Full-wave directional illumination analysis in the frequency domain

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1 GEOPHYSICS, VOL. 74, NO. 4 JULY-AUGUST 009 ; P. S85 S9, FIGS. 0.90/.8 Full-wave directional illumination analysis in the frequency domain Jun Cao and Ru-Shan Wu ABSTRACT Directional illumination analysis based on one-way wave equations has been studied extensively; however, its inherent limitations, e.g., one-way propagation, wide-angle error, and amplitude inaccuracy, can severely hinder its applications for accurate survey design and true-reflection imaging corrections in complex media. We have analyzed the illumination in the frequency domain using full two-way wave propagators considering the extensive computation and huge storage required for time-domain methods, and the fact that the illumination is frequency dependent. This full-wave analysis can provide frequency-dependent full-angle true-amplitude illumination not only for the downgoing waves but also for the upgoing waves, including turning waves and reflected waves. Two methods were considered to decompose the full wavefield into the local angle domain: a direct full-dimensional decomposition and more efficient split-step decomposition composed of three lower-dimensional decompositions. The results of illumination analysis demonstrated the advantages of this method. The two decomposition methods produced similar results. INTRODUCTION Local angle-domain LAD illumination analysis e.g., Wu and Chen, 00 studied the acquisition aperture and propagation-path effects. It has many applications, including survey design e.g., Li and Dong, 00, studying the influences of acquisition geometry and overburden structures on the image e.g., Jin and Walraven, 00; Wu and Chen, 00; Xie et al., 00, and image amplitude correction e.g., Wu et al., 004; Cao and Wu, 005, 008. Traditionally, illumination analyses have used ray-based methods e.g., Schneider and Winbow, 999; Bear et al., 000; Muerdter et al., 00; Muerdter and Ratcliff, 00a, 00b; Lecomte et al., 00, in which the directional information is inherently carried by the rays. However, the high-frequency asymptotic approximation and the caustics inherent in ray theory might limit severely its accuracy in complex regions e.g., Hoffmann, 00. Furthermore, the ray method cannot describe the finite-frequency effect of seismic waves. To obtain reliable and frequency-dependent illumination, we need a wave-theory-based method. One-way wave-equation-based propagators are widely used in illumination analysis. Although they neglect multiples, they can handle multiple forward-scattering phenomena, including focusing/defocusing, and diffraction. Unlike the ray-based methods, the wavefield obtained from wave-equation-based methods do not explicitly specify the directional information. Recently developed techniques, such as local slant-stack LSS e.g., Xie and Wu, 00 and beamlet decomposition e.g., Wu et al., 000, can decompose the wavefield into local plane waves, from which we can compute LAD illuminations e.g., Wu and Chen, 00, 00; Xie et al., 00, 00; Mao and Wu, 007; Cao and Wu, 009. However, the amplitude of the one-way wave propagator is inaccurate in complex models with sharp contrasts, even after corrections e.g., Zhang, 99; Zhang et al., 00; Kiyashchenko et al., 005; Wu and Cao, 005. Numerical implementations based on a one-way wave equation with the z-axis as the preferred propagation direction always have inherent limitations in wide-angle accuracy. Some one-way propagation-based methods can model backscattered waves or turning waves e.g., Wu, 99; Xie and Wu, 00; Kiyashchenko et al., 005; Jin et al. 00; Wu and Jia, 00; Xu and Jin, 00; Zhang et al., 00 ; however the amplitude accuracy for these propagators still is in question because of various factors. Full two-way wave equations solved by the finite-difference or finite-element method can simulate accurate and complete wave behavior in complex media. Therefore, full-wave equation-based illumination analysis should provide full-angle true-amplitude illuminations of all arrivals for survey design and image amplitude corrections. Both full-wave modeling and LAD illumination analysis can be implemented in the time or frequency domain. Xie and Yang 008 proposed an illumination method using a time-domain fullwave finite-difference method as the propagator and a time-domain local-slowness analysis method to derive the directional informa- Manuscript received by the Editor August 008; revised manuscript received October 008; published online 5 June 009. University of California, Department of Earth and Planetary Sciences, Institute of Geophysics and Planetary Physics, Santa Cruz, California, U.S.A. jcaogeo@gmail.com; wrs@pmc.ucsc.edu. 009 Society of Exploration Geophysicists. All rights reserved. S85

