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NASA NTRS1958

The Interaction of a Reflected Shock Wave with the Boundary Layer in a Shock Tube

Ideally, the reflection of a shock from the closed end of a shock tube provides, for laboratory study, a quantity of stationary gas at extremely high temperature. Because of the action of viscosity, however, the flow in the real case is not one-dimensional, and a boundary layer grows in the fluid following the initial shock wave. In this paper simplifying assumptions are made to allow an analysis of the interaction of the shock reflected from the closed end with the boundary layer of the initial shock afterflow. The analysis predicts that interactions of several different types will exist in different ranges of initial shock Mach number. It is shown that the cooling effect of the wall on the afterflow boundary layer accounts for the change in interaction type. An experiment is carried out which verifies the existence of the several interaction regions and shows that they are satisfactorily predicted by the theory. Along with these results, sufficient information is obtained from the experiments to make possible a model for the interaction in the most complicated case. This model is further verified by measurements made during the experiment. The case of interaction with a turbulent boundary layer is also considered. Identifying the type of interaction with the state of turbulence of the interacting boundary layer allows for an estimate of the state of turbulence of the boundary layer based on an experimental investigation of the type of interaction. A method is proposed whereby the effect of the boundary-layer interaction on the strength of the reflected shock may be calculated. The calculation indicates that the reflected shock is rapidly attenuated for a short distance after reflection, and this result compares favorably with available experimental results.

Mark, Herman↗
NASA NTRS2006

What does Reflection from Cloud Sides tell us about Vertical Distribution of Cloud Droplet Sizes?

Cloud development, the onset of precipitation and the effect of aerosol on clouds depend on the structure of the cloud profiles of droplet size and phase. Aircraft measurements of cloud profiles are limited in their temporal and spatial extent. Satellites were used to observe cloud tops not cloud profiles with vertical profiles of precipitation-sized droplets anticipated from CloudSat. The recently proposed CLAIM-3D satellite mission (cloud aerosol interaction mission in 3-D) suggests to measure profiles of cloud microphysical properties by retrieving them from the solar and infrared radiation reflected or emitted from cloud sides. Inversion of measurements from the cloud sides requires rigorous understanding of the 3-dimentional(3-D) properties of clouds. Here we discuss the reflected sunlight from the cloud sides and top at two wavelengths: one nonabsorbing to solar radiation (0.67 microns) and one with liquid water efficient absorption of solar radiation (2.1 microns). In contrast to the plane-parallel approximation, a conventional approach to all current operational retrievals, 3-D radiative transfer is used for interpreting the observed reflectances. General properties of the radiation reflected from the sides of an isolated cloud are discussed. As a proof of concept, the paper shows a few examples of radiation reflected from cloud fields generated by a simple stochastic cloud model with the prescribed vertically resolved microphysics. To retrieve the information about droplet sizes, we propose to use the probability density function of the droplet size distribution and its first two moments instead of the assumption about fixed values of the droplet effective radius. The retrieval algorithm is based on the Bayesian theorem that combines prior information about cloud structure and microphysics with radiative transfer calculations.

Marshak, A.↗
NASA NTRS2005

What Does Reflection from Cloud Sides Tell Us About Vertical Distribution of Cloud Droplet Sizes?

Cloud development, the onset of precipitation and the effect of aerosol on clouds depend on the structure of the cloud profiles of droplet size and phase. Aircraft measurements of cloud profiles are limited in their temporal and spatial extent. Satellites were used to observe cloud tops not cloud profiles with vertical profiles of precipitation-sized droplets anticipated from Cloudsat. The recently proposed CLAIM-3D satellite mission (cloud aerosol interaction mission in 3D) suggests to measure profiles of cloud microphysical properties by retrieving them from the solar and infrared radiation reflected or emitted from cloud sides. Inversion of measurements from the cloud sides requires rigorous understanding of the 3-dimensional (3D) properties of clouds. Here we discuss the reflected sunlight from the cloud sides and top at two wavelengths: one nonabsorbing to solar radiation (0.67 micrometers) and one with liquid water efficient absorption of solar radiation (2.1 micrometers). In contrast to the plane-parallel approximation, a conventional approach to all current operational retrievals, 3D radiative transfer is used for interpreting the observed reflectances. General properties of the radiation reflected from the sides of an isolated cloud are discussed. As a proof of concept, the paper shows a few examples of radiation reflected from cloud fields generated by a simple stochastic cloud model with the prescribed vertically resolved microphysics. To retrieve the information about droplet sizes, we propose to use the probability density function of the droplet size distribution and its first two moments instead of the assumption about fixed values of the droplet effective radius. The retrieval algorithm is based on the Bayesian theorem that combines prior information about cloud structure and microphysics with radiative transfer calculations.

