Toward High-Resolution, Rapid, Probabilistic Forecasting of the Inundation Threat from Landfalling Hurricanes

1304 MONTHLY WEATHER REVIEW VOLUME 141

ANDREW J. CONDON,Y.PETER SHENG, AND VLADIMIR A. PARAMYGIN Department of Civil and Coastal Engineering, University of Florida, Gainesville, Florida

(Manuscript received 16 May 2012, in ﬁnal form 28 September 2012)

ABSTRACT

State-of-the-art coupled hydrodynamic and wave models can predict the inundation threat from an approaching hurricane with high resolution and accuracy. However, these models are not highly efficient and often cannot be run sufficiently fast to provide results 2 h prior to advisory issuance within a 6-h forecast cycle. Therefore, to produce a timely inundation forecast, coarser grid models, without wave setup contributions, are typically used, which sacrifices resolution and physics. This paper introduces an efficient forecast method by applying a multidimensional interpolation technique to a predefined optimal storm database to generate the surge response for any storm based on its landfall characteristics. This technique, which provides a ‘‘digital lookup table’’ to predict the inundation throughout the region, is applied to the southwest Florida coast for Hurricanes Charley (2004) and Wilma (2005) and compares well with deterministic results but is obtained in a fraction of the time. Because of the quick generation of the inundation response for a single storm, the response of thousands of possible storm parameter combinations can be determined within a forecast cycle. The thousands of parameter combinations are assigned a probability based on historic forecast errors to give a probabilistic estimate of the inundation forecast, which compare well with observations.

1. Introduction constraints of a 6-h forecast cycle. Typically the NHC has roughly an hour at most from the time the most The extent of coastal inundation from a given hurri- recent track–intensity information is received to com- cane has proven to be difficult to forecast in an efficient plete storm surge forecasts for the next 36–120 h for manner. High-resolution, physics-based models such as inclusion in the latest advisory (J. R. Rhome 2011, Advanced Circulation (ADCIRC; Luettich et al. 1992; personal communication). Dietrich et al. (2012) show Weaver and Slinn 2006), Curvilinear-grid Hydrody- that coupled Simulating Waves Nearshore (SWAN) and namics in 3D–Storm Surge Modeling System (CH3D- ADCIRC simulations for Hurricane Katrina can take SSMS; Sheng et al. 2006, 2010a,b; Sheng and Paramygin between 10 and 2000 min of wall clock time per day of 2010), Princeton Ocean Model (POM; Peng et al. 2004; simulation depending on the computing resources (8192 Oey et al. 2006), and Finite Volume Coastal Ocean to 256 computational cores) and solver (implicit or ex- Model (FVCOM; Rego and Li 2009; Weisberg and plicit). For similar simulations of Hurricane Katrina, Zheng 2008) have all been proven to accurately simulate CH3D-SSMS runs at about 900 min of wall clock time coastal inundation from hurricanes. However, these per day of simulation on eight computational cores. Both models are all computationally expensive to run com- these examples demonstrate that either enormous compared to the Sea, Lake, and Overland Surges from putational resources or too much wall clock time are Hurricanes (SLOSH; Jelesnianski et al. 1992) model needed to develop inundation forecasts in a timely man- of the National Hurricane Center (NHC), which makes ner. In addition the National Research Council (NRC) forecasting much more difficult given the tight time report ‘‘Completing the Forecast’’ emphasizes the need for more probabilistic forecasts that involve an ensemble of storm simulations using a storm surge modeling system Corresponding author address: Y. Peter Sheng, Department of Civil and Coastal Engineering, University of Florida, 365 Weil (National Research Council 2006). Given the 1-h time Hall, P.O. Box 116580, Gainesville, FL 32607. window available to produce a hurricane storm surge E-mail: [email protected]fl.edu forecast, it is currently not possible to run an ensemble

