The daily precipitation generator is a Markov chain-skewed (Nicks, 1974) or Markov chain-exponential model (Williams, 1995). A first-order Markov chain is used to define the day as wet or dry. When a wet day is generated, a skewed distribution or exponential distribution is used to generate the precipitation amount. Table 1:3.1 lists SWAT+ input variables that are used in the precipitation generator.

With the first-order Markov-chain model, the probability of rain on a given day is conditioned on the wet or dry status of the previous day. A wet day is defined as a day with 0.1 mm of rain or more.

The user is required to input the probability of a wet day on day $i$ given a wet day on day $i-1,Pi-1(W/W)$, and the probability of a wet day on day $i$ given a dry day on day $i-1,P_i(W/D)$, for each month of the year. From these inputs the remaining transition probabilities can be derived:

$P_i(D/W)=1-P_i(W/W)$ 1:3.1.1

$P_i(W/W)=1-P_i(W/D)$ 1:3.1.2

where $P_i(D/W)$ is the probability of a dry day on day $i$ given a wet day on day $i-1$ and $P_i(D/D)$ is the probability of a dry day on day $i$ given a dry day on day $i-1$.

To define a day as wet or dry, SWAT+ generates a random number between 0.0 and 1.0. This random number is compared to the appropriate wet-dry probability, $P_i(W/W)$or $P_i(W/D)$. If the random number is equal to or less than the wet-dry probability, the day is defined as wet. If the random number is greater than the wet-dry probability, the day is defined as dry.

Numerous probability distribution functions have been used to describe the distribution of rainfall amounts. SWAT+ provides the user with two options: a skewed distribution and an exponential distribution.

The skewed distribution was proposed by Nicks (1974) and is based on a skewed distribution used by Fiering (1967) to generate representative streamflow. The equation used to calculate the amount of precipitation on a wet day is:

1:3.1.3

where is the amount of rainfall on a given day (mm HO), is the mean daily rainfall (mm HO) for the month, is the standard deviation of daily rainfall (mm HO) for the month, is the standard normal deviate calculated for the day, and is the skew coefficient for daily precipitation in the month.

The standard normal deviate for the day is calculated:

1:3.1.4

where and 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:

1:3.1.5

where is the amount of rainfall on a given day (mm HO), is the mean daily rainfall (mm HO) for the month, is a random number between 0.0 and 1.0, and is an exponent that should be set between 1.0 and 2.0. As the value of 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.

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:

1:3.3.1

,

where is the rainfall intensity at time (mm/hr), is the maximum or peak rainfall intensity during the storm (mm/hr), is the time since the beginning of the storm (hr), is the time from the beginning of the storm till the peak rainfall intensity occurs (hr), is the duration of the storm (hr), and and 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,

1:3.3.2

1:3.3.3

where the normalized rainfall intensity at time , is the rainfall intensity at time T (mm/hr), is the average storm rainfall intensity (mm/hr), is the time during the storm expressed as a fraction of the total storm duration (0.0-1.0), is the time since the beginning of the storm (hr), and is the duration of the storm (hr).

The normalized storm intensity distribution is:

1:3.3.4

,

where the normalized rainfall intensity at time , is the normalized maximum or peak rainfall intensity during the storm, is the time during the storm expressed as a fraction of the total storm duration (0.0-1.0), 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 are equation coefficients.

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

1:3.3.5

1:3.3.6

where is the equation coefficient for rainfall intensity before peak intensity is reached (hr), is the normalized equation coefficient for rainfall intensity before peak intensity is reached, is the equation coefficient for rainfall intensity after peak intensity is reached (hr), is the normalized equation coefficient for rainfall intensity after peak intensity is reached, and is the storm duration (hr).

Values for the equation coefficients, and , can be determined by isolating the coefficients in equation 1:3.3.4. At = 0.0 and at = 1.0,

1:3.3.7

1:3.3.8

where is the normalized equation coefficient for rainfall intensity before peak intensity is reached, is the normalized equation coefficient for rainfall intensity after peak intensity is reached, is the time during the storm expressed as a fraction of the total storm duration (0.0-1.0), 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 normalized rainfall intensity at time , and is the normalized maximum or peak rainfall intensity during the storm.

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.

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).

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 $\hat t_{peakU}$ .

SWAT+ requires daily values of precipitation, maximum and minimum temperature, solar radiation, relative humidity and wind speed. The user may choose to read these inputs from a file or generate the values using monthly average data summarized over a number of years.

SWAT+ includes the WXGEN weather generator model (Sharpley and Williams, 1990) to generate climatic data or to fill in gaps in measured records. This weather generator was developed for the contiguous U.S. If the user prefers a different weather generator, daily input values for the different weather parameters may be generated with an alternative model and formatted for input to SWAT+.

The occurrence of rain on a given day has a major impact on relative humidity, temperature and solar radiation for the day. The weather generator first independently generates precipitation for the day. Once the total amount of rainfall for the day is generated, the distribution of rainfall within the day is computed if the Green & Ampt method is used for infiltration. Maximum temperature, minimum temperature, solar radiation and relative humidity are then generated based on the presence or absence of rain for the day. Finally, wind speed is generated independently.

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).

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 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.

