Probabilistic assessment of flood risks using trivariate copulas

Abstract

In this paper, a copula-based methodology is presented for probabilistic assessment of flood risks and investigated the performance of trivariate copulas in modeling dependence structure of flood properties. The flood is a multi-attribute natural hazard and is characterized by mutually correlated flood properties peak flow, volume, and duration of flood hydrograph. For assessing flood risk, many studies have used bivariate analysis, but a more effective assessment can be possible considering all three mutually correlated flood properties simultaneously. This study adopts trivariate copulas for multivariate analysis of flood risks, and applied to a case study of flood flows of Delaware River basin at Port Jervis, NY, USA. On evaluation of various probability distributions for representation of flood variables, it is found that the flood peak flow and volumes can be best represented by Fréchet distribution, whereas flood duration by log-normal distribution. The joint distribution is modeled using four trivariate copulas, namely, three fully nested form of Archimedean copulas: Clayton, Gumbel–Hougaard, Frank copulas; and one elliptical copula: Student’s t copula. Based on distance-based performance measures, graphical tests, and tail-dependence measures, it is found that the Student’s t copula best representing the trivariate dependence structure of flood properties as compared to the other copulas. Similar results are found for bivariate copula modeling of flood variables pairs, where Student’s t copula performed better than the other copulas. The obtained copula-based joint distributions are used for multivariate analysis of flood risks, in terms of primary and secondary return periods. The resultant trivariate return periods are compared with univariate and bivariate return periods, and addressed the necessity of multivariate flood risk analysis. The study concludes that the trivariate copula-based methodology is a viable choice for effective risk assessment of floods.

Appendix

1.1 Chi plot

Chi-plot is a scatter plot of the pairs (λ _i , χ _i ), where it uses the data ranks (Fisher and Switzer 2001). The values are defined as,

$$ {\chi_i} = \frac{{{H_i} - {F_i}{G_i}}}{{\sqrt {{{F_i}\left( {1 - {F_i}} \right){G_i}\left( {1 - {G_i}} \right)}} }} $$

(A1)

where

$$ \begin{gathered} {H_i} = \frac{{\sum\limits_{{j \ne i}} {I\left( {{x_j} \leqslant {x_i},{y_j} \leqslant {y_i}} \right)} }}{{n - 1}};\,{F_i} = \frac{{\sum\limits_{{j \ne i}} I \left( {{x_j} \leqslant {x_i}} \right)}}{{n - 1}} \, ,and\,{G_i} = \frac{{\sum\limits_{{j \ne i}} I \left( {{y_j} \leqslant {y_i}} \right)}}{{n - 1}} \hfill \\ and\;{\lambda_i} = 4{{\text sgn}_i}\;\max \left\{ {{{\left( {{F_i} - 0.5} \right)}^2},{{\left( {{G_i} - 0.5} \right)}^2}} \right\} \hfill \\ \end{gathered} $$

(A2)

where ${{\text sgn}_i} = {\text sgn} \left\{ {({F_i} - 0.5)({G_i} - 0.5)} \right\}$The values of λ _i is a measure of the distance of the bivariate observation (x _i , y _i ) from the center of the data set and λ _i ∈ [−1,1]. If X and Y are positively associated, λ _i will tend to be positive and vice versa for negatively correlated variable. The chi-plot also include control limits that are placed at $\chi = \pm {{{{c_p}}} \left/ {{\sqrt {n} }} \right.}$, where c _p values are provided by Fisher and Switzer (2001). Using simulation studies, it is found that C _p values 1.54, 1.78, and 2.18 correspond to p values of 0.90, 0.95, and 0.99, respectively. In case of independence, the points lie within the control limits of the plot. When the scatter of data points are largely on the upper side of the control limits, it indicates positive dependence; whereas in case of negative dependence the scatter of data points are largely in the lower side of the control limits.

1.2 Kendall plot

The Kendall plots are analogous to Q–Q plots (Genest and Boies 2003). The Kendall plot, consists of plotting random variable W _i:n in ordinate against order statistics of the sample H _(i) in abscissa for i∈{1,_⋯,n}. W _i:n is the expected value of the ith order statistics of a random variable W = H(X,Y) = C(U,V) of sample size of n from distribution K _n and is defined as (Genest and Favre 2007)

$$ {W_{{i:n}}} = n\left( \begin{gathered} n - 1 \hfill \\ i - 1 \hfill \\ \end{gathered} \right)\int_0^1 {wk(w){{\left\{ {K(w)} \right\}}^{{i - 1}}}{{\left\{ {1 - K(w)} \right\}}^{{n - 1}}}dw} $$

(A3)

where $k = \frac{{dK(w)}}{{dw}};K(w) = P\left( {UV \leqslant w} \right) = \int_0^1 {P\left( {U \leqslant {{w} \left/ {v} \right.}} \right)} dv = w - w\log (w){\text{and}}\left( \begin{gathered} n - 1 \hfill \\ i - 1 \hfill \\ \end{gathered} \right)$ is a binomial coefficient. The deviation from the main diagonal is the indication of dependence in Kendall plots. The plot tends to be linear in case of independence whereas the curve tends to be $K(t) = w - w\log (w)$ in case of perfect positive dependence. In case of perfect negative dependence, all points will lie on x axis and a flat line will result with height zero.