Two-Stream Approximation to the Radiative Transfer Equation:A New Improvement and Comparative Accuracy with Existing Methods

Mathematical modeling of the interaction between solar radiation and the Earth's atmosphere is formalized by the radiative transfer equation(RTE), whose resolution calls for two-stream approximations among other methods. This paper proposes a new two-stream approximation of the RTE with the development of the phase function and the intensity into a third-order series of Legendre polynomials. This new approach, which adds one more term in the expression of the intensity and the phase function, allows in the conditions of a plane parallel atmosphere a new mathematical formulation of γparameters. It is then compared to the Eddington, Hemispheric Constant, Quadrature, Combined Delta Function and Modified Eddington, and second-order approximation methods with reference to the Discrete Ordinate(Disort) method(δ –128 streams), considered as the most precise. This work also determines the conversion function of the proposed New Method using the fundamental definition of two-stream approximation(F-TSA) developed in a previous work. Notably,New Method has generally better precision compared to the second-order approximation and Hemispheric Constant methods. Compared to the Quadrature and Eddington methods, New Method shows very good precision for wide domains of the zenith angle μ 0, but tends to deviate from the Disort method with the zenith angle, especially for high values of optical thickness. In spite of this divergence in reflectance for high values of optical thickness, very strong correlation with the Disort method(R ≈ 1) was obtained for most cases of optical thickness in this study. An analysis of the Legendre polynomial series for simple functions shows that the high precision is due to the fact that the approximated functions ameliorate the accuracy when the order of approximation increases, although it has been proven that there is a limit order depending on the function from which the precision is lost. This observation indicates that increasing the order of approximation of the phase function of the RTE leads to a better precision in flux calculations. However, this approach may be limited to a certain order that has not been studied in this paper.

Temperature and Water Vapor Weighting Functions from Radiative Transfer Equation with Surface Emissivity and Solar Reflectivity

Linearization of Radiative Transfer Equation (RTE) is the key step in physical retrieval of atmospheric temperature and moisture profiles from InfRared (IR) sounder observations. In this paper, the successive forms of temperature and water vapor mixing ratio component weighting functions are derived by applying one term variation method to RTE with surface emissivity and solar reflectivity contained. Retrivals of temperature and water vapor mixing ratio profiles from simulated Atmospheric Infrared Sounder (AIRS) observations with surface emissivity and solar reflectivity are presented.

An Easy Algorithm for Solving Radiative Transfer Equation in Clear Atmosphere

An accurate and rapid method for solving radiative transfer equation is presented in this paper. According to the fact that the multiple scattering component of radiance is less sensitive to the error of phase function than the single scattering component is,we calculate the multiple scattering component by using delta-Eddington approximation and the single scattering component by solving radiative transfer equation. On the ground, when multiple sattering component is small, for example, when the total optical depth T is small, the accurate radiance can be obtained with this method. For the need of the space remote sensing, the upward radiance at the top of the atmosphere is mainly studied, and an approximate expression is presented to correct the multiple scattering component. Compared with the more precise Gauss-Seidel method.the results from this method show an accuracy of better than 10% when zenith angle 0 < 50 掳 and T < 1. The computational speed of this method is, however, much faster than that of Gauss-Seidel method.

Laser-phase determination methods and transfer equations for direct temporal structure measurements of atto- and femtosecond XUV pulses

In this paper the laser-phase determination methods and transfer equations are presented to directly reconstruct the detailed temporal structures of ultra-short extreme ultraviolet （xuv） pulses from the measured photoelectron energy spectra （PES）. Each transfer equation includes one of PID （proportional-integral-differential） terms of PES. The intensity and instantaneous frequency of attosecond xuv can be retrieved from the integral term of PES. The intensity profiles of narrow bandwidth atto- and femtosecond xuvs can be rebuilt from the proportional and differential terms of PES respectively. The methods and equations may be used to improve time resolutions in measuring ultrashort pulses.

