Radiative exchange among non-Lambert surfaces.
Radiative exchange in enclosures, reflectance of whose surfaces can be approximated by sum of diffuse and specular component
SPECULAR REFLECTION↗
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Point spread functions in imaging a Lambert surface from zenith through a thin scattering layer
Analytical techniques for good spatial resolution in remotely sensed images of renewable resources, such as crops, are discussed for satellite multispectral radiometry. A model is developed for an optically and geometrically thin scattering layer to account for atmospheric scattering above the object pixel of fluxes reflected from adjacent areas. The cross radiance is explored as a spread function of a point source and as a spurious component of measured radiance, and an integration over large source areas is formulated. The Henyey-Greenstein (HG) phase function is defined for an integral over a sphere and point-spread functions are presented for HG scattering. Cross radiance limited spatial resolution is also determined for the HG phase function and boundaries between reflecting and black half-planes are considered in terms of the cross radiance.
Otterman, J.↗
Modeling the non-grey-body thermal emission from the full moon
The present series of thermophysical computer models for solid-surfaced planetary bodies whose surface roughness is modeled as paraboloidal craters of specified depth/diameter ratio attempts to characterize the nongrey-body brightness temperature spectra of the moon and of the Galilean satellites. This modeling, in which nondiffuse radiation properties and surface roughness are included for rigorous analysis of scattered and reemitted radiation within a crater, explains to first order the behavior of both limb-scans and disk-integrated IR brightness temperature spectra for the full moon. Only negative surface relief can explain lunar thermal emissions' deviation from smooth Lambert-surface expectations.
Vogler, Karl J.↗
Surface properties and effective resolution from ground-based observations of Mercury
A detailed study to evaluate ground-based photographs of Mercury has been carried out. Models of the surface scattering properties have been assumed and smeared with a Gaussian function for direct comparison with center-to-limb scans along Mercury's intensity equator. Data from a range of phase angles from 31 to 92 deg have been compared with smeared models assuming a Lambert surface, a surface which obeys the Lommel-Seeliger law and one which is Minnaertian, having a variable coefficient. Within the limits of the observations a lunar Minnaert surface yields the most consistent interpretation. An objective evaluation of the resolution of the photographs is obtained in terms of Gaussian half-widths.
Beebe, R. F.↗
The delta-four-stream approximation for radiative flux transfer
An approximate technique for solving the radiative transfer problem for fluxes reflected, transmitted, and absorbed by a homogeneous scattering layer is presented. The technique is a straightforward delta-function modification of a solution to the transfer equation in closed form via the discrete-ordinates method with four streams. By the use of four streams, significant improvement in accuracy is obtained for anisotropic phase functions over that obtained by similarly employed two-stream approximations with delta-functions. However, the computational effort is increased only slightly. Equations are developed for the general case of a layer underlaid by a reflecting Lambert surface. Graphical comparisons are given of fractional error resulting from the use of this method with that resulting from the use of typical four-stream and delta-two-stream techniques.
Cuzzi, J. N.↗
Azimuthal brightness variation and albedo measurements of the Uranian rings
The recent Voyager encounter with Uranus has produced new insights and observational constraints on the structure, composition, and evolution of the Uranian rings. Careful measurements have been made of the brightness variation as a function of longitude for the alpha, beta, gamma, delta, and epsilon rings. These measurements showed a substantial variation in azimuthal brightness because of the varying width of the rings. The single scattering albedo of the epsilon ring particles was found to be 3.9 + or - 0.6 percent (for the Lambert surface) or 2.3 + or - 0.4 percent assuming the 'lunar-type' phase function. The optical depth at apoapse of the epsilon ring was found to be 0.40 + or - 0.05.
Svitek, Tomas↗
Cryogenic storage on the moon.
