Introduction
Incoming shortwave solar radiation (SW↓) is an important component of the surface energy and water balances. As a result, there has been considerable effort toward obtaining insolation estimates at spatial and temporal resolutions appropriate for hydrologic and other distributed modeling applications (e.g., see Famiglietti and Wood 1994; Pinker et al. 1994; Dubayah 1994; Abdulla 1995; Dubayah et al. 1995). Two important modulators of SW↓ are clouds and topography. The spatial variability in SW↓ is dominated by cloud fields under partly cloudy conditions. Under uniformly clear or cloudy skies, however, the spatial variability in solar radiation at local and regional scales often is caused primarily by topography. Variations in slope, aspect, terrain reflectance, shadowing, and sky obstruction by nearby terrain all affect the amount of insolation incident at any point on the surface. Previous work has shown that, for clear-sky cases, the spatial autocorrelation of SW↓ modeled over topographic grids is quite short, usually less than 300 m and almost always less than 1000 m for a wide variety of landscapes (Dubayah et al. 1989, 1990; Dubayah 1992, 1994; Dubayah and van Katwijk 1992). Thus, it is not likely that any set of in situ observations, such as from pyranometers, will capture the spatial variability in radiation caused by topography. It further implies that coarse-resolution (greater than 1 km) satellite estimates of insolation will average most of this variability away. The main objective of this paper is to present a method for combining coarse-resolution satellite estimates of SW↓ with fine-resolution digital topographic data to produce insolation maps that better represent the influences of both clouds and topography.
The importance of topographic variability in hydrologic and biophysical processes has long been known (e.g., Dozier 1980; Gates 1980; Davis et al. 1992). As a common example, south-facing slopes in northern midlatitudes may receive more direct incident flux than north-facing slopes and, therefore, have different latent, sensible, and soil heat exchanges. Under cloudy skies, sky obstruction by nearby terrain, such as occurs at the bottom of deep valleys, significantly decreases the diffuse flux reaching the surface. To account for this variability, topographic models have been created that incorporate various topographic effects using digital elevation data (e.g., see Dozier 1980, 1989; Duguay 1993; Dubayah and Rich 1995). One of the reasons such models have not been used more widely for modeling applications is the difficulty in obtaining the required driving input parameters. Typically, several methods may be used to drive the models, including 1) point estimates of the scattering and absorbing properties of the atmosphere, such as the optical depth, single scattering albedo, and scattering asymmetry parameter (Dubayah et al. 1990; Dubayah 1991); 2) point measurements of the incoming field itself via pyranometers (Dubayah 1992, 1994; Dubayah and van Katwijk 1992); 3) climatological estimates of solar radiation, perhaps augmented by in situ observations of atmospheric state, such as total cloud cover (Abdulla 1995); and 4) estimates of atmospheric state under “potential” solar radiation conditions (Gates 1980; Hetrick et al. 1993). These suffer from various limitations, such as a lack of availability and limited spatial/temporal coverage, so that distributed modeling using these data is difficult.
One practical source for deriving SW↓ over large areas is from sensors aboard operational meteorological satellites, such as AVHRR (Advanced Very High Resolution Radiometer), Meteosat, and GOES VISSR (Visible–Infrared Spin Scan Radiometer), the last of which can combine high temporal resolution (30 min) with coarse spatial resolution (1 km and beyond) (Gibson 1984). However, the inability of even the finest-resolution GOES estimates to resolve important topographic variability was shown by Dubayah et al. (1993). For the FIFE [First ISLSCP (International Satellite Land Surface Climatology Project) Field Experiment] site (Sellers et al. 1988), clear-sky SW↓ derived from GOES showed almost no variability across the site, having spatial standard deviations of less than 3 W m−2. Radiation maps, made using a topographic model with a 30-m digital elevation model (DEM) that was driven by pyranometer estimates, had spatial standard deviations of about 30 W m−2, caused by topographic variations. Because the topographic model was driven by pyranometer estimates, which inherently spatially integrate the overlying hemisphere, it could not capture the instantaneous spatial variability caused by broken cloud fields (partly cloudy conditions). At these times, the spatial variability in the GOES fields was as large as 235 W m−2.
