Comparison of various drought indices to monitor drought status in Pakistan

Abstract

Various drought indices are normally used to monitor drought and its risk management. Precipitation, temperature and other hydro meteorological parameters are the essential parts to the identification of drought. For this purpose, several drought indices have been developed and are being used around the world. This study identifies the applicability and comparison of drought indices in Pakistan by evaluating the performance of 15 drought indices. The indices include standardized precipitation index (SPI), standardized precipitation temperature index, standardized precipitation evapotranspiration index (SPEI), China Z-Index, deciles index, modified CZI, Z-Score, rainfall variability index, standardized soil moisture anomaly index, weighted anomaly standardized precipitation index, percent of normal precipitation index, self-calibrated Palmer drought severity index, composite index, percentage area weighted departure and reconnaissance drought index (RDI). These indices are compared by utilizing long term data of 58 meteorological stations for the period 1951–2014. The performance, efficiency and significance are also tested by applying different statistical tests. The SPI, SPEI and RDI results showed a good capability to monitor drought status in Pakistan. The positive increasing trend (towards wetness) is noted by several of the aforementioned indices at 95% confidence level. In addition, historical drought years and intensity have been explored along with comparison of recent long episode of drought (1999–2002) and all the indices captured this period successfully.

Appendix: A description of drought indices

1.1 Standardized precipitation index (SPI)

SPI is calculated by fitting the long term precipitation data to the probability distribution (e.g., gamma distribution), which is then transformed to normal distribution to get the zero values for SPI mean (McKee et al. 1993; Edwards and McKee 1997). The SPI can be calculated at different time scales like 1-month, 3-months, 6-months, 12-months and so on. The SPI is not suitable at longer time scale as it reduces the sample size, even in the presence of long data (Guttman 1998). However, the SPI calculates the precipitation deficit at different time scales (i.e. 1–12) that may be helpful to identify drought conditions (e.g., soil moisture deficiency, reservoir storage, ground water and stream flow) over a region. Positive and negative values show above and below than the mean precipitation. SPI values show both dry and wet conditions. SPI drought part is randomly split into four classes, mild (− 0.49 to − 0.99), moderate (− 1.0 to − 1.49), severe (− 1.50 to − 1.99), and extreme (SPI < − 2.0) conditions. A drought event starts when SPI reaches to − 0.0 and ends when SPI become positive. This index is widely used throughout the world (WMO 2012). SPI shows a better performance to monitor drought in China and Iran by Wu et al. (2001) and Ansari (2003) respectively.

1.2 Standardized precipitation evapotranspiration index (SPEI)

SPEI calculation procedure is same as SPI. SPI uses the precipitation data only, whereas SPEI uses both rainfall and reference evapotranspiration (ET_o) (Vicente-Serrano et al. 2010) which helps to identify the drought types, severity and impacts over a region (Vicente-Serrano et al. 2013). The water balance (the difference of precipitation and potential evapotranspiration) is calculated (Thornthwaite 1948) at different time scales to get SPEI values. The Penmen Monteith (PM) equation is used to calculate PET (Allen et al. 1998) as follows;

$${\text{E}}{{\text{T}}_0}=\frac{{0.408~\Delta \left( {{R_n} - G} \right)+\gamma \left[ {\frac{{900}}{{{\text{T}}+273}}} \right]{u_2}\left( {{e_s} - {e_a}} \right)}}{{\Delta +\gamma \left( {1+0.34{u_2}} \right)}},$$

(6)

where ET_o represents evapotranspiration (mm/day); ∆ = saturated vapor pressure slope (kPa/°C); G = heat flux density of soil (MJ/m²/day); R_n = net radiation (MJ/m² per day); T = mean temperature (°C); u₂ = average daily speed of wind (m/s); e_s − e_a = deficit of vapor pressure; γ = psychrometric constant (kPa/°C).

The main advantage of SPEI is that it takes the PET data on a different time scales to monitor the drought conditions. These timescales are very useful to calculate the drought conditions in different hydrological sub-systems. SPEI is a standard variable and its average value is 0 and deviation is 1. According to Abramowitz and Stegun (1965), the classical approximation is used in the formula of SPEI as follows;

$${\text{SPEI}}=W - \frac{{{C_0}+{C_1}W+{C_2}{W^2}}}{{1+{d_1}W+{d_2}{W^2}+{d_3}{W^3}}}$$

(7)

where ${\text{W~=~}} - {\text{2ln~(Q)~for~Q~}} \leqslant {\text{~0}}{\text{.5}}$, Q is the probability of exceedance = ${\text{1}} - {\text{F}}\left( x \right){\text{~if~Q~>~0}}{\text{.5}}$, Q is replaced by 1 − Q and the resultant SPEI symbol is reversed. The C_o, C₁, C₂, d₁,d₂ and d₃ are constants.

