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.

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.

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:

$R_{day}=\mu_{mon}+2*\sigma_{mon}*(\frac{[(SND_{day}-\frac{g_{mon}}{{6}})*\frac{g_{mon}}{{6}}+1]^3-1}{g_{mon}})$ 1:3.1.3

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, $\sigma_{mon}$ is the standard deviation of daily rainfall (mm H$_2$O) for the month, $SND_{day}$ is the standard normal deviate calculated for the day, and $g_{mon}$is the skew coefficient for daily precipitation in the month.

The standard normal deviate for the day is calculated:

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