# 1:3.1.2 Amount of Precipitation

Numerous probability distribution functions have been used to describe the distribution of rainfall amounts. SWAT+ provides the user with two options: a skewed distribution and an exponential distribution.

The skewed distribution was proposed by Nicks (1974) and is based on a skewed distribution used by Fiering (1967) to generate representative streamflow. The equation used to calculate the amount of precipitation on a wet day is:

$$R\_{day}=\mu\_{mon}+2\*\sigma\_{mon}*(\frac{\[(SND\_{day}-\frac{g\_{mon}}{{6}})*\frac{g\_{mon}}{{6}}+1]^3-1}{g\_{mon}})$$                                                                             1:3.1.3

where $$R\_{day}$$ is the amount of rainfall on a given day  ($$mm\space H\_2O$$), $$\mu\_{mon}$$ is the mean daily rainfall              ($$mm\space H\_2O$$) for the month, $$\sigma\_{mon}$$ is the standard deviation of daily rainfall ($$mm\space H\_2O$$) for the month, $$SND\_{day}$$ is the standard normal deviate calculated for the day, and $$g\_{mon}$$ is the skew coefficient for daily precipitation in the month.

The standard normal deviate for the day is calculated:

$$SND\_{day}=cos(6.283*rnd\_2)*\sqrt{-2ln(rnd\_1)}$$                                                                                       1:3.1.4

where $$rnd\_1$$ and $$rnd\_2$$ are random numbers between 0.0 and 1.0.

The exponential distribution is provided as an alternative to the skewed distribution. This distribution requires fewer inputs and is most commonly used in areas where limited data on precipitation events is available. Daily precipitation is calculated with the exponential distribution using the equation:

$$R\_{day}=\mu\_{mon}\*(-ln(rnd\_1))^{rexp}$$                                                                                                                      1:3.1.5

where $$R\_{day}$$ is the amount of rainfall on a given day  ($$mm\space H\_2O$$), $$\mu\_{mon}$$ is the mean daily rainfall ($$mm\space H\_2O$$) for the month, $$rnd\_1$$ is a random number between 0.0 and 1.0, and $$rexp$$ is an exponent that should be set between 1.0 and 2.0. As the value of $$rexp$$ is increased, the number of extreme rainfall events during the year will increase. Testing of this equation at locations across the U.S. have shown that a value of 1.3 gives satisfactory results.

Table 1:3-1: SWAT+ input variables that pertain to generation of precipitation.

<table><thead><tr><th width="231">Definition</th><th>Source Name</th><th>Input Name</th><th>Input File</th></tr></thead><tbody><tr><td>Precipitation input: 'sim' for simulated or gage name</td><td>pgage</td><td>pcp</td><td><a href="../../../../introduction-1/climate/weather-sta.cli">weather-sta.cli</a></td></tr><tr><td><span class="math">P_i(W/D)</span>: probability of a wet day following a dry day in month</td><td>pr_wd</td><td>wet_dry</td><td><a href="../../../../introduction-1/climate/weather-wgn.cli">weather-wgn.cli</a></td></tr><tr><td><span class="math">P_i(W/W)</span>: probability of a wet day following a wet day in month</td><td>pr_ww</td><td>wet_wet</td><td><a href="../../../../introduction-1/climate/weather-wgn.cli">weather-wgn.cli</a></td></tr><tr><td><mark style="background-color:red;">Rainfall distribution code: 0-skewed, 1-exponential</mark></td><td></td><td>IDIST</td><td>file.cio</td></tr><tr><td><span class="math">rexp</span><mark style="background-color:red;">: value of exponent (required if IDIST = 1)</mark></td><td></td><td>REXP</td><td>file.cio</td></tr><tr><td>average amount of precipitation falling in month  (<span class="math">mm\space H_2O</span>)</td><td>pcpmm</td><td>pcp_ave</td><td><a href="../../../../introduction-1/climate/weather-wgn.cli">weather-wgn.cli</a> </td></tr><tr><td>average number of days of precipitation in month(<span class="math">\mu_{mon}</span>= PCPMM / PCPD)               </td><td>pcpd</td><td>pcp_days</td><td><a href="../../../../introduction-1/climate/weather-wgn.cli">weather-wgn.cli</a> </td></tr><tr><td><span class="math">\sigma_{mon}</span>: standard deviation for daily precipitation in month  (<span class="math">mm\space H_2O</span>)</td><td>pcpstd</td><td>pcp_sd</td><td><a href="../../../../introduction-1/climate/weather-wgn.cli">weather-wgn.cli</a> </td></tr><tr><td><span class="math">g_{mon}</span>: skew coefficient for daily precipitation in month</td><td>pcpskw</td><td>pcp_skew</td><td><a href="../../../../introduction-1/climate/weather-wgn.cli">weather-wgn.cli</a> </td></tr></tbody></table>