a study of the division of global irradiance into direct beam and diffuse irradiance at seven...

Pergamon 0038-092X(95)00088-7

Solar Energy Vol. 55, No. 6, pp. 493S504,1995 Copyright 0 1995 Elsevier Science Ltd

Printed in the U.S.A. All rights reserved 0038-092X/95 $9.50 + 0.M)

A STUDY OF THE DIVISION OF GLOBAL IRRADIANCE INTO DIRECT BEAM AND DIFFUSE IRRADIANCE AT SEVEN CANADIAN SITES

JOHN GARRISON and KAMRAN SAHAMI Physics Department, College of Sciences, San Diego State University, San Diego, CA 92182-0325, U.S.A

(Communicated by RICHARD PEREZ)

Abstract-Canadian hourly global and diffuse irradiation data and associated daily surface meterological data of humidity, temperature and snow depth for the years 1977-1984 are analyzed. These data have been measured at Edmonton, Goose Bay, Montreal, Port Hardy, Resolute, Toronto and Winnipeg. Hourly values of the clearness index k, and diffuse index k, are sorted into bivariate histograms according to their numerical values. Different histograms are established for different ranges of the three variables: solar elevation, atmospheric precipitable water, and snow depth for each station. Properties of the different histograms are compared using standard statistical procedures. It is found that the division of global irradiation into direct beam and diffuse irradiation is correlated with the four variables k,, precipitable water, solar elevation, and snow depth. It is also found that many, but not all, of the differences between data from the same station at different times and between different stations can be attributed to conditions associated with differences in these four variables. The data show evidence that the division of global irradiation into direct and diffuse irradiation can depend upon the properties of the clouds beyond how these clouds are characterized by the four variables

1. INTRODUCTION

This presentation studies the division of global irradiance into direct beam and diffuse irradi- ante for the seven Canadian stations of Edmonton, Goose Bay, Montreal, Port Hardy, Resolute, Toronto and Winnipeg. The data analyzed here consist of measurements of hourly values of global irradiation, diffuse irradiation, and corresponding daily measurements of temperature, humidity, and snow depth. These data have been obtained from the Atmospheric Environment Service, Downsview, Ontario, Canada. Table 1 lists the latitude, longitude, altitude and number of years of data for the seven stations. The data for the years prior to 1977 are not used in this initial paper. All the stations have seven or more years of irradiation and surface data. Data length is an important consideration in obtaining sufficient statistical precision, when the data are divided into the large number of categories considered in this study. The method of collection, numerical pro- cessing and accuracy of these data are discussed

Table 1. Station data

Station Lat Long Alt (m) Years

Edmonton 53.5 113.5 206 8.1 Goose Bay 53.3 60.4 13 22.6 Port Hardy 51.0 127.1 7.6 Montreal 45.5 73.6 17 20.2 Resolute 74.5 95.0 27.5 Toronto 43.7 79.4 35 17.4 Winnipeg 49.9 97.2 254 8.1

by Latimer (197%1980), Bristow (1980), Wilson (1980) and Hay and Wardle (1982).

Studies of these Canadian solar irradiation data have been done previously by Yamashita (1974), Ruth and Chant (1976), Hay (1976), Tuller (1976), Orgill and Hollands (1977), Polavarapu (1978), Iqbal (1979a,b; 1980), Hay and Wardle (1982), Davies and McKay (1982), Uboegbulam and Davies (1983), Freund (1983), Hollands and Crha (1987), Graham, Hollands and Unny (1988), Kierkus and Colborne (1989), and Graham and Hollands (1990). These studies indicate many important aspects of these Canadian data.

Liu and Jordan (1960) first showed that the division of global irradiation into direct and diffuse irradiation depends upon k,, the clearness index. Bugler (1977) and Iqbal ( 1980) were among the first to note that the division of global irradiation into direct and diffuse irradiation depends upon solar elevation, c(. Garrison (1984; 1985) has shown that the U.S. data exhibit a dependence upon c(. LeBaron and Dirmhirn (1983) and, more recently, Garrison (1985) and Kierkus and Colborne (1989) have shown that the large increases in surface albedo caused by snow have a strong effect on the division of global irradiation into direct and diffuse irradiation. A strong correlation of the division of global irradiation into direct and diffuse irradiation with atmospheric precipitable water, W, has been shown by Garrison (1985).

In this study, the division of hourly global

493

494 J. Garrison and K. Sahami

irradiation into direct and diffuse irradiation is related to the four primary variables: the clearness index, k,, the solar elevation, CI, atmospheric precipitable water, W, and surface albedo as affected by snow depth, S. The diffuse index, kd, and direct beam index, k,,, are used here, also. They are not entirely independent of the other variables, since knowledge of the values of k,, c(, W and S places some limit on the possible values of kd and k,,. It is anticipated that values of the variables k,, kd, a, W and S will provide some numerical measure of the amount and kind of cloud cover. It will be found that values of the variables k,, k,, CI, W and S are not sufficient to characterize completely the clouds as they affect the division of global irradiation into direct beam and diffuse irradiation. Reindl et al. (1990), have performed a statistical study of North American and European solar irradiation data searching for suitable variables such as those used here. Their study resulted in the selection of the four variables: clearness index, k,, solar elevation, ~1, mean ambient temperature and relative humidity. The method of analysis of this study is similar to earlier studies done by us, Garrison (1985) Garrison et al. (1991), Garrison and Sahami (1993).

Surface albedos for Canadian stations have been estimated by Hay (1976) and Gueymard (1993). Albedos for U.S. stations including stations near the Canadian border are found in a tabulation by Hoyt (1979). These studies indicate that when snow of sufficient depth is present the albedo has values in the range of about 0.5-0.8. In the absence of snow, the albedos for the six Canadian stations other than Resolute are much lower. They are expected to be about the same, in the range from about 0.10 to about 0.20. Resolute has no vegetation, light ground, and open landscape. It is expected to have a higher albedo than the other six stations, both with and without snow. In this study, only two categories with widely differing albedos, no snow and snow depth greater than five centime- ters, are used to characterize the data according to albedo. A category for snow with depths less than 5 cm is not considered because of a paucity of hours in this category and a greater variability of the albedo for these hours. It would be difficult to have more categories for surface albedo with no snow. The differences between stations and the changes with time of the surface albedo with no snow present are too small. They have not been measured.

