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See discussions, stats, and author profiles for this publication at:http://www.researchgate.net/publication/275670745
Combining Multiple Electrode Arrays for
Two-Dimensional Electrical Resistivity
Imaging Using the Unsupervised
Classification Technique
ARTICLE in PURE AND APPLIED GEOPHYSICS JUNE 2015
Impact Factor: 1.62 DOI: 10.1007/s00024-014-1007-4
READS
53
3 AUTHORS:
Kehinde S. Ishola
University of Science Malaysia
7PUBLICATIONS 1CITATION
SEE PROFILE
Mohd Nawawi
University of Science Malaysia
55PUBLICATIONS 70CITATIONS
SEE PROFILE
Khiruddin Abdullah
University of Science Malaysia
219PUBLICATIONS 176CITATIONS
SEE PROFILE
Available from: Mohd NawawiRetrieved on: 01 November 2015
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Combining Multiple Electrode Arrays for Two-Dimensional Electrical Resistivity ImagingUsing the Unsupervised Classification Technique
K. S. ISHOLA,1,2 M. N. M. NAWAWI,1 and K. ABDULLAH1
AbstractThis article describes the use ofk-means clustering,
an unsupervised image classification technique, to help interpret
subsurface targets. Thek-means algorithm is employed to combine
and classify the two-dimensional (2D) inverse resistivity models
obtained from three different electrode arrays. The algorithm is
initialized through theselection of the number of clusters, numberof
iterations and other parameters such as stopping criteria. Automat-
ically, it seeks to find groups of closely related resistivity values that
belong to the same cluster and are more similar to each other thanresistivity values belongingto other clusters. The approach is applied
to both synthetic and field data. The 2D postinversions of the
resistivity data werepreprocessed by resamplingand interpolating to
the same coordinate. Following the preprocessing, the three images
are combined into a single classified image. All the image prepro-
cessing, manipulation and analysis are performed using the PCI
Geomatics software package. The results of the clustering and
classification are presented as classified images. An assessment of
the performance of the individual and combined images for the
synthetic models is carried out using an error matrix, mean absolute
error and mean absolute percent error. The estimated errors show
that images obtained from maximum values of the reconstructed
resistivity for the different models give the best representation of the
true models. Additionally, the overall accuracy and kappa values
show good agreement between the combined classified images and
true models. Depending on the model, the overall accuracy ranges
from 86 to 99 %, while the kappa coefficient is in the range of
5498 %. Classified images with kappa coefficients greater than 0.8
show strong agreement, while images withkappa coefficients greater
than 0.5but less than 0.8give moderate agreement. Forthe field data,
the k-mean classifier produces images that incorporate structural
features of the three electrode array configurations. Consequently,
some clusters that overwhelmingly correspond to the lithologic units
of the investigated areas are better identified than the tomographic
images of each data set considered separately, underscoring the
relevance of the unsupervised classification technique in this study.
Key words: Unsupervised classification, electrode arrays,
k-means clustering, clusters, overall accuracy, lithologic units.
1. Introduction
In geophysical investigations involving hydroge-
ology, subsurface exploration, mining, geotechnical
and archaeological prospecting, electrical resistivity
imaging has remained a vital tool for some decades
(DAHLIN 1996; SEATON and BURBEY 2000; CANDANSA-YAR and BASOKUR 2001; LOKE et al. 2013). The
advancements made in the use of this tool by
allowing resistivity data to be collected and processed
in a short period of time using a suitable electrode
array are worthy of commendation. This is coupled
with the advent of multielectrode resistivity recording
systems that have popularized the use of electrical
resistivity imaging. However, one of the problems of
resistivity investigations is the choice of a suitable
electrode configuration that will give the best
response to the observed targets in the subsurface
(ZHOU and GREENHALGH 2000; ZHOU and DAHLIN
2003). Several electrode arrays have been used in
resistivity investigations, including, but not limited
to, the Wenner, Wenner-Schlumberger, dipole-
dipole, pole-pole and pole-dipole arrays (DAHLIN
1996; CHAMBERS et al. 1999; STORZet al.2000). Each
electrode array has its advantages and limitations
regarding field operations and interpretation capabil-
ities (LOKEet al.2003; DAHLINand ZHOU2004; ABER
and MESHIN CHIASL 2010). Also, the depth of inves-tigations (ROY and APPARAO 1971; BARKER 1989;
OLDENBURG and LI 1999; SZALAI and SZARKA 2008),
sensitivity to horizontal or vertical variations, and
signal strength (LOKE 2001; SEATON and BURBEY
2002; DAHLIN and ZHOU 2004; CANDANSAYAR 2008)
are some of the other factors needing to be taken into
consideration in using these arrays. In addition, the
geological structures to be mapped, heterogeneities of
the subsurface, sensitivity of the resistivity meter
1 Geophysics section, School of Physics, Universiti Sains
Malaysia, 11800 Pulau Pinang, Penang, Malaysia. E-mail: saidi-
[email protected]; [email protected];
[email protected] Department of Geosciences, University of Lagos, Akoka,
Lagos, Nigeria.
Pure Appl. Geophys.
2014 Springer Basel
DOI 10.1007/s00024-014-1007-4 Pure and Applied Geophysics
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7/24/2019 Ishola Et Al Pure and Applied Geophysics 2014
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(FURMAN et al. 2003), sensitivity of the arrays to
vertical and lateral variations in the resistivity of the
subsurface, horizontal data coverage and signal
strength of the array (DAHLINand ZHOU2004) should
be considered before embarking on field surveys.
In conducting an electrical resistivity survey over
a noisy site, the Wenner array is preferred because of
its high signal strength and high resolution of hori-
zontal structures, but it is less easily applicable when
mapping 3D structures where the dipole-dipole, pole-
dipole and pole-pole arrays are given preference
(DAHLINand LOKE1997; ZHOUand DAHLIN2003). The
Wenner array also has a good vertical resolution, thus
making it suitable for imaging horizontal structures
(IBRAHIMet al.2003). The dipole-dipole array is most
sensitive to resistivity variations beneath the elec-
trodes in each dipole length and is very sensitive tohorizontal variations. The dipole-dipole array is rel-
atively insensitive to vertical variations in the
subsurface resistivity (GRIFFITHS and BARKER 1993).
As a result, the dipole-dipole array is given priority
for mapping of vertical structures such as dykes and
cavities (LOKE 2001; EL-QADY et al. 2005). In prac-
tice, electrical resistivity imaging involves the use of
either four electrode arrays (e.g., the Wenner Sch-
lumberger and dipole-dipole) or the three electrode
type typical of the pole-dipole with a polarity-
reversed array for all possible currents and potential
electrode combinations. To gain knowledge of the
relative potentials of these electrode arrays, several
researchers have worked in this direction. SZALAI and
SZARKA (2008) classified about 100 electrode arrays
into eight classes based on three parameters; they are
superposition, focusing and collinearity. Also, STUM-
MERet al.(2004) opined that electrical resistivity data
acquired using a large number of four-point electrode
arrays gave substantial subsurface information com-
pared to the data sets obtained from both individualarrays, such as the Wenner, dipole-dipole or a com-
bination of the Wenner and dipole-dipole arrays.
