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The article is devoted to the research of changes in lighting terminology on the example of one of the main properties of colour – lightness. The article substantiates the importance of the qualitative definition of terms for the professional understanding of various aspects of colorimetry and possibility of usage colour appearance models (CAM). Examples and comparisons of the wording of theterms of the CIECAM02 system are given based on variants from International Lighting Vocabularies (ILV) different years of release. The next is an example of the history of the development of the terms associated with different conditions of perception and definition of brightness, in particular, such as “lightness” and “subjective brightness”. The importance of the clear agreement between the terms and their definitions for the understanding, use, and development of the modern models and systems of colour perception is substantiated. The English-language and the Russian-language scientific sources are considered. The analysis of which demonstrates the dynamics of changes in the understanding of the term “Lightness”. Special attention is paid to the fate of the term in the Russian-language lighting literature, discrepancies in the interpretation of the term in various sources are shown, which can lead to a misunderstanding of modern colour appearance models. The comparison of the interpretation of the term“Lightness” is reviewing based on the English-language and the Russian-language scientific sources. The comparison helps to identify the reasons that led to the problems of definitions and understanding of this term. In the conclusion, the authors invite experts to discuss lighting terminology and harmonize Russian-language lighting literature with English-language literature. Also substantiates the importance of the participation of lighting engineering scientists in translations of different scientific literary sources devoted to colour perception. It is proposed to continue work on improving, and possibly a special reprinting of translations for the better understanding in the Russian lighting engineering literature.

Reflection characteristics of asphalt pavements play one of the key roles in road lighting, as luminance is normalized and multiple reflections are absent. The values of the luminance factors of road surfaces can be found by carrying out measurements, but this is not always possible, since it requires the extraction of samples, or is laborious. Therefore, in this article it is proposed to model the spatialangular characteristics of reflection. The developed models (analytical and statistical) allow finding luminance factors for arbitrary angles of incidence and sighting. The models are based on a scattering planeparallel layer. The following optical properties of the medium serve as input data: single scattering albedo, optical thickness, phase function of the particles inside the layer. To validate the models, a comparison was made with the measurement results.

Light forms view of the environment and can influence the psychoemotional state in it. Human centric lighting was introduced after the receptor responsible for the effect of illumination on the circadian rhythm, that is, the nonvisual effect of light on the human body, was discovered. To form the concept of human centric lighting requires a holistic approach that takes into account both visual tasks and the psychoemotional state of users in this environment. The degree of discomfort may increase if there are glare light sources in the field of view. Insufficient monotonous illumination with sharp changes in luminance or contrast, which can be estimated using the spatialangular distribution of luminance, can also cause discomfort. The article proposes an experimental approach to assess the quality of illumination in laboratory conditions.

This article is devoted to the research of the influence of induction and crispening effects on the lightness of objects. The target of this research is to simplify and clarify the analysis of the light distributions derived from the results of lighting calculations and measurements, and also to form the recommendations for obtaining preferred lightness contrasts in the case of visual perception of objects on different backgrounds. To provide such possibilities, it is necessary to refine the model of the lightness calculation in modern colour appearance models, which, in turn, requires the results of new experiments to take into account the induction and crispening effects in the conditions of simultaneous contrast. The same experiment was established by Takasaki on visual perception of grey squares on grey backgrounds. The article substantiates the necessity to refine the results of the Takasaki experiment and conduct the new experimental approach in more strictly specified conditions and in the expanded luminance range to ensure conditions of the modern standards. Here we propose setting up an experiment that provides these refinements and considers the properties of the proximal field – the contour of objects. The search of the results uses multi-objective optimization methods. The first results of this experiment are presented, based on them, were formed proposals to determine the preferred luminance of objects to ensure the required levels of lightness contrast when working on specified backgrounds. The calculations of the lightness deviations obtained as the result of the experiment and in the CIECAM02 colour appearance model, are presented. The peculiarities of translations of lighting engineering literature in the field of modelling a light-colour environment are noted, which require clarification for a correct and reliable understanding of colour appearance models, for which is proposed to open a discussion on this subject.