Marshak, Alexander↗
NASA NTRS2005

Off-axis reflective optical apparatus

Embodiments of the present invention are directed to a simple apparatus and a convenient and accurate method of mounting the components to form an off-axis reflective optical apparatus such as a collimator. In one embodiment, an off-axis reflective optical apparatus comprises a mounting block having a ferrule holder support surface and an off-axis reflector support surface which is generally perpendicular to the ferrule holder support surface. An optical reflector is mounted on the off-axis reflector support surface and has a reflected beam centerline. The optical reflector has a conic reflective surface and a conic center. A ferrule holder is mounted on the ferrule holder support surface. The ferrule holder provides a ferrule for coupling to an optical fiber and orienting a fiber tip of the optical fiber along a fiber axis toward the optical reflector. The fiber axis is nonparallel to the reflected beam centerline. Prior to mounting the optical reflector to the off-axis reflector support surface and prior to mounting the ferrule holder to the ferrule holder support surface, the optical reflector is movable on the off-axis reflector surface and the ferrule holder is movable on the ferrule holder support surface to align the conic center of the optical reflector with respect to the fiber tip of the optical fiber, and the apparatus has at least one of the following features: (1) the optical reflector is movable on the off-axis reflector support surface to adjust a focus of the fiber tip with respect to the optical reflector, and (2) the ferrule holder is movable on the ferrule holder support surface to adjust the focus of the fiber tip with respect to the optical reflector.

Ames, Lawrence L.↗
NASA NTRS2008

On-Orbit Noise Characterization for MODIS Reflective Solar Bands

Since launch, the Moderate Resolution Imaging Spectroradiometer (MODIS) has operated successfully on-board the NASA Earth Observing System (EOS) Terra and EOS Aqua spacecraft. MODIS is a passive cross-track scanning radiometer that makes observations in 36 spectral bands with spectral wavelengths from visible (VIS) to long-wave infrared. MODIS bands 1-19 and 26 are the reflective solar bands (RSB) with wavelengths from 0.41 to 2.2 micrometers. They are calibrated on-orbit using an on-board solar diffuser (SD) and a SD stability monitor (SDSM) system. For MODIS RSB, the level 1B calibration algorithm produces top of the atmosphere reflectance factors and radiances for every pixel of the Earth view. The sensor radiometric calibration accuracy, specified at each spectral band's typical scene radiance, is 2% for the RSB reflectance factors and 5% for the RSB radiances. Also specified at the typical scene radiance is the detector signal-to-noise ratio (SNR), a key sensor performance parameter that directly impacts its radiometric calibration accuracy and stability, as well as the image quality. This paper describes an on-orbit SNR characterization approach developed to evaluate and track MODIS RSB detector performance. In order to perform on-orbit SNR characterization, MODIS RSB detector responses to the solar illumination reflected from the SD panel must be corrected for factors due to variations of the solar angles and the SD bi-directional reflectance factor. This approach enables RSB SNR characterization to be performed at different response levels for each detector. On-orbit results show that both Terra and Aqua MODIS RSB detectors have performed well since launch. Except for a few noisy or inoperable detectors which were identified pre-launch, most RSB detectors continue to meet the SNR design requirements and are able to maintain satisfactory short-term stability. A comparison of on-orbit noise characterization results with results derived from pre-launch calibration and characterization are also provided.

Xiong, X.↗
NASA NTRS2010

Alfven Wave Reflection Model of Field-Aligned Currents at Mercury

An Alfven Wave Reflection (AWR) model is proposed that provides closure for strong field-aligned currents (FACs) driven by the magnetopause reconnection in the magnetospheres of planets having no significant ionospheric and surface electrical conductance. The model is based on properties of the Alfven waves, generated at high altitudes and reflected from the low-conductivity surface of the planet. When magnetospheric convection is very slow, the incident and reflected Alfven waves propagate along approximately the same path. In this case, the net field-aligned currents will be small. However, as the convection speed increases. the reflected wave is displaced relatively to the incident wave so that the incident and reflected waves no longer compensate each other. In this case, the net field-aligned current may be large despite the lack of significant ionospheric and surface conductivity. Our estimate shows that for typical solar wind conditions at Mercury, the magnitude of Region 1-type FACs in Mercury's magnetosphere may reach hundreds of kilo-Amperes. This AWR model of field-aligned currents may provide a solution to the long-standing problem of the closure of FACs in the Mercury's magnetosphere. c2009 Elsevier Inc. All rights reserved.

