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 C 2009 The Authors C 2009 RAS 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), 1397 GJI Geomagnetism, rock magnetism and palaeomagnetism Geophys. J. Int. (2009) 179, 1397–1413 1398 L. Zhang, T. Hao and W. Jiang 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) C 2009 The Authors, GJI, 179, 1397–1413 C 2009 RAS Journal compilation Separation of potential field data 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 , 1399 (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. C 2009 The Authors, GJI, 179, 1397–1413 C 2009 RAS Journal compilation 1400 L. Zhang, T. Hao and W. Jiang 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). C 2009 The Authors, GJI, 179, 1397–1413 C 2009 RAS Journal compilation Separation of potential field data 1401 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. C 2009 The Authors, GJI, 179, 1397–1413 C 2009 RAS Journal compilation 1402 L. Zhang, T. Hao and W. Jiang 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. C 2009 The Authors, GJI, 179, 1397–1413 C 2009 RAS Journal compilation Separation of potential field data 1403 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. C 2009 The Authors, GJI, 179, 1397–1413 C 2009 RAS Journal compilation 1404 L. Zhang, T. Hao and W. Jiang 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. C 2009 The Authors, GJI, 179, 1397–1413 C 2009 RAS Journal compilation Separation of potential field data 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. 1405 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. C 2009 The Authors, GJI, 179, 1397–1413 C 2009 RAS Journal compilation 1406 L. Zhang, T. Hao and W. Jiang 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. C 2009 The Authors, GJI, 179, 1397–1413 C 2009 RAS Journal compilation Separation of potential field data 1407 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. C 2009 The Authors, GJI, 179, 1397–1413 C 2009 RAS 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. 1408 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. C 2009 The Authors, GJI, 179, 1397–1413 C 2009 RAS Journal compilation Separation of potential field data 1409 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 C 2009 The Authors, GJI, 179, 1397–1413 C 2009 RAS 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 1410 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. C 2009 The Authors, GJI, 179, 1397–1413 C 2009 RAS Journal compilation Separation of potential field data 1411 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 C 2009 The Authors, GJI, 179, 1397–1413 C 2009 RAS 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 1412 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. REFERENCES Abdelabaman, E.M. et al., 1985. 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