Henyey-Greenstein Phase Function Research Articles

МЕТОД МОНТЕ-КАРЛО ДЛЯ РАСЧЕТА ХАРАКТЕРИСТИК СВЕТОВЫХ ПОЛЕЙ В МОРСКОЙ ВОДЕ

The development of numerical methods for solving the integro-differential radiation transfer equation remains a relevant task. Among them, we can highlight the Monte Carlo method, which is in demand in various niches of modern ocean optics. The purpose of this work is a clear and concise presentation of the basics of the forward Monte Carlo method of light fields modeling in seawater, accompanied by a detailed description of its software implementation. The basics of the method are described, the procedures for choosing the type of interaction, the mean free path and the direction of photon motion are described. A simple case is considered, corresponding to an infinitely distant point source of unpolarized light, the absence of atmospheric influence, a smooth air-seawater interface, and the absence of stratification of inherent optical properties. In this case, realistic values of the absorption and scattering coefficients were used, calculated in accordance with the Case 1 model for a chlorophyll concentration of 1 μg/L, and a strongly elongated Henyey-Greenstein phase function with the parameter g = 0.95. The Fresnel reflection of light from the air-seawater interface was taken into account. The relative errors in the values of the diffuse attenuation coefficient for downward irradiance K d and the diffuse reflectance R, calculated in the spectral range of 400–700 nm using 106 photons, in comparison with the HydroLight results were 1.5 % and 0.4 %, respectively. Spectral calculation on one core of a 2017 Intel Core i5-8250U mobile processor in MATLAB takes 6 minutes. An assessment of the choice of the optimal number of photons required to obtain the desired quantities with a given accuracy was made. The implemented method is useful for becoming familiar with the basic principles used to numerically solve the radiative transfer equation in seawater using statistical methods and is used in the “Ocean Optics” course, taught by the author to 4th year students of the Department of Thermohydromechanics of the Ocean at MIPT.

Effects of the Earth curvature on Mie-scattering radiances at high solar-sensor geometries based on Monte Carlo simulations.

Given the importance of vector radiative transfer models in ocean color remote sensing and a lack of suitable models capable of analyzing the Earth curvature effects on Mie-scattering radiances, this study presents an enhanced vector radiative transfer model for a spherical shell atmosphere geometry by the Monte Carlo method (MC-SRTM), considering the effects of Earth curvature, different atmospheric conditions, flat sea surface reflectance, polarization, high solar and sensor geometries, altitudes and wavelengths. A Monte Carlo photon transport model was employed to simulate the vector radiative transfer processes and their effects on the top-of-atmosphere (TOA) radiances. The accuracy of the MC-SRTM was verified by comparing its scalar model outputs from Henyey-Greenstein (HG) phase function with the Kattawar-Adams model results, and the mean relative differences were less than 2.75% and 4.33% for asymmetry factors (g-values) of 0.5 and 0.7, respectively. The vector mode results of MC-SRTM for a spherical shell geometry with the Mie-scattering phase matrix were compared with the PCOART-SA model results (from Polarized Coupled Ocean-Atmosphere Radiative Transfer model based on the pseudo-spherical assumption), and the mean relative differences were less than 2.67% when solar zenith angles (SZAs) > 70 ∘ and sensor viewing zenith angles (VZAs) < 60 ∘ for two aerosol models (coastal and tropospheric models). Based on the MC-SRTM, the effects of Earth curvature on TOA radiances at high SZAs and VZAs were analyzed. For pure aerosol atmosphere, the effects of Earth curvature on TOA radiances reached up to 5.36% for SZAs > 70 ∘ and VZAs < 60 ∘ and reduced to less than 2.60% for SZAs < 70 ∘ and VZAs > 60 ∘. The maximum Earth curvature effect of pure aerosol atmosphere was nearly same (10.06%) as that of the ideal molecule atmosphere. The results also showed no statistically significant differences for the aerosol-molecule mixed and pure aerosol atmospheres. Our study demonstrates that there is a need to consider the Earth curvature effects in the atmospheric correction of satellite ocean color data at high solar and sensor geometries.

