3.1.1. Data Selection
 The Aerosol LIdar Validation Experiment (ALIVE) was a field campaign performed in North-central Oklahoma in September of 2005. This is the location of the Southern Great Plains Central Facility (SGP CF) of the Atmospheric Radiation Measurement (ARM) Program (Department of Energy). While the primary goals of the ALIVE campaign were not to investigate RSP surface characterization, the proximity to the suite of ground based instruments at the SGP site, simultaneous measurements of the aerosol profile from AATS-14, and a series of low altitude flights, provide an ideal data-set with which to investigate the surface characterization.
 Several short, low altitude flight segments were made in the vicinity of the SGP site, typically preceded or followed by a spiral maneuver used by the AATS-14 to determine an aerosol optical thickness altitude profile. Surface characteristics were typical for the rural mid-west of the United States in the fall. The ground was relatively flat, and covered by a patchwork of late-season crops, bare soil exposed by recent harvesting, and mixtures of trees and shrubs [Luo et al., 2003]. There are few buildings and the occasional paved road.
 Data from two flights were used. Table 1 presents these flight times, along with geometry, aerosol and weather conditions. Aerosol optical thickness at 500nm from an AERONET Project [Holben et al., 1998] sun photometer at the SGP site is also included to provide an understanding of aerosol properties from that day.
Table 1. Low Altitude ALIVE Flight Segments Used for Surface Characterization
|Start time, UTC||16:32:25||22:09:32|
|Number of RSP scans||270||41|
|J-31 Altitude above sea level||510 m||475 m|
|Relative sensor-solar azimuth||−45°||156°|
|Solar zenith angle||43°||62°|
|AERONET τa(λ = 500 nm)||0.07||0.05|
|AATS-14 τa(λ = 499 nm)||0.06||0.05|
 In Table 1, the tags in the first row (JRFx) identify the research flight. Start time for a data file within that flight is listed in UTC. Local time was five hours earlier. Scans were selected from a particular data file so that they included only low altitude, constant heading segments. Altitude is listed as meters above sea level. Ground height at the SGP site is about 315 m, so flights had an above ground height between 160 and 195 m. The (relative) azimuth is the instrument heading minus solar azimuth, in degrees. Aerosol optical thickness at 500 nm (τa) was measured by the AERONET Project with a ground sun photometer at the SGP site. Values from the time of flight are provided for comparison. AATS-14 optical thickness was measured on the J-31. Thus differences between AATS-14 and AERONET represent the optical thickness between the aircraft and the ground, a value which is well within the AERONET and AATS-14 uncertainty. The last row contains a visual description of the cloud scenario from the instrument operator.
 There were a total of twelve research flights as part of ALIVE, but only flights JRF3 and JRF4, listed above, were suitable for BRDF estimation. JRF3 and JRF4 were the only flights with low altitude segments when the sky was completely devoid of clouds. As we shall see later (see section 3.1.3), some effort was put into determining the diffuse downwelling irradiance while estimating the BRDF. This determination is only accurate, with our models, for clear skies. Including the effects of clouds under partially cloudy skies, even if implemented with a three dimensional radiative transfer code, would introduce large uncertainties, as there is limited informational to specify the vertical and horizontal cloud distribution.
3.1.2. Classification and Mixed Pixel Removal
 Before the Ross-Li BRDF models were fit to the data, we separated it into similar classes, and performed our fitting on each class individually. This was done to limit the dependence on data coverage differences from flight to flight, and also to facilitate comparisons with other lower spatial resolution data sets. Prior to image classification, gaseous absorption effects were removed as described above, and a metric describing the amount of vegetation (a vegetation index, see Appendix B) was calculated for each data point. Data were then split into “soil” and “vegetation” classes and extreme vegetation index values were removed. Boundary or mixed surfaces were also removed using an edge detection convolution kernel on the spatial image of vegetation index. In addition, analyses were performed on an “all” class which contains the entire data-set except for data that were removed as part of the quality control process. For each of these classes the data were then fit with the Ross-Li BRDF model.
