Identification of long-term annual pattern of meteorological drought based on spatiotemporal methods: evaluation of different geostatistical approaches

Abstract

Estimation and identification of long-term meteorological drought pattern play an important role in regional water management and dry land agricultural practices in arid and semiarid climates. In this work, Standardized Precipitation Index (SPI) has been selected as the main criterion for evaluating the severity of meteorological drought events. The purpose of this paper was to produce meteorological drought occurrence probability maps for different SPI classes by spatiotemporal analysis. Several statistical methods known as non-geostatistical approaches (such as Thiessen polygons, inverse distance-weighted, and spline-based) and geostatistical approaches (such as different types of kriging and Bayesian maximum entropy (BME)) are available, which can be used for the purpose of this study. In this study, ordinary kriging (OK) as a classical geostatistical method and BME as a modern geostatistical method have been used. The case study of this research has been the Namak Lake Watershed located in the central part of Iran with an area of approximately 90,000 km². This basin includes regions with significantly different climatic conditions ranging from very dry to very wet. The results of the case study include spatial distribution of SPI for dry SPI classes (moderately, severely, and extremely dry classes) and wet SPI classes (moderately, severely, and extremely wet classes) which can be used to locate vulnerable areas against drought. The selected geostatistical methods have been compared based on leave-one-out cross-validation procedure and spatiotemporal distribution of SPI values. The results of cross-validation have shown the superiority of BME over OK. BME maps of probability of occurrence have also been more realistic than OK maps.

Appendices

Appendix 1

Considering gamma distribution in this paper, the PDF of this distribution is defined as:

$$f(x) = \frac{1}{{\beta^{\alpha } \varGamma (\alpha )}}x^{\alpha - 1} {\text{e}}^{ - x/\beta } ,\quad x > 0$$

(12)

where x, α > 0, and β > 0 are the precipitation values, shape parameter, and scale parameter, respectively. Γ(α) is the gamma function which is defined as:

$$\varGamma (\alpha ) = \int\limits_{0}^{\infty } {y^{\alpha - 1} {\text{e}}^{ - y} {\text{d}}y}$$

(13)

Optimum α and β are calculated as follows:

$$\begin{gathered} \hat{\alpha } = \frac{1}{4A}\left[ {1 + \sqrt {1 + \frac{4A}{3}} } \right] \hfill \\ \hat{\beta } = \frac{{\bar{x}}}{{\hat{\alpha }}} \hfill \\ \end{gathered}$$

(14)

where $A = \ln (\bar{x}) - \frac{{\sum {\ln (x)} }}{n}$ and n is the number of observations. To calculate SPI values, cumulative distribution function (CDF) which is the integration of PDF should be obtained as follows:

$$F(x) = \int\limits_{0}^{\infty } {f(x)} {\text{d}}x = \frac{1}{{\hat{\beta }\;\varGamma (\hat{\alpha })}}\int\limits_{0}^{\infty } {x^{{\hat{\alpha } - 1}} {\text{e}}^{ - x/\beta } {\text{d}}x}$$

(15)

Since the gamma distribution is undefined at 0 and precipitation series may include zero entries, the CDF must be corrected as follows:

$$H(x) = q + (1 - q)F(x)$$

(16)

where q is the probability of zero precipitation.

The transformation is defined as:

$$\begin{gathered} Z = {\text{SPI}} = - \left( {t - \frac{{c_{0} + c_{1} t + c_{2} t^{2} }}{{1 + d_{1} t + d_{2} t^{2} + d_{3} t^{3} }}} \right),\quad 0 < H(x) \le 0.5 \hfill \\ Z = {\text{SPI}} = + \left( {t - \frac{{c_{0} + c_{1} t + c_{2} t^{2} }}{{1 + d_{1} t + d_{2} t^{2} + d_{3} t^{3} }}} \right),\quad 0.5 < H(x) < 1 \hfill \\ \end{gathered}$$

(17)

where

$$\begin{gathered} t = \sqrt {\ln \left[ {\frac{1}{{(H(x))^{2} }}} \right]} ,\quad 0 < H(x) \le 0.5 \hfill \\ t = \sqrt {\ln \left[ {\frac{1}{{(1 - H(x))^{2} }}} \right]} ,\quad 0.5 < H(x) < 1 \hfill \\ \end{gathered}$$

(18)

and the optimum parameters of Eq. (17) are c ₀ = 2.515517, c ₁ = 0.802853, c ₂ = 0.010328, d ₀ = 1.432788, d ₁ = 0.189269, and d ₂ = 0.001308. Actually, SPI represents deviation from average.

Appendix 2

• 1. Normalized mean square error (NMSE):

  $${\text{NMSE}} = \frac{1}{{S^{2} N}}\sum\limits_{i = 1}^{N} {\left\{ {X(\vec{s}_{i} ) - \hat{X}(\vec{s}_{i} )} \right\}^{2} } ,$$

  (19)

  S2: Variance of the observed values. N: number of observed values. $X(\vec{s}_{i} )$: observed values in location $\vec{s}_{i}$. $\hat{X}(\vec{s}_{i} )$: estimated values in location $\vec{s}_{i}$.

This index is estimated based on the variance of observed values.

• 2. Mean absolute error (MAE) or bias

  $${\text{MAE}} = \frac{1}{N}\sum\limits_{i = 1}^{N} {{\text{abs}}\left\{ {X(\vec{s}_{i} ) - \hat{X}(\vec{s}_{i} )} \right\}} .$$

  (20)

  abs: absolute value (B-2).

• 3. Mean square error (MSE):

  $${\text{MSE}} = \frac{1}{N}\sum\limits_{i = 1}^{N} {\left\{ {X(\vec{s}_{i} ) - \hat{X}(\vec{s}_{i} )} \right\}}^{2} .$$

  (21)