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Seismic data interpretation using the Hough transform and principal component analysis
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IOP PUBLISHING JOURNAL OF GEOPHYSICS AND ENGINEERING
J. Geophys. Eng. 8 (2011) 6173 doi:10.1088/1742-2132/8/1/008
Seismic data interpretation using theHough transform and principalcomponent analysis
M G Orozco-del-Castillo1, C Ortiz-Aleman1, R Martin2,
R Avila-Carrera1 and A Rodrguez-Castellanos1
1 Instituto Mexicano del Petroleo, Eje Central Lazaro Cardenas 152, Mexico, DF 07730, Mexico2 Universite de Pau et des Pays de lAdour, CNRS & INRIA Magique-3D, Laboratoire de Modelisation
et dImagerie en Geosciences UMR 5212, Avenue de lUniversite, 64013 Pau Cedex, France
E-mail: [email protected], [email protected], [email protected], [email protected] and
Received 19 February 2010
Accepted for publication 9 November 2010
Published 9 December 2010
Online at stacks.iop.org/JGE/8/61
Abstract
In this work two novel image processing techniques are applied to detect and delineate
complex salt bodies from seismic exploration profiles: Hough transform and principal
component analysis (PCA). It is well recognized by the geophysical community that the lack
of resolution and poor structural identification in seismic data recorded at sub-salt plays
represent severe technical and economical problems. Under such circumstances, seismic
interpretation based only on the human-eye is inaccurate. Additionally, petroleum fielddevelopment decisions and production planning depend on good-quality seismic images that
generally are not feasible in salt tectonics areas. In spite of this, morphological erosion, region
growing and, especially, a generalization of the Hough transform (closely related to the Radon
transform) are applied to build parabolic shapes that are useful in the idealization and
recognition of salt domes from 2D seismic profiles. In a similar way, PCA is also used to
identify shapes associated with complex salt bodies in seismic profiles extracted from 3D
seismic data. To show the validity of the new set of seismic results, comparisons between both
image processing techniques are exhibited. It is remarkable that the main contribution of this
work is oriented in providing the seismic interpreters with new semi-automatic computational
tools. The novel image processing approaches presented here may be helpful in the
identification of diapirs and other complex geological features from seismic images.
Conceivably, in the near future, a new branch of seismic attributes could be recognized by
geoscientists and engineers based on the encouraging results reported here.
Keywords: diapirs, Hough transform, image processing, feature extraction, salt bodies, seismic
exploration, profiles, pattern recognition
1. Introduction
The analysis of seismic data is important for understanding the
subsurface of the earth, but the process is usually not a simple
task. Because of the growing volume and resolution of seismic
data, digital image processing (DIP) is becoming an importantcomponent of this process, aiming to detect geological features
in seismic volumes (3D) without the full assistance of an
interpreter. Several advances in DIP have been made in
the computer science research area and in its application
to other areas, particularly medical images (Udupa 1999)
and detection, recognition and tracking of human features
(e.g. faces, heads, arms) (Turk and Pentland 1991). Theseadvances have not been fully applied to geophysical science,
1742-2132/11/010061+13$33.00 2011 Nanjing Geophysical Research Institute Printed in the UK 61
http://dx.doi.org/10.1088/1742-2132/8/1/008mailto:[email protected]:[email protected]:[email protected]:[email protected]:[email protected]:[email protected]:[email protected]:[email protected]://stacks.iop.org/JGE/8/61http://stacks.iop.org/JGE/8/61mailto:[email protected]:[email protected]:[email protected]:[email protected]:[email protected]://dx.doi.org/10.1088/1742-2132/8/1/008 -
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M G Orozco-del-Castillo et al
and we believe that automated and semi-automated seismic
interpretations are feasible and very useful.
Traditionally seismic analysis has been done by human-
eyed empirical interpretation of just some of the processed
2D slices of the whole 3D seismic volume. This usually
implies loss of information. Due to the increasing volume
and resolution of seismic data, along with the increasingcomputational power, direct processing and semi-automatic
interpretation of 3D seismic data are becoming more practical
(Jeong et al 2006). Seismic interpretation can be broadly
subdivided into two components (Cohen and Coifman 2002):
structural, which investigates the nature and geometry of the
subsurface structures, and stratigraphic, which investigates the
subsurface stratigraphy. A first step in seismic interpretation
usually consists of image segmentation, and it relies heavily on
the human visualization of sophisticated and complex images.
Seismic interpretation also involves feature discrimination and
visualization, both of which are fundamental to exploratory
data analysis in many other areas of science.
An early and significant contribution was the coherence
cube, proposed by Bahorich and Farmer (1995). Their work
has served other research areas related to the coherence
concept, like the robust coherence estimation algorithm based
on multiple traces with locally adapted similarity (semblance)
measures (Marfurt et al 1998). Another variant of the
coherence cube based on eigenanalysis of the covariance
matrix was proposed by Gersztenkorn and Marfurt (1999).
