Comparison of diffuse/global ratios calculated from one-minute, hourly and daily solar radiation data

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  • Solar Energy Vol. 32, No. 3. pp. 417~t23. 1984 0038-092X/84 $3.00 + .00 Printed in Great Britain, Pergamon Press Lid.



    P. J. SMIETANA, JR., R. G. FLOCCHINI, R. L. KENNEDY and J. L. HATFIELD Biosearch, Inc., 2980 Kerner Boulevard, San Rafael, CA 94901, U.S.A.

    (Received 9 October 1981; revision received 30 December 1982; accepted 21 April 1983)

    Abstract--One-minute values of direct, diffuse and global radiation have been continuously collected at Davis, California (38.5N, 121.1W) since 1 January, 1979. These datasets are quality controlled to insure the most accurate and reliable data possible. Analysis of one-minute data has provided an opportunity to evaluate some of the bias that may be inherent in statistical representation of solar radiation data. A simple mean and standard deviation do not adequately describe the variation in the data and we show that a more representative treatment includes the box and whisker analysis. In this the mean, median, first and third quartiles, and the maximum and minimum ranges are presented. It is possible to compute the variability between days more completely with this technique while the means may be very close. This has application to evaluation of solar collectors as a better method of evaluating theire efficiency. This is applied to diffuse/global ratios which show a seasonal dependence although some clear winter days have ratios close to clear summer values; however, the first and third quartile and median distinctly separate these days. Analysis of solar radiation data should be conducted with caution as shown by these results.

    A simple model is proposed to compute hourly global values from the integrated daily total. Comparisons of calculated with measured hourly values indicated less than a 10 per cent error between 0700 to 1600 with the maximum value being slightly underestimated. This procedure allows one to evaluate solar collectors with only daily values and presents a method for thoroughly evaluating our solar resources.


    Throughout the world solar flux measurements have been made in order to provide scientists and engineers with local solar data. These measurements are usually integrated hourly, daily or monthly values and from these data inferences pertaining to longer and sometimes shorter periods are made [I-14]. Many models have been developed to predict global, diffuse and direct solar radiation on a horizontal and inclined surface with various solar and meteorological components as in- dependent parameters [I-14]. These models have used simple linear regressions [9-14], multiple regressions [6, 7] and statistical Markovian [4, 7] techniques to arrive at general empirical relationships which have a variability of predictive use. However, it has been necessary to use mean hourly, daily and monthly values due to an insufficient number of reliable measurements. The datasets may have suffered from instruments which were not recalibrated at regular intervals, varying samp- ling time schedules and insufficient quality control. The latter involves, for example, making global and diffuse solar radiation measurements but not making simul- taneously corresponding direct or beam solar radiation measurements. Solar shortwave and longwave radiation measurements and meteorological parameters at the University of California at Davis, as part of a Solar Energy Research and Meteorological Training Site (SEMRTS) grant, have been recorded on a minute inter- val since January 1979. Davis is one of eight sites recording similar one-minute measurements and then archiving these measurements at the Solar Energy Research Institute (SERI) in Golden, Colorado [21]. These sites are located throughout the United States in

    fContribution from the California Agricultural Experiment Station. Research supported by the Department of Energy, Grant DE-FG03-79ET20187.

    areas which have different solar radiation and atmos- pheric conditions thus providing an extensive geo- graphical database. A quality control procedure, which is applied to each one-minute measurement, and an in- strument calibration procedure were established for the eight sites thus providing a reliable data base. A com- plete description of the instrumentation, data collection procedures and data quality control at Davis is given in Hatfield et a1.[15]. Tables of reduced one-minute measurements to hourly and daily integrated values and to monthly summaries for 1979 and 1980 are given in Hatfield et aL[16].

    The purpose of this paper is to provide a statistical comparison of one-minute values and ratios with hourly and daily integrated means and ratios. The results from the comparisons should provide a reliability or confidence level in the use of earlier recorded hourly or daily values. A method of generating hourly or smaller time divisions from a given integrated daily value is also explored. The extent of cloudiness on a day, which can be obtained from the presented statistics, is discussed.

    The amount of solar radiation approximates a sine curve [19] during the day suggesting that simple measurements throughout the day and calculation of an average will not adequately provide a complete and accurate description of the radiation flux for the day. This implies that averaging measurements taken within an hour will also not be valid since they would be biased toward the end of the hour in the morning and toward the beginning of the hour in the afternoon. The extent of error is analyzed and methods of presenting statistical analysis is part of the objectives of this paper.


