Exact Radiative Transfer Research Articles

Radiative transfer as a Bayesian linear regression problem

ABSTRACT Electromagnetic radiation plays a crucial role in various physical and chemical processes. Hence, almost all astrophysical simulations require some form of radiative transfer model. Despite many innovations in radiative transfer algorithms and their implementation, realistic radiative transfer models remain very computationally expensive, such that one often has to resort to approximate descriptions. The complexity of these models makes it difficult to assess the validity of any approximation and to quantify uncertainties on the model results. This impedes scientific rigour, in particular, when comparing models to observations, or when using their results as input for other models. We present a probabilistic numerical approach to address these issues by treating radiative transfer as a Bayesian linear regression problem. This allows us to model uncertainties on the input and output of the model with the variances of the associated probability distributions. Furthermore, this approach naturally allows us to create reduced-order radiative transfer models with a quantifiable accuracy. These are approximate solutions to exact radiative transfer models, in contrast to the exact solutions to approximate models that are often used. As a first demonstration, we derive a probabilistic version of the method of characteristics, a commonly-used technique to solve radiative transfer problems.

Multiphysics Simulation Combining Large-Eddy Simulation, Wall Heat Conduction and Radiative Energy Transfer to Predict Wall Temperature Induced by a Confined Premixed Swirling Flame

A multi-physics simulation combining large-eddy simulation, conjugate heat transfer and radiative heat transfer is used to predict the wall temperature field of a confined premixed swirling flame operating under atmospheric pressure. The combustion model accounts for the effect of enthalpy defect on the flame structure whose stabilization is here sensitive to the wall heat losses. The conjugate heat transfer is accounted for by solving the heat conduction within the combustor walls and with the Hybrid-Cell Neumann-Dirichlet coupling method, enabling to dynamically adapt the coupling period. The latter coupling procedure is enhanced to determine statistics (mean, RMS,. . .) in a permanent regime accurately and efficiently thanks to an acceleration technique which is derived and validated. The exact radiative heat transfer equation is solved with an advanced Monte Carlo method with a local control of the statistical error. The coupled simulation is carried out with or without accounting for radiation. Excellent results for the wall temperature are achieved by the fully coupled simulation which are then further analyzed in terms of radiative effects, global energy budget and fluctuations of wall heat flux and temperature.

A simple parameterization for the height of maximum ozone heating rate

It is well-known that the height of the maximum ozone heating rate is much higher than the height of the maximum ozone concentration in the stratosphere. However, it lacks an analytical expression to explain it. A simple theoretical model has been proposed to calculate the height of maximum ozone heating rate and further understand this phenomenon. Strong absorption of ozone causes the incoming solar flux to be largely attenuated before reaching the location of the maximum ozone concentration. By comparing with the exact radiative transfer calculations, the heights of the maximum ozone heating rate produced by the theoretical model are generally very close to the true values. When the cosine of solar zenith angle μ0=1.0, in US Standard atmosphere, the heights of the maximum ozone heating rate by the theoretical model are 41.4km in the band 0.204–0.233μm, 47.9km in the band 0.233–0.270μm, 44.5km in the band 0.270–0.286μm, 37.1km in the band 0.286–0.303μm, and 30.2km in the band 0.303–0.323μm, respectively. The location of the maximum ozone heating rate is sensitive to the solar spectral range. In band 1, the heights of the maximum ozone heating rate by the theoretical model are 52.3km for μ0=0.1, 47.1km for μ0=0.3, 44.6km for μ0=0.5, 43.1km for μ0=0.7, 41.9km for μ0=0.9, 41.4km for μ0=1.0 in US Standard atmosphere, respectively. This model also illustrates that the location of the maximum ozone heating rate is sensitive to the solar zenith angle.

