotropic earth model, electrical

current is not parallel to the electric field. However, if the

medium is isotropic, the conductivity is a scalar. In an

isotropic model, current flows in the same direction as

the applied electric field. In sedimentary sequences, the

conductivity can be assumed to be equal in the x and y

directions, while it is different in the z direction. In our

assumption, the formation is generally isotropic in the

horizontal plane (x–y). However, it is anisotropic in the

vertical plane (x–z). This kind of anisotropic model is

known as transverse isotropy in the vertical direction

(Figure 1). The conductivity tensor given by Eq. (2) can

be rotated through the Euler angles, which makes the

principal axes and the recording frame coincide with

each other. This diagonalization can be applied using the

following:

(3)

where R is the Euler matrix. The superscript T stands for

a transposition of the Euler matrix. Note that the Euler

matrix is an orthonormal matrix. Thus, the inverse of the

Euler matrix is equal to its transposition, which is RT=R-1.

Based on our assumption, the conductivity tensor can

be written as:

(4)

where the prime denotes the principal axis values. The

conductivity tensor can be easily transferred from the

primed to the unprimed or the unprimed to the primed

coordinate system through the Euler matrix without any

error with respect to the mathematical point of view, since

the Euler matrix is orthonormal. To perform this rotation,

Eq. (3) can be used. However, this requires a priori

information about the Euler angles, such as strike and dip

directions. In this study, we assume that the Euler angles

are known. More specifically, the strike and dip angles are

zeros, which means that the primed and the unprimed

coordinate systems coincide.

For anisotropic media, the forward response can be

obtained by a numerical solution of the following equation:

(5)

where v(x,y,z) is the potential field distribution in a 3D

domain, I is a point source with its location indicated by

subindex s,

is the conductivity tensor, and δ is the

Dirac delta function.

To develop the forward response of the isotropic

model, we used the theory introduced by Dey and

Morrison (1979). They solved Poisson’s equation in a 2D

arbitrarily shaped model by the FD method, assuming the

earth to be isotropic while employing a point source and

mixed boundary condition. In this paper, we follow the

same methodology for the forward modeling.

To solve Poisson’s equation using the FD method, we

need to discretize the corresponding domain in the x and

z directions (Figure 2). This discretization determines the

model size as well. In general, a finer mesh design gives

more accurate results. We cannot continuously increase

the number of cells in the x and z directions, since the

computational time increases (Pidlisecky and Knight,

2008). A typical value of the matrix dimension is 100 × 20

in our cases. This coarse mesh design can also be divided

into 2, 4, 8, etc. Our experience suggests that using 4

divisions of a single cell gives much better results, and it is

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PEKŞEN and YAS / Turkish J Earth Sci

Transverse isotropy in the vertical direction

σh

electrodes

0

z(m)

σh

-5

-10

40

σv

20

0

y(m)

-20

-40

-40

-30

-20

-10

10

0

x(m)

20

30

40

Figure 1. Sketch shows transversely isotropic medium. Red and blue colors represent various formations

of the corresponding earth models. Horizontal conductivities are in the x and y directions (σh). Vertical

conductivity is in the vertical direction (σv). Arrows show the location of potential and current electrodes

along the measuring profile.

X

Z

i=1

ΔZ1

ΔX1

i=2

ΔX2

ΔXN-1

i=N

j=1

j=2

ΔZM-1

j=M

Figure 2. The rectangular grid displays the discretization of the finite difference method.

suitable for us with respect to accuracy and computational

time. Thus, a cell was divided into 2 or 4 subcells to get a

finer grid in the 2D conductivity domain in this study.

