## 1. Introduction

The extent of coastal inundation from a given hurricane has proven to be difficult to forecast in an efficient manner. High-resolution, physics-based models such as Advanced Circulation (ADCIRC; Luettich et al. 1992; Weaver and Slinn 2006), Curvilinear-grid Hydrodynamics in 3D–Storm Surge Modeling System (CH3D-SSMS; Sheng et al. 2006, 2010a,b; Sheng and Paramygin 2010), Princeton Ocean Model (POM; Peng et al. 2004; Oey et al. 2006), and Finite Volume Coastal Ocean Model (FVCOM; Rego and Li 2009; Weisberg and Zheng 2008) have all been proven to accurately simulate coastal inundation from hurricanes. However, these models are all computationally expensive to run compared to the Sea, Lake, and Overland Surges from Hurricanes (SLOSH; Jelesnianski et al. 1992) model of the National Hurricane Center (NHC), which makes forecasting much more difficult given the tight time constraints of a 6-h forecast cycle. Typically the NHC has roughly an hour at most from the time the most recent track–intensity information is received to complete storm surge forecasts for the next 36–120 h for inclusion in the latest advisory (J. R. Rhome 2011, personal communication). Dietrich et al. (2012) show that coupled Simulating Waves Nearshore (SWAN) and ADCIRC simulations for Hurricane Katrina can take between 10 and 2000 min of wall clock time per day of simulation depending on the computing resources (8192 to 256 computational cores) and solver (implicit or explicit). For similar simulations of Hurricane Katrina, CH3D-SSMS runs at about 900 min of wall clock time per day of simulation on eight computational cores. Both these examples demonstrate that either enormous computational resources or too much wall clock time are needed to develop inundation forecasts in a timely manner. In addition the National Research Council (NRC) report “Completing the Forecast” emphasizes the need for more probabilistic forecasts that involve an ensemble of storm simulations using a storm surge modeling system (National Research Council 2006). Given the 1-h time window available to produce a hurricane storm surge forecast, it is currently not possible to run an ensemble of thousands of storms with a high-resolution modeling system. Other attempts to generate a timely estimate of the inundation response have been made.

The Saffir–Simpson hurricane scale (SSHS; Simpson 1974) was used by NHC to relate the storm surge hazard to hurricane intensity. Following the active Atlantic hurricane seasons of 2004 and 2005, it became obvious that storm surge hazard depends on other hurricane characteristics (e.g., size and forward speed) in addition to intensity. Irish et al. (2008) showed that storm size can cause variations of up to 30% in storm surge for a given storm intensity. Kantha (2006) and Powell and Reinhold (2007) developed storm surge classification schemes that look at hurricane characteristics beyond intensity to estimate the storm surge hazard posed by a particular hurricane. These scales represented an improvement over the SSHS as they accounted for storm size. However, storm surge is also dependent on the landfall location, track heading, and translational speed of the hurricane among other things for which these scales do not account (Jordan and Clayson 2008). Recently the NHC has officially removed storm surge information from the SSHS (NOAA/National Hurricane Center 2011a) because of the large differences that can develop in the surge response and inundation for storms with the same intensity but different other characteristics, and identical storms making landfall along different portions of the coast. The offshore bathymetry, coastline configuration, and topography of the affected area play a large role in dictating the extent of the inundation. Mildly sloping bathymetry has been shown to generate a larger surge response at the coast than steeper slopes (Irish et al. 2008). Likewise the landfall location can be important as demonstrated by Weisberg and Zheng (2008) for idealized storm surge simulations in the Tampa Bay, Florida, area. The topography of the area and roughness of the terrain will dictate the extent of the coastal inundation (Fletcher et al. 1995). Irish and Resio (2010) accounted for the local bathymetry in their hydrodynamics based scale, which gives the best quantitative results for the potential surge at the coast for 28 historical hurricanes compared to SSHS, Powell and Reinhold (2007), and Kantha (2006). However this scale lacks information regarding coastline configuration and topography, which is essential in determining the hazard from inundation.

In addition to the classification schemes described above to qualitatively estimate inundation hazard, more quantitative measures have been developed. The NHC uses the SLOSH (Jelesnianski et al. 1992) model operationally to produce hurricane forecasts. SLOSH is extremely efficient, with most approximately 100-h simulations taking under 1 min on a single computational core, and is typically accurate within 20% (Jelesnianski et al. 1992). The model does not account for dynamic effects of tides and waves, which other forecasting systems incorporate. The largest drawback to the SLOSH forecasts is the coarse resolution [Fort Myers, Florida, grid (efmy2) has an average resolution of 2 km] of the model domains compared to the other models mentioned. With a coarse grid many of the important small-scale topographic and bathymetric features are not captured in the model, and the effects of waves may not be accurate even if a wave model were coupled to SLOSH. Despite these drawbacks, SLOSH is used in the generation of forecasts and probabilistic products (P-Surge; Glahn et al. 2009; Taylor and Glahn 2008).

