On the relation between time-reversed acoustics and Green's function retrieval in space-variant and in time-variant materials

TL;DR

This paper generalizes the established link between time-reversed acoustics and Green's function retrieval from classical inhomogeneous, time-invariant media to the realm of homogeneous, time-variant materials. By identifying a formal space-time analogy in the wave equations, it derives homogeneous Green's function representations for both material classes and develops a propagator-matrix framework for time-variant media. It shows how time-reversed experiments and crosscorrelation-based retrievals correspond to distinct, but related, operations in time-variant contexts, including caustic-free, non-singular homogeneous Green's functions and their causality-constrained variants. The work highlights practical considerations, such as the need for two-source types in time-variant retrieval and the artefacts that can arise when applying ambient-noise methods, and it clarifies the deep connections between these inverse-scattering tools across space- and time-variant regimes.

Abstract

The methods of time-reversed acoustics and Green's function retrieval are traditionally deployed for classical inhomogeneous, time-invariant materials. The mutual relation between these methods is well-established. Recently, similar methods have been proposed for homogeneous, time-variant materials. Here we investigate their mutual relation and their relation with the corresponding methods in classical materials. For this analysis we make use of the fact that the wave equations for both types of material are similar, with the roles of time and space interchanged. However, the principle of causality holds for both types of material, hence, here the roles of time and space are not interchanged. We find that: (1) whereas classical time-reversed acoustics involves emission of a time-reversed single-component wave field from a (ideally closed) boundary into the inhomogeneous material, its idealized counterpart involves emission of a sign-reversed two-component wave field, recorded in a time-reversed material, from a single time instant into the actual time-variant material; (2) whereas classical Green's function retrieval involves temporal crosscorrelation of wave fields at two space locations in response to single-component sources on a (ideally closed) boundary, its counterpart involves spatial crosscorrelation of wave fields at two time instants in response to two-component sources at a single time instant.

Figures (7)

• Figure 1: Principle of time-reversed acoustics in an inhomogeneous, time-invariant material. (a) The response to a source at ${\bf x}_A$ and $t=0$ (indicated by the red dot), recorded by receivers at ${\bf x}'$. (b) Time-reversed response, fed to sources at ${\bf x}'$. The wave field emitted by these sources focuses at $t=0$ at the original source position ${\bf x}_A$. (c) After having focused, the wave field continues its propagation. The focused field at $t=0$ is a virtual source (indicated by the pink dot) for the field for $t>0$.
• Figure 2: Principle of Green's function retrieval in an inhomogeneous, time-invariant material. (a) Responses to impulsive sources at ${\bf x}'$ and $t=0$, recorded by receivers at ${\bf x}_A$ and ${\bf x}$. (b) Response to a virtual source at ${\bf x}_A$ (indicated by the pink dot), recorded by a receiver at ${\bf x}$, obtained from the temporal crosscorrelation of the responses in (a), integrated over all sources at ${\bf x}'$ on ${\cal S}$.
• Figure 3: 2D Green's function ${\cal G}_t({\bf x},{\bf 0},t,t'=0)$ (convolved with a spatial wavelet) in a piecewise constant, time-variant material. The red dot indicates the source at ${\bf x}={\bf 0}$ and $t'=0$. The dashed blue planes indicate the time boundaries.
• Figure 4: Snapshots of the Green's function of Figure \ref{['Figure3']} for constant $t$ as a function of ${\bf x}=(x,z)$. Movie available at https://www.keeswapenaar.nl/TimeMaterial/Green5.mp4
• Figure 5: Cross-section of the Green's function of Figure \ref{['Figure3']} for $z=0$ as a function of $(x,t)$. The red dot indicates the source at ${\bf x}={\bf 0}$ and $t'=0$. The dashed blue lines indicate the time boundaries.