Phase shift migration using orthogonal beamlet transforms

Abstract

The success of wavelet transforms for seismic data compression leads naturally to the idea of “compressing” numerical wavefield propagators. In spite of numerous efforts (i.e. Beylkin, 1992, Dessing and Wapenaar, 1995, Mosher, Foster, and Wu, 1996) application of orthogonal wavelet bases to wavefield propagation has had limited success. In this work, we examine implementations of phase shift migration using orthogonal wavelet transforms as an illustration of the limitations that orthogonality places on wavelet domain propagators. Since wave propagation has a simple representation in the frequency domain, frequency domain wavelet transforms provide a useful framework for studying wave propagation. In particular, we describe phase shift extrapolators for 2-dimensional wavefields that have been Fourier transformed over time and wavelet transformed over space. The wavelet transform over the space axis is implemented in the wavenumber-frequency domain by complex multiplication of low and high pass wavenumber filter functions to form wave packet trees. Spacewavenumber-frequency transforms are usually referred to as ‘beamlet transforms’ (Wu and Chen 2001), and are closely related to Gaussian beams (Hill, 2001, Albertin et al, 2001). The interaction of beamlet transform filter banks and phase shift wavefield extrapolators are simple complex multiplications. Wavefield propagation in the beamlet domain is complicated, however, by the digital implementation of decimation and upsampling operators used in orthogonal wavelet transforms. Unlike the filter functions, which can be viewed as diagonal matrix operators, the decimation and upsampling operators have significant off-diagonal terms. Since these operators do not commute with the filter and phase shift operators, the effects of the off-diagonal terms must be accounted for in the application of wave propagation operators. Use of filters designed for simple shape and compact support in the wavenumber domain reduces the domain of the interactions, resulting in implementations of phase shift extrapolators that have computational complexity comparable to traditional Fourier approaches. Compact support in the wavenumber domain, however, corresponds to poor localization in the space domain. Use of orthogonal wavelet bases for beam-based wavefield propagators results in a trade-off between computational complexity when the wavelet transform filters overlap, and poor localization in space when the overlap is limited. These results suggest that non-orthogonal transforms may provide a better domain for wave propagation. Introduction