Two-term Henyey-Greenstein Phase Function Research Articles

Spectrophotometry from Mars Hand Lens Imager goniometer measurements: Kimberley region, Gale crater

The light-scattering properties of surface materials on Mars are typically modeled using in situ spectrophotometric imaging sequences taken at multiple times of day to provide sufficient phase coverage. Herein, we report analyses of observations acquired at one time of day but under varying emission angles using the Mars Hand Lens Imager (MAHLI) on the robotic arm of the Mars Science Laboratory (MSL) rover as a goniometer. A multiple-viewpoint data set was acquired on Sol 544 by manipulating the arm to provide MAHLI images from 20 arm positions, all centered at the same location and from a near-constant distance of 1 m from the surface, permitting a phase angle coverage from ∼8 to 100°. From these images, we constructed a digital terrain model of the scene, and used it in combination with atmospheric scattering models to remove the diffuse components of radiance from surface units. Radiative transfer models using Hapke theory were conducted using the direct radiance components from specific rock and soil units. Despite the relatively limited number of observations, our modeled Hapke parameters were well-constrained for terrain types such as soils and certain types of rocks that were common within the scene, but the errors increased for rock types with fewer observations. Results from one-term and two-term Henyey-Greenstein phase functions show the soil units to be more backscattering than rocks, consistent with previous photometric studies of martian landing sites. Overall, we find that the MAHLI goniometer sequences can produce reasonable and consistent photometric results, providing a new and efficient means of acquiring spectrophotometric data by arm cameras on rovers such as MSL.

An Improved Accurate Solver for the Time-Dependent RTE in Underwater Optical Wireless Communications

Underwater optical wireless communication (UOWC) has been widely advocated as a viable way to satisfy these high-speed links constraints in the marine medium through the use of the visible spectrum. Nevertheless, UOWC faces several limitations, such as the path-loss due to the absorption and scattering phenomena, caused by underwater particles. Thus, quantifying this path-loss is of paramount importance in the design of futuristic UOWC systems. To this end, several approaches have been used in this regard, namely the Beer–Lambert’s law, Monte Carlo simulation, as well as radiative transfer equation (RTE). This last mentioned evaluates the optical path-loss of the light wave in an underwater channel in terms of the absorption and scattering coefficients as well as the scattering phase function (SPF). In this paper, an improved numerical solver to evaluate the time-dependent RTE for UOWC is proposed. The proposed numerical algorithm was improved based on the previously proposed ones, by making use of an improved finite difference scheme, a modified scattering angular discretization, as well as an enhancement of the quadrature method by involving a more accurate seven-point quadrature scheme in order to calculate the weight coefficients corresponding to the RTE integral term. Importantly, we applied the RTE solver to three different volume scattering functions, namely the single-term Henyey–Greenstein (HG) phase function, the two-term HG phase function, and the Fournier–Forand phase function, over both Harbor-I and Harbor-II water types. Based on the normalized received power evaluated through the proposed algorithm, the bit error rate performance of the UOWC system is investigated in terms of system and channel parameters. The enhanced algorithm gives a tightly close performance to its Monte Carlo counterpart by adjusting the numerical cumulative distribution function computation method as well as optimizing the number of scattering angles.

15-digit accuracy calculations of Ambartsumian-Chandrasekhar’s H $H$ -functions for four-term phase functions with the double-exponential formula

We have established an iterative scheme to calculate with 15-digit accuracy the numerical values of Ambartsumian-Chandrasekhar's H-functions for anisotropic scattering characterized by the four-term phase function: the method incorporates some advantageous features of the iterative procedure of Kawabata (2015) and the double-exponential integration formula~(DE-formula) of Takahashi and Mori (1974), which proved highly effective in Kawabata (2016). Actual calculations of the H-functions have been carried out employing 27 selected cases of the phase function, 56 values of the single scattering albedo $\varpi_0$, and 36 values of an angular variable $\mu(=\cos \theta)$, with $\theta$ being the zenith angle specifying the direction of incidence and/or emergence of radiation. Partial results obtained for conservative isotropic scattering, Rayleigh scattering, and anisotropic scattering due to a full four-term phase function are presented. As a sample application of the isotropic scattering H-function, an attempt is made in Appendix to simulate by iteratively solving the Ambartsumian equation the values of the plane and spherical albedos of a semi-infinite, homogeneous atmosphere calculated by Rogovtsov and Borovik (2016), who employed their analytical representations for these quantities and the single-term and two-term Henyey-Greenstein phase functions of appreciably high degrees of anisotropy, to find that our results are in satisfactory agreement with theirs.

