Fast acquisition aperture correction in prestack depth migration using ...

GEOPHYSICS, VOL. 74, NO. 4 共JULY-AUGUST 2009兲; P. S67–S74, 10 FIGS. 10.1190/1.3116284

Fast acquisition aperture correction in prestack depth migration using beamlet decomposition

Jun Cao1 and Ru-Shan Wu1

imaging makes ray-based methods more convenient than waveequation methods because angle information is embedded inherently in ray-based methods. The theory and method of true reflection/ amplitude imaging 共e.g., Newman, 1973; Hubral, 1983兲 have been developed based on high-frequency asymptotic theory 共ray theory兲 and are traditionally conducted using Kirchhoff migration 共e.g., Bleistein et al., 1987; Hubral et al., 1991; Hanitzsch, 1995; Xu et al., 2001; Audebert et al., 2002; Brandsberg-Dahl et al., 2003兲. However, the results may contain large errors in complex areas because of the high-frequency approximation and singularity problems. For wave-equation migration methods, Wu et al. 共2004兲 propose an angle-dependent image amplitude correction for acquisition aperture effects that include the acquisition configuration effect and propagation path effects through complex overburdens. Numerical examples 共Wu et al., 2004; Wu and Luo, 2005兲 show significant improvement in total strength of the images and angle-dependent reflection amplitudes after correction, demonstrating the significance of the acquisition aperture correction in true reflection imaging. However, the original implementation is inefficient because the wavefield is decomposed into the local angle domain 共LAD兲 using computationally demanding local slant stack 共LSS兲 共e.g., Xie and Wu, 2002兲. Even using a relatively short spatial window for decomposition, the computation to obtain the common-dip image 共summed image from all reflection angles for the local reflector with a certain dip兲 can be five times more expensive than space-domain imaging. This prohibits using the acquisition aperture correction in industrial migration processing. Our goal is to improve the efficiency of obtaining the LAD image and amplitude correction factors. To obtain the LAD image and amplitude correction factors, we need to decompose the extrapolated space-domain wavefield into local plane waves, which are simultaneously localized in both space and direction. In addition to the LSS, beamlet decomposition 共e.g., Wu and Chen, 2002, 2006兲 can also decompose a wavefield into local plane waves. A Gabor-Daubechies frame 共GDF兲 beamlet 共Wu et al., 2000; Chen et al., 2006兲 uniquely defines local direction information; however, GDF decomposition is more expensive to compute than local cosine-basis beamlet decom-

ABSTRACT Wave-equation-based acquisition aperture correction in the local angle domain can improve image amplitude significantly in prestack depth migration. However, its original implementation is inefficient because the wavefield decomposition uses the local slant stack 共LSS兲, which is demanding computationally. We propose a faster method to obtain the image and amplitude correction factor in the local angle domain using beamlet decomposition in the local wavenumber domain. For a given frequency, the image matrix in the local wavenumber domain for all shots can be calculated efficiently. We then transform the shot-summed image matrix from the local wavenumber domain to the local angle domain 共LAD兲. The LAD amplitude correction factor can be obtained with a similar strategy. Having a calculated image and correction factor, one can apply similar acquisition aperture corrections to the original LSS-based method. For the new implementation, we compare the accuracy and efficiency of two beamlet decompositions: Gabor-Daubechies frame 共GDF兲 and local exponential frame 共LEF兲. With both decompositions, our method produces results similar to the original LSS-based method. However, our method can be more than twice as fast as LSS and cost only twice the computation time of traditional one-way wave-equation-based migrations. The results from GDF decomposition are superior to those from LEF decomposition in terms of artifacts, although GDF requires a little more computing time.

INTRODUCTION Seismic imaging is most helpful when it estimates subsurface reflectivity/scattering strength, which is dependent on reflection/scattering angle. Therefore, imaging and image amplitude correction work best in the angle domain. This requirement of angle-dependent

Manuscript received by the Editor 2 June 2008; revised manuscript received 3 December 2008; published online 19 May 2009. 1 University of California, Department of Earth and Planetary Sciences/Institute of Geophysics and Planetary Physics, Santa Cruz, California, U.S.A. E-mail: [email protected]; [email protected]. © 2009 Society of Exploration Geophysicists. All rights reserved.

