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SolarEnerg3, Vol. 39, No. 5, pp. 445-4.53, 1987 00384)92X/87 $3.00 + .00 Printed in the U.S.A. Pergamon Journals Ltd.

MONTHLY AVERAGE SOLAR RADIATION IN P A N A M A - - DAILY AND HOURLY RELATIONS BETWEEN DIRECT

AND GLOBAL INSOLATION

P. BECKER* Smithsonian Tropical Research Institute, P.O. Box 2072, Balboa, Rep. of Panama

Abstract--Regressions are developed to estimate daily global and direct radiation and the hourly distribution of direct radiation for Barro Colorado Island, Panama from monthly mean values observed 35 km away at Chiva-Chiva. The ratio model of Liu and Jordan and the logarithmic model of Anderson for estimating direct from global radiation are compared. Both gave satisfactory results after accounting for "seasonal" variation, but the ratio model was preferred in this case for the smaller number of separate regressions required. The ratio model fitted for diffuse radiation at Chiva-Chiva agreed closely with regressions for stations at similar latitude. For a given value of the clearness index, the direct component of solar radiation was relatively (but not absolutely) reduced during the dry season compared with the wet season. A likely explanation for this unexpected result is increased marine and terrestrial aerosol during the dry season when offshore winds are stronger and burning of crop and wasteland occurs. The models of Whillier and of Gamier and Ohmura, which assume constant atmospheric transmittance throughout the day, gave unsatisfactory fits to the hourly distribution of direct radiation. They were also unable to mimic an observed morning/afternoon asymmetry that was strongest in the wet season. Hourly direct radiation was accurately estimated from hourly global radiation by quadratic polynomials fitted separately to the morning and afternoon data. The resulting regressions will enable estimation of radiation in forest understory from measurements of insolation in the open by computerized image analysis of hemispherical canopy photos.

1. INTRODUCTION

Quantification of solar radiation in forest understory has been a long-standing problem for ecologists and foresters due to extreme spatial and temporal variability, especially for the direct component seen as sunflecks. A promising approach is the analysis of hemispherical canopy photos to estimate radiation at forest sites from measurements of solar radiation in the open[l-2] . The necessary observations are monthly means of daily direct and global insolation and the hourly distribution of direct radiation on a horizontal surface. SYLVA, a newly developed, computerized image processing system, greatly facilitates application of this technique. Ecologists at the Smithsonian Tropical Research Institute 's field station on Barro Colorado Island, Panama, will apply it to correlate plant growth, survival, and distribution with insolation.

Several thousand canopy photos were taken during 1982-1985 on Barro Colorado, but measurements of direct radiation were unavailable for this period. Global radiation measurements were available for only some of the time. It was necessary, therefore, to obtain empirical relations to estimate the necessary radiation components for Barro Col- orado from more extensive observations made by the U.S. Army Met Team at Chiva-Chiva Antenna Farm 35 km to the southeast. These data are of general interest and there has been no previous pub- lication of solar radiation patterns in Panama. This

* Current address: % World Wildlife Fund, 1250 Twenty-fourth St., N.W., Washington, D.C. 20037, U.S.A.

article thus has the twofold purpose of preparing • the database for the canopy photo analyses on

Barro Colorado and of illustrating the relations among solar radiation components in Panama.

2. MONTItLY AVERAGES OF GLOBAL RADIATION AT

CHIVA-CHIVA AND BARRO COLORADO

Global radiation was measured at Chiva-Chiva Antenna Farm (9°02'N, 79°35'W, 16 m a.s.l, ele- vation) with a 15-junction Eppley pyranometer mounted horizontally at an open, sloping site cov- ered with grass. The sensor was calibrated yearly against one of the same type maintained for this purpose and traceable to an Eppley absolute cavity pyrheliometer. Monthly reports are filed at the Technical Information Center of the U.S. Army Tropic Test Center, Ft. Clayton, Panama. Figure 1 shows monthly and annual variation in mean daily global radiation at Chiva-Chiva during 1972-1985. The dry season typicaUy begins in December or January and ends in mid-April or early May. Global radiation is about 1.5 times greater during this period than in the cloudier months of the wet season.

Global radiation was measured at Barro Colo- rado Island (9°9'N, 79°51'W, 90 m a.s.1.) by the Smithsonian Institution's Environmental Sciences Program. Readings were made with a Li-Cor LI- 200S pyranometer mounted horizontally atop a tower about 8 m above the canopy of semiever- green, tropical moist forest. The tower is at the base of the Lutz catchment, 450 m from the shore of Lake Gatun. Data were available for March 1983 through May 1985, during which period several fac- tory-calibrated sensors were used without recall-

445

446

25

~0

N d 15

1o d F M A M d d A S 0 N O

MONTH

Fig. t. Tukey box plots of monthly mean daily global radiation at the Chiva-Chiva Antenna Farm, Panama, during 1972-1985. The median, interquartile range, and adjacent values are indicated by the central line, box, and whisker ends, respectively. More extreme points are plotted in- dividually. N = 13 for January (data unavailable for 1972);

N = 14 for remaining months.

bration. The LI-200S sensor is not spectrally ideal, but no correct ion was made as the error for daily totals is typically <-+5% [4]. The Li-Cor pyranometer, like the Eppley, is fully cosine corrected to a solar zenith angle of 80 °. There was excellent agreement between hourly measurements of global radiation by an Eppley PSP and a Li-Cor silicon cell pyranometer[27].

