The Joint Return Period Analysis of Natural Disasters Based on Monitoring and Statistical Modeling of Multidimensional Hazard Factors

Science of the Total Environment 538 (2015) 724–732

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Science of the Total Environment

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The joint return period analysis of natural disasters based on monitoring and statistical modeling of multidimensional hazard factors

Xueqin Liu a,b,c,NingLia,ShuaiYuanb,⁎,NingXub,WenqinShib, Weibin Chen b a State Key Laboratory of Earth Surface Processes and Resource Ecology, Beijing Normal University, Beijing 100875, China b National Marine Environmental Monitoring Center, State Oceanic Administration, Dalian 116023, China c School of Social Development and Public Policy, Beijing Normal University, Beijing 100875, China

HIGHLIGHTS GRAPHICAL ABSTRACT

• A method to estimate the multidimensional joint return periods is presented. • 2D function allows better ﬁtting results at the lower tail of hazard factors. • Three-dimensional simulation has obvious advantages in extreme value ﬁtting. • Joint return periods are closer to the reality than univariate return periods. • Copula method provides a new idea for multivariate analysis of natural disasters.

article info abstract

Article history: As a random event, a natural disaster has the complex occurrence mechanism. The comprehensive analysis of Received 5 April 2015 multiple hazard factors is important in disaster risk assessment. In order to improve the accuracy of risk analysis Received in revised form 2 August 2015 and forecasting, the formation mechanism of a disaster should be considered in the analysis and calculation of Accepted 16 August 2015 multi-factors. Based on the consideration of the importance and deficiencies of multivariate analysis of dust Available online xxxx storm disasters, 91 severe dust storm disasters in Inner Mongolia from 1990 to 2013 were selected as study Editor: Simon Pollard cases in the paper. Main hazard factors from 500-hPa atmospheric circulation system, near-surface meteorological system, and underlying surface conditions were selected to simulate and calculate the multidimensional joint Keywords: return periods. After comparing the simulation results with actual dust storm events in 54 years, we found Natural hazard that the two-dimensional Frank Copula function showed the better fitting results at the lower tail of hazard Hazard factor factors and that three-dimensional Frank Copula function displayed the better fitting results at the middle and Formation mechanism upper tails of hazard factors. However, for dust storm disasters with the short return period, three- Multidimensional return period dimensional joint return period simulation shows no obvious advantage. If the return period is longer than 10 Risk assessment years, it shows significant advantages in extreme value fitting. Therefore, we suggest the multivariate analysis method may be adopted in forecasting and risk analysis of serious disasters with the longer return period, such as earthquake and tsunami. Furthermore, the exploration of this method laid the foundation for the prediction and warning of other nature disasters. © 2015 Elsevier B.V. All rights reserved.

⁎ Corresponding author. E-mail address: [email protected] (S. Yuan).

