Amplification factors for extreme sea level frequency have problematic features as a metric of coastal hazard

The future projected frequency of a specified baseline extreme sea level (ESL), often called the amplification factor (AF), is extensively used as a metric of evolving coastal flood hazard with sea level rise (SLR). The baseline ESL is typically analyzed using extreme value analysis, and the SLR is added to the resulting distribution. In the presence of SLR uncertainty, it is natural to analyze AFs probabilistically. I derive probability density functions (PDFs) of AF, given uncertainty distributions of SLR. If the ESL distribution is modeled as Gumbel and the SLR distribution as normal, then the AF distribution is log normal. However, in active tropical cyclone regions, ESL often has a longer tail than Gumbel, and a Frechet (Type-II) Generalized Extreme Value (GEV) is more appropriate. In this case, I show that the AF distribution has a divergent mean, preventing its use as a hazard metric. In addition, I show that for Frechet ESL, the AF cannot even be defined for SLR above a threshold β/ξf0−ξ, where f 0 is the specified baseline frequency (e.g., f 0 = 0.01 yr−1 for the 100-year exceedance), β is the GEV scale parameter and ξ the shape parameter. This SLR threshold can be as low as 0.5 m in the southeast US and Caribbean, within reach mid to late century. Above the threshold, ESL at all frequencies exceeds the baseline reference frequency, preventing the calculation of AF. The resulting probabilistic distribution of AF is insensitive to SLR above the threshold. These features detrimentally impact the utility of AF as a hazard metric. Frechet distributions are appropriate and commonly used for ESL in tropical cyclone regions, but AFs applied to such distributions must interpreted with caution. In such regions, coastal risk managers should consider other flood hazard metrics, such as probabilistic estimates of flood depth.


