Surface Albedo Maps Research Articles

Constraining Thermal Emission of Pluto’s Haze from Infrared Rotational Lightcurves

The rotational lightcurves of the Pluto-Charon system were previously believed to be solely attributed to their surfaces. However, a proposed scenario of haze cooling suggests that the atmospheric haze of Pluto could significantly contribute to mid-infrared emission, which calls for a revisit of previous analyses. In this study, we employ a Bayesian retrieval approach to constrain the haze emission from the rotational lightcurves of the Pluto-Charon system. The lightcurves were observed by the Spitzer and Herschel telescopes at 24 and 70 μm, and were combined with the latest surface albedo maps of Pluto and Charon from the New Horizons spacecraft. Our results show that including the haze emission is consistent with all current observations, with the best-fit haze flux around 1.63 mJy. This is in agreement with the composition of Titan-like tholins. However, the “surface only” scenario, which excludes the haze contribution, can still explain the observations. We conclude that the current data at 24 μm cannot constrain Pluto’s haze emission due to the degeneracy with Charon’s surface emission. Regardless, some surface properties of Pluto are well constrained by the shape of the lightcurves, with a thermal inertia of approximately 8–10 MKS and a relatively low CH4 emissivity of 0.3–0.5. We suggest that observations by the JWST telescope at 18 μm, which can resolve Pluto from Charon, could directly probe the haze emission of Pluto due to the low surface emission at that wavelength.

Updated Radiative Transfer Model for Titan in the Near-infrared Wavelength Range: Validation against Huygens Atmospheric and Surface Measurements and Application to the Cassini/VIMS Observations of the Dragonfly Landing Area

We present an analysis of Titan data acquired by the Cassini Visual and Infrared Mapping Spectrometer (VIMS) at the landing site of the Dragonfly mission, using a new version of our radiative transfer model for Titan, with significant updates for the spectroscopic parameters of atmospheric gases and photochemical aerosols. Our updated radiative transfer model is validated against the in situ spectroscopic measurements of the Huygens probe during its descent and once landed. We confirm that aerosols with a fractal dimension of 2.3–2.4 provide the best fit to the observations. We apply our radiative transfer model to four VIMS data cubes over the Selk crater region including the Dragonfly landing and exploration areas, further validating our model by producing consistent aerosol population and surface albedo maps. These infrared albedo maps, further corrected from the photometry, enable us to study the Selk crater region in terms of surface composition, landscape formation, and evolution. Our results suggest that the Selk crater is in an intermediate state of degradation and that the mountainous terrains of the area (including the crater rim and ejecta) are likely to be dominated by fine grains of tholin-like sediment. This organic sediment would be transported to the lowlands (crater floor and surrounding plains), possibly with water ice particles, by rivers, and further deposited and processed to form the sand particles that feed the neighboring dune fields. These results provide information for the operational and scientific preparation of the Dragonfly mission, paving the way for future exploration of Titan’s surface composition and geology.

Photometric Modeling and VIS‐IR Albedo Maps of Tethys From Cassini‐VIMS

AbstractWe report about the derivation of visible (VIS) and infrared (IR) albedo maps and spectral indicators of Saturn's satellite Tethys from the complete Cassini‐Visual and Infrared Mapping Spectrometer (VIMS) data set. The application of a photometric correction is necessary to remove illumination and viewing effects from the I/F spectra, to compute spectral albedo and to correctly associate spectral variations to changes in composition or physical properties of the surface. In this work we are adopting the photometric correction proposed by Shkuratov et al. (2011, https://doi.org/10.1016/j.pss.2011.06.011) to derive albedo maps of Tethys from disk‐resolved Cassini‐VIMS data. After having applied a similar methodology to Dione's data (Filacchione et al., 2018, https://doi.org/10.1002/2017GL076869), we present here the results achieved for Tethys: surface albedo maps and photometric parameters are computed at five visible (0.35, 0.44, 0.55, 0.70, and 0.95 μm) and five infrared (1.046, 1.540, 1.822, 2.050, and 2.200 μm) wavelengths and rendered in cylindrical projection with a 0.5° × 0.5° angular resolution in latitude and longitude, corresponding to a highest spatial resolution of 4.7 km/bin. The 0.35‐ to 0.55‐ and 0.55‐ to 0.95‐μm spectral slopes and the water ice 2.050‐μm band depth maps are computed after having applied the photometric correction, in order to trace the leading‐trailing hemisphere dichotomy, to constrain the shape of the equatorial lens generated by the bombardment of high‐energy magnetospheric electrons on the leading hemisphere, and to observe the stronger water ice band depth and reddening within the floors of Odysseus and Penelope impact craters.

