Topographic Distribution of Clear-Sky Radiation · assess digital elevation grid spacing as a factor in radiation modeling. Concern over increasing human impacts on regional and global

WATER RESOURCES RESEARCH, VOL. 26, NO. 4, PAGES 679-690, APRIL 1990

Topographic Distribution of Clear-Sky Radiation Over the Konza Prairie, Kansas

RALPH DUBAYAH, JEFF DOZIER, AND FRANK W. D^ws

Department of Geography and Center for Remote Sensing and Environmental Optics, University of California, Santa Barbara

This research analyzes the topographic distribution of clear-sky incoming solar radiation over the tallgrass Konza Prairie, site of FIFE, the First ISLSCP (International Satellite Land Surface Climatology Program) Field Experiment. Using a two-stream atmospheric radiation model and digital elevation grids of 25-, 50-, and 100-m grid spacing, clear-sky radiation is simulated throughout the day for three dates: December 15, March 15, and June 15. Geostatistical analysis is used to characterize the spatial and temporal variability in modeled radiation at each grid spacing. The variance and spatial autocorrelation of simulated incoming radiation depend on Sun angle and elevation grid spacing. The behavior of the variance as a function of Sun angle, optical depth, and mean terrain slope can be explained by considering direct radiation variability on a simplified terrain model of uniform albedo where slopes are equal and azimuths are distributed uniformly in all directions. For this constant-slope model it can be shown analytically that the solar zenith angle at which variance is maximized is a function of optical depth only and is independent of elevation, slope, and aspect. Results from the two-stream simulations support this conclusion and suggest its applicability to real terrain.

Introduction

This paper explores the spatial and temporal distribution of incoming solar radiation over the tallgrass Konza Prairie in the Flint Hills, Kansas, the site for FIFE, the First !SLSCP (International Satellite Land Surface Climatology Program) Field Experiment. Our main objectives are to (1) model daily and seasonal variation in instantaneous incoming solar radiation in the visible and near-infrared wavelength regions, (2) characterize the spatiotemporal variability of this radiation through geostatistical analysis; and (3) assess digital elevation grid spacing as a factor in radiation modeling.

Concern over increasing human impacts on regional and global climate has focused attention on the use of satellite data for obtaining measurements of surface parameters, such as evapotranspiration and soil moisture, as inputs to regional and global climate models. FIFE has been undertaken to test the use of remote sensing for quantifying surface climate processes [Sellers et al., 1988a]. A central issue in FIFE is how processes operating at the scales of micrometers to meters integrate upward to affect interactions at atmospheric scales of tens to thousands of kilometers. Such knowledge is critical for validating and calibrating algorithms that convert remotely sensed observations to estimates of regional surface climate processes.

Incoming solar radiation is a key factor in most surface climate processes. Seasonal and latitudinal variations in incoming radiation are well understood and described, but only a few studies have analyzed topographic variability in diurnal and seasonal radiation patterns over a region or locale [e.g., Williams et al., 1972; Holland and Steyn, 1975; Dozier and Outcalt, 1979; Gates, 1980; Dozier, 1980; Kirk- patrick and Nunez, 1980; Dubayah et al., 1989]. Character- ization of local spatiotemporal variation in radiation is fundamental to understanding remotely sensed data on surface climate because significant spatiotemporal variation exists at

Copyright 1990 by the American Geophysical Union. Paper number 89WR03107. 0043-1397/90/89WR.03107505.00

scales below and above the resolutions of satellite-borne

sensors. Such scale-dependent variation should be ac- counted for in locating and integrating field measurements used to calibrate satellite-derived radiation estimates.

FIFE was designed to supply enough ground measurements to characterize the scale-dependent variation of surface climate parameters. Net or incoming solar radiation was measured at 16 sites over the 256-km 2 experiment site, a sampling density sufficient to account for atmospheric inho- mogeneities but still too coarse to capture topographically induced variability in radiation. The only means of describing this scale of variation is through modeling on topographic grids. By quantifying the spatial and temporal variability in modeled radiation we may also be able to infer variation in other climate parameters associated with radiation.

In modeling solar radiation over complex terrain, modeling scales are limited by the grid spacing and size of the grid. Given the cost and limited availability of digital elevation data at fine grid spacings (e.g., 30 m), the sensitivity of modeled radiation to grid spacing is of practical as well as theoretical interest. Therefore we have modeled and ana-

lyzed radiation over the FIFE site at several grid spacings. In addition to providing statistical descriptions of time-

dependent patterns in radiation, we have attempted to es- tablish the relative importance of some of the physical factors affecting the distribution of solar radiation at various scales. Specifically, how do solar geometry, terrain configuration, and atmospheric attenuation interact to produce radiation variability?

