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A comparison of methods to estimate daily global solar irradiation from other climatic variables on the Canadian prairies
Pergamon
0038-092X(95)00100-X
Solar Energy Vol. 56, No. 3, pp. 213-224, 1996 Copyright 0 1996 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0038-092X/96 $15.00+0.00
A COMPARISON OF METHODS TO ESTIMATE DAILY GLOBAL SOLAR IRRADIATION FROM OTHER CLIMATIC VARIABLES ON THE CANADIAN PRAIRIES ALAN Agriculture
G. BARR,* and Agri-Food
S. M. MCGINN Canada
Research
(Communicated
and SI BING CHENG+
Centre, Lethbridge,
Canada
TlJ 4Bl
by AMOS ZEMEL)
Abstract-Historic estimates of daily global solar irradiation are often required for climatic impact studies. Regression equations with daily global solar irradiation, H, as the dependent variable and other climatic variables as the independent variables provide a practical way to estimate H at locations where it is not measured. They may also have potential to estimate H before 1953, the year of the first routine H measurements in Canada. This study compares several regression equations for calculating H on the Canadian prairies. Simple linear regression with daily bright sunshine duration as the dependent variable accounted for 90% of the variation of H in summer and 75% of the variation of H in winter. Linear regression with the daily air temperature range as the dependent variable accounted for 45% of the variation of H in summer status (wet or dry) as and only 6% of the variation of H in winter. Linear regression with precipitation the dependent variable accounted for only 35% of the summer-time variation in H, but stratifying other regression analyses into wet and dry days reduced their root-mean-squared errors. For periods with sufficiently dense bright sunshine observations (i.e. after 1960), however, H was more accurately estimated from spatially interpolated bright sunshine duration than from locally observed air temperature range or precipitation status. The daily air temperature range and precipitation status may have utility for estimating H for periods before 1953, when they are the only widely available climatic data on the Canadian prairies. Between 1953 and 1989, a period of large climatic variation, the regression coefficients did not vary significantly between contrasting years with cool-wet, intermediate and warm-dry summers. They should apply equally well earlier in the century.
1. INTRODUCTION
Climatic impact studies in agronomy, hydrology and ecology are often limited by a lack of longterm solar irradiation data. Daily global solar irradiation (H) is required for many simulation models of crop growth and productivity (Ritchie and Otter, 1985; Williams et al., 1989; Amir and Sinclair, 1991) and evapotranspiration (Penman, 1948; Priestley and Taylor, 1972). However, H observations are sparse and relatively recent. Spatial interpolation is not feasible because the observations are sparse. Estimation by remote sensing is possible at high spatial and temporal resolution (Shmetz, 1989; Noia et al., 1993a, b), but only during the recent past. Efforts to estimate H with physically based models (Davies and McKay, 1982; Davies et al., 1984) have increased the density of historic H estimates, but the data requirements of these models still limit the spatial and historic coverage. A need exists to extend the H database *Atmospheric Environment Service, National Hydrological Research Centre, 11 Innovation Boulevard, Saskatoon, SK, Canada S7N 3H5. ‘Soil and Fertilizer Institute, Hebei Academy of Agricultural and Forestry Science, Shijiazhuang, Hebei, P.R. China. 213
backwards in time and to increase its spatial resolution. Statistical relationships between H and other climatic variables provide a practical way to calculate H at locations where it was not measured and to enlarge the database of H estimates in time and space. The use of a regression analysis to estimate daily H at locations with long time series of historic climatic data is attractive because it preserves the integrity of the observed time series and maintains the observed interannual variations. Methods which generate synthetic data using statistical methods (e.g. Richardson, 1981) may not adequately reproduce long-term variability and the interannual persistence of extreme climatic events such as drought. Global solar irradiation is highly correlated with surface daily climatic variables, including bright sunshine duration, S (Soler, 1990; Boisvert et al., 1990); cloud fraction, C (Norris, 1968; Davies and Hay, 1980); air temperature range, dT (Bristow and Campbell, 1984; De Jong and Stewart, 1993); precipitation (De Jong and Stewart, 1993); and precipitation status, P (McCaskill, 1990). Regression equations based on these variables have a strong physical basis.
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Day-time warming is driven in part by H, via the influence of H on the sensible heat budget of the lower atmosphere. Precipitation indicates cloudiness. Global solar irradiation is most highly correlated with S, as rz usually exceeds 0.85 for summer months. Although H is less highly correlated with ATand P, the correlations are strong enough to support the use of regression equations for estimating H. Using preradiation, cipitation, extraterrestrial pan evaporation and temperature range in a multiple regression equation accounted for 78% of the variation in H at one location in Georgia (Hook and McClendon, 1993). A non-linear relationship with ATaccounted for between 70 and 90% of the summer-time variation in H for the U.S. Pacific northwest (Bristow and Campbell, 1984). De Jong and Stewart (1993) found that a polynomial with AT and P accounted 54% of the summer-time variation in H on the Canadian prairies. McCaskill (1990) reported that H was significantly, negatively correlated with the precipitation status of the previous, current and following days in Australia. Because daily observations of AT and P are available at a much higher spatial density and for a longer historic period than S, it is important to explore their utility for calculating H. For instance, after excluding climatic stations with more than 20% of daily observations missing between 1960 and 1989, the climatic archive of the Atmospheric Environment Service of Canada contains 173 Canadian prairie locations with precipitation, 118 locations with maximum and minimum temperature, 24 locations with bright sunshine duration, and five locations with H observations. Regression equations which estimate H from other climatic variables have the potential to estimate H before 1953, the year of the first routine H observations. Daily temperature and precipitation and, to a lesser extent, bright sunshine duration, are available at many Canadian prairie locations between 1920 and 1953, and the daily climatic record at a few locations extends continuously back to the last century. However, because regression equations must be calibrated with recent data, they may not apply to earlier decades. A long-term variation in the regression coefficients could result, for instance, from changes in atmospheric aerosols (for S) or climatic warming [given the observed reduction in AT which has accompanied observations of warming (Skinner and Gullet, 1993)]. Before regression equations based on post-1953 data
can be applied to the pre-1953 period, the longterm stability of the regression coefficients must be evaluated. Estimates of H are often required at locations with no climatic data. Spatial interpolation of observed H is difficult because of low measurement density. Suckling (1985) reported relative errors of 33% (summer) and 48% (winter) when H was interpolated over distances of 100 km and 53% (summer) and 76% (winter) when H was interpolated over distances of 300 km. Canada has only sparse observations of H, and spatially interpolated estimates of H are inaccurate. However, H may be estimated at locations which lack climatic observations by first interpolating other climatic variables and then calculating H from the interpolated values. This study: (1) explored the utility of regression equations for calculating H from other climatic data on the Canadian prairies; (2) assessed the accuracy of H estimates from spatially interpolated climatic data; and (3) assessed the stability of the regression coefficients over time. 2. METHODOLOGY
2.1. Climatic data Historic hourly and daily climatic data were obtained for several sites in Alberta, and Manitoba during the Saskatchewan 1953-1989 period. Table 1 summarizes the site locations and their May-to-September climatic averages. Four sites had hourly data and seven sites had daily data. Daily global solar irradiation was expressed as H/Ho, the ratio of observed H to the value H, calculated for a horizontal plane at the top of the atmosphere (Iqbal, 1983). The H/Ho ratio varies between 0.0 and 1.0. Using H/H,-, rather than H in regression analysis made it possible to assess whether regression coefficients were independent of site and time of year. Daily bright sunshine duration was also scaled from 0.0 to 1.0, either as the ratio of actual to possible bright sunshine hours S/S, [where S is the observed bright sunshine duration and S, is the day length (Iqbal, 1983)] or as a weighted average S, of hourly observed bright sunshine duration s’: SW= z SiHi/Ho
(1)
n=l
where Hi/Ho is the fraction which hour i contributed to H,. No attempt was made to correct S
A comparison of methods to estimate daily global solar irradiation
215
Table 1. Climatic stations, including location, elevation, period of record for global solar irradiation and climatic normals for the period of record Period of record Location Beaverlodge CDA, Alberta Edmonton Municipal A, Alberta Edmonton Stony Plain, Alberta Suffield A, Alberta Bad Lake, Saskatchewan Swift Current CDA, Saskatchewan The Pas A, Manitoba Winnipeg Intl A, Manitoba
Latitude (“N)
Longitude (“W)
Elevation (m)
Daily data3
55.20
119.40
144
53.57
113.52
53.53 50.27
May-to-September
Hourly data4
daily meant
K (“C)
S (h)
H (MJ m-*)
1960-1990
12.4
8.4
18.3
668
1957-1989
14.5
8.7
19.6
114.10 111.18
765 768
1959-1990
13.6 15.9
8.6 9.3
18.8 20.1
51.32
108.42
637
1971-1986
50.27
107.73
823
1960-1989
1960-1989
14.9
8.6
20.3
53.97
101.10
268
1972-1990
1972-1989
13.5
8.5
17.9
49.90
97.23
238
1957-1990
1957-1989
15.8
8.9
19.7
1967-1989
20.1
TAveraged for the period of record. IDaily data were obtained from the Atmospheric Environment Service climatic archives. §Hourly data were obtained from the Canadian Weather and Engineering Data Set.
or S’ for the limitations of bright sunshine recorders at low solar elevations (Iqbal, 1983). The daily cloud fraction C was estimated as a simple average of day-time hourly observations, or as a weighted average C, of the hourly cloud fraction C’:
C,,,= E C'H'IH,.
maximum (TX) and minimum (T.) air temperatures. A second estimate (7’.‘ -. T,,) (Bristow and Campbell, 1984) used daily TXand the average T,from the same and following days (T,,). Other estimates were based on hourly values from 04.00, 16.00 and 24.00 LST. An analysis of Mayto-September showed that T,and TX occurred most frequently at 04.00 to 05.00 LST and 14.00 to 17.00 LST, respectively, with a shift to earlier T,, and later TXas H/H, increased (Fig. 1). Daytime warming was estimated as T16-T4,nighttime cooling- was estimated as T& - Tz4 and
(2)
n=l
The daily temperature variation ATwas estimated from hourly and daily measurements. The daily temperature range TX-T,used daily (a) H/H, c 0.2
(b) H/H, > 0.7
T
I
a-
1
4-
-8 0
4
8
12
18
20
24
0
4
8
12
16
20
24
Hour (LST) Fig. 1. The diurnal variation in surface air temperature at Winnipeg International Airport, June 1953-1989, for two classes of daily global solar irradiation. Temperature (7”) is offset to a daily mean of 0.0, and the error bars denote one standard deviation from the hourly mean.