Lyatsky, Wladislaw↗
NASA NTRS2010

The Effect of Incident Light Polarization on Vegetation Bidirectional Reflectance Factor

The Laboratory-based Bidirectional Reflectance Factor (BRF) polarization study of vegetation is presented in this paper. The BRF was measured using a short-arc Xenon lamp/monochromator assembly producing an incoherent, tunable light source with a well-defined spectral bandpass at visible and near-infrared wavelengths of interest at 470 nm and 870 nm and coherent light source at 1.656 microns. All vegetation samples were measured using P and S linearly polarized incident light over a range of incident and scatter angles. By comparing these results, we quantitatively examine how the BRF of the samples depends on the polarization of the incident light. The differences are significant, depend strongly on the incident and scatter angles, and can be as high as 120% at 67 deg incident and 470nm. The global nature of Earth's processes requires consistent long-term calibration of all instruments involved in data retrieval. The BRF defines the reflection characteristics of Earth surface. It provides the reflectance of a target in a specific direction as a function of illumination and viewing geometry. The BRF is a function of wavelength and reflects the structural and optical properties of the surface. Various space and airborne radiometric and imaging remote sensing instruments are used in the remote sensing characterization of vegetation canopies and soils, oceans, or especially large pollution sources. The satellite data is validated through comparison with airborne, ground-based and laboratory-based data in an effort to fully understand the vegetation canopy reflectance, The Sun's light is assumed to be unpolarized at the top of the atmosphere; however it becomes polarized to some degree due to atmospheric effects by the time it reaches the vegetation canopy. Although there are numerous atmospheric correction models, laboratory data is needed for model verification and improvement.

Georgiev, Georgi T.↗
NASA NTRS2010

Water Detection Based on Sky Reflections

This software has been designed to detect water bodies that are out in the open on cross-country terrain at mid- to far-range (approximately 20 100 meters), using imagery acquired from a stereo pair of color cameras mounted on a terrestrial, unmanned ground vehicle (UGV). Non-traversable water bodies, such as large puddles, ponds, and lakes, are indirectly detected by detecting reflections of the sky below the horizon in color imagery. The appearance of water bodies in color imagery largely depends on the ratio of light reflected off the water surface to the light coming out of the water body. When a water body is far away, the angle of incidence is large, and the light reflected off the water surface dominates. We have exploited this behavior to detect water bodies out in the open at mid- to far-range. When a water body is detected at far range, a UGV s path planner can begin to look for alternate routes to the goal position sooner, rather than later. As a result, detecting water hazards at far range generally reduces the time required to reach a goal position during autonomous navigation. This software implements a new water detector based on sky reflections that geometrically locates the exact pixel in the sky that is reflecting on a candidate water pixel on the ground, and predicts if the ground pixel is water based on color similarity and local terrain features

Rankin, Arturo L.↗
NASA NTRS2011

X-Ray Reflected Spectra from Accretion Disk Models. II. Diagnostic Tools for X-Ray Observations

We present a comprehensive study of the emission spectra from accreting sources. We use our new reflection code to compute the reflected spectra from an accretion disk illuminated by X-rays. This set of models covers different values of ionization parameter, solar iron abundance and photon index for the illuminating spectrum. These models also include the most complete and recent atomic data for the inner-shell of the iron and oxygen isonuclear sequences. We concentrate our analysis to the 2 - 10 keV energy region, and in particular to the iron K-shell emission lines. We show the dependency of the equivalent width (EW) of the Fe Ka with the ionization parameter. The maximum value of the EW is approx. 800 eV for models with log Epsilon approx. 1.5, and decreases monotonically as Epsilon increases. For lower values of Epsilon the Fe K(alpha) EW decreases to a minimum near log Epsilon approx. 0.8. We produce simulated CCD observations based on our reflection models. For low ionized, reflection dominated cases, the 2 -10 keV energy region shows a very broad, curving continuum that cannot be represented by a simple power-law. We show that in addition to the Fe K-shell emission, there are other prominent features such as the Si and S L(alpha) lines, a blend of Ar VIII-XI lines, and the Ca x K(alpha) line. In some cases the S xv blends with the He-like Si RRC producing a broad feature that cannot be reproduced by a simple Gaussian profile. This could be used as a signature of reflection.

