Snow Disaster Hazard Assessment on the Tibetan Plateau Based on Copula Function

Abstract

In the context of global climate change, the Tibetan Plateau is particularly susceptible to meteorological disasters, including snow disasters. This study utilized daily temperature and precipitation data from 44 meteorological stations on the Tibetan Plateau spanning from 1960 to 2018 to construct a snow event dataset. Optimal marginal distribution and the copula function were chosen to calculate the joint return period and joint probability, which effectively assess the hazard of snow disasters in the region. Additionally, the study analyzed the comprehensive risk of snow disasters under various return periods by integrating social and economic data. The results indicate the following: (1) Based on the five different Archimedean copula functions, the joint return period of an error rate of each station was calculated to be less than 36%, which is significantly lower than the recurrence interval of univariate analysis; (2) High-hazard areas are predominantly concentrated in the northwest region of the Tanggula Mountains and the eastern foothills of the Bayankara Mountains. As the return period increases, the spatial distribution of snow disaster hazard probability shifts gradually from “double-core” to continuous distribution; and (3) the northwestern Karakorum Mountains and Bayankara Mountains are two distinct high-risk areas for snow disasters. The range of high-risk areas in the region expands with an increase in the return period.

1. Introduction

Being a global climate-change-initiation zone, the Tibetan Plateau is highly vulnerable to natural disasters and extreme weather events caused by climate change, and the socio-economic ecosystem of this region is exceptionally sensitive to climate change [1]. Compared to the snow disasters that occur in developed countries in Europe and America [2,3,4], the subject of this research is the livestock snow disaster in the Tibetan Plateau. This type of disaster is a result of low temperatures and persistent snowfall during the winter and spring months. Snow blankets the plateau regions for extended periods, burying the grasslands and resulting in a significant number of livestock experiencing weight loss or death [5]. Consequently, snow disasters represent the most noteworthy meteorological catastrophe that adversely impacts the advancement of animal husbandry in the region [6].

The Tibetan Plateau is a unique ecological region with an average altitude of over 4000 m, characterized by a cold climate and the coexistence of glaciers, permafrost, and snow cover [7]. Due to the limitations of the alpine environment, the Tibetan Plateau region has limited resources and environmental carrying capacity, and its socio-economic development and urbanization level lag behind other areas [8]. Animal husbandry is the primary industry for economic development in the area, and the industrial structure is relatively simple [9]. Over the past 60 years, snow disasters have resulted in the death of over 12 million livestock in the region, causing severe economic losses [10]. The historical literature shows that the frequency and extent of snow disasters have significantly increased since the 1960s, with 366 snow disasters recorded from 1820 to 2009 [11,12]. With the instability of the climate system intensifying due to global climate change [13], the frequency and intensity of snow disasters on the Tibetan Plateau may further increase, exacerbating the region’s already fragile ecological system and socio-economic development [14].

Table 1. A list of significant historical snow disaster events and their respective losses in the Tibetan Plateau.

Years	Location	Losses: Livestock Lost (Numbers)	Reference
1956–1996	Qinghai Province	8,540,000	Li et al. (2018) [15]
1970	Wulan, Tianjun, Delingha of Qinghai	186,900	Shi et al. (2006) [16]
1974–2009	the eastern Tibetan Plateau	1,290,000	Ye et al. (2019) [17]
1975	Dulan, Wulan, Golmud County	100,000	Liu (2014) [18]
1983	Chengduo County	137,000
1983.10	Dari, Maqin County	350,000
1996	the southern Qinghai Province	1,080,000	Li et al. (2018) [15]
1997–1998	Naqu, Tibet	280,000	Wen (2008) [19]
2004	Tongde County of Qinghai Province	57,628	Wang et al. (2013) [20]
2018–2019	Yushu, Guluo and Haidong at the southern Qinghai–Tibetan Plateau	60,000	Liu et al. (2021) [11]

presents a list of significant historical snow disaster events and their respective losses in the Tibetan Plateau, lending strong support to the critical importance and value of snow disaster hazard assessment [15,16,17,18,19,20].

Snow disasters on the Tibetan Plateau are the result of a complex interplay of multiple factors, such as temperature, snowfall, snow depth, snow duration, and affected area, among others. These factors constitute a cluster and chain of hazards that pose a significant threat to the sustainable and stable development of socioeconomic systems [21]. To effectively mitigate regional disaster risk, it is crucial to identify the spatial patterns of snow disaster hazards and assess them through analysis of disaster factors and environmental conditions. Previous studies have used semi-quantitative methods, such as the analytic hierarchy process (AHP), fuzzy integrated evaluation, and Delphi method, to assess snow disaster hazards by selecting multiple causal factors [22,23,24,25]. However, these methods can only qualitatively reflect snow hazards and disaster magnitude. They cannot calculate the probability of snow hazards and losses, making them highly subjective. To quantitatively assess the hazard of snow disasters and facilitate decision makers in making targeted deployments [26], existing scholars have chosen to calculate the probability of disaster factors based on the depth and duration of snow. For instance, Yi evaluated the snow hazard in eastern Inner Mongolia by calculating the cumulative number of days exceeding the critical snow depth under different return periods [27]. Li analyzed the spatial distribution of snow hazard in Hebei Province using extreme value theory, based on annual maximum daily snow depth and snow pressure [28]. Bai used snow depth to represent the hazard of snow disasters and evaluated the risk of snow disaster loss in different return periods in Qinghai Province, combined with vulnerability [29]. Ye constructed a snow disaster event identifier–simulator to calculate the frequency and duration of snow damage based on historical event data to characterize snow hazard on the Tibetan Plateau [17]. While these studies quantitatively analyze snow hazards, most of them are characterized by single disaster factors or historical information, which cannot comprehensively reflect snow hazards. Therefore, to provide a more comprehensive assessment of snow hazards on the Tibetan Plateau, it is necessary to take into account multiple disaster factors and consider the joint probability of these factors.