DOI: 10.1175/MWR-D-12-00149.1

Ó 2013 American Meteorological Society Unauthenticated | Downloaded 10/04/21 01:29 PM UTC APRIL 2013 C O N D O N E T A L . 1305 of thousands of storms with a high-resolution modeling core, and is typically accurate within 20% (Jelesnianski system. Other attempts to generate a timely estimate of et al. 1992). The model does not account for dynamic the inundation response have been made. effects of tides and waves, which other forecasting sys- The Saffir–Simpson hurricane scale (SSHS; Simpson tems incorporate. The largest drawback to the SLOSH 1974) was used by NHC to relate the storm surge hazard forecasts is the coarse resolution [Fort Myers, Florida, to hurricane intensity. Following the active Atlantic grid (efmy2) has an average resolution of 2 km] of the hurricane seasons of 2004 and 2005, it became obvious model domains compared to the other models mentioned. that storm surge hazard depends on other hurricane With a coarse grid many of the important small-scale characteristics (e.g., size and forward speed) in addition topographic and bathymetric features are not captured to intensity. Irish et al. (2008) showed that storm size can in the model, and the effects of waves may not be ac- cause variations of up to 30% in storm surge for a given curate even if a wave model were coupled to SLOSH. storm intensity. Kantha (2006) and Powell and Reinhold Despite these drawbacks, SLOSH is used in the gener- (2007) developed storm surge classification schemes ation of forecasts and probabilistic products (P-Surge; that look at hurricane characteristics beyond intensity to Glahn et al. 2009; Taylor and Glahn 2008). estimate the storm surge hazard posed by a particular Irish et al. (2011) recently produced probabilistic hurricane. These scales represented an improvement maximum hurricane surge forecasts based on surge re- over the SSHS as they accounted for storm size. How- sponse functions (Irish et al. 2009; Resio et al. 2009), ever, storm surge is also dependent on the landfall lo- hurricane characteristics, and joint probability statistics. cation, track heading, and translational speed of the This approach uses high-resolution simulation results to hurricane among other things for which these scales do generate surge response functions for a given region that not account (Jordan and Clayson 2008). Recently the can determine the surge response for a set of meteoro- NHC has officially removed storm surge information logical parameters. This approach is very promising but from the SSHS (NOAA/National Hurricane Center has underlying assumptions that the influence of the 2011a) because of the large differences that can develop storm angle and forward speed can be neglected when in the surge response and inundation for storms with the compared to the storm intensity, size, and landfall lo- same intensity but different other characteristics, and cation. While their work shows that in most cases this is a identical storms making landfall along different portions fair assumption based on model results, there are outliers of the coast. The offshore bathymetry, coastline config- which can be important. As pointed out by Rego and Li uration, and topography of the affected area play a large (2009) and Jelesnianski (1972), neglecting the forward role in dictating the extent of the inundation. Mildly speed and angle of approach may not be appropriate as sloping bathymetry has been shown to generate a larger there is a ‘‘critical motion relative to a coast that gives the surge response at the coast than steeper slopes (Irish et al. highest possible surge.’’ Additionally the technique does 2008). Likewise the landfall location can be important not account for tides and wave setup, which can con- as demonstrated by Weisberg and Zheng (2008) for tribute significantly to the surge and inundation. idealized storm surge simulations in the Tampa Bay, This paper addresses the rapid generation of high- Florida, area. The topography of the area and rough- resolution probabilistic inundation forecasts. The opti- ness of the terrain will dictate the extent of the coastal mal storm generation and multivariate interpolation inundation (Fletcher et al. 1995). Irish and Resio (2010) technique of Condon and Sheng (2012a,b) is applied to accounted for the local bathymetry in their hydrody- a single storm to generate an estimate of the inundation namics based scale, which gives the best quantitative hazard for southwest Florida from Hurricanes Charley results for the potential surge at the coast for 28 historical (2004) and Wilma (2005). This is accomplished in an hurricanes compared to SSHS, Powell and Reinhold adaptive manner to improve accuracy with each forecast. (2007), and Kantha (2006). However this scale lacks in- The technique considers the effect of storm intensity, size, formation regarding coastline configuration and topog- landfall location, forward speed, and approach angle on raphy, which is essential in determining the hazard from the surge response. The optimal storm database, which inundation. includes wave effects on surge and inundation, is pro- In addition to the classification schemes described duced and can be combined with a simple tidal model to above to qualitatively estimate inundation hazard, more account for tidal effects. Analysis of the official NHC quantitative measures have been developed. The NHC forecast errors for the past five years (NOAA/National uses the SLOSH (Jelesnianski et al. 1992) model op- Hurricane Center 2011b; J. Franklin 2011, personal erationally to produce hurricane forecasts. SLOSH is communication) allows for efficient generation of high- extremely efficient, with most approximately 100-h sim- resolution probabilistic surge estimates as well for each ulations taking under 1 min on a single computational forecast period within the 1-h time constraints.

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2. Optimal storm generation and multivariate and drying, CH3D is used for simulating and forecasting interpolation storm surge and circulation in many coastal regions throughout Florida and the United States. Detailed For this study an optimal storm ensemble for the governing equations and boundary conditions for CH3D southwest Florida basin (NHC SLOSH efm2 basin) is are shown in appendix B. developed following the method presented in Condon CH3D has been dynamically coupled to a wave model and Sheng (2012b). In this method a dimension adaptive SWAN (Booij et al. 1999; Ris et al. 1999), using the same version of Smolyak’s algorithm (Smolyak 1963) is ap- curvilinear grid, to produce CH3D-SSMS (Sheng et al. plied to the storm surge problem to optimally select an 2006, 2010a,b; Sheng and Liu 2011). CH3D-SSMS uses ensemble of storms. The method is adapted from that of basin-scale models, such as the Hybrid Coordinate Ocean Agbley (2009) to make it more transportable and ac- Model (HYCOM; Halliwell et al. 1998, 2000; Bleck curate for storm surge estimation. This is done by using 2002), the Navy Coastal Ocean Model (NCOM; Barron the dimension adaptive sparse grid formulation of et al. 2006), and ADCIRC (Luettich et al. 1992) in a Gerstner and Griebel (2003) in the form of the spiniterp large-scale domain, to provide open boundary condi- MATLAB toolbox (Klimke and Wohlmuth 2005; tions for CH3D. To enable efficient simulation, this study Klimke 2007) and coupling with the SLOSH model couples the CH3D model, with a high-resolution coastal (Jelesnianski et al. 1992) to obtain the storm surge sim- grid, to the basin-scale model ADCIRC, which has a rel- ulations that provide the optimal recovery of the surge atively coarse grid in the offshore as well as coastal re- response for any given set of storm parameters. Multi- gions. To provide open boundary condition for SWAN, variate regression is used to build the response from the CH3D-SSMS uses the output of a large-scale wave model optimal simulation database. This is achieved with mul- such as WaveWatch-III (Tolman 1999, 2002). CH3D- tivariate adaptive regressive splines (MARS) as done by SSMS has been used extensively to simulate storm surge Friedman (1991). For additional details please see ap- and inundation due to various tropical storms including pendix A. Hurricanes Isabel (Sheng et al. 2010a), Charley (Sheng In Condon and Sheng (2012a,b) the hurricanes are et al. 2006; Davis et al. 2008, 2010), Ivan (Sheng et al. characterized by five parameters: the central pressure 2010b), and Wilma. Sheng and Paramygin (2010) com- deficit DP, the radius to maximum winds Rmax, the bined the baroclinic circulation element of CH3D with translational speed Vf, the storm heading u (angle of CH3D-SSMS to forecast the storm surge, inundation, and approach), and the landfall location Xland. These studies 3D baroclinic circulation in northeast Florida during determined the hazard to the region in present-day and Tropical Storm Fay. Using CH3D-SSMS, this study rep- future climates through an adapted version of the joint resents a marked improvement in model physics and probability method (JPM), which used probabilistic de- spatial resolution over a previous study (Condon and scriptions of these five variables combined with the surge Sheng 2012b) using SLOSH. response from 197 optimal storm simulations for the ba- For this study CH3D-SSMS uses the dynamically sin. The 197 high-resolution optimal simulations are coupled CH3D-SWAN models for the coastal domain. performed using CH3D-SSMS. CH3D is a hydrody- Open boundary conditions for CH3D are provided by namic model originally developed by Sheng (1987, 1990) a coarse grid ADCIRC model, while open boundary and has been significantly enhanced (e.g., Sheng et al. conditions for SWAN are provided by a coarse grid 2010a; Sheng and Kim 2009). The model can simulate basin-scale SWAN and WaveWatch-III models. Little 2D and 3D barotropic and baroclinic circulation driven difference (,0.01% in inundation) is found between the by tides, winds, waves, and density gradients. The model final results obtained using SWAN or WaveWatch-III in uses a boundary-fitted nonorthogonal curvilinear grid in the offshore region. Hence SWAN is used for both the the horizontal directions and terrain-following sigma offshore region and the coastal region. CH3D-SSMS grid in the vertical direction to allow accurate represen- uses wind fields developed by an analytic hurricane wind tation of the complex coastal and estuarine shorelines model based on Holland (1980) in which the central where forecasting of storm surge, waves, and in- pressure deficit, Holland B parameter, and radius to undation is needed. Based on the finite-volume method, maximum winds control the pressure and wind distri- CH3D is strictly conservative for momentum, water mass, bution. The winds are developed as straight-line tracks as well as for temperature and salinity. CH3D uses a ro- of constant intensity until landfall. For this study, the bust second-order closure model for calculating vertical storm intensity is dissipated following Vickery (2005) turbulent mixing (Sheng and Villaret 1989). In the hori- post landfall. To save computational cost, this study runs zontal direction, Smagorinsky-type turbulent diffusion the CH3D model in 2D (vertically averaged) mode. A coefficients are used. With its ability to simulate flooding spatially varying Manning’s n coefficient is developed

