Optimal Transport Map With Prescribed Direction Indicator for Seismic Full‐Waveform Inversion

Abstract

AbstractThe quadratic Wasserstein (W2) metric has been proposed as a promising misfit function to mitigate cycle‐skipping phenomena in full‐waveform inversion. Mathematically, we demonstrate that the smoothness of the W2‐based adjoint source is two orders of magnitude higher than that based on L2‐norm, which guarantees a larger convergence radius of related inverse problems. However, the oscillatory characteristics of seismic signals and subsequent operations of transforming them into probability densities would decrease the accuracy of the optimal transport map T(t) and exacerbate the nonconvexity of the misfit function. To tackle these challenges, we propose the concept of prescribed direction indicator, which indicates the properly matching direction from predictions to observations, in order to correct inaccurate T(t). 1D synthetic examples suggest that reasonable bijection can be constructed through the proposed method. Numerical experiments demonstrate that it works well during optimization procedures, including enlarging the convergence radius of the inverse problem, improving the computational efficiency and enhancing the reliability of inversion results.