GPU-Accelerated Monte Carlo Simulation for a Single-Photon Underwater Lidar

The Monte Carlo (MC) simulation, due to its ability to accurately simulate the backscattered signal of lidar, plays a crucial role in the design, optimization, and interpretation of the backscattered signal in lidar systems. Despite the development of several MC models for lidars, a suitable MC simulation model for underwater single-photon lidar, which is a vital ocean remote sensing technique utilized in underwater scientific investigations, obstacle avoidance for underwater platforms, and deep-sea environmental exploration, is still lacking. There are two main challenges in underwater lidar simulation. Firstly, the simulation results are significantly affected by near-field abnormal signals. Secondly, the simulation process is time-consuming due to the requirement of a high number of random processes to obtain reliable results. To address these issues, an algorithm is proposed to minimize the impacts of abnormal simulation signals. Additionally, a graphics processing unit (GPU)-accelerated semi-analytic MC simulation with a compute unified device architecture is proposed. The performance of the GPU-based program was validated using 109 photons and compared to a central processing unit (CPU)-based program. The GPU-based program achieved up to 68 times higher efficiency and a maximum relative deviation of less than 1.5%. Subsequently, the MC model was employed to simulate the backscattered signal in inhomogeneous water using the Henyey–Greenstein phase functions. By utilizing the look-up table method, simulations of backscattered signals were achieved using different scattering phase functions. Finally, a comparison between the simulation results and measurements derived from an underwater single-photon lidar demonstrated the reliability and robustness of our GPU-based MC simulation model.

Realistic deformable phantoms with optical properties of biological tissues for biomedical research applications

A matrix of poly (vinyl alcohol) (PVA) slime phantoms was investigated to replicate the published optical properties of the lung carcinoma and tumorous brain tissues at 630 nm wavelength. The effective attenuation coefficients (µeff) and penetration depths of light in the tissue phantoms were deduced from measurements of its optical properties and the anisotropy factor (g), obtained using the Henyey-Greenstein (H-G) phase function and Wolfram Mathematica. The mechanical response of the fabricated phantom materials was characterized and found to be flexible and mouldable into shapes and sizes comparable to desired tissue types.In the context of accuracy and stability, the phantoms were monitored for over two months and managed to preserve their optical properties with inconsiderable changes, possibly due to dehydration. Further data analysis is required to clarify if the optical properties would significantly change for long periods. We conclude that the fabrication of deformable tissue phantoms capable of uniformly incorporating any inserted lesion-like objects with relatively available materials and clear procedure is realizable. These phantoms are promising for testing and calibrating new optical imaging systems and developing disease diagnosis and treatment techniques.

Two-term scattering phase function for photon transport to model subdiffuse reflectance in superficial tissues.

For Monte Carlo simulations of light transport in a variety of diffuse scattering applications, a single-scattering two-term phase function with five adjustable parameters is sufficiently flexible to separately control the forward and backward components of scattering. The forward component dominates light penetration into a tissue and the resulting diffuse reflectance. The backward component controls early subdiffuse scatter from superficial tissues. The phase function consists of a linear combination of two phase functions [Reynolds and McCormick, J. Opt. Soc. Am.70, 1206 (1980)10.1364/JOSA.70.001206] that were derived from the generating function for Gegenbauer polynomials. The two-term phase function (TT) accommodates strongly-forward anisotropic scattering with enhanced backscattering and is a generalization of the two-term, three-parameter Henyey-Greenstein phase function. An analytical inverse of the cumulative distribution function for scattering is provided for implementation in Monte Carlo simulations. Explicit TT equations are given for the single-scattering metrics g 1, g 2, γ, and δ. Scattering data from previously published bio-optical data are shown to fit better with the TT than other phase function models. Example Monte Carlo simulations illustrate the use of the TT and its independent control of subdiffuse scatter.

Single scattering models for radiative transfer of isotropic and cone-shaped light sources in fog.

The simulation of rare edge cases such as adverse weather conditions is the enabler for the deployment of the next generation of autonomous drones and vehicles into conditions where human operation is error-prone. Therefore, such settings must be simulated as accurately as possible and be computationally efficient, so to allow the training of deep learning algorithms for scene understanding, which require large-scale datasets disallowing extensive Monte Carlo simulations. One computationally-expensive step is the simulation of light sources in scattering media, which can be tackled by the radiative transfer equation and approximated by analytical solutions in the following. Traditionally, a single scattering event is assumed for fog rendering, since it is the dominant effect for relatively low scattering media. This assumption allows us to present an improved solution to calculate the so called air-light integral that can be evaluated fast and robustly for an isotropic point source in homogeneous media. Additionally, the solution is extended for a cone-shaped source and implemented in a computer vision rendering pipeline fulfilling computational restrictions for deep learning uses. All solutions can handle arbitrary azimuthally symmetric phase functions and were tested with the Henyey-Greenstein phase function and an advection fog phase function calculated from a particle distribution using Mie's theory. The used approximations are validated through extensive Monte Carlo simulations and the solutions are used to augment good weather images towards inclement conditions with focus on visible light sources, so to provide additional data in such hard-to-collect settings.

