FAO56 Shortwave Method

The FAO56 Method implements the algorithm detailed by Allen, Pereira, Raes, and Smith (1998). The algorithm calculates the solar declination and solar angle for each time interval of the simulation, using the coordinates of the subbasin, Julian day of the year, and time at the middle of the interval. The solar values are used to compute the extra-terrestrial radiation for each subbasin. Total daylight hours are computed based on the Julian day and compared to the number of actual sunshine hours. Shortwave radiation arriving at the ground surface is then computed using the most common relationship accounting for reduction in sunshine hours due to cloud cover. Cloud cover is considered through a time-series of sunshine hours defined as the number of decimal hours per full hour where the shortwave radiation exceeds 120 watts per square meter.

The shortwave or solar radiation, Rs, is given by Equation 35 from Food and Agricultural Organization Paper No. 56 (FAO 56):

$\begin{array}{l}\displaystyle R_s = (a_s +b_s*n/N)*R_a\end{array}$

where

n is the actual duration of sunshine [hour],
N is the maximum possible duration of sunshine or daylight hours [hour],
n/N is the relative sunshine duration [-],
$\begin{array}{l}R_a\end{array}$ is the extraterrestrial radiation [MJ m-2 day-1],
$\begin{array}{l}R_s\end{array}$ is the fraction of extraterrestrial radiation reaching the earth on completely overcast days (when n=0)
$\begin{array}{l}a_s+b_s\end{array}$ is the fraction of extraterrestrial radiation reaching the earth on clear days (when n = N).

The duration of sunshine, n, is recorded with a Campbell Stokes sunshine recorder and can be input as a Sunshine Gage in HEC-HMS.

The extraterrestrial radiation for a given day of the year is estimated based on the solar constant, solar declination angle, time of year and the area of interest's location with Equation 21 from FAO56:

$\begin{array}{l}\displaystyle R_a= 24*60/\pi *G_sc*d_r*[ω_s *sin(\phi)*sin(\delta)+cos(\phi)*cos(d)*sin(ω_s)]\end{array}$

where

$\begin{array}{l}R_a\end{array}$ extraterrestrial radiation [MJ m-2 day-1],
$\begin{array}{l}G_sc\end{array}$ solar constant = 0.0820 MJ m-2 min-1,
$\begin{array}{l}d_r\end{array}$ inverse relative distance Earth-Sun (Equation 23),
$\begin{array}{l}\omega_s\end{array}$ sunset hour angle (Equation 25 or 26) [rad],
ϕ latitude [rad] = pi/180 * decimal degrees
δ solar declination (Equation 24) [rad].

The inverse relative distance Earth-Sun, $\begin{array}{l}d_r\end{array}$ , and the solar declination, δ, are given by Equations 23 and 24 in FAO56:

$\begin{array}{l}\displaystyle d_r = 1+0.033cos(\frac{2\pi}{365}J)\end{array}$

$\begin{array}{l}\displaystyle \delta = 0.409sin(\frac{2\pi}{365}J-1.39)\end{array}$

The variable J represents the number day of the year.

The sunset hour angle, $\begin{array}{l}\omega_s\end{array}$ , is given by Equation 25 in FAO56:

$\begin{array}{l}\displaystyle \omega_s=arccos[-tan(\phi)tan(\delta)]\end{array}$

FAO56 also provides methodology for computing $\begin{array}{l}R_a\end{array}$ on a sub-daily timestep based on solar angles at the beginning and ending of the time interval with FAO56 Equation 28:

$\begin{array}{l}\displaystyle R_a=\frac{12(60)}{\pi}G_{sc}d_r[(\omega_2-\omega_1)sin(\phi)sin(\delta)+cos(\phi)cos(\delta)(sin(\omega_2)-sin(\omega_1))]\end{array}$

where $\begin{array}{l}R_a\end{array}$ extraterrestrial radiation in the hour (or shorter) period [MJ m-2 hour-1],
$\begin{array}{l}G_s_c\end{array}$ is the solar constant = 0.0820 MJ m-2 min-1,
$\begin{array}{l}d_r\end{array}$ is the inverse relative distance Earth-Sun (Equation 23),
δ is the solar declination [rad] (Equation 24),
ϕ is latitude [rad],
$\begin{array}{l}\omega_1\end{array}$ solar time angle at beginning of period [rad] (Equation 29),
$\begin{array}{l}\omega_2\end{array}$ solar time angle at end of period [rad] (Equation 30)

The solar angle at the beginning of period, ω1, end of the period, ω2, and midpoint ω are provided by FAO56 Equations 29, 30 and 31:

$\begin{array}{l}\displaystyle \omega_1=\omega-\frac{(\pi)t_1}{24}\end{array}$

$\begin{array}{l}\displaystyle \omega_2=\omega+\frac{(\pi)t_1}{24}\end{array}$

$\begin{array}{l}\displaystyle \omega=\frac{(\pi)}{12}[(t+0.06667(L_z-L_m)+S_c)-12]\end{array}$

where

t is standard clock time at the midpoint of the period [hour].
$\begin{array}{l}L_z\end{array}$ is the longitude at the center of the local time zone [degrees west of Greenwich].
$\begin{array}{l}L_m\end{array}$ is the longitude of the measurement site [degrees west of Greenwich],
$\begin{array}{l}S_c\end{array}$ seasonal correction for solar time [hour]

The seasonal correction, $\begin{array}{l}S_c\end{array}$ , is given by FAO56 Equation 32 and 33:

$\begin{array}{l}\displaystyle S_c=0.1645sin(2b)-0.1255cos(b)-0.025sin(b)\end{array}$

$\begin{array}{l}\displaystyle b=\frac{(2\pi)(J-81)}{364}\end{array}$

Finally, daylight hours for a given latitude on earth, N, is given by FAO56 Equation 34:

$\begin{array}{l}N= 24/π * ω\end{array}$

Required Parameters

The only parameters required to utilize this method within HEC-HMS is the decimal degrees or degrees minute seconds of the central meridian of the local time zone and a sunshine gage in each subbasin assigned in the Meteorologic Model.