Preliminary analyses of the relation between daily global radiation at Barro Colorado and Chiva- Chiva indicated that from mid-August 1984 through February 1985 there was a sudden and substantial decline in the Barro Colorado values relative to the trend displayed in the remaining months. A similar pattern occurred when the Barro Colorado pyranometer data were plotted against PAR* (photosyn- thetically active radiation) data collected at the same site with a Li-Cor quantum sensor. Thus, it is nearly certain that the pyranometer data from Barro Colorado, rather than Chiva-Chiva, were in error. When data from the period in question were excluded, daily global radiation at Barro Colorado was linearly related to that at Chiva-Chiva with r 2 = 0.70 (data not shown). The modest coefficient of determination indicates that daily radiation at the two sites sometimes differed substantially, presum- ably due to different local weather patterns.

A linear, least squares regression was fitted to monthly mean daily global radiation for Barro Col- orado versus Chiva-Chiva with August 1984

* Althoug h desirable for studies of plant ecology and physiology, it was not possible to develop a regression relation between global PAR (0.4-0.7 i-tm) and shortwave radiation (0.285-2.8 gm) measured on Barro Colorado. The daily data were poorly correlated (r 2 < 0.8), and the PAR data were apparently affected by shifts in sensor re- s p o n s e . New PAR sensors have been installed at Barro Colorado, and conversion factors should eventually be available.

P. BECKER

through February 1985 excluded. A Wald-Wolfow- itz runs test[5] indicated significant (P < .05) temporal pattern in the residuals. This meant that regression error terms were not independent, which leads to biased estimates of the regression coefficients. Because the temporal pattern of the residuals was not seasonal and no other explanatory variable was available, the Cochrane-Orcutt method[6-7] was followed as a remedial measure. No significant chronological pattern remained in the residuals from this autoregressive model, which employs variables transformed with the estimated, first-order autocorrelation parameter. As it turned out, however, the ordinary and autoregressive models gave virtually identical predictions over the range of observed values in Fig. 2 where the regression line is shown for the latter model.

The pyranometers for Barro Colorado and Chiva-Chiva were not calibrated against each o t h e r or a common standard. Thus, there is no objective basis for deciding whether the slight nonequiva- lence of values indicated for these sites by the fitted Cochrane-Orcutt regression is real. The apparent discrepancy between stations was greatest at low values of mean daily global radiation. Even for the lowest Chiva-Chiva value in Fig. 2, however, the estimated Barro Colorado value was only 6% higher. This suggests that monthly global radiation at the two sites was nearly, if not actually, identical.

Table AI of the appendix lists monthly mean daily global radiation for 1982-1985 on Barro Col- orado, estimated by the Cochrane-Orcutt regression shown in Fig. 2 for those months where measured values were unavailable. The predictive accuracy of the regression was very good. Except for December with a difference of 19%, estimated values for Barro Colorado in Augus t -December 1985 differed by 5% or less from observed values that were not used to fit the regression. (June and July observations were not available due to prob- lems with a datalogger.) The discrepancy between

B4 ' i ' I ' I ' I I I '

22 MONTHLY MEAN OAILY ~ o / GLOBAL RADIATION. MJ m -2

2O 0

o o o ~ 0

16 o

14 S °

12

t ( , I , I , I , I , I , I , 12 14 i6 tB 20 22

CHIVA-CHIVA

Fig. 2. Relation between monthly mean daily global radiation on Barro Colorado Island and at Chiva-Chiva during March 1983-July 1984 and March-May 1985. Line fitted by linear, least squares regression according to the Cochrane-Orcutt method[6, 7] to correct for autocorre-

lation: Y = 1.95 + 0.895X.

24

Monthly average

estimated and observed values in December probably resulted from eight days of missing data for Chiva-Chiva at the beginning of the month. This led to an overestimate of the average daily radiation there in this transitional month between wet and dry seasons.

3. RELATION BETWEEN MONT"HLY MEAN DAILY DIRECT

AND GLOBAL RADIATION AT CHIVA-CHIVA

Over periods of at least a day, diffuse radiation is quite regularly related to the effective transmittance coefficient of the atmosphere, as first noted by Liu and Jordan[8]. This relation has been veri- fied at a large number of temperate and tropical sites[3, 9-11]. Because the direct and diffuse components of global radiation are complementary, a similar, but inverse, relation exists for direct radiation~ A lesser known model by Anderson[12] pro- posed a logarithmic relation between hourly or daily totals of direct and global radiation. The article ver- ifying this relation was apparently not published by the Quarterly Journal of the Royal Meteorological Society as cited by Anderson[12]. I therefore compared the Liu and Jordan and the b~nderson models, hereafter called the ratio and log models, respectively. For uniformity, I have generally adhered to the mathematical nomenclature of ref. [3]. For the sake of the canopy photo analyses, however, ref- erence is primarily made to the direct component of solar radiation instead of the traditional emphasis on the diffuse component.

Diffuse radiation was measured at Chiva-Chiva during February 1972 through November 1980 with a 15-junction Eppley pyranometer mounted horizontally under an Eppley shadow band. For this study, monthly mean daily direct radiation was calculated as the difference between monthly mean daily global and diffuse radiation. The latter was corrected for shadow band obstruction using the monthly coefficients for latitude 10°N in Table 1 of ref. [13].

According to the log model, mean daily direct radiation Ht, is related to mean daily global radiation Hh as

log Hb = a + b log nh (1)

where a and b are empirical coefficients[12]. In the ratio model, Hb/Hh is related to Hhlno, the ratio of terrestrial to extraterrestrial radiation, which is the monthly average clearness index Kh (sensu ref. [3]). Substituting the spectral coefficients for latitude 9°N in Table 2 of ref. [14], Ho (in ly day -a) as a harmonic series on the nth day of the year is given by

Ho = 847.6 - 58.5 cos(2rm/365)

- 35.7 cos(47rn/365) + 14.2 sin(2~rn/365) (2)

+ 14.0 sin(4rm/365)

solar radiation 447

Monthly means of Ho from (2) evaluated for every day of the year differed < 1% from values calculated for the fifteenth day of the month (solar constant = 1360 W m -2) by equation (5) ofref . [8]. The latter was, therefore, used as an excellent approximation to Ho in calculating Kh.