Photometric Modeling and VIS‐IR Albedo Maps of Dione From Cassini‐VIMS

AbstractWe report about visible and infrared albedo maps and spectral indicators of Dione's surface derived from the complete Visual and Infrared Mapping Spectrometer (VIMS) data set acquired between 2004 and 2017 during the Cassini tour in Saturn's system. Maps are derived by applying a photometric correction necessary to disentangle the intrinsic albedo of the surface from illumination and viewing geometry occurring at the time of the observation. The photometric correction is based on the Shkuratov et al. (2011, https://doi.org/10.1016/j.pss.2011.06.011) method which yields values of the surface equigonal albedo. Dione's surface albedo maps are rendered at five visible (VIS: 0.35, 0.44, 0.55, 0.7, and 0.95 μm) and five infrared (IR: 1.046, 1.540, 1.822, 2.050, and 2.200 μm) wavelengths in cylindrical projection with a 0.5° × 0.5° angular resolution in latitude and longitude, corresponding to a spatial resolution of 4.5 km/bin. Apart from visible and infrared albedo maps, we report about the distribution of the two visible spectral slopes (0.35–0.55 and 0.55–0.95 μm) and water ice 2.050 μm band depth computed after having applied the photometric correction. The derived spectral indicators are employed to trace Dione's composition variability on both global and local scales allowing to study the dichotomy between the bright‐leading and dark‐trailing hemispheres, the distribution of fresh material on the impact craters and surrounding ejecta, and the resurfacing of the bright material within the chasmata caused by tectonism.

Towards a comprehensive model of Earth's disk-integrated Stokes vector

AbstractA significant body of work on simulating the remote appearance of Earth-like exoplanets has been done over the last decade. The research is driven by the prospect of characterizing habitable planets beyond the Solar System in the near future. In this work, I present a method to produce the disk-integrated signature of planets that are described in their three-dimensional complexity, i.e. with both horizontal and vertical variations in the optical properties of their envelopes. The approach is based on Pre-conditioned Backward Monte Carlo integration of the vector Radiative Transport Equation and yields the full Stokes vector for outgoing reflected radiation. The method is demonstrated through selected examples inspired by published work at wavelengths from the visible to the near infrared and terrestrial prescriptions of both cloud and surface albedo maps. I explore the performance of the method in terms of computational time and accuracy. A clear strength of this approach is that its computational cost does not appear to be significantly affected by non-uniformities in the planet optical properties. Earth's simulated appearance is strongly dependent on wavelength; both brightness and polarization undergo diurnal variations arising from changes in the planet cover, but polarization yields a better insight into variations with phase angle. There is partial cancellation of the polarized signal from the northern and southern hemispheres so that the outgoing polarization vector lies preferentially either in the plane parallel or perpendicular to the planet scattering plane, also for non-uniform cloud and albedo properties and various levels of absorption within the atmosphere. The evaluation of circular polarization is challenging; a number of one-photon experiments of 109 or more is needed to resolve hemispherically integrated degrees of circular polarization of a few times 10−5. Last, I introduce brightness curves of Earth obtained with one of the Messenger cameras at three wavelengths (0.48, 0.56 and 0.63 μm) during a flyby in 2005. The light curves show distinct structure associated with the varying aspect of the Earth's visible disk (phases of 98–107°) as the planet undergoes a full 24 h rotation; the structure is reasonably well reproduced with model simulations.