2 S8 Cao and Wu tion. The method is particularly useful for providing illumination analysis for reverse time migration. We propose to analyze the full-wave illumination in the frequency domain for several reasons. First, the illumination is frequency dependent. Second, the frequency-domain analysis method requires less storage compared with the time-domain method. Third, the frequency-domain angle decomposition of the wavefield is more efficient than the time-domain method. We illustrate the idea in D media. First we describe the basic concept for full-wave equation-based LAD illumination. Then we describe the two proposed wavefield decomposition methods for fullwave propagators. Following that, we show some examples demonstrating the advantages of full-wave illumination in the frequency domain, including the illumination for upgoing waves, such as turning waves, and reflected waves. We also show the illumination analysis for the BP 004 benchmark model Billette and Brandsberg- Dahl, 005. In the discussion section, we show the influence of multiples on the illumination strength and compare the angle resolution of the two proposed angle-decomposition methods. FULL-WAVE EQUATION-BASED DIRECTIONAL ILLUMINATION ANALYSIS One-way wave-equation-based LAD illumination is discussed in many papers e.g., Wu and Chen 00; Xie et al., 00. Here we briefly describe the basic concepts of wave-equation-based LAD illumination. The computation procedure of LAD illumination can be summarized as follows. First, in a given model we put a unit-strength source at the shot/receiver location x s /x Rs and propagate the wavefield into the model space x x,z. Second, by local plane-wave decomposition techniques, e.g., LSS Xie and Wu, 00 and beamlet decomposition Wu et al., 000, we can obtain the Green s functions in the local source/receiving angle s, g domain, G x, s;x s, G x, g;x Rs. Third, with the LAD Green s functions from all shot and receiver locations, we can compute the LAD illuminations for this acquisition configuration. For a single frequency, the directional illumination is defined as e.g., Wu and Chen 00 DI x, s x s G x, s;x s /, which measures the incident-angle response of the source aperture in a given model. The acquisition dip response ADR is defined as e.g., Wu and Chen 00 ADR x, n DI x, s;x S DI x, g;x Rs /, x s x Rs r where n s g /, r s g / represent the local dip and reflection angle, respectively. The ADR measures the dip-angle response of the whole acquisition system, including the source and receiver apertures. All contributions from the various source-scattering angle pairs for the same dip are summed together to obtain the ADR for that dip. Because the Green s function is calculated by wave-theory-based one-way propagators, the illumination includes the path effects, including all forward-scattering phenomena. However, backscattering is excluded because of the one-way approximation of the propagator. For full-wave equation-based illumination, the first step, generating the space-domain wavefield, has been studied extensively. To obtain the frequency-domain wavefield, the frequency-domain modeling e.g., Marfurt, 984; Operto et al., 007 or time-domain modeling plus the running discrete Fourier transform DFT sum can be used e.g., Luo et al., 004; Nihei and Li, 007; Sirgue et al., 008. Nihei and Li 007 compared the requirements of storage and floating-point operations of the time-domain and frequency-domain finite-difference methods for D and D multiple-source frequency-response modeling. In the comparison, the time-domain method is an explicit scheme, and the frequency-domain method uses direct solution of the linear system equations by LU-factorization with the nested dissection reordering. Their comparison shows that, for most D problems, when there is ample memory the frequency-domain method can efficiently provide the frequency responses for multisource problems; however, for D problems, a better choice is time-domain modeling plus the running DFT sum over time marching see also Sirgue et al., 008. For the third step, illumination formulas originally defined for the one-way propagator can be used for the full-wave case, except that the local angle ranges are different full-wave illumination can obtain the illumination not only for downgoing waves, but also for upgoing waves. Therefore, the main task here is the second step: to decompose the space-domain full wavefield into LAD. WAVEFIELD DECOMPOSITION FOR FULL-WAVE PROPAGATORS Local plane-wave decomposition techniques, such as the LSS method and beamlet decomposition method, were applied along the horizontal coordinate s e.g., Xie and Wu, 00; Wu and Chen, 00; Xie et al., 00, which is appropriate to one-way propagators. First we summarize these two decomposition methods for one-way propagators. Then we describe two decomposition methods for fullwave propagators: direct D decomposition and split-step decomposition using D decompositions. Wavefield decomposition for one-way propagators D local slant stack for one-way propagators In D LSS, using a windowed Fourier transform along a horizontal coordinate, we can decompose the wavefield u z x, at depth z for frequency, obtained from any extrapolator, into local plane waves Figure, i x x k x sin u z x,, w x x u z x, e dx, where is the local plane-wave propagating angle with respect to the vertical direction, w is a D window in the horizontal direction and centered at x, and k x /V x ; V x is the local velocity. D beamlet decomposition for one-way propagators The wavefield u z x, also can be decomposed into beamlets by the following formula,