One of the important questions in lighting engineering is to determine the sensation of discomfort from lighting installations. There is no unified psychophysical scale for assessing the visual comfort of lighting (VCL) at any arbitrary distribution of luminance in space. This article considers a mathematical model of the scale based on a neural network (NN) as an ‘expert’ that trained to determine the comfort of perception of lighting depending on the light source’s luminance and background. The experimental data obtained at the lighting engineering department of the National Research University “MPEI” were used to train the NN. The experiment results presented in this article are consistent with the numerical scale for estimating the VCL proposed by Lakiesch and Holladay. A new model allows predicting the sensation of VCL with an accuracy of up to 70 %. This work allows formulating criteria for NN’s input and output parameters to choose a metric for evaluating NN’s performance, such as the confusion matrix, ROC curves, and a metric, such as the probability distribution for each sensation depending on the input parameters. It clearly follows, that amount of initial data is not allowed to make a final conclusion. One more experiment required considering the algorithm used to calibrate the experimental installation, instructions for observers, and the obtained results processing.

For centuries, light has been produced to enable the human visual system to operate but, today, light is being used for an increasing number of non-visual, technical purposes. Examples include plant growth, remote sensing, electricity generation, and communication. This paper discusses the theoretical foundation of such technical applications of light and shows that it is completely identical to the fundamentals of lighting. The foundation is the theory of the light field, which is generated by the interaction of the radiation field with a quadratic (energy) receiver. Within its range of applicability, the theory of the light field is complete and closed. As a result, the light field provides a unified theoretical base for lighting systems and the technical application of light. This creates the basis for combining them into a single section of science and technology, which will ensure their effective development and application.

The article formulates and proves a new approach to assessing the quality of lighting based on the luminance spatial-angular distribution (LSAD). Several aspects of assessing the quality of lighting are described and the main one is selected for further research. Visibility is considered as the first step to the formulation of the quality criterion. The validity of the new approach is proved in laboratory conditions on the example of carrying out a standard visual task and in real conditions in the Moscow metro.

The article proposes the classification of lenses with different symmetrical beam angles and offers a scale as a spot-light’s palette. A collection of spotlight’s images was created and classified according to the proposed scale. The analysis of 788 pcs of existing lenses and reflectors with different LEDs and COBs carried out, and the dependence of the axial light intensity from beam angle was obtained. A transfer training of new deep convolutional neural network (CNN) based on the pre-trained GoogleNet was performed using this collection. GradCAM analysis showed that the trained network correctly identifies the features of objects. This work allows us to classify arbitrary spotlights with an accuracy of about 80 %. Thus, light designer can determine the class of spotlight and corresponding type of lens with its technical parameters using this new model based on CCN.

The paper discusses the possibility of remote detection of a continuous laser beam propagating in a scattering continental and coastal atmosphere, when it is recorded outside the axial zone. In the single scattering approximation, estimates of the radianceat the registration site are carried out, which are compared with the threshold characteristics of existing photodetectors in the visible and IR spectral regions. It is shown that the laser radiation (LR) of the beam is reliably recorded in the range of angles (0–180)° at metrological range of visibility equal (5–20) km at night conditions. At twilight, under the same conditions, detection capabilities are significantly reduced. A significant increase of the LR beam radiance contrast with a decrease in its divergence has been shown experimentally in the field observations. At twilight, a decrease in the beam’s radiance contrast is seen. A beam with a divergence equal to 2 ceases to be distinguishable at angles equal to (80–90)°, and a beam with a divergence of 4 – at angles (60–70)°.In this case, the contrast difference reaches up to 10 times.

