.~dar Em,rgy Vol. 32. No. 2. pp. 307-3t)9. 19,~4 0038q392X,'84 S3.tX)*.0~q Printed in Great Britain. ~ 19S4 Pergamon Press Ltd.

T E C H N I C A L N O T E

E r r o r s in e s t i m a t i n g so l a r i r r a d i a n c e f r o m a n u m e r i c a l mode l

J. A. DAVIES, M. ABDEL-WAHAB and J. E. HOWARD Department of Geography. McMaster University. Hamilton. Ontario. Canada L8S 4K1

{Receired 20 April 1983)

I.Nq'RODUCTION This paper examines the errors and sources of error for the numerical model (Davies and MeKay[I]) used to calculate irradiances for the Canadian solar radiation data base (Hooper et al.[2]), and shows how these determine its performance on different time scales.

"r i le MODEL

Under n cloud layers the global irradiance G at the ground is given by

G = Go 1"I [(1 - C )+ t,Ci)l/(l - a/~) (1) i= l

where Go is a theoretical value for cloudless sky irradiance, C~ the cloud amount in layer i, ti the transmittance of the cloud in the layer, a mean landscape albedo and /3 the atmospheric backscatterance for surface-reflected radiation. Cloudless sky irradiance depends on extraterrestrial irradiance and transmit- tances determined by Rayleigh scattering, absorption by ozone and water vapour and extinction by aerosols. It is calculated from the solar constant, solar zenith angle, station pressure, ozone and preeipitable water amounts, and three aerosol vari- ables; the unit air mass transmittance k, bulk single scattering albedo o~ and ratio of forward to total scattering .r Details of the calculation procedures are given by Davies and McKay[l]. Irradiances are calculated hourly from observed cloud layer amounts and types, surface pressure, and dry-bulb and dew point temperatures. For the Canadian data base calculations aerosol effects were neglected and precipitable water was obtained from dew point temperatures using an empirical equation (Won[3]).

Since radiation-driven mechanisms may depend non-linearly on irradiance the model must be evaluated with actual hourly rather than mean hourly data. Calculated values can than be averaged over any required time interval.

SOURCES OF ERROR Model estimates contain errors due to inadequacies in the

model and input data. Since indices of model performance com- pare estimates with measurements they also contain the measurement error. In Canada this is - 5% for hourly integrated irradiances (Hay and Wardle[4]). The major sources of error in model estimates for real atmospheres arise from incorporating aerosol and cloud effects. Standard error analysis shows that expected uncertainties in input data needed to specify aerosol and cloud parameters generate nearly all the error in estimates. Following Bevington[5], the standard deviation of the difference

between calculated and measured irradiance (as a fraction of irradiance) can be predicted by

n ~( ' ) = [~-1/OG\2 2"]1/2

I , (2)

where an uncertainty AX, is allocated to each variable used in the model. "lhe true value of this standard deviation can be deter- mined directly from

a(E) z = [(RMSE) z- (MBE)"] (3)

where RMSE and MBE are the root-mean-squared error and

mean bias error defined by

R?,lSE = [ l ~=t ,i2]tl2 (4)

and

1 n

MBE = n ~= ~i. (5)

Fixed values for ozone amount (3.5 ram), the aerosol parameters k and e, (0.9 and 0.7) and dew point temperature (20~ were used to evaluate equation 2. Cloud transmittances from Davies and Howard[6] were averaged for high, middle and low cloud categories.

Uncertainties assigned to the variables used in calculating irradiance are given in Table I. Values for ~o,/', a and /3 are estimates based on scanty published data. Comparison of pre- cipitable water calculations from upper air data at Buffalo and Toronto with estimates from Won's formula yielded RMS error of about 45 per cent (of the mean value), which, at a dew point temperature of 20~ corresponds to an uncertainty of 0.02 in water vapour absorptivity. The value of 0.05 for Ak is half the range of values (0.9-1) used in previous studie.s. Uncertainty in cloud layer amounts increases with height in recoguitioa of a ground observers' difficulty in specifying cloud layers above the lowest and the procedures used to correct for layer overlap (Davies and McKay[l]). Cloud transmissivity uncertainties are standard deviations calculated from the values of Kasten and Gzeplak[7], Atwater and Ball[8] and Davies and Howard[6].