A practical survey of several variants of the coherence cube
algorithm can be found in Chopra (2002). Some other
approaches include a more efficient discontinuity measure
computation method using a normalized trace of a small
correlation matrix (Cohen and Coifman 2002). Applicationswith high-order statistics and supertrace techniques for more
accurate coherence estimationare presentedby Lu etal (2005).
Some efforts have been made to automatically detect and
classify geological features, such as seismic facies. Seismic
facies are groups of seismic reflections whose parameters(such
as amplitude, continuity, reflection geometry and frequency)
differ from those of adjacent groups (West et al 2002).
Seismic facies analysis involves two key steps: (a) their
classification and (b) their interpretation to produce a geologic
and depositional interpretation. West et al (2002) presented
an application of textural analysis to 3D seismic volumes,
combining image textural analysis with a neural networkclassification to quantitatively map seismic facies in 3D data.
A similar approach based on competitive neural networks for
the classification and identification of reservoir facies from
seismic data was presented by Saggaf et al (2003). Some
other automatic techniques for classification of seismic facies
include identification of the boundaries of rapidly varying sand
facies using a Bayesian linear decision function (Matlocket al
1985); determination of the sand/shale ratio of various zones
in the reservoir using discriminant factor analysis (Mathieu
and Rice 1969); identification of seismic facies using both
principal component analysis (PCA) and discriminant factor
analysis (Dumay and Fournier 1988); segmentation of a
seismic section basedon its texture through a knowledge-basedexpert system (Simaan 1991) and detection of anomalous
facies in data using a back-propagation neural network (Yang
and Huang 1991).Another line of investigation corresponds to faultsurfaces.
Where it is not required to extract the actual fault surfaces,some methods have been employed to enhance a fault structure
(Weickert 1999, Bakker etal 1999, Fehmers and Hocker 2003).
Despite the fact that filtering methods have proven useful to theinterpreters, there is still a significant human experience-basedinterpretation to be done after its application. Hence, there
has been a trend to automatically or semi-automatically detectfaults from seismic cubes in recent years. For example, Cohen
et al (2006) proposed a method for detecting and extracting
fault surfaces by creating and processing a volume of estimatesfrom seismic data which represents the likelihood that a given
point lays on a fault surface. Jeong et al (2006) developeda volumetric, seismic fault detection system aimed for an
interactive nonlinear 3D processing. This system was alsocombined with a graphics processing unit, showing thebenefits
over a CPU implementation.
Oil and gas prospecting has found a major challengein regions with complex geological settings, like areas withsalt tectonics. Broad areas exist in the world where seismicdepth imaging is a difficult task due to the progressive lack
of resolution beneath the presence of salt bodies. As oilexploration targets may be located close or below salt bodies,
in the underlying geologic structure, there is a growing interestin computational tools that can help seismic interpreters
to estimate geometry, position and depth distribution ofdiapirs from seismic profiles and volume data. Under
favourable circumstances, traditional seismic processing andinterpretation can provide an appropriate location of the top of
salt bodies. Nevertheless, estimation of the base of salt domes(and geometry distribution of salt at depth in general) is often adifficult task. In the application of standard seismic processing
techniques it is common to find complex wave diffractionpatterns giving rise to a significant lack of illumination near
and below salt bodies.A traditional and successful approach to seismic data
analysis has been the Radon transform (Moon et al 1986,Foster and Mosher 1992, Trad et al 2003). A very similar
approach in the field of DIP has been the Hough transform(Hough 1962) which, just like the Radon transform, is a
mapping from image space to a parameter space. The Houghtransform has been applied for error analysis (Aguado et al
2000, Jacquemin and Mallet 2005, Montana 1992, Niblackand Petkovic 1988, Rosenhahn et al 2001, Shapiro 1975,1978a, 1978b), reduction of the computational complexity
(Kiryati and Bruckstein 1992), extensions to other shapes(Aguado et al 1995, Ballard 1981, Van Ginkel 2002), choice
of the appropriate parameterization (Duda and Hart 1972, VanVeen and Groen 1981, Westin and Knutsson 1992), tracking
and pose estimation (Princen et al 1994), etc. The Houghtransform has also been successfully applied to seismology,
using a variant of this methodology based on a cascade oftwo Hough transforms and a specific backward transformation
to automatically extract faults from a 3D seismic cube(Jacquemin and Mallet 2005).