3 Full-wave directional illumination S87 u z x, û z x n, m, b mn x, m n 4 u x,, w x x u x, where b mn are the beamlets decomposition basis vectors located at space window x n and wavenumber window m, and where û z x n, m, are the corresponding decomposition coefficients. References in the introduction provide details on beamlet decomposition, including the formula to obtain the decomposition coefficients. We can obtain the local plane waves u z x, m, by partial reconstruction of the beamlet-domain wavefield, u z x, m, û z x n, m, b mn x. n By the dispersion relation of the wave equation, we also can convert these local plane waves from the wavenumber domain to the angle domain. Wavefield decomposition for full-wave propagators When the local Fourier transform or beamlet decomposition is applied only along the horizontal coordinate, the local plane waves u z x,, or u z x, m, include not only the waves with positive vertical wavenumbers propagating downward but also corresponding negative vertical wavenumbers propagating upward. In one-way propagators, the waves propagate along only one primary direction; therefore decomposition along only the horizontal coordinate is appropriate. However, the full-wave propagators usually include both the downgoing and upgoing waves. The D decomposition techniques above will mix the downgoing and upgoing waves, resulting in incorrect illumination amplitude and artificial interference patterns in the illumination map see, e.g., Figure ; and Luo et al., 004. We decompose the full waves using direct D decomposition and an efficient split-step decomposition. D decomposition for full-wave propagators One direct way to obtain the LAD wavefield for full waves is by using D local plane-wave decomposition for D problems. Similarly to D LSS, we can decompose the wavefield for a given frequency into local plane waves using D LSS, D window z θ Incident wave Figure. Basic geometry of D local angle-domain analysis for the D model. x 5 e ik x x x sin z z cos dx, where w is a D spatial window Figure. Split-step decomposition for full-wave propagators The D decomposition discussed above can directly obtain the wavefield for all directions; however, it costs much more than D decomposition. Here, we propose an efficient split-step decomposition method to obtain the LAD wavefield for full-wave propagators using D decomposition techniques. First we decompose the full wavefield along the vertical direction using the D technique to separate the downgoing and upgoing waves. Then we apply D decomposition along the horizontal direction to the downgoing and upgoing waves to obtain the wavefields in all directions. We need three D decompositions to decompose the full wavefields. This method can be extended to the D case with D decomposition along the vertical direction and D decomposition along the horizontal coordinates. For the first step, we need to separate the waves with positive and negative vertical wavenumbers. The Gabor-Daubechies frame and local exponential frame beamlet-decomposition methods are very efficient in providing the local wavenumber-domain wavefield with uniquely defined directional localization; hence, they can be used for this step. We use Gabor-Daubechies frame beamlet decomposition in this step because it can provide a more accurate directional wavefield than the local exponential frame beamlet decomposition Cao and Wu, 009. Appendix A summarizes the Gabor-Daubechies frame beamlet decomposition and partial reconstruction to obtain the local plane waves. For the second step, we can use either the LSS method or the more efficient method with the Gabor-Daubechies frame beamlet decomposition proposed by Cao and Wu 009. EXAMPLES OF LOCAL-ANGLE-DOMAIN FULL-WAVE ILLUMINATION ANALYSIS We demonstrate the advantages of full-wave equation-based illumination in the frequency domain, i.e., providing frequency-dependent full-angle true-amplitude illuminations for all arrivals. We Figure. Acquisition dip-response maps for 5 Hz using the fullwave equation with D LSS applied along horizontal direction for different dips: a 0 ; b 0. Note the interference patterns caused by the interaction between upgoing and downgoing waves.