The copula function, widely used in financial risk management, can portray linear and nonlinear relationships between variables and construct multivariate distributions flexibly, making it suitable for quantitively calculating multi-factor joint probabilities [30]. Therefore, the copula function has been gradually introduced into risk evaluation in the field of hydro-meteorological hazards in recent years. For example, Luo used the copula function to construct the joint distribution of sub-region precipitation and water level and to analyze the risk of flood disaster in the Taihu Lake Basin during the flood season [31]. Sneha used the copula function to build a joint probability model of daily rainfall and water level data and to analyze the comprehensive risk of flood in the tropical coastal district of Alappuzha [32]. Based on the copula function, Wang calculated the joint return period, representing the hazard of agricultural flood, by combining the total duration of extreme precipitation and the total extreme precipitation [33]. To better quantify snow hazards on the Tibetan Plateau, this study uses the copula function model to assess snow hazard from the perspective of the joint multidimensional disaster-causing factors under different return periods. This study focuses on the following three issues: (1) the accuracy of the copula function in assessing snow hazards on the Tibetan Plateau; (2) the spatial distribution characteristics of snow hazard on the Tibetan Plateau under different return periods; and (3) the comprehensive risk assessment of snow disasters on the Tibetan Plateau by combining multiple dimensions of socio-economic data, including livestock population, population density, and industrial structure.

To achieve the research objectives, this study adopted the following work steps: Firstly, multi-dimensional disaster-causing factor data, such as snowfall, temperature, etc., were collected from 44 meteorological stations in the Tibetan Plateau region during the period 1960–2018 and underwent data screening. Secondly, the copula function model in MATLAB2016 was utilized to combine various dimensions of disaster-causing factors, construct multivariate distribution, and quantify the probability and loss of snow disasters. Thirdly, ArcGIS10.2 was used to interpolate and analyze the spatial distribution characteristics of snow disaster hazard in the Tibetan Plateau region under different recurrence periods, describing the spatial distribution of snow disaster hazards. Ultimately, socio-economic data were integrated, and the analytical hierarchy process was applied to comprehensively evaluate the overall risk of snow disasters in the Tibetan Plateau region, providing a scientific basis for government decision makers to formulate disaster prevention and mitigation strategies.

2. Data and Methods

2.1. Study Area Overview

The Tibetan Plateau is situated in the southwest of China and spans from the Pamir Plateau in the west to the Hengduan Mountains in the east and from the southern edge of the Himalayas in the south to the northern side of the Kunlun–Qilian Mountains in the north. Its total area is 2.5724 × 10⁶ km² [34] and can be divided into nine geographic units based on its geological and geomorphological conditions, namely the Bayankara Mts, Hengduan Mts, Himalayas Mts, Karakoram, Kulun Mts, Nying Mts, Qilian Mts, Tanggula Mts, and Qiangtang (Figure 1a). The county boundaries of the Tibetan Plateau region are shown in the following diagram (Figure 1c). The snow-accumulation period on the plateau generally lasts from September to April and lasts for a long time [35]. Snow distribution is influenced by precipitation and topography, resulting in uneven spatial distribution [36]. The majority of snow is concentrated in the southern Himalayas, the Karakoram Mountains in the west, and the Nianqing Tangula Mountains in the southeast [37]. About 93% of the plateau area is covered by meadows, alpine shrubs, and alpine grasslands. Animal husbandry is the primary source of economic income, with approximately 70 million heads of livestock in the region, including about 13 million yaks and 50 million Tibetan sheep [38]. Currently, 96.88% of the towns in the Tibetan Plateau region have populations of less than 50,000, and large, medium, and small cities are underdeveloped and in the middle stage of urbanization [39].

2.2. Data Source and Pre-Processing

The precipitation and temperature data used in this study were obtained from the “China Ground Climate Data Daily Data Set (V3.0)” of the National Meteorological Science Data Center (http://data.cma.cn/, accessed on 2 June 2022), which underwent processing for missing value replacement, spatial outlier test, and uniformity test [40]. To separate snowfall from rainfall, the single-critical-temperature separation method was used after correcting the precipitation data [41].

According to the Chinese national standard for Snow Disaster Grades in Grazing Regions of China (GB/T20482-2017) [42], a snow disaster involves a weather process that includes snowfall, low temperature, and snow accumulation over a certain duration. In this study, we defined a snow event as occurring when the average temperature is less than −5.5 ℃ and the accumulated snowfall at one time is greater than 3 mm [43]. To analyze snow events on the Tibetan Plateau, the average temperature and accumulated snowfall data from October to the following April were extracted from 44 meteorological stations in the region from 1960 to 2018 using the annual maximum method (Figure 1b). The dataset for snow events was used to calculate the average annual frequency (F), average duration (T_AD), and maximum duration (T_MD). Additionally, socio-economic data, such as livestock quantity, gross product, and population density, were obtained from the 2021 statistical yearbooks of Sichuan, Qinghai, Gansu, Tibet, Xinjiang, and Yunnan.

