Plane-parallel Medium Research Articles

Exact Solutions for Radiative Transfer with Partial Frequency Redistribution

The construction of exact solutions for radiative transfer in a plane-parallel medium has been addressed by Hemsch and Ferziger in 1972 for a partial frequency redistribution model of the formation of spectral lines consisting in a linear combination of frequency coherent and fully incoherent scattering. The method of solution is based on an eigenfunction expansion of the radiation field, leading to two singular integral equations with Cauchy or Cauchy-type kernels, that have to be solved one after the other. We reconsider this problem, using as starting point the integral formulation of the radiative transfer equation, where the terms involving the coupling between the two scattering mechanisms are clearly displayed, as well as the primary source of photons. With an inverse Laplace transform, we recover the singular integral equations previously established and, with Hilbert transforms as in the previous work, recast them as boundary value problems in the complex plane. Their solutions are presented in detail for an infinite and a semi-infinite medium. The coupling terms are carefully analyzed and consistency with either the coherent or the incoherent limit is systematically checked. We recover the important result of the previous work that an exact solution exists for an infinite medium, whereas for a semi-infinite medium, which requires the introduction of half-space auxiliary functions, the solution is given by a Fredholm integral equation to be solved numerically. The solutions of the singular integral equations are used to construct explicit expressions providing the radiation field for an arbitrary primary source and for the Green’s function. An explicit expression is given for the radiation field emerging from a semi-infinite medium.

Application and comparative analysis of radiative heat transfer models for coal-fired furnace

Thermal radiation is a crucial mode of heat transfer in the industrial furnace. The present article compares various radiative heat transfer models for non-gray gases environments with typical application to the pulverized coal-fired furnace. An open source CFD software OpenFOAM is used to implement a three-dimensional photon Monte Carlo (PMC) method for radiation from gray gases and first-order spherical harmonics (P 1-method), discrete ordinates method (DOM) and zonal method for radiation from non-gray gases. The non-gray radiation properties are calculated using the weighted-sum-of-gray-gases (WSGG) and full-spectrum k-distribution models. The PMC method is employed as a benchmark for comparing the accuracy of the P 1-method and the DOM for gray gas environment, and the zonal method is employed as a benchmark for comparing the accuracy of P 1-method and DOM for the non-gray gas environment. The zonal method and PMC are validated against the exact solution for plane-parallel medium filled with a gray gas. The accuracy of the P 1-method and DOM is analyzed and compared with the zonal method for non-gray gas radiation inside a cubical box of 1 m sides. The effect of coal and ash particle’s radiative properties on the radiative heat transfer has been analyzed. The k-distribution model along with radiation due to coal and ash particles is presented. The variation in absorption and scattering efficiency due to particle size and the complex index of refraction is analyzed for different types of coal and ash particles. Isotropic, forward scattering, and δ-Eddington approximation are compared with the anisotropic scattering using Mie-theory. The coal and ash particle’s radiative properties are calculated using the Mie-theory and are averaged using Planck intensity distribution. The particle size of the coal and ash is a more critical parameter compared to the complex index of refraction. The slight variation in projected area per unit coal and ash particle leads to a significant change in the radiative heat source and radiative heat flux at the wall of industrial furnace.

Longwave radiative effect of the cloud–aerosol transition zone based on CERES observations