Irish et al. (2011) recently produced probabilistic maximum hurricane surge forecasts based on surge response functions (Irish et al. 2009; Resio et al. 2009), hurricane characteristics, and joint probability statistics. This approach uses high-resolution simulation results to generate surge response functions for a given region that can determine the surge response for a set of meteorological parameters. This approach is very promising but has underlying assumptions that the influence of the storm angle and forward speed can be neglected when compared to the storm intensity, size, and landfall location. While their work shows that in most cases this is a fair assumption based on model results, there are outliers which can be important. As pointed out by Rego and Li (2009) and Jelesnianski (1972), neglecting the forward speed and angle of approach may not be appropriate as there is a “critical motion relative to a coast that gives the highest possible surge.” Additionally the technique does not account for tides and wave setup, which can contribute significantly to the surge and inundation.

This paper addresses the rapid generation of high-resolution probabilistic inundation forecasts. The optimal storm generation and multivariate interpolation technique of Condon and Sheng (2012a,b) is applied to a single storm to generate an estimate of the inundation hazard for southwest Florida from Hurricanes Charley (2004) and Wilma (2005). This is accomplished in an adaptive manner to improve accuracy with each forecast. The technique considers the effect of storm intensity, size, landfall location, forward speed, and approach angle on the surge response. The optimal storm database, which includes wave effects on surge and inundation, is produced and can be combined with a simple tidal model to account for tidal effects. Analysis of the official NHC forecast errors for the past five years (NOAA/National Hurricane Center 2011b; J. Franklin 2011, personal communication) allows for efficient generation of high-resolution probabilistic surge estimates as well for each forecast period within the 1-h time constraints.

## 2. Optimal storm generation and multivariate interpolation

For this study an optimal storm ensemble for the southwest Florida basin (NHC SLOSH efm2 basin) is developed following the method presented in Condon and Sheng (2012b). In this method a dimension adaptive version of Smolyak’s algorithm (Smolyak 1963) is applied to the storm surge problem to optimally select an ensemble of storms. The method is adapted from that of Agbley (2009) to make it more transportable and accurate for storm surge estimation. This is done by using the dimension adaptive sparse grid formulation of Gerstner and Griebel (2003) in the form of the spiniterp MATLAB toolbox (Klimke and Wohlmuth 2005; Klimke 2007) and coupling with the SLOSH model (Jelesnianski et al. 1992) to obtain the storm surge simulations that provide the optimal recovery of the surge response for any given set of storm parameters. Multivariate regression is used to build the response from the optimal simulation database. This is achieved with multivariate adaptive regressive splines (MARS) as done by Friedman (1991). For additional details please see appendix A.

In Condon and Sheng (2012a,b) the hurricanes are characterized by five parameters: the central pressure deficit Δ*P*, the radius to maximum winds *R*_{max}, the translational speed *V _{f}*, the storm heading θ (angle of approach), and the landfall location

*X*

_{land}. These studies determined the hazard to the region in present-day and future climates through an adapted version of the joint probability method (JPM), which used probabilistic descriptions of these five variables combined with the surge response from 197 optimal storm simulations for the basin. The 197 high-resolution optimal simulations are performed using CH3D-SSMS. CH3D is a hydrodynamic model originally developed by Sheng (1987, 1990) and has been significantly enhanced (e.g., Sheng et al. 2010a; Sheng and Kim 2009). The model can simulate 2D and 3D barotropic and baroclinic circulation driven by tides, winds, waves, and density gradients. The model uses a boundary-fitted nonorthogonal curvilinear grid in the horizontal directions and terrain-following sigma grid in the vertical direction to allow accurate representation of the complex coastal and estuarine shorelines where forecasting of storm surge, waves, and inundation is needed. Based on the finite-volume method, CH3D is strictly conservative for momentum, water mass, as well as for temperature and salinity. CH3D uses a robust second-order closure model for calculating vertical turbulent mixing (Sheng and Villaret 1989). In the horizontal direction, Smagorinsky-type turbulent diffusion coefficients are used. With its ability to simulate flooding and drying, CH3D is used for simulating and forecasting storm surge and circulation in many coastal regions throughout Florida and the United States. Detailed governing equations and boundary conditions for CH3D are shown in appendix B.

