Atmospheric Science Component: State of Florida Hurricane Loss Projection Model _

Transcription

1 Atmospheric Science Component: State of Florida Public Hurricane Model Final report to: Florida International University and the Florida Department of Financial Services May, 2005 _

2 State of Florida Hurricane Loss Projection Model: Atmospheric Science Component Final Report to the Florida International University and the Florida Department of Financial Services Mark Powell a, George Soukup a, Steve Cocke b, Nirva Morisseau-Leroy d, Neal Dorst a, and Lizabeth Axe b a NOAA Hurricane Research Division, Miami, FL b Florida State University, Tallahassee, FL c Florida International University, Miami, FL d Cooperative Institute for Marine and Atmospheric Studies, University of Miami, Miami, FL

3 Executive Summary The State of Florida has developed an open, public model for the purpose of probabilistic assessment of risk to insured residential property associated with wind damage from hurricanes. The model comprises atmospheric science, engineering, and financial/actuarial components and is planned for submission to the Florida Commission on Hurricane Loss Projection Methodology. The atmospheric component includes modeling the track and intensity life cycle of each simulated hurricane within the Florida threat area. When a model storm approaches within a damage threshold distance of a Florida zip code location, the wind field is computed by a slab model of the hurricane boundary layer coupled with a surface layer model based on the results of recent GPS sonde research. A time series of open terrain surface winds is then computed for each zip code in the threatened area. Depending on wind direction, an effective roughness length is assigned to each zip code based on the upstream fetch roughness as determined from remotely sensed land cover/ land use products. Based on historical hurricane statistics, thousands of storms are simulated allowing determination of the wind risk for all residential zip code locations in Florida. The wind risk information is then provided to the engineering and loss models to assess damage and average annual loss, respectively. 1

4 1. Introduction The historical record for establishing the risk of hurricanes throughout the coastal United States is limited to a period of about 100 years, with relatively reliable and complete records [1]. Unfortunately this period is not sufficient to establish risk without large errors so alternative methods have been used since the 1970's (Ref [2, 3]). Hurricane risk models are currently used to conduct simulations of thousands of years of storms based on probability distributions of important historically observed parameters. This method is often referred to as the joint probability method since the probability of having an event is coupled with the probability that the event is of a given intensity. Commercial modeling interests have developed several versions of these which are used to advise the insurance industry for ratemaking. Unfortunately the models are proprietary so customers whose rates have increased on the basis of model calculations have no way of examining or questioning the results. The State of Florida is developing a public model to provide an understandable baseline for comparison to the commercial models. The model will be open and transparent, in which methods and results can be examined in great detail. The Hurricane Loss Projection Model consists (Fig. 1) of independent modules for atmospheric science, engineering, and actuarial components. Each component provides one way input to the next component in line until the end result (average annual loss per zip code) is achieved. A model flow chart (Fig. 2) describes the order of calculations for the atmospheric science component. State of Florida Hurricane Loss Projection Model Atmospheric Science Component I Storm genesis Movement Intensity Atmospheric Science Component II Wind Field Zip Code time series Terrain influences Probabilities of exceedence Engineering Component Inventory of residential construction Damage when wind load exceeds resistance Financial/Actuarial Component Compute insured losses Average annual loss by zip code Figure 1. Flow chart depiction of independent modules of the State of Florida Hurricane Loss Projection Model 2

5 State of Florida Hurricane Loss Projection Model Atmospheric Science Component Select number of years for simulation Select number of storms per year from PDF of annual hurricane occurrence rate Select genesis time and location of each storm from temporal and geographical PDFs Model storm track and intensity at 1h intervals using geographic motion and intensity change PDFs Determine wind model parameters for all Florida hurricane landfalls: Delta P, Rmax, Storm speed, Pressure profile Every 15 min, compute open terrain wind speed and direction at all zip codes within damage threshold distance of storm center Correct winds to actual terrain based on upstream fetch and roughness Determine wind speed exceedance probabilities at each zip code Compute damages at each zip code: Engineering damage model Compute average annual loss: Actuarial loss model Figure 2. Flow chart depiction of independent modules of the State of Florida Hurricane Loss Projection Model. 3

6 2. Threat area To focus on storms capable of causing residential property damage in Florida, a threat area is defined to best capture the statistical characteristics of historical tropical cyclones that have affected the state. The area within 1000 km of a location (26.0 N, 82.0W) off the southwest coast of Florida (Fig. 3) was chosen since this captures storms that can affect the panhandle, west, and northeast coasts of Florida, as well as storms that approach South Florida from the vicinity of Cuba and the Bahamas. FLORIDA HURRICANE THREAT AREA 1000 km Figure 3. Threat area map. All storms entering or developing within the threat area are considered for characteristics relating to formation, movement, and intensity. 4

7 3. Annual occurrence The model has the capability of simulating climate cycles and tropical cyclone activity according to different periods of the historical record (Ref [4]). The historical record for the Atlantic tropical cyclone basin (known as HURDAT ) is a six hourly record of tropical storm and hurricane positions and intensities [5], which are expressed as estimated maximum 1 min surface (10 m) winds and, when available, central sea level pressure. The period is the largest available but the period is most often used due to uncertainties about 19th century storms, especially for Florida due to a lack of population centers and meteorological measurements of hurricanes before the start of the 20th century. There are also uncertainties about the first half of the 20th century since aircraft reconnaissance only began in the 1940's so another choice in the period of record are the years Four additional choices are available which simulate the warm (El Nino, fewer hurricanes) and neutral or cold (La Nina, more hurricanes) inter annual climate cycles in tropical cyclone activity, as well as the cold or warm phases of the Multi-decadal climate cycles. These choices are primarily for research purposes and constrain the historical record to use only years with the specified climate cycle to fit annual tropical cyclone occurrence. For insurance applications the 1900-prior year period will be of most interest (e.g. for the 2005 season, the period would be of interest if the 2004 season official hurricane tracks are available). Two fits are tested to the annual hurricane occurrence frequency distribution: the negative binomial and the Poisson model. Chi-Square goodness of fit tests determine which fit is used for the subsequent simulations. The chosen fit is then randomly sampled to determine how many storms occur for each year of the simulation. Once the number of tropical cyclones within the threat area for a given year is determined, the date and time of genesis is computed from an empirical distribution based on historical storm tracks. Further details on the annual hurricane occurrence are contained in the use case appendix. 5

8 4. Storm genesis, movement, and intensity The threat area is divided into regions which contain the historical and seasonal characteristics of storm motion and intensity change. Initial location, intensity, and motion for each storm are based on the geographic probability distributions of each quantity for a given time within the season. Intensity is initially modeled as the gradient of pressure Δp, the difference between the central minimum sea level pressure and an outer peripheral pressure (assumed to be 1013 hpa). Ultimately (after the wind model is run) the intensity is defined according to the maximum surface wind speed in the storm. Hurricane tracks are modeled as a Markov process. We use a stochastic approach to model the storm genesis location and track and central pressure evolution. A probability distribution function (PDF) for the initial storm position is derived from the historical "genesis'' data. Here we define genesis as the time when a hurricane forms in or first appears in the threat area. The PDF is derived for 0.5 degree latitude/longitude box regions, as well as time of season (month). A (uniform) random error term is added so that the storm may form anywhere within the 0.5 degree box. Figure 4 shows a plot of the spatial PDF for storm genesis locations in the threat area for the month of August. 4. Geographic distribution of probability of a storm entering or developing within the threat area during the month of August based on the historical period We derive discrete PDFs based on historical data to provide the initial and subsequent motion and intensity of the storm. A storm is simulated by repeatedly sampling from these PDFs via a Monte Carlo approach. These PDFs are derived for variable-sized regions centered at every 0.5 degree latitude and longitude in the hurricane basin. The size of these regions is determined to be that which gives a robust probability density function (PDF) for the quantities of interest (speed, 6

