Journal of Earth Science  2018, Vol. 29 Issue (6): 1372-1379   PDF    
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Seismic Attribute Analysis with Saliency Detection in Fractional Fourier Transform Domain
Yuqing Wang1,2, Zhenming Peng1,2, Yan Han1,2, Yanmin He1,2    
1. School of Optoelectronic Information, University of Electronic Science and Technology of China, Chengdu 610054, China;
2. Center for Information Geoscience, University of Electronic Science and Technology of China, Chengdu 610054, China
ABSTRACT: Most image saliency detection models are dependent on prior knowledge and demand high computational cost. However, spectral residual (SR) and phase spectrum of the Fourier transform (PFT) models are simple and fast saliency detection approaches based on two-dimensional Fourier transform without the prior knowledge. For seismic data, the geological structure of the underground rock for-mation changes more obviously in the time direction. Therefore, one-dimensional Fourier transform is more suitable for seismic saliency detection. Fractional Fourier transform (FrFT) is an improved algo-rithm for Fourier transform, therefore we propose the seismic SR and PFT models in one-dimensional FrFT domain to obtain more detailed saliency maps. These two models use the amplitude and phase in-formation in FrFT domain to construct the corresponding saliency maps in spatial domain. By means of these two models, several saliency maps at different fractional orders can be obtained for seismic attribute analysis. These saliency maps can characterize the detailed features and highlight the object areas, which is more conducive to determine the location of reservoirs. The performance of the proposed method is assessed on both simulated and real seismic data. The results indicate that our method is effective and convenient for seismic attribute extraction with good noise immunity.
KEY WORDS: saliency detection    spectral residual    phase spectrum    fractional Fourier transform (FrFT)    attribute extraction    seismic data    

0 INTRODUCTION

In an image, the target area is usually different from its surroundings in shape, color, texture, brightness or other characteristics. These characteristics called saliency can make the target area stand out from the whole scene, and the corresponding target area is regarded as saliency area. The saliency detection aims at extracting saliency area from image, which is an important technique for target detection, and becomes a hotspot for image processing and computer vision. And the detection results are called saliency maps. Most saliency detection models are related to particular features, categories or other prior knowledge of the targets. These models are generally complex and computationally demanding.

Hou and Zhang (2007) proposed a spectral residual (SR) model, which is generated from the frequency domain point of view. The SR model is a fast and effective saliency detection algorithm without the prior knowledge, only using the amplitude and phase spectrum of Fourier transform of the image. On the basis of SR model, Guo et al. (2008) experimentally proved that the phase spectrum of image is an important feature to compute saliency map, thus proposed the phase spectrum of Fourier transform (PFT) model. Compared with SR model, the PFT model ignores the processing of amplitude spectrum information, reducing the computational cost. Moreover, Guo et al. (2008) extended PFT model using quaternion Fourier transform to obtain the spatio-temporal saliency map. These saliency detection models based on the frequency domain analysis have attracted much attention due to their simple calculation and good results (Qi et al., 2014; Li et al., 2013; Achanta et al., 2009; Yu et al., 2009; Ell and Sangwine, 2007).

Seismic attribute analysis as a key technology in seismic exploration, has wide applications in the structural interpretation, stratigraphic and lithologic interpretation, reservoir description and other fields (Wang and Peng, 2016; Wang C et al., 2015; Wang Y Q et al., 2015; Tian and Peng, 2014; Chen et al., 2013; Chopra and Marfurt, 2008, 2005; Goloshubin et al., 2008; Steeghs and Drijkoningen, 2001; Chen and Sidney, 1997). Seismic data also has some properties as saliency for extracting the region of interest. The saliency detection is used to highlight target areas and extract hidden information, which is conducive to seismic attribute analysis and reservoir interpretation. The fractional Fourier transform (FrFT) is an extension of Fourier transform in fractional domain. In this paper, we propose the seismic saliency detection models in FrFT domain by replacing two-dimensional Fourier transform with one-dimensional FrFT. Using these models, several saliency maps at different fractional orders can be obtained for more detailed attribute analysis.