Moreover, DE LAVEGAet al.(2003) used a combined
inversion algorithm for different electrode array data
sets with the combined inversion images outper-
forming the inversion results obtained from separate
electrode arrays. ATHANASIOUet al.(2007) introduced
a weighting factor to the two-dimensional (2D)
combined inversion algorithm, which prevented the
dominance of one array type over another as the
applied parameter allowed equal participation of the
data from an individual array.
In previous studies, attempts have been made at
comparing the ability of different electrode arrays to
resolve, map or identify subsurface targets (SEATON
and BURBEY 2002; DAHLIN and ZHOU 2004; CAN-
DANSAYAR 2008). Several researchers have compared
different electrode arrays individually on the basis of
their sensitivity analysis, depth of investigations, and
responses to resolving vertical or horizontal struc-
tures (SASAKI 1992; DAHLIN 2001; BENTLEY and
GHARIBI 2004; FIANDACA et al. 2005; CAPIZZI et al.
2007; BERGE and DRAHOR 2009; MARTORANA et al.
2009; NEYAMADPOUR et al. 2010; RUCKER 2012).
Meanwhile, in related studies, the use of joint
inversion techniques have been introduced and usedfor combining two or more geophysical data into a
single image for the cross gradients (GALLARDO and
MEJU 2003, 2004, 2007) and structural approach
(HABER and OLDENBURG 1997). The efficacy of the
joint inversion method hinges on the complementary
nature of the geophysical data sets (DOETSCH et al.
2010). In addition, concerted efforts have been geared
toward optimizing different electrode arrays for
electrical resistivity surveys in order to obtain as
much information as necessary in the detection and
imaging of subsurface structures within a short period
of time (STUMMER et al. 2002,2004; WILKINSONet al.
2006a, b; COSCIA et al. 2008; LOKE et al. 2010; HA-
GREY and PETERSEN 2011; HAGREY 2012). In view of
this, COSCIA et al. (2008) proposed an experimental
design method involving an independent comparison
between the information provided by one electrode
array with others using crosshole electrical resistivity
imaging. In another work, LOKEet al.(2010) adopted
four methods to automatically select an optimal set of
array configurations that provided maximum subsur-face model resolution for an electrical imaging
applied to both synthetic and field surveys.
In the present study, however, we focus on the
combination of the 2D post-inversion resistivity
models from three different electrode arrays, namely,
the dipole-dipole (Dpd), Wenner-Schlumberger
(Wsc) and pole-dipole (Pdp), reflecting a wide range
of geological situations, using an unsupervised image
classification technique. To this end, the capability
K. S. Ishola et al. Pure Appl. Geophys.
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and functionality of an image processing technique
are explored to generate a single integrated image for
both synthetic and real-field examples.
In this article, we present a review of the different
electrode arrays used in electrical resistivity imaging
with emphasis on the advantages and their limita-
tions. Also, a technique for combining the images
from some commonly used electrode array configu-
rations is mentioned. The remainder of the article is
organized into the traditional theoretical background,
method, results and discussion followed by directions
for further work. In Sect. 2, we introduce the image
classification technique, which emphasizes two broad
groups of classification techniques. Section3 dis-
cusses the methodology, which includes numerical
modeling, merging of post-resistivity inversion data
sets, clustering and classification procedures. Also,the other aspects of the methodology presented are
the assignment of resistivity to the classification
images and criteria for evaluating the accuracy of the
clustering technique. Section4presents the results of
the application of the clustering technique to both
synthetic and field examples. In addition, the dis-
cussion section interprets the results, and suggestions
for continued research are provided.
2. Image Classification Techniques
Mechanistically, image classification is a process
that translates the raw data into more meaningful
and understandable forms (TSO and OLSEN 2005).
The image classification technique involves cluster-
ing the pixels of an image, where a pixel is the
smallest unit of digital image data that can be
individually processed, into a relatively small set of
clusters, such that pixels in the same class have
similar statistical properties. Image classificationrelies on distinctive signatures or spectral contents
of the classes as well as the ability to reliably dis-
tinguish these signatures from others (EASTMAN
2003). The image classification techniques stand out
from other data integration methods, such as artifi-
cial neural networks, self-organizing maps and joint
inversion methods, to mention but a few. This is
because of its predictive abilities (KVAMME 2006) in
subsurface target detection using scattered fields
(CHATURVEDI and PLUMB 1995). In the joint inversion
method, an external constraint is imposed on the
models (e.g., PRIDE 1994; TRYGGVASON et al. 2002;
LINDE et al. 2006; LINDE and DOETSCH 2010) as the
approach seeks an existing empirical or mathemati-
cal relationship between the models, whereas the
image classification technique looks for a natural
grouping/pattern in the data sets.
Generally, image classification can be grouped as
either supervised or unsupervised classification, and
both groups are used especially in remote sensing
for image analyses. In supervised classification, the
pixel classes are controlled by the analyst through
the process of selecting training sites in advance in
order to train the classifier (CAMPBELL 2002; TSO and
OLSEN 2005). On the other hand, unsupervised
classification works by objectively extracting variousfeatures of an image. The detailed information
derived from an image is guided by the number of
clusters into which the image is partitioned. The
preference of unsupervised over supervised classifi-
cation schemes in this study is a result of knowledge
of the site to be classified being immaterial at the
initial separation of the image pixels. Meanwhile,
the classification process is less prone to human
error as an analyst is not at liberty to make as many
decisions during the classification process and the
classes in the data are not overlooked (LILLESAND
and KIEFER1994; EASTMAN 1995; ENDERLE and WEIH
2005).
Among the commonly used algorithms for parti-
tioning of image data sets in unsupervised
classification techniques are the k-means, iterative
self-organizing data analysis (ISODATA) and fuzzy
c-means. The ISODATA algorithm is similar to the k-
means with the distinct difference that it allows for
the minimum-maximum number of clusters, while
the k-means assumes that the number of clusters isknown a priori. On the other hand, the fuzzy c-means
is an alternative way of assigning data to specific
clusters with an absolute in or out value based on
degree of membership of the data into clusters (WARD
et al.2014). Each of the clustering algorithms uses an
iterative procedure for organizing objects into a pre-
defined fixed number of clusters while attempting to
optimize the distance between the observational data
set and cluster centers (CHENG 2003; KHAN and
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AHMAD 2004). Thus, unsupervised classification
techniques partition data sets into clusters by ensuring
that intercluster variability is maximized and mini-
mizing intraclass variability without recourse to field
knowledge (LILLESANDand KEIFER2000).
