New results in applied scattering theory: the physical-statistics approach, including strong multiple scatter versus classical statistical-physical methods* and the Born and Rytov approximations versus exact strong scatter probability distributions

Abstract

In order to carry out effective signal processing for signal detection and estimation in scatter-dominated environments, it is necessary to obtain the needed probability distributions and probability densities (PDFs) of the received scatter. In general, the received scatter is non-Gaussian and often strongly so. It is also dominated by multiple-scatter contributions. This is particularly the case for (radar) scatter off random interfaces at small angles, e.g.ocean wave surfaces and terrain, as well as acoustically (sonar) off the ocean surface and bottom. For ‘classical’ theory, based on statistical-physical (S-P) models and methods, the analytical construction of the required PDFs is beyond the general reach of S-P theory. The present paper reviews, amplifies and extends with new material the author's recently developed physical-statistical (P-S) alternative equivalent to classical (S-P) formulations. Here the fundamental innovation, starting with the basic Langevin equation of propagation in operational form, is to replace the explicit physical model, including boundary and initial conditions, with a purely statistical model based on a counting functional representation of the scattering process. A decomposition principle (DP) establishes the independence of different (k≥1) orders of multiple scatter. This in turn leads to the application of Poisson statistics, from which the characteristic functions of each scatter order and their totality can then be constructed. The appropriate Fourier and Hilbert transforms next provide the desired (first-order) PDFs and exceedance probabilities (EPs). Characteristic two-scale environments are next considered, in which the scattering ensemble is accordingly modulated slowly vis-á-vis the more rapidly decorrelating local scatter. For this it is shown that the modulation effect is physically well modelled by a Γ-PDF. The result is the well known Jakeman K-distribution. In addition, when there are a few ‘large’ scatterers (e.g.breaking waves, bubble patches, irregular terrain, etc), the author's canonical classA non-Gaussian noise model is indicated and results in the new KA-PDF. The latter is also a common phenomenon, as shown by sample data from radar and sonar experiments, and is well replicated by theory. Both ‘single-look’ and ‘multiple-look’ results are developed for the first-order PDFs and exceedance probabilities of the normalized envelope sums ϵL(=∑lϵl) under I.i.d.conditions. A number of equivalent analytic methods for evaluating their integral forms are also developed for the more complex cases including coherent components (signals) and accompanying ambient noise. Finally, using the classical operational form of the Langevin equation for the scattered field, one can readily show that the (nonlinear) Rytov approximation (∼ exp ψ1) contains all orders (k≥1) of scatter versus the weak-scatter Born approximation, which is linear and limited to single scatter (k=1). Furthermore, it is seen that the Rytov approximation is a quasi-weak approximation, since its magnitude is limited to |ψ1|≪2, but is still better than the Born approach, as expected. Under these conditions the Rytov approximation supports a log-normal (first-order) PDF. The paper concludes with a concise comparison of the various capabilities of the P-S and classical S-P approaches to scattering phenomena, along with the scope of the resulting PDFs in describing clutter and reverberation. *Aportion of thisworkwas presented in Session 5P7, Tuesday (PM), July 11, 2000, at PIERS 2000, MIT, Cambridge, MA 02138, USA.