Garcia, J.↗
NASA NTRS2013

Quantitative Comparison of the Variability in Observed and Simulated Shortwave Reflectance

The Climate Absolute Radiance and Refractivity Observatory (CLARREO) is a climate observation system that has been designed to monitor the Earth's climate with unprecedented absolute radiometric accuracy and SI traceability. Climate Observation System Simulation Experiments (OSSEs) have been generated to simulate CLARREO hyperspectral shortwave imager measurements to help define the measurement characteristics needed for CLARREO to achieve its objectives. To evaluate how well the OSSE-simulated reflectance spectra reproduce the Earth s climate variability at the beginning of the 21st century, we compared the variability of the OSSE reflectance spectra to that of the reflectance spectra measured by the Scanning Imaging Absorption Spectrometer for Atmospheric Cartography (SCIAMACHY). Principal component analysis (PCA) is a multivariate decomposition technique used to represent and study the variability of hyperspectral radiation measurements. Using PCA, between 99.7%and 99.9%of the total variance the OSSE and SCIAMACHY data sets can be explained by subspaces defined by six principal components (PCs). To quantify how much information is shared between the simulated and observed data sets, we spectrally decomposed the intersection of the two data set subspaces. The results from four cases in 2004 showed that the two data sets share eight (January and October) and seven (April and July) dimensions, which correspond to about 99.9% of the total SCIAMACHY variance for each month. The spectral nature of these shared spaces, understood by examining the transformed eigenvectors calculated from the subspace intersections, exhibit similar physical characteristics to the original PCs calculated from each data set, such as water vapor absorption, vegetation reflectance, and cloud reflectance.

Roberts, Yolanda, L.↗
NASA NTRS2013

Daily MODIS 500 m Reflectance Anisotropy Direct Broadcast (DB) Products for Monitoring Vegetation Phenology Dynamics

Land surface vegetation phenology is an efficient bio-indicator for monitoring ecosystem variation in response to changes in climatic factors. The primary objective of the current article is to examine the utility of the daily MODIS 500 m reflectance anisotropy direct broadcast (DB) product for monitoring the evolution of vegetation phenological trends over selected crop, orchard, and forest regions. Although numerous model-fitted satellite data have been widely used to assess the spatio-temporal distribution of land surface phenological patterns to understand phenological process and phenomena, current efforts to investigate the details of phenological trends, especially for natural phenological variations that occur on short time scales, are less well served by remote sensing challenges and lack of anisotropy correction in satellite data sources. The daily MODIS 500 m reflectance anisotropy product is employed to retrieve daily vegetation indices (VI) of a 1 year period for an almond orchard in California and for a winter wheat field in northeast China, as well as a 2 year period for a deciduous forest region in New Hampshire, USA. Compared with the ground records from these regions, the VI trajectories derived from the cloud-free and atmospherically corrected MODIS Nadir BRDF (bidirectional reflectance distribution function) adjusted reflectance (NBAR) capture not only the detailed footprint and principal attributes of the phenological events (such as flowering and blooming) but also the substantial inter-annual variability. This study demonstrates the utility of the daily 500 m MODIS reflectance anisotropy DB product to provide daily VI for monitoring and detecting changes of the natural vegetation phenology as exemplified by study regions comprising winter wheat, almond trees, and deciduous forest.

Vegetation↗
NASA NTRS2014

Design, Analysis, and On-Sun Evaluation of Reflective Strips Under Controlled Buckling

Solar concentrators are envisioned for use in a variety of space-based applications, including applications involving in situ resource utilization. Identifying solar concentrators that minimize mass and cost are of great interest, especially since launch cost is driven in part by the mass of the payload. Concentrators must also be able to survive the wide temperature excursions on the lunar surface. Identifying smart structures which compensate for changes in concentrator geometry brought about by temperature extremes are of interest. Some applications may benefit from the ability to change the concentrators focal pattern at will. This paper addresses a method of designing a single reflective strip to produce a desired focal pattern through the use of controlled buckling. Small variations in the cross section over the length of the reflective strip influence the distribution of light in the focal region. A finite element method of analysis is utilized here which calculates the curve produced for a given strip cross section and axial load. Varying axial force and strip cross section over the length of the reflective strip provide a means of optimizing ray convergence in the focal region. Careful selection of a tapered cross section yields a reflective strip that approximates a parabola. An array of reflective strips under controlled buckling produces a light weight concentrator and adjustments in the compression of individual strips provide a means of compensating for temperature excursions or changing the focal pattern at will.

