Comparison of the FWI-Adjoint and Time Reversal Methods for the Identification of Elastic Scatterers

Abstract

The paper falls into the category of computational methods for inverse scattering techniques for the identification of scatterers. We consider a linear elastodynamic problem and compare two popular methods for identifying a scatterer in the domain. Finite elements are employed with each of the two methods for spatial discretization. One method considered is Full Waveform Inversion using a gradient-based optimization and the adjoint method. In the adjoint procedure for calculating the gradient, we use the variant of discretizing the unknown parameters from the outset while all other variables remain continuous. Gradient optimization is performed in the examples using a quasi-Newton method. The other method compared is the computational Time Reversal technique, which is used in combination with an augmentation procedure to enhance performance. advantages and limitations of the two methods are outlined, and their performance is compared through an example from geophysics.

Single‐field identification of inclusions and cavities in an elastic medium

“…In References 9,30 it is suggested to split the cost function into two weighted parts. The first, denoted $$$ {D}_{shp} $$$, is the MHD of the obstacles when the centroids are rigidly translated to the same point (while the obstacles keep their orientation), and the result is normalized by the the characteristic length of the domain $$$ L $$$.…”

“…The total distance measure is thus $$$$ D=\xi {D}_{shp}+\left(1-\xi \right){D}_{cen}, $$$$ where $$$ \xi $$$ is the weighting factor. In References 9,30 it was shown that the value $$$ \xi =0.75 $$$ is a reasonable choice. With this value, good identification is obtained when $$$ D $$$ is in the range 0.009–0.036, and excellent identification is obtained for $$$ D\le 0.009 $$$.…”

“…Equivalently, use (11); take c and 𝜌 to be two independent unknowns. b Use (9); fix 𝜌, and take 𝜅 to be a single unknown variable. c Use (9); fix 𝜅, and take 𝜌 to be a single unknown variable.…”

“…b Use (9); fix 𝜌, and take 𝜅 to be a single unknown variable. c Use (9); fix 𝜅, and take 𝜌 to be a single unknown variable. d Use (9); set a linear relation between 𝜅 and 𝜌, that is, 𝜌 = 𝛼 2 𝜅 where 𝛼 is a known constant, and take 𝜅 (or 𝜌) to be a single unknown variable.…”

“…Four well‐known classes of computational methods for scatterer identification are linear sampling method (LSM), 4 arrival time imaging (ATI), 5 also called Kirchhoff migration, time reversal (TR) 6 and full waveform inversion (FWI) 7,8 . Rabinovich et al 9 compare between the TR and FWI‐adjoint methods through a sequence of numerical experiments and point to the relative advantages of each. The present work is based on the FWI‐adjoint technique, as will be described below.…”