Seatterplots of monthly mean daily direct and global radiation appropriate for the log and ratio models are shown in Figs. 3 and 4, respectively. The relation is slightly curvilinear in the log plot and linear in the ratio plot. Linear, least squares regressions (not shown) fitted to the data overall are: log Hb = -- 1.98 + 2.33 log-Hh (r z = 0.90) and HJHh = -- 0.14 + 1.26Kh (r z = 0.90). In both plots there is a clear tendency for the values of certain months to occur above or below the overall regression line. The coefficients a and b of (1) depend on the time of year and prevailing climate[12]. Sea- sonal variation in the ratio model has also been noted[3, 15].

Two approaches have been taken to account for seasonal variation in the ratio model- -grouping the data by sunset hour angle[3] and transforming the ratios to eliminate variation in multiple reflection of radiation between the earth 's surface and atmosphere[15]. At 9°N latitude the annual range of the sunset hour angle (evaluated on the fifteenth day of the month) is only 1.50-1.64 rad, which cor- responds to just one of the three classes employed in ref. [3]. Moreover, the monthly patterns of the data in Figs. 3 and 4 are apparently unrelated to variation in sunset hour angle. The second method was not feasible in this study because it requires information on percentage cloud cover, which was unavailable. Least squares regressions were therefore fitted to various combinations of months for the log and ratio models. My goal was to obtain the

~u

I E

-J

MONTHLY MEAN DAILY RADIATION _ ~ , 2 P - ~ s ~ l . a

+: 0 g o $ 5 " +4 s'4

Oo. ;.

n A I : , , , , I , , , , I , , , I I , , , ~'"i i.i i.E _t.3 1.4

LOGIo GLOBAL MJ m -~

Fig. 3. Relation between logarithms of monthly mean daily direct and global radiation at Chiva-Chiva during February 1972 through November 1980. Months are coded 1-9 for January-September, respectively; then O, N, D for Oc- tober-December, respectively• Lines fitted by linear regression of Iogto-transformed variables with April 1976 excluded as an outlier: Y = -2.07 + 2.40X, r 2 = 0.95 (Feb., May, June, Sep., Oct.); Y = -2.06 + 2.36X, r z = 0.97 (Mar., Apr., July, Aug.); Y = -2.24 + 2.63X, r 2 = 0.98 (Nov., Dee.); Y = -1.39 + 1.93X, r 2 = 0.92

(Jan.).

448 P. BECKER

O.B RATIOS OF MONTHLY .,,.i 0 7~ MEAN DAILY RADIATION _..,,22 .~,,,"--

• t ~ RAY-JUN. AUG-DEC ~.,l;Yi/" :

O.5F 5 070 _d-'3

O ~ s ' - : s - "(7.3 0.4 0.5 0.6 0.7 GLOBAL/EXTRATEPRESTRIAL

Fig. 4. Relation between ratios of direct/global and global/ extraterrestrial for monthly mean daily radiation at Chiva- Chiva during February 1972-November 1980. Months are coded as in Fig. 3. Lines fitted by linear, least squares regression with April 1976 excluded as an outlier: Y = - 0.224 + 1.39X, r 2 = 0.94 (Jan.-Apr., July); Y = - 0.218

+ 1.48X, r z = 0.86 (May-June, Aug.-Dee.).

smallest number of groups consistent with a linear relation and minimal monthly pattern in the residuals. The group regressions finally selected are de- picted in Figs. 3 and 4.

The present database was too small to allow des- ignation of independent subsets for development and validation of the empirical model. Instead, a limited but still useful evaluation was made by calculating the deviation between observed values and the corresponding regression est imates as

% difference = 100 [ predicted Hb

- observed Hb I/observed Hb

The results summarized in Fig. 5 show improved fit of the group compared with the overall regressions,

MONTHLY , , , . . . . . . . . ,

I I I T I t I i I I I = 0 20 40 60 BO 100

NUMBER OF MONTHS

g DIFFERENCE OF PREDICTED FROM OBSERVED

[ ] 0-5 [ ] 5-10 [ ] 10-35

1 2 0

Fig. 5. Frequency distribution of the absolute percentage difference between predicted and observed monthly mean daily direct radiation at Chiva-Chiva during February 1972-November 1980. Predictions were made by linear regressions for the ratio and log models fitted separately for each month, to the groups in Figs. 3 and 4, and to all 106 observations. April 1976 and October 1980 were excluded from the monthly regressions and April 1976 was

excluded from the group regressions as outliers.

especially for the log model. Additional improve- ments were obtained when regressions were fitted separately for each month (regression lines not shown or specified).

Reduced sample size in the separate monthly regressions made detection of outliers more diffi- cult, and points with high leverage may have unduly influenced the regressions. Therefore, the group regressions were preferred for predicting direct from global radiation outside the observation period. Although some monthly pattern remained in the group regressions (values for October in Fig. 3 and for January and November in Fig. 4 tend to occur above their group lines), estimated values differed by less than 10% from observed in more than 80% of the cases (Fig. 5). The group regressions of the ratio model were chosen over those of the log model to estimate direct radiation on Barro Colo- rado in 1982-1985 (Table A2 of the appendix). This was because their estimation accuracy was comparable (Fig. 5) and the sample sizes for the ratio model were larger due to the smaller number of groups. Although the ratio model was linear in this study, the log model may prove to be a preferable alternative when this is not the case.