1 METHOD 1.1 Saliency Detection Models

Hou and Zhang (2007) gave the SR model of image $I(x, y)$, by taking the two-dimensional Fourier transform $\mathit{\boldsymbol{\mathfrak{F}} }[ \cdot ]$ of image $I(x, y)$ to get Fourier spectrum $U(x, y)$

$U(x, y) = \mathit{\boldsymbol{\mathfrak{F}}}\left[ {I(x, y)} \right] $ (1)

calculating amplitude spectrum $A(x, y), $ phase spectrum $P(x, y)$ and log spectrum $L(x, y), $

$A(x, y) = \left| {U(x, y)} \right| $ (2)
$P(x, y) = {\rm{angle}}\left\{ {U(x, y)} \right\} $ (3)
$L(x, y) = \log A(x, y) $ (4)

then adopting a $n \times n$ local average filter $h{}_n$ to obtain the spectral residual $R(x, y), $

$R(x, y) = L(x, y) - {h_n} * L(x, y) $ (5)

where * represents convolution, and using inverse Fourier transform ${\mathit{\boldsymbol{\mathfrak{F}}}^{ - 1}}[ \cdot ]$ to construct the output image ${\rm{SR}}(x, y)$ in spatial domain, that is the saliency map of SR defined as

${\rm{SR}}(x, y) = g(m, \sigma) * {\left| {{\mathit{\boldsymbol{\mathfrak{F}}}^{ - 1}}\left[ {\exp \left({R(x, y) + j \cdot P(x, y)} \right)} \right]} \right|^2} $ (6)

where $g(m, \sigma)$ is a 2D gaussian filter (m denotes the size of filter and σ denotes standard deviation).

On the basis of SR, Guo et al. (2008) proposed the PFT model, which only uses the phase spectrum of the Fourier transform. The saliency map of PFT is given by

${\rm{PFT}}(x, y) = g(m, \sigma) * {\left| {{\mathit{\boldsymbol{\mathfrak{F}}}^{ - 1}}\left[ {\exp \left({j \cdot P(x, y)} \right)} \right]} \right|^2} $ (7)
1.2 Seismic Saliency Detection Models in Fractional Fourier Transform Domain

For seismic data, the information in the direction of time (or depth) is of great significance for reservoir prediction and hydrocarbon interpretation. It is required to adopt one-dimensional Fourier transform to get the SR and PFT models of seismic data. Furthermore, we extend these two models from the one-dimensional Fourier transform domain to the one-dimensional FrFT domain. Compared with the traditional Fourier transform, the FrFT has a fractional order variable, and can show all the changing characteristics of the data from the time domain to the frequency domain. In addition, the noise in the data can be better removed by using the rotation of the FrFT (Kutay et al., 1997).

In this paper, one-dimensional FrFT is used as a substitute for two-dimensional Fourier transform to give

${X_a}(u, q) = {F^a}\left[ {x(t, q)} \right] $ (8)
${A_a}(u, q) = \left| {{X_a}(u, q)} \right| $ (9)
${P_a}(u, q) = {\rm{angle}}\left\{ {{X_a}(u, q)} \right\} $ (10)
${L_a}(u, q) = \log {A_a}(u, q) $ (11)
${R_a}(u, q) = {L_a}(u, q) - {h_n} * {L_a}(u, q) $ (12)

where a is fractional order ($0 \le a < 2$) and ${F^a}[ \cdot ]$ denotes a-th order FrFT, which can be viewed as an operator that rotates any angle θ=aπ/2 counterclockwise from the time axis. For the special case where a=1, FrFT becomes traditional Fourier transform. Considering the horizontal continuity, we adopt the same order a for each single channel data.

According to Eqs. (6) and (7), the saliency detection models in FrFT domain can be easily obtained. Here we make a minor modification that filtering before computing the modulus square for more precise saliency map. Then the seismic SR model in FrFT domain (FrSSR) can be defined as

${\rm{FrSS}}{{\rm{R}}_a}(t, q) = {\left| {g(m, \sigma) * {F^{ - a}}\left[ {\exp \left({{R_a}(u, q) + j \cdot {P_a}(u, q)} \right)} \right]} \right|^2} $ (13)

and the seismic PFT model in FrFT domain (FrSPFT) can be defined as

${\rm{FrSPF}}{{\rm{T}}_a}(t, q) = {\left| {g(m, \sigma) * {F^{ - a}}\left[ {\exp \left({j \cdot {P_a}(u, q)} \right)} \right]} \right|^2} $ (14)

where m=5 and σ=2 are selected for Gaussian filter $g(m, \sigma)$ in this paper.

