Regularization and redatuming using least squares and conjugate gradients




Smith, Daniel Ryan

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Irregularity in source and receiver spacing along with dead traces and noise can result in incomplete data causing spatial aliasing problems. In addition, phase distortions from sampling near-surface velocity variations can cause lateral reector discontinuity which statics cannot handle. A method is developed to handle both these problems together as an inversion problem. Weighted, damped least squares are used to downward continue data by estimating the new wavefield at depth by minimizing the residual between the estimated wavefield and the observed wavefield and conjugate gradients are employed for optimization. The extrapolation operators are spatially varying phase-shifts applied within a Fourier integral operator. The Hessian in the least squares inversion is costly to compute, so conjugate gradients are employed to avoid computation of the Hessian as a matrix-matrix multiplication and instead reduce it to two matrix-vector multiplications. This reduces the total number of multiplication operations from O (n³) for the direct solution to O (n²) for the conjugate gradient method, where n is the number of trace locations. I use a synthetic example as well as a real data example to demonstrate the method's effectiveness. The synthetic data are from an exploding reflector model where the traces have been generated by finite differences. It is designed to simulate an irregular, horizontal recording array above a horizontal line source with five point sources and a laterally variable background velocity. The resulting data are a subhorizontal discontinuous reflector above steeply dipping diffractions. The method successfully removes the effects of the lateral velocity variations flattening the events and reconstructing the missing traces. The real data comes from the Alberta Foothills of the Canadian Rocky Mountains acquired by Husky Oil Ltd. where the shot spacing is very irregular and therefore common receiver gathers are quite irregular. Moreover, the near-surface is highly heterogeneous due to varying topography and laterally varying velocity so the data suffers from lateral reflector incoherency. The method successfully reconstructs the data, still with some artifacts due to actual spatial aliasing, and improves lateral continuity of the reflectors, also while suppressing ground roll. Overall, the computational efficiency of the method is improved by an order of magnitude when compared to the direct inversion methods


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