Accurate reconstruction of the optical parameter distribution in participating medium based on the frequency-domain radiative transfer equation

Reconstructing the distribution of optical parameters in the participating medium based on the frequency-domain radiative transfer equation （FD-RTE） to probe the internal structure of the medium is investigated in the present work. The forward model of FD-RTE is solved via the finite volume method （FVM）. The regularization term formatted by the generalized Gaussian Markov random field model is used in the objective function to overcome the ill-posed nature of the inverse problem. The multi-start conjugate gradient （MCG） method is employed to search the minimum of the objective function and increase the efficiency of convergence. A modified adjoint differentiation technique using the collimated radiative intensity is developed to calculate the gradient of the objective function with respect to the optical parameters. All simulation results show that the proposed reconstruction algorithm based on FD-RTE can obtain the accurate distributions of absorption and scattering coefficients. The reconstructed images of the scattering coefficient have less errors than those of the absorption coefficient, which indicates the former are more suitable to probing the inner structure.

3D Radiative Transfer Equation Coupled with Heat Conduction Equation with Realistic Boundary Conditions Applied on Complex Geometries

This paper presents the solution of coupled radiative transfer equation with heat conduction equation in complex three-dimensional geometries. Due to very different time scales for both physics, the radiative problem is considered steady-state but solved at each time iteration of the transient conduction problem. The discrete ordinate method along with the decentered streamline-upwind Petrov-Galerkin method is developed. Since specular reflection is considered on borders, a very accurate algorithm has been developed for calculation of partition ratio coefficients of incident solid angles to the several reflected solid angles. The developed algorithms are tested on a paraboloid-shaped geometry used for example on concentrated solar power technologies.

Tiêu Equation Experimentation of Understanding by Energy Transfer Quantum Mechanics

Background: The Tiêu equation has a ground roots approach to the process of Quantum Biology and goes deeper through the incorporation of Quantum Mechanics. The process can be measured in plant, animal, and human usage through a variety of experimental or testing forms. Animal studies were conducted for which, in the first day of the study all the animals consistently gained dramatic weight, even as a toxic substance was introduced as described in the introduction of the paper to harm animal subjects which induced weight loss through toxicity. Tests can be made by incorporating blood report results. Human patients were also observed to show improvement to their health as administration of the substance was introduced to the biological mechanism and plants were initially exposed to the substance to observe results. This is consistent with the Tiêu equation which provides that wave function is created as the introduction of the substance to the biological mechanism which supports Quantum Mechanics. The Tiêu equation demonstrates that Quantum Mechanics moves a particle by temperature producing energy thru the blood-brain barrier for example. Methods: The methods for the Tiêu equation incorporate animal studies to include the substance administered through laboratory standards using Good Laboratory Practices under Title 40 C.F.R. § 158. Human patients were treated with the substance by medical professionals who are experts in their field and have knowledge to the response of patients. Plant applications were acquired for observation and guidance of ongoing experiments of animals’ representative for the biologics mechanism. Results: The animal studies along with patient blood testing results have been an impressive line that has followed the Tiêu equation to consistently show improvement in the introduction of the innovation to biologic mechanisms. The mechanism responds to the substance by producing energy to the mechanism with efficient effect. For plant observations, plant organisms responded, and were seen as showing improvement thru visual observation.

THE ASYMPTOTIC PRESERVING UNIFIED GAS KINETIC SCHEME FOR GRAY RADIATIVE TRANSFER EQUATIONS ON DISTORTED QUADRILATERAL MESHES

In this paper,we consider the multi-dimensional asymptotic preserving unified gas kinetic scheme for gray radiative transfer equations on distorted quadrilateral meshes.Different from the former scheme [J.Comput.Phys.285（2015）,265-279] on uniform meshes,in this paper,in order to obtain the boundary fluxes based on the framework of unified gas kinetic scheme（UGKS）,we use the real multi-dimensional reconstruction for the initial data and the macro-terms in the equation of the gray transfer equations.We can prove that the scheme is asymptotic preserving,and especially for the distorted quadrilateral meshes,a nine-point scheme [SIAM J.SCI.COMPUT.30（2008）,1341-1361] for the diffusion limit equations is obtained,which is naturally reduced to standard five-point scheme for the orthogonal meshes.The numerical examples on distorted meshes are included to validate the current approach.