CH3D has been dynamically coupled to a wave model SWAN (Booij et al. 1999; Ris et al. 1999), using the same curvilinear grid, to produce CH3D-SSMS (Sheng et al. 2006, 2010a,b; Sheng and Liu 2011). CH3D-SSMS uses basin-scale models, such as the Hybrid Coordinate Ocean Model (HYCOM; Halliwell et al. 1998, 2000; Bleck 2002), the Navy Coastal Ocean Model (NCOM; Barron et al. 2006), and ADCIRC (Luettich et al. 1992) in a large-scale domain, to provide open boundary conditions for CH3D. To enable efficient simulation, this study couples the CH3D model, with a high-resolution coastal grid, to the basin-scale model ADCIRC, which has a relatively coarse grid in the offshore as well as coastal regions. To provide open boundary condition for SWAN, CH3D-SSMS uses the output of a large-scale wave model such as WaveWatch-III (Tolman 1999, 2002). CH3D-SSMS has been used extensively to simulate storm surge and inundation due to various tropical storms including Hurricanes Isabel (Sheng et al. 2010a), Charley (Sheng et al. 2006; Davis et al. 2008, 2010), Ivan (Sheng et al. 2010b), and Wilma. Sheng and Paramygin (2010) combined the baroclinic circulation element of CH3D with CH3D-SSMS to forecast the storm surge, inundation, and 3D baroclinic circulation in northeast Florida during Tropical Storm Fay. Using CH3D-SSMS, this study represents a marked improvement in model physics and spatial resolution over a previous study (Condon and Sheng 2012b) using SLOSH.

For this study CH3D-SSMS uses the dynamically coupled CH3D-SWAN models for the coastal domain. Open boundary conditions for CH3D are provided by a coarse grid ADCIRC model, while open boundary conditions for SWAN are provided by a coarse grid basin-scale SWAN and WaveWatch-III models. Little difference (<0.01% in inundation) is found between the final results obtained using SWAN or WaveWatch-III in the offshore region. Hence SWAN is used for both the offshore region and the coastal region. CH3D-SSMS uses wind fields developed by an analytic hurricane wind model based on Holland (1980) in which the central pressure deficit, Holland B parameter, and radius to maximum winds control the pressure and wind distribution. The winds are developed as straight-line tracks of constant intensity until landfall. For this study, the storm intensity is dissipated following Vickery (2005) post landfall. To save computational cost, this study runs the CH3D model in 2D (vertically averaged) mode. A spatially varying Manning’s *n* coefficient is developed based on land use data obtained from the U.S. Geological Survey (U.S. Geological Survey 2011) with an offshore value of 0.02. The coastal model domain features a minimum horizontal resolution of approximately 20 m in the coastal zone and an overall average grid size of ~100 m with a maximum of ~700 m offshore. The most up-to-date lidar topography data from the National Oceanic and Atmospheric Administration (NOAA) Coastal Services Center (NOAA/Coastal Services Center 2011), topography data from U.S. Geological Survey (U.S. Geological Survey 2011), and bathymetry data from NOAA/National Geophysical Data Center (NOAA/National Geophysical Data Center 2011), have been incorporated into the domain shown in Fig. 1.

In previous studies (Condon and Sheng 2012a,b; Toro et al. 2010a,b; Niedoroda et al. 2010) the astronomical tide, which can be an important component of the inundation, is included as an error term in the JPM formulation but not directly simulated. For this study the astronomical tidal amplitude and phase are considered in the selection of the optimal storm surge simulations. This region is characterized by mixed tides, so the approach of Lin et al. (2012) to use the characteristics over a single tidal cycle will not work. To account for the nonlinearities of the tides in this region a much more complicated model would be needed. This would lead to a very large increase in the number of simulations in the optimal database and decrease the value of the method. The workaround to this is to construct a simple one-component sinusoidal model that matches the amplitude of the tide at the time of landfall. The NOAA Naples, Florida, tide gauge (NOAA 2011) was examined to determine the range of expected tide levels. By including this range of expected tide levels a total of 265 optimal storms is needed to obtain similar estimated error as that with 197 storms and no tides. These are then adjusted to account for the phase by running simulations for both increasing (flood) and decreasing (ebb) tides for all intermediate tidal amplitudes. For amplitudes at the peak or trough of the tidal cycle, only one simulation is run. This resulted in a total of 448 simulations of the surge response in the optimal database. This is a substantial increase over the 197 responses without tides but far less than would be needed if all the possibilities of the mixed tides were included. This increase in the number of necessary optimal storm simulations will be analyzed below.

## 3. Forecast inundation application on southwest Florida coast

The southwest Florida coast experienced landfalls from two distinctly different hurricanes in just over one year’s time between August 2004 and October 2005. Hurricane Charley (2004) was a compact and intense hurricane that experienced a major shift in its forecast track just prior to landfall. Hurricane Wilma (2005) was a strong, large hurricane that was well forecasted with little change in track from forecast to forecast. These two storms show very different characteristics, which make them ideal for hindcast analysis using our interpolation method.

