A quick 2-D geoelectric inversion method using series expansion

APPGEO-01887; No of Pages 10
Journal of Applied Geophysics xxx (2010) xxx–xxx
Contents lists available at ScienceDirect
Journal of Applied Geophysics
j o u r n a l h o m e p a g e : w w w. e l s ev i e r. c o m / l o c a t e / j a p p g e o
A quick 2-D geoelectric inversion method using series expansion
Ákos Gyulai a, Tamás Ormos a, Mihály Dobroka a,b,⁎
a
b
Department of Geophysics, University of Miskolc, Hungary
MTA-ME Applied Geoscience Research Group, Department of Geophysics, University of Miskolc, Hungary
a r t i c l e
i n f o
Article history:
Received 6 February 2008
Accepted 24 September 2010
Available online xxxx
Keywords:
Geoelectric
Combined geoelectric inversion
1.5-D inversion
2-D inversion
a b s t r a c t
This paper presents the principles of a new inversion method used for determining 2-D geological structures.
The basis of the method is that horizontal changes in layer-thicknesses and resistivities of the geological
structure are discretized in the form of series expansion. The unknown expansion coefficients are determined
by linearized iterative least-squares (LSQ) inversion of data provided by surface geoelectric measurements. The
discretization of the 2-D model by means of series expansion gives the possibility to reduce the number of
model parameters. Thus, the resulting inverse problem becomes overdetermined and can be solved without
the application of additional regularization, e.g., by smoothness constraints, which is usually required for
traditional 2-D/3-D inversion. By knowing the expansion coefficients, the local layer parameters are calculated
along the profile, point by point. In conformity with the complexity of the model, 1-D forward modeling is
applied in the initial iteration steps, then as a continuation the direct problem is handled as a real 2-D problem
by means of a finite difference (FD) procedure—i.e., the forward modeling is combined, whereas the unknowns
(the expansion coefficients) are the same. This combination of the 1-D and 2-D forward modeling procedures
makes it possible to analyze quickly the geological models having considerable lateral variation.
© 2010 Elsevier B.V. All rights reserved.
1. Introduction
For the exploration of 2-D and 3-D geological structures, two
different kinds of measurement-interpretation methods are generally
used in practice of geoelectric measurements.
In case of models with slow lateral variations, the so-called traditional
vertical electric sounding (VES) measurement can be applied well. The
local layer parameters characterizing the model are frequently estimated
by using apparent resistivity data (sounding curves) determined at
relatively sparse stations (50–200 m) with individual 1-D inversion
methods (Koefoed, 1979). In order to ensure the fulfillment of a 1-D
assumption as much as possible, the measurement electrode arrays are
deployed in the strike direction of the geological structure. Then the
geological structure is determined from local layer parameters by means
of interpolation. In individual inversion, however, equivalence appearing
in three or more layered models commonly causes a large uncertainty in
parameter estimation, which is less acceptable in practice. This problem
can be reduced significantly by applying joint inversion methods (Vozoff
and Jupp, 1975; Hering et al., 1995; Haber and Oldenburg, 1997; Misiek et
al., 1997; Kis, 1998; Gallardo and Meju, 2004). In order to give better
approximation for geological models with slow lateral variations a new
inversion algorithm, the so-called 1.5-D inversion was introduced by
⁎ Corresponding author. MTA-ME Applied Geoscience Research Group, Department
of Geophysics, University of Miskolc, Hungary. Fax: +36 46 361 936.
E-mail addresses: [email protected] (Á. Gyulai), [email protected]
(T. Ormos), [email protected] (M. Dobroka).
Gyulai and Ormos (1999a). By the method all observation data
(measured with an electrode array) from each VES station along the
survey profile are integrated into one inversion procedure. The lateral
change of the model parameters (the dependence of layer resistivities
and thicknesses on the lateral coordinates) are approximated by series
expansion. In order to assure cost-effective computation the authors use
1-D forward modeling procedure. The efficiency of the 1.5-D method was
demonstrated both on synthetic and field data (Gyulai and Ormos, 1997,
1998, 1999a).
In solving 2-D or 3-D inversion problems with the FD forward
modeling method (Loke and Barker, 1996; Zhang et al., 2000), piecewise
constant resistivities defined on a rectangular grid of cells are assumed.
This results in a large number of unknowns normally leading to an
underdetermined inverse problem. In order to find a unique solution
various additional constraints have to be applied. These constraints may
assure the smoothness of the solution (Marquardt–Levenberg method)
or even focus on the existence of a boundary (Blaschek et al., 2008).
