Dependence between high sea-level and high river discharge increases flood hazard in global deltas and estuaries

We use mean daily discharge directly from GRDB. For sea-levels, we use hourly total sea-levels from GESLA-2, from which we extract daily maxima. We also use a time-series of daily maximum skew surge, i.e. vertical difference between predicted and observed high water in a tidal cycle (which may be with time-lag), extracted following Haigh et al (2016).

We then extract station combinations from GRDB and GESLA-2 that satisfy the following criteria: (1) minimum of 20 years overlapping data; (2) minimum completeness of 75% per year in both databases; (3) minimum upstream basin area least 1000 km²; (4) maximum Euclidean distance of 500 km between discharge station and tide gauge; and (5) tide gauge and river basin outlet within maximum distance of one 0.5° grid-cell from each other. For many tide gauges, this results in several discharge stations satisfying the criteria; in these cases the most downstream river stations are selected. Following this selection, there are 187 combinations of discharge stations and tide gauges (figure 1(b)), with mean length ∼39 years and median length 36 years.

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From the daily time-series, we extract time-series of annual maxima. To assess the sensitivity of the results to different sampling methods, we also use peaks over threshold (POT) (i.e. all values selected above a given percentile threshold). For each method, we first identify annual maximum discharge values (or POT) for each year, and then select the highest sea-level (total or skew surge) within ±1 days of this event. We used this ±1 days window as we do not have information on the exact timing of the discharge events, and are interested in discharge/surge events that occur within approximately a day of each other, rather than on the same calendar day per se. Throughout this paper, this is referred to as sea-level conditional on high discharge (SL_condQ) or skew surge conditional on high discharge (SS_condQ). Secondly, we identified annual maximum sea-levels (or POTs) for each year, and then selected the highest discharge value within ±1 days of this event. Throughout this paper, this is referred to as discharge conditional on high total sea-level (Q_condSL) or discharge conditional on high skew surge (Q_condSS). For each of the time-series of annual maxima or POT, we also extract time-series of the other variable using time-lags from −5 to +5 days. For example, for SL_condQ we identify annual maximum discharge values (or POT-series) for each year, and then select corresponding sea-levels with time lags of −5, −4, −3, −2, −1, 0, +1, +2, +3, +4, and +5 days.

For the POT method, we use both 95th and 99th percentiles. Events exceeding those percentiles are identified, whereby a 3 day window is used to ensure independent events (Haigh et al 2016). In other words, if discharge (or sea-level) exceeds the percentile threshold on more than 1 day within a 3 day period, this is identified as 1 event. Sensitivity was also assessed using 5 and 7-day windows; the results are very insensitive.

For each of these time-series, we calculate parameters of five marginal distributions: LogNormal, normal, exponential, Weibull, and generalized extreme value. Then, we identify the distribution best fitting each of the individual time-series, using a goodness-of-fit test comparing empirical and theoretical non-exceedance probabilities using root mean squared error. For more information on statistical methods see supplementary information available at stacks.iop.org/ERL/13/084012/mmedia.

As the discharge gauging stations used are up to 500 km from the coast, flow-times between gauges and coast can be several days. Therefore, we correct for this by estimating average flow-times between discharge gauging stations and the outlet in days, and then shift the daily time-series forward by this number of days. For example, if the flow-time is one day, we assume that observed discharge reaches the coastline one day later. Bankfull flow-time estimates are taken from Allen et al (2018), who applied a kinematic wave model to the global HydroSHEDS hydrography dataset, enhanced with information regarding river length, bankfull width, and bankfull depth. However, these estimates are not available for locations >60°N. For those locations, we applied a lagged cross-correlation analysis based on paired time-series of modelled discharge at the gauge and downstream river outlet to derive the flow-time. Discharge is modelled using the CaMa-Flood routing model (Yamazaki et al 2011) forced with specific runoff from the Watergap V2.2 WRR2 eartH2Observe re-analysis data (Müller Schmied et al 2016, Dutra et al 2017). We find the flow-time based on the lag with the highest correlation within a maximum window of 14 days. For stations where this approach returns a correlation coefficient >0.98, the number of days is either the same as in Allen et al (2018), or different by one day, for 83% of stations. For stations where the correlation coefficient is <0.98, we manually assign a 0-day flow-time as these locations are within ∼30 km of the coastline.

We measure dependence between discharge and total sea-level/skew surge using Kendall's rank correlation coefficient τ (Kendall 1938). Dependence is assessed for each of the lag-times in section 2.2, and significance assessed using α = 0.10 due to the relatively low number of years at some stations. We also carried out the analyses using α = 0.05.

