Potential Field Data Separation using 3-D PCA & Textural Analysis

Geophysical Journal International
doi: 10.1111/j.1365-246X.2009.04357.x
Separation of potential field data using 3-D principal component
analysis and textural analysis
Lili Zhang, Tianyao Hao and Weiwei Jiang
Key Laboratory of Petroleum Resources Research, Institute of Geology and Geophysics, Chinese Academy of Sciences, Beijing 100029, China.
E-mail: [email protected]
Accepted 2009 August 6. Received 2009 April 18; in original form 2007 November 29
SUMMARY
Potential field data represent the superposition of effects of all surface and underground
sources. A reliable interpretation of different gravity or magnetic anomalies greatly depends
on a reasonable separation between regional field and local anomalies. We present here a
novel separation method based on a 3-D principal component analysis (PCA) and textural
analysis. The PCA, used to decompose the potential field data into a linear superposition of
eigenimages, is performed not only on anomaly values but also on textural features, so as to
fully use the spatial distribution characteristics of the data and make the separated regional field
comprehensively account for the major variations of the data. In order to reduce subjectivity
and inaccuracy, we propose a texture-based criterion in separation result selection, which
measures the highlighted differences between the two kinds of anomaly by textural statistics
and select the first several eigenimages corresponding to the most important variability as the
region field when the differences reach maximum. The method is tested with two synthetic
models and two real data examples from the Huanghua area, located in Hebei Province, China.
Our tests suggest that the method provides a better separation of regional and local anomalies
than does the polynomial fitting technique. The separated regional fields and local anomalies
of the gravity and magnetic data coincide well with the geological structure of the Huanghua
area.
Key words: Image processing; Spatial analysis; Gravity anomalies and Earth structure;
Magnetic anomalies: modelling and interpretation.
1 I N T RO D U C T I O N
Gravity and magnetic field data measured for geophysical exploration comprise the superposition of effects of both internal and
external sources. For instance, a magnetic map may be composed
of regional, local and micro-anomalies. Small-amplitude, shortwavelength anomalies due to small-scale structures buried at shallow depths are superimposed on regional-field anomalies that come
from larger and/or deeper sources. In hydrocarbon exploration, we
are usually interested in shallow small-scale features, while in other
cases we use regional field anomalies to delineate deep structure.
Correct estimation and removal of the regional field from initial field
data yields the local anomalies. Evidently, the reliability of anomaly
interpretation and modelling depends on a reasonable regional-local
separation of the potential field data.
Commonly used separation methods (Dean 1958; Naudy et al.
1968; Garry & Clarke 1969; Syberg 1972; Blakely 1996), either
to anomaly profiles or contour maps, include graphical smoothing, polynomial fitting, four-pointed averaging, upward/downward
continuation and spectral filtering. The graphical smoothing is considered to be time-consuming and subjective (good and bad). The
polynomial fitting is a most popular method for separation, but its
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Journal compilation application efficiency is limited by the selection of polynomial parameters. Relatively objective selection rules have been developed
for the polynomial parameters (Abdelabaman 1985). The polynomial fitting method usually yields aliasing anomalies when potential
field anomalies are complicated and thus higher-order polynomials
are needed. The spectral filtering techniques cannot always give satisfactory results because of the spectral overlapping of the regional
field and local anomalies. To facilitate separation, data transformation and enhancement of one field component against the other
are used to sharpen anomalies or detect edges of anomalous bodies. The relevant methods include second derivatives (Elkins 1951;
Gupta & Ramani 1982), maximum gradient (Blakely & Simpson
1986), analytic signal (Roest et al. 1992) and so on. Such methods are useful for enhancing anomalies caused by shallow or deep
sources, but they are not able to make the separation. Moreover,
their applications require specification of a number of parameters.
In recent years, new separation methods have been developed, like
the frequency-domain Wiener filtering (Pawlowski & Hansen 1990)
which is preferable to a conventional bandpass filter in potentialfield anomaly separation, separation based upon the 3-D inversion of
magnetic data on multiscales (Li & Oldenburg 1996, 1998), wavelet
domain decomposition (Fedi & Quarta 1998; Hornby et al. 1999),
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preferential continuation (Pawlowski 1995), iterative lateral continuation (Boschetti et al. 2004), matched filtering based on the cosine
transform (Zhang et al. 2006), the model-based separation filtering
of magnetic data (Pilkington & Cowan 2006) and magnetic anomaly
separation based on the differential Markov random-field (DMRF)
method (Albora & Ucan 2006), etc. The methods have been proven
effective, thus providing basis for some later-developed separation
methods.
Elements of the anomaly data set are not independent of each
other but have spatial distribution patterns. In most cases, some
above-mentioned methods may have not made full use of the spatial
distribution of the potential field data; besides, separation results
have to be determined on the basis of several trials and prior cognitions of the field data. The two aspects affect to a certain extent
the effectiveness and efficiency of the field separation. This paper
hence presents a novel separation method aiming at making some
improvements.
The regional-field anomalies usually have smoother variations
than those of the local anomalies. The difference, which can be
considered as textural contrast, is a basis of many field separation
methods. When anomalies have no strong difference in amplitude
or when they overlap in frequency spectrum, they may be distinguishable due to their texture. Texture is a description of structural
arrangements and spatial variation of a data set or an image. To
mine spatial attributes of the field data, we use textural features
to differentiate the anomalies and then separate the anomalies by
a ‘3-D’ principal component analysis (PCA), which can decompose the data set or image into a linear superposition of eigenimages. We intend to find relations between the eigenimages and
the anomalies. The 3-D PCA is performed not only on anomaly
values but also on textural features so as to better decompose
the potential field. For the separation procedure, a judgment criterion based on textural differences of the anomalies is provided
to reduce subjectivity and thus a self-adaptive separation can be
realized.