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FIG. 1. High -resolution CH3D-SSMS domain for southwest Florida. based on land use data obtained from the U.S. Geo- The workaround to this is to construct a simple one- logical Survey (U.S. Geological Survey 2011) with an component sinusoidal model that matches the amplitude offshore value of 0.02. The coastal model domain fea- of the tide at the time of landfall. The NOAA Naples, tures a minimum horizontal resolution of approximately Florida, tide gauge (NOAA 2011) was examined to de- 20 m in the coastal zone and an overall average grid size termine the range of expected tide levels. By including of ;100 m with a maximum of ;700 m offshore. The this range of expected tide levels a total of 265 optimal most up-to-date lidar topography data from the National storms is needed to obtain similar estimated error as that Oceanic and Atmospheric Administration (NOAA) with 197 storms and no tides. These are then adjusted to Coastal Services Center (NOAA/Coastal Services Center account for the phase by running simulations for both 2011), topography data from U.S. Geological Survey increasing (ﬂood) and decreasing (ebb) tides for all in- (U.S. Geological Survey 2011), and bathymetry data from termediate tidal amplitudes. For amplitudes at the peak NOAA/National Geophysical Data Center (NOAA/ or trough of the tidal cycle, only one simulation is run. National Geophysical Data Center 2011), have been in- This resulted in a total of 448 simulations of the surge corporated into the domain shown in Fig. 1. response in the optimal database. This is a substantial In previous studies (Condon and Sheng 2012a,b; Toro increase over the 197 responses without tides but far less et al. 2010a,b; Niedoroda et al. 2010) the astronomical than would be needed if all the possibilities of the mixed tide, which can be an important component of the in- tides were included. This increase in the number of nec- undation, is included as an error term in the JPM for- essary optimal storm simulations will be analyzed below. mulation but not directly simulated. For this study the astronomical tidal amplitude and phase are considered 3. Forecast inundation application on southwest in the selection of the optimal storm surge simulations. Florida coast This region is characterized by mixed tides, so the approach of Lin et al. (2012) to use the characteristics over The southwest Florida coast experienced landfalls a single tidal cycle will not work. To account for the from two distinctly different hurricanes in just over one nonlinearities of the tides in this region a much more year’s time between August 2004 and October 2005. complicated model would be needed. This would lead to Hurricane Charley (2004) was a compact and intense a very large increase in the number of simulations in the hurricane that experienced a major shift in its forecast optimal database and decrease the value of the method. track just prior to landfall. Hurricane Wilma (2005) was