Regarding the properties of FrFT, the two models in the Eqs. (13)–(14) are symmetry at order a=1. Hence, we only consider the situation in which a belongs to [0, 1] in this paper. The saliency maps at different orders can highlight the target area in different directions θ=aπ/2 (θ indicates the counterclockwise angle to the time axis). When order a is close to 0, the saliency map shows more features in the time direction. When order a approaches 1, the saliency map takes advantage of more frequency spectral characteristic. FrSPFT model only uses phase spectrum, thus requires less computational complexity in comparison with FrSSR, and simultaneously will not weaken the detection effect. From the point of view, FrSPFT model is more superior to the FrSSR model.

The saliency feature can be regarded as a seismic attribute, using the above seismic saliency detection models in FrFT domain to extract the hidden information and obtain the saliency map of the entire seismic records for interpretation and analysis.

2 TEST AND ANALYSIS

In the paper, we compare the obtained saliency map with the S transform spectral decomposition results. S-transform, as a classical time-frequency analysis method, transforms the signal from the time domain to the time-frequency domain, so that the seismic profile at different frequencies can be obtained. And our method is to transform the signal from the time domain to the frequency domain, using its amplitude and phase information to reconstruct the signal and then transform back to the time domain to obtain the saliency map. Although these two methods are completely different attribute analysis methods, they are both used to characterize the structural features and locate reservoirs.

The simulated experiment with our method is carried out on a small part of the Marmousi 2 model (Martin, 2004). As shown in Fig. 1a, its structural model contains several different shales and two hydrocarbon layers: one shallow gas sand and one relatively shallow oil sand, while the oil layer information is mostly hidden in the two-dimensional profile of Fig. 1b. And a single channel at CDP=500 (bottom) is also displayed in Fig. 1b. The data comes from the website of Allied Geophysical Laboratory, University of Huston. The spectral decomposition technique based on time-frequency distribution is also an important approach for seismic attribute analysis. In order to better verify the performance of the proposed method, the S transform spectral decomposition results are given in Figs. 1c and 1d. Although the gas and oil layers are shown up in both profiles, the resolution is low. Moreover, we display the saliency maps of original SR and PFT models in Figs. 1e and 1f. It is obvious that both maps do not adequately characterize gas and oil layers.

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Figure 1. (a) Structural elements of a small part of the Marmusi 2 model; (b) the two-dimensional profile of a small part of the Marmusi 2 model (top) and its single channel at CDP=500 (bottom); (c) 40 Hz profile by S transform; (d) all frequencies cumulated profile by S transform; (e) saliency map of original SR model (Eq. 6); (f) saliency map of original PFT model (Eq. 7).

Figure 2 shows the saliency maps at order a=0.5 using different reconstruction forms (top) and single channel results at CDP=500 (bottom). The saliency maps in Figs. 2a and 2d are messy, not clear. Figures 2b and 2e are the Gaussian filtering results of the square of Figs. 2a and 2d, and Figs. 2c and 2f are the square of the Gaussian filtering results of Figs. 2a and 2d. These two approaches both improve the clarity of the saliency maps in Figs. 2a and 2d. The red numbers in the upper right corner of single channel results are the calculated kurtosis coefficients, which can reflect the sharpness of the data distribution. We can use the kurtosis coefficient to evaluate the salience of the target in the single channel. A larger kurtosis coefficient means that the targets are more salient. For single channel at CDP=500 in Fig. 1b, the targets are the red ellipse marked areas. Although Figs. 2b and 2e are smoother, Figs. 2c and 2f have larger kurtosis coefficients, representing better salience performance. This is because that Figs. 2b and 2e are more affected by the Gaussian filter and lose more details than Figs. 2c and 2f. The comparative experiment indicates that doing filter before the modulus square can lead to more salient and more precise result.