4 S88 Cao and Wu show illumination for downgoing waves with the D SEG/EAGE salt model Aminzadeh et al., 994; Aminzadeh et al., 995, illumination for turning waves with a V z model and reflected waves with a two-layer model, frequency-dependent illumination with a lens model, and final application in the BP 004 benchmark velocity model Billette and Brandsberg-Dahl, 005. We discuss the illumination of multiples in the following section. In examples below, we consider only a frequency of 5 Hz for illumination calculation except where otherwise specified. D window z θ θ Wave Wave Figure. Basic geometry for D local plane-wave decomposition for the D model. c) d) Split-step Split-step D LSS D LSS Figure 4. Acquisition dip-response maps for 5 Hz using the fullwave equation from the split-step decomposition method a-b and the D LSS method c-d for different dips: a, c 0 ; b, d 0. x Downgoing-wave illumination in the D SEG/EAGE salt model The acquisition geometry of synthetic data for this model consists of 5 shots with 7 left-side trailing receivers for each shot. The shot and receiver intervals are 0 feet 50 m and 80 feet 5 m, respectively. The ADR results for dips 0 and 0 from the split-step method are very similar to those obtained by the much more expensive D LSS method Figure 4. Interference patterns in the illumination maps using D decomposition along the horizontal direction Figure do not appear in these results. Turning-wave and reflected-wave illumination in simple models Turning waves and reflected waves can image overhung or vertical structures e.g., Jin et al. 00; Xu and Jin, 00; Zhang et al., 00; Jia and Wu, 007, which downgoing waves in traditional oneway propagators cannot image. The examples here are for a single shot. First we use a V z.5 0.5z km/s model to demonstrate the illumination by turning waves. The directional illumination for incident angle 5 from the split-step method is similar to that obtained by the D LSS method Figure 5. We notice that the illumination from the split-step method has a lower resolution; we discuss this result in the following section. The interference pattern is caused by the interaction of turning waves and a weak upgoing reflected wave. This reflection is produced by the sharp velocity-gradient change at the model bottom, where we pad the V z model with a constant velocity in the full-wave modeling. In the next example, we use a two-layer model to show the illumination by reflected waves. It consists of a 5-km-thick homogeneous layer with a velocity of.0 km/s and a half-space with a velocity of 4.5 km/ s. The directional illumination for an incident angle of 5 from the split-step method is very similar to that obtained by the D LSS method Figure. The interference pattern in the lower-right corner of the model is caused by the interaction of reflected waves and head waves. Frequency-dependent illumination To demonstrate frequency-dependent illumination, we use a lens model consisting of a homogeneous elliptical lens 000 m/s embedded in a homogeneous background 000 m/s. For 5-Hz Figure 5. Directional illumination at 5 Hz for an incident angle of 5 in a V z.5 0.5z km/s model: a from the split-step method; b from the D LSS method. In this and following plots, the star represents the source location.

