Perspectives on Compound Flooding in Chinese Estuary Regions

Since we are interested in estuary regions along the Chinese coast prone to coastal flooding, we need to select estuary locations for which catchment information, surge, and rainfall data are available. Hence, we simultaneously looked at the relative locations of catchment boundaries, rainfall gauges, and surge data. Then, we selected catchments discharging directly into the sea for which at least one rain gauge within the catchment boundary and one surge station in the vicinity of the catchment outlet could be found. Following this approach, we identified 26 hydrological catchments across the entire Chinese coastline (Fig. 1).

The selected catchments across the coastline have lengths up to 100 km and areas up to 1000 km², except catchment C5 that has a length of ~224 km and an area of ~2876 km² (Fig. 2a and Fig. 2b). China’s coastline covers approximately 14,500 km and a range of climates. The average annual temperature in the catchments ranges from 10.3 °C (north) to 24.8 °C (south) (Fig. 2c). Figure 2d shows that annual rainfall across the 26 catchments ranges from 575.6 mm (north) to 2183.5 mm (south). For those catchments containing more than one rainfall gauge (8 among the 26 selected), the weighted average of the station rainfall was considered. The weight was estimated based on the distance to the catchment outlet. More specifically, the closest gauge has the highest weight. A small variability between rainfall peaks of stations in the same catchments is observed.

To develop the copula-based framework, we constructed a dataset of pairs of surge peaks and corresponding rainfall potentially leading to compound flooding. First, we sampled surge peaks from the daily surge reanalysis data following the peak-over-threshold approach. To ensure independence between peaks, only peaks spaced at least three days apart were considered (Feng et al. 2018). Then, we ran a preliminary analysis on the correlation between surge peaks and accumulated rainfall over a window of time ranging from the day of the occurrence of the peak to seven days prior to it to investigate the lead time in the response of the catchments to a rainfall event.³ Based on this analysis, we selected the daily rainfall recorded on the day of the occurrence of the surge peak as the rainfall event, since this combination of surge peak and corresponding rainfall gives the highest correlation for most of the catchments. These pairs of surge and corresponding rainfall were then used to determine a copula-based framework for compound flooding.

We selected surge peaks following the peak over threshold (POT) approach. According to Pickands III (1975) and Coles et al. (2001), given a random variable X, the excess is \({X}_{ex}=x-u\), defined above a large enough threshold \(u\). In order to ensure the independence of the excesses, a declustering technique was also implemented to ensure the independence of excesses (Antonini et al. 2019). The threshold \(u\) was selected based on the empirical mean residual life (MRL) and dispersion index (DI). The MRL provides a visual diagnostic tool to select the threshold \(u\) in which the average mean of \({X}_{ex}\) is plotted as a function of \(u\). A suitable \(u\) was selected where the plot is approximately linear. On the other hand, the DI, being the ratio between the variance and the mean of the yearly occurrence of the peaks, was used to check whether the peaks above a given threshold \(u\) are independent and follow a Poisson distribution. The combination of MRL and DI can provide an indication on a suitable \(u\).

Copula functions have gained popularity in hydrological studies following the work of De Michele and Salvadori (2003). The advantage of copula functions is their ability to model the dependence between two (or more) random variables independently of their marginal distributions (Sklar 1973). Specifically, given two random variables \(X\) and \(Y\) with continuous marginal distributions \(F\left(x\right)\) and \(G\left(y\right)\), the joint distribution of \(X\) and \(Y\) is given bywhere \(C\) is the copula function and \(u=F(x)\) and \(v=G(y)\) are uniform margins.

In this study, we restricted our analysis to the most used copulas, namely Gaussian, Clayton, Frank, and Gumbel, as the candidate functions for modeling the joint probability distributions of compound events. This is because they are all one-parameter copulas that can cover different types of dependence behavior, that is, symmetry in the data (Gaussian and Frank copulas) or greater association between variables at the tails, that is, Clayton can capture lower tail dependence and Gumbel upper tail dependence. Moreover, when the sample size of investigated pairs is small, more complex copulas might not provide any advantage. The parameters of each copula function were estimated based on the maximum likelihood estimation (MLE) method (Haseeb 2013). In order to select the copula that best fits the data, the following metrics were considered: the Akaike information criterion (AIC), Bayesian information criterion (BIC), and root mean square error (RMSE).

