The RDI is expressed in three forms: the initial value (α_k), normalized RDI (RDIn), and standardized RDI (RDIst). The initial value (α_k) of RDI is presented in an aggregated form using a monthly timestep and is usually calculated for i-th year in a time basis of k of consecutive months using following equation: α_(`k)^((i))=(∑_(j=1)^k▒P_ij )/(∑_(j=1)^k▒〖ETo〗_ij ),i=1,…,N and j=1,…,k
Where pij and EToij are precipitation and ETo of the j-th month of the i-th year and N is the total number of years of the available data. The normalized RDI (RDIn) is estimated as follows:
〖RDI〗_(n(k))^((i))=(α_k^((i)))/(α_k ) ̅ -1
In which α ̅_k is the arithmetic mean of α_k values. Tsakiris et al. (2008) though analysis of various data from several location and different timescales shown that α_k values follow satisfactorily both gamma and lognormal distributions however gamma distribution shows the best fit in most timescales and locations. Therefore, the calculation of third form (RDIst) performed by fitting the gamma probability density function to the given frequency distribution of α_k . The following equations have been used to calculate the standardized RDI (RDIst).
The probability function of gamma distribution is defined as: g(x)=1/(β^γ Γ(γ)) x^(γ-1) e^(-x/β), for x>0
Where γ>0 is a shape factor, β>0 is a scale factor, and Γ(γ) is the gamma function. Parameters γ and β of the gamma function are estimated or each time scale (k) and for each location. Maximum