Nonlinear Interactions of Sea‐Level Rise and Storm Tide Alter Extreme Coastal Water Levels: How and Why?

Sea‐level rise (SLR) increasingly threatens coastal communities around the world. However, not all coastal communities are equally threatened, and realistic estimation of hazard is difficult. Understanding SLR impacts on extreme sea level is challenging due to interactions between multiple tidal and non‐tidal flood drivers. We here use global hourly tidal data to show how and why tides and surges interact with mean sea level (MSL) fluctuations. At most locations around the world, the amplitude of at least one tidal constituent and/or amplitude of non‐tidal residual have changed in response to MSL variation over the past few decades. In 37% of studied locations, “Potential Maximum Storm Tide” (PMST), a proxy for extreme sea level dynamics, co‐varies with MSL variations. Over all stations, the median PMST will be 20% larger by the mid‐century, and conventional approaches that simply shift the current storm tide regime up at the rate of projected SLR may underestimate the flooding hazard at these locations by up to a factor of four. Micro‐ and meso‐tidal systems and those with diurnal tidal regime are generally more susceptible to altered MSL than other categories. The nonlinear interactions of MSL and storm tide captured in PMST statistics contribute, along with projected SLR, to the estimated increase in flood hazard at three‐fourth of studied locations by mid‐21st century. PMST is a threshold that captures nonlinear interactions between extreme sea level components and their co‐evolution over time. Thus, use of this statistic can help direct assessment and design of critical coastal infrastructure.


