A flood damage allowance framework for coastal protection with deep uncertainty in sea-level rise

D.J. Rasmussen; Maya K. Buchanan; Michael Oppenheimer; Robert E. Kopp

REVIEW 3 major objections 6 minor 97 references

A flood damage allowance framework for coastal protection with deep uncertainty in sea-level rise

T0 review · 3 major / 6 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read To hold annual flood losses steady under sea-level rise, coastal defenses may need to be nearly twice as high as hazard-based allowances suggest.

desk verdict A useful and genuine extension of hazard allowances to damage-based design heights, with a sound core equation and some unvalidated choices that make the headline numbers illustrative rather than definitive. read the letter →

arxiv 1908.02844 v1 pith:MH535RTV submitted 2019-08-07 physics.ao-ph

D.J. Rasmussen , Maya K. Buchanan , Robert E. Kopp , Michael Oppenheimer This is my paper · ORCID

classification physics.ao-ph

keywords sea-levelrisecoastalfloodprotectiondamageallowanceannualaveragelossdeepuncertaintyAntarcticicesheetextremesealevelbenefit-costanalysis

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper proposes that the design height of coastal flood protection should be set by the financial damage it prevents, not just by the water level it blocks. It defines a 'damage allowance' as the height needed to keep the annual average flood loss at a chosen target, traditionally today's loss, under uncertain future sea-level rise. Using Manhattan as a test case, it shows the damage allowance can be about twice the conventional hazard allowance: a 1.70 m levee in 2070 versus a 0.86 m hazard allowance, because damages are nonlinear and protection structures can fail. The framework also shows that after mid-century under high emissions, the required height depends strongly on what one believes about Antarctic ice-sheet collapse, with 2100 levee allowances spanning roughly 1.8 to 3.4 m.

What carries the argument

The load-bearing object is the damage allowance A defined by Eq. (2), which balances the protected expected annual loss under sea-level rise against the current annual average loss. The protected damage functions encode strategy-specific behavior: levees and surge barriers fail with an exponential fragility curve; surge barriers leave gates open below a closure threshold; coastal retreat removes assets below A with a compliance parameter; elevation shifts the damage function upward. Uncertainty in sea-level rise is represented by an 'effective' distribution formed as a weighted average of two bounding cumulative distributions (a probability box), with weight $\beta_c$ reflecting belief in Antarctic ice-sheet collapse and with the upper tail truncated at a chosen AISmax. The allowance is the height that balances the two sides of the conservation equation.

What would settle it

Run the damage-allowance calculation on synthetic flood records drawn from a known sea-level distribution whose upper tail is longer than the chosen AISmax cutoff; if Eq. (2) fails to hold simulated annual losses at the target, the arbitrary truncation is load-bearing. Alternatively, track the Battery tide gauge over the coming decades: if realized sea level repeatedly falls outside the p-box bounds, allowances conditional on those bounds are biased.

Watch

Extended reading notes

Core claim

The central claim is that holding flood risk constant in financial terms requires solving a conservation-of-damage equation: the expected annual loss under future sea-level rise, after adding protection of height A, must equal the current annual average loss. The paper constructs protected damage functions for four strategies—elevation, levee, storm surge barrier, and coastal retreat—each with its own failure or compliance behavior, and solves Eq. (2) for A. For Manhattan's 2070 conditions with a levee, the resulting damage allowance is 1.70 m, nearly twice the 0.86 m hazard allowance that only keeps the frequency of extreme sea levels constant; by 2100 under RCP8.5 the levee allowance ranges from 1.8 m to 3.4 m depending on assumptions about Antarctic melt and collapse likelihood. The authors argue this makes the damage allowance a direct input to benefit-cost and cost-effectiveness analysis, because it quantifies avoided damages in dollars.

Load-bearing premise

The framework's headline numbers rest on an 'effective' sea-level distribution built by averaging two bounding projections with a subjective weight and cutting the upper tail at an arbitrary maximum Antarctic melt; the paper concedes that this truncation could change results significantly and is not tested.