9 direction, and central pressure), up to some maximum size. Once the storm has been given an initial condition, its subsequent evolution is governed by sampling the PDFs for changes in intensity, translation speed, and heading angle in 1 hour increments. Intensity change is modeled by using the observed geographic probability distribution of sixhour changes of central pressure as related to the relative intensity, a measure of how close the storm comes to meeting its maximum possible intensity (potential intensity) (Ref [6]). Potential intensity takes into account the concept of the hurricane as a heat engine constrained by the input (sea surface) and outflow (upper troposphere) temperatures. Intensity change is limited so as to not exceed the maximum observed change for a particular geographic region. When a storm center crosses the coastline (landfall) the intensity change follows a pressure decay model (discussed below). If the storm moves back over the sea, the former intensity change model is reinstated. A storm need not make landfall to be considered however; bypassing hurricanes are considered that never make landfall, or make landfall well before or well after producing damaging winds at a particular zip code. The PDFs for change in storm translation speed and direction depend on the current speed and direction (binned in discrete intervals), as well as geographic location (0.5 degree lat-lon location) and time of season (month). Figure 5 shows a PDF for change in direction for 8 possible direction intervals (45 degree intervals). The PDF indicates that the storm has a high probability for maintaining current direction, except for westward traveling storms which tend to turn right (northward) somewhat. 0.8 Probability of Change in Direction for 8 current direction intervals Probability Change in Direction 5. Probability of a change in direction of storm motion (over a 6 h time period) to the left or right of the current motion as a function of the current storm heading (in degrees from north), based on historical information from

10 Our approach has a great advantage over early models [2, 3] that considered a circular approach region surrounding coastal cities. Storms that parallel the coast or make several landfalls can be properly simulated with our method. 5. Storm decay The tropical ocean is typically warmer than the air above it, enabling a transfer of heat and moisture to the air. Inflow towards the center of the tropical cyclone transports this energy toward the eyewall, where it can help sustain convection, leading to a positive feedback of warming in the eye, lower minimum central pressure, stronger inflow, and more energy transport (Ref [7], [8]). Over the ocean this positive feedback loop may be slowed or reversed by ocean cooling, advection of relatively cold dry air with a history of travel over land, or strong wind shear that prevents the storm center from focusing heating in the eye (Refs. [9], [10]). As a hurricane makes landfall, the circulation traverses land, and the storm loses its source of energy (Ref. [11]). More and more dry and relatively cool air flows towards the center, causing the air to cool as it expands adiabatically while approaching lower pressures (Ref. [12]). The result is that the eye heating gradually decreases and the central pressure begins to increase or fill. Since the wind model depends on the specification of the pressure gradient, a method was needed to estimate the central pressure over land. As a starting point for a simple decay model we will use the exponential decay as a function of time after landfall developed by Vickery and Twisdale [13]. The HURDAT database and Ho et al. [14] contains documentation of storm decay and pressure filling for many hurricane landfall cases in the historical record. Vickery and Twisdale [13] developed and tested a model for the Florida peninsula based on nine landfalling hurricanes and found the model to be slightly conservative (larger values) within 3 h of landfall and slightly non-conservative (smaller values) beyond 3 h after landfall. The form of the model is:!!!!! Δp(t) = Δp0exp( at)!!! (1) where Δp(t) is the time dependent central pressure deficit and t represents the time after landfall. The filling rate constant is given as: a = ao + a1δpo + ε (2) where ε is a random error term with a normal distribution. The dependence of the filling rate constant on Δpo allows stronger storms to decay faster than weak storms; a characteristic observed in hurricane landfalls (Ref. [15]). The random error term allows for the possibility that some storms will decay slower or faster than average. For the Florida peninsula Vickery and Twisdale [13] use ao=0.006, a1= , and the standard deviation of = The Kaplan and DeMaria [16] model is also pertinent but it deals with wind decay rather than pressure decay and there is no well-established method to convert inland-decayed peak winds to central pressure. The advantage of the filling model is that it provides a starting point to invoke an intensity redevelopment for storms that exit the coastline and re-intensify over water. When a storm reemerges over water, the intensity is modeled along the track the same way it was before landfall using the decayed pressure as an initial value. 8

11 ! 6. Damage distance threshold To save computing resources, the wind field is not evaluated unless a hurricane passes within a distance from the storm at which damage might be possible, the damage distance threshold. Hurricanes come in a variety of sizes and shapes, so a fixed distance criterion may not be practical, since it may be unnecessarily large for small storms. As an alternative, we describe the damage distance threshold (D) as a function of the radius of maximum wind speed (Rmax). For small storms with small Rmax we will go outward from the storm center ~ 10 Rmax and for large storms we will go out a as far as 4 Rmax. Twelve hurricanes comprising a variety of storm structures and shapes were evaluated by comparing HRD Hurricane Wind Analysis System (H*Wind) analyses of the outermost radius of marine exposure 25 m/s sustained winds. For hurricanes with Rmax > 50 km, D is held constant at 4 Rmax. For Rmax < 48 km, D = Rmax (3) The 25 m/s sustained marine wind speed contour represents a 1 h mean marine wind of 21 m/s. This wind speed is assumed to represent the mean marine exposure wind speed at which light damage begins. The equivalent 1 h mean wind for roughness representative of Florida zip codes is ~ 14.5 m/s. Further details on the damage distance threshold are contained in the use case appendix. 7. Wind field model The model is based on the slab boundary layer concept originally conceived by Ooyama [17] and implemented by Shapiro [18]. Similar models based on this concept have been developed by Thompson and Cardone [19] and Vickery et al. [20, 21]. As in Ref. [18] the model is initialized by a boundary layer vortex in gradient balance. Gradient balance represents a circular flow caused by the balance of forces whereby the inward directed pressure gradient force is balanced by outward Coriolis and centripetal accelerations. The coordinate system translates with the hurricane vortex moving at velocity c. The vortex translation is assumed to equal the geostrophic flow associated with the large scale pressure gradient. As a possible future enhancement the large scale flow in which the vortex is embedded may be treated independently of the vortex motion as in Ref. [19]. In cylindrical coordinates that translate with the moving vortex, equations for a slab hurricane boundary layer under a prescribed pressure gradient (Ref. [18]} are: u u r v 2 r fv + v u r φ + p r K 2 u u r 2 u 2 r 2 + F c r,u φ ( ) = 0 = u t! (4)!! u v r + v + fu + v v r r φ K 2 v v r + 2 u 2 r 2 + F c r,v φ ( ) = 0 = v t! (5) 9

12 where u and v are the respective radial and tangential wind components relative to the moving storm, p is the sea-level pressure which varies with radius (r), f is the Coriolis parameter which varies with latitude, is the azimuthal coordinate, K is the eddy diffusion coefficient, and F(c,u), F(c,v) are frictional drag terms (discussed below). All terms are assumed to be representative of means through the boundary layer. The motion of the vortex is determined by the modeled storm track. Note that equations (4) and (5) represent a steady state solution. In order to solve the equations they must be integrated until the steady state assumption is satisfied either through time integration (e.g. Ref. [18, 19]) or numerical method (see appendix). More sophisticated multiple level models (e. g. Kurihara et al., [22]) include equations describing the thermodynamic processes including convection, cloud and precipitation microphysics, evaporation of sea spray, exchange of heat and moisture with the sea, etc. These processes interact to change the pressure and wind fields over time, but the computational requirements of such models make them poorly suited for risk assessment. An advantage of our approach is that the solution to (4) and (5) is straightforward and we do not have the computationally costly requirement to run the model to steady state each time we desire a solution. A limitation of our model (and all other Hurricane risk models) is the lack of physical representation of additional processes that may influence the wind field of a tropical cyclone. Further details on the wind field model are contained in the model equations appendix. 7.1 Surface pressure field The symmetric pressure field p(r) is specified as:!!!!! ( ) = po + Δpe p r R max r B!!!! (6) where po is the central minimum sea level pressure, B is the Holland [23] pressure profile shape parameter, R is the radius of maximum wind speed (in nautical miles), and Δp is the pressure deficit defined earlier. The central pressure is modeled according to the intensity modeling in concert with the storm track. The mean tangential gradient wind is determined primarily by (6). A recent publication by Willoughby and Rahn [25] of the NOAA-HRD database of research and reconnaissance aircraft measurements was supplemented with central pressure measurements and Rmax values adjusted for tilt between the surface and the 3 km level. These data were then cut to be more relevant to our model. We retained 201 profiles with maximum winds > 33 m/s, with flight levels below the 700 hpa pressure surface, and with Atlantic basin latitudes N and longitudes west of 60 degrees West. The resulting expression for B explains 20% of the variance: 10