Near-infrared spectroscopy of boulders with dust or patina coatings on the Moon: A two-layer radiative transfer model

Previous remote sensing studies focus on lunar surface regolith, which contains abundant mixtures of rock fragments and dust, making it hard to track the petrologic origin. Igneous boulders exposed on lunar surface, however, carry pristine mineralogy and chemistry since its formation, therefore are direct evidence of lunar thermal evolution events. High spatial-resolution remote sensing images and rover explorations of the Moon allow us to study the spectroscopy of igneous boulders. We modeled the optical scattering properties of rocks using the Legendre and Double Henyey–Greenstein phase functions, porosity parameter and grain size, and provided a modified radiative transfer model for rocks rather than powdered minerals. Considering that space weathering could generate a layer of dust or patina on the surface of boulders, we introduced a two-layer radiative transfer modeling algorithm to solve the spectroscopy of the substrate rock for dust- or patina-coated boulder. The modeled substrate rock spectra show less reddening, larger reflectance, and stronger absorption band depth compared to dust- or patina-coated rock, consistent with the measurements of Apollo rock samples. We applied this two-layer model on the dust-coated boulder detected by Yutu-2 rover and derived the spectrum of the substrate rock. Using Kaguya Multiband Imager data, we calculated the substrate rock spectra for an anorthosite boulder, and our result shows good consistency with laboratory measured anorthosite rock spectrum. This work is a beginning of understanding lunar boulder spectroscopy for a more precise interpretation of lunar thermal history.

Phase Function of Seawater Polluted by Oil Emulsion

Oil droplets suspended in the seawater will seriously affect the transmission characteristics of electromagnetic wave from ultraviolet and visible light to infrared. The scattering phase function can show the laser transmission characteristics in seawater polluted by oil. Water with oil emulsion has a phase function significantly different from that of natural ocean water. The accurate calculation of phase function can lay the foundation for laser propagation in the oil spill seawater, and is of great significance for protection of marine environment. This paper presents the modeling of phase function of seawater polluted by dispersed oil based on Mie scattering theory. Numerical calculation of scattering phase function of dispersed oil droplets in various times and at different wavelengths is conducted, and the numerical results of Henyey-Greenstein phase function and the modified Henyey-Greenstein phase function is given. The results show that the shapes of Mie scattering phase function of water with oil emulsion depend on wavelength and the oil emulsion ageing. The forward scattering attenuates with the time of oil emulsion ageing and the increase of incident wavelength, and the scattering gradually becomes homogeneously. Compared with Henyey-Greenstein the modified Henyey-Greenstein phase function can describe the scattering property of oil emulsion in the seawater better. But the fitting effects of both Henyey-Greenstein phase function and the modified Henyey-Greenstein phase function are undesirable for long time oil droplets, long wavelength and strong absorption ultraviolet light.

Phase function estimation from a diffuse optical image via deep learning

Objective. The phase function is a key element of a light propagation model for Monte Carlo (MC) simulation, which is usually fitted with an analytic function with associated parameters. In recent years, machine learning methods were reported to estimate the parameters of the phase function of a particular form such as the Henyey–Greenstein phase function but, to our knowledge, no studies have been performed to determine the form of the phase function. Approach. Here we design a convolutional neural network (CNN) to estimate the phase function from a diffuse optical image without any explicit assumption on the form of the phase function. Specifically, we use a Gaussian mixture model (GMM) as an example to represent the phase function generally and learn the model parameters accurately. The GMM is selected because it provides the analytic expression of phase function to facilitate deflection angle sampling in MC simulation, and does not significantly increase the number of free parameters. Main Results. Our proposed method is validated on MC-simulated reflectance images of typical biological tissues using the Henyey–Greenstein phase function with different anisotropy factors. The mean squared error of the phase function is 0.01 and the relative error of the anisotropy factor is 3.28%. Significance. We propose the first data-driven CNN-based inverse MC model to estimate the form of scattering phase function. The effects of field of view and spatial resolution are analyzed and the findings provide guidelines for optimizing the experimental protocol in practical applications.

