A discrete adjoint method for deterministic and probabilistic eikonal-equation-based inversion of traveltime for velocity and source location

TL;DR

This paper develops a discrete adjoint method for efficient joint inversion of seismic traveltimes to recover subsurface velocity and source locations, applicable in both deterministic and probabilistic frameworks.

Contribution

It introduces a formalism combining the eikonal equation with a discrete adjoint approach for joint inversion of velocity and source position, supporting both deterministic and Bayesian methods.

Findings

Effective gradient computation for joint inversion.

Successful application to synthetic 2D and 3D examples.

Compatibility with fast-marching and Bayesian algorithms.

Abstract

Seismic traveltime tomography represents a popular and useful tool for unravelling the structure of the subsurface across the scales. In this work we address the case where the forward model is represented by the eikonal equation and derive a formalism to solve the inverse problem where gradients are calculated efficiently using the discrete adjoint state method. Our approach provides gradients with respect to both velocity structure and source locations, allowing us to perform a consistent joint inversion. The forward problem is solved using a second-order fast-marching method, which provides a strategy to efficiently solve the adjoint problem. Our approach allows for arbitrary positions of both sources and receivers and for a refined grid around the source region to reduce errors in computed traveltimes. We show how gradients computed using the discrete adjoint method can be employed to perform either deterministic inversion, i.e., solving an optimization problem, or for a probabilistic (Bayesian) approach, i.e., obtaining a posterior probability density function. We show applications of our methodology on a set of synthetic examples both in 2D and 3D using the L-BFGS algorithm for the deterministic case and the Hamiltonian Monte Carlo algorithm for the probabilistic case.