Two-term Henyey-Greenstein Research Articles

Angle-dependent light scattering in tissue phantoms for the case of thin bone layers with predominant forward scattering.

The cochlea forms a key element of the human auditory system in the temporal bone. Damage to the cochlea continues to produce significant impairment for sensory reception of environmental stimuli. To improve this impairment, the optical cochlear implant forms a new research approach. A prerequisite for this method is to understand how light propagation, as well as scattering, reflection, and absorption, takes place within the cochlea. We offer a method to study the light distribution in the human cochlea through phantom materials which have the objective to mimic the optical behavior of bone and Monte-Carlo simulations. The calculation of an angular distribution after scattering requires a phase function. Often approximate functions like Henyey-Greenstein, two-term Henyey-Greenstein or Legendre polynomial decompositions are used as phase function. An alternative is to exactly calculate a Mie distribution for each scattering event. This method provides a better fit to the data measured in this work.

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.

Phase function effects on the retrieval of oceanic high-spectral-resolution lidar.

A semianalytic Monte Carlo model is developed to simulate oceanic high-spectral-resolution lidar (HSRL) signals with multiple scattering. The phase function effects on oceanic HSRL retrieval are studied, e.g., the effective particulate 180° volume scattering function (VSF) and lidar attenuation coefficient that describe characteristics of backscatter and attenuation, respectively. The results demonstrate that the particulate backward and forward phase functions both have a significant influence on δ1, which is the relative difference between the effective and true particulate 180° VSF. The values of |δ1| are typically quite small for all phase functions at the water surface and increase with depth up to ~17% for the Fournier and Forand (FF) phase function but up to ~40% for the two-term Henyey-Greenstein (TTHG) phase function and ~75% for the one-term Henyey-Greenstein (OTHG) phase function. The reason that δ1 is not zero is due to broadening of backscattering angles from 180° caused by multiple scattering and uneven backward phase function. Also, the reason that maximum TTHG and OTHG |δ1| are larger than FF is due to less sharply increasing feature of FF in the backward direction. In addition, the particulate forward phase functions are closely related to δ2, which is the relative deviation between the lidar attenuation coefficient and the sum of the absorption and backscattering coefficients. The values of δ2 are small for all phase functions at the water surface and increase with depth up to ~12% for TTHG but up to ~26% for FF and ~31% for OTHG, due to the less peaked forward phase functions that result in more angular spread of the beam with depth and therefore result in less photons within the field of view of the lidar.

Spectrophotometric modeling and mapping of Ceres

We report a comprehensive analysis of the global spectrophotometric properties of Ceres using the images collected by the Dawn Framing Camera through seven color filters from April to June 2015 during the RC3 (rotational characterization 3) and Survey mission phases. We derived the Hapke model parameters for all color filters. The single-scattering albedo of Ceres at 555 nm wavelength is 0.14 ± 0.04, the geometric albedo is 0.096 ± 0.006, and the bolometric Bond albedo is 0.037 ± 0.002. The asymmetry factors calculated from the best-fit two-term Henyey-Greenstein (HG) single-particle phase functions (SPPFs) show a weak wavelength dependence from −0.04 at 438 nm increasing to 0.002 at >900 nm, suggesting that the phase reddening of Ceres is dominated by single-particle scattering rather than multiple scattering or small-scale surface roughness. The Hapke roughness parameter of Ceres is derived to be 20° ± 6°, with no wavelength dependence. The phase function of Ceres presents appreciably strong scattering around 90° phase angle that cannot be fitted with a single-term HG SPPF, suggesting possible stronger forward scattering component than other asteroids previously analyzed with spacecraft data. We speculate that such a scattering characteristic of Ceres might be related to its ubiquitous distribution of phyllosilicates and high abundance of carbonates on the surface. We further grouped the reflectance data into a 1° latitude-longitude grid over the surface of Ceres, and fitted each grid independently with both empirical models and the Hapke model to study the spatial variations of photometric properties. Our derived albedo maps and color maps are consistent with previous studies [Nathues, A., et al., 2016, Planet. Space Sci. 134, 122–127; Schröder, S.E., et al., 2017, Icarus 288, 201–225]. The SPPF over the surface of Ceres shows an overall correlation with albedo distribution, where lower albedo is mostly associated with stronger backscattering and vice versa, consistent with the general trend among asteroids. On the other hand, the Hapke roughness parameter does not vary much across the surface of Ceres, except for the ancient Vendimia Planitia region that is associated with a slightly higher roughness. Furthermore, the spatial distributions of the SPPF and the Hapke roughness do not depend on wavelength. Based on the wavelength dependence of the SPPF of Ceres, we hypothesize that the regolith grains on Ceres either contain a considerable fraction of μm-sized or smaller particles, or are strongly affected by internal scatterers of this size.

