Novel Optimization Schemes for Full Waveform Inversion: Optimal Transport and Inexact Gradient Projection

Da Li

Full waveform inversion (FWI) is an important seismic inversion technique that provides high-resolution estimates of underground physical parameters. However, high-accuracy inverse results are not guaranteed due to the essential non-convexity characteristics of the FWI problem. This thesis focuses on designing novel optimization schemes for the FWI problem which improve the inverse results.

Applying optimal transport (OT) based distances to the FWI problem is popular because they provide additional geometric information. The OT distances are designed for positive measures with equal mass, and the unbalanced optimal transport (UOT) distance can overcome the mass equality condition. A mixed distance is constructed which can also overcome the mass equality condition, and the convex properties for the shift, dilation, and amplitude change are proved. Both UOT distance and the proposed distance are applied to the FWI problem with normalization methods transforming the signals into positive functions. Numerical examples show that the optimal transport based distances outperform the traditional L2 distance in certain cases.

The gradient projection methods are often used to solve constrained optimization problems, and the closed-form projection function is necessary since the projection has to be evaluated exactly. A constraint set expanding strategy is designed for the gradient projection methods such that the projection can be evaluated inexactly, which extends the application scope of the gradient projection methods. The convergence analysis is provided with proper assumptions.

A priori information of the model is important to improve the inverse result, and an optimization scheme is proposed for incorporating multiple a priori information into the FWI problem. The optimization scheme is a combination of the scaled gradient projection method and a projection onto convex sets algorithm. Also, the L-BFGS Hessian approximation and the above constraint set expanding strategy are used. Numerical examples show that the proposed optimization scheme is flexible for integrating multiple types of constraint sets such as total variation constraint, sparsity constraint, box constraint, and hyperplane constraint into the FWI problem.