Introduction
Extreme sea level (ESL), exceptionally high coastal water levels caused by tides and storm surge (Gregory et al 2019), is a major driver of coastal flooding.Sea-level rise (SLR) exacerbates flooding due to coastal storm tide by increasing the background water level on top of which the storm tide propagates.Research in projecting future storm tide levels with SLR is extremely active (e.g., Oppenheimer et al 2019, Kirezci et al 2020, Sweet et al 2022), and calculating coastal flood loss in response to storm tide and SLR is a crucial component of climate risk analysis.
A widely used metric of the evolving hazard is the future projected frequency in the exceedance of a specified historical ESL.For example, the specified ESL might be the level that was exceeded historically with an average annual frequency of 0.01 yr −1 (the 'hundred-year level'), while at a future SLR level that same ESL might be exceeded with a frequency of 0.03 yr −1 .The fractional change, in this case 3, is often termed the amplification factor (AF).The conclusions made here apply equally to the projected frequency and the AF, as their difference, multiplication by a constant reference frequency, is immaterial.Frequency and AF terminology are both used here.
Examples of the use of AF are numerous.Lin et al (2016) projected increased annual frequencies of exceeding Hurricane Sandy's storm-tide level on New York City.Vitousek et al (2017) estimated AFs globally to the baseline 50-year exceedances, including waves, and showed that regions with relatively smaller historical ESL variations have the greatest AFs.Rasmussen et al. (2018) use AFs to examine SLR impacts under several temperature stabilization targets.Ghanbari et al (2019) developed a statistical model encompassing both nuisance and extreme coastal flood on the US, using AFs as the metric for their projected increase in hazard.The IPCC 2019 report on oceans and the cryosphere summarized changes in coastal ESL in part with AFs (Oppenheimer et al 2019).Frederikse et al (2020) concluded that tropical coastal sites are especially susceptible to increased frequency of the historical 100-year ESL, with many such sites even having mean sea level above the 100-year reference level by 2100.Taherkhani et al (2020) concluded that many US coastal regions experience approximately exponential increase in AF of the baseline 1-in-50-year frequency.Rashid et al (2021) used AFs to diagnose the large impacts of interannual and decadal variability on ESL, while Tebaldi et al (2021) identified global warming levels beyond which coastal regions will experience historical 0.01 yr −1 frequencies daily.Finally, Hermans et al (2023) estimated future years at which AFs exceed regional flood protection standards.
SLR projections have many sources of uncertainty (Kopp et al 2023a), and this uncertainty impacts AF.All the studies listed above acknowledge the large uncertainty in SLR projections and, at least qualitatively, the impact that uncertainty has on AFs.A smaller number of studies have propagated probabilistic SLR descriptions to probabilistic AF.Buchanan et al. (2017) use uncertainty distributions of SLR to infer distributions of AF.Goodwin et al (2017) and Howard and Palmer (2020) analyze AF distributions and their means.Components of the work of Goodwin et al (2017) and Howard and Palmer (2020) are based on the theoretical work of Hunter (2012).Strictly speaking, the results from Hunter (2012) are limited to Gumbel ESLs, but Goodwin et al (2017) and Howard and Palmer (2020) generalize certain aspects beyond Gumbel.
Here, I address the impact of probabilistic descriptions of SLR on probabilistic descriptions of projected frequency (and therefore, AF).I show that AF has two features that may compromise its use in some cases as a coastal hazard metric.These features arise in regions with episodic intense ESL, such as tropical cyclone regions, where the ESL distribution is commonly and appropriately summarized with heavy-tailed Frechet (Type II) GEV distributions (e.g., Tebaldi et al 2012, Zervas 2013, Marcos and Woorworth (2017)).Firstly, in such regions, the AF has a distribution whose tail is heavy enough that the mean AF mean is infinite.Any practitioner of coastal-risk analysis who uses AF as a hazard metric under SLR and seeks to incorporate uncertainty probabilistically will be stymied in attempts to estimate mean AF.Secondly, for Frechet ESL, AF cannot be defined for SLR above certain thresholds determined by the GEV parameters, and these threshold SLRs are well within reach by late 21st century.For SLRs above threshold, ESL at all frequencies is higher than the baseline ESL of the specified reference return period.This feature prevents AFs from being able to diagnose the impact of SLR these thresholds, at least in active tropical cyclone regions.In such regions, coastal managers cannot base latecentury risk analysis on AFs.
The paper is organized as follows: First, the relationship between ESL and annual occurrence frequency in terms of GEV parameters is motivated.Then, I derive probability density functions (PDFs) of AF, assuming uncertainty distributions of SLR.In the case of Frechet (Type-II) GEV ESL, the mean of the AF PDF is divergent.In addition, it is shown that AF cannot be defined for SLR above a threshold value determined by the GEV parameters.I then conclude with discussion on the relevance of the results to diagnosing ESL hazard.

Occurrence frequency and GEV parameters
This section motivates the relationships between an ESL value, X, and the annual frequency of ESL exceeding X.These relationships can be found, for example, in Hunter (2012) in the Gumbel case and Palutikof et al (1999) in the GEV case.However, it is useful for subsequent development to derive the results in the current context.
The GEV distribution is derived from a Poisson process for an event having a maximum value X' exceeding X in interval τ.Thus, the probability of waiting longer than one year between events exceeding X is where f is the average number of events exceeding X in the interval τ.This is the equal to the probability that no event exceeds X in τ.We are interested in annual rates, and henceforth we use τ = 1 year henceforth.Consider the GEV distribution for the maximum annual exceedance level X.The probability of maximum X¢ less than X is the GEV CDF at X: where X 1 , ) μ is the location parameter, β is the scale parameter, and ξ is the shape parameter.The probability of annual maxima less than X equals the probability that no event exceeds X.Thus, (1) and (2) can be equated.Solving for X yields The timescale 1/f is not the return period.The return period is the reciprocal of the annual probability of at least one exceedance, and therefore cannot be less than one year.By contrast, 1/f is the average time between successive exceedances, sometimes called the average recurrence interval (ARI), and 1/f can assume any positive value.It may seem counterintuitive to consider values f >1 yr −1 (ARI < 1 yr) in the context of GEV for annual exceedances.The GEV is constructed from time series of annual maxima, and the GEV cumulative distribution function (CDF) is the probability p(X) that the maximum annual ESL is less than X.However, there is no contradiction.The GEV CDF at X is the probability that the maximum annual value X' is less than X, even when the underlying average frequency, f, of exceeding X, is high, including f > 1 yr −1 .That is, there is non-zero probability that the maximum X' in a year is less than X, even when the annual frequency of exceeding X is greater than 1 yr −1 .