The detection of geologic patterns on seismic profilescan be thought of as a complex problem, i.e. a problem
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Seismic data interpretation using Hough transform and PCA
that is best suited for a human than a machine, at least
in the traditional use of computing power. Interpreters of
seismic data are usually able to detect patterns successfully
despite changes in the images due to noise and variations
from one body to another and even from one seismic profile
to another. A main problem for the interpretation of seismic
profiles is being able to automate or semi-automate the abilitiesof an interpreter using a computational method. Much of
the work in automated pattern recognition ignores the issue
of which aspects of the pattern stimulus are important for
identification. For example, one of the most important goals of
seismic stratigraphy is to recognize and analyse seismic facies
regarding the geologic environment. Two main problems
become evident: to determine which seismic parameters are
relevant for characterizing the facies and to be sure that there is
a link between the seismic parameters and the geologic facies
we are investigating (Dumay and Fournier 1988). Seismic
pattern recognition and detection of geological structures are
also very high level operations for which the classification
according to detailed geometry or specific information may
be very difficult, inefficient, and either way, probably useless.
When trying to detect seismic patterns, the interpreter does
not focus his or her attention on the geometric or quantitative
characteristics of them, but rather on their global features,
which may or may not be directly related to our intuitive notion
of features. In other words, the interpreter is able to detect the
principal components of the patterns, but he is probably not
able to tell which those components are. Interpretation of
seismic data is a very high level task. By these means, seismic
pattern recognition is very similar to other pattern recognition
tasks, like speech or face recognition. Face recognition is also
a very high level task, where humans perform considerablybetter than computers, especially for quantitative and statistical
methodologies whichattempt to detect individual features, and
define a face model by the position, size and relationships
among these features.
In this work we study the feasibility of applying two
distinct pattern recognition approaches as auxiliary tools for
seismic methods to improve detection of salt bodies and
determination of their complex geometry: a Hough transform
mathematical morphology approach and a PCA approach.
These two methodologies, whilenot entirely automatic, clearly
have the potential to reduce the workload of the seismic
interpreter by asking him to define how the salt bodies look,by either establishing some initial parameters or manually
selecting some training images, instead of looking for the salt
bodies manually throughout the whole seismic cube.
2. The Hough transform mathematical morphologyapproach
2.1. Overview of the Hough transform methodology
Hough (1962) originally proposed his methodology, currently
referred to as Hough transform, to detect straight lines with the
intention of finding bubble tracks. Consider a point (xi , yi )
and the explicit general equation of a line, yi = hxi + k. Aninfinite number of lines pass through (xi , yi ), and all of them
satisfy yi = hxi + k for different values of h and k. However,
writing this equation as k = yi hxi and considering the hk
plane (also known as the parameter space), the equation for a
unique line for a determined pair (xi , yi ) is obtained. Besides,
a second point (xj , yj ) also has a line in the parameter space
associated with it, and this one intersects the line associated
with (xi , yi ) in (h
, k
), where h
is the slope and k
is the y-intercept of the line that contains (xi , yi ) and (xj , yj ) in the xy
plane. In fact, all of the points contained in this line have lines
in the parameter space that pass through (h, k).
The main attraction for the use of the Hough transform
comes from the subdivision of the parameter space in what
are denominated accumulator cells, where (hmax, hmin) and
(kmax, kmin) are the expected range values for the slope and
the y-intercept, respectively. The cells with coordinates
(i, j), with an accumulator value A(i, j), correspond to the
square associated with the coordinates in the parameter space
(hi , kj ). Initially, these cells are set to zero. Then, for each
point (xp, yp) of the image plane, the parameter h is fixed to
each and every one of the allowed values of subdivision for
the x axis, and k is obtained from solving the corresponding
equation k = yk hxk . After this procedure, a value ofM
in A(i, j) corresponds to M points of the xy plane located in
the line y = hi x + kj . The precision of the collinearity of
these points is determined by the number of subdivisions of
the hk plane. When subdividing the x axis in P increments,
for each point (xp, yp), the P values ofkcorrespond to the P
possible values ofh. With n image points, this method implies
nP operations. Therefore, this procedure is linear in n, and nP
operations are easily calculated using a standard computer.
The Hough transform was originally defined to detect
straight lines in black and white images. As it is trivial togeneralize the Hough transform to other shapes and grey-
value images, we describe it in its extended form. We
set up an N-dimensional accumulator array; each dimension
corresponding to one of the parameters of the shape looked for.
Each element of this array contains the number of votes for the
presence of a shape with the parameters corresponding to that
element. Of course, if a shape with certain fixed parameters
is present in the image, all of the pixels that are part of it will
vote for it, yielding a large peak in the accumulator array. For
this particular example, we generalize the Hough transform
to detect parabolas, which in comparison with lines consist of
three different parameters. The use of three parameters insteadof two implies a dimensional increase in the parameter space
and consequently in the accumulator array, now being 3D and
no longer 2D as in the case of line detection.