2 DECOMPOSITION OF THE PCA
2.1 1-D and 2-D PCA
The PCA is a multivariate procedure developed for the extraction
of maximal information from large data matrices containing numerous columns and rows (Mardia et al. 1979). This technique,
originally popularized by Hotelling (1933) and sometimes referred
to as the Karhunen–Loeve expansion (Fukunaga & Koontz 1970)
for continuous data, linearly transforms the matrix or data set into
eigenspaces in which the variables ordered by reducing variability
are uncorrelated and perpendicular.
The PCA can be used to find meaningful ‘directions’ in correlated
data, remove noise, compress data, reduce linear dimensionality and
perform face recognition (Kirby & Sirovich 1990). It has also been
applied to the processing of seismic-reflection data (Freire & Ulrych
1988; Biondi & Kostov 1989). The Karhunen–Loeve transform was
applied to sonic logging waveforms for wave component separation
and feature extraction (Hsu 1990).
The PCA usually involves a mathematical procedure that transforms a number of correlated variables into a smaller number of uncorrelated variables, which are called principal components (PCs).
The first PC accounts for as much of the variability in the data set
as possible, and each succeeding component accounts for as much
of the remaining variability as possible.
Let a sample A ∈ R n . If A is m rows by n columns, it must be
initially converted to a 1-D vector form [the size is (m × n) × 1]
in the classical 1-D PCA. Traditionally, the PCA is performed on a
square symmetric matrix of the sums of squares and cross products,
such as the covariance matrix, that is,
Cov = E{(A − A)(A − A)T },
(1)
where A is the mean value of the data set A. After the construction
of the covariance matrix, eigenvector decomposition is applied to
Cov to compute a sorted list of PCs in an orthonormal matrix U,
that is, Cov = U U T . Herein, U is the matrix whose columns
→
are eigenvectors −
u i for the covariance matrix, which are called
empirical eigenvectors. U T is the transposed matrix of U. =
diag(λ1 , λ2 , . . . λr ), are eigenvalues. The PCA representation of the
matrix data set A is defined as
Y = U T (A − A).
(2)
By convention, eigenvalues λi corresponding to the eigenvectors
−
→
u i are arranged in a non-increasing order. Let v be the number of eigenvectors. Then the projection Y is a v-dimensional
vector.
The PCs are the eigenvectors sorted in a descending order by their
related eigenvalue magnitudes λi . Eigenvectors that correspond to
big eigenvalues are the directions in which the data have large
variance. The eigenvector corresponding to the highest eigenvalue
is the first PC of the data set and retains the most significant amount
of information. The second PC corresponds to the second highest
eigenvalue and retains information of second significance, and so on
and so forth. The dot product u i u iT is the ith eigenimage of the data
set A. The data set can be decomposed into a linear combination of
the eigenimages.
Another representation or reconstruction of A is
A = UY + A =
v
σi u i u iT .
(3)
i=1
Owing to the orthonormality of the eigenvectors, the eigenimages
form an orthonormal basis, which may be used to reconstruct A
according to the above equation.
In the classic 1-D PCA, the 2-D data set must be initially converted to a 1-D vector. It is seen that the size of the covariance matrix
of eq. (1) is very large [to Am×n , the matrix size is (m ×n)×(m ×n)]
when the sample vector is very long. So it is difficult to estimate
the covariance matrix accurately. Furthermore, the projection Yi of
eq. (2) is a scale, which usually causes overcompression, that is, we
will have to use many PCs to approximate the data set A for a desired
quality. To solve the problems, the 2-D PCA (Yang & Yang 2002;
Yang et al. 2004) was developed and 2-D matrices can be directly
used to construct the corresponding covariance matrix instead of a
1-D vector set.
Eigenvectors, core of the PCA, can be calculated efficiently using the singular value decomposition (SVD) techniques (Sirovich
& Kirby 1987; Kirby & Sirovich 1990). There are many algorithms
that can compute very efficiently eigenvectors of a matrix. However, most of these methods can be very unstable in certain special
cases. The SVD, a method that is in general not the most efficient one, can be made numerically stable. In the 2-D PCA, the
size of the covariance matrix is m × m or n × n. If the number
of eigenvectors is v, then the projection Y consists of v row vectors. Sample A, A ∈ R m×n , is projected on a principal vector as
follows:
Yi = u iT (A − A), Yi ⊂ Y, u i ⊂ Uv , i = 1 . . . v.
(4)
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Clearly, the projection Yi is a row vector rather than a scale. Thus,
the overcompression problem is alleviated in this case. Furthermore,
2D-PCA can effectively overcome the dimensional problem for a
2-D data set.
As a result, the 2-D PCA has two important advantages over
the PCA. First, it is easier to evaluate the covariance matrix accurately. Second, less time is required to determine the corresponding
eigenvectors.
Our application of the PCA in the sections below is different from
common uses. For example, to obtain eigenvectors or eigenfaces for
human face recognition, the PCA is performed on some sample data
sets. In our use, one data set is input and the 2-D PCA is performed,
which thus can also be called SVD or Karhunen–Loeve transform
on such a case. The SVD is used to calculate eigenvectors for the
PCA.
2.2 Separation
To reconstruct the data set by leaving out the last less significant
eigenvectors may lose negligible information. For example, to the
first k eigenvectors or eigenimages, an approximation representation
of A, Ã, can be described as
à = Uk Yk + A,
(5)
where Uk is a matrix consisting of the first k eigenvectors of U,
corresponding to the projection Yk . The matrix à gives a filtered
version of A.
