#### 3.1.1. Statistics of Low Temperature Events

The frequency, maximum duration, and mean intensity of low temperature events for each station are presented in

Table 3. Clearly, there are regional difference in frequency. Zhaitang station, with 666 ECE occurrences, had the highest frequency. Generally, low temperature events occurred more often during the period 1978–1998 than during the period 1999–2015. For example, at Yanqing station, 393 events occurred before 1998, while 263 events occurred after 1998.

The maximum duration of ECEs at each station in Beijing averaged 10 d. The longest events were recorded at Miyun station, with a low temperature event lasting 14 days in 1978–1998, and at Pinggu station, with a low temperature event lasting 14 days in 1999–2015. There was little difference in the mean intensity of ECEs among stations. For example, the mean intensity of 148 events at Shunyi station from 1978 to 1998 was −11.53 °C, while the mean intensity of 170 events at Haidian station from 1978 to 1998 was −11.35 °C. For all stations, the mean intensity of events for the period 1999–2015 was lower than for the period 1978–1998. Both frequency and mean intensity were higher from 1978 to 1998 than from 1999 to 2015, which suggests that frequency and intensity of winter low temperature events are in decline.

#### 3.1.2. Joint Distribution of Winter Low Temperature Based on the Copula Function

Yanqing station (ID 54506) was selected as a case study, with the aim to verify computations using the copula function in this study. Yanqing was selected for illustration because some new venues planned for the 2022 Winter Olympic will be built here and the ECEs are important for the Olympic Games. The goodness-of-fit of the marginal distributions for duration and intensity of low temperature events at Yanqing station were evaluated (

Figure 2). Using the K–S test, the optimal probability distributions for duration and intensity for 20 stations in Beijing were determined. These distributions are listed in

Table 4.

There were 393 and 263 low temperature events recorded at Yanqing station during the periods 1978–1998 and 1999–2015, respectively. At a 90% confidence level, the K–S test yielded D values of 0.0822 and 0.1005, respectively [

25]. To model the intensity of these events, all probability distributions performed well, although the GEV distribution best fitted the observed data. However, duration of these events was only modeled by the Weibull distribution, based on the K–S test (

Figure 2).

Similar results were found for the other 19 stations in Beijing (

Table 4). Typically, for event duration, only one or two distributions passed the K–S test, with marginal distributions of duration best fitted by the Weibull distributions at the majority of stations. At a few stations, such as Miyun and Daxing, only the normal distribution passed the K–S test; while at Xiayunling station, only the gamma distribution passed the K–S test for the period 1978–1998. In the case of intensity, the GEV distribution performed well for most stations. At the Haidian station, an EV distribution performed better than other distributions in both study periods. In general, the marginal distributions of duration and intensity of low temperature events were best fitted by the Weibull and GEV distributions, respectively.

After selecting the best-fit marginal distribution functions, the parameters of each copula function were computed. The copula families were determined based on RMSE values. The copula that produced the least error (lowest RMSE) was selected (

Table 5). According to

Table 5, the GH copula provided the best-fit joint distribution for duration and intensity at most stations. The Frank copula was the best-fit for only three stations during the period 1978–1998, but four stations during the period 1999–2015. The Clayton copula distribution was selected for few stations, including Foyeding and Tanghekou stations, suggesting that the distributions of low temperature events may be regionally specific.

For the marginal distributions of the intensity and duration of low temperature events, the joint cumulative distribution function for the Yanqing station is shown in

Figure 3. In the next section, these joint distributions are used to predict the return periods of winter low temperature events.

#### 3.1.3. Return Period and Risk Analysis

Here, we consider only one scenario in our analysis of joint return periods, that is, the duration of events greater than 3 days duration with an intensity below −12 °C but above −15 °C. The results of the joint return period (

Figure 4) suggest there is spatiotemporal variation to the risk of low temperature events within the study area.

Larger joint return periods imply a smaller probability that low temperature events will occur, and vice versa.

Figure 4a,b show similar spatial patterns to return periods for both study periods. The highest low temperature risk was calculated for areas north and west of Beijing. Return periods exhibited a northeast–southwest trend, consistent with the topography of the Beijing region (

Figure 1). This indicates that under the influence of topography, low temperature events occur more frequently in the northern and western mountainous areas of Beijing and less frequently on the plain.

Comparison of

Figure 4a,b shows a general decrease in the risk of low temperature events after 1999, particularly in the central area of Beijing. Low temperature events were more prevalent during the period 1978–1998, with frequencies that were twice as high as during the period 1999–2015. Although there was still a high risk of low temperature events in Huairou, Miyun, and Pinggu from 1999 to 2015, the area of high risk has shrunk.

The spatial patterns for conditional return periods are shown in

Figure 5 and

Figure 6. Two scenarios were considered in our analysis of conditional return periods, that is, (1) with an intensity below −12 °C but above −15 °C, and given duration exceeding 3 days (

Figure 5); and (2) a duration greater than 3 days with a given intensity threshold below −12 °C but above −15 °C (

Figure 6).

The main concentration of low values for the conditional return period

${R}_{\{T<-12|D>3\}}$ was over Huairou, Yanqing, and Mentougou stations (

Figure 5), which means that events with an intensity in the range from −12 °C to −15 °C with a duration exceeding 3 days occurred frequently in these regions. The northern parts of Beijing experienced a shorter conditional return period compared with southeastern parts. The conditional return periods of

${R}_{\{T<-12|D>3\}}$ during the period 1999–2015 were larger than those during the period 1978–1998. The largest differential between study periods occurred at Tongzhou station, with

${R}_{\{T<-12|D>3\}}$ values ranging from 0.35 to 1.77 years.

The spatial distributions for

${R}_{\{T<-12|D>3\}}$(

Figure 5) and

${R}_{\{D>3|T<-12\}}$(

Figure 6) are relatively consistent, implying there was a high probability of concurrence of low temperature events in winter with longer duration and lower temperatures. The results for

${R}_{\{D>3|T<-12\}}$ showed that the southern areas of Beijing had an extended conditional return period, compared with northern areas. Hence, the northern areas of Beijing experienced recurrent events more frequently, compared with southern areas. Comparing

Figure 6a,b, a general increase in conditional return period of

${R}_{\{D>3|T<-12\}}$ after 1999 is observed, particularly at Tongzhou station. However, Miyun, Yanqing, and Mentougou stations experienced shorter return periods during the study period 1999–2015 than during the period 1978–1998, which contrasts with the conditional return period for

${R}_{\{T<-12|D>3\}}$. This indicates that the risk of a low temperature event lasting a long time was higher at Miyun, Yanqing, and Mentougou stations after 1998.

In addition, by comparing the spatial distributions for

${R}_{\{T<-12|D>3\}}$ (

Figure 5) and

${R}_{\{D>3|T<-12\}}$ (

Figure 6), it can be found that the conditional return period of

${R}_{\{D>3|T<-12\}}$ is smaller than that of

${R}_{\{T<-12|D>3\}}$. It indicates that the risk of the first scenario corresponding the conditional return period of

${R}_{\{T<-12|D>3\}}$ (

Figure 5) happening is lower than the risk of the second scenario corresponding to the conditional return period happening.