Underdetermined inversion algorithms can efficiently be stabilized with
different additional constraints. Using non-physical constraints can have
a strong influence on the inversion results.
In order to reduce the non-uniqueness of the problem and to achieve
a faster computer code, several authors have combined the use of 2-D
and 1-D calculations (e.g. Oldenburg and Ellis 1991; Smith and Booker
1991; Christiansen and Auken, 2004). Auken et al. (2005) presented
piecewise 1-D laterally constrained inversion procedure for processing
and interpreting very large data sets. The locally 1-D models are
connected laterally by requiring approximate identity between
0926-9851/$ – see front matter © 2010 Elsevier B.V. All rights reserved.
doi:10.1016/j.jappgeo.2010.09.006
Please cite this article as: Gyulai, Á., et al., A quick 2-D geoelectric inversion method using series expansion, J. Appl. Geophys. (2010),
doi:10.1016/j.jappgeo.2010.09.006
2
Á. Gyulai et al. / Journal of Applied Geophysics xxx (2010) xxx–xxx
neighboring parameters (typically resistivity and depth) within a
specified variance (Auken and Christiansen, 2004).
In this paper we propose an alternative way to reduce the nonuniqueness inherent in traditional 2-D inversion schemes. We develop
further the idea of the 1.5-D inversion presented by Gyulai and Ormos
(1999a). For the forward modeling calculations we use the FD algorithm
developed by Spitzer (1995). The essential part of the procedure is that
we make the parameterization of a 2-D geoelectric earth model in terms
of a series expansion of layer-thicknesses and resistivities and define the
expansion coefficients as the unknowns of the inverse problem. The use
of series expansion results in a sufficient reduction in the number of
unknowns compared to traditional FD or Finite Elements Method (FEM)
based inversion procedures. This gives the possibility to define the
inversion problem as an overdetermined one, without the need of any
additional regularization (or smoothness) constraint. In order to reduce
the computation time we apply the 1.5-D method (Gyulai and Ormos,
1999a) for the first few iteration steps and use the resultant model as an
initial one for the subsequent 2-D inversion. We call the above
procedure combined geoelectric inversion (CGI) method. Earlier a similar
inversion algorithm was developed to solve the 2-D seismic guidedwave inverse problem by Dobróka (1994) and Dobróka et al. (1995) and
a 2-D seismic refraction inverse problem by Bernabini et al. (1988) and
by Ormos (2002) and by Ormos and Daragó (2005), in which the lateral
variation of the model was discretized by using series expansion and the
inverse problem was formulated in terms of the expansion coefficients.
Another applications use the series expansion technique for
inverting well-logging, resistivity and IP data (Szabó, 2004; Turai,
2004; Drahos, 2005; Dobróka and Szabó, 2005; Dobróka et al., 2009).
2. Geoelectric inversion using series expansion
As a starting point, let us make a short overview of the 1.5-D
inversion method, because it is also based on the series expansion of the
model parameters. In the first phase of the CGI procedure we make use it
for finding an initial model for the subsequent 2-D inversion procedure.
The linearized CGI algorithm uses the LSQ method for the estimation of
the series expansion coefficients, with the help of which thicknesses and
resistivities can be derived.
2.1. The 1.5-D geoelectric inversion method
As mentioned in the Introduction, the 1.5-D inversion method was
developed for the interpretation of traditional VES measurements
(Gyulai and Ormos, 1997, 1998, 1999a). Using this, the lateral variation
of a 2-D geological model is described by series expansion using suitably
chosen basis functions. The inversion algorithm is an iterative method
supported by 1-D forward modeling procedure. The expansion
coefficients serve as the unknowns of the inverse problem.
As presented in Gyulai and Ormos (1999a), the series expansion
coefficients can be jointly determined from the data of all VES stations
along the profile. In this approach, the expansion coefficients, which are
the same along the entire profile, constitute the coupling between the
different 1-D models used in the forward modeling procedure. Our
studies of synthetic and field data demonstrate that the 1.5-D inversion
procedure gives accurate and reliable parameter estimation and – in
spite of 1-D forward modeling – the results are highly acceptable in the
field practice (Gyulai and Ormos, 1999a,b; Gyulai et al., 2000; Gyulai,
2001), too.
Previous investigations (Gyulai and Ormos, 1997, 1998, 1999a; Kis
et al., 1998; Ormos et al., 1999) showed several advantages of the 1.5-D
inversion method, i.e. speed, accuracy, stability etc. Based on these
results, Kis (1998) generalized the 1.5-D inversion method by using
other type of basis functions and studied its impact on the resolution of
the equivalence problem in detail.
