Empirical Phase Functions Research Articles

Quantitative Evaluation of the Phase Function Effects on Light Scattering and Radiative Transfer in Dispersed Systems

The light scattering properties of particles play important roles in radiative transfer in many dispersed systems, such as turbid atmosphere, ocean water, nanofluids, composite coatings and so on. As one of the scattering property parameters, the scattering phase functions of particles are strongly dependent on the particle size, size distribution, and morphology, as well as on the complex refractive indices of the particles and surrounding media. For the sake of simplicity, the empirical phase function models are widely used in many practical applications. In this work, we focus on the radiative transfer problem in dispersed systems composed of spherical particles, and give quantitative analyses of the impact of scattering phase functions on the radiative transfer process. We fit the scattering phase functions of four different types of practical dispersed systems using four previously proposed empirical phase function models, including the Henyey–Greenstein (HG) model, Cornette Shanks (CS) model, Reynold and McCormick (RM) model and two-term Reynolds–McCormick (TTRM) model. By comparing the radiative transfer characteristics (i.e., hemispherical reflectance, hemispherical transmittance and total absorptance) of dispersed layers calculated using the Monte Carlo method, the relative errors caused by using the empirical phase functions are systematically investigated. The results demonstrate that the HG, CS and RM models cause obvious errors in the calculation of hemispherical reflectance in many cases. Meanwhile, the induced errors show no obvious regularity, but are related to the particle size and layer optical thickness. Due to the good fitting effect in both forward and backward directions, the TTRM model provides significantly higher performances in fitting the phase functions of all considered cases than the widely used single-term parametrizations. Moreover, for different particle sizes and layer optical thicknesses, the induced errors of the TTRM model in radiative transfer characteristics are very small, especially for the case of polydisperse particles. Our results can be used to guide the design, analysis and optimization of dispersed systems in practical optics and photonics applications.

Two-term Reynolds-McCormick phase function parameterization better describes light scattering by microalgae and mineral hydrosols.

The presence of hydrosols, taken as suspension of micro- or macroscopic material in water, strongly alters light propagation and thus the radiance distribution within a natural or artificial water volume. Understanding of hydrosols' impacts on light propagation is limited by our ability to accurately handle the angular scattering phase function inherent to complex material such as suspended sediments or living cells. Based on actual quality-controlled measurements of sediments and microalgae, this Letter demonstrates the superiority of a two-term five-parameter empirical phase function as recently proposed for scattering by nanoparticle layers [Nanoscale11, 7404 (2019)NANOHL2040-336410.1039/C9NR01707K]. The use of such phase function parameterizations presents new potentialities for various radiative transfer and remote sensing applications related to an aquatic environment.

Photometric Normalization of Chang’e-4 Visible and Near-Infrared Imaging Spectrometer Datasets: A Combined Study of In-Situ and Laboratory Spectral Measurements

Until 29 May 2020, the Visible and Near-Infrared Imaging Spectrometer (VNIS) onboard the Yutu-2 Rover of the Chang’e-4 (CE-4) has acquired 96 high-resolution surface in-situ imaging spectra. These spectra were acquired under different illumination conditions, thus photometric normalization should be conducted to correct the introduced albedo differences before deriving the quantitative mineralogy for accurate geologic interpretations. In this study, a Lommel–Seeliger (LS) model and Hapke radiative transfer (Hapke) model were used and empirical phase functions of the LS model were derived. The values of these derived phase functions exhibit declining trends with the increase in phase angles and the opposition effect and phase reddening effect were observed. Then, we discovered from in-situ and laboratory measurements that the shadows caused by surface roughness have significant impacts on reflectance spectra and proper corrections were introduced. The validations of different phase functions showed that the maximum discrepancy at 1500 nm of spectra corrected by the LS model was less (~3.7%) than that by the Hapke model (~7.4%). This is the first time that empirical phase functions have been derived for a wavelength from 450 to 2395 nm using in-situ visible and near-infrared spectral datasets. Generally, photometrically normalized spectra exhibit smaller spectral slopes, lower FeO contents and larger optical maturity parameter (OMAT) than spectra without correction. In addition, the band centers of the 1 and 2 μm absorption features of spectra after photometric normalization exhibit a more concentrated distribution, indicating the compositional homogeneity of soils at the CE-4 landing site.