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FIG. 2. (left) Forecasted storm characteristics and (right) evolution of forecast tracks for Hurricanes (a) Charley and (b) Wilma. Dashed box shows outline of numerical model domain. Labeling corresponds to NHC forecast advisory (e.g., c12 is 12th forecast advisory for Hurricane Charley). a strong, large hurricane that was well forecasted with The multivariate interpolation method is applied to little change in track from forecast to forecast. These hurricane Charley in several different ways: with and two storms show very different characteristics, which without consideration of tidal effects, and in an adaptive make them ideal for hindcast analysis using our inter- and nonadaptive way. For the southwest Florida coast polation method. the tidal range is about 1.5 m during maximum spring tide conditions. During the tidal cycles prior to landfall a. Hurricane Charley the tidal range is generally about 1 m and landfall oc- Hurricane Charley made landfall near Cayo Costa, curred with a negative (20.4 m NAVD88) elevation Florida, just north of Captiva Island, Florida, around during ebb tide. Evaluation of the interpolation results 1945 UTC 13 August 2004 (Pasch et al. 2005). Charley are made with and without tide considerations. In ad- 2 was a very intense (240 km h 1 winds at landfall) and dition to the tidal considerations, two different analysis compact (Rmax of ;11 km) storm. The evolution of the methods are considered. The first is to develop the ex- hurricane forecast is shown in Fig. 2a along with the pected inundation based on the optimal storm database hurricane parameters used in the interpolation. From [197 storms without tides, 448 with tides; hereafter re- this it is seen that for most of the forecast period, ferred to as nonadaptive or (NON ADAP)]. The second Charley was forecasted to be heading toward Tampa is to fold the previous inundation forecast results into Bay. In the final forecast advisory before landfall the optimal storm database. This adaptive technique (Charley advisory 18, c18) the track took an abrupt [hereafter called adaptive (ADAP)] is employed by first change from previous advisories, with landfall forecast determining the expected inundation based on the over 100 km to the south. In addition, the size of Charley original optimal storm database for the initial forecast decreased considerably and the intensity forecast in- [NHC forecast advisory 12, for Charley (c12)] using the creased. Because of the small size, high intensity, and multivariate interpolation technique with the forecast shifting track Charley is a difficult storm to forecast. hurricane parameters near landfall. Simultaneously

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FIG. 3. Water elevation comparison between HWMs and simulated results for Hurricane Charley from (a) CH3D-SSMS with tides, (b) CH3D-SSMS without tides, (c) interpolated results using adaptive procedure including tides, (d) interpolated results using adaptive procedure without tides, (e) nonadaptive results including tides, and (f) nonadaptive results without tides.

CH3D-SSMS would be simulating the surge response make up the optimal storm databases. Figure 3a shows based on the forecast track and winds. For the next a comparison between the simulated CH3D-SSMS re- forecast advisory, the multivariate interpolation tech- sults using the best track winds and including the simple nique would be applied to the original optimal storm tide model and high water mark (HWM) data collected database plus the response from the previous CH3D- by the Florida Department of Environmental Protection SSMS forecast. In this way the results of the previous (Florida Department of Environmental Protection 2004) forecast are folded into the next forecast to improve following Hurricane Charley. Figure 3b shows the same accuracy. Assuming little change in forecast track and comparison for CH3D-SSMS without tides. There is intensity, this method will improve the forecast results little difference in the results between the two simula- by including past simulations in the optimal database, tions, with the simulations with tides showing a slightly which has input parameters that are very similar to the smaller average error and a larger correlation coefﬁcient. current forecast parameters. Both simulations show a positive average error indicating Both the ADAP and NON ADAP techniques can that the model tends to overestimate the surge height only be as good as the simulation results that are used to slightly.

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TABLE 1. Correlation coefﬁcient and average error (m) between results than the actual CH3D-SSMS simulations. The simulated (CH3D-SSMS) and interpolated (ADAP, NON ADAP) results including tides tend to be a little lower than the and observed high water marks for Hurricane Charley. results without tides, due to the negative tidal amplitude CH3D-SSMS ADAP NON ADAP at landfall. There is an improvement both with and Avg Avg Avg without tides when previous forecasts are included as is R2 error (m) R2 error (m) R2 error (m) the case in the ADAP results where simulations using With tides 0.6 0.14 0.47 20.08 0.47 20.064 NHC forecast advisories 12–18 are included in the opti- Without tides 0.6 0.18 0.55 0.19 0.55 0.23 mal storm database. In general, the results without tides show a little better correlation while the results with tides show a slightly smaller average error. It can take up to When the interpolated results are compared to the twice as long to obtain the interpolated results from the HWM data, they show that overall the average error optimal database with tides as it does to obtain the results remains small as demonstrated in Fig. 3 and summarized from the database without tides. in Table 1. Figure 3c shows the ADAP results with tides Figure 4 shows the envelope of high water (EOHW) included, Fig. 3d shows the ADAP results without tides, for Charley from the CH3D-SSMS simulation with tides Fig. 3e shows the NON ADAP results with tides, and (Fig. 4a), without tides (Fig. 4b), and the adaptive inter- Fig. 3f shows the NON ADAP results without tides. In polated results with (Fig. 4c) and without tides (Fig. 4d). all cases the results show a slight trend to more scattered The ﬁgure shows that the interpolated results do a good

FIG. 4. Envelope of high water (m) for best-track Hurricane Charley simulation in (a) CH3D-SSMS with tides, (b) CH3D-SSMS without tides, (c) adaptive interpolated results with tides, and (d) adaptive interpolated results without tides.

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FIG. 5. HWM comparison between USGS HWMs collected during Hurricane Wilma and (a) CH3D-SSMS results with tides, (b) CH3D-SSMS results without tides, (c) adaptive interpolated results with tides, (d) adaptive interpolated results without tides, (e) nonadaptive interpolated results with tides, and (f) nonadaptive interpolated results without tides. job of capturing the extent and height of the inundation. Fig. 2b. Hurricane Wilma made landfall in southwest The results without tides tend to produce slightly greater Florida on 24 October 2005 (Pasch et al. 2006) as a 2 inundation depths and extents due to the lack of a nega- category-3 hurricane with winds of 190 km h 1.Wilma tive tidal forcing. The adaptive results with tides are ob- was a very large hurricane (Rmax ; 65 km) and did not tained in 8 min, the adaptive results without tides are deviate much in track, intensity, size, forward speed, obtained in under 5 min, while the full CH3D-SSMS or approach angle from forecast to forecast. This is in simulation with and without tides takes approximately contrast to Charley, which had a large shift in track, in- 10 h each to complete with waves effects included. While tensity, and size during the forecast period. there is room for improvement in all the results, the in- The same approach used with Charley is applied to terpolated results do a good job of matching the model Wilma. Figure 5 and Table 2 show the HWM comparison simulated results in a timely manner. and summarized metrics for Hurricane Wilma. Figure 5a shows the comparison between the CH3D-SSMS results b. Hurricane Wilma using the best track winds and including tides and the In contrast to Hurricane Charley, Hurricane Wilma’s HWM data collected by USGS. Figure 5b shows the same track and intensity was well forecasted as shown in comparison but with CH3D-SSMS run without tides.