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position 共Wu et al., 2000; Wu et al., 2008兲. Local cosine-basis beamlets are orthonormal and the local cosine transform has a fast algorithm; however, these beamlets always have two symmetrical lobes with respect to the vertical axis, resulting from the inherent property of the cosine-basis function. Local exponential frame 共LEF兲 共Daubechies et al., 1991; Auscher, 1994; Mao and Wu, 2007兲 beamlet decomposition can eliminate the directional ambiguity in the local cosine-basis beamlets. LEF can be implemented using a combination of local cosine and sine transforms that have fast algorithms; therefore, it is efficient. The beamlet decompositions are more efficient than LSS; however, the decomposed wavefield is in the local wavenumber domain 共LWD兲 and not directly in the LAD. We propose a fast implementation to obtain the LAD image and amplitude correction factor by beamlet decomposition in the LWD. First, we summarize the beamlet decomposition method, limiting the discussion to the 2D framework. Then the new implementation of the acquisition aperture correction is described. We demonstrate the method with GDF and LEF beamlet decompositions for the 2D SEG/EAGE salt model 共Aminzadeh et al., 1994; Aminzadeh et al., 1995兲 and compare the results with those from the LSS method. We also discuss possible factors causing the difference in the results of GDF and LEF decompositions.

Beamlet decomposition 共e.g., Steinberg, 1993; Wu et al., 2000; Chen et al., 2006; Mao and Wu, 2007兲 provides a basis to localize a wavefield in the space and wavenumber domains simultaneously. The frequency-space-domain wavefield u共x,␻ 兲 at depth level z for frequency ␻ can be decomposed into beamlets by the following formula:

m

共1兲

n

where bmn are beamlets 共decomposition basis vectors兲 located at space window ¯xn and wavenumber window ¯␰ m and where ¯ n,¯␰ m,␻ 兲 are the decomposition coefficients for corresponding uˆ共x beamlets bmn. We can obtain the local plane waves u共x,¯␰ m,␻ 兲 by partially reconstructing the beamlet-domain wavefields:

¯ n,¯␰ m,␻ 兲bmn共x兲. u共x,¯␰ m,␻ 兲 ⳱ 兺 uˆ共x

共2兲

n

For a local plane wave with local wavenumber ¯␰ m, the corresponding propagating angle with respect to the vertical direction is

␪¯ m ⳱ sinⳮ1

¯␰ m , k共x兲

CALCULATING IMAGE AND AMPLITUDE CORRECTION FACTOR BY BEAMLET DECOMPOSITION For a subsurface imaging point, the traditional space-domain imaging condition 共Claerbout, 1971兲 for a single frequency can be written in the form of crosscorrelation of the incident source-side wave uS共x,␻ 兲 and the back-propagated receiver-side wave uRs共x,␻ 兲:

I共x,␻ 兲 ⳱ 兺 uS*共x,␻ 兲uRs共x,␻ 兲 S

冕

⳱ 兺 2GI*共x;xs兲 xs

dxg

⳵ GI*共x;xg兲 ⳵z

A共xg;xs兲

D共xg;xs兲, 共4兲

BEAMLET DECOMPOSITION OF THE WAVEFIELD

¯ n,¯␰ m,␻ 兲bmn共x兲, u共x,␻ 兲 ⳱ 兺 兺 uˆ共x

tio 共number of beamlet coefficients to number of coefficients in the space domain兲. Bell windows 共see, e.g., Mallat, 1999; Wu et al., 2008兲 and Gaussian windows 共see, e.g., Mallat, 1999; Wu and Chen, 2006兲 are used in LEF and GDF decomposition, respectively. The redundancy ratio R in GDF decomposition is adjustable; therefore GDF can provide an accurate, partially reconstructed local wavenumber-domain wavefield. However, R in the LEF representation is fixed to two, which can lead to error in that wavefield.

共3兲

where k共x兲 ⳱ ␻ / v共x兲 and v共x兲 is the local velocity. The beamlet used to obtain the LAD image and amplitude correction factor must have uniquely defined local direction information. We demonstrate our method with two of examples of this beamlet: the Gabor-Daubechies frame 共GDF兲 beamlet and the local exponential frame 共LEF兲 beamlet. Appendices A and B summarize the theory behind these decompositions. Both beamlets use windowed exponential harmonics as the decomposition basis. The differences are the windows used to modulate the harmonics and the redundancy ra-