2.1. The k-means Clustering Algorithm
Thek-means is a clustering algorithm (MACQUEEN
1967) that partitions multidimensional pixels in the
model space of the image, usually in a vector form
into prespecified k clusters. In the context of this
study, the clusters consist of resistivity values, and
the grouping is performed such that pixels within a
cluster are similar (i.e., intracluster variability) and
are dissimilar from pixels belonging to another
cluster (i.e., intercluster) with each cluster repre-sented by cluster centers. Thus, clustering techniques
are popularly used for finding interesting features or
patterns in a data set that may not be obvious (RANA
et al. 2010).
The theoretical background of the k-means algo-
rithm is as follows: suppose there are N input data
points such thatx1; x2; . . .; xm2 Rn is a finite number
of patterns existing in the model space of the model;
then the k-clustering aims at partitioning these data
points into kdisjoint subsets or clusters C1; . . .; Ck;
withk\Nsuch that optimization is performed based
on the clustering criterion. The criterion used for the
optimization is the sum of the squared of Euclidean
distances between each data point, xi, and the cluster
center, mk, of the subset Ck, which contains xi. This
optimization criterion is called the clustering error,
and it depends on the cluster centers m1, , mk. At
the k-th iterative step, the N observations are
partitioned into k clusters using the relation,
x2 Cjk,
provided x mjk \ x mikk k 1
for all i 1; 2; . . .; k; i 6j, where Cjk denotes the
set of observations whose cluster center is mj(k). In
the next step, a new cluster center mjk 1; j
1; 2; . . .; k is computed such that the sum of the
squared distances from all points (pixels) in Cj(k) to
the new cluster center is minimized. A measure that
minimizes this is the observation mean ofCj(k). Thus,
the new cluster center formed is given by:
mjk 1 1
Nj
Xx2Cjk
x; j 1; 2; . . .; k 2
where Nj is the number of pixels in the image. The
cluster centers are reference points used by the algo-
rithm to group the observation data points (pixels) into
clusters. The cluster centers, also called centroids, are
computed as the average of the observations in a
particular cluster. Being a point-based algorithm, the
k-means starts with the cluster centers that are initially
placed at arbitrary positions and proceeds by moving
the cluster centers at each step so that it minimizes the
clustering error. For clustering of the data set, a large
number of dimensions contains more information, and
the clusters obtained tend to increase with increasing
dimensionality in the data set.
The k-means algorithm calculates its clustercenters iteratively by initializing the centers in mkand decides membership of the pixels in one of thek-
clusters according to the closest neighbor, i.e., the
minimum distance from the cluster center. Then, it
calculates the new mkby repeating the steps above
until there are no changes in the cluster centers. Thus,
the goal is to minimize the sum of squares error
within each cluster represented by its Euclidean
distance as:
dx; mk XNi1
Xmj1
xi mj 2
3
where kk2 is a measure of the minimum distance
between a pixel, xi, and the cluster center, mj. In
Eq.3, high membership values are assigned to pixels
that are close to the cluster center for a particular
cluster, while low membership values are assigned to
pixels that are far from the centroid (PHAMand PRINCE
1999).
2.2. Selection of the Number of Clusters
In most clustering situations, an important step in
the implementation of the clustering procedures is to
specify the number of clusters into which the image is
to be partitioned. In this regard, an analyst is faced
with the dilemma of selecting the number of clusters
or partitions in the final solution as there is no
existing benchmark in the literature regarding the
selection of an optimal number of clusters. To
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overcome this nagging problem in clustering, a
variety of procedures has been proposed and used
for determining the number of clusters in data sets
(MILLIGAN and COOPER 1985; HALKIDI et al. 2001;
KVAMME 2006; WARD et al. 2014). Doing so will
assist in the final interpretation of the classification
image. In practice, the appropriate number of clusters
for a given data set is based on trial and error. The
subjective nature of deciding what constitutes correct
grouping by this method makes the selection difficult.
Another approach is to run the clustering algorithm
many times with the number of clusters gradually
being increased from a certain initial value to some
threshold values and the partitioning of data resulting
in the best validity measure being selected (HALKIDI
et al. 2001). WARD et al. (2014) used the number of
peaks obtained from the application of the kerneldensity estimation method to the data set to provide a
statistically grounded analysis of the data that in turn
was assumed to represent the appropriate number of
clusters required to group the data.
Grouping of the pixels into a few prespecified
clusters could result in a loss of important details
about the models during clustering of the data sets. In
order to ensure at the outset that no important
information in the image data sets is ignored,
clustering is often prepared with a large number of
clusters to reflect the specific characteristics of the
data sets (CIHLARet al. 1998; PHAMet al. 2004; GAO
2009). However, using a large number of clusters also
poses a problem, as too many details that could give
irrelevant interpretations of the targets may be
created (FIGUEIREDO and JAIN 2002; ERNENWEIN
2009). Therefore, finding an optimum number of
clusters in a data set could be very challenging since
it requires a priori knowledge of the data in some
cases. The performance of the k-means clustering
algorithm in terms of adequate interpretation of theclassification image depends on the selection of an
optimal number of clusters, types of attributes (i.e.,
measured parameters), types of data sets and scales of
attributes.
2.3. Stopping Rules of the k-means Algorithm
The convergence of the k-means algorithm is
guided by the cost function or sum of the square
error, which reduces to a local minimum. At this local
minimum, the correct number of cluster centers
converges into an actual cluster center as other initial
centers move away from the input data set (ZALIK
2008). A satisfactory clustering result is obtained
when the number of iterations as a prerequisite is
indicated before the user/analyst runs the algorithm.
A number of convergence conditions are possible in
the execution of clustering algorithms. These include:
(1) the search may stop when a given or defined
number of iterations is reached, (2) stopping when the
partitioning error is not reduced because of the
relocation of the centers is an indication that the
partition is locally optimal, (3) when there is no
exchange of data points between clusters, exceeding a
predefined number of iterations, and (4) when a
threshold value is attained. This convergence thresh-old corresponds to the maximum percentage of pixels
whose clusters remain unchanged from the previous
iteration (REIGBERet al. 2010; LI et al. 2013). At this
stage, the clusters are saved as the best cluster
solution for the clustering. Also, the threshold
parameter, T, typically a specific value, is stated or
selected by the user. Thek-means algorithm based on
a user-supplied parameter can modify the threshold
value by either increases or decreases depending on
the number of clusters required to represent the data
set.
3. Methodology
Subsurface imaging over a target using different
electrode configurations produces tomographic mod-
els of the target. Each inverse resistivity image gives
different information about the investigated area.
This is due to nonuniqueness in the inversion process
leading to ambiguity in the interpretation of theresults. To reduce the ambiguity, there is a need for
all the information from the different electrode con-
figuration images to be integrated into a single image.
To this end, an unsupervised classification technique
using the k-means algorithm that automatically par-
titions the 2D inverse resistivity model data sets into
some prespecified number of clusters with each
cluster linked to a particular geological or hydro-
geological (i.e., electrofacies) unit that makes up the
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resistivity image is employed. The procedure used in
this study can be grouped into four stages: (1)
numerical modeling, (2) clustering and classification
procedures, (3) assignment of resistivity and (4)
accuracy assessment.