Relating SLR distributions to projected frequency distributions
The PDF of the projected annual frequency of storm-tide induced coastal ESL is now derived, assuming a GEV PDF for the historical baseline period and normally distributed uncertainty on SLR.SLR has many sources of uncertainty, both quantifiable and unquantifiable (Kopp et al 2023b), and a number of studies argue compellingly for non-normal skewed SLR distributions driven in large part by uncertain ice sheet decline (De Winter et al 2017, Bamber et al 2019, Robel et al 2019).The purpose here is to make certain general observations about the impact on AF of the combination of SLR distributions and GEV ESL.The conclusions that, for a Frechet GEV ESL, the mean AF diverges is true for a range of SLR distributions, as discussed subsequently in section 5.Moreover, the conclusion that the AF cannot be defined for SLR above a certain threshold is independent of the SLR distribution.However, to illustrate with an analytic frequency PDF, a normal distribution of SLR is assumed.
The linearity between X and f ) in equation (3) allows a straightforward derivation of a frequency PDF.In figure 1(a), the baseline linear relationship between X and f ) is shown.Also shown is the reference ESL, X 0 , corresponding to the reference frequency, f 0 (e.g., f 0 = 0.01 yr −1 for the reference 100-year ESL exceedance).In figure 1(b), the baseline ESL curve is shifted upward by a mean SLR, μ SLR , and the new value, ) that corresponds to the same exceedance X 0 is shifted leftward.The linear geometry can be used to infer μ f .The slope of the curve, β from equation (3), can be seen from figure 1(b) to be μ SLR /Δ, where f 1 .
Equating μ SLR /Δ to β and solving for μ f yields In figure 1(c), the ESL curve is shifted upward in addition by one standard deviation, σ SLR , in SLR.The increase induces a further leftward shift in the f ) that corresponds to the same X 0 .Similar to the mean μ f , the standard deviation, σ f , in f ) corresponding to σ SLR can be inferred from the linear ) corresponding to the same baseline X 0 is indicated by the second dashed line.The shift in f C) A standard deviation in SLR, σ SLR , is added (red), shifting the curve further upward.The additional shift leftward, σ f , in f ) that corresponds to the same baseline X 0 , is indicated by the third dashed line.
geometry.From figure 1(c), the slope β is Because of the linearity of the ESL curve and the assumption of normally distributed SLR, the distribution of ) values that corresponds to X 0 is also normal.Thus, the PDF for the variable g f f ) is normal with mean μ f and standard deviation σ f .To convert to a PDF in f, note that Or, with the normal SLR distribution, In the limit 0, x  the GEV is Gumbel, and (8) is log-normal.Examples of PDF (8) are shown in figure 2.
Note that this PDF can readily be re-expressed in terms of AF by simply replacing f with AF f .0 4.

