Mitigating Cycle Skipping in Full Waveform Inversion by Using a Scaled-Sobolev Objective Function

Cycle skipping in the conventional full waveform inversion (FWI) objective function depends on the frequency content of the data, and on the error in the background velocity. The error in background velocity that can be tolerated without skipping cycles is determined by the half cycle criterion. However, the half cycle criterion is offset dependant so that far offsets in the data are more prone to cycle skipping than near offsets. This offset dependence of the half cycle criterion implies that the differentials of residuals with offset can be used as additional constraints in the objective function. In this study we introduce the scaled-Sobolev objective (SSO) that seeks to minimize a smooth version of the data residuals in addition to their derivatives in all data domain dimensions. The smoothing of the data is done using the scaled-Sobolev inner product (SSIP) in the data domain, resulting in an edge-preserving smoothing operator. In the absence of low frequencies, increasing the maximum order of derivatives in SSO is more important than the zeroth order scale factor. Initial results with synthetic data using the Marmousi model show that SSO can overcome a bulk shift in velocity of 30%, with a lowest frequency of 8 Hz.