3.1. Numerical Modeling of Synthetic Data
3.1.1 Forward Modeling
The synthetic models used in this study were created
through forward modeling. Forward modeling, also
known as the forward solver, involves a mapping
from the model space to the data space (SCHWARZBACH
et al. 2005). Forward modeling estimates/predicts
data on the basis of the known distribution of model
parameters and electrode configuration used. Theforward modeling for the DC potentials is accom-
plished using a finite difference (FD) technique to
solve the partial differential equation for charge
distribution. The program that performs this calcula-
tion is the RES2DMOD software package (GEOTOMO
2005). This forward solver is used to calculate the
apparent resistivity data for a user-defined 2D
subsurface model (i.e., it computes the electric
potential difference for a particular resistivity model).
In the FD approach, a resistivity model of the
subsurface is first discretized into several rectangularmesh or grids. Then, the FD method determines the
potentials at the nodes of the rectangular mesh in both
directions. The DC resistivity forward modeling was
performed with the synthetic data shown in Fig. 1ae
generated over the 2D resistivity structures. The
potential differences were computed for the three
electrode arrays, namely the Wenner-Schlumberger
(Wsc), dipole-dipole (Dpd) and pole-dipole (Pdp)
arrays. In all the simulations, 40 surface electrodes at
an electrode spacing of 1 m were used.
In this study, five synthetic models were used to
simulate a typical field survey and test the image
classification approach for the integration of different
electrode arrays images. These include:
1. A resistivity block prism (target) with a resistivity
value of 500 X-m (light blue) embedded in a
homogeneous medium of 10 X-m (deep blue)
(Fig.1a).
2. Two resistivity blocks having resistivity values of
100 X-m (green) and 300 X-m (orange) embedded
in a homogeneous medium of 10 X-m (deep blue)
(Fig.1b).
3. Three blocks with resistivity values of 100 X-m
(left), 300 X-m (middle) and 500 X-m (right) all
embedded in a homogeneous background of 10 X-
m (Fig.1c).
4. A vertical fault juxtaposing a conductive top layer
(hanging wall) with resistivity of 10 X-m (blue)
and a resistive bottom layer (foot wall) with
resistivity of 100 X-m (green) (Fig.1d)
5. A resistivity dyke of 500 X-m (light blue)intruding in a homogeneous background 100 X-m
(green) (Fig.1e).
3.1.2 Inverse Modeling
The determination of the model parameter (i.e.,
resistivity estimates) using the data space and model
space is known as inversion. Unlike forward model-
ing, inverse modeling is the mapping from the data
space to the model space (OLDENBURG and ELLIS
1991). Inversion is used to reconstruct the subsurface
resistivity distribution from the measurements of
voltage and current data. Inversions of the apparent
resistivity data sets from forward modeling were
carried out using the commercially available
RES2DINV software (GEOTOMO 2005). During the
inversion process, 5 % Gaussian noise was added to
reflect field conditions. The inversion of the apparent
resistivity data from forward modeling into 2D
resistivity models was carried out using RES2DINV,
a commercially available inversion program. Thissoftware program uses two different inversion rou-
tines for the generation of earth models from the
resistivity data. The inversion routines are based on
the blocky or L1-norm optimization (LOKE et al.
2003) and Gauss-Newton smoothness-constrained
least-squares method for L2-norm optimization (DEG-
ROOT-HEDLIN and CONSTABLE 1990; ELLIS and
OLDENBURG 1994). The L1-norm optimization method
Figure 1Synthetic resistivity models used for stimulation of a a resistive
block,b two resistive blocks, c three resistive blocks, d a vertical
fault and e a resistive dyke
c
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allows earth resistivity models with relatively sharp
variations across boundary resistivity to be recon-
structed as it tries to minimize the sum of absolute
values of the models resistivity. It is a better
optimization choice when geological discontinuities
are expected (SEATON and BURBEY 2002; LOKE et al.
2003). On the other hand, the L2-norm based on the
least-squares optimization method attempts to mini-
mize the sum of squares of the spatial variations in
the resistivity of the models and the data misfit
leading to earth resistivity models with gradual
transitions across zones of different resistivities
(LOKE et al. 2003). Although the default inversion
routine used by this program is based on an L2-norm,
both optimization techniques were used initially, but
the L2-norm was observed to have smearing effects
on the boundaries (LOKE et al. 2003) of the models.As a result, the L1-norm was preferred throughout the
inversion process in the synthetic examples.
3.1.3 Regularization Parameters
Inverse problems are ill posed because of the
nonuniqueness resulting from sparse noisy data and
the need to finely parameterize the earth so that
enough variation is allowed in the solution (MEJU
1994; HABER and OLDENBURG 2001). To overcome ill-
posedness in the inversion, regularization schemesare usually applied (DOESTCH et al. 2010). Examples
of the regularization constraints that allow for
stabilization of the inversion process are damping
and smoothing (BLOME et al. 2011). The common
perspectives in these regularization schemes are that
the data are given and inversion should produce a
single model that fits the data (ELLIS and OLDENBURG
1994). A 2D earth model is parameterized by means
of a grid of rectangular prisms, each having a uniform
resistivity. As a tradition, each block is made smaller
than the data resolution length so that the position of
the block boundaries does not affect the final model.
The regularization parameter is a trade-off between
data misfit and model roughness as a large value
results in a smooth model and weak data misfit, while
a small value leads to a highly rough model with
good data misfit (TIKHONOVand ARSENIN 1977; LOKE
et al. 2003). Several methods exist to compromise
data misfit and model constraints (FARQUHARSON and
OLDENBURG2004). In order to obtain a solution for the
inverse problem, an important factor to consider is
the damping factor (k), which depends among other
factors on the distribution of the resistivity data.
An estimation of the damping factor is based on
the trial and error method (LOKE and BARKER 1996;
OLAYINKAand YARAMANCI 2000). The selection of an
appropriate damping factor is important for subsur-
face imaging inversion in order to avoid local
minima. Using low damping values could lead to
low data misfit, but produce high instability in the
values of the estimated resistivity between adjacent
cells; this in turn could result in erroneous assump-
tions of the model resistivity in the area under study
(LOKEet al.2010). To establish an optimum value of
the damping factor that gives the best inversion
results after successive iterations, OLAYINKA andYARAMANCI (2000) suggest the use of a damping
factor in the range of 0.050.25 with a 20 %
increment between successive iterations. Also, Eas-
ton (1987) opined that higher damping values should
be used at the beginning and lower the damping
values on subsequent iterations.
In this article, the suggestion provided by OLAY-
INKA and YARAMANCI (2000) in respect to the
value(s) for the damping factor for models inversions
was adopted. The inversions were carried out with an
initial damping factor of k =0.25, which was
reduced to a minimum of 0.015 after successive
iterations. Tables1and2display the data points for
the three electrode arrays used for providing infor-
mation about the inverse models and parameters used
for the resistivity inversion with the RES2DINV
software package, respectively. The propagation
factor, n, for the dipole-dipole array increases from
1 to 6 (Table1). It is advisable not to use n values
greater than 6 for the dipole-dipole array in order to
increase the depth of investigation because of theweak signal strengths of this array (LOKE 2009).
