The following section describes the sources and characteristics of input data for the R-based method, an uncertainty analysis of the R-based method, and methodology for application to AIRS data.
3.1. Atmospheric Profiles
 In terms of accuracy, a major limiting factor for the R-based method is the requirement of an accurate independent set of atmospheric profiles (air temperature, relative humidity and geopotential height) that are required for the forward model. Radiosonde profiles may have the highest accuracy, but have limitations in terms of availability since they are usually only available for dedicated field campaigns. An alternative is to use numerical weather data such as those from the National Center for Environmental Prediction (NCEP) Global Data Assimilation System (GDAS) model, available at 1° × 1° resolution, with atmospheric fields produced 4 times daily. The disadvantage is that variables have to be interpolated between the 4 observation times (00, 06, 12, 18 UTC) to the sensor observation time, which could lead to errors in temperature and water vapor depending on time of day.Coll et al.  used both radiosonde and NCEP profiles to successfully validate the MODIS LST product over vegetated surfaces. In this study we used NCEP atmospheric profiles, since radiosonde profiles were not available at the 3 field sites used for validation.
 Wan and Li  proposed a quality check to assess the suitability of the atmospheric profiles by looking at differences between observed and calculated BTs in two nearby window regions with different absorption features. For example, the quality check for MODIS bands 31 and 32 at 11 and 12 μm is:
where: T11obs and T12obs are the observed brightness temperatures at 11 and 12 μm respectively, and T11calc and T12calcare the calculated brightness temperatures from the R-based simulation at 11 and 12μm respectively. If δ(T11 − T12) is close to zero, then the assumption is that the atmospheric temperature and water vapor profiles are accurately representing the true atmospheric conditions at the time of the observation, granted the emissivity is already well known. Because water vapor absorption is higher in the 12 μm region, negative residual values of ā(T11 − T12) imply the R-based profiles are overestimating the atmospheric effect, while positives values imply an underestimation of atmospheric effects. A simply threshold can be applied to filter out any unsuitable candidate profiles for validation. AlthoughWan and Li proposed a threshold of ±0.3 K for MODIS data, we performed a sensitivity analysis and found that a threshold of ±0.5 K resulted in a good balance between the numbers of profiles accepted and accuracy of the final R-based LST. Using the BT difference quality check, observations over long time periods are required to obtain reliable statistics since on average almost 1/3 of all observations do not pass the BT difference check.
3.2. Emissivity Spectra and Variability
 An essential requirement for the R-based method is accurate knowledge of surface emissivity in the 11–12μm window region. Emissivities for most terrestrial surfaces are relatively stable in this wavelength range, and only vary from ∼0.94–0.99 for most types of surfaces [Seemann et al., 2008; Snyder et al., 1998]. Large homogeneous sand dune sites have been demonstrated to be ideal targets for the validation of emissivity data due to their homogeneous mineralogy and consistent physical properties over long time periods [Hulley et al., 2009a; Hulley et al., 2009b]. One key advantage that sand dunes offer compared with more traditional ground sites such as playas, salt flats and claypans, is that the sand surface typically dries rapidly after a precipitation event, whereas playa surfaces remain damp for much longer periods and water beneath the surface can be wicked back to the surface through the clay particles that make up many playas resulting in emissivity variability from surface soil moisture [Mira et al., 2007]. Two large homogeneous sand seas in the Namib Desert, Namibia and the Gran Desierto, Mexico were chosen as two desert validation sites, and a dense stand of redwood trees in Redwood National Park, California to represent a vegetated site. The geographic locations of the sites are shown in Figure 1.
Figure 1. Site locations and images of the three Radiance-based validation sites–the Namib Desert, Namibia (24.5°S, 15.4°E); Gran Desierto, Mexico (32°N, 114.4°W); and Redwood forest, California (41.4°N, 123.7°W). The Namib image was produced by overlaying an ASTER visible image with a high resolution DEM (courtesy of ASTER science team).
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 The Namib Desert in Namibia is located in a hyper-arid region with very low annual rainfall and shifting dunes that are almost completely devoid of vegetation except for sparse perennial grasses [White, 1983]. The dune sand is primarily composed of quartz, with the degree of redness increasing from the coast to inland areas, and determined by a coating of hematite (iron-oxide mineral) adhering to the quartz grains [White et al., 2007].
 The Gran Desierto dune system constitutes the largest portion of the Sonoran Desert in Mexico and the largest and most active sand sea in North America. Scheidt et al.  showed that the central dune area consists of a mixture of approximately 90% quartz and 10% feldspar (plagioclase and potassium feldspar). The grain size, composition, texture, color and sorting have been well documented in previous studies [Blount and Lancaster, 1990; Lancaster, 1992]. Spatial variability in emissivity primarily occurs due to the distribution of quartz and feldspars across the central dune system via aeolian deposits [Scheidt et al., 2011].
 The Redwood National park located along the coast of northern California, USA covers roughly 133,000 acres in the Del Norte and Humboldt counties. The dominant tree species is the Coast Redwood, an evergreen tree with linear, sharp needle tips which sheds older needles in autumn. Redwoods are typically found in mixed forests with other evergreens and deciduous trees, but pure stands are often found.
