Maximum half-hour rainfall is required by SWAT+ to calculate the peak flow rate for runoff. When daily precipitation data are used by the model, the maximum half-hour rainfall may be calculated from a triangular distribution using monthly maximum half-hour rainfall data or the user may choose to use the monthly maximum half-hour rainfall for all days in the month. The maximum half-hour rainfall is calculated only on days where surface runoff has been generated.

The procedure used to generate daily values for maximum temperature, minimum temperature and solar radiation (Richardson, 1981; Richardson and Wright, 1984) is based on the weakly stationary generating process presented by Matalas (1967).

Relative humidity is required by SWAT+ when the Penman-Monteith equation is used to calculate potential evapotranspiration. It is also used to calculate the vapor pressure deficit on plant growth. Daily average relative humidity values are calculated from a triangular distribution using average monthly relative humidity. This method was developed by J.R. Williams for the EPIC model (Sharpley and Williams, 1990).

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).

For each month, users provide the maximum half-hour rain observed over the entire period of record. These extreme values are used to calculate representative monthly maximum half-hour rainfall fractions.

Prior to calculating the representative maximum half-hour rainfall fraction for each month, the extreme half-hour rainfall values are smoothed by calculating three month average values:

$R_{0.5sm(mon)}=\frac{R_{0.5x(mon-1)}+R_{0.5x(mon)}+R_{0.5x(mon+1)}}{3}$ 1:3.2.1

where $R_{0.5sm(mon)}$ is the smoothed maximum half-hour rainfall for a given month (mm H$_2$O) and $R_{0.5x}$ is the extreme maximum half-hour rainfall for the specified month (mm H$_2$O). Once the smoothed maximum half-hour rainfall is known, the representative half-hour rainfall fraction is calculated using the equation:

$\alpha_{0.5mon}=adj_{0.5\alpha}*[1-exp(\frac{R_{0.5sm(mon)}}{{\mu_{mon}}*ln*(\frac{0.5}{yrs*days_{wet}})})]$ 1:3.2.2

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.

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 user has the option of using the monthly maximum half-hour rainfall for all days in the month or generating a daily value. The variable ISED_DET in the basin input file (.bsn) defines which option the user prefers. The randomness of the triangular distribution used to generated daily values can cause the maximum half-hour rainfall value to jump around. For small plots or microwatersheds in particular, the variability of the triangular distribution is unrealistic.

The triangular distribution used to generate the maximum half-hour rainfall fraction requires four inputs: average monthly half-hour rainfall fraction, maximum value for half-hour rainfall fraction allowed in month, minimum value for half-hour rainfall fraction allowed in month, and a random number between 0.0 and 1.0.

The maximum half-hour rainfall fraction, or upper limit of the triangular distribution, is calculated from the daily amount of rainfall with the equation:

1:3.2.3

where is the largest half-hour fraction that can be generated on a given day, and is the precipitation on a given day (mm HO). The minimum half-hour fraction, or lower limit of the triangular distribution, , is set at 0.02083.

The triangular distribution uses one of two sets of equations to generate a maximum half-hour rainfall fraction for the day. If then

1:3.2.4

If then

1:3.2.5

where is the maximum half-hour rainfall fraction for the day, is the average maximum half-hour rainfall fraction for the month, is a random number generated by the model each day, is the smallest half-hour rainfall fraction that can be generated, is the largest half-hour fraction that can be generated, and is the average of , , and .

Table 1:3-2: SWAT+ input variables that pertain to generation of maximum half-hour rainfall.

The daily generated values are determined by multiplying the residual elements generated with equation 1:3.4.1 by the monthly standard deviation and adding the monthly average value.

1:3.4.10

1:3.4.11

1:3.4.12

where is the maximum temperature for the day (°C), is the average daily maximum temperature for the month (°C), is the residual for maximum temperature on the given day, is the standard deviation for daily maximum temperature during the month (°C), is the minimum temperature for the day (°C), is the average daily minimum temperature for the month (°C), is the residual for minimum temperature on the given day, is the standard deviation for daily minimum temperature during the month (°C), is the solar radiation for the day (MJ m), is the average daily solar radiation for the month (MJ m), is the residual for solar radiation on the given day, and is the standard deviation for daily solar radiation during the month (MJ m).