The regional field, whose spatial extent and variation distribution
are much larger than those of the local anomalies, can thus account
for the major variability of the data set and can be approximated by
the first PCs. Let A represent the total field. Then A is the superposition of the regional field and local anomalies. Another form of A
can be expressed as
Am×n =
v
yi u i + A
i=1
=
w
⎛
yi u i + A1 + ⎝
i=1
v
where Ā = A1 + A2 , v is the number of eigenvalues, w is to be
determined, threshold of PCA-based filtering, 1 ≤ w ≤ v, m and
n are the column and row number, respectively, of A; Rm×n is the
region field and L m×n represents the local anomalies, A1 is the mean
value of the matrix Rm×n and A2 is the mean value of L m×n .
It can be seen from eq. (6) that w, A1 and A2 , are the key to the
separation of A.
We apply the 2-D PCA to synthetic potential field data (Fig. 1).
Fig. 1(a) reveals the gravity effect of a sphere (Fig. 1b) superimposed
on a relatively deep-seated sloping plane (Fig. 1c). The given density
of the sphere is 0.5 g cm−3 , radius is 1 km, and depth is 3 km. The
contour interval is 1 mGal. The sphere anomaly is the local anomaly
and the linear background is considered as the regional gravity field.
The data in Fig. 1(a) vary from 6.7 to 27.7 mGal, in Fig. 1(b) from
6.6 to 26.8 mGal, in Fig. 1(c) from 1.1 to 8.6 mGal.
Figs 2(a) and (b) are reconstruction results of the data in Fig.
1(a) (with mean value removed) using the first eigenimage and the
first two eigenimages, respectively. According to the eigenvalues in
Fig. 2(c), it can be seen that four eigenimages are sufficient to
properly reconstruct the data with negligible loss of information.
Thus, w is given a value smaller than 4. Figs 2(a) and (b) reflect
only the reconstruction data, that is, the mean value A1 is not added
yet, so their amplitude range is different from that of the regional
field in Fig. 1(c). Moreover, their spatial distribution and shape are
also different from those of the regional field.
The test result suggests that the common 2-D PCA seems not
so effective in separating the data. Therefore, we need to use more
spatial features in separation besides the spatial-domain data set.
Texture is an indicator of spatial distribution and structural patterns.
In the next sections, we will present how to separate A into R and L
by integrating a ‘3-D’ PCA with textural analysis.
3 T E X T U R A L A N A LY S I S
3.1 Gray level co-occurrence matrix and its
statistical measures
⎞
y j u j + A2 ⎠
j=w+1
= Rm×n + L m×n ,
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(6)
Texture can be a comprehensive reflection of amplitude, spatial variation and distribution. Textural analysis is a procedure that extracts
textural features by image processing methods and thus obtains a
quantitative or qualitative description of texture like coarseness,
smoothness, and homogeneity. Textural analysis has been applied
Figure 1. A synthetic model of Gravity data. (a) Total field, reflecting gravity effect of a sphere (b) superimposed on a relatively deep-seated sloping plane (c).
The contour interval is 1 mGal. (b) Regional field, reflecting gravity effect of a relatively deep-seated sloping plane (c) Local anomaly, reflecting gravity effect
of a sphere with density of 0.5 g cm−3 , radius 1 km and depth 3 km.
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Figure 2. (a) Reconstruction of Fig. 1(a) using the first eigenimage. (b) Reconstruction using the first two eigenimages. (c) Eigenvalue curve.
to seismic data for processing and interpretation (West et al. 2002;
Gao 2003; Chopra & Alexeev 2006), and also to gravity data for
enhancing circular anomalies (Cooper 2004).
Gray level co-occurrence matrix (GLCM) is a well-known statistical method of extracting textural features from data sets or
images, and has been widely applied in object classification. The
co-occurrence matrix is the joint probability occurrence of element
values i and j for two matrix elements with a defined spatial relationship in an image. The spatial relationship is defined in terms of
distance d and angle θ, or, as a function of a displacement (x, y)
along the x and y direction.
P(i, j) or P(i, j, d,θ), the GLCM of the data set Am×n , is given as
P(i, j, d, θ) =
n
m x=1 y=1
√
ε=
δ(i, A(x, y))
x 2 +y 2
×δ( j, A(x + x, y + y)), i, j ∈ [min A, max A],
(7)
where (x, y) is element location, m is the width of A and n is the
height.
Let P(i. j) = P(i, j, d, θ )/R(d, θ), where R(.) is a normalization
constant that causes the entries of P(.) to sum to 1. Usually, R(.) is
the total number of co-occurring pairs.
For example, to the data set A, whose maximum element value
is 3 and minimum value is 1,
⎫
⎧
⎪
⎬
⎨1 1 2⎪
A= 3 2 2
⎪
⎭
⎩3 1 2⎪
3×3
to a given direction 0◦ , and one-element distance, the co-occurrence
matrix would be (before normalization)
(1)
(2)
(3)
(1)
1
0
1
(2)
2
1
1
can be calculated on the whole data set or in a small window centred on an element (x, y) scanning the data set. 20 s order or even
higher-order statistical measures of texture can be extracted from
the GLCM (Haralick et al. 1973; Haralick 1979), including entropy,
contrast, homogeneity, and so on. Such textural measures or statistics can be used to highlight differences of anomalies. Most of these
statistics are derived by weighting on either the matrix element value
or its spatial location.