MODELING SOLAR RADIATION

Two-Stream Model Structure

In the solar spectrum, slopes are irradiated from three sources: (1) Fs, direct irradiance from the Sun; (2) Fa, diffuse irradiance from the sky, where a portion of the overlying hemisphere is obscured by terrain; and (3) F t, direct and diffuse irradiance reflected from nearby terrain. The scattering and absorption of light by the clear atmosphere and by clouds are analyzed with a multiple-scattering

679

680 DUBAYAH ET AL.' TOPOGRAPHIC DISTRIBUTION OF CLEAR-SKY RADIATION

model, a two-stream approximation to the radiative transfer equation. The fundamental scattering properties of the water droplets or ice crystals in clouds and the aerosols in the atmosphere are calculated by the complex angular momen- tum approximation to the Mie equations [Nussenzveig and Wiscombe, 1980]. The LOWTRAN model [Kniezys et aI., 1988] is used to obtain values for molecular absorption in the atmosphere at the desired wavelengths.

The two-stream approximation to the radiative transfer equation is given by a pair of differential equations for upward and downward fluxes at any level in a homogeneous atmosphere [Meador and Weaver, !980]:

dF t(r) -------- = 3'iF ? (r) - T2F • (•') + 3'3t00S0 e-r/*'ø (la)

dF • ('r) d•' • = 3'2F ? (r) - 3'iF $ (*) + 3'4tooSo e-•/aø (lb)

F? and F, are upward and downward fluxes; So is the exoatmospheric parallel-beam flux, incident at angle arccos ix0; to o is the single-scattering albedo (i.e., the ratio of extinction by scattering to total extinction);, is the optical depth coordinate; and the T values parameterize the scattering phase function. The Mie equations are used to calculate the single-scattering albedo to 0 and the scattering asymmetry parameter g, and the T values are functions of w0, g, and •0 [Meador and Weaver, 1980, Table 1].

The usual top and bottom boundary conditions are that there is no diffuse flux at the top, i.e., F $ (0) = 0, and that the lower horizontal boundary, at optical depth %, is a diffuse reflector with reflectance R 0, i.e.,

F ? (to) = Ro[F $ (to) + tzoSoe The solution to the two-stream equations for the direction-

al-hemispherical transmittance Ts through the atmosphere, equivalent to what is measured with a level pyranometer, is

Ts = (F $ (to) + la, oSoe-*ø/•ø)/I. coSo = [2q•3,2(/.zorz ] +

+e-rd•ø(P+U- -P-U +)]/(P +V- -P-V +) (2)

where

p q- + - '- (3'1 q- •:)½ -

q = wo/(1 - •2/•)

Q-+ = V + e -•:vø(a 2 q- so'y3)/(1 q- •/z 0)

U -+ = 3'2 - [too(a2 + •3'3)/(1 + •/•0)]

V +- = T2-Ro(T1 + s c)

__ 2

Oil --' T13'4 + 'Y2T3

•2 -' T2T4 + YlT3

Once the transmittance is determined for a level surface, the direct irradiance as a function of slope angle and azimuth as well as Sun angle and azimuth can be computed. The direct irradiance on a slope is p,•.So e-•ø/*'ø, where •0 is the cosine of the solar illumination angle, 00, on a horizontal

0.0

..............

i i i i i i

0.0 0.2 0.4 0.6 0.8 1.0

Cosine of Solar Zenith Angle

Fig. 1. Radiation variability of the constant-slope model as a function of solar zenith angle and optical depth. Each solid line represents the radiation variance function of equation (7) normalized to (1/2)S• sin 2 S = 1, for a given optical depth. The optical depth increases incrementally by 0.1 for each unmarked line. The dashed line is derived from the maxima function of (8). The solar angle where this line intercepts the solid variance lines is the angle at which the variance is maximized for that optical depth.

surface and Ix s is the cosine of the solar illumination angle on the slope, O s , given by [Sellers, 1965]

•s = cos 0s =/zo cos S + sin 0o sin S cos (4•0- A)

where &0 and A are the azimuth of the Sun and the slope, and S is the slope angie.

To calculate the diffuse irradiance, a sky view factor the ratio of diffuse sky irradiance at a point to that on an unobstructed horizontal surface, is calculated from the slope, aspect, and horizon data. The sky view factor ac- counts for the slope and orientation of a point and the portion of the overlying hemisphere visible to the p6int. It can also be adapted to account for anisotropy in the diffuse irradiance, but the two-stream equations assume that diffuse irradiance is isotropic. Va on slope S with azimuth A is found by projecting each element of the sky onto the slope and integrating over the unobstructed hemisphere, i.e., from the zenith downward to the local horizon, through angle H•, for each direction •. For an unobstructed horizontal surface, HO = z-/2. The horizon can result either from "self- shadowing" by the slope itself or from adjacent ridges.

DUBAYAH ET AL.' TOPOGRAPHIC DISTRIBUTION OF CLEAR-SKY RADIATION 681

7500m

--< "' 7500m •

Fig. 2. Computer-simulated shaded-relief representation of the study site generated from the 25-m DEM. North is at the top of the image. The direction of sunlight is from the northwest.

