Distance (*40 ft) Depth (*40 ft) Profile A-A from SEG-EAEG salt model

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1 Proposal for a WTOPI Research Consortium Wavelet Transform On Propagation and Imaging for seismic exploration Ru-Shan Wu Modeling and Imaging Project, University of California, Santa Cruz August 27, 1996 Revised July, 1997 Objective The objective of this project is to develop the theory and algorithms of wave propagation and imaging in the wavelet domain for direct application to compressed seismic data. We will strive to nd optimum algorithms for both compression and processing of which imaging is the heart. Background and Approaches Processing huge seismic data sets has become of critical importance to the oil and gas industry in order to obtain high-resolution images of target areas beneath increasingly complicated structures. For imaging problems, the high-frequency ray-kirchho method and the nite-dierence wave equation method are commonly used. The former has low resolution and may notwork for highly complex media. The latter is prohibitively time-consuming and memory intensive for large-size 3D problems. Therefore, the computational eciency of wave propagation is vital to 3D imaging problems such as 3D prestack depth migration and inversion, or velocity analysis. The newly developed fast wavelet transform (WT) is considered to be a revolutionary breakthrough in signal analysis/processing. In the same time frame, there has been signicant progress in one-way wave propagation theory and algorithms, including the recently developed fast acoustic and elastic generalized screen propagators from our group (See our Technical report "Modeling and Imaging Project", No. 1 [1]). We propose to apply the wavelet transform to the generalized screen or other one-way wave propagation methods to develop ecient 3D imaging methods. The cross-breeding of these two new developments has the potential of revolutionizing modeling and imaging techniques for complex Earth media. Seismic data compression has achieved remarkable success. However, at present data compression is focused on data transportation and storage problems, and is assumed to be transparent to processing. Our approach, on the other hand, seeks to revolutionize the whole process, including compression and processing (imaging). The latter approach requires much dedicated theoretical study on wavelet transform and wave propagation. Seismic data compression is intended to nd ecient ways of representing data. Yet David Marr reminded us, "how information is presented can greatly aect how easy it is to do dierent things with it". Not all of the good compression algorithms suit seismic imaging. As Yves Meyer, a wavelet pioneer and expert, pointed out, "An algorithm that is optimal for compression can be disastrous for analysis". We need to nd the best algorithm not only for data compression but also for compression of imaging operators. We plan to study wave propagation operators, especially for one-way wave propagation, in the wavelet domain, and develop fast algorithms for W-domain propagators, such as the screen propagators, phase-shift propagators, beam or beamlet propagators, etc.. The generalized screen method for one-way wave propagation that we developed adopts a perturbation theory and a dual-domain implementation technique for fast computation. It has been shown that the method is 2{3 orders of magnitude faster than traditional nite-dierence wave equation methods for medium sized problems and can have huge computer memory saving which makes it capable of handling large volume problems prohibitive to other methods. The algorithm shuttles between the space domain and the wavenumber domain with a fast Fourier transform (FFT) algorithm. Since both