1.3 Standardized precipitation temperature index (SPTI) or S-index

Standardized precipitation temperature index (SPTI) or dryness index (Si) jointly considers the precipitation and temperature as proposed by Ped (1975).

$${S_i}=\frac{{DS}}{{{V_{DT}}}}~ - \frac{{DQ}}{{{V_{DQ}}}}$$

(8)

where ∆T and ∆P are the anomalies, σ_∆T and σ_∆P are the respective standard deviations at the station, calculated from long term data series of precipitation and temperature respectively. The values of the index range from very dry to very wet conditions.

1.4 Weighted anomaly standardized precipitation index (WASPI)

Lyon (2004) developed weighted anomaly standardized precipitation index (WASPI) to monitor the precipitation in tropical region. The monthly and annual precipitation data are used to calculate WASP. The values range from most severe (− 2.0) to severe wetness (2.0), which is well correlated with other drought indices. Basically, drought is termed as moisture deficit, therefore, generally 6 or 12 months’ rainfall accumulation is used.

$$WAS{P_N}=\frac{1}{{{\sigma _{WAS{P_N}}}}}.\mathop \sum \limits_{{i=1}}^{N} \left( {\frac{{{P_i} - {{\bar {P}}_l}}}{{{\sigma _i}}}.\frac{{{{\bar {P}}_l}}}{{{{\bar {P}}_A}}}} \right)~$$

(9)

where P_i, P_A is the monthly and annual rainfall, ${\bar {P}_l}$ and ${\bar {P}_A}$ is the monthly and annual rainfall climatology, ${\sigma _i}$ is the standard deviation of monthly rainfall where ${\sigma _{WAS{P_N}}}$ is the standard deviation of WASP, which is 0.44 for Pakistan.

1.5 Rainfall variability index (RVI)

It is the ratio between anomalies over the standard deviation of long period of rainfall data. According to Gocic and Trajkovic (2013), the following equation is used to determine RVI

$${\delta _i}=~\left( {{P_i} - \mu } \right)/\sigma$$

(10)

where ${\delta _i}$ represents RVI, P _i is annual rainfall for ith year, µ and σ are the annual mean and standard deviation of rainfall. Time series of rainfall are classified into different climatic regimes. In case, if $\delta$ is negative, then the year is known as drought year. The World Meteorological Organization (WMO 1975), classify the rainfall, according to their climate zones.

$$\left. {\begin{array}{*{20}{l}} {P<~\mu - 2 \cdot \sigma \quad\quad \quad\quad extreme~dry} \\ {\mu - 2 \cdot \sigma < P < \mu - \sigma\,\,\; dry} \\ {\mu - \sigma < P < \mu +\sigma \quad\,\,\;\; normal} \\ {P>\mu +\sigma \quad\quad \quad\quad\quad\,\, wet} \end{array}} \right\}$$

(11)

1.6 Standardized soil moisture anomaly index (SSMAI)

Monthly soil moisture data (1951–2014) produced by Leaky Bucket model at a horizontal resolution (0.5° × 0.5°) is used to calculate SSMAI. It is reasonably performed well against the limited observation in a different region with a spatial resolution (0.5° × 0.5°) from 1948 to-date (Fan and Dool 2004, 2008). A strong correlation is noted between the annual rainfall departure with 12-SPI (r = 0.97), rainfall departure with soil moisture (r = 0.80) and soil moisture with SPI (r = 0.77). Soil moisture anomaly index (SMAI) was developed by Bergman et al. (1988) in mid1980, a way to access global drought conditions in United States. Here a slight modification is incorporated to obtain standardized data. So, it could be easy to compare it with other drought indices.

$${\phi _j}=\frac{{{X_j} - \bar {X}}}{\sigma }~$$

(12)

where ${X_j}$ is the precipitation of jth month is, $\bar {X}$ is the mean precipitation and σ is the standard deviation.

