A procedure of designing the optimal finite-aperture extrapolators and performing the explicit 3D wavefield downward extrapolation by 2D spatial convolution of the wavefield with the extrapolator is proposed. The linear least square fitting of the 20 Fourier transform of the extrapolator with the desired 3D migration operator in the frequency-wavenumber(F-K) leads to a large-scale overdetermined complex linear system. The construction of the optimal operators are realized through the QR factorization and direct solving of the system. The procedure is unconditionally stable and can be applied recursively for depth extrapolation in a laterally varying media. No slight evidence of azimuthal distortion has been shown on the computational results. A global table consisting of all possible 2D extrapolators versus varying frequency and velocity ratio is precalculated and tabulated. The algorithm was designed to accommodate the hardware architecture, such that all frequency slices of the 3D wavefield are processed simultaneously on individual processing elements(PEs). The scheme has exactly equal number of floating point operations on each PE and the computation is thus perfectly balanced over the MPP. We have taken into account all macroarchitecture characteristics of the MPP systems to minimize communication overhead and maximize sustained computational rates. The timings of method running on the Cray T3D MPP are presented. As the long extrapolator is required for the imaging of steep reflectors, the influences of aperture size of the operators on the timing and numerical results of the 3D extrapolation process are analyzed. 1 I N T R O D U C T I O N The 3-D wavefield extrapolation in the frequency and wave-number domain can be implemented with forward and reverse propagation of wave along the depth axis as the multiplication of the wavefield with the extrapolator