13 ! B = Lat Rmax + ε!! (7)!! where Lat is the latitude and ε is a random term from a zero mean normal distribution with a standard deviation of B is censored to remain above 0.8 and below 2.2. Further details on the Holland B model are contained in the use case appendix. 7.2 Radius of maximum wind The radius of maximum wind is determined from a distribution of landfall values as a function of po and latitude. As in Ref. [21], a log normal distribution is assumed for Rmax with a mean value determined as a function of p and latitude. In developing the models for Rmax, we used the data from Ref. [14] for storms from ; NOAA-HRD archives of realtime surface wind analyses from ; an extended best track archive of the National Hurricane Center that was maintained by Dr. Mark DeMaria (now with NOAA s NESDIS at Colorado State University) for the years ; and an HRD archive of aircraft observations for the years To create a model to describe Rmax we considered Gulf of Mexico and U. S. Atlantic coast hurricane landfalls with latitudes as high as 34 degrees north in order to help fill a dearth of information on storms affecting the Northeast Florida coastline. The relationship between Rmax and p and latitude shows much scatter but a generalized linear model for the natural log of R (r 2 = 0.212) provides a useful estimation: lnrmax = Δp Δp Lat 2 + ε (8) where ε is a normal random variable with a mean of zero and a standard deviation of 0.3. Equation (7) describes the mean of the log normal distribution of R in nautical miles. When a simulated storm is close enough to land to become a threat, an R value is randomly chosen given the p and latitude. R is computed at each time step but the random error term is computed only once for each landfall. Rmax is censored to remain below 102 km and above 7.4 km. 7.3 Friction terms The frictional drag is in the direction opposite the total wind relative to the earth at the surface and represents the mean momentum flux from the atmosphere to the surface. The frictional terms in (3) and (4) may be specified in terms of the vertical gradient of stress:!!!! F( c r,u) = τ z!!!!! (9) The variation of with height over the depth of the slab boundary layer may be described by a function with properties supported by observations in the surface layer of tropical cyclones. The lower 250 m of the boundary layer is occupied by a surface layer in which stress is approxi- 11

14 mately constant. In reality the stress is greatest close to the surface, and is zero at the top of the boundary layer. By assuming stress as near constant over the surface layer we satisfy conditions for applying the logarithmic wind profile to describe the vertical variation of the mean wind speed with height. The fact that a mean logarithmic wind profile has been observed in tropical cyclone eyewalls (Ref. [26]) provides justification for the surface layer concept. The evaluation of (9) is approximated as 0.25 (τsfc / h), where τsfc is the surface stress and h represents the top of the boundary layer. Based on the mean height of the wind maximum in the hurricane eyewall in [26], an h value of 500 m is justified. The factor 0.25 is based on judgment and takes into account the fact that the average stress in the slab boundary layer is less than the mean stress of the surface layer. This factor is consistent with preliminary calculations of the stress profile in the boundary layer derived from GPS sonde observations described in [26]. Other modelers using the slab boundary layer approach have used 0.3 [13] and 1.0 [18]. At the surface, the stress may be expressed in terms of the earth-relative surface wind velocity U10 at a height of 10 m above the surface, and the neutral stability surface drag coefficient, Cd :!!!!! r τsfc = ρcdu 10 + c r r U 10 + c r ( )!!! (10) where!!!!! F ( c r,u) = τ z =.25 τ h!!!! (11) Cd is specified by Large and Pond [27] and U10 is the neutral stability surface (10 m, 32 ft) wind speed relative to the moving storm. 7.4 Eddy Diffusion The wind model describes the effects of horizontal turbulence following Shapiro [18]. In theory the role of eddy diffusion is to represent horizontal turbulent mixing in the radial and tangential directions. Vertical turbulent mixing (contained in the friction terms discussed earlier) is typically much larger in magnitude and is associated with a large body of research based on numerous field investigations. Horizontal mixing is most prominent in regions with strong horizontal gradients. Hence the greatest impact of eddy mixing will be in the eyewall where radial gradients are strong. In practice, this term primarily serves as a way to smooth out computational noise in the model results but future enhancements might include a dependence on horizontal shear or grid spacing. 7.5 Model implementation The hurricane wind field model is based on a fully two dimensional, time-independent, scaled version of the tangential and radial momentum equations (4 and 5) for the mean boundary layer wind components (see the appendix for a complete description). The model makes use of a polar coordinate representation grid centered on the moving cyclone. The nested circles are separated 12

15 from their inscribed and circumscribed neighbors by a radial separation of 0.1 R/Rmax; the azimuthal interval of the radial spokes is 10 degrees. Implementation proceeds according to the following steps: First, based on the input parameters, Rmax, po, and B, radial profiles of the radial and tangential winds are calculated based on a stationary cyclone over open water to provide an envelope with which to set the size of the cyclone vortex. The wind field produced by these profiles is radially symmetric. Azimuthal variation is introduced through the use of two form factors. The form factors multiply the radial and tangential profiles described above and provide a factorized ansatz for both the radial and tangential storm relative wind components. Each form factor contains three constant coefficients which are variationally determined in such a way that the ansatz constructed satisfies (as far as its numerical degrees of freedom permit) the scaled momentum equations for the storm-relative polar wind components. The azimuthal variable (ϕ) has its usual mathematical meaning such that increases from left to right with the rectangular X axis aligned ( =180, 0) and the Y axis aligned ( =270, 90) with Y increasing in the direction of storm translation. The storm translation vector is added to the storm-relative wind components in order to obtain the earth-relative wind field. The translational motion of the storm is incorporated in the surface friction terms in the momentum equations which depend on the and are specific for the direction of storm translation which is aligned with the Y axis. The wind field grid is then rotated so that the computational y axis coincides with the actual direction of motion of the cyclone center. The wind field thus far constructed (Fig. 6) usually shows the location of peak winds to be to the right or forward edge of the right-rear quadrant of the cyclone. 13

16 6. Horizontal distribution of mean surface wind speed (m s -1 ) relative to the earth for a Hurricane moving northward (top of page) at 5 m s -1. Horizontal coordinates are scaled by the radius of maximum wind. 7.6 Asymmetries in the wind field The solution of (4) and (5) exhibits a shift of the radius of maximum winds toward the center when compared with the gradient wind profile. The tangential winds are also super gradient due to the advection inward of angular momentum induced by the frictional convergence. Besides vortex translation, radial advection of tangential momentum, and differential friction, other factors affecting the asymmetric distribution of winds in a tropical cyclone include synoptic scale vertical shear of the horizontal wind, synoptic scale weather features (e.g. a strong high pressure ridge to one side of the hurricane), rain band convection, concentric eyewall cycles, mesoscale variations in air-sea temperature difference, and tertiary circulations associated organized linear flow features and turbulent eddies. In the simple model described here, only the motion and differential friction influences are taken into account. The remaining features are difficult to parameterize and may be computationally prohibitive to include in the model but play an important role in determining the azimuthal location of the peak wind. Future versions of the model will attempt to include enhancements that take into account asymmetric wind field mechanisms. 14