A technique is needed that simultaneously marries small-scale (coarse resolution) variability in irradiance caused by clouds with large-scale variability caused by topography over large regions. Such a technique has been proposed and developed (Dubayah and Loechel 1993; Loechel 1995), and its description and general implementation are the focus of this paper. We first review the physical basis and relevant equations underlying the topographic model and then describe the methodology for driving the model with GOES-derived insolation fields. An example of the technique is given using 8-km hourly GOES insolation fields and a 1° × 1° digital terrain model with 90-m grid spacing for a portion of the Rocky Mountains. Lastly, we discuss some of the limitations and sources of error that may occur both within the data and in the algorithms themselves.
Modeling topographic effects
The topographic model used in this study is based on Dozier (1980, 1989). Subsequent versions have been presented in detail elsewhere (Dubayah 1992, 1994; Dubayah and Rich 1995). It is necessary to briefly review its major components for an understanding of the method presented here. (Note that for convenience we have included some of the most important formulations from the above references in appendix A). Consider an arbitrary location on a landscape for which we wish to calculate the total incident solar radiation. This flux will be the sum of three components: diffuse irradiance from the sky, including clouds; direct irradiance from the sun; and direct and diffuse irradiance reflected from nearby terrain toward the location (Fig. 1).
The starting points for the topographic model are estimates of the direct and diffuse flux incident on a flat surface at the elevation of the location. Apart from the topographic influences that slope orientations cause, points at higher elevation have less atmosphere above them. Under clear skies, the optical depth of the atmosphere decreases with pressure, which in turn decreases with elevation, and therefore the irradiance on a flat surface will increase. For overcast skies, cloud optical thickness generally dominates the clear-air column optical thickness either above or below the clouds so that the effects of changes in the elevation of the surface are smaller than for clear conditions. Other factors affecting the fluxes are the particular scattering and absorbing properties of the atmosphere, the variations of these properties with height, the solar zenith angle, and the exoatmospheric flux. Once the direct and diffuse fluxes are known for a flat surface at the given elevation, they are then adjusted for other topographic effects.
Diffuse irradiance
In addition to those factors listed above, diffuse irradiance on a slope depends on surface reflectance (for multiple scattering between the atmosphere and the surface), anisotropy in the diffuse field, and sky obstruction by the slope itself and by nearby terrain. In general, diffuse irradiance, even under clear skies, is not isotropic (e.g., the sky is usually brighter near the sun and near the horizon). This is evident especially under partly cloudy conditions, where the diffuse flux coming from the sky varies depending on whether clouds are present in a section of the sky. This anisotropy can be further complicated by the local topography. A portion of the sky may be obstructed by topography, blocking diffuse irradiance from that direction. For example, we may have a largely obstructed view of the sky to the north and a relatively unobstructed view to the south, but sky conditions may be cloudy to the south and clear to the north. Not only is the diffuse field anisotropic north to south, but the sky obstruction by nearby terrain is nonuniform from north to south. We can account for both of these effects by knowing the angular distribution of diffuse radiation throughout the sky, as well as the amount of obstruction by topography in a particular viewing azimuth.
A sky view factor, Vd(ϕ), can be calculated that gives the total amount of unobstructed sky visible on a slope in a particular view direction azimuth ϕ, expressed as the ratio of diffuse irradiance at a point to that on an unobstructed horizontal surface. The sky view factor Vd(ϕ) varies from 1 (unobstructed) to 0 (completely obstructed). The sky view factor is derived using the angle to the local horizon in a given direction for each point on the digital elevation grid (Dozier and Frew 1990). The calculation of horizon angles, although not computationally intensive, is significant, and this may discourage its real-time calculation for very large grids. Precalculation of Vd(ϕ) avoids this problem but then requires considerable storage for even a limited set of viewing directions (Dubayah and Rich 1995).
