When radiation data are not available for a particular site, the amount of solar radiation reaching the surface of the earth is estimated by the following steps (List 1971, Campbell 1977). Note that this is different from the Udintsev (1939) procedure implemented by Budyko, but comparisons with charts of radiation reaching the Earth’s suface indicate that this procedure is reasonably accurate.

The amount of radiation reaching the top of the atmosphere for each hour of the day is estimated by the formula:

(1) Io = (Jo/R2) * sin È / 24;

where Io is the total radiation falling on the atmosphere for a unit of time, Jo is the solar constant (the default is 1360 W/m2-day1), R is the sun's radius vector (from Table 169 in the Smithsonian Meteorological Tables; List 1971), and sin È is the sun's elevation angle. sin È is calculated using the following formula:

(2) sin È = (sin ö * sin ä) + (cos ö * cos ä * cos h);

where ö is the latitude (supplied by the user), ä is the solar's declination (from the Nautical Almanac; current values are from the 1988 Nautical Almanac; U.S. Naval Observatory 1986) and h is the solar hour angle. h is calculated by:

(3) h = 15 * (ht - 12);

where ht is the current hour. If the sun is below the horizon, sin phi will be negative. For negative values of sin phi, no incoming solar radiation was assumed to be coming in for that time period (i.e., the period between dusk and dawn). The 60 in Equation (1) is convert the radiation estimate for one minute into an hourly estimate. This is necessary because of the units for the solar constant.

The amount of direct solar radiation falling at the earth's surface is then estimated by:

(4) Isr = (Jo/R2) * am * sin È / 24;

where Isr is the direct solar radiation at the surface, a is the atmospheric transmissivity (the default is 0.70), and m is a correction for optical path length (Campbell 1977). m is given by:

(5) m = (P/Po)/sin È;

where P is the atmospheric pressure at the site and Po is the atmospheric pressure at sea level. This program asks the user to provide the elevation of the site, and then estimates atmospheric pressure at the site (Hess 1959). The formula is:

(6) P = Po * e-(z/Ho);

where z is the elevation (in meters) and Ho is the height of the homogenous atmosphere (assumed to be 8,000 m; Hess 1959).

The amount of diffuse solar radiation reaching the surface is given by:

(7) Isf = ((0.91 * Io) - Isr)/2;

where Isf is diffuse solar radiation.

The direct and diffuse solar radiation amounts at the surface are summed to give the total radiation amount falling on the surface.

Hourly radiation estimates are summed to arrive at the daily estimate.

Copyright © 2003-2011, David M. Lawrence

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