One-dimensional solar radiative transfer: Perturbation approach and its application to independent-pixel calculations for realistic cloud fields

The radiative transfer perturbation theory (RTPT), which has already been introduced in atmospheric radiative transfer several years ago, is applied to cloud related problems. The RTPT requires the solution of the radiative transfer equation in the forward and the adjoint mode. The basic principles of this technique are presented as well as its extensions to isotropic surface reflection and its conjunction with the Hermite interpolation. This set of methods is applied to different atmospheric conditions including realistic cloud scenes. The results are compared with the usual (forward) independent-pixel calculations with respect to errors of individual pixels and domain-averaged values. The RTPT turns out to be sufficiently accurate in the case the clouds’ internal vertical variations remain moderate. It is also shown that, depending on the specific radiative transfer problem, the RTPT can offer some advantages on computational speed. However, the limitations of the RTPT with regard to realistic clouds are addressed as well.     

The reflection function of optically thick weakly absorbing turbid layers: a simple approximation

The paper is devoted to the derivation of correction terms for the asymptotic formulae of the radiative transfer theory, which allow to increase their accuracy for smaller optical thicknesses (typically, down to the optical thickness equal to 5). Most of the examples are limited to the case of water clouds. However, the main results are easily generalized on other types of disperse media.     

A simple new radiative transfer model for simulating the effect of cirrus clouds in the microwave spectral region

The upwelling atmospheric radiation in the millimeter wave spectral range is influenced by the presence of cirrus clouds. A plane parallel radiative transfer model which can take into account the effect of multiple scattering by ice particles in the cirrus has been developed and is used to simulate the brightness temperatures as they would be measured by a satellite instrument. The model uses an iterative procedure to solve the radiative transfer equation. The formulation of the model is such that it can easily be adapted to treat the full specific intensity vector instead of just the scalar total intensity. A convergence test for the model is explained and two cirrus cloud scenarios are simulated. The results illustrate the linearity of microwave radiative transfer for not too strong cirrus clouds in this frequency region.     

Impact of ice particle shape on short-wave radiative forcing: A case study for an arctic ice cloud

We used four different non-spherical particle models to compute optical properties of an arctic ice cloud and to simulate corresponding cloud radiative forcings and fluxes. One important finding is that differences in cloud forcing, downward flux at the surface, and absorbed flux in the atmosphere resulting from the use of the four different ice cloud particle models are comparable to differences in these quantities resulting from changing the surface albedo from 0.4 to 0.8, or by varying the ice water content (IWC) by a factor of 2. These findings show that the use of a suitable non-spherical ice cloud particle model is very important for a realistic assessment of the radiative impact of arctic ice clouds. The differences in radiative broadband fluxes predicted by the four different particle models were found to be caused mainly by differences in the optical depth and the asymmetry parameter. These two parameters were found to have nearly the same impact on the predicted cloud forcing. Computations were performed first by assuming a given vertical profile of the particle number density, then by assuming a given profile of the IWC. In both cases, the differences between the cloud radiative forcings computed with the four different non-spherical particle models were found to be of comparable magnitude. This finding shows that precise knowledge of ice particle number density or particle mass is not sufficient for accurate prediction of ice cloud radiative forcing. It is equally important to employ a non-spherical shape model that accurately reproduces the ice particle's dimension-to-volume ratio and its asymmetry parameter. The hexagonal column/plate model with air-bubble inclusions seems to offer the highest degree of flexibility.     

Comparison of single and multiple scattering approaches for the simulation of limb-emission observations in the mid-IR

The validity of single scattering radiative transfer calculations for simulation of limb-emission measurements of clouds in the mid-infrared spectral region was investigated by comparison with a multiple scattering model. For in limb direction optically thin clouds, like polar stratospheric clouds, errors of the single scattering scheme range below 3%. For optically thick clouds deviations are below 3% in case of low single scattering albedo (ω₀=0.24) increasing up to 10-30% for ω₀=0.84. Clouds which are optically thick in limb, but thin in nadir direction, can cause limb radiances which are by a factor of 1.7 higher than the blackbody radiance at cloud altitude.     