RESULTS Values of 6-(~) were computed for different optical air mass

and for cloud amounts of 0 .2,0.5 and 0.9 which represent the mean (0.5) and the bimodal peaks of the U-shaped distribution which characterizes cloud amount (Essenwanger[9]). Since con- tribution to 6"(6) from ozone absorption and Raylcigh scattering are negligible they have been omitted.

Values of ~(6) increase with cloud amount (Table 2) in reasonable quantitative agreement with hourly o(E) values com- puted from 9 years of pooled (but not averaged) data for Goose, Montreal and Winnipeg (Table 2). Because MBE values are small values of o-(r and RMSE are virtually identical. Uncertainties in cloud variables are the dominant contributors to 6-(6) 2 (Table 3). Aerosols become significant only at smaller cloudiness values and at larger optical air mass. Contributions to 6(E) 2 from un- certainties in cloud layer amounts dominate at small cloud amounts (near the lower modal value of cloud frequency dis- tributions) and those from uncertainties in layer transmissivities at large amounts (near the upper modal value).

Since contributions of aerosol and cloud variables comprise virtually all of 6"(r z the model is insensitive to substantial error (45%) in precipitable water and can be used confidently with empirical formulas such as Won's[3]. The lack of sensitivity was confirmed for dew point temperatures of 0 and - 20~

Hourly RMS errors for five Canadian cities and one sub-Arctic station (Table 4) are about 28% of the mean measured hourly irradiance for the year. An example of such errors is given in Fig. I where model values are superimposed upon one day of measurements for Montreal (April 14, 1978) selected by Gautier

307

308 Technical Note

Table 1. Assigned uncertainties for calculating o(E)

X a k ~ f ct B C C C t t t

I w h m L h m s

8X .02 .05 .20 .I0 .05 .05 .20 .15 .I0 .15 .11 .I0 i

Table 2. Values of (a) 6-(e)(percentages) from equation 2 and (b) MBE and ~(e) (percentages of mean measured irradiance G for given cloud amount) computed from pooled data (1968-76) for Goose, Montreal and Winnipeg.

(a)

Cloud amount 0.2 0.5 0.9 Air mass

(b) Cloud amount

1 14 22 43 2 15 22 44 3 17 23 44 4 21 27 46 5 23 29 47

0.2 0 .5 0.9

G HBE c(c) G HBE o(e ) G HBE o(c)

Goose 333 0.5 9 345 - 4 . 2 22 209 - . 02 44 Hontreal 392 -2 .6 12 359 -4 .3 22 234 1.7 41 Ninnipeg 378 1.3 9 363 - 4 . 2 19 255 1.5 38

Table 3. Percentage of d(E 2) due to cloud and aerosol variables. Contributions from cloud amount c and transmissivity t are also shown

Cloud Aerosol

C l o u d amount 0 . 2 0 . 5 0 . 9 0 . 2 0 . 5 0 . 9

c t c+t c t c+t c t c+t

Air mass 1 82 I t 93 57 40 97 34 65 99 4 2 < 1 2 73 9 82 54 37 91 34 64 98 15 7 2 3 61 8 69 50 34 84 33 63 96 27 14 4 4 37 5 42 37 26 63 30 57 87 56 36 12 5 31 4 35 33 23 56 29 55 84 63 43 16

900

800 t

7OO

60O

500

400

3OO

2OO

I00

0 5

/ / I I

i I

1 I

z v - i i i0 15 2o

Loca l apparent, t ime

Fig. 1. Diurnal variation of measured (shaded area) and cal- culated global irradiance for Montreal on 14 April 1978. The dashed line refers to the satellite model of Gautier et a/.[10] and

the solid line to the numerical model discussed in this paper.

et all10] to show the performance of their numerical model which uses satellite data. The model under discussion follows the diurnal variation of measured irradiance rather better than the satellite model and probably as well as can be expected for a model which uses surface data. The hourly RMSE is 30 per cent of the mean measured hourly irradiance for the year (which is similar to the mean measured hourly irradianee for the day).

If errors in the cloud variables, which determine the magnitude of hourly RMSE, are random, RMSE for averaged irradiances will decrease in inverse proportion to the square root of the averaging period. For monthly mean hourly data RMSE is less than 10 per cent and close to the values obtained if errors are random (Table 4).