In 1.5-D approximate inversion a series expansion technique is used
to describe the lateral change of the model parameters. We assume that
the laterally changing model parameters can be approximated with
sufficient accuracy by using the orthogonal Fourier series expansion
(Gyulai and Ormos, 1999a):
ρn ðsÞ =
Kn
Kn
1
2πs
2πs
+ ∑ dnk sin k
;
d + ∑ dnk cos k
2 no
S
Sp
k=1
k=1
p
ð1Þ
where n = 1; :::; N
hn ðsÞ =
Ln
Ln
1
2πs
2πs
+ ∑ cnl sin l
;
cno + ∑ cnl cos l
2
Sp
Sp
l=1
l=1
ð2Þ
where n = 1; :::; N−1
where ρn(s) is the resistivity function of the n-th layer, hn(s) is the
thickness function of the n-th layer and dnk, dn* k, cnl, cn* l denote the
expansion coefficients. N is the number of layers, and s is the lateral
coordinate along the profile of total length Sp. Maximum values of Kn and
Ln can be determined on the basis of the VES points as described by Gyulai
and Ormos (1999a). The inversion method is suitable for interpreting
most types of electrode arrays (e.g. Gradient, Wenner, dipole–dipole, and
pole–pole), and can also be considered as a constrained inversion
(Auken, et al., 2005, 2006; Pellerin and Wannamaker, 2005). In this sense,
the constraint is the coupling between the various data sets owing to the
fact that the expansion coefficients are the same for all of the calculated
data.
2.2. The combined geoelectric inversion method
First of all, for getting more accurate results in the case of abruptly
varying 2-D geological structures, it might be necessary to apply exact
forward modeling in the 2-D inversion. For this reason, we have further
developed the 1.5-D inversion method, because the 1-D forward
modeling represents an approximation in this case. In the new linearized
CGI inversion algorithm an initial model is produced by the 1.5-D
inversion method (first phase) meaning that in the very first iteration
steps 1-D forward modeling is applied. Then, in the subsequent iterations
(second phase) the slower but more accurate 2-D forward modeling FD
algorithm (Spitzer, 1995) is used for calculating the theoretical data and
the elements of the Jacobian matrix. The FD forward modeling procedure
of Spitzer is developed for 3-D calculations. Its use in a 2-D study results
in increased computation time. In spite of this fact we still used Spitzer's
procedure in order to ease the programming effort in our later studies
when we extend the investigations to 3-D earth models.
In our CGI method the coefficients of the series expansion serve as
unknown model parameters (see Fig. 1).
When using series expansion, the layer-thicknesses and resistivities
are expressed as continuous functions of the lateral coordinates.
Ordinary 2-D modeling assumes piecewise constant resistivities which
are defined on a rectangular grid of cells. A mapping from the series
expansion onto the grid is necessary in each forward calculation.
Namely, for each VES station an individual grid is used, which spacing is
logarithmical both in horizontal and vertical direction, according to the
electrode spacing, and the expected resolution of the model. The
resistivities at the grid points are determined by the series expansion of
the resistivity in the given layer, therewith in case of grid points located
in the near vicinity of a layer a weighting is also used. We can formulate
the basic idea of the CGI procedure in detail as
– as a result of the first phase a model is quickly obtained from 1.5-D
approximation after some iteration steps, which is considered as
the initial model for the subsequent 2-D inversion procedure,
– in the second phase the theoretical data and the elements of the
Jacobian matrix are calculated by using 2-D forward modeling. The
initial model (found in the first phase) is usually quite near to the
solution, so that only a few 2-D iteration steps are enough for
acceptable parameter estimation in the second phase. The great
Please cite this article as: Gyulai, Á., et al., A quick 2-D geoelectric inversion method using series expansion, J. Appl. Geophys. (2010),
doi:10.1016/j.jappgeo.2010.09.006
Á. Gyulai et al. / Journal of Applied Geophysics xxx (2010) xxx–xxx
Field
data
Field
parameters
1,5-D start
model parameters
3
A priori
information
Computing of
start function
coeffcients
1. phase: 1.5-D inversion
Y
Diff.
between
computed and
field data
Forward
modeling
1-D
MIN
?
Correction of function
coefficients
Result of 1,5-D
equal to startmodel
of 2-D
2. phase: 2-D inversion
Y
Diff.
between
computed and
field data
MIN
Forward
modeling
2-D
Computing of
model parameters,
and model
discretization
?
Correction of function
coefficients
End of combined
inversion
Fig. 1. Algorithm and flow chart of the combined inversion method (CGI).
advantage of the series expansion technique is that the number of
unknowns is much less than that of the measurement data, which
makes the inverse problem overdetermined. Thus, we can avoid the
use of any additional non-physical constraints.