Editorial extensions

If this is right

Hazard-based allowances understate the vertical protection needed when the goal is to stabilize dollar losses, because damage grows faster than water level and defenses can fail.
Design heights should be reported as ranges over Antarctic ice-sheet assumptions rather than as single values; under RCP8.5 the 2100 levee allowance spans 1.8 to 3.4 m.
Strong emissions reductions make allowances far less sensitive to ice-sheet beliefs: under RCP2.6 the spread across assumptions is at most about 0.6 m.
The allowance can be used directly as the benefit side of a benefit-cost or cost-effectiveness analysis, because it specifies the avoided annual loss.
Different strategies imply different allowances: coastal retreat needs less height than a levee because it removes assets rather than defending them, while a surge barrier needs more than a levee because gates remain open up to the closure threshold.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The relative gap between damage and hazard allowances is likely to vary by city: places with steep depth-damage curves or concentrated assets just above the waterline would need proportionally taller defenses, a testable extension of the Manhattan result.
The p-box weighting could be interpreted as a set of priors; combining the resulting allowance ranges with robust decision rules, such as minimax regret over the allowance set, would avoid committing to a single $\beta_c$.
The 'frozen city' assumption may understate allowances where protection encourages development behind the levee; re-running the model with endogenous asset growth would test the size of this levee effect.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Short version: this is a real contribution, worth a serious referee, but treat the Manhattan numbers as conditional illustrations until a few loose ends are tightened. The damage-allowance concept - design heights that hold annual average loss constant rather than event frequency - is not in the earlier hazard-allowance literature, and the four-strategy treatment is a genuine extension. Equation 2 is algebraically sound given its stated assumptions, and the strategy-specific damage functions are clearly laid out. The authors also deserve credit for acknowledging several caveats in the discussion, including the arbitrary AISmax truncation and the limitation of a frozen city. The soft spots are three. First, the AISmax truncation is admittedly arbitrary and can significantly affect the late-century spread; the paper says this, but the headline tables lean on it heavily. Second, the reported allowances are point values with no propagated uncertainty, even though GPD parameter uncertainty and SLR projection uncertainty are both sampled elsewhere in the chain; that makes the 0.1 m precision in Tables 1 and 2 look stronger than the inputs support. Third, the depth-damage cubics in Sec. A.4 are fit over roughly 0-3 m of inundation depth, and the stated polynomials turn negative above about 6.3 m. No clipping or renormalization is given. This is a real defect in the written formulation, and it should be fixed and quantified. That said, the stress-test claim that all reported allowances are artifacts is too strong: Fig. 1B shows a monotone D(z) to 7 m, so the practical effect of the unclipped tail is probably small, but the authors need to confirm it rather than leave it implicit. The circularity concern from the reader does not land. The SLR inputs come from published Kopp et al. projections, beta_c is a user input, and nothing is fitted to reproduce the allowances. Self-citation appears because two authors are on the projection papers, but that is not a flaw here. The code release is incomplete, with a placeholder in the acknowledgments and no versioned workflow; that should be fixed. Bottom line: the framework is coherent and the central idea holds up. I would send this to peer review, asking for a clipped or reformulated damage function with a monotonicity check, uncertainty propagation or an explicit statement that the tables are best-estimate illustrations, and a complete code archive. It is a decision-support template worth engaging with, not a final design manual.

Referee Report

3 major / 6 minor

Summary. The paper proposes a 'damage allowance' framework for sizing coastal protection under uncertain sea-level rise: the design height A is the vertical adjustment that keeps the annual average flood loss (AAL) at a chosen target, most often the current AAL, when the full distribution of future sea-level rise is considered. The central balance is Eq. (2), equating the current AAL to the AAL under future sea level with a strategy-modified damage function. Strategy-specific damage functions are given for elevation, levee, storm surge barrier, and coastal retreat in Secs. 2.1.1–2.1.3. Deep uncertainty in Antarctic ice-sheet behavior is represented by a p-box whose edges are the Kopp et al. (2014) and Kopp et al. (2017) local projections, combined by a user-specified weight βc and an upper truncation AISmax (Sec. 2.2, Eq. A.6). The framework is applied to Manhattan, with results including a 2070 levee allowance of 1.70 m versus a hazard allowance of 0.86 m (Sec. 3.1) and 2100 levee allowances of 1.8–3.4 m across AIS assumptions (Table 1).