Introduction
Sea-level rise (SLR) is accelerating, and given the current trajectory of warming, a rise in global mean sea-level exceeding 1 m is possible during the 21st century (Dangendorf et al., 2019;IPCC, 2019;Palmer et al., 2020).This rise reduces the freeboard between high tidal datum and flood stage, which introduces uncertainties in flood risk allowances that need to be considered in coastal risk and adaptation assessments (Arns et al., 2017;Boumis et al., 2023a;Buchanan et al., 2016;Hunter, 2012;Nicholls et al., 2021).Currently, 250 million people live on a land below annual coastal flood levels; without protective measures; this number may reach 340 million by 2050 (Kulp & Strauss, 2019).This estimate of coastal flooding exposure is based, however, on shifting today's storm tide regime to a higher elevation for a given SLR scenario, ignoring nonlinear interactions between tidal and non-tidal components of coastal water level and future mean sea levels (MSLs) (Hinkel et al., 2014;Kulp & Strauss, 2019;Neumann et al., 2015).We show that this approach under-estimates coastal flood hazard in some regions, while over-estimating it in others.
Extreme coastal water level (ECWL) is determined as the combination of MSL, harmonics tides (HT), and the non-tidal residuals (NTR).MSL refers to the arithmetic mean of hourly heights observed over a relatively long period (preferably several years).HT, is the most predictable component of ECWL (Gregory et al., 2019;Parker, 2007); it is the sum of astronomical tides and overtides (NOAA-COOPS, 2000).NTR refers to the difference between observed still water level and estimated HT thus, NTR contains both surge, driven by wind and pressure, and the tide-surge interactions (Fernández-Montblanc et al., 2019).The sum of NTR and HT is called storm tide (NOAA-COOPS, 2000); thus: ECWL = MSL + storm tide = MSL + HT + NTR (1) The spatio-temporal variability of ECWL will change in a warming climate at global and regional scales (Calafat et al., 2022;Goodwin et al., 2017;Marcos et al., 2015;Mawdsley & Haigh, 2016;Muis et al., 2016Muis et al., , 2019Muis et al., , 2020;;Rashid et al., 2019).NTR, composed of intra-annual/interannual variability and a high-frequency residual related to storm surge due to atmospheric pressure anomalies and wind setup, is expected to be altered by climate change (Serafin et al., 2017).There is ample evidence that trends in the climate factors have significantly affected wave characteristics over the past few decades and that shifts in oceanic currents will occur in a warming climate (Bromirski & Cayan, 2015;Johnson & Lyman, 2020;Melet et al., 2018;Morim et al., 2020;Semedo et al., 2011;Tebaldi et al., 2021).Moreover, altered terrestrial inflows to coastal regions will likely change NTR through quadratic bottom friction and their steric contribution to sea level (Guo et al., 2015;Hoitink & Jay, 2016;Matte et al., 2013;Xiao et al., 2021).While changes will be partially due to climate forcing (Gori et al., 2022;Marsooli et al., 2019;Wahl & Chambers, 2016), nonlinear interactions among different components of ECWL are also important (Arns et al., 2020).Further, HT characteristics (amplitudes and phases) are subject to change due to non-astronomical factors (Haigh et al., 2019;Woodworth, 2010).HTs are shallow-water waves and react to changes in MSL through barotropic effects, including frictional effects and altered tidal resonance (Talke & Jay, 2020), and a variety of internal processes (Idier et al., 2017).These MSL-HT interactions are nonlinear, and often highly variable in a single oceanic region (Devlin et al., 2014;Wahl, 2017).MSL-NTR interactions are also hard to characterize, because wind patterns may change, and SLR may either amplify the NTR via reduced friction or dampen it via reduced surface wind stress on the water column (Arns et al., 2020;De Dominicis et al., 2020;Familkhalili & Talke, 2016).NTR also interacts with HT via nonlinear phase and amplitude alterations (Proudman, 1955a;Rossiter, 1961).Thus, nonlinear HT-NTR interactions contribute to altered ECWL dynamics in a changing climate (Familkhalili et al., 2020;Idier et al., 2012) and modulate coastal flooding regime associated with ECWL (Bilskie et al., 2016(Bilskie et al., , 2022;;S. L. Dykstra & Dzwonkowski, 2021;S. Li et al., 2021;Rashid et al., 2021).
Historically, scientists have often failed to connect SLR with an exacerbated coastal flooding risk, for various reasons.In certain regions, MSL has actually decreased (e.g., parts of Northern Europe and the Northeast Pacific Ocean), or there has been an inconsequential/negative correlation between the fluctuation in MSL and the height of storm tides (Pickering et al., 2012(Pickering et al., , 2017;;Woodworth et al., 2007).However, recent studies have explored regional patterns in MSL-HT interactions, and reported significant correlation between HT and MSL in gauges across Pacific and Atlantic oceans (Jay, 2009;Devlin et al., 2019;Devlin, Jay, Talke, et al., 2017;Devlin, Jay, Zaron, et al., 2017;D. Idier et al., 2017;Pickering et al., 2012Pickering et al., , 2017;;Ray & Talke, 2019;P.L. Woodworth, 2010), as reviewed by Jay et al. (2021).Thompson et al. (2021) studied the combined effects of SLR and nodal cycle modulations of tidal amplitude on frequency of high tide flooding around the United States, and found that there might be a rapid increase in the frequency of high tide flooding in multiple US coastal regions after mid-2030s.In freshwater-influenced systems like estuaries and bays, however, the process is more complicated, because MSL is influenced by freshwater influx from riverine tributaries as well (Hoitink & Jay, 2016;Jay et al., 2015;Khojasteh et al., 2023).In these systems, frictional interactions of the main tidal constituents with each other and with freshwater influx gives rise to the overtides and dampen tides (Guo et al., 2015;Proudman, 1955b).Thus, a higher MSL is not necessarily associated with higher flooding hazard.Moreover, time scale matters.At San Francisco, MSL and the amplitude of the largest lunar tidal constutuent M 2 have both increased over the last 160 years, but on a seasonal scale, fluctuations in MSL and M 2 are anti-correlated (Moftakhari et al., 2013;Talke & Jay, 2020).
Finally, increasing or decreasing inflow to the coast may raise or lower local steric heights (e.g., Devlin et al., 2018), thereby affecting tides.
Here, we integrate the contributions of various tidal and non-tidal components of coastal water level to flooding, using a proxy for ECWL fluctuations around MSL called Potential Maximum Storm Tide (PMST).PMST is a coastal flood hazard upper limit or threshold that captures the nonlinear dynamics between MSL, NTR and HA, and is helpful in characterizing the co-evolution of these components of ECWL with SLR.We then use hourly water level data from global tide gauges (Woodworth et al., 2016) to quantify how tidal and non-tidal components of PMST interact with MSL around the globe.PMST generally represents the highest NTR on top of a perigean spring tide (Wood, 1978), or in media jargon, a king tide.An extreme example occurred during Hurricane Sandy -a coincidence of a storm surge and the peak of king tide reached a water level significantly greater than the previous record and resulted in a "significant change" in exceedance probability calculations (W.Sweet et al., 2013;Zervas, 2013).To calculate PMST, we take the sum of amplitude for the five major diurnal and semidiurnal tidal constituents plus three overtides and the largest NTR (more details in the Methods section).The probability of PMST occurring is low, because it ignores the phase difference between tidal constituents and the lag time between NTR and greater diurnal/semidiurnal fluctuations.Nonetheless, major storms occur on king or perigean spring tides often enough to be culturally important; for example, Wood (1978) describes a considerable number of such events in North America, starting in 1635.Next, using site-specific probabilistic relative SLR projections (Kopp et al., 2014(Kopp et al., , 2017)), we estimate how the nonlinear interactions among flood hazard drivers affect estimated PMST in the upcoming decades.