Cryogenic fluid storage on moon surface, considering Lambert reflection law, thermal insulation performance, incident heat flux and boil-off rates
LUNAR SURFACE↗
Competition between roughness and strength for scale-dependent surfaces
Rocks famously have scale-dependent strength, yet the actual dependence is notoriously hard to measure or incorporate into any theoretical framework. Natural rough surfaces present an opportunity to solve the problem. Surfaces sliding in shear evolve as protrusions collide. These asperities can deform or break, thus creating a new surface shape. In particular, natural surfaces have roughness at all scales as well as scale-dependent strength. Based on a scaling analysis, we have previously suggested that the scale-dependent aspect ratio of steady-state surfaces should be proportional to the scale-dependent shear strain at yield. If true, scale-dependent strength could easily be inferred from natural surfaces. Thus, moving beyond the scaling argument to a rigorous treatment of scale-dependent strength for multiscale rough surfaces in shear is important. However, analytic frameworks for analyzing multiscale problems are challenging, as conventional continuum mechanics typically involves a single value for a material property across scales. Here, in this work, we build on the formalism of Persson (2001) that presents a method to compute contact area for rough surfaces with a prescribed topographic spectrum using a stochastic differential equation. The Persson formalism allows for plastic yield under normal loading of otherwise elastic materials and leaves open the possibility of scale-dependent yield stress. In this study, we pursue this route to develop a theory and numerical results for the yielding of a rough, elastoplastic surface with scale-dependent yield stress. Here, we examine surfaces for which the power spectrum of the topography 𝐶 and yield stress 𝑌 follow power laws as a function of scale 𝜆, such that 𝐶∼𝜆 −𝑚 and 𝑌∼𝜆 −𝑛 , respectively. In this formal treatment of the problem, we focus on surfaces in contact and the resulting yield and do not impose shear. Numerical solutions show that the deviation from the elastic scaling solution is bounded as expected by the prior 1D heuristic scaling argument that anticipates the Hurst exponent as 1−𝑛. We also show that the plasticity is expected to erode the contacts if 𝑚 is lower than 𝑛−3, which corresponds to a Hurst exponent lower than 1−𝑛/2. This result is rigorously sound for 2D, i.e., realistic surfaces, and quantitatively different than the prior scaling argument. The theory now permits a correspondingly quantitative approach to interpreting natural surfaces.
elasticity↗
Lunar surface roughness - Shadowing and thermal emission.
Model for lunar surface roughness effect on emission of thermal IR radiation and casting of shadows in sunlight
THERMAL EMISSION↗
Cryogenic storage on the moon.
Cryogenic fluid storage on moon surface, considering Lambert reflection law, thermal insulation performance, incident heat flux and boil-off rates
Gabron, F.↗
Investigation of the effect of atmospheric dust on the determination of total ozone from the earth's ultraviolet reflectivity measurements, 1
Two computer algorithms are described. These algorithms were used for computing the aximuth-independent component of the intensity of the monochromatic radiation emerging at the top of a pseudo-spherical atmosphere with arbitrary vertical distribution of ozone, and with any arbitrary height distribution of up to two different kinds of aerosol. This atmospheric model was assumed to rest on a surface obeying Lambert's law of reflection.
Dave, J. V.↗
Atmospheric effects on radiometry from zenith of a plane with dark vertical protrusions
Effects of an optically thin plane-parallel scattering atmosphere on radiometric imaging from the zenith of a specific surface-type are analyzed. The surface model was previously developed to describe arid steppe, where the sparse vegetation forms dark vertical protrusions from the bright soil-plane. The analysis is in terms of the surface reflectivity to the zenith r sub p for the direct beam, which is formulated as r sub p = r sub i exp (-s tan theta sub 0), where v sub i is the Lambert law reflectivity of the soil, the protrusions parameters s is the projection on a vertical plane of protrusions per unit area and theta sub 0 is the zenith angle. The surface reflectivity r sub p is approximately equal to that for the global irradiance (which is directly measured in the field) only for a narrow range of the solar zenith angles. The effects of the atmosphere when imaging large uniform areas of this type are comparable to those in imaging a Lambert surface with a reflectivity r sub p. Thus, the effects can be approximated by those in the case of a dark Lambert surface (analyzed previously), inasmuch as r sub p is smaller than the soil reflectivity r sub i for any off-zenith illumination. The surface becomes effectively darker with increasing solar zenith angle. Adjacency effects of a reflection from one area and scattering in the instantaneous field of view (object pixel) are analyzed as cross radiance and cross irradiance.