基于粒子群优化的近似散射相函数拟合方法

The scattering phase function is an important parameter for studying the optical transmission characteristics in aerosols. Four approximate scattering phase functions commonly used in Monte Carlo simulations in atmospheric radiation propagation were compared. Aiming at the problem that the parameters of the Two-Term Henyey-Greenstein (TTHG) phase function were difficult to determine, a TTHG scattering phase function based on particle swarm optimization was proposed. This function can well fit the Mie scattering phase function, especially at backscatter angles greater than 90. Compared with the phase functions such as HG, HG* and RHG, the phase function proposed in this paper can better approximate the actual scattering and obtain more accurate Monte Carlo simulation results.

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.

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.

Estimation of anisotropy coefficient of swine pancreas, liver and muscle at 1064nm based on goniometric technique.

Optical properties of tissues are required for theoretical modeling of Laser Ablation in tumor therapy. The light scattering characteristic of tissues is described by the anisotropy coefficient, g. The relationship between the angular distribution of scattered light and g is given by the Henyey-Greenstein (HG) phase function. This work describes the estimation of anisotropy coefficients of ex vivo swine pancreas, liver and muscle at 1064 nm. The intensities of scattered light at fixed angles were measured under repeatability conditions. Experimental data were fitted with a two-term HG, estimating the anisotropy coefficients for the forward (e.g., 0.956 for pancreas, 0.964 for liver and 0.968 for muscle) and the backward (e.g., -0.481 for pancreas, -0.414 for liver and -0.372 for muscle) scattering. Experimental set up employed to estimate the anisotropy coefficient of biological tissues. The image on the left depicts the holder used to house tissue, laser fiber and photodetector; on the left an example of scattered light beam is shown, as well as the effect due to Snell's law.

Estimation of the effective phase function of bulk diffusing materials with the inverse adding-doubling method.

The accuracy of optical simulations including bulk diffusors is heavily dependent on the accuracy of the bulk scattering properties. If no knowledge on the physical scattering effects is available, an iterative procedure is usually used to obtain the scattering properties, such as the inverse Monte Carlo method or the inverse adding-doubling (AD) method. In these methods, a predefined phase function with one free parameter is usually used to limit the number of free parameters. In this work, three predefined phase functions (Henyey-Greenstein, two-term Henyey-Greenstein, and Gegenbauer kernel (GK) phase function) are implemented in the inverse AD method to determine the optical properties of two strongly diffusing materials: low-density polyethylene and TiO₂ particles. Using the presented approach, an estimation of the effective phase function was made. It was found that the use of the GK phase function resulted in the best agreement between calculated and experimental transmittance, reflectance, and scattered radiant intensity distribution for the LDPE sample. For the TiO₂ sample, a good agreement was obtained with both the two-term Henyey-Greenstein and the GK phase function.

Comparison study of several underwater light scattering phase functions

Scattering phase function is assumed to be one of the most significant factors in the inherent optical properties (IOPs) of natural water. According to three criteria proposed for assessment, several commonly used empirical phase functions are compared with their related practical or theoretical scattering distributions in terms of fitting errors under the circumstances of typical seawater and single-component polydisperse systems. The optimal factors corresponding to the minimum fitting errors are also calculated. It is found that both the one-term Henyey-Greenstein (OTHG) and two-term Henyey-Greenstein (TTHG) phase functions agree well with the theoretical ones for small particles, while the Fouriner-Forand (FF) phase function can be used in the case of suspensions with large suspended particles. The fitting accuracy of OTHG is the worst, FF is better and TTHG is the best.

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.

One-parameter two-term Henyey-Greenstein phase function for light scattering in seawater.

A one-parameter two-term Henyey-Greenstein (TTHG) phase function of light scattering in seawater is proposed. The original three-parameter TTHG phase function was reduced to the one-parameter TTHG phase function by use of experimentally derived regression dependencies between integral parameters of the marine phase functions. An approach to calculate a diffuse attenuation coefficient in the depth of seawater is presented.

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.

Variation of atmospheric optical depth for remote sensing radiance calculations

An analysis of the radiative transfer processes in the atmosphere suggests that the ratio between the diffuse radiance flux and the directly transmitted flux may be a useful parameter for investigating the variation of atmospheric optical depth. The radiative parameter is derived for an observer at the top of the atmosphere in the position of a satellite. One of the important variables required in the computations of the radiative parameter is the aerosol phase function which can be satisfactorily fitted by the two-term Henyey-Greenstein function. When the surface reflectance equals zero, representing the atmospheric radiance alone without any influence from an oceanic surface, the study shows that the variation of the optical depth is associated with the changing of aerosol size distribution. The optical depth increases with the presence of large aerosol particles. The results indicate that inhomogeneities in aerosol optical properties, such as the variation of the scattering phase function due to the changes in aerosol size distribution, can affect the magnitude of the optical depth.

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.