216
A. G. Barr et al.
T16-(T4 + T,,)/2 combined the two. (The subscripts denote the hour in LST.) The daily precipitation status P was set to 0 (dry) when daily total precipitation was less than 0.2 mm and 1 (wet) when daily total precipitation was greater than or equal to 0.2 mm.
Table 3 summarizes the groupings of independent variables. Based on a preliminary assessment of several relationships, an inverse-square relationship was used for the H/H, versus C (and H/H,, versus C,) regression analyses: HJHo=a+bCp2
2.2. Regression analyses Simple linear [eqn (3)], multiple linear [eqn (4)] and non-linear regression analyses [eqns (5)-( 7)] used the least-squares method (Genstat, 1987; SAS, 1989). The simple linear regression analysis was: H/H0 = a0 + a,x
Several non-linear relationships were evaluated for use in the S/S, versus d T non-linear regression analysis, including logistic, exponential and Gompertz equations (Genstat, 1987) and the Bristow and Campbell (1984) equation. A preliminary test showed that the Bristow-Campbell equation gave the highest r2. The Bristow-Campbell equation is:
(3)
where a, and a, are the linear regression coefficients and x is the independent variable. See Table 2 for a list of independent variables used in this study. In some cases, data were stratified into wet (P= 1) and dry (P=O) days, and wetand dry-day regression analyses were performed separately. The multiple linear regression analysis was: H/H0 = b, + blx, + bzxz... + b,x,
H/H0 = a( 1- exp (- b( TX- T,,“)‘))
(4)
H/H, = a( 1 - exp (- b(d + T16- ( T4 + $,)/2)‘))
Table 2. Simple linear regression coefficients [a0 and a,, eqn (3)], coefficients of determination (r*) and root-meansquared errors (rmse) for regression of several climatic variables versus daily solar irradiation (expressed as H/H,), averaged for May-to-September at four locations on the Canadian prairies (Table 1, stations with hourly data)
(7) in
$0 K-T, K-T,, P TX---T,(P = 1) TX--T, (P = 0) TN--T, T&s- 7-L &-(T4+ WI2
a,,
aI
rz
0.213 0.221 0.200 0.179 0.635 0.132 0.443 0.311 0.292 0.249
0.510 0.556 0.0283 0.0299 -0.224 0.0294 0.0138 0.0237 0.0317 0.333
0.90 0.89 0.48 0.44 0.35 0.47 0.22 0.51 0.41 0.57
order
to
permit
negative
values
of
T16-(T4 + T,,)/2. The value of d in eqn (7) was
fixed at the mean estimate of 5.0 in order to avoid occasional convergence problems when solving for the coefficients. Because the regression coefficients for all equations were dependent on time of year, they were estimated monthly. When applying the regression equations, monthly coefficients were assumed to represent the midpoint of each month. The significance of geographic, monthly, and among regression inter-annual differences coefficients was assessed by the analysis of
rmse variable
(6)
where a, b and c are empirical coefficients. In applying eqn (6), Bristow and Campbell (1984) first conditioned TX-- T,, by multiplying wetday values of TX- T,, by a factor 0.75. We did not find this to be useful on the Canadian prairies. We substituted T,, for T,, in eqn (6) because this increased the r2 value. In addition, eqn (6) was modified by adding the variable d:
where b,,-b, are the multiple regression coefficients and x1-x, are the independent variables.
Independent
(5)
-mean
0.103 0.110 0.236 0.248 0.264 0.297 0.152 0.23 1 0.252 0.215
Table 3. Multiple linear regression coefficients [b, to b,, eqn (4)], coefficients of determination (R2) and root-mean-squared errors (rmse) for regressions of daily climatic variables versus daily solar irradiation (expressed as H/H,), averaged for May-to-September at four locations on the Canadian prairies (Table 1, stations with hourly data). The blank cells indicate that the independent variable was not included in the analysis b, to b, Independent
variables
SIS,T, - T#.p T,--T,, P %-+‘I,+ T24)/2r P P, p, P,
b.
S/S,
AT
0.218 0.328
0.498
0.00377 0.0220 0.027 1
0.649
PO
P
- 0.0376
- 0.0234 -0.125 -0.102 -0.199
P,
-0.0231
R2
rmsefmean
0.90 0.58 0.63 0.36
0.106 0.211 0.199 0.261
A comparison
of methods
to estimate
covariance. The use of regional and seasonal versus local and monthly regression coefficients was also assessed by “jackknifing”. In the jackknifing test, the local regression coefficients were estimated using only data from neighbouring locations. The coefficients were estimated either prairie-wide or by inverse-distance-squaredweighted interpolation of the four nearest estimates. The accuracy of estimated H (fi) was calculated as the root-mean-squared (rms) error of estimate: Jc
daily global
point and the neighbouring observation. Nearest neighbours were chosen by distance only, with no attempt made to select neighbours in all directions. It was not critical to select neighbours in all directions because the station distribution was not clumped (Bill Hogg, Atmospheric Environment Service, Downsview, personal communication, 1994). Kriging used a daily semivariogram fitted by a spherical model (Isaaks and Srivastava, 1989). 3. RESULTS
(fi - H)‘/n.
The rms error was further subdivided into mean bias, regression and random components using the method of Allen and Racktoe (1981). The bias component determined the fraction of the rms error due to differences in mean H and fi. The regression component determined the fraction of the rms error due to departures of the H and fi data from a one-to-one relation. The random component determined the fraction of the rms error due to random variations in H and fi. The long-term stability of the H/H,, versus S/S, and H/H, versus TX- T, regression coefficients was assessed by analysis of covariance of: (a) inter-annual variation; and (b) three contrasting 1953-1989 year groupings. The year groupings were based on the May-to-September climate (Jones, 1991): (1) cool-wet (1954-1956, 1961, 1964, 1967-1968, 1977, 1980-1981, 1984 and 1988-1989); (2) warm-dry (1957, 1965-1966, 1971-1972, 1975-1976, 1983 and 1986); and (3) intermediate (1963, 1958-1960, 1962-1963, 1969-1970, 1973-1974, 1978-1979, 1983, 1985 and 1987). 2.3. Spatial interpolation We also evaluated global solar irradiation estimates from spatially interpolated values of S/S, and TX- T,. The S/S, and TX- T. variables were chosen because they were dense enough for spatial interpolation and had proven utility for estimating H. The simple linear regression coefficients for the spatially interpolated values were estimated in two ways, either prairie-wide (after excluding the location in question) or by inverse-distance-squared weighted interpolation of the four nearest estimates. Two methods of spatial interpolation were compared: inverse-distance-squared weighting and kriging. Distance-weighting used eight nearest neighbour values, weighted by the inverse distance squared between the interpolation
217
solar irradiation
AND DISCUSSION
3.1. Regression analyses
3.1.1. Bright sunshine duration. Table 2 summarizes the results of simple linear regression analyses for May-to-September at four Canadian prairie locations (stations with hourly data, Table 1). Table 3 summarizes the results of multiple linear regression analyses for the same period and locations. The weighted daily bright sunshine duration S, accounted for 90% of the variation in H/Ho, a slight improvement over S/S, (r2 =0.89) (Table 2). The corresponding rms errors were 10.3% for S, and 11.0% for S/S,. Adding AT and P to S/S, or S, in a multiple linear regression analysis, or stratifying by precipitation status, resulted in a slightly higher coefficient of determination and smaller rms error (Table 3). A similar small increase in R2 was found when C or C, were added to S/S0 or S, in a multiple linear regression analysis. The H/H, versus S/S,, regression coefficients varied considerably but consistently with season (Table 4, Fig. 2) and weakly with location (Table 5). The greatest differences occurred between winter and summer. The mean prairiewide regression coefficients, for the periods of Table 4.. Simple linear regression coefficients for daily values of H/H, versus S/S,, by month, averaged for seven locations on the Canadian prairies (Table 1, stations with daily data) Month Jan. Feb. Mar. Apr. May Jun. JUl. Aug. Sep. Oct. Nov. Dec.
a0
aI
rz
0.377 0.390 0.364 0.291 0.241 0.220 0.225 0.223 0.228 0.244 0.280 0.334
0.396 0.409 0.490 0.515 0.545 0.565 0.535 0.528 0.564 0.505 0.453 0.407
0.73 0.77 0.78 0.84 0.89 0.90 0.88 0.89 0.89 0.86 0.77 0.69
rmse/mean 0.161 0.140 0.147 0.126 0.106 0.093 0.100 0.098 0.125 0.141 0.189 0.190
A. G. Barr et al.
218
(a) January
(b) June 1.0
0.8
0.6
0
$
0.4
.:-.