Once the mesh design was completed, we assigned each

intersection of the discretized domain with a conductivity

σn(x,z) value (Figure 3a). Therefore, a capacitance matrix

can be set up properly by employing a proper boundary

condition such as Neumann, Dirichlet, or mixed. Using

matrix notation, a discrete system of equation can be

written as:

Cv=s

(6)

where C is the capacitance matrix, v is a vector with

unknown potential values in the modeling domain,

154

and s is a point source vector. The capacitance matrix

depends on the model geometry and physical properties

of the domain. Eq. (5) can be solved by the Cholesky

decomposition method (LU), or it may be solved with

other methods, such as QR factorization (Haber and

Oldenburg, 2000; Candansayar, 2008). For each source

position, Eq. (5) must be solved.

Even though we assume that a geological subsurface

model is 2D, the distribution of potential is in the 3D

domain. Domain transformation from 3D to 2D can be

achieved by integration in the Fourier domain (Dey and

Morrison, 1979; Pidlisecky and Knight, 2008; Xu et al.,

2000). Finally, from this 2D model’s potential values,

PEKŞEN and YAS / Turkish J Earth Sci

ΔXi-1

ΔXi

ΔXi-1

i,j-1

ΔZj-1

ΔZj

i-1,j

i,j

σi-1,j

i,j-1

σi,j-1

σi-1,j-1

ΔXi

i+1,j

σi,j

λΔZj-1

λΔZj

i-1,j

v

i-1,j-1

σ

i,j

v

σi,j-1

h

s i-1,j

i+1,j

σi,jh

v

i,j

σ

σvi-1,j

i,j+1

i,j+1

(a)

h

σi,j-1

h

σi-1,j-1

(b)

Figure 3. (a) A rectangular grid discretization of a cell, assuming the earth to be electrically isotropic. (b) A rectangular grid discretization

of a cell, assuming the earth to be electrically anisotropic.

one can calculate apparent resistivity for any electrode

configuration. Apparent resistivity calculations can be

classically achieved by multiplication of a geometric

factor, with potential difference normalized by injected

currents (Dey and Morrison, 1979). Furthermore, we

also normalize these apparent resistivity values by a

homogeneous response in the model introduced by

Marescot et al. (2006) as:

,

(7)

where Δv is the potential differences measured in the field,

ΔvH is the potential differences in a homogeneous model,

and ρH is the resistivity value of a homogeneous model,

e.g., 1 ohm-m (Marescot et al., 2006).

In our anisotropic model, we assume that the earth

consists of arbitrary shapes with electrical anisotropy.

Figure 3b shows a cell in an anisotropic medium. Based

on our assumption that the horizontal conductivity values

are in the x direction and the vertical conductivity values

are in the z direction, the vertical conductivity values

are always smaller than the horizontal value in each cell

(Maillet, 1947). The geometric mean of the conductivity

values is given as (Maillet, 1947):

,

(8)

where σh(x,z) and σv(x,z) are conductivities in the horizontal

and vertical directions, respectively. The subindices h and

v stand for horizontal and vertical directions. Note that

. From these 2 conductivity values, one

can calculate the electrical anisotropy coefficient of each

cell of the medium. Thus, the anisotropy coefficient can

be calculated with the following equation (Maillet, 1947;

Grant and West, 1965):

(9)

Laboratory measurements show

in most

sedimentary rocks. The range of the coefficient of the

anisotropy value is between 1.0 and 7.5 (Carmichael,

1989; Negi and Saraf, 1989). Without electrical anisotropy

assumption, interpretation may be erroneous (Nguyen

et al., 2007). Electrical anisotropy should be considered

even when the anisotropy coefficient is

(Wiese et

al., 2009). Electrical anisotropy can even be detected in

alluvium (Greenhalgh et al., 2010).

Based on our assumption given above, the forward

response of an isotropic earth model can be extended to

an anisotropic earth model by replacing the conductivity

tensor with the geometric mean of the conductivity values.

Thus, we have:

(10)

where v(x,y,z) is the potential field distribution in the 3D

domain, I is a point source with its location indicated by a

subindex s, σn(x,y) is the geometric mean of conductivity

values calculated by Eq. (8), and δ is the Dirac delta function.

The FD method requires a mesh designed by dividing

rectangular cells in the region as mentioned previously.