2.2. Mathematical morphology
The field of mathematical morphology provides a wide range
of operators to image processing, all based around a few
simple mathematical concepts from set theory. The operators
are particularly useful for the analysis of binary images and
common uses include edge detection, noise removal, image
enhancement and image segmentation.
The two most basic operations in mathematicalmorphology are erosion and dilation. Both of these operators
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(a)
(b)
Figure 1. Original seismic profile (a) and the binarized segment of the profile used in this section, showing three distinct salt bodies (b).
take two pieces of data as inputs: an image to be eroded or
dilated, and a structuring element (also known as a kernel).
The two pieces of input data are each treated as representing
sets of coordinates in a way that is slightly different for binary
and greyscale images.
For a binary image, white pixels are normally
used to represent foreground regions, while black pixelsdenote background (note that in some implementations this
convention is reversed, as is the case in this work for visual
purposes). Then the set of coordinates corresponding to
that image is simply the set of two-dimensional Euclidean
coordinates of all the foreground pixels in the image, with a
coordinates origin normally taken in one of the corners so that
all coordinates have positive elements, i.e. each element of a
set belongs to N2.
Let A and B be two sets of N2, with components
a = (a1, a2) and b = (b1, b2), respectively. The translation of
A through x = (x1, x2), represented as (A)x is defined as
(A)x = {c|c = a + x, a A}. (1)
Let A denote a binaryimage andB denote a structuring element
(usually a square mask of values equal to 1; in this paper a
3 3 mask was used). Then the erosion ofA by B is given by
A B = {x|(B)x A}. (2)
So, the erosion of A by B is the set of all the points x so
that B, translated through x, is contained in A. This means
that, only when B is completely contained inside A, the pixel
values are retained, else they get deleted, i.e. they get eroded.
Although equation (2) is not the only definition for erosion, it
is normally the most adequate in practical implementations ofmorphology.
2.3. Detection of salt bodies using the Hough transform
In the segment of the seismic profile shown in figure 1, several
salt bodies with parabolic-like shapes can be observed. The
aim of this work was to develop a way to extract them from the
rest of the profile by featuring them as parabolas with defined
parameters. The proposed methodology consists of digitallyprocessing the original seismic profile using mathematical
morphology, such that the final product is appropriate to
analysis through the Hough transform.
2.3.1. DIP of the original seismic profile. The values in
this profile (figure 1(b)) were initially modified so it could
be seen as a greyscale image of 8 bits (0255), and later
binarized (thresholded to have values of 0 and 1) because the
techniques used further on for segmentation and recognition
are designed for binary images. For visual purposes, the black
pixels correspond to the value 1, while the white pixels to 0.
The standard generalized Hough transform can theoreticallybe applied to this image, but the quantity of information in
the image would considerably affect the time and the results
of the analysis, mostly because a great part of the information
does not represent the objects we are trying to extract. It was
therefore needed to apply an erosion operation that would
diminish the quantity of non-useful information, i.e. that
information not representing the salt bodies. When taking
into account that the non-useful areas are not so large in terms
of size (quantity of pixels), then the erosion operation results
are even more appropriate. In other words, the segments
of useful information are formed by large accumulations of
pixels, while the others are mostly formed by narrow lines or
small accumulations. Accounting for this, one should erodethe image until the non-useful information has disappeared;
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Seismic data interpretation using Hough transform and PCA
(a)
(b)
(c)
Figure 2. (a) The binarized seismic profile image after two binary erosion operations, ( b) the result of the region growing on it, and ( c) thefinal results of the region growing applied in (b) with the centroids calculated from (a).
we experimentally determined that nine erosion operations
are needed to achieve this (figure 2(a)). The inconvenience is
that, by its nature, the erosion operation also erases some ofthe relevant information. To avoid this issue, we decided to
implement and apply a region growing algorithm to recover the
lost informationdue to the erosion operations. Region growing
is a commonly used procedure that groups pixels or sub-
regions to greater ones. The simplicity of this method resides
in the pixel aggregation that starts with a set of generating
points (seeds) from which the regions grow by adding to each
of these pointsthe neighbour pointsthat have similar properties
(e.g. grey level, texture, colour) (Gonzalez and Woods 1992).
Since we are dealing with binary images, the region growing
algorithm we used is the simplest of its possible variations; it
simply looks for all of the black pixels that are connected
to the seed. The seeds were calculated as the centroids
of the remaining black areas after nine erosion operations
(figure 2(a)).