Table 1
Modeling parameters for the three electrode arrays used
Parameter/electrode array Dpd Wsc Pdp
Number of data points 308 253 319
Number of model layers 6 8 8
Number of blocks 178 204 240
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3.2. Assignment of Resistivity Without Clustering
The reconstructed inverse resistivity values of the
three electrode array configurations were combined
using Eqs.4,5,6and7. This combination yields four
new images (i.e., minimum, maximum, median and
average), which are illustrated as:
Rminxi;yi minRdpd;Rwsc;Rpdp 4
Rmaxxi;yi maxRdpd;Rwsc;Rpdp 5
Rmedxi;yi medRdpd;Rwsc;Rpdp 6
Ravgxi;yi Rdpd Rwsc Rpdp
3
7
where Rdpdxi;yi; Rwscxi;yi andRpdpxi;yi are the
resistivity values for the dipole-dipole, Wenner-Sch-
lumberger and pole-dipole electrode arrays. Also, for
the models images, the mean resistivity values
denoted by Rmax, Rmin, Rmed and Ravg for the maxi-mum, minimum, median and average images,
respectively, were assigned to the blocks and back-
ground of the models.
3.3. Clustering and Classification Procedures
The 2D post-inverse resistivity models saved in
text files format, i.e., as xyz, where x represents the
electrode position,y is the depth of investigation, and
z represents the resistivity values, were exported into
an image-processing environment using the PCI
Geomatics software for further manipulations and
analysis. The PCI Geomatics used throughout this
article has a variety of functionality that enables users
or analysts to get more from imagery in support of a
wide range of geospatial applications. It is used in the
remote sensing environment for imagery processing,
geographic information systems and photogrammetry
(PCI2001).
To implement the k-means algorithm, the follow-
ing input parameters were used in addition to the
three input images from the electrode configurations:
(1) the number of clusters, which is usually a priori
defined, was 16. To avoid the difficulty in the
selection of an appropriate number of clusters, in this
article the prior information obtained during the 2Dinversion process was used to specify the number of
clusters, k, for the clustering. By this an objective
evaluation measure to suggest suitable values for the
number of clusters, thus avoiding the need for trial
and error, is implied. Moreover, the use of prior
knowledge from the inversion results provided an
automatic decision rule eliminating the problems of
human subjectivity. This was to ensure that adequate
information needed for the interpretation of the
model was obtained: (2) the stopping criteria (i.e.,
the number of iterations and the convergence thresh-
old were set to 20 and 0.01, respectively). After
convergence, thek-means algorithm creates an output
image file with a thematic raster layer as a result of
clustering, and this is stored in the PCIDSK program.
The PCIDSK is a data structure for holding images
and related data. When this convergence threshold is
attained, the program terminates. Prior to classifica-
tion of the combined images, we masked the region
of the models images that did not have data points
manually. This was necessary to ensure that only thecoregistered part of the images was classified.
3.3.1 Assignment of Resistivity to Clusters
Each cluster formed is associated with three resistiv-
ity values from the three electrode array
configurations. These resistivity values are denoted
by Rc1, Rc2 and Rc3 in contrast to the resistivity
assigned to the images obtained without clustering in
Table 2
Summary of parameters used during 2D resistivity inversions
Initial damping factor 0.25
Minimum damping factor 0.015
Convergence limit 1
Minimum change in absolute error \10 %
Number of iterations 38Jacobian matrix is recalculated for the first
two iterations
Increase of the damping factor with depth 1.05
Robust data inversion constraint is used with
the cutoff factor
0.05
Robust model inversion constraint is used with
the cutoff factor
0.005
Extended model is used
Effect of side blocks is not reduced
Normal mesh is used
Finite difference method is used
Number of nodes between adjacent electrodes 4
Logarithm of the apparent resistivity used
Reference resistivity used is the average of minimum
and maximum values
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Sect.3.2. To assign a single resistivity value, we used
the minimum, maximum, median and average ofRc1,
Rc2 and Rc3 represented by Rcmin; Rcmax; Rcmed and
Rcavg, respectively, using Eqs. 8, 9, 10and11.
Rcminxi;yi minRc1;Rc2;Rc3 8
Rcmaxxi;yi maxRc1;Rc2;Rc3 9
Rcmedxi;yi medRc1;Rc2;Rc3 10
Rcavgxi;yi Rc1 Rc2 Rc3=3: 11
Also, overall resistivity values denoted by
Rcmin; Rcmax; Rcmed and Rcavg were assigned to the
blocks and background of the classified images. The
final step in the classification campaign was merging
of clusters.
Another important stage in image classification
process involves aggregation or merging of clusters in
the classified images and then ascribing the merged
clusters to desirable classes according to similarity in
their spectral properties. To this end, merging was
confined to spatially adjacent clusters by comparison
of one cluster with its neighbor(s) and assigning it to
the same class on the basis of their pixel values. This
was carried out through the class merging submenu
available at the Geomatica Focus window of the PCI
software. Also, the spatial context (i.e., boundary) of
the individual pixels can be used. This is particularlyrelevant when the spectral signatures of the clusters are
reasonably similar. Thus, as the clusters were merged,
the data structure of the classified image was updated
by reestimating the resistivity value of each class.
3.4. Accuracy Assessment
After assigning resistivity values to the clusters,
an assessment of the accuracy/performance of the
classified images was carried out. This was carried
out relative to the true model images. There are many
criteria for assessing the performance of models
images in the literature. These include: R-square,
semi-partial R-square, absolute error (AE), root-
mean-square, standard deviations, mean absolute
error (MAE) and mean absolute percentage error
(MAPE). In this article, however, accuracy assess-
ment being an important step in the classification
process is necessary in order to evaluate the quality of
the classified images. With this end in view, the error
matrix or contingence tables were used for quantita-
tive evaluation of the images. The error matrix table
displays statistics for assessing the accuracy of
classification images using the number of pixels that
represent the features in the image (SINGH 1989;
CONGALTON and GREEN 1999; LILLESAND and KEIFER
2000; LIUet al. 2003; MELGANIand BRUZZONE 2004).
To construct the error matrix table, every pixel in the
classified images was compared with the pixel in the
reference images (i.e., true models) for the blocks and
background of the models images. Then, percentages
obtained from the table gave the accuracies of the
classification (CONGALTON 1991; LILLESAND and KEIF-
ER 2000; LIU et al. 2003; ENDERLE and WEIH 2005;
HASMADI et al.2009; PERUMALand BHASKARAN 2010).
In interpreting classification accuracy, it is importantnot only to note the percentage of correctly classified
pixels but also to determine the nature of errors of
omission and commission on a pixel-by-pixel basis.