Estimation in linear errors-in-variables models with unknown error distribution

Summary Parameter estimation in linear errors-in-variables models typically requires that the measurement error distribution be known or estimable from replicate data. A generalized method of moments approach can be used to estimate model parameters in the absence of knowledge of the error distributions, but it requires the existence of a large number of model moments. In this paper, parameter estimation based on the phase function, a normalized version of the characteristic function, is considered. This approach requires the model covariates to have asymmetric distributions, while the error distributions are symmetric. Parameters are estimated by minimizing a distance function between the empirical phase functions of the noisy covariates and the outcome variable. No knowledge of the measurement error distribution is needed to calculate this estimator. Both asymptotic and finite-sample properties of the estimator are studied. The connection between the phase function approach and method of moments is also discussed. The estimation of standard errors is considered and a modified bootstrap algorithm for fast computation is proposed. The newly proposed estimator is competitive with the generalized method of moments, despite making fewer model assumptions about the moment structure of the measurement error. Finally, the proposed method is applied to a real dataset containing measurements of air pollution levels.

A novel phase function describing light scattering of layers containing colloidal nanospheres.

Light scattering from small particles exhibit unique angular scattering distributions, which are strongly dependent on the radius to wavelength ratio as well as the refractive index contrast between the particles and the surrounding medium. As the concentration of the particles increases, multiple scattering becomes important. This complicates the description of the angular scattering patterns, and in many cases one has to resort to empirical phase functions. We have measured the angle dependence of light scattering from a polymer layer containing sub-micron metallic and dielectric particles. The samples exhibited strongly forward and backward peaked scattering patterns, which were fitted to a number of empirical approximative phase functions. We found that a novel two-term Reynolds-McCormick (TTRM) phase function gave the best fit to the experimental data in all cases. The feasibility of the TTRM approach was further validated by good agreement with numerical simulations of Mie single scattering phase functions at various wavelengths and sizes, ranging from the Rayleigh scattering regime to the geometrical optics regime. Hence, the widely adaptable TTRM approach is able to describe angular scattering distributions of different kinds of nanospheres and nanocomposites, both in the single scattering and multiple scattering regimes.

Density estimation in the presence of heteroscedastic measurement error of unknown type using phase function deconvolution.

It is important to properly correct for measurement error when estimating density functions associated with biomedical variables. These estimators that adjust for measurement error are broadly referred to as density deconvolution estimators. While most methods in the literature assume the distribution of the measurement error to be fully known, a recently proposed method based on the empirical phase function (EPF) can deal with the situation when the measurement error distribution is unknown. The EPF density estimator has only been considered in the context of additive and homoscedastic measurement error; however, the measurement error of many biomedical variables is heteroscedastic in nature. In this paper, we developed a phase function approach for density deconvolution when the measurement error has unknown distribution and is heteroscedastic. A weighted EPF (WEPF) is proposed where the weights are used to adjust for heteroscedasticity of measurement error. The asymptotic properties of the WEPF estimator are evaluated. Simulation results show that the weighting can result in large decreases in mean integrated squared error when estimating the phase function. The estimation of the weights from replicate observations is also discussed. Finally, the construction of a deconvolution density estimator using the WEPF is compared with an existing deconvolution estimator that adjusts for heteroscedasticity but assumes the measurement error distribution to be fully known. The WEPF estimator proves to be competitive, especially when considering that it relies on minimal assumption of the distribution of measurement error.

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.

Phase functions of mixed clouds

A model of the phase function of a mixed cloud is proposed. It is based on Mie calculations for spherical water droplets and empirical phase function of a crystalline cloud measured in situ. The simple parameterization for the crystalline cloud phase function has been developed.