Otterman, J.↗
The effect of surface anisotropy on the accuracy of total ozone estimates from satellite observations
The total amount of ozone in a vertical column of the earth's atmosphere is being derived from satellite measurements of the intensity of ultraviolet sunlight scattered by the earth-atmosphere system. The algorithm for deriving the ozone amount utilizes the assumption that the earth's surface reflects the incident light isotropically according to Lambert's law. Natural surface reflection deviates more or less from this law. Two extreme examples of anisotropic reflection from dark ocean and from bright snow are analyzed by means of models for their effects on the derived values of ozone.
Fraser, R. S.↗
Lageos orbit and the albedo problem
The objective was to obtain an analytic expression for the radiation pressure force on a satellite due to sunlight reflected from the Earth. The Lageos satellite undergoes unexplained along-track accelerations. These accelerations are believed to be due mainly to terrestrial radiation pressure. The effect of sunlight reflected off the surface of the Earth must thus be modeled to insure an accurate orbit for Lageos. An accurate orbit is necessary for carrying out Lageos' mission of measuring tectonic plate motion, polar motion, and Earth rotation. The present investigation focuses on a spherical harmonic approach to the problem. An equation for the force was obtained by assuming the Earth's surface reflects sunlight according to Lambert's law. The equation is an integral over the whole Earth's surface. Expressions occurring inside the integral are expressed in terms of spherical harmonics. The problem is thus reduced to integrating products of spherical harmonics.
Rubincam, D. P.↗
Custom surface reflectance, shade mask, and equivalent water thickness maps for the Colorado Headwaters Ecological Spectroscopy Study (2025)
This dataset contains land surface reflectance estimates and additional derived products generated from NEON Imaging Spectrometer (NIS) data collected in the Upper Gunnison river basin during June and July of 2025. Data was collected over three domains: the Upper East River (CRBU), Almont Triangle (ALMO), and the Upper Taylor Basin (UPTA). These products were derived from radiance and LiDAR data collected by the NEON Airborne Observation Platform (AOP) campaign funded by the Colorado Headwaters Ecological Spectroscopy Study (CHESS) (doi:10.15485/3017965). Products include per-pixel surface reflectance (rfl) and reflectance uncertainty (rfl_unc), observational data (obs), canopy equivalent water thickness (ewt), and shade masks. Atmospheric correction was performed per flightline using the ISOFIT (Imaging Spectrometer Optimal FITting) optimal estimation framework to estimate surface reflectance and the associated per-band reflectance uncertainty. Reflectance retrievals achieved a mean absolute error of 1.5% across diverse validation surfaces (see validation report.pdf). Equivalent water thickness was calculated from surface reflectance using the Beer–Lambert absorption of liquid water. Shade masks were generated based on the geometry between the sun angle, ground surface, and sensor at the time of flight. Data products are provided per-flightline and as mosaics for each domain. Flightline data products are provided as ENVI-formatted binary files (rfl, rfl_unc, ewt) and GeoTIFFs (shade). Reflectance and uncertainty mosaics are provided as tiled NetCDFs, while all other mosaicked products are provided as cloud-optimized GeoTIFFs. These formats are supported by common geospatial software (e.g., QGIS, ArcGIS, ENVI) and programmatic libraries in Python (e.g., rasterio, xarray, spectral, netCDF4) and R (e.g., terra, ncdf4). Processing workflows were designed to be equivalent to those used to generate the 2018 CHESS campaign airborne imaging spectroscopy data products (doi:10.15485/3013527). All outputs were co-registered to a common spatial grid to support time series analyses. CHESS Project Description: The Colorado Headwaters Ecological Spectroscopy Study (CHESS) comprised a multi-week airborne remote sensing and field observation campaign in the Upper Gunnison Basin, Colorado, conducted in June and July of 2025. Airborne remote sensing was conducted by the National Ecological Observatory Network Airborne Observation Platform (NEON AOP), concurrent with a field campaign run by the Rocky Mountain Biological Laboratory (RMBL), the Lawrence Berkeley National Laboratory (LBNL) and SLAC National Accelerator Laboratory Watershed Function Science Focus Area (SFA), and NASA-JPL (Jet Propulsion Laboratory) Earth Surface Mineral Dust Source Investigation (EMIT) program. Between June 10 and July 18, 2025, the NEON AOP flight team collected high-resolution aerial imaging spectroscopy and Light Detection and Ranging (LiDAR) data over three domains: the Upper East River (CRBU), Almont Triangle (ALMO), and the Upper Taylor Basin (UPTA). In coordination with the flights, a field campaign acquired ground-truth observations, including observations of vegetation composition, foliar traits, forest demography, and subsurface properties in 18 core sampling areas within the domains. Additional surface water observations were taken at over 380 point locations. All CHESS campaign datasets can be found within the CHESS ESS-DIVE data portal: https://data.ess-dive.lbl.gov/portals/chess. Funding Acknowledgment: Data acquisition was performed under a grant from the National Aeronautics and Space Administration (80NSSC24K1005). Computational research was carried out at the Jet Propulsion Laboratory, California Institute of Technology, under a contract with the National Aeronautics and Space Administration (80NM0018D0004) and was funded by EMIT Extended Mission Phase E Science.
