1:3.3.3 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:

$R_T ={R_{Tpeak}-i_{mx}*\delta_1*(1-exp[(\frac{(T-T_{peak})}{\delta_1})] , {R_{Tpeak}+i_{mx}*\delta_2*(1-exp[\frac{(T_{peak}-T)}{\delta_2}])}}$ 1:3.3.12

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

where $R_T$ is the cumulative amount of rain that has fallen at time $T$(mm H$_2$O), $R_{Tpeak}$ is the amount of rain that has fallen at time $T_{peak}$ (mm H$_2$O), $i_{mx}$ is the maximum or peak rainfall intensity during the storm (mm/hr), $\delta_1$ is the equation coefficient for rainfall intensity before peak intensity is reached (hr), $\delta_2$ is the equation coefficient for rainfall intensity after peak intensity is reached (hr), $T_{peak}$ is the time from the beginning of the storm till the peak rainfall intensity occurs (hr), and $T_{dur}$ is the storm duration (hr). The time to peak intensity is defined as

$T_{peak}=\hat t_{peak}*T_{dur}$ 1:3.3.13

where $T_{peak}$ is the time from the beginning of the storm till the peak rainfall intensity occurs (hr), $\hat t_{peak}$ 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 $T_{dur}$ is the storm duration (hr). The cumulative volume of rain that has fallen at $T_{peak}$ is

$R_{Tpeak}=\hat t_{peak} *R_{day}$ 1:3.3.14

where $R_{Tpeak}$ is the amount of rain that has fallen at time $T_{peak}$ (mm H$_2$O), $\hat t_{peak}$ 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 $R_{day}$ is the total rainfall on a given day (mm H$_2$O).

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

$R_{day}=i_{mx}*(\delta_1+\delta_2)=i_{mx}*T_{dur}*(d1+d2)$ 1:3.3.15

where $R_{day}$ is the rainfall on a given day (mm H$_2$O), $i_{mx}$ is the maximum or peak rainfall intensity during the storm (mm/hr), $\delta_1$ is the equation coefficient for rainfall intensity before peak intensity is reached (hr), $\delta_2$ is the equation coefficient for rainfall intensity after peak intensity is reached (hr), $d1$ is the normalized equation coefficient for rainfall intensity before peak intensity is reached, $d2$ is the normalized equation coefficient for rainfall intensity after peak intensity is reached, and $T_{dur}$ is the storm duration (hr). This equation can be rearranged to calculate the storm duration:

$T_{dur}=\frac{R_{day}}{i_{mx}*(d_1+d_2)}$ 1:3.3.16

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