The data in Fig. 4 show a strong tendency for the ratio of direct to global radiation HJHh to increase with increasing clearness Kk, as is typically observed[3, 8-11, 15]. The dry season (December or January to mid-April) tends to be clearer than the wet season. Surprisingly, however, the group regression for January-Apr i l and July lies below that for the remaining months of the year. This relatively minor " seasona l " pattern runs counter to the expectation of reduced scattering and an increase in the direct component with decreasing cloudiness[15]. A possible explanation is that extensive burning of crop and wasteland during the dry season in Panama increases atmospheric particulates. This then leads to increased scattering of solar radiation. On exceptionally clear days or days with high total radiation (Kh = 0.76) at Kew, England, depletion of direct radiation by suspended matter accounted for 21-46% of the total depletion[23] so this effect could be important. It would be consistent with December being distinct from the remaining months of the dry season because burning does not begin until the rains have ceased for some time. I cannot, however, account for July ' s align- ment with the dry season group. Unfortunately, there is no published data on seasonal variation in total atmospheric particulates near Chiva-Chiva to test this idea.

Another factor contributing to increased aerosol during the dry season is salt spray. Winds are stronger throughout the dry season than in the wet season[22]. Even at an inland station such as Chiva- Chiva I0 km from the Pacific coast, monthly mean saltfall collected on wet gauze wicks is much higher in December -Apr i l than in the remaining m o n t h s - -

4

Monthly average solar radiation 449

O . B ' ' ' 1 ' ' ' ' I ' ' ' ' l ' ' ' ' 1 ' ' ' ' 1 ' ' ' ' ,

RATIOS OF MONTHLY 0 . 7 ,%%~ MEAN DAILY RADIATION

CHIVA-CHIVA. 9* N 0.6 ,..,. ~.-~

0.5 KEW, 5 t* : ~ ~ . ¢ . . - STANLEYVI LLE. t* .N. ~i'~_~.'~. ~ NEW OR_Hi 29 N

o., ~. " ~ . ~ /LEOPOLD- . . . . . . . , . A ~,L~"~I~. VILLE. 4"S

_ _ _ ~ . " ; " N , " , , , I , , , , I , , , , I , , , , I , , , , I , , ,

° 'E~ 0.~ 0.4 0.5 0.~ 0.7 0.~ GLOBALIEX~ATEP~ESTBIAL

Fig. 6. Relation between ratios of diffuse/global and global/ extraterrestrial for monthly mean daily radiation at seven stations. Record lengths were 12-106 months. Results are from Fig. 13 of ref. {10l for New Delhi, from the present study for Chiva-Chiva, and from Table 3 of ref. [9] for the remaining stations. Observed K#, was 0.40-0.72 for New Delhi, 0.32-0.66 for Chiva-Chiva, and not specified for the remaining stations. Lines fitted by linear, least squares regressions: Y = 1.06 - i.14X, r z = ? (New Delhi); Y = I.t4 - 1.26X, r z = 0.90 (Chiva-Chiva); Y = 1.07 - 1.16X, r z = 0.86 (Stanleyville); Y = i.08 - 1.21X, r z = 0.92 (Leopoldville); Y = i.10 - 1.43X, r2 = 0.94 (Dur- ban); Y = i.07 - 1.26X, r z = 0.86 (Capetown); Y = 0.94

- 1.03X, r z = 0.96 (Kew).

about 25 vs. 3.5-10 mg CI m -2 d-][25]. Finally, when the s u n itself is not screened by a cloud (as is most likely in the dry season), solar radiation may be reflected in substantial quantities offclouds else- where in the sky[24]. These increases in diffuse radiation should produce a relative decrease in the direct component.

Comparison of the ratio model for different stations is complicated by the choice of different time intervals for the insolation data. Also, some inves- tigators fail to specify whether diffuse radiation was corrected for shadow band obstruction. Both these factors significantly affect the relationship[3]. Fig- ure 6 shows the ratio model for diffuse radiation H a fitted to data from all available studies meeting the following criteria: (I) diffuse radiation was corrected for shadow band obstruction, (2) an overall, least squares regression was fitted to data from a single station (as it turned out, all relations were fitted by linear regressions), (3) the ratios H d l H h and H h / H o were calculated from monthly mean daily radiation, and (4) stations occurred at an el- evation <500 m a.s.1. Except for the present study, a solar constant of 1395 W m -2 was used to calculate Ho, but this discrepancy has only a slight effect on the regressions. The relations in Fig. 6 weakly confirm the claim in ref. [16] that the diffuse component of solar radiation is higher in the tropics than the temperate zone. The available data indicate relatively uniform relations for the ratio model in the tropics and greater variability in the temperate and subtropical zones.

4. ItOURLY DISTRIBUTION OF DIRECT RADIATION AT CHIVA-CHIVA

The possibility of estimating the hourly distribution of direct radiation from daily direct radiation is attractive for its utility at stations where the for- mer is unknown. Unfortunately, the hourly distribution of direct radiation can be quite irregular from year to year. This is illustrated in Fig. 7 for Chiva- Chiva during May, the month with the greatest annual variation. A semiempirical and a theoretical approach to this problem were explored with unsatisfactory results. Ultimately, empirical relations were used to estimate hourly direct from hourly global radiation.

Mean hourly direct radiation was calculated as the difference between mean hourly diffuse and global radiation on a monthly basis, without cor- recting diffuse radiation for shadow band obstruction. The shadow band correction factors were developed for daily, not hourly, radiation totals[13]. Their application during the early and late hours of the day would sometimes have led to negative values for direct radiation. For the ratio rb of hourly to daily direct radiation, the numerator was the monthly mean and the denominator was calculated as the sum of the mean hourly values. This was done for convenience and gives a slightly different value than if the daily values of the ratio for each hour were averaged by month. Hourly radiation data at Chiva-Chiva was analyzed for January 1973 through December 1978. April 1976 and June 1973 were excluded due to substantial missing data.

