A computationally feasible approximate resolution matrix for seismic inverse problems

SUMMARY Seismic inversion produces model estimates which arc at most unique in an average sense. The model resolution matrix quantifies the spatial extent over which the estimate averages the true model. Although the resolution matrix has traditionally been defined in terms of the singular value decomposition of the discretized forward problem, this computation is prohibitive for inverse problems of realistic size. Inversion requires one to solve a large normal matrix system which is best tackled by an iterative technique such as the conjugate gradient method. The close connection between the conjugate gradient and Lanczos algorithms allows us to construct an extremely inexpensive approximation to the model resolution matrix. Synthetic experiments indicate the data dependence of this particular approximation. The approximation is very good in the vicinity of large events in the data. Two large linear viscoelastic inversion experiments on p-τ marine data from the Gulf of Mexico provide estimates of the elastic parameter reflectivities corresponding to two different seismic sources. Traditionally, one evaluates the accuracy of the two reflectivity estimates by comparing them with measured well logs. The approximate model resolution matrices agree with the well-log ranking of the two models and provide us with a way to compare different model estimates when, for example, such well-log measurements are not available.