Optimising Seismic Imaging via Bilevel Learning: Theory and Algorithms

Abstract

Seismic inversion is the inverse problem of determining properties of the Earth’s subsurface from measurements of waves propagating through it. A standard algorithm for solving this inverse problem is called full waveform inversion (FWI). FWI computes a model describing the subsurface by minimising the misfit between actual measurements and numerically-predicted data plus one or more regularisation terms, which are added to deal with the ill-posedness of the inverse problem. The implementation of FWI requires the a priori choice of a number of parameters, including the positions of sensors for the wave measurement and the regularisation parameters. A problem of great practical interest, which is not considered in the standard approach to FWI, is the optimal positioning of the sensors in order to obtain the best outcome from the seismic imaging process. In this thesis, it is shown that, given a set of training models of realistic wave velocities, one can learn the optimal sensor positions and regularisation parameters, thus optimising the performance of the standard FWI reconstruction algorithm. We establish a novel fundamental theory underpinning the solution to this sensor optimisation problem by placing it in the framework of bilevel learning. In our formulation, the upper-level objective function measures the misfit in the reconstruction of the training models via FWI, so that FWI itself constitutes the lower-level optimisation problem. We propose to solve the bilevel problem with a gradient-based optimisation method. Our chosen forward problem is the acoustic wave equation, which we solve in the frequency domain (via the Helmholtz equation).

This thesis contains contributions both in the theory and application of this bilevel learning problem. In particular, for the theory, this thesis contains the following novel contributions:

• We give sufficient conditions, in terms of the regularisation parameters in the lower-level/FWI problem, for the lower-level problem to have a unique solution.
• Wederive a formula for the gradient of the upper-level objective function and show that this requires solving systems involving the Hessian of the lower-level problem, for which ill-conditioning is mitigated by the choice of lower-level regularisation.
• We prove smoothness properties of the upper-level objective function by exploiting the theoretical properties of the partial differential equations modelling the propagation of acoustic waves in the frequency domain.
• We show that, under assumptions on the symmetry of the domain, model and source positions, the optimal set of sensor positions is symmetric.

Our main novel contributions to the application aspect of this bilevel problem are the following:

• We design a bilevel learning algorithm for optimising sensor positions and the Tikhonov regularisation parameter in FWI.
• We give a complexity analysis for the bilevel algorithm, involving a study of the number of forward solves needed by the algorithm.
• We propose a bilevel frequency continuation strategy to improve the performance of the bilevel algorithm.
• We propose a preconditioning strategy for the systems involving the Hessian which have to be solved at each step of the upper-level gradient descent.
• We implement the bilevel algorithm and provide illustrations of the algorithm on test problems.

Date of Award	22 Jun 2022
Original language	English
Awarding Institution	University of Bath
Supervisor	Euan Spence (Supervisor), Ivan Graham (Supervisor) & Silvia Gazzola (Supervisor)