The corresponding 2D region consists of 2 dimensions

(assuming that the conductivity does not change along the

y direction in a Cartesian coordinate system). This region

can be divided into Nx by Mz cells (Figure 2). Here, Nx is the

number of cells in the x direction with step size Δx. Similarly,

Mz is the number of cells in the z direction with step size

Δz. In cases where electrical anisotropy exists, the vertical

step size is divided by the electrical anisotropy coefficient

with Δz / λ. However, we do not know initially what the

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PEKŞEN and YAS / Turkish J Earth Sci

anisotropy coefficients are. Thus, we need to determine

the anisotropy coefficient so that the mesh design can then

be corrected. Our inversion algorithm iteratively finds

not only horizontal but also vertical resistivity values. At

the end of the inversion process, we also need to calculate

electrical anisotropy coefficients by using Eq. (9) so that

the electrical anisotropy correction can be employed. Note

that any type of FD discretization requires mesh design as

mentioned above. In our model, the mesh does not change

during iteration as usual. The pseudosection is represented

by using median depth of investigation (Edwards, 1977;

Loke, 2016). This is a crucial point: if electrical anisotropy

exists in a geological formation, interpretation using 2D

inversion code may be erroneous without considering

anisotropy.

We developed a new inversion code by using MATLAB.

The code can be used for inverting resistivity sections of

2D isotropic and anisotropic earth models. Note that here

we assume that the measurement profile and layers of the

formation are parallel to each other. The newly developed

forward code was validated against 1D analytical solutions

on anisotropic examples (Pekşen et al., 2014). Comparison

of the results for an analytic solution for a 2-layered earth

model and a 2D DC FD anisotropic model are illustrated

in Figure 4. The results are very similar. The purpose of

numerical comparison with analytic results is to be a

useful control for numerical modeling (Greenhalgh et al.,

2009a.; Yan et al., 2016).

2.2. Inversion

Estimation of subsurface conductivity distribution

from measurement requires inversion. There are many

algorithms and methods available in the literature for this

purpose (e.g., Pain et al., 2003; Herwanger et al., 2004;

LaBrecque et al., 2004; Kim et al., 2006; Greenhalgh et al.,

2009b; Wiese et al., 2015). The forward 2D DC FD method

can be formulated as the following matrix notation:

d=G(m)

(11)

where G is a nonlinear forward operator, m is a

parameter vector, and d is an observation vector (Meju,

1994). Various approximations can be followed in order

to estimate model parameters from observations. In

this study, we applied the singular value decomposition

(SVD) method for solving Eq. (11). The sensitivity matrix

of the system is calculated for each iteration to solve Eq.

(11). The partial derivative of the sensitivity matrix was

calculated by using the method proposed by Tripp et

al. (1984). The sensitivity matrix of the model consists

of the partial derivative of the model parameters; the

dimensions of the matrix are N by M. N is the number

of measurements and M is the number of parameters.

Depending on the condition (N > M, N < M, or N=M),

we may meet some difficulties while inverting the matrix.

Problems based on the number of parameters and

observation points can be categorized as follows: when N

> M, it is called an overdetermined problem; when N=M,

the problem is known as evenly determined; finally, when

Figure 4. Comparison of the results of an analytic solution for a 2-layered earth model and a 2D DC FD anisotropic model.

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PEKŞEN and YAS / Turkish J Earth Sci

N < M, it is called an underdetermined one (Menke, 1989).

This problem is ill-posed from the mathematical point

of view; the solution may not be unique. These kinds of

inversion problems can be coped with by using different

algorithms, such as a damped-least squares method (Meju,

1994), Tikhonov regularization (Tikhonov and Arsenin,

1977; Zhdanov, 2002), or the Occam method (DeGrootHedlin and Constable, 1990; Aster et al., 2005).