0.2
0.2 - ”
0.0 0.0
0.2
0.4
0.6
0.8
1.0
0.0
0.2
I
/
0.4
0.6
’
I
0.8
’
1.0
S/S, Fig. 2. Daily global solar irradiation fraction versus bright sunshine duration fraction for January and June 1953-1989, at four Canadian prairie locations (Table 1, stations with hourly data). The line was fit by least-squares simple linear regression. Table 5. Simple linear regression coefficients for daily values of H/H, versus S/S,, by location, for the month of June Location Beaverlodge CDA Edmonton Municipal Suffield A Bad Lake Swift Current CDA The Pas A Winnipeg Intl A
A
a0
al
12
0.211 0.224 0.232 0.233 0.261 0.192 0.201
0.0579 0.0585 0.0549 0.0538 0.0522 0.0582 0.0573
0.91 0.88 0.90 0.91 0.88 0.92 0.92
November-to-March and May-to-September, respectively, were: 0.35 and 0.22 for a,; 0.43 and 0.56 for a,; and 0.75 and 0.89 for r2. Analysis of covariance showed that the differences in the May-to-September regression coefficients among locations and months were statistically significant in some cases. However, the differences were small, and the use of a regional versus local or monthly versus seasonal (Mayto-September) regression analysis did not appreciably alter a,,, aI, r2 or the rms of estimate. For the month of June at seven locations (stations with daily data, Table l), the mean rms error was 9.8% when a, and aI were estimated locally, 10.1% when a, and a, were estimated regionally and 10.2% when a,, and aI were estimated locally by interpolation of the nearest neighbour estimates. The Canadian, prairie-wide, May-toSeptember a0 and a, estimates of 0.22 and 0.56 were similar to the estimates of 0.23 and 0.54 obtained for European locations between 39
and 60”N (Soler, 1990). However, the Canadianprairie, November-to-March a, and a, estimates of 0.35 and 0.43 differed from the European estimates of 0.19 and 0.63 (Soler, 1990). The Canadian winter estimates for a, and a, of Halouani et al. (1993) and Boisvert et al. (1990) were similar to those of Soler (1990) for locations outside the prairie region, but similar to this study for prairie locations. It is possible that the winter estimates of a, and a, were affected more by continentality than latitude. If, for instance, continentality influenced the diurnal timing of cloud cover, it would have altered a, and a,. Simple linear regression with S/S, performed nearly as well as the more detailed MAC3 model (Davies and McKay, 1982). The mean annual MAC3 rms errors at Winnipeg were 11.1% (Davies and McKay, 1982) compared with 12.3% in this study when the simple linear regression coefficients were estimated locally, and 12.6% when the coefficients were estimated regionally based on all locations excluding Winnipeg. 3.1.2. Cloudfraction. Figure 3 shows the relationship between daily solar irradiation (expressed as H/Ho) and the weighted cloud fraction C,. Equation (5) with C, as the independent variable accounted for 74% of the variation in H/Ho between May-and-September at four locations (Table 1, hourly stations), a small improvement over eqn (5) with C as the independent variable (r2 = 0.71) (Table 6). The
A comparison
of methods
to estimate
daily global
219
solar irradiation
Table 8. Simple linear regression coefficients for daily values of H/H, versus TX- T., by location, for the month of June a0
4,
r2
0.075 0.102 0.174 0.177 0.117 0.109 0.126
0.0368 0.0406 0.0273 0.0270 0.0342 0.0351 0.0325
0.59 0.53 0.37 0.47 0.56 0.43 0.41
Location Beaverlodge CDA Edmonton Municipal Suffield A Bad Lake Swift Current CDA The Pas A Winnipeg Intl A
0.0
-1 0.0
0.2
0.4
0.6
0.6
1.0
cw Fig. 3. Daily global solar irradiation fraction versus weighted cloud fraction [eqn (2)] for June 1953-1989, at four Canadian prairie locations (Table 1, stations with hourly data). The line was fit by eqn (5). Table 6. Regression coefficients (a and b), coefficients of determination (r’) and root-mean-squared errors (rmse) for eqn (5), averaged for May-to-September at four locations on the Canadian prairies (Table 1, stations with hourly data) Independent variable
G C-I
a
b
rz
0.761 0.747
- 0.470 -0.521
0.74 0.71
rmse/mean 0.165 0.176
corresponding rms errors were 16.5% (for C,) and 17.6% (for C). These results were intermediate between those for S/S, and dT(Table 3). A similar analysis by Norris (1968) confirmed that estimates of H from S were more accurate than estimates of H from C. 3.1.3. Temperature range. Table 2 and Tables 7-12, and Figs 4-6 show the results of regression analyses based on the daily temperature variation. The use of TX- T,, rather than TX- T. did Table 7. Simple linear regression coefficients for daily values of H/Ho versus TX- T, by month, averaged for seven locations on the Canadian prairies (Table 1, stations with daily data) Month
al
r2
rmse/mean
0.0000 0.0074 0.0154 0.0188 0.0276 0.0312 0.029 1 0.0268 0.0273 0.0227 0.0151 0.0011
0.00 0.04 0.13 0.23 0.43 0.46 0.43 0.45 0.48 0.34 0.13 0.000
0.300 0.273 0.276 0.276 0.245 0.232 0.204 0.219 0.276 0.312 0.376 0.340
A
Table 9. Coefficients of determination (r’) and root-meansquared errors (rmse) for the Bristow-Campbell equation, averaged for May-to-September at four locations on the Canadian prairies (Table 1, stations with hourly data) Regression equation
Independent variables c--T, Tr.s- (T4 + T24)/2 = 1)
TX-T,(P TX-T,(P
= 0)
(6) (7)
0.56 0.63
0.213 0.195
Equation Equation
(6) (6)
0.58 0.30
0.205 0.171
b
c
rz
0.701 0.683 0.688 0.685 0.732
0.0493 0.0328 0.0336 0.0358 0.0699
1.48 1.64 1.67 1.61 1.20
0.53 0.57 0.56 0.57 0.56
Table 11. Regression coefficients four locations on the Canadian with hourly Month May
Jun. Jul. Aug. Sep.
for eqn (6) by month, for prairies (Table 1, stations data)
a
Month May
rmse/mean
Equation Equation
Table 10. Regression coefficients four locations on the Canadian with hourly
Jun. Jul. Aug. Sep.
rz
for eqn (7) by month, for prairies (Table 1, stations data)
a
b
c
dt
r2
0.712 0.708 0.716 0.700 0.693
0.0111 0.0046 0.0104 0.0069 0.0095
1.91 2.02 1.95 2.11 1.98
5.00 5.00 5.00 5.00 5.00
0.60 0.63 0.60 0.64 0.64
td was fixed at a value of 5.00. Jan. Feb. Mar. Apr. May Jun. Jul. Aug. Sep. Oct. Nov. Dec.
0.516 0.488 0.427 0.338 0.178 0.143 0.186 0.187 0.156 0.214 0.300 0.454
Table 12. Regression coefficients for eqn (6), stratified by precipitation status, for June at four locations on the Canadian prairies (Table 1, stations with hourly data) Precipitation status Wet(P = 1) Dry (P = 0)
a
b
c
r2
0.622 0.688
0.049 0.061
1.44 1.52
0.58 0.29
220
A. G. Barr et al.
(a) January
(b) June
0.8 0.6
0.6
0.4
0.4
0 B
0.2
0
I
I
I
I
5
10
15
20
0.0 25
i 0
5
10
15
20
25
T, - T, WI Fig. 4. Daily global solar irradiation versus temperature range for January and June 1953-1989, at four Canadian prairie locations (Table 1, stations with hourly data). The line in (b) was fit by eqn (6).
regional versus local regression coefficients did not appreciably alter a,, a,, r2 or the rms error. Unlike S/S,, TX- T, accounted for almost none of the November-to-March variation in H/H, ( r2 = 0.06) (Fig. 4, Table 7). Global solar irradiation apparently contributed more significantly to the warming of the lower atmosphere in summer than winter on the Canadian prairies. The estimates of day-time warming from hourly temperature observations (7& - T4 and T16- ( T4 + T2J/2) accounted for a higher fraction of the observed variation in H/H, than did TX- T, (Table 2, Figs 4 and 5). In order of increasing r2 values, TX--T, (r2=0.48)
0.8 i 0.6 1
0.0
11 -5
0 7;~&+
5
10 T&2
15
20
WI
Fig. 5. Daily global solar irradiation versus diurnal warming for June 1953-1989, at four Canadian prairie locations (Table 1, stations with hourly data). The line was fit by eqn (7).
not improve the simple linear regression estimates of H (Table 2). The mean r2 values were 0.48 (for TX- T,) and 0.46 (for TX-- T,,) for Mayto-September at four locations (Table 1, hourly stations). The mean May-to-September rms error was 23.6%. Regression coefficients varied significantly but subtly among months for Mayto-September (Table 7). Regional differences were greater among the H/H, versus TX- T, regression coefficients (Table 8) than the H/H,, versus S/S, regression coefficients (Table 5). However, the use of seasonal versus monthly or
< TM-T, (r2=0.58).
(r2=0.51)
< T16-(T4+
T2,)/2
A variation of one or two hours in the times used to estimate AT had little effect upon r2 or the rms error. Apparently, radiationdriven, diurnal warming was better characterized by an hourly-based AT than by TX- Tn. Furthermore, estimating the lower hourly temperature from two observations (the average of 04.00 and 24.00 LST) rather than one (at 04.00 LST) helped to eliminate non-radiative sources of warming from AT and further increased I’. In contrast, the TX and T, values depended not only on H but also on surface energy partitioning and sensible heat advection in the lower atmosphere. Night-time cooling ( T,6 - T2,J accounted for the lowest fraction of the observed variation in H/Ho (r2 = 0.42). Tables 9-12 summarize the results of H/H, versus AT non-linear regression analyses. Tables 10 and 11 show the May-to-September
A comparison
of methods
to estimate
daily global
(a) P= 1
(b)P=O
I
0
221
solar irradiation
-
5
10
15
20
25
0
5
10
15
20
25
Fig. 6. Daily global solar irradiation versus temperature range, stratified by precipitation status, for June 1953-1989, at four Canadian prairie locations (Table 1, stations with hourly data). The line was fit by eqn (6).
regression coefficients for eqns (6) and (7). The use of non-linear regression increased the utility of dT for estimating H. Equation (6) (Table 9, Fig. 4) outperformed simple linear regression for May-to-September at four locations (Table 1, hourly stations) by increasing the mean r2 value from 0.48 to 0.56 and reducing the rrns error from 23.6 to 21.3%. Equation (7) (Table 9, Fig. 5) further improved the analysis, increasing the r2 value to 0.63 and decreasing the rms error to 19.5%. However, eqn (6) did not perform as well for the Canadian prairies (r* = 0.45-0.76) as it did for the U.S. Pacific northwest (r* = 0.70-0.90, Bristow and Campbell, 1984). The Canadian prairie estimate of a, eqn (6), was similar to the summer-time Bristow and Campbell (1984) estimate (0.70) but the Canadian prairie b was higher (0.04 versus 0.004) and c was lower ( 1.6 versus 2.4). These differences in b and c were mutually compensating, and if c was constrained to 2.4, b was estimated to be near 0.004 as in Bristow and Campbell (1984). Unlike the U.S. Pacific northwest, eqn (6) accounted for almost none of the Canadian prairie winter-time variation in H/H,. In the period of greatest utility for the Canadian prairies (May-to-September), the coefficients from eqn (6) were nearly constant with month and location. 3.1.4. Precipitation. Table 2, 3,9 and 12 summarize the results of regression analyses which included the precipitation status P. Simple linear
regression with P as the independent variable (Table 2) accounted for 35% of the May-toSeptember variation in global solar irradiation at four locations (Table 1, hourly stations). The mean rain-day and non-rain-day values of H/H, were 0.411 and 0.635, respectively. Only a small increase in R* (to 0.36) occurred when the P values of the previous and subsequent days (PO and P2) were added to P in a multiple linear regression analysis (Table 3). As with TX- T,, P accounted for almost none of the November-to-March variation in H/H,,. In a similar study in Australia, McCaskill (1990) found that P,, P and P, values of 1 (wet) reduced H by 1.8, 4.7 and 2.1 MJ m-*, respectively, with rms errors of 3.4 MJ m-* (dry days) and 5.2 MJ me2 (wet days). We conclude that TX- T, was more useful than P for estimating H for two reasons. The rms error of H estimated from P was higher than that of H estimated from TX- T,, and TX- K and P were available at a similar spatial density and for a similar period of record. Precipitation status was useful for estimating H when used in conjunction with TX- T,, either as an additional variable in multiple regression (Table 3) or to stratify the data (Tables 2 and 12). When P was added to TX- T, in a multiple linear regression analysis, the mean coefficient of determination increased from r* = 0.48 (TX- T, alone) to R2 = 0.58, and the mean rms error decreased from 23.6 to 21.1% (Table 3).
