Fast 2D full-waveform modeling and inversion using the Schur complement approach

In most full-waveform inversion (FWI) problems, sufficient prior information is available to constrain the velocity of certain parts of the model, e.g., the water column or, in some cases, near-surface velocities. We take advantage of this situation and develop a fast Schur-complement-based forward modeling and inversion approach by partitioning the velocity model into two parts. The first part consists of the constrained zone that does not change during the inversion, whereas the second part is the anomalous zone to be updated during the inversion. For this decomposition, we partially factorize the governing system of linear equations by computing a Schur complement for the anomalous zone. The Schur complement system is then solved to compute the fields in the anomalous zone, which are then back substituted to compute the fields in the constrained region. For each successive modeling steps with new anomalous zone velocities, the corresponding Schur complement is easily computed using simple algebra. Because the anomalous part of the model is comparatively smaller than the whole model, considerable computational savings can be achieved using our Schur approach. Additionally, we showed that the Schur complement method maintains the accuracy of standard frequency-domain finite difference formulations, but this comes at a slightly higher peak memory requirement. Our FWI workflow shows reduced runtime by 15%-57% depending upon the depth of the water column without losing any accuracy compared to the standard method.