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:

1:3.4.1

where is a 3 × 1 matrix for day whose elements are residuals of maximum temperature (), minimum temperature () and solar radiation (), ) is a 3 × 1 matrix of the previous day’s residuals, is a 3 × 1 matrix of independent random components, and and are 3 × 3 matrices whose elements are defined such that the new sequences have the desired serial-correlation and cross-correlation coefficients. The and matrices are given by

1:3.4.2

1:3.4.3

where the superscript denotes the inverse of the matrix and the superscript T denotes the transpose of the matrix. and are defined as

1:3.4.4

1:3.4.5

is the correlation coefficient between variables and on the same day where and may be set to 1 (maximum temperature), 2 (minimum temperature) or 3 (solar radiation) and is the correlation coefficient between variable and with variable lagged one day with respect to variable . 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 and matrices become

1:3.4.6

1:3.4.7

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

1:3.4.8

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 volume of rain is related to rainfall intensity by:

1:3.3.11

where is the amount of rain that has fallen at time (mm HO) and is the rainfall intensity at time (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:

1:3.3.12

where is the cumulative amount of rain that has fallen at time (mm HO), is the amount of rain that has fallen at time (mm HO), is the maximum or peak rainfall intensity during the storm (mm/hr), is the equation coefficient for rainfall intensity before peak intensity is reached (hr), is the equation coefficient for rainfall intensity after peak intensity is reached (hr), is the time from the beginning of the storm till the peak rainfall intensity occurs (hr), and is the storm duration (hr). The time to peak intensity is defined as

1:3.3.13

where is the time from the beginning of the storm till the peak rainfall intensity occurs (hr), 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 is the storm duration (hr). The cumulative volume of rain that has fallen at is

1:3.3.14

where is the amount of rain that has fallen at time (mm HO), 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 is the total rainfall on a given day (mm HO).

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

1:3.3.15

where is the rainfall on a given day (mm HO), is the maximum or peak rainfall intensity during the storm (mm/hr), is the equation coefficient for rainfall intensity before peak intensity is reached (hr), is the equation coefficient for rainfall intensity after peak intensity is reached (hr), is the normalized equation coefficient for rainfall intensity before peak intensity is reached, is the normalized equation coefficient for rainfall intensity after peak intensity is reached, and is the storm duration (hr). This equation can be rearranged to calculate the storm duration:

1:3.3.16

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

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

1:3.5.1

where is the average relative humidity for the month, is the actual vapor pressure at the mean monthly temperature (kPa), and is the saturation vapor pressure at the mean monthly temperature (kPa). The saturation vapor pressure, , is related to the mean monthly air temperature with the equation:

1:3.5.2

where is the saturation vapor pressure at the mean monthly temperature (kPa), and 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, , and the mean minimum monthly temperature, .

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:

1:3.5.3

where is the actual vapor pressure at the mean month temperature (kPa), and is the average dew point temperature for the month (°C).

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, , and average daily solar radiation, , 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:

1:3.4.14

where is the average daily maximum temperature for the month (°C), are the total number of days in the month, is the average daily maximum temperature of the month on wet days (°C), are the number of wet days in the month, is the average daily maximum temperature of the month on dry days (°C), and 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 ():

1:3.4.15

where is the average daily maximum temperature of the month on wet days (°C), is the average daily maximum temperature of the month on dry days (°C), is a scaling factor that controls the degree of deviation in temperature caused by the presence or absence of precipitation, is the average daily maximum temperature for the month (°C), and is the average daily minimum temperature for the month (°C). The scaling factor, , 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 :

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:

1:3.4.17

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

1:3.4.18

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

1:3.4.19

where is the average daily solar radiation for the month (MJ m), are the total number of days in the month, is the average daily solar radiation of the month on wet days (MJ m), are the number of wet days in the month, is the average daily solar radiation of the month on dry days (MJ m), and 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:

1:3.4.20

where is the average daily solar radiation of the month on wet days (MJ m), is the average daily solar radiation of the month on dry days (MJ m), and is a scaling factor that controls the degree of deviation in solar radiation caused by the presence or absence of precipitation. The scaling factor, , 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 :

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:

1:3.4.22

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

1:3.4.23

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

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:

1:3.5.8

where is the average relative humidity for the month, are the total number of days in the month, is the average relative humidity for the month on wet days, are the number of wet days in the month, is the average relative humidity of the month on dry days, and 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:

1:3.5.9

where is the average relative humidity of the month on wet days, is the average relative humidity of the month on dry days, and is a scaling factor that controls the degree of deviation in relative humidity caused by the presence or absence of precipitation. The scaling factor, , 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 :

1:3.5.10

To reflect the impact of wet or dry conditions, SWAT+ will replace with on wet days or 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 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:

1:3.5.4

where is the largest relative humidity value that can be generated on a given day in the month, and 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:

1:3.5.5

where is the smallest relative humidity value that can be generated on a given day in the month, and 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 then

1:3.5.6

If then

1:3.5.7

where is the average relative humidity calculated for the day, is a random number generated by the model each day, is the average relative humidity for the month, is the smallest relative humidity value that can be generated on a given day in the month, is the largest relative humidity value that can be generated on a given day in the month, and is the mean of and .