Application of beamlet propagator to migration amplitude correction Shengwen Jin1, Mingqiu Luo1,2, Ru-Shan Wu2 , and David Walraven3

1 Screen Imaging Technology, Inc., Houston, TX 77074 2 Institute of Geophysics and Planetary Physics, University of California, Santa Cruz, CA, 95064 3 Anadarko Petroleum Corporation, The Woodlands, TX 77380

Summary

Beamlet migration provides robust imaging in the presence of strong velocity contrasts by combining local perturbation theory with wavelet transforms. Source and receiver wavefields are decomposed into beamlets during the wavefield extrapolation. Each beamlet propagates with a local reference velocity in which the local velocity perturbation is small, resulting in accurate and efficient wave propagation. It simultaneously extracts local information in both space and angle. Such information can be further applied to the computation of imaging amplitude corrections using directional illumination in the local angle domain. Synthetic examples show the local angle imaging properties of beamlets and the improved amplitude balance of the depth image after acquisition aperture correction.

Introduction

In most Fourier based migration propagators, such as SSF (Stoffa et al., 1990), FFD (Ristow & Ruhl, 1994) and GSP (Wu, 1994; de Hoop et al. 2000), a single global reference velocity is used along the whole window length at each depth level during the wavefield extrapolation. In a strong contrast velocity medium, the global velocity perturbation could be very large at some locations and can thus tend to degrade the imaging quality of steeply dipping events. A windowed Fourier transform (WFT) was introduced to phase-space propagators (Jin & Wu, 1999) where the local reference velocity is selected in each window. The WFT, however, is a non-orthogonal basis making computation of the inverse WFT very expensive for wavefield reconstruction.

Based on the localization characteristics and orthogonal basis of wavelet transform as opposed to WFT, a beamlet migration was proposed using local perturbation theory by Wu et al. (2000). The Gabor-Daubechies frame and local cosine basis beamlets are two of the most frequently used propagators for prestack depth migration (Wu & Chen, 2002; Wang & Wu, 2002; Luo & Wu, 2003). In these methods, the velocity windows are adaptively partitioned to make the local velocity perturbation small. Within each window, waves propagate with beamlets that can simultaneously provide localized information in both the space and angle domains. Such information can be further used for directional illumination analysis (Wu & Chen, 2002; Xie et al., 2003; Jin & Walraven, 2003) and true-reflection imaging in local space-angle domain

using acquisition aperture correction (Wu et al., 2004).

In this paper, we will briefly describe the implementation of beamlet migration and its localized characteristics of wave propagation. Applications to both directional illumination analysis and migration amplitude correction will be demonstrated.

Implementation of beamlet migration

In most traditional wavenumber domain implementations, such as PSPI, SSF and GSP, global reference velocities and a global FFT are used along the whole window length for wavefield extrapolation at each depth step. Based on local perturbation theory, beamlet migration adaptively partitions the local velocity windows in which the local velocity perturbation is smaller than the global perturbation. Within each window, the wavefield is decomposed into beamlets witheach beamlet propagating with its appropriate local reference velocity. The corresponding thin-lens term correction is then used to account the effects of small local velocity perturbations. Thus, beamlet propagation for a given depth step is composed of three procedures: 1) Wavefield decomposition into beamlets; 2) Propagation of each beamlet to the next depth step across the local windows; 3) Wavefield reconstruction from all propagated beamlets by the inverse wavelet transform. Unlike plane wave migration, beamlets are coupled during the extrapolation process.

Beamlet migration simultaneously extracts local information in both the space and angle domains due to its wavelet transform formulation. Figure 1 shows the wave propagation of an individual beamlet in a homogeneous medium. Each beamlet propagates along a specific direction or angle. For the same propagation direction, a higher-frequency beamlet has better localization characteristics than does a lower-frequency beamlet.

Imaging condition in local angle domain

To obtain the true reflection depth image, we decompose the total wavefield into beamlets that correspond to the true incident and scattered waves at the local image points. Adeconvolution imaging condition is then applied to the decomposed incident and the scattered wavefields in the local space and angle domain. Such an imaging condition can be expressed as (Wu et al, 2004)

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Application of beamlet propagator to migration amplitude correction

);();,(

);,(2),;()(

*

gsss

xA

ggIgssIgs xxu

z

xxGdxxxGxI

g

vv

vv

vvv

∑ ∫ ∂∂

⋅=θ

θθθ

where, ),,( zyxx =v denotes a three-dimensional vector,

sθ and gθ are source and receiver angles at the local image

point, respectively. );,( ssI xxG θ is the beamlet

decomposition of Green’s function );( sI xxG at the local

image point. *IG is the complex conjugate of IG and

)( gxA is the spatial receiver aperture. From the above

imaging condition, we see that the contributions to the migration amplitude arising from different pairs of incident and scattering waves are different in the local angle domain. This is in contrast to the more traditional space domain imaging condition shown below

);();(

);(2)()(

*

gsss

xA

gI

gsI xxuz

xxGdxxxGxI

g

vv

vv

vvv

∑ ∫ ∂∂

⋅=

where no local angle information is extracted during the wavefield propagation process. This implies that migration amplitude corrections will be independent of wavenumber which can have the effect of boosting certain noises along with the signal – an undesirable occurrence.

