Abstract
The coastal destruction wreaked by Hurricane Sandy in 2012 prompted motivation to estimate the event’s return period. The Clustered Separated Peaks-over-threshold Simulation (CSPS) method for estimating return periods uses a Monte Carlo simulation of storm surge activity based on statistics derived from tidal gauge data at the Battery, New York in lower Manhattan. The data are separated into three independent components (storm surge, tidal cycle and sea level rise) because different physical processes govern different components of water level. Peak storm surge heights are fit to the generalized Pareto distribution, chosen for its ability to fit a wide tail to limited data. The algorithm incorporates the evolution of storm surge over surge duration. The CSPS suggests that the return period of Hurricane Sandy’s peak water level is 103 years (95% confidence interval 38–452 years), significantly lower than previously published return periods. The estimated 100-year water level is 5.23 m above the station datum (or 3.39 m above the North American Vertical Datum of 1988, or 3.45 m above mean sea level). With 1 m of sea level rise (holding all other climatological conditions constant), this water level would become the 28-year event. Although the method’s exclusion of surge-tide interaction and its reliance on a 90-year tidal gauge time history may limit the reliability of high return period estimates, application of the CSPS method to lower Manhattan suggests that storm surge hazard in the New York Harbor has, until now, been underestimated.
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Notes
Each NOAA tidal gauge station contains a fixed elevation datum called the “Station Datum” to which elevation measurements can be referenced. On the other hand, the commonly used North American Vertical Datum of 1988 (NAVD88) is a fixed vertical datum whose elevation is constant across all stations. To convert measurements referenced to the STND to the NAVD88 scale at the Battery, NY, one must simply subtract 1.85 m.
This estimate for storm surge return period is less useful for practical application than estimates for total water level return periods because it neglects the probability of a surge occurring at different times within the tidal cycle.
This formulation neglects any presence of tide-surge interaction, which would preclude treatment of the data as separable. For the remainder of this paper, all analysis is performed assuming tide and surge are independent, though the authors acknowledge that tide-surge interaction has been identified at the Battery, NY as well as other locations (Lin et al. 2012).
The notation conventions used herein are based on the following distinctions: deterministic values that are constant in time contain no subscript, e.g., \(M\); deterministic functions of time \(t_i\) are expressed as such, e.g., \(R(t_i)\); values representing stochastic quantities measured at discrete points in time are given the subscript \(i\) directly, e.g., \(X_i\). A detailed list of notation used in this paper can be found in “Appendix”.
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Appendices
Appendix: Notation
1.1 NOAA data notation
\(a\) | Number of years of data contained in dataset |
\(i\) | Index variable to identify datapoints (used as right subscript \([-]_i\)) |
\(n\) | Number of datapoints in dataset such that \(i=1,2,\ldots ,n\) |
\(t_i\) | Time in hours of \(i\)th datapoint measured from the start of the record |
\(X_i\) | Predicted water level at time \(t=t_i\) from NOAA tidal gauge data |
\(Y_i\) | Observed water level at time \(t=t_i\) from NOAA tidal gauge data |
\(T(t)\) | Component of water height due to tides estimated from harmonic constituents |
\(j\) | Index variable to identify harmonic constituents (used as right subscript \([-]_j\)) |
\(A_j, a_j, \alpha _j\) | Harmonic constituent parameters (from NOAA) |
\(M\) | Mean sea level at tidal gauge station (constant in time) |
\(R(t)\) | Estimated amount of SLR since start of data record |
\(\beta _1, \beta _2\) | Linear least squares parameters for estimating SLR \(R(t)\) |
\(S_i\) | Residual surge height at time \(t=t_i\) computed from NOAA tidal gauge data |
\(W(t)\) | Instantaneous mean water level due to SLR such that \(W(t) = M + R(t)\) |
1.2 Simulation notation
\(c\) | Index variable to identify exceedance clusters (used as left subscript \(_c [-]\)) |
\(C\) | Number of clusters in the dataset such that \(c=1,2,\ldots ,C\) |
\(\bar{C}\) | Target number of clusters in the dataset such that \(\bar{C} = \bar{\lambda }a\) |