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.

1:3.4.13

where is the standard deviation for daily solar radiation during the month (MJ m), is the maximum solar radiation that can reach the earth’s surface on a given day (MJ m), and is the average daily solar radiation for the month (MJ m).

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 , maximum time to peak intensity expressed as a fraction of total storm duration , minimum time to peak intensity expressed as a fraction of total storm duration 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 then

1:3.3.9

If then

1:3.3.10

where 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),is the average time to peak intensity expressed as a fraction of storm duration, is a random number generated by the model each day, is the minimum time to peak intensity that can be generated,is the maximum time to peak intensity that can be generated, and is the mean of and .

The triangular distribution used to generate daily relative humidity values requires four inputs: mean monthly relative humidity, maximum relative humidity value allowed in month, minimum relative humidity value allowed in month, and a random number between 0.0 and 1.0.

The maximum relative humidity value, or upper limit of the triangular distribution, is calculated from the mean monthly relative humidity with the equation:

$R_{hUmon}=R_{hmon}+(1-R_{hmon})*exp(R_{hmon}-1)$ 1:3.5.4

where $R_{hUmon}$ is the largest 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 minimum relative humidity value, or lower limit of the triangular distribution, is calculated from the mean monthly relative humidity with the equation:

$R_{hLmon}=R_{hmon}*(1-exp(-R_{hmon}))$ 1:3.5.5

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 $R_{hUmon}$.

Relative humidity is defined as the ratio of the actual vapor pressure to the saturation vapor pressure at a given temperature:

$R_{hmon}=\frac{e_{mon}}{e^o_{mon}}$ 1:3.5.1

where $R_{hmon}$ is the average relative humidity for the month, $e_{mon}$ is the actual vapor pressure at the mean monthly temperature (kPa), and $e^o_{mon}$ is the saturation vapor pressure at the mean monthly temperature (kPa). The saturation vapor pressure,$e^o_{mon}$ , is related to the mean monthly air temperature with the equation:

$e^o_{mon}=exp[\frac{16.78*\mu tmp_{mon}-116.9}{\mu tmp_{mon}+237.3}]$ 1:3.5.2

where $e^o_{mon}$ is the saturation vapor pressure at the mean monthly temperature (kPa), and $\mu tmp_{mon}$ is the mean air temperature for the month (°C). The mean air temperature for the month is calculated by averaging the mean maximum monthly temperature, $\mu mx_{mon}$, and the mean minimum monthly temperature, $\mu mn_{mon}$.

The dew point temperature is the temperature at which the actual vapor pressure present in the atmosphere is equal to the saturation vapor pressure. Therefore, by substituting the dew point temperature in place of the average monthly temperature in equation 1:3.5.2, the actual vapor pressure may be calculated:

$e_{mon}=exp[\frac{16.78*\mu dew_{mon}-116.9}{\mu dew_{mon}+273.3}]$ 1:3.5.3

where $e_{mon}$ is the actual vapor pressure at the mean month temperature (kPa), and $\mu dew_{mon}$ is the average dew point temperature for the month (°C).

Residuals for maximum temperature, minimum temperature and solar radiation are required for calculation of daily values. The residuals must be serially correlated and cross-correlated with the correlations being constant at all locations. The equation used to calculate residuals is:

$\chi_i(j)=A{\chi_{i-1}}(j)+B{\varepsilon_i}(j)$ 1:3.4.1

where $\chi_i(j)$ is a 3 × 1 matrix for day $i$ whose elements are residuals of maximum temperature ($j=1$), minimum temperature ($j=2$) and solar radiation ($j=3$), $\chi_{i-1}(j)$) is a 3 × 1 matrix of the previous day’s residuals, $\varepsilon_i$ is a 3 × 1 matrix of independent random components, and $A$ and $B$ are 3 × 3 matrices whose elements are defined such that the new sequences have the desired serial-correlation and cross-correlation coefficients. The $A$ and $B$ matrices are given by

