The Monte Carlo atmospheric radiative transfer model McArtim: Introduction and validation of Jacobians and 3D features

A new Monte Carlo atmospheric radiative transfer model is presented which is designed to support the interpretation of UV/vis/near-IR spectroscopic measurements of scattered Sun light in the atmosphere. The integro differential equation describing the underlying transport process and its formal solution are discussed. A stochastic approach to solve the differential equation, the Monte Carlo method, is deduced and its application to the formal solution is demonstrated. It is shown how model photon trajectories of the resulting ray tracing algorithm are used to estimate functionals of the radiation field such as radiances, actinic fluxes and light path integrals. In addition, Jacobians of the former quantities with respect to optical parameters of the atmosphere are analyzed. Model output quantities are validated against measurements, by self-consistency tests and through inter comparisons with other radiative transfer models.     

An inverse source problem in radiative transfer

The spherical-harmonics method is used to develop a solution to an inverse source problem in radiative transfer. It is assumed that, with the exception of the inhomogeneous source term, all aspects of the radiation-transport problem are known, and we seek to determine the inhomogeneous source term from specified angular distributions of radiation exiting the two surfaces of a homogeneous plane-parallel medium. Anisotropic scattering is included in the monochromatic radiative-transfer model and general reflecting boundary conditions are considered.     

ARTS, the atmospheric radiative transfer simulator

ARTS is a modular program that simulates atmospheric radiative transfer. The paper describes ARTS version 1.0, which is applicable in the absence of scattering. An overview over all major parts of the model is given: calculation of absorption coefficients, the radiative transfer itself, and the calculation of Jacobians. ARTS can be freely used under a GNU general public license.Unique features of the program are its scalability and modularity, the ability to work with different sources of spectroscopic parameters, the availability of several self-consistent water continuum and line absorption models, and the analytical calculation of Jacobians.     

A linearized two-stream radiative transfer code for fast approximation of multiple-scatter fields

Performance is an issue for radiative transfer simulations in hyper-spectral remote sensing backscatter retrieval algorithms. 2-Stream models are often used to speed up flux and radiance calculations. Here we present a linearized 2-stream multiple-scatter code, with the ability to generate analytic weighting functions with respect to any atmospheric or surface property. We examine 2-stream accuracy for the satellite intensity diffuse field, and the corresponding Jacobians for total ozone column and surface albedo, for an application in the ozone UV Huggins bands.     

An efficient and robust reconstruction method for optical tomography with the time-domain radiative transfer equation

An efficient and robust method based on the complex-variable-differentiation method (CVDM) is proposed to reconstruct the distribution of optical parameters in two-dimensional participating media. An upwind-difference discrete-ordinate formulation of the time-domain radiative transfer equation is well established and used as forward model. The regularization term using generalized Gaussian Markov random field model is added in the objective function to overcome the ill-posed nature of the radiative inverse problem. The multi-start conjugate gradient method was utilized to accelerate the convergence speed of the inverse procedure. To obtain an accurate result and avoid the cumbersome formula of adjoint differentiation model, the CVDM was employed to calculate the gradient of objective function with respect to the optical parameters. All the simulation results show that the CVDM is efficient and robust for the reconstruction of optical parameters.     

An Hermitian method for the solution of radiative transfer problems

A new method for the solution of the radiative transfer equation is developed. The resulting system, like the ordinary difference equations is tridiagonal in form and easy to use. Numerical examples are given which show the superiority of the new technique to previous differential and integral equation methods.     

An efficient diffusion approximation for 3D radiative transfer parameterization: application to cloudy atmospheres

The three-dimensional (3D) diffusion radiative transfer equation, which utilizes a four-term spherical harmonics expansion for the scattering phase function and intensity, has been efficiently solved by using the full multigrid numerical method. This approach can simulate the transfer of solar and thermal infrared radiation in inhomogeneous cloudy conditions with different boundary conditions and sharp boundary discontinuity. The correlated k-distribution method is used in this model for incorporation of the gaseous absorption in multiple-scattering atmospheres for the calculation of broadband fluxes and heating rates in the solar and infrared spectra. Comparison of the results computed from this approach with those computed from plane-parallel and 3D Monte Carlo models shows excellent agreement. This 3D radiative transfer approach is well suited for radiation parameterization involving 3D and inhomogeneous clouds in climate models.     

