A robust waveform inversion using a global comparison of modeled and observed data

A high-resolution model of the subsurface is the product of a successful full-waveform inversion (FWI) application. However, this optimization problem is highly nonlinear, and thus, we iteratively update the subsurface model by minimizing a misfit function that measures the difference between observed and modeled data. The L2-norm misfit function provides a simple, sample-by-sample comparison between the observed and modeled data. However, it is susceptible to local minima in the objective function if the low-wavenumber components of the initial model are not accurate enough. We review an alternative formulation of FWI based on a more global comparison. A combination of Radon transform and utilizing a matching filter allows for comparisons beyond sample to sample. We combine two recent developments to suggest the following algorithm for optimal inversion: (1) we compute the matching filter between the observed and modeled data in the Radon domain, which helps reduce the crosstalk introduced in the deconvolution step of computing the matching filter, and (2) we use Wasserstein distance to measure the distance between the resulting matching filter in the Radon domain and a representation of the Dirac delta function, which provides us with the optimal transport between the two distribution functions. We use a modified Marmousi model to show how this Radon-domain optimal-transport-based matching-filter approach can mitigate cycle skipping. Starting from a rather simplified v(z) media as the initial model, the proposed method can invert for the Marmousi model with considerable accuracy, while standard L2-norm formulation is trapped in a local minimum. Application of the proposed method to an offshore data set further demonstrates its robustness and effectiveness.