Abstract. This study presents an approach for the quantification of cloud–aerosol transition-zone broadband longwave radiative effects at the top of the atmosphere (TOA) during daytime over the ocean, based on satellite observations and radiative transfer simulation. Specifically, we used several products from MODIS (MODerate Resolution Imaging Spectroradiometer) and CERES (Clouds and the Earth's Radiant Energy System) sensors for the identification and selection of CERES footprints with a horizontally homogeneous transition-zone and clear-sky conditions. For the selected transition-zone footprints, radiative effect was calculated as the difference between the instantaneous CERES TOA upwelling broadband longwave radiance observations and corresponding clear-sky radiance simulations. The clear-sky radiances were simulated using the Santa Barbara DISORT (DIScrete Ordinates Radiative Transfer program for a multi-Layered plane-parallel medium) Atmospheric Radiative Transfer model fed by the hourly ERA5 reanalysis (fifth generation ECMWF ReAnalysis) atmospheric and surface data. The CERES radiance observations corresponding to the clear-sky footprints detected were also used for validating the simulated clear-sky radiances. We tested this approach using the radiative measurements made by the MODIS and CERES instruments on board the Aqua platform over the southeastern Atlantic Ocean during August 2010. For the studied period and domain, transition-zone radiative effect (given in flux units) is on average equal to 8.0 ± 3.7 W m−2 (heating effect; median: 5.4 W m−2), although cases with radiative effects as large as 50 W m−2 were found.

The Sobolev approximation for radiation transport with line overlap and continuous opacity

The radiation transport problem in the plane–parallel medium with the large velocity gradient is considered. The Sobolev approximation is used. The effects of continuum absorption and line overlap are taken into account. The photon loss probability functions are calculated and tabulated. Two calculations are performed – for the Gaussian spectral line profile and for the rectangular profile. It is shown that at particular choice of the rectangular profile width the results of the calculations are very close. The evaluated photon loss probability functions may be used in the calculations of energy level populations of OH molecule in the interstellar gas flows.

Physical Parameterization of Hyperspectral Reflectance in the Oxygen A-Band for Single-Layer Water Clouds

Previous studies have shown that it is feasible to retrieve multiple cloud properties simultaneously based on the space-borne hyperspectral observation in the oxygen A-band, such as cloud optical depth, cloud-top height, and cloud geometrical thickness. However, hyperspectral remote sensing is time-consuming if based on the precise radiative transfer solution that counts multiple scatterings of light. To speed up the radiation transfer solution in cloud scenarios for nadir space-borne observations, we developed a physical parameterization of hyperspectral reflectance in the oxygen A-band for single-layer water clouds. The parameterization takes into account the influences of cloud droplet forward-scattering and nonlinear oxygen absorption on hyperspectral reflectance, which are improvements over the previous studies. The performance of the parameterization is estimated through comparison with DISORT (Discrete Ordinates Radiative Transfer Program Multi-Layered Plane-Parallel Medium) on the cases with solar zenith angle θ, the cloud optical depth τc, and the single-scattering albedo ω in the range of 0 ≤ θ ≤ 75, 5 ≤ τc ≤ 50, 0.5 ≤ ω ≤ 1. The relative error of the cloud reflectance is within 5% for most cases, even for clouds with optical depths around five or at strong absorption wavelengths. We integrate the parameterization with a slit function and a simplified atmosphere to evaluate its performance in simulating the observed cloud reflection at the top of the atmosphere by OCO-2 (Orbiting Carbon Observatory-2). To better visualize the possible errors from the new parameterization, gas molecular scattering, aerosol scattering, and reflection from the underlying surface are ignored. The relative error of the out-of-band radiance is less than 4% and the relative error of the intra-band radiance ratio is less than 4%. The radiance ratio is the ratio of simulated observations with and without in-cloud absorption and is used to assess the accuracy of the parameterization in quantifying the in-cloud absorption. The parameterization is a preparation for rapid hyperspectral remote sensing in the oxygen A-band. It would help to improve retrieval efficiency and provide cloud geometric thickness products.