Gueymard (1994) recently reported monthly

average atmospheric turbidity values for many Canadian stations which increase significantly the published information on turbidity at Canadian stations. The turbidities tabulated by Gueymard do not include the seven stations studied here. It would be useful if turbidity could be used as one of the variables for separating the data into different groups. However, the nature and amount of the solar irradiation data analyzed here do not allow the incorporation of turbidity as one variable to be used in the grouping of the data. Turbidity and precipitable water are both seasonal and vary from higher values in the summer to lower values in the winter. Separating the data for each station according to different values of the variable W is, to some extent, also a separation according to turbidity.

2. METHOD OF ANALYSIS

2.1. Sources of the values of the variables k,, k,, k,, S, Wand cx

Hourly values of k,, the clearness index, k,, the diffuse index, and k,, the direct beam index, are calculated from the hourly global and diffuse irradiation using the relations first introduced by Liu and Jordan (1960)

k,= G dt/ I, cos 8, dt (clearness index) (1) s s

kd= Gd dt/ I, cos 8, dt (diffuse index) (2) s 5

kb = k, - kd (direct beam index). (3)

In eqns (l)-( 3), I, is the extraterrestrial solar irradiance whose average value is taken to be 1367 W mm2 (solar constant). 1, is assumed to vary sinusoidally over the year with an amplitude of 45 W mm2 and a maximum on 1 January. 0, is the zenith angle, G is the total irradiance on a horizontal surface (global irradiance) and G, is the diffuse irradiance on a horizontal surface. Integration is over an hour. The numer- ators in eqns (1) and (2) are the hourly global irradiation and diffuse irradiation provided by the Atmospheric Environment Service of Canada.

Daily values of snow depth are part of the data provided by the Atmospheric Environment Service of Canada. The daily values of snow depth, S, are used for each hour of the day.

Mean atmospheric precipitable water, W, for each day is calculated using the method of

A study of the division of global irradiance 495

Garrison and Adler (1990), but using daily, rather than monthly values of mean surface humidity and temperature. The daily value of W is used for every hour of the day. Studies by Bolsenga (1965) and Wright et al. (1989) provide additional justification for using this procedure for daily values. A comparison of monthly averages of the daily precipitable water obtained this way for these data are in good agreement with the results of Gueymard (1994).

Solar elevation, c(, and solar zenith angle, 6, = 90” -a, are calculated from the hour of the day and day of the year using standard formulas (see Duffie and Beckman, 1991, for example). The values of a used in this analysis are calculated at the midpoint of each hour.

2.2. The bivariate histograms Hours of irradiation data with values of k,,

and k,, and associated values of S, W, and c( are sorted into bivariate histograms according to the values of these variables. Each histogram has 24 intervals of 0.04 in k, covering the range o-0.96, and 19 intervals of 0.04 in kd covering the range o-0.76. A different histogram is established for each of two different ranges of snow depth, for each of three different ranges of atmospheric precipitable water, for each of five ranges of solar elevation angle, for each different station, and for each of the four two year intervals for the eight years from 1977 to 1984. The ranges for the intervals in S, W and c( are given in Table 2.

Lower case symbols for S, W, and a are used for indices to characterize each histogram and quantities obtained from them. Snow depth uses s= 1 for no snow and s=2 for snow depths greater than 5 cm. Precipitable water uses w = 1, 2 and 3 to represent the three ranges of precipitable water in order of increasing precipitable water. Solar elevation uses CC= 1, 2, 3, 4 and 5 to represent the five ranges of solar elevation in order of decreasing solar elevation (increasing zenith angle). The values of the indices used with each range of the variables S, W and tl are included in Table 2.

Table 3 shows the number hours of data

in each interval of k, and kd in a histogram for s= 1, w =2, and a= 3. Table 3 is for Montreal for the 8 yr from 1977 to 1984, combined. The values of these indices indicate that the hours sorted into this histogram have S, W, and a in the ranges: S=O, 1~ W < 2 cm, and 30” < CI < 43”. Zero hours of data in an interval of k, and kd are omitted from Table 3. All the information concerning the division of global irradiation into direct and diffuse irradiation is contained in the distribution of the hours of data among the different intervals of kd and k,. For low values of k, when the sky is overcast, k, should equal k,. In Table 3, a few values of kd are in histogram intervals one greater or one smaller than the corresponding k, interval. This provides some measure of the uncertainty in the measurements.

2.3. Quality control of the data Although these data have already been sub-

jected to tests of validity, the values of k, and kd are examined by computer to test whether they satisfy certain conditions. No hourly irradiation data with solar elevation angle less than 9” at the midpoint of the hour are used. No hours are used in which data are not available for any one or more of the four quantities: global irradiation, diffuse irradiation, precipitable water, and snow depth. The majority of the hours of data are complete in the data for all four of these quantities. Values of k, and k, outside the range covered by the histograms are rejected. Values of kd greater than 1.15 k, are also rejected.

An examination of the histograms for all years of data for each station and each set of values of w, s, and CY shows no values for k, or k, which are well outside the region of the histogram where most of the values are found.