 Sand samples were collected at the Namib Desert and Gran Desierto during separate field campaigns and measured in a controlled laboratory environment to determine the spectral emissivity. During November 2008 ten sand samples were collected over dune and inter-dune areas within the Namib-Naukluft Park in Namibia at Sossussvlei, a dry salt/clay pan extending into the southern part of the dunes. The directional hemispherical reflectance of the samples was measured in the lab at Jet Propulsion Laboratory (JPL) using a Nicolet 520 Fourier transform infrared (FT-IR) spectrometer at 4 cm−1 spectral resolution from 2.5 to 15 μm, and converted to emissivity using Kirchhoff's law. The uncertainty associated with the FT-IR lab emissivities is 0.002 (0.2%) [Korb et al., 1999]. During Dec 2004, 76 sand samples were collected at different locations in the Gran Desierto desert in Mexico [Scheidt et al., 2011]. These samples were measured with a Nicolet Nexus 670 instrument at 2 cm−1 spectral resolution from 5 to 25 μm. The thermal emission was then converted to emissivity spectra using the method by Ruff et al. . For the Redwood forest site (41.4°N, 123.7°W), a conifer spectrum from the ASTER spectral library was used to represent the effective emissivity of the conifer canopy for the AIRS pixel at 45 km resolution [Baldridge et al., 2009]. The conifer spectrum measurement was made at Johns Hopkins University (JHU) and reduced in reflectance by a factor of two to account for canopy scattering effects. Norman and Becker found that the emissivity of canopies normally varies from 0.97 to 0.99 for a leaf emissivity of 0.95. Therefore, the conifer spectrum from JHU which has a minimum emissivity of 0.989 and min-max difference of 0.003 over the measured spectrum gives a good approximation for the effective tree-canopy emissivity.
 Analyzing the spatial homogeneity of the validation sites is important for the R-based method since the average emissivity spectra derived from the field sample locations must be representative of the effective emissivity measured at the remote sensing scale, i.e., 45 km for AIRS. Sampling locations for each field campaign at these two sites are shown inFigure 2, along with the corresponding mean emissivity spectra and spatial variation shown as solid error bars in the 10–13 μm range. Spatial variations in 11 μm emissivity were 0.0057 (0.57%) for the Gran Desierto sample sites, and 0.0044 (0.44%) for the Namib sample sites. These variations can be considered small and are equivalent to ∼0.3 K for a material at 300 K. Spatial variability in emissivity at each validation site was further analyzed using high resolution (100 m) ASTER emissivity data covering an area corresponding to one AIRS pixel (∼45 km) shown in Figure 3. The ASTER emissivities represent a mean composite of all scenes ever acquired since 2000, part of a Level-3 gridded ASTER Global Emissivity Map (ASTER-GEM) currently being produced at JPL, and previously released as the North American ASTER Land Surface Emissivity Database (NAALSED) [Hulley and Hook, 2009]. Figure 3shows ASTER-GEM band 13 (11.3μm) emissivities at each site representing one AIRS pixel (∼45 km). The ASTER emissivities have low variability over the Gran Desierto and Namib sites with spatial standard deviations of 0.0017 (0.17%) and 0.0015 (0.15%) respectively. The Namib site variability primarily occurs due to differences in mineralogy between dune crests (primarily pure quartz) and interdune areas (higher clay content) previously discussed in Hulley et al. [2009b]. The Redwood site spatial variability was higher than the desert sites at 0.0043 (0.43%), mainly due to access roads and areas of open bare ground. Again these variations can be considered low variability, and equivalent to <0.3 K for a material at 300 K.
Figure 2. (left) Sampling site locations and (right) corresponding lab-measured emissivity spectra in the 10–13μm range for the (top) Namib Desert validation site and (bottom) Gran Desierto. Mean spectra are shown as solid lines and spatial standard deviation are shown as solid error bars.
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Figure 3. Mean ASTER band 13 (11.3 μm) emissivity centered at each validation site location corresponding to one AIRS pixel (∼45 km). The ASTER data represent a mean composite of all ASTER data acquired since 2000 at 0.001° (∼100 m) spatial resolution. Spatial variations in ASTER emissivity were 0.17% (Gran Desierto), 0.15% (Namib), and 0.43% (Redwood).
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 Lab-measured emissivity spectra are currently available for more than ten sand dune sites in the southwestern USA, the Namib and the Kalahari Desert (http://emissivity.jpl.nasa.gov). However, the majority of these sites are too small to be representative of the AIRS spatial footprint. This currently limits the R-based method to the largest sites where emissivity spectra are available (Namib and the Gran Desierto).