Another difficulty is the dynamic nature of the diffuse irradiance field. Because clouds move with time, we do not know cloud position relative to obstructions on the ground at any point other than at the satellite overpass time, and therefore the use of directional sky view factors is not straightforward for time-integrated fluxes. A related problem arises from the mismatch between sensor field of view and the hemispheric (though possibly obstructed) view of a terrain element (or of a pyranometer). A single GOES pixel of, say, 8-km resolution would not capture the atmospheric conditions of the entire overlying hemisphere visible to an individual terrain element, either instantaneously or through time. There may be a large cloud mass at zenith above the terrain element but clear skies elsewhere. The flux reaching the element comes from the entire hemisphere, including both the clear and cloudy areas.
Direct irradiance
Reflected irradiance
Total irradiance on a slope
Methodology
With the basis of the topographic model given, we next present a detailed methodology for driving the model using coarse-resolution satellite data. Our goal is to use SW↓ fields derived from GOES or other meteorological satellites to obtain the three fluxes given in (4) over large areas at high temporal resolution. Most researchers probably want to avoid the task of generating SW↓ fields directly from raw GOES imagery and instead would prefer to obtain surface solar radiation datasets generated by others. However, existing solar radiation archives, such as found in ISLSCP Initiative I (Meeson et al. 1995; Sellers et al. 1995) or the Surface Radiation Budget (SRB) program (Darnell et al. 1996), for example, often have spatial and temporal aggregate resolutions that are too coarse for many distributed modeling applications. The recent availability of relatively simple algorithms for deriving SW↓ from meteorological satellite data (but without topographic adjustment) opens the possibility for users to derive the fields for themselves at resolutions most appropriate for their work. These algorithms are well documented and will, therefore, not be discussed in detail (e.g., Gautier et al. 1980; Pinker and Ewing 1985; Pinker and Laszlo 1992). Here we briefly outline the algorithm used in this research, the “2001” code (Gautier and Landsfeld 1997), an improved version of earlier radiative transfer models (Gautier et al. 1980; Diak and Gautier 1983; Frouin et al. 1989; Frouin and Gautier 1990).
The 2001 model is physically based and accounts for scattering and absorption by molecules, aerosols, and clouds. The algorithm first uses a threshold value derived from the satellite minimum top-of-atmosphere (TOA) brightness to classify each pixel as clear or cloudy. Depending on the classification, either a clear-sky or cloudy-sky radiative transfer model is then applied. In the clear-sky case, satellite input is used only to determine the surface reflectance (for multiple scattering between the surface and the atmosphere) with atmospheric constituents such as ozone and water vapor, and their scattering and absorbing properties, found from standard climatologies. In the cloudy-sky case, the clear-sky formulation is modified to include the effects of cloud absorption and reflection, which are assumed to occur in one plane-parallel layer. Cloud albedo is derived from TOA reflectances, with cloud absorption a function of cloud albedo. The final product is a coarse-resolution field of surface (sea level pressure) SW↓.
The surface reflectance is derived during the processing of GOES data by finding the smallest clear-sky reflectance over a given number of days at a particular hour. This produces a set of minimum reflectances, one for each hour, which are then used in the radiative transfer calculations for the period. The time period must be long enough to ensure a clear view of the surface, but not so long that the surface reflectance or solar illumination changes from seasonal effects. For bright surfaces, such as snow, this method will result in errors in retrieved reflectances because the effect of the atmosphere is to create planetary (TOA) reflectances that are smaller than surface reflectances. For dark surfaces the opposite is true—+anetary reflectances are larger than surface reflectances. One way around this problem is to use reflectance maps generated from other sources. For example, the National Weather Service uses AVHRR imagery (bands 1, 3, and 4) to classify images into snow, land, and clouds (Hartman et al. 1996). Snow maps also can be used to differentiate between snow and clouds on GOES imagery, so that the appropriate SW↓ retrieval algorithm—that is, clear or cloudy, may be used. Clear-sky reflectances may also be determined by using other methods depending on snow cover and surface type [e.g., for the SRB project, see Darnell et al. (1992)].
Once the SW↓ and reflectance fields are obtained, there are four modeling steps needed to produce topographic SW↓ fields: 1) spatial integration of coarse-resolution GOES SW↓, 2) direct–diffuse partitioning of SW↓, 3) elevation correction, and 4) topographic correction. The processing chain is illustrated in Fig. 2.