Producers accuracy (PA) is obtained by dividing
the number of correctly classified pixels in each
category by the total number of pixels in the
corresponding column. Users accuracy (UA) is
computed by dividing the number of correctly
classified pixels by the total number of pixels in the
corresponding row (i.e., it is concerned with what
percentage of the cluster has been correctly classi-
fied). The PA and UA were obtained using Eqs. 12
and13, respectively, as follows:
PA % 100 error of omission % 12
UA % 100 error of commission % : 13
In these equations, both the PA and UA are
related to errors of omission and commission,
respectively. Pixels that are incorrectly excluded
from a particular cluster are defined as an error ofomission, while pixels that are incorrectly assigned to
a particular cluster that actually belong in other
classes are defined as an error of commission
(BOSCHETTI et al. 2004).
Also, the percentage of overall accuracy (OA)
was estimated using Eq. 14given by:
O:AU
W 100 % 14
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where U is the total number of correctly classified
pixels in the same category (i.e., sum of diagonal
pixels) and W is the total number of pixels (i.e., sum
of pixels in a row or column) in the classified image.
There are situations where the correct classification is
made by chance. In this case the most widely used
statistics for the estimation of the effect of change
agreement is known as the kappa coefficient (j). The
kappa coefficient was calculated using Eqs.15 and
16a,16bas follows:
ja1 a2
1 a215
a1
PNi1
xij
N16a
and
a2
PNi1
yizi
N2 16b
wherea1 is the overall accuracy, a2is the percentage
of items that has been classified correctly by chance,
xij is the number of pixels in the ijth cell of the
contingence table, yi marginal total of row,
zi =marginal total of column, N =the total number
of pixels in the image under consideration.
Furthermore, the AE was computed by making
pixel-by-pixel comparison between the true models
and the combined images. With this end in view,
both MAE and MAPE were calculated. First, the
absolute error (AE) of the model features (i.e., for
the blocks and background) was computed. This was
carried out by estimating the difference in resistivity
between the true model and classified image on a
pixel-by-pixel basis. For all the pixels within the
blocks, for instance, MAE was estimated as the
average of all AEs. Also, the MAPE was estimatedfor the blocks and background of the models
images. AE, MAE and MAPE are given in Eqs.17,
18, 19 as follows:
A:E xi; yi qi qij j 17
MAE 1
N
XNi1
qi qij j 18
MAPE 1
N
XNi1
qi qiqi
100 19
whereqi is the true resistivity of the block(s) and qi is
the calculated resistivity of the block; N is the total
number of pixels in the image.
4. Results and Discussion
4.1. Synthetic Results
A tabulated set of reconstructed resistivity values
for each block and background of the models in
Fig.2 are presented in Table3. The images
obtained using Eqs.4, 5, 6 and 7 to merge the
resistivity values of the three individual electrode
arrays (i.e., the dipole-dipole, pole-dipole and Wen-
ner-Schlumberger) are shown in Fig.2dg. It is
observed from the table that for all the models, the
reconstructed resistivity values of the blocks are
underestimated compared to the resistivity of the
true models, while for the background the recon-
structed resistivity values are an overestimation. In
essence, the amplitude of resistivity has been
dampened and is a known feature inherent in
resistivity imaging (RUCKER 2012). For the individ-
ual electrode arrays, it is observed that the dipole-dipole images give better results than Wenner-
Schlumberger and pole-dipole images. This means
that the reconstructed blocks resistivity for dipole-
dipole images is closer to the true models resistivity
than the resistivity for Wenner-Schlumberger or
pole-dipole images. Also, for the combined images,
the reconstructed resistivity of the maximum images
is higher than that of the remaining images (i.e.,
minimum, median or average). Thus, the maximum
outperforms any of the other combination as well as
the individual images in attempting to reproduce thetrue models.
The classification results after the pixels (i.e.,
resistivity values) had been grouped into 16 clusters
by the classifier algorithm together with their mean
resistivity values are presented in Table4. It is seen
that each cluster has three mean resistivities associ-
ated with it. These resistivity values show that each
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cluster contains information about the three electrode
configurations used. Furthermore, a gradual increase
in resistivity from one cluster to another is noted.
This gradual increase might be due to the electrical
contrast between the blocks and the background of
the models images as the classifier recognizes
certain patterns in resistivity that are grouped to aparticular cluster and other patterns to another cluster.
Also, the classification results of the combined
classified images obtained for the model of a single
block embedded in a homogeneous background are
shown in Fig.3ad. The rest of the classified images
containing thematic information are shown in Figs. 4,
5, 6and7.
The aggregation of the clusters into classes shows
that, for a single block model, cluster 1 is assigned to
the background and clusters 216 to the block leading
to a model with two classes. Class 1 corresponds to
the background and class 2 to the block (Fig. 3). For
the two-block model, clusters 24 are assigned to the
first block (left), clusters 516 to the second block
(right) and cluster 1 to the background. As a result
three classes are produced, and each class corre-sponds to a particular feature of the model (Fig. 4).
The three-block model images are such that cluster 1
is assigned to the background, clusters 24 are
assigned to the left block, while clusters 5 through
10 are assigned to the middle block; clusters 1116
are assigned to the right block with a total of four
classes representing this model. Class 1 represents the
background, while classes 24 represent the blocks of
the model (Fig.5). In Fig.6, clusters 13 are
Figure 2Two-dimensional inverse images of a block model for individual arrays: a Dpd,b Pdp,c Wsc and combined images,d max,e min,fmed and
g avg
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assigned to the conductive top layer, and clusters
416 are assigned to the resistivity bottom layer. As a
result, two classes are obtained for the fault model:
class 1 is typical of the top layer, and class 2
corresponds to the bottom layer of the model. In the
resistivity dyke model, clusters 216 are assigned to
the intrusive block, while cluster 1 belongs to the
background. This implies that only two classes are
desirable to represent these models. Again, class 1
represents the background, while class 2 represents
the intruding block (Fig.7). In consequence, the
number of clusters aggregated/merged into a partic-
ular class depends on the structural features identified
in the models image.The overall resistivity obtained after assigning the
mean resistivity values obtained in Table3 to the
blocks and background of the classified images is
summarized in Table5. From this table, two deduc-
tions could be drawn. For instance, for a one-block
model type (1) the resistivity values range from a
minimum of 180 X-m to a maximum of 340 X-m.