Applying region growing with these seeds to the originalimage recovers both useful and, due to the connectivity of the
original regions, non-useful information. To solve this issue,
we need to break the connectivity of the useful areas from the
non-useful ones in the original image. As we did before, this
can be achieved through erosion. At this stage, the erosion
operation is not intended to remove unwanted areas in the
image, but rather to break the connectivity of the useful from
the non-useful areas, so that the region growing algorithm can
be successfully applied. We experimentally determined that
two erosion operations are sufficient to achieve this effect, as
shown in figure 2(b). It is in this image in which we applied
region growing with the centroids calculated from figure 2(a),
and the final result is shown in figure 2(c). Notice how alarge portion of the non-useful information from the original
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M G Orozco-del-Castillo et al
Figure 3. Workflow of the Hough transform approach.
image has been eliminated, leaving the three areas of the image
corresponding to the three salt bodies that we intend to extract.
There remains a group of pixels in the lower part of the image
which do not correspond to a salt body. However, the curve
recognition is not affected by means of the Hough transform
nature. The process is illustrated in figure 3.
2.3.2. The Hough transform analysis for the detection of theparabolic-like top of salt bodies. We originally described
the Hough transform methodology in its original and most
commonly used form, applied for the detection of lines (whose
equation is given by yi = hxi + k). To achieve the recognition
of parabolas, the modification of the usual form of the Hough
transform is needed. The general equation of a parabola is
given by
y k =1
4m(x h)2, (3)
where the vertex is located at (h, k), focused at (h, k+ m) and
directrix y = k m, with m being the distance from the vertex
to the focus. For simplicity purposes, it is possible to rewriteequation (3) as
y k = a(x h)2, (4)
where a = 1/4m.
It is obvious that the parameter space (or Hough space)
that represents parabolas is a three-dimensional space formed
by the parameters (a, h, k), instead of the two-dimensional
space representing lines, formed by the parameters (h, k). This
implies that, to accomplish the spatial transformation, we need
to clear one of the variables and change the values of the other
two from a minimum to a maximum value for each one. For
example, clearing kfrom equation (4) we obtaink = y a(x h)2. (5)
Table 1. Parameters of the three parabolas found using the Houghtransform.
a h k
1 0.0025 170 1302 0.0025 534 1743 0.0047 1260 287
It is now necessary to vary the parameters a and h, from
amin to amax and hmin to hmax, respectively. Since the point
(h, k) represents the vertex of the parabola in the image, it is
obvious that hmin = 1 and hmax is the number of columns.
The values of amin and amax that control the aperture of the
parabola were determined experimentally (according to the
possible apertures of the pseudo-parabolas representing the
salt bodies) as amin = 0.002 and amax = 0.03, and the
intermediate values were distributed logarithmically, unlike h
and kwhich were distributed linearly. With these values, and
the possible values for k(from 1 to the number of rows in the
image), a three-dimensional parameter space is created with
accumulator cells where all possible parabolas from the black
pixels in the image shown in figure 2(c) are stored. After the
transformation process, a cube with dimensions corresponding
to the rows, columns and the number of intervals for a was
obtained. This cube contains in each cell the amount of
votes that the parabola received, with parameters given by
the location of the cell in the space. To extract those cells
with the greater amount of votes without extracting nearby
cells that represent different parabolas, but the same salt body,
a minmax clustering was applied. The detection of the
peaks in this 3D space is not a very complicated task since
the values are clearly above those in cells not representingparabolas; however, it could be possible that the extraction of
the parabolas from the parameter space is more complicated,
so some other kind of search could be needed. Montana (1992)
proposed a genetic search of a generalized Hough transform
space for the detection and tracking of a class of sonar signals,
which yielded very narrow peaks not too far above the random
background variations. He concluded that the genetic search
required far fewer evaluations to outperform an exhaustive
search algorithm.
The result of the clustering yielded the three parabolas
with the greatest number of votes. The parameters of these
parabolas are shown in table 1. These parabolas were overlaid
in the original images as shown in figure 4.
The application of this methodology to one seismic
profile yields the parameters for three different parabolas that
resemble the top of their corresponding salt body. Since the
salt bodies show similar characteristics across the different
profiles of the seismic cube, the successive application of the
aforementioned procedure would result in a set of parameters
for each salt body.
3. The PCA approach
We consider that the automatic detection of seismic patterns
should emulate how a human interpreter actually does it,i.e. a qualitative approach instead of a quantitative one. By
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these means, seismic pattern detection is very similar to other
high-level pattern detection tasks like face detection. One of
the most successful approaches to automatic qualitative face
detection was proposed by Turk and Pentland (1991). Using
a PCA approach, their near-real-time computer system was
able to locate and track a subjects head, and then recognize
the person by comparing characteristics of the face to those ofknown individuals. This is why we believe that an information
theory approach like PCA of coding and decoding seismic
patterns, in this particular case salt bodies images, maygive an
insight into the information content of them, emphasizing their
significant local and global features, rather than quantitative
ones.
