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).

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.

$T_{mx}=\mu mx_{mon} + \chi_i(1)*\sigma mx_{mon}$ 1:3.4.10

$T_{mn}=\mu mn_{mon} + \chi_i(2)*\sigma mn_{mon}$ 1:3.4.11

$H_{day}=\mu rad_{mon} + \chi_i(3)*\sigma rad_{mon}$ 1:3.4.12

where $T_{mx}$ is the maximum temperature for the day (°C), $\mu mx_{mon}$ is the average daily maximum temperature for the month (°C), $\chi_i(1)$ is the residual for maximum temperature on the given day, $\sigma mx_{mon}$ is the standard deviation for daily maximum temperature during the month (°C), $T_{mn}$ is the minimum temperature for the day (°C), $\mu mn_{mon}$ is the average daily minimum temperature for the month (°C), $\chi_i(2)$ is the residual for minimum temperature on the given day, $\sigma mn_{mon}$ is the standard deviation for daily minimum temperature during the month (°C), $H_{day}$ is the solar radiation for the day (MJ m$^{-2}$), $\mu rad_{mon}$ is the average daily solar radiation for the month (MJ m$^{-2}$), $\chi_i(3)$ is the residual for solar radiation on the given day, and $\sigma rad_{mon}$ is the standard deviation for daily solar radiation during the month (MJ m$^{-2}$).

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$^{-2}$).

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

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 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.

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.