One approach to estimating the hourly distribution of direct radiation from its daily total is to evaluate the summation approximation developed by Gamier and Ohmura[17] for the flux of direct radiation on a horizontal surface. Hourly totals ob-

0.t5

~ 0.!

i 0.05

0

, i ~ i , i , i , I , I , I , I ' 1 ' I ' 1 '

,,7:":'i~.mli i : ', ": O~UFIA

i , * / ,z-,---...\i-'~ / /#, i OBSERVED "\ 41, .--,~, ,, '~,I- MIN. MEAN & MAX "~%%~",~,~,~

/iS'," " i t973-78 "~itt~'; ?k • ; - \ ",

,?:N HOUR (TRUE SOLAR TIME)

Fig. 7. Hourly distribution of mean daily direct radiation at Chiva-Chiva during May: (a) 6-year grand means of observations for 1973-1978 (solid line), (b) minimum and maximum values for 1973-1978 (light dashed lines), (c) estimates by Gamier and Ohmura[17, 19] method (dotted line), (d) estimates by Whillier [21] method (heavy dashed line), and (e) grand means of estimates for 1982-1985 by

quadratic equations in Figure 8 (stars).

450 P. BECKER

tained in preliminary trials using time intervals of 6 and 20 min were nearly identical so 20-min intervals were used. Monthly mean solar declination was calculated as in [16]. The relative optical air mass was calculated as in [18] with corrections for curvature of the atmosphere and atmospheric re- fraction. Finally, monthly values of equivalent atmospheric transmittance were obtained by the two- step process described in [19]. They ranged from 0.237-0.543 and were lowest during the wet season.

Limitations of space prevent a complete descrip- tion of the results here, but the pattern shown for May in Fig. 7 is typical. The hourly distribution of direct radiation estimated by the Garnier and Ohmura method was more peaked than that observed. The persistence of this tendency throughout the year is illustrated in Table 1 by a comparison of the observed and estimated proportions for the two central hours of the day, which account for about one-third of the daily direct radiation. Fur- thermore, the symmetrical distribution generated by the Garnier and Ohmura equation did not cor- respond with the empirical tendency for direct radiation in the morning to exceed that in the afternoon (Fig. 7 and see below). Finally, irregular dips, peaks, and shoulders were displayed in the observed hourly distributions of direct solar radiation. These patterns cannot be mimicked when atmospheric transmittance is assumed constant throughout the day. The Garnier and Ohmura method accurately estimates daily direct radiation[17, 19-20], but it was inadequate for estimating hourly distributions in this study.

A second approach to estimating the ratio of hourly to daily direct radiation is suggested by Whillier's theoretical model[21], which again as- sumes constant atmospheric transmittance throughout the day: A plot of the distribution computed by equation (8) of Whillier[21] is shown in Fig. 7 for May 15. This was typical of other months in that the Whillier distribution tended to be too broad relative to the observed hourly distribution of direct radiation (Table 1). Like the Garnier- Ohmura model, Whillier's model is symmetrical around solar noon. It is further limited in being unable to model annual variation. The Whillier model is more commonly associated with estimating the hourly distribution of diffuse radiation, which it does very well[3, 8].

Because the theoretical models proved inadequate, a simple and direct empirical approach was

taken. Figure 8 shows a tight but slightly curvilinear relation between rb and rh, the respective ratios of monthly mean hourly to daily totals for direct and global radiation. Logarithmic transformations of rb and of both rb and rh did not eliminate this curvi- linearity so polynomial, least squares regressions were fitted. A quadratic polynomial fitted to the overall data showed distinct patterns for morning and afternoon residuals. Therefore, separate regressions were fitted for these periods and are plotted in Fig. 8. Since residuals tended to increase with rh, arcsin-square root transformations of rb and rh were fitted with various ordered polynomials. A cubic polynomial fitted to transformed rb and untransformed rh eliminated residual pattern. Over the observed range of rh, its estimates were nearly identical to those of the quadratic polynomial fitted to untransformed variables. The latter model was therefore retained for its simplicity.

Despite some pattern in the residuals and a slight tendency to overestimate (caused mostly by an irregular trend in the data around rh = 0 . 0 3 ) , the simple quadratic polynomials fitted the data very well when negative estimates were converted to zero. Only 10% of the estimated rb differed > ___ 0.01 from observed values; less than 1% differed >_+0.02. The daily sums of rb estimated by the quadratic polynomials for 70 months in 1973-1978 ranged from 0.99-1.01, with three exceptions that ex- tended the range to 0.98-1.02. Hence, scaling to achieve a daily sum of 1.00 was unnecessary. The quadratic polynomials were clearly superior to the Whillier and the Garnier-Ohmura models in estimation accuracy (Table 1).

To estimate the unobserved hourly distribution of direct radiation during 1982-1985, it was first necessary to compensate for a switch from True Solar Time (TST) to Local Standard Time (LST) in radiation records beginning in January 1981. The procedure followed is best illustrated by example. According to a Met Team table based on the Amer- ican Ephemeris and NauticaI Abstract, 15 min must be added to LST to obtain TST on 15 May at Chiva- Chiva. Thus, the period 06:00-07:00 h (TST) com- prises 45 min from 06:00-07:00 h (LST) and 15 min from 07:00-08:00 h (LST). An estimate of monthly mean hourly global radiation on the TST scale can be obtained by linear interpolation as

lh for 6-7 (TST) = (45/60) [lh for 6-7 (LST)]

+ (15/60) [lh for 7-8 (LST)]

Table 1. Grand means of observed and estimated proportions of monthly mean daily direct radiation occurring during 11:00-13:00 h at Chiva-Chiva in 1973-1978

Month Model Jan. Feb. Mar. Apr. May June July Aug. Sep. Oct. Nov. Dec.

