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

The 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. Plain Language Summary Full waveform inversion serves as a powerful tool to enhancing our understanding of Earth's interior structure. However, the traditional L2‐norm‐based method often falls into a local minimum due to the well known cycle‐skipping problem. To address this issue, a novel technique named the Quadratic Wasserstein Metric has been proposed, which yields smoother and more reliable results. Nevertheless, a challenge persists regarding the low precision in calculating the optimal transport map. To counteract this problem, we have introduced a tool called the “Prescribed Direction Indicator,” which allows us to correct our calculations of the transport map and assist us in identifying the appropriate alignment between predictions and observations. With numerical synthetic experiments, we prove its superiority to the traditional Wasserstein metric in handling seismic inversions across a variety of scales, from small and local to global problems. Key Points We mathematically prove W2‐based adjoint source's smoothness is two orders higher than L2‐based We propose a concept of prescribed direction indicator to correct inaccurate optimal transport map T(t) Numerical experiments demonstrate the superiority of our proposed method compared to the traditional W2 metric