Rapidly convergent series for high-accuracy calculation of the Voigt function

A rapidly convergent series, based on Fourier expansion of the exponential multiplier, is presented for highly accurate approximation of the Voigt function (VF). The corresponding algorithm enables the rapid calculation, required for its implementation as a subprogram in an interpolation approach. The numerical analysis of this VF approximation suggests that it may be more accurate than 10⁻⁹ in the Humlí?ek regions 3 and 4.     

A new procedure for obtaining the Voigt function dependent upon the complex error function

This work proposes a new method for obtaining the differential equation of the Voigt function and, from this equation, expressing the Voigt function as dependent upon the complex error function. In addition, the integral expression of the successive derivatives of the Voigt function is given, and from this a method is generalized which permits the representation, also, of other functions depending on the complex error function. This enables us to simplify other functions which are the convolution of a Gaussian function with rational polynomial functions. Moreover, the relationship between the Lorentzian (w_L), Gaussian (w_G) and Voigt (w_V) widths at half maximum for the function is given, which is of great interest in diverse branches of physics, such as plasma spectroscopy, astrophysics, nuclear magnetic resonance, etc.     

Comment on “A new procedure for obtaining the Voigt function dependent upon the complex error function”

The numerical solution of Eq. (1) relating the Voigt, Lorentzian, and Gaussian full-widths at half-maximum, as shown in Fig. 2 of the article A new procedure for obtaining the Voigt function dependent upon the complex error function by Luque, Calzada and Saez (JQSRT 2005; 94:151-161), is not valid in some parts of the computational domain. The problem resides within a physical domain that is of current spectroscopic interest for the diagnostics of low temperature (∼300 K), high pressure (∼1 atm) dielectric barrier discharge plasmas. The purpose here is to identify the deficiency and to demonstrate a well-conditioned numerical approach for this domain of interest.     

Rapid computation of the Voigt and complex error functions

A simple and efficient method is proposed for the calculation of the complex error function, including its real part, which is the Voigt spectrum line profile. This procedure is based on the observation by Harstad that a rational approximation to the error function, which is the value of the complex error function on the imaginary axis, analytically continues off this axis upon replacement of y by y-ix. The error involved in this approximation can be minimized by choice of an optimal set of coefficients according to methods developed by Curtis and Osborne. An analysis shows that the results are sufficiently accurate for most applications over the whole complex plane, but with the error increasing toward the real axis. The procedure is simpler and faster than previous methods, requiring only two arithmetic statements for coding in FORTRAN. Comparisons are presented of the time required for various methods on various IBM computer systems which supports this conclusion.     

Finite element method for the two-dimensional atmospheric radiative transfer

The finite element method is applied to the solution of the two-dimensional atmospheric radiative transfer. The analysis is mainly focussed on the derivation of the cell or element equation. The Galerkin method and several hybrid methods using the integral and finite difference form of the radiative transfer equation are employed to obtain the cell equation. The assembled system of equations relating the radiances at the lower and upper boundary of the domain is solved by a direct method.     

Rapid and accurate calculation of the Voigt function

Two general techniques significantly improve both the accuracy and speed of spectral line shapes that use only the complex Voigt function. Such line shape functions include the Voigt line profile, which is closely related to the real part of this function. The first technique is a new algorithm, which trades a small amount of RAM (at most 1.49 megabytes) for a considerable gain in speed and accuracy. The accuracy is one part in 10⁶ of the function itself. In addition, the derivatives of the function with respect to x and y are returned with an accuracy of 0.5%. The algorithm is up to nine times faster than the Drayson or Humlí?ek algorithms and about two orders of magnitude more accurate. A second independent improvement is a set of simple criteria to decide when an evaluation is required and when a value of zero can be assumed. This section alone can reduce the computation time by three orders of magnitude or more and it can be applied to algorithms using any Voigt approximation.     

Microwave radiative transfer in scattering atmosphere: Effects of complex permittivity of ice spherical crystals

This paper presents the effects of ice particle's complex permittivity uncertainties on the scattering properties and upwelling brightness temperatures of cloudy atmospheres at the Advanced Microwave Sounding Unit-B (AMSU-B) channel frequencies of 89, 150 and 183 GHz. We investigated the mean deviations of ice particle's optical efficiencies and asymmetry parameters due to the uncertainties in the real and imaginary parts of its complex permittivities. We assumed that the true values of ice particle's permittivity are, respectively, within ±20% for the imaginary part and ±5% for the real part of the permittivity values given by the model of Hufford. Microwave radiative transfer calculations were performed to estimate the absolute errors of brightness temperatures due to uncertainties in ice particle's permittivities. Ice particles were taken to be spherical and their diameters were chosen in the range of 40-4000 μm. Gamma-size distribution was employed in computing volume scattering properties and the effective diameters were 70, 100 and 150 μm with an effective variance being 0.25. We found that ±20% uncertainty in the imaginary part of ice particle's permittivity resulted in only about 10% mean deviations in the absorption efficiencies at the three AMSU-B channel frequencies. However, an uncertainty of ±5% in the real part resulted in more than 15% mean deviations in both scattering and extinction efficiencies, especially significant at the frequency of 183 GHz. The absolute variations of the emerging brightness temperature from the cloudy atmosphere due to uncertainties in the permittivity were found to be more than 1 K, which is already significant compared with the sensitivities achieved with today's technology for millimeter wave radiometers.     