17 ! 7.7 Marine surface layer Once the mean PBL motion field is determined, the surface wind is estimated through surface layer modeling. Monin-Obukov heights provide an estimate of the importance of shear- or mechanically-produced turbulence to buoyancy-produced turbulence. The large values of Monin-Obukov heights computed in tropical cyclones by Moss and Rosenthal [28] and Powell [29] are consistent with shear-induced turbulence associated with neutral, well-mixed surface layers. Hence, a neutral stability surface layer is assumed to exist. In these conditions we can specify the surface stress and friction velocity in terms of a drag coefficient, and use the wellknown log profile to describe the variation of wind speed with height. The mean boundary layer (MBL) depth is assumed to be 500 m, and the MBL wind speed is assumed to apply to the midpoint of this layer or 250m. The MBL height assumption is consistent with mean GPS sonde wind speed profiles [26] which depict the maximum winds at 500 m. A log profile for neutral stability is assumed to apply from the surface (10 m) to 250 m. Recent research on marine boundary layer wind profiles in tropical cyclones (Ref. [26]) support this assumption to at least 150 m. The mean surface wind for marine exposure is assumed to be 80% of the slab boundary layer wind, in accordance with recent results from boundary layer wind profiles measured in tropical cyclones (Ref. [26]): r U V! (12)!!! There is considerable variability in the factor used to estimate the mean surface wind speed. The standard deviation of the reduction factor in (12) is about 10% but this is further complicated by profile failures in extreme conditions and the fact that the profiles are representative of open ocean conditions, rather than coastal onshore flow. When sufficient eyewall wind profiles become available near the coast, it is conceivable that a more sophisticated approach would involve sampling from a statistical distribution of coastal marine boundary layer reduction factors. The height of the model surface wind is 10 m in accordance with the American Society for Testing and Materials (ASTM) standard practice for characterizing surface wind [30] and is assumed to represent a mean over a 1 h time period (Ref. [13]). We should note that the appropriate time period to assign the model wind speed is not well known and could be adjusted based on how well the model compares to observations. Other hurricane risk models assign wind speed averaging time periods of 20 to 60 min (see Refs [13, 19]), while real-time hurricane wind analyses using surface winds estimated with model-adjusted flight-level or GPS sonde MBL measurements typically assume 5-10 min averaging periods. Marine roughness is modeled using the Large and Pond [27] drag coefficient (10) to compute friction velocity given the mean surface wind speed, and then solving the neutral stability log law for Zo. The Large and Pond [27] expression for drag coefficient was found to compare well with open-ocean measurements in hurricanes for wind speeds up to hurricane force. For higher 15

18 wind speeds, recent hurricane measurements [Ref. 26] suggest that the drag coefficient and roughness decrease with wind speed over the open ocean. However, sea state conditions during the landfall of a hurricane are very different than those over the open ocean. Anctil and Donelan [31] suggest that shoaling conditions in the shallow water adjacent to the coastline cause increased roughness and drag coefficient. An extrapolation of the Large and Pond [27] expression to hurricane wind speeds yields CD values larger than those observed over the open ocean. This behavior is consistent with additional surface layer turbulence associated with coastal wave breaking and shoaling conditions. Further justification for our approach comes from preliminary results of coastal roughness measurements within 200 m of the shore recently obtained during the landfall of Hurricane Isabel at Cape Hatteras by the State of Florida Coastal Monitoring Program (Reinhold and Gurley) [32]. These unique measurements (Fig. 7) in onshore flow document roughness values measured by the turbulent intensity method at 5 m and 10 m heights, in addition to the profile method. Although there is considerable scatter among the different methods, the results are consistent with the Large and Pond [27] estimates and more than an order of magnitude larger than open-ocean roughness [26] in similar wind speeds. Further measurements are needed to see if this behavior persists for winds greater than a Saffir-Simpson category one hurricane, and for onshore flow without possible effects of intervening dunes. Figure 7. Dependence of coastal roughness on maximum 1 min sustained wind speed for onshore flow. Measurements from Cape Hatteras in the eyewall of Hurricane Isabel, courtesy of the State of Florida Coastal Monitoring Program. Isabel measurements [29], from 5 m (squares), 10 m (x), profile method (triangles), Large and Pond s [24] drag coefficient relationship (diamonds), and open ocean measurements (Y) from other hurricanes [23]. 16

19 8. Land friction influences To standardize observations for a common terrain (Ref. [33]), the mean surface wind for marine conditions is converted to open terrain conditions over land using the expression given in Simiu and Scanlan [34]. Improved techniques for converting between large and small roughness regimes are the subject of a future model enhancement. For each 10 min segment of storm motion, the open terrain exposure surface wind speed and direction is determined for all populationweighted zip code centroid locations within the damage threshold distance from the storm center. The open terrain wind at each zip code centroid is corrected to the observed terrain using a fetch-dependent effective roughness for that particular direction and zip code. The effective roughness takes into account the flow over upstream flow obstacles and assumes that internal boundary layer development prevents the flow from reaching complete equilibrium with its surroundings (Refs. [33, 35]). The flow is most influenced by the roughness of the terrain 3 km upstream of the zip code centroid, but the flow is still influenced by terrain further upstream. The approach we use is based on the Source Area Model (SAM) described in Schmidt and Oke [36] and implemented by Axe [37]. SAM takes into account turbulence created by patchy terrain and determines the relative importance of the turbulence source area on a downstream wind sensor located at the zip code centroid. This approach is an improvement over current models that consider zip code roughness constant for all wind directions. Our method is especially advantageous for coastal zip code locations since flow with an upstream fetch over the sea can be significantly stronger than flow over a constant land roughness. The geographic distribution of roughness (Fig. 8) is associated with a classification of the land use / land cover (LU/LC) in a particular region according to high resolution (50 m) LANDSAT imagery used to develop the National Multi-Resolution Land Cover database (Ref. [38]). Figure 8. Distribution of aerodynamic roughness (m) for the State of Florida determined from multi- resolution land-use land-cover classifications. 17

20 Determination of the roughness for each LU/LC classification was developed by the National Institute for Building Sciences for FEMA s multi hazard damage mitigation model (HAZUS). A limitation of the LU/LC method is that only 21 classifications are available with only two corresponding to developed residential areas. Florida Water Management District LU/LC databases include additional classifications but only three residential classifications. Detailed analysis of airborne lidar data is expected to provide more accurate distributions of surface roughness in the future. A gust factor (Refs. [33, 39]) is used to convert the mean surface wind for the appropriate fetchdependent roughness to a maximum sustained one min wind as required by the State of Florida Commission on Hurricane Loss Projection, and to a peak 3s gust as required for the engineering component damage calculations. The same effective roughness is used for the mean wind and gusts. We should caution that the equilibrium fetch lengths for gusts and mean winds are not the same and a more sophisticated fetch modeling approach may be needed to address this in the future. An example of the model wind field for three 2004 hurricanes affecting Florida Hurricane are shown in Fig. 9. At the end of a simulation, time series of wind speed and direction exist for all zip codes in Florida for which hurricanes (or hurricanes that have decayed to a weaker status) have passed within the damage threshold distance. An advantage of our approach over other models is that the complete time series of the wind may be retained at high resolution. Retaining this information makes possible the determination of additional damage-relevant parameters such as duration of winds exceeding hurricane force and wind steadiness. Powell et al., [40] showed that the highest damage observed in Hurricane Andrew was associated with large values of duration and small values of wind direction steadiness. These parameters capture the physical torques of thousands of gust-lull cycles acting on a structure. In addition, given the susceptibility of residential buildings to damage at roof corners and gables, the more the wind direction changes during a strong wind event, the greater the chance that a given wind direction will occur for which a structure is susceptible. Currently the time series capability is available only for scenario simulations. A future enhancement of the model will relate the duration and wind direction unsteadiness to the vulnerability functions contained in the engineering damage component of the model. 9. Validation The track and intensity model will be validated by comparison to historical distributions of landfalls by intensity, translation speed, and Saffir-Simpson category along the Florida coastline. The wind model accuracy is validated by comparisons of model fields to observation-based wind field analyses created by the Hurricane Research Division s Hurricane Wind Analysis System (H*Wind, Fig. 9, [9, 42] ) as well as to maximum wind speeds from hurricanes that have affected Florida (based on the National Hurricane Center s Best Track, and the HURDAT database. 18