1.7 China Z-index (CZI)

Wilson-Hilferty cube root transformation is used to calculate CZI (Kendall and Stuart 1977). CZI formula is as follows;

$$CZ{I_j}=~\frac{6}{{{C_S}}}{\left. {\left( {\frac{{{C_S}}}{2}~+~1} \right.} \right)^{1/3}} - \frac{6}{{{C_S}}}~+\frac{{{C_S}}}{6}$$

(13)

$${C_S}={\text{~}}\frac{{\mathop \sum \nolimits_{{{\text{j}}=1}}^{{\text{n}}} {{\left( {{{\text{X}}_{\text{j}}} - {\bar{\text {X}}}} \right)}^3}}}{{{\text{n~}} \times {\text{~}}{\sigma ^3}}}$$

(14)

$${\phi _{\text{j}}}=\frac{{{{\text{X}}_{\text{j}}} - {\bar{\text {X}}}}}{\sigma }$$

(15)

where j is the current month, C_s is the coefficient of skewness, n is the total number of months in the record, σ is the standard variant also called Z-Index and x_j is precipitation of j month.

1.8 Modified CZI (MCZI)

To calculate MCZI, Wu et al. (2001) used a median of precipitation instead of mean precipitation. This attempt minimized the difference between MCZI and SPI.

1.9 Z-Score

According to Triola (1995a), the equation of CZI is used to calculate Z-Score. Furthermore, the data fitting adjustment Pearson type III or Gamma distribution is not requiring by Z-Score and it might not be represented in shorter duration as well as the SPI (Edwards and McKee 1997).

1.10 The decile index (DI)

The decile index is widely used in Australia to monitor drought (Coughlan 1987). Long term monthly precipitation data is ranked in descending order to make cumulative frequency distribution (Gibbs and Maher 1967). The distribution depicts decile index. The first decile precipitation amount should not exceed by the lowest 10% of the total. The second decile is the amount between 10 and 20% of the total and so on. The severity of the drought can be assessed by comparing the amount of precipitation in a month or over a period of several months with the cumulative distribution of precipitation over a long term period. The 20% of lowest precipitation falls is termed to be much below normal (decile 1 and 2). Decile 3 and 4 (20–40%) shows below normal, decile 5 and 6 (40–60%) indicate near normal and so on. The Box-Cox transformation is used to normalize the monthly rainfall time series (McMahon 1986).

1.11 Percent of normal precipitation index (PNPI)

Morid et al. (2006) and Masoudi and Hakimi (2014) used the following equation to monitor drought in the region.

$${\text{PNPI}}={\text{~}}\frac{{{{\text{P}}_{\text{i}}}}}{{\text{P}}} \times 100$$

(16)

where Pi is the total precipitation of each year, P is the average climatology for a period from 1951 to2014.

1.12 Palmer drought severity index (PDSI)

It is the most significant meteorological index used in the United States to monitor drought (Heim 2002). Palmer (1965) considers precipitation, temperature and soil water content to calculate PDSI. Several studies (e.g., Karl 1986; Alley 1984; Zoljoodi and Didevarasl 2013) described the calculation procedure of this index. The final equation of PDSI is as follows;

$${\text{PDS}}{{\text{I}}_{{\text{i~}}}}=0.897{\text{PDS}}{{\text{I}}_{{\text{i}} - 1{\text{~}}}}+{\text{~}}\frac{1}{3}{{\text{Z}}_{\text{i}}}$$

(17)

where PDSI is dry and wet period of initial month, Z anomaly of the Palmer moisture index. The constant values of 0.987 and 1/3 are derived from the linear slope of line accessed from the extreme droughts. According to Wells et al. (2004), the development of Sc-PDSI resolved many deficiencies experienced earlier in PDSI and its data is freely available on the Koninklijk Netherlands Meteorological Institute (KNMI) website of Climate Research Unit (CRU) (http://climexp.knmi.nl/select.cgi).The data set has been calculated globally for a period more than 110 years (1901–2012) with the horizontal resolution of 0.5° × 0.5° on monthly basis.