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TABLE 2. As in Table 1, but for Hurricane Wilma. the results. For this region the tidal inﬂuence is generally rather small, however, inclusion of tides may be more CH3D-SSMS ADAP NON ADAP important in other basins. In general there is a much Avg Avg Avg better correlation between all the results and the ob- R2 error (m) R2 error (m) R2 error (m) served HWMs than for Charley. As mentioned Wilma is 2 2 With tides 0.8 0.16 0.91 0.013 0.86 0.25 a much better forecast storm, which is shown in the im- Without tides 0.81 0.24 0.88 0.0002 0.63 20.096 provement in the adaptive results compared to the nonadaptive results as previous forecasts are included in the optimal storm database. Figures 5c,d show the adaptive interpolated comparison Figure 6 shows the EOHW for Wilma from the with and without tides, respectively. Figures 5e,f show the CH3D-SSMS simulation with tides (Fig. 6a), without nonadaptive interpolated comparison with and without tides (Fig. 6b), and the adaptive interpolated results with tides, respectively. As is the case with Charley, the in- (Fig. 6c) and without tides (Fig. 6d). As is the case with clusion of tides does not seem to have much of an effect Charley, the interpolated results do a very good job on the success of the model or interpolation scheme to capturing the inundation extent. The simulations and produce results comparable to the HWM data. Wilma the interpolated results both capture the surge response made landfall with a very slightly negative tidal elevation well in the populated areas around Sanibel, Florida, and (20.086 m NAVD88) during ebb tide. This small tidal Captiva Island. The interpolated results tend to be a little inﬂuence likely explains the lack of a difference between low at the peak of the surge in the Florida Everglades.

FIG. 6. As in Fig. 4, but for Hurricane Wilma.

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While this underestimate is certainly a concern for the storm size. To determine a probability distribution emergency managers it can be explained by considering for the error in this term, the model of Willoughby and potential errors in the input characteristics. The storm Rahn (2004) is used to determine the Rmax from the characteristics (DP, Rmax, Vf, and u) vary across the latitude and wind intensity. It is seen in Fig. 7 that for domain as the storm propagates toward shore. To de- the 0-h forecast the error is very well confined to a small velop the single set of test parameters used to determine range and expands as the forecast advances in time. the inundation response, the central pressure deficit, Depending on how far out landfall is from the current radius to maximum winds, and forward speed of the time, the appropriate probability distribution is applied hurricane are averaged over the 6-h period prior to to each parameter. landfall. The landfall location and angle of approach are Each parameter is discretized into a number of values taken as those at landfall. This set of test parameters has and the response for each parameter combination is error built into them since in reality they do vary but determined. That response is assigned the joint prob- a single set is necessary to run the interpolation. Since our ability of the parameter combination based on the method can rapidly develop the high-resolution in- following: undation response, it is better to develop the response for 5 : a number (thousands) of possible storm characteristics to P PD P P PuP (1) storm P Rmax Vf Xland develop a better estimate of the inundation response as is done next. In this case the tides are not included since the response can be computed in less time without tides and they did not contribute any significant improvement to 4. Generation of high-resolution probabilistic the results. However, as is done by Niedoroda et al. inundation response estimates (2010) and Condon and Sheng (2012a,b) the tides are The interpolation method has been shown to serve as included as a secondary error term in the determination a good first estimate of the inundation hazard from an of the probabilistic inundation. For each grid cell in the approaching hurricane. The method is based on deter- domain a histogram of accumulated storm probability is mining the inundation response from an optimal storm constructed consisting of 500 2-cm-wide elevation bins database characterized by five (or seven in case of tides) spanning the range from 0 to 12 m. These histograms hurricane parameters (DP, Rmax, Vf, u, and Xland). The represent approximations of the surge height density response is built by specifying a test set of the five (or distributions. An error function based on the local tide seven for tides) forecasted parameters for the storm of (with standard deviation of 0.2 m) and precision of interest. There is some error built in since the optimal CH3D-SSMS (standard deviation ; 0.15 3 surge height) storm database is built using straight-line tracks of is redistributed over the bins in the histogram creating constant hurricane characteristics until landfall, while a modified version of the original histogram. This is then in reality these parameters change prior to and after summed from the highest bin down to the lowest bin to landfall. As mentioned a 6-h-averaging period is used give an estimate of the cumulative inundation distribu- to develop the test set for the above results. Since the tion for the grid cell. With the CDF of the inundation, the method can rapidly produce the inundation response it inundation for any probability can be interpolated from is possible to include thousands of parameter combi- the curve. The inundation response that is 90%, 75%, nations, each with a probability of occurrence, into the 50%, 25%, and 10% likely to occur is determined. These test set to develop a probabilistic inundation response responses can be determined within the 1-h time con- inatimelymanner. straint on a single computational core. The test set for each storm is expanded to include Figure 8a shows the adaptive interpolated results likely values based on historical forecast errors. The without tides for hurricane Charley using the single set official NHC forecast track (along and cross track) and of test parameters as shown in Fig. 4d. Figures 8b–f show intensity errors are obtained for the past five years the inundation heights with a 90%, 75%, 50%, 25%, and (NOAA/National Hurricane Center 2011b; J. Franklin 10%, respectively, chance of occurrence based on the 2011, personal communication). These are analyzed and best-track simulation and 0-h probabilities. The figure fit to a normal distribution to associate a probability with demonstrates that there is some variance in the inunda- each (Fig. 7). To determine the central pressure deficit tion extents and depths; however, it is not that great since from the wind intensity error, the model of Knaff and the storm probabilities for 0-h forecast are well confined Zehr (2007) is used. The landfall location, forward near the actual forecast value so the new test set does speed, and storm heading can be directly computed not feature a very large spread in the storm parameters. from the data. The only variable that is not forecasted is Figure 9 shows the same panel of plots as Fig. 8, but for

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FIG. 7. (a) Forecast errors for track (landfall location), (b) intensity, (c) storm size, (d) forward speed, and (e) approach angle based on 2005–09 NHC Atlantic basin forecasts.