In order to validate the accuracy of the inversion results various
tools can be used. In the data space, based on the L2 norm, the
normalized data distance d is defined
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
0
1ffi
u
ðobservedÞ
ðcalculatedÞ 2
u I
ρa;i
−ρa;i
u1
A 100%
d=t ∑ @
ðcalculatedÞ
I i=1
ρa;i
where Q is the total number of model parameters.
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
JðkÞ JðkÞ
∑ ∑ fΨki ðxm Þ*Ψkj ðxm Þ*COVij g
ð3:aÞ
where I denotes the total number of measured apparent resistivity
data ρa, i. In case of inversion of synthetic data calculated on an exactly
⇀ðexactÞ ), the relative model
known model (with model parameters m
distance is essential to be computed
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
!
u
ðestimatedÞ
ðexact Þ 2
u1 Q
mq
−mq
t
*100%
∑
D=
ðexact Þ
Q q=1
mq
The accuracy of the parameter estimation is often characterized by
means of the variances, which are derived from the diagonal elements
of the parameter covariance matrix (Menke, 1984). We determined
for the case of series expansion the elements of the covariance matrix
at each VES stations
ð3:bÞ
σkm = σk ðxm Þ =
i=1 j=1
pk ðxm Þ
ð4:aÞ
where σk(x) denotes the estimation error of the kth model parameter
(i.e., resistivity or thickness), and σkm the same at the mth VES station
(at x = xm). K is the total number of the pk(x) model parameters
(k = 1,2,…,K), and M is the number of VES stations along the profile
(m = 1,2,…,M). J(k) is the number of elements in the basis function in
the series expansion describing the kth model parameter, Ψki(x) and
Ψkj(x) are the ith and jth basis functions and COVij the covariance
matrix elements of the estimated expansion coefficients (defined in
Menke, 1984; Salát et al., 1982).
Please cite this article as: Gyulai, Á., et al., A quick 2-D geoelectric inversion method using series expansion, J. Appl. Geophys. (2010),
doi:10.1016/j.jappgeo.2010.09.006
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Á. Gyulai et al. / Journal of Applied Geophysics xxx (2010) xxx–xxx
In order to give an overall characteristic of the parameter estimation
for the whole model the mean (percentage) estimation error F is
introduced as
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
M
1 K
∑ ∑ σ 2 100%
F=
KM k = 1 m = 1 km
S=
ð4:bÞ
In order to characterize the degree of correlation among the
estimated model parameters the correlation matrix is extensively used
COVij
CORRij = qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi :
COVii COVjj
Because of the large number of the matrix elements, it is useful to
introduce a single scalar
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2
P
P 1
∑ ∑ CORRij −δij
P ðP−1Þ j = 1 i = 1
ð5Þ
called the mean spread (Menke, 1984; Salát et al., 1982), as a general
characteristic value describing the model correlation (P denotes the
total number of the model parameters, δij is the Kronecker delta
symbol).
The smaller the values of D, F and S characterizing the whole inverted
section, the reliable the results of the 1.5-D and CGI inversion are. We
Fig. 2. Finding the optimal number of coefficients. a.)–n.) Results of CGI-s using different number of coefficient regarding to the second layer thickness (white boxes). d.) The optimal
coefficient number is 9. (See also in Fig 3a).
Please cite this article as: Gyulai, Á., et al., A quick 2-D geoelectric inversion method using series expansion, J. Appl. Geophys. (2010),
doi:10.1016/j.jappgeo.2010.09.006
Á. Gyulai et al. / Journal of Applied Geophysics xxx (2010) xxx–xxx
5
can use only F and S values for characterization of the inversion results in
field cases. For investigations using synthetic data we can additionally
use the model distance D, too. These values can be applied for the
determination of the optimal number of coefficients.
2.3. Optimal number of coefficients
In case of series expansion based inversion methods it is very
important to specify the optimal number of expansion coefficients. They
are depending on the geological structure and the noise, but the
structure itself can only be revealed after the inversion. The investigations of 1.5-D and CGI inversions (both with synthetic and field data)
show that it can be found many resulting models defined by different
number of coefficients at the same minimum of the normalized data
distance d. Therefore, the choosing of the best resulting model should be
based on the minimum of mean estimation error F and on the model
distance D in the synthetic case. The determination of the optimal
number of coefficients can be treated as an optimization procedure. We
made it by executing a series of CGI program runs for different number
of coefficient.