Electron Density Structure of the Local Galactic Disk

Abstract Pulsar dispersion measures (DMs) have been used to model the electron density of the interstellar medium (ISM) in the Galactic disk as a plane-parallel medium, despite significant scatter in the DM-distance distribution and strong evidence for inhomogeneities in the ISM. We use a sample of pulsars with independent distance measurements to evaluate a model of the local ISM in the thick disk of the Galaxy that incorporates turbulent fluctuations, clumps, and voids in the electron density. The latter two components are required because about one-third of the lines of sight are discrepant from a strictly plane-parallel model. A likelihood analysis for smooth components of the model yields a scale height kpc and a mid-plane density n 0 = 0.015 ± 0.001 cm−3. The scatter in the DM-distance distribution is dominated by clumps and voids but receives significant contributions from a broad spectrum of density fluctuations, such as a Kolmogorov spectrum. The model is used to identify lines of sight with outlier values of DM. Three of these pulsars, J1614−2230, J1623−0908, and J1643−1224, lie behind known H ii regions, and the electron density model is combined with Hα intensity data to constrain the filling factors and other substructure properties of the H ii regions (Sh 2–7 and Sh 2–27). Several pulsars also exhibit enhanced DM fluctuations that are likely caused by an intersection of their lines of sight with the superbubble GSH 238+00+09.

Light scattering as a Poisson process and first-passage probability

A particle entering a scattering and absorbing medium executes a random walk through a sequence of scattering events. The particle ultimately either achieves first-passage, leaving the medium, or it is absorbed. The Kubelka–Munk model describes a flux of such particles moving perpendicular to the surface of a plane-parallel medium with a scattering rate and an absorption rate. The particle path alternates between the positive direction into the medium and the negative direction back towards the surface. Backscattering events from the positive to the negative direction occur at local maxima or peaks, while backscattering from the negative to the positive direction occur at local minima or valleys. The probability of a particle avoiding absorption as it follows its path decreases exponentially with the path-length λ. The reflectance of a semi-infinite slab is therefore the Laplace transform of the distribution of path-length that ends with a first-passage out of the medium. In the case of a constant scattering rate the random walk is a Poisson process. We verify our results with two iterative calculations, one using the properties of iterated convolution with a symmetric kernel and the other via direct calculation with an exponential step-length distribution. We present a novel demonstration, based on fluctuation theory of sums of random variables, that the first-passage probability as a function of the number of peaks n in the alternating path is a step-length distribution-free combinatoric expression involving Catalan numbers. Counting paths with backscattering on the real half-line results in the same Catalan number coefficients as Dyck paths on the whole numbers. Including a separate forward-scattering Poisson process results in a combinatoric expression related to counting Motzkin paths. We therefore connect walks on the real line to discrete path combinatorics.

Angular reflectance of a highly forward scattering medium at grazing incidence of light.

We study the angular distribution of light diffusely reflected from a turbid medium with large (compared to the light wavelength) inhomogeneities. Using Monte Carlo radiative transfer simulations, we calculate the azimuthally averaged bidirectional reflectance for an optically thick plane-parallel medium and analyze its dependence on the parameters of the scattering phase function. To model single scattering in the medium, we take advantage of the Reynolds-McCormick phase function. For grazing angles of incidence, we find that the angular distribution of reflected light becomes very sensitive to the angular profile of the scattering phase function. The more elongated the phase function, the more pronounced the peak that arises around the specular reflection angle. Comparison of our numerical results with an analytic solution of the radiative transfer equation is performed, and it is shown that the bidirectional reflectance can be decomposed into two contributions, namely, the diffusion contribution and the contribution from light experiencing multiple scattering through small angles. The latter relates directly to the angular profile of the scattering phase function and is responsible for the peak in the angular distribution of reflected light. An explicit analytic formula for the azimuthally averaged bidirectional reflectance is obtained.

Chebyshev collocation spectral method for vector radiative transfer equation and its applications in two-layered media

The Chebyshev collocation spectral method (CCSM) is developed for solving the vector radiative transfer equation (VRTE) in participating media and then extended to polarized radiative transfer problems in two-layered media. The derivation of the CCSM discretization for the VRTE is presented and the numerical performance of the CCSM is investigated. The accuracy of the CCSM for solving the VRTE is first verified by comparing the CCSM solutions for the polarized radiative transfer in a plane-parallel medium with published results. Afterward, the CCSM is applied to polarized radiative transfer problems in inhomogeneous media, including two-layered media with different scattering characteristics and two-layered media with different refractive indices. Finally, taking both effects of different medium properties and reflective/refractive interface into account, we investigated the polarized radiative transfer in a realistic atmosphere-ocean system via the CCSM and discussed the distributions of the Stokes vector components. Results show that the CCSM is accurate and efficient for solving the VRTE and the distributions of Stokes vector for a realistic atmosphere-ocean system are presented in detail.