### a. Hurricane Charley

Hurricane Charley made landfall near Cayo Costa, Florida, just north of Captiva Island, Florida, around 1945 UTC 13 August 2004 (Pasch et al. 2005). Charley was a very intense (240 km h^{−1} winds at landfall) and compact (*R*_{max} of ~11 km) storm. The evolution of the hurricane forecast is shown in Fig. 2a along with the hurricane parameters used in the interpolation. From this it is seen that for most of the forecast period, Charley was forecasted to be heading toward Tampa Bay. In the final forecast advisory before landfall (Charley advisory 18, c18) the track took an abrupt change from previous advisories, with landfall forecast over 100 km to the south. In addition, the size of Charley decreased considerably and the intensity forecast increased. Because of the small size, high intensity, and shifting track Charley is a difficult storm to forecast.

(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).

Citation: Monthly Weather Review 141, 4; 10.1175/MWR-D-12-00149.1

(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).

Citation: Monthly Weather Review 141, 4; 10.1175/MWR-D-12-00149.1

(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).

Citation: Monthly Weather Review 141, 4; 10.1175/MWR-D-12-00149.1

The multivariate interpolation method is applied to hurricane Charley in several different ways: with and without consideration of tidal effects, and in an adaptive and nonadaptive way. For the southwest Florida coast the tidal range is about 1.5 m during maximum spring tide conditions. During the tidal cycles prior to landfall the tidal range is generally about 1 m and landfall occurred with a negative (−0.4 m NAVD88) elevation during ebb tide. Evaluation of the interpolation results are made with and without tide considerations. In addition to the tidal considerations, two different analysis methods are considered. The first is to develop the expected inundation based on the optimal storm database [197 storms without tides, 448 with tides; hereafter referred to as nonadaptive or (NON ADAP)]. The second is to fold the previous inundation forecast results into the optimal storm database. This adaptive technique [hereafter called adaptive (ADAP)] is employed by first determining the expected inundation based on the original optimal storm database for the initial forecast [NHC forecast advisory 12, for Charley (c12)] using the multivariate interpolation technique with the forecast hurricane parameters near landfall. Simultaneously CH3D-SSMS would be simulating the surge response based on the forecast track and winds. For the next forecast advisory, the multivariate interpolation technique would be applied to the original optimal storm database plus the response from the previous CH3D-SSMS forecast. In this way the results of the previous forecast are folded into the next forecast to improve accuracy. Assuming little change in forecast track and intensity, this method will improve the forecast results by including past simulations in the optimal database, which has input parameters that are very similar to the current forecast parameters.

Both the ADAP and NON ADAP techniques can only be as good as the simulation results that are used to make up the optimal storm databases. Figure 3a shows a comparison between the simulated CH3D-SSMS results using the best track winds and including the simple tide model and high water mark (HWM) data collected by the Florida Department of Environmental Protection (Florida Department of Environmental Protection 2004) following Hurricane Charley. Figure 3b shows the same comparison for CH3D-SSMS without tides. There is little difference in the results between the two simulations, with the simulations with tides showing a slightly smaller average error and a larger correlation coefficient. Both simulations show a positive average error indicating that the model tends to overestimate the surge height slightly.

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.

Citation: Monthly Weather Review 141, 4; 10.1175/MWR-D-12-00149.1

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.

Citation: Monthly Weather Review 141, 4; 10.1175/MWR-D-12-00149.1

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.

Citation: Monthly Weather Review 141, 4; 10.1175/MWR-D-12-00149.1

When the interpolated results are compared to the HWM data, they show that overall the average error remains small as demonstrated in Fig. 3 and summarized in Table 1. Figure 3c shows the ADAP results with tides included, Fig. 3d shows the ADAP results without tides, Fig. 3e shows the NON ADAP results with tides, and Fig. 3f shows the NON ADAP results without tides. In all cases the results show a slight trend to more scattered results than the actual CH3D-SSMS simulations. The results including tides tend to be a little lower than the results without tides, due to the negative tidal amplitude at landfall. There is an improvement both with and without tides when previous forecasts are included as is the case in the ADAP results where simulations using NHC forecast advisories 12–18 are included in the optimal 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 twice as long to obtain the interpolated results from the optimal database with tides as it does to obtain the results from the database without tides.

Correlation coefficient and average error (m) between simulated (CH3D-SSMS) and interpolated (ADAP, NON ADAP) and observed high water marks for Hurricane Charley.

Figure 4 shows the envelope of high water (EOHW) for Charley from the CH3D-SSMS simulation with tides (Fig. 4a), without tides (Fig. 4b), and the adaptive interpolated results with (Fig. 4c) and without tides (Fig. 4d). The figure shows that the interpolated results do a good job of capturing the extent and height of the inundation. The results without tides tend to produce slightly greater inundation depths and extents due to the lack of a negative tidal forcing. The adaptive results with tides are obtained in 8 min, the adaptive results without tides are obtained in under 5 min, while the full CH3D-SSMS simulation with and without tides takes approximately 10 h each to complete with waves effects included. While there is room for improvement in all the results, the interpolated results do a good job of matching the model simulated results in a timely manner.

