Total Rainfall and Duration

The volume of rain is related to rainfall intensity by:

$R_T=\int_0^T i dT$ 1:3.3.11

where $R_T$ is the amount of rain that has fallen at time $T$ (mm H$_2$O) and $i$ is the rainfall intensity at time $T$ (mm/hr).

Using the definition for rainfall intensity given in equation 1:3.3.1, equation 1:3.3.11 can be integrated to get:$R_T ={R_{Tpeak}-i_{mx}*\delta_1*(1-exp[(\frac{(T-T_{peak})}{\delta_1})] , {R_{Tpeak}+i_{mx}*\delta_2*(1-exp[\frac{(T_{peak}-T)}{\delta_2}])}}$

$0 \le T \le T_{peak} , T_{peak}<T\le T_{dur}$ 1:3.3.12

where is the cumulative amount of rain that has fallen at time (mm HO), is the amount of rain that has fallen at time (mm HO), is the maximum or peak rainfall intensity during the storm (mm/hr), is the equation coefficient for rainfall intensity before peak intensity is reached (hr), is the equation coefficient for rainfall intensity after peak intensity is reached (hr), is the time from the beginning of the storm till the peak rainfall intensity occurs (hr), and is the storm duration (hr). The time to peak intensity is defined as

1:3.3.13

where is the time from the beginning of the storm till the peak rainfall intensity occurs (hr), is the time from the beginning of the storm till the peak intensity expressed as a fraction of the total storm duration (0.0-1.0), and is the storm duration (hr). The cumulative volume of rain that has fallen at is

1:3.3.14

where is the amount of rain that has fallen at time (mm HO), is the time from the beginning of the storm till the peak intensity expressed as a fraction of the total storm duration (0.0-1.0), and is the total rainfall on a given day (mm HO).

The total rainfall for the day can be defined mathematically by integrating equation 1:3.3.11 and solving for the entire storm duration:

1:3.3.15

where is the rainfall on a given day (mm HO), is the maximum or peak rainfall intensity during the storm (mm/hr), is the equation coefficient for rainfall intensity before peak intensity is reached (hr), is the equation coefficient for rainfall intensity after peak intensity is reached (hr), is the normalized equation coefficient for rainfall intensity before peak intensity is reached, is the normalized equation coefficient for rainfall intensity after peak intensity is reached, and is the storm duration (hr). This equation can be rearranged to calculate the storm duration:

1:3.3.16

Table 1:3-3: SWAT+ input variables that pertain to generation of maximum half-hour rainfall.