where x ⳱ 共x,z兲 is the imaging point coordinates, * stands for complex conjugate, GI is the propagator used in the imaging process, and the integral is a back-propagation Rayleigh integral of the recorded scattered wavefield D共xg;xs兲 共see, e.g., Berkhout, 1987兲, in which A共xg;xs兲 is the receiver aperture for a given source. Subsurface reflectivity/scattering strength is reflection/scattering angle dependent; therefore, the conventional space-domain imaging condition 4 must be extended to the LAD 共or beamlet domain兲 共e.g., Wu and Chen, 2002, 2003; Chen et al., 2006兲. Then the image obtained at each imaging point is no longer a scalar but a matrix, called the local image matrix L共x,␪ s,␪ g兲, where ␪ s and ␪ g are the source and receiving angles, respectively 共Figure 1兲. The local image matrix is a distorted estimate of the local scattering matrix because of

Source

Receiver

z n θs θn

θr

θg

x

Reflec

tor

i Figure 1. Local angle definition in a scattering experiment for a local planar reflector. The terms ␪ s and ␪ g are the local source and receiving angles, respectively; ␪ n is the reflector normal angle 共equal to dip angle兲 and ␪ r is the reflection angle.

Fast acquisition aperture correction acquisition aperture limits and propagation path effects. The local scattering matrix is the intrinsic property of the scattering medium: it contains information about the local structure and elastic properties 共Wu et al., 2004兲. The task of true reflection/amplitude imaging is to restore the true local scattering matrix from the distorted local image matrix by applying image amplitude corrections. The LAD image amplitude correction proposed by Wu et al. 共2004兲 can improve the image significantly; however, the original implementation based on LSS is inefficient because LSS is expensive to compute. Because it is much more efficient to obtain the wavefield in the LWD than in the LAD, we propose an efficient way to obtain the LAD image by partial imaging in the LWD. First, the image for all shots at a single frequency can be formed in the LWD:

L共x,¯␰ m,¯␰ j,␻ 兲 ⳱ 兺 uS*共x,¯␰ m,␻ 兲uRs共x,¯␰ j,␻ 兲,

共5兲

S

where ¯␰ m and ¯␰ j are the local source and receiving wavenumbers respectively, and where uS共x,¯␰ m,␻ 兲 and uRs共x,¯␰ j,␻ 兲 are incident 共source兲 and back-propagated 共receiver兲 local plane waves, respectively. The summed LWD image for all shots at a given frequency L共x,¯␰ m,¯␰ j,␻ 兲 in imaging condition 5 is a real number that can be interpolated easily to form the LAD image L共x,␪ s,␪ g,␻ 兲. We only need to transform this shot-summed real image matrix from the LWD to the LAD by interpolation, so the computation time to obtain the LAD image for all shots is reduced significantly. The LAD images from different frequencies can then be summed to form the multifrequency image. With derivation similar to that for obtaining the amplitude correction factor directly from the LAD wavefield 共Wu et al., 2004兲, we obtain the LWD amplitude correction factor for the corresponding imaging condition 5:

Fa共x,¯␰ m,¯␰ j兲 ⳱ 兺 兩GI*共x,¯␰ m;xs兲GF共x,¯␰ m;xs兲兩 xs

⫻

再冕

dxg兩GF共x,¯␰ j;xg兲兩2

A共xg;xs兲

冎

1/2

共6兲

,

where GF is the Green’s function used in forward modeling. In equation 6, we make the following approximation from the energy conservation point of view:

冏

2

冕

dxgGF共x,¯␰ j;xg兲

⳵ GI*共x,¯␰ j;xg兲

A共xg;xs兲

⳱

冕

dxg兩GF共x,¯␰ j;xg兲兩2 .