To show this procedure we used two synthetic models. For the first
case we chose a three-layered 2-D model with straight line
boundaries (drawn with white lines in Fig. 2) (Gyulai et al., 2007).
In this investigation only the layer-thicknesses were varying laterally
in the model. The model parameters of Fig. 2 are ρ1 = 10 Ωm,
ρ2 = 5 Ωm, and ρ3 = 20 Ωm. The layer-thicknesses were described
with trigonometric series. In case of the first layer the number of
coefficients was 3. For the approximation of the second layer-thickness a
varying number of coefficients were chosen from 3 to 29 (marked with
white boxes in Fig. 2). Models were estimated, separately, for each
coefficient number by using the CGI method. The results are shown in
Figs. 2 and 3 (Gyulai et al., 2007). It can be seen in Fig. 3a that the
normalized data distance d attains its minimum value rapidly and does
not change any more as a function of the number of coefficients. It
practically means that there exist many resulting models (with different
number of coefficients) at the same data fitting value d (green dots and
fitting line in Fig. 3a). Both the mean estimation error F and the relative
model distance D (red and blue dots and fitting lines in Fig. 3a) reach a
minimum at 9 as the optimum number of coefficients. The results at the
best fitting can also be seen graphically in Fig. 2.
The correlation between F and D is near to 1 (see Fig. 3b).
Therefore, in case of field applications (where the relative model
distance D is not possible to be determined) the optimum number of
coefficients is to be found at the minimum of d and F, simultaneously.
We got similar results for a different types of synthetic models.
For the second case a high resistivity “block” placed in a
homogeneous half space (white boxes in Fig. 4) was chosen as a
model. The description of this type of model with continuous functions
expanded in trigonometric series is not favourable. (The model
parameters, and the synthetic data used in Fig. 4 are given later in the
text). During this investigation all model parameters (thicknesses and
resistivities) were described with trigonometric series using 3–3
coefficients. As an exception, only the coefficient number of resistivity
in the 2nd layer was changed from 7 to 15 in five steps (white boxes
with coefficient number in Fig. 4) in a series of CGI inversions.
The normalized data distance d as a function of coefficient number
reached a minimum at 2.1%. This value remained constant for all
resulting models having been estimated by more than 7 coefficients.
The minimum value of mean estimation error F was 22.4% at 9
coefficients. (In case of this model type we cannot use the relative
model distance D, because the layer-thicknesses in a homogenous half
space outside of the “block-model” were indefinable). The optimal
number of coefficients can be found as a result of a “trial and error”
optimization procedure, which can be done manually at the minimum
of d and F, simultaneously. The fast CGI procedure allows us to
perform many inversions for trying different coefficient numbers in a
Fig. 3. a.) The optimal number of coefficients can be found at the minimum of d, F and D.
See also in Fig 2d. b.) The main estimation error (F) and the relative model distance (D)
are almost entirely correlated.
relatively short time. The optimal number of coefficients can differ in
the presence of different amount and type of noise even if the model is
the same.
3. Investigations on CGI method with synthetic data
In order to discuss the most important features of the proposed
inversion methods, synthetic data sets generated on a model
published previously by Loke and Barker (1996) were used. In this
model a 2-D rectangular (20 × 100m) body with ρ = 50 Ωm was
embedded in a medium of 10 Ωm at a depth of 25 m. The same model
was used at the investigations in Fig. 4. Synthetic apparent resistivity
data were calculated for Schlumberger electrode configurations at 16
stations with AB/2 = 3.2–200 m by means of the FD code of Spitzer
(1995). In Fig. 5 the model (with box) and pseudo section of the
calculated Schlumberger apparent resistivities are illustrated.
This data set was used as input data of the 1.5-D and CGI methods,
which did not contain any noise. The 2-D model to be investigated
Please cite this article as: Gyulai, Á., et al., A quick 2-D geoelectric inversion method using series expansion, J. Appl. Geophys. (2010),
doi:10.1016/j.jappgeo.2010.09.006
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Fig. 4. Finding the optimal number of coefficients. a.)–e.) Results of CGI-s using different number of coefficients regarding to the second layer resistivity (white boxes). b.) The
optimal coefficient number is 9, which is to be found at the minimum of d, and F.
was constructed with three layers with model parameters varying
continuously in the lateral direction. Both the layer-thicknesses and
layer resistivities were discretized by means of trigonometric series
expansion. All layer-thicknesses and the 1st and 3rd resistivities were
described with 3–3 coefficients, for the 2nd resistivity with 17.
Fig. 6a shows the result of the 1.5-D inversion after 35 iteration
steps. This was the first phase of the inversion procedure (see Fig. 1).
