Spectral-Element and Adjoint Methods in Seismology

TLDR

The paper introduces the spectral‑element method for seismology, discusses using discrepancies between synthetic and observed seismograms to refine Earth models, and highlights adjoint methods as a practical approach to the inverse problem. It reviews seismic wave equations and explains how the spectral‑element method numerically solves the forward problem, while also outlining adjoint methods for tackling the inverse problem. Synthetic seismograms generated with the spectral‑element method are shown to match Global Seismographic Network observations.

Abstract

We provide an introduction to the use of the spectral-element method (SEM) in seismology. Following a brief review of the basic equations that govern seismic wave propagation, we discuss in some detail how these equations may be solved numerically based upon the SEM to address the forward problem in seismology. Examples of synthetic seismograms calculated based upon the SEM are compared to data recorded by the Global Seismographic Network. Finally, we discuss the challenge of using the remaining differences between the data and the synthetic seismograms to constrain better Earth models and source descriptions. This leads naturally to adjoint methods, which provide a practical approach to this formidable computational challenge and enables seismologists to tackle the inverse problem.