$A=M_1*M_0^{-1}$ 1:3.4.2

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

Maximum temperature and solar radiation will be lower on overcast days than on clear days. To incorporate the influence of wet/dry days on generated values of maximum temperature and solar radiation, the average daily maximum temperature, $\mu mx_{mon}$, and average daily solar radiation, $\mu rad_{mon}$, in equations 1:3.4.10 and 1:3.4.12 are adjusted for wet or dry conditions.

The continuity equation relates average daily maximum temperature adjusted for wet or dry conditions to the average daily maximum temperature for the month:

$\mu mx_{mon}*days_{tot}=\mu Wmx_{mon}*days_{wet}+\mu Dmx_{mon}*days_{dry}$

1:3.4.14

where $\mu mx_{mon}$ is the average daily maximum temperature for the month (°C), $days_{tot}$ are the total number of days in the month, $\mu Wmx_{mon}$ is the average daily maximum temperature of the month on wet days (°C), $days_{wet}$ are the number of wet days in the month, $\mu Dmx_{mon}$ is the average daily maximum temperature of the month on dry days (°C), and $days_{dry}$ are the number of dry days in the month.

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

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

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

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.

Wind speed is required by SWAT+ when the Penman-Monteith equation is used to calculate potential evapotranspiration. Mean daily wind speed is generated in SWAT+ using a modified exponential equation:

$\mu _{10m}=\mu wnd_{mon}*(-ln(rnd_1))^{0.3}$ 1.3.6.1

where $\mu _{10m}$ is the mean wind speed for the day (m s$^{-1}$), $\mu wnd_{mon}$ is the average wind speed for the month (m s$^{-1}$), and $rnd_1$ is a random number between 0.0 and 1.0.

Table 1:3-6: SWAT+ input variables that pertain to generation of wind speed.

To incorporate the effect of clear and overcast weather on generated values of relative humidity, monthly average relative humidity values can be adjusted for wet or dry conditions.

The continuity equation relates average relative humidity adjusted for wet or dry conditions to the average relative humidity for the month:

$R_{hmon}*days_{tot}=R_{hWmon}*days_{wet}+R_{hDmon}*days_{dry}$ 1:3.5.8

where $R_{hmon}$ is the average relative humidity for the month, $days_{tot}$ are the total number of days in the month, $R_{hWmon}$ is the average relative humidity for the month on wet days, $days_{wet}$ are the number of wet days in the month, $R_{hDmon}$ is the average relative humidity of the month on dry days, and $days_{dry}$ are the number of dry days in the month.

The wet day average relative humidity is assumed to be greater than the dry day average relative humidity by some fraction:

$R_{hWmon}=R_{hDmon}+b_H*(1-R_{hDmon})$ 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 continuity equation relates average daily solar radiation adjusted for wet or dry conditions to the average daily solar radiation for the month:

$\mu rad_{mon}*days_{tot}=\mu Wrad_{mon}*days_{wet}+\mu Drad_{mon}*days_{dry}$ 1:3.4.19

where $\mu rad_{mon}$ is the average daily solar radiation for the month (MJ m$^{-2}$), $days_{tot}$ are the total number of days in the month, $\mu Wrad_{mon}$ is the average daily solar radiation of the month on wet days (MJ m$^{-2}$), $days_{wet}$ are the number of wet days in the month, $\mu Drad_{mon}$is the average daily solar radiation of the month on dry days (MJ m$^{-2}$), and $days_{dry}$ are the number of dry days in the month.

The wet day average solar radiation is assumed to be less than the dry day average solar radiation by some fraction:

$\mu Wrad_{mon}=b_R*\mu Drad_{mon}$ 1:3.4.20

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.