Mathematically, we wish to find the principal components
of the distribution of the salt bodies, or the eigenvectors
of the covariance matrix of a training set of salt bodies
images, treating an image as a point (or vector) in a very
high dimensional space. The eigenvectors should then be
ordered, with each one accounting for a different amount of
the variation among the salt bodies images according to their
respective eigenvalues. These can be thought of as a set of
features that together characterize the variation between salt
bodies images. Each individual body in thetrainingset (minus
the average salt body) can be represented exactly in terms of a
linear combination of the eigenvectors. Each body can also be
approximated usingonly the best eigenvectors, i.e. those with
the largest associated eigenvalues, which therefore account for
the most variance within the set of salt body images. The
best M eigenvectors span an M-dimensional subspace of all
possible images (salt-body-space). The construction of this
PCA approach system consists of the following.
(1) Acquiring an initial set of salt bodies images (the training
set).
(2) Calculating the eigenvectors from the covariance matrix
of the training set, and keeping only the M images that
correspond to theMhighest eigenvalues. TheseMimages
define a subspace of the salt-body-space.
Once the construction has been achieved, the following steps
are then used to detect new salt bodies.
(1) Tracking of a seismic profile capturing input images.
(2) Projecting the input image onto the salt-body-space.
(3) Determining if the image is a seismic body by measuring
the distance of the original image (as a vector) to thesalt-body-space.
3.1. Theoretical background
Let a salt body image I(x,y) be a two-dimensional N N
array of intensity values. An image may also be considered as
a vector of dimension N2. A set of images maps to a collection
of points in this space. Images presenting the same pattern, in
our case salt bodies, will not be randomly distributed in this
space since they have a similar overall configuration, and can
be described by a lower dimensional subspace. The main idea
of the principal component analysis is to find the vectors that
best account for the distribution of salt body images within theentire image space. These vectors define the subspace of salt
body images, which we call salt-body-space. Each vector is
of length N2 and is a linear combination of the original salt
bodies images. Because these vectors are the eigenvectors of
the covariance matrixcorresponding to the original salt bodies
images, and because they are salt-body-like in appearance, we
refer to them as eigenbodies.
Let the training set of salt bodies images beB1, B2, B3, . . . , BM. The average body of the set is defined by
=1
M
M
n=1
Bn. (6)
Each body differs from the average by the vector
i = Bi . (7)
This set of very large vectors is then subject to principal
component analysis. We need to calculate the eigenvalues
and eigenvectors of the covariance matrix
C =1
M
M
n=1
nTn = AA
T, (8)
where matrix A = [1 2 ... M]. Matrix C, however, is N2
N2, and determining the N2 eigenvectors and eigenvalues
is an intractable task for typical image sizes. The problem is
solved by Turk and Pentland by calculating the eigenvalues
and eigenvectors of the M M matrix given by L = ATA,
instead of C = AAT, where the first (most significant)
eigenvectors and eigenvalues, are indeed the same. This
alternative methodology enables the near-real-time calculation
of the eigenvectors and eigenvalues. These vectors v determine
linear combinations of the M training set images to form the
eigenbodies ul ,
ul =M
k=1
vlk k l = 1, . . . , M . (9)
The eigenbody images calculated from the eigenvectors of Lspan a basis set with which to describe salt bodies images. In
this framework, identification becomes a pattern recognition
task. The eigenbodies span an M-dimensional subspace of
the original N2 image space. The M significant eigenvectors
of the L matrix are chosen as those with the largest associated
eigenvalues.
A new salt body image B is transformed into its
eigenbodies components (projected into salt-body-space)
by a simple operation,
k = uTk (B ), (10)
for k = 1, . . . , M . The weights form a vector
[1, 2, . . . , M ] that describes the contribution of each
eigenbody in representing the input salt body image, treating
the eigenbodies as a basis set for salt body images. This
process is equivalent to projecting the original salt body image
onto the low-dimensional salt-body-space. The distance
between the new input image and the salt-body-space is
simply the squared distance (Euclidean norm) between the
mean-adjusted input image and the salt-body-space f =Mi=1 i ui :
= (B ) f. (11)
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Figure 4. Original image with overlaid resulting parabolas from table 1.
Figure 5. Set of the 15 initial training images obtained manually from several profiles of the seismic cube after applying the Gaussianfunction in equation (12).
3.2. Application to seismic profiles
The purpose is to detect the salt bodies present in a seismic
cube. A profile of this cube is shown in figure 1(a). A
closer look at the seismic profile makes three salt bodies quite
evident (as shown in figure 1(b)). The detection of salt bodies
is sensitive to the background, i.e. the system is not able to
determine the limit of a salt body, so it is also taken into
consideration by the detection and recognition processes. To
deal with this problem without having to solve other difficult
image-processing problems such as the segmentation of the
salt body (which would also mean having to detect the body to
segment it, and having it segmented for its detection), we
performed a pointwise product of the input images (both
the training images and the new input images) by a two-
dimensional Gaussian window centred on the salt body, thus
diminishing the background and accentuating the middle of
the body. The two-dimensional Gaussian is given by
h(x,y) =e
(x2 +y2 )
2
2 2(12)
wherex andy correspond to the values of the rows and columns
of the image respectively, ranging between 1 and M(230), and
is the standard deviation or aperture of the Gaussian, which
was experimentally determined as 2.5. Greater values for
would tend to leave the image intact, while smaller ones
would drastically obscure the entire salt bodies, instead of just
the corners of the images which do not account to the bodies.