2 the Fourier transform (FT) and the inverse FT involve global operations, the background media (reference velocities) must also be global. This global nature in space or wavenumber domain hampers the further development and optimization of the generalized screen method. The wavelet transform (WT) has some specic features which make it more suitable for wave propagation problems than the FT. One is the exibility of time-frequency (or space-wavenumber) localizations the other is the multiscale/multiresolution nature. Due to these features, wave propagation can be optimized by using some special bases so that the computation time can be minimized while the accuracy and stability can be increased. A straightforward and simple way to get a exible space-wavenumber localization is the use of windowed FT (Fourier transform). An example of the application of windowed FT to our generalized screen propagator (GSP) method for a poststack migration using the synthetic data for the SEG-EAEG salt model A-A'prole (Courtesy of Amoco) is shown in Figure 1 (model) and 2 (results and comparison). We see that not only most of the subsalt structure is reconstructed clearly by our method, but also the nearly vertical reectors, including the steeply dipping interfaces and the reecting faults which are missing in the f-x and Kirchho migrated images, are correctly imaged. Our task is to rst derive one-way propagators in the wavelet domain for dierent wavelet bases. To demonstrate the principle an example is shown in Figure 3 for the case of homogeneous media. The upper panel (a) and (b) are the matrix representations of the Kirchho operator and the compressed beamlet operator. The latter is the wavelet decomposition of the Kirchho operator using the Coiet 5 basis. The Kirchho matrix is a dense matrix, while the compressed beamlet matrix is a highly sparse matrix. For the lower two rows: On the right, (d) and (f) are the migrated image and the residual image (after subtracting the reference image which uses the full-aperture Kirchho operator, and amplied by a factor of 1) by the compressed C5 beamlet migrator (compression ratio = 17:1), respectively On the left, (c) and (e) are the corresponding images by a 33 points width Kirchho migrator which has a similar amount of computation time of matrix operation as the compressed C5. This demonstrates the high eciency of the beamlet migration which retains the eective wide-aperture and therefore the good image quality. The hard but more exciting task in front of us is the research of the case of laterally varying media and using the multi-dimensional wavelet bases. We will adopt primarily two approaches. One is the local trigonometric transform or best-basis decomposition of the waveeld the other is the multiscale decomposition of both the medium and the waveeld. For the rst approach, theoretical and numerical study for beam diraction and scattering in strongly heterogeneous media will be conducted. For the second approach, wave propagation and cross-coupling between beamlets and scales can be formulated using wave propagation theory and wavelet transform theory. The research goal is to select proper wavelet bases so as to optimize the waveeld decomposition and minimize the cross-coupling. Some recent progress in WT on solving partial dierential equations and integral equations can be utilized for the theoretical derivation and algorithm design. Research Team The UCSC seismic Modeling and Imaging group (two professors, two researchers and three postdoctoral researchers) has been established as one of the foremost group in the world on one-way wave propagation and imaging. We are teaming up with other collaborators, such as Marteen de Hoop From CWP, Colorado School of Mines, and industrial researchers, such as C. Mosher, D. Foster and others. The joint eort of the team will increase signicantly the speed of research progress. Sponsorship The fee for a sponsor of Oil & Gas company or geophysical service company is $2,/year. Computer and software companies contributing hardware, software and other facilities can become associate sponsors. References UCSC Modeling and Imaging Project, Periodical report No. 1 (1996) and No. 2 (1997). Foster, D.J., Lane, F.D., Mosher, C.C. and Wu, R.S., 1997, Wavelet transforms for seismic data processing, Expanded Abstracts of the Technical Program, SEG 67th Annual Meeting 2

3 Distance (*4 ft) Depth (*4 ft) 2 Profile A-A from SEG-EAEG salt model Figure 1: A-A' Prole of the SEG-EAEG salt model crosses many of the more dicult structural elements in the model (Courtesy of O'Brien and Gray, Amoco) Mosher, C.C, Foster, D.J. and R.S. Wu, 1996, Phase shift migration with wave packet algorithms, "Mathematical methods in geophysical imaging IV", (ed. Hassanzadeh, S.), Proc. SPIE v. 2822, Wu, R.S. and S. Jin, 1997, Windowed GSP (Generalized Screen Propagators) Migration Applied to SEG-EAEG Salt Model Data, Expanded Abstracts of the Technical Program, SEG 67th Annual Meeting Wu, R.S., Yang, F., Wang, Z.L. and L. Zhang, 1997, Migration operator compression by wavelet transform: beamlet migrator, Expanded Abstracts of the Technical Program, SEG 67th Annual Meeting 3

4 F E D GSP Depth Migration Depth (*4 ft) 2 Distance (*4 ft) Figure 2: Comparison of F-X migration (upper panel), Kirchho migration (middle panel) and windowed GSP (generalized screen propagator) migration (bottom panel) on the synthetic data of A-A' prole of SEG-EAEG salt model (The upper two panels are from O'Brien and Gray, Jan., 1996, "Leading Edge") 4

5 (a) (b) Depth (m) Depth (m) (c) Kir_33-image (d) C5-image Depth(m) Depth(m) (e) Kir_33-residue (f) C5-residue Figure 3. Performance comparison of compressed beamlet migrators and limited-aperture Kirchho migrator. (a): Matrix representation of Kirchho Migration operators in space domain (f = 5.9 Hz) (b): Matrix representation of compressed Migration operators in beamlet domain (using coiet 5) (f = 5.9 Hz) On the right: (d) and (f) are the migrated image and the residual image (amplied by a factor of 1) by the compressed C5 beamlet migrator (compression ratio = 17:1), respectively On the left: (c) and (e) are the corresponding images by a 33 points width Kirchho migrator. 5