Optimal inference of the scattering cross-section through the phase decoherence

Abstract

From a random walk model introduced by Jakeman, Field and Tough derived the stochastic dynamics accounting for the scattering of a wavelike field from a random medium, and showed how the scattering cross-section was observable through the fluctuations of the scattered field phase. In the context of K-scattering, we pursue this strategy by deriving an explicit analytical expression of the deviation between the exact (underlying) and the inferred (observed) crosssection. We then deduce a condition to optimize the inference. Simulation results assert the viability of our theoretical formulae. Their consequences are discussed. IntroductionStochastic models, originating from electromagnetic scattering, but appropriate for more general situations (e.g., spin dynamics in nuclear magnetic resonance (NMR)) have recently encountered growing interest. In the same vein, this paper derives a new result, closely related to the previous contribution of [1], concerning the inference of the cross-section from the received amplitude. In the context of Jakeman's random walk model with step number fluctuations (cf. [2]) for a component phase diffusion model, it was shown in [1] that the scattering cross-section could be obtained from the (smoothing over a sample window of the) intensity-weighted instantaneous (squared) fluctuations of the phase. This result extended for arbitrary populations the anterior theoretical model of [3], whose pertinence had been established in [4] for anomaly detection. However, the closeness (in terms of the correlation coefficient) between the hidden population (exact cross-section) and the population obtained through the highly volatile phase decoherence (inferred cross-section) was heavily influenced by the smoothing process, i.e. over how many pulses the phase decoherence was averaged. Here lies the pertinency of our new result. Having recalled the stochastic model described in these previous papers and its experimental implications, we derive an expression for the error between the hidden and the estimated cross-sections. The concept of optimal smoothing, defined as the lower bound of this smoothing error, is then introduced. Moreover, we provide analytical formulae for the optimal window length (a value to be used for experimental situation) and for the corresponding error. Emphasis is placed on K-scattering and the extension to more general populations is discussed. The adequacy of this development is illustrated by simulation data, which compares the analytical error (derived in this

Geometrical implications of the compound representation for weakly scattered amplitudes

“…Equation (45) suggests that the discrepancy between the exact and inferred cross-sections is caused either by a sensitivity to the sampling process or by the loss of information arising whilst averaging the left-hand side of (44). This situation was investigated in detail in [14,19] for a strongly scattered amplitude, where it was shown that the mean square error (MSE) between the exact and inferred cross-sections reads…”

“…[13,14]). This paper aims to extend this procedure, established for a strongly scattered amplitude, to the weak scattering case.…”

“…(Color online) Experimental (dots) and analytical (46) (solid line) MSE between the exact cross-section and the cross-section inferred through the phase fluctuations (45) for a generalized weakly scattered amplitude with an underlying inverse Gamma texture whose amplitude random walk structure conforms to(14). (For parameter values ¼ 20, A ¼ 10 À3 , B ¼ 10 À2 .…”

Optimal inference of the inverse Gamma texture for a compound-Gaussian clutter

“…However, due to the highly volatile phase variations, this inference process was heavily influenced by the smoothing process (i.e., over how many pulses the phase decoherence was averaged). In the K-distributed case, the error arising in this procedure was analytically studied and a condition, on the smoothing window length, was derived to minimize it as described in [6]. Whilst focusing on an inverse Gamma texture, which is also of interest in radar applications, this paper aims to derive a corresponding analytical expression for the error arising during this inference process and optimize the smoothing (thus extending the strategy followed in [6]).…”

Discrete Models for Scattering Populations

“…Besides the state parameters, knowledge of the characteristic frequency constant A is also required to fully determine the population model. This quantity may also be extracted from the observed time series of the intensity (as discussed in [6]). As a result, the discrete population model driving the speckle can be entirely inferred from the intensity time series alone.…”