Statistics of Echoes From Mixed Assemblages of Scatterers With Different Scattering Amplitudes and Numerical Densities

In this paper, an exact solution is derived, evaluated, and numerically validated for describing the statistics of echoes from mixed assemblages of scatterers. Here, a “mixed assemblage” involves the geometry in which there is more than one type of scatterer spatially interspersed and uniformly distributed within the analysis window, which is much larger than the resolution cell of the system (i.e., there are many independent samples per window). The scatterers are generally not resolvable and the signals are narrowband. The scattering geometry is in the backscattering direction in the direct-path case in which there are no interfering echoes from neighboring boundaries. The probability density functions (pdfs) of echo envelopes in such cases can be highly non-Rayleigh and possess heavy tails, and the shape of the pdf curves contains information for characterizing and discriminating the composition of mixed assemblages. The general formulation is based on characteristic functions (CFs; hereafter referred to as the CF-based mixed assemblage pdf) and incorporates effects due to the scatterers being randomly located in the beam. Comparisons are made between the performance of the CF-based mixed assemblage pdf and the commonly used mixture model for simulated cases involving two different types of scatterers, arranged either interspersed or segregated in the analysis window. Both models can be made to fit the shape of the echo pdf of simulated data in some conditions. However, mismatch between model assumptions and the actual physical scattering processes can lead to order of magnitude errors in the inferred numerical density and backscattering amplitude of each type of scatterers.