How time-discretization can break the asymptotics of inverse scattering

Abstract

Inverse scattering by the Marchenko or Gelfand–Levitan equations consists of two steps. The first step consists of solving an integral equation to retrieve the wavefield in the interior of a scattering medium from waves recorded outside of the scattering region. The second step consists of estimating the scattering potential from this wavefield. The estimation of the potential can be justified either by taking a high-frequency limit, or by evaluating the wavefield an infinitesimal time before or after the direct wave. Waveforms obtained in experiments are discretized in time with a sampling interval d t . We show an example that this time discretization precludes the extraction of the potential. The reason is that the frequencies of discretized waveforms are below the Nyquist frequency. This limitation precludes taking the high-frequency limit that is used for estimating the potential from the waveforms in the interior of the scattering region. We present an alternative method to estimate the potential from wavefields in the interior of the scattering region. This method can be used even when the waves are in the strong scattering regime.