5 Full-wave directional illumination S89 waves, the directional illumination from one shot right above the lens Figure 7a and d shows obvious illumination shadows and wavefield focusing features below the lens because of the low-velocity anomaly. For lower frequencies 5 and.5 Hz here, the illuminations below the lens are quite different from the illumination for 5 Hz: they seem to provide more even illumination, and the shadow zones shrink with decreasing frequency Figure 7. Illumination in the 004 BP benchmark velocity model Finally we apply the full-wave-equation illumination to the complicated 004 BP benchmark model. One challenge in this model is to delineate the vertical and overhung salt flanks e.g., circled area in Figure 8a. These targets can be imaged by the turning/reflected waves. The ADR map from a single source-receiver acquisition on the surface see Figure 8a shows that this simple acquisition system can illuminate the vertical salt flank well Figure 8d, although it does not illuminate the overhung salt flank well Figure 8e. For comparison, we also compute theadrs in the BP model without salt Figure 9. The illuminations are more uniform in space and very different from those in the exact model. However, the wave can still illuminate the potential vertical structures very well Figure 9d. DISCUSSION Influence of multiples on the illumination strength The LAD wavefield-decomposition methods for the full wavefield can separate the downgoing and upgoing waves. However, the Max Figure. Directional illumination at 5 Hz for an incident angle of 5 in a two-layer model: a from the split-step method; b from the D LSS method c) d) e) f) 4 Figure 7. Frequency-dependent single-shot directional illumination in a low-velocity lens black ellipse area in the figure model. The left and right columns are for 0 and 0 incidence angles, respectively. The rows from top to bottom are for frequencies of 5 Hz a, d,5 Hz b, e, and.5 Hz c, f, respectively. downgoing waves include not only the primary incident waves but also multiples; the primary incident waves include the first arrival and multiarrivals. The ADR maps for the most energetic waves by the split-step method show some illumination holes Figure 0. Comparison with theadrs for the full downgoing waves Figure 4a and b shows that the other arrivals i.e., multiarrivals and multiples provide extra illumination to the subsurface. The migration methods based on one-way propagators use not only the first arrival but also c) d) e) Figure 8. Acquisition dip response from a single source-receiver acquisition for 5-Hz waves in the 004 BP benchmark model: a part of the exact model used; b-e ADR for different dips: b 0, c 40, d 90, e 0. In this and following plots, the triangle represents the receiver location.

6 S90 Cao and Wu multiarrivals; therefore the multiarrivals should be included in the illumination analysis for survey design and true-reflection imaging corrections. However, multiples, especially internal multiples, should be eliminated because most migration methods based on the one-way propagator do not use them. We cannot separate multiples from other arrivals in the frequency domain. In the time domain, it is also hard to do because both might have similar traveltimes in complex media. Therefore, it is difficult to evaluate the relative illumination strength from multiples and multiarrivals in general media. c) d) e) Figure 9. Acquisition dip response from a single source-receiver acquisition for 5-Hz waves in the 004 BP benchmark model without salts: a part of the model used; b-e ADR for different dips: b 0, c 40, d 90, e 0. The one-way and one-return boundary element method in the frequency domain He and Wu, 007 can calculate the primary transmitted waves and multiples for a layered model or an inclusion model. This method can handle strong velocity contrasts. Here, we investigate the influence of internal multiples on illumination strength with a simplified SEG/EAGE salt model, in which a homogeneous salt body 4480 m/s with the same shape as that of the original D SEG/EAGE salt model is embedded in a homogeneous background medium 80 m/s. This is a scalar-wave model. For the acousticwave model, the salt internal multiples will be weaker because the density of the salt usually is lower than that of the background sediment. Directional illumination maps from the primary transmitted waves and salt internal multiples in the subsalt region for incident angles 0 and 0 show that the maximum amplitude of the illumination of primaries is more than six times stronger than that of the multiples for these two angles Figure. Therefore the contribution of the multiples to illumination strength could be considered as a secondary effect here. However, we can also notice the difference in the spatial distribution of the illumination between results for multiples and those for primaries. For example, for the 0 incident angle, the primaries strongly illuminate only the left part of the subsalt area; however, the multiples illuminate the whole subsalt area more evenly and provide extra illumination to the shadow in the illumination by primaries right part in subsalt area. Comparison of the D and D local slant-stack methods Previous results show that the split-step method based on D decompositions might produce a lower resolution result than the D LSS method Figure 5. Here we show a theoretical analysis and numerical investigation. For D decomposition, from the dispersion relation we have d d k 0 cos, where k 0 is the wavenumber. We can further obtain the angular resolution as a function of wavelength, window length L win, and the angle, Figure 0. Acquisition dip-response maps for the most energetic waves at 5 Hz with the split-step method for different dips: a 0 ; b 0.