Migration amplitude correction on SEG salt model

The SEG salt model was used as an example to test the migration amplitude correction in the local angle domain. The synthetic data for this model has 361 shots with off-end acquisition geometry. The distance between shots is 160 feet. Each shot has 176 receivers separated by 80 feet. Figure 2a shows the velocity model. Figure 2b is the beamlet migration result without amplitude correction. Figure 2c is the enlarge part of subsalt area. AGC is applied to the image. Most structures are well imaged in general. But imaging shadows

are present beneath the salt body as shown by arrow A. The amplitudes of some events are weak as shown by arrow B. Some events are truncated as shown by arrow C.

For the given acquisition system and velocity model, the directional illumination strength gives the detecting power of a specific source-receiver pair to the target reflector, while the total illumination sums the directional illumination from all possible dipping events. Figure 3 illustrates both total and directional illumination for three cases: horizontal events, 50 degree dipping events and -30 degree dipping events with respect to the horizontal direction. From these results, it is evident that the illumination strengths vary for different dipping events. The imaging shadow of horizontal events on the basement of the model (as shown by arrow A of Figure 2c) correspond very well to the low illumination of horizontal events as shown in Figure 4a. Similarly, the weak image amplitude of the steep event (as shown by arrow B) has an excellent correlation with the low illumination for such dipping events as shown in Figure 3b. The truncation of the event (as shown by arrow C) is consistent with the illumination distribution as shown in Figure 3b. In Figure 3c, strong illuminations of 50 degree dips are present in the same region. One might expect that certain dip ranges which are well illuminated would also be well imaged. However, it can be difficult to distinguish the illumination strength among different dipping events from the total illumination result as shown in Figure 4d.

Due to the limits of the acquisition system, only part of the scattering waves can be recorded. Thanks to the good correlations between the image quality and directional illumination, we can perform a migration amplitude correction to partially compensate for the acquisition aperture effect in the local angle domain. A local angle imaging condition is applied in this case. After amplitude correction, the image quality in the subsalt area is significantly improved as shown in Figure 4, especially for those events denoted by A, B and C. Furthermore, various

0

1. 2

(a) (b)

(c) (d)

0 2.4 km

Figure 1: Beamlets propagate in a homogeneous medium with velocity of 2000m/s. (a) Vertical beamlet, (b) small oblique angle beamlet, (c) large oblique angle beamlet, and (d) vertical beamlet for low frequency. The frequency of (a), (b) and (c) is 25Hz, while the frequency of (d) is 10Hz.

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Application of beamlet propagator to migration amplitude correction

Figure 2: Beamlet migration on SEG salt model. (a) is the velocity model, (b) is the beamlet migration result without amplitude correction. (c) is the enlarge part of subsalt area.

(a)

(b)

A

BC

(c)

(a)

(b)

(c)

(d)

Figure 3: Directional illuminations of SEG salt model. (a) is the directional illumination for horizontal events; (b) is the directional illumination for -30 degree dipping events with respect to horizontal direction; (c) is for +50 degree dipping events; and (d) is the total illumination from all dipping events.

types of noise have been attenuated. As a comparison, Figure 5 shows the image result using the traditional deconvolution imaging condition in the space domain that corrects the migration amplitude using the total illumination. The image result looks similar to that of AGC applied as shown in Figure 2b and 2c.

Conclusion

The beamlet method decomposes a wavefield into beamlets and propagates each beamlet using a local reference velocity based on small, local velocity perturbations. This formulation makes beamlet migration suitable for the imaging of complex media where large local velocity gradients and short wavelength velocity features may be present. Each localized beamlet bears both space and angle information. It provides directional illumination and local angle domain

images. The acquisition aperture effects can be partially eliminated through the directional illumination correction in the local angle domain. The image quality is significantly improved for the areas with low illumination after amplitude correction.

Acknowledgements:

The authors are grateful for Xiao-Bi Xie and Shiyong Xu for the beneficiary discussions and helps.

