Exact results in modeling planetary atmospheres—III.: The general theory applied to the Earth's semi-gray atmosphere

We apply the semi-gray model of our previous paper to the particular case of the Earth's atmosphere, in order to illustrate quantitatively the inverse problem associated with the direct problem we dealt with before. From given climatological values of the atmosphere's spherical albedo and transmittance for visible radiation, the single-scattering albedo and the optical thickness in the visible are inferred, while the infrared optical thickness is deduced for given global average surface temperature. Eventually, temperature distributions in terms of the infrared optical depth will be shown for a terrestrial atmosphere assumed to be semi-gray and, locally, in radiative and thermodynamic equilibrium.     

K. Schwarzschild's problem in radiation transfer theory

We solve exactly the problem of a finite slab receiving an isotropic radiation on one side and no radiation on the other side. This problem—to be more precise the calculation of the source function within the slab—was first formulated by K. Schwarzschild in 1914. We first solve it for unspecified albedos and optical thicknesses of the atmosphere, in particular for an albedo very close to 1 and a very large optical thickness in view of some astrophysical applications. Then we focus on the conservative case (albedo=1), which is of great interest for the modeling of grey atmospheres in radiative equilibrium. Ten-figure tables of the conservative source function are given. From the analytical expression of this function, we deduce (1) a simple relation between the effective temperature of a grey atmosphere in radiative equilibrium and the temperature of the black body that irradiates it, (2) the temperature at any point of the atmosphere when it is in local thermodynamical equilibrium. This temperature distribution is the counterpart, for a finite slab, of Hopf's distribution in a half-space. Its graphical representation is given for various optical thicknesses of the atmosphere.     

Thin cadmium sulphide film for radiative cooling application

The present work reports the possibility of a specific shield (CdS) to provide passive cooling for the purpose of reducing the use of classical active method. The ideal radiation shield would completely block solar radiation, but allow complete transmission in the “atmospheric-window” region.Chemical solution deposition of the thin film CdS (1 mm) for radiative cooling is described and optical properties of the thin film were measured by an OL-750 Spectroradiometer. The radiative properties of the shield improved optical properties of cooling purposes; which indicates that it has very low IR band reflectance and is transparent across the full 8-13 μm.     

Influence of the vertical profile of Saharan dust on the visible direct radiative forcing

The vertical profile of Saharan dust in the atmosphere is generally characterized by a large aerosol concentration in the mid troposphere, differently from the climatological distribution of other types of particles, that show a peak at the surface and a rapid decrease with height. Saharan dust is also characterized by particles of relatively large size of irregular shape, and variable values of the single scattering albedo (the ratio between radiation scattering and extinction). The dust's peculiar vertical distribution is expected to produce an effect on the calculation of the direct aerosol radiative forcing at the surface and at the top of the atmosphere. This effect is investigated by comparing estimates of aerosol direct visible radiative forcing at the surface and at the top of the atmosphere for dust vertical profiles measured in the Mediterranean, and for the climatological profile. The radiative forcing is estimated by means of an accurate radiative transfer model, and for the ocean surface. The sensitivity of the results on the solar zenith angle, aerosol optical depth, and aerosol absorption is also investigated. The aerosol radiative forcing at the surface shows a very small dependency on the aerosol vertical profile. At the top of the atmosphere, the radiative forcing is weakly dependent on the vertical profile (up to 10% variation on the daily average forcing) for low absorbing particles; conversely, it shows a strong dependency (the daily radiative forcing may vary up to 100%) for absorbing particles. The top of the atmosphere visible radiative forcing efficiency produced by dust having single scattering albedo <0.7 is higher by 4 W m⁻² when the observed vertical profile instead of the standard profile is used in the calculations (i.e. it produces a lower cooling). For values of the single scattering albedo around 0.67, the sign of the forcing depends on the vertical profile. The influence of the vertical distribution on the radiative forcing is largest at small values of the solar zenith angle, and at short wavelengths.     

