We present a new seismic depth migration algorithm using the Gabor transform, also termed as the windowed Fourier transform, over the lateral spatial coordinates and the discrete Fourier transform over time. These transforms enable a wavefield depth extrapolation by laterally variable, frequency and wavenumber dependent, phase shift. The Gabor transform is implemented as a windowed discrete Fourier transform where the windows are confined to form a partition of unity (POU), meaning that they sum to one. For efficiency, an adaptive partitioning scheme that relates window width to the lateral velocity variation is developed, and defines an adaptive Gabor imaging scheme. Within each window, the Gabor method uses the familiar split-step Fourier technique. The construction of the adaptive partition of unity is guided by an accuracy threshold that constrains the spatial positioning error, for each depth step. The spatial positioning error is estimated by comparing the Gabor method to a nonstationary phase shift that changes according to the local velocity at each position. We present the details of building the adaptive POU for both 2D and 3D imaging. The performance of Gabor depth imaging using this partitioning algorithm is illustrated with imaging results from prestack depth migration of the Marmousi dataset.
Seismic migration by phase shift (also referred to as wavefield extrapolation with phase shift) was proposed as an accurate and efficient method that is considered to be unconditionally stable provided that there are no lateral velocity variations. An approximate extension to lateral velocity variations, called PSPI (phase-shift-plus-interpolation) was presented by Gazdag and Sguazzero (1984), who proposed spatial interpolation between a set of constant velocity reference extrapolated wavefields. This method has proven popular and, although it is no longer unconditionally stable (e.g., Etgen, 1994), it is more stable than a typical explicit space-frequency method (Margrave et al., 2006). Other ways of extending the concept of a phase-shift include the split-step Fourier method (Stoffa et al., 1990), the various phase-screen methods (Wu and Huang, 1992; Roberts et al., 1997; Rousseau and de Hoop, 2001; Jin et al., 2002), the generalized phase-shift-plus-interpolation (GPSPI) and nonstationary phase-shift methods (NSPS) (Margrave and Ferguson, 1999), and the Gabor method (Grossman et al., 2002; Ma and Margrave, 2005). Margrave and Ferguson (1999) showed that the GPSPI Fourier integral is the limit of PSPI in the extreme case of using a distinct reference velocity for each output location. Writing outside the typical seismic literature, Fishman and McCoy (1985) derived the GPSPI formula as a high frequency approximation to the exact wavefield extrapolator for a laterally variable medium. They refer to the GPSPI formula as the locally homogeneous approximation (LHA) and we adopt that nomenclature. While the LHA is called a high frequency approximation it is still much more accurate than raytracing because the approximation is done at a different place in the theoretical development (see Fishman and McCoy for more discussion). In fact, it can be shown that virtually all explicit depth-stepping methods in practice today, including those mentioned above, are approximations to the LHA formula.