Divergent mean for frechet GEV
For the Frechet GEV realm (0 < ξ < 1), the frequency PDF (8) has an infinite mean, which prevents its use as a hazard metric.The mean's divergence can be seen by noting that, in the limit of large f, the argument in the exponent approaches a finite value.Meanwhile, the pre-factor varies as f .
) Thus, in computing the mean, the integrand of the first moment, f p( f ), limits to f − ξ, which results in divergence for ξ < 1.The long tail is illustrated by comparing figure 3, the Gumbel ESL case, and figure 4, the Frechet case.Both figures show a series of SLR increments (the parallel curves) and the corresponding series of frequencies that result in the same baseline ESL (the vertical dashed lines).If the SLR distribution is assumed normal, then the distribution of log( f ) for Gumbel ESL (figure 3) is also normal, as the ln( f ) values corresponding to X 0 (vertical dashed lines) are equally spaced.For Frechet (figure 4), however, the ln( f ) corresponding to the same X 0 (vertical dashed lines) bunch together at small ln( f ) and expands at large ln( f ), resulting in a longer tail than the Gumbel case.The  8).Black is the Gumbel-derived frequency PDF (log-normal, ξ = 0), red is derived from GEV with ξ = 0.2 (Frechet), and blue is derived from GEV with ξ = −0.2(Weibull).The corresponding black and blue vertical dashed lines indicate the mean frequencies, 2.4 yr −1 (Gumbel) and 6.5 −1 years (Weibull).For Frechet, the mean of the frequency PDF is infinite.The reference frequency in this example is f 0 = 0.01 yr −1 .3, here the relationship is not log-linear, but is instead increasingly shallower with f.As SLR increases, the frequencies that correspond to the baseline ESL exceedance X 0 are increasingly more widely spaced, and the log-frequency distribution is skewed to heavy tails, as illustrated in the distribution at bottom.In addition, there is a critical SLR increment above which the ESL curve never reaches as low as X 0 .At SLR above this threshold (about 1.18 m in this example) future ESL is higher than the reference ESL at all frequencies.ESL curves above this threshold are indicated with dashed lines.A truncation is shown in the normal SLR distribution at left, indicating the SLR values to which the frequency is not sensitive.
projected frequencies (and hence AF) increase rapidly enough that the mean of the PDF diverges.(As shown below in section 5, the divergence occurs for a wide range of SLR distributions, not just normal.) Several studies have hinted at the divergence of mean AF.Vitousek et al (2017) and Buchanan et al (2017) noted the high sensitivity of AF to GEV parameters, especially the shape parameter.They also note the fact that mean of the AF distribution is greater than the AF in response to the mean SLR.Buchanan et al (2017) calculated a mean AF in response to the SLR uncertainty distributions of Kopp et al (2014), showing strong sensitivity of the mean to the SLR uncertainty.Rasmussen et al (2018) estimated extremely high AFs in Kushimoto, Japan, where the ESL distribution is heavy tailed.For example, AF = 1462 by 2100 for the reference 10-year ESL and AF = 41479 for the reference 500-year ESL (2.0 °C stabilized global mean surface temperature increase).Rasmussen et al (2018) further note that their AF estimates are sensitive to truncations in the uncertainty sampling scheme and may not converge within their truncated range.
I am not aware, however, of the divergence of the mean AF for the Frechet GEV being made explicit previously.The divergence is a consequence of analyzing the distributional frequency response to an uncertainty distribution of SLR, combined with a positive shape parameter (Frechet) in the ESL GEV distribution.Frechet distributions must be interpreted with caution.As noted by Vitousek et al (2017), positive shape parameter for ESL does not respect physical upper bounds on surge and tide.Statistical fits of storm tide with positive shape parameter are driven by the presence of outlier ESL in finite time series, for example, from rare and extreme tropical cyclones.With physical upper bounds enforced, the tail of the ESL distribution would be truncated, and the mean AF would be finite.However, the mean would still depend on the poorly-known storm-tide upper bounds.