2.4. Selection of parameters to characterize the division of global irradiation into direct and d@use irradiation

For each interval of k, in each histogram, the value of (kd ), the mean value of kd for this interval of k, is determined. For each interval of k, in each histogram, the value of (T, the standard

Table 2. Range of values of variables

Variable Range Units

Snow depth s=o 5>s cm Prec. water O<W<l l<W<2 2< W<4 Solar elevation 601~ 43<m<60 30<a<43 18!x<30 9<%<18 Index 1 2 3 4 5


Table 3. Bivariate distribution of k,,kd values for Montreal (S=O cm, 1~ IV<2 cm, 30~~~43”)

0.54 0.50 1 0.46 3 2 3 1 1 0.42 5 17 16 6 6 3 3 0.38 12 31 21 16 20 12 5 4 1 1 0.34 25 34 12 15 23 11 17 9 5 2 2 1 0.30 44 39 24 17 11 12 15 25 22 17 7 3

k, 0.26 1 62 24 10 5 9 8 12 16 13 30 31 24 5 3 0.22 3 76 12 5 3 4 6 4 9 8 18 28 31 35 9 4 2 0.18 106 6 1 2 2 4 1 6 9 32 71 46 9 1 0.14 194 9 1 3 4 8 91 104 19 1 0.10 2125 7 1 41 226 34 0.06 2 136 6 1 37 15 0.02 80 6

0.02 0.06 0.10 0.14 0.18 0.22 0.26 0.30 0.34 0.38 0.42 0.46 0.50 0.54 0.58 0.62 0.66 0.70 0.74 0.78 0.82 k

deviation of k, for this interval of k, is deter- sponding to k, less than about 0.3, when k, = kd mined. The values of (kd) and u are the first and kb is zero. Also, there are few hours of data and second moments of the distribution of the with k, above the range of values for clear hours. values of kd for each interval of k, in each Thus, tests for significant differences between histogram. The distributions of kd are not histograms (histograms with different s, w, and normal distributions. These two moments do a; different stations; or different year group) not completely characterize the distributions. using values of (kd ) are performed best in the However, they are assumed to be sufficient for partly cloudy interval of k, near and below the indicating the properties of the division of global clear hour values of k, and above k, =0.3. The irradiation into direct beam and diffuse irradia- mean value of (kd ) for the seven intervals of k, tion and for testing similarities and differences from 0.44 to 0.72 is given the symbol (kd)range. between the histograms used in this study. (kd Lge is used to test for differences between

Figure 1 shows (kd) as a function of k,, for the bivariate histogram of Table 3. k, is the value of k, at the mid-point of each k, interval. This is close to the average value of k,, for the small intervals used here. Also included in Fig. 1 is a curve of (kb) as a function of k, to show the relation of (kb) to k, and (kd). (k,,) is the average value of k, for each interval of k,.

histograms. The seven intervals of k, are selected so as to cover the approximate region of k, where differences in (kd ) for different stations are expected to be most significant.

In order to test whether differences between different values of (kd)range for different histograms are significant the standard deviation of

(kd )range for each histogram must be determined.

2.5. Method of comparison of histograms A plot of (kd) versus k, is the same for all

histograms for the overcast condition corre-

Similarly, the mean value of c for the seven intervals of k, from 0.44 to 0.72, (cr),ange, and its standard deviation are obtained. The division of the data into four groups of 2 yr each for the 8 yr 1977-1984 allows estimation of the standard deviations of (kd)range and (g)range from the fluctuations of these quantities from year group to year group.

MONTREAL s=l w=2 IJ= 3

0.6

0 0.25 0.5 0.75 1

Kt

Fig. 1. A plot of (kd) versus k, and a plot of (kb) versus k, for Montreal for s = 1 (no snow), w = 2 (precipitable water between 1 and 2 cm) and u = 3 (solar elevation between 30

and 43”) for the years 197771984.

The values of (kd)range and (C)_,~~ and their standard deviations for all the different sets s, w, and tl are presented in tabular form below. This allows a numerical comparison of the values of (kd)range and (U)~_~ and their standard deviations for all the different sets s, w, and CI to see how they vary from station to station and how they vary as a function of S, W, and CC. It is of particular interest to see if the values of (kd)range and (c),,,~~ can be treated the same for the different stations when the


values of s, w, and CI are the same. If values of

(Mrange and <B),,,~~ can be treated the same for the different stations when the values of s, w, and CI are the same, then characterizing the hours of data by their associated values of k,, kd, s, w, and CI is sufficient for modeling solar irradiation. The values of (kd)range and (G)_~~ for each set of s, w, and CI are examined for differences between stations. The differences in values for pairs of stations are used with their standard deviations to determine whether the differences are statistically significant.

2.6. Instrument errors and data uncertainties

In making the comparison between stations, an estimate of the instrument error must be made. It has been stated that the instrument errors for these data “are in the order of +5% under the best of circumstances” (Latimer, 1980). Systematic error arising from changes in sensitivity, when internal to the instrument, should be corrected during recalibration. External changes in sensitivity can arise from, for example, deposition of dust, dew, rain, ice, snow or frost on the generally clean pyrano- meter windows, for example. Both these types of errors should be, for the most part, included in the standard deviations obtained for (kd)range and (gLange. The added error arising from differences in instrument response for the two stations compared must be combined with the standard deviations for each station to obtain the standard deviation for the differences in the values of (kd)range and (cJ)~~“~~ for two stations. The calibration error for these instruments is stated to be about f2% (Latimer, 1980). Part of this calibration error must be common to all the stations, since all the stations except Resolute use the same models of instruments and are calibrated in the same manner. The added instrument error to be combined with the standard deviations of (kd)range should lie between the values of about +2% (or less) and *5% (or more). For this study, the added instrument error is taken to be +3% for use in comparison of the values of (kd)range of two stations.

3. RESULTS OF ANALYSIS

3.1. Average k, in the range of k, from 0.44 to 0.72

Table 4 shows the average values of (kd)range and their standard deviations, for the seven Canadian stations, separately. These are for the

different values of s, w, and a considered here, and for the 8 yr 1977-1984, inclusive. The 82

values of (kd Ange and their standard deviations are obtained from 82 x 4 = 328 histograms, with an average of roughly 600 h of data in each histogram. Each value of (kd)range in the table is the mean of four values obtained from four histograms containing 2 yr of data each for the 8 yr from 1977 to 1984. Each standard deviation

of (kd )range is the standard deviation of the mean for the four values.

The standard deviations in Table 4 include that part of the instrument error which involves changes in the response of the detectors over time for each station, but not the part which involves differences between detectors at different stations or calibration errors. The added independent instrument error of +3% beyond that already included in the standard deviations of Table 4 is combined with the standard deviations of Table 4 for each station for the statistical tests presented below.

