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.     

Exact results in modeling planetary atmospheres—II. Semi-gray atmospheres

We solve the radiative transfer equation for a semi-gray planetary atmosphere in radiative equilibrium, in an attempt to define an entirely analytical non-gray model atmosphere of finite optical thickness. The salient feature of the model is that the incident solar radiation is partitioned between two adjacent spectral domains—the “visible” and the “infrared”—in each of which the atmosphere's (effective) opacity is assumed to be independent of frequency (the semi-gray assumption). We envisage a plane-parallel atmosphere illuminated by a beam of parallel radiation and bounded below by a partially reflecting and emitting ground. The former emits infrared radiation, induced by the absorption of radiation both visible and infrared, deriving from the external irradiation as well as from the emission of the planet's surface layer. For an atmosphere with given single-scattering albedos and optical thicknesses in both the visible and infrared domains, we compute the temperature at every depth of the atmosphere, as well as the ground's temperature.     

Formation of spectral lines in planetary atmospheres—V. Collision narrowed profiles of quadrupole lines in hydrogen atmospheres

The interpretation of a hydrogen quadrupole spectrum is complicated by the collision narrowing phenomenon exhibited by these lines. The profile must be accurately computed for use within any interpretative technique since the spectroscopic features of line profile and center to limb variation of the equivalent width are sensitive to the quadrupole line shape.The Dicke profile, which assumes that the quadrupole line shape is Lorentzian with a half-width that varies inversely as the pressure has been suggested as a simple alternative to the Galatry profile. In this paper, we present comparisons of the line profile, equivalent width and center-to-limb variation of the equivalent width of quadrupole lines for a characteristic scattering and nonscattering hydrogen atmosphere, where the line profile has been computed by the Dicke and Galatry shapes. Our computations show that the results produced by the Dicke profile may be very different from the features predicted by the Galatry profile, which has been shown to give results in good agreement with experiment.We therefore conclude that it is invalid to use the Dicke approximation for computing collision-narrowed profiles of hydrogen quadrupole lines at visible and near infrared spectroscopic wavelengths in any hydrogen atmosphere.     

Atmospheric weighting functions and surface partial derivatives for remote sensing of scattering planetary atmospheres in thermal spectral region: general adjoint approach

An approach to formulation of inversion algorithms for remote sensing in the thermal spectral region in the case of a scattering planetary atmosphere, based on the adjoint equation of radiative transfer (Ustinov (JQSRT 68 (2001) 195; JQSRT 73 (2002) 29); referred to as Papers 1 and 2, respectively, in the main text), is applied to the general case of retrievals of atmospheric and surface parameters for the scattering atmosphere with nadir viewing geometry. Analytic expressions for corresponding weighting functions for atmospheric parameters and partial derivatives for surface parameters are derived. The case of pure atmospheric absorption with a scattering underlying surface is considered and convergence to results obtained for the non-scattering atmospheres (Ustinov (JQSRT 74 (2002) 683), referred to as Paper 3 in the main text) is demonstrated.     

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.     

The resolution of adjacent spectral lines in the photoelectric recording of Raman spectra

Under the assumption that systematic errors are small, the problem of resolving two lines while scanning a spectrum is discussed. The errors resulting from equipment inertia, from the slit, and from the detector noise are discussed. The relationships determined permit one to choose the slit width of the spectrometer and the time constant of the recording system so that optimum line resolution for various distances between the lines and for various line-width and line-intensity ratios is obtained.     

The scattering of line radiation—III. The source function

After discussing the problems encountered in any attempt to interpret frequency-dependent source functions in terms of the population numbers, a method is developed for doing this to a certain class of frequency-dependent source functions. It is pointed out that, unlike the frequency-independent source functions, an analysis of this nature on the frequency-dependent source functions is not generally practicable and can be done in this case only by virtue of the particular form assumed for the absorption coefficient. Solutions in the form of source functions are presented and discussed for a semi-infinite uniform atmosphere for five different forms of frequency redistribution during the scattering event. The function ∫^∞₀S(ν)φ(ν) dν is then calculated and subsequently is used to obtain the departure parameters of the populations of the energy states. The different population distributions through the atmosphere, that can be deduced from these quantities, are compared for the different cases of frequency redistribution used. Because of the nature of the results obtained, the feasibility of setting up iterative schemes to obtain frequency-dependent source functions for more complex problems is also considered.     

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.     