Hurricane Wilma. By adjusting the test set to include with the 0-h forecast error data. This figure demonstrates some variation in the storm parameters the resulting how the surge response changes with each forecast based EOHW looks very similar to the actual model results on the input parameters and the forecast error proba- for the 90% probabilistic response. bilities. The progression of forecasts show an increase Figures 8 and 9 show that by considering a larger test in surge as the storm is forecast to become more in- set the interpolated inundation response can better match tense. The spatial extent of the inundation is largest in the actual model results. Those figures are constructed the earlier forecasts where the uncertainty in landfall using the 0-h forecast probabilities that feature little location is greatest. As the forecast becomes more re- variance around the actual forecast. Figure 10 demon- fined and the forecast error decreases, the surge re- strates the evolution of a complete forecast of Hurricane sponse becomes more focused on the area of landfall in Wilma with 90% chance of occurrence. Figure 10a shows the Everglades. the inundation response based on Wilma forecast advisory 32. This advisory has landfall forecast approximately 5. Summary 24 h out, so the 24-h probabilities are used. Figure 10b shows the inundation response based of forecast advisory The rapid evaluation of the inundation threat is nec- 33 and 24-h probabilities, Fig. 10c is the response for essary for disaster planning when a hurricane is forecast advisory 34 and 12-h probabilities, Fig. 10d is based on to affect a coastal area. Current state-of-the-art numerical advisory 35 and 12-h probabilities, Fig. 10e shows the modeling systems provide accurate estimates of this re- response for advisory 36 and 0-h probabilities, and sponse, but require a large computational cost to run and Fig. 10f shows the response from the best-track winds may not be able to produce a forecast in the 6-h window

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FIG. 8. (a) Adaptive interpolation forecast for Hurricane Charley using best-track forecast parameters. Probabi- listic forecast for Hurricane Charley with (b) 90%, (c) 75%, (d) 50%, (e) 25%, and (f) 10% chance of occurrence based on 0-h forecast errors and best-track parameters.

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FIG. 9. As in Fig. 8, but for Hurricane Wilma. between forecast advisories. With this in mind a tech- selection of optimal storms for a basin. A database of nique for the rapid and high-resolution evaluation of the the surge response for the optimal storms can be built inundation hazard has been developed and presented. using a state-of-the-art storm surge modeling system This technique utilizes Smolyak’s algorithm for the for the basin. When a hurricane is forecast to affect the

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FIG. 10. Evolution of adaptive inundation response with 90% chance of occurrence for Hurricane Wilma based on forecast advisory (a) 32, (b) 33, (c) 34, (d) 35, (e) 36, and (f) the best track. basin, multivariate interpolation can be used to esti- This technique is tested in southwest Florida for mate the surge response given the storm characteris- Hurricanes Charley and Wilma, both with and without tics (DP, Rmax, Vf, u,andXland) of the approaching a simple tidal model. For this basin the use of the tide hurricane. model did not improve the results much, but does nearly