    Solar radiation and meteorological data are collected on a one-minute basis at Davis, California (38.5N, 121.1VO. Data collected include direct, diffuse and glo-


  • 418

    bal radiation with Eppley Normal Incidence Pyr- heliometer and Precision Spectral Pyranometers, res- pectively. Diffuse radiation is measured by means of a shading disk which continually provides a blockage of the direct beam from the pyranometer.

    The hourly and daily values are computed by per- forming a trapazoidal numerical integration on the ori- ginal one-minute measurements. This method was chosen because of the sinusoidal nature of the data. The general trapazoidal integral equation is

    Area=(Xo+X~o)12+(X ,+X2+. . . + X~9) (1)

    where Xo and X,o are the Values for the starting minutes of two consecutive hours and Xt-Xs9 are the intervening 59 rain. The daily integral is calculated by summing of the individual hourly integrals. The programs for these conversions are given in Hatfield et al. [15].

    Statistical calculations were performed using BMDP (1979) computer programs (17). Box and whisker diagrams (Tukey [20]) are used to graphically present a compact statistical summary of one-minute measure- ments.


    3.1 Statistical comparisons Daily global radiation values (calculated from the one-

    minute measurements) for 1980 are illustrated in Fig. 1. The envelope of this curve follows the predicted extra- terrestrial radiation curve but shows considerable day-to- day variation caused by cloudiness, changing air mass and aerosols. Tables 1-3 present a summary of yearly per cent sunshine statistics for years 1979-1981. These values were computed using the daily total number of minutes of sunshine as measured by a Campbell-Stokes Sunshine Recorder and dividing by the number of minutes between sunrise and sunset. In Table 1 it should be noted that the median is approximately 18 percentage points above the mean which indicates that Davis has predominantly sunny days. Table 2 presents the frequency of percent sunshine values in range intervals of 5 per cent. This table shows that approximately 10 per cent of the days have less than 5 per cent sunshine, approximately 20 per cent of the days have between 5 to 70 per cent sunshine and the remaining 70 per cent have


    19::-:0 JI_IL I All BAT

    Fig, 1. Daily global shortwave radiation for 1980 from one- minute measurements and predicted extraterrestrial solar flux on

    a horizontal surface.

    P. J. SMIETANA, JR. et al.

    Table 1. Yearly per cent sunshine statistics for Davis, California

    1979 1980 1981

    Maximum 99.00 99.50 98.10

    3rd Quartile 95.30 94.40 92.90

    Median 88.80 88.20 83.10

    Mean 70.30 70.60 65.90

    Ist Quartile 51.10 52.90 38.00

    Minimum .00 .00 .00 i

    Table 2. Yearly frequency distribution of daily per cent sunshine values in five per cent range intervals for Davis, California


    Interval Frequency Number Percent 1979 1980 1981

    I 0 - 5 31 32 a3 2 5 - I0 5 5 I0 3 10 - 15 4 5 3 4 15 - 20 6 3 4 5 2O - 25 4 6 7 6 2 5 - 3 O 3 3 6

    7 3 0 - 3 5 3 8 7

    8 35 - 40 6 3 6 9 40 - 45 3 10 4 I0 45 - 5O 7 6 9 11 50 - 55 8 5 4 12 55 - 60 4 8 7 13 60 - 65 5 10 2 14 65 - 70 5 3 8 15 70 - 75 17 9 13 16 75 - 80 13 18 19 17 80 - 85 17 17 30 18 85 - 90 16 34 27 19 90 - 95 62 82 85 20 95 -100 81 7O 45

    Number of Days 300 337 339

    greater than 70 per cent sunshine with it being heavly skewed to the 90 per cent value. Figure 2 pictorially illustrates the 1980 daily per cent sunshine variability and similar figures were obtained for 1979 and 1981. As shown in Fig. 2 cloudy days (less than 70 per cent sunshine) exist predominantly in the winter months, but they also occur in the summer months.

    In Table 3 the mean per cent sunshine values were determined for each hour of the day. It can be seen that the values cluster about the yearly mean; however, the skewed data in Table 2 suggest that the median value would have been a better measurement statistic. These

    Table 3. Hourly mean per cent sunshine values determined from Campbell-Stokes sunshine recorder data for Davis, California 197~ ~9 6~ 6~ 70 7~ 72 73 7~ 76 7~ 7~ rO 64 63

    1~aO 7J 61 71 77 rB 78 T8 79 79 ~9 79 r3 68 ~8

    1981 69 ~9 66 72 7~ 7~ 75 76 76 76 7~ 68 59 56

  • Comparison of diffuse/global ratios 419

    i F:

    . . . . . . I '1" ' ! j ;i

    19: . : :0 . h J l i~tn I)~t~

    Fig. 2. Daily per cent sunshine values for 1980 determined from Campbell-Stokes sunshine recorder data.