Regularization of inverse boundary design radiative heat transfer problems

Inverse boundary design heat transfer problems, including only radiation, are considered. Variational methods of regularization are used for solving these (mathematically ill-posed) problems, they give possibility to take into account various a priori information about the desired solution. For minimizing the discrepancy functional we use the adjoint problem method; however, we use it not only in iterative regularization, but in Tikhonov and parametric ones as well. We use all the available a priori information about desired solutions in all the techniques; this allows to obtain physically feasible solutions in all the cases.     

Numerical test of an inverse polarized radiative transfer algorithm

A procedure is tested with which to determine the single-scattering albedo from polarization measurements of the angle-dependent intensity at two locations within, or on the boundaries of, a homogeneous finite or infinite atmosphere that scatters radiation according to the Rayleigh law with true absorption.     

An efficient numerical method for time-dependent NLTE radiation transfer

We present a numerical method for performing coupled time-dependent radiation transfer and atomic rate equation calculations. The method incorporates the desirable features of time-dependent, non-local thermodynamic equilibrium (NLTE) radiation transfer calculations. It has the advantage that it allows existing time-dependent calculations to be adapted to model time dependent effects. The approximations which are used are discussed.Calculations have been performed for a highly ionised silicon plasma with a single optically thick line. The line is taken to have a top-hat profile which is independent of position. Although the model is simple, it serves to illustrate some important features of time-dependent calculations.     

Variational method for solving radiative transfer problems

The variational principle is used to solve two problems of radiative transfer. The first one is the temperature distribution and radiative heat flux for a plane layer of ceramic material. While the second is the calculation of the integral blankness degree of a sphere filled with dust of an arc steel-melting furnace. Numerical results obtained shows good agreement with the published data.     

On the numerical characteristics of an inverse solution for three-term radiative transfer

Certain numerical characteristics of an inverse formulation for three-term scattering radiative transfer are investigated. Specifically, approximate solutions to the direct problem are constructed by the F_N and Monte Carlo methods, allowing approximation of the various surface angular moments and related quantities needed for the inverse calculation. Several numerical schemes are employed in order of demonstrate the computational characteristics for some specific phase functions. The numerical results indicate that the single-scatter albedo can be calculated fairly consistently and accurately, but higher order coefficients of the scattering law are more difficult to obtain by this method.     

An inverse radiative transfer problem of simultaneously estimating profiles of temperature and radiative parameters from boundary intensity and temperature measurements

A parallel-plane space filled with absorbing, emitting, isotropically scattering, gray medium is studied in this paper. The boundary intensity and boundary temperature profiles are calculated for the inverse analysis. For the simultaneous estimation of temperature, absorption and scattering coefficient profiles in the medium, the sum of residuals of boundary intensity and temperature after being weighted by a balance factor is minimized through using a Newton-type iteration algorithm and the least-squares method. To avoid over-updating for the parameters, the relative updating magnitude during the iteration process is constrained not to be >0.5. It is shown that the boundary intensity measurement alone is not enough to estimate simultaneously the temperature (source) and the radiative properties (both absorption and scattering coefficients) when the measurement data contain sensitive random errors. The boundary temperature measurement can serve as a necessary supplementation to the boundary intensity to make this kind of inverse radiative transfer problem resolvable. It was shown that a compensation relationship between absorption and scattering coefficients makes it difficult to fix them accurately. Parabolic profiles for the three parameters are used to validate the estimation method. When the optical thickness approaches 4.0, the results for the radiative properties are not acceptable, although the result for temperature profile is reasonable. This means the method needs further improvements.     

Spectral element method for vector radiative transfer equation

A spectral element method (SEM) is developed to solve polarized radiative transfer in multidimensional participating medium. The angular discretization is based on the discrete-ordinates approach, and the spatial discretization is conducted by spectral element approach. Chebyshev polynomial is used to build basis function on each element. Four various test problems are taken as examples to verify the performance of the SEM. The effectiveness of the SEM is demonstrated. The h and the p convergence characteristics of the SEM are studied. The convergence rate of p-refinement follows the exponential decay trend and is superior to that of h-refinement. The accuracy and efficiency of the higher order approximation in the SEM is well demonstrated for the solution of the VRTE. The predicted angular distribution of brightness temperature and Stokes vector by the SEM agree very well with the benchmark solutions in references. Numerical results show that the SEM is accurate, flexible and effective to solve multidimensional polarized radiative transfer problems.     