Vd = •-•--• [cos S sin 2 H&

+ sin S cos (4• - A)(Hrk - sin H& cos H&)] d4• •4) For each point, reflected radiation from surrounding ter-

rain is also estimated by calculating an average reflected radiation term and adjusting this by a terrain configuration factor. The configuration factor Ct includes both the anisotropy of the radiation and the geometric effects between a point and each of the other terrain points which are mutually •isible. The contribution of each of these terrain elements to

the configuration factor could be computed [Siegel and Howell, 1981], but this is a formidable computational problem. Rigorous calculation is difficult because it is necessary to consider every terrain facet visible from a point. In contrast to the sky radiation, the isotropic assumption is unrealistic because considerable anisotropy results from geometric effects even if the surrounding terrain is a Lam- bertian reflector. We therefore note that Va for an infinitely 10ng slope is (1 + cos S)/2, and we approximate Ct by

C t •-• [(1 + cOS S)/2] - Va (5)

Simplified Radiation Model

The above discussion of the two-stream model shows that incoming solar radiation at a point is determined by a complex interaction of the atmosphere, terrain, and solar geometry. At any particular time, for a spatially homogeneous atmosphere, most of the variance in incoming radia-

tion will be due to variations in direct solar irradiance caused

by terrain orientation. Unless there are large variations in the sky and terrain view factors, variations in the diffuse sky component and the diffuse and direct terrain component will make relatively little contribution to the overall site variance as compared with variability in the direct solar component. By considering a radiation model composed only of a direct solar beam component on a simplified terrain we can derive an analytical expression which describes the variation in radiation variance as a function of slope, Sun angle, and optical depth.

Consider a terrain model composed of many slopes, each with a fixed angle of S, but with azimuths distributed uniformly in all directions. Recall that the direct irradiance on a slope is given by

TABLE 1. Two-Stream Radiation Model Simulation Parameters

Parameter December 15 March 15 June 15

Optical depth 0.20 0.20 0.20 Single-scattering albedo

VIS 0.90 0.90 0.90 NIR 0.75 0.75 0.75

Scattering asymmetry parameter VIS 0.55 0.55 0.55 NIR 0.65 0.65 0.65

Reflectance of substrate VIS 0.12 0.09 0.05 NIR 0.20 0.30 0.40

VIS (visible radiation) = 0.35-0.75/xm. NIR (near-infrared radiation) = 0.75-2.8/xm.


Fs = !a, sSo e-rø/cosøø = So e-talcøsøø [cos 00 cos S

+ sin 00 sin S cos (•b0- A)]

Since aspect is uniformly distributed in our model over the range (-rr, ,r), its probability density function is 1/2,r, and the mean direct flux over all slopes is therefore

P = • Soe -•ø/•ø [cos 00 cos S + sin 00 sin S cos

-A)] dA = So e-*ø/cøsøø cos S cos 00 (6)

25m • ............. 50m •

300 350 400 450 Elevation [m]

The variance can be calculated in a similar fashion, integrating over all azimuths:

0-•,• = • (Fs- p)2 dA

1

S• e -2'ø/cøsøø sin 2 S sin 2 00 f_•• cos (4)0- A) 2 dA = « S02e -2*ø/cøsøø sin 2 S sin 2 00 (7)

If we assume a constant optical depth, then the critical points of this variance function with respect to solar zenith angie can be found by taking its derivative and setting the result equal to 0, i.e., Ocr•2s/aO o = 0. This leads to a unique maximum at a value of 00 such that

cos 3 00 •- r0 = 0 (8) sin 2 00

In summary, for a constant-slope terrain model, the magni- tude of the variance depends on slope, but the Sun angle at which the maximum variance occurs is independent of slope, being completely determined by the optical depth. For any given optical depth there is a solar zenith angle at which variance is a maximum.

We emphasize that this result is derived solely from consideration of the direct solar component and not from the full two-stream model and assumes a constant albedo, optical depth, and terrain slope. Note that/x s is negative when the slope is self-shaded. We have implicitly constrained/x s to be nonnegative in the derivations of the mean and variance, which means that the Sun must be above the angle where shadows can occur. This will not affect the result

because for most landscapes the incoming beam is heavily attenuated when the Sun is low enough for shadowing to Occur.

Figure 1 plots the variance function of (7) normalized to (1/2)S• sin 2 S = 1 versus cos 00, for several optical depths along with the maxima function of (8).

DATA

The study site covers a 56-km 2 portion of the FIFE site including the Konza Prairie Long Term Ecological Research Site near Manhattan, Kansas. The Konza Prairie occupies the northwest 6 km x 6 km corner of the study site and is composed mainly of native tallgrass prairie vegetation. Man- agement within the prairie consists of controlled treatments

.--..

/oø

[ , ! T

15 20 25 30 Slope [deg]

, [

-100 0 100

Aspect [deg]

Fig. 3. Cumulative frequency distributions for elevation, slope, and aspect data obtained from the 25-, 50-, and 100-m resolution digital elevation models. Aspect is measured in degrees from south and increases positively to the east. Legend at top of figure applies to all three plots.

of burned and unburned, grazed and ungrazed areas in various annual rotations.

