A hurricane surge risk assessment framework using the joint probability method and surge response functions

Abstract

Hurricane surge events have caused devastating damage in active-hurricane areas all over the world. The ability to predict surge elevations and to use this information for damage estimation is fundamental for saving lives and protecting property. In this study, we developed a framework for evaluating hurricane flood risk and identifying areas that are more prone to them. The approach is based on the joint probability method with optimal sampling (JPM-OS) using surge response functions (SRFs) (JPM-OS-SRF). Derived from a discrete set of high-fidelity storm surge simulations, SRFs are non-dimensional, physics-based empirical equations with an algebraic form, used to rapidly estimate surge as a function of hurricane parameters (i.e., central pressure, radius, forward speed, approach angle and landfall location). The advantage of an SRF-based approach is that a continuum of storm scenarios can be efficiently evaluated and used to estimate continuous probability density functions for surge extremes, producing more statistically stable surge hazard assessments without adding measurably to epistemic uncertainty. SRFs were developed along the coastline and then used to estimate maximum surge elevations with respect to a set of hurricane parameters. Integrating information such as ground elevation, property value and population with the JPM-OS-SRF allows quantification of storm surge-induced hazard impacts over the continuum of storm possibilities, yielding a framework to create the following risk-based products, which can be used to assist in hurricane hazard management and decision making: (1) expected annual loss maps; (2) flood damage versus return period relationships; and (3) affected business (e.g., number of business, number of employees) versus return period relationships. By employing several simplifying assumptions, the framework is demonstrated at three northern Gulf of Mexico study sites exhibiting similar surge hazard exposure. The framework results reveal Gulfport, MS, USA is at relatively more risk of economic loss than Corpus Christi, TX, USA, and Panama City, FL, USA. Note that economic processes are complex and very interrelated to most other human activities. Our intention here is to present a methodology to quantify the flood damage (i.e., infrastructure economic loss, number of businesses affected, number of employees in these affected businesses and sales volume in these affected businesses) but not to discuss the complex interactions of these damages with other economic activities and recovery plans.

Appendix

1.1 Surge response function algebraic form

The algebraic SRF form used in this study is modified from Song et al. (2012a, b) to include the influence of forward speed v _f and approach angle θ (open-coast shore normal is zero). Though, we note the relative impact of including the additional influence of forward speed and approach angle within the SRF is small compared to other parameters (see Taylor et al. 2015) and their influence may best be incorporated as uncertainty when uncertainty is included in the analyses. The SRF form used here is:

$$ \zeta ' = \frac{\zeta \gamma }{\Delta p} + m\left( {\frac{\Delta p}{{\Delta p_{ \hbox{max} } }}} \right)^{\beta } \left( {\frac{{\frac{{R_{p} }}{{L_{30} }}}}{{\left( {\frac{{R_{p} }}{{L_{30} }}} \right)_{\text{ref}} }}} \right) + F_{{L_{30} }} \left( {\frac{{v_{f} }}{{v_{{f - {\text{ref}}}} }}} \right) - { \cos }\theta $$

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$$ x' = \frac{{x - x_{o} }}{{R_{p} }} - \tilde{\lambda } - cH\left[ {\frac{{x - x_{o} }}{{R_{p} }} - \tilde{\lambda } - 1} \right]H\left[ {\frac{{L_{30} }}{{L_{{30 - {\text{ref}}}} }}} \right]\left( {\frac{{R_{p} }}{{L_{30} }}} \right) - F_{R} H\left[ {1 - \frac{{R_{P} }}{{R_{\text{thres}} }}} \right] $$

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$$ \tilde{\lambda } = \frac{{x_{{\zeta_{ \hbox{max} } }} - x_{o} }}{{R_{p} }} $$

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$$ F_{{L_{{30}} }} = \left\{ {\begin{array}{ll} {\frac{{L_{{30}} }}{{L_{{30 - {\text{ref}}}} }}} \hfill & {\quad {\text{when }}L_{{30}} < L_{{30 - {\text{ref}}}} } \hfill \\ 1 \hfill & {\quad {\text{when }}L_{{30}} > L_{{30 - {\text{ref}}}} } \hfill \\ \end{array} } \right. $$

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$$ F_{R} = {\text{ }}\left\{ {\begin{array}{ll} {a_{1} \left( {1 - \frac{{R_{p} }}{{R_{{{\text{thres}}}} }}} \right) + b_{1} } \hfill &\quad {{\text{for }} - \tilde{\lambda } \le x^{\prime} \le 0} \hfill \\ {a_{2} \left( {1 - \frac{{R_{p} }}{{R_{{{\text{thres}}}} }}} \right) + b_{2} } \hfill &\quad {{\text{for }}0 \le x^{\prime} \le \tilde{\lambda }} \hfill \\ 0 \hfill &\quad {{\text{for}}\left| {x^{\prime}} \right| > \tilde{\lambda }} \hfill \\ \end{array} } \right. $$

