Time-reversal acoustics can be understood in a straightforward way from the scattering theory presented so far. First, define a complex vector relating scattering at point with all M of the acoustic sensors in a sensor array located at positions . 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 , indexed by a general location in the model space , such that
H_r^T = G_0(r,x_1), & ..., & G_0(r,x_M). We will term a ``trial vector'' at , and 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 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 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 is complex and symmetric.