The article presents the experiment in studying a new criterion of lighting quality based on spatial and angular distribution of luminance proposed by the Lighting Engineering sub-department of NIU MPEI. The experiment studies correlation between expert evaluations of lighting quality at 21 stations of the Moscow Metro with analysis based on the criterion of quality of RAW-format luminance photographs of the stations made by means of a camera and adjusted according to luminance measured by a luminance meter. The obtained photos were processed using the proposed criterion. The article presents design of station models and calculations made by means of DIALux software and the programme developed (as part of the work) on the basis of local evaluations. It is demonstrated that the proposed criterion allows us to take account of extended veiling reflections and may be considered as enhancement of the unified glare rating UGR.

The article describes the role of polarisation in calculation of multiple reflections. A mathematical model of multiple reflections based on the Stokes vector for beam description and Mueller matrices for description of surface properties is presented. On the basis of this model, the global illumination equation is generalised for the polarisation case and is resolved into volume integration. This allows us to obtain an expression for the Monte Carlo method local estimates and to use them for evaluation of light distribution in the scene with consideration of polarisation. The obtained mathematical model was implemented in the software environment using the example of a scene with its surfaces having both diffuse and regular components of reflection. The results presented in the article show that the calculation difference may reach 30 % when polarisation is taken into consideration as compared to standard modelling.

The currently applied unified glare rating (UGR) takes into account discomfort from glare in complex scenes by breaking up arbitrarily shaped sources with non-uniform luminance that differ from small-angle ones into small-angle ones and assuming their luminance distribution to be uniform. Therefore, the UGR may turn to be insufficiently informative in rooms containing materials with high reflectivity, the lighting quality in which should be assessed taking into account the luminance at each point of the scene in all spatial directions. There is a need to broaden the URG content for such scenes and to introduce a new lighting quality criterion based on the physical sense of the discomfort concept from brilliant sources. Since the feeling of discomfort can only be determined experimentally, an experimental setup based on LEDs was constructed at the NRU MPEI Department of Lighting Engineering for studying the luminance at the comfort-discomfort boundary from small-angle sized and arbitrarily shaped sources. A conclusion has been drawn from the results of the accomplished experiments that the developed experimental setup and technique can be used for further investigations of the lighting quality criterion based the spatial-angular distribution of luminance in the observer's field of view

The article describes the problem of replacement of active fluorescent lamp lighting installations of an assembly line of a car assembly plant with LED LDs including a comparison of the gained lighting and economic indicators. Therefore, several LED-based LDs by different manufacturers were selected. Based on LI computer modelling using DIALux Evo, an optimal option in terms of light engineering and economy was found. Lighting characteristics of the active LI and areas of the assembly line with the application of LED-based LDs were determined experimentally. The results of the study allow assessing relevant changing of visual performance of shop workers and to compare the pay-off periods of LED and fluorescent lamps-based lighting devices.

The physical model of the radiance field is similar in some aspects to the elementary particle transport theory under the assumptions of the classical mechanics. Disregarding the differences in the used nomenclatures, it can be shown that the transport equations for the radiance field are identical to those for the particle flux density. Since the end of the 19th century, both theories have been developing in parallel, thereby enriching each other. In other words, a breakthrough, which has been made in one theory, readily contributes to the significant progress in another one. Nowadays the accuracy achieved in the experiments with particles is close to the limit, which allows validating the relationships derived within the light scattering theory. Besides, the experiments with particles are free from uncertainties in the scattering medium, which are typical for atmospheric remote sensing applications. In this paper, a new algorithm is described, which is derived by analogies between these theories. It is applied for calculating the electron flux elastically scattered by plane-parallel layers of a solid with the strongly forward peaked phase functions. The calculations are compared against the experimental angular distributions of electrons, which are elastically reflected by the two-layer solid samples.

