Acoustic beam scattering from submerged bulk and layered elastic structures is of interest for applications ranging from determining material properties to locating and identifying interior defects. Nonspecular reflection, which occurs when the incident beam is phase matched to leaky waves (LW) supported by the structure, constitutes an effective sensor for such applications. Forward frequency domain modeling based on a robust asymptotic hybrid beam-LW algorithm [Zeroug and Felsen, 1992] has shown that the nonspecular data, which is established by interference between specular reflection and LWs, depends strongly on the collimation of the incident beam, the number and leakage strengths of the excited LWs, and the curvature of the insonified structure. The present contribution addresses the inverse problem of extracting this phenomenology via wave-oriented processing, which is implemented by subjecting the scattered field data to a Gaussian-windowed Fourier transform (GWT) along spatial tracks parallel to the fluid-structure interface. For the fluid-immersed elastic configurations of: 1) solid half-spaces, 2) plates, 3) solid cylinders, and 4) cylindrical shells, the GWT-generated local wavenumber phase-space distributions footprint the correct wave physics, but with resolutions that are limited by the configuration-spectrum tradeoff. Examples demonstrate how the resolution is influenced by the GWT window size. The paper also includes preliminary results on application of a Prony superresolution algorithm for extraction of LW phase velocities and leakage rates.