21 Figure 9 Comparisons of model surface wind field to observation based H*Wind analyses of 2004 Hurricanes Charley, Frances, and Ivan. Wind fields represent maximum sustained (1 min ) surface wind speeds (m/s) for marine exposure. Scaled horizontal coordinate is R / Rmax, North is at top. 19

22 In the model validation that follows, we have chosen 14 cases of recent hurricane landfalls in which H*Wind analyses were constructed based on all available observations. Comparisons of H*Wind maximum sustained wind speeds for marine exposure are compared to the wind field model (Fig. 10). Observed Validation5_3_2005.jmp: (H*Wind) vs Fit Y Model by X max sustained surface wind speeds Page 1 of 2 Bivariate Fit of H*wind Marine wind (mph) By Model Marine wind 1 H*wind Marine wind (mph) Linear Fit Linear Fit Model Marine wind 1 minute (mph) H*wind Marine wind (mph) = Model Marine wind 1 minute (mph) Summary of Fit 4, 5 RSquare RSquare Adj Root Mean Square Error Mean of Response Observations (or Sum Wgts) 8 9 6, 7 99 % confidence interval Andrew 2. Charley 3. Georges (Keys) 4. Frances 5. Jeanne 6. Ivan 7. Opal 8. Erin (NW FL) 9. Georges (MS) 10. Danny (LA/MS) 11. Earl 12. Danny (AL) 13. Earl 14. Erin (Central FL) Fig. 10 Comparison of model peak sustained surface wind speeds to H*Wind objectively analyzed peak winds. The blue line is the line of perfect fit and the red dashed lines are the 99 % confidence limits on the fit. Arrows indicate typical uncertainty of the model and observed peak winds. A fit to the data show that the model explains over 80% of the variance in the observed maximum winds. These comparisons show a tendency to predict higher winds than observed for intense storms and lower than predicted for the weaker storms. It should be kept in mind that with the exception of ARA s model these types of plots are not generally available for the proprietary models so we do not have a benchmark for how well this type of model should be reproducing the peak winds for the 14 cases. We should also point out that we have not adjusted the input files to attempt to tune the fit. The input data for Rmax, storm motion, minimum central pressure and storm position are as observed. In most cases, the Holland B parameter was esti- 20

23 mated based on observations from NOAA and Air Force reconnaissance aircraft. In addition, although the H*Wind analyses are considered objective and state of the art, alternative peak wind speed estimates are available for the same hurricane landfalls from the National Hurricane Center s Best Track (BT) and the Official Hurricane Base Set published in the 2004 Commission Report of Activities. Comparisons of these data with H*Wind show that the BT contains a high bias for storms with peak winds over 110 mph while all but one base set value exceed H*Wind. Throughout the range of wind speeds each alternative estimate contains a high bias when compared to the peak observed winds in the storm from the H*Wind analysis. Bivariate Fit of H*wind Marine wind (mph) B 180 H*wind Marine wind (mph) Linear Fit Linear Fit Base Set wind at landfall (mph) H*wind Marine wind (mph) = Base Set wind at landfall (mph) Summary of Fit RSquare RSquare Adj Root Mean Square Error Mean of Response Observations (or Sum Wgts) Fig. 11 H*Wind objective analyses vs. NHC Best track (left) and Official Hurricane Base Set estimates (Right). The base set does not include 2004 Hurricanes Charley, Frances, Ivan and Jeanne which are included in the H*Wind cases. A comparison of the model to BT (Fig. 12, left) shows less tendency for the model to overpredict wind speeds for winds over 110 mph but tendency towards under-prediction for weak storms, and a preponderance for most of the points to fall above the blue line of perfect fit. Points plotted above the blue line represent model winds that are less than BT. For the Official Hurricane Base Set, many more points cluster on either side of the line of perfect fit throughout the wind speed range but there is considerable scatter as shown by a smaller amount of variance explained by the fit (65%). An important aspect of the public model is to disclose the shortcomings as well as the highlights. Based on the Official Hurricane Set it might look like the model is performing reasonably well but based on the H*Wind analyses we are overestimating winds in the strong hurricanes and underestimating in the weaker hurricanes. 21

24 Base Set wind at landfall (mph) Linear Fit Linear Fit Model Marine wind 1 minute (mph) Base Set wind at landfall (mph) = Model Marine wind 1 minute (mph) Summary of Fit RSquare RSquare Adj Root Mean Square Error Mean of Response Observations (or Sum Wgts) Fig. 12 As in Fig. 10 but for the BT values (left) and Official Hurricane Base Set values (right). 9.1 Uncertainty It is instructive to evaluate the model according to NIST guidelines for evaluating and expressing uncertainty [44]. The most basic measure is Type A standard uncertainty, which is the standard deviation of the mean errors from a series of independent model simulations. Here we evaluate the Type A standard uncertainty by evaluating the comparisons of the model to the H*Wind and Official Hurricane base set. 22

25 Fig. 13 Distribution of differences between model and H*Wind peak sustained wind speeds. Fig. 14 Distribution of differences between model and Official Hurricane Base Set peak sustained wind speeds. Looking at the histogram of differences between the model and H*Wind in Fig. 13, a small negative bias is shown (due to a larger number of relatively weak storms) and the standard deviation (Type A Uncertainty) is 14 mph. The differences between the model and the Official Hurricane Base Set (Fig. 15) shows a larger negative bias of almost 2 mph and a similar standard deviation (Type A Uncertainty) near 14 mph. Expressed as a percentage of the mean from the Official Hurricane Base set, the uncertainty is 14%. This uncertainty is similar to that expressed in Vickery and Twisdale (1995) [13] for fastest mile winds over 90 mph. However, our model uncertainty is larger than that expressed by Vickery et al., 2000, [20] which suggested values of 9%. One should keep in mind that there is also uncertainty in the observations used as the truth. For H*Wind analyses, the uncertainty is 7-12 % depending on what observing platforms are used. Uncertainty increases to > 20% if the maximum wind is estimated by a reduction factor as was done for Hurricane Andrew. If individual wind speed records are used (as in Vickery s investigations), the uncertainties are similar to H*Wind since both methods use similar standardization processes. Some of the outliers in both histograms may be reduced by a detailed evaluation of the available storm track, Rmax, minimum central pressure and Holland B information that are required to run the model. Most of this information comes from the HURDAT database which (especially for early 20th century storms) has many missing as well as questionable data. We have already spent considerable time preparing 1 hour resolution input files for each hurricane in the official base set as well as the 2004 season. In preparing these files we have had to deal with missing HURDAT central pressure values, or HURDAT containing unrealistic changes in wind relative to pressure. If Rmax and Holland B data were not available we used the same Rmax and B models that are used for the stochastic version of the hurricane track and intensity model. 23