⳵z

冏

2

共7兲

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we approximate GF with a one-way propagator. With a scheme similar to obtaining the LAD image, we can interpolate the LWD amplitude correction factor to that in the LAD, Fa共x,␪ s,␪ g兲. Having obtained the LAD image matrix L共x,␪ s,␪ g兲 and amplitude correction factor matrix Fa共x,␪ s,␪ g兲, we can apply similar acquisition aperture correction as in the LSS-based method, such as correcting for common reflection-angle imaging and total strength imaging 共Wu et al., 2004兲. Compared with traditional space-domain imaging, extra computations to obtain LAD images include obtaining the local plane waves and applying the LAD/LWD imaging condition. We compare the number of complex floating-point operations in these extra computations for the original LSS-based method and our method 共with LEF decomposition as an example兲. We also illustrate with the operation number for one depth level with N spatial points. For the LSS method, the operation number of the local planewave construction is about N ⫻ NangLwin and for LAD imaging is N 2 ⫻ Nang , where Lwin is window size and Nang is angle sample number. For our method by LEF decomposition with the same window size Lwin, the operation numbers are about N ⫻ 2共log2共Lwin / 4兲 Ⳮ Lwin兲 and 2 N ⫻ Lwin . Setting Nang ⳱ Lwin, the imaging procedure needs the same number of operations for both methods; but for local plane-wave construction, our method needs much fewer operations than LSSbased method. Furthermore, in the LWD, horizontal wavenumbers for some local plane waves are larger than the wavenumber k共x兲. These waves correspond to evanescent waves. We can neglect them for the imaging, saving some computation for local plane-wave construction and imaging. If we use local windowed FFT to obtain the LWD plane waves, we need about N ⫻ Lwin log2 Lwin operations, which is usually more than for LEF beamlet decomposition.

EXAMPLES We demonstrate the new implementation of the LAD acquisition aperture correction using the 2D SEG/EAGE salt model 共Figure 2兲 data set 共Aminzadeh et al., 1994; Aminzadeh et al., 1995兲. The acquisition geometry of this model consists of 325 shots with 176 leftside trailing receivers for each shot; the shot and receiver intervals are 49 m 共160 ft兲 and 24 m 共80 ft兲, respectively. We use a local cosine-basis propagator 共Wu et al., 2000; Luo and Wu, 2003; Wang et al., 2003; Wu et al., 2008兲 for the wavefield extrapolation. The redundancy ratio R in the GDF decomposition is set to four except when specified otherwise. Common-dip images obtained from the new implementation by LEF and GDF decompositions are similar to those from the original LSS-based method 共Figure 3兲. Subtle differences exist among these images, e.g., the artifacts around the steep salt top in the image for Ⳮ30° dip obtained from LEF decomposition are stronger than those from the original method and from the new method with GDF de-

A共xg;xs兲

Depth (km)

The back-propagation integral on the left side represents a refocusing process for the Green’s function in the LAD. At the scattering point, we assume that all scattered energy received by the receiver array has been refocused so the energy conservation law can be applied. This approximation neglects the possible edge diffraction of the receiver array. For sufficiently large receiver apertures, the edge leakage should be small for most of the Green’s beamlets 共Green’s function in the LAD兲. 共For a detailed derivation, refer to Wu et al. 关2004兴.兲 When calculating the acquisition aperture correction factor,

0

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Surface location (km) 6 8 10

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Figure 2. 2D SEG/EAGE salt model.

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Cao and Wu with the same window length, the value is only two, making it more than twice as fast as the LSS-based method. With GDF decomposition, the new method requires a little more computation time than LEF.

composition. The common-dip amplitude correction factors from the new method by both beamlet decompositions are also similar to those from the LSS method 共Figure 4兲. However, some stripelike artifacts appear in the results obtained by LEF decomposition. The final total strength of the image after acquisition aperture correction from both beamlet decompositions is similar to that from the LSS method 共Figure 5兲. Predictably, subtle differences exist among the final corrected images: the artifacts in the subsalt area and around the steep salt top obtained from the new method with LEF decomposition are stronger than those from the original LSS-based method and the new method with GDF decomposition. As for efficiency, if we assign a hypothetical value of one to the computation cost of conventional space-domain imaging, the cost of the LSS-based method to form the LAD image will be a value of about five, even using a relatively short spatial window for decomposition. For the new method using either beamlet decomposition

Depth (km)

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The GDF results are superior to the LEF results, e.g., the amplitude correction factor map obtained by GDF decomposition has no stripe-like artifacts, such as those that appear in the LEF map. The differences in LEF and GDF beamlet decomposition are the window functions 共bell window versus Gaussian window; see Figure 6兲 and redundancy ratio. The redundancy ratio is fixed at two in LEF decomposition but is adjustable in GDF decomposition 共we used four兲. Here, we investigate these factors to find what may lead to the difference in results. We use directional illumination 共e.g., Wu and Chen,

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Figure 3. Common dip images for 共a-c兲 Ⳮ30° and 共d-f兲 ⳮ30° obtained from our method using 共a, d兲 LEF and 共b, e兲 GDF decomposition compared with 共c, f兲 the original LSS-based method.