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.

Citation: Monthly Weather Review 141, 4; 10.1175/MWR-D-12-00149.1

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.

Citation: Monthly Weather Review 141, 4; 10.1175/MWR-D-12-00149.1

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.

Citation: Monthly Weather Review 141, 4; 10.1175/MWR-D-12-00149.1

### b. Hurricane Wilma

In contrast to Hurricane Charley, Hurricane Wilma’s track and intensity was well forecasted as shown in Fig. 2b. Hurricane Wilma made landfall in southwest Florida on 24 October 2005 (Pasch et al. 2006) as a category-3 hurricane with winds of 190 km h^{−1}. Wilma was a very large hurricane (*R*_{max} ~ 65 km) and did not deviate much in track, intensity, size, forward speed, or approach angle from forecast to forecast. This is in contrast to Charley, which had a large shift in track, intensity, and size during the forecast period.

The same approach used with Charley is applied to Wilma. Figure 5 and Table 2 show the HWM comparison and summarized metrics for Hurricane Wilma. Figure 5a shows the comparison between the CH3D-SSMS results using the best track winds and including tides and the HWM data collected by USGS. Figure 5b shows the same comparison but with CH3D-SSMS run without tides. Figures 5c,d show the adaptive interpolated comparison with and without tides, respectively. Figures 5e,f show the nonadaptive interpolated comparison with and without tides, respectively. As is the case with Charley, the inclusion of tides does not seem to have much of an effect on the success of the model or interpolation scheme to produce results comparable to the HWM data. Wilma made landfall with a very slightly negative tidal elevation (−0.086 m NAVD88) during ebb tide. This small tidal influence likely explains the lack of a difference between the results. For this region the tidal influence is generally rather small, however, inclusion of tides may be more important in other basins. In general there is a much better correlation between all the results and the observed HWMs than for Charley. As mentioned Wilma is a much better forecast storm, which is shown in the improvement in the adaptive results compared to the nonadaptive results as previous forecasts are included in the optimal storm database.

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.

Citation: Monthly Weather Review 141, 4; 10.1175/MWR-D-12-00149.1

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.

Citation: Monthly Weather Review 141, 4; 10.1175/MWR-D-12-00149.1

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.

Citation: Monthly Weather Review 141, 4; 10.1175/MWR-D-12-00149.1

Figure 6 shows the EOHW for Wilma from the CH3D-SSMS simulation with tides (Fig. 6a), without tides (Fig. 6b), and the adaptive interpolated results with (Fig. 6c) and without tides (Fig. 6d). As is the case with Charley, the interpolated results do a very good job capturing the inundation extent. The simulations and the interpolated results both capture the surge response well in the populated areas around Sanibel, Florida, and Captiva Island. The interpolated results tend to be a little low at the peak of the surge in the Florida Everglades. While this underestimate is certainly a concern for emergency managers it can be explained by considering potential errors in the input characteristics. The storm characteristics (Δ*P*, *R*_{max}, *V _{f}*, and θ) vary across the domain as the storm propagates toward shore. To develop the single set of test parameters used to determine the inundation response, the central pressure deficit, radius to maximum winds, and forward speed of the hurricane are averaged over the 6-h period prior to landfall. The landfall location and angle of approach are taken as those at landfall. This set of test parameters has error built into them since in reality they do vary but a single set is necessary to run the interpolation. Since our method can rapidly develop the high-resolution inundation response, it is better to develop the response for a number (thousands) of possible storm characteristics to develop a better estimate of the inundation response as is done next.

As in Fig. 4, but for Hurricane Wilma.

Citation: Monthly Weather Review 141, 4; 10.1175/MWR-D-12-00149.1

As in Fig. 4, but for Hurricane Wilma.

Citation: Monthly Weather Review 141, 4; 10.1175/MWR-D-12-00149.1

As in Fig. 4, but for Hurricane Wilma.

Citation: Monthly Weather Review 141, 4; 10.1175/MWR-D-12-00149.1

## 4. Generation of high-resolution probabilistic inundation response estimates

The interpolation method has been shown to serve as a good first estimate of the inundation hazard from an approaching hurricane. The method is based on determining the inundation response from an optimal storm database characterized by five (or seven in case of tides) hurricane parameters (Δ*P*, *R*_{max}, *V _{f}*, θ, and

*X*

_{land}). The response is built by specifying a test set of the five (or seven for tides) forecasted parameters for the storm of interest. There is some error built in since the optimal storm database is built using straight-line tracks of constant hurricane characteristics until landfall, while in reality these parameters change prior to and after landfall. As mentioned a 6-h-averaging period is used to develop the test set for the above results. Since the method can rapidly produce the inundation response it is possible to include thousands of parameter combinations, each with a probability of occurrence, into the test set to develop a probabilistic inundation response in a timely manner.