Transmission of light is one of the key optical processes in the Earth’s atmosphere and natural waters, and transmittance (T) is an optical parameter showing the rate of change of irradiance with the optical depth. A knowledge of T or another optical parameter, diffuse attenuation coefficient, Kd, steady connected with the T, allows many practical tasks to be solved regarding the ocean and atmospheric optics, such as water quality, primary production, and atmospheric correction. Therefore, knowledge of the reliable relationships between T (or Kd) and such parameters as incident illumination angle, cloud coverage, diffuseness of irradiance, and inherent optical properties (such as the scattering phase function, backscattering probability, scattering asymmetry parameter, and single-scattering albedo) is crucial. We have analyzed the impact of these parameters on the T and Kd. We computed T and Kd using a synthetic dataset covering any possible values of parameters by the numerical method (MDOM) and 21 analytical models and compared results with the MDOM solutions. An analysis of individual models has shown that the best of them yield average errors for T and Kd better than 10% for the majority of real optical conditions in the Earth’s atmosphere and natural waters.

The wavelength is that natural scale that determines the applicability domains of the ray approximation and the wave approximation of light. If the change of the radiation power spatial density is significant at the wavelength scale, then we deal with the light diffraction phenomenon, which is a subject to the wave optics. Consider the diffraction phenomenon at the diaphragm. It is possible to distinguish the near zone with significant wave inhomogeneities (i.e. the Fresnel zone) and the far Fraunhofer diffraction zone, in which the wave becomes close to homogeneous (the so-called quasi-homogeneous) and the ray approximation is possible. The problem is that there is no explicit relationship between the radiance of the rays before and after diaphragm. Method for determining the boundary conditions for the radiance in the Fraunhofer zone through the radiance of the incident radiation is proposed in the paper. This approach for computing the radiance field in the Fraunhofer zone can be generalized to other problems of optics, thereby providing the possibility of using computationally efficient ray-approximation-based methods to determine the light fields.

The article discussed two important problems, which have not been addressed previously: – The conceptual interpretation of lighting engineering, and the expansion of its definition beyond illumination, which has became widely accepted. An expanded concept of lighting engineering proposed by the authors includes all fields of application of optical radiation (light): illumination, irradiation, phototherapy (light therapy), light alarm systems, light location, light design, etc; – The problem of developing the theoretical foundation of Lighting Engineering. Differences between three existing light theories are summarised. The first one is quantum theory, explaining all known light phenomena. The second one is wave theory based on the provision that light spreads in waves. The third theory is beam and photometric based on the light field theory, which is the most widely applied and simple for practical calculations and simulation.

The article is devoted to the creation of a surface radiance factor mathematical model. The basis of the model is the solution of the boundary value problem of the radiative transfer equation (RTE). The surface is considered as a structure consisting of several turbid layers, each of which is characterized by its optical parameters. The top of the structure is randomly rough, uncorrelated, Fresnel. The lower boundary reflects perfectly diffusely. The complexity of solving the RTE boundary value problem for real layers is due to the fact that the suspended particles in each layer are always much longer than the wavelength. This leads to a strong anisotropy of the radiance angular distribution according to Mie theory. The solution comes down to a system of equations by the discrete ordinates method that consists of several hundred of differential equations. Subtraction of the anisotropic part from the solution based on an approximate analytical solution of the RTE allows avoiding this problem. The approximation is based on a slight decrease in the anisotropic part of the angular spectrum. The matrix-operator method determines the general solution for a complex multilayer structure. The calculation speed can be increased without compromising the accuracy of the solution with the help of the synthetic iterations method. The method consists of two stages: the first one repeats the described one with a small number of ordinates; on the second one the iteration of it is performed. The model is realised in the Matlab software.

In this article, we describe the mathematical model of the reflective surface as a scattering layer with the diffuse substrate and randomly rough Fresnel boundary. This model opens the way for a physically correct description of the light reflection processes with polarization account and hence enables engineers and designers to obtain much more precise results in their work. The algorithm of Fresnel boundary modeling based on the method of mathematical expectations reduces calculation time by constructing the randomly rough surface only at the ray trajectory nodes instead of constructing realizations of a random field. As a part of the complete reflective surface model, the algorithm made it able for us to model the effect of the average lens emergence.