ANALYSIS OF A NUMERICAL METHOD FOR RADIATIVE TRANSFER EQUATION BASED BIOLUMINESCENCE TOMOGRAPHY

In the bioluminescence tomography （BLT） problem, one constructs quantitatively the bioluminescence source distribution inside a small animal from optical signals detected on the animal＇s body surface. The BLT problem is ill-posed and often the Tikhonov regularization is used to obtain stable approximate solutions. In conventional Tikhonov regularization, it is crucial to choose a proper regularization parameter to balance the accuracy and stability of approximate solutions. In this paper, a parameter-dependent coupled complex boundary method （CCBM） based Tikhonov regularization is applied to the BLT problem governed by the radiative transfer equation （RTE）. By properly adjusting the parameter in the Robin boundary condition, we achieve one important property： the regularized solutions are uniformly stable with respect to the regularization parameter so that the regularization parameter can be chosen based solely on the consideration of the solution accuracy. The discrete-ordinate finite-element method is used to compute numerical solutions. Numerical results are provided to illustrate the performance of the proposed method.

Diffusion Limit of 1-D Small Mean Free Path of Radiative Transfer Equations in Bounded Domain

In this paper, we consider the diffusion limit of the small mean free path for the radiative transfer equations, which describe the spatial transport of radiation in material. By using asymptotic expansions, we prove that the nonlinear transfer equa- tion has a diffusion limit as the mean free path tends to zero, and moreover we study the boundary layer problem and mixed layer problem in bounded domain [0,1]. Then we show the validity of their asymptotic expansions relies only on the smoothness of boundary condition, and remove the Fredholm alternative and centering condition.

Discrete Duality Finite Volume Discretization of the Thermal-P_(N) Radiative Transfer Equations on General Meshes

The discrete duality finite volume method has proven to be a practical tool for discretizing partial differential equations coming from a wide variety of areas of physics on nearly arbitrary meshes.The main ingredients of the method are:(1)use of three meshes,(2)use of the Gauss-Green theorem for the approximation of derivatives,(3)discrete integration by parts.In this article we propose to extend this method to the coupled grey thermal-P_(N) radiative transfer equations in Cartesian and cylindrical coordinates in order to be able to deal with two-dimensional Lagrangian approximations of the interaction of matter with radiation.The stability under a Courant-Friedrichs-Lewy condition and the preservation of the diffusion asymptotic limit are proved while the experimental second-order accuracy is observed with manufactured solutions.Several numerical experiments are reported which show the good behavior of the method.

Quantitative Evaluation of Bioluminescence Imaging Based on Radiative Transfer Equation

B iolum inescence in aghg is a khd ofem erghg detection technology at cellular, m olecu lar and genetic level. The most popular b io lum m inescence in aghg model is diffusion approxin ation （DA）. However, because of the ill-posedness of the D A -based hverse problem and the instability of reconstruction algorithm s, the location accuracy of the reconstucted sources is low. Radiative transfer equation （RTE）, which considers the direction of the photon m igration and the effect of absorption and scattering in tissues, can accurately express the transmission ofbiolum hescent photns through the tissues. In this paper, we studied the biolum hescence inaging based on the RTE. 2D sinuiations were performed, and quantitative evaluation was given by the absolute source position error, the relative source area error and the m h in um bound hg box. The resu Its of the experin ent showed that the in aging quality based on R TE was beer than thatone based on D A.

On the combined heat transfer of natural convection, radiation and conduction in the non-gray semitransparent thermal fibrous insulation

A detailed numerical modeling is performed to investigate heat transfer in high-porous, high-temperature non-gray semitransparent silica insulation materials. Radiation between fibers, conduction within fibers and convection from the fibers to the surrounding fluid are considered. Macroscopic (porous media) modeling is used to determine the velocity, pressure and temperatures fields for fibrous insulation with a random packing geometry under natural convection. Based on a non-gray application of the solution to the radiative transfer equation, the value of the refractive index(n,m)is used to generate macroscopic average radiative properties such as extinction coefficient, scattering albedo and phase function. Key features of the macroscopic model include two-dimensional effects,non-gray radiative exchange, and the relaxation of the local thermodynamic non-equilibrium. The effectiveness of this numerical model is validated by the previous experimental data.