卷云短波红外辐射特性

 采用离散纵标法耦合大气分子吸收，模拟计算了卷云大气的反射特性。研究了在短波波段卷云辐射性质随波长、卷云光学厚度、卷云有效尺度、云高和卷云中冰晶粒子形状等的变化关系，分析了卷云对大气红外背景辐射的影响。结果表明：在2.7 μm的水汽强吸收带上，卷云的出现明显增强了该波段的大气背景辐射，反射率随光学厚度和云高增大而增大。     

Exact results in modeling planetary atmospheres—I. Gray atmospheres

An exact model is proposed for a gray, isotropically scattering planetary atmosphere in radiative equilibrium. The slab is illuminated on one side by a collimated beam and is bounded on the other side by an emitting and partially reflecting ground. We provide expressions for the incident and reflected fluxes on both boundary surfaces, as well as the temperature of the ground and the temperature distribution in the atmosphere, assuming the latter to be in local thermodynamic equilibrium. Tables and curves of the temperature distribution are included for various values of the optical thickness. Finally, semi-infinite atmospheres illuminated from the outside or by sources at infinity is dealt with.     

Radiation field in a semi-infinite homogeneous atmosphere with internal sources

The equation of radiative transfer in a semi-infinite homogeneous atmosphere with different internal sources is solved by the method of kernel approximation—the kernel in the equation for the Sobolev resolvent function is approximated by a Gauss-Legendre sum. Then the obtained approximate equation can be solved exactly and the solution is a weighted sum of exponentials. All the necessary coefficients of the solutions may be easily found. Since the resolvent function is closely connected with the Green function of the integral radiative transfer equation, the radiation field for different internal sources can be found by simple integration. For the considered cases the formulas for the radiation field are obtained and the respective accuracy estimated. The package of codes in Fortran-77 is given at http://www.aai.ee/∼viik/homogen.for.     

Relationship between different approaches to derive weighting functions related to atmospheric remote sensing problems

The paper is devoted to the investigation of the relationship between different methods used to derive weighting functions required to solve numerous inverse problems related to the remote sensing of the Earth's atmosphere by means of scattered solar light observations. The first method commonly referred to as the forward-adjoint approach is based on a joint solution of the forward and adjoint radiative transfer equations and the second one requires the linearized forward radiative transfer equation to be solved. In the framework of the forward-adjoint method we consider two approaches commonly used to derive the weighting functions. These approaches are referenced as the “response function” and the “formal solution” techniques, respectively. We demonstrate here that the weighting functions derived employing the formal solution technique can also be obtained substituting the analytical representations for the direct forward and direct adjoint intensities into corresponding expressions obtained in the framework of the response function technique. The advantages and disadvantages of different techniques are discussed.     

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.     

Many polarized radiative transfer models

This note is an introduction to the reprint of the 1991 JQSRT article “A new polarized atmospheric radiative transfer model” by K.F. Evans and G.L. Stephens. We discuss the significance of the article, how our two plane-parallel polarized radiative transfer codes came about, how our codes have been used, and more recent developments in polarized radiative transfer modeling.     

Resolution of the uncertainties in the radiative forcing of HFC-134a

HFC-134a (CF₃CH₂F) is the most rapidly growing hydrofluorocarbon in terms of atmospheric abundance. It is currently used in a large number of household refrigerators and air-conditioning systems and its concentration in the atmosphere is forecast to increase substantially over the next 50-100 years. Previous estimates of its radiative forcing per unit concentration have differed significantly ∼25%. This paper uses a two-step approach to resolve this discrepancy. In the first step six independent absorption cross section datasets are analysed. We find that, for the integrated cross section in the spectral bands that contribute most to the radiative forcing, the differences between the various datasets are typically smaller than 5% and that the dependence on pressure and temperature is not significant. A “recommended' HFC-134a infrared absorption spectrum was obtained based on the average band intensities of the strongest bands. In the second step, the “recommended' HFC-134a spectrum was used in six different radiative transfer models to calculate the HFC-134a radiative forcing efficiency. The clear-sky instantaneous radiative forcing, using a single global and annual mean profile, differed by 8%, between the 6 models, and the latitudinally-resolved adjusted cloudy sky radiative forcing estimates differed by a similar amount. We calculate that the radiative forcing efficiency of HFC-134a is .     