2.3. Methods

2.3.1. Fitting the Edge Distribution Function

To construct the copula joint function, it is essential to select the optimal marginal distribution function for the indicator. In this study, we chose the most commonly used marginal distributions of the GEV (generalized extreme value), EV (extreme value), Gamma, and Weibull in the field of hydrology to fit the temperature and snowfall at each station. To determine the optimal marginal distribution function, we conducted a Kolmogorov–Smirnov (K-S) test with a significance level of 0.05. The parameter estimation was then performed using the maximum likelihood method.

2.3.2. Constructing the Two-Dimensional Copula Function

“Copula” means “connected together” and is a function that connects one-dimensional marginal distributions to form a multivariate distribution on (0,1], which is also a measure of the dependence function of multivariate extreme value theory. The copula function is based on Sklar’s theorem and can be expressed as follows: let x,y be a continuous random variable with marginal distribution functions of $F_X(x)$ and $F_Y(y)$.The joint distribution function is $F(x,y)$. If the marginal distribution functions are continuous, there exists a unique function copula function such that:

$$F(x,y)=C_{\theta }([F_X(x),F_Y(y)])$$

(1)

where $F(x,y)$ is the joint distribution function. $C_{\theta }$ is the copula function with the parameters to be determined as θ. The result of its calculation is the probability of the copula joint distribution function, which is used to characterize the snow hazard (hazard). And in this study, the snow hazard is denoted by H.

Many classes of copula functions are broadly classified into three types: Archimedean, elliptic, and quadratic. Archimedean copula functions are widely used in current research because of their convenient solution and simple construction. The most commonly used ones include Clayton, Frank, Gumbel, Gaussian, and t-copula, which are the five most common copula functions (Table 2).

Table 2. Commonly used two-dimensional copula function expressions.

Name	${\mathit{C}}_{\mathit{\theta }}(\mathit{u},\mathit{v})$
Clayton copula	${(u^{-\theta }+v^{-\theta }-1)}^{-1/\theta }$
Frank copula	$-\frac{1}{\theta }\ln \left[1+\frac{(e^{-\theta u}-1)(e^{-\theta v}-1)}{e^{-\theta }-1}\right]$
Gumbel copula	$\exp \left(-{\left[{(\ln u^{\theta })}^{\theta }+{(-\ln v)}^{\theta }\right]}^{1/\theta }\right)$
Gaussian copula	${\displaystyle {\int }_{-\infty }^{{\Phi }^{-1}(u)}}{\displaystyle {\int }_{-\infty }^{{\Phi }^{-1}(v)}}\frac{1}{2\pi \sqrt{(1-{\rho }^2)}}\exp \left[-\frac{x^2-2\rho xy+y^2}{2(1-{\rho }^2)}\right]dxdy$
t-copula	${\displaystyle {\int }_{-\infty }^{t_{\delta }{}^{-1}(u)}}{\displaystyle {\int }_{-\infty }^{t_{\delta }{}^{-1}(v)}}\frac{1}{2\pi \sqrt{(1-{\rho }^2)}}{\left[1+\frac{x^2-2\rho xy+y^2}{\delta (1-{\rho }^2)}\right]}^{-(\delta +2)/2}dxdy$

Table 2. Commonly used two-dimensional copula function expressions.

Name	${\mathit{C}}_{\mathit{\theta }}(\mathit{u},\mathit{v})$
Clayton copula	${(u^{-\theta }+v^{-\theta }-1)}^{-1/\theta }$
Frank copula	$-\frac{1}{\theta }\ln \left[1+\frac{(e^{-\theta u}-1)(e^{-\theta v}-1)}{e^{-\theta }-1}\right]$
Gumbel copula	$\exp \left(-{\left[{(\ln u^{\theta })}^{\theta }+{(-\ln v)}^{\theta }\right]}^{1/\theta }\right)$
Gaussian copula	${\displaystyle {\int }_{-\infty }^{{\Phi }^{-1}(u)}}{\displaystyle {\int }_{-\infty }^{{\Phi }^{-1}(v)}}\frac{1}{2\pi \sqrt{(1-{\rho }^2)}}\exp \left[-\frac{x^2-2\rho xy+y^2}{2(1-{\rho }^2)}\right]dxdy$
t-copula	${\displaystyle {\int }_{-\infty }^{t_{\delta }{}^{-1}(u)}}{\displaystyle {\int }_{-\infty }^{t_{\delta }{}^{-1}(v)}}\frac{1}{2\pi \sqrt{(1-{\rho }^2)}}{\left[1+\frac{x^2-2\rho xy+y^2}{\delta (1-{\rho }^2)}\right]}^{-(\delta +2)/2}dxdy$

where x and y are random variables and u and v are their corresponding edge distribution function. $C_{\theta }(u,v)$ is the copula function of different parameters ρ, δ and θ.

2.3.3. Fitting Test Methods

Based on the selection of univariate marginal distributions, a combination of qualitative and quantitative methods is employed to construct a two-dimensional joint distribution model and assess the fit of the copula function. The graphical evaluation analysis method is used to visually examine the deviation between the empirical and theoretical frequencies and to qualitatively evaluate the copula function’s fitting effect. In addition, this study uses the Akaike information criteria (AIC), Bayesian information criteria (BIC), and mean square error (MSE) to quantitatively evaluate the goodness-of-fit of the copula function. Smaller values of the AIC, BIC, and MSE indicate a better fit.

