Advances in Meteorology

Volume 2017, Article ID 4650284, 11 pages

https://doi.org/10.1155/2017/4650284

## Exploration of Use of Copulas in Analysing the Relationship between Precipitation and Meteorological Drought in Beijing, China

^{1}College of Water Sciences, Beijing Normal University, Beijing Key Laboratory of Urban Hydrological Cycle and Sponge City Technology, Beijing 100875, China^{2}Agricultural Water Conservancy Department, Changjiang River Scientific Research Institute, Wuhan 430015, China^{3}Environmental Science Division, Argonne National Laboratory, Lemont, IL 60439, USA^{4}State Key Laboratory of Simulation and Regulation of Water Cycle in River Basin, China Institute of Water Resources and Hydropower Research, Beijing 100038, China

Correspondence should be addressed to Hongrui Wang; nc.ude.unb@gnawyrneh

Received 15 December 2016; Revised 21 February 2017; Accepted 14 March 2017; Published 16 May 2017

Academic Editor: Stefano Dietrich

Copyright © 2017 Linlin Fan et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Drought risk analysis is essential for regional water resource management. In this study, the probabilistic relationship between precipitation and meteorological drought in Beijing, China, was calculated under three different precipitation conditions (precipitation equal to, greater than, or less than a threshold) based on copulas. The Standardized Precipitation Evapotranspiration Index (SPEI) was calculated based on monthly total precipitation and monthly mean temperature data. The trends and variations in the SPEI were analysed using Hilbert-Huang Transform (HHT) and Mann-Kendall (MK) trend tests with a running approach. The results of the HHT and MK test indicated a significant decreasing trend in the SPEI. The copula-based conditional probability indicated that the probability of meteorological drought decreased as monthly precipitation increased and that 10 mm can be regarded as the threshold for triggering extreme drought. From a quantitative perspective, when mm, the probabilities of moderate drought, severe drought, and extreme drought were 22.1%, 18%, and 13.6%, respectively. This conditional probability distribution not only revealed the occurrence of meteorological drought in Beijing but also provided a quantitative way to analyse the probability of drought under different precipitation conditions. Thus, the results provide a useful reference for future drought prediction.

#### 1. Introduction

Drought is a recurrent extreme climate event that affects every aspect of the natural environment, as well as human lives [1]. It is defined as a natural temporary imbalance in water availability, which may be related to persistently lower-than-average precipitation. Droughts have uncertain frequencies, durations, and severities, and their occurrence is difficult to predict. Additionally, the water resource availability and carrying capacity of an ecosystem are diminished [2, 3]. Based on these factors, drought analysis has been a challenging topic in water resource management.

A variety of studies have examined the different aspects of drought phenomena by developing different drought indices [4–9], analysing the spatiotemporal patterns of drought [10–12], and predicting future drought risks [13, 14]. Regarded as a complex hydrologic event, drought is a type of multivariate event that is characterized by a few correlated random variables. Thus, a number of studies have focused on constructing multivariate distributions of drought characteristics, such as the intensity, duration, frequency, and recurrence [15–18]. Copulas are effective tools in multivariate distribution construction. They use a nonlinear approach to establish the joint probability distribution of two or more related variables and are able to model the dependence structure between random variables independent of their marginal distributions. Since Sklar [19] first introduced copulas, they have been widely used in water-related research fields. When using copulas for drought analysis, a joint probability distribution has commonly been used between different characteristics of drought. Shiau [20] used two-dimensional copulas to construct a joint drought duration and severity distribution for the first time. Song and Singh [21] used three-dimensional metaelliptical copulas to construct a joint drought duration, severity, and interarrival time distribution. Similar studies [22–27] have shown that copulas are useful in exploring the associations among correlated drought variables.

Apart from the joint probability distribution, copulas are also able to establish conditional probability distributions. Unlike joint probability, conditional properties can provide the probability distributions of drought under different conditions. Saghafian and Mehdikhani [28] investigated the conditional return periods of drought severity, peak, and duration based on a conditional probability distribution established by copulas. Additionally, Madadgar and Moradkhani [29] applied conditional probability for drought prediction. By incorporating copulas into a Bayesian framework, they developed a probabilistic forecasting model for predicting future spring drought status given the drought conditions in the previous season. Hao et al. [30] proposed a hydrological drought prediction method based on a meta-Gaussian model of conditional probability. The model considered the drought persistence and the prior meteorological drought conditions.