The test set for each storm is expanded to include likely values based on historical forecast errors. The official NHC forecast track (along and cross track) and intensity errors are obtained for the past five years (NOAA/National Hurricane Center 2011b; J. Franklin 2011, personal communication). These are analyzed and fit to a normal distribution to associate a probability with each (Fig. 7). To determine the central pressure deficit from the wind intensity error, the model of Knaff and Zehr (2007) is used. The landfall location, forward speed, and storm heading can be directly computed from the data. The only variable that is not forecasted is the storm size. To determine a probability distribution for the error in this term, the model of Willoughby and Rahn (2004) is used to determine the *R*_{max} from the latitude and wind intensity. It is seen in Fig. 7 that for the 0-h forecast the error is very well confined to a small range and expands as the forecast advances in time. Depending on how far out landfall is from the current time, the appropriate probability distribution is applied to each parameter.

(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.

Citation: Monthly Weather Review 141, 4; 10.1175/MWR-D-12-00149.1

(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.

Citation: Monthly Weather Review 141, 4; 10.1175/MWR-D-12-00149.1

(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.

Citation: Monthly Weather Review 141, 4; 10.1175/MWR-D-12-00149.1

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 the results. However, as is done by Niedoroda et al. (2010) and Condon and Sheng (2012a,b) the tides are included as a secondary error term in the determination of the probabilistic inundation. For each grid cell in the domain a histogram of accumulated storm probability is constructed consisting of 500 2-cm-wide elevation bins spanning the range from 0 to 12 m. These histograms represent approximations of the surge height density distributions. An error function based on the local tide (with standard deviation of 0.2 m) and precision of CH3D-SSMS (standard deviation ~ 0.15 × surge height) is redistributed over the bins in the histogram creating a modified version of the original histogram. This is then summed from the highest bin down to the lowest bin to give an estimate of the cumulative inundation distribution for the grid cell. With the CDF of the inundation, the inundation for any probability can be interpolated from the curve. The inundation response that is 90%, 75%, 50%, 25%, and 10% likely to occur is determined. These responses can be determined within the 1-h time constraint on a single computational core.

Figure 8a shows the adaptive interpolated results without tides for hurricane Charley using the single set of test parameters as shown in Fig. 4d. Figures 8b–f show the inundation heights with a 90%, 75%, 50%, 25%, and 10%, respectively, chance of occurrence based on the best-track simulation and 0-h probabilities. The figure demonstrates that there is some variance in the inundation extents and depths; however, it is not that great since the storm probabilities for 0-h forecast are well confined near the actual forecast value so the new test set does not feature a very large spread in the storm parameters. Figure 9 shows the same panel of plots as Fig. 8, but for Hurricane Wilma. By adjusting the test set to include some variation in the storm parameters the resulting EOHW looks very similar to the actual model results for the 90% probabilistic response.

(a) Adaptive interpolation forecast for Hurricane Charley using best-track forecast parameters. Probabilistic 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.

Citation: Monthly Weather Review 141, 4; 10.1175/MWR-D-12-00149.1

(a) Adaptive interpolation forecast for Hurricane Charley using best-track forecast parameters. Probabilistic 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.

Citation: Monthly Weather Review 141, 4; 10.1175/MWR-D-12-00149.1

(a) Adaptive interpolation forecast for Hurricane Charley using best-track forecast parameters. Probabilistic 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.

Citation: Monthly Weather Review 141, 4; 10.1175/MWR-D-12-00149.1

As in Fig. 8, but for Hurricane Wilma.

Citation: Monthly Weather Review 141, 4; 10.1175/MWR-D-12-00149.1

As in Fig. 8, but for Hurricane Wilma.

Citation: Monthly Weather Review 141, 4; 10.1175/MWR-D-12-00149.1

As in Fig. 8, but for Hurricane Wilma.