Limit on SLR sensitivity for frechet GEV
The Frechet GEV has a minimum allowed ESL exceedance value.In the limit of high frequency, equation (3) has the minimum X of In the frequency PDF (8), as f ranges from zero to infinity, the associated X ranges from infinity to X min .As seen in figure 4, when X min exceeds the historical baseline reference ESL, X 0 , no projected future frequency exists that generates the ESL X 0 .This implies an upper threshold value of SLR, above which a projected frequency corresponding to X 0 cannot be found, an AF cannot be defined, and further SLR cannot influence the frequency (or AF) PDF.To obtain this threshold, consider the reference ESL from equation (3) where f 0 is the reference frequency, e.g., f 0 = 0.01 yr −1 for the 100-yr exceedance.With SLR, the location parameter shifts from its baseline value, μ, to μ+ μ SLR , and X min becomes X min increases with SLR.Once X min exceeds X 0 , there is no projected frequency that can produce X 0 .Above that threshold value of X min , all frequencies result in X above X 0 .The upper portion of the SLR distribution, where X min > X 0 , does not impact on the frequency (or AF) distribution.The threshold X min occurs when Solving for the threshold SLR, Finally, substituting from (10) for X 0 yields: This threshold SLR, above which no AF can be defined and no contributions are made to the frequency distribution, depends only on the baseline GEV scale and shape parameters.Figure 4 illustrates the threshold magnitude with GEV parameter values from table A of Zervas (2013) for Charleston, South Carolina: β = 0.094 m and ξ = 0.234.For the 100-yr reference ( f 0 = 0.01 yr −1 ), this gives μ SLR = 1.18 m.Further SLR above 1.18 m does not contribute to the AF PDF.As another example, for Vaca Key, Florida, Zervas (2013) estimates β = 0.057 m and ξ = 0.325, which yields μ SLR = 0.78 m From figure 3 of Vitousek et al (2017), the North Caribbean, Florida, and the US Gulf and southeast coastlines have ξ approximately 0.2 and β in the range 0.04 m to 0.06 m, implying a threshold SLR in the range 0.5 m to 0.8 m.Thus, for typical observationallyestimated Frechet GEV parameters, a threshold SLR, above which AF cannot be defined, is well within range of projected values for mid to late century in certain regions.For example, NASA's IPCC AR6 SLR projection tool at sealevel.nasa.gov/data_tools/17(Fox-Kember et al (2021), Kopp et al 2023b, Garner et al 2021) indicates SLR reaching 0.5 m on Key West FL in about 2070 under the scenario SSP3-7.0.Note that in the Gumbel limit of (3) ( 0 x  ), the threshold SLR is infinite.All SLR contribute to the Gumbel frequency distribution.The threshold SLR issue does not arise, as X = μ − β ln(f) has no lower bound on X.

Other SLR distributions
The existence of an upper SLR threshold of frequency-distribution sensitivity does not depend on the nature of the SLR distribution.The to obtain the threshold (14) depend only on the existence of a minimum ESL, X min .The divergence of the mean projected frequency is also largely independent of the SLR distribution.To see this, note that in equation ( 7), as f increases, X limits to X min and g(f) limits to −1/ξ.Therefore, if the SLR PDF is nonzero at X min , then p(g(f)) is nonzero, and the PDF (7) limits to f − ξ −1 .As before, the integrand of the first moment then limits to f − ξ, which diverges for 0 < ξ < 1.In other words, if the ESL is Frechet distributed and there is non-zero probability of SLR above f , ) then the mean projected frequency is infinite.

Discussion and conclusions
In many coastal regions, such as those with tropical cyclone activity, extreme sea levels (ESL) are best described by a Frechet GEV distribution, which has a longer tail than the Gumbel distribution.Sea level rise (SLR) increases the frequency of ESLs above a given threshold, with the fractional frequency change termed the amplification factor (AF). SLR has uncertainty, and this translates to an uncertainty distribution of AF.I have derived certain properties of AFs with Frechet ESL distributions that compromise AF as a coastal hazard metric in this context: 1. AF cannot be defined when SLR is above the value f , ) where f 0 is the baseline frequency specifying the reference ESL (e.g., f 0 = 0.01 yr −1 for the 100-year exceedance), β is the GEV scale parameter and ξ the shape parameter.For higher SLR, all frequencies result in ESL above the reference ESL.In effect, when the threshold SLR is exceeded, the location in question is underwater all the time.No baseline reference frequency can match permanent submersion, and hence no AF can be defined.