1. Introduction weighted comprehensive analysis of multivariate variables. Dust storm occurrence mechanism or hazard factors are not considered Dust storm is one kind of weather-based disaster, in which wind from the perspective of disaster. raises a large amount of dust, makes air particularly turbid, and de- As a random event, a dust storm is the complex interaction process creases horizontal visibility below 1 km. Dust storm is one of the stron- among atmosphere, soil, and land surface from its occurrence to devel- gest wind and sand activities. It causes health problems, pollution and opment and ending. In the previous studies of dust storm disasters, haz- huge economic losses in downstream regions, thus leading to direct or ard factors considered relatively simple and mainly selected from the indirect climatic effects (Hamadneh et al., 2015; Tam et al., 2012). The underlying surface. It is because observation data near-surface can ac- arid area in China is one of the regions with the highest frequency and quire easier, and the multifactor study method of dust storm disasters the largest intensity of dust storms in Central Asia (Yang et al., 2009). was not mature. According to the findings of dust storm mechanism, a The majority of northwest China, the entire north China, and the west- dust storm has three basic formation conditions: high wind speed, ern region in northeast China belong to the area with frequent dust abundant dust sources, and unstable atmospheric stratification. The pe- storms. However, the area affected by dust storms is much larger than riodic change, frequency, and intensity of dust storms are closely related the above regions. Especially, severe dust storms characterized by to the large-scale circulation background, local weather systems, and strong wind speed, low visibility, and high dust content can cause seri- underlying surface conditions. The sand and dust exposed on the sur- ous damages to the environment and human beings within a very short face is the dust source. A dust storm requires a strong wind. Meanwhile, time (Hsieh and Liao, 2013). Under the background of the overexploita- the unstable stratification state of atmosphere is an important local tion of land resources, global warming, and shortage of water resources, thermodynamic condition. The three predisposing factors are comple- dust storm has seriously affected local socio-economic development mentary and indispensable (Wang et al., 2005). Therefore, dust storms and ecological environment construction in northwest China. It is an involve several hazard factors. In the study of dust storms, in order to ecological environment problem that cannot be ignored, and also a objectively, accurately, and quickly calculate the return period, enhance hot topic in current research fields of atmospheric science, disaster sci- its extension prediction capability, and improve the accuracy and depth ence, resources, and environmental science. (Natsagdorj et al., 2003; of risk assessment of severe dust storms, it is necessary to trace the dust Wang et al., 2005; Zhou et al., 2013). At present, the risk assessment source and establish the joint distribution model of multi-hazards study and return period calculation of severe dust storms are still diffi- through the long-term observation and the exploration of the relation- cult because severe dust storm disasters are characterized by the com- ship and interaction among various hazard factors. Currently, more dust plex formation mechanism, large influencing area, and huge loss. In storm researchers recognized the importance of multivariate analysis order to reduce economic loss caused by severe dust storm disasters and achieved some results. Based on the comprehensive consideration and improve corresponding risk management level, it is necessary to of multi-hazards, Xu and Chen (2003) calculated the risk index of dust evaluate the return periods timely, accurately, quickly, and objectively, storms in the Tarim Region and applied the index in the study of dust especially for extremely severe dust storms, and establish an effective storm disasters. With the data of dust storms recorded from 1954 to early warning system and dust storm control measures based on dust 2001, Wang et al. (2008) calculated the annual risk of severe dust storm hazard mechanism and monitoring data (Frans et al., 2006). storms in eastern northwest China. J.L. Liu et al. (2012) analyzed the Disaster return period refers to the average recurrence interval of risk of spring sand-dust storm disasters in northwestern China based certain event repeated for many times in several tests. The occurrences on information diffusion method. X.Q. Liu et al. (2012) performed mul- of an independent event Q for different magnitudes q are illustrated in tivariate analysis with the underlying surface, meteorology, and other Fig. 1,LetL denote the time period between any two successive events factors and supplemented the mechanism and risk assessment of dust without consideration of the magnitude, called the interarrival time. storms in northern China. Because hazard factors of dust storms belong The events with a magnitude equal to or greater than any value q, to different distribution types, these hazard factors show the non-linear Q ≥ q, are denoted by ● in Fig. 1, while the events with a magnitude correlation relationship. The non-linear correlation was not fully less than any value q, Q b q, are denoted by ○ in Fig. 1.Hence,the considered in previous studies. time period between two ● is the recurrence interval, denoted by TQ, Using copula theory is a good way to solve these problems. The pre- which is equal to the summation of the interarrival time for all events vious application studies of storms, floods, and droughts with the copula between them. TQ is called the return period for Q ≥ q (Shiau, 2003; theory were focused on the disaster characteristic variables during or Loaiciga and Leipnik, 1996; Shiau et al., 2007). The return period of an after natural disasters, such as storm peak, storm amount and discharge, event is calculated based on the records of the event or an analytical volume and duration of flood, duration and severity of drought. They function, using the statistics of extreme values. Return period allows mainly use disaster characteristic variables to calculate return period the analyst to assess the frequency of occurrence and magnitude of (Schӧlzel and Friederichs, 2008; Shiau and Modarres, 2009; Wong the long-term tendency of the event's extremes (Coles, 2001). Because et al., 2010; Zhang and Singh, 2007a,b). We have discussed the impor- of frequent severe disasters in recent years, the return period becomes tance and shortage of multivariate analysis in nature disasters and pre- a key reference value in disaster risk evaluation and medium- and sented a method to estimate the bivariate joint probability for the long-term risk forecasting and warning (Sorensen, 2000; Aven, 2012). return period estimation and risk analysis (Li et al., 2013). This paper The return period is also a valuable indicator for the development of was focused on the critical hazard factors. From the perspective of the countermeasures of disaster prevention and mitigation, engineering this, we selected the critical hazard factors from 3 spatial levels (under- design, risk management, and even the disaster-related insurance lying surface conditions, 500-hPa atmospheric circulation system and (Botzen et al., 2010; Fekete et al., 2010; Felix, 2014; Gerrard and near-surface meteorological system) to analyze the return period and Petley, 2013). Traditional dust storm return period is calculated based risk of dust storm disasters for the first time. Based on previous results on records of historical dust storm events, univariate analysis or the and the mechanism of dust storm disasters, we expanded the bivariate

