Risk Assessment of Droughts in Gujarat Using Bivariate Copulas

Abstract

This study presents risk assessment of hydrologic extreme events droughts in Saurashtra and Kutch region of Gujarat state, India. Drought is a recurrent phenomenon and risk assessment of droughts can play an important role in proper planning and management of water resources in the study region. In the study, drought events are characterized by severity and duration, and drought occurrences are modeled by Standardized Precipitation Index (SPI) computed on mean areal precipitation, aggregated at a time scale of 6 months for the period 1900–2008. After evaluating several distribution functions, drought variable—severity is best described by non-parametric kernel density, whereas duration is best fitted by exponential distribution. Considering the extreme nature of drought variables, the upper tail dependence copula families including two Archimedean—Gumbel-Hougaard, BB1 and one elliptical—Student’s t copulas are evaluated for modeling joint distribution of drought variables. On evaluating their performance using various goodness-of-fit measures, Gumbel-Hougaard copula is found to be the best performing copula in modeling the joint dependence structure of drought variables. Also, while comparing with traditional bivariate distributions, the copula based distributions are resulted in better performance as compared to bivariate log-normal and the logistic model for bivariate extreme value distributions. Then joint and conditional return periods of drought characteristics are derived, which can be helpful for risk based planning and management of water resources systems in the study region.

Appendix

1.1 A1 Simulation from Bivariate Archimedean Copulas

Simulation of random pair (u, v) from distribution function Copula C(•) is performed using following steps (Joe 1997):

• 1. Simulate sequence of two independent uniform random variates u and q from U (0, 1).

• 2. Evaluate the inverse of conditional distribution function at v, i.e.,

  v = C ⁻¹(q|u), where $C\left( {v|u;\theta } \right) = \frac{{\partial C\left( {u,v;\theta } \right)}}{{\partial u}}$ is the first derivative of the bivariate marginal of copula function, C(u, v; θ).

• 3. Then the pair (u, v) are uniformly distributed random variables drawn from the respective copula families C(u, v; θ). The simulated observations from copulas can be obtained by computing $\left( {x,y} \right) = \left[ {F_X^{{ - 1}}(u),F_Y^{{ - 1}}(v)} \right]$, where $F_X^{{ - 1}}\left( \bullet \right)$ and $F_Y^{{ - 1}}\left( \bullet \right)$ are the inverse of cumulative distribution functions of X and Y respectively.

1.2 A2 Simulation from Bivariate Student’s t Copula

• 1. Generate random vector z ₁, z ₂ with ϑ degrees of freedom and P _i,j as correlation matrix after calibration of Student’s t copula, i.e., (z ₁, z ₂) ≈ t _d (ϑ, P _i,j ); where t _d (•) is multivariate t distribution.

• 2. Set (u, v) = (t _ϑ (z ₁), t _ϑ (z ₂))^′, where t _ϑ (•) is the univariate Student’s t distribution with ϑ degrees of freedom.

• 3. Simulated observations from copula based joint distribution can be obtained by computing $\left( {x,y} \right) = \left[ {F_X^{{ - 1}}(u),F_Y^{{ - 1}}(v)} \right]$.