Direct prediction of petrophysical and petroelastic reservoir properties from seismic and well-log data using nonlinear machine learning algorithms

An analytical comparison of seismic inversion with several multivariate predictive techniques is made. Statistical data reduction techniques are examined that incorporate various machine learning algorithms, such as linear regression, alternating conditional expectation regression, random forest, and neural network. Seismic and well-log data are combined to estimate petrophysical or petroelastic properties, like bulk density. Currently, spatial distribution and estimation of reservoir properties is leveraged by inverting 3D seismic data calibrated to elastic properties (VP, VS, and bulk density) obtained from well-log data. Most commercial seismic inversions are based on linear convolution, i.e., one-dimensional models that involve a simplified plane-parallel medium. However, in cases that are geophysically more complex, such as fractured and/or fluid-rich layers, the conventional straightforward prediction relationship breaks down. This is because linear convolution operators no longer adequately describe seismic wavefield propagation due to nonlinear energy absorption. Such nonlinearity is also suggested by the seismic nonstationarity phenomenon, expressed by vertical and horizontal changes in the shape of the seismic wavelet (amplitude and frequency variations). The nonlinear predictive operator, extracted by machine learning algorithms, makes it possible in certain cases to estimate petrophysical reservoir properties more accurately and with less influence of interpretational bias.

On sampling of scattering phase functions

Abstract Monte Carlo radiative transfer, which has been demonstrated as a successful algorithm for modelling radiation transport through the astrophysical medium, relies on sampling of scattering phase functions. We review several classic sampling algorithms such as the tabulated method and the accept–reject method for sampling the scattering phase function. The tabulated method uses a piecewise constant approximation for the true scattering phase function; we improve its sampling performance on a small scattering angle by using piecewise linear and piecewise log-linear approximations. It has previously been believed that certain complicated analytic phase functions such as the Fournier–Forand phase function cannot be simulated without approximations. We show that the tabulated method combined with the accept–reject method can be applied to sample such complicated scattering phase functions accurately. Furthermore, we introduce the Gibbs sampling method for sampling complicated approximate analytic phase functions. In addition, we propose a new modified Henyey–Greenstein phase function with exponential decay terms for modelling realistic dust scattering. Based on Monte Carlo simulations of radiative transfer through a plane-parallel medium, we also demonstrate that the result simulated with the new phase function can provide a good fit to the result simulated with the realistic dust phase function.

Wavelength-scale gas-filled cuboid acoustic lens with diffraction limited focusing

Abstract Per the laws of geometrical optics, a plane-parallel medium does not possess focusing properties. However, in this paper, we demonstrate for the first time that an acoustic cuboid particle filled with CO2 is able to focus sound despite its flat surface. It is worth noting that traditional lenses focus sound through their curved surfaces. We report both numerically and experimentally the acoustic focusing of a cuboid of 2λ side, where λ is the wavelength in air. From these results, it can be derived that its focusing capabilities are close to the diffraction limit: at frequency of 5000 Hz, acoustic beam waist is about 0.5 wavelength and the gain is 8.8 dB. The flat structure lens proposed open new possibilities to design and built new lightweight and low cost acoustic lenses for different applications.

