Bivariate frequency analysis of flood and extreme precipitation under changing environment: case study in catchments of the Loess Plateau, China

Abstract

Floods have changed in a complex manner, triggered by the changing environment (i.e., intensified human activities and global warming). Hence, for better flood control and mitigation in the future, bivariate frequency analysis of flood and extreme precipitation events is of great necessity to be performed within the context of changing environment. Given this, in this paper, the Pettitt test and wavelet coherence transform analysis are used in combination to identify the period with transformed flood-generating mechanism. Subsequently, the primary and secondary return periods of annual maximum flood (AMF) discharge and extreme precipitation (Pr) during the identified period are derived based on the copula. Meanwhile, the conditional probability of occurring different flood discharge magnitudes under various extreme precipitation scenarios are estimated using the joint dependence structure between AMF and Pr. Moreover, Monte Carlo-based algorithm is performed to evaluate the uncertainties of the above copula-based analyses robustly. Two catchments located on the Loess plateau are selected as study regions, which are Weihe River Basin (WRB) and Jinghe River Basin (JRB). Results indicate that: (1) the 1994–2014 and 1981–2014 are identified as periods with transformed flood-generating mechanism in the WRB and JRB, respectively; (2) the primary and secondary return periods for AMF and Pr are examined. Furthermore, chance of occurring different AMF under varying Pr scenarios also be elucidated according to the joint distribution of AMF and Pr. Despite these, one thing to notice is that the associate uncertainties are considerable, thus greatly challenges measures of future flood mitigation. Results of this study offer technical reference for copula-based frequency analysis under changing environment at regional and global scales.

Appendix 1

Procedures of the maximum likelihood method are given as follows.

The joint distribution of (x, y) is defined as

$$H\left( {x,y;\xi } \right) = C\left( {F\left( {x;\alpha_{x} } \right),F\left( {y;\alpha_{y} } \right);\varTheta } \right)$$

(19)

where $\xi = \left( {\alpha_{x} ,\alpha_{y} ,\varTheta } \right)$.

Hence, the $h\left( {x,y;\xi } \right)$ can be written as (Cherubini et al. 2004, p. 154):

$$h\left( {x,y;\xi } \right) = c\left( {F\left( {x;\alpha_{x} } \right),F\left( {y;\alpha_{y} } \right);\varTheta } \right)f\left( {x;\alpha_{x} } \right)f\left( {y;\alpha_{y} } \right)$$

(20)

where $c\left( {F\left( {x;\alpha_{x} } \right),F\left( {y;\alpha_{y} } \right);\varTheta } \right) = \frac{{\partial \left( {F\left( {x;\alpha_{x} } \right),F\left( {y;\alpha_{y} } \right);\varTheta } \right)}}{{\partial F\left( {x;\alpha_{x} } \right)\partial F\left( {y;\alpha_{y} } \right)}}$ denotes the density function of copula.

Let $\left( {x_{1} ,y_{1} } \right), \ldots ,\left( {x_{T} ,y_{T} } \right)$ be the random sample of size T from copula C. The log-likelihood function is expressed as

$$l\left( \xi \right) = \sum\limits_{t = 1}^{T} {\log \left( {h\left( {x,y;\xi } \right)} \right)} = \sum\limits_{t = 1}^{T} {\log \left( {c\left( {F\left( {x;\alpha_{x} } \right),F\left( {y;\alpha_{y} } \right);\varTheta } \right)f\left( {x;\alpha_{x} } \right)f\left( {y;\alpha_{y} } \right)} \right)}$$

(21)

Then, maximizing the log-likelihood function $l\left( \xi \right)$ to generate $\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\xi }_{MLE} = \left( {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\alpha }_{{x\left( {MLE} \right)}} ,\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\alpha }_{{y\left( {MLE} \right)}} ,\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\varTheta }_{{\left( {MLE} \right)}} } \right).$