26 The way the wind model is currently configured, it is difficult to determine the cause of the model behavior. Over-prediction of strong storms could be related to difficulties in specifying the pressure profile parameter from fits of the hurricane reconnaissance observations. Underprediction of weaker hurricanes could be caused by an incorrect specification of boundary layer depth. The current configuration of the model combines the boundary layer depth and frictional stress terms into a term that is constrained to preserve the input Rmax. In order to investigate model behavior at the low and high ends, the frictional terms will need to be unbundled which in turn will allow the radius of maximum surface wind to migrate inward from the Rmax specified in the model input file. That work is expected to be completed during the summer of At this point the model appears to reproduce reasonably well the peak sustained winds speeds of the Official Hurricane Set. 10. Conclusions A Monte Carlo simulation model has been constructed to estimate average annual loss from hurricane wind damage to residential properties. The model is open in the sense that all results will be accessible to the public and the methodology will be completely documented and available for examination. The model incorporates results of many recent research advances while keeping basic enough to run in a reasonable amount of time. In order to reduce sampling error, a very large simulation (~55,000 years of activity) is prescribed. It is expected that the model would be run once per year to take advantage of the latest historical data to assess annual wind exceedance probabilities at each zip code in Florida. Such information can then be provided to the engineering and actuarial components of the model to assess average annual loss for any given residential property exposure portfolio. At present the wind model calculation requires minutes of computation time per storm, requiring the simulations to be distributed among a number of powerful workstations. The primary user will be the Florida Department of Financial Services, insurance companies, and homeowners in the state of Florida. There are also many research uses for the model and it is expected that annual enhancements will be required to keep up with the state of the art. The model will reside at Florida International University s International Hurricane Research Center in Miami. Complete documentation of the model algorithms and code will be available for public examination. Given that there may be one Florida hurricane landfall per simulation year, a large number of storms will be contained in our database. Each storm may effect as many as 50 zip codes so it is expected that the database could contain several million records for track, intensity, and landfall wind field information on each storm. The database could then be queried for details on simulated storms and wind exceedance probability distributions relevant to a given zip code or county in Florida. 24

27 Acknowledgments This research is supported by the State of Florida through a Department of Financial Services grant to the Florida International University International Hurricane Research Center. We thank the Department of Homeland Security (Federal Emergency Management Agency) and the National Institute for Building Sciences for their assistance with the roughness classifications. Dr. Chris Landsea of NOAA-AOML provided the climate cycle information used for the annual hurricane occurrence portion of the model. Drs. Hugh Willoughby and Frank Marks, former and current directors of the Hurricane Research Division, respectively, provided valuable guidance and support. We appreciate helpful comments by Drs. Robert Rogers, Matthew Eastin, Chris Landsea, and the anonymous reviewers of our manuscript submitted to the Journal of Wind Engineering and Industrial Aerodynamics. We also appreciate the assistance of Professor Gary Barnes, University of Hawaii, who conducted two site visits to examine and review the atmospheric science component of the model. This work benefited from an outstanding collaboration of university and federal government scientists under the guidance of Professor Shahid Hamid, the principal investigator. We greatly appreciate the assistance of Professors Jean Paul Pinelli (FIT), Kurtis Gurley (UF), Sneh Gulati (FIU), and Shu-Ching Chen (FIU), together with Dr. Emil Simiu of NIST. The outstanding assistance of FIU Computer science students Min Chen, Na Zhao, Xiaosi Zho, and Kasun Wickramaratna is gratefully acknowledged. 25

28 Appendix A: Hurricane Model Equations and Integration 1. Definitions rmax = Radius of maximum surface wind speed, specified ct = storm translation speed, specified from storm track cdir = storm translation direction compass heading, specified from storm track Δp = Central pressure deficit, specified ( ) r = po + Δpe max r p r B = sea level pressure B = Holland profile parameter Φ = Azimuthal coordinate, measured counterclockwise from east s = r/rmax normalized radial coordinate vg(s) = Gradient wind: s + r max fvg = 1 p ρ s vg 2 f =2 Ω sinθ Coriolis parameter Θ latitude of storm center Ω earth angular acceleration vo(s) = normalized gradient wind (symmetric) = vg(s) /vgmax where vgmax is the maximum gradient wind in the radial profile f = r max f vg max normalized Coriolis parameter v(s,φ) = v vg max normalized storm-relative tangential wind component 26

29 u(s,φ) = u vg max normalized storm-relative radial wind component α= friction coefficient, r maxcd h h= mean boundary layer height Cd = Drag Coefficient ct c = vg max normalized translation speed g(s) = 2vo(s)s 1 + f (A1) d(s) = v o + vos 1 + f (A2) where a dot represents a derivative with respect to s, g(s) and d(s) depend only on v0 and f σ(s,φ) = v(s,φ) vo(s) normalized departure from gradient balance 2. Scaling of the governing equations prior to implementation. Substituting the terms from the above definitions and changing the radial coordinate from r to s, the steady-state form of the governing equations (4) and (5) become: u s u + s 1 (v o +σ ) φ u σ ( g + s 1 σ ) +α(u +csinφ)(w c) = 0 (A3) u s σ + s 1 (v o +σ ) φ σ + u(d + s 1 σ ) +α(v o +σ + ccosφ)(w c) = 0 (A4) w = (u +csinφ) 2 +(v o +σ + ccosφ) 2 (A5) where w is the total normalized earth-relative wind. In the event that c vanishes, so that the cyclone is stationary, these equations reduce to the ordinary differential equations: uu σ ( g + s 1 σ ) +αuw = 0 (A6) u( σ + s 1 σ + d) +α(v o +σ )w = 0 (A7) w = u 2 +(v o +σ ) 2 (A8) 27

30 for the radial profiles u(s) and σ(s). Here,. indicates differentiation with respect to s. 3. Wind model implementation Equations A3 and A4 supplemented by A5, constitute two, coupled, time independent partial differential equations for the storm relative radial velocity u and the storm relative departure from gradient balance s. The storm relative tangential wind is then given by v=vg+s Unfortunately, the direct numerical solution of A3 and A4 is time consuming even though the equations are time-independent because the non-linear coupling of the terms necessitates an iterative numerical approach. However, equations A6 and A7, can readily be numerically integrated to furnish a completely symmetric wind field fully described by the radial profiles u(s) and v(s)=vg(s)+ σ(s). The functions u(s) and s(s) so obtained can serve as radial profiles for the construction of basis functions for a more realistic attack on A3 and A4. Namely, we put forth the ansatz: u(s,φ) = ffu(φ)u(s) (A9) σ (s,φ) = ffσ (φ)σ (s) (A10) where the azimuthal dependence is introduced through the form factors: ffu(φ) = a 0 + a 1 cosφ + a 2 sinφ (A11) ffσ(φ) = b 0 + b 1 cosφ + b 2 sinφ (A12) Now the six coefficients a0, a1, a2 and b0, b1, b2 can be variationally determined by substituting A9 and A10 into the left hand sides of A3 and A4, supplemented by A5 to form the "residuals" RA3 and RA4. We then form the functional: J(a, b) = RA3 + RA4 NGRID (A13) where the sum is taken over every spatial point for which the profiles and trigonometric functions are known (polar grid) and NGRID is the total number of such grid points. J then depends solely on the unknown coefficients a0,a1,a2 and b0,b1,b2. These coefficients are chosen to minimize J and so furnish us with an approximate solution for u,(s,ϕ)) and σ(s,ϕ), from which we form the storm relative radial and tangential wind components ur and vt, namely: 28

31 ur(s,ϕ)= u(s,ϕ) and vt(s,ϕ)=vg(s,ϕ)+ σ(s,ϕ) (A14) By adding the translational velocity c (in polar coordinates) to ur and vt we obtain the earthrelative components of the windfield uer and ver : uer(s,ϕ)=ur(s,ϕ)+c sin ϕ (A15) ver(s,ϕ)=vt(s,ϕ)+c cos ϕ (A16) where c is the normalized translation speed c = ct/ vgmax. Finally, since A3, A4 and A5 refer to a cyclone moving along the y axis, the entire generated wind field grid must be rotated so that the y axis of the calculation coincides with the actual compass direction of motion of the translating cyclone. 29

32 Appendix C: Use Cases for the Hurricane Model C1. Annual Hurricane Occurrence (AHO) was developed in collaboration with the statistical expert (Professor Gulati) and is described in the statistical and computer science materials. C2. Hurricane Genesis was developed in collaboration with Dr. Cocke and the statistical expert (Professor Gulati) and is described in the statistical and computer science materials. Note: The wind speed probability use case was developed in collaboration with Drs. Gulati, Simiu, and Pinelli and is described in the statistical and computer science materials. 30