Due to redundancy in these statistics, we choose four measures,
energy, entropy, contrast, and homogeneity, which generate the desired discrimination without redundancy. When we test the measures
on the potential field data, we find that entropy and contrast are relatively sensitive to the differences between the local anomalies and
regional field.
Entropy is a measure of complexity or randomness of matrix
element values, defined as
(3)
0
.
0
0
If there are n different element magnitudes in the data set, then the
co-occurrence matrix will be n × n elements in size. The rows and
columns represent the set of possible element values.
The GLCM is based on the repeated occurrence of some grey
level configuration in texture; this configuration varies rapidly with
distance in fine textures and slowly in coarse textures. The GLCM
1 P(i, j) log2 P(i, j).
2 i=0 j=0
s
E(A) = −
s
(8)
Contrast is a measure of the amount of local variation. The contrast
is,
D(A) =
s s
(i − j)2 P(i, j),
(9)
i=1 j=1
where P(i, j) is the occurrence probability of pair (i, j), as defined
in eq. (7) and A is the data set.
To facilitate integration of GLCM-based entropy and contrast
into the PCA procedure, eqs (8) and (9) are normalized to the range
[0, 1].
Complex textures tend to have high entropy, and low values for
smooth images. When all entries in P(i, j) are equal, the entropy
is the highest. A low-value contrast usually results from uniform
images, whereas images with large variation or with quickly varying
magnitude produce a high value. The contrast value may get high
when the potential field is abundant in distortion and abrupt gradient
belts, as is often the case.
The displacement value d, orientation θ, and computation window size will affect feature values. The larger the window size is, the
poorer the spatial resolution of the resulting GLCM will be (Cooper
2004).
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Figure 3. Textural features of Fig. 1. (a) GLCMs of the entire data set of Fig. 1(a). (b) GLCMs of the regional field data of Fig. 1(b). (c) GLCMs of the local
anomaly data of Fig. 1(c). In (a), (b) and (c), from up to down, d = 1 and 10, respectively. (d) Contrast data in scanning windows throughout the data set,
window size is 3 × 3. (e) Entropy data in scanning windows throughout the data set, window size is 3 × 3. (f) Curve of contrast values of the entire data set,
the regional field and the local anomaly. (g) Curve of entropy values of the entire data set, the regional field and the local anomaly. In (f) and (g), data type
index = 1, the total field; 2, the regional field; 3, the local anomaly.
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3.2 Textural properties of the potential field data
To demonstrate textural properties of the potential field data, we use
the synthetic models (Figs 1a and 4a) and calculate their GLCMs and
statistical measures. For the potential field data, computing GLCM
texture measures at one location (x, y) yields localized features at
that point. By repeating the computation of these measures in a
sequential manner or in scanning windows throughout the data set,
the data are transformed into a textural feature matrix. The GLCMs
are calculated for different d values. To reduce the influence of θ,
we use a synthesis of eight choices for θ , 0◦ , 45◦ , 90◦ , 135◦ , 180◦ ,
225◦ , 270◦ and 315◦ , which is considered as the isotropic GLCM
(Kapur et al. 1985).
In Fig. 3, the local anomaly differs obviously from the region
field and the total field in GLCMs (d = 1, 10) and GLCM-based
measures in spatial distribution and value range. To other d values,
the difference trend generally keeps the same. In the contrast and
entropy images (contrast and entropy are calculated in scanning
windows throughout the data set) of Figs 3(b) and (c), distortion,
abrupt variation, or gradient belts, appear as distinct high values.
In Figs 3(f) and (g), the contrast and entropy value (calculated on
the whole data set) of the local anomaly is the smallest. To other d
values, the trend almost keeps the same.
In Fig. 4(a), the synthetic model is the superposition effect of
magnetic anomalies of a tabular mass and two spheres.
To the synthetic magnetic model, the differences between the
local anomalies and the regional field (see Fig. 5) are similar to the
difference trends in Fig. 3. Textural statistics is implicitly related
to spectral analysis, so a little overlapping sometimes can be seen
between the GLCMS of regional field data and local anomaly data
when the entire data set is complicated, though the overlapping is
not as serious as that of the commonly used spectral separation.
Due to small value range and texture scale, the entropy of the local anomaly is the lowest, and the contrast is also the lowest though
it has sharper spatial variation, which demonstrates the particularity
of the potential field data. While in the contrast and entropy images,
the contrast and entropy get high to abrupt changes. It is seen that
the GLCM-based contrast and entropy are efficient to the potential field data in enhancing abrupt changes like edges, lineaments,
sharp gradient belts and contorted anomalies. Combined with our
test results of other data samples, it is found that the contrast is more
effective than the entropy in enhancing lineaments and distortion,
and that the entropy highlights area-based ‘equality’ or homogeneity. The differences of the anomalies indicate that the GLCM and
its statistics are promising in differentiating the anomalies.
The distance d and window size influence the GLCM measures.
Although the potential field data are usually complicated in texture,
by a lot of tests we find that the increase of d or window size will
result in decreased resolution but at the same time help us learn
better of the spatial homogeneity of the data on a larger scale.
Figure 4. A synthetic model of magnetic data. (a) Total field, that is, superposition of (b), (c) and (d). (b) Regional field. (c) Local anomaly A. (d) Local
anomaly B.