New Immersed Boundary Method on the Adaptive Cartesian Grid Applied to the Local Discontinuous Galerkin Method

Currently, many studies on the local discontinuous Galerkin method focus on the Cartesian grid with low computational e ciency and poor adaptability to complex shapes. A new immersed boundary method is presented, and this method employs the adaptive Cartesian grid to improve the adaptability to complex shapes and the immersed boundary to increase computational e ciency. The new immersed boundary method employs different boundary cells(the physical cell and ghost cell) to impose the boundary condition and the reconstruction algorithm of the ghost cell is the key for this method. The classical model elliptic equation is used to test the method. This method is tested and analyzed from the viewpoints of boundary cell type, error distribution and accuracy. The numerical result shows that the presented method has low error and a good rate of the convergence and works well in complex geometries. The method has good prospect for practical application research of the numerical calculation research.

On Urban Heat Island of Beijng Based on Landsat TM Data

Land surface temperature (LST) of Beijing area was retrieved from Landsat TM thermal band data utilizing a radiative transfer equation and the urban heat island (HUI) effects of Beijing and its relationship with land cover and normalized difference vegetation index (NDVI) were discussed. The result of LST showed that the urban LST was evidently higher than the suburban one. The average urban LST was found to 4.5 ℃ and 9 ℃ higher than the suburban and outer suburban temperature, respectively, which demonstrated the prominent UHI effects in Beijing. Prominent negative correlation between LST and NDVI was found in the urban area, which suggested the low percent vegetation cover in the urban area was the main cause of the urban heat island.

Review on retrieval of lunar regolith thickness by active and passive microwave measurements

It is one of the important methods to retrieve lunar regolith thickness using active and passive microwave techniques.The retrieval of lunar regolith thickness is based on microwave radiation transfer process simulation in the regolith media.The lunar regolith model is first introduced,and the features of the involved physical parameters are indicated thereafter,such as dielectric constants,surface roughness,particle size and thermal grads of the lunar regolith.The time delay and the migration of the radar echoes from the different interfaces is the key problem for active microwave measurement.And the simulation of the microwave radiative transfer in the regolith media is the important technique for the passive microwave measurement.The important parameters and the physical mechanism for the two measurements are also presented.

Research on a Class of Non-linear Programming Problems

A detailed study of some simple forms which have a given special structure have been solved, in this paper, we research the extension of this kind of special structure problems.

Use of photoelectron laser phase determination method for attosecond measurements with quantum-mechanical calculations

This paper calculates quantum-mechanically the photoelectron energy spectra excited by attosecond x-rays in the presence of a few-cycle laser. A photoelectron laser phase determination method is used for precise measurements of the pulse natural properties of x-ray intensity and the instantaneous frequency profiles. As a direct procedure without any previous pulse profile assumptions and time-resolved measurements as well as data fitting analysis, this method can be used to improve the time resolutions of attosecond timing and measurements with metrological precision. The measurement range is half of a laser optical cycle.

Quantum-mechanical analysis of pulse reconstruction for a narrow bandwidth attosecond x-ray pulse

The photoelectron energy spectra （PESs） excited by narrow bandwidth attosecond x-ray pulses in the presence of a few-cycle laser are quantum-mechanically calculated. Transfer equations are used to reconstruct the detailed temporal structure of an attosecond x-ray pulse directly from a measured PES. Theoretical analysis shows that the temporal uncertainties of the pulse reconstruction depend on the x-ray bandwidth. The procedure of pulse reconstruction is direct and simple without making any previous pulse assumption, data fitting analysis and time-resolved measurement of PESs. The temporal measurement range is half of a laser optical cycle.

The Radiation Loss of a Cylindrical Methane-Argon Plasma Arc

Calculation of the net radiation emitted by a CH4-Ar mixture, in a temperature range of 5,000-30,000 K with the assumption of local thermodynamic equilibrium （LTE）, is conducted. Continuum and line emissions are taken into account. The radiative transfer of each line is calculated by means of an escape factor depending on the shape and broadening of the line. Assuming a cylindrical, homogeneous, and isothermal plasma, the net emission coefficient is calculated for different pressures between 1 atm and 10 atm and arc radia of 0 mm to 1 mm. Results show that the argon presence in the CH4-Ar mixture has a significant effect on the total radiation emitted for the temperature above 17,000 K and the results for pure argon agree with those of BAUDER and EVANS.