Observed 0.29 0.29 0.28 0.27 0.29 0 .31 0 .31 0.30 0 .33 0.30 0.32 0.30 Garnier-Ohmura 0.33 0.33 0 . 3 3 0.34 0.35 0.36 0.35 0.36 0.36 0 .37 0.37 0.34 Whillier 0.27 0.27 0.26 0.26 0.25 0.25 0.25 0.25 0.26 0.26 0.27 0.27 Fig. 8 equations 0.30 0.30 0.29 0 .28 0.28 0 .31 0.30 0.30 0.32 0 .31 0.32 0.31

solar radiation 451

0.2

o.t5

. . o . t

H g

~ 0.05

Monthly average

' ' I ' ' I ' ' I ' ' I ' ' I ] c ~ ; o x ,~

-:"~- MDRNIN5

x AFTE~OON ~ _ ' % " '

two x

° ° ° ~ x

• It

~. x x

0.03 0.06 0.09 0.12 0 . t5 O.tB HOURLY/X DAILY: GLOBAL RADIATION

Fig. 8. The relation between monthly mean hourly proportions of direct and global daily radiation at Chiva-Chiva during January 1973 through December 1978 excluding April 1976 and June 1973. Lines fitted by quadratic polynomials: Y = -0.006 + 1.05X + 0.343X 2, r 2 = 0.99 (morning); Y = -0.003 + 0.741X + 2.45X 2, r 2 = 0.98 (afternoon). N = 420 for each group (70 months × 6

hours).

This approximation is most subject to error where the tangent to the hourly distribution curve changes slope rapidly.

Monthly mean hourly global radiation was adjusted as in the example, using the time correction for longitudinal displacement from the standard me- ridian and the equation of time on the fifteenth day of each month. Hourly values were then summed to calculate mean daily global radiation, and rb was estimated from adjusted rh by the quadratic poly- nominals in Fig. 8.

There was close correspondence between the grand means of rb as observed in 1973-1978 and estimated in 1982-1985. Provided that there have been no long-term changes in solar radiation patterns, this generates confidence in the predictive accuracy of the quadratic polynomials and the validity of the linear interpolation for conversion from LST to TST. The good agreement shown for May in Fig. 7 is typical of all months. Differences in the grand means of rb for the observed and estimated periods exceeded -+0.01 in just 6 out of 144 cases (12 months × 12 hours) and never exceeded -+0.02. The validity of using these estimates of the hourly distribution of direct radiation at Chiva-Chiva for Barro Colorado (Table A3 of the appendix) is pres- ently unverifiable. It is perhaps more questionable than is the case for daily values. At least, however, allowance is made for monthly and annual variation, which is important.

The inequality of morning and afternoon totals of direct radiation observed in 1973-1978 was also exhibited by the estimated values for 1982-1985 (Fig. 9). This asymmetry is most pronounced in the wet season and probably results from afternoons tending to be cloudier than mornings. Rainfall is indeed more likely during afternoon than morn-

*ing[26], but there are no comparable published data for cloudiness. A similar asymmetry exists for

0.7 >-J-i

~ o = 0.s LL I.-i

z ~ 0.55 I:a t:~ * - - 1 4 I - - G :

~ 0 .5 c:~ lad t~: t:l:

n . ~ 0.45 d F M A M d d A S 0 N D

MDNTH

Fig. 9. Monthly variation in proportion of mean daily direct radiation occurring during the morning at Chiva-Chiva as observed during 1973-1978 (open) and estimated for 1982-1985 (stippled) by quadratic polynomials in Fig. 8. Conventions for Tukey box plots are as in Fig. 1. N = 5 for April and June due to exclusion of 1976 and 1973, respectively; N = 6 for remaining months in 1973-1978, and

N = 4 for all months in 1982-1985.

global radiation (not shown). It may be large enough to affect the responses of plants that are energy- limited, especially since it is strongest when radiation is lowest (Fig. 1). A possible effect is that growth, survival, or phenology of plants in the western halves of forest gaps may differ from that of conspecifics in the eastern halves. Observations in conjunction with the canopy photos taken on Barro Colorado should allow a test of this predic- tion.

Acknowledgments--This work was supported by grant BSR-82-14915 from the National Science Foundation to P. W. Rundel and by a grant to A. P. Smith from the Smithsonian Environmental Sciences Program. I thank the staff of the U.S. Army Tropic Test Center, especially W. Dement, A. Guerriero, B. Lanoue, R. Malone, and D. Weingarten, as well as P. Robert, J. Sager, and D. Windsor of the Smithsonian Institution's Environmental Sciences Program for their cooperation in providing radiation and other data. I am grateful to J. Hay, K. Hogan, and R. Kamada for their helpful comments on an earlier draft of this article.

NOMENCLATURE

The solar constant is expressed as radiation on a surface normal to the sun's beam. All other radiation values refer to a horizontal surface. The subscripts follow the conventions of ref. [3]. Bars over symbols indicate mean values, always calculated on a monthly basis in this study.

Ho extraterrestrial irradiation (daily total) Hb direct irradiation (daily total) Hh global irradiation (daily total) Ha diffuse irradiation (daily total) I~ direct irradiance

! h ~obAl irradiance Kh _Hh/_Ho = average clearness index rb IblHb = ratio of hourly to daily total for direct com-

ponent rh ]h/nh = ratio of hourly to daily total for global ra-

diation

452 P. BECKER

a, b regression parameters n day of year (starting 1 January)

REFERENCES

1. M. C. Anderson, Studies of the woodland light climate. I. The photographic computation of light con- ditions. J. Ecol. 52, 27 (1964).

2. G. C. Evans and D. E. Coombe, Hemispherical and woodland canopy photography and the light climate. J. Ecol. 47, 103 (1959).

3. M. Collares-Pereira and A. Rabl, The average distribution of solar radiation--Correlations between diffuse and hemispherical and between daily and hourly insolation values. Solar Energy 22, 155 (1979).

4. J. P. Kerr, G. W. Thurtell, and C. B. Tanner, An integrating pyranometer for climatological observer stations and mesoscale networks. J. Appl. Meteorol. 6, 688 (1967).