Efficient unbiased variance reduction techniques for Monte Carlo simulations of radiative transfer in cloudy atmospheres: The solution

We present five new variance reduction techniques applicable to Monte Carlo simulations of radiative transfer in the atmosphere: detector directional importance sampling, n-tuple local estimate, prediction-based splitting and Russian roulette, and circum-solar virtual importance sampling. With this set of methods it is possible to simulate remote sensing instruments accurately and quickly. In contrast to all other known techniques used to accelerate Monte Carlo simulations in cloudy atmospheres - except for two methods limited to narrow angle lidars - the presented methods do not make any approximations, and hence do not bias the result. Nevertheless, these methods converge as quickly as any of the biasing acceleration techniques, and the probability distribution of the simulation results is almost perfectly normal. The presented variance reduction techniques have been implemented into the Monte Carlo code MYSTIC (“Monte Carlo code for the physically correct tracing of photons in cloudy atmospheres”) in order to validate the techniques.     

Effects of collisional interference of lines in the spectra of symmetric top molecules: II. Self- and foreign-gas-broadening of rotational spectral lines

The theory outlined in the first part of the study is applied to an analysis of self-broadening and broadening by foreign gases of spectral lines for a number of symmetric top molecules with a high potential barrier to the inversion vibration. In all cases considered, the results obtained are in better agreement with the experimental data than previous theoretical results; however, in some cases, the discrepancy is rather large. Possible reasons are discussed and the acute need for experimental data obtained with modern equipment is noted.     

A translational heat bath model for foreign gas broadening of spectral lines—III. W.K.B. approximation for satellite bands

The W.K.B. approximation for the line-wing profile is developed by using a translational heat bath model. A general expression is developed for the semiclassical shape of satellite bands. The results are compared with other works.     

A translational heat-bath model for foreign gas broadening of spectral lines—II. The T approximation

The theory of foreign-gas broadening is developed for a central potential by using the T approximation, in a Green's functions formalism. The translational degrees of freedom of the absorbing and foreign gases are separated from the internal degrees of freedom by a translational heat bath model. The line profile is expressed by a Lorentzian in which the half width and shift are functions of frequency. A general method is developed for calculating the half width as a function of frequency. The Lorentzian line profile obtained for a hard-core potential is compared with the explicit line profile obtained for weak potentials by the Born approximation. The results of the present paper are compared with those of other theoretical studies.     

Effects of collisional interference of lines in the spectra of symmetric top molecules: III. Broadening of rotational transitions with a hyperfine structure

The peculiarities of the broadening of rotational transitions of molecules in the case where the spectrum exhibits a hyperfine structure caused by the nuclear quadrupole interaction are considered. The relaxation parameters of the components of a number of hyperfine multiplets of CH₃I molecules are calculated taking into account self-broadening, and calculations are compared with experiment.     

The use of complex wiggle-beat patterns for the estimation of small splittings in NMR spectra

The analysis of the ‘wiggle-beat’ patterns produced by rapid passage through simple NMR multiplets is modified and extended to permit analysis of beats from multiplets where adjacent signals are not equally separated and where the intensities are not necessarily in the ratio of the appropriate binomial coefficients. The simple example of mono-silane-d₁ (SiH₃D) is illustrated: J _HD = 0·427 ± 0·003 c/s. Allowance is also made for the finite time of passage through the signal and the usefulness of this demonstrated by analysing the beat pattern from the asymmetric spectrum of a mixture of SiH₃D and SiH₄. In this way the SiH₄-SiH₃D chemical shift is estimated to be 0·0080 ± 0·0005 p.p.m., the SiH₄ resonance being to low field.     

Necessary and sufficient conditions for the existence of a Lagrangian in field theory. III. Generalized analytic representations of tensorial field equations