A digital elevation model (DEM) for the U.S. Geological Survey Swede Creek 7.5 arc min quadrangle was produced by the U.S. Army Corps of Engineers. Five-meter contours were digitized and then interpolated to a 25-m grid. This grid was subsampled by picking every second or fourth point to produce models with 50- or 100-m grid spacings, respectively. No aggregation or averaging was performed. Figure 2 shows a shaded-relief representation of the 25-m data. Ele- vations range from 306 m to 466 m with a mean of 395 m.

Slope and aspect data were derived by taking the directional derivative of the elevation data for each grid. The equations for slope and aspect are given by

tan S --- Vz{ = [(Oz/Ox) 2 + (c3z/Oy)2] 1/2 (9)

-az/Oy tan A = • (10)

-Oz/ax

DUBAYAH ET AL.'. TOPOGRAPHIC DISTRIBUTION OF CLEAR-SKY RADIATION 683

b

•o i 25 m , i ! ! ] . ]1 i

500 1000 1500 2000 0 500 1000 1500 2000

.>, 50m .>'

I i [ i i i [ i

0 500 1000 1500 2000 0 500 1000

50 m

! l

1500 2000

100 m

(:3

0 500 1000 1500 2000 0 500 1000 1500

lOO m

!

2000

Distance [m] Distance [m]

Fig. 4. (a) Aspect and (b) slope semivariograms at different grid spacings. The semivariance at a particular lag is given by each triangle. The solid lines represent the best fit exponential model through the points. Dashed vertical lines give the range of spatial dependence as determined from the exponential model.

where S is the slope, A the aspect, and z the elevation [Dozier and Frew, 1989]; 0z/0 x and Oz/Oy are calculated by finite differences. For a DEM with a grid spacing of A h, at point (i, j) on the grid,

Oz zi+ ]j- zi- ij (11) Ox 2Ah

OZ Zij+ l -- Zij-1

oy 2Ah (12) Note that the elevation zid is ignored in calculating slopes. Therefore slope is found in the x and y directions using data points whose centers are separated by twice the grid spacing.

The angle to the local horizon for 16 compass directions was also calculated using efficient numerical techniques [Dozier eta!., 1981; Dozier and Frew, 1989]. The horizon angles are necessary to determine shadowing effects and for

estimating what portions of the sky and terrain are visible to a point.

METHODS

Generating Radiation Images

Clear-sky radiation was simulated for three dates, Decem- ber 15, March 15, and June 15. On each date, 15 times were picked, corresponding to the intervals for Kronrod quadra- ture between sunrise and sunset [Piessens et al., 1983]. The two earliest and latest times on each day were excluded. At these times the Sun is very close to the horizon, and almost the entire site is in shadow, so that the amount of incoming radiation is small and mostly diffuse. The instantaneous global shortwave radiation (0.35-0.75 gin) and near-infrared radiation (0.75-2.8/xm) were calculated separately, resulting in 11 radiation images per day for each wavelength range. The shortwave and near-infrared images were summed to-

684 DUBAYAH ET AL.: TOPOGRAPHIC DISTRIBUTION OF CLEAR-SKY RADIATION

TABLE 2. Radiation Image Statistics

25 m 50 m Local

Time, 00,* Variance, Mean, Variance, CST deg Mean, W m -2 [W m-2] 2 W m -2 [W m-2] 2

100 m

Mean, Variance, W m -2 [W rn-212

December 15

0720 83.9 63 323 62 288 62 228 0754 78.8 153 1800 153 1449 153 945 0838 73.0 268 3303 269 2566 270 1656 0928 67.7 380 4070 380 3196 381 2102 1023 63.8 460 4278 461 3371 463 2292 1121 62.3 490 4203 491 3324 493 2317 1219 63.8 463 4073 464 3230 465 2267 1314 67.7 384 3757 385 2996 386 2062 1404 73.0 273 3022 274 2393 275 1616 1447 78.8 157 1657 157 1348 158 930 1522 83.9 65 314 65 277 65 232

March 15

0730 80.8 114 1187 114 973 114 667 0813 72.6 275 3521 276 2688 276 1598 0908 62.8 473 4625 475 3543 476 2130 1012 52.7 664 4495 665 3445 667 2133 1122 44.5 803 3577 805 2841 806 1849 1235 41.2 855 2888 857 2426 859 1691 1348 44.5 807 3370 808 2667 810 1815 1458 52.7 671 4128 672 3170 674 2080 1602 62.8 482 4294 483 3285 484 2083 1657 72.6 282 3 344 283 2514 284 1568 !740 80.8 119 1148 119 935 119 637

June 15

0604 86.2 132 1367 132 1108 132 717 0658 79.6 326 3584 326 2694 327 1623 0807 69.5 573 4277 574 3257 576 1910 0927 56.4 82! 3275 822 2489 824 1437 1054 41.0 1007 1577 1009 1211 1011 718 1226 25.0 1079 547 1080 535 1082 332 1358 41.0 1012 1384 1013 1! 10 1016 700 1526 56.4 829 3089 830 2354 832 1399 1645 69.5 583 4240 583 3182 586 1865 1754 79.6 334 3588 335 2683 336 1585 1848 86.2 137 1420 137 1147 138 698

CST is central standard time. *Solar zenith angle.

gether at each time to give the total instantaneous clear-sky radiation.

