Trivariate frequency analysis of droughts using copulas under future climate change over Vidarbha region in India

Abstract

Mankind is currently facing a crucial challenge in terms of climate change which may result in severe drought in many areas across the world. The present study analysed the joint return periods of meteorological drought events for historical and future projections across the Vidarbha region of India. To describe the drought occurrences in the area, the study used the standardized precipitation evapotranspiration index. The properties of each drought event, viz., duration, severity, and peak, were identified to model the multivariate drought risks. Several copula families were evaluated using statistical tests for joint dependency modelling of drought properties. The obtained joint distribution and selected univariate marginals are further used for estimating trivariate return periods for the historical and future climate change scenarios. The drought properties exhibited good interdependency justifying the use of a trivariate framework. The observed historical data of rainfall, maximum and minimum temperature were collected from the India Meteorological Department. The latest future projections of climate variables from coupled model intercomparison project phase 6 (CMIP6) datasets were acquired from NASA Earth Exchange Global Daily Downscaled Projections (NEX-GDDP-CMIP6) for six general circulation models. The copula-based trivariate frequency analysis was performed to compare the return periods associated with ‘moderate’ and ‘above moderate drought’ (i.e., combination of severe and extreme drought) events for the reference period (1981–2020) and future periods (2021–2100) under two SSP scenarios, SSP2-4.5 and SSP5-8.5. The ‘AND’ condition return period showed high drought risk during historical period covering a larger area. The findings of the study suggested that for ‘AND’ condition the trivariate frequency of above moderate droughts would increase in the future across Vidarbha region. The analysis showed that during near future more area will be under the risk of above moderate drought conditions compared to the far future, and the high-risk zones will be more under SSP5-8.5 than SSP2-4.5 scenario. The frequency of moderate drought episodes will be higher than the above moderate drought events, but the percentage area in the high-risk zone will be lower. The ‘OR’ condition return period projected alleviation of drought risk in the future, while the far future projected more area under drought risk compared to near future for the SSP scenarios.

Appendices

Appendix

Appendix 1 PET estimation using Hargreaves Method

There exist several methods for estimating PET such as Thornthwaite, Hargreaves-Samani, Penman–Monteith, etc. The applicability of each method depends on the availability of the meteorological variables data.

Initially Vicente-Serrano et al. (2010) applied the Thortnthwaite method for estimating PET, but later Beguería et al. (2014) suggested the Penman–Monteith method as the primary approach and recommended Hargreaves and Thornthwaite methods as first and second choice methods respectively for estimating the PET. The Hargreaves method is one of the most commonly used methods recommended by FAO when weather data are limited or unavailable. In this analysis, the Hargreaves approach is adopted for estimating PET, and the equation is given as (Zohrab 2000):

$$PET = 0.0135\left( {KT} \right)R_{a} \left( {T_{d} } \right)^{0.5} \left( {T_{a} + 17.8} \right)$$

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where $R_{a}$ is the water equivalent of the extra-terrestrial solar radiation (mm/day), $T_{d}$ is the difference of maximum and minimum temperature (°C), $T_{a}$ is mean daily temperature (°C), and KT = empirical coefficient. The empirical coefficient can be expressed as,

$$KT = 0.00185T_{d}^{2} - 0.0433T_{d} + 0.4023$$

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Depending on latitude and month, the $R_{a}$ can be expressed as follows (Allen et al. 1998):

$$R_{a} = \frac{{24\left( {60} \right)}}{\pi }G_{sc} d_{r} \left[ {\omega_{s} \sin \left( \varphi \right)\sin \left(\Delta \right) + \cos \left( \varphi \right)\sin \left(\Delta \right)\sin \left( {\omega_{s} } \right)} \right]$$

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The $G_{sc}$ is the solar constant, and its value is 0.0820 MJ m⁻² min⁻¹, $d_{r}$ is the inverse relative distance Earth-Sun, $\omega_{s}$ is the sunset hour angle, $\varphi$ is the latitude (rad), $\Delta$ is the solar declination (rad). The $R_{a}$ in the above equation is multiplied by 0.408 to convert the unit from MJ m⁻² day⁻¹ to mm day⁻¹.

Appendix 2 Fitting 3-parameter log-logistic distribution for the time series of X

Vicente-Serrano et al. (2010) recommended to apply a three-parameter log-logistic distribution in order to standardize the water-balance series as they can contain negative values. Vicente-Serrano et al. (2010) applied Kolmogorov–Smirnov (KS) test and found that the three-parameter log-logistic distribution performed better all over the globe. The probability density function (pdf) of the log-logistic distribution is given as follows:

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

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where α, β, and γ are scale, shape, and origin parameters, respectively. The parameters were obtained using the method of L-moments as follows:

$$\beta = \frac{{2s_{1} - s_{0} }}{{6s_{1} - s_{0} - 6s_{2} }}$$

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$$\alpha = \frac{{\left( {s_{0} - 2s_{1} } \right)\beta }}{{\Gamma \left( {1 + \frac{1}{\beta }} \right)\Gamma \left( {1 - \frac{1}{\beta }} \right)}}$$

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$$\gamma = s_{0} - \alpha \Gamma \left( {1 + \frac{1}{\beta }} \right)\Gamma \left( {1 - \frac{1}{\beta }} \right)$$

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where $\Gamma \left( \beta \right)$ is the function of $\beta$, and $s_{0}$, $s_{1}$ & $s_{2}$ are the probability-weighted moments computed based on an unbiased estimator (Hosking 1986; Beguería et al. 2014) as:

$$s_{r} = \frac{1}{n}\frac{{\sum\limits_{i = 1}^{n} {\left( \begin{gathered} n - i \hfill \\ \,\,\,\,r \hfill \\ \end{gathered} \right)} \,\delta_{i} }}{{\left( \begin{gathered} n - 1 \hfill \\ \,\,\,\,r \hfill \\ \end{gathered} \right)}},\,\quad \forall r = 0,1,2$$

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where n is the number of observations, and δ_i is the matrix of observations in decreasing sequence.