A new analytical scattering phase function for interstellar dust

Context. Properly modelling scattering by interstellar dust grains requires a good characterisation of the scattering phase function. The Henyey-Greenstein phase function has become the standard for describing anisotropic scattering by dust grains, but it is a poor representation of the real scattering phase function outside the optical range. Aims. We investigate alternatives for the Henyey-Greenstein phase function that would allow the scattering properties of dust grains to be described. Our goal is to find a balance between realism and complexity: the scattering phase function should be flexible enough to provide an accurate fit to the scattering properties of dust grains over a wide wavelength range, and it should be simple enough to be easy to handle, especially in the context of radiative transfer calculations. Methods. We fit various analytical phase functions to the scattering phase function corresponding to the BARE-GR-S model, one of the most popular and commonly adopted models for interstellar dust. We weigh the accuracy of the fit against the number of free parameters in the analytical phase functions. Results. We confirm that the Henyey-Greenstein phase functions poorly describe scattering by dust grains, particularly at ultraviolet (UV) wavelengths, with relative differences of up to 50%. The Draine phase function alleviates this problem at near-infrared (NIR) wavelengths, but not in the UV. The two-term Reynolds-McCormick phase function, recently advocated in the context of light scattering in nanoscale materials and aquatic media, describes the BARE-GR-S data very well, but its five free parameters are degenerate. We propose a simpler phase function, the two-term ultraspherical-2 (TTU2) phase function, that also provides an excellent fit to the BARE-GR-S phase function over the entire UV-NIR wavelength range. This new phase function is characterised by three free parameters with a simple physical interpretation. We demonstrate that the TTU2 phase function is easily integrated in both the spherical harmonics and the Monte Carlo radiative transfer approaches, without a significant overhead or increased complexity. Conclusions. The new TTU2 phase function provides an ideal balance between being simple enough to be easily adopted and realistic enough to accurately describe scattering by dust grains. We advocate its application in astrophysical applications, in particular in dust radiative transfer calculations.

Temperature dependent radiative properties of semi-transparent fiberglass-epoxy composite materials from 20 °C to 200 °C

The radiative properties of polymer-reinforced composite materials such as laminates may change according to the temperature. The knowledge of the relationship between radiative properties and temperature becomes crucial when designing structural components subjected to thermal stresses, induced for example by a laser irradiation. This study deals with the analysis of the absorption and scattering properties of fiberglass-epoxy composites from room temperature to 200 °C. From the point of view of thermal radiation, the two-phase system (polymer matrix and fibers reinforcements) is treated as an equivalent homogeneous medium characterized by volumetric radiative properties, namely the absorption coefficient, the scattering coefficient, and the scattering phase function. The light interaction with the rough boundaries of the sample is modeled by boundary scattering coefficients. The aim is to determine these volumetric and boundary scattering properties as function of the temperature. We use a robust inverse method based on a Gauss-Newton algorithm to solve the nonlinear least squares problem. It minimizes the sum of the squared difference between the theoretical and experimental data of bidirectional and normal-hemispherical reflectances and transmittances. Scattering measurements are carried out using a visible and near-infrared spectrometer (400 to 2500 nm spectral range) equipped with a goniometric system and a heated sample holder. Theoretical results are retrieved by employing the Monte Carlo method for solving the Radiative Transfer Equation (RTE). We report that reflectance are insensitive to the temperature rise below 150 °C, and increase above this temperature up to 200 °C. Transmittances increase with temperature, reach a maximum around 105–125 °C near the glass transition temperature of the resin (assessed by a differential scanning calorimetry analysis), and decrease until 200 °C. Upon others, this study reveals that radiative properties are strongly temperature-dependent above 100 °C, first because of the dependence of the resin refractive index to the temperature, but also due to the onset of the polymer chemical degradation. The mismatch between the glass fibers and the polymer refractive indexes is at its lowest near the glass transition temperature of the resin, and explains the peak of transmittances and the trough on the extinction coefficient and the scattering albedo. The onset of the resin chemical degradation above 150 °C increases the absorption coefficient. The asymmetry parameter of the Henyey Greenstein phase function decrease notably from 0.94 to 0.3 above 150 °C, due to the enhancement of backscattered light within the sample.

Angular distribution of photon density waves radiance in media with different scattering anisotropy.

Statistical modeling of pulsed frequency responses of the light field radiance by an isotropic point source was performed by Monte Carlo technique. Scattering properties of the medium were simulated by the Henyey-Greenstein phase function with different anisotropy factor values. Angular distributions of the pulsed field and amplitudes of the photon density waves in a certain range of parameters were shown to have a qualitatively different character for media with quasi-isotropic and strongly anisotropic scattering. A comparison of the impulse and frequency characteristics was performed for media with strongly anisotropic scattering with different scattering phase functions yet the same anisotropy factor. The main difference in the angular distributions of the fields is observed in the rear hemisphere.