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Figure 5. Textural features of Fig. 3. (a) GLCMs of the total field data of Fig. 1(a). (b) GLCMs of the regional field data of Fig. 1(b). (c) GLCMs of the local
anomaly data of Figs 3(c) and (d). In (a), (b) and (c), from up to down, d = 1, d = 10, respectively. (d) Contrast data (not normalized to [0, 1]) in scanning
windows throughout the data set, window size is 3 × 3. (e) Entropy data (not normalized to [0, 1]) in scanning windows throughout the data set, window size is
3 × 3. (f) Curve of contrast values of the entire data set, the regional field and the local anomaly. (g) Curve of entropy values of the entire data set, the regional
field and the local anomaly. In (f) and (g), data type index = 1, the total field; 2, the regional field; 3, the local anomaly.
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4 S E PA R AT I O N B A S E D O N A ‘ 3 - D ’ P C A
AND A TEXTURE-BASED CRITERION
The key problems of a PCA-based field separation primarily lie in
two aspects: one is how to make the limited PCs comprehensively
represent the major tendency of the anomalies; the other is the
number of PCs to be used, that is, w (see eq. 6). We integrate the
PCA with the textural analysis and hence provide solutions to the
problems.
4.1 The 3-D PCA
Recently, the PCA technique has been extended to 3-D and even an
arbitrary n-dimensional space. The new nD-PCA is applied directly
to n-order tensors (n ≥ 3) rather than 1-D vectors or 2-D matrices
(Yu & Bennamoun 2005). An important property of the nD-PCA is
the use of higher-order SVD (Lathauwer et al. 2000).
However, the ‘3-D’ PCA proposed in this paper has a different
meaning. The data set is parametrized in terms of amplitude and
texture. That is, this PCA is performed on both anomaly values and
textural features so that the regional anomalies can represent main
variations of not only the spatial data but also their textures.
We transform the data set Am×n into the textural measures of the
matrix size, Dm×n and E m×n . After finding the mean values for A,
D, and E, then calculate the autocovariance functions of matrices,
C(I, J ). I, J refer to A, D and E that is, CA−D , CA−E . The obtained
eigenvectors are sorted in a descending order by their corresponding
eigenvalue magnitudes. Using the eigenvectors as rows, the eigenvector matrix Um is obtained. After that, the eigenvector matrix is
used to transform the data set A. Use A T = Um A and obtain a
new matrix integrating amplitude and textural features. The abovementioned 2-D PCA is performed on the matrix and anomalies are
separated according to eq. (6).
The 3-D PCA identifies patterns in the data and highlights their
similarities and differences. In this way, the anomalies are decomposed into meaningful eigen parts according to their enhanced differences of both amplitude and texture. In the ‘3-D’ PCA, the textural data perform as weighting factors, that is, the locals with large
contrast or entropy values are relatively weakened in the PCA, and
thus their influences to the extraction of maximal information in
Figure 6. Results of Fig. 1. (a) The regional field separated by the 3-D PCA method. (b) The regional field separated by the polynomial fitting method.
(c) Eigenvalue curve of the 3-D PCA.
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important directions in PCA can be reduced to a certain degree. In
this way, the resultant regional field can be better separated.
The two mean values, A1 and A2 (see eq. 6), can be determined
according to a measure of difference deviation (Zhang 2003). A
self-adaptive separation will be realized if w can be determined
automatically, that is, by maximizing a given cost function or by
giving a criterion.
4.2 Maximum criterion for separation
Determination of w or threshold of PCA-based filtering has been
documented in some literatures. One commonly used approach is
to select w such that the first w eigenvectors of A capture important
appearance variations in the data set, that is,
w
i=1
v
λi
λi
≥ T,
(10)
i=1
where the threshold T is close to, but less than, unity. For example,
T is given 0.8.
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Some famous information-theory principles, like the Akaike information criterion (AIC) (Karhunen et al. 1997), the minimum
description length (MDL) (Wax & Kailath 1985), and the Bayesian
information criterion (BIC; Minka 2001), have been used to set a
threshold between signals and noise. The AIC is a measure of the
goodness of fit of an estimated statistical model. It is grounded in
the concept of entropy, in effect offering a relative measure of the
information lost when a given model is used to describe reality and
can be said to describe the trade-off between bias and variance in
model construction. The MDL provides a criterion for the selection
of models, which dictates that the best hypothesis for a given set
of data is the one that leads to the largest compression of the data.
The BIC can measure the efficiency of the parametrized model in
terms of predicting the data and it is exactly equal to Minimum
Description Length Criterion but with negative sign. These criteria are very sensitive to the signal-to-noise ratio and data sample
number. Besides, they are derived on the condition that the data
set A obeys the Gaussian distribution. Montagne & Vasconcelos
(2006) applied thermodynamic-like extremum criteria (minimum
energy and maximum entropy) in the Karhunen–Loeve transform
to suppress coherent noise in seismic data.
Figure 7. Results of Fig. 3. (a) The regional field separated by the 3-D PCA method. (b) The regional field separated by the polynomial fitting method. (c)
Eigenvalue curve of the 3-D PCA.
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The above-mentioned criteria mainly aim at removing noise. As
we know, however, the local anomalies cannot be classified as noise.
Take the local anomalies as object and the region field as background, criteria for features are needed to help differentiate object and background. In a wavelet-based separation, Fedi & Quarta
(1998) used the criterion of minimum entropy compactness to select
wavelet bases to ensure optimum wavelet decomposition.
In image segmentation (an image processing technique), object
and background are sometimes separated by selecting an optimum
threshold based on maximum Shannon entropy or minimum crossentropy criteria (Kapur et al. 1985; Pal & Pal 1989; Brink & Pendock
1996; Sahoo et al. 1997). The total second-order entropy of the
object and background is written as
Ht (gi ) = Ho (gi ) + Hb (gi ) , i < L ,
(11)
where H o is the entropy of the object, H b is the entropy of the
background, gi is the ith grey level of the image set and L is the grey
level number. When both the object and background have best inner
consistency, the total entropy reaches its maximum. The grey level
corresponding to the maximum of Ht (gi ) gives the optimal threshold
for object-background separation. This is called the thresholding
criterion of the maximum entropy.