222
A. G. Barr et al.
Stratifying by precipitation status (Fig. 6) also decreased the rms error, from 23.6 to 20.1% for simple linear regression and from 21.3 to 18.6% for eqn (6). [These values are combined rms errors for the two stratified samples. Individual wet- and dry-day values of eqn (6) coefficients and rms errors are given in Tables 9 and 12.1 These results compare favourably with the De Jong and Stewart (1993) equation:
which gave a mean monthly May-to-September r2 of 0.54 and rms error of 22.1% at five Canadian prairie locations. For the period of November-to-March, eqn (9) outperformed both eqn (6) and simple linear regression, with mean monthly R2 of 0.18 and rms errors of 28.6%. However, these values were too low to justify the application of the De Jong and Stewart (1993) equation in winter.
plete overlap for the period of S record was 123 km away) to 22.8% (for The Pas, where the nearest station was 267 km away). These rms errors were small enough to recommend the use of spatially interpolated S/S,, for estimating H/H, on the Canadian prairies. For periods when S/S,, was available at high density (e.g. after 1960), H/H,, was estimated more accurately from spatially interpolated S/S, than locally observed TX- T. (Tables 2, 10 and 11) or T,- T, and P (Table 3). The rms error of H estimated from spatially interpolated TX- T. was identical to the rms error of H estimated from locally observed TX- T. (24.5%). Observed and interpolated TX- T, were equally effective for predicting H, apparently because of the high density of TX- T. observations, with several proximate observations typically available within a 50 km radius.
3.2. Application
3.3. Application to historic data
H/H, = a( T, - T,Jb( 1 + CP + dP2)
to spatially interpolated
(9)
data
In a preliminary assessment of kriging and distance-weighted interpolation for the month of June 1970-1972, the two methods produced nearly identical mean rms errors. The rms error for interpolated S/S, was 29% of S/S, by both methods. Kriging was attractive because it enabled the uncertainty of the interpolated values to be estimated. However, distance-weighted interpolation was simpler to implement and avoided the difficulties associated with the variability and anisotropy of S/S, and TX- T, semivariograms. Because the operationally simpler distance-weighted interpolation performed as well as kriging, it was used in the subsequent assessment of estimating H/H, from interpolated S/S,, and TX- T,. Zelenka et al. (1992) reached a similar conclusion for interpolated H based on an intensive comparison of weighted interpolation and kriging in the U.S. northeast and Switzerland, two regions with dense solar irradiation networks. They found that weighted interpolation and kriging were equally effective in most situations, and recommended weighted interpolation because of its operational simplicity. The mean rms error of H estimated from spatially interpolated S/S, was 14.7%, compared with 10.1% for H estimated from observed S/S, at seven locations (Table 1, daily stations) between 1960 and 1989. On an individual station basis, the rms error of H estimated from interpolated S/S, varied from 10.5% (for Edmonton, where the nearest station with com-
The analysis of covariance of annual differences in the H/H, versus S/S, and H/H, versus (TX- T.) regression coefficients showed significant year-to-year differences. However, the differences were not due to climatic variation; no significant differences were found in the coefficients of three climatically contrasting year groupings: (1) cool-wet; (2) intermediate; (3) warm-dry. These results indicated that the H/H,, versus S/S, and H/HO versus (TX- T,J regression coefficients were stable over time between 1953 and 1989. Because the climatic variation on the Canadian prairies between 1953 and 1989 was as great as any observed in this century (Ken Jones, Atmospheric Environment Service, Saskatoon, personal communication, 1994), the regression coefficients should apply equally well to periods earlier in the century. 4. SUMMARY
AND CONCLUSIONS
Simple linear regression with daily bright sunshine duration as the independent variable accounted for more of the observed variation in global solar irradiation on the Canadian prairies than simple linear regression with the daily temperature range or precipitation status. The bright sunshine duration accounted for almost as much of the observed variation in global solar irradiation as the more detailed MAC3 model. No appreciable increase in the coefficient of determination or decrease in the rms errors were observed when sunshine was supplemented
A comparison
of methods
to estimate
by temperature range and precipitation status in a multiple linear regression analysis. In all cases, the regression coefficients varied strongly with season but only weakly with location. For the months of May-to-September, the variation of the coefficients with location and month was small. The coefficients of determination and rms errors were not appreciably degraded when the coefficients were estimated regionally for May-to-September inclusive, or locally by spatial interpolation of nearest neighbour values. The utility of the daily temperature range and precipitation status for estimating irradiation on the Canadian prairies was improved by: (1) multiple linear regression; (2) non-linear regression; (3) separating wet and dry days; and (4) estimating AT from hourly observations. The regression coefficients were found to be stable over time between 1953 and 1989 on the Canadian prairies. Independent assessment of cool-wet, warm-dry and intermediate year groupings showed that the irradiation versus sunshine and irradiation versus temperature range simple linear regression coefficients were independent of inter-annual climatic variation. This justifies their application before 1953 when irradiation was not measured and calibration of the regression coefficients was not possible. Global solar irradiation was also estimated at locations with limited or no local climatic observations, by spatial interpolation of nearest neighbour observations. For the post-1960 period, when sunshine duration was available at a high spatial density, spatially interpolated sunshine had greater utility for estimating irradiation on the Canadian prairies than locally measured temperature or precipitation. Acknowledgements-This research was conducted during the development of a daily climatic database for the Canadian prairies for use in the study “The Influence of Climate Variability of Agricultural Sustainability in Alberta”, funded by the Nat Christie Foundation of Calgary, Alberta. We are grateful to Dr Tim Goos, Atmospheric Environment Service, Edmonton, for providing the daily climatic data used in this study.
NOMENCLATURE Variable definitions
C daily average cloud fraction (O-1) C’ observed hourly cloud fraction (O-l) C, daily average cloud fraction (O-l), weighted according to eqn (2) H observed daily global solar irradiation ( MJme2)
daily global
solar irradiation
223
fi estimated daily global solar irradiation ( MJ m-‘) H, daily global solar irradiation calculated for a horizontal surface at the top of the atmosphere ( MJ m-*) H/H,, daily global solar irradiation, scaled from 0 to 1 P precipitation status: 0 dry (< 0.2 mm); 1 wet (2 0.2 mm) P,, P, the P values of the previous and subsequent days R2 multiple regression coefficient of determination S observed dailv bright sunshine duration (h) S’ observed hourly blight sunshine duration from hour i (h) S, maximum possible daily bright sunshine duration (h) S, observed daily bright sunshine fraction, weighted according to eqn (l), and scaled from 0 to 1 S/S, daily bright sunshine duration, scaled from 0 to 1 T. observed daily minimum air temperature (K) T,, observed minimum air temperature, average of same and following days’ T. (K) TX observed daily maximum air temperature (K) &, q6, G4 observed hourly air temperature at 04.00, 16.00 and 24.00 LST (K) AT observed daily temperature variation, denoting T, - T. (the daily temperature range), TX- G,, T6--T4, T6--% or q6--(T4+T2.,)/2 (K) aO. a, simple linear regression coefficients b,, b,... b, multiple linear regression coefficients a, b, c, d non-linear regression coefficients rz simple linear and non-linear regression coefficient of determination
REFERENCES Allen 0. B. and Racktoe B. L. Accuracy analysis with special reference to the prediction of grassland yield. Biom. J. 23, 371-388 (1981). Amir J. and Sinclair T. R. A model for temperature and solar radiation effects on spring wheat growth and yield. (1991). Field Crops Res. 28,47-58 Boisvert J. B., Hayhoe H. N. and Dubt P. A. Improving the estimation of global radiation across Canada. Agric. For. Meteorol. 52, 275-286 (1990). Bristow K. L. and Campbell G. S. On the relationship between incoming solar radiation and daily maximum and minimum temperature. Agric. For. Meteorol. 31, 159-166 (1984). Davies J. A and Hay J. E. Calculation of the solar radiation incident on a horizontal surface. In Hay J.E. and Won T.K. (Eds), Proc. First Can. Solar Radiation Data Workshop, Minister of Supply and Services Canada, pp. 32-58 (1980). Davies J. A. and McKay D. C. Estimating solar irradiation and components. Solar Energy 29, 55-64 (1982). Davies J. A., Abdel-Wahab M. and McKay D. C. Estimating solar irradiation on horizontal surfaces. Int. J. Solar Energy 2, 405-424 (1984). De Jong R. and Stewart D. W. Estimating global solar radiation from common meteorological variables in western Canada. Can. J. Plant Sci. 73, 509-518 (1993). Genstat 5 Committee of the Statistical Department of the Rothamsted Experimental Station. Genstat 5 Reference Manual, p. 749. Oxford University Press, Oxford (1987). Halouani N., Nguyen C. T. and Vo-Ngoc D. Calculation of monthly average global solar radiation on horizontal surfaces using daily hours of bright sunshine. Solar Energy 50, 247-258 (1993).