Influence of spatial heterogeneity of local surface albedo on the area-averaged surface albedo retrieved from airborne irradiance measurements

Abstract. Spectral airborne upward and downward irradiance measurements are used to derive the area-averaged surface albedo. Real surfaces are not homogeneous in their reflectivity. Therefore, this work studies the effects of the heterogeneity of surface reflectivity on the area-averaged surface albedo to quantify how well aircraft measurements can resolve the small-scale variability of the local surface albedo. For that purpose spatially heterogeneous surface albedo maps were input into a 3-dimensional (3-D) Monte Carlo radiative transfer model to simulate 3-D irradiance fields. The calculated up- and downward irradiances in altitudes between 0.1 and 5 km are used to derive the area-averaged surface albedo using an iterative retrieval method that removes the effects due to atmospheric scattering and absorption within the layer beneath the considered level. For the case of adjacent land and sea surfaces, parametrizations are presented which quantify the horizontal distance from the coastline that is required to reduce surface heterogeneity effects on the area-averaged surface albedo to a given limit. The parametrization which is a function of altitude, aerosol optical depth, single scattering albedo, and the ratio of local land and sea albedo was applied for airborne spectral measurements. In addition, the deviation between area-averaged and local surface albedo is determined for more complex surface albedo maps. For moderate aerosol conditions (optical depth less than 0.4) and a wavelength range between 400 and 1000 nm, the altitude and the heterogeneity of the surface albedo are the dominant factors determining the mean deviation between local and area-averaged surface albedo. A parametrization of the mean deviation is applied to an albedo map that was derived from a Landsat image of an area in East Anglia (UK). Parametrization and direct comparison of local and area-averaged surface albedo show similar mean deviations (20% vs. 25%) over land.

Análise quantitativa de parâmetros biofísicos de bacia hidrográfica obtidos por sensoriamento remoto

O objetivo deste trabalho foi avaliar quantitativamente os parâmetros biofísicos obtidos por sensoriamento remoto, para a área de abrangência da Bacia Hidrográfica do Rio Tapacurá, em Pernambuco. Utilizaram-se imagens do TM‑Landsat 5 de 10/7/1989, 6/7/2005 e 29/8/2007. As imagens foram registradas pela correção geométrica polinomial de primeira ordem. Foram realizadas as etapas de calibração radiométrica, reflectância, albedo planetário e transmissividade e, subsequentemente, geraram-se cartas temáticas de albedo e de temperatura da superfície, e do índice de vegetação melhorado ("enhanced vegetation index", EVI). O albedo da superfície apresentou valores médios crescentes entre as imagens obtidas em 1989 e 2005, o que indica expansão territorial urbana. A imagem de 29/8/2007 mostrou maior temperatura da superfície, seguida das temperaturas mostradas nas imagens de 10/7/1989 e 6/7/2005, e os maiores valores foram os das malhas urbanas. A imagem de 1989 mostrou o maior valor médio de EVI, o que indica ter havido, naquela data, maior presença de vegetação.

A method to account for surface albedo heterogeneity in single‐column radiative transfer calculations under overcast conditions

A simple parameterization to derive the broadband effective albedo over highly reflecting surfaces under overcast conditions is presented. High spatial variability in the surface albedo affects the downwelling solar irradiance in neighboring regions via the multiple reflections of light between the surface and the cloud base. The effective albedo is defined as the albedo of a homogeneous surface that would result in the same downwelling irradiance as observed at the observation point in the presence of a heterogeneous surface. The proposed method parameterizes the effective albedo using the cloud base height and a surface albedo map as inputs. The parameterization is based on the spatial distribution of surface reflections contributing to the downwelling irradiance at the observation site, which is approximated with a gamma distribution. The parameterization was validated against reference values of effective albedo derived from three‐dimensional backward Monte Carlo and one‐dimensional DISORT radiative transfer calculations for four idealized surface albedo maps and various specifications of cloud properties. It gave values of effective albedo very close to the reference calculations, performing substantially better than any other approach tested, also when applied to the retrieval of cloud optical depth. The method can be implemented into one‐dimensional radiative transfer models or used to interpret broadband irradiance measurements in Polar coastal regions, in the marginal sea ice zones, or in patchy terrain with forests and snow‐covered fields.

Monitoring ice-capped active Volcán Villarrica, southern Chile, using terrestrial photography combined with automatic weather stations and global positioning systems

