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

where $\alpha_{0.5mon}$ is the average half-hour rainfall fraction for the month, $adj_{0.5\alpha}$ is an adjustment factor, $R_{0.5sm}$ is the smoothed half-hour rainfall amount for the month (mm H$_2$O), $\mu_{mon}$ is the mean daily rainfall (mm H$_2$O) for the month, $yrs$ is the number of years of rainfall data used to obtain values for monthly extreme half-hour rainfalls, and $days_{wet}$ are the number of wet days in the month. The adjustment factor is included to allow users to modify estimations of half-hour rainfall fractions and peak flow rates for runoff.

where $R_{hLmon}$ is the smallest relative humidity value that can be generated on a given day in the month, and $R_{hmon}$ is the average relative humidity for the month.

The triangular distribution uses one of two sets of equations to generate a relative humidity value for the day. If $rnd_1 \le (\frac{R_{hmon}-R_{hLmon}}{R_{hUmon}-R_{hLmon}})$ then

$R_h=R_{hmon}*\frac{R_{hLmon}+[rnd_1*(R_{hUmon}-R_{hLmon})*(R_{hmon}-R_{hLmon})]^{0.5}}{R_{hmon,mean}}$ 1:3.5.6

If $rnd_1>(\frac{R_{hmon}-R_{hLmon}}{R_{hUmon}-R_{hLmon}})$ then

$R_h=R_{hmon}*\frac{R_{hUmon}-(R_{hUmon}-R_{hmon})*[\frac{R_{hUmon}(1-rnd_1)-R_{hLmon}(1-rnd_1)}{R_{hUmon}-R_{hmon}}]^{0.5}}{R_{hmon,mean}}$ 1:3.5.7

where $R_h$ is the average relative humidity calculated for the day, $rnd_1$ is a random number generated by the model each day, $R_{hmon}$ is the average relative humidity for the month, $R_{hLmon}$ is the smallest relative humidity value that can be generated on a given day in the month, $R_{hUmon}$ is the largest relative humidity value that can be generated on a given day in the month, and $R_{hmon,mean}$ is the mean of $R_{hLmon},R_{hmon},$and $R_{hUmon}$.

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

$B*B^T=M_0-M_1*M_0^{-1}*M_1^T$ 1:3.4.3

where the superscript $-1$ denotes the inverse of the matrix and the superscript T denotes the transpose of the matrix. $M_0$ and $M_1$ are defined as

$M_0=\left[\begin{array}{ccc} 1 & \rho_0(1,2) & \rho_0(1,3) \\ \rho_0(1,2) & 1 & \rho_0(2,3) \\ \rho_0(1,3) & \rho_0(2,3) & 1 \end {array} \right ]$ 1:3.4.4

$M_1=\left[\begin{array}{ccc} \rho_1(1,1) & \rho_1(1,2) & \rho_0(1,3) \\ \rho_1(2,1) & \rho_1(2,2) & \rho_1(2,3) \\ \rho_1(3,1) & \rho_1(3,2) & \rho_1(3,3) \end {array} \right ]$ 1:3.4.5

$\rho_0(j,k)$ is the correlation coefficient between variables $j$ and $k$ on the same day where $j$ and $k$ may be set to 1 (maximum temperature), 2 (minimum temperature) or 3 (solar radiation) and $\rho_1(j,k)$ is the correlation coefficient between variable $j$ and $k$ with variable $k$ lagged one day with respect to variable $j$. Correlation coefficients were determined for 31 locations in the United States using 20 years of temperature and solar radiation data (Richardson, 1982). Using the average values of these coefficients, the $M_0$ and $M_1$ matrices become

$M_0=\left[\begin{array}{ccc} 1.000 & 0.633 & 0.186 \\ 0.633 & 1.000 & -0.193 \\ 0.186 & -0.193 & 1.000 \end {array} \right ]$ 1:3.4.6

$M_1=\left[\begin{array}{ccc} 0.621 & 0.445 & 0.087 \\ 0.563 & 0.674 & -0.100 \\ 0.015 & -0.091 & 0.251 \end {array} \right ]$ 1:3.4.7

Using equations 1:3.4.2 and 1:3.4.3, the A and B matrices become

$A=\left[\begin{array}{ccc} 0.567 & 0.086 & -0.002 \\ 0.253 & 0.504 & -0.050 \\ -0.006 & -0.039 & 0.244 \end {array} \right ]$ 1:3.4.8

$B=\left[\begin{array}{ccc} 0.781 & 0 & 0 \\ 0.328 & 0.637 & 0 \\ 0.238 & -0.341 & 0.873 \end {array} \right ]$ 1:3.4.9

The A and B matrices defined in equations 1:3.4.8 and 1:3.4.9 are used in conjunction with equation 1:3.4.1 to generate daily sequences of residuals of maximum temperature, minimum temperature and solar radiation.