Unsteady Radiative Heat Transfer Model of a Ceria Particle Suspension Undergoing Solar Thermochemical Reduction

Unsteady radiative heat transfer is analyzed numerically in a directly irradiated plane-parallel medium containing a suspension of ceria particles undergoing nonstoichiometric thermal reduction. The micrometer-sized ceria particles are assumed homogenous, nongray, absorbing, emitting, and anisotropically scattering, whereas the overall medium is of nonuniform temperature and composition. The unsteady mass and energy conservation equations are solved using the finite volume method and the Shampine–Gordon time integration scheme. Radiative transport is modeled using the energy-portioning Monte Carlo ray-tracing method with radiative properties obtained from the Mie theory. Increasing the particle volume fraction and decreasing the particle diameter both increase the optical thickness of the particle suspension, resulting in increasing peak temperature and nonstoichiometry at steady state. For -diam particles under 1000 suns irradiation, the peak temperature at steady state ranges from 1855 K for a particle volume fraction of to 2092 K for ; the temperature nonuniformity ranges from 9 to 622 K. For a fixed volume fraction of , decreasing the particle diameter from 20 to increases the peak temperature at steady state from 1734 to 2162 K; the temperature nonuniformity increases from 9 to 61 K.

The algorithms for calculating synthetic seismograms from a dipole source using the derivatives of Green’s function

The problem of calculating complete synthetic seismograms from a point dipole with an arbitrary seismic moment tensor in a plane parallel medium composed of homogeneous elastic isotropic layers is considered. It is established that the solutions of the system of ordinary differential equations for the motion–stress vector have a reciprocity property, which allows obtaining a compact formula for the derivative of the motion vector with respect to the source depth. The reciprocity theorem for Green’s functions with respect to the interchange of the source and receiver is obtained for a medium with cylindrical boundary. The differentiation of Green’s functions with respect to the coordinates of the source leads to the same calculation formulas as the algorithm developed in the previous work (Pavlov, 2013). A new algorithm appears when the derivatives with respect to the horizontal coordinates of the source is replaced by the derivatives with respect to the horizontal coordinates of the receiver (with the minus sign). This algorithm is more transparent, compact, and economic than the previous one. It requires calculating the wavenumbers associated with Bessel function’s roots of order 0 and order 1, whereas the previous algorithm additionally requires the second order roots.

Radiative transfer equation and direct simulation prediction of reflection and absorption by particle deposits

Two methods for computing the normal incidence absorptance and hemispherical reflectance from plane parallel layers of wavelength–sized spherical particles are presented. The first method is based on an exact superposition solution to Maxwell's time harmonic wave equations for a system of randomly–positioned spherical particles excited by an incident plane wave. The second method is based upon the scalar radiative transport equation (RTE) applied to a plane parallel medium. Comparisons are made using five values of particle refractive index, sphere size parameters ranging from 1 to 4, and particle volume concentrations ranging from 0.05 to 0.4. The results indicate that the multiple sphere T matrix method (MSTM) and RTE predictions of hemispherical reflectance and absorptance converge when particle volume fraction becomes small. At higher volume fractions the RTE can yield results for hemispherical reflectance that, depending on the particle size and refractive index, significantly depart from the exact predictions. On the other hand, RTE and MSTM predictions of absorptance have a much closer agreement which is largely independent of the sphere optical properties and volume concentration.

Markov chain formalism for generalized radiative transfer in a plane-parallel medium, accounting for polarization

A Markov chain formalism is developed for computing the transport of polarized radiation according to Generalized Radiative Transfer (GRT) theory, which was developed recently to account for unresolved random fluctuations of scattering particle density and can also be applied to unresolved spectral variability of gaseous absorption as an improvement over the standard correlated-k method. Using Gamma distribution to describe the probability density function of the extinction or absorption coefficient, a shape parameter a that quantifies the variability is introduced, defined as the mean extinction or absorption coefficient squared divided by its variance. It controls the decay rate of a power-law transmission that replaces the usual exponential Beer-Lambert-Bouguer law. Exponential transmission, hence classic RT, is recovered when a→∞. The new approach is verified to high accuracy against numerical benchmark results obtained with a custom Monte Carlo method. For a<∞, angular reciprocity is violated to a degree that increases with the spatial variability, as observed for finite portions of real-world cloudy scenes. While the degree of linear polarization in liquid water cloudbows, supernumerary bows, and glories is affected by spatial heterogeneity, the positions in scattering angle of these features are relatively unchanged. As a result, a single-scattering model based on the assumption of subpixel homogeneity can still be used to derive droplet size distributions from polarimetric measurements of extended stratocumulus clouds.