The empirical joint distribution probabilities of the sample points can be denoted as follows [44]:

$$F(xi,yi)=P(x\le xi,y\le yi)=\frac{{\displaystyle \sum _{I=1}^i{\displaystyle \sum _{k=1}^i{Q}_{Ik}-0.44}}}{Q+0.12}$$

(2)

where (x_i, y_i), i = 1, 2, …, Q is the observed sample, and Q_IK is the number of x_ij ≤ x_li, x_2j ≤ x_2i after rearranging the sample points.

The goodness-of-fit test formula is written as follows:

$$MSE=\frac{1}{n-1}{\displaystyle \sum _{i=1}^n{(F(xi,yi)-C(ui,vi))}^2}$$

(3)

$$\mathrm{AIC}=-2\log (L)+2k$$

(4)

$$B\mathrm{IC}=-2\log (L)+2k\log (M)$$

(5)

C(u_i,v_i) is the probability of the copula joint distribution function. L is the maximum likelihood function; k is the number of independent parameters in the copula function; and M is the number of samples.

2.3.4. Return Periods

For the extracted snowfall events, we define E(L) as the average interval of events, where N is the length of the time series of snowfall events and n is the number of events.

$$E(L)=\frac{N}{n}$$

(6)

The return period is the average number of years that a similar snow disaster recurs and is generally calculated from the snow disaster causation factor that responds to the disaster, which is used to measure the frequency of snow disaster occurrence. It is generally expressed in T, and the unit is “year”.

The univariate return period is written as follows:

$$T_x=\frac{E(L)}{1-F(x)}$$

(7)

$$Ty=\frac{E(L)}{1-F(x)}$$

(8)

The bivariate joint return period is written as follows:

$$T_{xy}=\frac{E(L)}{1-P(x,y)}=\frac{E(L)}{1-C(F_X(x),F_Y(y)}=\frac{E(L)}{1-C(u,\nu )}$$

(9)

where F_X, F_Y are the marginal distribution functions of precipitation and temperature, respectively; T_x, T_y are the univariate return periods of precipitation and temperature, respectively; T_xy are the joint return periods; and C is the joint distribution function.

The accuracy of the return period directly affects the quality of the risk assessment. Using the snow event dataset, the 75% quantile of temperature and snow values was defined as the threshold for extreme events. The average interval of past snow events was defined as the true recurrence period (T_real) to test the correlation results. The formula for calculating the error rate is as follows:

$$\mathrm{E}\mathrm{r}\mathrm{r}\mathrm{o}\mathrm{r}\mathrm{R}\mathrm{a}\mathrm{t}\mathrm{e}=\frac{T_{or}-T_{real}}{T_{real}}\times 100\%$$

(10)

where T_or represents the joint recurrence period and T_real is the average interval of historical events.

2.3.5. Risk Assessment Method of Snow Disasters

Step 1: Theoretical basis of the snow-disaster-risk-assessment model. Based on the theory of natural disaster risk formation, disaster risk is the result of the combined effects of hazard, sensitivity of disaster-causing environment, vulnerability of affected bodies, and disaster prevention and mitigation capability. The expression is as follows:

$$D=f(QH,QE,QV,QR)$$

(11)

where D, Q_H, Q_E, Q_V, and Q_R represent disaster risk, hazard, sensitivity of disaster-causing environment, vulnerability of affected bodies, and emergency response and recovery capability, respectively.

Step 2: The comprehensive weighted assessment method comprehensively considers the influence of various indicators on the comprehensive evaluation object, synthesizes the degree of the effect of each specific indicator, and quantifies it with a unified indicator. The specific formula is as follows:

$$Q(H,E,V,R)={\displaystyle \sum _{i=1}^{n}Wi\times Dij}$$

(12)

where H, E, V, and R represent the factor indexes of hazard, exposure, vulnerability, and Emergency response and recovery capability, respectively; j is the number of evaluation factors; D_ij is the normalized value of index i for factor j; n is the number of evaluation indicators; and W_i is the weight of indicator i, which comprehensively considers the impact of each indicator on the evaluation factors. There are many methods to determine it. This article uses the analytic hierarchy process (AHP) to determine it.

Step 3: Construction of a comprehensive snow-disaster-risk-assessment model. Finally, by using the natural disaster risk mathematical formula and combining the snow disaster evaluation index system on the Tibetan Plateau, the following comprehensive snow disaster risk index model is established using the comprehensive weighted assessment method:

$$SDRI={HW}_1+{EW}_2+{VW}_3-{RW}_4$$

(13)

SDRI is the snow disaster risk index used to express the degree of snow disaster risk. The larger its value, the greater the level of disaster risk. H, E, V, and R represent the factor indexes of hazard, exposure, vulnerability, and emergency response and recovery capability, respectively. W₁, W₂, W₃, and W₄ represent the weights of each evaluation factor (Figure 2).

Finally, this article uses methods such as Kriging interpolation, raster calculation, and natural break classification in the GIS spatial analysis module to delineate the risk level zones of snow disasters on the Tibetan Plateau (TP) based on the distribution of disaster risk index. The size of the weights is determined by the analytic hierarchy process.

3. Result

3.1. Spatial Distribution of Snow Event Frequency and Duration

The average annual frequency (F) of snow events varied across the TP, ranging from 1.0 to 9.9, with a high value (9.89) in the central region. The F decreased from the east-central part of the plateau to the north and south, indicating a clear spatial pattern. In the northwestern Hengduan Mountains and the eastern foothills of the Bayankara Mountains, six stations had an F of more than five times per year, while twenty-five stations had an F of less than three times per year, mainly located in the southern part of the Hengduan Mountains, Qilian Mountains, and the eastern foothills of the Kunlun Mountains (Figure 3a).

