1. Wide-Angle One-Way and One-return Elastic Wave Propagation Theory and Applications:
    The method of Wide-Angle One-Way Elastic Wave Propagation is a practical way to calculate synthetic seismograms for long range propagation in 3D arbitrarily heterogeneous elastic media even with parallel supercomputers. Based on his previous work on elastic wave Rayleigh-integral and elastic wave Born scattering theory, he derived a thin-slab formula and a Elastic Complex-Screen (ECS) (or generalized phase-screen) formula for elastic wave one-way propagation (Wu 1994). Compared with earlier methods on elastic wave one-way propagation, this newly developed method can correctly handle the converted waves between P and S waves. Collaborated with X.B. Xie, a 3D algorithm of the method has been coded and many 3D numerical tests and applications of the method have been conducted (Wu and Xie, 1993, 1994). By comparison with exact solutions and 3D finite difference simulations for an elastic sphere, the elastic complex-screen (ECS) method showed good agreement with those exact methods. However, the ECS method uses dual domain implementation (shuttling between the space and wavenumber domains by FFT) and is much faster than the full wave finite difference program. For a medium size 3D problem, the time saving is more than 2-3 orders of magnitude. Because it needs to store the medium parameters only one grid-plane for each step, the tremendous computer memory saving makes it capable of handling large 3D problem prohibitive to other methods.

    To model reflection seismograms, the one-way approximation is extended to one-return approximation based on the De Wolf approximation (Multiple-Forescattering-Single-Backscattering: MFSB) (Wu, 1996; Wu, Huang and Xie, 1995; Xie and Wu, 2001; Wu, 2002b; Wu and Wu, 2002). Both the thin-slab propagator and complex-screen propagator have been developed and tested. 2D numerical simulations have shown the validity, feasibility and advantages of the one-return methods. For moderate contracts the method is very fast and reasonably accurate. The method has been applied to calculations of synthetic seismograms and AVO (Amplitude vs. Offsets) analysis (Wu and Wu, 2002, 2002 submitted).

  2. Development of GSP (Generalized Screen Propagators) Subsurface Imaging Methods:
    Based on the work on elastic complex screen (Wu, 1994), many variants of the GSP (Generalized Screen Propagators) for modeling and imaging has been developed in the last decade. The GSP method filled the gap between the full-wave finite difference method and the ray method, and is several orders of magnitude faster than the full-wave methods. In the same time it keeps all the essential features of the full-wave methods and can deliver even better image quality than the full-wave methods. Collaborated with LANL (Fehler and Huang), CWP (de Hoop), MIT and colleagues in our Lab (Xie, Jin, Wu and others) different versions of the GSP has been developed in the past few years including the WGSP (windowed generalized screen propagator) (Wu and Jin, 1997; Jin and Wu, 1998, 1999), pseudo-screen propagators (Huang and Wu, 1996; Jin and Wu 1999; Jin at al., 2000, 2002), the wide-angle Pade-screen (Xie and Wu, 1998, 1999; Xie et al., 2000), and the GSP with the generalized screen expansion (De Hoop et al., 2000). These methods have been tested using both synthetic data sets, such as the SEG-EAEG salt model, Marmousi overthrust model, and field data sets provided by Shell and Conoco, and showed excellent results. The comparison with the traditional Ray-Kirchhoff method and F-X method showed the improved accuracy and efficiency of the screen methods.

    Based on the new theory and method, joint projects (with Los Almos National Laboratory, MIT, and industrial partners), have been supported by the ACTI (Advanced Computational Technology Initiative) and the Office of Basic Energy Sciences of DOE (The 3D Seismic Model ing and Imaging Project (Modeling and Imaging Laboratory) at UCSC ). A Research Consortium (WTOPI) ( The WTOPI Research Consortium at UCSC (Wavelet Transform On Propagation and Imaging applied to seismic exploration) ) has been also started and now is in its 6th years.

  3. Modeling Regional Wave Propagation (Including Lg) in Heterogeneous Crustal Waveguides:
    Collaborated with Jin and Xie and X. Wu, the theory and method of half-space screen propagators for the 2D SH case has been developed to model high-frequency (up to 25 Hz), long distance ( greater than 1000 km) Lg wave simulations in complex crustal waveguides including macro structures of the crust, small-scale random heterogeneities and rough interfaces (Wu, Jin, Xie, 2000a,b). Detailed accuracy verification and tests of the method have been conducted by comparing the results with the reflectivity and finite difference numerical methods. Excellent agreement between these methods demonstrated the high accuracy and efficiency of the half-space GSP method. For a medium size Lg problem, The GSP method is 300 times faster than the finite difference method. For large size problems, the saving could be much greater. The GSP method for SH Lg waves been extended to including the rough surface topography (collaborated with Xianyun Wu) (Wu and Wu, 2001). Some preliminary results of P-SV Lg wave propagation in the crust has also been obtained (in draft).

    Boundary element (BE) method has also been applied to model Lg wave propagation for the case of rough free surface. Collaborated with L.Fu, the calculation of absorbing boundary condition for BE has been improved (Fu and Wu, 2000) and a connection formula has be developed for the hybrid BE-GSP method (Fu and Wu, 2001). Scattering attenuation due to rough surface has been investigated by the method (Fu et al., 2002).