AbstractVolcán Villarrica (39°25′12″ S, 71°56′27″ W; 2847 m a.s.l.) is an active ice-capped volcano located in the Chilean lake district. The surface energy balance and glacier frontal variations have been monitored for several years, using automatic weather stations and satellite imagery. In recent field campaigns, surface topography was measured using Javad GPS receivers. Daily changes in snow-, ice-and tephra-covered areas were recorded using an automatic digital camera installed on a rock outcrop. In spite of frequently damaging weather conditions, two series of consecutive images were obtained, in 2006 and 2007. These photographs were georeferenced to a resampled 90 m pixel size SRTM digital elevation model and the reflectance values normalized according to several geometric and atmospheric parameters. The resulting daily maps of surface albedo are used as input to a distributed glacier-melt model during a 12 day mid-summer period. The spatial pattern of cumulative melt is complex and controlled by the distribution of airfall and wind-blown tephra, with extremely high melt rates occurring downwind of the crater and exposed ash banks. The camera images are also used to visualize the pattern of glacier crevassing. The results demonstrate the value of terrestrial photography in understanding the energy and mass balance of the glacier, including the generation of meltwater, and the potential value of the technique for monitoring volcanic activity and potential hazards associated with ice–volcano interactions during eruptive activity.

Photometric analysis of 1 Ceres and surface mapping from HST observations

The highest resolution (pixel scale 30 km) images of Ceres to date have been acquired by the Advanced Camera for Surveys onboard Hubble Space Telescope, through three wide band filters, centered at 535, 335, and 223 nm, covering more than one rotation of Ceres. The lightcurve at 535 nm agrees with earlier observations at V-band [Tedesco, E.F., Taylor, R.C., Drummond, J., Harwood, D., Nickoloff, I., Scaltriti, F., Schober, H. J., Zappala, V., 1983. Icarus 54, 23–29] in terms of magnitude, amplitude, and shape. The 0.04 magnitude lightcurve amplitude cannot be matched by Ceres' rotationally symmetric shape, and is modeled here by albedo patterns. The geometric albedos at the above three wavelengths are measured to be 0.087 ± 0.003 , 0.056 ± 0.002 , and 0.039 ± 0.003 , respectively. V-band geometric albedo is calculated to be 0.090 ± 0.003 , consistent with earlier observations [Tedesco, E.F., 1989. In: Binzel, R.P., Gehrels, T., Matthews, M.S. (Eds.), Asteroids II. Univ. of Arizona Press, Tucson, pp. 1090–1138]. A strong absorption band (30%) centered at about 280 nm is observed, but cannot be identified with either laboratory UV spectra or the spectra of Europa or Ganymede. The single-scattering albedo has been modeled to be 0.070 ± 0.002 , 0.046 ± 0.002 , and 0.032 ± 0.003 , respectively. The photometric roughness of Ceres' surface is found to be about 44 ° ± 5 ° from photometric modeling using Hapke's theory, consistent with earlier radar observations [Mitchell, D.L., Ostro, S.J., Hudson, R.S., Rosema, K.D., Campbell, D.B., Velez, R., Chandler, J. F., Shapiro, I.I., Giorgini, J.D., Yeomans, D.K., 1996. Icarus 124, 113–133]. The first spatially resolved surface albedo maps of Ceres at three wavelengths have been constructed from HST observations, as well as the corresponding color maps. Eleven surface albedo features are identified, ranging in scale from 40–350 km. Overall the range of these albedo and color variations is small compared to other asteroids and some icy satellites.

Spatially complete global spectral surface albedos: value-added datasets derived from Terra MODIS land products

Recent production of land surface anisotropy, diffuse bihemispherical (white-sky) albedo, and direct-beam directional hemispherical (black-sky) albedo from observations acquired by the Moderate Resolution Imaging Spectroradiometer (MODIS) instruments aboard the National Aeronautics and Space Administration's Terra and Aqua satellite platforms have provided researchers with unprecedented spatial, spectral, and temporal information on the land surface's radiative characteristics. Cloud cover, which curtails retrievals, and the presence of ephemeral and seasonal snow limit the snow-free data to approximately half the global land surfaces on an annual equal-angle basis. This precludes the MOD43B3 albedo products from being used in some remote sensing and ground-based applications, climate models, and global change research projects. An ecosystem-dependent temporal interpolation technique is described that has been developed to fill missing or seasonally snow-covered data in the official MOD43B3 albedo product. The method imposes pixel-level and local regional ecosystem-dependent phenological behavior onto retrieved pixel temporal data in such a way as to maintain pixel-level spatial and spectral detail and integrity. The phenological curves are derived from statistics based on the MODIS MOD12Q1 IGBP land cover classification product geolocated with the MOD43B3 data. The resulting snow-free value-added products provide the scientific community with spatially and temporally complete global white- and black-sky surface albedo maps and statistics. These products are stored on 1-min and coarser resolution equal-angle grids and are computed for the first seven MODIS wavelengths, ranging from 0.47-2.1 /spl mu/m and for three broadband wavelengths 0.3-0.7, 0.3-5.0, and 0.7-5.0 /spl mu/m.