The AIC and BIC are defined as follows:where \(K\) is the number of estimated parameters in the model including the intercept and \(L(\overline{\vartheta }| y)\) is the log-likelihood at its maximum point of the estimated model; \(n\) is the sample size. The smaller the AIC and BIC are, the better the fit (Burnham and Anderson 2004).

The RMSE is defined as:where \(n\) is the number of pairs \((u,v)\); \({C}_{C}\) is the theoretical copula; and \({C}_{n}\) is the empirical copula. A low RMSE value indicates that the simulated and observed data are close to each other, showing a better accuracy.

A probabilistic dependence model, like copula function, can be used to infer a design value based on the probability of occurrence of the event of interest. When dealing with dependent variables, the (statistical) occurrence of an event, for example, a flood event, is defined in relation to the occurrence of its physical drivers, that is, \(X\) and \(Y\) (or \(U\) and \(V\) in the copula domain).

In the bivariate case, the probability of occurrence of the flood event of interest depends on whether both the physical drivers exceed their respective thresholds \({u}_{d}\) and \({v}_{d}\) (AND scenario, Eq. 5) or whether either one or both physical drivers exceed their respective thresholds (OR scenario, Eq. 6) (Salvadori and De Michele 2004). The “AND” scenario is frequently employed in the studies of compound flooding, as flooding is often caused by a combination of excessive runoff and high sea levels, rather than by either factor alone.

For a fixed probability of exceedance, or return period, the pair of variables \(\left({u}_{d},{v}_{d}\right)\) can be inferred following the “most-likely method,” which selects the pair with the highest density, hence the method’s name.

However, in some cases, one of the two physical drivers is the most important from a design perspective. For example, higher storm surges, which determine hydraulic loads on coastal and offshore structures, often result from intense typhoons in southeastern China. Higher sea water reduces the discharge capabilities of inland drainage systems, potentially causing pluvial flooding. In such a case, a dependent model can be used to infer the design value of one physical driver, for example, rainfall, when the other one, for example, storm surge, is fixed via conditional probability.

We selected surge peaks greater than the thresholds \(u\) based on the MRL and DI plots. The thresholds (0.33–1.39 m) in all the catchments range between the 99th and 99.4th percentile of observed surges. The threshold \(u\) around Hainan Island is generally lower than other stations because the western side of Hainan Island (South China), and the mainland across the strait, are sheltered by Hainan Island itself, reducing the surge (Zhang et al. 2017). In addition, the eastern side of Hainan Island has a narrower continental shelf than the rest of the Chinese coast. Surge is proportional to the width of the continental shelf over which the wind blows. In general, a narrow continental shelf results in a smaller surge than a wider shelf. The highest threshold obtained is around Shanghai, at 1.3 m. A declustering time of three days is considered (Feng et al. 2018) to ensure independence between events selected. In the end, we obtained an average of 53 excesses in the 26 catchments over a period of 36 years. The average amount of peaks/pairs is 1.5 per year for all catchments. The maximum number of peaks/pairs is located in catchment D7 and the minimum number of peaks/pairs is in catchment A3. For each excess, we selected the total daily rainfall recorded on the same day to form pairs of surge and rainfall. We quantified the dependence between entire excess pairs of surge peak and rainfall via Kendall’s rank correlation coefficient \(\tau \) significance test (Hauke and Kossowski 2011) (Fig. 3a). We detected a significant correlation (with a level of significance \(\alpha \) = 0.05) in 10 of the 26 catchments analyzed, especially in southern China (Fig. 3b). Catchments A2, A3, A4, A5 (south), B1, B2, B3 (center), C4, D3 and D4 (north) show a significant correlation between surge and rainfall (with a level of significance \(\alpha \) = 0.05). Eastern Hainan Island (catchment A2) shows the highest correlation (\(\tau \) = 0.35). This is reasonable since Hainan Island is highly affected by typhoons and this implies that extreme surge conditions are often accompanied by high rainfall during typhoon events. We also find statistical dependence between rainfall and surge in three catchments around the Bohai Sea (C4, D3, and D4 gauges, Fig. 3b). Northern China is affected by extra-tropical storms outside the typhoon season (mainly in the late winter and early spring), which can often cause flooding and damage.