With the separation of the hours of irradiation data into different bivariate histograms according ranges of S, W, and LX, histograms with the same ranges of S, W, CI for different stations are expected to have close to the same values of

(kd )range and <ojrange. Differences in values of

<kdjrange and <o)range among the stations for the same ranges of S, W, c( will be attributable to differences in surface albedo, turbidity, instrument response, differences in the types of clouds beyond that characterized by the four variables: k,, S, W, and tl, and statistical fluctuation. The averages of Wand IX for hours within each range for W and CI may also differ somewhat for the different stations.

The following statements can be made concerning Table 4: ?? The values of (kd)range show a monotonic

decrease with decreasing solar elevation, which can be attributed to increased atmospheric attenuation of the solar radiation as the air mass traversed by the solar radiation increases.

?? The values of (kd)range for higher solar elevations, for the most part, show a monotonic decrease with increasing atmospheric precipitable water (turbidity is also generally increasing). This also can be attributed to increased atmospheric attenuation of the solar radiation. Port Hardy is an exception which is discussed below. The cases for which a monotonic decrease in (kd)range with increasing W do not occur are associated


Table 4. (kd) for 0.44 < k, < 0.72

s w c( Edmonton Goose Bay Montreal Port Hardy Resolute Toronto Winnipeg

1 1 2 0.319~0.007 0.321 kO.010 0.363 +0.004 0.344 & 0.007 1 1 3 0.32OkO.007 0.318 kO.008 0.264~0006 0.389iO.036 0.331* 0.007 0.330~0.009 1 1 4 0.276&0.011 0.269+0.010 0.266&0.013 0.262+0.009 0.338*0.016 0.283 0.005 k 0.275+0.013 1 1 5 0.241kO.009 0.235+0.010 0.225+0.012 0.220&0.011 0.285&0.015 0.240 0.010 k 0.247kO.012 1 2 2 0.302)0.004 0.308 k 0.003 0.336 + 0.006 0.317~0.002 0.300 0.006 & 1 2 3 0.29OkO.005 0.281 k 0.005 0.298 k 0.003 0.288 0.004 + 0.284 k 0.009 1 2 4 0.255kO.005 0.249 + 0.005 0.267 k 0.006 0.256 0.005 k 0.252 0.014 k 1 2 5 0.229+0.009 0.227 k 0.008 0.241 0.007 f 0.226 0.007 k 0.232 k 0.002 1 3 2 0.290+0.004 0.305 k 0.003 0.358 0.009 + 0.314+0.001 0.289 +0.003 1 3 3 0.278 k 0.003 0.280 kO.004 0.305 k 0.005 0.293 0.002 f 0.267 kO.006 1 3 4 0.244kO.005 0.245 f0.003 0.254+ 0.009 0.254 0.005 f 0.240 0.007 & 1 3 5 0.238kO.008 0.235+0.012 0.217 0.239 0.258 kO.018 2 1 3 0.433 kO.013 0.429+0.012 0.377kO.008 0.529&0.010 0.444 0.005 + 0.474f0.015 2 1 4 0.384+0.010 0.377kO.010 0.352&0.003 0.45OkO.012 0.397 kO.013 0.410~0.015 2 1 5 0.310+0.006 0.295kO.005 0.276&0.011 0.361 kO.004 0.304+0.010 0.338 k 0.019

with air which has an increased scattering of radiation that more than compensates for the added attenuation of radiation associated with higher values of W and turbidity. This can occur if the nature of the clouds changes and/or if the increased attenuation by the aersols is caused by more scattering by these aersols. Diffuse, thin clouds can yield much more diffuse radiation than thicker scattered clouds for the same value of k,, for example. The effect of increasing W on (k, )range, which represents partly cloudly skies, is seen here to be mixed. However, there is a definite, monotonic decrease in clear hour global and direct beam radiation with increasing W. For the s= 1 and w=2, 3 cases for all solar elevations, the Port Hardy values are almost all higher than the values for the other stations. The higher values for Port Hardy are similar to what is observed in the U.S. data for the west coast stations of Los Angeles, California; Medford, Oregon; and Seattle, Washington in the summer season (Garrison, 1985). This could be due to the common occurence of a marine cloud layer during the night and part of the morning on the west coast of the U.S. and Canada, which evapo- rates as the air temperature rises. Evaporation of the cloud layer generally occurs by a thinning of the marine layer leading to a generally brighter sky with associated high values for the diffuse radiation. For each set of ranges of S, W, and c( for Resolute, where data are available, the values of <kd )range are considerably higher than the corresponding values for all of the other stations. This has been noted previously by Tuller (1976). Several factors may contribute to this increase in diffuse radiation: (1) reso-

lute is expected to have higher surface albedo, both with and without snow; (2) the top of the troposphere in the Arctic is about 10 km. This is below that at lower latitudes where it is as high as 15 km. The difference in height of the troposphere between Resolute and the other stations may reduce the attenuation of radiation through the atmosphere and increase the illumination of the clouds; (3) cloud conditions may differ at this high latitude and may provide more diffuse radiation than what is observed at the other stations. For example, thin fog or cloud layers would provide more diffuse radiation; (4) atmospheric precipitable water averages are lower at Resolute for the w= 1 cases (the only prevalent data for Resolute) than at the other stations.

3.2. A statistical comparison of the values of (k,Jrange of the stations

The values of (kd)range and (u)*_~ obtained from histograms with the same values of s, w and CI are similar for the different stations. This similarity is tested here numerically by assuming that values of (kd)range from histograms with the same, s, w and CI for all stations are drawn from the same parent population. The central limit theorem of statistics indicates that the

values of <kd)range listed in Table 4 can be considered to be normally distributed. The differences in values of (kd)range for each pair of stations will be normally distributed also.