Fig. 1. Illustration of the definition of the return period. The occurrences of event Q, ○ denoting Q b q,and● denoting Q ≥ q. 726 X. Liu et al. / Science of the Total Environment 538 (2015) 724–732 joint probability to three-dimensional simulation to improve the calcu- function, three-dimensional copula function is difficult to be established lation accuracy and prediction capability for return period of dust and calculated (Cherubini et al., 2003; Nelsen, 1998). storms. Moreover, we studied severe dust storms in Inner Mongolia, ex- plored the non-linear relationship among hazard factors, and 2.2. Calculation of the return period established the joint distribution of multi-hazards with Frank Copula function. We simulated and analyzed return periods of severe dust N is time sequence length of an event sample, year; n is the number storms and compared the simulated return periods with historic re- of events; L is the intervals time between events, years; E(L)istheaver- cords. The paper supplemented multivariate analysis methods of dust age intervals time of an event within a long time sequence, E(L)=N/n. storms from the perspective of hazard mechanism. The study results For severe dust storms that have occurred, the return period is can increase the calculation accuracy of the return period of dust calculated as storm disasters. In the near future, through the real-time monitoring the factors causing disasters, the study results can provide the basis ðÞ ¼ EL ; ðÞ¼ ½≤ : ð Þ for forecasting and early warning of disasters, especially for extreme di- TX FX x Pr X x 2 1−FX ðÞx sasters, thus facilitating sandstorm disaster prevention and emergency response. For multidimensional joint return period, according to the definition 2. Methods of Copula function, the joint exceedance probability of the two- dimensional variables is calculated as 2.1. Copula function method PXðÞ≥ x; Y ≥ y In this paper, the structure and joint distribution of multi-hazards ¼ 1−FX ðÞx −FY ðÞþy CFðÞX ðÞx ; FY ðÞy ð3Þ were simulated with Copula theory and method. Previous studies indi- ¼ 1−u−v þ CuðÞ; v : cated that with Copula coupling function, various marginal distribution fl structures could be selected according to the situation to establish ex- The two-dimensional joint return period is calculated as ible multivariate distribution and to describe non-linear and asymmetric relationships among various variables. The multivariate analysis ELðÞ ELðÞ ELðÞ method has been applied in multivariate modeling and risk manage- TxðÞ¼; y ¼ ¼ PXðÞ≥ x ∪ Y ≥ y 1−FxðÞ; y 1−CFðÞðÞx ; F ðÞy ment in financial and insurance fields, stochastic simulation and multi- X Y ELðÞ variate analysis of the flood process, the calculation of return period of ¼ : ð4Þ 1−CuðÞ; v major droughts, and the design of marine engineering under extreme sea conditions (Goda and Ren, 2010; Klein et al., 2010; Wong et al., 2010). In this paper, based on previous results of the two-dimensional Joint exceedance probability and return period of three-dimensional model of the coupling function, we carried out the simulation of variables are respectively calculated as follows: three-dimensional Copula function with multi-hazards of dust storms.

PXðÞ≥x; Y ≥y; Z ≥z ¼1−FX ðÞx −FY ðÞy −FZ ðÞþz CFðÞX ðÞx ; FY ðÞy 2.1.1. Basic forms of coupling function þCFðÞþX ðÞx ; FZ ðÞz CFðÞY ðÞy ; FZ ðÞz Copula function was ﬁrst proposed in 1959 and its theoretical foun- −CFðÞX ðÞx ; FY ðÞy ; FZ ðÞz ð5Þ dation was the famous Sklar Theorem (Sklar, 1959): ¼ 1−u−v−w þ CuðÞþ; v CuðÞþ; w CvðÞ; w Firstly, n random variables uniformly distributed within [0, 1] are de- −CuðÞ; v; w ﬁned as X1, X2, …, Xn and their marginal distribution functions are F1(x1), F2(x2), …, Fn(xn). Then the joint distribution function F(x , x , …, x ) can be expressed with an n-Copula function C: ELðÞ ELðÞ 1 2 n TxðÞ¼; y; z ¼ PXðÞ≥x∪Y ≥y∪Z ≥z 1−FxðÞ; y; z ðÞ ðÞ ¼ EL ¼ EL : ð Þ FxðÞ¼1; x2; …; xn CθðÞðF1ðÞx1 ; F2ðÞx2 ; …; FnðÞxn 1Þ 6 1−CFðÞXðÞx ; FYðÞy ; FZðÞz 1−CuðÞ; v; w where Cθ is copula function; θ is the parameter to be determined. If

