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Wavelet transforms have a simple representation in the frequency domain (Daubchies, 1992; Veterlli and Herley, 1992; Mosher and Foster, 1995). Since wave propagation also has a simple representation in the frequency domain, frequency domain wavelet transforms provide a useful framework for studying the nature of wave propagation in the wavelet domain. In this paper, we study 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. To differentiate this operation from time-frequency wavelet transforms, we refer to the space-wavenumber-frequency transform as the 'beamlet transform.' 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 non-diagonal operators must be accounted for in the application of wave propagation operators. A simple (but unsatisfying) solution would be to apply forward-inverse transforms at each extrapolation step. Beamlet transforms with compact support in the wavenumber domain (Mosher and Foster, 1995) provide an alternate solution. Analysis of phase shift migration in the beamlet domain yields a simple matrix representation defining the interaction of filters, phase operators, and decimation/upsampling. The effects of decimation/upsampling are represented by simple folding operations. Use of filters designed for simple shape and compact support in the wavenumber domain reduces the domain of the interactions, resulting in efficient implementations of phase shift extrapolators that compare favorably with traditional Fourier approaches. Coupled with data compression, implementations of phase shift migration with multi-dimensional wavelet/beamlet transforms that exceed traditional implementations in computational efficiency may be possible.
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