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.     

Discrete ordinate and Monte Carlo simulations for polarized radiative transfer in a coupled system consisting of two media with different refractive indices

We present an algorithm for polarized radiative transfer in a vertically stratified system consisting of two plane-parallel media with different refractive indices. It is based on the discrete ordinate method and includes multiple elastic scattering, thermal radiation, Fresnel reflection and transmission, incident parallel-beam or isotropic radiation at the top of the upper medium and bidirectional reflection at the bottom of the lower medium. Comparisons with results from Monte Carlo simulations show that the discrete-ordinate code provides accurate results for all four elements of the Stokes vector (I, Q, U, and V) at a speed that is orders of magnitude faster.     

考虑地表反射类型的气溶胶散射偏振特性

采用逐次阶散射法求解矢量辐射传输方程来研究气溶胶在不同地表反射模型下的散射偏振特性.首先,选取单一地表反射模型和耦合地表反射模型两种地表反射模型.然后,根据地表反射模型计算得到相应的地表反射率,进而采用逐次阶散射法对矢量辐射传输方程进行求解,得到散射光的Stokes矢量.最后,由Stokes矢量计算得出散射光的偏振度.仿真结果表明,两种地表反射模型下气溶胶单次散射的散射辐射强度和线偏振度均相等;耦合地表反射模型的总散射辐射强度和线偏振度总是大于单一地表反射模型;单一地表反射模型的气溶胶单次散射相对总散射的贡献总是大于耦合地表反射模型.研究结果对气溶胶光学特性的反演具有一定意义.     

Evaluation of errors from neglecting polarization in the forward modeling of O2A band measurements from space, with relevance to CO2 column retrieval from polarization-sensitive instruments

Sensitivity studies have been performed to evaluate the errors resulting from ignoring polarization in analyzing spectroscopic measurements of the O₂A band from space, using the Orbiting Carbon Observatory (OCO) as a test case. An 11-layer atmosphere, with both gas and aerosol loading, and bounded from below by a lambertian reflecting surface, was used for the study. The numerical computations were performed with a plane-parallel vectorized discrete ordinate radiative transfer code. Beam and viewing geometry, surface reflectance and aerosol loading were varied one at a time to evaluate and understand the individual errors. Different behavior was observed in the line cores and the continuum because of the different paths taken by the photons in the two cases. The errors were largest when the solar zenith angle was high, and the aerosol loading and surface reflectance low. To understand the effect of neglecting polarization on CO₂ column retrievals, a linear error analysis study was performed on simulated measurements from the OCO spectral regions, viz. the 1.61 and 2.06 μm CO₂ bands and the O₂A band. It was seen that neglecting polarization could introduce errors as high as 10 ppm, which is substantially larger than the required retrieval precision of ∼2 ppm. A variety of approaches, including orders of scattering, spectral binning and the use of lookup tables are being explored to reduce the errors.     

Polarization of multiple scattered light in planetary atmospheres from solution of the coupled linear integral equations derived for mixed rayleigh-isotropic scattering

A set of coupled integral equations describing nonconservative multiple scattering for a mixed isotropic and Rayleigh single scattering phase function in inhomogeneous, plane-parallel planetary atmospheres is derived. The equations are applicable for the frequency redistributions MRE (monochromatic radiative equilibrium), CFR (complete frequency redistribution), or PFR (partial frequency redistribution). Solution of the equations permits one to calculate the intensity and degree of polarization in an arbitrary direction outside or inside the plane-parallel scattering medium. The equations are readily adaptable to more complicated geometries. Solutions for several cases are presented to demonstrate the versatility and validity of the method. These include a calculation of MRE pure Rayleigh scattering of sunlight in an optically-thick planetary atmosphere to demonstrate agreement with the results of Coulson⁽¹⁾et al. (1960), calculations of the altitude profile of the degree of polarization of the earth's Ly-α 1216A and helium 584A dayglow, and the center to limb variation of the degree of polarization of the sunlight diffusely reflected from a distant planet such that the scattering is pure Rayleigh and conservative at the top of the planet's atmosphere, varying smoothly to conditions of nonconservative and pure isotropic scattering deep in the planet's atmosphere. Tables of functions that one might utilize (without resort to a digital computer) to obtain solutions in the escape function approximation are also given.     

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

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

Impulse-approximation correction in pion-nucleus optical potential

The local density approximation is used in this paper to calculate the first-order pion-nucleus optical potential. TheπN scattering matrix in nuclear medium is computed by employing separableπN scattering matrix. This nuclear-mediumπN scattering matrix, which includes impulse-approximation correction is then used to construct the pion-nucleus optical potential.π-¹²C elastic scattering results obtained by using this potential are compared with the impulse-approximation potential results.     