大气辐射传输模型的比较研究

 讨论了三种通用的大气辐射传输模型的特点和使用限制，用辐射传输定律作了数值检验，并与实验测量资料作了比较。结果表明，氧碘激光和氟化氢泛频20P4激光谱线大气透过率的计算值与实验测量值吻合，氟化氢泛频20P5却出现严重偏差。还研究了大气气溶胶种类对大气透过率计算和测量的严重影响。     

An efficient method for evaluation of the complex probability function: The Voigt function and its derivatives

An efficient method is developed to evaluate the function w(z)=e^-z2(1+(2i/√π)∫^z₀e^t2dt) for the complex argument z = x + iy. The real part of w(z) is the Voigt function describing spectral line profiles; the imaginary part can be used to compute derivatives of the spectral line shapes, which are useful, e.g. in least-squares fitting procedures. As an example of the method a simple and fast FORTRAN subroutine is listed in the Appendix from which w(z) in the entire y ? 0 half-plane can be calculated, the maximum relative error being less than 2 × 10^-6 and 5 × 10^-6 for the real and imaginary parts, respectively.     

佛克脱光谱线型干涉图的理论及其应用研究

在一般压强的条件下,佛克脱光谱线型(高斯光谱线型和洛伦兹光谱线型的卷积)是最接近实际的光谱线型,但由于其没有具体的表达式,所以在实际应用中常采用各种近似来表示佛克脱光谱线型。文章未采用任何近似,计算了佛克脱光谱线型干涉图的具体表达式,表明虽然佛克脱光谱线型没有具体表达式,但其干涉图是可以严格求解的。而且其变换结果为一实函数,与高斯光谱线型和洛伦兹光谱线型的变换结果的表达式非常相似,因而可以进行谱线综合展宽的各种计算,而不是像有的文献所预言的有实部和虚部两部分,给计算带来困难。计算结果与实验结果一致。作为应用的一个实例,作者指出了其在大气风场测量中的重要应用。此计算对气体展宽在傅里叶变换光谱仪中的应用方面具有非常重要的意义。     

Impulsive generalized function synchronization of complex dynamical networks

This Letter investigates generalized function synchronization of continuous and discrete complex networks by impulsive control. By constructing the reasonable corresponding impulsively controlled response networks, some criteria and corollaries are derived for the generalized function synchronization between the impulsively controlled complex networks, continuous and discrete networks are both included. Furthermore, the generalized linear synchronization and nonlinear synchronization are respectively illustrated by several examples. All the numerical simulations demonstrate the correctness of the theoretical results.     

Determination of nonlinear optical properties using the Voigt function: Stochastic considerations

In this work, the absorptive and dispersive profiles of a molecular system have been calculated through a methodology based on the averaging of the coherence obtained by the resolution of the optical stochastic Bloch equations. To this aim, a generalized Lorentzian approximation to the Voigt function has been used as a probability distribution, which allows a more generalized analysis of the interactions between the solvent and the molecule. This has been modeled with the parameters of the standard organic dye Green Malaquite, which exhibits a nonlinear behavior under the interaction with a high intensity electric field.     

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.     

波动光学MTF计算误差估计和新的计算方法

用数理统计方法,推导出波动光学MTF数值计算的误差估计式,它适用于对不同的数值计算方法进行自相关积分所求得的MTF值进行误差估计.本文根据波动光学的基本性质,提出了新的MTF数值计算方法,它具有较高的数值精度,更可观的计算量大大减少.     

The kinetic equation and the equation of momentum transfer for a plasma

An attempt is made to derive a simple form of the collision integral of the kinetic equation for a plasma, by using Rostoker’s equation which expresses the pair correlation function in terms of the distribution functions of the particles, and the conditional probability of one particle shielding the other. The conditional probability function is assumed to be given by its equilibrium value. By taking first order velocity-moment of the resulting kinetic equation, the equation of momentum transfer has been obtained.     