Unauthenticated | Downloaded 10/04/21 01:29 PM UTC 1318 MONTHLY WEATHER REVIEW VOLUME 141 double the time needed to build the surge response. An a NOAA IOOS grant titled ‘‘A Regional Storm Surge adaptive method that includes previous forecasts and and Coastal Inundation Model Testbed.’’ simulated surge responses into the optimal database is shown to improve the results, especially when the storm APPENDIX A is well forecasted, with little variation in intensity and track over previous forecasts, like Wilma. For Wilma the Development of Optimal Storms and Generation forecast track did not shift much, leading to a better in- of Interpolated Response terpolated inundation response. An HWM comparison shows an improvement in the correlation from 0.86 to 0.91 A multivariate interpolation scheme is used to de- and from 0.63 to 0.88 with and without tides, respectively. velop the surge response for any storm in a basin. First The average error decreases from 20.25 to 20.013 and an optimal set of storms for a basin must be developed. from 20.096 to 0.0002 for the case with tides and without This is achieved by use of dimension adaptive sparse tides, respectively. For Charley there is little difference grids. The storms are characterized by five parameters between the adaptive and nonadaptive techniques since (i.e., DP, Rmax, Vf, u,andXland) that control the surge the final forecast is very different from previous forecasts. response. There are an infinite number of possible com- The correlation did not change at all in HWM analyses binations of these parameters. To focus the work to the and the average error changed no more than 0.04 m. southwest Florida basin and with an emphasis on storms The largest source of error in the method is the pa- that contribute a significant level of surge we review the rameterization of a hurricane by a single set of five storm historic climatology and restrict these parameters to characteristics. These characteristics are not constant 33 # DP # 113 hPa, 13 # Rmax # 78 km, 2.7 # Vf # 21 throughout the forecast, but the method builds the surge 10.7 m s , 222.58 # u # 908,and2222 # Xland # 370 km response based on the optimal storm database that is of a central reference point defined as Fort Myers Beach constructed using straight-line tracks of constant char- for this study. To interpolate the response of any com- acteristics. To minimize this error a number of likely bination of these parameters in multidimensional space changes to the storm characteristics are considered. This on a regular grid requires that support nodes (i.e., DP, is done by examining the official NHC forecasts error Rmax, Vf, u, and Xland) must be specified and regularly data and applying a range of variations to each parameter spaced, which leads to a large number of necessary nodes. based on this data. Each discrete parameter value is given To work around this the support nodes can be specified a probability based on the error data so that the joint on a sparse grid to drastically reduce the required number probability of each parameter set can be determined. In of nodes. this way inundation probabilities can be determined for Sparse grid interpolants are based on Smolyak’s algo- the storm. By computing the interpolated results in this rithm (Smolyak 1963) and involve the careful combina- way, a greater range of possibilities is considered and tion of one-dimensional formulas such that multivariate potential errors are minimized. For operational use we functions can be optimally recovered (Agbley 2009). emphasize using this probabilistic method. We choose the dimension adaptive sparse grid scheme The technique presented provides a quick way to de- of Gerstner and Griebel (2003), which has been im- termine the expected inundation from an approaching plementedinMATLAB(spinterp)byKlimkeand hurricane, by utilizing a database of high-resolution Wohlmuth (2005) and Klimke (2007). The dimension optimal storm inundation responses generated by a state- adaptive scheme eliminates the isotropic construction of-the-art numerical modeling system. An ensemble ap- of a traditional sparse grid by placing more nodes in proach is adopted to account for errors in hurricane track the dimensions that minimize the calculated interpolation and the simplified parameterizations of the method. This error. The dimension adaptive aspect means that the technique will provide emergency managers the inunda- MATLAB toolbox must be coupled to a numerical model. tion data they need, as well as the probability associated The toolbox will generate a set of nodes (DP, Rmax, Vf, u, with the inundation in a timely manner so that proper and Xland), a hurricane track based on these nodes is then disaster preparation plans can be made. This technique developed and run in the storm surge model and returns will be further tested for storm events in separate basins in the surge response to the toolbox. After a few iterations future studies. the sensitivity to the different parameters is identified and more emphasis is placed on those parameters which Acknowledgments. AJC was supported by DoD and minimize the estimated interpolation error. Once a de- the Office of Naval Research through a National Defense sired level of estimated error is achieved the optimal Science and Engineering Graduate (NDSEG) Fellow- storm set is obtained. In this study that consisted of 197 ship, 32 CFR 168a. YPS and PVA were supported by storms (parameter combinations).

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The 197 optimal storms are simulated in the fully › › › y › ›S ›S u 1 uu 1 u 1 uw 1 1 xx 1 1 xy coupled CH3D-SSMS and their inundation response is › › › › r › r › t x y z w x w y recorded. To generate the response of any storm from the ›§ 1 ›P ›2u ›2u recorded response of the 197 optimal storms, multivariate 52g 2 a 1 f y 1 A 1 ›x r ›x H ›x ›y regression is needed. A number of multivariate regres- w sion techniques were tried and the method of Friedman › ›u 1 A , (B2) (1991) was used to find multivariate adaptive regression ›z V ›z splines (MARS) to fit the data. This was chosen because the MARS approach has the advantage of producing ›y › y ›yy ›y ›S ›S 1 u 1 1 w 1 1 yx 1 1 yy continuous regression functions, which makes it reliable › › › › r › r › t x y z w x w y for a number of function types and well suited for im- ›§ 1 ›P ›2y ›2y plementation on sparse grids (Agbley 2009). The inter- 52g 2 a 2 fu 1 A 1 ›y r ›y H ›x ›y polation is performed by defining the training set as the w 197 optimal tracks and the test set as the storm(s) of in- › ›y 1 A , (B3) terest characterized by the five parameters. The MARS ›z V ›z algorithm will then build the surge response of the test set based on the recorded values in the training set but in where u(x, y, z, t), y(x, y, z, t), and w(x, y, z, t) are the a fraction of the time of a full model simulation. velocity vector components in x-, y-, and z-coordinate directions, respectively; t is time; §(x, y, t) is the free APPENDIX B surface elevation; g is the acceleration of gravity; AH and AV are the horizontal and vertical turbulent eddy Governing Equations and Boundary Conditions for coefficients, respectively; Sxx, Sxy, Syy are radiation CH3D stresses; Pa is atmospheric pressure; and f is the Coriolis parameter. The A is calculated by the vertical turbu- In Cartesian coordinate system, the governing equations V lence model described in Sheng and Villaret (1989), and for water continuity, X-momentum, and Y-momentum A by a Smargorinsky-type formula. equations are H Following Sheng (1987, 1990), the nondimensional ›u ›y ›w 1 1 5 0, (B1) form of above equations in curvilinear, boundary-fitted ›x ›y ›z grid system can be written as

›z b › pffiffiffiffiffi › pffiffiffiffiffi ›Hv 1 pffiffiffiffiffi ( g Hu) 1 ( g Hy) 1 b 5 0, (B4) › ›j 0 ›h 0 ›s t g0

! 1 ›Hu ›z ›z ›P ›P g g 52 11 1 12 2 11 1 12 1 p12ffiffiffiffiffi 1 p22ffiffiffiffiffiy › g ›j g ›h g ›j g ›h u H t g0 g0 R › pffiffiffiffiffi pffiffiffiffiffi › pffiffiffiffiffi pffiffiffiffiffi 2 0 x y g S 1 y g S 1 y g S 1 y g S g h ›j j 0 jj h 0 jh ›h j 0 jh h 0 hh 0 › pffiffiffiffiffi pffiffiffiffiffi › pffiffiffiffiffi pffiffiffiffiffi 2 y x g S 1 x g S 1 x g S 1 x g S h ›j j 0 jj h 0 jh ›h j 0 jh h 0 hh R › pffiffiffiffiffi pffiffiffiffiffi › pffiffiffiffiffi pffiffiffiffiffi 2 0 x y g Huu 1 y g Huy 1 y g Huy 1 y g Hyy g H h ›j j 0 h 0 ›h j 0 h 0 0 › pffiffiffiffiffi pffiffiffiffiffi › pffiffiffiffiffi pffiffiffiffiffi ›Huv 2 y x g Huu 1 x g Huy 1 x g Huy 1 x g Hyy 2 g h ›j j 0 h 0 ›h j 0 h 0 0 ›s E › ›u 1 y A 1 E A (horizontal diffusion of u) H2 ›s y›s H H ð ð R 0 ›r ›r ›H ›H 0 2 0 H g11 1 g12 ds 1 g11 1 g12 r ds 1 sp , (B5) 2 ›j ›h ›j ›h Fr s s