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where H[arg] is the Heaviside function where H = 1 when arg ≥ 0 and H = 0 when arg < 0. The non-dimensional SRF is described in two dimensions as ζ ^′ is the dimensionless surge versus the non-dimensional geographic position x’. Physical parameters are defined as follows: ζ is the dimensional surge, γ is the specific weight of seawater, Δp is the central pressure deficit calculated as the central pressure c _p less the far-field (standard atmospheric) pressure, Δp _max is a constant theoretical minimum central pressure (e.g., Tonkin et al. 2000), R _p is the pressure radius, L ₃₀ is the cross-shore distance between open coast and the 30-m offshore depth contour, x is the geographic location, x _o is the landfall location, and $x_{{\zeta_{max} }}$ is x at the open-coast location of maximum along-coast surge. The following are user-specified study-area reference and scaling constants for hurricane and physical parameters, constrained by parameter limits and selected to minimize error in SRF estimates: (R _p /L ₃₀)_ref, v _f-ref, L _30-ref, R _thres and c. Finally, the following are location-specific fit constants, evaluated separately for x′ ≥ 0 and x′ < 0: m, β, a ₁,a ₂,b ₁ and b ₂.

Root-mean-square error (RMSE) between SRF predictions and ADCIRC simulations are generally between 0.3 to 0.5 m (e.g., Song et al. 2012a, b; Udoh 2012; Taylor et al. 2015); Taylor et al. reports RMSE of 0.5 m or less at 96% of bay locations, when heading and forward speed are omitted in SRF development but included in the simulation set. Taylor et al. (2015) showed use of SRFs marginally adds to overall surge height epistemic uncertainty, increasing it from 0.7 m (Resio et al. 2013) to 0.8 to 0.9—on the order of 1 m—depending on location.

1.2 Probability density function used in the joint probability method

The northern Gulf of Mexico probability density function, f, is from Irish et al. (2011) using the JPM-OS-SRF, following the approach introduced in Resio et al. (2009). Because of the importance of storm size, the meteorological distributions were developed from the observational record including storm size, for hurricanes (Category 1 and higher) making landfall along the US northern Gulf of Mexico from 1941 to 2010. Observations are predominantly from HURDAT (e.g., Landsea et al. 2005) supplemented with additional storm radii information (e.g., Powell and Reinhold 2007; A. Cox, Oceanweather). These observations are summarized in Irish et al. (2008) and Irish and Resio (2010). To simplify the analysis, we generalized the meteorological probability distributions as uniform along the northern US Gulf of Mexico coast.

$$ f\left( {c_{p} ,R_{p} , v_{f} ,\theta ,x_{o} } \right) = \Lambda_{1} \Lambda_{2} \Lambda_{3} \Lambda_{4} \Lambda_{5} $$

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$$ \Lambda_{1} = f\left( {c_{p} |x_{o} } \right) = \frac{1}{{a_{1} \left( {x_{o} } \right)}}{ \exp }\left[ { - \frac{{\Delta p - a_{o} \left( {x_{o} } \right)}}{{a_{1} \left( {x_{o} } \right)}}} \right]{ \exp }\left\{ { - { \exp }\left[ { - \frac{{\Delta p - a_{o} \left( {x_{o} } \right)}}{{a_{1} \left( {x_{o} } \right)}}} \right]} \right\} $$

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$$ \Lambda_{2} = f\left( {R_{p} |c_{p} } \right) = \frac{1}{{\sigma \left( {\Delta p} \right)\sqrt {2\pi } }}{ \exp }\left\{ { - \frac{{\left[ {\mu \left( {\Delta p} \right) - R_{p} } \right]^{2} }}{{2\sigma^{2} \left( {\Delta p} \right)}}} \right\} $$

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$$ \Lambda_{3} = f\left( {v_{f} |\theta } \right) = \frac{1}{{\sigma \left( \theta \right)\sqrt {2\pi } }}{ \exp }\left\{ { - \frac{{\left[ {\mu \left( \theta \right) - v_{f} } \right]^{2} }}{{2\sigma^{2} \left( \theta \right)}}} \right\} $$

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$$ \Lambda_{4} = f\left( {\theta |x_{o} } \right) = \frac{1}{{\sigma \left( {x_{o} } \right)\sqrt {2\pi } }}{ \exp }\left\{ { - \frac{{\left[ {\mu \left( {x_{o} } \right) - \theta } \right]^{2} }}{{2\sigma^{2} \left( {x_{o} } \right)}}} \right\} $$

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$$ \Lambda_{5} = {\text{ rate of storm landfall occurrence per unit distance alongshore}} $$

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where a ₁ and a ₂ are Gumbel coefficients and μ and σ are the Gaussian mean and standard deviation. Equation 2 is applied separately at each study area. To simplify the analysis given the relatively weak dependence of surge on v _f and θ at our sites, we assumed $f\left( {v_{f} |\theta } \right) = f\left( {v_{f} } \right)$ and $f\left( {\theta |x_{o} } \right) = f\left( \theta \right)$.

Because we assume alongshore uniformity in the joint probability distributions, some error is introduced into the statistics. Nonetheless, our results are in good agreement with those from the FEMA DFIRM work in Panama City (S. Hagen, Louisiana State University, personal communications), within 5–15% over the range of AEP from 0.1 to 1%.