 Simple thresholding of the Aerosol Resistant Vegetation Index (ARVI, see Appendix B for details on how it is computed) was used to split the data into a “soil” and “vegetation” class. ARVI values between −0.25 and 0.075 were classified as “soil”, while ARVI values between 0.375 and 0.775 were classified as “vegetation”. Such narrow ARVI regions centered on the modes of each surface type were chosen to avoid mixed pixel data and focus on the properties of generic “soil” and “vegetation”. Figure 1 is a histogram of the ARVI for each flight. Peaks for both classes are pronounced, and vertical dashed lines show the regions used for each class. The histogram computed using different small segments of the view angle range (not pictured here) is similar to the one shown here indicating an absence of significant BRDF effects on this index. Each flight flew over slightly different areas in the region of the SGP CF, so there are some differences between the histograms of each flight. In particular, JRF03 has a third peak at about an ARVI of 0.25, possibly indicating post-harvest, sparsely vegetated, fields that were not observed during JRF04. Since that surface type is not present in JRF04 data, it was omitted in the study. Furthermore, JRF04 contains less data than JRF03, as the length of flight time dedicated to the low altitude segment was less in JRF04. Few measurements passed the “soil” class criteria. After additional screening, described below, only the “vegetation” class remained from JRF04. Finally, a third, “all” class was created for ARVI values between −0.25 and 0.775. For consistency, additional screening procedures described below were applied to this class as well.
Figure 1. Histograms of the Atmospherically Resistant Vegetation Index (ARVI) for each flight. Vertical dotted lines represent the boundaries of the “soil” and “vegetation” class thresholds.
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 To further restrict our data to generic land types, the ARVI spatial image is used to identify data that is in the middle of a patch of soil or vegetation. An edge detection algorithm was applied that uses a discrete convolution with a 3 × 3 spatial kernel. This kernel is applied as a multiplier to each pixel and its neighbors in the image, and the summed result of each multiplication forms the value of that pixel in the resulting image. This is a discrete version of the gradient and the technique (when this mask is added to the original image to enhance boundaries) is also called unsharp masking [Gonzalez and Woods, 1992]. The gradient image is used to remove boundary and mixed pixels. A threshold, η = 0.1, was chosen such that pixels satisfying ∣∇(ARVI)∣ > η are excluded. η was selected arbitrarily, but it is of the same magnitude as the ARVI range for the soil class, thus adjacent pixels containing as much variability as the narrowest class (or more) are removed. Figure 2 illustrates the classification and imagery from flight JRF3. Classification results (in part 2d) compare favorably with intuition from the imagery (part 2a), and take boundary and edge pixels (part 2c) into account. Note also the small quantity of data available for analysis. For safety reasons, the aircraft was flown at low altitude for only short segments near the SGP site.
Figure 2. Data from flight JRF03 was used to show (from left to right), (a) Ground reflectance, where the 670 nm band is displayed in the red channel, the 865 nm band in the green channel, and the 470 nm band in the blue channel, (b) Atmospherically Resistant Vegetation Index (ARVI), with the color bar at the right, (c) gradient of ARVI, used to remove boundary and edge pixels with its color bar at the right, and (d) classification results, where red indicates “soil” type, green indicates “vegetation”, and black are unclassified areas.
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 Several geometric screening criteria were also applied. Data with view zenith angles greater than 65° were removed. This was done to avoid view angles at the extremes of measurement capability. Since the RSP is scanning in the direction of aircraft motion, data from turns were excluded because those scans may not represent a single ground location. Thus, data from aircraft headings 3° greater or less than an average heading were removed. The effects of both of these geometric screening criteria are evident in Figure 2d, where data at the right of the image have been removed because their zenith angle was too large, and data at the top removed because this is were the aircraft began banking into its spiral ascent for the next segment of the flight.