Materials and Methods
Hourly water level data from a global tide gauge set were obtained from the GESLA-2 database (Woodworth et al., 2016).We analyzed 446 sites with at least 19 years of observations, and less than 20% gaps over the entire record.Where there were multiple data files for the same station in different time periods,the files were merged to provide the longest possible record.A harmonic analysis package, t_tide, was used to calculate harmonic constituents (phase and amplitude) of the observed tides (Leffler & Jay, 2009;Pawlowicz et al., 2002).We applied t_tide using a 40-day moving window over the length of each record in steps of three days, requiring at least 98% complete data at each step.Thus, there is ∼92% overlap in the time period for each estimate of MSL and HT.Only constituents with signal-to-noise ratio greater than two were used in the further analyses.Astronomically forced "nodal" variability of constituents (with 4.4, 8.8, and 19.6-yr periodicities) was accounted for using the nodal correction feature of t_tide (Pawlowicz et al., 2002).While the actual nodal variability at some stations departs somewhat from that estimated by t_tide, this approach removes most of astronomically driven constituent variability.The calculated tidal constituents for each 40-day window were then used to estimate HT over the same window.The timeseries of NTR, defined as the difference between observed water level and the estimated HT over the calculating window, was estimated as: where WL(τ) is the timeseries of observed water levels over the length of a calculating window.MSL is the arithmetic average of observed water level over the 40-day window, as: and the harmonic tides are reconstructed as: where, n is the number of significant constituents in harmonic analysis, and A i ,ω i and φ i are the amplitude, frequency and phase of constituent i.Then the largest NTR(τ) over ±1 day around the central day of each window was stored as NTR associated with that window.So while the MSL and tidal constituents are calculated over a 40day moving window the effective temporal resolution of the PMST estimates is 3-day.
To calculate PMST, we take the sum of amplitude for the five major tidal constituents, namely main lunar diurnal (K 1 and O 1 ) and semidiurnal (M 2 , N 2 , and S 2 ) tides (Pugh & Woodworth, 2014) plus three important overtides (M 4 , MK 3 , and MS 4 ) calculated over a 40-day window and the largest NTR calculated over the central three days of the same calculating window, as: PMST is a statistic that is broadly applicable to diverse diurnal, semidiurnal and mixed tidal regimes.It is an upper limit, and upper limits or thresholds on complex phenomena are useful in many engineering contexts (Cheung et al., 2011;Faber & Stewart, 2003;Karamouz et al., 2022;Yim et al., 2015).In this situation, we need a parameter that: (a) can be used globally, regardless of the tidal characteristics at a given location; and (b) provides a solid upper limit.PMST achieves both of these objectives, because it is an upper limit that is applicable to diverse diurnal, semidiurnal and mixed tidal regimes.While PMST linearizes complex nonlinear processes, it compensates for this in a conservative manner by adding constituent amplitudes.A simple measure of this sort is the only practical possibility for a global analysis, because there is no single pattern of MSL-HT-NTR interactions that applies even regionally (e.g., Devlin et al., 2014).
Further, PMST is not unduly conservative.To use an example particularly relevant to semidiurnal and mixed tidal regimes: it might happen only once every few centuries that a storm would occur at any given location within a few days of a perigean tide at a time when the moon is particularly close to the earth (representing an especially close conjunction of M 2 , S 2 , N 2 , and their overtides).But it is not at all improbable that a major storm will occur somewhere on earth in a semidiurnal tidal regime within a few days of this event.An example of this is the "Saxby Gale" of October 1869 (Wood, 1978).As it happened, an especially close conjunction of perigee and syzygy occurred on 5 October 1869, and was accompanied by a hurricane.The two together caused major devastation throughout the Canadian Maritime Provinces.Moreover, the major constituents do not have to be exactly in phase -maximum water levels are high for several days around a spring tides.In this regard, Wood (1978) defines a perigean spring tide as one in which perigee (closest approach of the earth and moon) and syzygy (alingment of the sun and moon) coincide within 36 hr; that is, within a 72 hr window.A pseudo-perigean spring tide, which can also cause very high waters, occurs with when perigee and syzygee coincide within 84 hr, providing a 7-day window.Diurnal tidal regimes have different periodicities, but the same principles applies.In summary, PMST is a conservative upper limit or threshold, useful for estimating the potentially highest storm tide based on the available record and as input for the design of critical infrastructure.To further demonstrate the applicability of PMST in design practices and its conservativeness we have provided a time-series of PMST annual max and the associated cumulative distribution function (CDF) in panels a and b of Figures 2-5, respectively.The CDF curves are drawn based on maximum likelihood estimates of the parameters of the generalized extreme value distributions at each gauge, and the still water level plus the associated annual maxima for PMST at the gauge.From these CDF curves, the design flood magnitude of a return period of interest can be obtained, that is, the CDF = 0.90 is associated with the height of a 10 year flood (annual exceedance probability of 0.1).We implement regression analysis to analyze how HTs and NTR have varied with MSL over the length of observation.For this purpose, we employ a robust linear regression scheme with reweighted least squares and a bisquare weighting function (Street et al., 1988).Such a robust method helps ensure the detected trend is not as sensitive to outliers, as it is in ordinary least square methods (Yu & Yao, 2017).The regression analysis implemented for each tidal component and NTR with MSL, one at a time.We assume that the estimated slope out of this robust regression analysis remains valid within the range of projected MSL in the following decades.Here, we also calculate Kendall rank correlation coefficient (CC) as a measure of dependence between variables (Kendall & Gibbons, 1990) and assume a significant correlation between variables if the p-value is <0.05.
We use MSL projections from Kopp et al. (2017) that integrate a probabilistic framework with a process model to estimate the likelihood of relative MSL rise above a reference (i.e., MSL in year 2000) at each location.These estimates are based on the framework proposed by Kopp et al. (2014) that uses a joint probability distribution for global mean thermal expansion and regional ocean dynamics.The regional contributions of nonclimatic factors are based upon a spatiotemporal statistical model of tide-gauge observations (Kopp et al., 2017).Here, we use the near-term projections, that is, for year 2050, under representative concentration pathway (RCP) 8.5, at which time the different RCPs (e.g., 2.6, 4.5, and 8.5) have not greatly deviated from one another (W.V. Sweet et al., 2022;van Vuuren et al., 2011).Realistic projection of ECWL and flooding beyond 2050 requires a comprehensive understanding of anthropogenic activities and protective measures to be implemented in the following decades.Also, MSL is not the only non-astronomical factor that modulate tides, and for a more comprehensive analysis of trends in coastal flooding beyond the mid-21st century, other factors contributing to ECWL like tectonic activity, shoreline position, hydrologic regime, and harbor modification should be taken into account (Haigh et al., 2019;Talke & Jay, 2020;Woodworth et al., 2019).