33 C3. Use Case for Zip Code Distance Criterion Mark Powell, HRD Summary: The wind field model captures the peak wind speed and associated direction for all zip codes within a criterion-specified distance from the storm circulation center. The distance is specified in a scaled R/Rmax radial coordinate. For example a criterion of R/Rmax of 10.0 is ten times the value of Rmax. If Rmax is 25 miles, then all zip codes within 250 miles radially outward from the storm center would be considered for damage calculations. 2. References: The HRD H*Wind Surface Wind Analysis database: Powell, M. D. et al., 3. Data: A sample of 12 Hurricanes analyzed by the HRD Real Time Hurricane Wind Analysis System were used to describe the normalized distance from the radius of maximum wind to the location of the outermost extent of marine exposure 50 kt (57.5 mph) winds. This analysis included Florida Hurricanes Andrew (1992) and 2004 Hurricanes Charley, Frances, Ivan, and Jeanne. 4. Documentation: A. Minimum wind speeds for modeled damage Since damage is modeled probabilistically, damage can be produced at wind speeds below hurricane force, although with a very small chance of occurrence. For example, a peak wind gust of 55 mph corresponds to a 0.6% probability of < 2.0% damage to a wood frame residential structure in Florida. For a 3 sec gust of 50 mph there is a 0.6 % probability of <1.0% damage and a peak gust of 60 mph has a 2% chance of < 1 % damage. For purposes of computational efficiency, it is practical to restrict wind and damage calculations to zip codes that have a reasonable chance of receiving damaging winds from a particular hurricane. B. Zip code distance Criterion Since hurricanes come in a variety of sizes and shapes, a fixed distance criterion is not practical, since it may be unnecessarily large for small storms and consume valuable computer resources. 31

34 As an alternative to a fixed distance criterion, we have chosen to describe the distance criterion as a function of the radius of maximum wind speed (Rmax). For small storms with small Rmax we will go outward from the storm center ~ 10 Rmax and for large storms we will go out a as far as 4 Rmax. Sensitivity testing with zip code distance criterion with values of 3 R/RMAX and 5 R/Rmax resulted in underestimates of the extent of damaging winds. This was noticed during evaluations of the model against observed wind fields in Hurricane Charley of Twelve hurricanes comprising a variety of storm structures and shapes were evaluated by comparing the outermost radius of marine exposure 50 kt (57.5 mph) sustained winds in terms of the normalized radial coordinate, R/Rmax. The 57.5 mph sustained marine wind speed contour represents a 10 min mean marine wind of mph. Converting the 50 mph, 10 min mean marine wind speed to open terrain results in a 40.6 mph 10 min wind speed and a sustained wind speed of mph (using a gust factor of for the peak 1 min wind over 10 min in open terrain), and a peak 3s gust of 57.2 mph. This gust wind speed is close to the speed at which damage begins (2% probability of 1 % damage to a wood residential structure in Florida). Since most of the residential structures in Florida comprise rougher terrain (e.g. ~99% of the zip code roughness values exceed open terrain), our R/ Rmax criterion is conservative. A residential exposure corresponding to the 10% quantile for Florida zip codes (Fig. 1) is a roughness of 0.15 m. Converting the marine 10 min mean wind speed of 50 mph to this exposure results in a residential 10 min wind speed of 32 mph and a peak 3 sec gust of 42 mph. This peak gust corresponds to a <1% probability of <1 % damage to a wood residential structure for more than 90% of the zip codes in Florida. 32

35 Fig. 1 Distribution of Florida zip code roughness lengths looking upwind over a 45 deg segment NE from the zip code centroid out to a distance of 20 km and weighting the values with a Gaussian filter with a 3 km half power point. For Rmax > or equal to 30 sm: A plot of R50/Rmax vs Rmax (sm) shows that a value of 4 R/ Rmax is adequate. For Rmax < 30 sm the Criterion distance (Z) in sm is described by the linear relationship: Z (R/Rmax) = Rmax (sm) 5. Implementation: The above relationship will be coded into the wind model. 33

36 C4. Rmax Use Case Mark Powell HRD Sneh Gulati FIU 1. Summary: The Rmax value is computed from information contained in the track for each storm of the simulation. Rmax is the radius of maximum surface winds and is a parameter required for input to the wind field model. Based on generalized linear model fit by S. Gulati References: Demaria, M., J. Pennington, and K. Williams, 2002: Description of the Extended Best track file (EBTRK1.4) version 1.4. Available from NESDIS/CIRA/Regional and Mesoscale Meteorology Team, Colorado State University, Fort Collins, CO. Ho, F. P., J. C. Su, K. L. Hanevich, R. J. Smith, and F. P. Richards, 1987: Hurricane climatology for the Atlantic and Gulf coasts of the United States. NOAA Tech Memo NWS 38, NWS Silver Spring, MD. Powell, M., G. Soukup, S. Cocke, S Gulati, N. Morisseau-Leroy, Shahid Hamid, N. Dorst and L. Axe, 2005: State of Florida Hurricane Loss Projection Model: Atmospheric Science Component, Submitted to Journal of Wind Engineering and Industrial Aerodynamics. 3. Documentation from JWEIA paper: Radius of maximum wind Rmax:! The radius of maximum wind is determined from a distribution of values as a function of po and latitude. As in Ref. [21], a log normal distribution is assumed for Rmax with a mean value determined as a function of Delta p (in mb) and Latitude (in decimal degrees). In developing the models for Rmax, we used the data from Ho et al., 1987 for storms from ; NOAA-HRD archives of realtime surface wind analyses from ; an archive of the National Hurricane Center that was maintained by Dr. Mark DeMaria (now with NOAA s NESDIS at Colorado State University) for the years ; and an HRD archive of aircraft observations for the years To create a model to describe Rmax we considered U. S. Atlantic and Gulf of Mexico basin hurricane landfalls with latitudes as high as 34 degrees north in order to help fill a dearth of information on storms affecting the Northeast Florida coastline. The relationship between Rmax and Delta p and latitude is shows much scatter but a generalized linear model for the natural log of Rmax (r2 = 0.212) provides a useful estimation: 34

37 Ln Rmax = Delta p Delta p^ Lat^2 +epsilon (1) where epsilon is a normal random variable with a mean of zero and a variance of This equation is for the natural log of Rmax in Nautical miles, it must be converted to statute miles for any output shown to the commission or Pro team. The max and min values of Rmax are censored: If Rmax > 55 nm (63.5 sm) set Rmax = 55 nm (63.5 sm) If Rmax < 4 nm (4.6 sm) set Rmax = 4 nm (4.6 sm) 4. Use Case Steps: A. Using landfall storm center latitude and Po from the track and intensity information in the database. B. Compute Delta P = Po C. User generates a random variable (epsilon) with a mean of zero and a variance of D. User evaluates equation (1) E. New Rmax in nautical miles is calculated as follows: Rmax = e ^(Ln Rmax) F. To simplify the model, this value of Rmax is assigned to all time entries of the track/intensity database, even though the storm may go to sea again and make a second landfall. G. For output display purposes, Rmax may be converted from kilometers to statute miles by multiplying by Note: The Best set track files contain Rmax values at landfall for Hurricanes that have made landfall in Florida. These data are to be used for Scenario runs of the model. 5. Sample Calculation: Input: Latitude= 25.5 degrees, Po = 963 mb Delta P = 50 mb 35