 Image data were rearranged so that each scan represents a set of view angles about a single ground location, rather than the actual order of measurements (which represent a set of view angles about an airborne location). While this has no effect on the actual data, classification results for each scan were checked for consistency. This final screening criteria (which is not displayed in Figure 2) required that 50% of the data in a scan must have passed all previous screening criteria and were grouped into a single class. The result is a set of data that is of a consistent surface type over most of the view angle range and that has had any outlier measurements, which may represent noise, surface boundary or mixed pixel effects removed.
3.1.3. Determination of Ground Reflectance and the Diffuse Effect
 Measurement of ground reflectance from an aircraft requires adequate compensation for atmospheric effects. During ALIVE, a high quality characterization of the atmospheric scattering was provided by the combination of polarized RSP measurements (above the aerosol layer) with the vertical profile of aerosol optical thicknesses from the AATS-14. This cannot be used to determine the ground reflectance directly because of the multiple scattering that occurs between the atmosphere and the surface. The atmospheric correction is therefore performed using an iterative process, where initial estimates of surface reflectance are adjusted until the surface-atmosphere scattering model reproduces the reflectance measured by the RSP.
 The atmospheric-surface model uses the doubling and adding method [Lacis and Hansen, 1974; Hansen and Travis, 1974], and produces a reflectance to compare to RSP data, given aerosol and other atmospheric properties together with solar and instrument geometry and kernel values for the Ross-Li BRDF reflectance model. The observed reflectance can be separated into an atmospheric and a surface component:
where ρo is the reflectance at the altitude of the observations, ρa is the reflectance due to atmospheric scattering of radiance into the instrument field of view without interacting with the surface (path radiance) and S includes all surface interaction terms. In what follows we are primarily interested in S and the correction for diffuse and multiple interaction terms, since we have an accurate and comprehensive characterization of the atmosphere from high altitude RSP measurements that allows us to calculate ρa. We will differentiate between measurements and model calculations by using a caret for those quantities that are direct observations. The surface interaction term, S, can be calculated using the expression
where ρg is the surface reflectance, t is the direct solar transmittance, and T is the diffuse transmittance, with the arrows indicating whether they apply to transmission from the sun to the ground () or from the ground to the observational altitude (). The star symbol, *, indicates that integrations over zenith and azimuth are performed for diffuse interactions. As is usually the case for scattering problems with no preferred azimuthal plane, that integration is actually implemented using a Fourier decomposition and re-summation [Hovenier, 1971; Hansen and Travis, 1974; de Haan et al., 1987]. The function Σ is used in the calculation of multiple surface atmosphere interaction terms and is given by the formula
where ρa is the reflectance of the atmosphere illuminated from below. The implementation of this summation is described in [Hovenier, 1971; Hansen and Travis, 1974; de Haan et al., 1987].
 In equation (8), ρo is measured by the RSP and calculated with the doubling-adding model, while ρa, t, T and Σ are determined from the model based on the atmospheric state that is prescribed by AATS-14, AERONET and high altitude RSP data. The model includes the effects of both Rayleigh (molecular) and aerosol scattering. In order to find an estimate of ρg that has the effects of diffuse transmission and multiple surface-atmosphere scattering removed, we use the following iteration
where p is the iteration index, ρg,k is the kernel fit to the latest estimate of surface reflectance and is the observation corrected for path radiance. We have implicitly defined the function γp, which is the ratio of measurement to model S, to adjust the surface reflectance until the model calculated reflectance matches the observations. This iteration is similar to that introduced by Chahine  for atmospheric sounding. The kernel fit to the reflectance uses a least mean square estimate of the kernel coefficients, so the vector of kernel coefficients, f, is given by the expression
and the kernel estimate for the surface reflectance at the observed viewing geometry is
with K being the 3 × N reflectance kernel matrix formed from the isotropic, volumetric and geometric kernels (cf. equation (5)) and N is the number of view angles for the given viewing geometry. K and f depend on the same set of wavelength (λ) and other geometric parameters (θs, ϕ), so those subscripts are omitted from the above equations. The iteration is initialized with the value
 In an atmosphere with no scattering this initial value gives the atmospherically corrected surface reflectance, and no further iterations are therefore necessary. Otherwise, equations (11) through (13) are iterated until γ is close to unity. If there is scattering in the atmosphere, we can determine that γ < 1 from equation (9) for the first step in the iteration. This is a necessary condition for the convergence of this iteration [Twomey, 1977]. Convergence also requires that the matrix associated with the estimate of the kernel parameters is diagonally dominant (Dubovik and King , Appendix C). Since we are interested in the convergence of the estimation of the weights associated with the kernels we define matrix M
where we use the convention that there is a summation over repeated subscripts. The average degree of diagonal dominance of that matrix, dd, is
 As expected (see Figure 3), dd is largest at shortest wavelengths, where the effects of scattering are largest, and smallest at the longer wavelengths, where the effects of scattering are negligible.