Spatial integration
As discussed above, some type of spatial integration of the GOES estimates is required given the dynamic nature of the diffuse field with time, the mismatch between GOES-observed instantaneous fluxes and the (usually) desired time-integrated quantities, and the sensor field of view versus the implicit hemispheric integration of flux incident on any terrain element (i.e., the element may observe the entire overlying hemisphere). Another important reason is that there may be considerable error in the registration of the GOES imagery relative to the ground. GOES imagery is navigated and usually projected to some map form. The derived SW↓ fields may then have to be further reprojected to match that of the digital elevation data [e.g., the 30-m DEM data of the U.S. Geological Survey (USGS) are in Universal Transverse Mercator; the 3" USGS products are in geographic coordinates]. Precise registration to these data is difficult given the coarse resolution of the GOES fields relative to the digital elevation grid spacing.
For these reasons the GOES SW↓ are averaged. We choose a window size of approximately 50 km × 50 km, based on previous work (Pinker et al. 1994) and our own empirical study, presented below. Rather than simply averaging the GOES observations, however, we use a moving average window centered around each digital elevation grid cell. The mean insolation within this window is found, and the window is then moved to the next grid cell until the entire grid is covered. This produces a global insolation field at the grid spacing of the elevation data (e.g., 90 m), where the value at each cell is a 50 km × 50 km average of some group of complete and partial GOES SW↓ pixels. The advantage of a moving window approach is that it preserves some of the spatial variability inherent in the original GOES imagery and is also more physically plausible than a static aggregation. The procedure also removes the sharp borders or discontinuities between adjacent 8-km GOES pixels with different irradiances. Such discontinuities are problematic when transferred to a modeling grid of much higher resolution, as they lead to unrealistically sharp gradients in SW↓ relative to the digital grid cell spacing. To avoid any “ringing” that may be caused by using a moving average window with equal weights (equivalent to a perfect low-pass filter), some form of central weighting should be applied, so that the GOES pixels closest to the center of the window count more heavily in the averaging.
Direct–diffuse partitioning
Most existing solar radiation products do not partition total irradiance into diffuse and direct components, although this is an integral part of the algorithms generating them. If such components are available, they can be used directly. Otherwise, several climatological formulas have been developed for splitting total insolation into direct and diffuse fluxes. One such formula has been developed by Erbs et al. (1982) and modified by Olyphant (1984) for mountainous conditions (see appendix B). These can only approximate the real distribution of fluxes, as this will vary with specific atmospheric conditions, and in general will perform better over longer periods (i.e., monthly compared to hourly fluxes). For flat regions, the partitioning is not important, as there will be no topographic effects other than elevation adjustment. For more complex surfaces, the partitioning is significant because the nature of the topographic correction changes as the direct–diffuse proportion changes in (4). The partitioning results in two images, direct and diffuse sea level insolation.
Elevation correction
Each of the sea level direct and diffuse fluxes must then be adjusted for changes in elevation. For the direct flux profile, a pressure-dependent decrease in direct transmittance is assumed, where pressure decreases with elevation according to the hydrostatic equation. The direct transmittance at sea level is found by ratioing the beam irradiance from the partitioned GOES insolation and the exoatmospheric solar flux.
The vertical diffuse profile is obtained using an empirical formulation of Lowry (1980), which, as modified by Dubayah and van Katwijk (1992), has reasonable application for the clear-air column below clouds (see appendix C). For cloudy conditions we assume that we never enter or get above the cloud deck. For areas of high elevation (over 4000 m) this may be a poor assumption, but a more accurate treatment requires information on cloud-base and cloud-top heights, of which the former of which is difficult to infer remotely from space. The profiling method is applied to each elevation grid cell to produce two images, direct and diffuse irradiance on a flat surface at the elevation of the grid cell.
Topographic correction
Once the elevation-corrected fields of direct and diffuse radiation are derived, the remaining topographic effects as given in (4) are modeled. These require images of the cosine of the illumination angle cosθi (from slope and aspect), sky view factor Vd, and terrain configuration factor Ct, all derived from digital elevation data. The direct and diffuse fluxes from the previous step give B̄x↓(z) and F̄x↓(z), respectively, in (4). The local reflectances R̄x are found by averaging the reflectances from the GOES imagery over the same areas as in the spatial integration above. If the GOES reflectances are not available, other reflectance data must be used. With this, all quantities in (4) are known, and the algorithm is complete.