F1(x1), F2(x2), …, Fn(xn) is continuous, then C has a unique value. 3. Materials According to Eq. (1), the results of various joint probability distributions can be obtained through numerical integration or theoretical 3.1. Study area derivation. Inner Mongolia is located in the arid and semi-arid region with the 2.1.2. Archimedean coupling function high latitude in central Asia. Severe dust storm is one of the most serious Because one-parameter Archimedean coupling function can be eas- natural disasters in Inner Mongolia (Wang et al., 2005). Most of the sur- ily extended to N-variate scenario, it has the widest application scope. face area in Inner Mongolia is arid and covered by sparse vegetation. Various flexible one-parameter Archimedean coupling functions are Nine deserts and sand areas are distributed from west to east in Inner available. During the simulation, optimal Copula functions can be select- Mongolia and provide the favorable underlying surface conditions for ed according to the correlation among variables and the goodness of fit the formation and development of dust storms. Moreover, Mongolian (Wei and Zhang, 2008; Zhang and Singh, 2007a; Grimaldi and Serinaldi, cyclone activity is frequent in the Midwest of Inner Mongolia and is 2006). Table 1 lists four Archimedean functions often used in the natural the only way of the cold air from northwest to south. Frequent strong sciences and engineering fields. Gumbel and Clayton Copula Functions winds provide favorable dynamic conditions for the formation of dust can only be used to describe the positive correlation among variables, storms (Wang et al., 2010; Shen, 2008; Zhou and Wang, 2002). There- while Frank and AMH Copula Functions can be used to describe the pos- fore, the study of risk analysis and prediction of severe dust storms in itive correlation and negative correlation among variables, AMH Func- Inner Mongolia on the basis of the occurrence and development mech- tion is not applicable to describe the highly positive correlation or anism is of great significance to dust storm disaster prevention and negative correlation. Compared with the two-dimensional coupling effective risk management. X. Liu et al. / Science of the Total Environment 538 (2015) 724–732 727

Table 1 Deﬁnition of the four one-parameter of Archimedean Copula Function.

Copula Two-dimensional Copula Three-dimensional Copula

Cθ(u,v) Parameter spaceθ Cθ(u,v,w) Parameter spaceθ

θ θ θ − − 1/ − 1/ θ θ θ 1 ∞ Gumbel exp( [( ln u) +( ln v) ] ) (0, 1] expð−½ð− ln uÞ þð− ln vÞ þð− ln wÞ θ Þ [1, ) −θ −θ − θ − 1/ ∞ −1 − ∞ Clayton (u + v 1) (0, ) ½ð −θ þ −θ þ −θ− Þ θ ; [ 1, ]\{0} max u v w 2 0 ð −θu − Þð −θv − Þ −∞ ∞ ð −θu − Þð −θv − Þð −θw − Þ −∞ ∞ Frank − 1 þ e 1 e 1 ( , )\{0} − 1 ð þ e 1 e 1 e 1 Þ ( , )\{0} ln 1 −θ θ ln 1 2 θ e −1 ðe−θ −1Þ uv − uvw − AMH 1−θð1−uÞð1−vÞ [ 1, 1) 1−θð1−uÞð1−vÞð1−wÞ [ 1, 1) u, v,andw are the marginal distribution functions of variables. According to the joint distribution function expressions of Cθ(u,v)andCθ(u,v,w), corresponding joint probability density 2 3 ∂ Cθ ðu;vÞ ∂ Cθ ðu;v;wÞ θð ; Þ¼ θð ; ; Þ¼ function may be deduced as: c u v ∂u∂v and c u v w ∂u∂v∂w .

3.2. Determination standard of severe dust storms hazard factors are selected and analyzed: 500-hPa atmospheric longitudinal circulation index (I), maximum wind speed in the 10-m high near To explore typical characteristics and hazards of severe dust storms, ground (S), and surface soil moisture (M), which respectively indicate it is necessary to determine whether dust storms show regional charac- 500-hPa atmospheric circulation system, near-surface meteorological teristics through studying the weather process. According to foreign system, and underlying surface conditions. The three hazard factors grading standards of dust storms (Joseph et al., 1980; Middleton, are used to simulate multidimensional joint probability distribution 1986) and combined with actual conditions in China, Xu and Hu and calculate return periods. (1996) and Qian et al. (1997) have brought forward the classification Longitudinal circulation index was first proposed by Rossby in 1939. standard of dust storms in northwest China, which has already been He defined the average geostrophic wind speed in the sea area between verified to be suitable for the operation of China meteorological depart- 35°N and 55°N as the circulation index and applied it in the ment. Therefore, in this paper, we adopted it to judge severe dust storms. aerographical chart. Then, the height difference between the two defi- For the dust storm observed and recorded at only one station, we nite altitude circles on isobaric surface can be calculated to indicate judge its grade according to the classification standard in Table 2. the intensity of the circulation index. When longitudinal circulation For the dust storm observed and recorded at more than one station, index is high, the western longitudinal circulation is the dominant circu- we determine whether it belongs to severe dust storm according to any lation and cold air activities are frequent. This situation is prone to bring one of the following supplementary criteria. Other cases will not be more windy weather and leads to frequent dust storms. Regions A (75°– accepted in the severe dust storm series. 85°E,40°–50°N), B (85°–95°E,50°–60°N) and C (95°–105°E,45°–55°N) shown in Fig. 3 are respectively the generation, accumulation, and en- 1) During the same weather process, there are three or more stations hancement regions of cold air in western path, northwestern path and where severe dust storms break out and recorded simultaneously. northern path. For each severe dust storm event, according to its occur- 2) During the same weather process, there are only two stations where rence path, the geopotential height difference on isobaric surface among severe dust storms break out and recorded, and three or more ten latitudes in the upper air region (A, B or C) within 12 h before the stations where moderate dust storms break out and recorded occurrence of dust storm were calculated to indicate the longitudinal simultaneously. circulation index I (dimensionless) of this severe dust storm process. 3) During the same weather process, there are only one station where The statistical data of longitudinal circulation index are from the “Dust severe dust storms break out and recorded, and five or more stations where moderate dust storms break out and recorded simultaneously.