7 Full-wave directional illumination S9 d L win cos. Therefore, the angular resolution is angle and window-length dependent for a given. For the D LSS method, d L win, which is angle independent. This can explain previous differences of angular resolution in illumination results. Next we compare the methods numerically. We investigate the decomposed local plane waves for incident global plane waves along different directions, using the D LSS method and the D LSS method along the horizontal direction. Results show that the D LSS method gives the same angular resolution for the plane-wave incident along any direction Figure a. The D LSS method yields angular resolution decreasing with the increase of the propagating angle of the global plane wave Figure b, and it gives the best resolution red line when the decomposition direction is parallel to the wavefront c) d) Figure. Directional illumination for the primary transmitted waves and salt internal multiples at 5 Hz in a simplified SEG/ EAGE salt model: a transmitted waves for 0 ; b multiples for 0 ; c transmitted waves for 0 ; d multiples for The numerical results show that for waves propagating within about 0 from the decomposition direction, the angular resolution of the D LSS method is similar to that of the D LSS method. The direction of the local plane wave having maximum energy from the D LSS decomposition still is consistent with the true incident direction of the global plane wave from any direction. CONCLUSIONS We have analyzed the local-angle domain LAD illumination in the frequency domain with full-wave propagators using two wavefield decomposition methods: one is the direct D/D local planewave decomposition; the other is the more efficient split-step decomposition. The methods can provide frequency-dependent fullangle true-amplitude illumination analysis for all arrivals. They are more efficient and storage saving compared with the time-domain angle analysis method. They can be used for accurate survey design and true-reflection imaging corrections. Results of illumination analysis from both decomposition methods are very similar, although the split-step method might produce a lower resolution result at wide angles. We conclude that the proposed methods provide efficient and accurate tools for the LAD wave-theory-based illumination analysis in complex models. ACKNOWLEDGMENTS The authors thank the associate editor and reviewers for their valuable comments that greatly improved this manuscript. We acknowledge Xiao-Bi Xie for helpful discussion on LSS, and Yaofeng He for generating the primary and multiple data with the one-way and one-return boundary element method. This research is sponsored by the Wavelet Transform on Propagation and Imaging for seismic exploration Research Consortium WTOPI at the University of California, Santa Cruz. We also thank BP and Frederic Billette for providing the 004 BP D benchmark velocity model. APPENDIX A 80 Wavefield amplitude Decomposed angle ( ) Decomposed angle ( ) Figure. Decomposed local plane waves for incident global plane waves from different directions dashed lines show the true incident directions using the a D LSS and b D LSS methods GABOR-DAUBECHIES FRAME BEAMLET DECOMPOSITION The Gabor-Daubechies frame GDF beamlets e.g., Wu et al., 000; Wu and Chen, 00; Chen et al., 00 have uniquely defined and good localization information available after decomposition. The GDF beamlets for the beamlet decomposition equation 4 are Gaussian function windowed exponential harmonics, b mn x g x x n e i m x, A- where m m is the wavenumber sampling interval, and g x is a Gaussian window function, g x s /4 exp x s, A-

