Stochastic Transfer Equation Research Articles

A probabilistic study of the influence of parameter uncertainty on solutions of the radiative transfer equation

The influence of uncertainty in the absorption and scattering coefficients on the solution and associated parameters of the radiative transfer equation is studied using polynomial chaos theory. The uncertainty is defined by means of uniform and log-uniform probability distributions. By expanding the radiation intensity in a series of polynomial chaos functions we may reduce the stochastic transfer equation to a set of coupled deterministic equations, analogous to those that arise in multigroup neutron transport theory, with the effective multigroup transfer scattering coefficients containing information about the uncertainty. This procedure enables existing transport theory computer codes to be used, with little modification, to solve the problem. Applications are made to a transmission problem and a constant source problem in a slab. In addition, we also study the rod model for which exact analytical solutions are readily available. In all cases, numerical results in the form of mean, variance and sensitivity are given that illustrate how absorption and scattering coefficient uncertainty influences the solution of the radiative transfer equation.

The meso-structured magnetic atmosphere

We present a general radiative transfer model which allows the Zeeman diagnostics of complex and unresolved solar magnetic fields. Present modeling techniques still rely to a large extent on a-priori assumptions about the geometry of the underlying magnetic field. In an effort to obtain a more flexible and unbiased approach we pursue a rigorous statistical description of the underlying atmosphere. Based on a Markov random field model the atmospheric structures are characterized in terms of probability densities and spatial correlations. This approach allows us to derive a stochastic transport equation for polarized light valid in a regime with an arbitrary fluctuating magnetic field on finite scales. One of the key ingredients of the derived stochastic transfer equation is the correlation length which provides an additional degree of freedom to the transport equation and can be used as a diagnostic parameter to estimate the characteristic length scale of the underlying magnetic field. It is shown that the stochastic transfer equation represents a natural extension of the (polarized) line formation under the micro- and macroturbulent assumption and contains both approaches as limiting cases. In particular, we show how in an inhomogeneous atmosphere asymmetric Stokes profiles develop and that the correlation length directly controls the degree of asymmetry and net circular polarization (NCP). In a number of simple numerical model calculations we demonstrate the importance of a finite correlation length for the polarized line formation and its impact on the resulting Stokes line profiles.

Stochastic radiative transfer and causality condition

We consider the stochastic radiative transfer problem in random media with scattering by using a causality condition. Three classes of models are proposed to describe this problem. In low order models we assume that the solutions of the stochastic transfer equation satisfy the causality condition themselves. Only the solutions of the closure equations satisfy this condition in higher order models. In the highest order models both the stochastic transfer equation and the closure equations (or only the closure equations) have the form of the series satisfying the causality condition. Our results describe nonstationary scattering and apply to any geometry of medium.

Statistical Description of Radiation Transfer in Clouds

Abstract The statistical characteristics of simulated cloud fields constructed based on Poisson point fluxes are studied. The input parameters of mathematical models of cloudiness include the cloud fraction and the mean horizontal size of clouds experimentally determined; the closure problem is solved. The methods for calculating the statistical characteristics of shortwave radiation are developed based on the stochastic transfer equation. Estimates of precision of equations for mean radiance are derived, and comparison between experimental and theoretical statistical characteristics of cloud fields and radiation is made.

Turbulence influenced line formation in stellar atmospheres

AbstractThe influence of stochastic velocity fields with finite correlation lengths on the formation of spectral lines is taken into account without restrictions to specific velocity models. To construct a perturbation theory treating the influence of stochastic motion on the radiative transfer we start with the stochastic transfer equation for plane‐parallel atmospheres and expand it to an infinite hierarchical system. An appropriate cut‐off of the infinite system admits to achieve exactly the micro‐ and macroturbulent limit at every level of approach. The formalism is derived for n‐point correlations of the absorption coefficient and specialised for 2‐point correlations.