The wavelength-averaged parameters used in the simulations are summarized in Table 1. Optical depth, single- scattering albedo, and scattering asymmetry parameters were derived from theoretical studies by Dozier and Marks [1978] and measurements collected during FIFE [Sellers et al., 1988b]. These atmospheric parameters were held constant over the three dates. Seasonal substrate albedo estimates were derived from experimental radiometer measurements made over the prairie [Asrar et al., 1988; Irons et al., 1988; Sellers et al., 1988b].

It should be stressed that although these average values are realistic, they are not actual values for the particular days studied. However, we are more interested in describing relative spatial variation of radiation than in estimating actual radiation values for validation and calibration pur- poses. By holding atmospheric conditions constant we can isolate the interactions of topography, Sun position, and grid spacing.

Calculating Semivariograms

For each of the radiation images a subset of pixels was picked at random. For the 25-, 50-, and 100-m digital elevation models, 3000, 2000, and 1500 points were chosen, respectively. Initial analyses of the terrain and radiation data identified no significant anisotropic behavior. Therefore the isotropic semivariogram was calculated on the basis of all possible interpoint distances [Journel and Huijbregts, 1978; Oliver and Webster, 1986]. The total number of paired comparisons used to estimate each variogram was over 4,000,000, but the number of samples varied as a function of sample separation. For a lag of one pixel on the 25-m DEM there was an average of 400 samples, i.e., 400 pairs of points within 25 m of each other. At a lag of 1000 m the sample size was almost 10,000. The maximum lag was at 10,425 m. Even though the semivariograms were calculated to this extreme lag, we limited our analysis to a lag of 2000 m on all grids. Semivariograms for slope and aspect were calculated in a similar fashion.

DUBAYAH ET AL.i TOPOGRAPHIC DISTRIBUTION OF CLEAR-SKY RADIATION 685

25 rn

0.0 0.2 0.4 0.6 0.8 1.0

50 m

!

0.0 0.2 0.4 0.6 0.8 1.0

0.0 0.2 0.4 0.6 0.8 1.0

Cosine of Solar Zenith Angle

Fig. 5. Radiation variability of the constant-slope model as compared to radiation variability obtained from the two-stream model at each grid spacing. The solid line is the estimated variability (equation (7)) normalized to 1 for an optical depth of 0.2. The points represent the variance in radiation across the entire site, also normalized to 1, as simulated using the two-stream atmospheric model with the 25-, 50-, and 100-m elevation grids (see Table 2). The maximum variance for the constant-slope model occurs at a zenith angle whose cosine is 0.53 (58ø).

Each semivariogram was then fitted to a variety of para- metric curves using a least squares criterion [Caceci and Cacheris, 1984]. An exponential curve of the form

y(h) = C(1 - e -h/r) (13)

fit the data best, where y(h) is the semivariance at a distance h, C is the sill variance, and r is a distance parameter related to the range, a measure of spatial dependence. Since the sill is reached asymptotically in the exponential model, the range is not definite. However, other researchers have defined a working distance of a = 3r at which point the variance is approximately 95% of the sill value [Oliver and Webster, 1986].

Time-Dependent Spatial Variation

We are interested not only in how radiation varies across space for any particular instant but also in the time- dependent nature of this variation. Therefore we define an isotropic time-dependent semivariogram for a variable F as

3/(h, t)= «E{[F(x + h, t)- F(x, t)] 2} (14)

where t is a time variable and x is a position vector. This statistic is useful for summarizing the spatiotemporal char- acteristics of the data by effectively showing the behavior of the range and sill variance through time. The reason it has not been used previously for radiation is that while concep- tually straightforward it demands a considerable amount of computation.

RESULTS

Terrain Geomorphometry

Figure 3 shows cumulative frequency distributions of elevation, slope, and aspect for each DEM. The frequency distribution of elevation is unaffected by grid spacing. As- pects are distributed uniformly in all directions on each grid. Mean slopes are 6.0 ø, 5.2 ø, and 4.1 ø, and maximum slopes are 29 ø, 24 ø, and 18 ø for the 25-, 50-, and 100-m models, respectively. As grid spacing increases from 25 to 100 m the percentage of slopes less than 10 ø increases from 50% to 8O%.

Isotropic semivariograms for aspect and slope are shown in Figures 4a and 4b. Aspect semivariograms are very similar to each other. The sill variance decreases slightly at 100 m resolution. The range, as determined from least squares fit of an exponential semivariogram model, is 255 m for the 25-m grid, 300 m for the 50-m grid, and 363 m for the 100-m grid. In contrast, the slope semivariograms strongly depend on grid spacing. The range of spatial dependence for slopes is approximately 700 m on the 25- and 50-m grids. At 100-m grid spacing, the range increases to 1200 m. Increasing the grid spacing to 100 m also reduces the sill variance considerably.