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Hook J. E. and McClendon R. W. Estimation of solar radiation from long-term meteorological records. Agron. J. 84, 739-742 (1993). Iqbal M. An Introduction to Atmospheric Radiation, p. 390. Academic Press, New York (1983). Isaaks E. H. and Srivastava R. M. Applied Geostatistics, p. 561. Oxford University Press, New York (1989). Jones K. H. Drought on the mairies. In Wall G. (Ed.), Svmp. Impacts of Climatic Change and Variability dn the dre& Plains, pp. 125-129. Department of Geography Publication Series, Occasional Paper No. 12, University of Waterloo (1991). McCaskill M. R. Prediction of solar radiation from rainday information using regionally stable coefficients. Agric. For. Meteorol. 51, 247-255 (1990). Noia M., Ratto C. F. and Festa R. Solar irradiation estimation from geostationary satellite data-I. Statistical models. Solar Energy 51,449-456 (1993a). Noia M., Ratto C. F. and Festa R. Solar irradiation estimation from geostationary satellite data-II. Physical models. Solar Enerav 51, 457-465 (1993b). Norris D. J. Correlationof sblar radiation with clouds. Solar Energy 12, 107-112 (1968). Penman H. L. Natural evaporation from open water, bare soil, and grass. Proc. R. Sot. Land. A 193,120-146 (1948). Priestley C. H. B. and Taylor R. J. On the assessment of surface heat flux and evaporation using large-scale parameters. Mon. Wea. Reo. 100, 81-92 (1972). Richardson C. W. Stochastic simulation of daily precipitation, temperature and solar radiation. Water Resour. Res. 17, 182-190 (1981).
Ritchie J. R. and Otter S. Description and performance of CERES-WHEAT: a user oriented wheat yield model. USDA-ARS, ARS. 38, 159-175 (1985).
SAS Institute Inc. SASISTAT Version 6, 4th edn, p. 1686. SAS Institute Inc., Cary, NC. (1989). Shmetz J. Towards a surface radiation climatology: retrieval of downward irradiations from satellites. A&&. Res. 23, 287-321 (1989). Skinner W. i. and Gullet D. W. Trends of daily maximum and minimum temperature in Canada during the past century. Climatol. Bull. 21, 63-77 (1993). Soler A. Statistical comparison for 77 European stations of seven sunshine-based models. Solar Energy 45, 365-370 (1990).
Suckling P. W. Estimating daily solar radiation values in selected mid-latitude regions by extrapolating measurements from nearby stations. Solar Energy 35, 491-495 (1985).
Williams J. R., Jones C. A., Kinity J. R. and Spanel D. A. The EPIC model crop growth-model. Trans. ASAE 32, 497-511 (1989). Zelenka A., ezepaik G., D’Agostino V., Josefsson W., Maxwell E., Perez R., Noia M., Ratto C. and Fest R. Techniques for Supplementing Solar Radiation Network Data,
Vol. 1, p. 261. International Energy Agency, Solar Heating and Cooling Programme, ENET, Swiss Federal Office of Energy, Bern (1992).
0038-092X(95)00100-X
Solar Energy Vol. 56, No. 3, pp. 213-224, 1996 Copyright 0 1996 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0038-092X/96 $15.00+0.00
A COMPARISON OF METHODS TO ESTIMATE DAILY GLOBAL SOLAR IRRADIATION FROM OTHER CLIMATIC VARIABLES ON THE CANADIAN PRAIRIES ALAN Agriculture
G. BARR,* and Agri-Food
S. M. MCGINN Canada
Research
(Communicated
and SI BING CHENG+
Centre, Lethbridge,
Canada
TlJ 4Bl
by AMOS ZEMEL)
Abstract-Historic estimates of daily global solar irradiation are often required for climatic impact studies. Regression equations with daily global solar irradiation, H, as the dependent variable and other climatic variables as the independent variables provide a practical way to estimate H at locations where it is not measured. They may also have potential to estimate H before 1953, the year of the first routine H measurements in Canada. This study compares several regression equations for calculating H on the Canadian prairies. Simple linear regression with daily bright sunshine duration as the dependent variable accounted for 90% of the variation of H in summer and 75% of the variation of H in winter. Linear regression with the daily air temperature range as the dependent variable accounted for 45% of the variation of H in summer status (wet or dry) as and only 6% of the variation of H in winter. Linear regression with precipitation the dependent variable accounted for only 35% of the summer-time variation in H, but stratifying other regression analyses into wet and dry days reduced their root-mean-squared errors. For periods with sufficiently dense bright sunshine observations (i.e. after 1960), however, H was more accurately estimated from spatially interpolated bright sunshine duration than from locally observed air temperature range or precipitation status. The daily air temperature range and precipitation status may have utility for estimating H for periods before 1953, when they are the only widely available climatic data on the Canadian prairies. Between 1953 and 1989, a period of large climatic variation, the regression coefficients did not vary significantly between contrasting years with cool-wet, intermediate and warm-dry summers. They should apply equally well earlier in the century.
1. INTRODUCTION
Climatic impact studies in agronomy, hydrology and ecology are often limited by a lack of longterm solar irradiation data. Daily global solar irradiation (H) is required for many simulation models of crop growth and productivity (Ritchie and Otter, 1985; Williams et al., 1989; Amir and Sinclair, 1991) and evapotranspiration (Penman, 1948; Priestley and Taylor, 1972). However, H observations are sparse and relatively recent. Spatial interpolation is not feasible because the observations are sparse. Estimation by remote sensing is possible at high spatial and temporal resolution (Shmetz, 1989; Noia et al., 1993a, b), but only during the recent past. Efforts to estimate H with physically based models (Davies and McKay, 1982; Davies et al., 1984) have increased the density of historic H estimates, but the data requirements of these models still limit the spatial and historic coverage. A need exists to extend the H database *Atmospheric Environment Service, National Hydrological Research Centre, 11 Innovation Boulevard, Saskatoon, SK, Canada S7N 3H5. ‘Soil and Fertilizer Institute, Hebei Academy of Agricultural and Forestry Science, Shijiazhuang, Hebei, P.R. China. 213
backwards in time and to increase its spatial resolution. Statistical relationships between H and other climatic variables provide a practical way to calculate H at locations where it was not measured and to enlarge the database of H estimates in time and space. The use of a regression analysis to estimate daily H at locations with long time series of historic climatic data is attractive because it preserves the integrity of the observed time series and maintains the observed interannual variations. Methods which generate synthetic data using statistical methods (e.g. Richardson, 1981) may not adequately reproduce long-term variability and the interannual persistence of extreme climatic events such as drought. Global solar irradiation is highly correlated with surface daily climatic variables, including bright sunshine duration, S (Soler, 1990; Boisvert et al., 1990); cloud fraction, C (Norris, 1968; Davies and Hay, 1980); air temperature range, dT (Bristow and Campbell, 1984; De Jong and Stewart, 1993); precipitation (De Jong and Stewart, 1993); and precipitation status, P (McCaskill, 1990). Regression equations based on these variables have a strong physical basis.
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A. G. Barr et al.
Day-time warming is driven in part by H, via the influence of H on the sensible heat budget of the lower atmosphere. Precipitation indicates cloudiness. Global solar irradiation is most highly correlated with S, as rz usually exceeds 0.85 for summer months. Although H is less highly correlated with ATand P, the correlations are strong enough to support the use of regression equations for estimating H. Using preradiation, cipitation, extraterrestrial pan evaporation and temperature range in a multiple regression equation accounted for 78% of the variation in H at one location in Georgia (Hook and McClendon, 1993). A non-linear relationship with ATaccounted for between 70 and 90% of the summer-time variation in H for the U.S. Pacific northwest (Bristow and Campbell, 1984). De Jong and Stewart (1993) found that a polynomial with AT and P accounted 54% of the summer-time variation in H on the Canadian prairies. McCaskill (1990) reported that H was significantly, negatively correlated with the precipitation status of the previous, current and following days in Australia. Because daily observations of AT and P are available at a much higher spatial density and for a longer historic period than S, it is important to explore their utility for calculating H. For instance, after excluding climatic stations with more than 20% of daily observations missing between 1960 and 1989, the climatic archive of the Atmospheric Environment Service of Canada contains 173 Canadian prairie locations with precipitation, 118 locations with maximum and minimum temperature, 24 locations with bright sunshine duration, and five locations with H observations. Regression equations which estimate H from other climatic variables have the potential to estimate H before 1953, the year of the first routine H observations. Daily temperature and precipitation and, to a lesser extent, bright sunshine duration, are available at many Canadian prairie locations between 1920 and 1953, and the daily climatic record at a few locations extends continuously back to the last century. However, because regression equations must be calibrated with recent data, they may not apply to earlier decades. A long-term variation in the regression coefficients could result, for instance, from changes in atmospheric aerosols (for S) or climatic warming [given the observed reduction in AT which has accompanied observations of warming (Skinner and Gullet, 1993)]. Before regression equations based on post-1953 data
can be applied to the pre-1953 period, the longterm stability of the regression coefficients must be evaluated. Estimates of H are often required at locations with no climatic data. Spatial interpolation of observed H is difficult because of low measurement density. Suckling (1985) reported relative errors of 33% (summer) and 48% (winter) when H was interpolated over distances of 100 km and 53% (summer) and 76% (winter) when H was interpolated over distances of 300 km. Canada has only sparse observations of H, and spatially interpolated estimates of H are inaccurate. However, H may be estimated at locations which lack climatic observations by first interpolating other climatic variables and then calculating H from the interpolated values. This study: (1) explored the utility of regression equations for calculating H from other climatic data on the Canadian prairies; (2) assessed the accuracy of H estimates from spatially interpolated climatic data; and (3) assessed the stability of the regression coefficients over time. 2. METHODOLOGY
2.1. Climatic data Historic hourly and daily climatic data were obtained for several sites in Alberta, and Manitoba during the Saskatchewan 1953-1989 period. Table 1 summarizes the site locations and their May-to-September climatic averages. Four sites had hourly data and seven sites had daily data. Daily global solar irradiation was expressed as H/Ho, the ratio of observed H to the value H, calculated for a horizontal plane at the top of the atmosphere (Iqbal, 1983). The H/Ho ratio varies between 0.0 and 1.0. Using H/H,-, rather than H in regression analysis made it possible to assess whether regression coefficients were independent of site and time of year. Daily bright sunshine duration was also scaled from 0.0 to 1.0, either as the ratio of actual to possible bright sunshine hours S/S, [where S is the observed bright sunshine duration and S, is the day length (Iqbal, 1983)] or as a weighted average S, of hourly observed bright sunshine duration s’: SW= z SiHi/Ho
(1)
n=l
where Hi/Ho is the fraction which hour i contributed to H,. No attempt was made to correct S
A comparison of methods to estimate daily global solar irradiation
215
Table 1. Climatic stations, including location, elevation, period of record for global solar irradiation and climatic normals for the period of record Period of record Location Beaverlodge CDA, Alberta Edmonton Municipal A, Alberta Edmonton Stony Plain, Alberta Suffield A, Alberta Bad Lake, Saskatchewan Swift Current CDA, Saskatchewan The Pas A, Manitoba Winnipeg Intl A, Manitoba
Latitude (“N)
Longitude (“W)
Elevation (m)
Daily data3
55.20
119.40
144
53.57
113.52
53.53 50.27
May-to-September
Hourly data4
daily meant
K (“C)
S (h)
H (MJ m-*)
1960-1990
12.4
8.4
18.3
668
1957-1989
14.5
8.7
19.6
114.10 111.18
765 768
1959-1990
13.6 15.9
8.6 9.3
18.8 20.1
51.32
108.42
637
1971-1986
50.27
107.73
823
1960-1989
1960-1989
14.9
8.6
20.3
53.97
101.10
268
1972-1990
1972-1989
13.5
8.5
17.9
49.90
97.23
238
1957-1990
1957-1989
15.8
8.9
19.7
1967-1989
20.1
TAveraged for the period of record. IDaily data were obtained from the Atmospheric Environment Service climatic archives. §Hourly data were obtained from the Canadian Weather and Engineering Data Set.