According to the weather process, all of the 11,275 original meteorological records of 50 stations in Inner Mongolia from 1990 to 2013 are checked in accordance with the above criteria. The result indicates that there are 91 severe dust storms during the period from 1990 to 2013 and that each year has 4 severe dust storms. The occurrence of sandstorm area is related to the synoptic system movement path. In these 91 severe dust storms events, there are three synoptic system movement paths: the western path, the northwestern path, and the northern path, which respectively corresponded to 47, 34, and 10 severe dust storm events and accounted for 51.6%, 37.4%, and 11.0% of the total storm events.

3.3. Factor selection and data sources

Based on the formation mechanism and inﬂuencing mechanism of hazard factors of dust storm disasters (Fig. 2), the following three

Table 2 The classiﬁcation standard of dust storms.

Dust storm Instantaneous maximum wind speed Minimum visibility V (m)

grades fmax (m/s)

Light fmax ≥ 10; or classes 4–6 500 b V ≤ 1000

Moderate fmax ≥ 17; or classes 6–8 200 b V ≤ 500

Severe fmax ≥ 20; or ≥ class 8 V ≤ 200 Note: The 0–12 classes of wind force scale is Beaufort scale. Fig. 2. Illustration of the mechanism of dust storm disaster. 728 X. Liu et al. / Science of the Total Environment 538 (2015) 724–732

Fig. 3. 500-hPa atmospheric statistical area of dust storms on different paths in Inner Mongolia. A, B, and C are the 500-hPa upper statistical area for the western path, northwest path and the north path of dust storms. a: Xilingaole Station b: Wushenzhao Station c: Xianghuangqi Station, 1: Badain Jaran Desert 2: Tengger Desert 3: Mu Us Sandland the 4: Kubuqi Desert 5: Hunshandake Sand.

Weather Process Maps of China” of each year and NCEP 500-hPa same day were collected between 1990 and 2013. Through ADF test Geopotential Height Grid Data of China Meteorological Administration by EViews software, the T values of the variables passed the significant (Kang et al., 2009; Hu, 1990–2012; Jiao, 2000–2012). level of 0.05. In severe dust storm sequence (91 severe dust storm S is the maximum wind speed at the 10-m height near the ground on events in 24 years), longitudinal circulation index I, maximum wind the occurrence day of severe dust storm. The data of severe dust storm speed S, and soil moisture M are continuous random variables and occurrence and corresponding maximum wind speeds are from the their marginal distributions are FI(i), FS(s)andFM(m), respectively. Ac- “Chinese Severe Dust Storm Case Sequence Data Set” in China Meteorolog- cording to the empirical decision, parameter estimation and hypothesis ical Data Sharing Service System, the dust storm spectrum of National testing of Anderson–Darling goodness-of-fit test under the significant Meteorological Information Center and The Dust Storm Spectrum of level of 0.01, their optimal probability distributions and marginal distri- Inner Mongolia Meteorological Bureau from 1990 to 2013. bution types are determined and shown in Table 3. The parameter M is the soil moisture. It refers to the humidity of 10-cm deep surface values were calculated by maximum likelihood estimation method soil and is expressed with the mass fraction of water weight to dry soil (Dubois, 2010). Corresponding marginal distribution curves of the weight. As shown in Fig. 3, the soil moisture values of severe dust storms three variables are shown in Fig. 4. on the west path are from Xilingaole Meteorological Station which located on the path; the values of severe dust storms on the northwest 4.2. Correlation analysis and establishment of the joint distribution path are from Wushenzhao Meteorological Station; the values of severe dust storms on the north path are from Xianghuangqi Meteorological 4.2.1. Correlation analysis Station. Different Copula functions can be used to describe different relevant structures of variables. There are often different correlations among 4. Results and discussion hazard factors of disaster. When Copula coupling function model is used to establish the joint distribution, it is necessary to measure the 4.1. Determination of the marginal distribution of hazard factors correlation between different hazard factors firstly. Correlation coefficients of three hazard factors are shown in Table 4. Through Pearson Dust storm events are stochastic in nature. Statistical modeling is an correlation analysis, all of the three variables passed the significant appropriate way to assess the characteristics of severe dust storms. level of 0.05. The correlation between longitudinal circulation index Severe dust storm events recorded by the meteorological stations and and maximum wind speed passed the significant level of 0.01 by the values of corresponding hazard factor variables recorded on the Kendall's τ rank correlation analysis. Pearson correlation analysis can

Table 3 The probability distribution and marginal distribution of I, S,andM.