Transfer matrix based layered materials rendering

A statistical multi-lobe approach was recently introduced in order to efficiently handle layered materials rendering as an alternative to expensive general-purpose approaches. However, this approach poorly supports scattering volumes as the method does not account for back-scattering and resorts to single scattering approximations. In this paper, we address these limitations with an efficient solution based upon a transfer matrix approach which leverages the properties of the Henyey-Greenstein phase function. Under this formalism, each scattering component of the stack is described through a lightweight matrix, layering operations are reduced to simple matrix products and the statistics of each BSDF lobe accounting for multiple scattering effects are obtained through matrix operators. Based on this representation, we leverage the versatility of the transfer matrix approach to efficiently handle forward and backward scattering which occurs in arbitrary layered materials. The resulting model enables the reproduction of a wide range of layered structures embedding scattering volumes of arbitrary depth, in constant computation time and with low variance.

A Layered Debris Disk around M Star TWA 7 in Scattered Light

Abstract We have obtained Hubble Space Telescope (HST) coronagraphic observations of the circumstellar disk around M star TWA 7 using the Space Telescope Imaging Spectrograph (STIS) instrument in visible light. Together with archival observations, including HST/NICMOS using the F160W filter and Very Large Telescope/SPHERE at the H-band in polarized light, we investigate the system in scattered light. By studying this nearly face-on system using geometric disk models and Henyey–Greenstein phase functions, we report a new discovery of a tertiary ring and a clump. We identify a layered architecture: three rings, a spiral, and an ≈150 au2 elliptical clump. The most extended ring peaks at 28 au, and the other components are on its outskirts. Our point-source detection-limit calculations demonstrate the necessity of disk modeling in imaging fainter planets. Morphologically, we witness a clockwise spiral motion, and the motion pattern is consistent with both solid body motion and local Keplerian motion; we also observe underdensity regions for the secondary ring that might result from mean-motion resonance or moving shadows: both call for re-observations to determine their nature. Comparing multi-instrument observations, we obtain blue STIS-NICMOS color, a STIS-SPHERE radial distribution peak difference for the tertiary ring, and a high SPHERE-NICMOS polarization fraction; these aspects indicate that TWA 7 could retain small dust particles. By viewing the debris disk around M star TWA 7 at a nearly face-on vantage point, our study allows for the understanding of such disks in scattered light in both system architecture and dust property.

Effect of scattering anisotropy on the properties of photon density waves.

The frequency characteristics of spherical photon density waves excited in media with different degrees of scattering anisotropy are studied. Statistical modeling of the frequency and phase responses of the spatial irradiance of the light field emitted by a point-sized isotropic source were performed employing the Monte Carlo technique. The scattering anisotropy of the medium was determined by the Henyey-Greenstein phase function with different values of the mean scattering cosine. It is shown that the scattering anisotropy factor determines the frequency range, in which the effect of the photon path length distribution on the magnitude of the photon density wave dispersion is maximal. In media with quasi-isotropic scattering, dispersion effects are manifested at lower frequencies as compared to those for anisotropic media. The simulation results are compared with the analytical solution for the asymptotic regime of the light field in an isotropically scattering medium.

Periodic brightening of Kepler light curves: investigating the possibility of forward scattering due to dust clouds

ABSTRACT Dedicated transiting surveys, such as the Kepler space telescope, have provided the astronomy community with a rich data set resulting in many new discoveries. In this paper, we look at eight Kepler objects identified by Wheeler &amp; Kipping with a periodic, broad increase in flux, that look distinctly different from intrinsic star variability. We consider two physical phenomena as explanations for these observed Kepler light curves; the first being the classical explanation while the second being an alternative scenario: (i) tidal interactions in a binary star system, and (ii) forward scattering from an optically thin cloud around an exoplanet. We investigate the likelihood of each model by modelling and fitting to the observed data. The binary system qualitatively does a good job of reproducing the shape of the observed light curves due to the tidal interaction of the two stars. We do, however, see a mismatch in flux right before or after the peak brightness. We find that six out of the eight systems require an F-type primary star with a K-type companion with large eccentricities. At the same time, we find that optically thin discs, modelled using a Henyey–Greenstein phase function are also able to generate these broad brightening events. Five of the eight observed objects can be described with this new hypothesis in the absence of RV observations. As the other three are not well-described by the disc model, we conclude that they are indeed heartbeat stars.

