For simulations where the timing of rainfall within the day is required, the daily rainfall value must be partitioned into shorter time increments. The method used in SWAT+ to disaggregate storm data was taken from CLIGEN (Nicks et al., 1995).

A double exponential function is used to represent the intensity patterns within a storm. With the double exponential distribution, rainfall intensity exponentially increases with time to a maximum, or peak, intensity. Once the peak intensity is reached, the rainfall intensity exponentially decreases with time until the end of the storm.

The exponential equations governing rainfall intensity during a storm event are:

$i(T)={i_{mx}*exp[\frac{T-T_{peak}}{\delta_{1}}], i_{mx}*exp[\frac{T_{peak}-T}{\delta_2}}]$ 1:3.3.1

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

where $i$ is the rainfall intensity at time $T$ (mm/hr), $i_{mx}$ is the maximum or peak rainfall intensity during the storm (mm/hr), $T$ is the time since the beginning of the storm (hr), $T_{peak}$ is the time from the beginning of the storm till the peak rainfall intensity occurs (hr), $T_{dur}$ is the duration of the storm (hr), and $\delta_1$ and $\delta_2$ are equation coefficients (hr).

The maximum or peak rainfall intensity during the storm is calculated assuming the peak rainfall intensity is equivalent to the rainfall intensity used to calculate the peak runoff rate. The equations used to calculate the intensity are reviewed in Chapter 2:1 (section 2:1.3.3).

The rainfall intensity distribution given in equation 1:3.3.1 can be normalized to eliminate units. To do this, all time values are divided, or normalized, by the storm duration and all intensity values are normalized by the average storm intensity. For example,

$\hat i =\frac{i}{i_{ave}}$ 1:3.3.2

$\hat t=\frac{T}{T_{dur}}$ 1:3.3.3

where $\hat i$ the normalized rainfall intensity at time $\hat t$, $i$ is the rainfall intensity at time T (mm/hr), $i_{ave}$ is the average storm rainfall intensity (mm/hr),$\hat t$ is the time during the storm expressed as a fraction of the total storm duration (0.0-1.0), $T$ is the time since the beginning of the storm (hr), and $T_{dur}$ is the duration of the storm (hr).

The normalized storm intensity distribution is:

$\hat i(\hat t)={\hat i_{mx}*exp[\frac{\hat t - \hat t_{peak}}{d_1}] , \hat i_{mx}*exp[\frac{\hat t_{peak}-\hat t}{d_2}]}$ 1:3.3.4

$0 \le \hat t \le \hat t_{peak}$ , $\hat t_{peak} < \hat t< 1.0$

where $\hat i$ the normalized rainfall intensity at time $\hat t$, $\hat i_{mx}$ is the normalized maximum or peak rainfall intensity during the storm,$\hat t$ is the time during the storm expressed as a fraction of the total storm duration (0.0-1.0), $\hat t_{peak}$ 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), $d_1$ and $d_2$ are equation coefficients.

The relationship between the original equation coefficients and the normalized equation coefficients is:

$\delta_1=d_1*T_{dur}$ 1:3.3.5

$\delta_2=d_2*T_{dur}$ 1:3.3.6

where $\delta_1$ is the equation coefficient for rainfall intensity before peak intensity is reached (hr), $d_1$is the normalized equation coefficient for rainfall intensity before peak intensity is reached, $\delta_2$ is the equation coefficient for rainfall intensity after peak intensity is reached (hr), $d_2$ is the normalized equation coefficient for rainfall intensity after peak intensity is reached, and $T_{dur}$ is the storm duration (hr).

Values for the equation coefficients, $d_1$ and $d_2$, can be determined by isolating the coefficients in equation 1:3.3.4. At $\hat t$ = 0.0 and at $\hat t$= 1.0, $\frac{\hat i}{\hat i_{mx}} \approx 0.01$

$d_1=\frac{\hat t-\hat t_{peak}}{ln(\frac{\hat i}{\hat i_{mx}})}=\frac{0-\hat t_{peak}}{ln(0.01)}=\frac{\hat t_{peak}}{4.605}$ 1:3.3.7

$d_2=\frac{\hat t_{peak}-\hat t}{ln(\frac{\hat i}{\hat i_{mx}})}=\frac{\hat t_{peak}-1}{ln(0.01)}=\frac{1.0-\hat t_{peak}}{4.605}$ 1:3.3.8

where $d_1$ 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, $\hat t$ is the time during the storm expressed as a fraction of the total storm duration (0.0-1.0), $\hat t_{peak}$ 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), $\hat i$ is the normalized rainfall intensity at time $\hat t$ , and $\hat i_{mx}$ is the normalized maximum or peak rainfall intensity during the storm.

The normalized time to peak intensity is calculated by SWAT+ using a triangular distribution. The triangular distribution used to generate the normalized time to peak intensity requires four inputs: average time to peak intensity expressed as a fraction of total storm duration $(\hat t_{peakM})$, maximum time to peak intensity expressed as a fraction of total storm duration $(\hat t_{peakU})$, minimum time to peak intensity expressed as a fraction of total storm duration $(\hat t_{peakL})$ and a random number between 0.0 and 1.0.

The maximum time to peak intensity, or upper limit of the triangular distribution, is set at 0.95. The minimum time to peak intensity, or lower limit of the triangular distribution is set at 0.05. The mean time to peak intensity is set at 0.25.

The triangular distribution uses one of two sets of equations to generate a normalized peak intensity for the day. If $rnd_1\le[\frac{\hat t_{peakM} - \hat t_{peakL}}{\hat t_{peakU} - \hat t_{peakL}}]$ then

$\hat t_{peak}=\hat t_{peakM} * \frac{\hat t_{peakL}+[rnd_1*(\hat t_{peakU}-\hat t_{peakL})*(\hat t_{peakM}-\hat t_{peakL})]^{0.5}}{\hat t_{peak,mean}}$ 1:3.3.9

If $rnd_1>[\frac{\hat t_{peakM} - \hat t_{peakL}}{\hat t_{peakU}-\hat t_{peakL}}]$ then

$\hat t_{peak}=\hat t_{peakM}*\frac{\hat t_{peakU}-(\hat t_{peakU}-\hat t_{peakM})*[\frac{\hat t_{peakU}(1-rnd_1)-\hat t_{peakL}(1-rnd_1)}{\hat t_{peakU}-\hat t_{peakM}}]^{0.5}}{\hat t_{peak,mean}}$ 1:3.3.10

where $\hat t_{peak}$ 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), $\hat t_{peakM}$is the average time to peak intensity expressed as a fraction of storm duration, $rnd_1$ is a random number generated by the model each day, $\hat t_{peakL}$ is the minimum time to peak intensity that can be generated, $\hat t_{peakU}$is the maximum time to peak intensity that can be generated, and $\hat t_{peak,mean}$ is the mean of $\hat t_{peakL} , \hat t_{peakM}$ and $\hat t_{peakU}$ .

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.