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Figure 2. The fractional saliency maps at a=0.5 using different reconstruction forms (top) and single channel results at CDP=500 (bottom). Red number: Kurtosis coefficient. (a) The reconstruction result without filtering and squaring using amplitude and phase spectrum (i.e., $\left| {{F^{ - a}}\left[ {\exp \left({{R_a}(u, q) + j \cdot {P_a}(u, q)} \right)} \right]} \right|$); (b) the reconstruction result with squaring before filtering using amplitude and phase spectrum (i.e., $g(m, \sigma) * {\left| {{F^{ - a}}\left[ {\exp \left({{R_a}(u, q) + j \cdot {P_a}(u, q)} \right)} \right]} \right|^2}$); (c) the reconstruction result with filtering before squaring using amplitude and phase spectrum (i.e., FrSSR model: ${\left| {g(m, \sigma) * {F^{ - a}}\left[ {\exp \left({{R_a}(u, q) + j \cdot {P_a}(u, q)} \right)} \right]} \right|^2}$); (d) the reconstruction result without filtering and squaring using phase spectrum (i.e., $\left| {{F^{ - a}}\left[ {\exp \left({j \cdot {P_a}(u, q)} \right)} \right]} \right|$); (e) the reconstruction result with squaring before filtering using phase spectrum (i.e., ${\left| {g(m, \sigma) * {F^{ - a}}\left[ {\exp \left({j \cdot {P_a}(u, q)} \right)} \right]} \right|^2}$); (f) the reconstruction result without filtering before squaring using phase spectrum (i.e., FrSPFT model: $g(m, \sigma) * {\left| {{F^{ - a}}\left[ {\exp \left({j \cdot {P_a}(u, q)} \right)} \right]} \right|^2}$). Ra stands for residual in Eq. (12), Pa stands for phase in Eq. (10), m=5 and σ=2.

The saliency maps of simulated data are obtained by FrSSR model at order a=0.3, 0.6, 0.9 (Figs. 3a3c (top)) and FrSPFT model at order a=0.3, 0.6, 0.9 (Figs. 3d3f (top)). These saliency maps at different orders can be used for a more detailed analysis of simulated model. As seen from figures, the saliency maps at a=0.3 give more different shales formation information, while the saliency maps at a=0.9 focus on reservoir regions. The salience performance of these maps can be analyzed and compared through the single channel results at CDP=500 (Fig. 3 (bottom)). As the value of order a increases, the Kurtosis coefficient (red number) of saliency map becomes larger. From this point of view, a=1 is the best order selection to obtain a clear saliency map regardless of detailed seismic events and the irregularity of the target area. However, if the target area is irregular, such as winding underground river, use only saliency map at a=1 can not show its changes in all directions, and the fusion of saliency maps at multi-orders may have better effect. At the same order a, the results of FrSPFT model are similar to those of FrSSR model, only slightly different in amplitude. This slight difference will lead to differences in the kurtosis coefficient values. We also give the single channel time cost (millisecond) of these two models as the green numbers shown in the upper right corner. Obviously, FrSPFT model is faster than FrSSR model at the same order a.

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Figure 3. The simulated data's fractional saliency maps (top) and single channel results at CDP=500 (bottom). Red number: kurtosis coefficient, green number: time cost (millisecond). (a) Saliency map and single channel result by FrSSR model at a=0.3; (b) saliency map and single channel result by FrSSR model at a=0.6; (c) saliency map and single channel result by FrSSR model at a=0.9; (d) saliency map and single channel result by FrSPFT model at a=0.3; (e) saliency map and single channel result by FrSPFT model at a=0.6; (f) saliency map and single channel result by FrSPFT model at a=0.9.

To verify the anti-noise performance of the proposed method, the Marmousi 2 model is added with white Gaussian noise, giving different signal-to-noise ratio (SNR) values. Figures 4a, 4d and 4g show the noisy model data with the SNR equal to 8 dB, 3 dB and -5 dB, respectively. And their corresponding FrSSR and FrSPFT saliency maps at order a=0.6 are also given in Fig. 4. As the SNR decreases, the interference of the noise to the model data becomes more serious. However, from the single channel results of the saliency maps, it can be seen that the noise is effectively suppressed. Simultaneously, the results of FrSPFT model have higher kurtosis coefficients and less computation time compared to FrSSR model.