The article proposes a new criterion for lighting quality that extends the existing UGR. The proposed criterion is tested in an experiment on real lighting installations - Moscow metro stations. At metro stations, the spatial-angular distribution of luminance is measured using a camera that shoots in RAW format and a luminance meter. A luminance meter is used to normalize the image to absolute luminance values. In the experiment, observers evaluate the quality of station lighting and compare the results with the calculation of a new criterion. A high correlation of the results obtained by observers and calculated using the new quality criterion is shown. For a few stations, a 3D model is built, and the criterion is calculated according to the model. The article concludes that the quality criterion is applicable to the assessment of lighting quality. Moreover, the proposed criterion can be considered an extension of the existing UGR.

It has been shown that the regular part of the solution (RPS) which remains after separating the anisotropic part of the solution (APS) in the small-angle modification of the spherical harmonics method (SHM) is a smooth quasi-isotropic function with individual peaks in the angular distribution. The smooth part of the RPS without peaks can be determined in the two-streaming or diffuse approximation. The first iteration of the angular distribution of the radiance significantly refines the solution and allows one to restore the abovementioned angular peaks. The quasi-diffusion approximation—separation of the APS on the basis of the SHM, the determination of the RPS in the diffusion approximation, and the refinement of the solution on the basis of the first iteration—does not depend on the symmetry of the problem and, therefore, can be generalized to the case of an arbitrary geometry of the medium.

2017 has been a very productive and interesting year for our journal Light & Engineering/Svetotekhnika. It was marked by the publication of a large series of analytical reviews on the current state and prospects for the development of a number of important areas of lighting engineering (a total of 12 reviews), the publication of the regional volume of the Light & Engineering Journal (No. 3) devoted to solar energy technology in China (250 pages, 33 articles), further expansion of publications by international authors in Light & Engineering (since 2010, 120 articles by 230 authors from 23 countries have been published).

The purpose of this article is to construct a psychophysical scale of visual perception from lighting scene based on a direct propagation neural network using for assessment of real or synthesized images with spatial brightness distribution. Visual perception assessments of different scenes were obtained for 10 observers at the experimental installation of the Department of lighting engineering of the MPEI (NRU). These results were checked and found out agreed with the numerical scale of visual perception proposed by Lekish and Holladay. Neural network was trained to predict a sensation at the level of 40-70%, depending on the scale category. For more careful prediction level in each of 5 categories of scale a new experiment should be done with new calibration and with tested instructions and with more observers involved. The novelty consists in using a neural network as an expert to assess the degree of comfort of the lighting scene.

One of the most critical problems of realistic visualization of the real-world objects is physically adequate modeling of their reflection of light. Reflection of light by objects occurs both from the surface and the bulk of matter (scattering). Accounting for the light reflection from the surface of objects was solved almost a century ago based on its representation as a Fresnel randomly rough surface. Scattering by a bulk of matter is the subject of radiation transfer theory, which has only recently received its known completion in the form of discrete transfer theory. Strict analytical methods for solving the radiation transport equation (RTE) are often not highly effective for calculating the radiance factor. For a long time, in the absence of effective numerical methods for solving problems and the unavailability of high-speed computers for practical calculations, approximate methods for solving RTE were developed. However, their accuracy and applicability limits were poorly defined. The discrete transfer theory allowed us to evaluate the existing approximate methods for solving the UPI, their accuracy, and the efficiency of application for calculating the radiance factor. It is shown that the most effective method is the method of synthetic iterations. The method is based on the selection of the solution anisotropic part based on a small-angle approximation of the RTE solution. The solution regular part can be calculated by any approximation. Then a simple iteration from the complete solution is performed to refine the angular distribution of the radiance factor.