Changes in the Storm Tide Regime Due To SLR
Relative MSL has risen in most (79%) locations studied here.The median rate of SLR among the gauges with a positive trend is 2.5 mm/yr (90% range of variability [i.e., 5th and 95th percentiles] of {0.4,7.5 mm/yr}).The East and Gulf coasts of United States, southwestern coasts of Europe, East Asia and northern coasts of Australia generally show strong positive SLR rates.In contrast, gauges on the coasts of Alaska, Canada and in northern Europe, including Scandinavian Peninsula, generally show negative MSL change trends over the last few decades (Figure 1a).The positive SLR trend for majority of gauges across tropical Pacific Ocean found here is consistent with previous studies and particularly important as altered ECWL due to SLR and its associated flooding pose serious threat to the inhabitants of these areas (Church et al., 2006).
The locations that show significant SLR and a strongly positive relationship between PMST and MSL variations exhibit a variety of tidal/nontidal patterns.Metrics like tide-to-NTR ratio (Figure 1c) and overtide generation (Figure 1d) can help characterize these patterns and understand the underlying mechanisms deriving higher likelihood for ECWL.For example, at Wyndham (Northern Australia) there is a significant positive correlation between MSL and PMST; the CC is 0.28; with a 95% confidence bound of {0.27, 0.33}, caused mainly by the positive correlation between MSL and the solar semidiurnal tide S 2 (CC = 0.20; {0.18, 0.22}) and to a lesser extent by MSL and NTR (CC = 0.10; {0.08, 0.13}).Other components show a relatively weak relationship with MSL (Figure 2).Thus, at Wyndham a combination of rapid SLR (6.87 mm/yr), large tides with increasing S 2 amplitude, and intensified surge dynamics may increase flood hazard in the future.Given the strong interactions between various ECWL components in this location (and similar locations marked by red circles in both Figures 1a and 1b), flood hazard projections that ignore altered storm tide regime at a higher MSL are subject to a significant error (Boumis et al., 2023a).There are also, however, locations with positive relative SLR trends at which there is a negative relationship between MSL and other components of ECWL (i.e., East coast of United States and East Asia).In such cases, the approach based on linear combinations of current storm tide regime and the projected SLR yields an overly conservative estimates of flood hazard.The extent to which altered tidal amplitude at higher MSLs influences flooding dynamics depends on the tide-to-NTR ratio (Figure 1c).In Lvsi, Jiangsu (China) a relatively large HT to NTR ratio makes the lunar semidiurnal tide M 2 the dominant component that governs the relationship between PMST and MSL (Figure 3).In this case, despite a negative correlation (CC = 0.15; { 0.18, 0.13}) between NTR and MSL, a strong positive correlation between M 2 and MSL (CC = 0.65; {0.63-0.66})and to a lesser extent K 1 and MSL (CC = 0.19; {0.17, 0.22}) produces a positive correlation between PMST and MSL (CC = 0.14; {0.11, 0.16}).
There are locations at which a relatively small HT to NTR ratio (i.e., <1) means that the relationship between MSL and NTR dominates the PMST regime.Baltimore, MD (USA) is an example.Despite a positive relationship between MSL and major HTs (e.g., K 1 and M 2 ), a negative correlation between MSL and NTR (CC = 0.15; { 0.17, 0.13}) controls the PMST regime; suggesting an insignificant contribution of nonlinearities in coastal flooding at higher MSLs (Figure 4).In cases like Baltimore with relatively small HT to NTR ratio (marked by bluish circles in Figure 1c), adding SLR projections to the estimated storm tide regime is adequate for planning purposes at present, though the storm tide regime may change in the future (Gori et al., 2022).