Frequency integration for radiative transfer problems involving homogeneous non-gray gases: The inverse transmission function

It is shown that narrow-band frequency integration for radiative-transfer problems, involving homogeneous non-gray gases, can be accomplished by an integration with respect to the absorption coefficient. The kernel of the absorption-coefficient integral is a property of the gas and can be obtained directly from the absorption coefficient for a simple dependence of the absorption coefficient on frequency. It is shown that the kernel (called the ‘inverse transmission function’) can also be obtained as the inverse Laplace transform of the transmission function, without explicit knowledge of the absorption coefficient. The kernel has been obtained using the inverse transmission-function method for two statistical band models: the Goody model and the Malkmus model. The kernel for the Malkmus model offers a simple, accurate, straightforward method for narrow-band frequency integration.     

Scattering by random rough surfaces: Study of direct and inverse problem

In order to study the problems of scattering by rough metallic surfaces, we have used Maxwell’s equations in covariant form within the framework of a non-orthogonal coordinates system adapted to the geometry of the problem. Electromagnetic fields are written in Fourier’s integral form. The solution is found by using a perturbation method applied to the smooth surface problem; this is fully justified when the defects are of small magnitude.For the direct problem, the mean value of diffraction intensity is obtained for random rough surfaces of finite conductivity by computer simulation.In the case of the inverse problem, the reconstruction of the profile of the metal surface from values of the diffraction intensity, obtained by simulation, is found using an iterative algorithm.     

Calculation of radiative heat transfer in argon arc plasmas

Radiation emission and absorption in arc plasmas are important energy transfer processes. Exact calculations, though possible in principle, are usually impossible in practice because of the need to treat a large number of spectral lines and also the continuum radiation in the whole spectrum range. Recently, we have used an approximate method of partial characteristics to evaluate the radiation intensities, radiation fluxes and the divergence of radiation fluxes for SF₆ arc plasma with cylindrical symmetry. In this paper, we have extended our calculations toargon arc plasmas for the plasma pressures of 0.1, 0.5 and 1.0 MPa. We have calculated the coefficients of absorption for Ar plasmas at temperatures from 300 to 35 000 K, and have used these coefficients to calculate the partial characteristics. Both the continuum and the line spectra have been included in calculations. We have taken into account the radiative photo-recombination and bremsstrahlung for the continuous spectrum, and over 500 spectral lines for the discrete spectra.The method of partial characteristics has been applied to three-dimensional calculations of radiative heat transfer — i.e. radiation intensity, radiation flux and its divergence — in simplified temperature profiles. Conclusions have been made concerning validity and utilization of the method of partial characteristics in general gas dynamics problems.     

SASKTRAN: A spherical geometry radiative transfer code for efficient estimation of limb scattered sunlight

The inversion of satellite-based observations of limb scattered sunlight for the retrieval of constituent species requires an efficient and accurate modelling of the measurement. We present the development of the SASKTRAN radiative transfer model for the prediction of limb scatter measurements at optical wavelengths by method of successive orders along rays traced in a spherical atmosphere. The component of the signal due to the first two scattering events of the solar beam is accounted for directly along rays traced in the three-dimensional geometry. Simplifying assumptions in successive scattering orders provide computational optimizations without severely compromising the accuracy of the solution. SASKTRAN is designed for the analysis of measurements from the OSIRIS instrument and the implementation of the algorithm is efficient such that the code is suitable for the inversion of OSIRIS profiles on desktop computers. SASKTRAN total limb radiance profiles generally compare better with Monte-Carlo reference models over a large range of solar conditions than the approximate spherical and plane-parallel models typically used for inversions.     

Finite element method for the two-dimensional atmospheric radiative transfer

The finite element method is applied to the solution of the two-dimensional atmospheric radiative transfer. The analysis is mainly focussed on the derivation of the cell or element equation. The Galerkin method and several hybrid methods using the integral and finite difference form of the radiative transfer equation are employed to obtain the cell equation. The assembled system of equations relating the radiances at the lower and upper boundary of the domain is solved by a direct method.     

Extension of the discrete-ordinate algorithm and efficient radiative transfer calculation

As an accurate and efficient algorithm, the discrete-ordinate method (DOM) has been used to solve the radiative transfer problem of plane-parallel scattering atmosphere illuminated by a parallel beam, an idealized case of the sun, from above the atmosphere. In this paper, we extend this algorithm so that radiative problems of more general sources, such as parallel surface sources that illuminate with a parallel beam in any direction from any vertical position, and general surface sources that illuminate continuously in a hemisphere, can be solved. For a problem where intensity distributions are sought for a number of different sources within the same atmosphere-surface system, the intrinsic properties of DOM are used so that the time required for the solution for extra sources is reduced to a substantially small amount. In the case of parallel surface sources, numerical testing has shown that the amount can be reduced to as little as 15% of a full solution. Examples of applications are presented.