Estimates of radiation over clouds and dust aerosols: Optimized number of terms in phase function expansion

The bulk-scattering properties of dust aerosols and clouds are computed for the community radiative transfer model (CRTM) that is a flagship effort of the Joint Center for Satellite Data Assimilation (JCSDA). The delta-fit method is employed to truncate the forward peaks of the scattering phase functions and to compute the Legendre expansion coefficients for re-constructing the truncated phase function. Use of more terms in the expansion gives more accurate re-construction of the phase function, but the issue remains as to how many terms are necessary for different applications. To explore this issue further, the bidirectional reflectances associated with dust aerosols, water clouds, and ice clouds are simulated with various numbers of Legendre expansion terms. To have relative numerical errors smaller than 5%, the present analyses indicate that, in the visible spectrum, 16 Legendre polynomials should be used for dust aerosols, while 32 Legendre expansion terms should be used for both water and ice clouds. In the infrared spectrum, the brightness temperatures at the top of the atmosphere are computed by using the scattering properties of dust aerosols, water clouds and ice clouds. Although small differences of brightness temperatures compared with the counterparts computed with 4, 8, 128 expansion terms are observed at large viewing angles for each layer, it is shown that 4 terms of Legendre polynomials are sufficient in the radiative transfer computation at infrared wavelengths for practical applications.     

A retrospective view on “3-dimensional radiation in absorbing, emitting and scattering media using discrete-ordinates approximation” by J.S. Truelove

Discrete-ordinates (DO) approximations to the radiative transfer equation in three-dimensional enclosures have extensively been used during the last three decades. The 1988 paper by Truelove [1] is one of the pioneering works in this field wherein traditional DO formulations were adapted to radiative transfer problems, and has impacted both the science and the technology related to large-scale combustion chambers since it was published. The following is a short introduction to this seminal JQSRT paper.     

A retrospective view on “algorithms for radiative intensity calculations in moderately thick atmospheres using a truncation approximation” by Teruyuki Nakajima and Masayuki Tanaka (1988)

I explain the motivation behind our paper “Algorithms for radiative intensity calculations in moderately thick atmospheres using a truncation approximation” (JQSRT 1988;40:51-69) and discuss our results in a broader historical context.     

The Hanle effect with partial frequency redistribution. Construction of a frequency-dependent polarization matrix and numerical solution by a PALI method

The polarized approximate lambda iteration (PALI) technique developed for the weak field Hanle effect relies on the decomposition of the Stokes parameters I, Q and U into six cylindrically symmetrical components. It has been applied to complete and partial frequency redistribution with redistribution matrices in which frequency redistribution is decoupled from scattering polarization. For partial frequency redistribution, the decoupling is obtained by an adequate decomposition of the frequency space into several domains. By angle-averaging frequency-dependent terms in the exact weak field Hanle redistribution matrix for a normal Zeeman triplet, we construct a redistribution matrix with coupling between frequency redistribution and polarization and no domain decomposition. The coupling is contained in generalized frequency redistribution functions that depend on the magnetic field. The redistribution matrix is expanded in the Landi Degl’Innocenti spherical tensors for polarimetry and the Stokes parameters are decomposed into cylindrically symmetrical components. A PALI method is set up for the calculation of these components. The Stokes parameters are calculated for different simple atmospheric models. The positive Q direction corresponds to the linear polarization perpendicular to the solar limb. It is shown that the frequency space decomposition may induce large errors on Stokes U in the transition region between line core and line wings but can safely be used for Stokes I and Q, the errors staying less than 1% at all the frequencies.     