or S’ for the limitations of bright sunshine recorders at low solar elevations (Iqbal, 1983). The daily cloud fraction C was estimated as a simple average of day-time hourly observations, or as a weighted average C, of the hourly cloud fraction C’:
C,,,= E C'H'IH,.
maximum (TX) and minimum (T.) air temperatures. A second estimate (7’.‘ -. T,,) (Bristow and Campbell, 1984) used daily TXand the average T,from the same and following days (T,,). Other estimates were based on hourly values from 04.00, 16.00 and 24.00 LST. An analysis of Mayto-September showed that T,and TX occurred most frequently at 04.00 to 05.00 LST and 14.00 to 17.00 LST, respectively, with a shift to earlier T,, and later TXas H/H, increased (Fig. 1). Daytime warming was estimated as T16-T4,nighttime cooling- was estimated as T& - Tz4 and
(2)
n=l
The daily temperature variation ATwas estimated from hourly and daily measurements. The daily temperature range TX-T,used daily (a) H/H, c 0.2
(b) H/H, > 0.7
T
I
a-
1
4-
-8 0
4
8
12
18
20
24
0
4
8
12
16
20
24
Hour (LST) Fig. 1. The diurnal variation in surface air temperature at Winnipeg International Airport, June 1953-1989, for two classes of daily global solar irradiation. Temperature (7”) is offset to a daily mean of 0.0, and the error bars denote one standard deviation from the hourly mean.
216
A. G. Barr et al.
T16-(T4 + T,,)/2 combined the two. (The subscripts denote the hour in LST.) The daily precipitation status P was set to 0 (dry) when daily total precipitation was less than 0.2 mm and 1 (wet) when daily total precipitation was greater than or equal to 0.2 mm.
Table 3 summarizes the groupings of independent variables. Based on a preliminary assessment of several relationships, an inverse-square relationship was used for the H/H, versus C (and H/H,, versus C,) regression analyses: HJHo=a+bCp2
2.2. Regression analyses Simple linear [eqn (3)], multiple linear [eqn (4)] and non-linear regression analyses [eqns (5)-( 7)] used the least-squares method (Genstat, 1987; SAS, 1989). The simple linear regression analysis was: H/H0 = a0 + a,x
Several non-linear relationships were evaluated for use in the S/S, versus d T non-linear regression analysis, including logistic, exponential and Gompertz equations (Genstat, 1987) and the Bristow and Campbell (1984) equation. A preliminary test showed that the Bristow-Campbell equation gave the highest r2. The Bristow-Campbell equation is:
(3)
where a, and a, are the linear regression coefficients and x is the independent variable. See Table 2 for a list of independent variables used in this study. In some cases, data were stratified into wet (P= 1) and dry (P=O) days, and wetand dry-day regression analyses were performed separately. The multiple linear regression analysis was: H/H0 = b, + blx, + bzxz... + b,x,
H/H0 = a( 1- exp (- b( TX- T,,“)‘))
(4)
H/H, = a( 1 - exp (- b(d + T16- ( T4 + $,)/2)‘))
Table 2. Simple linear regression coefficients [a0 and a,, eqn (3)], coefficients of determination (r*) and root-meansquared errors (rmse) for regression of several climatic variables versus daily solar irradiation (expressed as H/H,), averaged for May-to-September at four locations on the Canadian prairies (Table 1, stations with hourly data)
(7) in
$0 K-T, K-T,, P TX---T,(P = 1) TX--T, (P = 0) TN--T, T&s- 7-L &-(T4+ WI2
a,,
aI
rz
0.213 0.221 0.200 0.179 0.635 0.132 0.443 0.311 0.292 0.249
0.510 0.556 0.0283 0.0299 -0.224 0.0294 0.0138 0.0237 0.0317 0.333
0.90 0.89 0.48 0.44 0.35 0.47 0.22 0.51 0.41 0.57
order
to
permit
negative
values
of
T16-(T4 + T,,)/2. The value of d in eqn (7) was
fixed at the mean estimate of 5.0 in order to avoid occasional convergence problems when solving for the coefficients. Because the regression coefficients for all equations were dependent on time of year, they were estimated monthly. When applying the regression equations, monthly coefficients were assumed to represent the midpoint of each month. The significance of geographic, monthly, and among regression inter-annual differences coefficients was assessed by the analysis of
rmse variable
(6)
where a, b and c are empirical coefficients. In applying eqn (6), Bristow and Campbell (1984) first conditioned TX-- T,, by multiplying wetday values of TX- T,, by a factor 0.75. We did not find this to be useful on the Canadian prairies. We substituted T,, for T,, in eqn (6) because this increased the r2 value. In addition, eqn (6) was modified by adding the variable d:
where b,,-b, are the multiple regression coefficients and x1-x, are the independent variables.
Independent
(5)
-mean
0.103 0.110 0.236 0.248 0.264 0.297 0.152 0.23 1 0.252 0.215
Table 3. Multiple linear regression coefficients [b, to b,, eqn (4)], coefficients of determination (R2) and root-mean-squared errors (rmse) for regressions of daily climatic variables versus daily solar irradiation (expressed as H/H,), averaged for May-to-September at four locations on the Canadian prairies (Table 1, stations with hourly data). The blank cells indicate that the independent variable was not included in the analysis b, to b, Independent
variables
SIS,T, - T#.p T,--T,, P %-+‘I,+ T24)/2r P P, p, P,
b.
S/S,
AT
0.218 0.328
0.498
0.00377 0.0220 0.027 1
0.649
PO
P
- 0.0376
- 0.0234 -0.125 -0.102 -0.199
P,
-0.0231
R2
rmsefmean
0.90 0.58 0.63 0.36
0.106 0.211 0.199 0.261
A comparison
of methods
to estimate
covariance. The use of regional and seasonal versus local and monthly regression coefficients was also assessed by “jackknifing”. In the jackknifing test, the local regression coefficients were estimated using only data from neighbouring locations. The coefficients were estimated either prairie-wide or by inverse-distance-squaredweighted interpolation of the four nearest estimates. The accuracy of estimated H (fi) was calculated as the root-mean-squared (rms) error of estimate: Jc
daily global
point and the neighbouring observation. Nearest neighbours were chosen by distance only, with no attempt made to select neighbours in all directions. It was not critical to select neighbours in all directions because the station distribution was not clumped (Bill Hogg, Atmospheric Environment Service, Downsview, personal communication, 1994). Kriging used a daily semivariogram fitted by a spherical model (Isaaks and Srivastava, 1989). 3. RESULTS
(fi - H)‘/n.
The rms error was further subdivided into mean bias, regression and random components using the method of Allen and Racktoe (1981). The bias component determined the fraction of the rms error due to differences in mean H and fi. The regression component determined the fraction of the rms error due to departures of the H and fi data from a one-to-one relation. The random component determined the fraction of the rms error due to random variations in H and fi. The long-term stability of the H/H,, versus S/S, and H/H, versus TX- T, regression coefficients was assessed by analysis of covariance of: (a) inter-annual variation; and (b) three contrasting 1953-1989 year groupings. The year groupings were based on the May-to-September climate (Jones, 1991): (1) cool-wet (1954-1956, 1961, 1964, 1967-1968, 1977, 1980-1981, 1984 and 1988-1989); (2) warm-dry (1957, 1965-1966, 1971-1972, 1975-1976, 1983 and 1986); and (3) intermediate (1963, 1958-1960, 1962-1963, 1969-1970, 1973-1974, 1978-1979, 1983, 1985 and 1987). 2.3. Spatial interpolation We also evaluated global solar irradiation estimates from spatially interpolated values of S/S, and TX- T,. The S/S, and TX- T. variables were chosen because they were dense enough for spatial interpolation and had proven utility for estimating H. The simple linear regression coefficients for the spatially interpolated values were estimated in two ways, either prairie-wide (after excluding the location in question) or by inverse-distance-squared weighted interpolation of the four nearest estimates. Two methods of spatial interpolation were compared: inverse-distance-squared weighting and kriging. Distance-weighting used eight nearest neighbour values, weighted by the inverse distance squared between the interpolation
217
solar irradiation
AND DISCUSSION
3.1. Regression analyses
3.1.1. Bright sunshine duration. Table 2 summarizes the results of simple linear regression analyses for May-to-September at four Canadian prairie locations (stations with hourly data, Table 1). Table 3 summarizes the results of multiple linear regression analyses for the same period and locations. The weighted daily bright sunshine duration S, accounted for 90% of the variation in H/Ho, a slight improvement over S/S, (r2 =0.89) (Table 2). The corresponding rms errors were 10.3% for S, and 11.0% for S/S,. Adding AT and P to S/S, or S, in a multiple linear regression analysis, or stratifying by precipitation status, resulted in a slightly higher coefficient of determination and smaller rms error (Table 3). A similar small increase in R2 was found when C or C, were added to S/S0 or S, in a multiple linear regression analysis. The H/H, versus S/S,, regression coefficients varied considerably but consistently with season (Table 4, Fig. 2) and weakly with location (Table 5). The greatest differences occurred between winter and summer. The mean prairiewide regression coefficients, for the periods of Table 4.. Simple linear regression coefficients for daily values of H/H, versus S/S,, by month, averaged for seven locations on the Canadian prairies (Table 1, stations with daily data) Month Jan. Feb. Mar. Apr. May Jun. JUl. Aug. Sep. Oct. Nov. Dec.
a0
aI
rz
0.377 0.390 0.364 0.291 0.241 0.220 0.225 0.223 0.228 0.244 0.280 0.334
0.396 0.409 0.490 0.515 0.545 0.565 0.535 0.528 0.564 0.505 0.453 0.407
0.73 0.77 0.78 0.84 0.89 0.90 0.88 0.89 0.89 0.86 0.77 0.69
rmse/mean 0.161 0.140 0.147 0.126 0.106 0.093 0.100 0.098 0.125 0.141 0.189 0.190
A. G. Barr et al.
218
(a) January
(b) June 1.0
0.8
0.6
0
$
0.4
.:-.