The average duration of snow events (T_AD) at each station (Figure 3b) was less variable than the F, ranging from 1.0 to 3.15 days. There were 23 stations with a T_AD between 1.5 and 2.0 days, most of which were in the eastern part of the TP. The T_AD of seven stations, including Chengduo County, Maduo County, Gonghe County, and Wuwei City, exceeded 2.5 days and were located in the northwestern part of the Hengduan Mountains, the eastern part of the Kunlun Mountains, and the southeastern part of the Qilian Mountains. The maximum duration of snow events (T_MD) at each site (Figure 3c) ranged from 2 to 29 days, with only two stations exceeding 20 days, located in Dingri County in the central Himalayas, Tianzhu County in the eastern Qilian Mountains, and Kargil County in the western part of the Gangdise Mountains. The spatial distribution of T_AD was similar to that of T_MD.

3.2. Optimal Copula Selection

Four stations were selected as examples, namely Qingshuihe in Qinghai, Songpan in Sichuan, Zuogong in Tibet, and Shangri-La in Yunnan. At Qingshuihe, for instance, the largest p-values for temperature and snowfall were 0.98 and 0.73, respectively (Figure 4a,b); at Songpan, the largest p-values for temperature and snowfall were 0.44 and 0.71, respectively (Figure 4c,d); and at Shangri-La, the largest p-values for temperature and snowfall were 0.49 and 0.36, respectively (Figure 4g,h). Therefore, the optimal marginal distribution functions for both temperature and snowfall at these three stations were GEV. For Zuogong, the maximum p-values for the probabilistic fits of the temperature and snowfall distributions were 0.81 and 0.99, respectively (Figure 4e,f). Hence, the GEV and Gamma functions were chosen to fit the temperature and snowfall, respectively. Overall, the GEV distribution function was optimal for temperature at all 44 stations, while the optimal distribution function for snowfall was GEV at 36 stations, and Gamma distributions were used for the rest of the stations.

The scatter plot in Figure 5 compares the theoretical joint frequency values with the empirical joint frequency values. The distribution of the frequency points is even around the 45° line, which indicates that there is good agreement between the theoretical and empirical frequency values.

Table 3. Parameters and goodness-of-fit tests for the copula function.

Sites	Copula	Parameters	AIC	BIC	Preferred Copula
Qingshuihe, Qinghai (56034)	Gaussian	0.9886	14.9897	16.3220	Frank
	t-copula	0.9885	16.9865	19.6509
	Gumbel	1.0000	15.3615	16.6937
	Clayton	0.0000	15.3615	16.6937
	Frank	−0.8249	14.8092	16.1414
Songpan, Sichuan (56182)	Gaussian	0.9035	13.2123	14.9500	Gumbel
	t-copula	0.9017	15.1592	18.6345
	Gumbel	1.2064	12.9385	14.6761
	Clayton	0.6347	13.7011	15.4388
	Frank	1.7411	13.1449	14.8825
Zuogong, Tibet (56331)	Gaussian	0.9436	12.5612	13.1262	Gaussian
	t-copula	0.9428	14.5591	15.6890
	Gumbel	1.0000	12.7084	13.2734
	Clayton	0.0000	12.7084	13.2734
	Frank	−1.3921	12.6459	13.2109
Shangri-La, Yunnan (56543)	Gaussian	0.7249	12.1527	13.1972	Clayton
	t-copula	0.7294	14.1678	16.2569
	Gumbel	1.5232	12.1210	13.1655
	Clayton	0.8380	11.9078	12.9523
	Frank	3.3284	12.0473	13.0918

summarizes the AIC and BIC values of different copula functions for the joint distribution between temperature and snowfall at four example stations. The Frank copula function has the smallest AIC and BIC values at Qingshuihe station, indicating that it provides a better quantitative description of the joint distribution than the other copulas. At Songpan, Zuogong, and Shangri-La stations, the optimal copula functions are the Gumbel, Gaussian, and Clayton copulas, respectively, based on their smallest AIC and BIC values. Among the 44 stations studied, 24, 13, 4, 2, and 1 stations had the optimal copula functions of Clayton, Gumbel, Frank, Gaussian, and t-copula, respectively.

3.3. Error Rate Test

Table 4. Error rate and snow event thresholds.

Sites	Snow Event Thresholds		Return Period		Error Rate/%
	Temperature/°C	Snowfall/mm	${\mathit{T}}_{\mathit{r}\mathit{e}\mathit{a}\mathit{l}}$	${\mathit{T}}_{\mathit{o}\mathit{r}}$
Qingshuihe, Qinghai	−21.00	16.31	1.86	2.24	20.64
Songpan, Sichuan	−6.30	5.37	1.74	1.48	14.71
Zuogong, Tibet	−6.00	4.96	4.57	4.75	3.85
Shangri-La, Yunnan	−6.30	7.24	3.62	3.09	14.44

summarizes the error rates of joint return periods calculated using the copula functions for snow risk assessment at 44 meteorological stations on the TP. The highest error rate of 20.64% was found at Qinghai Qingshuihe, while the lowest error rate of 3.85% was observed at Zuogong in Tibet. Only five stations had joint return periods that were greater than the true return period. Across all 44 stations, the joint return period error rate was below 36%, indicating that the copula functions constructed for snow risk assessment on the TP are highly accurate.

