Adaptive Gabor imaging using the lateral position error criterion
Yongwang Ma, Gary F. Margravé · 2007
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.