solar collectors↗
NASA NTRS2014

Design, Analysis, and On-Sun Evaluation of Reflective Strips Under Controlled Buckling

Solar concentrators are envisioned for use in a variety of space-based applications, including applications involving in situ resource utilization. Identifying solar concentrators that minimize mass and cost are of great interest, especially since launch cost is driven in part by the mass of the payload. Concentrators must also be able to survive the wide temperature excursions on the lunar surface. Identifying smart structures which compensate for changes in concentrator geometry brought about by temperature extremes are of interest. Some applications may benefit from the ability to change the concentrator's focal pattern at will. This paper addresses a method of designing a single reflective strip to produce a desired focal pattern through the use of controlled buckling. Small variations in the cross section over the length of the reflective strip influence the distribution of light in the focal region. A finite element method of analysis is utilized here which calculates the curve produced for a given strip cross section and axial load. Varying axial force and strip cross section over the length of the reflective strip provide a means of optimizing ray convergence in the focal region. Careful selection of a tapered cross section yields a reflective strip that approximates a parabola. An array of reflective strips under controlled buckling produces a light weight concentrator and adjustments in the compression of individual strips provide a means of compensating for temperature excursions or changing the focal pattern at will.

buckling↗
NASA NTRS2014

CLARREO Approach for Reference Intercalibration of Reflected Solar Sensors: On-Orbit Data Matching and Sampling

The implementation of the Climate Absolute Radiance and Refractivity Observatory (CLARREO) mission was recommended by the National Research Council in 2007 to provide an on-orbit intercalibration standard with accuracy of 0.3% (k = 2) for relevant Earth observing sensors. The goal of reference intercalibration, as established in the Decadal Survey, is to enable rigorous high-accuracy observations of critical climate change parameters, including reflected broadband radiation [Clouds and Earth's Radiant Energy System (CERES)], cloud properties [Visible Infrared Imaging Radiometer Suite (VIIRS)], and changes in surface albedo, including snow and ice albedo feedback. In this paper, we describe the CLARREO approach for performing intercalibration on orbit in the reflected solar (RS) wavelength domain. It is based on providing highly accurate spectral reflectance and reflected radiance measurements from the CLARREO Reflected Solar Spectrometer (RSS) to establish an on-orbit reference for existing sensors, namely, CERES and VIIRS on Joint Polar Satellite System satellites, Advanced Very High Resolution Radiometer and follow-on imagers on MetOp, Landsat imagers, and imagers on geostationary platforms. One of two fundamental CLARREO mission goals is to provide sufficient sampling of high-accuracy observations that are matched in time, space, and viewing angles with measurements made by existing instruments, to a degree that overcomes the random error sources from imperfect data matching and instrument noise. The data matching is achieved through CLARREO RSS pointing operations on orbit that align its line of sight with the intercalibrated sensor. These operations must be planned in advance; therefore, intercalibration events must be predicted by orbital modeling. If two competing opportunities are identified, one target sensor must be given priority over the other. The intercalibration method is to monitor changes in targeted sensor response function parameters: effective offset, gain, nonlinearity, optics spectral response, and sensitivity to polarization. In this paper, we use existing satellite data and orbital simulationmethods to determinemission requirements for CLARREO, its instrument pointing ability, methodology, and needed intercalibration sampling and data matching for accurate intercalibration of RS radiation sensors on orbit.

radiometry↗
NASA NTRS2017

Methods for Reducing Singly Reflected Rays on the Wolter-I Focusing Figures of the FOXSI Rocket Experiment

In high energy solar astrophysics, imaging hard X-rays by direct focusing offers higher dynamic range and greater sensitivity compared to past techniques that used indirect imaging. The Focusing Optics X-ray Solar Imager (FOXSI) is a sounding rocket payload which uses seven sets of nested Wolter-I figured mirrors that, together with seven high-sensitivity semiconductor detectors, observes the Sun in hard X-rays by direct focusing. The FOXSI rocket has successfully flown twice and is funded to fly a third time in Summer 2018. The Wolter-I geometry consists of two consecutive mirrors, one paraboloid, and one hyperboloid, that reflect photons at grazing angles. Correctly focused X-rays reflect twice, once per mirror segment. For extended sources, like the Sun, off-axis photons at certain incident angles can reflect on only one mirror and still reach the focal plane, generating a pattern of single-bounce photons that can limit the sensitivity of the observation of faint focused X-rays. Understanding and cutting down the singly reflected rays on the FOXSI optics will maximize the instrument's sensitivity of the faintest solar sources for future flights. We present an analysis of the FOXSI singly reflected rays based on ray-tracing simulations, as well as the effectiveness of different physical strategies to reduce them.