Simulation of surface meteorology at the Pathfinder and VL1 sites using a Mars mesoscale model

The Fifth‐Generation Penn State/NCAR Mesoscale Model (MM5) has been adapted to study circulations in the Martian atmosphere. The NASA Ames Mars general circulation model (GCM) provides initial and boundary conditions. The meteorology of this Mars MM5 (the OSU MMM5) is compared with Pathfinder and Viking Lander 1 (VL1) data for late northern summer. The MMM5 uses an equator‐crossing semiglobal polar stereographic mother domain, significantly reducing the boundary reflections inherent in Martian mesoscale simulations. Using two‐way nests, simultaneous simulations of two regions are performed: (1) Chryse Planitia (Mars Pathfinder (MPF)/VL1) and (2) the central chasmas of Valles Marineris. Simulations are hydrostatic and dry. The MMM5 uses the same atmospheric radiation package as the GCM but a much enhanced near‐surface vertical resolution and a different Planetary Boundary Layer (PBL) scheme. Model topography, thermal inertia, and surface albedo maps have all been developed using the most recent data from the Mars Global Surveyor (MGS) Mars Orbiter Laser Altimeter (MOLA) and Thermal Emission Spectrometer (TES) experiments. The diurnal cycles of surface air temperature, surface pressure, and surface wind all show improvement in comparison with the GCM. For certain regions the local surface pressure tidal amplitudes are strongly dependent on the resolution of the model and/or the topography; VL1 and MPF are in such a region. The diurnal cycles of wind are complex, and for many locations the near‐surface winds are dominated by slope flows of multiple scales. Comparison with data indicates that resolving these slope flows is very important for simulating the diurnal wind cycle. At specific locations these slope flows dramatically influence the diurnal surface pressure cycle.

Images of Pluto Generated by Matrix Lightcurve Inversion

We use a new indirect imaging method, matrix lightcurve inversion (MLI), to generate surface albedo maps of Pluto from the rotational photometric lightcurves. The mathematical basis of MLI is established and a constrained iterative regularization method used to invert the underdetermined system of equations. If the secular dimming of Pluto observed in all phases of the rotational lightcurves from 1954 to 1982, followed by an overall brightening of the planet in later years, could be ascribed to a static surface albedo distribution, then an albedo map can be generated using all the available lightcurves. However, Stern et al. (1988, Icarus 75, 485-498) have suggested that an extensive covering of methane frost sublimates as Pluto approaches perihelion, exposing dark layers of photolyzed methane in a spatially nonuniform pattern, and then freezes out again as Pluto passes perihelion. Assuming that Stern et al. (1988, Icarus 75, 485-498) are correct in their interpretation that Pluto undergoes atmospheric sublimation and freezeout cycles, we present an image that represents a snapshot of Pluto as it might have appeared during the interval 1980 to 1986 by using lightcurves measured in these years. Another image using the lightcurves between 1954 and 1986 shows very extensive and bright polar caps though this image appears to be highly unrealistic. We propose that the inability to fit these lightcurves (to even remotely within the noise level in the data) furnishes evidence to the hypothesis of large-scale and dynamic atmospheric volatile transport over Pluto's surface between the mid 1950s and the mid 1980s.

衛星リモートセンシングを用いたHEIFE領域における諸過程のスケーリング・アップ

The use of satellite remote sensing is indispensable to improving the understanding of the processes of land surface-atmosphere interactions on a scale of 10 000 kmr in HEIFE. Because of their high spatial resolution, Landsat TM observations can be compared directly with surface measurements and be used for this purpose. The relevant data set and a state-of-the-art data processing methodology are discussed. A summer scene of TM was used to produce a set of maps of surface albedo, vegetation index, surface temperature, net radiation and other basic energy balance components of the whole area. Statistical analysis based on these maps revealed some quantitatively significant land surface parameters. Future developments are discussed.