Citation: Monthly Weather Review 141, 4; 10.1175/MWR-D-12-00149.1

Figures 8 and 9 show that by considering a larger test set the interpolated inundation response can better match the actual model results. Those figures are constructed using the 0-h forecast probabilities that feature little variance around the actual forecast. Figure 10 demonstrates the evolution of a complete forecast of Hurricane Wilma with 90% chance of occurrence. Figure 10a shows the inundation response based on Wilma forecast advisory 32. This advisory has landfall forecast approximately 24 h out, so the 24-h probabilities are used. Figure 10b shows the inundation response based of forecast advisory 33 and 24-h probabilities, Fig. 10c is the response for advisory 34 and 12-h probabilities, Fig. 10d is based on advisory 35 and 12-h probabilities, Fig. 10e shows the response for advisory 36 and 0-h probabilities, and Fig. 10f shows the response from the best-track winds with the 0-h forecast error data. This figure demonstrates how the surge response changes with each forecast based on the input parameters and the forecast error probabilities. The progression of forecasts show an increase in surge as the storm is forecast to become more intense. The spatial extent of the inundation is largest in the earlier forecasts where the uncertainty in landfall location is greatest. As the forecast becomes more refined and the forecast error decreases, the surge response becomes more focused on the area of landfall in the Everglades.

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.

Citation: Monthly Weather Review 141, 4; 10.1175/MWR-D-12-00149.1

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.

Citation: Monthly Weather Review 141, 4; 10.1175/MWR-D-12-00149.1

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.

Citation: Monthly Weather Review 141, 4; 10.1175/MWR-D-12-00149.1

## 5. Summary

The rapid evaluation of the inundation threat is necessary for disaster planning when a hurricane is forecast to affect a coastal area. Current state-of-the-art numerical modeling systems provide accurate estimates of this response, but require a large computational cost to run and may not be able to produce a forecast in the 6-h window between forecast advisories. With this in mind a technique for the rapid and high-resolution evaluation of the inundation hazard has been developed and presented. This technique utilizes Smolyak’s algorithm for the selection of optimal storms for a basin. A database of the surge response for the optimal storms can be built using a state-of-the-art storm surge modeling system for the basin. When a hurricane is forecast to affect the basin, multivariate interpolation can be used to estimate the surge response given the storm characteristics (Δ*P*, *R*_{max}, *V _{f}*, θ, and

*X*

_{land}) of the approaching hurricane.

This technique is tested in southwest Florida for Hurricanes Charley and Wilma, both with and without a simple tidal model. For this basin the use of the tide model did not improve the results much, but does nearly double the time needed to build the surge response. An adaptive method that includes previous forecasts and simulated surge responses into the optimal database is shown to improve the results, especially when the storm is well forecasted, with little variation in intensity and track over previous forecasts, like Wilma. For Wilma the forecast track did not shift much, leading to a better interpolated inundation response. An HWM comparison shows an improvement in the correlation from 0.86 to 0.91 and from 0.63 to 0.88 with and without tides, respectively. The average error decreases from −0.25 to −0.013 and from −0.096 to 0.0002 for the case with tides and without tides, respectively. For Charley there is little difference between the adaptive and nonadaptive techniques since the final forecast is very different from previous forecasts. The correlation did not change at all in HWM analyses and the average error changed no more than 0.04 m.

The largest source of error in the method is the parameterization of a hurricane by a single set of five storm characteristics. These characteristics are not constant throughout the forecast, but the method builds the surge response based on the optimal storm database that is constructed using straight-line tracks of constant characteristics. To minimize this error a number of likely changes to the storm characteristics are considered. This is done by examining the official NHC forecasts error data and applying a range of variations to each parameter based on this data. Each discrete parameter value is given a probability based on the error data so that the joint probability of each parameter set can be determined. In this way inundation probabilities can be determined for the storm. By computing the interpolated results in this way, a greater range of possibilities is considered and potential errors are minimized. For operational use we emphasize using this probabilistic method.

The technique presented provides a quick way to determine the expected inundation from an approaching hurricane, by utilizing a database of high-resolution optimal storm inundation responses generated by a state-of-the-art numerical modeling system. An ensemble approach is adopted to account for errors in hurricane track and the simplified parameterizations of the method. This technique will provide emergency managers the inundation data they need, as well as the probability associated with the inundation in a timely manner so that proper disaster preparation plans can be made. This technique will be further tested for storm events in separate basins in future studies.

## Acknowledgments

AJC was supported by DoD and the Office of Naval Research through a National Defense Science and Engineering Graduate (NDSEG) Fellowship, 32 CFR 168a. YPS and PVA were supported by a NOAA IOOS grant titled “A Regional Storm Surge and Coastal Inundation Model Testbed.”

