Acoustic time-reversal data analysis

Time-reversal acoustics can be understood in a straightforward way from the scattering theory presented so far. First, define a complex vector ${\bf H}_n$ relating scattering at point ${\bf y}_n$ with all M of the acoustic sensors in a sensor array located at positions ${\bf x}_1,{\bf x}_2,\ldots,{\bf x}_M$. Then,

H_n^T = G_0(y_n,x_1), & ..., & G_0(y_n,x_M).   By analogy to (Hn), we define a general vector of the same form ${\bf H}_r$, indexed by a general location in the model space ${\bf r}$, such that

H_r^T = G_0(r,x_1), & ..., & G_0(r,x_M).   We will term ${\bf H}_r$ a ``trial vector'' at ${\bf r}$, and ${\bf H}_n$ one of the set of N ``solution vectors.''

[Note that, for inhomogeneous media, if we use G instead of G₀ in (Hn) and (Hr), then we have the method known as ``matched field processing.'' Similarly, if we replace G₀ by the appropriate fundamental solution for the dipole (instead of the monopole) term, then the analysis proceeds again essentially as follows, but the dipole modes can then be used for imaging.]

With these definitions, the fundamental solution in the Born approximation can be rewritten for $m,m' = 1,\ldots,M$ as

G(x_m,x_m') G_0(x_m,x_m') + K(x_m,x_m'),   where the ``response matrix'' (or transfer matrix)

K = _n=1^N q_nH_nH_n^T.   Elements of the matrix ${\bf K}$ are given by

K_m,m' = K(x_m,x_m') = _n=1^N q_n G_0(x_m,y_n)G_0(y_n,x_m')   Clearly, the response matrix ${\bf K}$ is complex and symmetric.