    ~ T T ~

    T J - L

    T i T

    - J- T

    J U L 1 , 1 9 7 9 H o u r

    Fig. 4. Direct solar radiation on a horizontal surface for ], July 1980 from one-minute measurements.

    means also suggest little variability since the bimodal distribution has been averaged out. They could however be used as lower bound estimates of per cent sunshine or direct radiation.

    As an example of the daily variability the global values for July, 1980 are shown in Fig. 3 where the data are from one-minute measurements. The mean values for each day have approximately the same value typical for July; however, the maximum and minimum ranges exhi- bit large diurnal variations with some variation between days for the maximum values. The major part of the daily variation is explained by the diurnal cycle as illus- trated in the plot of hourly mean, maximum and mini- mum values for the direct component of global solar radiation which were calculated from one-minute measurements for 1 July, 1980 (Fig. 4). It is shown that maximum and minimum values should be reported along with mean values when the effect of the diurnal cycle is present. This also applies to monthly mean values since the beginning and ending predicted extraterrestrial solar radiation values for a month vary from as little as 290 KJ/M 2 (June) to as much as 7200 K JIM 2 (March).

    Since all solar radiation measurements fluctuate due to a variety of changes, e.g. intermittent clouds, diurnal and seasonal sun elevation changes, these could cause the ratios of any radiation measurements to also fluctuate with any of these temporal variations. This suggests that traditional methods of analysis may not be adequate to describe the daily variations in relationships between parameters. One of the ratios most frequently used to describe the solar radiation components is the diffuse/global ratio. During the winter with numerous cloudy and foggy days, ratios near 1.0 are evident (Fig. 5)

    and during the summer with typical clear days, ratios near 0.15 are evident. However, it should be noted that on clear winter days in Davis ratios near 0.15 are also present. Ratios greater than 1.0 are due to inclusion of measurements at low sun elevation.

    Ratios of diffuse to global solar radiation remove the diurnal variation in the data and thus calculating averages and standard deviations would seem applicable descrip- tors. However, a simple mean and standard deviation may not be adequate to completely describe the variation in these data. It has been found that calculating quartiles and plotting them along with means, medians, maximums and minimums provides a more meaningful description of data calculated from diffuse/global ratios [20]. This representation utilizes the box and whisker diagram which not only presents the maximum and minimum range bars (whiskers), but one can quickly see the clus- tering of measurements about the mean and median by the size of the box determined by the 1st and 3rd quartiles. The relative positions of the mean and median are also not static since the mean is sensitive to ranges during the day indicating its value alone could be quite misleading. As an illustration of this technique, Box and Whisker diagrams are presented for January and June in 1980 (Figs. 6.1 and 6.2) where one-minute data with the sun elevation less than 5 degrees were excluded. A complete set of these graphs for all months in 1980 is given in Hatfield et al. [15]. The winter months tend to produce more days with ratios near one, but on clear winter days the ratios are nearly the same as those during the summer months; however, on these clear days the mean and median still exhibit a wide separation as compared to clear days in the summer. Another pre-


    lllliilliiiiili ! J U L Y 198~3 Julian Bays

    Fig. 3. Daily global shortwave radiation for July 1980 from one-minute measurements.


    Fig. 5. Daily diffuse/global ratios for 1980 from one-minute measurements.

  • 420 P.J. SMIETANA, JR. et al.

    D '~I IIII T'i , I JAl l 19::':0 DAYS

    Fig. 6. I.

    30 31

    - , - - - _

    J l J I l I '7:~0 BA'/S

    Fig. 6.2. Fig. 6. Box and Whisker diagrams of diffuse/global radiation for January and June of 1980 calculated from one-minute measure-


    dominant feature is that during the summer months the minimum ratios are close to the mean and median values as shown for June (Fig. 6.2).

    Even though two days (e.g. 2 January and 10 January or 20 January and 30 January) have similar maximums and minimums their 50 percent clustering, means and

    c, .d t~

    D u .


    i~urt 7 . 1

    l -

    \ oo


    i ,

    I i : re>

    JQHI.IQRY 1 6 , 1 9 8 0 HOUR



    F i ~ r e 7.2

    Fig, 7. Time of day plots using one-minute measurements on 16, January 1980: (1)Diffuse/global ratios calculated from one- minute measurements and hourly integrated...


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