(In Fig. 6a and b the number of expansion coefficients 2L1 + 1, 2L2 + 1,
2K1 + 1, 2K2 + 1, 2K3 + 1 found in Eqs. 1 and 2 are shown. For example,
“Number of coefficients: 3, 3, 3, 15, and 3” means that the two
thicknesses were described by using 3 expansion coefficients while the
layer resistivity functionsρ1(s), ρ2(s),ρ3(s)were approximated by means
of 3, 15 and 3 coefficients in the Fourier series expansion, respectively).
The CGI procedure was continued with 2-D forward modeling in
the second phase (Fig. 1), of which result after 4 iteration steps is
shown in Fig. 6b. For the comparison between the results of 1.5-D and
CGI methods, the same numbers of expansion coefficients was used. In
comparing the values of d, F and S, it can be stated that improvements
Fig. 5. 2-D rectangular prism model in a homogeneous half space (white box), and its apparent resistivity response calculated by FD method for Schlumberger array at 16 stations.
Please cite this article as: Gyulai, Á., et al., A quick 2-D geoelectric inversion method using series expansion, J. Appl. Geophys. (2010),
doi:10.1016/j.jappgeo.2010.09.006
Á. Gyulai et al. / Journal of Applied Geophysics xxx (2010) xxx–xxx
Fig. 6. Comparison of inversion results using synthetic data of Fig 5. a.) 1.5-D inversion
with noise-free data; b.) CGI with noise-free data; c.) CGI with 2% noise contaminated data.
7
are significant, because the d = 1.7%, F = 51.6% and S = 0.41 values of
1.5-D inversion were reduced to d = 0.2%, F = 5.1% and S = 0.26 in the
CGI procedure. The relatively high value of d in the first phase of CGI
(in case of noiseless data) is the consequence of the 1-D forward
modeling used in the first part of combined inversion. The shape of the
resistivity anomaly approximates almost exactly the rectangular
inhomogeneity, and the resistivity values are still close to the actual
ones (ρ1 = 10 Ωm, ρ2 = 50 Ωm, and ρ3 = 10 Ωm). In the inversion the
total number of unknowns (expansion coefficients) was 27.
It is important to see, how the CGI method works in case of noisy
data, so we contaminated the previous data set (Fig. 5) with 2%
random noise of Gaussian distribution. As it is shown in Fig. 6c the
combined inversion method gives similar results even in the case of
noisy data. In this case – in the presence of noise – the optimum
number of coefficients of the resistivity function in the second layer
was 9. This procedure and the best model are shown in the previous
section in Fig 4b. The relative data distance is 2.1% (nearly the same as
the percentage error of the input data set). The mean estimation error
(defined in Eq. (4a)) F = 22.4% is relatively large, which was caused by
the undulation of the resistivity in the 2nd layer. The amount of F at
this type of model is acceptable. The mean spread characterizing the
correlated nature of the set of expansion coefficients was S = 0.27,
which shows slightly correlated (i.e. reliable) results. In case of 1-D
inversion the value of S can reach 0.7 in a normal field case.
We found important to compare the accuracy of the 1.5-D
approximation is in case of a series of geological models with different
variability. To do this, six models of similar shape were defined (see
Fig. 7). Each model consists of three layers, where the boundary
between the second and third layers is composed of straight line
segments. The length of the model, i.e. profile distance varies from
70 m to 2800 m in 6 steps, while the vertical size of all models is
unchanged.
Fig. 7. Test of the 1.5-D inversion method on models with different length, using noisy synthetic data. a.)–b.) In the case of relatively short models the 1-D approximation can be used
only with considerable error; c.)–f.) in the case of the long models, which can be treated as nearly 1-D ones, the using 1-D approximation results in acceptable errors.
Please cite this article as: Gyulai, Á., et al., A quick 2-D geoelectric inversion method using series expansion, J. Appl. Geophys. (2010),
doi:10.1016/j.jappgeo.2010.09.006
8
Á. Gyulai et al. / Journal of Applied Geophysics xxx (2010) xxx–xxx
Thus we had a series of models, which was varying between 2-D
(Fig. 7a) and nearly 1-D (Fig. 7f). The synthetic data were computed
by using the FD method for the Schlumberger array at 29 stations
along the profile with current electrode spacing (AB/2) between 1.6
and 50 m. Gaussian noise of 2% was added to the synthetic data.
During the tests of the 1.5-D method, the Fourier series expansion
was used for discretization. The first thickness was described with 3
coefficients, and the second with 29 coefficients. For seeing clearly this
effect, the resistivities did not change here laterally; i.e., they were
described with one coefficient for each. Thus the numbers of
expansion coefficients were chosen as 3, 29, 1, 1, and 1.