8 S9 Cao and Wu s R N, A- 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 beamlet representation A- into the beamlet decomposition equation 4, we can obtain the partially reconstructed local plane waves, u z x, m, e i m x g x x n û z x n, m,, n with the decomposition coefficients A-4 û z x n, m, u z x,,b mn x u z x, b mn * x dx, A-5 where stands for inner product, * stands for complex conjugate, and b mn x are the dual GD frame atoms b mn x g x x n e i m x, A- with g x being the dual-window function of g x. The dual-window function can be calculated by pseudoinversion of the original window function Qian and Chen, 99; Mallat, 998; Wu and Chen, 00. From equation 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 u z x, u z x, m,. m REFERENCES A-7 Aminzadeh, F., N. Burkhard, T. Kunz, L. Nicoletis, and F. Rocca, 995, -D modeling project: rd report: The Leading Edge, 4, 5 8. Aminzadeh, F., N. Burkhard, L. Nicoletis, F. Rocca, and K. Wyatt, 994, SEG/EAGE -D modeling project: nd update: The Leading Edge,, Bear, G., C. Liu, R. Lu, and D. Willen, 000, The construction of subsurface illumination and amplitude maps via ray tracing: The Leading Edge, 9, Billette, F. J., and S. Brandsberg-Dahl, 005, The 004 BP velocity benchmark: 7th Conference & Exhibition, EAGE, ExtendedAbstracts, B05. Cao, J., and R. S. Wu, 005, Influence of propagator and acquisition aperture on image amplitude: 75th Annual International Meeting, SEG, Expanded Abstracts, , 008, Amplitude compensation for one-way wave propagators in inhomogeneous media and its application to seismic imaging: Communications in Computational Physics,, 0., 009, Fast acquisition aperture correction in prestack depth migration using beamlet decomposition: Geophysics, this issue. Chen, L., R. S. Wu, and Y. Chen, 00, Target-oriented beamlet migration based on Gabor-Daubechies frame decomposition: Geophysics, 7, no., S7 S5. He, Y. F., and R. S. Wu, 007, One-return boundary element method and salt internal multiples: 77th Annual International Meeting, SEG, Expanded Abstracts, Hoffmann, J., 00, Illumination, resolution, and image quality of PP- and PS-waves for survey planning: The Leading Edge, 0, Jia, X. F., and R. S. Wu, 007, Imaging steep salt flanks by super-wide angle one-way method: 77thAnnual International Meeting, SEG, ExpandedAbstracts, 5 9. Jin, S., and D. Walraven, 00, Wave equation GSP prestack depth migration and illumination: The Leading Edge,, 0 0. Jin, S., S. Xu, and D. Walraven, 00, One-return wave equation migration: Imaging of duplex waves: 7th Annual International Meeting, SEG, ExpandedAbstracts, 8 4. Kiyashchenko, D., R.-E. Plessix, B. Kashtan, and V. Troyan, 005, Improved amplitude multi-one-way modeling method: Wave Motion, 4, Lecomte, I., H. Gjøystdal, and Å. Drottning, 00, Simulated prestack local imaging: A robust and efficient interpretation tool to control illumination, resolution, and time-lapse properties of reservoirs: 7rd Annual International Meeting, SEG, ExpandedAbstracts, Li, P. M., and L. G. Dong, 00, Optimal seismic survey design based on seismic wave illumination: 8th Conference & Exhibition, EAGE, Extended Abstracts, F09. Luo, M. Q., J. Cao, X. B. Xie, and R. S. Wu, 004, Comparison of illumination analyses using one-way and full-wave propagators: 74th Annual International Meeting, SEG, ExpandedAbstracts, Mallat, S., 998, Awavelet tour of signal processing: Academic Press. Mao, J., and R. S. Wu, 007, Illumination analysis using local exponential beamlets: 77th Annual International Meeting, SEG, Expanded Abstracts, 5 9. Marfurt, K. J., 984, Accuracy of finite-difference and finite-element modeling of the scalar and elastic wave equations: Geophysics, 49, Muerdter, D., M. Kelly, and D. Ratcliff, 00, Understanding subsalt illumination through ray-trace modeling: Part : Dipping salt bodies, salt peaks, and nonreciprocity of subsalt amplitude