In this study, we used the SVD method. The details of

the SVD method can be found in the work of Golub and

Loan (1996). The problem can be solved iteratively with

the SVD algorithm given by Meju (1994) in the following

matrix notation:

(12)

,

(13)

where Δp is a vector consisting of model parameters, V and

U are orthonormal matrices, λi is the ith eigenvalue of the

system, and ε is a constant. m stands for a parameter vector

and k denotes an iteration number. To clarify the inversion

steps we have used here, flowcharts of the 2 algorithms for

isotropic and anisotropic media are shown in Figures 5 and

6. We have followed these inversion steps. As initial values,

we use the geometric mean of the apparent resistivities of a

profile. Since the initial value is a single apparent resistivity

value, it is a sort of homogeneous model for isotropic cases.

As for the anisotropic model, the initial value we begin

with for an electrical anisotropy coefficient is 1.1. This is a

kind of threshold value for inversion with the anisotropic

model. It can be between 1.1 and 3. Without a small

perturbation of anisotropy, the method suffers in finding

the correct model. Using a classical inversion algorithm,

the horizontal and vertical resistivity values cannot be

distinguished from surface resistivity measurement, due to

the principle of equivalence. This can also be seen in Eq. (8).

To overcome this problem, we use the constraints

during iterations. We also apply a Laplacian

and

filter operator to the data as a smoothing factor (Sasaki,

1989). Note that the problem is an underdetermined one,

since the number of parameters is smaller than the number

of observations. In spite of these difficulties, we have found

the subsurface resistivity distribution of the isotropic and

anisotropic models. Once the inversion algorithm reaches a

predetermined RMS threshold value (e.g., 3%), the process

ends. For an anisotropic model, we further calculate the

electrical anisotropy coefficient, and then we finalize the

process by incorporating the vertical shift due to electrical

anisotropy into the mesh discretization by Δz/λ.

3. Micro- and macroanisotropy

In general, electrical anisotropy can be classified as macro

or micro. For geological studies, macroanisotropy exists in

homogeneous stratified rocks. The presence of successive

parallel and different conductivities with long linear

structures causes macroanisotropy. The layered internal

structure of shale results in microanisotropy (Negi and

Saraf, 1989). Electrical anisotropy is a scaling problem.

An anisotropic layer can equally be represented by a

stack of isotropic layers (Maillet, 1947). Furthermore, a

number of isotropic layers can cause pseudoanisotropy.

Elongated small particles in the sedimentation such as

clay or sand prefer certain directions, which also gives

rise to electrical anisotropy (Kunz and Moran, 1958;

Rey and Jongmans, 2007).

4. Numerical example

Figure 7 shows a synthetic 2D model. The background

of the model is anisotropic. The anisotropy coefficient is

2 with

. The resistivity of the buried

body is 200 ohm-m in the horizontal and vertical

directions. Therefore, it is an isotropic block. The depth

of the upper side of the block is 1.5 m. The inverted

resistivity sections with isotropic and anisotropic

models are given in Figures 8a and 8b.

The white color shows the exact position of the block

in the middle of Figures 8a and 8b. The vertical shift

can be seen with and without incorporating electrical

anisotropy when comparing both results. Based on

Figures 8a and 8b, one can state that inversion without

electrical anisotropy can make interpretation erroneous

regarding the depth of the anomalous body’s location.

Inversion results of the isotropic and anisotropic

assumptions give some differences in the vertical body

position. These differences should be related to the

anisotropy coefficient of the medium. Thus, we can say

that if any geological formation has electrical anisotropy

properties, we should use the 2D inversion process with

incorporated electrical anisotropy.

The second synthetic model consists of 2 electrically

anisotropic blocks embedded in a model. The exact

model is shown in Figure 9. The anisotropy coefficients

of the blocks are 2 and 2.5, respectively. We assume that

the background of the model is isotropic in this synthetic

example. Figures 10a and 10b show the inverted results

with and without anisotropy assumptions. It can be

seen from Figures 10a–10c that there is no significant

vertical shift in the depth, since the background of the

model is isotropic. Our inversion algorithm successfully

found the upper block close to the surface; however, the

lower block was not located precisely. Comparison of

both inverted results with and without incorporating

anisotropy shows a relatively small enhancement of the

result with anisotropy (see Figures 10a and 10b).

