Edge Illumination and Imaging of Extended Reflectors

We use the singular value decomposition of the array response matrix, frequency by frequency, to image selectively the edges of extended reflectors in a homogeneous medium. We show with numerical simulations in an ultrasound regime, and analytically in the Fraunhofer diffraction regime, that information about the edges is contained in the singular vectors for singular values that are intermediate between the large ones and zero. These transition singular vectors beamform selectively from the array onto the edges of the reflector cross-section facing the array, so that these edges are enhanced in imaging with travel-time migration. Moreover, the illumination with the transition singular vectors is done from the sources at the edges of the array. The theoretical analysis in the Fraunhofer regime shows that the singular values transition to zero at the index $N^\star(\om) = |{\cal A}||{\cal B}|/(\lambda L)^2 $. Here $|{\cal A}|$ and $|{\cal B}|$ are the areas of the array and the reflector cross-section, respectively, $\omega$ is the frequency, $\lambda$ is the wavelength, and $L$ is the range. Since $(\lambda L)^2/|{\cal A}|$ is the area of the focal spot size at range $L$, we see that $N^\star(\om)$ is the number of focal spots contained in the reflector cross-section. The ultrasound simulations are in an extended Fraunhofer regime. The simulation results are, however, qualitatively similar to those obtained theoretically in the Fraunhofer regime. The numerical simulations indicate, in addition, that the subspaces spanned by the transition singular vectors are robust with respect to additive noise when the array has a large number of elements.