With this range, it shows that after classification of
the images, the overall resistivity values obtained for
Table 3
Summary of reconstructed resistivity for the different models
without classification
Model
type
True model Reconstructed mean resistivity (X-m)
Resistivity
(X
-m)
Dpd Pdp Wsc Rmax Rmin Rmed Ravg
Oneblock
Block (500) 320 250 160 330 160 260 250Background
(10)14 13 12 11 13 12 12
Twoblocks
Block 1(100)
110 100 87 120 89 110 110
Block 2(300)
360 330 260 370 260 330 320
Background(10)
17 18 15 19 15 17 17
Threeblocks
Block 1(100)
59 57 53 61 55 59 58
Block 2(300)
170 130 84 180 83 140 130
Block 3
(500)
240 150 120 250 130 150 180
Background(10)
18 16 14 18 13 15 16
Fault Hanging wall(10)
13 12 13 14 12 13 13
Foot wall(100)
95 93 92 100 86 92 93
Dyke Block (500) 470 460 460 470 460 460 460Background
(100)100 100 100 100 100 100 100
Table4
Summaryofclustering
forsyntheticmodelswithmeanresistivity
inX-m
ModeltypeArray
type
Cluster
1
Cluster
2
Cluster
3
Cluster
4
Cluster
5
Cluster
6
Cluster
7
Cluster
8
Cluster
9
Cluster
10
Cluster
11
Cluster
12
Cluster
13
Cluster
14
Cluster
15
Cluster
16
Oneblock
Dpd
9
29
58
90
120
150
190
220
300
340
390
340
380
420
430
460
Pdp
9
24
45
68
92
120
140
170
170
220
240
260
290
320
340
360
Wsc
8
27
41
56
72
85
100
120
130
160
160
180
200
210
220
220
Twob
locks
Dpd
9
37
71
110
140
190
230
280
320
360
400
450
480
520
560
590
Pdp
10
41
72
100
130
180
220
260
300
340
370
410
440
470
500
530
Wsc
9
29
51
76
100
120
160
190
220
250
290
310
350
380
410
440
Three
blocks
Dpd
10
10
29
52
68
90
120
140
170
180
210
220
250
280
310
340
Pdp
10
10
25
43
63
60
82
100
130
120
160
140
160
170
180
140
Wsc
10
11
21
34
59
41
61
77
79
110
100
120
130
130
140
140
Fault
Dpd
11
19
30
41
52
62
72
82
91
99
110
100
110
110
110
110
Pdp
12
17
25
33
42
51
59
67
76
86
95
110
120
120
120
130
Wsc
11
16
24
32
40
48
56
64
73
82
91
110
120
130
140
140
Dyke
Dpd
100
120
150
180
200
230
260
290
310
340
360
390
420
440
470
500
Pdp
100
120
150
170
200
230
260
290
310
350
370
400
430
460
480
490
Wsc
99
130
150
180
210
240
270
290
320
350
370
400
430
470
480
490
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the maximum images are closer to the true models
resistivity (i.e., 500 X-m) than the resistivity values
obtained for the rest of the models images. Also, the
rest of the models images show similar trends, and
(2) in comparison with the models images obtained
without classification, it is noted that there is anincrease in resistivity values of the combined classi-
fied images resulting from the unsupervised
technique. This could suggest that the unsupervised
classification technique finds patterns or features
residing in the original models that might not be
obviously known by any of the individual electrode
array configurations before the unsupervised classi-
fication technique is utilized (ERNENWEIN2009; RANA
et al. 2010).
Furthermore, the error matrices generated for
statistical evaluation of the performance of the
classified images relative to the true models are
presented in Tables6,7,8,9and10. PA indicates the
degree to which the reference pixels are classified
(LILLESAND and KEIFER 2000). UA indicates theprobability that a given cluster is actually present
on the ground (LILLESAND and KEIFER 2000). Gener-
ally, both the users and producers accuracies
indicate good classification of the models as the
errors of commission and omission during the
classification were less than 40 %. The accuracy
assessment from the unsupervised classification tech-
nique further shows that the overall percentage
accuracy is greater than 70 %. Overall accuracy
Figure 3Two-dimensional integrated classified images for a resistivity block model: a max, b min, c med and d avg
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evaluates the percentage of cases correctly classified,
i.e., the probability that a pixel is classified correctly
in the classified image. This indicates a good
agreement between the classified images and the true
model images used as reference data. According to
LANDIS and KOCH (1977), the ranking of the kappa
coefficient (j) ranges from 1 to 1. As a result, three
groups can be obtained: (1) j greater than 0.8
represents strong agreement between the classifica-
tion and the true models images; (2) j between 0.4
and 0.8 represents moderate agreement; (3) j less
than 0.4 represents poor agreement between the
classified and the true model images. By this
standard, j shows that for the one-, two- and three-
block images, substantial agreement exists between
the classified images and true models with kappa
accuracy from 76 to 83 %, while for the fault and
dyke models, the j values are 90 and 93 %,
respectively. This implies that there is perfect agree-
ment between the classified and true model. Thus, j
compensates for chance agreement in the classifica-
tion and provides a measure of how much better the
classification performs in the probability of randomly
assigning pixels to their correct classes.
Figure 4Two-dimensional integrated classified images for the resistivity two-block model: a max, b min, c med and d avg
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In addition, an assessment of the accuracy of the
combined images, employing both MAE and MAPE,
shows that for all the synthetic models, the maximum
resistivity values assigned to the models give the least
errors, as presented in Tables 11and12, respectively.Thus, the maximum approach is considered the best
representative of the true models as it reproduces the
true model with the least possible errors compared to
the other models images. Although our focus in this
article was on classified images, more details on the
quality of images obtained without clustering in
terms of MAE and MAPE for the synthetic models
can be found in ISHOLAet al. (2014).
4.2. Application to Field Examples
As feasibility tests of the application of the
aforementioned unsupervised classification tech-
nique on synthetic data, we also extended the
clustering technique to real field data. In fieldexample 1, electrical resistivity imaging was con-
ducted to delineate subsurface lithological units that
control the hydrogeology of the study area for
groundwater resources development. Also, in field
example 2, electrical resistivity imaging was applied
to demarcate regions of leachate infiltration in a
septic field.
Figure 5Two-dimensional integrated classified images for the resistivity three-block model: a max, b min, c med and d avg
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4.2.1 Field Example 1
A 2D electrical resistivity survey was performed at
Universiti Sains Malaysia Engineering campus, Ni-
bong Tebal, in Penang, Malaysia. The ABEM Lund
Imaging system comprising a Terrameter SAS 4000
supplemented with an automatic multielectrode
selector ES 10-64C system was used in the acquisi-
tion of electrical resistivity data. A resistivity profile
of about 420 m was established. Along this profile,
four cable reels, each 100 m long and trending in the
west-east direction, were used. Sixty-two takeouts
electrodes were deployed at 5-m spacing for the two
inner cable sets (100300 m), i.e., short (S) and 10-m
electrode spacing for the outer cable sets, i.e., long
(L) corresponded to distances of 0100 m and
300400 m on the standard 400 m (ADIAT et al.
2013). Two cable joints were used to link the cables to
ensure continuity. The first cable joint was used to
connect cables 1 and 2, while the second cable joint
connected cables 3 and 4. The other terminals ofcables 2 and 3 were connected to the Terrameter used
for the measurements. The three electrode array
configurations used were the dipole-dipole (LS),
pole-dipole (LS) and Schlumberger (LS). A forward
modeling subroutine was used to calculate the appar-
ent resistivity values, and a nonlinear least-squares
optimization technique was used for the inversion.