Color Variations on Eros from NEAR Multispectral Imaging

NEAR multispectral imaging was obtained at seven wavelengths (450–1050 nm) to characterize hemispheric and regional color properties of Eros. The highest-resolution whole-disk data, 180 m/pixel, were obtained during the last pre-orbit insertion sequence on 12 February 2000. The same areas were imaged again in color at 10–20 m/pixel from high orbit in March–April 2000, and selected targets have been studied in color at resolutions as high as 4 m/pixel from low orbit. Whole-disk spectra are in close agreement with ground-based observations. These and the disk-resolved measurements show little variation in visible-wavelength color, but they do reveal spatial variation of several percent in the 950-nm/760-nm reflectance ratio, used here as a proxy for depth of the 1-μm olivine-pyroxene absorption band. After photometric correction to i=30° e=0° using both a Hapke correction and a modified empirical phase function, the disk-resolved images show reproducible spatial variations in albedo and 950-nm/760-nm reflectance ratio. The northern hemisphere exhibits average reflectances at i=30°, e=0° of 0.136±0.007 at 760 nm and 0.115±0.006 at 950 nm, equivalent to geometric albedos of 0.30±0.02 and 0.26±0.02, respectively. There is more than factor of 2 spatial variation in reflectance, but only about 10% variation in the 950-nm/760-nm reflectance ratio. Reflectance and color ratio are highly correlated, with the highest reflectances in discrete, sharp-edged patches on slopes exceeding 20°, consistent with material being exposed by downslope movement. Eros is also conspicuously deficient in small, bright, spectrally distinctive craters which are found on the other two S asteroids, Gaspra and Ida, imaged by spacecraft at close range. Eros exhibits a larger range of albedos than other S asteroids, but its color variations are much more subtle: variation in the 950-nm/760-nm reflectance ratio with 760-nm reflectance is several times less than on those asteroids or in the lunar maria. Of the different mechanisms possibly responsible for reflectance and color differences on Eros, spatial differences in the extent of regolith aging by space weathering are most consistent with Eros's observed properties. However, the effects of this process are both qualitatively and quantitatively different than in the lunar maria.

Aircraft measurements of the solar and infrared radiative properties of cirrus and their dependence on ice crystal shape

We present aircraft measurements of the radiative transfer properties of thin cirrus cloud sampled off the east coast of Scotland on November 9, 1995. Downwelling radiances were measured from below the cirrus at 0.87, 1.61, 3.7, 8.55, and 11.0 μm, thereby covering a large range of size parameter and ice refractive index and enabling information on cirrus optical thickness, effective crystal size, and scattering phase function to be deduced. The sensitivity of these quantities to the ice crystal shape assumed in the calculations is examined, and the results are compared with in situ data. We find that a randomized polycrystal shape produces effective sizes that are consistent with the in situ data across all wavelengths considered and performs better in this respect than the other crystal shapes analyzed. However, the optical thicknesses retrieved from the 0.87 μm radiances for this shape are considerably less than those derived from the 11.0 μm data, implying that the phase function at solar wavelengths is in error for this shape over a significant portion of the full scattering angle range. An empirical phase function derived from laboratory measurements produces optical thicknesses which are more consistent with the 11.0 μm and in situ data and matches the angular distribution of scattered radiance more accurately than that calculated using any of the model crystal shapes. The anomalous diffraction approximation is found to produce good agreement with the measurements at 8.55 and 11.0 μm for the crystal sizes relevant to the present case study.

Single scattering by red blood cells

A highly diluted suspension of red blood cells (hematocrit 0.01) was illuminated with an Ar or a dye laser in the wavelength range of 458-660 nm. The extinction and the angle-resolved intensity of scattered light were measured and compared with the predictions of Mie theory, the Rayleigh-Gans approximation, and the anomalous diffraction approximation. Furthermore, empirical phase functions were fitted to the measurements. The measurements were in satisfactory agreement with the predictions of Mie theory. However, better agreement was found with the anomalous diffraction model. In the Rayleigh-Gans approximation, only small-angle scattering is described appropriately. The scattering phase function of erythrocytes may be represented by the Gegenbauer kernel phase function.

Estimation of the Aerosol Phase Function in Backscatter from Simultaneous Satellite and Sun-Photometer Measurements

Abstract The backscatter part of the aerosol phase function PA(χ), where χ is the scattering angle, is difficult to measure from the ground. Experimental data for χ > 120° are not reported in the literature. Customarily, PA(χ) is calculated from Mie theory using an aerosol size distribution either prescribed or estimated by inversion of spectral or almucantar/aureole measurements. These results clearly require validation using direct measurements. In this paper, an empirical phase function of atmospheric aerosol over the ocean is estimated in backscatter (χ > 130°) from coincident measurements of upward radiance in channel 1 (0.63 μm) of the Advanced Very High Resolution Radiometer (AVHRR) on board National Oceanic and Atmospheric Administration satellites and sun-photometer aerosol optical thickness, δASP. This study uses 31 sun-photometer measurements, collected during two oceanic cruises over the North Atlantic in 1989 and 1991. The accuracies of both satellite radiances and sun-photometer δASP are wel...

On the definition of asteroid magnitudes

AbstractThe problem of a sharp definition of two photometric parameters for asteroids in the presence of empirical phase functions is discussed. In a rational magnitude system advantage should be taken of the nearly linear phase curve.