Can AERONET data be used to accurately model the monochromatic beam and circumsolar irradiances under cloud-free conditions in desert environment?

Abstract. Routine measurements of the beam irradiance at normal incidence include the irradiance originating from within the extent of the solar disc only (DNIS), whose angular extent is 0.266° ± 1.7 %, and from a larger circumsolar region, called the circumsolar normal irradiance (CSNI). This study investigates whether the spectral aerosol optical properties of the AERONET stations are sufficient for an accurate modelling of the monochromatic DNIS and CSNI under cloud-free conditions in a desert environment. The data from an AERONET station in Abu Dhabi, United Arab Emirates, and the collocated Sun and Aureole Measurement instrument which offers reference measurements of the monochromatic profile of solar radiance were exploited. Using the AERONET data both the radiative transfer models libRadtran and SMARTS offer an accurate estimate of the monochromatic DNIS, with a relative root mean square error (RMSE) of 6 % and a coefficient of determination greater than 0.96. The observed relative bias obtained with libRadtran is +2 %, while that obtained with SMARTS is −1 %. After testing two configurations in SMARTS and three in libRadtran for modelling the monochromatic CSNI, libRadtran exhibits the most accurate results when the AERONET aerosol phase function is presented as a two-term Henyey–Greenstein phase function. In this case libRadtran exhibited a relative RMSE and a bias of respectively 27 and −24 % and a coefficient of determination of 0.882. Therefore, AERONET data may very well be used to model the monochromatic DNIS and the monochromatic CSNI. The results are promising and pave the way towards reporting the contribution of the broadband circumsolar irradiance to standard measurements of the beam irradiance.

Goniometric measurement for the estimation of anisotropy coefficient of human and animal pancreas.

Estimation of optical properties of biologic tissues is determinant for laser dosimetry in medical applications. Tissues highly absorb and scatter the light in near infrared spectrum, where the laser provides therapeutic effects. Novel frontiers of clinical practice, e.g., the employment of laser light for the treatment of pancreatic cancer, require information about pancreas-laser interaction, which are crucial for therapy management. The property of biological tissues to scatter the light traveling through is described by the anisotropy coefficient (g). The relationship between g and the angular distribution of the scattered light is described by Henyey-Greenstein phase function. The measurement of angular distribution of scattered light is performed by the goniometric technique. This paper describes the estimation of g of ex vivo pancreas at 1064 nm, performed by a goniometric-based system, where a photodetector measures intensities of scattered light at fixed angles between -120° and 120°. A two-term Henyey-Greenstein phase function has been employed to estimate anisotropy coefficient for forward (gfs) and backward scattering (gbs). Experimental trails were performed to assess the repeatability of measurement system: percentage value of standard deviation is generally lower than 8% for angles higher (lower) than 13° (13°). Measurements were performed for the first time on healthy swine pancreas, aiming to investigate the influence of coagulation temperature: gfs decreases from 0.94 (at 25 °C) to 0.93 (at 80 °C). Afterwards, the same set up has been employed for the estimation of g of human pancreas affected by neuroendocrine tumor, which presented an estimated values for gfs of 0.89.

Estimation of anisotropy coefficient and total attenuation of swine liver at 850 nm based on a goniometric technique: influence of sample thickness.