Significance. If the numerical results are robust, the paper makes a useful contribution by extending hazard allowances to financial risk and by making the decision-maker's subjective AIS assumptions explicit. The strengths include the transparent conservation-of-damage equation, the clear strategy-specific damage functions, the use of published probabilistic SLR projections, an open-source code base, and an honest statement of the p-box truncation's acknowledged limitations. The framework is a reasonable reduced-form input to benefit-cost or cost-effectiveness analysis, which is the paper's stated aim.

major comments (3)

[Sec. A.4 (also Eq. 2, Tables 1-2)] The depth-damage polynomials φ_hrise and φ_res are least-squares fits to USACE data over roughly x ∈ [0,3] m (Figs. A.3 and A.4), but they are used without any clip or renormalization for all flood depths in Eq. (A.3) and hence in Eq. (2). Since φ_hrise(x) = 0.142 + 0.0541x − 0.00368x² − 0.00133x³ becomes negative for x above about 6.3 m (and φ_res turns negative above about 6.7 m), the aggregate damage function D(z) is not guaranteed to be non-negative or monotone, and with the 2100 p-box extending to roughly 3 m of SLR, the z-integration in Eq. (2) samples this invalid regime. The reported allowances in Tables 1–2 and Fig. 5, including the headline 1.70 m levee versus 0.86 m hazard allowance, may therefore be affected by an extrapolation artifact. Please re-estimate or clip the depth-damage functions to a physically valid range and re-run the allowance calculations, or demonstrate numerically that the results are insensitive to this choice.
[Sec. A.8, Eq. A.6] The effective SLR distribution P̃(βc, AISmax, t) = βc·Phigh + (1−βc)·Plow is built on an arbitrarily truncated AIS contribution, and the paper acknowledges in Sec. A.8 that this truncation 'could impact results in a significant way, but is not investigated here.' Because Tables 1–2 and the associated figures condition every headline allowance on AISmax, this acknowledged limitation is load-bearing for the main numerical claims. A simple sensitivity sweep over AISmax values does not substitute for an analysis of the truncation's effect, since the reported spread across AISmax in Table 1 is itself the quantity being used to communicate deep uncertainty. Please provide a quantitative statement of how the allowance ranges change if the truncation is removed or replaced with a defined-tail distribution, or clearly re-label the results as illustrative of the method rather than as design recommendations.
[Tables 1-2 and Sec. 3.3] The headline allowance tables report point estimates only, although the underlying calculations include GPD parameter uncertainty (Sec. A.2, with 1000 Latin hypercube samples) and SLR projection uncertainty (Table B.1). For a decision-support tool whose purpose is to communicate uncertainty, the absence of any interval or sensitivity measure for the allowances makes it difficult to know whether differences across AISmax and βc, such as the 1.8 m versus 3.4 m levee allowances in Table 1, are larger than the numerical uncertainty in the calculation. Please report at least a 5–95% range or an indication of the parameter-uncertainty contribution for the main allowance values, or state explicitly why they are omitted.

minor comments (6)

[Notation throughout] The notation for the Antarctic ice-sheet limit is inconsistent: the text uses AISmax, AIS max, and AIS_max (e.g., Secs. 2.2, 3.2, and A.8); please choose one symbol and define it once.
[Sec. 2.1.2 and Fig. 2B] Eq. (4) defines the levee-protected damage function with the failure probability p_f, but the caption of Fig. 2B states that the levee curve assumes 'zero probability of structural failure and no freeboard'; please clarify whether that curve is the limit p_f→0 or a different construct.
[Abstract and Sec. 2] The phrase 'average flood damage in a given year' is a bit loose; the paper actually uses annual average loss, which is an expectation over all events, so consider using 'expected annual flood damage' consistently.
[Eq. A.3] The lower limit of integration e_min in Eq. (A.3) is not defined precisely; please specify the lowest first-floor elevation in the data or set e_min = 0 for clarity.
[Sec. 3.1] When reporting that 0.5 m of SLR increases the AAL from roughly $0.1 billion/yr to roughly $0.7 billion/yr, please state the rounding convention and confirm that all values are in 2017 USD.
[Acknowledgments] The acknowledgments contain the placeholder text '[ADD OTHERS]'; please complete the data and code availability statement before publication.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the damage allowance is solved from Eq. 2 using external damage, tide-gauge, and SLR inputs; self-citations are data sources, not load-bearing premises.