Image Statistics

Table 2 gives summary statistics for each of the modeled radiation images. Mean radiation increases with decreasing zenith Sun angie and does not vary significantly with grid spacing. This is expected since the mean slope across the site changes only 1.8 ø from the 25- to 100-m grids.

Variance changes with both resolution and solar zenith angie. The maximum variance on all grids occurs at a zenith angle of about 60 ø . In December the Sun does not get this high, and the variance stays relatively constant from late morning to early afternoon. In March and June, however, the Sun reaches this angle in early morning and late afternoon with corresponding peaks in variance. Equation (8) estimates a maximum variance at a solar zenith angle of 58 ø for an optical depth of % = 0.2, near the peak variances observed in the simulated radiation data obtained from the two-stream model. Figure 5 shows the variance of the simulated radiation data, normalized to 1, plotted against the normalized variance function of (7) for each DEM grid.

The variance drops considerably with larger grid spacings.


TABLE 3. Parameters for Exponential Variogram Model, 7(h) = C(1 - e -h/r)

25 m 50 m 100 m

00,* C, r, u, C, r, u, C, r, deg [W m-2] • m % [W m-2] 2 m % [W m-2] 2 - m %

December 15

83.9 310 74 3.6 272 83 9.6 215 95 39.3 78.8 1787 67 3.4 1398 79 8.6 849 110 9.5 73.0 3254 69 3.2 2418 81 7.9 1515 116 10.4 67.7 3960 73 3.0 2995 83 5.9 1902 117 1!.7 63.8 4150 77 2.8 3141 88 3.9 2108 122 10.9 62.3 4061 84 2.1 3152 95 3.7 2188 126 9.7 63.8 4015 88 2.2 3147 100 4.7 2202 127 9.1 67.7 3804 87 2.5 3042 100 5.3 2066 128 9.5 73.0 3067 83 2.5 2491 98 5.5 1645 125 11.0 78.8 1688 8! 2.4 1404 96 5.5 942 120 1!.6 83.9 304 81 2.6 279 86 8.7 237 111 29.7

March 15

80.8 1192 64 3.2 984 83 7.0 644 95 17.1 72.6 3580 64 4.4 2658 81 11.1 1507 110 10.0 62.8 4707 65 4.3 3517 80 11.1 1964 108 9.6 52.7 4442 68 4.2 3317 80 10.2 1935 111 10.8 44.5 3512 74 3.2 2663 84 6.0 1673 117 12.6 41.2 3016 85 2.2 2324 95 4.0 1592 126 10.1 44.5 3390 87 2.5 2676 103 5.7 1809 129 10.0 52.7 4233 80 2.6 3364 98 6.0 2119 123 13.1 62.8 4471 74 2.6 3560 91 6.2 2148 115 15.7 72.6 3463 70 2.9 2724 86 7.4 1608 112 13.9 80.8 1194 72 2.4 1012 80 8.2 664 101 17.0

June 15

86.2 1412 68 2.7 1132 89 8.5 706 110 16.8 79.6 3692 68 3.2 2841 88 8.0 1606 ! 12 15.1 69.5 4465 67 3.5 3392 85 9.1 1855 110 13.7 56.4 3397 65 4.1 2553 81 11.0 1374 106 11.5 41.0 1551 66 4.5 1153 78 ! 1.9 652 105 11.2 25.0 621 88 3.0 478 94 4.9 332 120 10.2 41.0 1483 78 2.7 1170 95 6.8 701 119 13.7 56.4 3327 69 3.1 2593 85 8.0 1424 109 17.1 69.5 4465 66 3.9 3465 80 10.5 1865 102 16.7 79.6 3717 64 4.6 2881 77 12.4 1567 98 16.0 86.2 1460 62 4.5 1208 69 13.7 693 95 17.2

Here C is the sill variance, r is a distance parameter related to the range, a measure of spatial dependence, and u is the percent variance unexplained by fitted exponential model (i.e., the ratio of the residual sum of squares and the total sum of squares expressed as a percent).

*Solar zenith angle.

The drop in variance between grids agrees with that estimated by (7) when the mean slope for each grid is substituted for S. From (7) the variances for two terrains with mean slopes S• and S2 are related by

o-• sin 2 •1 .-.- .

cr 2 sin 2 •q2 (15) F, .

where o-•2• and cry22 are the radiation variances over each grid for a given Sun angle. To test whether this relation held for real terrain data, we calculated two linear least squares fits using the two-stream radiation variances for the 25-m grid as the independent variable and the two-stream radiation variance results for the 50- and 100-m grids, respectively, as the dependent variables. For each fit there were 33 observations, corresponding to the variances listed in Table 2. From (15) the slopes of these two regression lines should be equal to

sin 2 (5.2 ø) sin 2 (4.1 ø) = 0.76 = 0.47

sin 2 (6.0 ø) sin 2 (6.0 ø)

for the 50- and 100-m grids, respectively. Regression results yield factors of 0.75 (r 2 = 0.98) and 0.47 (r 2 - 0.94). It appears therefore that the reduction in variance with grid spacing for the two-stream results can be attributed to the reduction in mean slope for each grid.