Unauthenticated | Downloaded 10/04/21 01:29 PM UTC 1320 MONTHLY WEATHER REVIEW VOLUME 141 ! 1 ›Hy ›z ›z ›P ›P g g 52 21 1 22 2 21 1 22 2 p11ffiffiffiffiffi 1 p21ffiffiffiffiffiy › g ›j g ›h g ›j g ›h u H t g0 g0 R › pffiffiffiffiffi pffiffiffiffiffi › pffiffiffiffiffi pffiffiffiffiffi 2 0 x y g S 1 y g S 1 y g S 1 y g S g j ›j j 0 hj h 0 hh ›h j 0 hj h 0 hh 0 › pffiffiffiffiffi pffiffiffiffiffi › pffiffiffiffiffi pffiffiffiffiffi 2 y x g S 1 x g S 1 x g S 1 x g S h ›j j 0 jj h 0 hj ›h j 0 hj h 0 hh R › pffiffiffiffiffi pffiffiffiffiffi › pffiffiffiffiffi pffiffiffiffiffi 2 0 x y g Huy 1 y g Hyy 1 y g Huy 1 y g Hyy g H j ›j j 0 h 0 ›h j 0 h 0 0 › pffiffiffiffiffi pffiffiffiffiffi › pffiffiffiffiffi pffiffiffiffiffi ›Hyv 2 y x g Huu 1 x g Huy 1 x g Huy 1 x g Hyy 2 g h ›j j 0 h 0 ›h j 0 h 0 0 ›s E › ›y 1 y A 1 E A (horizontal diffusion of y) H2 ›s y›s H H ð ð R 0 ›r ›r ›H ›H 0 2 0 H g21 1 g22 ds 1 g21 1 g22 (r ds 1 sp) , (B6) 2 ›j ›h ›j ›h Fr s s

where Boundary conditions for the coastal surge model CH3D j, h, and s are the transformed coordinates; u, y, w are the nondimensional contra-variant veloc- The boundary condition at the free surface is calcu- j h s lated using pffiffiffiffiffiities in curvilinear grid ( , , ). g0 is the Jacobian of horizontal transformation; tw 5 r C u W , (B7) 11 22 x a d w s g , g , g11, g12, g22 are the matric coefficients of coordinate transformations; tw 5 r y y aCd wWs , (B8) b is the nondimensional parameter; z is the water level. where uw and yw are wind speed components, and Ws is the total wind speed. The drag coefficient C is calcu- It can be shown that the wave-averaged Eqs. (B4)–(B6) d lated using Garratt (1977) formulation: are valid for two regions: the region between the free surface (mean sea level) and the wave trough, as well as C 5 0:001 3 (0:75 1 0:067W ): (B9) the region between the wave trough and the bottom. In d s the region above the wave trough, the wave-averaged When waves are present, the drag coefficient is calcu- horizontal currents include the mean currents and the lated following the Donelan et al. (1993) formula de- Stokes drift, while the radiation stress includes the ver- scribed in Eqs. (1) and (2) of Sheng et al. (2010a). tically uniform radiation stress according to Longuet- The boundary condition at the bottom is expressed in Higgins and Stewart (1964) plus that due to the surface terms of bottom stress given by the quadratic law: roller (Svendsen 1984; Haas and Svendsen 2000). Below qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi the wave trough, the Stokes drift is zero, while the radi- t 5 r 2 1 y2 ation stress does not have the surface roller contribution. bx wCdub ub b , (B10) Equations (B4)–(B6) can be solved numerically using the qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi conjugate gradient algorithm modified from that used by t 5 r y 2 1 y2 by wCd b ub b , (B11) Casulli and Cheng (1992) for Cartesian grids, given suf- ficient boundary conditions (wind stresses, river inflows, where ub and yb are bottom velocities and Cd is the drag precipitation, and open boundary water elevation), initial coefficient, which is defined using the formulation by conditions (water level), and other data (bathymetry Sheng (1983): and topography). In practical applications, it is possible k 2 to solve only the vertically integrated 2D version of the 5 Cd , (B12) CH3D model, instead of solving the complete 3D equa- ln(z1/z0) tions. The 2D model generally results in significant saving in computational time and comparable water level sim- where k is the von Ka´rmań constant. The formulation ulation in shallow coastal regions. states that the coefficient is a function of the size of the

Unauthenticated | Downloaded 10/04/21 01:29 PM UTC APRIL 2013 C O N D O N E T A L . 1321 bottom roughness z0, and the height at which ub is Donelan, M. A., F. W. Dobson, S. D. Smith, and R. J. Anderson, 1993: On the dependence of sea surface roughness on wave measured z1 is within the constant flux layer above the bottom. The size of the bottom roughness can be ex- development. J. Phys. Oceanogr., 23, 2143–2149. Florida Department of Environmental Protection, 2004: Hurricane pressed in terms of the Nikuradse equivalent sand grain Charley: Post-storm beach conditions and coastal impact with 5 size ks using the relation z0 ks/30. recommendations for recovery and modifications of beach In the two-dimensional mode, the bottom bound- management strategies. 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