Daily RMSE ( ~ 16 per cent) also decreases in proportion to the square root of the averaging period to below 10 per cent for a week and below 5 per cent for a month. For monthly averages the RMSE approaches the MBE for individual months. Random errors are then small. On a monthly basis the model's per- formance compares favourably with the best results published for other models.

CONCLUSION This discussion of sources, magnitudes and behaviour of errors

in irradiance estimates from a numerical model may be useful to users of the Canadian solar radiation data base. Most users of solar irradiance data probably require average values. Because errors are mainly random RMSE decreases rapidly and predict- ably with increasing averaging period for bolh mean hourly and

Technical Note

Table 4. Errors for hourly and daily irradianees for six Canadian Stations (1968-76). Errors are expressed as percentages of mean measured irradiance. N is the number of hours in an averaging period

Station Goose Charlotte Montreal Toronto Winnipeg Vancouver -fowl%

HOURLY IRK/~IAMCES

Mean measured s (W/m 2) 228 273 267 294 290 258

MBE (hourly and daily) 0.2 0.2 I.O -3.3 1.1 -0.8

RMSE Hourly 34 29 29 27 25 29 Monthly mean hourly 10 8 7 7 7 9

Hourly//-N 7 6 6 5 5 6

DALLY IRRADIANCES

Mean measured Irradlances

(MJ/m2/d) 10.54 12.54 12.15 13.38 13.28 11.76

RMSE Daily 19 17 13 15 15 17 Averaging period

(days) 5 10 9 8 8 7 9

30 5 4 4 5 3 5

309

mean daily irradiances. Therefore, error for a given averaging period is easily estimated or an averaging period can be selected to ensure a required level of accuracy.

Hourly RMSE for this model can be accounted for by random errors of about 0.1-0.15 in both cloud layer amounts and trans- missities. It is unlikely that a model which uses surface data can reduce the RMSE significantly. This may be possible with a model which incorporates satellite reflectivity data since these include the combined effect of variations in cloud amount and transmissivity. However, comparisons between the results of Gautier et al.[10] and results with this model (Davies and Abdel- Wahab[ll]) using data for the same days at Montrdal, Ottawa and Toronto showed little difference in either hourly (5 days) or daily (45 days) performance.

Acknowledgements--The research was supported by a contract with the Canadian Atmospheric Environment Service. We are grateful to Dr. D. C. McKay for his encouragement and assis- tance in this study and to Mr. Michael Yu for programming assistance.

REFERENCES 1. J. A. Davies and D. C. McKay, Estimating solar irradiance

and components. Solar Energy 29, 55---64 (1982). 2. F. C. Hooper, C. R. Attwater, A. F. Brunger, J. A. Davies, J.

E. Hay, D. C. McKay and T. K. Won, The Canadian solar radiation data base. ASHRAE Trans. 85, 497-506 (1979).

3. T. K. Won, The simulation of hourly global radiation from hourly reported meteorological parameters--Canadian Prairie area. Paper presented at the 3rd Conf. Canadian Solar Energy Society Inc., 23 p. Edmonton, Alberta (1977).

4. J. E. Hay and D. I. Wardle, An assessment of the uncertainty in measurements of solar radiation. Solar Energy 29, 271-278 (1982).

5. P. R. Bevington. Data Reduction and Error Analyses for tire Physical Sciences, 336 p. McGraw-Hill, New York (1969).

6. J. A. Davies and J. E. Howard, Determination of cloud transmittance parameters for Canada. Report to the Canadian Atmospheric Enrironment Sen'ice, 73 p. (1981).

7. F. Kasten and G. Czeplak, Solar and terrestrial radiation dependence on the amount and type of cloud. Solar Energy M, 177-190 (1980).

8. M. A. Atwater and J. T. Ball, A surface solar radiation model for cloudy atmospheres. Mon. Wea. Rer. 109, 878-888 (1981).

9. O. Essenwanger, Applied Statistics in Atmospheric Science. Part A. Frequencies and Curre Fitting; Developments in Atmospheric Science, 4A, 412 p. Elsevier, New York (1976).

10. C. Gautier, G. Diak and S. Masse, A simple physical model to estimate incident solar radiation at the surface from GOES satellite data. J. Appl. Meteorol. 19, 1005-1012 (1980).

11. J. A. Davies and M. Abdel-Wahab, Evaluation of models for simulating solar radiation incident on horizontal surfaces. Report to the Canadian Atmospheric Em'iromnent Senice, 63 p. (1982).