3.4. Snow Hazard Analysis

Based on the copula function’s joint cumulative probability, the distribution of snow hazards was analyzed under different joint return periods, and spatial differences were examined. For example, during the 5-year return period, the overall hazard was found to be low, located in the southeast of Qinghai Province, northwestern Sichuan, and part of the northwest of the Tanggula Mountains at the eastern foothills of the Bayankara Mountains. The hazard index (H) for these areas ranged between 0.6 and 0.7, indicating these areas to be relatively high hazard, and the total area was approximately 3.7 × 10⁵ km². The surrounding areas had a gradually decreasing H, with most areas in the central-eastern and western edges of the TP having an H from 0.4 to 0.6, accounting for 47.74% of the region’s area. The H less than 0.4 was mainly located on the edge of the TP and a narrow strip of the mid-west (Figure 6a).

Under the 10-year return period, the H increased to 0.7–0.8 in Dieme and Jiuzhaigou, situated at the eastern foothills of the Bayankara Mountains, making it high hazard with an area coverage of approximately 1 × 10⁴ km². The range of lower-hazard areas with H between 0.5 and 0.7 expanded significantly and encompassed most of the TP, accounting for 77% of the total area. However, the low-hazard areas with H below 0.4 were only distributed in the northern and southern margins of the TP, with an area of about 1.04 × 10⁵ km² (Figure 6b).

Under the 20-year return period, the overall snow hazard continued to increase. Nanping, Diebe, and Zhouqu, located at the eastern foothills of the Bayankara Mountains, had H values of 0.9 or more, making them high hazard, covering an area of about 7500 km². Areas with H between 0.8 and 0.9 were mainly distributed in the central Tibetan Plateau, including the vicinity of the Bayankara Mountains, Tanggula Mountains, and Qiangtang Plateau, accounting for about 74% of the total area. Lower-hazard areas were only distributed in the northern Tibetan plateau and southern Himalayan fringe areas. Among them, the lowest hazard was found in some areas of Lushui County, Yunnan, where H was only 0.5 (Figure 6c).

Under the 50-year return period, the snow hazard on the TP is generally very high, with high-hazard areas covering about 90% of the total area, with an H value of 0.9 or more. The higher-hazard areas are only distributed in the northern city of Yumen and the southern fringe of the TP, where the lowest H is also greater than 0.8 (Figure 6d).

In general, during winter and spring, water vapor flows from the southwest and southeast and converges at low elevations above the regions around 30°N, before spreading out. The southwestern water vapor flow moves across the Hengduan Mountains and then rises against the Bayankara Mountains [14]. Therefore, the northwestern Tanggula Mountains and the eastern foothills of the Bayankara Mountains are more prone to snowfall, which is distributed in two high-value areas. The hazard is lowest in the central and western Tibetan Plateau, the Qaidam Basin, and the southern edge of the TP. As the return period increases, the spatial distribution of snow hazard probability gradually shows a continuous distribution from a “double-core” to an entire higher-hazard area with H greater than 0.82 in the 50-year return period.

3.5. Snow Disaster Risk Analysis

3.5.1. Exposure Analysis

For snow disasters, both humans and livestock are vulnerable groups. Snow disasters often cause frostbite to herdsmen, and severe snow disasters can also cause them to suffer from snow blindness, influenza, and gastrointestinal diseases [45]. At the same time, grass is buried to varying degrees by snow disasters, causing a rapid decline in the condition of livestock. In combination with low temperature and cold weather, this leads to a significant increase in the mortality rate of livestock [46]. Usually, in areas with larger population density and livestock numbers, the E value is higher. Therefore, population density (E₁) and number of livestock (E₂) were selected as indicators of E and divided into five levels to obtain zoning results [14] (Figure 7).

$$E=0.05E_1+0.05E_2$$

(14)

The results showed that areas with higher exposure were mainly concentrated in Xining City on the southeastern edge of the Qilian Mountains; Xiahe, Hezuo, Tongren, and Maqin on the eastern foothills of the Bayan Har Mountains; Jianchuan in the southeastern part of the Hengduan Mountains; and places to the north of the Himalayas, such as Jiangzi, Bailang, Renbu, and Sangzhuzi District. The E value gradually decreased in the surrounding areas, with the lowest mainly distributed in parts of Qaidam Basin, Kunlun Mountains, Tanggula Mountains, and Qiangtang Plateau.

3.5.2. Vulnerability Analysis

Under unchanged climatic conditions, when terrain and snow disasters combine, the terrain and landform conditions of the disaster-prone environment and the occurrence of secondary disasters can be exacerbated or weakened in most cases. Factors such as altitude, aspect, slope, and terrain shielding have a more significant impact [15,47]. After comprehensive consideration of the terrain and landform of the TP, altitude (V₁), slope (V₂), and aspect (V₃) were selected as vulnerability indicators and divided into five levels, and the zoning results were obtained (Figure 8).

$$V=0.06V_1+0.02V_2+0.02V_3$$

(15)

The areas with a high V level are mainly concentrated near the Himalayas and Karakoram Mountains, with relatively higher levels near the Qiangtang Plateau. The Qaidam Basin and the marginal areas in the eastern and southeastern parts of the TP belong to the low-level areas. The results show that areas with high V have higher altitudes, followed by areas with medium V, and areas with low V have the lowest altitude in the entire region. This indicates that high-altitude areas are more susceptible to snow disaster hazards, while low-altitude areas are less susceptible.