In this paper we first study the equivalence transformations of class C², regular, tensorial, quasi-linear systems of field equations which (a) preserve the continuity, regularity, and quasi-linear structure of the systems; and (b) occur within a fixed system of Minkowski coordinates and field components. We identify, among the transformations of this class, those which either induce or preserve a self-adjoint structure of the field equations and we term them genotopic and isotopic transformations, respectively. We then give the necessary and sufficient conditions for an equivalence transformation of the above type to be either genotopic or isotopic. By using this methodology, we then extend the theorem on the necessary and sufficient condition for the existence of ordered direct analytic representations introduced in the preceding paper to the case of ordered indirect analytic representations in terms of the conventional Lagrange equations; we introduce a method for the construction of a Lagrangian, when it exists, in this broader context; and we explore some implications of the underlying methodology for the problem of the structure of the Lagrangian capable of representing interactions within the framework of the indirect analytic representations. Some of the several aspects which demand an inspection prior to the use of this analytic approach in actual models are pointed out. In particular, we indicate a possible deep impact in the symmetries and conservation laws of the system generated by the use of the concept of indirect analytic representation. As a preparatory step prior to the analysis of these problems, we study some methodological aspects which underlie the generalized Lagrange equations postulated in the first paper of this series for the case when they are regular, namely, when they are simple equivalence transformations of the conventional Lagrange equations. We first introduce a generalization of the action principle capable of inducing the generalized as well as the conventional equations. In this way we establish that the former equations are “bona fide” analytic equations. Finally, as our most general analytic framework for the case of unconstrained field equations, we work out the necessary and sufficient condition for the existence of ordered direct analytic representations of quasi-linear systems in terms of the generalized analytic equations and study their relationship to the conventional representations.     

The use of photoinduced light scattering for the evaluation of photoelectric fields in Lithium Niobate Crystals

The photovoltaic and diffusion fields in nominally pure single crystals of stoichiometric composition (R = Li/Nb = 1) grown from the melt with 58.6 mol % of Li₂O (LiNbO₃ stoich), in the nominally pure single crystals of congruent composition (LiNbO₃), and in congruent single crystals doped with Cu²⁺, Zn²⁺, and Gd³⁺ are found from the parameters of the photoinduced light scattering indicatrix obtained with the use of a 60-mW He-Ne laser.     

Using chemometric methods for overlap correction of sodium–zinc spectral lines generated by wavelength dispersive X‐ray fluorescence in mineral samples

Partial least squares, principal component regression and support vector machine multivariate methods were used for overlap correction of sodium–zinc (Na(Kα)–Zn(Lα)) spectral lines generated by means of wavelength dispersion X‐ray fluorescence (WDXRF) combined with standard‐less software (IQ+) technique for the analyses of mineral samples. This methodology uses one scan channel using PX1 analyzer crystal, 550‐µm collimator, flow detector (Ar + CH₄), and rhodium (Rh) tube for determination of Na and Zn in mineral compositions in minimum time. The calibration matrix was made up of 35 samples containing different amounts of Na₂O and ZnO. The considered concentration ranges were 0–5% for both Na₂O and ZnO. The values for 2θ angle were recorded between 25° and 29.9° at every 0.1°. Variable tube powers (kV ? mA) were used to investigate the effect of tube power on the analyses of elements. The validation of the multivariate methods was realized by analyzing soil samples. Atomic absorption and flame photometry methods were used as reference methods for analyzing Zn and Na in the soil samples, respectively. The results of using chemometric methods, WDXRF (standard‐less software) and reference method determined partial least squares and support vector machine models obtained more acceptable results for Na₂O in presence of ZnO than those of WDXRF (standard‐less software). Copyright © 2013 John Wiley & Sons, Ltd.     

The use of the galerkin method for radiation transfer in an anisotropically scattering slab with reflecting boundaries

Radiation transfer in an absorbing, emitting, anisotropically scattering, plane-parallel medium with diffusely reflecting boundaries is solved by application of the Galerkin method. With this approach, the radiation heat flux, angular distribution of radiation intensity, and the divergence of the radiation heat flux anywhere in the medium can be determined highly accurately. For optical thicknesses up to about 10, exact results are also readily obtainable if sufficient number of terms are considered in the expansion. Numerical results are presented for representative cases.     

光散射聚集速率测定中T矩阵方法的应用

聚集速率是评估胶体体系特性及稳定性的关键参数, 静态光散射和动态光散射则是测量聚集速率的两个重要方法. 然而, 用静态光散射和动态光散射测量聚集速率时, 需要知道有关单粒子和双粒子聚集体光散射特性的数据. 为此, 通常需要把动、静两种方法结合, 才能消去这个数据. 以前各种近似理论曾用来解决这个问题, 但因粒子尺寸和形状的限制, 结果并不理想. 而T矩阵方法可以不受粒子大小和形状的限制计算其光散射特性. 本工作用T矩阵方法直接计算静态光散射和动态光散射所必须的粒子散射特性, 并将该法得到的聚集速率与动静态光散射结合法得到的聚集速率进行了比较, 两者结果很接近. 本工作为简化静态光散射和动态光散射测量聚集速率, 扩展其应用范围开辟了新途径. 关键词： T矩阵')" href="#">T矩阵 光散射法 聚集速率