Each entry in Table 5 gives the probability that the absolute value of a difference between pairs of stations equal or exceed the observed difference. It is assumed that the difference in values of (kd)range for a particular set of values of s, w, and LX and pair of stations is drawn from

A study of the division of global irradiance

Table 5. The probability that a (k,) difference between two stations be larger than the observed difference

499

Stationlswcc 113 114 115 122 123 124 125 132 133 134

Edmonton compared to: Montreal 0.893 0.636 Port Hardy 0.000 0.431 Toronto 0.528 0.698 Winnipeg 0.568 0.937

Montreal compared to: Port Hardy 0.001 0.819 Toronto 0.452 0.376 Winnipeg 0.492 0.705

Port Hardy compared to: Toronto 0.000 0.180 Winnipeg 0.000 0.508

Toronto compared to: Winnipeg 0.973 0.651

0.349 0.215 0.934 0.766

0.810 0.062 0.212 0.182 0.336 0.002 0.077 0.536 0.404 0.506 0.599 0.578 0.934 0.512 0.316 0.461 0.255 0.625 0.836 0.880 0.701 0.223 0.311 0.701

0.261 0.200 0.473 0.433 0.268 0.011 0.395 0.989 0.155 0.027 0.371 0.409 0.458 0.000 0.008 0.375

0.715 0.266 0.795 0.804 0.608 0.057

0.662

0.273 0.941

0.531 0.630 0.904 0.253 0.865 0.912 0.550 0.391 0.421 0.000 0.051 0.489 0.913 0.940 0.836 0.070 0.233 0.414 0.724 0.847 0.819 0.958 0.387 0.784

0.306

a parent population which is normally distributed with zero mean and which has a standard deviation equal to the standard deviation of the difference. High probabilities indicate that the observed difference between stations is smaller than the standard deviation.

The following statements can be made concerning Table 5:

Under the assumptions made above, approximately one-third of the probabilities in Table 5 should lie near or below 0.3. This is clearly not the case for the four stations of Edmonton, Montreal, Toronto and Winnipeg, which have higher probabilities indicating that the differences in (kd)range for these four stations are smaller than would be expected from each parent distribution standard deviation. Probably the main part of the explanation for this is that the values of

<kd )range for the four stations are correlated in time, with values for all stations increasing in years with larger amounts of volcanic aerosol in the stratosphere and decreasing in years with smaller amounts of volcanic aerosol in the stratosphere. The change in

(kd Lange from years of low amounts of volcanic aerosol in the stratosphere to the high amounts in the year or so following the eruption of El Chichon in Mexico in early 1982 is of the order of 8-lo%, for the lowest two ranges of W. The changes in volcanic aerosol in the stratosphere will increase the standard deviation of (kd)range for all the stations without a corresponding increase in the differences between stations. It can be concluded that the four stations of Edmonton, Montreal, Toronto and Winnipeg have values of (kd)range which can be considered the same within the accuracy of their

determination when including the changes which can occur due to volcanic eruptions. The probabilities in Table 5 obtained when comparing Port Hardy with the four stations of Edmonton, Montreal, Toronto and Winnipeg are generally lower than the probabilities obtained when comparing the four stations with each other. This reflects the fact that the values for the s= 1, w = 1 cases for Port Hardy are lower than all the other stations in Table 4, while the s = 1, w = 2 or 3 cases are higher than the other stations. The results for all stations with s = 2 are not included in Table 5 because of the larger variability of the surface albedo with snow on the ground. The stations at Goose and Resolute do not appear in Table 5, since the s= 1 no snow data for these stations have a very limited number of hourly k,, kd pairs. The case for each station with s= 1, w = 3, and tl= 5 is not included in Table 5 and should not be considered as significant in Table 4 because the clear hour values of k, for some stations lie mostly well below the upper end of the range of (kt)range for 1983, following the eruption of El Chichon in Mexico.

3.3. Average G in the range of k, from 0.44 to 0.72

Table 6 shows values of (G)_~~, and their standard deviations, for the seven Canadian stations. These are for the different values of s, w, and a considered here, and for the 8 yr 1977-1984, inclusive. The added instrument error has not yet been included in the standard deviations of Table 6.



Table 6. (a) for 0.44 <k, < 0.72

s w G( Edmonton Goose Bay Montreal Port Hardy Resolute Toronto Winnipeg

1 1 2 0.071* 0.004 0.054 + 0.005 0.049 + 0.005 0.063 & 0.008 1 1 3 0.068 k 0.004 0.066 + 0.007 0.057 kO.006 0.057 + 0.005 0.066 + 0.007 0.060 & 0.007 1 1 4 0.072 kO.003 0.059 +0.003 0.066 + 0.005 0.062 +O.OlO 0.072 + 0.006 0.067 & 0.003 0.064 + 0.003 1 1 5 0.074 kO.002 0.056 kO.012 0.058 k 0.003 0.069 kO.005 0.072 k 0.011 0.069 k 0.002 0.061 k 0.003 1 2 2 0.072 &O.OOl 0.067 kO.001 0.077+0.004 0.063 k 0.001 0.064 k 0.002 1 2 3 0.070 * 0.002 0.067 kO.001 0.077+0.003 0.059 * 0.002 0.066 * 0.001 1 2 4 0.068 k 0.002 0.063 k 0.001 0.080 k 0.004 0.060 k 0.002 0.062 +O.OOl 1 2 5 0.074 + 0.003 0.062 kO.003 0.085 + 0.004 0.066 + 0.003 0.058 +0.004 1 3 2 0.069 + 0.003 0.066 kO.003 0.074 & 0.005 0.061 If: 0.001 0.062 + 0.003 1 3 3 0.065 k 0.005 0.059 kO.003 0.077 * 0.003 0.058 * 0.002 0.059 kO.002 1 3 4 0.067 k 0.004 0.056 kO.002 0.084 & 0.005 0.055 * 0.001 0.060 & 0.002 1 3 5 0.068 k 0.002 0.059 f0.007 0.080 k 0.000 0.090 0.054 * 0.003 2 1 3 0.086 k 0.007 0.067 T 0.004 0.66 * 0.003 0.053 kO.004 0.058 * 0.011 0.061 k 0.004 2 1 4 0.085 + 0.003 0.078 i 0.004 0.71 kO.005 0.078 f0.003 0.073 * 0.003 0.079 k 0.006 2 1 5 0.093 & 0.005 0.079 & 0.006 0.68 kO.008 0.089 +0.008 0.073 + 0.007 0.090 + 0.006

The values of (B),,,,~~ for s= 1, and w=2 and 3 for Port Hardy are higher than those of all other stations. The marine cloud layers found along the west coast of the continental United States and Canada provide a possible explanation for this, as they did for the high values

of (kd Lange for Port Hardy discussed in Section 3.1. The values of (o)*_~ for Edmonton are higher than those of all other stations except for the s = 1, w = 2 and 3 cases for Port Hardy.

the standard deviation of the difference. High probabilities indicate that the observed difference between stations is smaller than the standard deviation.


