Bangyu Wu

Seismic wave propagation and imaging using time‐space wavelets

Using a tensor product of a local exponential frame vector as the time-frequency atom (a drumbeat) and a local cosine basis function as the space-wavenumber atom (a beamlet), we construct a localized ( t ω − , x ξ − ) atom (a drumbeatbeamlet). The imaging condition in the local t ω − domain is introduced and the propagator matrix in the local ( t ω − , x ξ − ) domain is derived. The compression of seismic data using the new decomposition is tested and simple numerical tests on the propagator matrix are performed. The sparseness in both data and propagator decompositions is demonstrated through these examples. The method has potential application in efficient representation of seismic data and wave-theory based seismic imaging using compressed data.

Physical wavelet defined on an observation plane and the Dreamlet

Summary Wavefield or seismic data are special data sets. They cannot fill the 4-D space-time in arbitrary ways. The timespace distributions must observe causality which is dictated by the wave equation. Wave solutions can only exist on the light cone in the 4D Fourier space. Physical wavelet is a localized wave solution by extending the light cone into complex causal tube. In this study we establish the link between the physical wavelet defined by Kaiser using AST (analytic signal transform) and the dreamlet (drumbeatbeamlet). We prove that dreamlet can be considered as a type of physical wavelet defined on an observation plane (earth surface or a plane at depth z during extrapolation). Causality (or dispersion relation) built into the wavelet (dreamlet) and propagator is a distinctive feature of physical wavelet which is advantageous for applications in wavefield decomposition, propagation and imaging. One example of dreamlet decomposition on seismic data is given.

Dreamlet prestack depth migration using Local Cosine Basis and Local Exponential Frames

Dreamlet migration using Local Cosine Basis (LCB) for spatial decomposition and Local Exponential Frames (LEF) for temporal decomposition is further developed and tested. Based on the beamlet local perturbation theory, dreamlet method is formulated with a local background velocity and local perturbations for each space window of the wavefield decomposition. The background dreamlet propagator based on LCB and LEF are obtained analytically. The numerical test using the 2D SEG/EAGE A-A’ and SmaartJV Sigsbee2A model demonstrate the accuracy and imaging quality of this method. The application in target oriented migration is discussed and tested through numerical examples.

Imaging in Compressed domain using Dreamlets

Summary The objective of this paper is to further investigate the theory and algorithm of wave propagation and imaging in the dreamlet domain for direct application of compressed seismic data. In our former work, dreamlet shows great application potential in efficient representation of seismic data. Based on the combination of dreamlet compression and wave propagation theory, we discuss one-way dreamlet wave propagation and imaging in terms of data processing in the compressed domain, that allows us to process only a few percentage of the decomposition coefficients of the whole data set to get the accurate imaging result. By using a 5-layer velocity model as numerical example, we show that the dreamlet coefficients of the seismic data is also decreasing with the depth of migration in the receiver side, meanwhile the imaging quality using the compressed data remains in a similar accuracy.

A Carleman estimate of some anisotropic space-fractional diffusion equations

Abstract This paper is concerned with Carleman estimates for some anisotropic space-fractional diffusion equations, which are important tools for investigating the corresponding control and inverse problems. By employing a special weight function and the nonlocal vector calculus, we prove a Carleman estimate and apply it to build a stability result for a backward diffusion problem.

Source-receiver Prestack Depth Migration Using Dreamlets

Summary The dreamlet method adds the time-frequency localization to the frequency domain beamlet method by an additional wavelet transform along the time axis. This method has been applied to shot-profile prestack depth migration and shows great potential on wavefield compression and good imaging quality. However, in shot domain migration, the point source excited wavefield expands during migration on the time-space plane and then increases the amount of dreamlet coefficients and therefore, the computation time. The survey sinking migration not only downward continues the geophones but also the shots. Combining the survey sinking and dreamlet decomposition can achieve higher compression ratio on the whole data volume. For the SEG/EAGE model, the dreamlet survey sinking algorithm is 3 times faster compared with the shot domain dreamlet method with similar imaging quality.