Modeling example: Solar radiation over the Rocky Mountains
To illustrate the application of this methodology, we give an example for the Rocky Mountains. A 4-yr monthly solar radiation topoclimatology for the hydrologic years from 1987 to 1990 has been produced previously for a nearby area, the Rio Grande basin (Dubayah 1994). This topoclimatology was derived by driving the topographic model with hourly measurements from a pyranometer over a topographic grid. A single pyranometer observation cannot be expected to adequately capture the spatial variability in SW↓ for a basin this size (3500 km2); yet such a sparse distribution of in situ data is not uncommon and highlights the advantage of using remotely sensed observations.
The modeling area is centered at about 38.5°N, 106.5°E and covers an area of 1.25° × 1.25° of latitude and longitude (roughly 9500 km2). The digital elevation data covering the region were obtained from the USGS and are the 3" (1:250 000) product; that is, the grid spacing size is 3", approximately 90 m in the north–south direction at this latitude (Fig. 3a). The size of the grid is 1500 × 1500 cells. The region is high and rugged, with a mean elevation of 2970 m, an elevation range from 1890 to 4400 m, and an average slope of 11°, as determined from the elevation grid.
Hourly GOES-7 visible observations (not solar radiation data, but raw digital counts) at nominal 8-km spatial resolution were obtained from the GOES Pathfinder project for August 1988 (Young 1995). These digital data were calibrated and then navigated and projected using the Man–Computer Interactive Direct Access System (McIDAS-X) image processing software (see acknowledgments). The images were next passed through the 2001 solar radiation algorithm described above (Gautier and Landsfeld 1997) to yield SW↓ at sea level pressure. Finally, these SW↓’s were used to drive the topographic model.
Figures 3, 4, and 5 show the various input and modeled images at various stages in the processing chain needed to produce one hourly topographic radiation map. The elevation data are used to produce illumination angle (Fig. 3b), sky view factor (Fig. 3c), and terrain configuration factor (Fig. 3d) images. Notice that even at a small scale, variability in these is considerable. Also note the differences between illumination angle and sky view factor. Spatial variability in solar radiation will be controlled by some combination of these two, depending on the proportion of direct to diffuse radiation.
The 8-km sea level SW↓ field (Fig. 4a) is spatially integrated, producing the field shown in Fig. 4b. This image is then partitioned into direct and diffuse fluxes and each of these corrected for the effects of elevation (Figs. 4c and 4d). The remaining topographic effects are applied giving direct, diffuse, and reflected terrain components, the three terms in (4), and these are then summed to produce the final radiation map (Fig. 5). Note the interaction of clouds with topography in Fig. 5 and the presence of various topographic effects simultaneously. For clearer areas, most of the irradiance is direct and illumination angle effects dominate, while a relatively short distance away, the irradiance is mostly diffuse and sky view factor is most important.
Discussion
The method we have outlined has clear advantages over in situ techniques for distributed modeling applications because it uses remotely sensed data. Our approach is based on fields of SW↓ that are presumed correct at sea level, enabling the use of previously generated solar radiation data and thus avoiding the task of processing raw GOES data. Even if such processing is done, as in the example given in the previous section, because it does not require initially separate fields of diffuse and direct insolation, an existing radiation code can be used without modification. However, the decoupling of the sea level estimates of SW↓ from the topographic correction creates several problems, some of which involve unresolved research issues that may limit its accuracy. There are also additional sources of error in both data and method that have been discussed elsewhere in part (Dubayah and Rich 1995). For example, there are errors associated with digital elevation data, such as pits and mistagged contours (Brown and Bara 1994). We note that the issue of validation is particularly difficult. Comparisons with pyranometer data validate only the GOES shortwave retrieval algorithm and the elevation correction. This is because pyranometers are usually oriented horizontally, regardless of underlying slope. Although there have been some attempts at validation of our algorithm (Dubayah 1992), a more thorough validation should be conducted using field data aimed specifically at testing various aspects of the topographic correction in mountainous terrain.