0.2
0.2 - ”
0.0 0.0
0.2
0.4
0.6
0.8
1.0
0.0
0.2
I
/
0.4
0.6
’
I
0.8
’
1.0
S/S, Fig. 2. Daily global solar irradiation fraction versus bright sunshine duration fraction for January and June 1953-1989, at four Canadian prairie locations (Table 1, stations with hourly data). The line was fit by least-squares simple linear regression. Table 5. Simple linear regression coefficients for daily values of H/H, versus S/S,, by location, for the month of June Location Beaverlodge CDA Edmonton Municipal Suffield A Bad Lake Swift Current CDA The Pas A Winnipeg Intl A
A
a0
al
12
0.211 0.224 0.232 0.233 0.261 0.192 0.201
0.0579 0.0585 0.0549 0.0538 0.0522 0.0582 0.0573
0.91 0.88 0.90 0.91 0.88 0.92 0.92
November-to-March and May-to-September, respectively, were: 0.35 and 0.22 for a,; 0.43 and 0.56 for a,; and 0.75 and 0.89 for r2. Analysis of covariance showed that the differences in the May-to-September regression coefficients among locations and months were statistically significant in some cases. However, the differences were small, and the use of a regional versus local or monthly versus seasonal (Mayto-September) regression analysis did not appreciably alter a,,, aI, r2 or the rms of estimate. For the month of June at seven locations (stations with daily data, Table l), the mean rms error was 9.8% when a, and aI were estimated locally, 10.1% when a, and a, were estimated regionally and 10.2% when a,, and aI were estimated locally by interpolation of the nearest neighbour estimates. The Canadian, prairie-wide, May-toSeptember a0 and a, estimates of 0.22 and 0.56 were similar to the estimates of 0.23 and 0.54 obtained for European locations between 39
and 60”N (Soler, 1990). However, the Canadianprairie, November-to-March a, and a, estimates of 0.35 and 0.43 differed from the European estimates of 0.19 and 0.63 (Soler, 1990). The Canadian winter estimates for a, and a, of Halouani et al. (1993) and Boisvert et al. (1990) were similar to those of Soler (1990) for locations outside the prairie region, but similar to this study for prairie locations. It is possible that the winter estimates of a, and a, were affected more by continentality than latitude. If, for instance, continentality influenced the diurnal timing of cloud cover, it would have altered a, and a,. Simple linear regression with S/S, performed nearly as well as the more detailed MAC3 model (Davies and McKay, 1982). The mean annual MAC3 rms errors at Winnipeg were 11.1% (Davies and McKay, 1982) compared with 12.3% in this study when the simple linear regression coefficients were estimated locally, and 12.6% when the coefficients were estimated regionally based on all locations excluding Winnipeg. 3.1.2. Cloudfraction. Figure 3 shows the relationship between daily solar irradiation (expressed as H/Ho) and the weighted cloud fraction C,. Equation (5) with C, as the independent variable accounted for 74% of the variation in H/Ho between May-and-September at four locations (Table 1, hourly stations), a small improvement over eqn (5) with C as the independent variable (r2 = 0.71) (Table 6). The
A comparison
of methods
to estimate
daily global
219
solar irradiation
Table 8. Simple linear regression coefficients for daily values of H/H, versus TX- T., by location, for the month of June a0
4,
r2
0.075 0.102 0.174 0.177 0.117 0.109 0.126
0.0368 0.0406 0.0273 0.0270 0.0342 0.0351 0.0325
0.59 0.53 0.37 0.47 0.56 0.43 0.41
Location Beaverlodge CDA Edmonton Municipal Suffield A Bad Lake Swift Current CDA The Pas A Winnipeg Intl A
0.0
-1 0.0
0.2
0.4
0.6
0.6
1.0
cw Fig. 3. Daily global solar irradiation fraction versus weighted cloud fraction [eqn (2)] for June 1953-1989, at four Canadian prairie locations (Table 1, stations with hourly data). The line was fit by eqn (5). Table 6. Regression coefficients (a and b), coefficients of determination (r’) and root-mean-squared errors (rmse) for eqn (5), averaged for May-to-September at four locations on the Canadian prairies (Table 1, stations with hourly data) Independent variable
G C-I
a
b
rz
0.761 0.747
- 0.470 -0.521
0.74 0.71
rmse/mean 0.165 0.176
corresponding rms errors were 16.5% (for C,) and 17.6% (for C). These results were intermediate between those for S/S, and dT(Table 3). A similar analysis by Norris (1968) confirmed that estimates of H from S were more accurate than estimates of H from C. 3.1.3. Temperature range. Table 2 and Tables 7-12, and Figs 4-6 show the results of regression analyses based on the daily temperature variation. The use of TX- T,, rather than TX- T. did Table 7. Simple linear regression coefficients for daily values of H/Ho versus TX- T, by month, averaged for seven locations on the Canadian prairies (Table 1, stations with daily data) Month
al
r2
rmse/mean
0.0000 0.0074 0.0154 0.0188 0.0276 0.0312 0.029 1 0.0268 0.0273 0.0227 0.0151 0.0011
0.00 0.04 0.13 0.23 0.43 0.46 0.43 0.45 0.48 0.34 0.13 0.000
0.300 0.273 0.276 0.276 0.245 0.232 0.204 0.219 0.276 0.312 0.376 0.340
A
Table 9. Coefficients of determination (r’) and root-meansquared errors (rmse) for the Bristow-Campbell equation, averaged for May-to-September at four locations on the Canadian prairies (Table 1, stations with hourly data) Regression equation
Independent variables c--T, Tr.s- (T4 + T24)/2 = 1)
TX-T,(P TX-T,(P
= 0)
(6) (7)
0.56 0.63
0.213 0.195
Equation Equation
(6) (6)
0.58 0.30
0.205 0.171
b
c
rz
0.701 0.683 0.688 0.685 0.732
0.0493 0.0328 0.0336 0.0358 0.0699
1.48 1.64 1.67 1.61 1.20
0.53 0.57 0.56 0.57 0.56
Table 11. Regression coefficients four locations on the Canadian with hourly Month May
Jun. Jul. Aug. Sep.
for eqn (6) by month, for prairies (Table 1, stations data)
a
Month May
rmse/mean
Equation Equation
Table 10. Regression coefficients four locations on the Canadian with hourly
Jun. Jul. Aug. Sep.
rz
for eqn (7) by month, for prairies (Table 1, stations data)
a
b
c
dt
r2
0.712 0.708 0.716 0.700 0.693
0.0111 0.0046 0.0104 0.0069 0.0095
1.91 2.02 1.95 2.11 1.98
5.00 5.00 5.00 5.00 5.00
0.60 0.63 0.60 0.64 0.64
td was fixed at a value of 5.00. Jan. Feb. Mar. Apr. May Jun. Jul. Aug. Sep. Oct. Nov. Dec.
0.516 0.488 0.427 0.338 0.178 0.143 0.186 0.187 0.156 0.214 0.300 0.454
Table 12. Regression coefficients for eqn (6), stratified by precipitation status, for June at four locations on the Canadian prairies (Table 1, stations with hourly data) Precipitation status Wet(P = 1) Dry (P = 0)
a
b
c
r2
0.622 0.688
0.049 0.061
1.44 1.52
0.58 0.29
220
A. G. Barr et al.
(a) January
(b) June
0.8 0.6
0.6
0.4
0.4
0 B
0.2
0
I
I
I
I
5
10
15
20
0.0 25
i 0
5
10
15
20
25
T, - T, WI Fig. 4. Daily global solar irradiation versus temperature range for January and June 1953-1989, at four Canadian prairie locations (Table 1, stations with hourly data). The line in (b) was fit by eqn (6).
regional versus local regression coefficients did not appreciably alter a,, a,, r2 or the rms error. Unlike S/S,, TX- T, accounted for almost none of the November-to-March variation in H/H, ( r2 = 0.06) (Fig. 4, Table 7). Global solar irradiation apparently contributed more significantly to the warming of the lower atmosphere in summer than winter on the Canadian prairies. The estimates of day-time warming from hourly temperature observations (7& - T4 and T16- ( T4 + T2J/2) accounted for a higher fraction of the observed variation in H/H, than did TX- T, (Table 2, Figs 4 and 5). In order of increasing r2 values, TX--T, (r2=0.48)
0.8 i 0.6 1
0.0
11 -5
0 7;~&+
5
10 T&2
15
20
WI
Fig. 5. Daily global solar irradiation versus diurnal warming for June 1953-1989, at four Canadian prairie locations (Table 1, stations with hourly data). The line was fit by eqn (7).
not improve the simple linear regression estimates of H (Table 2). The mean r2 values were 0.48 (for TX- T,) and 0.46 (for TX-- T,,) for Mayto-September at four locations (Table 1, hourly stations). The mean May-to-September rms error was 23.6%. Regression coefficients varied significantly but subtly among months for Mayto-September (Table 7). Regional differences were greater among the H/H, versus TX- T, regression coefficients (Table 8) than the H/H,, versus S/S, regression coefficients (Table 5). However, the use of seasonal versus monthly or
< TM-T, (r2=0.58).