5. R. R. Sokal and F. J. Rohlf, Biometry. 2nd ed. Free- man, San Francisco (1981).

6. S. Chatterjee and B. Price, Regression Analysis by Example. Wiley, New York (1977).

7. J. Neter and W. Wasserman, Applied Linear Statis- tical Models. Richard D. Irwin, Homewood, III. (1974).

8. B. Y. H. Liu and R. C. Jordan, The interrelationship and characteristic distribution of direct, diffuse, and total solar radiation. Solar Energy 4, 1 (1960).

9. J. K. Page, The estimation of monthly mean values of daily total short wave radiation on vertical and in- clined surfaces from sunshine records for latitudes 40°N-40°S. Proc. UN Conf.--New Sources of Energy 4, 378-389 (1961).

10. N. K. D. Choudhury, Solar radiation at New Delhi. Solar Energy 7, 44 (1963).

11. D. W. Ruth and R. E. Chant, The relationship ofdif- fuse radiation to total radiation in Canada. Solar En- ergy 18, 153 (1976).

12. M. C. Anderson, Interpreting the fraction of solar radiation available in forest. Agric. Meteorol. 7, 19 (1970).

13. A. J. Drummond, Comments on Sky radiation mea- surement and corrections. J. Appl. Meteorol. 3, 810 (1964).

14. E. C. McCullough, Total daily radiant energy available extraterrestrially as a harmonic series in the day of the year. Arch. Meteorol. Geophys. Bioklimatol. Series B 16, 129 (1968).

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

16. O. A. Bamiro, Empirical relations for the determination of solar radiation in Ibadan, Nigeria. Solar En- ergy 31, 85 (1983).

17. B. J. Gamier and A. Ohmura, The evaluation of surface variations in solar radiation income. Solar Energy 13, 21 (1970).

18. E. C. McCullough and W. P. Porter, Computing clear day solar radiation spectra for the terrestrial ecolog- ical environment. Ecology 52, 1008 (1971).

19. B .J . Gamier and A. Ohmura, A method of calculating the direct shortwave radiation income of slopes. J. Appl. Meteorol. 7, 796 (1968).

20. L. D. Williams, R. G. Barry and J. T. Andrews, Ap- plication of computed global radiation for areas of high relief. J. Appl. Meteorol. 11, 526 (1972).

21. A. Whillier, The determination of hourly values of total solar radiation from daily summations. Arch Me- teorol. Geopl,ys. Bioklimatol., Series B 7, 197 (1956).

22. W. E. Dietrich, D. M. Windsor and T. Dunne, Ge- ology, climate, and hydrology of Barro Colorado Is- land. Tile Ecology of a Tropical Forest, pp. 21-46. Smithsonian Institution Press, Washington, D. C. (1982).

23. J. M. Stagg, Solar radiation at Kew Observatory. Geo- phys. Mere. Lond. 11(86), 1-37 (1950).

24. J. F. Orgill and K. G. T. Hollands, Correlation equation for hourly diffuse radiation on a horizontal surface. Solar Energy 19, 357 (1977).

25. Anonymous, Materiel Testb~g bz the Tropics. 6th ed. U. S. Army Tropic Test Center, Fort Clayton, Panama (1979).

26. A. S. Rand and W. M. Rand, Variation in rainfall on Barro Colorado Island. The Ecology of a Tropical For- est, pp. 47-59. Smithsonian Institution Press, Wash- ington, D. C. (1982).

27. A. Weiss and J. M. Norman, Partitioning solar radiation into direct and diffuse, visible and near-infrared components. Agric. For. Meteorol. 34, 205 (1985).

APPENDIX

Table AI. Monthly mean daily global radiation (MJ m -2) on Barro Colorado Island during 1982-1985. Values for March 1983-July 1984, March-May 1985, and August-December 1985 were measured on Barro Colorado. Those for the remaining months were estimated

from observations at Chiva-Chiva using the regression in Fig. 2.

Month Year Jan. Feb. Mar. Apr. May June July Aug. Sep. Oct. Nov. Dec.

1982 17.5 20.5 21.5 18.6 15.7 14.8 15.7 15.9 13.3 13.2 16.2 17.1 1983 18.3 19.1 18.9 19.6 15.0 14.6 14.9 15.8 13.2 14.3 15.0 12.7 1984 19.2 19.5 22.2 20.2 16.1 11.8 12.9 14.1 14.6 13.8 15.7 18.6 1985 19.6 20.8 20.5 20.2 15.1 12.7 14.2 14.8 15.4 13.4 15.0 14.8

Table A2. Monthly mean daily direct radiation (MJ m -z) on Barro Colorado during 1982- 1985 as estimated from the values in Table AI of the appendix using the regressions in Fig. 4.

Month Year Jan. Feb. Mar. Apr. May June July Aug. Sep. Oct. Nov. Dec.

1982 9.4 12.2 12.6 8.6 6.4 5.7 5.8 6.6 4.2 4.4 8.2 10.1 1983 10.4 10.3 9.2 9.8 5.7 5.5 5.1 6.5 4.1 5.4 6.8 4.8 1984 11.7 10.8 13.6 10.5 6.8 3.1 3.4 4.8 5.3 4.9 7.6 12.3 1985 12.3 12.6 11.2 10.5 5.8 3.8 4.5 5.5 6.1 4.5 6.8 7.1

Monthly average solar radiation

Table A3. Proportion of monthly mean daily direct radiation occurring during specified hour at Chiva-Chiva as estimated from hourly distribution of global radiation

using regressions in Fig. 8. For each month, lines 1-4 are 1982-1985, respectively.