HELIOS: AN OPEN-SOURCE, GPU-ACCELERATED RADIATIVE TRANSFER CODE FOR SELF-CONSISTENT EXOPLANETARY ATMOSPHERES

We present the open-source radiative transfer code named HELIOS, which is constructed for studying exoplanetary atmospheres. In its initial version, the model atmospheres of HELIOS are one-dimensional and plane-parallel, and the equation of radiative transfer is solved in the two-stream approximation with non-isotropic scattering. A small set of the main infrared absorbers is employed, computed with the opacity calculator HELIOS-K and combined using a correlated-$k$ approximation. The molecular abundances originate from validated analytical formulae for equilibrium chemistry. We compare HELIOS with the work of Miller-Ricci & Fortney using a model of GJ 1214b, and perform several tests, where we find: model atmospheres with single-temperature layers struggle to converge to radiative equilibrium; $k$-distribution tables constructed with $\gtrsim 0.01$ cm$^{-1}$ resolution in the opacity function ($ \lesssim 10^3$ points per wavenumber bin) may result in errors $\gtrsim 1$-10 % in the synthetic spectra; and a diffusivity factor of 2 approximates well the exact radiative transfer solution in the limit of pure absorption. We construct "null-hypothesis" models (chemical equilibrium, radiative equilibrium and solar element abundances) for 6 hot Jupiters. We find that the dayside emission spectra of HD 189733b and WASP-43b are consistent with the null hypothesis, while it consistently under-predicts the observed fluxes of WASP-8b, WASP-12b, WASP-14b and WASP-33b. We demonstrate that our results are somewhat insensitive to the choice of stellar models (blackbody, Kurucz or PHOENIX) and metallicity, but are strongly affected by higher carbon-to-oxygen ratios. The code is publicly available as part of the Exoclimes Simulation Platform (ESP; exoclime.net).

Estimation of daily photosynthetically active radiation (PAR) in presence of low to high aerosol loads: application to OLCI-like satellite data.

Estimation of daily photosynthetically active radiation (PAR) is of primary importance for monitoring the ocean primary production and the subsequent production of carbon by phytoplankton at global scale from remote sensing ocean color sensors. On the other hand, aerosol abundance and composition play a critical role in the modulation of PAR. In this study, an original algorithm, so-called OLCIPAR, is proposed for routinely determining the daily PAR from optical satellite sensors such as the OLCI sensor aboard Sentinel-3 (ESA). The OLCIPAR algorithm has been developed to overcome some of the limitations of the current existing methods. In particular, multiple scattering effects induced by the atmospheric layer are taken into account based on exact radiative transfer calculations. Another advantage of OLCIPAR method is to consider a great variety of aerosol models to better account for their optical variability as observed in real world conditions. The OLCIPAR algorithm was applied to the archive of MERIS data, whose sensor is similar to OLCI. The validation of the retrieved daily PAR was carried out based on comparison with the time series acquired by the BOUSSOLE oceanographic buoy moored in the Mediterranean Sea. Results show a regression slope of 1% and an accuracy within 10% which confirms the robustness of the algorithm. The comparison of OLCIPAR retrievals with the products routinely distributed by NASA shows that estimates of PAR differ by up to 20% in the subtropical Atlantic Ocean where important amounts of dust aerosols are present. The improvements brought by OLCIPAR method for deriving the daily PAR could thus permit to better assess the impact of aerosols on reduction of PAR with implications on the estimation of oceanic primary production.

Cyanide/isocyanide abundances in the interstellar medium - III. The excitation of Al and Mg compounds

Cyanide species are the most common metal-containing molecules in the circumstellar gas. The exact determination of their abundances requires accurate molecular data. For this purpose, collisional excitation rate coefficients are essential in order to simulate the excitation processes and hence to model the emission lines. In this paper, we use the aluminium cyanide (AlCN), aluminium isocyanide (AlNC), magnesium cyanide (MgCN) and magnesium isocyanide (MgNC) collisional data recently computed to simulate the excitation of these molecules in the circumstellar gas. We perform radiative transfer calculations for typical physical conditions encountered in the circumstellar gas. We obtain the brightness and excitation temperatures of selected lines frequently observed towards the circumstellar envelopes. In circumstellar gas, we have found that local thermodynamic equilibrium conditions are not fulfilled for these species and that radiative transfer calculations are needed in order to accurately determine their abundances. The calculations also show that the estimations of the cyanides/isocyanides abundance ratios deduced from line intensities ratios lead to large differences compared to exact radiative transfer calculations. Finally, the results confirm that AlCN and MgCN are significantly less abundant than AlNC and MgNC, respectively. This shows again the evidence of selective cyanide chemistry.