River Influence
Freshwater-influenced systems (e.g., estuaries, bays and deltas) behave differently from open coasts.Along a typical open coast, MSL rise implies deeper water, less friction and smaller overtides.As MSL increases, HT and 1.6337), a macro-tidal system with a semi-diurnal tidal regime.Red lines represent the linear regression line between variables in each panel.
NTR will (barring a major change in resonance) increase along with MSL, the increased greater diurnal tidal range needs to be considered in coastal flooding projections (Arns et al., 2017;Devlin, Jay, Talke, et al., 2017).Furuögrund (Sweden) is an example, at which tidal and non-tidal components are positively correlated with MSL variations.Nantes on the Loire Estuary (France), on the other hand, provides an example of the system where the interactions of river discharge and coastal water level tend to ameliorate flood hazard.Here, as in other similar systems (marked by red circles in Figure 1d), overtide ratio M 4 /M 2 2 is positively correlated with MSL (Figure 5).
Fort Pulaski, GA (USA) is another example where an analysis the underlying dynamics and their compounding effects is necessary to understand how freshwater influx characteristics might change the ECWL regime under SLR (Muñoz et al., 2020(Muñoz et al., , 2021)).San Francisco Bay, CA is a location with relatively large HT to NTR ratio (reddish circles in Figure 1c) where overtides are positively correlated with MSL (reddish circles in Figure 1d), at which modeling the interactions between freshwater influx and coastal ocean water level is critical for accurate estimation of future coastal flooding.In such systems, understanding of oceanic processes is not sufficient for accurate estimation of SLR impacts on coastal flooding, and a coupled hydrologic-hydrodynamics modeling framework is vital for accurate estimation of flood risk (Bakhtyar et al., 2020;Bilskie et al., 2021;Moftakhari et al., 2019;Santiago-Collazo et al., 2019;Ye et al., 2020).

Extreme Water Level Dynamics Changes With SLR
The median projected relative SLR by mid-century (under RCP 8.5) suggests that global median PMST will increase 20% relative to year 2000 (Figure 6a; 90% range of variability { 30%, +84%}).This increase is regionally variable and more pronounced (24% larger; 90% range of variability of {9%, 59%}) in the tropics (latitude 30°to +30°).To quantify the contribution of nonlinear interactions in altered ECWL regime, we calculate the normalized difference between projected PMST with and without adjustments for the nonlinear interactions shown in Figures 6b-6d.The median difference under 50th percentile SLR scenario across all studied gauges is 10% with a strong regional variability (90% range of variability { 6%, +41%}).This range of variability is much wider in temperate and higher latitudes (latitude >45°), where one-fourth of gauges are located and median normalized difference under 50th percentile SLR scenario is 3.5% (90% range of variability { 84%, +59%}).Thus, ignoring nonlinear interactions between tidal and non-tidal components of ECWL under SLR may yield significant error in estimated flooding hazards by mid-century, but results vary regionally and locally.
The contribution of nonlinear interactions becomes even more important under extreme SLR scenarios (e.g., 5th and 95th percentiles).The median of aforementioned normalized difference under the 95th percentile of SLR scenario increases to 20%, with strong regional patterns (90% range of variability {5%, 286%}), and a wider range of variability in latitudes >45°with 15% normalized difference and 90% range of variability { 6%, 448%}.These results indicate that: (a) including nonlinear interactions generally yields a higher estimated PMST increase; and (b) the increase may be up to a factor of four relative to a conventional approach, where the current storm tide regime is moved up at the rate of projected SLR.

