#### 2.2. Model Development

McKee et al. [

13] introduced the Standardized Precipitation Index (

SPI) as an indicator for drought assessment. If the probability distribution of monthly total rainfall data is normal, then

SPI is given as follows

where

$P$ is the precipitation,

$\overline{P}$ is the sample mean precipitation, and

$\sigma $ is the sample standard deviation of precipitation.

Table 3 provides the

SPI classifications based on

SPI values.

Equation (1) requires normalization for the precipitation time series. The normalization procedure aims to give

SPI a uniform measure of drought in different climatic regimes [

24]. Stagge et al. [

25] evaluated candidate probability distributions suitable for the normalization procedure. They recommended the two-parameter gamma distribution

where

α and

β represent the shape and location parameters for gamma distribution, and Γ(α) represents gamma function. Although the gamma distribution is skewed to the right with a lower bound of zero much like a precipitation frequency distribution, Sienz et al. [

24] showed that the distribution can fail to represent drought in many cases, especially at short timescales. The main interest in drought applications is not in the

$g\left(x\right)$ but in the integral form that provides the probability

It is known that $G\left(x\right)$ cannot be found except as an expansion in series or continued fractions, which is also the case for $\mathsf{\Gamma}\left(\alpha \right)$ when $\alpha $ is a real number.

Another statistical problem in gamma distribution is realized in the estimation of the parameters

α and

β from a sample precipitation record. For gamma distribution, the population mean

$\mu $ and variance

${\sigma}^{2}$ are given by

which are expressed in terms of probability function as

To overcome the difficulty in performing the numerical integration, Thom [

26] provided

α and

β estimates from maximum likelihood solutions

where

$\overline{x}$ is the sample mean

In this procedure, the sample mean $\overline{x}$ is used to estimate the parameters α and β, because it is considered an unbiased estimator of the population mean, converging to the correct value as the number of data points grow arbitrarily large. Determining the variance is usually considered less straightforward. However, determining the exact probability distribution for the data is impossible, although the assumption of gamma is useful.

Another approximation is considered by realizing that Equation (3) is undefined for zero measurements. The following mixed distribution function of zeros and continuous precipitation has been employed

where

$q$ represents the probability of having zero measurement, such that if

$m$ is the number of zero precipitation measurements in the data series of

$n$ records then

$q=m/n$. That is if

$x=0$, then

$G\left(0\right)=0$ and

$H\left(0\right)=q$ as it should be. The cumulative probability

$H\left(X\right)$ can then be converted to the standard random variable

$Z$ using the typical standardized normal cumulative probability table found in statistical textbooks or the rational approximation provided by Abramowitz and Stegun [

27]

where

with the coefficients being equal to

${c}_{0}=2.515517$,

${c}_{1}=0.802853$,

${c}_{2}=0.010328$,

${d}_{1}=1.432788$,

${d}_{2}=0.189269$ and

${d}_{3}=0.001308$. Apparently, the approximation in handling zero rainfall events is problematic as it is associated with assigning higher

SPI values for that period. The improper use of probabilistic distribution will bias the drought index value and lead to misleading drought severity characterizations [

25].

An alternative criterion suggested here is to use a precipitation index (

$PI$) analogous to the error equation obtained by subtracting the mean precipitation from the precipitation value and then dividing by the mean precipitation, such as

In this simple expression, the extreme condition of

$P=0$ produces

$PI=-1$. The average precipitation

$P=\overline{P}$ results with

$PI=0$. For

$P=2\overline{P}$ the expression becomes

$PI=+1$.

Table 3 shows the proposed

$PI$ classifications. This criterion assesses precipitation deficit in a manner similar to the original

SPI, but it is not uniquely tied to a probability distribution. While the original

SPI is defined to be equivalent to the

Z-score in normal probability deviate,

PI evaluates precipitation deficit and divide that deficit with a long-term average to relax the requirement for normalization.

The drought computation based on Equation (18) will be straightforward with timescales 12 and 24 months, which may be convenient for groundwater drought assessments [

7]. For timescales less than 12 months, Equation (18) can still be used but with seasonal mean

$\overline{P}$ accounting for the average precipitation within the last

N months

For example, short term drought analysis using

N = 1 month is appropriate for assessing soil moisture deficit and crops stress. An analysis based on

N = 9 months may be relevant for other agricultural applications [

7].

Two important drought measures are of interest here. Drought severity

$S$ is a measure for the accumulated drought magnitude. This parameter is calculated by adding

$PI$ values less than zero for consecutive occurrences

where

x is the total number of consecutive occurrences. The negative sign in Equation (20) is added for convenience to render

S positive. Even though

Table 3 defines drought to occur with

PI less than −0.5, the severity calculation is carried out when

PI is less than zero. The drought duration

d is a measure for the drought temporal extent, defined as the time within which the drought severity occurs. Similar to the severity, drought duration is calculated starting from

PI less than zero. For many cases with long timescales, droughts of one-month durations are neglected, as they occur frequently just before the start of a rainfall season.

The assessment of drought impact on hydrologic systems based on the two variables of drought severity and duration can be accomplished using bivariate probability analysis. Here, the joint distribution for two independent variables

$A$ and

$B$ is

If the two variables are dependent, then the joint probability becomes

In this case, the estimation of the conditional probability

$P\left(B|A\right)$ is not straightforward, because it requires fitting parameters from the data. However, the mathematical difficulty can be handled by using copulas (e.g., [

4,

27,

28,

29]), which are multivariate cumulative probability functions used to describe the dependence between random variables while separating the effect of dependence without using sophisticated joint probability modeling [

30]. The asymmetric Archimedean copula known as a Clayton copula is appropriate for drought simulation and will thus be used here, because the tail structure of droughts is well-reflected using Clayton copula [

31,

32,

33]. The Clayton copula is given by

where

A and

B stand for the marginal probability for drought severity and duration, and

$\tau $ is Kendall’s Tau. The coefficient

$\tau $ will address the dependency requirement. That is, if the agreement between the two variables is perfect, then

$\tau $ will be equal to 1; if the disagreement between the two variables is perfect, then it will be −1. However, if the two variables are independent, then

$\tau $ will be close to zero.