For each station combination with statistically significant dependence, we apply copula theory (De Michele and Salvadori 2003, Grimaldi and Serinaldi 2006, Nelsen 2006) to assess the dependence structure. We use the maximum pseudo-likelihood estimator (Kojadinovic and Yan 2010) to calculate parameters for three different copulas: Gumbel (upper tail dependence), Frank (no tail dependence), and Clayton (lower tail dependence). We then use these parameters to simulate 1000 pairs of ranks for each copula model. The copula model most suitable for simulating the dependence structure for each station combination is selected by comparing non-parametric tail dependence coefficients (Schmidt and Stadmüller 2006) above a threshold of 0.8, derived from the observed and simulated rank pairs. Finally, we use the Cramer-von-Mises test (Genest et al 2009) to assess goodness-of-fit between observations and simulations. In this letter we only apply copula models for which the null hypothesis is accepted that the samples are from the same underlying distribution.

For station combinations with significant dependence, and a copula model whose simulated values are statistically similar to observed values, we assess the joint probability of floods exceeding the design discharge and design sea-level simultaneously or in close succession. To do this, we need an estimate of the design standard of river and coastal flood protection measures for the river stretch or coastline closest to each discharge station and tide gauge. For river flood protection, we use protection standards from the FLOPROS database (Scussolini et al 2016). For coastal flood protection, standards are derived from the DIVA database (Hallegatte et al 2013, Sadoff et al 2015). Given the high uncertainty of protection standards in FLOPROS and DIVA, we also assess the joint probability of sea-levels and discharge exceeding a return period of 10 years.

We use the marginal distributions identified in section 2.4 to estimate the probability of discharge or sea-level exceeding design levels. Then, we calculate the joint exceedance probability assuming: (1) full independence, for which the joint exceedance probability is the probability of discharge exceeding design discharge multiplied by the probability of sea-level exceeding design water level; and (2) dependence, as defined by the copula model. Finally, we calculate the factor difference in the joint exceedance probability when using the dependence compared to independence assumption.

First, we examine dependence between discharge and skew surge using a 0-day lag-time. We find statistically significant dependence for 41 stations (22% of stations) for SS_condQ and 67 stations (36%) for Q_condSS; τ values are shown in figure 2. Therefore, there are more locations with statistically significant dependence for Q_condSS than SS_condQ; this is especially the case in UK and Japan. However, across all locations there is no clear signal of difference in the strength of dependence between the SS_condQ and Q_condSS cases; τ values are higher for SS_condQ than Q_condSS at 52% of locations. Moreover, the mean τ values for SS_condQ and Q_condSS are 0.09 and 0.08 respectively (no significant difference; t-test, p = 0.366).

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In figure 3 we show dependence for the lag-time (−5 to +5 days) for which τ is highest; corresponding lag-times are shown in figure S1. When lags are included, we find statistically significant dependence between discharge and skew surge for 104 stations (56%) for SS_condQ and 101 stations (54%) for Q_condSS. The results show that it is more common for high discharge events to follow high surge events than vice versa; for SS_condQ, this occurs in 70% of locations (i.e. locations with negative lag in figure S1), and for Q_condSS, this occurs in 76% of locations (i.e. locations with positive lag in figure S1).

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Comparing results for skew surge (figure 3) with total sea-levels (figure S2), geographical patterns are similar, with regional differences. For example, in western Britain our skew surge results (figure 3) show significant dependence for most stations, yet this is not the case for SL_condQ. Here, where the tidal range is large, the tidal influence on overall sea-level is stronger than in locations with low tidal range. Haigh et al (2016) showed that most extreme sea-levels around the UK are generated by moderate storm surge coinciding with high spring tides; this is probably the reason dependencies are lower for total sea-level.

We examine sensitivity to using the POT method for event selection. Results are shown for skew surge with thresholds of 95% (figure S3) and 99% (figure S4). Using a 99% threshold, results are very similar to those using annual maxima. Using a 95% threshold, the number of significant results decreases, as do the τ values. We also carried out the analyses using a confidence limit of α = 0.05 to assess the significance of the τ values; the results for dependence for 0-day time-lags (figure S5) and with time-lags (figure S6) are very similar to those using α = 0.10.

In this section, we discuss regional patterns of dependence, and interpret results with reference to existing literature. The discussion is limited to regions with a large number of stations, as it is not possible to identify general patterns for those regions with limited observations.

