Understanding the Adjoint Method in Seismology: Theory and Implementation in the Time Domain

TL;DR

This paper clarifies the mathematical foundations of the adjoint method in seismology using variational principles, proposes a new time domain implementation, and introduces a novel misfit function to improve seismic imaging accuracy.

Contribution

It provides a rigorous variational derivation of the adjoint wavefield, addresses inconsistencies in previous theories, and develops a new travel-time misfit function for better seismic inversion.

Findings

The adjoint wavefield should be defined via variational principles, not ad hoc assumptions.

The new misfit function resolves the zero sensitivity issue along the ray path.

Travel-time Frechet kernels show an extremum along the ray path without data-synthetics similarity assumptions.

Abstract

The adjoint method is a popular method used for seismic (full-waveform) inversion today. The method is considered to give more realistic and detailed images of the interior of the Earth by the use of more realistic physics. It relies on the definition of an adjoint wavefield (hence its name) that is the time reversed synthetics that satisfy the original equations of motion. The physical justification of the nature of the adjoint wavefield is, however, commonly done by brute force with ad hoc assumptions and/or relying on the existence of Green's functions, the representation theorem and/or the Born approximation. Using variational principles only, and without these mentioned assumptions and/or additional mathematical tools, we show that the time reversed adjoint wavefield should be defined as a premise that leads to the correct adjoint equations. This allows us to clarify mathematical inconsistencies found in previous seminal works when dealing with visco-elastic attenuation and/or odd-order derivative terms in the equation of motion. We then discuss some methodologies for the numerical implementation of the method in the time domain and to present a variational formulation for the construction of different misfit functions. We here define a new misfit travel-time function that allows us to find consensus for the long-standing debate on the zero sensitivity along the ray path that cross-correlation travel-time measurements show. In fact, we prove that the zero sensitivity along the ray-path appears as a consequence of the assumption on the similarity between data and synthetics required to perform cross-correlation travel-time measurements. When no assumption between data and synthetics is preconceived, travel-time Frechet kernels show an extremum along the ray path as one intuitively would expect.