Effects of Tidal Range and Regime on Variability
Tidal statistics can help characterize the potential response of ECWL to SLR.These systems can be classified into three categories based on their tidal amplitude, namely: micro-tidal (<0.5 m), meso-tidal (0.5-1.75 m), and macro-tidal (>1.75 m) (Passeri et al., 2015).Micro-and meso-tidal systems are more susceptible to SLR than macro-tidal systems (Figure 7).Based on the historic records, the median increase in tidal range in micro-and meso-tidal systems has been 9% (with 25th and 75th percentiles of 0% and 15%, respectively); in contrast to macro-tidal systems at which no significant change is detected (Figure 7a).
Following the NOAA tidal glossary (NOAA-COOPS, 2000) we also use F, the amplitude ratio of K 1 + O 1 to M 2 + S 2 , to classify tidal regime in studied systems.We detect a significant difference between systems with different tidal regimes.Systems with diurnal (F > 3.0) and mixed-mainly diurnal (1.5 < F < 3.0) regime are more sensitive to MSL change than systems with semidiurnal (F < 0.25) and mixed-mainly semidiurnal (0.25 < F < 1.5) tidal regime.Diurnal systems have been the most susceptible systems to SLR with median increased tidal range of 11% (with 25th and 75th percentiles to be 4 and 19%, respectively) over the past few decades (Figure 7b).
The tidal range and regime classification provides good insight towards expected change in PMST over the next few decades.The expected change in PMST under 50th percentile of projected SLR by 2050 in micro-and mesotidal systems (median = 17%) does not seem to be significantly different from macro-tidal systems (median = 18%); however the range of variability is wider for systems with smaller tidal amplitude (Figure 7c), probably because a particular absolute change represents a higher percentage change in these systems.Also, while the median change in PMST under projected SLR by 2050 ranges between 12% and 24% in the four different tidal regimes, systems with diurnal regime show more variability compared with semidiurnal ones (Figure 7d).

Utility of PMST
For planning purposes, it is vital to know how a harbor or region will respond to the next Saxby tide or Hurricane Sandy.Accordingly, PMST provides an estimate of an upper limit threshold on inundation in coming decades, useful for estimating the potentially highest storm tide associated with a given forcing or event.It can, therefore, inform design of critical infrastructure, without needing or intending to be the only necessary measure.
Numerical modeling is an alternative or complementary approach to data-driven analyses, like the one proposed here.Indeed, use of numerical models is a critical aspect of many coastal scientific/engineering endeavors.However, validation of coastal modeling, especially in areas with complex bathymetry/topography and dynamics, presents inherent challenges.In some places, despite advancements in technology such as Lidar, obtaining precise topographical data remains challenging, even when benchmarks are available for reference.Bathymetric data are often insufficient in quality and quantity to provide model results of the quality needed for practical needs.We emphasize that use of a data-driven method is not intended to replace traditional modeling approaches, but rather to supplement them.PMST methodology provides a valuable and cost-effective alternative, particularly for obtaining a preliminary understanding of the coastal dynamics.While it does not replace detailed studies necessary for constructing facilities, it serves as an efficient and economical means for an initial assessment.Thus, the choice between modeling approaches is not binary; rather, it is contingent on the specific objectives and circumstances.A data-driven methods like ours, by virtue of its cost-effectiveness, is well-suited for providing an initial overview, allowing researchers and stakeholders to identify areas that necessitate more detailed investigation.This approach is particularly advantageous in scenarios where resource constraints or time limitations may impede the feasibility of extensive modeling endeavors.

