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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

[email protected]

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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(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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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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(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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