Montreal, Toronto and Winnipeg, and perhaps Edmonton, have differences in values of

(0) range which are smaller than the standard deviations here would lead us to expect, as indicated by the probabilities in Table 7. This is related most likely to the correlated changes in values of (c)ran_ with changes in the concentration of volcanic aerosols in the stratosphere for the four stations, like those noted for (kd)range above. The probabilities of Table 7 indicate that Port Hardy has values of (o)*_~ which are consis- tent with the values of the other stations, for the standard deviations found here. Thus, all five stations have individual values of (g)range which can be considered the same when including the effects of volcanic eruptions on

3.4. A statistical comparison of the values of

(0) *ange of the stations

Each entry in Table 7 gives the probability that the absolute value of a difference in values

of (cLange between pairs of stations equal or exceed the observed difference. It is assumed that the difference in values of (cr)range for a particular set of values of s, w, and CI and pair of stations is drawn from a parent population which is normally distributed with zero mean and which has a standard deviation equal to

Table 7. The probability that a (u) difference between two stations be larger than the observed difference

Station/swcc 113 114 115 122 123 124 125 132 133 134

Edmonton compared to: Montreal 0.914 0.593 Port Hardy 0.467 0.500 Toronto 0.926 0.647 Winnipeg 0.610 0.514

Montreal compared to: Port Hardy 0.572 0.826 Toronto 0.991 0.927 Winnipeg 0.705 0.932

Port Hardy compared to: Toronto 0.571 0.762 Winnipeg 0.878 0.876

Toronto compared to: Winnipeg 0.701 0.852

0.129 0.691 0.763 0.683 0.257 0.832 0.661 0.323 0.646 0.755 0.586 0.314 0.302 0.726 0.396 0.155 0.609 0.478 0.375 0.486 0.456 0.548 0.601 0.28 1 0.248 0.541 0.717 0.574 0.163 0.592 0.660 0.519

0.314 0.499 0.395 0.158 0.034 0.598 0.188 0.016 0.316 0.753 0.545 0.762 0.709 0.705 0.931 0.918 0.762 0.836 0.947 0.874 0.752 0.753 0.987 0.711

0.971 0.337 0.157 0.096 0.083 0.388 0.161 0.013 0.498 0.382 0.365 0.120 0.020 0.419 0.177 0.036

0.508 0.910 0.593 0.883 0.505 0.946 0.915 0.638


the data, even though statistically significant differences in the values of (G),._~ for Port Hardy and Edmonton from those of the other stations appear to exist when considering all values of (0)range.

3.5. The dependence of the variation of (kd) and a with k, upon S, W, and u

Values of (k,,), for each interval of k, for the different histograms are found to be rather similar for the four stations: Edmonton, Montreal, Toronto, and Winnipeg. The histograms for these four stations have been combined for each set of values of s, w, and a for the 8 yr from 1977 to 1984 to yield Figs 2, 3, and 4 which illustrate representative examples of the dependence of (kd) upon k, for the different sets of values of s, w, and a.

Figure 2 shows a plot of (kd) versus k, for

0.4 r

0.3

h

2 0.2 ”

0.1

0

0 0.2 0.4 0.6 0.8 1

Kt

Fig. 2. A plot of (kd) versus k,. This is an average for the four stations of Edmonton, Montreal, Toronto and Winnipeg for s= 1 (no snow) and w=2 (precipitable water between 1 and 2cm) for the years 197771984. The four curves are for the four solar elevation ranges with indices

a=2, 3, 4, and 5.

0 ’ 0 0.2 0.4 0.6 0.8 1

Kt

Fig. 3. A plot of (kd) versus k,. This is an average for the four stations of Edmonton, Montreal, Toronto and Winnipeg for s = 1 (no snow) and a= 3 (solar elevation between 30 and 43”) for the years 1977-1984. The three curves are for the three ranges of precipitable water with

indices w= 1, 2, and 3.

0.5 1

0.1 t/

- s=z \

Id A

0 0 0.2 0.4 0.6 0.8 1

Kt

Fig. 4. A plot of (kd) versus k,. This is an average for the four stations of Edmonton, Montreal, Toronto and Winnipeg for w= 1 (precipitable water less than 1 cm) and a=3 (solar elevation between 30 and 43”) for the years 1977-1984. The two curves are for the two ranges of snow depth with indices s= 1 (no snow) and s= 2 (snow depth

greater than 5 cm).

s = 1, w = 2. The four curves are for four different ranges of solar elevation: CI = 2, 3, 4 and 5. The LX= 1 case is omitted because Edmonton has no hours in this category and the other stations have few hours. In Fig. 2 for the range of k, from 0.44 to 0.72, (kd) decreases as CI increases, i.e. with decreasing solar elevation.

Figure 3 shows a plot of (kd) versus k, for s= 1 and a= 3. The three curves are for the three different ranges of atmospheric precipitable water: w = 1,2, and 3. There is little difference between two of the three curves. Apparently, two changes occur with increasing precipitable water for these two s, w, and cx cases which leave the values of (kd) almost unchanged. Increasing precipitable water increases the attenuation of the solar radiation by water vapor and aerosols and reduces the values of k,. At the same time, the atmospheric scattering by the clouds and/or aerosols apparently increases, increasing the values of k,. Together these two changes apparently keep (kd) for each interval of k, about the same. The curve for w= 1 is somewhat higher than the other two curves. This may indicate a higher surface albedo during the colder weather when there is no snow, but when the color of vegetation is considerably changed from that for warmer weather. Curves like those of Fig. 3 for the higher solar elevations of a= 1 and 2 show somewhat lower (kd) values when W is higher and the sky is partly cloudy.