38 Random selection of epsilon = Ln Rmax = Ln Rmax = Rmax = nm 6. Potential enhancement areas: 1. Rmax accommodates seafall and subsequent landfalls such that a new value of Rmax would be computed after seafall and apply through the next landfall (until a subsequent seafall). Publication with table of all Rmax values. Continue to update landfall database with landfall Lat and Lon, Rmax, Po information using H*Wind analyses. Revision History: MP I noticed in an dated that the random term seemed to be too large. Sneh said she would check. I mentioned that trial and error suggested a value of epsilon as from a population with a variance of instead of The use case is revised to reflect a variance of in the random term MP Rmax is also censored to limit the maximum value of Ln (Rmax) to be less than or equal to ~2.2 standard deviations above the mean observed value in the Rmax database ( 55 nm or 63 sm). The largest value in the Rmax dataset for Florida and vicinity is 45 nm (52 sm). For the lower end limit of Rmax, we use ~3.5 standard deviations below the mean (4 nm or 4.6 sm). The lower limit is consistent with data from 2004 Hurricane Charley. The use case is revised accordingly Above paragraph edited by MP MP Fixed error in text noticed by Dr. Gulati to we considered U. S. Atlantic and Gulf of Mexico basin hurricane landfalls with latitudes as high as 34 degrees north in order to help fill a dearth of information on storms affecting the Northeast Florida coastline MP added units for input parameters, corrected error in peripheral pressure using in equation 4b. Changed Po in sample calculation from 960 to 963 mb. Added Rmax conversion from kilometers to statute miles. Moved notes to revision history section Corrected mistake in upper censor value (55 nm or 63.5 sm). 36

39 Mean Rmax is 22.2 nm, sigma is 8.89 nm. An upper censor value of 55 nm is ( ) = 32.8 / 8.89 = 3.68 sigma above the mean, so upper censor (55 nm) is a rare event. However, if you go by Ln Rmax, the standard deviation (from the output of the Ln (Rmax) model fit) is and the value of 55 nm is 2.25 standard deviations above the mean Ln Rmax of (e^3.011 = 20.3 nm). As a compromise, we reduce the value of the Ln Rmax standard deviation of the random error term to LN(55) = 4.007, which is 3 sigma above the mean of 3.01 if we use the smaller value of 0.3 for sigma. 37

40 C5. Use case for Holland Beta term in pressure profile equation Mark Powell HRD 1. Summary: The Holland B parameter is an exponent value that specifies the shape of the radial pressure gradient and is an important input parameter for the wind field model. 2. References: Vickery, P. J., P. F. Skerjl,, and L. A. Twisdale, 2000b: Simulation of hurricane risk in the United States using an empirical storm track modeling technique, Journal of Structural Engineering., 126, Powell, M., G. Soukup, S. Cocke, S Gulati, N. Morisseau-Leroy, Shahid Hamid, N. Dorst and L. Axe, 2005: State of Florida Hurricane Loss Projection Model: Atmospheric Science Component, Submitted to Journal of Wind Engineering and Industrial Aerodynamics. Willoughby, H. E. and M. E. Rahn, 2004: Parametric representation of the primary hurricane vortex. Part I: Observations and evaluation of the Holland (1980) model. Mon. Wea. Rev., 132, Documentation: Willoughby and Rahn (2004) used HRD Hurricane Field Program flight level measurements and Aif Force Reconnaissance data to assemble a database of 606 mean flight-level profiles, one profile per flight-level of a particular flight. From aircraft sorties were included (most in the 1990 s) with 113 failing detailed QC. Locations of Mean flight level profiles from Willoughby and Rahn (2004) The QC was associated with: 38

41 1. Rmax > half of radial extent of profile 2. Min wind at center not sampled well (> 1/2 Vmax) 3. A/c didnt pass within 1/2 Rmax of the wind center Most failed QC due to 1, therefore the sample does not reflect well storms with large Rmax relative to the size of the leg Limitations of Willoughby and Rahn (2004) data set relevant to Cat Modeling: 1) FL Rmax is biased outward from sfc Rmax (use SFMR to correct)\ 2) Ht gradient should be weaker at 700 mb than at 500 m level. Exceptions are Andrew, Gilbert, Inez 3) The fits do not accommodate storms with outer wind maxima By correcting the FL Rmax to Sfc, and using the Sfc Delp from Hurdat, we can better estimate B(FL) from sfc quantities Willoughby and Rahn (2004) state that due to the tendency for Holland to show a broader wind max in the eyewall, in extreme storms it will have winds> 50 m/s about 50% more often than really occurs. However, this is B at Flt level, which should be less than what it should be at the surface since the Rmax is closer to the center at the surface...except Vmax sfc is also smaller so not sure how this would play out. Key question is how much B would vary with height if a given storm were sampled at several levels. Sonde data from CBLAST could help with this. B model based on Willoughby and Rahn (2004): Based on 45 radial profiles of SFMR measured surface winds from altitudes between 2-4 km, the Rmax is at RmaxFL on the left side and RmaxFL on the right side of the storm. The mean Rmax surface adjustment is: Rmax = RmaxFL (1) Vickery and Twisdale 2000 only gets 2-7% explained variance with their old fit B = ( * DelP * Rmax) (2) For DelP > 25 and FL <= 850 rsq = 0.026** We used this one previously Their R square went up if they used higher level data: B = ( * DelP * Rmax) (3) 39

42 For DelP > 25 and FL <= 700 rsq = We filtered the Willoughby and Rahn (2004) data set as follows: 1) by Height of flight-level pressure surface <= 700, 2) Longitude >60 < 98 degrees west, 3) Storm relative flight level Vmax > 30 m/s, 4) Latitude > 15 < 35 We also added surface Pmin values according to HURDAT for each mean flight level profile in the database. Note: about 200/493 (41%) mean profiles had outer maxima present but these do not affect the fit which is constrained to pass through the center and the Vmax. For our filtered data set, secondary outer maxima were indicated in 47/231 profiles (~ 20%). Distribution of B values according to the above filtering for flight level data from Willoughby and Rahn (2004). B model: B = Lat DelP^ Rmax + Epsilon (4) Censor values: Upper: 2.2 Lower: 0.5 This fit describes ~ 15% of the variance in B. Here Rmax represents the open ocean surface Rmax in km and Epsilon is the random error term chosen from a normal distribution with a mean of zero and a standard deviation of Note that the samples used for Rmax above differs from those used to model our Rmax use case. The former is an open ocean flight-level measurement corrected to a surface value while the lat- 40

43 ter is modeled from a database of coastal landfall measurements. We assume that both samples of Rmax come from the same population. 4. Use Case Steps: Note: The code for this calculation is a part of the track and intensity model as coded by Dr. Steve Cocke of FSU. A. Input from track information output within the track and intensity model: Landfall Rmax (km), Pmin (mb), latitude (decimal degrees north) B. Compute Delta P = Po C. Generate a random variable (epsilon) with a mean of zero and a variance of (Standard deviation of 0.286). The random term is computed only once per simulated storm track. D. Evaluate equation (4) 41

44 5. Sample Calculation: Input: Rmax = 25 km, Po = 960 mb, Latitude = 26 degrees north Random selection of epsilon = 0.2 DelP = 53 mb B = Lat DelP^ Rmax + Epsilon B = (26) (53)^ (25) B = B = Potential enhancement areas: 1. Possibly replace Holland B parameter with new fitting methods being developed by Willoughby if some method to correct B for variation with height can be developed (perhaps based on GPS sonde data). 7. Problems with low B values On we notices that some tracks of relatively weak storms ( mb Pmin) caused the wind model to stall if B values were < 0.8. The lower bound censor value for the above model is 0.5. We are investigating this. A further filtering of the Willoughby and Rahn data set suggests that if we constrain ourselves to mean flight level winds of at least hurricane force (33 m/s), then the smallest B is 0.8. We may decide to fix our lower censor at 0.8 accordingly. We may also revise the model accordingly, reaching a new R squared of 20% with a new equation as follows: B = Lat Rmax + Epsilon Where epsilon is a random error term from a zero mean normal distribution with a standard deviation of Upper censor = 2.2 Lower censor =

45 43