Regularization and redatuming using least squares and conjugate gradients

Daniel R. Smith, Mrinal K. Sen, Robert James Ferguson

Irregular spacing of sources and receivers, and dead traces plus noise result in incomplete data. Moreover, phase distortion from a complex near-surface can cause lateral reflector discontinuity that statics cannot handle. As a remedy, we have developed a method to handle irregular data and near-surface complexity as one inversion problem. In particular, we use conjugate gradients for optimization where a weighted, damped least squares approach is used to downward continue data. Data that correspond to the new wavefield at depth are generated by minimization of the residual between the given wavefield and the estimated wavefield. The required extrapolation operators are implemented as spatially variable phase-shifts applied within a Fourier integral operator. The resultant Hessian is extremely costly to compute, so we use the method of conjugate gradients (CG) to avoid direct computation of the Hessian. Our CG approach reduces the total number of operations from O(n3) for direct computation of the Hessian, to O(n2) for the CG method, where n is the number of trace locations.

We use a synthetic data example plus a real data example to demonstrate our simultaneous inversion. The synthetic data are the result of an exploding reflector model where the traces are generated by finite differences. Our setup simulates an irregular, horizontal recording aperture above a line source in which a series of point sources are embedded - the design of the sources results in a flat reflection event and a series of steep diffractions with conflicting dips. The near-surface correction aspect of our CG inversion removes the lateral velocity effects in the synthetic data, and the trace interpolation aspect reconstructs the missing traces.

Our real data example was acquired by Husky Oil Ltd in the Alberta Foothills of the Canadian Rocky Mountains. Shot spacing for these data is very irregular, and common receiver gathers suffer from incomplete trace coverage as a result. Further, the near-surface is highly heterogeneous due to significant topographic variation and lateral velocity variation, and reflector continuity is compromised as a result. CG inversion of these data successfully reconstructs the data, with some remaining artifacts due to aliasing, and lateral continuity of the reflectors is improved. As a side benefit, because our extrapolation operator is implemented in the temporal and spatial Fourier domain, ground roll is suppressed.

In all instances, we find that the efficiency of the method is improved by an order of magnitude when compared to direct inversion.