## APPENDIX A

### Development of Optimal Storms and Generation of Interpolated Response

A multivariate interpolation scheme is used to develop the surge response for any storm in a basin. First an optimal set of storms for a basin must be developed. This is achieved by use of dimension adaptive sparse grids. The storms are characterized by five parameters (i.e., Δ*P*, *R*_{max}, *V _{f}*, θ, and

*X*

_{land}) that control the surge response. There are an infinite number of possible combinations of these parameters. To focus the work to the southwest Florida basin and with an emphasis on storms that contribute a significant level of surge we review the historic climatology and restrict these parameters to 33 ≤ Δ

*P*≤ 113 hPa, 13 ≤

*R*

_{max}≤ 78 km, 2.7 ≤

*V*≤ 10.7 m s

_{f}^{−1}, −22.5° ≤ θ ≤ 90°, and −222 ≤

*X*

_{land}≤ 370 km of a central reference point defined as Fort Myers Beach for this study. To interpolate the response of any combination of these parameters in multidimensional space on a regular grid requires that support nodes (i.e., Δ

*P*,

*R*

_{max},

*V*, θ, and

_{f}*X*

_{land}) must be specified and regularly spaced, which leads to a large number of necessary nodes. To work around this the support nodes can be specified on a sparse grid to drastically reduce the required number of nodes.

Sparse grid interpolants are based on Smolyak’s algorithm (Smolyak 1963) and involve the careful combination of one-dimensional formulas such that multivariate functions can be optimally recovered (Agbley 2009). We choose the dimension adaptive sparse grid scheme of Gerstner and Griebel (2003), which has been implemented in MATLAB (spinterp) by Klimke and Wohlmuth (2005) and Klimke (2007). The dimension adaptive scheme eliminates the isotropic construction of a traditional sparse grid by placing more nodes in the dimensions that minimize the calculated interpolation error. The dimension adaptive aspect means that the MATLAB toolbox must be coupled to a numerical model. The toolbox will generate a set of nodes (Δ*P*, *R*_{max}, *V _{f}*, θ, and

*X*

_{land}), a hurricane track based on these nodes is then developed and run in the storm surge model and returns the surge response to the toolbox. After a few iterations the sensitivity to the different parameters is identified and more emphasis is placed on those parameters which minimize the estimated interpolation error. Once a desired level of estimated error is achieved the optimal storm set is obtained. In this study that consisted of 197 storms (parameter combinations).

The 197 optimal storms are simulated in the fully coupled CH3D-SSMS and their inundation response is recorded. To generate the response of any storm from the recorded response of the 197 optimal storms, multivariate regression is needed. A number of multivariate regression techniques were tried and the method of Friedman (1991) was used to find multivariate adaptive regression splines (MARS) to fit the data. This was chosen because the MARS approach has the advantage of producing continuous regression functions, which makes it reliable for a number of function types and well suited for implementation on sparse grids (Agbley 2009). The interpolation is performed by defining the training set as the 197 optimal tracks and the test set as the storm(s) of interest characterized by the five parameters. The MARS algorithm will then build the surge response of the test set based on the recorded values in the training set but in a fraction of the time of a full model simulation.

## APPENDIX B

### Governing Equations and Boundary Conditions for CH3D

*X*-momentum, and

*Y*-momentum equations are

*u*(

*x*,

*y*,

*z*,

*t*),

*υ*(

*x*,

*y*,

*z*,

*t*), and

*w*(

*x*,

*y*,

*z*,

*t*) are the velocity vector components in

*x*-,

*y*-, and

*z*-coordinate directions, respectively;

*t*is time;

*ς*(

*x*,

*y*,

*t*) is the free surface elevation;

*g*is the acceleration of gravity;

*A*and

_{H}*A*are the horizontal and vertical turbulent eddy coefficients, respectively;

_{V}*S*

_{xx},

*S*

_{xy},

*S*

_{yy}are radiation stresses;

*P*is atmospheric pressure; and

_{a}*f*is the Coriolis parameter. The

*A*is calculated by the vertical turbulence model described in Sheng and Villaret (1989), and

_{V}*A*by a Smargorinsky-type formula.

_{H}*ξ*, η, and σ are the transformed coordinates;*u*,*υ*,*w*are the nondimensional contra-variant velocities in curvilinear grid (ξ, η, σ).is the Jacobian of horizontal transformation; are the matric coefficients of coordinate transformations; *β*is the nondimensional parameter;*ζ*is the water level.

#### Boundary conditions for the coastal surge model CH3D

*u*and

_{w}*υ*are wind speed components, and

_{w}*W*is the total wind speed. The drag coefficient

_{s}*C*is calculated using Garratt (1977) formulation:

_{d}*u*and

_{b}*υ*are bottom velocities and

_{b}*C*is the drag coefficient, which is defined using the formulation by Sheng (1983):

_{d}*κ*is the von Kármán constant. The formulation states that the coefficient is a function of the size of the bottom roughness

*z*

_{0}, and the height at which

*u*is measured

_{b}*z*

_{1}is within the constant flux layer above the bottom. The size of the bottom roughness can be expressed in terms of the Nikuradse equivalent sand grain size

*k*using the relation

_{s}*z*

_{0}=

*k*/30.

_{s}*C*is the Chezy friction coefficient defined as

_{z}*R*is the hydraulic radius that can be approximated by the total depth given in centimeters, and

*n*is Manning’s

*n*.

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