  A fundamental challenge in acoustic data processing is to separate a measured
time series into relevant phenomenological components. A given measurement is
typically assumed to be an additive mixture of myriad signals plus noise whose
separation forms an ill-posed inverse problem. In the setting of sensing
elastic objects using active sonar, we wish to separate the early-time returns
(e.g., returns from the object's exterior geometry) from late-time returns
caused by elastic or compressional wave coupling.
  Under the framework of Morphological Component Analysis (MCA), we compare two
separation models using the short-duration and long-duration responses as a
proxy for early-time and late-time returns. Results are computed for Stanton's
elastic cylinder model as well as on experimental data taken from an in-Air
circular Synthetic Aperture Sonar (AirSAS) system, whose separated time series
are formed into imagery. We find that MCA can be used to separate early and
late-time responses in both cases without the use of time-gating. The
separation process is demonstrated to be robust to noise and compatible with
AirSAS image reconstruction. The best separation results are obtained with a
flexible, but computationally intensive, frame based signal model, while a
faster Fourier Transform based method is shown to have competitive performance.