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Figure 4. Image amplitude correction factors for dip angle 共a-c兲 Ⳮ30° and 共d-f兲 ⳮ30° obtained from the new method using 共a, d兲 LEF and 共b, e兲 GDF decomposition compared with 共c, f兲 those from the original LSS-based method.

Fast acquisition aperture correction

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Surface location (km) 2

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Figure 6. Window functions used in LEF and GDF beamlet decompositions. Solid line — steep bell window function used in previous LEF decomposition; dotted line — smooth bell window function for LEF decomposition; dashed line — Gaussian window function used in GDF decomposition.

Depth (km)

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Figure 8. Image amplitude correction factors for a dip angle of Ⳮ30° obtained using LEF decomposition with a less steep bell window function 共dotted line in Figure 6兲 but different lateral sampling interval of the frame ⌬N: 共a兲 ⌬N ⳱ 8, 共b兲 ⌬N ⳱ 16.

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Figure 5. 共a兲 Image without correction compared with final total strength of the image after acquisition aperture correction from the new method using 共b兲 LEF and 共c兲 GDF decomposition and from 共d兲 the original LSS-based method.

2006; Xie et al., 2006兲 in LWD from a single shot for investigation because it represents the amplitude of the decomposed directional wavefield from beamlet decomposition. We also investigate the amplitude correction factor for the entire acquisition geometry. First, we investigate the influence of the window function. In previous calculations with LEF decomposition, we used a relatively steep bell function with frame-sampling interval ⌬N ⳱ 8 共solid line in Figure 6兲. With a less steep bell function 共dotted line, Figure 6兲, the LWD single-shot directional illumination shows improvement compared with the result using the original window 共Figure 7兲. Results also show that the directional illumination maps get smoother with larger ⌬N 共Figure 7兲. With the new window, the striped artifacts are invisible in the amplitude correction factor for a dip angle of Ⳮ30° in the subsalt and right part of the model 共Figure 8a兲; however, artifacts in the left part of the model are similar to those with the steep bell function in Figure 4a. With ⌬N ⳱ 16, the artifacts are weaker, but we also lose the spatial resolution in some degree in the subsalt area 共Figure 8b兲. Thus, with a longer and less steep bell window, the artifacts in the LAD results by LEF decomposition weaken but cannot be removed completely. Next, we investigate the effect of R in GDF decomposition. Using R ⳱ 4, LWD single-shot directional illumination maps are very clean 共Figures 9a and b兲. With R ⳱ 2, directional illumination results

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Figure 7. LWD single-shot directional illumination maps in the SEG/EAGE salt model by LEF decomposition using bell windows with different steepness and lateral sampling interval of the frame ⌬N. 共a兲 With a less steep bell window 共dotted line in Figure 6兲 and ⌬N ⳱ 8; 共b兲 with same less steep bell window but ⌬N ⳱ 16; 共c兲 with a steep bell window 共solid line in Figure 6兲 and ⌬N ⳱ 8; 共d兲 with same steep bell window but ⌬N ⳱ 16.

Figure 9. LWD single-shot directional illumination maps in the SEG/EAGE salt model by GDF decomposition with different R and lateral sampling interval of the frame ⌬N: 共a兲 R ⳱ 4, ⌬N ⳱ 4; 共b兲 R ⳱ 4, ⌬N ⳱ 8; 共c兲 R ⳱ 2, ⌬N ⳱ 4; 共d兲 R ⳱ 2, ⌬N ⳱ 8.

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support from the WTOPI 共Wavelet Transform on Propagation and Imaging for seismic exploration兲 Research Consortium and the U.S. Department of Energy/ Basic Energy Sciences Project at University of California, Santa Cruz. We thank Jian Mao for providing the LEF decomposition algorithm and helpful discussion on it. Contribution 498 of the CSIDE, IGPP, University of California, Santa Cruz.