Three-dimensional polarized Monte Carlo atmospheric radiative transfer model (3DMCPOL): 3D effects on polarized visible reflectances of a cirrus cloud

A polarized atmospheric radiative transfer model for the computation of radiative transfer inside three-dimensional inhomogeneous mediums is described. This code is based on Monte Carlo methods and takes into account the polarization state of the light. Specificities introduced by such consideration are presented. After validation of the model by comparisons with adding-doubling computations, examples of reflectances simulated from a synthetic inhomogeneous cirrus cloud are analyzed and compared with reflectances obtained with the classical assumption of a plane parallel homogeneous cloud (1D approximation). As polarized reflectance is known to saturate for optical thickness of about 3, one could think that they should be less sensitive to 3D effects than total reflectances. However, at high spatial resolution (80 m), values of polarized reflectances much higher than the ones predicted by the 1D theory can be reached. The study of the reflectances of a step cloud shows that these large values are the results of illumination and shadowing effects similar to those often observed on total reflectances. In addition, we show that for larger spatial resolution (10 km), the so-called plane-parallel bias leads to a non-negligible overestimation of the polarized reflectances of about 7–8%.     

Numerical simulations of single and multiple scattering by fractal ice clusters

We consider the scattering model in the form of a vertically and horizontally homogeneous particulate slab of an arbitrary optical thickness composed of widely separated fractal aggregates built of small spherical ice monomers. The aggregates are generated by applying three different approaches, including simulated cluster-cluster aggregation (CCA) and diffusion-limited aggregation (DLA) procedures. Having in mind radar remote-sensing applications, we report and analyze the results of computations of the backscattering circular polarization ratio obtained using efficient superposition T-matrix and vector radiative-transfer codes. The computations have been performed at a wavelength of 12.6 cm for fractal aggregates with the following characteristics: monomer refractive index m=1.78+i0.003, monomer radius r=1 cm, monomer packing density p=0.2, overall aggregate radii R in the range 4≤R≤10 cm and fractal dimensions D_f=2.5 and 3.We show that for aggregates generated with simulated CCA and DLA procedures, the respective values of the backscattering circular polarization ratio differ weakly for D_f=2.5, but the differences can increase somewhat for D_f=3, especially in case of an optically semi-infinite medium. For aggregates with a spheroidal overall shape, the dependence of the circular polarization ratio on the cluster morphology can be quite significant and increases with increasing the aspect ratio of the circumscribing spheroid.     

Emergent intensity from a plane-parallel medium exposed to a Gaussian laser beam: A comparison of Rayleigh and isotropic scattering

The focus of this two-dimensional study is the radially varying intensity emergent from a plane-parallel scattering medium exposed to a collimated, Gaussian laser beam directed perpendicular to the upper surface. The method of analysis is the integral transform technique. Specifically, this work uses the generalized reflection and transmission functions from a previous study to construct the emergent intensity with the use of an inverse Hankel transform. Radially varying backscattered and transmitted intensities are calculated for media with isotropic and Rayleigh scattering phase functions and optical thicknesses that range from 0.125 to 8.0. The behavior of the emergent radiation inside and outside the beam is investigated for both narrow and wide beams. A new integration method is implemented to compute the emergent intensity at the beam center. The emergent intensity at the beam center is used to quantify when a one-dimensional model may be used. As expected, for small optical thicknesses and near the beam the phase function has significant influence, while far from the beam multiple scattering reduces the influence of the Rayleigh phase function. Results from this study will be useful in understanding and interpreting more complicated situations, such as those that include polarization.     

Maxwell's equations, radiative transfer, and coherent backscattering: A general perspective

This tutorial paper provides a general overview of the hierarchy of problems involving electromagnetic scattering by particles and clarifies the place of the radiative transfer theory and the theory of coherent backscattering in the context of classical electromagnetics. The self-consistent microphysical approach to radiative transfer is compared with the traditional phenomenological treatment.     