Variables Distribution function Parameters h i α− α I Weibull distribution ð j ; ; αÞ¼α ði−mÞ 1 −ð−ði−mÞÞ m =0 f i m s s s exp s s = 19.9114 α− h i α − 1 α ð j ; ; αÞ¼∫ i m −ð−ði−mÞÞ α = 2.1512 FI i m s exp s di s h s i S Extreme value type I distribution ð jμ; λÞ¼1 −ðs−μÞ − −ðs−μÞ μ = 20.6879 f S s λ exp λ exp λ h i λ = 2.8085 ð jμ; λÞ¼∫ 1 −ðs−μÞ − −ðs−μÞ FS s λ exp λ exp λ ds − − M Gamma distribution f ðmjk; s; rÞ¼∫s−r ðy−kÞr 1 exp − y k ΓðrÞ k =0 s − − s = 0.6792 F ðmjk; s; rÞ¼∫s−r ðy−kÞr 1 exp − y k ΓðrÞdm M s r = 9.9256 X. Liu et al. / Science of the Total Environment 538 (2015) 724–732 729

Fig. 4. Comparison between the observed values and the fitted marginal distribution of the three variables. only represent the linear relationships among different variables. Frank joint distribution function is shown in Table 1 and corresponding Kendall's τ rank correlation analysis can not only reveal the linear three-dimensional joint probability density function is provided as relationship, but also reflect the nonlinear correlation (Klein et al., 2010). The significant levels of Kendall's τ correlation coefficients in 3 ∂ CθðÞu; v; w cuðÞ¼; v; w Table 4 indicate that there is nonlinear relationship between the three ∂ ∂ ∂ u v w hi 2 2 variables. θ2e−θðÞuþvþw e−θ−1 e−θ−1 − e−θu−1 e−θv−1 e−θw−1 ð7Þ ¼ hi ðÞe−θ−1 2 þ ðÞe−θu−1 ðÞe−θv−1 ðÞe−θw−1 4.2.2. Establishment and verification of joint distribution Because negative correlation exists among the three hazard factors, according to the applicable conditions of different types of Archimedean where u = FI(i), v = FS(s), w = FM(m). Copula functions, two kinds of Copula functions (Frank Copula and AMH Quantile–Quantile (Q–Q) plots are a popular technique for checking Copula) were selected for fitting. Maximum likelihood method and dis- the fit of stationary extremal distributions, as they are particularly effec- tribution estimation method were used to estimate parameters and cal- tive at highlighting apparent discrepancies in the extreme upper tail culate the evaluation index of goodness-of-fit (RMSE and AIC values). (Katz, 2013). The Q–Q diagrams show the fitting effects of two- The smaller RMSE and AIC values indicate the higher goodness of fitof dimensional Frank Copula and three-dimensional Frank Copula corresponding functions (Goda and Ren, 2010). According to calculation (Fig. 5). Lateral axis is the joint cumulative probability of hazard factors results, RMSE and AIC obtained with Frank function are respectively calculated based on observed data. Longitudinal axis is the theoretical 0.02 and −601.63, while RMSE and AIC obtained with AMH function value of joint cumulative probability calculated by Frank Copula func- are respectively 0.11 and −339.99. Therefore, Frank function has the tion. If the sample points are distributed around the diagonal, it is indi- better fitting results for the three hazard factors of severe dust storms. cated that the fitting results are good. According to the R2 values between the joint cumulative probability based on observed data and the theoretical data calculated by Frank Copula function, the two- Table 4 dimensional fitting results obtained with Frank Copula function are ρ τ Pearson correlation and Kendall's correlation matrix of longitudinal circulation index, slightly better than three-dimensional fitting results obtained with maximum wind speed, and soil moisture. Frank Copula function and the lower tail part shows the better fitting re- ρ ISMτ ISM sults. However, in the middle and upper tail parts, three-dimensional I 1 0.268⁎ −0.266⁎ I 1 0.231⁎⁎ −0.183⁎ fitting results are significantly better than the two-dimensional fitting S 0.268⁎ 1 −0.261⁎ S 0.231⁎⁎ 1 −0.177⁎ results, indicating that three-dimensional Frank Copula allows the − ⁎ − ⁎ − ⁎ − ⁎ M 0.266 0.261 1 M 0.183 0.177 1 better fitting results for the occurrence probability of extreme events. ⁎ Indicates the significant level of 0.05. Therefore, the two-dimensional joint probability distribution can be ⁎⁎ Indicates the significant level of 0.01. established for risk analysis of frequent common disaster events. 730 X. Liu et al. / Science of the Total Environment 538 (2015) 724–732

Fig. 5. Comparison of ﬁtting results obtained with different dimensions of coupling functions.