3.5.3. Emergency Response and Recovery Capability Analysis

Disaster prevention and mitigation capacity refers to the strength of disaster-prone areas’ ability to resist snow disasters and recover after disasters. In this paper, per capita GDP (R₁) and the first industry output value (R₂) were used to characterize the region’s economic development level and disaster resistance ability [48], and the results were divided into five levels to obtain zoning results (Figure 9). The larger the per capita GDP and the first industry output value, the higher the level of economic development and disaster-resistance ability of the region; this is in contrast to weak disaster-resistance ability [49].

$$R=0.05R_1+0.05R_2$$

(16)

The results showed that R was higher in the northern and eastern parts of the TP and in some parts of the northwest region. The highest level was found in Mangya City and the Hainan Tibetan Autonomous Prefecture south of the Kunlun Mountains, followed by Golmud City, Delingha City, Maqin, and Ledu in the northeastern direction of Bayankara Mountains, and sporadically distributed in Sangzhuzi District north of the Himalayas and in Yecheng near the Karakoram Mountains. R gradually decreased in the surrounding areas, and economic development in vast areas of the Qiangtang Plateau in the central part of the TP was slow, resulting in lower R.

3.5.4. Comprehensive Snow Disaster Risk Analysis

SDRI is the snow disaster risk index, which is used to represent the degree of snow disaster risk. The greater the value, the higher the risk of disasters. The weights of each evaluation factor, namely W₁, W₂, W₃, and W₄, were 0.7, 0.1, 0.1, and 0.1, respectively, determined by the analytic hierarchy process. The risk results were divided into five levels to obtain the snow disaster risk distribution map (Figure 10).

The results show that, under a return period of 5 years, the higher-risk-level areas with the SDRI in the range of 0.6–0.7, are mainly distributed in Wuchia and Aktau counties near the Karakoram Mountains, as well as north of the Tanggula Mountains, the eastern part of the Qiangtang Plateau, and around the Bayankara Mountains, covering an area of approximately 34.8%. Risk gradually decreases towards the edge and the areas with a moderate-risk level and with the SDRI in the range of 0.5–0.6, as well as those with a low-risk level and with the SDRI in the range of 0.4–0.5, covering 44.8% and 17.2% of the total area, respectively. Low-risk areas are mainly concentrated in the marginal areas of the northwest and southeast parts of the TP, covering an area of only 5.84 × 10⁵ km².

Under a return period of 10 years, a high-risk zone with the SDRI exceeding 0.7 appears in Wuchia and Aktau counties near the Karakoram Mountains, covering an area of approximately 4.6 × 10⁴ km². The high-risk and moderate-risk areas change significantly with the area of risk level ranging from 0.6 to 0.65 expanding to the western edge of the Himalayas, covering about 64.4% of the total area. The area of moderate and low-risk level decreases gradually in the southeastern and western edges, covering about 32% of the area.

Under a return period of 20 years, the range of low-risk areas is almost zero. High-risk level areas are mainly distributed in Wuchia and Aktau counties near the Karakoram Mountains, with little change in range. The areas with a relatively higher risk level to the west of the Himalayas expand. The range of the moderate-risk level areas decreases from 7.3 × 10⁵ km² to 6.4 × 10⁵ km².

Under a return period of 50 years, most areas are in the higher risk level. The risk of snow disasters further expands in Cona County to the east of the Himalayas and in Shangri-La City, Yunnan.

To put it briefly, there are two areas with a high risk of snow disasters situated in the northwest of the Karakoram Mountains and the surrounding region of the Bayankala Mountains. These areas include counties such as Maqin, Chengduo, Madou, Golmud, and Wuqia. The higher risk around the Bayankala Mountains is mainly due to the elevated hazard of snow disasters, while in the northwest of the Karakoram Mountains, the risk arises from higher exposure and vulnerability. As the return period increases, the range of areas in the low-risk category gradually decreases, mainly distributed along the western and southeastern margins of the TP. Overall, the risk in inland high-altitude areas is much higher than that in marginal regions.

4. Discussion

Snow disasters are among the most significant meteorological hazards on the TP, and their spatial and temporal characteristics, as well as the factors that influence them, have become increasingly recognized in recent years. In this study, a dataset of snow disaster events spanning 59 years was developed based on temperature and precipitation data gathered from 44 meteorological stations. By identifying the optimal univariate and bivariate joint distribution models for each station, and by taking into account factors such as exposure, vulnerability, and disaster prevention and mitigation capacity, the snow disaster risk was comprehensively assessed. Analysis of the spatial distribution of historical snow disaster events revealed an increased frequency of these disasters near the Bayankala Mountains. Additionally, snow disasters with longer durations were also found to be more common around the Bayankara and Karakoram Mountains, which corresponds with the distribution of snow disaster risk. In these expansive areas, water vapor flows from both the southwest and southeast and converges at low elevations during winter and spring, especially in regions around 30° N latitude, before spreading outwards. The southwestern flow of water vapor moves across the Hengduan Mountains and then collides against the high Bayankara Mountains. Consequently, snow disasters are more likely to happen at the southern edge and eastern foothills of the Bayankara Mountains [14]. The location variation of the subtropical high in winter may be a significant factor in influencing snow disasters in these areas [50]. Both the frequency and intensity of historical snow disasters are therefore relatively high in this region. In contrast, locations in the hinterlands of the Qiangtang Plateau and Qaidam Basin receive little water vapor due to the encompassing high mountains, resulting in lower snow coverage and relatively lower hazards [51,52].