A primary source of error is the accuracy of the GOES SW↓ fields themselves. Although previous validations have shown that such algorithms have small rms errors at daily and monthly time integrations, the instantaneous measurements can sometimes be quite far off, especially during cloudy conditions (Pinker et al. 1994; Gautier and Landsfeld 1997). Part of this problem may be caused by the mismatch between the temporal and spatial integration sizes of the GOES observations versus those of pyranometers, as discussed earlier. Pinker et al. (1994) have shown that approximately 50-km averages may be more representative of the hourly fluxes as measured by pyranometers than finer-resolution GOES estimates for a small (10 km × 15 km) watershed in Arizona. To explore this issue further for mountainous regions, where cloud conditions change rapidly, we calculated the rms error between pyranometer observations and GOES-derived insolations at various window sizes. Three pyranometer times series were used located at Alamosa (elevation 2286 m), Durango (elevation 2225 m), and Montrose (elevation 1774 m), all in Colorado. The 8-km GOES pixel containing each station was found, and then subsequent averages of pixels were taken around this for squares with sides of length 16, 24, 32, 40, and so on up to 320 km. These averages were then compared with the pyranometer measurements for 1–15 August 1988. Figure 6 plots the rms error for each site as a function of window size. A window size of about 50 km yields the lowest rms error for Montrose and Durango, with a larger window of about 100 km for Alamosa. Although not conclusive given the short span of the observation period, the results are consistent with Pinker et al. (1994) and suggest that 8-km SW↓ derived from the GOES Pathfinder dataset may be at too fine a resolution for accurately retrieving hourly fluxes using standard navigation in such terrain. More research is required to determine the most appropriate size and form for the integration window.
Apart from these issues, the accurate retrieval of hourly integrated surface fluxes under some types of cloudy conditions is a difficult radiative transfer problem and appears to be at the edge of the capability of current algorithms. The magnitude of this error is illustrated in Fig. 7, which plots 50-km GOES estimates, corrected for elevation effects, against the observations acquired from the three pyranometers. Although the relationship shown is reasonable and consistent with those of other validations, errors at individual times may be large.
It is possible that inaccuracies in any retrieval code may swamp topographic variability at short timescales when solar radiation is aggregated over large areas, relative to the digital elevation grid spacing. For example, suppose topographic SW↓ is modeled using a 90-m elevation grid, and these 90-m estimates are then aggregated to the resolution of the GOES data at 8 km. The change in mean irradiance that the topographic corrections make to this aggregated estimate may be small, smaller than the error inherent in an uncorrected (surface) GOES SW↓ estimate. The reason for this is that topographic effects generally decrease as the aggregation area increases (Dubayah et al. 1989). However, the goal is to provide inputs to distributed models, usually at relatively fine grid spacings, and not to correct coarse scale means for topographic effects, which is a separate issue. Even over a distributed modeling grid with spacing as coarse as 1 km, the change in irradiance between a south- and north-facing slope can be hundreds of watts per square meter at this modeling scale. Thus, errors in excess of those in the surface GOES SW↓ fields would result if topographic effects were ignored. This is most evident during clear conditions, when, in general, GOES SW↓’s are most accurate and topographic effects are greatest.
Another problem concerns the accuracy of the methods used to partition global irradiance into direct/diffuse components, and the subsequent elevation correction for each of these. The partitioning algorithm, as it is based on a climatological relationship, is only approximate. In areas of rugged terrain, a correct partitioning is important because the type and magnitude of the topographic correction depend on these quantities. However, errors in the direct/diffuse ratio, if kept moderately low, should lead to only small errors in total irradiance on a slope. This is illustrated by considering a 30° slope facing directly toward the sun at a solar zenith angle of 60°, with a sky view factor of 0.8 and no other topographic effects. If the direct/diffuse ratio is 0.7/0.3, then a 20% error in the direct portion (i.e., the direct amount is 20% larger or smaller than estimated) would lead to only a 7% error in total slope irradiance. Ideally, the direct and diffuse portions should be obtained directly from the radiative transfer algorithm generating the surface SW↓ and saved as separate fields. However, this may require some modification of the radiation code.