For the west coast of the USA, we find significant dependence between high discharge and skew surge (figure 3) and total sea-level (figure S2) at most stations. For two locations (La Jolla, California, in the southwest, and Neah Bay, Washington, in the northwest) we show composite analyses of atmospheric conditions on the dates of annual maximum discharge and annual maximum skew surge in figures S7 and S8 respectively. The data for the composites are derived from the NOAA-CIRES 20th Century Reanalysis Version 2c dataset (NOAA 2015), which has a 2° × 2° spatial resolution and four times daily temporal resolution from 1851–2014. To cover post 2014 events, we use the NCEP-NCAR reanalysis dataset (Kalnay et al 1996), interpolated from its native 2.5° × 2.5° resolution to 2° × 2°. Figures S7 and S8 are representative of atmospheric conditions in most of the studied locations along the US West Coast, and show that atmospheric conditions are similar on the dates of annual maximum discharge and skew surge. Moreover, most of the basins studied on this coast (except the Columbia River) are relatively small and steep with fast catchment response times. Together, these aspects can explain the similar patterns in dependence for SS_condQ and Q_condSS. In contrast, Wahl et al (2015) found few locations with significant dependence between surge and precipitation along this coastline. Our composite analyses do show elevated precipitable water content (especially for Q_condSS) in the catchments upstream from the tide gauges (albeit relatively weak for La Jolla for SS_condQ). The west coast of the USA is typified by a topography in which mountain ranges lie close to the coast, which can lead to orographic rainfall. In Wahl et al (2015), precipitation gauges were used if they are within 25 km radius of the tide gauges. Hence, these may be located closer to the coastline than the first mountain ranges, meaning that coastal rainfall may not show dependence with sea-level, but that river discharge does.

In eastern USA, our results for discharge and skew surge are largely similar to those of Wahl et al (2015). We find significant dependence for most station combinations, although for Q_condSS we find no significant dependence for stations in northeastern USA. Figure S9 shows a composite analysis of atmospheric conditions on the days of annual maximum skew surge for Portland (Maine). The figure shows surge events driven by strong low pressure systems to the southeast of the tide gauge, causing strong easterlies and northeasterlies. Whilst precipitable water content is elevated for this location, there is no strong dependence with the discharge of the Kennebec River (to the north). Further investigation shows that 86% of the annual maximum skew surge events occur between October and February, whilst peak discharges occur in the boreal spring. Hence, seasonality has an important influence on dependence in such rivers with strong seasonal discharge and surge characteristics; very similar composite analysis plots are found for other locations in the region.

For the Gulf Coast, Wahl et al (2015) found significant dependence for most stations. For discharge, we only find this relationship at one station combination in each case (figure 3). For the combination of Grand Isle (near New Orleans)/Mississippi River, this can be explained by catchment size (>3 000 000 km²). The other basins studied here are also relatively large, except the Guadeloupe River (ca. 13 000 km²) and Sabine River (ca. 24 000 km²); the latter are the rivers for which we find dependence. Moreover, highest surges in the region typically occur during hurricane season, when the discharge of the region's rivers are at their lowest.

Results for western Japan show strong dependence, especially for Q_condSS and Q_condSL, with strongest dependence for either 0-day or small positive time-lag (i.e. high surge or high total sea-level leading discharge). Similarly to the west coast USA, this region is typified by steep topography and relatively short distances to the coastline, meaning that precipitation from surge-causing storms can be enhanced by orographic effects (Negri and Adler 1993) and the resulting discharge can rapidly reach the coast.

In Europe, several local studies have been carried out in the UK and the Netherlands. For stations in the western Netherlands, we find no statistically significant dependence. Klerk et al (2015) found significant dependence between surge at Hoek van Holland and discharge at Lobith on the Rhine for a 6-day lag, which exceeds the maximum lag in our study. They relate the lagged response to precipitation delay between the Dutch coast and upstream in Germany, and typical travel times of the Rhine of four days.

