Potential evapotranspiration (PEI) is a key variable to calculate the SPEI drought index. Following Thornthwaite

^{[11]}method, PET is calculated from the dataset as follows:

\(PET=\left\{\begin{array}{c} 0 T<0\\ 16\left(\frac{N}{12}\right)\left(\frac{NDM}{30}\right){\left(\frac{10T}{I}\right)}^{m} 0\le T<26.5\\ -415.85+32.24T-0.43{T}^{2} T\ge 26.5\end{array}\right\\) (1)

where *T *is the monthly average temperature, *N* is the maximum daily sunshine duration, *NDM *is the number of days every month, *I *is the annual heat index, which is obtained by summation of the 12 monthly heat indexes per year. The annual heat index is calculated as below:

\(I=\sum _{i=1}^{12}{\left(\frac{T}{5}\right)}^{1.514}\mathrm{ }\mathrm{ }\mathrm{ }\mathrm{T}>0\) (2)

*m* is a coefficient related to *I*, and can be obtained Equation (3):

*m* =6.75×10^{-7}*I*^{3} -7.71×10^{-5}*I*^{2} +1.79×10^{-2}*I* +0.492 (3)

The estimation of this parameter requires less computed variables for the calculation of PET by the Thornthwaite method, and as the method is simple and easy, it has been widely applied.

**2.2.3 Calculation of standardized precipitation evapotranspiration index (SPEI)**

The effect of increasing temperature on drought has been increasingly apparent over the past few decades in relation with the global warming, and the SPEI is designed to take into account the impact of temperature variations on drought. Its estimation is made via the CSIC software from Spain (http://digital.csic.es/), and the computational procedure incorporates the following 4 steps

^{[9]} :

(1) Calculation of climate level

The climate level *D*_{i } is the difference between the amount of precipitation *P*_{i } and the potential evapotranspiration *PET*_{i} ,

*D*_{i} =*P*_{i}*－PET*_{i} (4)

PET was calculated via the Thomthwaite method as delineated in section 1.2.2.

(2) Establishment of an accumulated sequence of climate level on different time scales:

\({D}_{n}^{k}=\sum _{i=0}^{k-1}\left({P}_{n-i}-{PET}_{n-i}\right) , n\ge k\) (5)

In Eq. (5), *k *is the time scale(generally monthly), and *n *is the period over which the calculation is conducted.

(3) Construction of data series by log-logistic probability density function fitting:

\(f\left(x\right)=\frac{\beta }{\alpha }{\left(\frac{\chi -\gamma }{\alpha }\right)}^{\beta -1}{\left[1+{\left(\frac{\chi -\gamma }{\alpha }\right)}^{\beta }\right]}^{-2}\) (6)

In the formula, \(\alpha \) is the scale factor, \(\beta \) is the shape factor, and \(\gamma \) is the Origin parameter, which can be obtained by estimation of the L-moment parameter. Thus, on a given time scale, the cumulative probability is:

\(F\left(x\right)={\left[1+{\left(\frac{\mathrm{\alpha }}{\mathrm{\chi }-\mathrm{\gamma }}\right)}^{\mathrm{\beta }}\right]}^{-1}\) (7)

(4) Acquisition of the corresponding time variation sequence of SPEI through the transformation of standard normal distribution by cumulative probability density:

\(SPEI=W-\frac{{C}_{0}+{C}_{1}W+{C}_{2}{W}^{2}}{1+{d}_{1}W+{d}_{2}{W}^{2}+{d}_{3}{W}^{3}}, \) (8)

In Eq. (8), *W* is a parameter equal to \(\sqrt{-2ln\left(P\right)}。\). *P *is the probability to exceed a certain water surplus and deficit. When *P*≤0.5, *P*=1−*F*(x), while as *P*＞0.5, *P*=1−*P*, and the symbol of *SPEI* is reversed. The other constant terms are *C*_{0} =2.515517, *C*_{1} =2.515517, *C*_{2} =2.515517, *d*_{1} =2.515517, *d*_{2} =2.515517 and *d*_{3} =2.515517, respectively.

To sum up, not only does SPEI possess the feature of multiple timescales, it also considers the effect of temperature sensitivity. This presents obvious advantages in the analysis of dry and wet climate under warming conditions

^{[6]}^{[9] } .

**2.2.4 Selection of time scale and interpretation of results**

Drought is a multiscale phenomenon, and differences exist in the effects of the respective time scales on an affected area. Thus, different time scales can reflect different drought conditions. For example, the 3-month time scale shows the meteorological drought, whereas the 6-month scale reflects the agro-ecological drought and the 12-month scale represents the hydrologic drought

^{[6]}. Therefore, the time scale of 1, 3, 6, 12, 24, and 48 months are selected in the dataset to calculate the monthly drought indexes on different time scales at each station. Table 1 shows the international classification standard of drought grade based on SPEI. The monthly drought variations at any given station can then be determined following this standard

^{[6]}.

**Table 1
**Drought grade based on SPEI | Extreme drought | Moderate drought | Light drought | Normal | Lightly moist | Moderately moist | Extremely moist |

SPEI | ≤−2.0 | −2.0～−1.0 | −1.0～−0.5 | −0.5～0.5 | 0.5～1.0 | 1.0～2.0 | ≥2.0 |