3.3. Uncertainty Analysis
 The uncertainty in the R-based LST estimate (LSTR-based) was calculated by perturbing the atmospheric temperature and water vapor profiles, and by varying the surface emissivity. Atmospheric effects were simulated by first increasing the relative humidity at each NCEP level by 10%, and then by increasing the air temperature by 1 K at each level. The effect on the accuracy of LSTR-based was estimated as the calculated LST difference between the original and the perturbed profiles for the 11 μm window region. The results are summarized in Table 1. Using a standard profile with total column water vapor of 2 cm, the absolute LST differences where 0.35 K for the water vapor variation (10%), and 0.19 K for the air temperature variation (1 K), resulting in a total atmospheric effect of ±0.39 K. Using an emissivity perturbation of 0.005 (0.5%), which represents the maximum spatial variation found from the lab measured spectra and ASTER data at each site, resulted in an absolute LST difference of 0.23 K. Validation of the Stand-Alone AIRS Radiative Transfer Algorithm (SARTA) with in situ data have shown accuracies approaching 0.2 K depending on the wave number region [Strow et al., 2006], and this uncertainty was considered negligible The total combined root mean square error (RMSE) for the uncertainty in LSTR-based based on estimated atmospheric profile, emissivity and radiative transfer model errors was ±0.47 K. This is within the 1 K accuracy requirement for typical in situ measurements of LST [Hook et al., 2007].
Table 1. Uncertainty Analysis Results Showing How Perturbations in Emissivity, Air Temperature and Relative Humidity Affect the Relative Accuracy of the R-Based LST Derivation
|Parameter||Perturbation||R-Based LST Change|
|Emissivity||ε + 0.005||0.23 K|
|Air Temperature||Tair + 1 K||0.19 K|
|Relative Humidity||RH + 10%||0.35 K|
 Further, since air temperature and water vapor errors (and emissivity) typically cancel each other out and may have different signs at different levels, the simulated error of 0.47 K is most likely an overestimate, i.e., a ‘worse-case-scenario’. Also, using the brightness temperature profile quality check would most likely filter out the majority of unsuitable profiles.
3.4. Application of the R-Based Method to AIRS Data
 The R-based method was applied to the AIRS v5 LST product using the standard L2 cloud-cleared radiance product for all available observations at each site from 2002 to 2010 with surface retrieval quality set to good or best (Quality Assurance (QA) = [0,1]). The closest AIRS pixel for each observation was matched to the sampling points for the desert sites, while an AIRS pixel at 41.4°N, 123.7°W over the Redwood forest was chosen for the vegetation site matchup.
 The simulated TOA BTs for AIRS channels in the 11–12 μm region were calculated using the Stand-Alone AIRS Radiative Transfer Algorithm (SARTA) with input NCEP atmospheric profiles. SARTA is a stand-alone version of the full AIRS operational retrieval system which parameterizes atmospheric transmittances in 100 pressure layers [Strow et al., 2003a] using the AIRS spectral response functions (SRFs) [Strow et al., 2003b]. In this study SARTA V107 was used, which is available upon request from http://asl.umbc.edu/pub/rta/sarta. Validation with in situ data has shown accuracies approaching 0.2 K depending on the wave number region [Strow et al., 2006].
 The LSTR-based was calculated from the 11 μm window region using the average radiance of 3 channels between 904 and 912 cm−1, while the average of 27 channels between 848 and 856 cm−1 were used in the 12 μm region for the profile quality check. The LST accuracy is estimated as the difference between the retrieved LST product and the calculated LSTR-based. For each observation, the quality of the NCEP profile was checked using the ā(T11 − T12) BT criterion, which is the difference between the AIRS observed and calculated T11 − T12 BT values as discussed in section 3.1. Any profiles that did not meet the 0.5 K > ā(T11 − T12) > − 0.5 K criterion were discarded. Profiles that met this criteria were considered accurate enough for the simulation. By applying this quality metric, roughly 1/3 of all observations were discarded at each site, although this did not have a big impact on statistical analysis since 9 years of observations were used. Figure 4 shows an example of the differences between calculated and observed AIRS brightness temperatures from 840 to 920 cm−1 (10.9–11.9 μm) for an observation over the Redwood forest. Indicated on the plot are locations of the channels used for the R-based temperature inversion (904–912 cm−1) in a clear window region, and the profile quality test (27 channels between 848–856 cm−1) in a more opaque region of the atmosphere due to water vapor absorption.
Figure 4. Differences between calculated and observed AIRS brightness temperatures from 840 to 920 cm−1 (10.9–11.9 μm) for Redwood forest, USA. Locations of the channels used for the R-based temperature inversion (904–912 cm−1) in a clear window region, and the profile quality test (27 channels between 848–856 cm−1) in a more opaque region are also indicated on the plot.
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 The biggest assumption in the R-based method over long time periods is that the emissivity stays relatively constant at the site, both temporally and spatially. Soil moisture data from the Advanced Microwave Scanning Radiometer (AMSR-E) were used to confirm that no significant rainfall events occurred preceding each AIRS observation at the two desert sites in a similar manner as described inHulley et al. . Regarding the spatial uniformity, the largest variability at the Namib site occurs due to differences in mineralogy between dune crests (primarily pure quartz) and interdune areas (higher clay content) [Hulley et al., 2009b], while at the Desierto site the largest variability occurs due to the distribution of quartz and feldspars across the central dune system [Scheidt et al., 2011]. At both desert sites multiple samples were collected in these different zones and the lab spectra were averaged in order to be representative of the effective AIRS pixel at 45 km.