In connection with the appearance and spread of COVID-19, lighting equipment manufacturers have prepared product lines and marketing programs to promote irradiators for air disinfection for domestic and industrial needs In this regard, the CIE issued an official request for the use of ultraviolet radiation to fight the spread of COVID-19 Nevertheless, the penetration force of ultraviolet rays is small The action of rays is limited only to the surface of the irradiated object, and the method of calculating and simulating irradiators is based on obtaining normalized values of irradiation on the surface However, the main goal of designing bactericidal systems is to distribute UV radiation uniformly in all directions of engineering designs, regardless of their type However, now, there is no computer graphics program allows you to get the values of normalized values at each point in space to check the effectiveness of the bactericidal system This article presents a new method for solving the problem of modeling irradiation installations with the ability to analyze the normalized values at each point of the volume of air being treated © 2020 for this paper by its authors Use permitted under Creative Commons License Attribution 4 0 International (CC BY 4 0)

Now days regulatory documents for non-special lighting systems, the illuminance and various parameters derived from it are normalized as a quantitative characteristic. In most cases, all calculations are carried out for illuminance on the floor of the room or on an imaginary working plane located at the height of the table. However, illuminance is an integral characteristic of incident light, while the human eye responds to light reflected from the surface. That is, if we take a completely black surface with a reflection coefficient equal to zero, then formally you can get the required illuminance on it, while visually we will not see anything, since nothing will be reflected from the surface. In terms of the human eye, luminance must be normalized instead of illuminance. Recently, the calculation and measurement of luminance was an extremely difficult task, so the it is understandable, that current regulatory documents describes almost illuminance normalization, but not luminance. This paper aims to modeling luminance spatial-angular distribution, which enables us to run the assessment of the lighting quality.

The article is devoted to the modelling of light reflection from real surfaces. Most of the existing models are often created to solve a certain task and therefore can not be used in other realms. To create a model useful for different purposes, we propose a reflective surface representation as a scattering layer bounded by the diffuse bottom and a randomly rough Fresnel upper boundary. Such an approach allows one to account for light polarization in the scattering layer and slope correlation of the randomly rough boundary which paves the way for the observation of some physical effects that take place in nature (for instance, statistical lens emergence). The first results of the light reflection modelling were taken at such initial parameters as to compare to those obtained in other research. Former occurred to be qualitatively of the same form as the latter. The model needs to undergo further validation in numerous experiments to prove its serviceability for different types of reflective surfaces.

This article shows a new mathematical model for calculation of multiple reflections based on the Stokes vector and Mueller matrices. The global illumination equation and local estimations method were generalized on the polarization case. The results of the calculation of multiple reflections using the local estimations method show a difference of more than 30% between the standard calculation and the polarization-accounting one. A way to describe the surface reflection with polarization account is proposed.

The transmission of light is one of the key optical processes in the terrestrial environment (the atmosphere and underlying surfaces). The dependence of light transmittance on the illumination/observation conditions and optical properties of the atmosphere–underlying system can be studied using the integro-differential radiative transfer equation. However, for numerous applications a set of analytical equations is needed to describe the transmitted light intensity and flux. In this paper, we describe various analytical techniques to study light transmittance through light scattering and absorbing media. A physical significance and improved mathematical accuracy of approximations are provided using the analytical models for the diffusion exponent, average cosine of the light field, spherical and plane albedos. The accuracy of various approximations is studied using exact radiative transfer calculations with various scattering phase functions, single-scattering albedos, observational conditions, and optical depths.