Our method for obtaining the vertical profiles of direct and diffuse radiation under cloudy skies is also approximate. Figure 8 shows a comparison of our profiling method with the LOWTRAN 7 radiative transfer code. The height of the surface is varied within LOWTRAN 7, and the fluxes at that height are found. Figure 8a compares total irradiance (which is mostly direct) using a clear-sky, midlatitude, summer atmosphere. The agreement is good, although at the highest elevation (5 km) and largest solar zenith angle (about 60°) the difference approaches 10%. (This elevation is higher than the highest point in the contiguous United States, Mount Whitney, at 4500 m). Figure 8b uses a midlatitude summer atmosphere, but with an altostratus cloud deck, where the total irradiance is entirely diffuse. The height of the the cloud deck varies between 2.4 and 3.0 km, depending on the height of the surface, hence we limit our range of elevation to avoid entering the cloud. Up to the height of the cloud, there is excellent agreement. However, note that we are comparing the radiative transfer properties of a prescribed climatological atmosphere in LOWTRAN 7 with our own empirical/climatological formulation. The actual profile for any particular instance will deviate from these. Short of inputting actual lapse profiles of the major atmospheric scatterers and absorbers directly into a GOES radiative transfer code (e.g., from sounding data or numerical weather forecast models) and then coupling the topographic model directly to these, improved empirical relationships may be the best that can be hoped for.
Conclusions
Issues related to scale and scaling have become increasingly dominant in earth system science, especially in terms of the effect of land surface heterogeneity on the energy and mass transfer between the surface and the atmosphere. As a result, distributed energy and water balance models are being used to explore the importance of finescale variability for surface–atmosphere interactions. Two of the primary controls on the distribution of one key driving variable, solar radiation, are clouds and topography. However, obtaining insolation fields that reflect the contributions of both these effects at temporal and spatial resolutions appropriate for such modeling has been difficult. We have developed an algorithm for obtaining topographically varying insolation fields that combines the coarse-scale variability in solar radiation captured by GOES observations with finescale variability caused by topography. Although many refinements are envisioned, the algorithm in its present form should prove useful for a variety of distributed modeling applications.
Acknowledgments
We thank Catherine Gautier and Marty Landsfeld at the Department of Geography and Institute for Computational Earth System Science (ICESS), University of California, Santa Barbara, for providing the 2001 radiation code, as well as other generous assistance. We also thank Rachel Pinker at the University of Maryland at College Park for useful discussions. Inquiries regarding the 2001 radiation code should be directed to the Earth Space Research Group at ICESS (http://www/esm.ucsb.edu/esrg.html). Information on obtaining the topographic code is available from R. Dubayah (rdubayah@geog.umd.edu). The McIDAS-X image processing system is available from the UCAR Unidata Program Center, P.O. Box 3000, Boulder, CO, 80307-3000 (support@unidata.ucar.edu; http://www.unidata.ucar.edu). Information about the Goes Pathfinder Project, as well as details on obtaining data, is available from the Space Science Engineering Center at the University of Wisconsin—Madison (goesprods@ssec.wisc.edu; http://www.ssec.wisc.edu/research/pathfinder/PFHomepage.html). This work was supported by National Aeronautics and Space Administration Grants NAGW-2928 and NAGW-4554, and the University of Maryland Institute for Advanced Computer Studies.
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APPENDIX A
Equations for Topographic Modeling
APPENDIX B
Direct–Diffuse Partitioning
APPENDIX C
Diffuse Irradiance Profile
The three sources of energy on a slope: 1) direct or beam irradiance from the sun; 2) diffuse irradiance from the sky, where a portion of the overlying hemisphere may be obstructed; and 3) diffuse and direct irradiance reflected off of nearby terrain (after Dubayah 1994).
Citation: Journal of Applied Meteorology 36, 2; 10.1175/1520-0450(1997)036<0141:MTSRUG>2.0.CO;2
The processing chain from the GOES insolation field to the final topographic solar radiation map.