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1 2

GABOR-DAUBECHIES FRAME BEAMLET DECOMPOSITION

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Figure 10. Image amplitude correction factors for a dip angle of Ⳮ30° obtained using GDF decomposition with R ⳱ 2 but different lateral sampling interval of the frame ⌬N: 共a兲 ⌬N ⳱ 4, 共b兲 ⌬N ⳱ 8. 共Figures 9c and d兲 show subtle striped artifacts that are much weaker than those by LEF decomposition 共Figure 7兲. With the same framesampling interval as in Figure 4 共⌬N ⳱ 4兲 but a smaller R 共 ⳱ 2兲, the amplitude correction factor map for a dip angle of Ⳮ30° by GDF decomposition 共Figure 10a兲 now displays striped artifacts in the left part of the model, similar to those in LEF decomposition 共Figure 4a and Figure 8兲. However, the artifacts are much weaker and narrower than by LEF decomposition 共Figure 4a兲 because stripe width is related to the frame sampling interval. Using ⌬N ⳱ 8 in GDF decomposition, the result 共Figure 10b兲 shows artifacts as wide as those by LEF decomposition 共Figure 4a and Figure 8a兲 but much weaker. These results demonstrate that GDF decomposition with lower redundancy ratio 共e.g., R ⳱ 2兲 could give LAD directional illumination and amplitude correction factor with subtle stripelike artifacts. However, with higher R 共e.g., four兲, it could yield very clean results.

CONCLUSIONS We propose a fast method to obtain the LAD image and amplitude correction factor using LWD beamlet decomposition. We first obtain the LWD image and factor for all shots at a given frequency; then we transform them to the LAD. Having obtained the image and correction factor, acquisition aperture corrections can be applied similar to the original LSS-based method. The results for the 2D SEG/EAGE salt model illustrate that the new method with LEF and GDF beamlet decompositions produces results similar to the original LSS-based method. GDF decomposition can provide results superior to LEF in terms of artifacts, although it requires a little more computation time. As for efficiency, the new method can be more than twice as fast as LSS-based and only costs about twice the computation time of traditional one-way wave-equation-based migration. This makes it possible to apply wave-equation-based LAD processing, e.g., illumination analysis and acquisition aperture correction, to the vast quantities of data in industrial processing.

GDF beamlets 共e.g., Wu et al., 2000; Chen et al., 2006兲 define localization information uniquely. The GDF beamlets for decomposition equation 1 are Gaussian function windowed exponential harmonics: ¯

bmn共x兲 ⳱ g共x ⳮ¯xn兲ei␰ mx,

共A-1兲

where ¯␰ m ⳱ m⌬␰ 共⌬␰ is the wavenumber sampling interval兲 and g共x兲 is a Gaussian window function,

冉 冊

g共x兲 ⳱ 共␲ s2兲ⳮ1/4 exp ⳮ

s2 ⳱

x2 , 2s2

R⌬N2 , 2␲

共A-2兲

共A-3兲

where s is the scale of the Gaussian window, R is the redundancy ratio, and ⌬N is the lateral sampling interval of the frame. Substituting the GDF beamlets 共equation A-1兲 into the partial reconstruction equation 2, we can obtain the local plane waves, ¯ ¯ n,¯␰ m,␻ 兲, u共x,¯␰ m,␻ 兲 ⳱ ei␰ mx 兺 g共x ⳮ¯xn兲uˆ共x

共A-4兲

n

with the decomposition coefficients

˜ 共x兲典 ⳱ ¯ n,¯␰ m,␻ 兲 ⳱ 具u共x,␻ 兲,b uˆ共x mn

冕

˜ * 共x兲dx, u共x,␻ 兲b mn 共A-5兲

where 具,典 stands for inner product, * stands for complex conjugate, and ˜bmn共x兲 are the dual GDF atoms,

˜b 共x兲 ⳱ ˜g共x ⳮ¯x 兲ei¯␰ mx, mn n

共A-6兲

ACKNOWLEDGMENTS

with ˜g共x兲 the dual-window function of g共x兲. The dual-window function can be calculated by pseudoinversion of the original window function 共Qian and Chen, 1996; Mallat, 1999兲. From the partially reconstructed wavefield A-4, we can see that the local plane wave is a weighted average of the windowed beamlets with the same wavenumber from neighboring windows. The total space-domain wavefield can be written as

The authors thank the associate editor and reviewers for their valuable comments that improved this manuscript. We acknowledge

u共x,␻ 兲 ⳱ 兺 u共x,¯␰ m,␻ 兲. m

共A-7兲

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u共x,␻ 兲 ⳱ 兺 u共Ⳮ兲共x,¯␰ m,␻ 兲 Ⳮ 兺 u共ⳮ兲共x,¯␰ m,␻ 兲.