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PEKŞEN and YAS / Turkish J Earth Sci

Figure 5. The flow chart of inversion steps, assuming the earth to be electrically isotropic.

5. Field example

Bathonea is a prehistoric city in İstanbul. Figure 11 shows

the location of the Küçükçekmece Lake district in İstanbul.

158

Many archaeological objects and structures in Bathonea

have been found and identified, such as harbor structures,

roads, buildings, and smaller artifacts. The periods of these

PEKŞEN and YAS / Turkish J Earth Sci

Figure 6. The flow chart of inversion steps, assuming the earth to be electrically anisotropic.

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PEKŞEN and YAS / Turkish J Earth Sci

0

0

10

20

30

Depth (m)

ρh=200Ω.m

2

λ =1

3

5

6

50

1.5 m

1

4

40

ρh=2.5Ω.m

ρv=10Ω.m

λ =2

ρvv =200Ω.m

2m

5m

7

Figure 7. A synthetic 2D resistivity model is shown. There is a rectangular body (electrically isotropic) in the

middle of the model. The background of the resistivity model is assumed to be electrically anisotropic.

Figure 8. (a) Estimated 2D model obtained from 2D inversion; the model is assumed to be electrically isotropic. The

white rectangular shape in the inverted resistivity section shows the exact location of the body. (b) Estimated 2D model

obtained from 2D inversion assuming the model to be electrically anisotropic. The white rectangular shape in the inverted

resistivity section shows the exact location of the body.

findings are Hellenistic, Roman, and Byzantine. Excavations

and research in the ancient city continue to the present day

(Aydıngün, 2007). More information about and photos of

Bathonea can be found at http://www.bathonea.org.

Here, we show 3 ERT profiles measured with 30

electrodes. Each profile was collected with a multielectrode

instrument. The profile length was 58 m, with 2-m electrode

spacing. As an array, we used the Wenner alpha electrode

configuration. Apparent resistivity pseudosections are

displayed in Figures 12a–12c for each profile. Each profile

was inverted by using the presented inversion methods

with and without electrical anisotropy assumptions.

Inverted resistivity sections are given in Figures 13a–13c

without electrical anisotropy assumption. Figures 14a–14c

shows the inverted resistivity sections for each profile

160

incorporating electrical anisotropy. The vertical shift can

be seen when we compare Figures 13a–13c and 14a–14c.

Figure 15 shows the field before and after excavations.

At the beginning, we interpreted the resistivity section

without considering electrical anisotropy. We found some

anomalies that were related to structures such as ancient

remnants, possible ancient water channels, and some

remains of ancient walls. However, after excavation, we

realized that there was a difference between the interpreted

depth based on 2D inversion without electrical anisotropy

and the archaeological findings. The application of the 2D

inversion method with electrical anisotropy gave much

better results in this field study.

We estimated the location of the archaeological wall

based on our inversion results. The archaeologist found

PEKŞEN and YAS / Turkish J Earth Sci

0

10

0

20

Depth (m)

1

1.4m

2

3

6

40

50

ρ =100Ω.m

2.9m

h

ρϖv=625Ω.m λ =2.5

5.5m

4

5

30

1.1m

ρ =100Ω.m

h

ρv =100Ω.m

λ =1

ρ =10Ω.m

h

ρϖv=40Ω.m

2.5m

λ =2.0

5.5m

7

Figure 9. A synthetic 2D resistivity model is shown. There are 2 rectangular bodies with electrical

anisotropy embedded in the model. The background of the resistivity model is assumed to be

electrically isotropic.