Citation: Journal of Applied Meteorology 36, 2; 10.1175/1520-0450(1997)036<0141:MTSRUG>2.0.CO;2
Digital elevation and other data derived from USGS 3" (approximately 90 m) product: (a) elevation field, with elevation ranges from 1889 m (black) to 4389 m (white); (b) cosine of the illumination angle cosθi for the time of the satellite overpass (ranges from 0 to 1); (c) sky view factor, black is 0.22 (nearly completely obstructed) and white is 1 (unobstructed); and (d) terrain configuration factor, black is 0 (no terrain contribution) and white is 0.55. Note in (c) the large amount of sky obstruction that occurs in the bottom of valleys. For overcast conditions, where the surface reflectance is not large, such sky obstruction will dominate the spatial variability of incoming solar radiation. In contrast, under clear conditions, the spatial variability will be dominated by cosθi.
Citation: Journal of Applied Meteorology 36, 2; 10.1175/1520-0450(1997)036<0141:MTSRUG>2.0.CO;2
Images produced at intermediate processing steps for a portion of the Rocky Mountains: (a) original 8-km GOES insolation field at sea level, (b) spatially integrated insolation field using 50-km moving average window, (c) direct insolation field corrected for elevation effects, and (d) diffuse insolation field corrected for elevation effects. The range of insolation values occuring on each image is given. The gray levels in images (a) and (b) are directly comparable; however, images (c) and (d) have been histogram equalized separately to highlight detail, and therefore direct comparison of gray levels between these is misleading. Note the appearance of major river drainage features in the southwest corner of (c), reflecting the decrease in direct insolation caused by increased atmospheric attenuation as elevation decreases (see Fig. 3a). The opposite effect occurs in (d), through the central portion of the image. Because this image gives diffuse flux, as we descend downward into river valleys, the increased direct attenuation leads to an increase in diffuse flux.
Citation: Journal of Applied Meteorology 36, 2; 10.1175/1520-0450(1997)036<0141:MTSRUG>2.0.CO;2
Final topographic solar radiation map combining all topographic effects and cloud conditions from GOES. Range of values on the map is from 50 W m−2 (black) to 1018 W m−2 (white), with a mean irradiance of 587 W m−2. Insets are enlarged areas showing detail and have been histogram equalized separately. Note that for the clear portions of the map, the illumination angle effects dominate (left inset), whereas in the cloudier portions, the sky view factor is the controlling element (right inset).
Citation: Journal of Applied Meteorology 36, 2; 10.1175/1520-0450(1997)036<0141:MTSRUG>2.0.CO;2
Root-mean-square error between instantaneous GOES insolation estimates at various levels of aggregation and hourly pyranometer measurements at three mountainous locations. Windows of increasing size, from 8 km to over 300 km, surrounding the known location of a pyranometer were taken, and the average irradiance in that window was found. Because of navigational uncertainties, the mismatch in timescale between instantaneous and hourly integrated fluxes, and other considerations, the lowest rms errors do not occur at the finest resolution, but at coarser aggregations, from 50 to 100 km.
Citation: Journal of Applied Meteorology 36, 2; 10.1175/1520-0450(1997)036<0141:MTSRUG>2.0.CO;2
Instantaneous hourly GOES SW↓ estimates, corrected for elevation effects, versus hourly integrated pyranometer measurements from three sites for 1–15 August 1988 (see Fig. 6). The GOES irradiances are 50-km averages centered around each pyranometer location. The dashed line shows the regression, and the solid line is the 1:1 ratio.
Citation: Journal of Applied Meteorology 36, 2; 10.1175/1520-0450(1997)036<0141:MTSRUG>2.0.CO;2
Comparison of radiation profiling technique with radiation profiles obtained from LOWTRAN 7 for two solar zenith angles θ0 under (a) clear conditions and (b) an altostratus cloud deck. In (a), global irradiance (but mostly direct) is given; in (b), the irradiance is entirely diffuse. The profile in (b) stops at 3 km to avoid entering the cloud deck in LOWTRAN 7. For both cases, a midlatitude summer atmosphere with 23-km surface visibility is used in LOWTRAN 7.
Citation: Journal of Applied Meteorology 36, 2; 10.1175/1520-0450(1997)036<0141:MTSRUG>2.0.CO;2