Semivariogram Analysis

Table 3 summarizes the fitted exponential model parameters for the radiation semivariograms. Note that in this model, C, the sill variance, is also an estimate of the variance and therefore agrees closely with the variances listed in Table 2. Figure 6 is an example of four of the semivariograms and corresponding fitted exponential models for March 15 using the 25-m DEM. The semivariograms are very similar in form for most times. The range (a = 3r) is consistently between 200 m and 300 m on the 25-m data set. At 100-m grid spacing the range increases to between 300 m and 400 m, and the sill variance decreases because of decreased mean slope.

Time-dependent semivariograms are shown in Figures 7a, 7b, and 7c. The raw semivariance data were smoothed using


o O _ O

O O _ O

O -

AA A A A A A 9:08 AM

A A A 8:13 AM

A A

12:35 PM

7:30 AM

i I i i i

0 500 1000 1500 2000

Lag [m]

Fig. 6. Examples of radiation semivariograms for March 15. Each triangle is the semivariance at a particular lag as calculated from radiation images generated by the two-stream model using the 25-m DEM. Solid lines represent the best fit exponential model through the points. Times are central standard.

a nonlinear running median filter and interpolated to form a surface [Becker et al., 1988, pp. 586, 475]. These clearly show the March and June peaks in variance, the overall similarity in variogram form, and the reduction in variance with larger grid spacings.

DISCUSSION

Factors Affecting Radiation Variability

Slope. Our results suggest that for a given Sun angle and exoatmospheric irradiance, spatial variance in incoming solar radiation is a function of the mean slope of the terrain. In (7), radiation variance increases proportionally to sin 2 S. We derived this result from a constant-slope model which considers only direct solar irradiance, but the relationship seems to hold using the full two-stream atmospheric radiation model [Dubayah et al., 1989]. In addition, since the sill variance of a semivariogram is an estimate of the variance, the height of the sill also should vary as sin 2 S.

The variance of the simulated two-stream data is greater than would be estimated from (7). The primary reason for this is that there is no variance in the slope distribution of a constant-slope model. Real terrains always have slope variability that must necessarily contribute to radiation variability. Exactly how these are related is a subject for further research.

Optical depth. For a terrain with a constant slope and albedo, and a uniform distribution of aspects, time- dependent spatial variability in radiation can be estimated as a function of optical depth. Variance decreases as optical depth increases. This reflects the greater contribution of diffuse sky irradiance, which is isotropic in the two-stream model and therefore independent of terrain orientation. In

addition, the solar angle at which variance is a maximum depends exclusively on optical depth.

At larger optical depths the maximum variance occurs at smaller solar zenith angles. When the Sun is low in the sky, the attenuation due to the atmosphere is large enough that the difference between shadowed and sunlit slopes is less than occurs when slopes are differentially illuminated by more intense radiation. As the Sun reaches a certain angle, the variance between slopes is maximized and then decreases as the Sun continues higher in the sky. There is a trade-off between shading and atmospheric attenuation. If the Sun is too low, the increased effects of shading are offset by increased attenuation. If the Sun is too high, the decreased attenuation is offset by decreased shading. As optical depth increases, the increased attenuation means that the angle at which the variance is maximal must decrease to achieve the same illumination.

Grid spacing. Spatial variability in modeled radiation depends critically on DEM grid spacing. There are two main effects: (1) radiation variance decreases with increasing grid spacing due to decreasing average slope; and (2) radiation spatial autocorrelation increases with increasing grid spacing. Recall that our method of producing elevation models with coarser grid spacing drops out elevation points, sampling every second point for the 50-m DEM and every third point for the 100-m DEM. When a coarser sampling interval is used, an aliasing effect is introduced: terrain features with spatial wavelengths shorter than twice the sampling distance (grid spacing) are aliased as longer-wavelength features [Ternpill, 1980; Castleman, 1979, p. 230]. Specifically, slopes with a wavelength shorter than 200 m (for example, ridge to valley bottom to ridge distance) cannot be represented by a 100-m DEM. In addition, our method of obtaining slopes


25 m 50 m

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Fig. 7. Time-dependent semivariograms of clear-sky solar irradiance as simulated with the two-stream model for the different DEM data: (a) 25-m grid spacing, (b) 50-m grid spacing, and (c) 100-m grid spacing. Note the midday drop in semivariance in March and June, and the overall decrease in semivariance with grid spacing. The decrease in semivariance with increased grid spacing is related to a corresponding decrease in mean slope.

introduces a further sampling effect since data points used to calculate a slope are separated by twice the grid spacing. For the Konza Prairie the net result of these operations is an elimination of shorter slopes and an increase in longer, less steep ones, as shown in the cumulative frequency distribution for slope. This causes the mean slope to decrease and leads to a corresponding decrease in radiation variance.

