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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.