Based on the risk assessment results, the areas with high risks are located in the northwest of the Karakoram Mountains, the northern part of the Tanggula Mountains, and the surrounding region of the Bayankara Mountains. The western part of the Himalayas is the second highest risk zone, while the risks in the southeastern edge of the Qaidam Basin are relatively lower. These findings are in agreement with previous research results [5,10,14,53,54]. Apart from the direct impact of local climatic conditions [55], factors such as atmospheric circulation are also pertinent to the incidence of snow disasters in the TP. Regarding the impact of weather conditions, the Arctic Oscillation (AO) has the most notable control over the TP region during spring and winter seasons [56]. For example, when the AO is positively modulated, the weakening of the Baikal Ridge results in the southward movement of cold air, which, upon encountering warm and moist air from low-latitude regions in the eastern part of the TP, generates an environment conducive to increased snowfall and a higher likelihood of snow disasters [57]. Additionally, the mid-latitude westerly jet stream is another crucial factor controlling the seasonal snow accumulation on the plateau, particularly in the northern and western TP. Over long-term scales from November to the subsequent February, the southwestward shift of the upper tropospheric westerly jet may have favored the development of more intense surface cyclones over the TP, which is favorable for heavier snowfall, leading to an increase in snow depth over the TP. However, the interannual variation of snowfall in the TP region is also regulated by the ocean–atmosphere coupling system, such as El Niño-Southern Oscillation (ENSO) and sea-surface temperature (SST), which have an impact on the occurrence of snow disasters [57,58,59]. Furthermore, research has found that as the pure income of herders’ families increased and infrastructure such as shelters improved, continuous socioeconomic development led to a decrease in annual livestock losses caused by snow disasters during the period from 1981 to 2015, at the rate of −7% year⁻¹ [17]. Undoubtedly, socioeconomic development will have an impact on the results of snow disaster risk assessment [15].

Accurately assessing the risk of snow disasters relies heavily on establishing a systematic assessment indicator system and constructing a suitable risk assessment model [60]. This article employs copula functions to calculate the probability of snow disaster risk and determine the optimal distribution for each site to better express the regional site differences and enhance the accuracy of probability calculations for selecting snow-disaster-event-distribution references. In addition, by selecting historical snow disaster events and comparing the joint return period with the historical return period, it was found that the joint return period greatly improved the accuracy of return period estimates at various stations. Therefore, copula functions should be considered for the multivariate evaluation of snow disaster risk in future evaluations.

When conducting a snow risk assessment, the hazard index usually consists of multiple factors. For instance, Wang et al. used annual snow occurrence probability, snow cover duration, and the number of days with low temperatures as indicators for snow risk assessment [61]. Similarly, Yang et al. used snow frequency, snow intensity, maximum annual snow days, and maximum annual snow depth as snow hazard indicators, and utilized expert scoring to assign weights for calculating the results [62]. In comparison, Ma selected six meteorological factors to measure snow disasters [63]. However, the combined effect of snowfall, snowpack, and temperature is the primary cause of snowstorms [64]. Therefore, it is more accurate to comprehensively assess snow hazard by combining relevant indicators of snowfall, snow accumulation, and temperature. Although copula functions were used in this study to objectively evaluate snow hazard, it only considered two factors of snowfall and low temperature, without incorporating snow-accumulation-related indexes [65,66,67]. Including these indexes would lead to more comprehensive snow hazard analysis, and the use of three-dimensional copulas, such as floods and droughts, could further enhance the accuracy of snow risk assessments.

5. Conclusions

This study used historical meteorological data combined with copula functions to calculate the probability of snow disasters occurring under different return periods. By using the analytic hierarchy process combined with the comprehensive risk formula, based on hazard, exposure, vulnerability, and emergency response and recovery capability, risk assessment analysis was carried out, and the following main conclusions were drawn:

(1)

The average annual occurrence frequency of snow disasters at each station on the TP is between 1.0 and 9.9, decreasing from the central-eastern region of the plateau to the north and south. The average duration of snow disasters and the historical maximum duration are between 1.0 and 3.15 days and between 2 and 29 days, respectively, with the overall spatial distribution of the two being relatively consistent.

(2)

Compared with the univariate return period, the joint return period calculated using the copula function has practical value and can improve the accuracy of hazard assessment. The true return period and joint return period were calculated for 44 stations, and the conclusion was reached that the error rate of joint return period for each station was less than 36%.

(3)

The northwest region of the Tanggula Mountains and the eastern foothills of the Bayankara Mountains are high-hazard snow disaster areas, while the central and western parts of the TP, Qaidam Basin, and the southern edge of the TP are the areas with the lowest hazard of snow disasters. As the return period increases, the spatial distribution of snow disaster probability gradually changes from “dual-core” to continuous distribution. In the case of a return period of 50 years, the whole region is considered high hazard.

(4)

Generally speaking, the risk of inland high-altitude areas is much higher than that of marginal areas. This study identified two high-risk areas around the northwestern Kunlun Mountains and the Bayankara Mountains. As the return period increases, the area of low-risk areas gradually decreases, mainly concentrated in the western and southeastern marginal areas of the TP.

The introduction of the copula model for snow disaster hazard calculation in this study is innovative, and the return period of the copula model can provide scientific support for the production and life of local herdsmen, improve disaster prevention capabilities, and formulate risk response strategies to alleviate potential disasters. However, there are still some deficiencies. For example, in variable selection, copula functions can be used for research on multivariate snow disasters from a three-dimensional or even higher-dimensional perspective in the future. In terms of research methods, more detailed indicators and data will be supplemented, and fine risk assessment methods will be further explored.