The spatial autocorrelation in radiation is primarily a function of the spatial autocorrelation in direct solar irradiance. For a given optical depth this is a function of slope, aspect, and solar position. Since the spatial autocorrelation in both slope and aspect changes with grid resolution, it will change for radiation as well. For most terrains the spatial autocorrelation in radiation should be dominated by the

autocorrelation in aspect. This is true for our study site, where at all grid spacings the radiation semivariograms are much more similar in shape and range of spatial dependence to aspect semivariograms than to slope semivariograms. In general changes in aspect over short distances can be much larger than the corresponding changes in slope. For example, the 25-m slope and aspect semivariograms show that the average change in slope at a distance of 250 m is 3.5 ø, while the average change in aspect is over 70 ø .

We should note that the importance of DEM grid spacing as a factor in radiation modeling depends on the spatial scales of interest, the particular geomorphology of the terrain, and the methods used to create the elevation, slope, and aspect grids. If a different method of elevation grid


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Fig. 7. (continued)

interpolation were used, perhaps by averaging 25-m DEM data within a 100-m 2 area to produce a 100-m grid, or if slopes and aspects were calculated using more or fewer elevation points, the resulting terrain geomorphometry would be different.

Other factors. The agreement between the variance of simulated radiation data and that estimated by the constant- slope model may not seem surprising given that the constant- slope model is derived from the two-stream model. We find it noteworthy that such an extensive simplification leads to an equation that describes radiation variability as simulated by the more realistic two-stream model. However, there are factors not included in the simplified model that can contribute to radiation variability. The effect of nonconstant slope distribution on variance was previously discussed. While such a distribution would lead to a greater variance in

radiation, simulations here and on other terrains suggest it has a limited effect on the solar zenith angle at which variance is maximized [Dltbayah et al., 1989]. For rugged terrain there may also be considerable variability in diffuse sky irradiance because of variation in the sky view factor, for example. between the top of a ridge and a valley bottom. Likewise, if there is significant variability in albedo such as that caused by intermittent snow cover, the variability in reflected direct and diffuse terrain radiation could also be

important. In general, such variability will contribute to the total variation across a region. If the terrain has a strong spatial alignment, i.e., nonuniform distribution of aspects, the variance will then depend not only on slope, optical depth, and zenith Sun angle but also on solar azimuth, for example, when the Sun is aligned along or across a directional trend in a ridge and valley system. Furthermore, this also could cause the solar zenith angle at which the variance peaks to shift.

CONCLUSION

We have presented a method for exploring the topographic modulation of clear-sky incoming radiation. A two-stream atmospheric radiation model together with digital elevation data can be used to obtain the temporal and spatial distribution of this energy. Using this method for the Konza Prairie, we found that the variance and spatial autocorrelation of the simulated radiation data changed with Sun angle and elevation grid spacing. As grid spacing increased, variance decreased, and spatial autocorrelation increased. Variance was largest at a particular solar zenith angle (about 60ø), independent of grid spacing.

An analytical expression describing the behavior of the variance as a function of Sun angle, optical depth, and mean terrain slope can be derived by considering direct solar radiation variability on a constant-slope terrain having a uniform albedo and a uniform distribution of slopes. For a given zenith Sun angle, optical depth, and exoatmospheric irradiance, variance is scaled as a function of slope. Further- more, the solar zenith angle at which variance is maximized is determined by the optical depth. Spatial autocorrelation in simulated two-stream radiation is mainly a function of aspect autocorrelation. We attribute the decrease in mean slope and the increase in slope and aspect autocorrelation to a loss of some of the shorter-wavelength terrain features at coarser grid spacings.

Further work is needed to relate the spatial distribution of radiation to terrain variability. How does slope and aspect variability control radiation variability? If we can find appro- priate measures of terrain variability as related to actual radiation variances, then it might be possible to predict the semivariogram a priori from terrain considerations, without having to model the radiation explicitly. For a given type of terrain (alpine, prairie, coastal, etc.) a few elevation transects, perhaps obtained from contour maps, might be enough to characterize the terrain variability of these landscapes (for example, by providing estimates of mean slope, autocorrelation, variance, and frequency distributions of slope and aspect). These could then be used to develop an efficient sampling and stratification scheme with a minimum of radiation modeling and without obtaining a complete digital elevation model.

690 DUBAYAH ET AL.: TOPOGRAPHIC DISTRIBUTION OF CLEAR-SKY RADIATION

Acknowledgments. This work benefited from discussions with David Simonnett, Michael Goodchild, Jim Frew, and Sud Menon and especially from the comments of the anonymous reviewers. Many thanks also go to Dave Lawson and Kirsten Zecher for their help in figure preparation. This research was supported by NASA grant NAG 5-917.

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(Received January 13, 1989; revised September 29, 1989; accepted October 2, 1989.)

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Topographic Distribution of Clear-Sky Radiation · assess digital elevation grid spacing as a factor in radiation modeling. Concern over increasing human impacts on regional and global