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Application of beamlet propagator to migration amplitude correction

References

de Hoop, M.V., Wu, R.S., and le Rousseau, J., 2000, Generalization of the phase screen approximation for the scattering of acoustic waves, Wave Motion, 31, 43-70.

Jin, S., and Wu, R.S., 1999, Depth migration with a windowed screen propagator, J. Seismic Exploration, 8,27-38.

Jin, S., and Walraven, D., 2003, Wave equation GSP prestack depth migration and illumination, The Leading Edge, 22, 604-610.

Luo M. and Wu R.S., 2003, 3D beamlet prestack depth migration using the local cosine basis propagator, Expanded Abstracts, SEG 73th Annual Meeting, 985-988.

Luo M., Wu R.S., and Xie X.B., 2004, Beamlet migration using local cosine basis with shifting windows, Expanded abstracts, SEG 74th Annual Meeting, 945-948.

Ristow, D., and Ruhl, T., 1994, Fourier finite-difference migration, Geophysics, 59, 1882-1893.

Stoffa, P.L., Fokkenma, J.T., de Luna Freire, R.M., and Kessinger, W.P., 1990, Split-step Fourier migration, Geophysics, 55, 410-421.

Wang, Y. and Wu, R.S., 2002, Beamlet prestack depth migration using local cosine basis propagator, Expanded Abstracts, SEG 72th Annual Meeting, 1340-1343.

Wu, R.S., 1996, Synthetic seismograms in heterogeneous media by one-return approximation, Pure Appl. Geophs, 148, 155-173.

Wu, R.S. Wang, Y. and Gao, J.H., 2000, Beamlet migration based on local perturation theory, Expanded Abstracts, SEG 70th Annual Meeting, 1008-1011.

Wu, R.S. and Chen L., 2001, Beamlet migration using Gabor-Daubechies frame propagator, Expanded Abstracts, EAGE 63th Annual Meeting, 74-78.

Xie, X.B., Jin, S., and Wu, R.S., 2003, Three-dimensional illumination analysis using wave equation based propgator, 73rd SEG Meeting, Expanded Abstracts, 989-992.

Figure 5: Migration amplitude correction using the total illumination.

Figure 4: Migration amplitude correction using the directional illumination in local angle domain.

A

B

C

A

B

C

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EDITED REFERENCES Note: This reference list is a copy-edited version of the reference list submitted by the author. Reference lists for the 2005 SEG Technical Program Expanded Abstracts have been copy edited so that references provided with the online metadata for each paper will achieve a high degree of linking to cited sources that appear on the Web. Application of beamlet propagator to migration amplitude correction REFERENCES de Hoop, M. V., R. S. Wu, and J. le Rousseau, 2000, Generalization of the phase screen

approximation for the scattering of acoustic waves: Wave Motion, 31, 43-70. Jin, S., and R. S. Wu, 1999, Depth migration with a windowed screen propagator: Journal

of Seismic Exploration, 8, 27-38. Jin, S., and D. Walraven, 2003, Wave equation GSP prestack depth migration and

illumination: The Leading Edge, 22, 604-610. Luo, M., and R. S. Wu, 2003, 3D beamlet prestack depth migration using the local cosine

basis propagator: 73rd Annual International Meeting, SEG, Expanded Abstracts, 985-988.

Luo, M., R. S. Wu, and X. B. Xie, 2004, Beamlet migration using local cosine basis with shifting windows: 74th Annual International Meeting, SEG, Expanded Abstracts, 945-948.

Ristow, D., and T. Ruhl, 1994, Fourier finite-difference migration: Geophysics, 59, 1882-1893.

Stoffa, P. L., J. T. Fokkema, R. M. de Luna Freire, and W. P. Kessinger, 1990, Split-step Fourier migration: Geophysics, 55, 410-421.

Wang, Y., and R. S. Wu, 2002, Beamlet prestack depth migration using local cosine basis propagator: 72nd Annual International Meeting, SEG, Expanded Abstracts, 1340-1343.

Wu, R. S., 1996, Synthetic seismograms in heterogeneous media by one-return approximation: Pure and Applied Geophysics, 148, 155-173.

Wu, R. S., Y. Wang, and J. H. Gao, 2000, Beamlet migration based on local perturbation theory: 70th Annual International Meeting, SEG, Expanded Abstracts, 1008-1011.

Wu, R. S., and L. Chen, 2001, Beamlet migration using Gabor-Daubechies frame propagator: 63rd Annual Conference, EAGE, Extended Abstracts, 74-78.

Xie, X. B., S. Jin, and R. S. Wu, 2003, Three-dimensional illumination analysis using wave equation based propagator: 73rd Annual International Meeting, SEG, Expanded Abstracts, 989-992.