A Method for the Estimate of Broadband Directional Surface Albedo from a Geostationary Satellite

Surface albedo can be inferred from geostationary satellite measurements as long as the effects due to the atmosphere, the spectral response of the sensor, and the angular anisotropy of the reflected field are corrected. In this paper, we developed a method which includes ad hoc correction procedures for the three categories of effects. An application of the method is conducted over the Sahara and the African Sahel using METEOSAT radiances together with auxiliary data derived from other satellites (Tiros-N and Nimbus-7) and standard meteorological observations. The surface albedo maps are estimated over this region, at a spatial resolution compatible with one used in climate models, for 2 days representative of the dry and the wet seasons, respectively. The observed seasonal surface albedo change and the relationships between the surface and the planetary albedos are discussed in order to examine the validity of the method and the correction procedures.

Atmospheric effects in satellite imaging of mountainous terrain

It is possible to obtain useful maps of surface albedo from remotely sensed images by eliminating effects due to topography and the atmosphere, even when the atmospheric state is not known. A simple phenomenological model of earth radiance that depends on six empirically determined parameters is developed given certain simplifying assumptions. The model incorporates path radiance and illumination from sun and sky and their dependencies on surface altitude and orientation. It takes explicit account of surface shape, represented by a digital terrain model, and is therefore especially suited for use in mountainous terrain. A number of ways of determining the model parameters are discussed, including the use of shadows to obtain path radiance and to estimate local albedo and sky irradiance. The emphasis is on extracting as much information from the image as possible, given a digital terrain model of the imaged area and a minimum of site-specific atmospheric data. The albedo image, introduced as a representation of surface reflectance, provides a useful tool to evaluate the simple imaging model. Criteria for the subjective evaluation of albedo images are established and illustrated for Landsat multispectral data of a mountainous region of Switzerland. The method exposes some of the limitations found in computing reflectance information using only the image-forming equation.

The seasonal and global behavior of water vapor in the Mars atmosphere: Complete global results of the Viking Atmospheric Water Detector Experiment

The water vapor content of the Mars atmosphere was measured from the Viking Orbiter Mars Atmospheric Water Detectors (MAWD) for a period of more than 1 Martian year, from June 1976 through April 1979. Results are presented in the form of global maps of column abundance for 24 periods throughout each Mars year. The data reduction incorporates spatial and seasonal variations in surface pressure and supplements earlier published versions of less complete data. Column abundances vary between 0 and about 100 precipitable microns (pr μm), depending on location and season, while the entire global abundance varies seasonally between an equivalent of about 1 and 2 km3 of ice. The first appearance of vapor at nonpolar latitudes as northern summer approaches and the drop in abundance at mid‐latitudes as summer ends both strongly imply the existence of a seasonal reservoir for water within the regolith. There appear to be no net annual sources away from the poles that contribute significant amounts of water. However, the strong annual gradient of vapor from north to south implies a net annual flow of vapor toward the south. This southward flow may be balanced by a northward flow during the global dust storms, by transport in the form of clouds or adsorbed onto dust grains, or during other years. The perenially cold nature of the south polar residual cap, along with the relatively large summertime vapor abundances over the cap, implies a net annual condensation of vapor onto the cap. Comparison with earlier (Earth‐based) observations of vapor in the south during the local summer indicates that all of the seasonal CO2 cap may sublime away in some years to reveal the water ice cap which must lie underneath, with corresponding large southern summer vapor abundances. The global distribution of the annual average abundance of vapor correlates well with Martian topography, as might be expected for a uniform constant atmospheric mixing ratio. If this topographic effect is divided out, the resulting residual map correlates well with maps of surface albedo and thermal inertia; this correlation may be related to the control exerted by the surface and subsurface temperatures on the adsorption/desorption process and on the atmospheric temperature profile and, hence, the vapor holding capacity of the atmosphere.

A Global Surface Albedo Model

Abstract We have shown in the past that the radiative-convective energy balance atmospheric model used at the General Motors Research Laboratories in our atmospheric physics research is extremely sensitive to the input value of surface albedo used. In response to this we developed a new global model of the surface albedo based on a detailed analysis of land use. The model has been used to determine the average surface albedo of the earth's surface in latitudinal bands for the four seasons of the year (January-March, April-June, July-September and October-December). In addition, global maps of surface albedo for elements 10° latitude by 10° longitude are provided. Our model involves: assigning one of 49 different types of surface to each of 77 040 areas for each of the four seasons. Each type of surface is assigned a different albedo for each season. The average surface albedo for a given latitudinal band is obtained by averaging the albedos for the areas in the desired latitudinal band. The average surfac...