# 1:1.2.4 Hourly Solar Radiation

The extraterrestrial radiation falling on a horizontal surface during one hour is given by the equation:

$$I\_0=I\_{SC}E\_0(\sin\delta\sin\phi+\cos\delta\cos\phi\cos\omega t)$$                                                                                              1:1.2.8        

where $$I\_0$$ is the extraterrestrial radiation for 1 hour centered around the hour angle $$\omega t$$.

An accurate calculation of the radiation for each hour of the day requires a knowledge of the difference between standard time and solar time for the location. SWAT+ simplifies the hourly solar radiation calculation by assuming that solar noon occurs at 12:00pm local standard time. 

When the values of $$I\_0$$ calculated for every hour between sunrise and sunset are summed, they will equal the value of $$H\_0$$. Because of the relationship between $$I\_0$$ and $$H\_0$$, it is possible to calculate the hourly radiation values by multiplying $$H\_0$$ by the fraction of radiation that falls within the different hours of the day. The benefit of this alternative method is that assumptions used to estimate the difference between maximum and actual solar radiation reaching the earth’s surface can be automatically incorporated in calculations of hourly solar radiation at the earth’s surface.

SWAT+ calculates hourly solar radiation at the earth’s surface with the equation:

$$I\_{hr}=I\_{frac} H\_{day}$$                                                                                                                                                    1:1.2.9

where $$I\_{hr}$$  is the solar radiation reaching the earth’s surface during a specific hour of the day                    ($$MJ m^{-2}hr^{-1}$$), $$I\_{frac}$$ is the fraction of total daily radiation falling during that hour, and $$H\_{day}$$ is the total solar radiation reaching the earth’s surface on that day.

The fraction of total daily radiation falling during an hour is calculated

$$I\_{frac}=\frac {\displaystyle(\sin\delta\sin\phi + \cos\delta\cos\phi\cos\omega t\_i)} {\displaystyle\sum\_{t=SR}^{SS}(\sin\delta\sin\phi+\cos\delta\cos\omega t)}$$                                                                                                   1:1.2.10

where $$t\_i$$ is the solar time at the midpoint of hour $$i$$.  


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