FOXSI Rocket Experiment↗
NASA NTRS2017

Impact of MODIS Sensor Calibration Updates on Greenland Ice Sheet Surface Reflectance and Albedo Trends

We evaluate Greenland Ice Sheet (GrIS) surface reflectance and albedo trends using the newly released Collection 6 (C6) MODIS (Moderate Resolution Imaging Spectroradiometer) products over the period 2001-2016. We find that the correction of MODIS sensor degradation provided in the new C6 data products reduces the magnitude of the surface reflectance and albedo decline trends obtained from previous MODIS data (i.e., Collection 5, C5). Collection 5 and 6 data product analysis over GrIS is characterized by surface (i.e., wet vs. dry) and elevation (i.e., 500-2000 m, 2000 m and greater) conditions over the summer season from 1 June to 31 August. Notably, the visible-wavelength declining reflectance trends identified in several bands of MODIS C5 data from previous studies are only slightly detected at reduced magnitude in the C6 versions over the dry snow area. Declining albedo in the wet snow and ice area remains over the MODIS record in the C6 product, albeit at a lower magnitude than obtained using C5 data. Further analyses of C6 spectral reflectance trends show both reflectance increases and decreases in select bands and regions, suggesting that several competing processes are contributing to Greenland Ice Sheet albedo change. Investigators using MODIS data for other ocean, atmosphere and/or land analyses are urged to consider similar re-examinations of trends previously established using C5 data.

radiative forcing↗
NASA NTRS2018

E-Beam Generated Plasma Etching for Developing High-Reflectance Mirrors for Far-Ultraviolet Astronomical Instrument Applications

Astronomical space telescopes to study astrophysical phenomena from the far-ultraviolet (FUV) to the near infrared (NIR) will require mirror coatings with high reflectance over this entire spectral region. While coatings for the optical and NIR part of the spectrum are fairly well developed with proven performance, the FUV presents significant challenges. The U.S. Naval Research Laboratory (NRL) has developed a processing system based on an electron beam-generated plasma that provides for controlled fluorination and/or etching of surfaces with near monolayer precision and minimal changes to surface morphology. In this paper, we report recent results of samples treated in the NRL Large Area Plasma Processing System (LAPPS) where restoration of the high intrinsic reflectance in the FUV spectral range have been observed of aluminum (Al) mirrors protected with a magnesium di-fluoride (MgF2) overcoat. This paper will also extend these studies to other Al mirrors protected with aluminum tri-fluoride (AlF3) in order to realize the high intrinsic reflectance Al down to FUV wavelengths (100−200 nm), while still maintaining the high reflectance in the optical and NIR spectral regions. Laboratory test data and optical diagnostic techniques used to verify surface scattering and durability of selected coatings will be presented. Finally, we will discuss the scalability of the LAPPS etching process in order to realize these high-reflectivity coatings on mirror segments as large as those proposed for the Large Ultraviolet, Optical, and Infrared (LUVOIR) astronomical telescope system (1+meter class).

LUVOIR↗
NASA NTRS2017

Iterative Discrete Ordinates Solution of the Equation for the Surface-Reflected Radiance

This paper presents a new method of numerical solution of the integral equation for the radiance reflected from an anisotropic surface. The equation relates the radiance at the surface level with BRDF and solutions of the standard radiative transfer problems for a slab with no reflection on its surfaces. It is also shown that the kernel of the equation satisfies the condition of the existence of a unique solution and the convergence of the successive approximations to that solution. The developed method features two basic steps: discretization on a 2D quadrature, and solving the resulting system of algebraic equations with successive over-relaxation method based on the Gauss-Seidel iterative process. Presented numerical examples show good coincidence between the surface-reflected radiance obtained with DISORT and the proposed method. Analysis of contributions of the direct and diffuse (but not yet reflected) parts of the downward radiance to the total solution is performed. Together, they represent a very good initial guess for the iterative process. This fact ensures fast convergence. The numerical evidence is given that the fastest convergence occurs with the relaxation parameter of 1 (no relaxation). An integral equation for BRDF is derived as inversion of the original equation. The potential of this new equation for BRDF retrievals is analyzed. The approach is found not viable as the BRDF equation appears to be an ill-posed problem, and it requires knowledge the surface-reflected radiance on the entire domain of both Sun and viewing zenith angles.

Alexander Radkevich↗