response: The Leading Edge, 0, Muerdter, D., and D. Ratcliff, 00a, Understanding subsalt illumination through ray-trace modeling: Part : Simple -D salt models: The Leading Edge, 0, , 00b, Understanding subsalt illumination through ray-trace modeling: Part : Salt ridges and furrows, and the impact of acquisition orientation: The Leading Edge, 0, Nihei, K. T., and X. Li, 007, Frequency response modeling of seismic waves using finite difference time domain with phase sensitive detection TD- PSD : Geophysical Journal International, 9, Operto, S., J. Virieux, P. Amestoy, J.-Y. L Excellent, L. Giraud, and H. Ben HadjAli, 007, D finite-difference frequency-domain modeling of viscoacoustic wave propagation using a massively parallel direct solver: A feasibility study, Geophysics, 7, no. 5, SM95 SM. Qian, S., and D. Chen, 99, Joint time-frequency analysis, methods, and applications: Prentice Hall PTR. Schneider, W. A., and G. A. Winbow, 999, Efficient and accurate modeling of -D seismic illumination: 9thAnnual International Meeting, SEG, ExpandedAbstracts,. Sirgue, L., J. T. Etgen, and U. Albertin, 008, D frequency domain waveform inversion using time domain finite difference methods: 70th Conference & Exhibition, EAGE, ExtendedAbstracts, F0. Wu, R. S., 99, Synthetic seismograms in heterogeneous media by one-return approximation: Pure andapplied Geophysics, 48, Wu, R. S., and J. Cao, 005, WKBJ solution and transparent propagators: 7th Conference & Exhibition, EAGE, ExtendedAbstracts, P7. Wu, R. S., and L. Chen, 00, Beamlet migration using Gabor-Daubechies frame propagator: rd Conference & Exhibition, EAGE, Expanded Abstracts, , 00, Mapping directional illumination and acquisition-aperture efficacy by beamlet propagators: 7nd Annual International Meeting, SEG, ExpandedAbstracts, 5 55., 00, Directional illumination analysis using beamlet decomposition and propagation: Geophysics, 7, no. 4, S47 S59. Wu, R. S., and X. Jia, 00, Accuracy improvement for super-wide angle one-way waves by wavefront reconstruction: 7th Annual International Meeting, SEG, ExpandedAbstracts, Wu, R. S., M. Q. Luo, S. C. Chen, and X. B. Xie, 004, Acquisition aperture correction in angle-domain and true-amplitude imaging for wave equation migration: 74th Annual International Meeting, SEG, Expanded Abstracts, Wu, R. S., Y. Wang, and J. H. Gao, 000, Beamlet migration based on local perturbation theory: 70th Annual Meeting, SEG, Expanded Abstracts, Xie, X. B., S. W. Jin, and R. S. Wu, 00, Three-dimensional illumination analysis using wave-equation based propagator: 7rd Annual International Meeting, SEG, ExpandedAbstracts, , 00, Wave-equation-based seismic illumination analysis: Geophysics, 7, no. 5, S9 S77. Xie, X. B., and R. S. Wu, 00, Modeling elastic wave forward propagation and reflection using the complex screen method: Journal of the Acoustical Society ofamerica, 09, 9 5., 00, Extracting angle domain information from migrated wave field: 7nd Annual International Meeting, SEG, Expanded Abstracts, 0. Xie, X. B., and H. Yang, 008, A full-wave equation based seismic illumination analysis method: 70th Conference & Exhibition, EAGE, Extended Abstracts, P84. Xu, S., and S. Jin, 00, Wave equation migration of turning waves: 7thAnnual International Meeting, SEG, ExpandedAbstracts, 8.

9 Full-wave directional illumination S9 Zhang, G. Q., 99, System of coupled equations for upgoing and downgoing waves: Acta MathematicaeApplicatae Sinica,, 5. Zhang, Y., S. Xu, and G. Q. Zhang, 00, Imaging complex salt bodies with turning-wave one-way wave equation: 7th Annual International Meeting, SEG, ExpandedAbstracts,. Zhang, Y., G. Q. Zhang, and N. Bleistein, 00, True amplitude wave equation migration arising from true amplitude one-way wave equations: Inverse Problems, 9, 8.