We derive the stochastic dynamics of the complex valued amplitude resulting from coherent scattering from a random population of scatterers when this becomes asymptotically large. Considerations of a random walk model, introduced by Jakeman, are used to derive stochastic differential equations for the amplitude and corresponding intensity and phase stochastic processes. An analysis of the correlation structure in the fluctuations is provided and interpreted geometrically in terms of the gauge invariant properties of the field and the Markov property. A Fokker–Planck description for the evolution of the probability density is given and the equilibrium and detailed balance conditions shown to hold. Expressions for the intensity autocorrelation function and power spectral density are provided in closed form. The practical implications of the stochastic theory are discussed.

The dynamics of a spin system is usually calculated using the density matrix. However, the usual formulation in terms of the density matrix predicts that the signal will decay to zero, and does not address the issue of individual spin dynamics. Using stochastic calculus, we develop a dynamical theory of spin relaxation, the origins of which lie in the component spin fluctuations. This entails consideration of random pure states for individual protons, and how these pure states are correctly combined when the density matrix is formulated. Both the lattice and the spins are treated quantum mechanically. Such treatment incorporates both the processes of spin-spin and (finite temperature) spin-lattice relaxation. Our results reveal the intimate connections between spin noise and conventional spin relaxation.

Acoustic propagation through anisotropic internal wave fields: Transmission loss, cross-range coherence, and horizontal refraction

We present a classification of continuous-time diffusion processes for the local scattering cross-section and scattered intensity for an electromagnetic field scattering from a random medium, such that the asymptotic marginal distribution for the intensity is the K-distribution. These processes are represented as Ito stochastic differential equations, which enables identification of their stochastic volatilities with certain free functions that serve to calibrate the model. A study is made of the volatility behaviour of the electromagnetic intensity scattered from a random phase screen, which establishes the form of the volatility function to a high accuracy. This model is used to derive a stochastic differential equation for the complex-valued amplitude, which is then applied to radar scattering from the sea surface for the purpose of anomaly detection.

The estimation of hydrologic transit times in a catchment provides insights into the integrated effects of water storage, mixing dynamics, and runoff generation processes. There has been limited effort to estimate transit times in southern boreal Precambrian Shield landscapes, which are characteristically heterogeneous with surface cover including till, thin soils, bedrock outcrops, and depressional wetland features that play contrasting hydrologic roles. This study presents approximately 3.5 years of precipitation and streamflow water isotope data and estimates mean transit times (MTTs) and the young water fraction (p y ) across six small catchments in the Muskoka-Haliburton region of south-central Ontario. The main objectives were to define a typical range of MTTs for headwater catchments in this region and to identify landscape variables that best explain differences in MTTs/p y using airborne light detection and ranging and digital terrain analysis. Of the transit time distributions, the two parallel linear reservoir and gamma distributions best describe the hydrology of these catchments, particularly because of their ability to capture more extreme changes related to events such as snowmelt. The estimated MTTs, regardless of the modelling approach or distribution used, are positively associated with the percent wetland area and negatively with mean slope in the catchments. In this landscape, low-gradient features such as wetlands increase catchment scale water storage when antecedent conditions are dryer and decrease transit times when there is a moisture surplus, which plausibly explains the increases in MTTs and mean annual runoff from catchments with significant coverage of these landscape features.