Symmetry analysis in neutron diffraction studies of magnetic structures: 5. Polarization effects in the scattering of neutrons by magnetic structures

It is shown that both the cross section of elastic scattering by a magnetic structure and the polarization vector of the scattered neutrons beam are described by a set of complex axial vectors M^L. They depend on the vectors of atomic magnetic moments in a primitive cell of the crystal and the wave-vector star of the magnetic structure. Particularly, the intensity of a magnetic Bragg reflection and the polarization vector of the scattered beam of that reflection are determined only by a vector M^L connected with an arm star contribution of the magnetic structure and corresponding to the wave vector k_L of a star {k}. A single magnetic reflection allows one to determinean arm contribution of the magnetic structure if one uses polarization effects and an expansion of the magnetic structure over the basis functions of the irreducible representation of the crystal space group. In the general case minimal number of necessary reflections for the total determination of a magnetic crystal is equal to the arm number of the wave-vector star. All analyses here have been done for the single domain magnetic structure. The polarization phenomena will not be observed for many cases where there is a uniform distribution of domains.     

Variational methods in radiative transfer problems

An efficient variational-iterative method is applied to the problem of diffuse reflection by a plane-parallel inhomogeneous atmosphere with isotropic scattering. The emergent intensity I(τ = 0; μ, μ₀) with μ = μ₀ corresponds to the maximum of an associated functional. It is, however, shown that I(τ = 0; μ, μ₀) computed by the variational method alone has relatively large errors when μ ≠ μ₀. Such deficiencies are removed by a combined variational-iterative method. The interdependence of the iterative and variational methods is also investigated. They are shown to play a complementary role to each other. The proper choice of trial functions is emphasized in light of computational efficiency and flexibility. Two distinct classes of trial functions: the polynomials, and the step functions are investigated as possible choices of trial functions. The latter choice is shown to be far more efficient in computation. Numerical results for both approximate emergent intensities and source functions are presented and found to be in good agreement with the exact solutions. Simple analytic two-step function approximations of the source function and intensities are also presented for the case of a two-layer inhomogeneous model.     

A comparison of numerical and analytical radiative-transfer solutions for plane albedo of natural waters

Several numerical and analytical solutions of the radiative transfer equation (RTE) were compared for plane albedo in a problem of solar light reflection by sea water. The study incorporated the simplest case—a semi-infinite one-dimensional plane—parallel absorbing and scattering homogeneous layer illuminated by a monodirectional light beam. Inelastic processes (such as Raman scattering and fluorescence), polarization and air-water surface refraction-reflection effects, were not considered. Algorithms were based on the invariant imbedding method and two different variants of the discrete ordinate method (DOM). Calculations were performed using parameters across all possible ranges (single-scattering albedo ω₀ and refracted solar zenith angle θ₁), but with a special emphasis on natural waters. All computations were made for two scattering phase functions, which included an almost isotropic Rayleigh phase function and strongly anisotropic double-peaked Fournier-Forand-Mobley phase function. Models were validated using quasi-single-scattering (QSSA) and exponential approximations, which represent the extreme cases of ω₀→0 and ω₀→1, respectively. All methods yielded relative differences within 1.8% for modeled natural waters. An analysis of plane albedo behavior resulted in the development of a new extended QSSA approximation, which when applied in conjunction with the extended Hapke approximation developed earlier, resulted in a maximum relative error of 2.7%. The study results demonstrated that for practical applications, the estimation of inherent optical properties from observed reflectance can best be achieved using an extended Hapke approximation.     

Analytical solution of the direct problem of ellipsometry for some profiles of optical constants of inhomogeneous layers

Differential equations that describe the reflection of polarized light from an optically inhomogeneous medium are considered. In the approximation of small variations of the refractive index, analytical expressions for the reflection coefficients are obtained for both types of polarization for the exponential and harmonic profiles of the optical constants. The accuracy of the obtained expressions is estimated by numerical simulation. It is found that analytical formulas describe well the behavior of the ellipsometric parameters of periodic structures based on Hg_1–xCd_xTe with a sinusoidal profile of the refractive index.     

A vector radiative transfer model for coupled atmosphere and ocean systems with a rough interface

We report on an exact vector (polarized) radiative transfer (VRT) model for coupled atmosphere and ocean systems. This VRT model is based on the successive order of scattering (SOS) method, which virtually takes all the multiple scattering processes into account, including atmospheric scattering, oceanic scattering, reflection and transmission through the rough ocean surface. The isotropic Cox–Munk wave model is used to derive the ref and transmission matrices for the rough ocean surface. Shadowing effects are included by the shadowing function. We validated the SOS results by comparing them with those calculated by two independent codes based on the doubling/adding and Monte Carlo methods. Two error analyses related to the ocean color remote sensing are performed in the coupled atmosphere and ocean systems. One is the scalar error caused by ignoring the polarization in the whole system. The other is the error introduced by ignoring the polarization of the light transmitted through the ocean interface. Both errors are significant for the cases studied. This code fits for the next generation of ocean color study because it converges fast for absorbing medium as, for instance, ocean.     