Therefore, the multiplied image would correspond to
Gij =Iij e
(i2 +j 2 )
2
2 2, (13)
where i and j correspond to the values of the rows and columns
respectively of the image, and I is the original image. We
assembled a set of 15 training images obtained manually from
several profiles of the seismic cube. Figure 5 shows these after
applying the Gaussian function to every one of them. While
these images were obtained manually by selecting areas of
different profiles showing salt bodies, the results from the
Hough transform methodology could be used as inputs for this
procedure to improve the automation of the detection of thesalt bodies throughout the seismic cube.
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Figure 6. The 15 best eigenbodies (ordered), corresponding to the 15 eigenvectors with the largest eigenvalues associated with them, ascalculated in equation (9).
(a) (b)
(d)(c)
Figure 7. Four different images and their projection on the salt-body-space. The left images on (a) and (c) show two images correspondingto salt bodies, and the left images on (b) and (d) correspond to other images in the same seismic cube. Notice how in ( a) and (c) theright-hand-side images are similar, while those on the left-hand-side, in (b) and (d), considerably differ.
The first step is to calculate the mean image of the
set of images, what we can call the average salt body.
Once we have the mean image, according to the procedure
described before, we are able to obtain the 15 (due to the
number of training images) greatest eigenvalues and their
associated (mostrepresentative) eigenvectors of the covariance
matrix given by equation (9). These vectors determine linear
combinations of the 15 training set salt bodies images to form
the eigenbodies shown in figure 6. Notice how the images in
the first row represent the general form of the bodies, while
those in the last row represent their details.
These eigenbodies are used to project an image into the
salt-body-space by the operation described in equation (10).
According to how an image is projected into this space, we can
know if the image corresponds to an element of the space, i.e.
represents a salt body. When a salt body image is projectedinto this space, the result is a very similar image, but if not
the projected image differs considerably from the original,
as shown in figure 7. The images in figures 7(a) and (c)
correspond to salt bodies and their projection on the salt-body-
space, meanwhile the others in figures 7(b) and (d) correspond
to the areas of the seismic profile that do not contain these
bodies, and their respective projections.
Since not all of the eigenvectors, and consequently the
eigenbodies, have the same contribution in representing the
input body image, it is advisable to take into consideration
only M of them because of processing-time issues, where M
is less or equal to the number Mof training images. The effect
ofM to an input image is shown in figure 8. The first image
is the original image; the following four are projections of this
image on the salt-body-space using values of 1, 5, 10 and 15,
respectively for M. It can be seen that the projections areadequate and fairly similar in all cases; therefore, the choice
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Figure 8. Different projections of the same image. The image on the left shows the original image, while the four images on the rightcorrespond to projections of the original image onto the salt-body-space using values of 1, 5, 10 and 15 for M, i.e. using 1, 5, 10 and 15eigenvectors, respectively.
(a)
(b)
(c)
Figure 9. Results of the image tracking. The seismic profile on which the methodology was applied is shown in ( a). The results of imagetracking are shown in (b) and (c), for values of 2 and 15 respectively for M. Notice how in (c) the bright spots corresponding to salt bodiesare more evident and there appear fewer bright areas than in ( b).
ofM is not critical in terms of performance, but is important
in terms of processing time.
3.3. Image tracking and parameter determination
The procedure explained so far allows recognizing whether an
image is a salt body or not, but in order to semi-automate
the recognition in seismic profiles/cubes it is needed to
show the system every possible image in them. There is
an obvious tradeoff between precision and time, but, not being
the main goal of this work, we decided to track a complete
seismic profile pixel by pixel, capturing all of the possible
images present in it and measuring the distance (according to
equation (11)) of its projection onto the salt-body-space
(equation (10)) to the salt-body-space itself. We do this ina horizontally rotated seismic profile, distinct from those used
for the training of the system (figure 9(a)), but similar enough
for the salt bodies to be evident in it. The results are shown in
figures 9(b) and (c); the first image shows the seismic profile,
while the other two show the results using values of 2 and
15 for M, respectively. Bright areas correspond to areas
where the appearance of salt bodies is more probable, i.e.
areas where the value of as calculated in equation (13) are
lower. The image using M = 2 shows a greater number of
bright areas overall than the one for M = 15 but, despite
these differences, they both show three very distinct bright
spots, each one corresponding to the salt bodies present in the
seismic profile.