With increasing of the accuracy of measuring equipment for the optical remote sensing in recent years the requirements for speed and accuracy of the algorithms for satellite data processing has greatly increased. It became necessary accurately to account all of the known factors, which affect the signal significantly. At each time, more than half of the planet is covered with clouds, so it is almost always necessary to take measurements into breaks in clouds. Cloudiness is among those factors which affect significantly the signal and its neglect in extreme cases can lead to an error of 140%. Here we propose a new solution of the radiative transfer equation (RTE) for a slab of a turbid medium with consideration of broken clouds. We use the classical approach to solving RTE: complete solution is represented as the sum of the anisotropic and regular parts. We express anisotropic part using small-angle modification of the spherical harmonics method. For the regular part we propose to use quasi two-stream approximation. This method is a special case of the synthetic iterations method. The method is based on splitting the ordinary iteration into two stages. At the first step one of approximate methods is used, and on the second step one ordinary iteration is used. We use two-stream approximation as an approximate method. In this paper we proposed a solution for the simplest case of broken clouds - cylindrical hole in the slab. Comparison of the algorithm was performed with established program MDOM, and showed good agreement.

Scattering in random structure is a typical problem for different media: colloids, clouds, liquids in coagulation stage, etc. In these structures, the group formation leads to interference effects in the scattered phase function due to the significant static structure factor. This work includes a computational study of the group formation impact on the radiation transfer in such media, when a point isotropic source is used. Rigorous calculations of a group scattering phase function for structures with different particle׳s size are made. A Monte-Carlo approach is applied to solve the radiative transfer equation in these media. To avoid singularities in the direct Monte-Carlo solution, appeared due to the radiation detection in a single point, the double-local Monte-Carlo estimation was used. The impact of group scattering phase function and a structure factor on the calculated scattered field radiance are discussed in this work. It is shown that for the point isotropic source the inhomogeneity of the group scattering phase function appears in the radiative transfer effects.

The article deals with the local and double local estimation of the Monte Carlo method for solving the equation of global illumination. The local estimation allows calculating the illumination at any point at the approximation of diffuse reflection, whereas the double local estimation allows calculating directly the luminance at a given point in a given direction. The article presents the mathematical basis of local estimations and the basic stages of the software implementation. The representation of three-dimensional objects in the basis of spherical functions and the possibility of using them in the local estimations are also considered.

An efficient approach to solving the equation of radiation transfer in a turbid medium with strongly anisotropic scattering is proposed based on the small-angle approximation of the theory of radiation transfer with allowance for the dispersion of the radiation propagation path length, as well as with allowance for polarization. The approach is applied to the solution of the problem of the field of a pointlike isotropic radiation source in a two-dimensional medium.

The paper suggests a new improved approach to calculating the anisotropic part of a diffuse light field in a turbid medium. We discuss some specific applications related to specific problems of radiative transfer theory, which demonstrate the accuracy and flexibility of this approach in practice.

This article covers the analytical solution of the discretized radiative transfer equation in the matrix form. The equation is discretized according to the discrete ordinates method. The solution is based on the representation of the light field in a scattering medium as a superposition of an anisotropic and a smooth regular parts. The first of them is calculated analytically using the smoothness of the solution angular spectrum. The regular part is obtained from a radiative transfer equation boundary problem with the anisotropic part as a source function by discrete ordinates method with a scaling transformation and a matrix-operator method applied. There is no limitation of the scattering law in a medium.

The problem of the propagation of the short light pulse in the scattering medium is considered within the radiative transfer theory. The novel approach to the accounting for the propagation path length dispersion in the small scattering angle approximation limit is proposed. The new type of the approximate solution for the medium with the high anisotropy of the scattering is formulated. The results of the numerical simulations of the proposed transient solution are presented. The application of the new solution to the polarized radiation problems is discussed.

The method of matrix Green’s functions in the classical theory of electromagnetic waves is stated. This method allows to obtain a closed equation system in the presence of the random media for the calculation both coherent, and incoherent (fluctuating) components of radiation. The density and heterogeneity of scattering media can be arbitrary. The coherent channel is calculated independently. The fluctuating radiation distribution in the medium is developed initially by an interference pattern generated by the coherent channel. The limitations of the processes speed are absent. The theory embraces such phenomena as the boson peak, flickering noise, memory effect, backscattering processes and also conventional radiative transfer equation and Fresnel’s formulae.