An efficient approach to estimate the transmittance and reflectance of a mixture of aerosol components

Atmospheric aerosols are formed by a mixture of different chemical components that changes rapidly with time and space. The refined characterization of this mixture is crucial to meet the accuracy requirements of satellite products derived from passive sensor data in the shortwave wavelengths. This article proposes an efficient analytical approach to estimate two key radiative terms in aerosol remote sensing: the transmittance and reflectance of an aerosol mixture. This study demonstrates that these terms can be approximated by a simple weighted average of the individual radiative counterparts related to each aerosol component. Weights are the optical depths resulting from each aerosol component separately. The proposed approach is very fast and is exact for the first order of scattering. Its accuracy is assessed against exact radiative transfer calculations for a broad range of aerosol scenarios. For typical aerosol conditions (optical depth lower than 1.0 and solar zenith angle lower than 70°), the average error of estimated transmittances is 0.6%. Reflectances are affected by a higher average error of 7.6% due to their higher sensitivity to multiple scattering orders. The proposed approach may advantageously replace the use of sophisticated radiative transfer codes at the cost of a slight accuracy decrease to better answer the needs of the near real time constraint required by the remote sensing community.

Radiative Heating of a Glass Plate

This paper aims to prove existence and uniqueness of a solution to the coupling of a nonlinear heat equation with nonlinear boundary conditions with the exact radiative transfer equation, assuming the absorption coefficient κ(λ) to be piecewise constant and null for small values of the wavelength λ as in the paper of N. Siedow, T. Grosan, D. Lochegnies, E. Romero, “Application of a New Method for Radiative Heat Tranfer to Flat Glass Tempering”, J. Am. Ceram. Soc., 88(8):2181-2187 (2005). An important observation is that for a fixed value of the wavelength λ, Planck function is a Lipschitz function with respect to the temperature. Using this fact, we deduce that the solution is at most unique. To prove existence of a solution, we define a fixed point problem related to our initial boundary value problem to which we apply Schauder theorem in a closed convex subset of the Banach separable space L 2 (0,t f ;C([0,l])). We use also Stampacchia truncation method to derive lower and upper bounds on the solution.

OBSERVATIONS AND MODELING OF H2FLUORESCENCE WITH PARTIAL FREQUENCY REDISTRIBUTION IN GIANT PLANET ATMOSPHERES

Partial frequency redistribution (PRD), describing the formation of the line profile, has negligible observational effects for optical depths smaller than ~10^3, at the resolving power of most current instruments. However, when the spectral resolution is sufficiently high, PRD modeling becomes essential in interpreting the line shapes and determining the total line fluxes. We demonstrate the effects of PRD on the H_2 line profiles observed at high spectral resolution by the Far-Ultraviolet Spectroscopic Explorer (FUSE) in the atmospheres of Jupiter and Saturn. In these spectra, the asymmetric shapes of the lines in the Lyman (v"- 6) progression pumped by the solar Ly-beta are explained by coherent scattering of the photons in the line wings. We introduce a simple computational approximation to mitigate the numerical difficulties of radiative transfer with PRD, and show that it reproduces the exact radiative transfer solution to better than 10%. The lines predicted by our radiative transfer model with PRD, including the H_2 density and temperature distribution as a function of height in the atmosphere, are in agreement with the line profiles observed by FUSE. We discuss the observational consequences of PRD, and show that this computational method also allows us to include PRD in modeling the continuum pumped H_2 fluorescence, treating about 4000 lines simultaneously.

Retrieval of warm cloud optical properties using simple approximations

A new technique relying on SimpLe Approximations for cLOudy Media (SLALOM) for the retrieval of cloud optical and microphysical parameters from optical satellite data during daytime is introduced. The technique is based on simple yet highly accurate approximations of the asymptotic solutions of the radiative transfer theory which have already been implemented in the forward radiative transfer model CLOUD. These approximations enable a solution of the equations of the corresponding backward model during runtime leading to a very fast computation speed. Since these asymptotic solutions are generally applicable to weakly absorbing media only, pre-calculated look-up tables for the reflection function of a semi-infinite cloud (and also the escape function) are used to overcome this restriction within this new retrieval. SLALOM is capable of retrieving the cloud optical thickness, the effective cloud droplet radius, the liquid and ice water paths, the particle absorption length as well as some other properties of water and ice clouds. The comparison of SLALOM with both exact radiative transfer computations and the NASA MODIS cloud property product shows a very good agreement. A Fortran implementation of both CLOUD and SLALOM is available for download under the Creative Commons Attribution-Noncommercial-Share Alike 3.0 license (see http://creativecommons.org/licenses/by-nc-sa/3.0) at http://www.klimatologie.uni-bayreuth.de.

