Copula-based risk evaluation of droughts across the Pearl River basin, China

Abstract

Daily precipitation data for the period of 1960–2005 from 42 precipitation gauging stations in the Pearl River basin were analyzed using the Mann–Kendall trend test and copula functions. The standardized precipitation index method was used to define drought episodes. Primary and secondary return periods were also analyzed to evaluate drought risks in the Pearl River basin as a whole. Results indicated that: (1) in general, the drought tendency was not significant at a 95 % confidence level. However, significant drought trends could be found in November, December, and January and significant wetting trends in June and July. The drought severity and drought durations were not significant at most of the precipitation stations across the Pearl River basin; (2) in terms of drought risk, higher drought risk could be observed in the lower Pearl River basin and lower drought risk in the upper Pearl River basin. Higher risk of droughts of longer durations was always corresponding to the higher risk of droughts with higher drought severity, which poses an increasing challenge for drought management and water resources management. When drought episodes with higher drought severity occurred in the Pearl River basin, the regions covered by higher risk of drought events were larger, which may challenge the water supply in the lower Pearl River basin. As for secondary return periods, results of this study indicated that secondary return periods might provide a more robust evaluation of drought risk. This study should be of merit for water resources management in the Pearl River basin, particularly the lower Pearl River basin, and can also act as a case study for determining regional response to drought changes as a result of global climate changes.

Appendix

The Copula family is described as the follows.

• 1. Gumbel–Hougaard (GH) family

  The GH family can be formulated as:

  $$ C\left( {u,v} \right) = \exp \left\{ { - {{\left[ {{{\left( { - \ln u} \right)}^\theta } + {{\left( { - \ln v} \right)}^\theta }} \right]}^{1/\theta }}} \right\},\theta \in \left[ {1,\infty } \right) $$

  (9)

  where u and v are the marginal distribution functions of two hydrological variables; θ is the parameter of the GH family; C is the copula having the function of capturing the essential features of the dependence among random variables. Parameters u, v, and θ in the following equations have the same meaning as those in Eq. (9), so no further explanations are provided thereafter. The copula generator of the GH family (Zhang and Singh 2007) is:

  $$ \phi (t) = {\left( { - \ln t} \right)^\theta } $$

  (10)

  t is the u ₁ or v ₁, being specific values of u and v in Eq. (9) varying from 0 to 1. The relationship between parameter, θ, and Kendall’s coefficient of correlation, τ, between X and Y is τ = 1 − θ ⁻¹. The GH copula performs well in the description of the correlation structure of two variables characterized by a positive correlation.

• 2. Clayton family

  The Clayton family can be formulated as:

  $$ C\left( {u,v} \right) = {\left( {{u^{ - \theta }} + {v^{ - \theta }} - 1} \right)^{ - 1/\theta }},\theta \in \left( {0,\infty } \right) $$

  (11)

  The copula generator of Clayton family is

  $$ \varphi (t) = \left( {{t^{ - \theta }} - 1} \right)/\theta, \theta \in \left( {0,\infty } \right) $$

  (12)

  The relation between parameter, θ, and Kendall’s coefficient of correlation, τ, between X and Y is:

  $$ \tau = \frac{\theta }{{2 + \theta }},\theta \in \left( {0,\infty } \right) $$

  (13)

  The Clayton family is good for describing two variables with a positive correlation.

• 3. Frank family

  The Frank family can be formulated as:

  $$ C\left( {u,v} \right) = - \frac{1}{\theta }\ln \left[ {1 + \frac{{\left( {{e^{ - \theta u}} - 1} \right)\left( {{e^{ - \theta v}} - 1} \right)}}{{\left( {{e^{ - \theta }} - 1} \right)}}} \right],\theta \in R $$

  (14)

  The copula generator of the Clayton family is

  $$ \varphi (t) = - \ln \frac{{{e^{ - \theta t}} - 1}}{{{e^{ - \theta }} - 1}},\theta \in R $$

  (15)

  The relation between parameter, θ, and Kendall’s coefficient of correlation, τ, between X and Y is:

  $$ \tau = 1 + \frac{4}{\theta }\left[ {\frac{1}{\theta }\int\limits_0^\theta {\frac{t}{{{e^t} - 1}}dt - 1} } \right],\theta \in R $$

  (16)

  The Frank family performs well for describing two variables with a positive correlation.

• 4. t copula family

  The t family can be formulated as:

  $$ C_{v,R}^t\left( {u,v} \right) = {\int_{ - \infty }^{t_v^{ - 1}(u)} {\int_{ - \infty }^{t_v^{ - 1}(v)} {\frac{1}{{2\pi {{\left( {1 - {\theta^2}} \right)}^{\frac{1}{2}}}}}\left[ {1 + \frac{{{s^2} - 2\theta st + {t^2}}}{{v\left( {1 - {\theta^2}} \right)}}} \right]} }^{ - \frac{{\left( {v + 2} \right)}}{2}}}dsdt $$

  (17)

  $t_{v,R}^n$ is the R-dimensional standard t distribution with degree of freedom of v. $t_v^{ - 1}$ is the inverse function of $t_{v,R}^n$.

  The relation between parameter, θ, and Kendall’s coefficient of correlation, τ, between X and Y is:

  $$ \theta = \sin \left( {\frac{{\pi \tau }}{2}} \right) $$

  (18)

Conditional copula

Let X and Y be random variables with marginal distribution as u = F _X (x) and v = F _Y (y). The conditional distribution function of X given Y = y can be expressed by the copula method as:

$$ \begin{array}{*{20}{c}} {H\left( {X < x\left| {Y = y} \right.} \right) = {{C}_{\theta }}\left( {u\left| {v = {{v}_{1}}} \right.} \right) = \mathop{{\lim }}\limits_{{\Delta v \to 0}} \frac{{{{C}_{\theta }}\left( {u,v + \Delta v} \right) - {{C}_{\theta }}\left( {u,v} \right)}}{{\Delta v}}} \\ { = \frac{\partial }{{\partial v}}{{C}_{\theta }}\left( {u,v} \right)\left| {v = {{v}_{1}}} \right.} \\ \end{array} $$

(19)

An equivalent formula for the conditional distribution function for Y given X = x can be obtained. The conditional distribution function of X given Y ≥ y can be expressed by the copula method as

$$ {{C}_{\theta }}\left( {u\left| {v < {{v}_{1}}} \right.} \right) = \frac{{{{C}_{\theta }}\left( {u,v} \right)}}{v} $$

(20)

Similarly, an equivalent formula for the conditional distribution function for Y given X < x can be obtained.