1.13 Composite index (CI)

Zhang et al. (2006) developed this index to monitor the drought in China. The same index was employed to analyze the spatial and temporal characteristics of drought in Pakistan at annual basis. CI uses the data of standardized precipitation index (Z) and relative moisture index (M) as follows;

$${\text{CI}}=a{Z_{{\text{30}}}}+{\text{b}}{{\text{Z}}_{{\text{90}}}}+{\text{c}}{{\text{M}}_{{\text{30}}}}$$

(18)

where Z₃₀ and Z₉₀ indicates the SPI index for the 1-month and 3-months, respectively, M₃₀ shows the monthly moisture index and a, b, c are the coefficients, whose values are 0.47, 0.36, and 0.96 respectively.

$${M_{{\text{30}}}}=\frac{{P - PE}}{{PE}}$$

(19)

Here, P is the precipitation, and PE is the potential evapotranspiration on monthly basis. The potential evapotranspiration was calculated by the Food and Agriculture Organization (FAO)-Penman–Monteith method (Allen et al. 1998).

1.14 Percentage area weighted departure (PAWD)

According to Rhynsburger (1973), it is commonly used in the fields of meteorology and hydrology. The PAWD is a very good tool to monitor drought conditions in Sindh province of Pakistan and its adjoining region. Thiessen Polygon Method (1911) is used to determine the area factor of precipitation in Pakistan. First the area factor is multiplied by the monthly or annual rainfall then the following method is used to calculate PAWD.

$${\text{PAWD}}=\left[ {\frac{{\left( {{{\text{X}}_{\text{i}}} - {\bar{\text {X}}}} \right)}}{{{\bar{\text {X}}}}}} \right]\; \times \;100{\text{~}}$$

(20)

where x_i is the precipitation of ith month, $\bar {X}$ is the normal precipitation.

1.15 Reconnaissance drought index (RDI)

Reconnaissance drought index (RDI) is a meteorological index that is used for drought assessment. It is expressed as initial, normalized and standardized values (Eq. 21). The initial value (α_k) is based on the ratio between precipitation and potential evapotranspiration (PET). The PET is very helpful for identification and assessment of drought events. RDI does not depend upon the PET calculation method (Vangelis et al. 2013). The normalized RDI_n is the arithmetic mean of α_k values, whereas the standard RDI_st uses the assumption that α_k values follow the log normal distribution. So, RDI is computed by using the following equations as suggested by Tsakiris and Vangelis (2005).

$$\alpha _{{\text{k}}}^{{\left( {\text{i}} \right)}}=~\frac{{\mathop \sum \nolimits_{{j=1}}^{k} {P_{ij}}}}{{\mathop \sum \nolimits_{{j=1}}^{k} PE{T_{ij}}}},~i=1~to~N$$

$${\text{RDI}}_{{\text{n}}}^{{\left( {\text{i}} \right)}}={\text{~}}\frac{{\alpha _{{\text{k}}}^{{\text{i}}}}}{{{{\bar {\alpha }}_{\text{k}}}}} - 1$$

$${\text{~RDI}}_{{{\text{st}}}}^{{\left( {\text{i}} \right)}}=~\frac{{{{\text{y}}_{\text{k}}} - {{{\bar{\text {y}}}}_{\text{k}}}}}{{{\alpha _{{\text{yk}}}}}}$$

(21)

where P_ij and PET_ij are the precipitation and potential evapotranspiration of the jth month of the ith year, N is the total number of years, ${\bar {\alpha }_k}$ is the arithmetic mean of $\alpha _{{\text{k}}}^{{\text{i}}}$. y_k is the ln(α_k ⁽ⁱ⁾), ${{\bar{\text {y}}}_{\text{k}}}$ is arithmetic mean and σ_yk is standard deviation (see Table 2).

Table 2 Different statistical error and performance test name along with their equations where xi and yi shows the values of SPI and the other indices respectively, ${\bar{\textbf{X}}}$ and ${\bar{\textbf{Y}}}$ shows the long-term data average

The CZI, the SPEI, the MCZI, the Z-Score, the S Index, the RVI, the WASPI, the SSMAI and the SPI have similar numerical value range (Table 3). Therefore, it is easy to compare these items. However, the range of Decile (DI), PNPI and %AWD values has been categorized into similar classes (Table 3). Here, the DI classes of 5 to 6 is classified as normal 4–5 slightly below normal, 3–4 below normal, 2–3 is much below normal, 1–2 very much below normal. The value excess of 90% is considered as normal period, the value less than 60% is considered as very much below normal, 70% much below normal, 80% below normal, 90% slightly below normal.

Table 3 Drought indices values and their classes