# 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 $$T$$ ($$mm\space H\_2O$$) 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  

&#x20;$$0 \le T \le T\_{peak} , T\_{peak}\<T\le T\_{dur}$$                                                                                                                  &#x20;

where $$R\_T$$ is the cumulative amount of rain that has fallen at time $$T$$$$T$$ ($$mm\space H\_2O$$), $$R\_{Tpeak}$$ is the amount of rain that has fallen at time $$T\_{peak}$$ ($$mm\space H\_2O$$), $$i\_{mx}$$ 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 $$T\_{peak}$$ ($$mm\space H\_2O$$), $$\hat t\_{peak}$$ 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 ($$mm\space H\_2O$$).

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

$$R\_{day}=i\_{mx}\*(\delta\_1+\delta\_2)=i\_{mx}*T\_{dur}*(d\_1+d\_2)$$                                                                                   1:3.3.15

where $$R\_{day}$$ is the rainfall on a given day ($$mm\space H\_2O$$), $$i\_{mx}$$ 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 ($$hr$$), $$d\_1$$ is the normalized equation coefficient for rainfall intensity before peak intensity is reached, $$d\_2$$ 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.

| Definition                                                               | Input Name | Input File                                                                                              |
| ------------------------------------------------------------------------ | ---------- | ------------------------------------------------------------------------------------------------------- |
| $$R\_{day}$$: amount of rain falling on a given day ($$mm\space H\_2O$$) | pcp        | [.pcp](https://swatplus.gitbook.io/io-docs/introduction-1/climate/pcp.cli-and-precipitation-data-files) |