Obviously, this ‘separation’ procedure is quite different from the
separation of the potential field. To the potential field, the object is
to be subtracted from the background, while in the image segmentation, the object is only kept apart from the background. Never-
theless, the idea of measuring the differences between object and
background before and after separation by entropy-related features
can be borrowed.
The GLCM-based entropy (eq. 8) presented in the paper is different from the Shannon entropy and cross entropy. The larger the
w value is, the larger entropy or contrast the resultant region field
will have. However, the field obviously contains too many local details and thus is not a successful separation. A balance between the
object and background is needed.
As mentioned before, gravity or magnetic images have complicated textures. So there are several factors affecting the textural
measures, and the discriminating effect of a single statistics may
be weak. Each statistical measure represents one or multi textural property and has its own enhancement capacity. For example,
high amplitude continuous anomalies generally have relatively high
contrast and low entropy. Low amplitude, small-scale anomalies
have low contrast. With more than one textural measure taken into
account in the separation, it will not only increase the discriminating ability but also reduce the influence of d and window size to a
certain degree.
Taking these aspects into account, we combine the GLCM-based
entropy with contrast to form a cost function, defined as
(t) = s1 ∗ (E 2 (t) + E 1 (t)) + s2 ∗ (D2 (t) − D1 (t)), t ∈ [1, v]
(12)
Figure 8. Topography of the study area.
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herein,
E 2 (t) = E( f 2 ),
E 1 (t) = E( f 1 ),
D2 (t) = D( f 2 ),
D1 (t) = D( f 1 ),
f2 =
v
yi u i ,
i=t+1
f1 =
t
yi u i .
i=1
In eq. (12), v is the number of eigenvalues, as mentioned in
eq. (6); s1 and s2 are weighted factors, related to standard deviation within a localized neighbourhood (Zhang 2003); the function
E2 and E1 represents the GLCM-based entropy of the matrix f 2 and
f 1 , respectively. D2 and D1 represents the GLCM-based contrast of
the matrix f 2 and f 1 , respectively. The functions E and D in eqs (8)
and (9) have been normalized to be of similar order-of-magnitude
change. To each t ∈ [1, v], calculate (t). When the function
reaches its maximum, the t value is then the right w (see eq. 6)
we try to determine; combined with A1 and A2 , the anomalies are
considered optimally separated.
A problem of the criterion is the excessive computation burden of
the GLCMs and measures. We thus use the grey level co-occurrence
integrated algorithm (GLCIA; Clausi & Zhao 2003), which can
make a dramatic improvement on the normal GLCM implementations. The GLCIA integrates the grey level co-occurrence hybrid
structure and the grey level co-occurrence hybrid histogram.
4.3 Synthetic model test
The separation method based the 3-D PCA and textural analysis
is referred to simply as the 3-D PCA method in the following.
The separation method is applied to the synthetic models shown in
Figs 1(a) and 4(a) to assess its separating ability. The polynomial
fitting method is also applied. For the synthetic model of Fig. 1(a)
is simple, s1 = 1, s2 = 1; d = 3, and the window size is 3 × 3; w
= 1; the polynomial degree is 1, coefficients ρ0 = 6.916315, ρ1 =
1.726731E − 02, ρ2 = 4.071759E − 02. To the data of Fig. 4(a),
s1 = 1, s2 = 1; d = 3, and the window size is 3 × 3; w = 2.
The polynomial degree is 1, and coefficients are ρ0 = 86.69552,
ρ1 = −1.840482 and ρ2 = −3.501539.
The simple model in Fig. 1(a) can be separated very well by the
polynomial fitting method. The separation result (Fig. 6a) of the
3-D PCA method differs from that of the polynomial fitting method
(Fig. 6b) in amplitude and shape. But the result approaches the
model regional field a little better in amplitude than the polynomial
fitting result does. The eigenvalues in Fig. 6(c) indicate that the
first eigenimage contain more information than the eigenimage in
Fig. 2(c) does.
The model in Fig. 4(a) has one more anomaly superposed. The
separation results in Figs 7(a) and (b) show that the 3-D PCA outperforms the polynomial fitting in amplitude and shape. The eigenvalues in Fig. 7(c) indicate that the first two eigenimages contain
most of the information.
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Journal compilation Figure 9. Tectonic map of the study area
5 R E A L D ATA E X A M P L E S
The field separation has been applied to a real data set obtained
from the gravity and magnetic exploration of Huanghua and its
adjacent areas. The surveyed area (see rectangles in Fig. 8, including the gravity survey area and the magnetic survey area) covers
several cities and counties like Tianjin and Huanghua, where lies
Dagang Oilfield. The area spreads over three main tectonic units:
the Chengning Uplift, the Huanghua Depression, and the Cangxian
Uplift. A geological map (Fig. 9) shows main second-order tectonic
units and discordogenic faults in the study area. The Huanghua
Depression lies in the northcentral part of the Bohai Bay Basin in
Eastern China, to the south of the Yanshan Foldbelt, east of the
Cangxian Uplift, and west of the Chengning Uplift. The Depression is an asymmetric downfaulted basin formed since Mesozoic
era, extending mainly in the NNE–NE direction. It consists of 16
secondary-order structural units like the Qikou Sag, the Banqiao
Sag, the Beitang Sag, the Kongdian Upheaval and the Yanshan Sag.
