# 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:$$R_T ={R_{Tpeak}-i_{mx}*\delta_1*(1-exp[(\frac{(T-T_{peak})}{\delta_1})] , {R_{Tpeak}+i_{mx}*\delta_2*(1-exp[\frac{(T_{peak}-T)}{\delta_2}])}}$$               &#x20;

&#x20;$$0 \le T \le T_{peak} , T_{peak}<T\le T_{dur}$$                                                  1:3.3.12

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).

&#x20;                     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.

| Variable Name | Definition                                                       | File Name |
| ------------- | ---------------------------------------------------------------- | --------- |
| PRECIPITATION | $$R_{day}$$: amount of rain falling on a given day (mm H$$_2$$O) | .pcp      |

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