MUSIC and variations

In the present application, ${\bf r}$ is a vector ranging over all or some discrete subset of the positions in the model space (usually a set of grid points). Then, there are several functionals we could plot in order to produce an ``image'' of the scatterers. The most common choice is the MUSIC classification functional

^2({V_n},H_r) = 11-_n=1^N^2(V_n,H_r).   Another closely related possibility that has similar characteristics (but does not require normalization of ${\bf H}_r$ in some implementations) is

^2({V_n},H_r) = _n=1^N^2(V_n,H_r) 1-_n=1^N^2(V_n,H_r).   The interpretation of these functionals as cosecants and cotangents in the subspaces determined by the eigenvectors should now be clear. By plotting these functionals, we find that the targets are located at those points where the denominators approach zero, and therefore in locations where the trial vector in entirely in the range of the scattering operator ${\bf K}$, or equivalently in the range of ${\bf T}$.

Now we can ask the question, how do we make use of these ideas if the data available to us are limited? In particular, it might happen that some of the nonzero eigenvalues are quite small compared to the others, and we do not know whether to include the corresponding eigenvectors in the set $\{{\bf V}_n\}$ or not. In this case, we can use a variation on the MUSIC scheme by only considering a subset of the eigenvectors, say $n = 1,\ldots,N' \le N$. In this case, either of the two schemes just described is easily modified by restricting the sums to

^2({V_n}',H_r) = 11-_n=1^N'^2(V_n,H_r),   and

^2({V_n}',H_r) = _n=1^N'^2(V_n,H_r) 1-_n=1^N'^2(V_n,H_r).   This approach can then be used to test whether certain eigenvectors are really in the range or not by replotting these functions for different values of N'. The scheme just described could also be used to do crude imaging if only a single eigenvector is known, as might happen if we have used time-reversal processing in the time domain and had found only the first eigenvector. When viewing eigenvectors as measurements, we see that using fewer eigenvectors will result in poorer resolution, as less information is then available to constrain the images.