Photometric properties of lunar regolith revealed by the Yutu-2 rover

Context.The surface composition of the Moon has mainly determined based on the visible and near-infrared spectra achieved from orbits and/or landing sites, and the spectroscopic analysis is based on photometric properties of the topmost lunar regolith. However, the lack of a ground truth for the photometric parameters of the undisturbed lunar surface has limited accurate applications of spectral observations.Aims.Here we report the photometric properties of the small-scale (i.e., centimeter level) undisturbed lunar regolith around the Chang’E-4 landing site, determined from a series of photometric experiments conducted by the rover Yutu-2.Methods.The simplified Hapke model was used to derive the photometric properties. The micro-topographic effect on the spectral measurements was corrected for the first time in thein situphotometric investigations on the Moon, which improves the accuracy of the derived photometric parameters.Results.The single-scattering albedowand two parameters (b,c) of the Henyey-Greenstein phase function were derived, and they show a wavelength dependence. The regolith at the Chang’E-4 landing site exhibits strong forward scattering according to the retrievedcvalues, and the higher asymmetry parameter indicates that the regolith here is more strongly forward scattering than the Apollo lunar soil samples. The derived photometric parameters can serve as ground truth and can be used in the radiative transfer modeling analysis of the orbital remote-sensing data.

Simultaneous reconstruction of absorption, scattering and anisotropy factor distributions in quantitative photoacoustic tomography

We present for the first time the simultaneous reconstruction of three optical parameters distributions of biological tissues namely, the absorption μa and scattering μs coefficients, as well as the anisotropy factor g of the Henyey-Greenstein phase function as a new optical contrast. The 2D images are obtained from the simulation experiments and multi-source quantitative photoacoustic tomography with the radiative transfer equation (RTE) as light transport model. The image reconstruction method is based on a gradient-based optimization scheme. The adjoint method applied to the RTE is used to efficiently compute the gradient of the objective function. The results show simultaneous reconstructions of the three optical properties even with noisy data. The crosstalk problem between the three parameters is highlighted. Superior quality images are obtained for μa compared to those of μs and g. Moreover, our algorithm allows reconstructing inserts-like heterogeneities with very good spatial resolution and qualitative accuracy.

Subdiffuse scattering model for single fiber reflectance spectroscopy.

.To detect small-scale changes in tissue with optical techniques, small sampling volumes are required. Single fiber reflectance (SFR) spectroscopy has a sampling depth of a few hundred micrometers. SFR spectroscopy uses a single fiber to emit and collect light. The only available model to determine optical properties with SFR spectroscopy was derived for tissues with modified Henyey–Greenstein phase functions. Previously, we demonstrated that this model is inadequate for other tissue phase functions. We develop a model to relate SFR measurements to scattering properties for a range of phase functions, in the absence of absorption. Since the source and detector overlap, the reflectance cannot be accurately described by diffusion theory alone: SFR measurements are subdiffuse. Therefore, we describe the reflectance as a combination of a diffuse and a semiballistic component. We use the model of Farrell et al. for the diffuse component, solved for an overlapping source and detector fiber. For the semiballistic component, we derive a new parameter, , which incorporates the integrals of the phase function over 1 deg in the backward direction and 23 deg in the forward direction. Our model predicts the reflectance with a median error of 2.1%, compared to 9.0% for the currently available model.

The δ-six-stream spherical harmonic expansion adding method for solar radiative transfer

A δ-six-stream spherical harmonic expansion adding method (δ-6SDA) for solar radiative transfer is proposed. δ-6SDA has similar accuracy as δ-six-stream discrete ordinates method (δ-6DOM) but it is almost three orders of magnitude faster than δ-6DOM in flux calculation. In addition, δ-6SDA is extended to calculate the azimuthally dependent intensity in this study. The truncated-plus-single-scattering (TMS) correction is introduced to improve accuracy. Two single-layer cases with the Cloud C1 phase function and Henyey-Greenstein phase function are considered to investigate the accuracy of δ-6SDA for intensity calculation. The relative errors of these two cases for δ-6SDA are generally less than 2% and 5%, respectively. The root-mean-square errors for δ-6SDA are generally smaller than δ-6DOM. The computing time of δ-6SDA is 3–4 orders of magnitude less than δ-6DOM. Due to its high accuracy and efficiency, δ-6SDA can be widely used in remote sensing for simulating satellite observations.