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 - ? , x - ? ) atom (a drumbeatbeamlet). The imaging condition in the local t - ? domain is introduced and the propagator matrix in the local ( t - ? , x - ? ) 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.
To overcome the fundamental limitation of global wave propagators, such as the f-x domain finite difference method, Fourier-finite difference method and generalized screen propagators, in strong-contrast heterogeneous media, efforts have been made to develop wavefield decomposition and extrapolation methods with localization in both space and direction. Localized wave propagators can be easily tailored for local heterogeneities and to specific directions (Steinberg, 1993; Steinberg and McCoy, 1993; Wu et al., 2000; Wu and Chen, 2002; Chen et al., 2006; Wu et al., 2008). However, these local propagators have only space-direction localization, but without timefrequency localization. Recently curvelet transform (Candès & Donoho., 2002; Candès & Demanet, 2005), which is a complete space-time localization, has been applied to wave propagation and seismic imaging using a map migration method (Douma & De Hoop, 2007; Chauris & Nguyen, 2008). In these methods, high-frequency asymptotic approximation was invoked to propagate the curvelets in smoothly inhomogeneous media. For complicated structures with high-contrast inclusions, such as those involved with salt domes, high-frequency asymptotic methods have very limited success. In this paper, we apply a multi-dimensional local harmonic transform to wavefield on a surface, such as on the recording surface of data acquisition or on the transversal plane of one-way wave propagation. Specifically, we apply the x - ? localization using the local cosine basis (LCB) (Coifman and Meyer, 1991), where x and ? are the position and wavenumber for the 2D wave propagation case, or 2-D vectors for the 3D wave propagation case, and the t - ? localization using the local exponential tight frame (Auscher, 1994; Wu and Mao, 2007; Mao etal., 2007). Unlike the curvelet transform, here the t - ? localization is treated separately and differently from the x - ? localization. These two sets of coefficient are not totally independent parameters, but related by the dispersion relation. This is an important feature of our treatment for wavefield decomposition and propagation.
First we discuss the time-frequency analysis and introduce the imaging condition in the localized t - ? domain. Then we discuss the wavefield decomposition using localized ( t - ? , x - ? ) atoms. With these preparation, we derive the formulation of wave propagation in the localized ( t - ? , x - ? ) domain.