The Generalized SLW Model

The Generalized SLW Method is presented, formulating the SLW method with the help of both the ALBDF and the Inverse ALBDF. The result is two equivalent symmetric models: the SLW Model and the Inverse SLW Model. The advantage of the unified dual formulation and of application of the ALBDF and the Inverse ALBDF is in more efficient implementation of the model and the elimination of the solution of the implicit equations for the absorption cross-sections in the construction of the spectral model in the case of nonisothermal media. The generalized approach explores all possibilities of the SLW method under both direct and inverse formulations including its limiting cases: the minimal one clear gas-one gray gas SLW-1 model, and the case when the number of gray gases approaches infinity termed the Exact SLW model. The present work outlines the steps in a unified construction of the generalized SLW model in isothermal and non-isothermal media, and compares different forms of the modelled radiative quantities in plane parallel media: directional total radiative flux, total emissivity, Planck mean and Rosseland mean absorption coefficients.

On Galerkin method for solving radiative heat transfer in finite slabs with spatially-variable refractive index

The problem of radiative heat transfer through a finite plane-parallel medium which possesses a continuous transverse spatial variation in refractive index has been solved. Galerkin method is used to have the analytical solution of the integral form of the radiative heat transfer equation, which gives the analytical forms of both the radiation energy density and the net radiant heat flux through the medium. Numerical results are obtained for the reflection and transmission coefficients at the medium boundaries. These results are carried out for isotropic scattering with different thicknesses and different ranges of refractive index. Some obtained results are compared with those calculated by using the Pomraning-Eddington approximation which was previously published, and gave good agreement.

Solving nonlinear heat transfer problems using variation of parameters

Nonlinear problems arise in many heat transfer applications, and several analytical and numerical methods for solving these problems are described in the literature. Here, the method of variation of parameters is shown to be a relatively simple method for obtaining solutions to four specific heat transfer problems: 1. a radiating annular fin, 2. conduction-radiation in a plane-parallel medium, 3. convective and radiative exchange between the surface of a continuously moving strip and its surroundings, and 4. convection from a fin with temperature-dependent thermal conductivity and variable cross-sectional area. The results for each of these examples are compared to those obtained using other analytical and numerical methods. The accuracy of the method is limited only by the accuracy with which the numerical integration is performed. The method of variation of parameters is less complex and relatively easy to implement compared to other analytical methods and some numerical methods. It is slightly more computationally expensive than traditional numerical approaches. The method presented may be used to verify numerical solutions to nonlinear heat transfer problems.

Approximate solution to vector radiative transfer in gradient-index medium

Within a gradient-index medium, the radiative rays propagate in curved paths, which makes polarized states change continuously and the solution to the radiative transfer be thus more complex and difficult. In this paper, an arbitrary multilayer model is developed to approximately simulate vector (polarized) radiative transfer in a gradient-index plane-parallel medium. The gradient-index medium is divided into an arbitrary number of sublayers, and each sublayer has a uniform refractive index and two virtual Fresnel's interfaces where only transmission (refraction) is considered. Thus the polarization caused by the curved propagation of lights is approximated by that resulting from refraction on the interfaces. Radiative transfer with consideration of polarization caused by particle scattering and refraction (reflection) on the interfaces (surfaces) in the multilayer model is solved by the MC method. The grid independence of results obtained by the multilayer model for vector radiative transfer in gradient-index medium shows that the convergent solution of Stokes vector will be achieved provided that the sublayer number is large enough. The results for apparent emissivity of gradient-index medium and Stokes vector for two-layer medium are compared well with those in published literatures. Finally, we investigate polarized behaviors of radiative transfer in Rayleigh scattering slabs with linear and sinusoidal gradient indexes and present angular distributions of Stokes vector. Results show that total reflection inside the gradient-index medium resulting from the curved paths traveled by the photons affects greatly the angular distribution characteristics of Stokes vector.