Possible Limitations
Some of the general conclusions drawn here may be affected by regional biases in the GESLA-2 data base, which has incomplete coverage along the African coasts and South America and southwest Asia, as compared to nearly full coverage of North America, Europe and East Asia.This regional bias yields, for example, no or fewer samples from diurnal macro tidal stations (e.g., Sea of Okhotsk).Another factor contributing to regional bias is the insufficient number of inland tide gauges installed in estuarine systems, in contrast to coastal areas.This lack of comprehensive coverage could result in uncertainties regarding our comprehension of freshwater-influenced tidal systems and their response to SLR (Khojasteh et al., 2023).A fuller range of stations would provide additional understanding of HT-NTR-MSL interactions and might change some of our interpretations of broad patterns.However, better data coverage would not change the interpretation of results at stations analyzed, or future use of the tool for practical purposes.
The timeseries length threshold used here was 19 yr.This is sufficient for various tidal analysis purposes, but records lengths less than about 50 yr that are not integral lengths of tidal periodicities such as 4.4, 8.8, and 18.6 yrs can yield biases in constituent trends.To minimize this problem, we have used the nodal variation feature of t_tide.Moreover, use of the 19-yr threshold, ensures inclusion of a large number of stations.Thus, while some bias associated with record length may affect results at individual stations, especially at those with larger influence of storm surge in PMST dynamics, these are likely to be averaged out regionally.Moreover, use of a longer timeseries and/or advanced statistical methods that enable sharing information across gauged locations (Boumis et al., 2023b;Rashid et al., 2024) would help with the problems regarding the regional biases mentioned above.
There is a potential uncertainty in using nodal correction feature of t_tide.Many stations, especially those in shallower coasts, do not follow the theoretical nodal cycle pattern used by t_tide (Feng et al., 2015;Woodworth, 2010).There are studies that have found estimates based on standard nodal corrections close enough to those based on empirical corrections (Ray & Foster, 2016;Ray & Talke, 2019).Thus, here we assume that this potential source of uncertainty does not significantly influence the global patterns recognized and so the conclusions remain valid.
With a focus on tidal constituents and their variability with SLR at Global scale, our analysis does not take important local processes, that is, rainfall/runoff regime, baroclinic effects, wave setup, steric effects and other anthropogenic effects (i.e., dredging) that significantly affect the tidal regime, especially in estuarine systems (Dykstra et al., 2022;Familkhalili et al., 2020;Khojasteh et al., 2021Khojasteh et al., , 2022;;Passeri et al., 2016) and in southeast Asia (Devlin et al., 2018).Thus, while our results are helpful for understanding the behavior of tides in the face of SLR, accurate prediction of extreme sea level at local scale requires consideration of local processes and engineering alterations.
Seasonal patterns are important in ECWL estimation in many locations.They have not been considered in our long-term Global analysis, in part because these patterns are variable between stations.In fact, seasonality in tidal and non-tidal components and even in their dependence structure can pose a significant uncertainty in ECWL estimates (D'Arcy et al., 2021;Devlin et al., 2018;Nasr et al., 2021).This could be dealt with in future studies in several ways, for example, use of a probabilistic approach that allows inclusion of seasonal effects, or use of seasonal regression models.Because seasonal patterns vary widely between stations and regions, such an approach was not practical here and better carried out at a local or regional scale.
Here, the determination of the PMST does not rely on establishing an upper limit for the NTR at each step of the calculation.Achieving such an upper limit necessitates an exhaustive hydrometeorological analysis, coupled with detailed ocean circulation models, and the generation of a large number of ensembles that reasonably represent potential scenarios (H.Li et al., 2023).Instead, PMST is derived based on the actual magnitude that this stochastic variable has taken at any given step.The surge activity is also assumed to be changing only with SLR.In fact the surge dynamics are subject to future change in a warming climate as its atmospheric drivers vary (Gori et al., 2022;Grinsted et al., 2013;Marsooli et al., 2019).Our analysis method, however, captures both changes in NTR directly related to MSL rise, and those that are driven by atmospheric proceses that are correlated with, but not caused by, MSL rise.Moreover, PMST is deliberately conservative, and we do not extend the analysis beyond 2050, because of the likelihood that NTR and MSL dynamics will change in ways that cannot be predicted with our methods.Thus, lack of a model for how NTR will change in the future is a conceptual limitation, but up to 2050, probably not a practical one.
Finally, the correlations between the various components of the water level spectrum that the PMST procedure reveals are also practically important in themselves, not just for their contribution to the PMST statistic.If an engineer knows that, for example, tides and/or surge increase at a particular location as MSL rises, this suggests both more detailed investigations, and a conservative design approach.This would be especially true in situations where the water level record is short and storm surges large.Thus, the PMST calculation may have value even when the sea level record is undesirably short.