APPENDIX B LOCAL EXPONENTIAL FRAME BEAMLET DECOMPOSITION Similar to the GDF beamlet, the LEF beamlet 共e.g., Daubechies et al., 1991; Auscher, 1994; Mao and Wu, 2007兲 can decompose the wavefield into LWD with uniquely defined direction information. It eliminates the directional ambiguity in the local cosine-basis beamlet. The LEF beamlets for the decomposition equation 1 are bell function windowed exponential harmonics:

bmn共x兲 ⳱

冑

2 Bn共x兲exp共i¯␰ m共x ⳮ¯xn兲兲, Ln

共B-1兲

where Bn共x兲 is a smooth bell 共window兲 function, Ln ⳱¯xnⳭ1 ⳮ¯xn is 1 the nominal length of the window, and ¯␰ m ⳱ ␲ 共 m ⳮ 2 兲 / Ln 共m ⳱ ⳮM Ⳮ 1,. . ., ⳮ 1,0,1,. . .,M兲. The LEF beamlets described by equation B-1 can be grouped into the right- and left-propagating beamlets 共Ⳮ兲 共ⳮ兲 bmn 共x兲 and bmn 共x兲:

冦

共Ⳮ兲 bmn 共x兲 ⳱

冑

2 Bn共x兲exp共Ⳮ i¯␰ m共x ⳮ¯xn兲兲 Ln

共c兲 共s兲 共x兲 Ⳮ ibmn 共x兲 ⳱bmn 共ⳮ兲 bmn 共x兲 ⳱

冑

2 Bn共x兲exp共ⳮ i¯␰ m共x ⳮ¯xn兲兲 Ln

共c兲 共s兲 共x兲 ⳮ ibmn 共x兲 ⳱bmn

where

冦

共c兲 bmn 共x兲 ⳱ 共s兲 共x兲 ⳱ bmn

冑 冑

冧

共B-2兲

,

2 Bn共x兲cos共¯␰ m共x ⳮ¯xn兲兲 Ln 2 Bn共x兲sin共¯␰ m共x ⳮ¯xn兲兲 Ln

冧

,

共B-3兲

共c兲 共s兲 with m ⳱ 1,2,. . .,M and where bmn 共x兲, bmn 共x兲 are the local cosineand sine-basis beamlets, respectively. Substituting the LEF beamlets B-2 into the partial reconstruction equation 2, we can obtain the right- and left-propagating local plane waves u共Ⳮ兲共x,¯␰ m,␻ 兲 and u共ⳮ兲 ⫻共x,¯␰ m,␻ 兲:

冦

u共Ⳮ兲共x,¯␰ m,␻ 兲 ⳱ exp共Ⳮ i¯␰ mx兲 ⫻兺 n

u

共ⳮ兲

冑

2 ¯ n,¯␰ m,␻ 兲exp共ⳮ i¯␰ m¯xn兲 Bn共x兲uˆ共Ⳮ兲共x Ln

共x,¯␰ m,␻ 兲 ⳱ exp共ⳮ i¯␰ mx兲

⫻兺 n

冑

2 ¯ n,¯␰ m,␻ 兲exp共Ⳮ i¯␰ m¯xn兲 Bn共x兲uˆ共ⳮ兲共x Ln

再

with the decomposition coefficients 共Ⳮ兲 ¯ n,¯␰ m,␻ 兲 ⳱ 具u共x,␻ 兲,bmn 共x兲典 uˆ共Ⳮ兲共x 共ⳮ兲 ¯ n,¯␰ m,␻ 兲 ⳱ 具u共x,␻ 兲,bmn uˆ共ⳮ兲共x 共x兲典

The space-domain total wavefield can be written as

冧

冎

.

,

共B-4兲

共B-5兲

共B-6兲

m

m

Using the relation between the LEF and local cosine- and sine-basis beamlets of expression B-2, the LEF decomposition coefficients 共Ⳮ兲 共ⳮ兲 uˆmn and uˆmn of expression B-5 can be calculated using the local co共c兲 共s兲 sine- and sine-basis decomposition coefficients uˆmn and uˆmn :

冦

共Ⳮ兲 uˆmn ⳱

共c兲 共s兲 ⳮ iuˆmn uˆmn 4

共ⳮ兲 ⳱ uˆmn

共c兲 共s兲 Ⳮ iuˆmn uˆmn 4

冧

.

共B-7兲

The local cosine- and sine-basis decomposition coefficients can be obtained by fast algorithms, such as the one introduced by Malvar 共1992兲 共see also Mallat, 1999兲, which makes LEF decomposition efficient.

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