  4. Wavelet Transform Applied to Wave Propagation and Imaging (Beamlet Imaging)
    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 significant progress in one-way wave propagation theory and algorithms, including the recently developed fast acoustic and elastic generalized screen propagators from our group. The cross-breeding of these two new developments has the potential of revolutionizing modeling and imaging techniques for complex Earth media. Collaborated with Postdoctoral researchers and students Wu has done some investigation on propagator decomposition and compression and comparison has been made for adapted wavelet packet transform (best basis), standard wavelet transform, and adapted local cosine transform (Wu and Yang, 1997; Wu and Wang, 1998; Wang and Wu, 1998, 2000). It is shown that the one-way wave propagator in wavelet domain, which is called the beamlet (applying WT to the transversal space-coordinates) propagator, is a highly sparse matrix and has both the space and direction (wavenumber) localizations. Propagation and imaging in wavelet domain operate in local phase-space, and can adjust the resolutions in both space and directions, calculate and correct illumination and aperture effects in angle-domain (Wu and Chen, 2002), perform local AVA analysis and local inversion. Wu and his colleagues have developed a local perturbation theory for beamlet imaging (Wu, Wang and Gao, 2000) and implemented the theory for two kinds of beamlets. One is Gabor-Doubechies frame (GBF) beamlet (Weyl-Heisenberg coherent state) (Wu and Chen, 2001,2002; Chen and Wu, 2002); the other is the local cosine basis (LCB) beamlet (Wu,Wang and Gao, 2000; Wang and Wu, 2002; Wu, Chen and wang, 2002). Beamlet imaging has been applied to the SEG/EAGE salt model and Marmousi model. The imaging quality is superior to the traditional Kirchhoff migration and comparable to that of GSP. The Research Consortium (WTOPI) ( The WTOPI Research Consortium at UCSC (Wavelet Transform On Propagation and Imaging applied to seismic exploration) ) supported by the Oil/Gas and geophysical companies ( WTOPI Sponsors ) has been started to advance the theory and its applications, and now is in its 6th years.


  5. Scattering Attenuation and its Separation from Absorption:
  6. Fractal Heterogeneous Media and Seismic Wave Scattering:
    Introduced three methods from seismic wave scattering to determine the fractal dimension of a fractal heterogeneous medium. Results were derived for some regions (Wu and Aki 1985c, Wu 1986a, b).
  7. Direct Measurement of Power Spectra and Aspect Ratio of Crustal Heterogeneities.
    For the first time, a power-law spectrum of crustal heterogeneities was directly observed in the spatial frequency range from less than one meter to a few kilometers using the well-log data of the German super-deep continental holes (KTB) (Wu, Xu and Li, 1994). The slope of the 1D power spectrum is about -1.1. In the same time, the aspect ratio of crustal heterogeneities was estimated using a cross-spectrum method applied to the sonic logs of the two holes (main hole: 6 km, pilot hole: 4 km in depth, separated by 200 m), to be around 2.5. These direct observation and measurements will help the understanding and give some constraints for spectral inversion of crustal heterogeneities.
  8. Multi-Frequency Holography, Diffraction Tomography and MFBT:
    Developed multi-frequency holography as a method of geophysical imaging (Wu et al. 1977, Wu and Xu 1978, 1979, 1980) and later introduced Diffraction tomography to geophysical application based on T. Devaney's work (single frequency) (Wu and Tokso\*:z 1987). Recently he developed the MFBT (Multi-Frequency Backscattering Tomography) (Wu, Araujo and Huang, 1994; Huang adn Wu, 1994). The method is formulated for both the constant and vertically varying backgrounds. Compared with single-frequency diffraction tomography, the method has much higher resolution and image quality, but with similar computation speed. For the case of constant background, the computation speed is even faster than the single frequency case due to a FFT reconstruction scheme.
  9. Joint Coherence Analysis of Seismic Array Data and Stochastic Tomography
    Introduced a New type of coherence function ACF (Angular Coherence Function) for travel-time and amplitude fluctuations across an array with S.M. Flatte\*' and inverted the NORSAR data to derive a new spectral model of lithospheric heterogeneities under the array (Flatte\*' and Wu 1988, Wu and Flatte\*' 1990, Wu, 2002a). Later He developed a more general joint coherence function JTACF (Joint Transverse-Angular coherence Function) which has much better depth resolution and makes the stochastic tomography feasible (Wu 1989, Wu and Flatte\*' 1990, Wu and Xie 1991).
    Stochastic tomography is a method which inverts different coherence functions of travel-time and amplitude fluctuations across a seismic array for spectral images of heterogeneities at different depths in the crust and mantle under the array. At present the emphasis is put on the application of stochastic tomography to lithospheric heterogeneities. This method may become an effective way of characterizing different tectonic systems related to lithospheric dynamics. Some preliminary results have been obtained using the data from NORSAR and the Southern California Seismic Network. Distinct random layers at 100 km, 220 km have been discovered. The possible connections with crustal and mantel dynamic processes are under examination.
  10. Elastic Wave Scattering Theory and Applications:
    Derived the elastic wave Born scattering in both deterministic and stochastic heterogeneous media. The theory has been applied to lithospheric heterogeneities and established: (1) the importance of relative scale-lengths of heterogeneities in forward and backward scattering, (2) the multi-scale nature of the lithospheric heterogeneities, especially in tectonically active regions. (Wu and Aki 1985a, b, Wu 1989b, c)
    Elastic Wave Rayleigh Integral:
    Derived the elastic wave Rayleigh integrals which do not require the stress field known in the given surface and make the elastic wave extrapolation more efficient and applicable (Wu 1989a, e).

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