Conclusions
Overall, we detect a significant correlation between MSL variability and PMST at 89% of locations studied around the globe.In 42% of these locations (37% of the total studied gauges) MSL is rising over time, and there is a significant positive correlation between MSL variation and PMST.Thus, ignoring nonlinear interactions between SLR and storm tide may cause underestimation of coastal flooding risk in the future at almost one-third of studied locations (e.g., in western Pacific, northern Atlantic, and northern coasts of Australia).
In 72% of the studied locations, we expect a higher ECWL by the mid-century (where PMST shows positive trends), though the rates and pattern of these increased flood hazards are different between various systems.Among all studied sites, the 27% of systems with micro-and meso-tidal range and with diurnal tidal regime are most sensitive to SLR.This means there is no single recipe for SLR adaptation around the world.Depending on the local MSL trend, the tidal range and regime (diurnal vs. semidiurnal) of the system of interest, and the strength of interactions between various ECWL components (MSL, HT, and NTR), different preventive measures should be implemented to ensure resilience in the face of increasing flood hazards.

Figure 1 .
Figure 1.Observed changes in the drivers of coastal flooding.(a) rate of mean sea level (MSL) rise (size of the circle at each location represents the length of available record at that gauge); (b) rate of change in Potential Maximum Storm Tide (PMST) per millimeter (mm) change in MSL (size of the circle at each location represents the correlation coefficient (CC) between PMST and MSL); (c) average tide (harmonics tides) to non-tidal residual (non-tidal residuals) ratio (size of the circle at each location represents the average sum of amplitude for the five major tidal constituents; K 1 , O 1 , M 2 , N 2 , and S 2 , and the three overtides M 4 , MK 3 , and MS 4 ); (d) rate of change in amplitude of overtides per mm change in MSL (size of the circle at each location represents the CC between M 4 /M 2 2 and MSL).Black circles in panels b and d represent the points with insignificant rank correlation ( p-value > 0.05).

Figure 2 .
Figure 2. Trends and relationships at Wyndham, Australia.(a) Historic trends in mean sea level (MSL) and observed annual maximum still water level versus annual maximum Potential Maximum Storm Tide (PMST); (b) cumulative distribution function based on observed annual maxima and PMST estimates; (c-h) the relationships of MSL with PMST, non-tidal residuals and diurnal/semidiurnal tidal constituents and over tides at Wyndham, Australia (Lat: 15.4533, Lon: 128.1017), a macro-tidal system with a mixed (mainly Semi-Diurnal) tidal regime.Red lines represent the linear regression line between variables in each panel.

Figure 3 .
Figure 3. Trends and relationships at Lvsi, China.(a) Historic trends in mean sea level (MSL) and observed annual maximum still water level versus annual maximum Potential Maximum Storm Tide (PMST); (b) cumulative distribution function based on observed annual maxima and PMST estimates; (c-h) the relationships of MSL with PMST, non-tidal residuals and diurnal/semidiurnal tidal constituents and over tides at Lvsi, Jiangsu, China (Lat: 32.1333, Lon: 121.6167), a macro-tidal system with a semi-diurnal tidal regime.Red lines represent the linear regression line between variables in each panel.

Figure 4 .
Figure 4. Trends and relationships at Baltimore, MD.(a) Historic trends in mean sea level (MSL) and observed annual maximum still water level versus annual maximum Potential Maximum Storm Tide (PMST); (b) cumulative distribution function based on observed annual maxima and PMST estimates; (c-h) the relationships of MSL with PMST, non-tidal residuals and diurnal/semidiurnal tidal constituents and over tides at Baltimore, MD, USA (Lat: 39.2667, Lon: 76.5783).Red lines represent the linear regression line between variables in each panel.

Figure 5 .
Figure 5. Trends and relationships at Nantes, France.(a) Historic trends in mean sea level (MSL) and observed annual maximum still water level versus annual maximum Potential Maximum Storm Tide (PMST); (b) cumulative distribution function based on observed annual maxima and PMST estimates; (c-h) the relationships of MSL with PMST, non-tidal residuals and diurnal/semidiurnal tidal constituents and overtides at Nantes Usine Bluree, France (Lat: 47.1935, Lon:1.6337), a macro-tidal system with a semi-diurnal tidal regime.Red lines represent the linear regression line between variables in each panel.

Figure 6 .
Figure 6.Projected changes in the drivers of future coastal flooding.(a) Estimated change in coastal extreme water level under projected Sea-level rise (SLR) by 2050 (SLR from Ref (Kopp et al., 2017)).(b-d) Normalized difference between the scenarios with and without considering nonlinear interactions between still water level components under 50th, 5th, and 95th percentile of SLR projection in year 2050, respectively.Size of the circles in panel b shows the estimated extreme water level in the future above mean sea level in year 2000.

Figure 7 .
Figure 7. Response of different tidal systems to Sea-level rise (SLR).Estimated historic (panels a and b) and projected (panels c and d) change in coastal extreme water level.Future estimates are under 50th percentile of projected SLR by 2050 (SLR under representative concentration pathway 8.5 from Kopp et al. (2017)).