The determination of dust cloud altitudes from a satellite using hyperspectral measurements in the gaseous absorption band

An approach to derive dust layer optical thickness and top height using top-of-atmosphere (TOA) reflectance in the oxygen A-band is introduced. The algorithm is similar to that developed by the authors for the case of water clouds. It is based on the fitting of spectral TOA reflectance measurements in the narrow band around 760 nm using results of the exact radiative transfer calculations for a given dust layer model. The accuracy of the technique with respect to the uncertainty in a priori assumption of the dust single-scattering albedo is discussed. The algorithm is applied to satellite hyperspectral measurements over the Atlantic Ocean.

The uncertainties of satellite DOAS total ozone retrieval for a cloudy sky

This paper aimed to the study of the influence of the uncertainties of satellite Differential Optical Absorption Spectroscopy (DOAS) total ozone retrieval scheme due to unconstrained parameters of the plane-parallel scattering cloud model. The work is based on the exact radiative transfer calculations for a realistic atmospheric model described in the paper. In particular, the accuracy of the DOAS technique based on hyperspectral measurements in the ozone absorption band 315–340 nm is investigated. The general weighting functions approach to the calculation of the error of total ozone introduced in the paper can be used for studies of errors in the retrieved concentration of other trace gases (e.g., NO 2, SO 2, etc.) as well. It was found that the uncertainties in the cloud geometrical and optical characteristics can produce absolute errors in the retrieved total ozone column of up to 6%. This calls for the introduction of accurate cloud retrieval algorithms in ozone satellite retrieval schemes.

Scattered light corrections to Sun photometry: analytical results for single and multiple scattering regimes

Analytical equations for the diffused scattered light correction factor of Sun photometers are derived and analyzed. It is shown that corrections are weakly dependent on the atmospheric optical thickness. They are influenced mostly by the size of aerosol particles encountered by sunlight on its way to a Sun photometer. In addition, the accuracy of the small-angle approximation used in the work is studied with numerical calculations based on the exact radiative transfer equation.

Line profile variations in WR+O binary systems

We compute the P Cyg line profiles formed by the stellar winds of binary systems containing a Wolf-Rayet and an O-type star, incorporating the effects due to wind eclipses and wind-wind collisions. The contribution from both stellar winds to the P Cygni emission lines is modeled for different orbital phases. The opacity and the source function are calculated assuming a simplified atom and the Sobolev approximation, and the emission-line profile is calculated by exact radiative transfer through the 3D geometry wind. We analyze the cases of a P Cygni line that is formed only in the WR wind, and the case of a line formed in both the WR and the O-star winds. The line-profile variations that are predicted by this model are presented. When compared with observations, the synthetic profiles and their phase-dependent variability provide an estimate for the opening angle of the WWC shock cone and the velocity law of both stellar winds. Ultraviolet (UV) observations of the binary system Vel are used to illustrate how the model predictions can be applied to the observational data.

Parameterization of Cloud–Radiation Processes in the UCLA General Circulation Model

Abstract A contemporary radiation parameterization scheme has been implemented in the University of California, Los Angeles (UCLA), atmospheric GCM (AGCM). This scheme is a combination of the delta-four-stream method for solar flux transfer and the delta-two-and-four-stream method for thermal infrared flux transfer. Both methods have been demonstrated to be computationally efficient and at the same time highly accurate in comparison with exact radiative transfer computations. The correlated-k distribution method for radiative transfer has been used to represent gaseous absorption in multiple-scattering atmospheres. The single-scattering properties for ice and water clouds are parameterized in terms of ice/liquid water content and mean effective size/radius. In conjunction with the preceding radiative scheme, parameterizations for fractional cloud cover and cloud vertical overlap have also been devised in the model in which the cloud amount is determined from the total cloud water mixing ratio. For radiati...

