# Compound Extremes in Hydroclimatology: A Review

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## Abstract

**:**

## 1. Introduction

## 2. Compound Extremes

#### 2.1. Definition of Compound Extremes

“(1) two or more extreme events occurring simultaneously or successively, (2) combinations of extreme events with underlying conditions that amplify the impact of the events, or (3) combinations of events that are not themselves extremes but lead to an extreme event or impact when combined.”

“A compound event is an extreme impact that depends on multiple statistically dependent variables or events.”

#### 2.2. Typical Compound Extremes

## 3. Statistical Approaches

#### 3.1. Empirical Approach

#### 3.2. Multivariate Distribution

#### 3.2.1. Copula Approach

#### 3.2.2. Joint Probability

#### 3.2.3. Conditional Probability

_{0}) or range (e.g., u < u

_{0}). The conditional probability of V ≤ v given U = u

_{0}can be expressed with a copula C as [143]:

_{0}can be expressed as [128,138]:

#### 3.3. Indicator Approach

_{x}, M

_{y}), where M

_{x}and M

_{y}are two extremes (e.g., combined thunderstorm and tornado) [149,150,151,152]. Among different ways of developing indicators, the multivariate distribution has been commonly explored for characterizing compound extremes from a multivariate perspective, since it completely describes the joint behavior of two or more variables [54,60,153]. The joint probability or percentile P

_{1}= P(X ≤ x, Y ≤ y) can be employed as the measure of the compound extreme of both variable X and Y lower than certain thresholds (e.g., 10th percentile). Similarly, the joint probability P

_{2}= P(X ≤ x, Y > y) can be used as a measure of the compound extreme with X lower than a specific threshold (e.g., 10th percentile) and Y exceeding a higher threshold (e.g., 90th percentile), such as the compound drought and hot extremes.

#### 3.4. Quantile Regression

_{τ}(Y|X). Specifically, for any quantile τ in (0,1), the quantile regression can be expressed as:

_{τ}and β

_{τ}are the parameters associated with quantile τ. By changing quantile τ, the relationship between Y and X can be explored.

#### 3.5. Markov Chain Model

_{t}(t > 0) with each random variable taking values in the set S = (1, 2, …, m) is a Markov Chain if [161],

_{ij}the transition probability of the Markov chain. The transition matrix with element p

_{ij}can be estimated as:

_{t}(e.g., persistence, recurrence time). For example, the persistence of X

_{t}stays in the state j and will reside in the same state j in the following time step can be expressed as p

_{jj}. Recently, the Markov Chain model has been employed to study compound extremes (e.g., heavy precipitation- cold in winter and hot- dry days in summer) for central Europe under changing climate [87].

## 4. Discussion

#### 4.1. Comparison of Approaches

#### 4.2. High Dimensional Modeling

#### 4.3. Compound Extremes Under Climate Change

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A. Types of Compound Extremes

Type of Compound Extremes | Combined Variables/Events/Extremes | Approaches |
---|---|---|

Compound drought and hot extreme | Drought and heat wave (hot days or months, high temperature) [37,38,39,65,90,138,156,157,159] | Empirical approach, Quantile regression, Multivariate distribution |

Compound precipitation and temperature extreme | Heavy precipitation and cold/warm condition [47,87,184,190] | Empirical approach, Markov Chain approach |

Low precipitation and high temperature [29,179,181,191] | Empirical approach, Multivariate distribution | |

Dry-warm/dry-cold/wet-warm/wet-cold condition [29,77,80,192] | Empirical approach | |

Compound flood | Storm surge and high rainfall [35,57,185,193] | Multivariate distribution |

Storm surge and high discharge/runoff [48,55,56,60,61,136] | Empirical approach, Indicator approach, Multivariate distribution | |

Storm surge and sea level [73,178] | Empirical approach, Indicator approach | |

Sea levels and rainfall/river flow [59,97,136,194] | Multivariate distribution | |

Compound drought | Deficit from precipitation, soil moisture, runoff or other variables [54,153,195] | Indicator approach, Multivariate distribution |

Combined drought, moisture surplus, precipitation/temperature, and other extremes | Drought indices, precipitation extremes and temperature extremes [146,147,148,196] | Indicator approach |

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**Figure 2.**Number of the compound dry-warm extreme for each year (and 5-year running average) during 1901–2016 for Melbourne, Australia.

**Figure 4.**Comparison of individual and compound drought based on the percentile of precipitation and runoff (6-month time scale) for the period 1932–2011 using copula. The circled point represents the data pairs during 2011.

**Figure 5.**Conditional distribution of hydrological drought (represented with SRI) conditioned on the meteorological drought (represented with SPI).

**Figure 6.**Comparison of drought condition based on SPI, SRI and MSDI in the bivariate case for the period from 1932-2011. (

**a**) 1932–1971; (

**b**) 1972–2011.

**Figure 7.**Quantile regression of hot extremes (number of hot days, NHD) with respect to the drought indicator SPI.

**Figure 8.**The change of the transition probability to the compound dry and hot extreme during 1960–2016 for Melbourne, Australia.

Copulas | C(u,v) | Parameter |
---|---|---|

Gaussian | ${\mathsf{\Phi}}_{2}({\mathsf{\Phi}}^{-1}(u),{\mathsf{\Phi}}^{-1}(v))$ * | $\theta \in [-1,1]$ |

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

Frank | $-\frac{1}{\theta}\mathrm{ln}\left[1+\frac{\left({e}^{-\theta u}-1\right)\left({e}^{-\theta v}-1\right)}{{e}^{-}{}^{\theta}-1}\right]$ | $\theta \in (-\infty ,\infty )$ |

Gumbel | $\mathrm{exp}\left\{-{\left[{(-\mathrm{log}u)}^{-\theta}+{\left(-\mathrm{log}v\right)}^{-\theta}\right]}^{-1/\theta}\right\}$ | $\theta \in [1,\infty )$ |

_{2}represent the standard normal distribution in the univariate and bivariate case.

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Hao, Z.; Singh, V.P.; Hao, F.
Compound Extremes in Hydroclimatology: A Review. *Water* **2018**, *10*, 718.
https://doi.org/10.3390/w10060718

**AMA Style**

Hao Z, Singh VP, Hao F.
Compound Extremes in Hydroclimatology: A Review. *Water*. 2018; 10(6):718.
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**Chicago/Turabian Style**

Hao, Zengchao, Vijay P. Singh, and Fanghua Hao.
2018. "Compound Extremes in Hydroclimatology: A Review" *Water* 10, no. 6: 718.
https://doi.org/10.3390/w10060718