On the multi-frequency obstacle reconstruction via the linear sampling method

This paper investigates the possibility of multi-frequency reconstruction of sound-soft and penetrable obstacles via the linear sampling method involving either far-field or near-field observations of the scattered field. On establishing a suitable approximate solution to the linear sampling equation and making an assumption of continuous frequency sweep, two possible choices for a cumulative multi-frequency indicator function of the scatterer's support are proposed. The first alternative, termed the 'serial' indicator, is taken as a natural extension of its monochromatic companion in the sense that its computation entails space-frequency (as opposed to space) L²-norm of a solution to the linear sampling equation. Under a set of assumptions that include experimental observations down to zero frequency and compact frequency support of the wavelet used to illuminate the obstacle, this indicator function is further related to its time-domain counterpart. As a second possibility, the so-called parallel indicator is alternatively proposed as an L²-norm, in the frequency domain, of the monochromatic indicator function. On the basis of a perturbation analysis which demonstrates that the monochromatic solution of the linear sampling equation behaves as O(|k²-k² _*|^-m), m ≥ 1 in the neighborhood of an isolated eigenvalue, k^{2 *} , of the associated interior (Dirichlet or transmission) problem, it is found that the 'serial' indicator is unable to distinguish the interior from the exterior of a scatterer in situations when the prescribed frequency band traverses at least one such eigenvalue. In contrast the 'parallel' indicator is, due to its particular structure, shown to be insensitive to the presence of pertinent interior eigenvalues (unknown beforehand and typically belonging to a countable set), and thus to be robust in a generic scattering configuration. A set of numerical results, including both 'fine' and 'coarse' frequency sampling, is included to illustrate the performance of the competing (multi-frequency) indicator functions, demonstrating behavior that is consistent with the theoretical results.