For the UK, where rivers are shorter, we find many sites with statistically significant dependence. On the south coast, the Environment Agency (2000) found simultaneous dependence between discharge and total sea-level at Brockenhurst, Hampshire. In this region, our 0-day lag results for Portsmouth/Stour river (west of Brockenhurst) (figure S10) also show relatively strong dependence (SL_condQ: τ = 0.358, p = 0.014; Q_condSL: τ = 0.284, p = 0.043). More broadly for Britain, for Q_condSS we find significant dependence in most regions (figure 3(b)), except the southeast. Similarly, Svensson and Jones (2002) found significant dependence of discharge conditional to surge north of the Firth of Forth and little dependence southwards, particularly south of the Humber Estuary. They state that during winter, high-surge events are often associated with cyclones to the north of Scotland, which are associated with precipitation from a south-westerly/westerly circulation. As the area north of the Firth of Forth is not sheltered from south-westerly winds by major topographical barriers, orographic rainfall can be enhanced when these airflows meet the hilly terrain north of the Firth, leading to high discharge. To the south, southwesterly winds have already encountered the Southern Uplands, Pennines, or Welsh mountains. Moreover, in the southeast, catchments are generally permeable, and therefore respond slowly to rainfall. In western Britain, Q_condSS shows significant dependence at all stations (figure 3(b)), with highest dependence for positive lag-times (i.e. when high surge precedes high discharge; figure S1). Svensson and Jones (2004) also found significant dependence at stations along most of this coastline. Most of the station combinations studied here are south or west facing, with fast catchment response times and orographically precipitation, meaning that discharge can arrive at the coast on the same day or shortly after a high surge event. For SS_condQ, we find significant dependence along large parts of the coastline, except for several western stations and one station in northeastern Scotland. Using composite analyses, Hendry et al (2018) showed that the storms that generate high discharge at eastern UK sites differ from those generating large surges. Discharge causing storms on the East coast typically tracked over the centre of Britain, with the low-pressure system being located over central Britain at the time of peak discharge. Meanwhile storms generating surges tracked north of Scotland, with the low-pressure system located over Scandinavia at the time of peak surge.

For Australia, we find significant dependence between discharge and total sea level for the majority of stations (figure 3). In Victoria, there are several stations where the dependence is not significant. This can probably be explained by lack of dependence in the storm-causing atmospheric processes in this region, since Zheng et al (2013) found significant dependence between surge and precipitation in most areas of Australia, but not in those parts of Victoria for which we find no dependence between discharge and skew surge.

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An understanding of the simultaneous joint occurrence is particularly important for designing flood protection infrastructure and drainage, whereas lagged dependence can be relevant for hazard mapping and planning emergency responses. In figure 4 we show the difference in joint exceedance probability of discharge and total sea-level exceeding design level on the same day, assuming dependence versus independence. In figure 5 we show the same results for the ranked correlations with the best lag-time. We only show results where there is significant dependence and where we identify a copula model with significant goodness-of-fit; copula models used are shown in figures S11 and S12 respectively. The results show the importance of considering the dependence structure on the estimated joint exceedance probability. For SL_condQ, significant dependence and copula models are found for 20% of stations for simultaneous occurrence and 55% of stations for lagged occurrences. For Q_condSL, significant dependence and copula models are found for 25% of stations for simultaneous occurrence and 42% of stations for lagged occurrences. The differences in joint exceedance probabilities when assuming dependence versus independence are large. For those station combinations for which significant dependence and copula models are identified, the factor difference exceeds +2 in 68% of cases for SL_condQ and 74% of cases for Q_condSL for the zero lag analysis, and in 59% of cases for SL_condQ and 67% of cases for Q_condSL for the lagged analysis. In these locations, compound river and coastal flood events are more than twice as likely to occur than if they were independent. To assess the sensitivity of the results to the estimated flood protection standards in FLOPROS and DIVA, figures S13 and S14 show the same results, but using a return period of 10 years for both river and coastal protection. The spatial patterns are robust, though the factor differences are generally lower, since the protection standards are greater than 10 years in most of the locations studied.

When assessing flood risk, it is vital to know flood impacts across a whole spectrum of exceedance probabilities. In figure 6, we show two examples of differences in joint exceedance probability, assuming dependence versus independence, for different combinations of exceedance probabilities of the two marginal distributions (Santa Monica and Santa Clara River in California, USA; and Komatsushima and Yodo River Japan); note that the colour bars for the two sub-figures have different scales. For both examples, the (non)-inclusion of dependence has an increasing influence on the joint exceedance probability, as the marginal exceedance probabilities become lower. Hence, when assessing the influence of compound floods in a risk assessment, it is not sufficient to assume a linear shift in the exceedance probability loss curve, but the change in exceedance probability should be assessed along the entire curve. Moreover, this shows that differences in joint exceedance probability for the dependence versus independence assumptions increase as the marginal exceedance probabilities decrease (i.e. with higher return periods). This is an important finding for flood risk management in a changing world where flood protection standards are generally being increased.