Abstract
The seismic inverse problem appears in the exploration geophysics applications and consists of determining an underneath region's geophysical information using the data measured on the surface. It is formulated as a PDE constrained inverse medium problem: given the values of a family of solutions in a hyperbolic system, find the system's coefficients.
A fitting data scheme is usually employed to solve it in mathematics and engineering.
In most realistic situations, solving this inverse problem is challenging due to the large scale and ill-posedness of the problem (noisy measurement, lack of stability, and non-uniqueness).
Mathematicians and geophysicists have extensively studied the analysis and numerical scheme of this nonlinear problem in the past decades. The main mathematical obstacle remaining is on the appearance of many local minima in the data-misfit function. I will show two approaches to mitigate the ill-posedness. One is via a generalized version of the Total Variation regularization, and the other is to use the quadratic Wasserstein metric.
TV has become one of the most popular and successful methodologies for image processing. It is widely used to restore images from blurry or nosy observations with the sharp interfaces, edges and discontinuities preserved. We apply a generalized TV regularization method to FWI with the aid of the a priori information from other geological investigation. A skew field is introduced to guide the update using the wavefront information. The algorithm is efficiently implemented using the split Bregman method. It has been successfully applied to a realistic model with over $10$ billion degrees of freedom.
The second approach is to change how we measure the data difference by switching from least squares to the Wasserstein distance. The Wasserstein distance is a natural metric for comparing two histograms or probability distributions and becomes a prevalent measure of similarity in computer vision and machine learning. It bears the robustness to small shifts and noise in histograms and provides a larger convex zone compared to the usual least-squares distance. We employ the quadratic Wasserstein space for mitigating the cycle-skipping problem. The approach is based on solving an optimal transport problem and modifying the adjoint source using the Kantorovich potential.
