A review of misfit functions for adjoint full waveform inversion in seismology

Abstract

SUMMARY In seismological full waveform inversion, the choice of misfit functions plays a critical role in quantifying the discrepancy between observed and synthetic data, affecting convergence rate and also final results. We revisit and compare six commonly used misfit functions, including cross-correlation time-shift (CC), least-square waveform difference (L2), multitaper time-shift (MT), exponentiated phase shift (EP), time–frequency phase shift (TF) and zero-lag cross-correlation coefficient (CCC), with respect to their definitions, adjoint sources and misfit kernels for velocity perturbations. Synthetic tests are performed for several canonical models. First, we simulated wave propagation in a model with a single rectangular anomaly with sharp boundaries and a smoothed variant of that model. We analysed the resulting misfit kernels first for the P-wave phase, which is highly distorted in the sharp model due to strong heterogeneities, and mostly experiences traveltime perturbations in the smooth model. Second, we considered a model where a laterally limited region is subject to layered anomalies (low velocity in the middle crust and high velocity in the lower crust) and determine misfit kernels for S and surface waves in this model. Based on these two simplified seismological scenarios, we further perform iterative test inversions using different misfit functions. Combining the features of misfit kernels and synthetic inversion results, we find that CCC, L2 and EP are the most effective at identifying the sharpness of velocity anomalies from the direct body waves and their scattered phases. Consequently, inversion based on those misfit measures yielded the best recovery in the inversion test. For surface and S waves from crustal sources, TF appears to be the most effective in constraining the heterogeneous structure in the crust but needs more iterations for convergence than other misfit functions.