In this paper seven vector radiative transfer codes are inter-compared for the case of underlying black surface. They include three techniques based on the discrete ordinate method (DOM), two Monte-Carlo methods, the successive orders scattering method, and a modified doubling-adding technique. It was found that all codes give very similar results. Therefore, we were able to produce benchmark results for the Stokes parameters both for reflected and transmitted light in the cases of molecular, aerosol and cloudy multiply scattering media. It was assumed that the single scattering albedo is equal to one. Benchmark results have been provided by several studies before, including Coulson et al. [22], Garcia and Siewert [7,8], Wauben and Hovenier [10], and Natraj et al. [11] among others. However, the case of the elongated phase functions such as for a cloud and with a high angular resolution is presented here for the first time. Also in difference with other studies, we make inter-comparisons using several codes for the same input dataset, which enables us to quantify the corresponding errors more accurately.

The development of the small angle approximation of the radiative transfer equation (RTE) solution for the slab taking into account the photon path distribution function in the forward hemisphere of sighting directions is offered. The solution is based on the Taylor series expansion of the sighting angle cosine up to the 2nd order without the RTE conversion to the diffusion approximation. The obtained solution practically coincides with the exact solution in the forward hemisphere of directions up to the deep regime.

The angular distributions of elastically reflected electrons are described on the basis of the transfer equation for a radiation particle. The exact solution of the problem is obtained. This solution is compared with experimental spectra of elastically reflected electrons and with the approximate solutions.

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 ω 0 and refracted solar zenith angle θ 1 ), 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 ω 0 →0 and ω 0 →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.

The efficiency of the proposed method of solution for the spin-considered highly anisotropic multiple scattering problem is based, following optics, upon the admission that the complete solution of the vectorial transport equation boundary problem is the superposition of the anisotropic and smooth parts, calculated separately. We use a vectorial small angle modification of the spherical harmonics method to evaluate the anisotropic part and the vectorial discrete ordinates method to obtain a smooth one. Some calculation examples for reflected radiation are given.

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.

The development of aerosol radiation scattering models for the purposes of remote sensing is related to the essential account of the polarization properties (the most complete) of the scattered radiation. The smoothness of the spatial spectrum of the light field distribution vector function caused by mathematical singularities of the boundary conditions for the vectorial radiative transfer equation (vectorial radiative transfer equation) boundary problem allows us to restrict the light-field general spherical functions expansion coefficients in a Taylor series with respect to a discrete zenith-number (the order) of the general spherical function to two terms. This gives relatively simple expressions for the matrix general spherical function expansion coefficients of the vectorial radiative transfer equation solution-the vectorial small angle modification of the spherical harmonics method (vectorially modified spherical harmonics). The subsequent determination of the solution smooth non-small angle part using the boundary problem for the vectorial radiative transfer equation with the vectorially modified spherical harmonics as the source function gives the complete solution to the vectorial radiative transfer equation and prevents some calculation problems caused by ill conditions of the matrices.

The authors developed a numerical method of the boundary-value problem solution in the vectorial radiative transfer theory applicable to the turbid media with an arbitrary three-dimensional geometry. The method is based on the solution representation as the sum of an anisotropic part that contains all the singularities of the exact solution and a smooth regular part. The regular part of the solution could be found numerically by the finite element method that enables to extend the approach to the arbitrary medium geometry. The anisotropic part of the solution is determined analytically by the special form of the small-angle approximation. The method development is performed by the examples of the boundary-value problems for the plane unidirectional and point isotropic sources in a turbid medium slab.

A numerical comparison of algorithms for solving the radiative transfer equation by the Monte-Carlo method is performed for the direct simulation and local estimations. The problems of radiative transfer through a turbid medium slab in the scalar and vector case is considered. The case of reflections from the boundaries of the medium is analyzed. The calculations are performed in a wide variation of parameters of the medium. It is shown that the calculation time with the same accuracy for the local estimation method is less than one - two orders of magnitude.