Isotropic Section

Figure 10. (a) Estimated 2D model obtained from 2D inversion; the model is assumed to be electrically isotropic. The white rectangular

shapes in the inverted resistivity section show the exact location of the bodies. (b) Estimated 2D model obtained from 2D inversion

assuming that the model is electrically anisotropic. The white rectangular shapes in the inverted resistivity section show the exact

location of the body. (c) The exact model.

the wall and some other archaeological objects as we

expected. The depth of the wall was approximately between

0.75 m and 2.00 m. Before any excavations, it is vitally

important to know the exact location of any buried objects

in an archaeological area with respect to the depth. Our

numerical and field experiences suggest that if electrical

anisotropy exists in an area, we should always expect that

the depth of layer boundaries or buried objects appearing

on an inverted resistivity section is closer to the earth’s

surface than what the actual depth is.

6. Conclusions

In this paper, we have developed and investigated an

inversion method in electrically anisotropic media. In

general, we do not expect electrical anisotropy to have

an effect on the data acquired in an archaeological area.

However, the background formation of archaeological

objects can be electrically anisotropic due to sedimentation.

Some small elongated particles prefer a certain direction,

which is a well-known fact.

161

PEKŞEN and YAS / Turkish J Earth Sci

Küçükçekmece

Lake

Küçükçekmece

Lake

Profile-2

Profile-3

Profile-1

Figure 11. Location of the study area in Küçükçekmece district in Turkey. A circle in the upper left corner of the figure indicates the

study area in Turkey. The aerial photo in the background shows Küçükçekmece Lake (Image Landsat Data SIO, NOAA, US Navy, NGA,

GEBCO © 2016 Başarsoft, Google Earth). Electrical resistivity tomography (ERT) profiles are illustrated on the right of the figure as

profiles 1–3 (http://www.google.com/maps).

Figure 12. Pseudosections of profile-1 (a), profile-2 (b), and profile-3 (c).

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PEKŞEN and YAS / Turkish J Earth Sci

Figure 13. Estimated 2D model obtained from 2D inversion assuming electrically isotropic earth for profile-1 (a), profile-2 (b), and

profile-3 (c). The lines with white in the sections indicate that the areas were excavated by archaeologists.

Figure 14. Estimated 2D model obtained from 2D inversion assuming electrically anisotropic earth for profile-1 (a), profile-2 (b), and

profile-3 (c). The lines with white in the sections indicate that the areas were excavated by archaeologists.

163

PEKŞEN and YAS / Turkish J Earth Sci

Lake

Lake

Profile-2

Profile-3

Profile-1

Lake

Figure 15. Photo of the ERT profiles in the field before and after excavations next to Küçükçekmece Lake. There are some archaeological

findings shown: wall and channels (to see more pictures: http://www.bathonea.org).

The study shows that the depth estimated based on

inversion without electrical anisotropy can be erroneous.

Thus, interpreting the data with consideration of electrical

anisotropy gives much better results regarding the actual

depth of archaeological finds.

When considering geological formation complexity, it

is difficult to say that a formation is electrically isotropic.

To enhance the interpretation of an ERT profile, we may

incorporate the electrical anisotropy effect to the data.

We show that it is possible to estimate the anisotropy

coefficient of any cells in the FD mesh. The mesh can

then be corrected by using the anisotropy coefficient.

This can be done after inversion is completed. After the

vertical mesh step size is corrected by dividing all of them

by the electrical anisotropy coefficient, we finally get the

corrected mesh. From our synthetic examples and field

164

study, we can say that the proposed method can be useful

for interpretation of ERT data. This approximation was

successfully tested on synthetic and field datasets. The

method we used here requires running inversion codes

twice. If one finds the two sets of results to be very close

to each other, one may say that the medium is isotropic.

Otherwise, the medium may be electrically anisotropic.

Acknowledgments

We thank Prof Dr Şerif Barış for permission to use his

data. We also thank Associate Prof Dr Şengül G Aydıngün

for providing us with archeological information from

Bathonea and for her support during the fieldwork. We

appreciate the efforts of Demirkan Baylar, Erdinç Duman,

and Güngör Doğan during the fieldwork.

PEKŞEN and YAS / Turkish J Earth Sci

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