## Solar radiation at the Earth's surface

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### INTRODUCTION

This program predicts the amount of solar radiation reaching the surface of the Earth. The solar radiation is estimated by the following steps (List 1971, Campbell 1977).

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

(1) I_o = (J_o/R^2) * sin phi / 24where I_o is the total radiation falling on the atmosphere for a unit of time, J_o is the solar constant (the default is 1360 W m^-2 day^-1), R is the Sun's radius vector (from Table 169 in the Smithsonian Meteorological Tables; List 1971), and sin phi is the Sun's elevation angle. sin phi is calculated using the following formula:

(2) sin phi = (sin LAT * sin DECL)

+ (cos LAT * cos DECL * cos h)where LAT is the latitude (supplied by the user), DECL is the solar's declination and h is the solar hour angle. h is calculated by:

(3) h = 15 * (h_t - 12)where h_t 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.

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

(4) I_sr = (J_o/R^2) * a^m * sin phi / 24where I_sr is the direct solar radiation at the surface, a is the atmospheric transmissivity (the default is 0.84), and m is a correction for optical path length (Campbell 1977). m is given by:

m = (P/P_o)/sin phiwhere P is the atmospheric pressure at the site and P_o 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:

P = P_o * 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).

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

(5) I_sf = ((0.91 * I_o) - I_sr)/2where I_sf is diffuse solar radiation.

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

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

6) Note that the solar declination is calculated by the following formula:

(6) DECL = 23.45 * sin[360/365 * (284 + day of year)]I don't have a reference for this yet, but it's simpler than my old way of calculating declination.

7) The slope radiation calculations, based on algorithms from the program, MTCLIM (Hungerford et al. 1989) seem to work fine now. [DML 09 Dec 2006]

### REFERENCES

Campbell, G.S. 1977. An introduction to environmental biosphysics. Springer-Verlag, New York, New York, USA.

Hess, S.L. 1959. Introduction to theoretical meteorology. Robert E. Krieger Publishing Company, Malabar, Florida, USA.

Hungerford, R.D., R.R. Nemani, S.W. Running and J.C. Coughlan. 1989. MTCLIM: A Mountain Microclimate Simulation Model. USDA Forest Service, Intermountain Research Station, Research Paper INT-414.

List, R.J. 1971. Smithsonian meteorological tables, 6th ed. Smithsonian Institution Press, Washington, D.C., USA.

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