Figure 4 shows plots of (kd) versus k, for w = 1 and CI = 3. The two curves are for the two different ranges of snow depth: s= 1 (lowest (kd )) and for s = 2 (highest ( kd )). These curves indicate that snow can have a large effect on


the amount of diffuse irradiation. It is difficult distribution of k, for each interval of k, as to quantify this numerically in terms of an represented by (kd)range and (cr)__ can be accurate value of the effective surface albedo, considered the same statistically for the four since the presence or absence of trees, roads and stations of Edmonton, Montreal, Toronto, buildings can reduce or increase the effect of Winnipeg, and perhaps also for Port Hardy, snow reflection. for the same ranges of S, W, a.

Values of 0 for each interval of k, are similar for the four stations of Edmonton, Montreal, Toronto, and Winnipeg. The histograms for these four stations for the eight years from 1977 to 1984 are combined. Figure 5 shows a plot of c versus k, for s= 1 and w = 2, for the four stations of Edmonton, Montreal, Toronto and Winnipeg combined, for the 8 yr from 1877 to 1984, inclusive. The four curves are for a=2, 3, 4 and 5. 0 is approximately independent of CL Similarly, it is found that 0 is approximately independent of W. A curve of 0 for s= 2 with snow on the ground will have significantly larger values of c. This is expected, since snow reflection lighting the sky and clouds is expected to amplify all the effects seen without snow.

Small systematic differences can be seen in (kd)range and (G)~_~ between the four stations of Edmonton, Montreal, Toronto and Winnipeg when considering data for all sets of s, w, and a. In most cases these differences probably can be accounted for by differences in surface albedo and atmospheric turbidity. The common presence of a coastal marine layer of cloud and perhaps the increased concentration of sea salt aerosol at Port Hardy makes atmospheric conditions at Port Hardy differ more from those at the other four stations than the atmospheric conditions differ among the four stations. This probably accounts for the larger differences in (kd)range and (~Lge seen between Port Hardy data

(kd) should equal k, and CJ should be zero in the overcast region for k, below about 0.3. The fact that this is not generally so arises from measurement error. Values of CJ increase rapidly as k, increases above 0.3 until a maximum is reached for k, about equal to 0.6 and is the about the same for all cases with no snow. The curves are quite erratic above the clear hour values of k, because of intermittant cloud reflec- tance and a limited number of k,, kd pairs in this region.

and that of the other four stations for w= 2 and 3. This information implies that the division of global irradiation into direct beam and diffuse irradiation cannot be predicted by the variables, k,, S, W, and M alone for all stations. Specification of the type of clouds beyond their characterization by k,, S, W, and CI may be required for some stations to predict the division of global irradiation into direct beam and diffuse irradiation. The effect of changes in scattering by volcanic aerosols, and, in particular, the large increase in scattering following the eruption of El Chichon in Mexico in 1982, increase the standard deviations of (kd)__ and (c),,,,~~. These increases occur without significantly increasing the differences between (kd)range for different stations, or increasing the differences between (o)__ for different stations. This is because the changes in scattered radiation due to changes in volcanic aerosol occur with about the same time variation and amplitude for all the stations. The value of (kd)range decreases monotoni- cally with decreasing solar elevation a for all stations because of atmospheric attenuation of the solar radiation for all stations. The decrease with decreasing c( is not as great as that for the U.S. data, which generally have higher values of W and turbidity (Garrison, 1985). The changes in (kd)range with changes in atmospheric precipitable water W are small

4. CONCLUSIONS

?? The results presented here indicate that the

0.12

1 - a=2 P

0.09

2 I? 0.06 vr

0.03

0

0 0.2 0.4 0.6 0.8 1

Kt

Fig. 5. A plot of D versus k,. This is an average for the four stations of Edmonton, Montreal, Toronto and Winnipeg for s = 1 (no snow) and w = 2 (precipitable water between 1 and 2 cm) for the years 197771984. The four curves are for the

four solar elevation ranges with indices c( = 2, 3, 4, and 5.


for the four stations of Edmonton, Montreal, Toronto and Winnipeg for the values of W which occur for Canadian data. Quite significant changes in (kd)range with changes in W occur for the higher values of W that occur for some U.S. stations at lower latitudes (Garrison, 1985). The value of (kd)range depends strongly upon surface albedo for all Canadian stations, for the kind of changes in surface albedo associated with changing from no snow to snow depth greater than 5 cm. The value of (o),,_,~ is approximately independent of c1 and W for the five stations of Edmonton, Montreal, Port Hardy, Toronto and Winnipeg. (c)ran_ increases significantly for the four stations of Edmonton, Montreal, Toronto and Winnipeg for the change in surface albedo which occurs in changing from no snow to snow depth greater than 5 cm. Port Hardy has very few hours of solar irradiation data with snow depths greater than zero.

Acknowledgments-We are particularly indebted to the many individuals, unknown to us, who were involved in the careful collection and analysis of the Canadian solar irradiation data and who put this set of data in the suitable form in which it came to us from the Canadian Climate Centre. David Wardle and Bruce McArthur of the Climate and Atmospheric Research Directorate of Canada kindly provided additional information on procedures and errors of measurement. Bruce McArthur and Chris Gueymard reviewed the manuscript prior to submission for publication and provided many valuable comments and suggestions.

NOMENCLATURE

Gd diffuse irradiance G global irradiance I, extraterrestrial irradiance kb direct beam index kd diffuse index k, clearness index

(kd) average k, for each 0.04 interval of k, <kd Lange average k, for k, in the range 0.44-0.72

S snow depth s snow depth index

W atmospheric precipitable water w atmospheric precipitable water index A solar elevation angle OL solar elevation index o standard deviation of the distribution of k, for

each 0.04 interval of k, <g> range average o for k, in the range 0.440.72

0, solar zenith angle

REFERENCES

Bolsenga S. J. The relationship between total atmospheric water vapor and surface dew point on a mean daily and hourly basis. J. appl. Meteorol. 4, 430-432 (1965).