Thefinal results forM = 3 anda pixel-by-pixel tracking of
the seismic profile are shown in figure 10(a). As stated before,
the bright areas correspond to areas where there is a greaterpossibility of finding a salt body. Despite several of these
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Seismic data interpretation using Hough transform and PCA
(a)
(b)
Figure 10. Image tracking results as an image and as a 2D function. The image in ( a) corresponds to the grey-level image (0255) obtainedby a pixel-by-pixel tracking using M = 3. The image in (b) shows the same results, but as a 2D function. Brighter spots in (a) correspond to
greater values in (b).
areas, there clearly appear three bright spots corresponding to
the three salt bodies in the seismic profile. The results are more
obvious if this image is plotted as a two-dimensional function,
as shown in figure 10(b). In this new image, the three bright
spots, corresponding to the three salt bodies, appear as peaks
in a relatively smooth image. The determination of how strong
or how great the peaks should be to represent a salt body and
how to find them in a dataset is a pattern recognition technique
that will depend on the task at hand. In this case, a simple
thresholding of the image successfully isolates the location of
the peaks from the rest of the image; for other cases, wherethe contrast of the peaks corresponding to the location of salt
bodies to the rest of the image could not be as obvious, an
image segmentation or clustering technique should be used.
This also holds true for the future automation of the salt body
detectionprocess in a seismic cube. This task of peak detection
is very similar to the detection of the cell with the greater
amount of votes in the Hough parameter space, which was
addressed earlier on. Peak detection in data is a relevant matter
of research, where several techniques have been proposed and
proved useful. Some approaches use shape information of
the histogram of the data to achieve multilevel thresholding
(Prewitt and Mendelsohn1966, Weszka etal 1974, Otsu 1979).
Kanungo et al (2006) proposed a segmentation techniquebased on both thresholding the data and applying genetic
algorithms for the location of peaks and valleys. Otherapproaches for peak detection correspond to data mining, suchas hierarchical clustering, partitioning relocation clustering,and density-based partitioning (Berkhin 2002). The choiceof which peak detection technique should be used in theautomation of the salt body detection process throughout thewhole seismic cube will depend on the characteristics of thedata obtained from the application of the PCA methodology,and is out of the scope of this work.
3.4. Extraction of salt bodies
For an appropriate interpretation of seismic profiles, it ishelpful to display the recognized patterns in three dimensions.In the previous sections we introduced a novel approach tosalt dome identification by using the Hough transform andPCA. By using these approaches we are able to identify theapproximate location of salt bodies inside a seismic profile.This location is used to infer coordinates of points inside thesalt bodies that will be used as seeds for 3D region growing in aseismic cube. Once the salt bodies are identified in the seismiccube and in order to obtain always closed forms for regiongrowing in 3D, we used conventional seismic interpretationin several profiles. Results from the application of region
growing to the extraction of the estimated salt bodies aredepicted in figure 11.
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Figure 11. Three-dimensional geometry of the recognized saltbodies using region growing on the seismic volume.
4. Conclusions
The Hough transform was successfully applied to seismic
profiles to detect parabolic shapes, which were associated with
the presence of salt bodies. We believe that this methodology
could be useful for the detection of other geologic structures,
including other salt bodies which do not present parabolic
shapes. The Hough transform could be generalized to detect
more complicated geometric structures like hyperbolas or
high-order polynomials; the biggest disadvantage would bethat the parameter space resulting from the application of the
generalized Hough transform would be represented by a higher
dimensional space. This can be prohibitive when generated
and/or processed by a conventional PC. We also applied PCA
for the semi-automatic recognition of salt bodies in the same
seismic profile, where only the manual selection of images
containing them is needed. The method appears to be capable
of extracting conspicuous geological features from the data.
A PCA system working directly with 3D data should improve
the accuracy of the detections, although it would also increase
significantly the time required for the training of the system
and the detection process itself. The detection process of both
the Hough transform and PCA methodologies yielded similar
results, but with the first one the top of salt body (vertex of
the parabola) is found, while the bright points obtained from
the PCA methodology correspond to the centre of the salt
bodies. As we find these results encouraging, we believe that
more complex patterns of geological units could be recognized
and extracted by using other generalizations of the Hough
transform and PCA processing.
Acknowledgments
We would like to acknowledge Jim Spurlin for his critical
reading and suggestions to refine this work. We also thankSandra Pineda for her detailed revision of the grammar in this
paper. Special thanks are given to K Marfurt for his fruitful
discussions on feature extraction and pattern recognition
methods applied to oil exploration. This contribution was
supported by project IMP/D.000475, D.00468, SENER-
Conacyt 128376. We also thank the French CNRS for the
financial support by grant 94154.
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