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Figure 4. The fractional saliency maps at a=0.6 of noisy model data (top) and single channel results at CDP=500 (bottom). Red number: Kurtosis coefficient, green number: time cost (millisecond). (a) Noisy model data with SNR=8 dB; (b) the saliency map and single channel result by FrSSR model of (a); (c) the saliency map and single channel result by FrSPFT model of (a); (d) noisy model data with SNR=3 dB; (e) the saliency map and single channel result by FrSSR of (d); (f) the saliency map and single channel result by FrSPFT model of (d); (g) noisy model data with SNR=-5 dB; (h) the saliency map and single channel result by FrSSR model of (g); (i) the saliency map and single channel result by FrSPFT model of (g).
3 SEISMIC ATTRIBUTE ANALYSIS

To evaluate the performance of the proposed method, a real seismic data derived from Sichuan Basin in Fig. 5a is analyzed. The seismic data consists of 101 seismic traces with the sampling interval of 2 ms, and has a gas-containing area marked with the red ellipse. And Figs. 5b and 5c represent the S transform spectral decomposition results of the seismic data. These two slices depict the gas-containing area, but the energy is divergent and the resolution is relatively low.

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Figure 5. (a) A real seismic data profile; (b) 35 Hz profile by S transform; (c) 45 Hz profile by S transform.

FrSSR model and FrSPFT model are applied to the seismic data to construct the fractional saliency maps as shown in Fig. 6. The top figures are saliency maps and bottom figures are single channel results at CDP=350. In the saliency maps of Figs. 6a and 6d, there are more horizontal features for stratum structure. With the increase in fractional order, kurtosis coefficient become larger, and the saliency maps weaken some detailed seismic events and highlight reservoir regions. Using these fractional saliency maps at different orders, we can get more structural information and reservoir information.

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Figure 6. The fractional saliency maps of seismic data (top) and single channel results at CDP=350 (bottom). Red number: Kurtosis coefficient, green number: time cost (millisecond). (a) Saliency map and single channel result by FrSSR model at a=0.3; (b) saliency map and single channel result by FrSSR model at a=0.6; (c) saliency map and single channel result by FrSSR model at a=0.9; (d) saliency map and single channel result by FrSPFT model at a=0.3; (e) saliency map and single channel result by FrSPFT model at a=0.6; (f) saliency map and single channel result by FrSPFT model at a=0.9.

Another experiment with our method is carried out on the real seismic data with additive white Gaussian noise with SNR of 8 dB, 3 dB and -5 dB in Figs. 7a, 7d and 7g. The remaining pictures in Fig. 7 represent the FrSPFT and FrSSR saliency maps at a=0.6 for each noisy seismic data. The seismic data is heavily corrupted by noise when the SNR is low (see Fig. 7g), hindering the observation of important information. Of course, this causes issues for the seismic signal processing. However, the proposed method can suppress noise interference to obtain clean saliency map even under low SNR condition (see Figs. 7h7i). And FrSPFT model has better salience performance and lower computational cost than FrSSR model. Professional interpreters can analyze stratigraphic structures and locate reservoir areas based on these saliency maps and subsurface information.

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Figure 7. The fractional saliency maps at a=0.6 of noisy seismic data (top) and single channel results at CDP=350 (bottom). Red number: kurtosis coefficient, green number: time cost (millisecond). (a) Noisy seismic data with SNR=8 dB; (b) the saliency map and single channel result by FrSSR model of (a); (c) the saliency map and single channel result by FrSPFT model of (a); (d) noisy seismic data with SNR=3 dB; (e) the saliency map and single channel result by FrSSR of (d); (f) the saliency map and single channel result by FrSPFT model of (d); (g) noisy seismic data with SNR=-5 dB; (h) the saliency map and single channel result by FrSSR model of (g); (i) the saliency map and single channel result by FrSPFT model of (g).
4 CONCLUSIONS

In this paper, we proposed the fractional saliency detection models and applied them to seismic data. FrSSR and FrSPFT models are mainly introduced as the extension of SR and PFT models in fractional Fourier domain. Using these two models, several saliency maps at different fractional orders can be obtained to analyze the structure of the input data effectively. The comparative experiments showed that FrSPFT model is better and faster than FrSSR model.

The test on simulated and real seismic data demonstrated that the fractional saliency detection models are effective and feasible for seismic attribute extraction. The obtained fractional saliency maps provide more precise structural features and highlight the object areas, which is important in the analysis of hydrocarbon reservoirs. In addition, we also showed that these two models have strong ability to suppress noise interference.

Furthermore, the saliency maps at different fractional orders can be fused according to certain rules for better representation, which will be studied in future work.

ACKNOWLEDGMENTS

This work was supported by the National Natural Science Foundation of China (Nos. 61571096, 61775030, 41274127, 41301460, and 40874066). The final publication is available at Springer via https://doi.org/10.1007/s12583-017-0811-z.


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