2018 NEON and 2025 CHESS Campaigns↗
A Kinetic Model for Predicting Trace Gas Uptake and Reaction
A model is developed to describe trace gas uptake and reaction with applications to aerosols and microdroplets. Gas uptake by the liquid is formulated as a coupled equilibria that links gas, surface, and bulk regions of the droplet or solution. Previously, this framework was used in explicit stochastic reaction-diffusion simulations to predict the reactive uptake kinetics of ozone with droplets containing aqueous aconitic acid, maleic acid, and sodium nitrite. With the use of prior data and simulation results, a new equation for the uptake coefficient is derived, which accounts for both surface and bulk reactions. Lambert W functions are used to obtain closed form solutions to the integrated rate laws for the multiphase kinetics; similar to previous expressions that describe Michaelis-Menten enzyme kinetics. Together these equations couple interface and bulk processes over a wide range of conditions and do not require many of the limiting assumptions needed to apply resistor model formulations to explain trace gas uptake and reaction.
37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
Effect of aerosols on the estimation of total ozone in an atmospheric column from the measurements of its ultraviolet radiance
The problem of estimating the total ozone amount of an atmospheric column in the presence of stratospheric and tropospheric aerosols is studied by means of simulated spectral measurements for the five-wavelength Nimbus 4 BUV configuration and the six-wavelength Nimbus G TOMS configuration, and by analyzing such measurements with appropriate total ozone estimation procedures. The simulated spectral measurements were generated for forty different atmospheric models arrived at by varying the total ozone content and various aerosol parameters. A total of ten different values of the solar zenith angle and eight different values of the Lambert reflectivity of the underlying surface were used. It was found that the values of total ozone in an atmospheric column estimated by the five-wavelength procedure can be in error by about + or -2% as a result of particulate contamination encountered in the terrestrial atmosphere under average conditions. Errors due to tropospheric particulate contamination can be reduced significantly by taking appropriate advantage of the additional spectral measurement offered by the Nimbus G TOMS instrument.
Dave, J. V.↗
Voyager disk resolved photometry of the Saturnian satellites
The scattering properties of the icy satellites of Saturn are studied. Scans of intensity along the photometric equator near opposition are analyzed, and disk-resolved measurements of intensity are fitted to simple photometric functions. Histograms of the albedo distribution across the satellite disks are constructed, and the geometric albedos are determined as a function of wavelength. It is found that, at low phase angles, Rhea and Dione closely follow lunar behavior with almost no limb darkening. Mimas, Tethys, and especially Enceladus show significant limb darkening at low phase angles, which suggest multiple scattering is important for their surfaces. No satellite's surface can be described by Lambert's law. The darker satellites are redder, and the darker areas of each satellite are redder. A longitudinal dark stripe which represents a 15 percent decrease in albedo is situated near the center of the trailing side of Tethys. Mimas and Enceladus both have remarkably flat spectral reflectances into the ultraviolet.
Buratti, B. J.↗