Heat transfer by radiation and conduction in fibrous media without axial symmetry

Coupled radiative and conductive heat transfer in a fibrous medium formed by silica fibres is investigated in this paper by not taking account of the axial symmetry for the distribution of fibres or the boundary conditions. Radiative properties of the medium are calculated by using the Mie theory. The model obtained depends only on optical parameters (indices of silica) and on morphological parameters (diameter and orientation of the fibres, density of the medium). Simulations make it possible to study the strongly anisotropic behaviour of the scattering of the radiation by a fibre and to study the influence of various parameters on the radiative properties of the medium. The results of the Mie theory make possible the simulation of the heat transfer coupled by radiation and conduction. To do this, we introduce a new numerical scheme able to simulate heat transfer in the lack of axial symmetry. With this model, we can show the effects of distribution of fibres and temperature on the thermal behaviour of the medium as well as showing the importance of the phenomenon of scattering in fibrous media.     

Electromagnetic scattering by a morphologically complex object: Fundamental concepts and common misconceptions

Following Keller (Proc Symp Appl Math 1962;13:227-46), we classify all theoretical treatments of electromagnetic scattering by a morphologically complex object into first-principle (or “honest” in Keller's terminology) and phenomenological (or “dishonest”) categories. This helps us identify, analyze, and dispel several profound misconceptions widespread in the discipline of electromagnetic scattering by solitary particles and discrete random media. Our goal is not to call for a complete renunciation of phenomenological approaches but rather to encourage a critical and careful evaluation of their actual origin, virtues, and limitations. In other words, we do not intend to deter creative thinking in terms of phenomenological short-cuts, but we do want to raise awareness when we stray (often for practical reasons) from the fundamentals. The main results and conclusions are illustrated by numerically-exact data based on direct numerical solutions of the macroscopic Maxwell equations.     

Modeling single-scattering properties of small cirrus particles by use of a size-shape distribution of ice spheroids and cylinders

In this study, we model single-scattering properties of small cirrus crystals using mixtures of polydisperse, randomly oriented spheroids and cylinders with varying aspect ratios and with a refractive index representative of water ice at a wavelength of 1.88 μm. The Stokes scattering matrix elements averaged over wide shape distributions of spheroids and cylinders are compared with those computed for polydisperse surface-equivalent spheres. The shape-averaged phase function for a mixture of oblate and prolate spheroids is smooth, featureless, and nearly flat at side-scattering angles and closely resembles those typically measured for cirrus. Compared with the ensemble-averaged phase function for spheroids, that for a shape distribution of cylinders shows a relatively deeper minimum at side-scattering angles. This may indicate that light scattering from realistic cirrus crystals can be better represented by a shape mixture of ice spheroids. Interestingly, the single-scattering properties of shape-averaged oblate and prolate cylinders are very similar to those of compact cylinders with a diameter-to-length ratio of unity. The differences in the optical cross sections, single-scattering albedo, and asymmetry parameter between the spherical and the nonspherical particles studied appear to be relatively small. This may suggest that for a given optical thickness, the influence of particle shape on the radiative forcing caused by a cloud composed of small ice crystals can be negligible.     

Plane Wave and Gaussian Beam Scattering by Long Dielectric Cylinders: 2.5D Simulations versus Measurements

For the development of millimeter wave imaging systems, it is important to be able to simulate some representative scattering configurations. Typically, Gaussian beams are used in active imaging systems. Since these beams only illuminate a spatially limited region, many objects can be treated as infinitely long 2D (in)homogenous cylinders. However, the incident Gaussian beams have a 3D character. Therefore, a dedicated 2.5D scattering simulator was developed. In this paper, simulation results obtained with this simulator are compared to measurements obtained from a bi-static microwave set-up and from a W-band millimeter wave set-up. Comparison of simulations and measurements proves that the 2.5D algorithm is a good simulation tool to study scattering of long inhomogeneous cylinders, illuminated by 3D plane waves or 3D Gaussian beams under different elevation angles.     