However, because the extreme values have the higher fitting precision periods. The larger values of the hazard factors indicate the larger differ- requirements for extreme disaster events, it is necessary to establish ence between multivariate return period and univariate return period. multidimensional joint probability distribution for risk analysis. Meanwhile, as shown in Fig. 7, when the return period is less than 10 years, the difference between the calculated three-dimensional return periods and two-dimensional return periods is not significant. However, 4.3. Calculation of multidimensional return period when the return period is more than 10 years, the calculated three- dimensional return period is less than the calculated two-dimensional According to Eq. (4) for joint return period, two-dimensional joint 00 return period. return periods T (I ≥ i ∪ S ≥ s), T' (I ≥ i ∪ M ≤ m)andT ðS≥s∪M≤mÞ 2 2 2 are calculated with the following three two-factor combinations 4.4. Verification (I and S; I and M; S and M). Three-dimensional joint return period

T3(I ≥ i ∪ S ≥ s ∪ M ≤ m) is calculated with the factor combination (I, S, According to the meteorological statistics from 50 meteorological and M) according to Eq. (6). Fig. 6(a) shows the slice map of three- stations in Inner Mongolia in 54 years (1960–2013), there are 236 se- dimensional return period plotted under the conditions: I = 20, I = vere dust storms in total. The consequences caused by different dust 40; S =25m/s,S =40m/s;M = 15%, M = 25%. In the slice map of lon- storm events of the same grade may be quite different. Because the gitudinal circulation index (I = 40), with the increase of wind speed and dust storm events of the same grade may have different hazard bearing the decrease of soil moisture, the joint return period becomes longer. bodies. A dust storm event will cause fewer losses if occurred in sparsely That is to say, under the inﬂuence of the same atmospheric circulation populated areas. While the same dust storm event, if occurred in a place system, the larger near-surface wind speed and the smaller soil mois- with developed economy, dense population and property, will result in ture of the underlying surface indicate the longer corresponding joint huge losses. This is the disaster character of dust storm. For the purpose return period. Other slices obey the same law. Fig. 6(b) shows the of considering the disastrous of severe dust storm events during dust change of two-dimensional joint return periods when soil moisture is storm analysis, it is necessary to comprehensively consider the occur- 15%. rences of severe dust storms, direct economic losses, and ratings criteria Under the given sample values of three hazard factors, the calculated of natural disasters (Zhao and Ma, 1993; Leitch, 2010; Birkmann et al., multivariate joint return period (including two-dimensional and three- 2010). Also, in order to verify the accuracy of the return period calcula- dimensional joint return periods) are smaller than the univariate return tion results of severe dust storms, we need consider the severe dust

Fig. 6. (a) The three-dimensional joint return periods, (b) the two-dimensional joint return periods when M = 15%. X. Liu et al. / Science of the Total Environment 538 (2015) 724–732 731

the calculated three-dimensional return period is closer to the actual return period. The overestimation of return period of extremely severe dust storms easily leads to a reduction of the government's and public's awareness. Hence, their efforts in disaster prevention and mitigation will be weakened, thus negatively inﬂuencing the long-term disaster prevention and mitigation (Apostolakis, 2004; Melanie et al., 2012). Therefore, for the severe disasters with longer return period, the three-dimensional return period have more application values.