 The final iteration products are the kernel values of the Ross-Li surface reflectance model. The iteration was repeated between 5–9 times for each band until the change in kernel values for each iteration was smaller than 10−5.
3.1.4. Spectral to Broadband Albedo Computation
 DHR and BHR, as calculated in the previous section, represent surface properties in a set of narrow instrument bands. Data from these narrow bands must be spectrally interpolated if they are to be compared to broadband ground radiometer data such as that from BEFLUX. In MODIS products, this is done according to the methodology of Liang  and validated in Liang et al. . Liang  used libraries of surface reflectance spectra and model simulations to create a set of coefficients that are applied to scene albedo values to approximate a broadband DHR or BHR. These coefficients are applied uniformly across the entire MODIS dataset. Our RSP-ALIVE dataset comprises a single day with a well known atmospheric scenario. In this sense, we are fortunate in that we can utilize knowledge about the atmosphere in our broadband albedo computation, and we do so as follows.
 Conversion of DHR(Λ, θs) to the broadband version DHRbb(θs) involves the spectral integration of the DHR weighted by the downwelling solar irradiance. Irradiances are computed using a hyperspectral version of the doubling and adding model applied in section 3.1.3 [Cairns et al., 2003]. Spectral DHR's are created using linear interpolations of Ross-Li kernel weights, f, determined in section 3.1.3. This is then normalized by the total downwelling solar irradiance appropriate for a particular day at all wavelengths
where Eo(λ) is the exo-atmospheric irradiance and t(λ) is the direct solar transmittance. Spectrally dependent DHR(λ) is created by linear interpolation of kernel weights and application of the DHR parameterization described in Lucht et al. [2000b].
g parameters from Lucht et al. [2000b] are in Table 2. BHRbb is computed in a similar fashion, where spectrally interpolated kernel weights are applied to Lucht et al. [2000b]'s BHR parameterization. This parameterization is then integrated in a weighted manner.
Table 2. Lucht et al. [2000b]DHR Parameters
 For consistency, broadband BHRbb and DHRbb are computed using the same method for both RSP and MODIS, and are thus integrated over the same spectral range (400 nm to 2500 nm). It should be noted that this is not the same spectral range as the standard MODIS broadband albedo products, but it matches the range of BEFLUX radiometers. We also analyzed the radiative effect of albedo beyond this spectral range using the hyperspectral doubling and adding model from Cairns et al.  and estimates of surface albedo by extrapolating the RSP “vegetation” and “soil” data from within the measured spectral range. We found that the relative error in estimation of DHR (of the entire radiative system) using our restricted spectral range was 3.0% for “vegetation” and 0.7% for the darker “soil” data. The absolute bias for a surface with an albedo of 0.2 is 0.0058 and 0.0014 for “vegetation” and “soil”, respectively. This estimation accounted for Rayleigh (molecular) scattering alone. Aerosols and absorbing gases would have the effect of further reducing the out of band radiance, and thus decreasing the above errors.