As it is shown in Fig. 7a–f, the relative distance between the exact and
the estimated models (the model distance D defined in Eq. (3b))
appreciably decreases with increasing model length. As it was expected,
the 1.5-D inversion method (containing 1-D forward modeling) gives
better estimates for long models (Fig. 7c–f), where 1-D modeling is a
proper approximation. However, in the case of models of Fig. 7a–b, the
1-D approximation can be used only with considerable error.
Because of this reason we continued the calculation with 5 CGI
iterations for improving the estimated model. The given accuracy for
the most rapidly varying model (Fig. 7a) in case of combined inversion
is due to the performance of 2-D forward modeling. The result of the
CGI inversion regarding to this model can be seen in Fig. 2d too. We
got the value of D = 2.1% for CGI and D = 11.4% (Fig. 7a) for the 1.5-D
inversion. The values of d are practically the same in both cases.
As a comparison, Fig. 8a shows the 2-D model derived by the CGI
procedure (see also Fig. 3d), where d = 1.9%, D = 2.1%, F = 2.3%, and
S = 0.26 were obtained. Using the smoothness constrained inversion
software called RES2DINV (Geotomo Software), which is commonly
applied in the geoelectric practice, results represented in Fig 8b–c
were obtained. (Here in order to find the best result we used vertical
to horizontal flatness filter ratio = 0.25 (Fig. 8b) and 1.0 (Fig. 8c), ratio
of maximum number of model blocks to data points = 50. The rms
value calculated by the RES2DINV software was 1.71 for both Fig. 8b
and c). Data sets in Fig. 8a–c were contaminated with 2% Gaussian
noise.
In Fig. 9a similar model to the one represented by Fig. 8 can be seen
with this difference that we can see a series of faults in the second layer.
Using CGI method d = 1.9%, D = 5.1%, F = 4.9%, and S = 0.28 were
obtained. (In case of RES2DINV, we used vertical to horizontal flatness
filter ratio = 0.25 (Fig. 9b) and 1.0 (Fig. 9c), ratio of maximum number of
model blocks to data points = 50. The rms value calculated by the
RES2DINV software was 1.64 for Fig. 9b and 1.71 for Fig. 9c). Data sets in
Fig. 9a–c were charged with 2% Gaussian noise.
As the above comparison shows the CGI method solving an
overdetermined inverse problem (without the use of any non-physical
constraints or smoothing) produces sharper boundaries than RES2DINV,
which gives a smooth geological model.
The values of former slowly varying model parameters (i.e. resistivity
and layer-thicknesses) were changed randomly by 10% in order to form
a new model shown in Fig. 10. The input apparent resistivity data for the
inversion were calculated by FD modeling by the Spitzer method, which
were then contaminated with 2% Gaussian noise. The corresponding
theoretical model can be seen in Fig. 10a. In Fig. 10b–c the CGI inversion
Fig. 8. Comparison of inversion results of noisy synthetic data. a.) Result of CGI; b.) result of
RES2DINV using vertical to horizontal filter ratio of 0.25; c.) result of RES2DINV using
vertical to horizontal filter ratio of 1.0.
Fig. 9. Comparison of inversion results of noisy synthetic data on a fault model. a.) Result of
CGI; b.) result of RES2DINV using vertical to horizontal filter ratio of 0.25; c.) result of
RES2DINV using vertical to horizontal filter ratio of 1.0.
Please cite this article as: Gyulai, Á., et al., A quick 2-D geoelectric inversion method using series expansion, J. Appl. Geophys. (2010),
doi:10.1016/j.jappgeo.2010.09.006
Á. Gyulai et al. / Journal of Applied Geophysics xxx (2010) xxx–xxx
9
Fig. 11. Comparison of 1.5-D and CGI inversion of VES data, measured over a waste
deposit. a.) The 1.5-D inversion results high values of d, F and S; b.) the CGI gives
considerable less values for of d, F and S, yielding more reliable results.
Fig. 10. Comparison of CGI results of noisy synthetic data on a model with horizontal
rapidly changing parameters. a.) True (theoretical) model; b.) result of CGI using lower
number of coefficients; c.) result of CGI applying higher number of coefficients.
results can be found. The optimal numbers of coefficients were chosen as
29, 17, 29, 7, and 7. Thus d = 1.9%, D = 10.3%, F = 11.6% and S = 0.15
were obtained. It can be seen in Fig. 10b that the faster the variation of
the resistivity, the higher the number of coefficients should be used in
case of the CGI procedure. But, as it is also seen in Fig. 10c, the
unreasonable increase of the number of coefficients will degrade the
quality of the inversion results, which can be avoided by choosing the
optimal number of coefficients.