Wherein, the Qikou Sag is one of the deepest, largest sag in the
Bohai Bay Basin, and it has become a ‘hotspot’ for hydrocarbon
exploration.
The Qikou Sag is a large-scale composite sag after chasmic stage
and subsidence since the Oligocene Epoch. The Banqiao Sag lies in
the northwestern part of the Huanghua Depression, southeast of the
Cangxian Uplift and north of northern Dagang. Under the control
of the Cangdong fault, the Banqiao Sag is a half graben-like fault
subsidence, with axial trend parallel to the Cangdong fault zone.
The Beitang Sag northeast of the Huanghua Depression lies in the
included-angle area between the Cangxian Uplift and the Yanshan
Foldbelt. The sag is a marginal trough.
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L. Zhang, T. Hao and W. Jiang
Gravity and magnetic data of the research area were supplied by
Institute of Geology and Geophysics, Chinese Academy of Sciences,
and Geophysical Research Institute of Dagang Oilfield.
5.1 Gravity field separation
Fig. 10(a) illustrates the gravity field of the surveyed area. The data
used are Bouguer gravity data gridded with 0.9 km intervals. The
Bouguer gravity anomalies vary from –68 to –1 mGal. The gravity
field of the area indicates an alternate structural pattern of uplift
and depression, with a general trending in the NE-NNE direction.
The gravity field can be divided into four zones: Wuqing low-valueanomaly zone, Cangxian high-value-anomaly zone, Huanghua low-
value-anomaly zone and Chengning high-value-anomaly zone.
The Wuqing low-value-anomaly zone lies at the northwest part of
the study area, revealing a NE-trending, low-value gradient zone.
The anomalies decrease gradually from southeast to northwest. The
Cangxian high-value anomaly zone is between the Wuqing lowvalue zone and the Cangdong fault. The Cangdong fault shows a
NE-trending gradient zone. To the south of Tianjin lies a local lowvalue trap, which is deduced to be caused by a local small-scale
relief above the Cangxian Uplift. A high-value anomaly zone at
the northern part of the area is where the Yanshan Foldbelt lies. In
the Huanghua low-value anomaly zone, major low-value anomalous
traps correspond to the Banqiao Sag, the Qikou Sag and the Beitang
Sag, while local high-value traps are related to structural highs in
Figure 10. (a) Bouguer gravity anomaly map with major tectonic elements superimposed on. The data are from Huanghua and its adjacent areas. (b) (t)
value curve. Let t ∈ [1, 15], (t) is defined in eq. (12). (c) Curve of contrast values of the entire data set, the regional field and the local anomaly. (d) Curve
of entropy values of the entire data set, the regional field and the local anomaly. In (c) and (d), data type index = 1, the total field; 2, the regional field; 3, the
local anomaly. (d) Separation result of the 3-D PCA method, regional field. (f) Separation result of the 3-D PCA method, local anomalies. (g) Separation result
of the polynomial fitting method, regional field. (h) Separation result of the polynomial fitting method, local anomalies.
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Figure 10. (Continued.)
between the sags. The anomalies generally extend in the NE direction. The high-value anomalous zones in the southeast or eastern
part of the research area are reflection of the Chengning Uplift and
the Shaleitian Uplift. In the Chenning Uplift, anomalies are almost
NE trending. In the eastern sea area, high-value anomalies mainly
extend in the EW direction. The major density interface between
Cretaceous and Jurassic strata lies in the gravity basement of the
area (Xu 2007; Hao et al. 2008). The regional field separated from
the total data set can be used for gravity basement inversion.
We use the new method presented in the paper and also the polynomial fitting to separate the gravity field. The results are shown
in Figs 10(e)–(h). Through tests we choose a polynomial that most
approximate the regional geological structure of the study area:
the polynomial degree is 6. In the calculation of textural statistics,
d = 3, and the window size is 3 × 3. In Fig. 10(b), when t is
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Journal compilation 7, (t) reaches its maximum, so w is given 7 based on the maximum criterion; s1 = 1, s2 = 1. Figs 10(b) and (c) illustrate the
contrast and entropy measures of the separated anomalies by the
PCA method and the polynomial fitting, respectively. The difference values of both contrast and entropy between the regional field
and local anomalies indicate that the separation results of the PCA
method coincide with the maximum criterion and that the resultant
regional field account for the major variability of the gravity field
data as assumed.
According to the regional gravity field (Fig. 10e) obtained by the
PCA method, the gravity anomalies vary from –74 to –1 mGal,
assumed to approximate the gravity field of deep sources. The
regional field varies smoothly and has characteristics in agreement with the structural pattern. For example, the anomalies in the
Qikou Sag mainly show as relatively low gravity values while the
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L. Zhang, T. Hao and W. Jiang
anomalies in the Chengning Uplift have high values. In the local anomaly map (Fig. 10f), the local details almost coincide
with the known geological features of the study area, in location, shape, and trending. The local anomalies vary from –12 to
16 mGal.
The regional field separated by the polynomial fitting
(Fig. 10g) looks much smoother but has less structural information. In Fig. 10(c), the contrast measure of the regional field is even
smaller than that of the local anomalies (Fig. 10h). The field anomalies vary from –76 to –1 mGal. The local anomalies vary from –10 to
18 mGal. A notable difference between the two regional fields lies
in the Cangxian Uplift. To the south of Tianjin, it is indicative of
a NNE-trending fault in Fig. 10(e) (see the white arrow), which
cannot be indicated in Fig. 10(g). Correspondingly, there are distinct gradient belts at the same location in the local anomaly map
(Fig. 10f). This case just accords with the proved geological con-
dition that there exists a major basement fault with large cutting
depth.