5. Conclusions

Through the mechanism analysis of severe dust storm disasters and multi-dimensional simulation of return period based on monitoring data, we drew the following conclusions: In common Archimedean coupling functions, Frank Copula function can be used in the simulation of the positive correlation of hazard factors and the analysis of the negative correlation condition. Frank function is applicable for the establishment of the multi-hazards of severe dust storm, indicating that copula function method can be extended to the application of three-dimensional simulation of severe dust storms. Copula function method also provides a new idea for exploring the multivariate analysis method of natural disasters. Fig. 7. The fitting curve of two-dimensional joint return period and three-dimensional For severe dust storms, two-dimensional Frank coupling function joint return period. allows the better fitting results in the lower tail part and three- dimensional Frank coupling function has the better fitting results in storm events with their direct economic losses and relevant disaster the middle and upper tail parts. For severe dust storm disasters, when grades. The economic loss statistics work of dust storms is very difficult the return period is less than 10 years, the difference between two- because of its wide affected area and multiple losses. Therefore, some dimensional return period and three-dimensional return period is not severe dust storm events can't find any losses records. In this paper, significant. However, when the return period is longer than 10 years, we collected the severe dust storm events with relatively complete two-dimensional return period is relatively longer and three- loss records to verify the accuracy of the return period calculation dimensional return period is more close to the actual return period. If results. the return period is overestimated, the frequency of disasters will be In the 24 years (1990–2013) for the establishment of the distribu- underestimated, thus weakening the government's and public's disaster tion of multi-hazard factors, three severe dust storms reached the prevention and mitigation efforts. major disaster level, respectively on April 6–8, 2001, April 9–10, 2001, In China, several dust storms occurred every year, but fewer ex- and April 9–11, 2006. The average return period is 8 years according tremely severe dust storms occurred. Generally, dust storm disasters to the actual situation. The average value of two-dimensional return pe- belong to the disasters with short return period in China. The fitting ac- riods calculated based on three combinations (I and S; I and M; S and M) curacies of two-dimensional joint distribution and three-dimensional are respectively 8.37 years, 8.53 years, and 8.57 years. The average value joint distribution of severe dust storms show no significant difference. of three-dimensional return periods calculated based on the hazard fac- However, compared with the two-dimensional coupling function, tor combination (I, S,andM) is 8.33 years. The difference between the three-dimensional Copula function is more difficult to be established above calculated values and the actual return period was not significant and calculated. Thus, for the dust storm disasters with short return pe- (Table 5). riod (less than 10 years) and small losses, the accuracy of two- In the 24 years, only one extremely severe dust storm on May 5–6, dimensional joint return period simulation can meet the requirements 1993 reached the mega-disaster level and led to economic loss of 280 of risk management. Three-dimensional joint return period simulation million CNY (Wang, 1994). Two mega-disaster level dust storms oc- has no obvious advantage. For the dust storm disasters with the return curred within 54 years (1960–2013). Therefore, the actual return period period longer than 10 years (like mega-disaster level dust storm disas- of extremely severe dust storm in this level was 27 years. The average ters and other extreme disaster events), three-dimensional simulation value of two-dimensional return periods calculated based on three has obvious advantages in extreme value fitting and the result has the combinations (I and S; I and M; S and M) are respectively 35.10 years, better accuracy. Although the parameter estimation and establishment 36.05 years, and 32.40 years. The average value of three-dimensional re- of the distribution structure with three-dimensional joint function are turn period calculated based on the hazard factors (I, S,andM) is 27.85 more complex, three-dimensional return periods still have more appli- years. For extremely severe dust storms with long return period, the cation values for extremely severe natural disasters in the prediction calculated two-dimensional return periods are relatively longer, while and warning work. Therefore, we suggest that for the disasters with

Table 5 Multidimensional return periods calculated based on different hazard factor combinations.

Dust storm disaster events Actual return periods (year) Two-dimensional return periods (year) Three-dimensional return periods (year)

I&SI&MS&MI&S&M

Major disaster level Apr. 6–8, 2001 8 8.2 8.7 8.3 8.5 Apr. 9–10, 2001 8 9.1 8.5 8.8 8.3 Apr. 9–11, 2006 8 7.8 8.4 8.6 8.2 Average 8 8.37 8.53 8.57 8.33 Mega-disaster level Apr. 10–12, 1979 27 38.2 35.5 29.7 28.3 May 5–6, 1993 27 32.0 36.6 35.1 27.4 Average 27 35.10 36.05 32.40 27.85 732 X. Liu et al. / Science of the Total Environment 538 (2015) 724–732 the longer return period and bigger destructive power, such as earth- Li, N., Liu, X.Q., Xie, W., Wu, J.D., Zhang, P., 2013. The return period analysis of natural disasters with statistical modeling of bivariate joint probability distribution. Risk quake and tsunami, can simulate the return period according to this Anal. 33 (1), 134–145. multivariate analysis method for forecasting and risk analysis. Liu, J.L., Zhou, S.W., Niu, T., Sun, J., Wang, Y., 2012a. Risk analysis of spring sand–dust storm disasters in northwestern China based on information diffusion method. Meteorol.Sci.Technol.40(5),858–864. Acknowledgments Liu, X.Q., Li, N., Xie, W., Wu, J.D., Zhang, P., Ji, Z.H., 2012b. The return periods and risk assessment of severe dust storms in Inner Mongolia with consideration of the main This work was supported by “National Natural Science Foundation of contributing factors. Environ. Monit. Assess. 184 (9), 5471–5485. fl fl China (Grant Nos. 41301583, 41171401, 41306091, and 41306087)”, Loaiciga, H.A., Leipnik, R.B., 1996. Stochastic renewal model of low- ow stream ow sequences. Stoch. Hydrol. Hydraul. 10 (1), 65–85. “Project Supported by the State Key Laboratory of Earth Surface Process- Melanie, S.K., Margreth, K., Kirsten, E., Thomas, G., 2012. Challenges of analyzing es and Resource Ecology (Grant No. 2013-KF-09)”, “the Fundamental multi-hazard risk: a review. Nat. Hazards 64, 1925–1958. ” “ Middleton, N.J., 1986. A geography of dust storms in south-west Asia. J. Climatol. 6, Research Funds for the Central Universities , Postdoctoral Science – ” 183 196. 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