Simultaneous retrieval of aerosols and ocean properties: A classic inverse modeling approach. I. Analytic Jacobians from the linearized CAO-DISORT model

In this paper and the sequel, we investigate the application of classic inverse methods based on iterative least-squares cost-function minimization to the simultaneous retrieval of aerosol and ocean properties from visible and near infrared spectral radiance measurements such as those from the SeaWiFS and MODIS instruments. Radiance measurements at the satellite are simulated directly using an accurate coupled atmosphere-ocean-discrete-ordinate radiative transfer (CAO-DISORT) code as the main component of the forward model. For this kind of cost-function inverse problem, we require the forward model to generate weighting functions (radiance partial derivatives) with respect to the aerosol and marine properties to be retrieved, and to other model parameters which are sources of error in the retrievals.In this paper, we report on the linearization of the CAO-DISORT model. This linearization provides a complete analytic differentiation of the coupled-media radiative transfer theory, and it allows the model to generate analytic weighting functions for any atmospheric or marine parameter. For high solar zenith angles, we give an implementation of the pseudo-spherical (P-S) approach to solar beam attenuation in the atmosphere in the linearized model. We summarize a number of performance enhancements such as the use of an exact single-scattering calculation to improve accuracy. We derive inherent optical property inputs for the linearized CAO-DISORT code for a simple 2-parameter bio-optical model for the marine environment coupled to a 2-parameter bimodal atmospheric aerosol medium.     

Simultaneous derivation of intensities and weighting functions in a general pseudo-spherical discrete ordinate radiative transfer treatment

The retrieval of atmospheric constituents from measurements of backscattered light requires a radiative transfer forward model that can simulate both intensities and weighting functions (partial derivatives of intensity with respect to atmospheric parameters being retrieved). The radiative transfer equation is solved in a multi-layer multiply-scattering atmosphere using the discrete ordinate method. In an earlier paper dealing with the upwelling top-of-the-atmosphere radiation field, it was shown that a full internal perturbation analysis of the plane-parallel discrete ordinate solution leads in a natural way to the simultaneous generation of analytically-derived weighting functions with respect to a wide range of atmospheric variables. In the present paper, a more direct approach is used to evaluate explicitly all partial derivatives of the intensity field. A generalization of the post-processing function is developed for the derivation of weighting functions at arbitrary optical depth and stream angles for both upwelling and downwelling directions. Further, a complete treatment is given for the pseudo-spherical approximation of the direct beam attenuation; this is an important extension to the range of viewing geometries encountered in practical radiative transfer applications. The numerical model LIDORT developed for this work is able to generate intensities and weighting functions for a wide range of retrieval scenarios, in addition to the passive remote sensing application from space. We present a number of examples in an atmosphere with O₃ absorption in the UV, for satellite (upwelling radiation) and ground-based (downwelling radiation) applications. In particular, we examine the effect of various pseudo-spherical parameterizations on backscatter intensities and weighting functions with respect to O₃ volume mixing ratio. In addition, the use of layer-integrated multiple scatter output from the model is shown to be important for satellite instruments with wide-angle off-nadir viewing geometries.     

Matrix formulations of radiative transfer including the polarization effect in a coupled atmosphere-ocean system

A vector radiative transfer model has been developed for a coupled atmosphere-ocean system. The radiative transfer scheme is based on the discrete ordinate and matrix operator methods. The reflection/transmission matrices and source vectors are obtained for each atmospheric or oceanic layer through the discrete ordinate solution. The vertically inhomogeneous system is constructed using the matrix operator method, which combines the radiative interaction between the layers. This radiative transfer scheme is flexible for a vertically inhomogeneous system including the oceanic layers as well as the ocean surface. Compared with the benchmark results, the computational error attributable to the radiative transfer scheme has been less than 0.1% in the case of eight discrete ordinate directions. Furthermore, increasing the number of discrete ordinate directions has produced computations with higher accuracy. Based on our radiative transfer scheme, simulations of sun glint radiation have been presented for wavelengths of 670 nm and 1.6 μm. Results of simulations have shown reasonable characteristics of the sun glint radiation such as the strongly peaked, but slightly smoothed radiation by the rough ocean surface and depolarization through multiple scattering by the aerosol-loaded atmosphere. The radiative transfer scheme of this paper has been implemented to the numerical model named Pstar as one of the OpenCLASTR/STAR radiative transfer code systems, which are widely applied to many radiative transfer problems, including the polarization effect.     