Since climate change and the increasing global warming trend have affected hydrological events in recent years, it is important to consider climatic variables when analysing drought. Generally, drought originates from a precipitation deficit, which first causes meteorological drought. Zhang et al. [31] studied how precipitation and temperature affect drought occurrence. They concluded that the similarity coefficient between the drought probability and low precipitation probability in China was 0.9732, reflecting a strong relationship between drought and precipitation. Copulas can be used to combine precipitation and drought indexes and to provide conditional probability distributions (with precipitation as the condition) between them. From this conditional probability distribution, we can quantify the probability of drought occurrence under different precipitation conditions.

Beijing is located in the dry northern part of China, which experienced water crises caused by drought from 1980 to 1985 and from 1999 to 2007 [32]. Cai et al. [33] evaluated the spatiotemporal characteristics of drought in the Beijing-Tianjin-Hebei metropolitan region. Liu et al. [34] proposed a vegetation-dependent temperature-vegetation dryness index model for analysing regional drought disasters in the Beijing-Tianjin-Hebei metropolitan region. Li et al. [35] analysed the joint probability and return period of the drought severity and duration during the growth period of winter wheat in Beijing. Such studies have focused more on the spatiotemporal characteristics of drought and drought variables. They found that Beijing experienced quite frequent moderate and severe droughts, especially during the periods of 1965–1973 and 1997–2002. In addition, the risk of drought during winter was very high, and the area affected by drought was >70,000 ha. However, similar studies have focused less on how the climatic variables can affect drought conditions.

In this study, a conditional probability distribution was applied for meteorological drought analysis based on copulas. Monthly total precipitation and monthly mean temperature data from 1951 to 2013 were collected from one meteorological station to calculate the Standardized Precipitation Evapotranspiration Index (SPEI). After a temporal analysis of the SPEI based on the Hilbert-Huang Transform (HHT) and Mann-Kendall (MK) trend test, the meteorological drought probability was analysed under three different precipitation conditions (precipitation equal to, greater than, or less than a threshold).

#### 2. Materials and Methods

##### 2.1. Data and SPEI Calculation

The SPEI was used to represent meteorological drought. A meteorological station in Beijing (latitude 39.80°N, longitude 116.47°E) was chosen as the study site. The monthly total precipitation (mm) and monthly mean air temperature (°C) data from 1951 to 2013 were collected from the National Climate Center of the China Meteorological Administration (http://data.cma.cn/). The SPEI describes the effects of drought on vegetation and agricultural practices on short-term scales and represents a broad proxy for water resource management on long-term scales [36]. In this study, we calculated both short-term (1 month) and long-term (12 months) SPEI values for meteorological drought analysis. According to the definition of the SPEI, the -month SPEI is based on precipitation and evapotranspiration total for the previous months. Thus, a 12-month SPEI represents conditions over the full calendar year, while a 1-month SPEI represents conditions within a single month. The aim of analysing the temporal characteristics of the SPEI is to explore the changing patterns in meteorological drought conditions between different years and provide a holistic view. Thus, we chose the 12-month SPEI for this part of the analysis. Meanwhile, the goal of the copula-based analysis of precipitation and SPEI is to obtain the conditional probability distributions of meteorological drought given different precipitation conditions. Conditional probability distributions can be used for meteorological drought prediction in the future. Because short-term drought prediction is more practical than long-term prediction, we used the 1-month SPEI for the copulas-based analysis.

The calculation process can be divided into the following three steps [8].

Calculate the water balance for month according to the following equation:where is monthly total precipitation and is potential evapotranspiration, which was calculated using the Thornthwaite method [37]. The calculated values are aggregated at different time scales. The difference in month and year depends on the chosen time scale . For example, the accumulated difference for one month in a particular year with a 12-month time scale is calculated as follows:where is the difference in month and year* i*, in millimetres.

Use a three-parameter log-logistic probability density function to fit the established series. Then, obtain the cumulative probability function as follows:where is the probability density function of series; is the probability distribution function of series; and *α*, *β*, and *γ* are scale, shape, and origin parameters, respectively, for values, which can be obtained using an* L*-moments approach.

Obtain the SPEI as the standardized values of :where is the probability of exceeding a determined value and . If , then is replaced by , and the sign of the resultant SPEI is reversed. The constants are as follows: = 2.515517, = 0802853, = 0.010328, = 1.432788, = 0.189269, and = 0.001308.

The SPEI follows the same classification criteria as the Standardized Precipitation Index (SPI) because of the similarity in the calculation principles of SPEI and SPI [11]. Table 1 reports the climate classification according to the SPEI [5].