addition, the values obtained by the use of 

 Black's (1956) monthly charts were examined. 



The equations are as follows: 

 Kimball (1928): 



Q = Q^ (0.29 + 0.71 [1.0 - C]) (1) 



where Q is the calculated average daily total of 

 solar radiation received on a horizontal surface 

 (direct and diffuse), Qq is the average daily 

 total of solar radiation with a cloudless sky, 

 and C is the proportion of sky covered by 

 clouds. 



Savino-Angstrbm (Budyko, 1956; Burdeki, 

 1958): 



(Q +q) = (Q +q)„ (1 - [1 -K]n) (2) 



where (Q + q), (Q + q)o, and n are identical 

 with the Kimball quantities Q, Qq, and C, and 

 K is the ratio between the actual radiation 

 under overcast sky conditions and the possible 

 radiation (Budyko, 1956; p. 31). K varies with 

 the region and with latitude, and diurnally and 

 seasonally. Values of the coefficient K have 

 been calculated and averaged for various lati- 

 tudes by Budyko (1956; table 2). 

 Black (1956): 



Q = Q^ (0.803 - 0.340C - 0.4580^) (3) 



where Q = Qq= (Q + q), C = n = C above, and 

 Qf^ is the maximum possible radiation in the 

 absence of an atmosphere. 

 Laevastu (1960): 



Qs = 0.014 Aj^td (1 - O.OOO6C3) (4) 



where Qg = (Q + q) = Q above, Aj, is the noon 

 sun altitude, t^ the day length in minutes, and 

 C the cloudiness (in tenths). 



These equations were developed with dif- 

 ferent purposes in mind. The Kimball and 

 Savino-Angstrdm equations have been used by 

 oceanographers in heat-budget studies and for 

 estimating the amount of solar energy avail- 

 able for photosynthesis. The Black equation 

 was developed in a recent study of the distri- 

 bution of solar energy on the earth's surface at 

 monthly intervals. Laevastu has attempted to 

 devise a formula which is suitable for the com- 

 putation of insolation on a daily basis. Although 

 more sophisticated equations have been pro- 

 posed which take into account the cloud cover 

 at different altitudes, water vapor content of 



the atmosphere, (see Bortovskii, 1961), 



they cannot be evaluated here because the 

 necessary data are not available. 



Each of the four equations given above is 

 different because somewhat different theoreti- 

 cal considerations attended their development. 

 Kimball's equation is based on the assumption 

 that the amount of solar energy reaching the 

 earth's surface is proportional to cloudiness 

 and that the transmission of the atmosphere 

 under cloudless conditions approaches a con- 

 stant. In the Savino-Angstrom equation the 

 underlying assumptions are similar, except 

 that a coefficient K is introduced to take into 



consideration the seasonal and latitudinal vari- 

 ations in the effect of cloudiness. Black chose 

 to evaluate the effects of clouds by considering 

 the relation between cloudiness and the amount 

 of radiation that would reach the earth's sur- 

 face in the absence of an atmosphere, and 

 obtained a regression which takes a quadratic 

 form. Laevastu assumed that the amount of 

 daily insolation on the ocean surface under a 

 clear sky depends mainly on day length and 

 the noon altitude of the sun, but also provided 

 a correction for cloudiness. 



The summarized data used in these compu- 

 tations and calculated daily radiation totals on 

 the basis of the five methods are presented in 

 table 2. It should be noted that two series of 

 figures are tabulated for the Black nnethod; 

 one series was obtained from the monthly 

 charts prepared by Black, and the other was 

 computed from the Black equation (3) given 

 above. The value of Qq in the Kimball equation 

 was taken from the tabulation of Qo values 

 (i.e., (Q + q)o) of Budyko (1956, table 1) rather 

 than from the tabulation of Kimball (1928, 

 table 3). 



To obtain a quantitative estimate of the error 

 arising from the use of climatological equa- 

 tions in the eastern Pacific, a comparison by 

 linear regression techniques was made be- 

 tween the measured values and those obtained 

 with the four climatological equations. In each 

 comparison between the measured and com- 

 puted values the correlation was significant 

 (p < 0.01). The regression coefficients were 

 highly significant, and the sample standard 

 deviations ranged from 81.9 to 120 g. 

 cal./cm. /da. The intercept differed signifi- 

 cantly from zero only in the Laevastu com- 

 parison. 

 ^ o 



Both the Kimball and Savino-Angstrbm equa- 

 tions yielded estimates that were roughly 6 to 

 7 percent higher than the observed value, 

 whereas the Black map and equation under- 

 estimated the mean value by a somewhat greater 

 amount. The Laevastu equation overestimated 

 the mean value considerably, and the failure 

 of the regression line to pass through the 

 origin indicates additional bias. Of course, 

 neither the Kimball, Savino-Angstrom or Black 

 equations, nor the Black map, were designed 

 for the estimation of a given daily radiation 

 total, and their use for such a purpose is not 

 recommended unless a relatively large stand- 

 ard error of estimate can be tolerated. 



On the few occasions when cloudiness was 

 essentially lacking during the day, the Kimball, 

 Savino-Angstrom, and Black equations yielded 

 daily total estimates that were in excess of the 

 observed daily radiation total. This disagree- 

 ment suggests that the tropical air mass is 

 somewhat more opaque to solar energy under 

 a clear sky than the equations predict. Unfortu- 

 nately, the data for clear skies are sofew that 

 the correctness of this conclusion must be re- 

 examined after more data become available. 