Like in the synthetic example, the RES2DINV
program was employed. In the dipole-dipole array,
Figure 6Two-dimensional integrated classified images for a resistivity fault model: a max, b min, c med and d avg
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with a minimum electrode spacing of 5 m, a total of
770 data points were used for measurements of
resistivity (i.e., 414 data points for L, 356 data points
for S), for the pole-dipole array 792 data points (516
for L, 276 for S) and for the Schlumberger array 748data points (418 for L, 330 S). For all three arrays,
after the seven iterations, the inversion converged
with misfit error less than 20 %. The misfit errors were
for the dipole-dipole (17 %), pole-dipole (13 %) and
Schlumberger (10 %) arrays. Inversion results for the
noise-contaminated data sets were saved in text file
format in order to allow for further manipulation and
analysis using the image processing program package
of the PCI Geomatics 10.3 software.
The reconstructed tomographic images of the
three electrode array configurations are shown in
Fig.8ac. From a geophysical viewpoint, the geo-
logical structures delineated behave like a three-layer
model for all the three arrays. With the dipole-dipolearray, the inversion image (upper panel) shows that a
low resistivity layer is detected close to the surface.
The low resistivity zone has a surface layer thickness
between 3 and 10 m. This zone is underlain by a layer
with higher resistivity, while the third layer, which
seems to extend throughout the model, is character-
ized by an intermediate resistivity. Using the pole-
dipole array, the inverse image (middle panel) reveals
that the first layer is a low-resistivity pocket-like
Figure 7Two-dimensional integrated classified images for a resistivity dyke model: a max, b min, c med and d avg
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structure haloed by a more resistive layer that extends
downward to a depth of about 60 m. This layer
appears to be separated by a less conductive layer
with depth from 15 to 40 m. The Schlumberger array
image (lower panel) shows that the first layer is a
conductive zone that extends through the model. It
surrounds a more conductive pocket-like structure.
Also, a high-resistivity structure of finite extent is
observed, while the deeper portion of the inverse
image is a zone that corresponds with an intermediate
resistivity. For resistivity imaging, the difference in
the imaging abilities of the three electrode arrays
when applied to the geological model is underscored.
The difference in the imaging capability is controlled
by the array parameters electrode spacing (a) and
propagation factor (n) (DAHLIN and ZHOU 2004). We
should also mention that the differing effects of the
electrode arrays on the 2D inverse images are due to
the arrangement of the electrodes, spreading patterns
and density of the data.
The aforementioned electrical imaging empha-
sizes the nonuniqueness of the inversion process
(TARANTOLAand VALETTE1982; NARAYANet al.1994).
To reduce the ambiguity in the tomographic images,
the three 2D inverse resistivity images were com-bined using an unsupervised image classification
technique. The three images in Fig. 8ac were used
as inputs to the k-means algorithm alongside other
initial parameters used in the synthetic modeling
section. Also, like the synthetic examples, the number
of clusters into which the combined images should be
partitioned was set at 16, while the number of
iterations and convergence threshold where 20 and
0.01, respectively. Again, the prior knowledge of the
inversion results for these models guided the choice
of the number of clusters used. The clustering result
is presented in Table13. Following the assignment of
resistivity values to the clusters, the clusters were
merged into six desirable classes. The resulting
classification image with a superimposed borehole
lithologic log for the assignment of lithologies is
shown in Fig. 9.
Cluster 1 was assigned to class 1; clusters 2 and 3
were merged to class 2; clusters 46 were merged to
class 3. Also, clusters 7 and 8 were merged to class 4,
and clusters 912 were merged to class 5, whileclusters 1316 were assigned to class 6. This
aggregation/merger of the clusters corresponded with
the subsurface geological units of the study area.
Though the unavailability of resistivity log data (i.e.,
as ground truth) made it impossible to evaluate the
accuracy of the classified image using the error
matrix table as in the synthetic examples, neverthe-
less a qualitative approach involving the available
borehole lithologic log was used. The electrical
Table 5
Summary of overall resistivity for classified images
Model type Model description
(X-m)
Overall resistivity (X-m)
Rcmax Rcmin Rcmed Rcavg
One block Block 1 (500) 340 180 260 260Background (10) 14 11 12 13
Two blocks Block 1 (100) 130 92 120 110
Block 2 (500) 380 260 330 320
Background (10) 19 15 17 16
Three blocks Block 1 (100) 74 59 67 67
Block 2 (300) 200 90 140 150
Block 3 (500) 250 150 160 190
Background (10) 12 5 8 8
Fault Hanging wall (10) 14 11 12 13
Footwall (100) 100 86 93 93
Dyke Block 1 (500) 480 460 470 470
Background (100) 100 100 100 100
Table 6
Error matrix/contingency table for a one-block classified image on
a pixel-by-pixel basis
j = 0.80 Block 1 Background Row total UA
Block1 108,658 6,568 115,226 0.94
Background 51,351 751,003 802,354 0.99
Column total 160,009 757,571 917,580
PA 0.68 0.99 O.A = 0.94
Table 7
Error matrix/contingency table for two-block classified image on a
pixel-by-pixel basis
j = 0.76 Block
1
Block
2
Background Row
Total
UA
Block 1 54,965 11,775 37,360 104,100 0.53
Block 2 0 56,136 10,298 66,434 0.85
Background 2,035 89 690,060 692,184 0.99
Column
total
57,000 68,000 737,718 862,718
PA 0.96 0.83 0.94 O.A = 0.93
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contrast in the geologic units delineated, on corrob-
oration with the representative borehole lithologic log
information, showed that the geology of the area
consists of clay, sandy clay, clayey sand and sand.
The range of resistivity values to the depth of
investigation was between 5 and 55 X-m. The
relationship that links the various classes with the
litho-resistivity values for the subsurface character-
ization of the study area is summarized in Table 12.
A conductive anomalous body that presumes to be aniron material was identified as class 1. This material
is buried in a less conductive layer that corresponds
with clay (class 2). Class 3 corresponds with sandy
clay. Because it is less cohesive, the groundwater
potential of this layer might be low. The promising
zones for probable groundwater exploitation in the
area are clayey sand (i.e., class 4) and sand units
belonging to classes 5 and 6. In general, the
Table 8
Error matrix/contingency table for the three-block classification model on a pixel-by-pixel basis
j = 0.83 Block 1 Block 2 Block 3 Background Row total UA
Block 1 28,381 18,562 10,989 55,014 112,946 0.25
Block 2 0 20,475 25,256 2,260 47,991 0.43
Block 3 0 2,892 7,255 0 10,147 0.71Background 1,619 571 0 689,447 691,637 0.99
Column total 30,000 42,500 43,500 746,721 862,721
PA 0.95 0.48 0.17 0.92 O.A = 0.86
Table 9
Error matrix/contingency table for a fault classified image on a
pixel-by-pixel basis
j = 0.90 Top layer Bottom layer Row total UA
Top layer 37,408 35,050 72,458 0.52
Bottom layer 2