低太阳高度角条件下的天空偏振模式模拟及大气折射影响研究

为模拟低太阳高度角条件下的天空偏振模式,自主开发了考虑大气球形几何及大气折射效应的辐射传输模式VSPART,并将其运用于漫射光偏振特性仿真.在模式中,基于射线追踪法实现了光线传播轨迹的追踪和入射光偏振态及透过率的计算,基于矩阵算法实现了辐射传输方程的求解.将VSPART模拟结果与基准值、SPDISORT模拟值进行了比较,验证了模型的准确性.在瑞利散射大气和含气溶胶大气条件下,模拟并分析了漫射光偏振度及偏振方向的分布特征,讨论了大气球形几何及折射效应对天空偏振度的影响.结果表明,低太阳高度角条件下,随着波长增加,瑞利散射大气对应的偏振度整体随之增强,中性点向大天顶角方向移动;气溶胶的存在并不改变天空偏振度分布特征,但对偏振方向影响显著,随着光学厚度的增加,天空偏振度值迅速降低;中性点的偏移可能与低阶散射过程紧密相关;大气球形几何和折射效应的主要影响区域为地平线区域、两中性点附近及天顶区域;瑞利散射大气条件下,随着波长增加,大气球形几何及折射效应的影响逐步减弱,特别在中性点附近及天顶区域,其影响逐步消失;随着气溶胶光学厚度的增加,其影响随之增强.     

Elastic and inelastic scattering of protons from Li between 25 and 45 MeV

The elastic and inelastic scattering of protons from ⁶Li has been studied at incident energies of 25.9, 29.9, 35.0, 40.1 and 45.4 MeV. The 2.18 MeV (3⁺, T = 0) first excited state of ⁶Li was found to be strongly excited, but the 3.56 MeV (0⁺, T = 1) second excited state was quite weakly excited. Angular distributions for excitation of the 2.18 MeV level were measured at all five energies, while angular distributions for excitation of the 3.56 MeV level were extracted only at 25.9 and 45.4 MeV. To test the applicability of the optical model for the scattering of protons from such a light nucleus the elastic scattering angular distributions have been analyzed using the eleven-parameter search code SEEK. Available polarization angular distributions were included in the analysis. Reasonable fits to the data have been obtained with an average geometry potential. Theoretical estimates of the real part of the optical potential and the inelastic scattering differential cross sections have been made using the microscopic model for proton-nucleus scattering. Both phenomenological and realistic forces have been considered and the necessary nuclear transition densities have been extracted from experimental elastic and inelastic electron scattering data. An estimate of a possible spin-spin term in the optical potential has also been made.     

海洋水色及水温扫描仪精确瑞利散射计算

瑞利散射计算精度对海洋水色遥感大气校正的精度起着关键作用,为了提高瑞利散射的计算精度,需要求解平行分层大气矢量辐射传输方程。详细地推导了加倍法解大气矢量辐射传输方程的基本关系式,并阐述了利用加倍法解矢量辐射传输方程的原理。通过与海视宽视场遥感器(SeaWiFS)精确瑞利散射查找表计算结果的比较,证明利用加倍法计算瑞利散射的精度优于0．5％,完全能够满足海洋水色遥感大气校正对瑞利散射计算精度的要求,并可以用来生成海洋水色及水温扫描仪精确瑞利散射查找表。最后,利用开发的加倍法解矢量辐射传输方程软件包,生成了海洋水色及水温扫描仪的精确瑞利散射查找表。开发的加倍法解矢量辐射传输方程软件包可直接用于我国第二代高性能海洋水色遥感器精确瑞利散射查找表的生成。     

Fast weighting functions for retrievals from limb scattering measurements

The satellite-borne UV-visible-NIR spectrometers SCIAMACHY and OSIRIS perform operational measurements of the Earth's limb radiance with global coverage. The vertically resolved atmospheric composition is retrieved from the measurements. We have computed synthetic limb measurements and weighting functions (WFs) with several orders of scattering and surface reflection. Comparisons reveal the wavelength-dependent contributions of single scattering and the second orders of scattering and surface reflection. We have also performed test retrievals of the and profiles with the program package SCIARAYS. They prove that the single scattering approximation is sufficient for the calculation of the WFs during the retrieval process. We conclude that algorithms for the analysis of limb scattering measurements can be accelerated by neglecting higher orders of scattering in the WF calculations.     

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.     

Three-dimensional radiative transfer using a fourier-transform matrix-operator method

The three-dimensional equation of transfer for a scattering medium with planar geometry is solved by using a spatial Fourier transform and extending matrix-operator techniques developed previously for the one-dimensional equation. Doubling and adding algorithms were derived by means of an interaction principle for computing the fourier-transformed radiation field. The resulting expressions fully describe the radiative transfer process in a scattering medium, inhomogeneous in the x-, y- and z-directions, illuminated from above by an arbitrarily general intensity field and bounded from below by a surface with completely general reflection properties.