To better understand the influence of seasonality, we count the number of events in the typhoon season (July to October) and the non-typhoon season (November to June of the following year) separately. Figure 3c shows the number of compound events in the typhoon season and the non-typhoon season. After an analysis of the surge peak events, we found that in the south, most of the compound events occur in the typhoon season while in the north, most of the compound events occur during the non-typhoon season.

To further analyze the dependence between surge and rainfall and the implication of such dependence on infrastructure design, we focus the following analysis on two locations: Hainan Island (catchment A2) and the Bohai Sea (catchment C4). These locations show positive and significant correlations between storm surge and rainfall. However, in catchment A2 (\(\tau \) = 0.35) the highest number of surge peaks (46) were observed during typhoon season, while in catchment C4 (\(\tau \) = 0.24) the highest number of surge peaks were observed during the non-typhoon season (36). Based on the results in terms of AIC, BIC, and RMSE (Table 1), the copulas that best fit the observations are Frank for catchment A2 and Gumbel for catchment C4. Here, empirical margins are assumed.

Following the most-likely approach and based on the “AND” hazard scenario, that is, both rainfall and surge should be above their respective thresholds, Fig. 4 shows the pairs of surge and rainfall corresponding to return periods of 50, 100, and 200 years. The design events (yellow squares in Fig. 4) are given by the point of maximum relative probability density (most-likely method) on the isoline associated with a fixed return period. The processes of simulating samples from the fitted copulas, estimating the relative likelihood along the isolines, and extracting the most-likely event, was then repeated to determine the three design events associated with the 50, 100, and 200 year return periods. The most extreme observed compound event with rainfall of 123.8 mm and a surge peak of 1.4 m (close to 200 year return period) in Fig. 4b was during typhoon Meari (June 2011). This typhoon caused floods in 17 counties of three provinces on the east coast of China.

In coastal areas, coastal flooding is driven by sea water level, which depends on both storm surge and astronomical tide. However, sea level might also affect pluvial flooding, since high sea level might prevent inland water discharge by gravity. Hence, we used the dependence model obtained above to assess changes in rainfall probability distribution when information of storm surge is known in relation to the probability distribution of annual maximum rainfall.

From the dependence model (Frank copula for catchment A2 and Gumbel copula for catchment C4) we randomly sampled 10,000 dependent surge and rainfall pairs (red dots in Fig. 5). Then, we selected pairs for which the surge level is close to the 50 year return period event in the univariate case, that is, 0.86 m (0.835–0.885 m) for catchment A2 and 1.45 m (1.425–1.475 m) for catchment C4 (Fig. 5), to obtain the distribution of rainfall conditioned on the 50 year return period surge event.

Figure 6a shows that for catchment A2 (eastern Hainan Island, South China), where the majority of surge peaks occurred during typhoon season, we observe a shift in the rainfall median when rainfall is conditioned on storm surge. Specifically, the median of the rainfall condition on the 50 year return period storm event (258.76 mm/day) is about 40.8% higher than the median of annual maxima rainfall (183.69 mm/day). This result implies that extreme rainfall events during typhoon season are generally more intense compared to most annual maxima events, requiring extra attention when developing a flood protection intervention. For catchment C4 (the Bohai Sea, North China), where the majority of surge peaks occurred outside the typhoon season, we observe the opposite behavior: the median rainfall event conditioned on the 50 year return period surge event (49.43 mm/day) is smaller than the median of the annual maxima rainfall event (88.15 mm/day).