基于ARTS的傅里叶红外高光谱计算模型研究及其影响因素分析

在基于红外高光谱辐射数据进行大气遥感方面的研究中,准确模拟红外高光谱数据是很重要的一步。分析了红外高光谱辐射仪的测量原理,建立了基于Atmospheric Radiation Transfer Simulator(ARTS）的考虑仪器干涉图截断与离散化处理过程的正向模型。在该正向模型中,首先采用高光谱辐射传输模式ARTS模拟得到离散化理想光谱,通过逆傅里叶变换将理想光谱转化为干涉图,对干涉图加窗截断处理,模拟仪器响应函数对干涉图的影响,最后采用傅里叶变换得到仪器测量光谱。在这一过程中,窗口函数的选择取决于仪器的干涉图截断方式。未经过切趾处理的仪器,其对应的窗口函数为矩形窗口;经过切趾函数处理,可以减少干涉图截断造成的能量泄露现象。逆傅里叶变换与傅里叶变换过程中必须满足Nyquist采样定律。基于已建立的正向模型,模拟了Atmospheric Emitted Radiance Interferometer (AERI)在Southern Great Plains (SGP)站点的108组晴空辐射数据,并与AERI的实测结果进行比较分析,结果发现理想光谱与AERI实测光谱在吸收线上差异较大,最大残差达到35 mW·sr-1·m-2·(cm-1)-1（简称RU）以上,增加干涉图截断过程后,模拟光谱与实测光谱的最大残差减小到10 RU以内。截断过程的增加对模拟光谱的精度有明显提高,尤其在吸收线上,模拟光谱明显被平滑,模拟精度显著提高。进一步分析六种常用窗口函数截断处理的结果与AERI实测数据的残差,结果发现,模拟过程中选择窗口函数为矩形窗口时,模拟光谱与AERI实测数据残差最小,基本可以约束在5 RU以内,确定了AERI的干涉图截断方式可以近似看作矩形截断。另外,在理想光谱转换为干涉图的过程中,理想光谱分辨率的选择决定了干涉图信息的采样率以及ARTS的计算效率,因此综合考虑模型计算精度和模型计算效率,确定最佳的理想光谱分辨率对于提高模型计算效能是非常必要的;基于此,本文模拟了不同理想光谱分辨率下的仪器测量光谱,对比分析了模拟光谱与AERI实测光谱的残差分布,并讨论了光谱分辨率对模型计算耗时的影响。结果表明,对于AERI,在对应的正向模型中设置理想光谱分辨率为0.241 1 cm-1时,可在保证模型准确度的前提下,最大化模型计算效率。     

VLIDORT: A linearized pseudo-spherical vector discrete ordinate radiative transfer code for forward model and retrieval studies in multilayer multiple scattering media

We describe a new vector discrete ordinate radiative transfer model with a full linearization facility. The VLIDORT model is designed to generate simultaneous output of Stokes vector light fields and their derivatives with respect to any atmospheric or surface property. We develop new implementations for the linearization of the vector radiative transfer solutions, and go on to show that the complete vector discrete ordinate solution is analytically differentiable for a stratified multilayer multiply scattering atmospheric medium. VLIDORT will generate all output at arbitrary viewing geometry and optical depth. The model has the ability to deal with attenuation of solar and line-of-sight paths in a curved atmosphere, and includes an exact treatment of the single scatter computation. VLIDORT also contains a linearized treatment for non-Lambertian surfaces. A number of performance enhancements have been implemented, including a facility for multiple solar zenith angle output. The model has been benchmarked against established results in the literature.     

A numerical scheme for the solution of axisymmetric radiative transfer problems in irregular domains filled by media with opaque and transparent diffuse and specular boundaries

This paper presents a new numerical scheme of the discrete ordinates method for the solution of axisymmetric radiative transfer problems in irregular domains filled by media with opaque and transparent diffuse and specular (Fresnel) boundaries and interfaces. New test problems of radiative transfer, which describe radiative transfer in domains with Fresnel interfaces, are proposed in this paper. These problems admit analytic solutions and can be used as benchmark ones. The proposed scheme is applied to the solution of the problems. Numerical results show that the presence of Fresnel interfaces leads to an appreciably larger error in numerical solution. This is connected with the “discontinuity” of the Fresnel reflectivity, which, through numerical diffusion, leads to the distortion of numerical solution. Modification of the scheme allows to reduce the numerical error.