4. Case study
The applicability of the proposed 1.5D and the CGI inversion
methods was studied in an environmental geophysical problem. The
target area was a waste site near to Miskolc, North-east Hungary.
Along the profile 198 data were collected in 9 VES stations. In both
cases the numbers of expansion coefficients were the same: 9, 9, 9, 9,
9, 1, and 1. As it can be seen in Fig. 11, the 1.5-D method gave an
estimate for the model with extremely large mean estimation error
(F = 150%) and relatively high internal correlation among the
expansion coefficients (S = 0.46). On the contrary, the CGI method
resulted in significantly decreased data distance and mean parameter
estimation error (F = 16.2%). Though the number of unknowns was
the same in both inversion problems, but the mean spread (defined in
Eq. 5.) was obtained sufficiently smaller in case of the CGI procedure
(S = 0.27) indicating less correlated (i.e. more reliable) parameter
estimation.
The next field case was the investigation of an aquifer situated at
Tisza River, East Hungary. The geologic target was an inhomogeneous
gravel sequence of strata with a clayey basement. The structure was
approximated by a four-layered model, in which both layer-thicknesses
and resistivities were allowed to vary rapidly. The optimal number of
coefficients for this case 13, 11, and 13 for the thicknesses and 13, 9, 7,
and 2 for the resistivities, respectively. Along the profile 312 data were
collected in 13 VES stations. In Fig. 12 the 1.5-D and CGI inversion results
are compared. It is also seen in the figure that this technique may
visualize the outcrop of layers with thin layer-thickness or nearly the
same value of resistivities in adjacent layers. This can be seen, for
instance, at s = 1200m. Comparing 1.5-D and CGI inversion methods,
the mean parameter estimation error decreased from 29% to 19% and
the mean correlation from 0.35 to 0.29.
5. Conclusion
In conclusion, it can be stated that the new combined geoelectric
inversion procedure is promising in determining two-dimensional
geoelectric structures. Due to the discretization by means of series
expansion, the number of unknowns is relatively small in comparison
with common inversion schemes, in which each element of the grid
represents an unknown model parameter. It was shown, that because of
solving an overdetermined inverse problem (without the use of any
non-physical constraints or smoothing) synthetic 2-D geological models
can be reconstructed by the CGI method with high accuracy. In order to
reduce the computation time, the CGI method consisted of two phase. At
first 1.5-D inversion method (using 1-D forward modeling) was applied
to calculate an appropriate initial model for the second phase, where
then 2-D forward modeling was applied in the form of a linearized
inversion procedure.
At a given noise level, it was shown that more complicated models
can only be reconstructed with lower accuracy than a simpler one
with the same physical parameters (see Figs. 8a and 9a).
Besides, the unreasonable increase of the number of coefficients
degrades the quality of the inversion results, which can be avoided by
choosing the optimal number of coefficients (see Figs. 2–4 and 10).
Better inversion results can only be obtained by using joint inversion
methods integrating additional data sets and/or borehole information.
The 2-D CGI method can easily be generalized for evaluating 3-D
geological structures. We think that it would be useful to undertake
further research on more complicated models for different observation
arrays as well as other types of basis functions in the series expansion
and test it on several 3-D field models as an approximation method.
Please cite this article as: Gyulai, Á., et al., A quick 2-D geoelectric inversion method using series expansion, J. Appl. Geophys. (2010),
doi:10.1016/j.jappgeo.2010.09.006
10
Á. Gyulai et al. / Journal of Applied Geophysics xxx (2010) xxx–xxx
Fig. 12. Comparison of 1.5-D and CGI inversion of VES data measured over an aquifer. a.) 1.5-D inversion; b.) CGI. It is also seen in both figures that the technique expansion series
may visualize the outcrop of layers with thin layer-thickness or nearly the same value of resistivities in adjacent layers.
Acknowledgments
Our research concerning the investigation of near-surface geoelectric structures was carried out in the frame of the projects of the
Hungarian Scientific Research Fund (OTKA T042686 and T062416). As
a member of the MTA-ME Applied Geoscience Research Group, one of
the authors thanks also the Hungarian Academy of Science for the
support. The authors are grateful to the two referees and the Editor A.
Hördt for their suggestions, which appreciably improved the quality
of the paper.
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Please cite this article as: Gyulai, Á., et al., A quick 2-D geoelectric inversion method using series expansion, J. Appl. Geophys. (2010),
doi:10.1016/j.jappgeo.2010.09.006