The physical parameterization of the top-of-atmosphere reflection function for a cloudy atmosphere—underlying surface system: the oxygen A-band case study

The paper is devoted to the physical parameterization of the top-of-atmosphere reflection function. The accuracy of the parameterization is checked against exact radiative transfer calculations in a cloudy atmosphere for various cloud-top-heights, cloud optical and geometrical thicknesses, solar illumination and surface reflection conditions. It was found that the error of approximation is smaller than 5% for most cases studied at the wavelength interval 758 –768 nm , which corresponds to the oxygen A-band. This band is routinely used in cloud-top-height retrievals. The model proposed can be used for cloud-top-height and cloud geometrical thickness retrievals. This allows to avoid a standard look-up-table retrieval scheme, involving complex numerical procedures.

Photon density wave for imaging through random media

The passage of a photon density wave through random media has been investigated extensively for medical imaging based on the diffusion approximation. In this paper, the photon density wave is studied based on the exact time-dependent vector radiative transfer theory. Both continuous and pulse photon density waves are analysed in a plane parallel medium using Mie scattering and the discrete ordinates method. The photon density wave shows superior properties over regular waves in several aspects. It has a narrower angular spectrum and maintains the original pulse shape. It also preserves the degree of polarization and increases the cross-polarization discrimination. These properties of a photon density wave suggest its potential for improving imaging. Thus, we apply the photon density wave to an imaging problem and show that it improves the quality of the images compared to other conventional imaging techniques.

Modeling maize canopy 3D architecture: Application to reflectance simulation

The aim of this study is to develop a 3D model of maize ( Zea mays) canopy structure for accurate reflectance simulations. We focus on fully developed maize plants without paying attention to the reproductive organs. Several experiments are used to describe the dimension, shape, position and orientation of the leaves and stems. They correspond to a wide range of cultural practices. Empirical models are proposed to derive the position, size, and shape of the leaves and stems as a function of three main variables: the final number of leaves, the final height of the canopy and the cumulated leaf area per plant that is considered as a vigor index. Leaf orientation is described through the curvature of the main rib and the cross section used to represent the undulation of the lamina. The statistical distributions of the parameters of each structure sub-models are investigated. It allows to generate realistic computerized plant and canopy 3D architecture representation. The canopy structure is then used to compare the SAIL reflectance model to the reflectance simulated with PARCINOPY which is a monte-carlo ray tracing model. The combined use of detailed canopy architecture models and quasi exact radiative transfer models such as PARCINOPY appears to be a very convenient tool to evaluate more simple reflectance models applied to specific vegetation types. Results show a general good agreement between the two models, except for the exponential hot-spot function used in the SAIL model, and the directionality of the multiple scattering.

Transient Thermal Analysis of Parallel Translucent Layers by Using Green's Functions

An analysis is developed for computing transient thermal behavior in a composite of two parallel translucent layers with a space between them. The radiative two-flux equation is coupled with the transient energy equation, and both equations are solved by using Green's functions. Illustrative results are provided for transient heating of two parallel translucent layers that exchange radiation. The external boundaries of the two-layer combination are each exposed to a radiative environment, and all boundaries can be convectively heated or cooled. The layer refractive indices are larger than one, and internal reflections are included with boundaries assumed diffuse. The analysis includes internal emission, absorption, isotropic scattering, and heat conduction. Transient results from the present method are verified for a single gray layer by comparison with a finite difference solution that incorporates a numerical solution of the exact radiative transfer equations for the radiative heat source. Illustrative transient temperature distributions show the effects of an insulating gap between layers, when one side of the twolayer composite is subjected to heating by radiation and convection, while the other side is being cooled.

Physically based parameterizations of the short-wave radiative characteristics of weakly absorbing optically thick media: application to liquid-water clouds

We propose the physically based parameterization of the radiative characteristics of liquid-water clouds as functions of the wavelength, effective radius, and refractive index of particles, liquid-water path, ground albedo, and solar and observation angles. The formulas obtained are based on the approximate analytical solutions of the radiative transfer equation for optically thick, weakly absorbing layers and the geometrical optics approximation for local optical characteristics of cloud media. The accuracy of the approximate formulas was studied with an exact radiative transfer code. The relative error of the approximate formula for the reflection function at nadir observations was less then 15% for an optical thickness larger than 10 and a single-scattering albedo larger than 0.95.