full rationale

The central quantity, the damage allowance A, is defined by Eq. 2 as the protection height that makes the future annual average loss (AAL) equal to the current AAL. This is a defining balance equation, not a hidden restatement of the inputs: the current AAL is computed from the tide-gauge-based extreme sea level distribution f(z) and the USACE-based depth-damage functions, while the future AAL is computed by convolving the same damage model with the user-supplied SLR distribution P(Δ). The allowance is then solved for, not fitted. The key SLR distributions are taken from published work (Kopp et al., 2014; Kopp et al., 2017), and the paper explicitly labels these as illustrative choices and exposes the user-controlled parameters βc and AISmax in Eq. A.6; these parameters are not tuned to reproduce the reported allowances. The depth-damage polynomials come from USACE (2015), the extreme value model follows standard GPD methodology, and the levee fragility curve is attributed to Wolff (2008). Self-citations appear (Buchanan et al., 2016; Rasmussen et al., 2018; Kopp et al., 2014, 2017), but they are used as sources for methods and projection data, not as an authority that forces the damage-allowance result. The admitted arbitrary truncation of the AIS contribution in Sec. A.8 is a stated limitation and robustness concern, not a circular step. The skeptic concern about the depth-damage cubic becoming negative outside its fitted range is a correctness/extrapolation issue, not circularity: it does not make any predicted quantity equal to an input by construction. Overall, the derivation chain is self-contained conditional on its transparently stated inputs.

Assumptions & free parameters 9 free parameters · 7 assumptions · 0 invented entities

The central framework has no hidden physical entities. However, the specific Manhattan allowances depend on many fitted or hand-chosen inputs: damage curves, property exposure, GPD parameters, levee fragility, and the p-box mixing and truncation. The deepest load-bearing choices are the linear p-box mixture and the arbitrary AISmax truncation, both acknowledged by the authors as illustrative or unresolved.

free parameters (9)

beta_c (AIS collapse likelihood weight) = 0, 0.25, 0.5, 0.75, 1 in example runs
User-selected weight linearly mixes the bounding CDFs to create the effective SLR distribution; no empirical calibration is provided.
AISmax (2100 AIS contribution cap) = 0.25, 0.5, 1.0, 1.5, 1.75 m
Arbitrary truncation of the upper tail of Antarctic contribution; acknowledged in Sec. A.8 and strongly affects post-mid-century allowances.
tolA (failure probability at design height) = 0.10
Assumed minimum threshold for Dutch flood defenses; determines the levee and storm surge barrier fragility curves.
Freeboard Fb = 0.5 m
Assumed for levee and storm surge barrier examples; adds directly to design height.
Barrier closure threshold zclose = 1.0 m above MHHW
Chosen so the barrier closes about once every 10 years, following the Maeslant Barrier design.
Compliance alpha for retreat and elevation = 1 (perfect compliance) for headline results
Tables 1 and 2 assume perfect compliance; alpha is varied only in supplementary figures.
Depth-damage polynomial coefficients = phi_hrise = 0.142 + 0.0541x - 0.00368x^2 - 0.00133x^3; phi_res = 0.18 + 0.178x + 0.0233x^2 - 0.00778x^3
Least-squares fits to USACE depth-damage data; directly shape the damage function and hence the allowances.
Property exposure fit coefficients = y = 0.99x - 0.61x^2 for x <= 3; y = 2.01 + 7.25x for x > 3
Piecewise fit to NYC tax-assessed building value by first-floor elevation; shapes the damage function for Manhattan.
GPD shape and scale parameters = xi = 0.19 (0.05 to 0.33), sigma = 0.13 (0.10 to 0.15) m
Maximum-likelihood estimates at the Battery tide gauge; GPD uncertainty is sampled for return curves but not propagated into Tables 1 and 2.

assumptions (7)

domain assumption Extreme sea levels at the Battery follow a generalized Pareto distribution above the 99th percentile.
Standard peaks-over-threshold method, but single-station representativeness for all of Manhattan is assumed.
domain assumption The Battery tide gauge ESL distribution represents all of Manhattan.
Stated in Sec. A.2; ignores spatial variation in bathymetry, topography, and coastal protection.
domain assumption Manhattan is protected by a uniform 1.0 m bulkhead above MHHW.
Sec. 2 states this simplification, while recognizing the actual bulkhead height varies around the island.
domain assumption Building stock, population, and infrastructure remain fixed in time.
The "frozen city" assumption is stated in Sec. 2 and used for all damage projections.
domain assumption Levee and storm surge barrier failure probability increases exponentially with water load, calibrated by tolA and freeboard.
Sec. 2.1.2 adopts an exponential fragility based on Wolff (2008) and Dutch design standards.
ad hoc to paper The effective SLR distribution is the convex combination beta_c * Phigh + (1 - beta_c) * Plow of two bounding CDFs.
Eq. A.6 uses linear pooling as a modeling choice; no empirical or theoretical derivation is given for this mixture.
ad hoc to paper Truncating the upper tail of AIS contribution at AISmax yields a valid conditional SLR distribution.
Sec. A.8 truncates the distributions to avoid overlapping p-box boundaries and admits the cutoff is arbitrary.