Bristow G. E. Canadian solar radiation data archives, Hay J. and Won T. (Eds), Proc. First Can. Solar Radiation Workshop. Canadian Atmospheric Environment Service, Downsview, Ontario, pp. 74-93 (1980).

Bugler J. W. The determination of hourly insolation on an inclined plane using a diffuse irradiance model based on hourly measured global horizontal insolation. Solar Energy 19, 477-491 (1977).

Davies J. and McKay D. Estimating solar irradiance and components. Solar Energy 29, 55-64 (1982).

Duffie J. and Beckman W. Solar Engineering of Thermal Processes. Wiley, New York (1991).

Freud J. Aerosol optical depth in the Canadian Arctic. Atmo- sphere-ocean 21, 158-167 (1983).

Gamble D. W. Atmospheric Environment Service, Winni- peg, Manitoba (private communication) (1990).

Garrison J. D. A study of solar radiation data for six sites. Solar Energy 32, 237-249 (1984).

Garrison J. D. A study of the division of global irradiance into direct and diffuse irradiance at thirty three U.S. sites. Solar Energy 35, 341-351 (1985).

Garrison J. D. and Adler G. P. Estimation of precipitable water over the United States for application to the division of solar radiation into its direct and diffuse components. Solar Energy 44, 225-241 (1990).

Garrison J. D., Sahami K. and Hutton G. A study of the division of global irradiance into direct and diffuse irradi- ante at seven Canadian sites, Arden M., Burley S.. Cole- man M. (Eds), 1991 Solar World Congr., Proc. Biennial Conf. Int. Solar Energy Sot., Denver, Colorado (1991).

Garrison J. and Sahami K. Analysis of clear sky solar radiation for seven Canadian stations, Burley S and Arden M. (Eds), Proc. Solar ‘93, The 1993 Am. Solar Energy Sot. Ann. Co& Washington, D.C., pp. 445-450 (1993).

Graham V. A., Hollands K. G. T. and Unny T. E. A time series model for k, with application to global synthetic weather generation. Solar Energy 40, 83-92 (1988).

Graham V. A. and Hollands K. G. T. A method to generate synthetic hourly solar radiation globally. Solar Energy 44, 333-341 (1990).

Gueymard C. Mathematically integrable parameterization of clear-sky beam and global irradiances and its use in daily irradiation applications. Solar Energy 50, 385-397 (1993).

Gueymard C. Analysis of monthly average atmospheric precipitable water and turbidity in Canada and the North- ern United States. Solar Energy 53, 57-71 (1994).

Hay J. E. A revised method for determining the direct and diffuse components of the total short-wave radiation. Atmosphere 14, 278-287 (1976).

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

Hollands K. G. T. and Crha S. J. A probability density function for the diffuse fraction, with applications. Solar Energy 38, 237-245 (1987).

Hoyt D. V. Theoretical calculations of the true solar noon atmospheric transmission. In Appendix V of Solmet Vol. 2-Final Report, Hourly solar radiation-surface meteorological observations, U.S. Department of Com- merce Report TD-9724 (1979).

Iqbal M. A study of Canadian diffuse and total solar radiation data--I. Monthly average daily horizontal radiation. Solar Energy 22, 81-86 (1979).

Iqbal M. A study of Canadian diffuse and total solar radiation data--II. Monthly average hourly horizontal radiation. Solar Energy 22, 87-90 (1979).

Iqbal M. Prediction of hourly diffuse solar radiation from measured hourly global radiation on a horizontal surface. Solar Energy 24, 491-503 (1980).

Kierkus W. T. and Colborne W. G. Diffuse solar radiation- daily and monthly values as affected by snow cover. Solar Energy 42, 143-147 (1989).

‘(PL61) 8lS-LOS ‘8 ‘UoJ -!nu~~ alaydsourlp- .oluo~o~ 1~ ~uaumo_~!ma pml e pm usqIn u1? u! d$!p!q_m$ JO Kpnls ag~s.wduro~ v ‘s )?~!ysmn?~

‘(6861) P6E-L8E ‘ZP LzJauT JVS .suo!l!PuO3 amls!om pm uo~~e~osu! ql!~ suoye!x3i\ :axmpey 33aup JO he3ga snou!tunT 'f d~sp2qx~ pm 'a zalad “1 zqE?!IM

'(()$$jI)E6-PL .dd ‘O!~~~u~‘M~~ASUMO~‘a~~A~aS $UaUIUOI

-!Au~ D!laqdsourlv uwpeuy) ‘doysyAoM uo~gmpa~ .mlos Ut%X?Ut7~ lS.UJ ‘30.&j ‘(SptJ) ‘L UOM pUV 'f dl?H ‘U+ap

pm $uatussasse ?IoMlau uoye!pw .~e[os 't) y uosI!M

'(E861) '$'oN ‘ZZ ~[O~OalVV WJWl3 'f 'EpVuQ u.Ia)saa u! bl!p!qInL y 'l sa!ar?a pue '3 'L unqnq8aoqfl

‘(9L61)

ml0.y ‘axyns p2)uozyoq B uo uogwpw asn#!p bpnoq IOJ uo!)enba UO!$QlaIlO3 'L 'f) '>I SpU"IIOH pUV ‘d '1 @()

‘(E861) ZLlkl91 ‘IS ~~J='?l JVS ‘uo!W!P=‘! l”‘loI% UIOIJ asngp au!umlap 0~ lapotu uep.xor pm ng aql JO suoyz~~mg put2 [email protected] ‘1 unquma pue ‘8 uo.waa7

‘(0861) ~6-v~ ,dd ‘O!IE~UO ‘Ma!asuMoa ‘a+Ias ~uaumo.+u~ cgaqdsourlv ue!peue~ ‘doysylo,$j uo~ltnpt~~ .qos ump -0Ut73 JS.l!J ‘301d ‘(Spa) ‘J_ UOM pUt2 '1 k?H ‘UO!)E?!ptQ

.yos Bupol!uo~ 10~ sampa3oId wxpeutg ‘8 ‘1 lays1

Fmeqes ‘2 pm UOS!IIE~ '1 POS

a study of the division of global irradiance into direct beam and diffuse irradiance at seven...

Documents