Estimation of optical properties of biologic tissue is crucial for theoretical modeling of laser treatments in medicine. Tissue highly absorbs and scatters the light between 650 nm and 1300 nm, where the laser provides therapeutic effects. Among other properties, the characteristic of biological tissues to scatter the light traveling trough, is described by the anisotropy coefficient (g). The relationship between g and the distribution of the scattered light at different angles is described by Henyey-Greenstein phase function. The measurement of angular distribution of scattered light is performed by the goniometric technique. This paper describes the estimation of g and attenuation coefficient, μt, of swine liver at 850 nm, performed by an ad hoc designed goniometric-based system, where a spectrometer measures intensities of scattered light at fixed angles (0°, 30°, 45°, 60, 120°, 135° and 150°). Both one-term and two-term Henyey-Greenstein phase function have been employed to estimate anisotropy coefficient for forward (gfs) and backward scattering (gbs). Measurements are performed on samples of two thicknesses (60 um and 30 urn) to investigate the influence of this factor on g, and repeated 6 times for each thickness. The estimated values of gfs were 0.947 and 0.951 for thickness of 60 μm and 30 μm, respectively; the estimations of gfs were -0.498 and -0.270 for thickness of 60 μm and 30 μm, respectively. Moreover, μt of liver has been estimated (i.e., 90±20 cm(1)), through Lambert-Beer equation. The comparison of our results with data reported in literature encourages the use of the ad hoc designed tool for performing experiments on other tissue, and at other wavelengths.

Successive order, multiple scattering of two-term Henyey-Greenstein phase functions

An analytic solution to the problem of determining photon direction after successive scatterings in an infinite, homogeneous, isotropic medium, where each scattering event is in accordance with a two-term Henyey-Greenstein phase function, is presented and compared against Monte Carlo simulation results. The photon direction is described by a probability density function of the dot product of the initial direction and the direction after multiple scattering events, and it is found that such a probability density function can be represented as a weighted series of one-term Henyey-Greenstein phase functions.

Simulation of spectral planetary reflectance of targets

Um modelo físico é estabelecido para simular a reflectância planetária de diferentes alvos de uma região do nordeste do Brasil. O modelo simula o sinal detectado por um sensor como uma função de sucessivas ordens de interação da radiação solar com o sistema terra-atmosfera. Uma precisa expressão analítica é estabelecida para determinar a reflectância aparente de uma superfície uniforme isotrópica, incluindo-se os efeitos de absorção de gases e espalhamento molecular e de aerossóis. A atmosfera é considerada como uma mistura de ar e aerossóis. O método "single scattering" é utilizado para determinar a reflectância intrínseca da atmosfera. A parametrização dupla de Henyey-Greenstein é utilizada para representar a função de fase dos aerossóis. Os parâmetros ópticos atmosféricos são obtidos da literatura, exceto a profundidade óptica dos aerossóis que foi avaliada através de dados piranométricos. O modelo é aplicado para simular a reflectância planetária no intervalo espectral de 0,4 a 1,1 <FONT FACE="Symbol">m</font>m e também para simular o sinal detectado nas bandas 1, 2, 3 e 4 do sensor TM do satélite Landsat-5, de áreas caracterizadas espectralmente (água, solo e vegetação). Os resultados mostram que a reflectância da atmosfera é predominante na reflectância planetária para alvos de baixa reflectância, mesmo quando a profundidade óptica dos aerossóis é pequena. No caso de alvos com média reflectância (solo e vegetação), a contribuição da atmosfera é pouco notada, pois a reflectância planetária é predominantemente representada pela reflectância do alvo.

Reduction of a particle transfer problem in finite plane-parallel media to a problem of semi-infinite media. II

For pt.I see ibid., vol.13, p.1811, 1980. A recently developed method, which reduces a particle transfer problem in finite plane-parallel media to a problem of a semi-infinite media by means of invariant embedding is used to find the reflection and transmission functions of the finite medium, assuming that the surface of the medium is subjected to an angular flux of the type psi (0, mu )= mu beta mu >0 beta =0,1,2, ... Calculations for the semi-infinite slab show that the Eddington approximation (the diffusion-type model) used here is adequately accurate for thick slabs. When applied to finite slabs with beta =0, the accuracy is within 1% for slabs thicker than one mean free path. When applied for beta >0, accuracy of better than 1% is achieved in slabs of only one mean free path. The same approach is applied to a synthetic kernel approximating the two-term Henyey-Greenstein phase function. The modelled scattering law allows a combination of forward scattering, back scattering and linearly anisotropic scattering.