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Cite this review

Pith. "Pith review of A flood damage allowance framework for coastal protection with deep uncertainty in sea-level rise." pith.science (2026). https://pith.science/paper/MH535RTV

@misc{pith2026190802844,
  author       = {Pith},
  title        = {Pith review of: A flood damage allowance framework for coastal protection with deep uncertainty in sea-level rise},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/MH535RTV}},
  note         = {Machine review of arXiv:1908.02844}
}

read the original abstract

Future projections of Antarctic ice sheet (AIS) mass loss remain characterized by deep uncertainty (i.e., behavior is not well understood or widely agreed upon by experts). This complicates decisions on long-lived projects involving the height of coastal flood protection strategies that seek to reduce damages from rising sea levels. If a prescribed margin of safety does not properly account for sea-level rise and its uncertainties, the effectiveness of flood protection will decrease over time, potentially putting lives and property at greater risk. We develop a flood damage allowance framework for calculating the height of a flood protection strategy needed to ensure that a given level of financial risk is maintained (i.e., the average flood damage in a given year). The damage allowance framework considers decision-maker preferences such as planning horizons, preferred protection strategies (storm surge barrier, levee, elevation, and coastal retreat), and subjective views of AIS stability. We use Manhattan (New York City)\textemdash with the distribution of buildings, populations, and infrastructure fixed in time\textemdash as an example to show how our framework could be used to calculate a range of damage allowances based on multiple plausible AIS outcomes. Assumptions regarding future AIS stability more strongly influence damage allowances under high greenhouse gas emissions (Representative Concentration Pathway [RCP] 8.5) compared to those that assume strong emissions reductions (RCP2.6). Design tools that specify financial risk targets, such as the average flood damage in a given year, allow for the calculation of avoided flood damages (i.e., benefits) that can be combined with estimates of construction cost and then integrated into existing financial decision-making tools, like benefit-cost or cost-effectiveness analyses.

Figures

Figures reproduced from arXiv: 1908.02844 by the authors.

**Figure 1.** A. Expected number of extreme sea level (ESL) events per year as a function of ESL height (m above mean higher high water [MHHW]) at the Battery (Manhattan, New York City) for historical mean sea level (grey lines), 0.5 m of sea-level rise (SLR; red line), and projected SLR in 2070 (blue line). Thin grey lines are the historical ESL height return curves for the 17/50/83 percentiles of the generalized Pareto distribu… view at source ↗

**Figure 2.** A. Schematics illustrating each flood protection strategy for an arbitrary protection height (A1). B. Time-invariant damage functions for Manhattan that relate extreme sea level (ESL; meters) to total direct damage due to flooding (billions of USD$; Sec. A.3). The thick grey line is the unprotected damage function (i.e., no flood protection strategy) that assumes no existing bulkhead around Manhattan; the dashed red… view at source ↗

**Figure 3.** A. Top Row: probability boxes (’p-boxes’; solid lines) for 2100 local sea-level rise (SLR) in Manhattan (located at the Battery tide gauge) under the representative concentration pathway (RCP) 8.5 climate forcing scenario. Effective cumulative distribution functions (CDFs) of local SLR (dashed lines) are generated within each p-box by averaging the edges using weights (βc ∈ [0,1]) that reflect a user’s belief of AIS… view at source ↗

Figures from the paper (2 more)

**Figure 4.** Figure 4: A. Flow chart illustrating how to apply the coastal flood damage allowance framework B. An example application for Manhattan seeking to maintain the current annual average loss from flood damages using a storm surge barrier. –19– [PITH_FULL_IMAGE:figures/full_fig_p01…

**Figure 5.** Figure 5: A. Top Row: Levee damage allowances (meters above the current protection height) over time (2000–2100) for protecting Manhattan under different maximum 2100 Antarctic ice sheet (AIS) contribution thresholds (AISmax, relative to 2000), different subjectively perceived l…

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