If an SLR uncertainty probability density function (PDF)
is used to derive a corresponding uncertainty PDF for AF, then the AF PDF is not sensitive to SLR values above the threshold f .) are well within range for the end of century, e.g., an estimated threshold of 1.18 m in Charleston, SC.Therefore, these features may limit the utility of projected frequency (and AF) as a diagnostic of coastal flood hazard metric under SLR, at least in regions of large episodic ESL, such as tropical cyclone regions, where Frechet GEV is appropriate.A divergent mean AF is merely inconvenient.An infinite mean of a distribution is not a particularly useful summary, but other summaries, such as the mode and median, are still available.The threshold SLR above which AF cannot be defined is more problematic.Any coastal-risk practitioner using AFs in such a situation would find that AF values get arbitrarily large before the threshold SLR of interest is reached, and after the threshold is exceeded, no AF can be defined.One would like a diagnostic of coastal hazard under SLR to be sensitive to the full plausible range of SLR.If it is not, the diagnostic has limited utility.In such situations, coastal risk managers are advised to avoid AFs and rely instead on direct estimates of flood depth (preferably probabilistic, with uncertainty).Flood depth, though more difficult to calculate due to the requirements for high-resolution elevation data, is not subject to these problematic features.
AFs have other clear limitations as coastal hazard metrics.As highlighted by Rasmussen et al 2022, AFs are derived exclusively from the physical hazard, extreme sea level.They don't take into account the exposure and vulnerability necessary to estimate risk.Even as a hazard metric, AFs alone don't indicate the absolute ESL value, only its frequency change.For example, a 100-year AF of 10, indicating a 10-fold increase in frequency of the reference 100-year ESL exceedance, may have no bearing on hazard if that reference ESL amounts to negligible flood depth.This is partly because AFs only measure change in ESLs at the coast, without taking into account flow pathways and elevation (e.g., Rasmussen et al 2022).The theoretical analysis here has shown that, when applied to Frechet (long-tailed) ESL distributions, AFs have additional limitations based purely on their mathematical properties.Caution is therefore advised in the use of AFs as a hazard diagnostic in active tropical cyclone regions, where Frechet GEV ESLs are appropriate.

Figure 1 .
Figure 1.Graphical illustration relating ESL with SLR to frequency.ESL is plotted as its linear function of f 1 x -x -/ ( ) from equation (3), with slope β. (A) The baseline relationship (blue), with the ESL, X 0 , corresponding to the reference frequency f 0 indicated by dashed lines.(B) A mean value of SLR, μ SLR , is added to the baseline (orange), shifting the curve upward.The new value, μ f , of f 1 x -x -/ () corresponding to the same baseline X 0 is indicated by the second dashed line.The shift in f 1 x -

Figure 2 .
Figure 2. Examples of frequency PDF of equation (8).Black is the Gumbel-derived frequency PDF (log-normal, ξ = 0), red is derived from GEV with ξ = 0.2 (Frechet), and blue is derived from GEV with ξ = −0.2(Weibull).The corresponding black and blue vertical dashed lines indicate the mean frequencies, 2.4 yr −1 (Gumbel) and 6.5 −1 years (Weibull).For Frechet, the mean of the frequency PDF is infinite.The reference frequency in this example is f 0 = 0.01 yr −1 .

Figure 3 .
Figure 3. ESL versus log frequency for a Gumbel fit by Zervas (2013) for Boston, MA (μ 0 = 0.764 m, β = 0.133 m).The baseline ESL curve is shown blue, and projected SLR in 0.25 m increments from the baseline (including negative SLR) are shown red.The SLR values are imagined to be distributed normally, as indicated schematically by the distribution on the right.The tall vertical black line indicates the reference frequency, f 0 = 0.01 yr −1 .The horizontal black line indicates the baseline reference ESL, i.e., the ESL X 0 that is exceeded annually with frequency f 0 , here about 1.34 m.The short vertical lines indicate the projected frequencies that correspond to X 0 for the various SLR increments.Due to the log-linearity of the relationship, the frequency distribution is also normal in ln( f ), as indicated schematically by the distribution below; i.e., it is log-normal in f.

Figure 4 .
Figure 4.As in figure3, but here for an example of Frechet GEV ESL, taken fromZervas (2013) for Charleston, SC (μ 0 = 0.526 m, β = 0.094 m, and ξ = 0.234), with X 0 = 1.30 m.Compared to the Gumbel example of figure 3, here the relationship is not log-linear, but is instead increasingly shallower with f.As SLR increases, the frequencies that correspond to the baseline ESL exceedance X 0 are increasingly more widely spaced, and the log-frequency distribution is skewed to heavy tails, as illustrated in the distribution at bottom.In addition, there is a critical SLR increment above which the ESL curve never reaches as low as X 0 .At SLR above this threshold (about 1.18 m in this example) future ESL is higher than the reference ESL at all frequencies.ESL curves above this threshold are indicated with dashed lines.A truncation is shown in the normal SLR distribution at left, indicating the SLR values to which the frequency is not sensitive.