\(N\) | Number of years of simulated surge activity in CSPS |
\(\Delta\) | Random variable modeling waiting time between exceedance occurrences |
\(\delta\) | “Dummy” variable used in exponential distribution for waiting time \(\Delta\) |
\({_c \varPi }\) | Random variable modeling peak surge height for cluster \(c\) |
\(_c \pi (t)\) | Function used to estimate surge profile for all \(t\) for cluster \(c\) |
\({_c\pi _1(t)}, {_c\pi _2(t)}\) | Exponential rise and fall functions used to estimate surge profile for cluster \(c\) |
\({_c \varUpsilon }\) | Time of peak surge for cluster \(c\) (measured in hours since beginning of record) |
\({_c \tau _a}, {_c \tau _b}\) | Start and end times (respectively) for the exceedance profile of cluster \(c\) |
\({_c \varGamma }\) | Duration of surge for cluster \(c\) (measured in hours) |
\(_c[\chi _1,\chi _2,\omega _1,\omega _2]\) | Exponential function coefficients used in \({_c\pi _1(t)}, {_c\pi _2(t)}\) for cluster \(c\) |
\({_c \varPhi }\) | Rise ratio for cluster \(c\) |
\(\beta _1', \beta _2'\) | Linear least squares parameters for mean of duration lognormal distribution |
\(\beta _1'', \beta _2''\) | Linear least squares parameters for variance of duration lognormal distribution |
\(\bar{\varGamma }(\varPi )\) | Mean of lognormal distribution for duration as a function of surge height |
\(V(\varPi )\) | Variance of lognormal distribution for duration as a function of surge height |
\(B\) | Number of bootstrap resamplings |
\(\alpha\) | Parameter used to establish confidence interval (CI \(=1-2\alpha\)) |
1.3 Probability distribution parameters
\(\theta\) | Threshold parameter for GPD |
\(\xi\) | Shape parameter for GPD |
\(\sigma\) | Scale parameter for GPD |
\(G_S(s)\) | GPD CDF for surge value \(S\) |
\(f_{\Delta }(\delta )\) | Exponential density function for waiting time \(\Delta\) |
\(\lambda\) | Poisson parameter (average number of events per year) |
\(\bar{\lambda }\) | Target Poisson parameter value used in threshold selection stage |
1.4 Miscellaneous
\(x\) | Return period for generic event |
\(c, w\) | Season identifiers where \(c=\) cold, \(w=\) warm (used as right superscript \([-]^c, [-]^w\)) |
\(R_{SS}(\tau )\) | Autocorrelation function for surge time series \(S_i\), \(i = 1,2,\ldots ,n\) |
\(\tau\) | Time lag for autocorrelation function for surge time series |
\(\theta _0\) | Initial threshold parameter used in threshold selection method |
\(\theta _1, \theta _2\) | Threshold values used to illustrate event selection method |
\(C_0\) | Number of clusters identified using \(\theta _0\) |
\(\varPi _0\) | Peak surge height vector using \(\theta _0\) |
\(\varOmega\) | Average number of days per season per year |
\(d\) | Block length used in block maxima method |
\(m\) | Number of blocks used in block maxima method |
Appendix: Surge data quality issues
As mentioned in Sect. 3, data quality issues were identified in the NOAA tidal gauge data for the Battery, NY from 1942 to 1945. The predicted and observed values (\(X_i\) and \(Y_i\)) appear to be offset by a few hours during these years, resulting in artificial oscillations in the computed surge values \(S_i\). Figure 17 shows an overview of the years in which these issues are present. Upon zooming in on the beginning and end of this period of concern, it is clear precisely where the issues begin and end (see Fig. 18a, b). Figure 19a and b show, respectively, a period of surge data during the period of quality issues and a period of “normal” surge residual data. There is strong frequency content apparent in the period with issues that is not present in the “normal” surge data. For this reason, the years 1942 through 1945 were excluded from the CSPS.
Overview of surge data quality issues in tidal gauge data for the Battery, NY
Beginning (a) and end (b) of surge data quality issues in tidal gauge data for the Battery, NY. It is clear from zooming in on the surge data that the quality issues begin in early 1942 and extend to near the end of 1945
Period of surge data with quality issues (a) and with no quality issues (b) from tidal gauge data for the Battery, NY. An apparent oscillatory component of the surge data is observed in the 1944 data but not in the 1950 data
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Lopeman, M., Deodatis, G. & Franco, G. Extreme storm surge hazard estimation in lower Manhattan. Nat Hazards 78, 355–391 (2015). https://doi.org/10.1007/s11069-015-1718-6
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DOI: https://doi.org/10.1007/s11069-015-1718-6