5.2 Magnetic field separation
Fig. 11(a) illustrates the magnetic field of the surveyed area. The
data used are magentic data gridded with 1 km intervals. The magnetic anomalies vary from –173 to 593 nT. We used the reductionto-the-pole (RTP) processing to the magnetic field data. The basic
parameters of the magnetic field are: the region magnetic dip angle
is 55◦ ; the region magnetic declination is –5.6◦ . The anomalies have
more obvious zonal features and clearer strike after the RTP processing. The regional aeromagnetic anomalies of the study area are
largely caused by the Archaeozoic magnetic basement (Xu 2007;
Hao et al. 2008). The overlying strata are almost nonmagnetic.
Local magnetic anomalies are mainly generated by shallow magnetic substance like igneous rocks.
Figure 11. (a) Magnetic anomaly map with major tectonic elements superimposed on. The data are from Huanghua and its adjacent areas. (b) (t) value
curve. Let t ∈ [1, 15], (t) is defined in eq. (12). (c) Contrast value curve. (d) Entropy value curve. In (c) and (d), data type index = 1, the total field; 2, the
regional field; 3, the local anomaly. (e) Separation result of the 3-D PCA method, regional field. (e) Separation result of the 3-D PCA method, local anomalies.
(f) Separation result of the polynomial fitting method, regional field. (h) Separation result of the polynomial fitting method, local anomalies.
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Figure 11. (Continued.)
The magnetic field can be divided into three anomalous zones: the
northern high-value-anomaly zone, the central low-value-anomaly
zone, and the southeastern high-value-anomaly zone. The northern
anomaly zone can be further divided into two parts according to
its two main strikes, corresponding to the Cangxian Uplift and the
Beitang Sag. The anomalies vary from 100 to 400 nT, with sparse
and smooth anomalies in between. The central anomaly zone almost
lies in the sag and distributes in a SW-trending triangle shape, with
anomalies varying from –100 to 100 nT. In the Shaleitian Uplift,
anomalies have a relatively large variation in features, revealing a
low-high-low alternating feature in the NS direction. The maximum
anomaly value is up to 300 nT in the uplift. The southeastern highvalue anomaly zone can be divided into two parts according to
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Journal compilation its strike: one part is of NE strike, and the other is NW. The two
parts are approximately orthogonal, and their location is almost in
accord with the Chengning Uplift. In the anomaly zone, there are
small-length-scale, block-like, high-value magnetic anomalies with
dense gradient belts, which reflects that the local structures here
are characteristic of relatively steep occurrence and acute lateral
variation.
Figs 11(e)–(h) are the separation results of our method and the
polynomial fitting. The polynomial degree is 6. In the calculation
of textural statistics, d = 3, and the window size is 3 × 3. According to Fig. 11(b), w is given 8 based on the maximum criterion;
s1 = 1, s2 = 0.8. In Figs 11(b) and (c), the difference values of both
contrast and entropy between the regional field and local anomalies
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L. Zhang, T. Hao and W. Jiang
indicate that the separation results of the PCA method coincide with
the maximum criterion and that the resultant regional field account
for the major variability of the magnetic field data.
According to the regional magnetic field (Fig. 11b) obtained by
our method, the magnetic anomalies vary from –100 to 480 nT.
The local magnetic anomalies (Fig. 11c) vary from –160 to 215
nT. The regional field data separated by the polynomial fitting (Fig.
11d) vary from –120 to 359 nT, and the local magnetic anomalies
(Fig. 11e) vary from –237 to 445 nT. In Fig. 11(c), the contrast
measure of the regional field is even smaller than that of the local
anomalies (Fig. 11h).
The differences between the PCA method and polynomial results
not only lie in the amplitudes but also in the structural details. The
results of the method show that they fit with the structure of the area
better than the results of the polynomial fitting. The comparison
suggests our method works better.
6 C O N C LU S I O N S
Our method can be described as the procedure in which the
potential-field data set combined with its GLCM-based entropy
and contrast matrices are decomposed into eigenimages by the 3-D
PCA and then the regional field is obtained by the reconstruction
of the first w eigenimages based on the maximum criterion of textural differences. The proposed field separation method utilizes the
spatial characteristics and major variations of the potential field
data, and allows the separation to be performed in a self-adaptive
manner. The method can reduce the inaccuracy and subjectivity
caused by lack of a priori information or geological cognitions.
Through the tests with the real gravity and magnetic data from the
Huanghua area, the method is proven to obtain better results than
the polynomial fitting technique does. Moreover, the application
of textural analysis is promising for the potential field data and
PCA.
We have to point out that the textural statistics is considered
implicitly related to spectral analysis and that slight overlapping
may happen when the data set is very complicated, which may be
the case with almost all data-domain based separation methods.
The limitation could be solved if we continue an in-depth study on
textural analysis and try more textural statistics for the potential
field data in future work.
AC K N OW L E D G M E N T S
The financial support and data supply for this project are provided
by the NSFC projects (grant No. 40674046, 40620140435 and
40704013), ‘973’ National Key Fundamental Research Plan (No.
2007CB411701), and ‘863’ Research Plan (No. 2006AA09Z359).
We thank the Editor and reviewers of this paper for their constructive
suggestions and valuable comments. We thank Miss Sylvia Hales
for her kindness and help. Thank Dr Xu Ya, Li Jun, Tu Guanghong
and Huang Song, for their help and data.
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