Temporal breakdown and Borel resummation in the complex Langevin method

We reexamine the Parisi–Klauder conjecture for complex e^iθ/2?⁴$e^{i\theta /2}{?}^4$ measures with a Wick rotation angle 0≤θ/2≤π/2$0\le \theta /2\le \pi /2$ interpolating between Euclidean signature and Lorentzian signature. Our main result is that the asymptotics for short stochastic times t$t$ encapsulates information also about the equilibrium aspects. The moments evaluated with the complex measure and with the real measure defined by the stochastic Langevin equation have the same t→0$t\to 0$ asymptotic expansion which is shown to be Borel summable. The Borel transform correctly reproduces the time dependent moments of the complex measure for all t$t$, including their t→∞$t\to \infty$ equilibrium values. On the other hand the results of a direct numerical simulation of the Langevin moments are found to disagree from the ‘correct’ result for t$t$ larger than a finite t_c$t_c$. The breakdown time t_c$t_c$ increases powerlike for decreasing strength of the noise’s imaginary part but cannot be excluded to be finite for purely real noise. To ascertain the discrepancy we also compute the real equilibrium distribution for complex noise explicitly and verify that its moments differ from those obtained with the complex measure.     

The 2009 edition of the GEISA spectroscopic database

The updated 2009 edition of the spectroscopic database GEISA (Gestion et Etude des Informations Spectroscopiques Atmosphériques; Management and Study of Atmospheric Spectroscopic Information) is described in this paper. GEISA is a computer-accessible system comprising three independent sub-databases devoted, respectively, to: line parameters, infrared and ultraviolet/visible absorption cross-sections, microphysical and optical properties of atmospheric aerosols. In this edition, 50 molecules are involved in the line parameters sub-database, including 111 isotopologues, for a total of 3,807,997 entries, in the spectral range from 10⁻⁶ to 35,877.031 cm⁻¹.The successful performances of the new generation of hyperspectral sounders depend ultimately on the accuracy to which the spectroscopic parameters of the optically active atmospheric gases are known, since they constitute an essential input to the forward radiative transfer models that are used to interpret their observations. Currently, GEISA is involved in activities related to the assessment of the capabilities of IASI (Infrared Atmospheric Sounding Interferometer; http://smsc.cnes.fr/IASI/index.htm) on board the METOP European satellite through the GEISA/IASI database derived from GEISA. Since the Metop-A (http://www.eumetsat.int) launch (19 October 2006), GEISA is the reference spectroscopic database for the validation of the level-1 IASI data. Also, GEISA is involved in planetary research, i.e., modeling of Titan's atmosphere, in the comparison with observations performed by Voyager, or by ground-based telescopes, and by the instruments on board the Cassini-Huygens mission.GEISA, continuously developed and maintained at LMD (Laboratoire de Météorologie Dynamique, France) since 1976, is implemented on the IPSL/CNRS (France) “Ether” Products and Services Centre WEB site (http://ether.ipsl.jussieu.fr), where all archived spectroscopic data can be handled through general and user friendly associated management software facilities. More than 350 researchers are registered for on line use of GEISA.     

The spatial polarization distribution over the dome of the sky for abnormal irradiance of the atmosphere

The paper deals with the polarized radiative transfer within a slab irradiated by a collimated infinitely wide beam of arbitrary polarized light. The efficiency of the proposed analytical solution lies in the assumption that the complete vectorial radiative transfer solution is the superposition of the most anisotropic and smooth parts, computed separately. The vectorial small-angle modification of the spherical harmonics method is used to evaluate the anisotropic part, and the vectorial discrete ordinates method is used to obtain the smooth one. The azimuthal expansion is used in order to describe the light field spatial distribution for the case of abnormal irradiance and to obtain some known neutral points in the sky especially useful for polarized remote sensing of the atmosphere.     

A quasi-analytic solution of the radiative transfer equation for three-dimensional atmospheres

In this study, we present a new solution of the three-dimensional (3-D) radiation transfer equation (RTE). The solution employs a discretization technique to separate the independent variables involved in the 3-D RTE, and the doubling-adding method to solve the RTE explicitly and quasi-analytically. The remarkable feature of the present solution is the application of scaling-function expansion to those terms that are dependent on horizontal coordinates. Scaling-function expansion is suitable for representing irregular horizontal inhomogeneities with small-scale variations. By applying scaling-function expansion, the 3-D RTE can be formulated in the form of a vector-matrix differential equation; matrices involved in the equation are generally sparse and dominantly diagonal matrices, and this considerably reduces the labor involved in matrix calculations. We tested the performance of the present solution via radiative transfer calculations of solar radiation in horizontally inhomogeneous two-dimensional cloud models. The calculated results indicate that even if the resolution of the scaling-function expansion is too coarse in regions around small-scale variations, the influence does not spread problematically to other regions far from the variations; this illustrates the advantage of the scaling-function expansion. The present solution can be used to investigate quantitatively and to estimate the effects of cloud spatial inhomogeneity on the corresponding radiation field.