453

Hour Ending (True Solar Time) Month 7 8 9 10 11 12 13 14 15 16 17 18

Jan. 0.01 0.04 0.08 0.12 0.15 0.15 0.14 0.12 0.09 0.06 0.03 0.01 0.00 0.03 0.08 0.12 0.14 0.16 0.15 0.13 0.10 0.06 0.02 0.00 0.00 0.04 0.08 0.I1 0.14 0.15 0.15 0.13 0.10 0.07 0.03 0.00 0.00 0.04 0.08 0.11 0.13 0.15 0.16 0.14 0.I0 0.06 0.03 0.00

Feb. 0.01 0.04 0.08 0.12 0.14 0.16 0.15 0.13 0.10 0.06 0.03 0.00 0.00 0.03 0.07 0 .11 0.14 0.15 0.15 0.14 0.11 0.06 0.03 0.00 0.01 0.04 0.09 0.12 0.15 0.16 0.15 0.12 0.08 0.05 0.02 0.00 0.00 0.04 0.08 0.I1 0.14 0.15 0.15 0.13 0. I0 0.07 0.03 0.00

Mar. 0.01 0.04 0.08 0.11 0.14 0.15 0.15 0.13 0.10 0.06 0.03 0.00 0.01 0.04 0.08 0.12 0.14 0.15 0.15 0.13 0.10 0.06 0.03 0.01 0.01 0.04 0.08 0.11 0.14 0.15 0.15 0.14 0.10 0.06 0.03 0.01 0.01 0.04 0.08 0.11 0.14 0.15 0.15 0.13 0.10 0.06 0.03 0.00

Apr. 0.01 0.05 0.08 0.11 0.13 0.15 0.14 0.12 0.I0 0.06 0.03 0.01 0.01 0.04 0.09 0.12 0.14 0.15 0.14 0.11 0.09 0.06 0.03 0.01 0.01 0.05 0.09 0.12 0.14 0.15 0.12 0.I1 0.09 0.06 0.04 0.01 0.01 0.05 0.08 0 .11 0.14 0.15 0.14 0.12 0.09 0.06 0.03 0.01

May 0.01 0.05 0.09 0.12 0.13 0.14 0.14 0.12 0.09 0.06 0.03 0.01 0.01 0.05 0.09 0.13 0.14 0.15 0.14 0.11 0.08 0.06 0.03 0.01 0.01 0.05 0.10 0.13 0.14 0.14 0.13 0.12 0.09 0.05 0.02 0.01 0.01 0.06 0.10 0.13 0.15 0,15 0.12 0.10 0,07 0.05 0.03 0.01

June 0.01 0.05 0.I0 0.14 0.17 0.16 0.14 0 .10 0.06 0.04 0.02 0.01 0.02 0.05 0.10 0.13 0.15 0.16 0.15 0.11 0.06 0.04 0.03 0.01 0.02 0.05 0.I0 0.13 0.13 0.14 0.14 0.11 0.08 0.05 0.03 0.01 0.02 0.06 0.12 0.15 0.16 0.16 0.13 0.08 0.05 0.04 0.02 0.01

July 0,01 0.05 0.09 0.13 0.15 0.15 0.14 0.11 0.07 0.05 0,03 0.01 0.01 0.05 0.09 0.12 0.15 0.16 0.14 0.11 0.07 0.05 0.03 0.01 0.02 0.06 0.10 0.14 0.15 0.15 0.14 0.10 0.07 0.04 0.02 0.01 0.01 0.05 0.08 0. t l 0.14 0.16 0.15 0.12 0.08 0.05 0.03 0.01

Aug. 0.01 0.05 0.I0 0.14 0.16 0.16 0.14 0.10 0.06 0.04 0.02 0.01 0.01 0.05 0.09 0.13 0.15 0.16 0.14 0.10 0.07 0.04 0.03 0.01 0.01 0.05 0.09 0.12 0.15 0.15 0.13 0.12 0.09 0.05 0.02 0.00 0.01 0.05 0.10 0.13 0.16 0.16 0.13 0.10 0.07 0.04 0.03 0.01

Sep. 0.01 0.05 0.10 0.14 0.17 0.18 0.15 0.09 0.06 0.04 0.02 0.00 0.01 0.05 0.10 0.14 0.16 0.17 0.14 0.I0 0.07 0.04 0.02 0.00 0.01 0.06 0.10 0.14 0.16 0.15 0.12 0.10 0.08 0.05 0.02 0.00 0.01 0.05 0.10 0.14 0.16 0.16 0.15 0.12 0.06 0.04 0.02 0.00

Oct. 0.00 0.04 0.10 0.14 0.17 0.17 0.16 0.12 0.06 0.03 0.02 0.00 0.00 0.04 0.08 0.13 0.16 0.15 0.16 0.14 0.09 0.04 0.01 0.00 0.00 0.05 0.10 0.14 0.16 0.16 0.14 0.I1 0.08 0.04 0.02 0.00 0.01 0.05 0.09 0.14 0.17 0.16 0.14 0.I1 0.07 0.05 0.02 0.00

Nov. 0.00 0.03 0.08 0.12 0.14 0.16 0.17 0.13 0.09 0.06 0.02 0.00 0.00 0.04 0.09 0.13 0.17 0.17 0.17 0.11 0.07 0.04 0.02 0.00 0.00 0.04 0.08 0.12 0.16 0.16 0.15 0.13 0.09 0.05 0.02 0.00 0.00 0,04 0.I0 0.13 0.15 0.15 0.14 0.12 0.08 0.05 0.03 0.00

Dec. 0.00 0.04 0.08 0.12 0.14 0.15 0.15 0.13 0.11 0.06 0.03 0.00 0.00 0.04 0.09 0.12 0.14 0.16 0.16 0.12 0.09 0.06 0.02 0.00 0.00 0.03 0.07 0.11 0.14 0.15 0.16 0.14 0.10 0.06 0.03 0.00 0.00 0.04 0.08 0.12 0.14 0.15 0.15 0.13 0.I0 0.06 0.03 0.00