(r2=0.51)
< T16-(T4+
T2,)/2
A variation of one or two hours in the times used to estimate AT had little effect upon r2 or the rms error. Apparently, radiationdriven, diurnal warming was better characterized by an hourly-based AT than by TX- Tn. Furthermore, estimating the lower hourly temperature from two observations (the average of 04.00 and 24.00 LST) rather than one (at 04.00 LST) helped to eliminate non-radiative sources of warming from AT and further increased I’. In contrast, the TX and T, values depended not only on H but also on surface energy partitioning and sensible heat advection in the lower atmosphere. Night-time cooling ( T,6 - T2,J accounted for the lowest fraction of the observed variation in H/Ho (r2 = 0.42). Tables 9-12 summarize the results of H/H, versus AT non-linear regression analyses. Tables 10 and 11 show the May-to-September
A comparison
of methods
to estimate
daily global
(a) P= 1
(b)P=O
I
0
221
solar irradiation
-
5
10
15
20
25
0
5
10
15
20
25
Fig. 6. Daily global solar irradiation versus temperature range, stratified by precipitation status, for June 1953-1989, at four Canadian prairie locations (Table 1, stations with hourly data). The line was fit by eqn (6).
regression coefficients for eqns (6) and (7). The use of non-linear regression increased the utility of dT for estimating H. Equation (6) (Table 9, Fig. 4) outperformed simple linear regression for May-to-September at four locations (Table 1, hourly stations) by increasing the mean r2 value from 0.48 to 0.56 and reducing the rrns error from 23.6 to 21.3%. Equation (7) (Table 9, Fig. 5) further improved the analysis, increasing the r2 value to 0.63 and decreasing the rms error to 19.5%. However, eqn (6) did not perform as well for the Canadian prairies (r* = 0.45-0.76) as it did for the U.S. Pacific northwest (r* = 0.70-0.90, Bristow and Campbell, 1984). The Canadian prairie estimate of a, eqn (6), was similar to the summer-time Bristow and Campbell (1984) estimate (0.70) but the Canadian prairie b was higher (0.04 versus 0.004) and c was lower ( 1.6 versus 2.4). These differences in b and c were mutually compensating, and if c was constrained to 2.4, b was estimated to be near 0.004 as in Bristow and Campbell (1984). Unlike the U.S. Pacific northwest, eqn (6) accounted for almost none of the Canadian prairie winter-time variation in H/H,. In the period of greatest utility for the Canadian prairies (May-to-September), the coefficients from eqn (6) were nearly constant with month and location. 3.1.4. Precipitation. Table 2, 3,9 and 12 summarize the results of regression analyses which included the precipitation status P. Simple linear
regression with P as the independent variable (Table 2) accounted for 35% of the May-toSeptember variation in global solar irradiation at four locations (Table 1, hourly stations). The mean rain-day and non-rain-day values of H/H, were 0.411 and 0.635, respectively. Only a small increase in R* (to 0.36) occurred when the P values of the previous and subsequent days (PO and P2) were added to P in a multiple linear regression analysis (Table 3). As with TX- T,, P accounted for almost none of the November-to-March variation in H/H,,. In a similar study in Australia, McCaskill (1990) found that P,, P and P, values of 1 (wet) reduced H by 1.8, 4.7 and 2.1 MJ m-*, respectively, with rms errors of 3.4 MJ m-* (dry days) and 5.2 MJ me2 (wet days). We conclude that TX- T, was more useful than P for estimating H for two reasons. The rms error of H estimated from P was higher than that of H estimated from TX- T,, and TX- K and P were available at a similar spatial density and for a similar period of record. Precipitation status was useful for estimating H when used in conjunction with TX- T,, either as an additional variable in multiple regression (Table 3) or to stratify the data (Tables 2 and 12). When P was added to TX- T, in a multiple linear regression analysis, the mean coefficient of determination increased from r* = 0.48 (TX- T, alone) to R2 = 0.58, and the mean rms error decreased from 23.6 to 21.1% (Table 3).
222
A. G. Barr et al.
Stratifying by precipitation status (Fig. 6) also decreased the rms error, from 23.6 to 20.1% for simple linear regression and from 21.3 to 18.6% for eqn (6). [These values are combined rms errors for the two stratified samples. Individual wet- and dry-day values of eqn (6) coefficients and rms errors are given in Tables 9 and 12.1 These results compare favourably with the De Jong and Stewart (1993) equation:
which gave a mean monthly May-to-September r2 of 0.54 and rms error of 22.1% at five Canadian prairie locations. For the period of November-to-March, eqn (9) outperformed both eqn (6) and simple linear regression, with mean monthly R2 of 0.18 and rms errors of 28.6%. However, these values were too low to justify the application of the De Jong and Stewart (1993) equation in winter.
plete overlap for the period of S record was 123 km away) to 22.8% (for The Pas, where the nearest station was 267 km away). These rms errors were small enough to recommend the use of spatially interpolated S/S,, for estimating H/H, on the Canadian prairies. For periods when S/S,, was available at high density (e.g. after 1960), H/H,, was estimated more accurately from spatially interpolated S/S, than locally observed TX- T. (Tables 2, 10 and 11) or T,- T, and P (Table 3). The rms error of H estimated from spatially interpolated TX- T. was identical to the rms error of H estimated from locally observed TX- T. (24.5%). Observed and interpolated TX- T, were equally effective for predicting H, apparently because of the high density of TX- T. observations, with several proximate observations typically available within a 50 km radius.
3.2. Application
3.3. Application to historic data
H/H, = a( T, - T,Jb( 1 + CP + dP2)
to spatially interpolated
(9)
data
In a preliminary assessment of kriging and distance-weighted interpolation for the month of June 1970-1972, the two methods produced nearly identical mean rms errors. The rms error for interpolated S/S, was 29% of S/S, by both methods. Kriging was attractive because it enabled the uncertainty of the interpolated values to be estimated. However, distance-weighted interpolation was simpler to implement and avoided the difficulties associated with the variability and anisotropy of S/S, and TX- T, semivariograms. Because the operationally simpler distance-weighted interpolation performed as well as kriging, it was used in the subsequent assessment of estimating H/H, from interpolated S/S,, and TX- T,. Zelenka et al. (1992) reached a similar conclusion for interpolated H based on an intensive comparison of weighted interpolation and kriging in the U.S. northeast and Switzerland, two regions with dense solar irradiation networks. They found that weighted interpolation and kriging were equally effective in most situations, and recommended weighted interpolation because of its operational simplicity. The mean rms error of H estimated from spatially interpolated S/S, was 14.7%, compared with 10.1% for H estimated from observed S/S, at seven locations (Table 1, daily stations) between 1960 and 1989. On an individual station basis, the rms error of H estimated from interpolated S/S, varied from 10.5% (for Edmonton, where the nearest station with com-
The analysis of covariance of annual differences in the H/H, versus S/S, and H/H, versus (TX- T.) regression coefficients showed significant year-to-year differences. However, the differences were not due to climatic variation; no significant differences were found in the coefficients of three climatically contrasting year groupings: (1) cool-wet; (2) intermediate; (3) warm-dry. These results indicated that the H/H,, versus S/S, and H/HO versus (TX- T,J regression coefficients were stable over time between 1953 and 1989. Because the climatic variation on the Canadian prairies between 1953 and 1989 was as great as any observed in this century (Ken Jones, Atmospheric Environment Service, Saskatoon, personal communication, 1994), the regression coefficients should apply equally well to periods earlier in the century. 4. SUMMARY
AND CONCLUSIONS
Simple linear regression with daily bright sunshine duration as the independent variable accounted for more of the observed variation in global solar irradiation on the Canadian prairies than simple linear regression with the daily temperature range or precipitation status. The bright sunshine duration accounted for almost as much of the observed variation in global solar irradiation as the more detailed MAC3 model. No appreciable increase in the coefficient of determination or decrease in the rms errors were observed when sunshine was supplemented
A comparison
of methods
to estimate
by temperature range and precipitation status in a multiple linear regression analysis. In all cases, the regression coefficients varied strongly with season but only weakly with location. For the months of May-to-September, the variation of the coefficients with location and month was small. The coefficients of determination and rms errors were not appreciably degraded when the coefficients were estimated regionally for May-to-September inclusive, or locally by spatial interpolation of nearest neighbour values. The utility of the daily temperature range and precipitation status for estimating irradiation on the Canadian prairies was improved by: (1) multiple linear regression; (2) non-linear regression; (3) separating wet and dry days; and (4) estimating AT from hourly observations. The regression coefficients were found to be stable over time between 1953 and 1989 on the Canadian prairies. Independent assessment of cool-wet, warm-dry and intermediate year groupings showed that the irradiation versus sunshine and irradiation versus temperature range simple linear regression coefficients were independent of inter-annual climatic variation. This justifies their application before 1953 when irradiation was not measured and calibration of the regression coefficients was not possible. Global solar irradiation was also estimated at locations with limited or no local climatic observations, by spatial interpolation of nearest neighbour observations. For the post-1960 period, when sunshine duration was available at a high spatial density, spatially interpolated sunshine had greater utility for estimating irradiation on the Canadian prairies than locally measured temperature or precipitation. Acknowledgements-This research was conducted during the development of a daily climatic database for the Canadian prairies for use in the study “The Influence of Climate Variability of Agricultural Sustainability in Alberta”, funded by the Nat Christie Foundation of Calgary, Alberta. We are grateful to Dr Tim Goos, Atmospheric Environment Service, Edmonton, for providing the daily climatic data used in this study.
NOMENCLATURE Variable definitions
C daily average cloud fraction (O-1) C’ observed hourly cloud fraction (O-l) C, daily average cloud fraction (O-l), weighted according to eqn (2) H observed daily global solar irradiation ( MJme2)
daily global
solar irradiation
223
fi estimated daily global solar irradiation ( MJ m-‘) H, daily global solar irradiation calculated for a horizontal surface at the top of the atmosphere ( MJ m-*) H/H,, daily global solar irradiation, scaled from 0 to 1 P precipitation status: 0 dry (< 0.2 mm); 1 wet (2 0.2 mm) P,, P, the P values of the previous and subsequent days R2 multiple regression coefficient of determination S observed dailv bright sunshine duration (h) S’ observed hourly blight sunshine duration from hour i (h) S, maximum possible daily bright sunshine duration (h) S, observed daily bright sunshine fraction, weighted according to eqn (l), and scaled from 0 to 1 S/S, daily bright sunshine duration, scaled from 0 to 1 T. observed daily minimum air temperature (K) T,, observed minimum air temperature, average of same and following days’ T. (K) TX observed daily maximum air temperature (K) &, q6, G4 observed hourly air temperature at 04.00, 16.00 and 24.00 LST (K) AT observed daily temperature variation, denoting T, - T. (the daily temperature range), TX- G,, T6--T4, T6--% or q6--(T4+T2.,)/2 (K) aO. a, simple linear regression coefficients b,, b,... b, multiple linear regression coefficients a, b, c, d non-linear regression coefficients rz simple linear and non-linear regression coefficient of determination
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