Linearization of radiative transfer with respect to surface properties

The linearization of radiative transfer with respect to surface properties in the UV and visible part of the solar spectrum is presented. The proposed method is a rigorous extension of the radiative perturbation theory with respect to surface properties. Given the forward and adjoint intensity field, analytical expressions are provided for the linearization of any observable related to the radiation field with respect to surface properties characterized by Minnaert's and Lambertian bidirectional reflection distribution function. For the considered surface reflection characteristics, we also discuss an extension of the reduction approach of Chandrasekhar as an alternative linearization method. The suitability of both approaches for the combined retrieval of trace gas and surface properties from the backscattered sunlight in the UV and visible part of the spectrum is discussed. The authors come to the conclusion that the perturbation theory, for this purpose, represents the superior method because of its general applicability to any parameter characterizing the optical properties of the atmosphere and the underlying surface.     

Inverse problem for particle size distributions of atmospheric aerosols using stochastic particle swarm optimization

As a part of resolving optical properties in atmosphere radiative transfer calculations, this paper focuses on obtaining aerosol optical thicknesses (AOTs) in the visible and near infrared wave band through indirect method by gleaning the values of aerosol particle size distribution parameters. Although various inverse techniques have been applied to obtain values for these parameters, we choose a stochastic particle swarm optimization (SPSO) algorithm to perform an inverse calculation. Computational performances of different inverse methods are investigated and the influence of swarm size on the inverse problem of computation particles is examined. Next, computational efficiencies of various particle size distributions and the influences of the measured errors on computational accuracy are compared. Finally, we recover particle size distributions for atmospheric aerosols over Beijing using the measured AOT data (at wavelengths λ=0.400, 0.690, 0.870, and 1.020 μm) obtained from AERONET at different times and then calculate other AOT values for this band based on the inverse results. With calculations agreeing with measured data, the SPSO algorithm shows good practicability.     

Integration of Chandrasekhar's integral equation

We solve Chandrasekhar's integration equation for radiative transfer in the plane-parallel atmosphere by iterative integration. The primary thrust in radiative transfer has been to solve the forward problem, i.e., to evaluate the radiance, given the optical thickness and the scattering phase function. In the area of satellite remote sensing, our problem is the inverse problem: to retrieve the surface reflectance and the optical thickness of the atmosphere from the radiance measured by satellites. In order to retrieve the optical thickness and the surface reflectance from the radiance at the top-of-the atmosphere (TOA), we should express the radiance at TOA “explicitly” in the optical thickness and the surface reflectance. Chandrasekhar formalized radiative transfer in the plane-parallel atmosphere in a simultaneous integral equation, and he obtained the second approximation. Since then no higher approximation has been reported. In this paper, we obtain the third approximation of the scattering function. We integrate functions derived from the second approximation in the integral interval from 1 to ∞ of the inverse of the cos of zenith angles. We can obtain the indefinite integral rather easily in the form of a series expansion. However, the integrals at the upper limit, ∞, are not yet known to us. We can assess the converged values of those series expansions at ∞ through calculus. For integration, we choose coupling pairs to avoid unnecessary terms in the outcome of integral and discover that the simultaneous integral equation can be deduced to the mere integral equation. Through algebraic calculation, we obtain the third approximation as a polynomial of the third degree in the atmospheric optical thickness.     

Approximate polarization calculations with scalar radiative transfer theory

The Pomraning phase function can be used to perform approximate polarized Rayleigh transfer calculations with a scalar radiative transfer equation. The approximation is numerically tested for the albedo problem consisting of azimuthally independent radiation incident on a homogeneous semi-infinite atmosphere. The numerical tests were carried out with the same approach used by Viik (JQSRT 68 (2000) 319-326) to numerically test the approximate phase function for solving the Milne problem. Away from the surface the Pomraning phase function gives marginally better results for the diffuse radiation than the usual scalar Rayleigh phase function because it was derived from an asymptotic limit more appropriate for deeper locations in an atmosphere. For optical depths less than unity, though, the scalar Rayleigh approximation is better than the Pomraning approximation.     

Radiative transfer with internal reflection via the converged discrete ordinates method

In response to the challenge of establishing highly accurate solutions to the plane layer radiative transfer equation with the simplest of methods, the converged discrete ordinates method is presented. With this algorithm of only finite difference, quadrature and acceleration, we show how to obtain highly accurate intensities for radiative transfer in a finite layer with internal surface reflection. The method features angular smoothing and angular interpolation through “faux” quadrature. In addition, a manufactured solution demonstrates the high accuracy of the method for forward peaked scattering. We consider scattering in a heterogeneous medium as a final demonstration.