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 wet day average maximum temperature is assumed to be less than the dry day average maximum temperature by some fraction of ($\mu mx_{mon}-\mu mn_{mon}$):

$\mu Wmx_{mon}=\mu Dmx_{mon}-b_T*(\mu mx_{mon}-\mu mn_{mon})$ 1:3.4.15

where $\mu Wmx{mon}$ is the average daily maximum temperature of the month on wet days (°C), $\mu Dmx_{mon}$ is the average daily maximum temperature of the month on dry days (°C), $b_T$ is a scaling factor that controls the degree of deviation in temperature caused by the presence or absence of precipitation, $\mu mx_{mon}$ is the average daily maximum temperature for the month (°C), and $\mu mn_{mon}$ is the average daily minimum temperature for the month (°C). The scaling factor, $b_T$, is set to 0.5 in SWAT+.

To calculate the dry day average maximum temperature, equations 1:3.4.14 and 1:3.4.15 are combined and solved for $\mu Dmx_{mon}$:

$\mu Dmx_{mon}=\mu mx_{mon}+b_T*\frac{days_{wet}}{days_{tot}}*(\mu mx_{mon}-\mu mn_{mon})$ 1:3.4.16

Incorporating the modified values into equation 1:3.4.10, SWAT calculates the maximum temperature for a wet day using the equation:

$T_{mx}=\mu Wmx_{mon}+\chi_i(1)*\sigma mx_{mon}$ 1:3.4.17

and the maximum temperature for a dry day using the equation:

$T_{mx}=\mu Dmx_{mon}+\chi_i(1)*\sigma mx_{mon}$ 1:3.4.18

$SND_{day}=cos(6.283*rnd_2)*\sqrt{-2ln(rnd_1)}$ 1:3.1.4

where $rnd_1$ and $rnd_2$ are random numbers between 0.0 and 1.0.

The exponential distribution is provided as an alternative to the skewed distribution. This distribution requires fewer inputs and is most commonly used in areas where limited data on precipitation events is available. Daily precipitation is calculated with the exponential distribution using the equation:

$R_{day}=\mu_{mon}*(-ln(rnd_1))^{rexp}$ 1:3.1.5

where $R_{day}$ is the amount of rainfall on a given day (mm H$_2$O), $\mu_{mon}$ is the mean daily rainfall (mm H$_2$O) for the month, $rnd_1$ is a random number between 0.0 and 1.0, and $rexp$ is an exponent that should be set between 1.0 and 2.0. As the value of $rexp$ is increased, the number of extreme rainfall events during the year will increase. Testing of this equation at locations across the U.S. have shown that a value of 1.3 gives satisfactory results.

Table 1:3-1: SWAT+ input variables that pertain to generation of precipitation.

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.

where $\mu Wrad_{mon}$ is the average daily solar radiation of the month on wet days (MJ m$^{-2}$), $\mu Drad_{mon}$ is the average daily solar radiation of the month on dry days (MJ m$^{-2}$), and $b_R$ is a scaling factor that controls the degree of deviation in solar radiation caused by the presence or absence of precipitation. The scaling factor, $b_R$, is set to 0.5 in SWAT+.

To calculate the dry day average solar radiation, equations 1:3.4.19 and 1:3.4.20 are combined and solved for $\mu Drad_{mon}$:

$\mu Drad_{mon}=\frac{\mu rad_{mon}*days_{tot}}{b_R*days_{wet}+days_{dry}}$ 1:3.4.21

Incorporating the modified values into equation 1:3.4.12, SWAT+ calculated the solar radiation on a wet day using the equation:

$H_{day}=\mu Wrad_{mon}+\chi_i(3)*\sigma rad_{mon}$ 1:3.4.22

and the solar radiation on a dry day using the equation:

$H_{day}=\mu Drad_{mon}+\chi_i(3)*\sigma rad_{mon}$ 1:3.4.23

Table 1:3-4: SWAT+ input variables that pertain to generation of temperature and solar radiation.

$R_{hWmon}=R_{hDmon}+b_H*(1-R_{hDmon})$ 1:3.5.9

where $R_{hWmon}$ is the average relative humidity of the month on wet days, $R_{hDmon}$ is the average relative humidity of the month on dry days, and $b_H$ is a scaling factor that controls the degree of deviation in relative humidity caused by the presence or absence of precipitation. The scaling factor, $b_H$, is set to 0.9 in SWAT+.

To calculate the dry day relative humidity, equations 1:3.5.8 and 1:3.5.9 are combined and solved for $R_{hDmon}$:

$R_{hDmon}=(R_{hmon}-b_H*\frac{days_{wet}}{days_{tot}})*(1.0-b_H*\frac{days_{wet}}{days_{tot}})^{-1}$ 1:3.5.10

To reflect the impact of wet or dry conditions, SWAT+ will replace $R_{hmon}$ with $R_{hWmon}$ on wet days or $R_{hDmon}$ on dry days in equations 1:3.5.4 through 1:3.5.7.

Table 1:3-5: SWAT+ input variables that pertain to generation of relative humidity.

The user is required to input standard deviation for maximum and minimum temperature. For solar radiation the standard deviation is estimated as ¼ of the difference between the extreme and mean value for each month.

$\sigma rad_{mon}=\frac{H_{mx}-\mu rad_{mon}}{4}$ 1:3.4.13

where $\sigma rad_{mon}$ is the standard deviation for daily solar radiation during the month (MJ m$^{-2}$), $H_{mx}$ is the maximum solar radiation that can reach the earth’s surface on a given day (MJ m$^{-2}$), and $\mu rad_{mon}$ is the average daily solar radiation for the month (MJ m$^{-2}$).