The most common way for estimating the loss to single family homes caused by natural hazards is by using vulnerability functions. The vulnerability function takes wind speed as an input and produces a damage ratio, which is defined as a ratio of the repair cost of damage over the total replacement cost of the house. Following the state-of-the-art reviews (Walker, 2011; Pita et al., 2014), vulnerability estimation can be classified into two categories.

The first category is function approximation based on past insurance claims data (Leicester and Reardon, 1976; Friedman, 1984; Sparks et al., 1994). The biggest disadvantage of functions developed by this method is a generalization of all houses without considering structural differences. In addition, those functions may become obsolete as building code and materials change over time.

The second category is an engineering approach. Based on the characterization of the house structure such as house geometry, location of the house, and capacities of components (e.g., roof cover, roof-to-wall connection, roof sheathing, windows, doors, walls sheathing), a computational model is developed to simulate the loss. Many studies have been conducted to improve the engineering approach. Some of the key studies are briefly summarized. Vickery et al. (2006a) and Vickery et al. (2006b) described the overall algorithms based on different damage states, which are used in Hazus (HAZUS MHMR, 2003). Gurley et al. (2005) developed the Florida Public Hurricane Loss Projection Model, which answered many questions on the subject of engineering-based vulnerability estimation. Pinelli et al. (2011) specified the importance of different house components in loss estimation. Pita et al. (2013) underlined the importance of utilizing geographic information systems for loss predictions. Ellingwood et al. (2004) and Li and Ellingwood (2006) developed fragility functions of house components.

In this research, we followed the engineering approach for vulnerability estimation. A Matlab program was developed, pursuing the overall algorithm described in Figure 1. The input values of the program were the house characteristics (house dimensions, type of roof, site location, surrounding environment) and capacities of house components (windows, doors, garage door, roof cover, roof sheeting, roof-to-wall connections, walls). The capacities of all house components were adopted from the work of previous researchers or experimental studies. To describe the uncertainties in components, the capacities were randomly generated based on statistical distributions. A particular element was marked as damaged when the load exceeded the capacity of the element.

The Monte Carlo simulations were repeated m times, where value of m was determined as 10,000 by a convergence test. For every simulation, the wind speed was generated from 0 to 120 m/s with an increment of 1 m/s. In order to convert the wind speed into the loads that act on the house, formulas from ASCE 7 (2010) were utilized.

The wind pressure is characterized by Eq. 1: (1)p=qh(GCp−GCpi)[kPa] where q_h is velocity pressure evaluated at the mean roof height of the building, G is the gust-effect factor, C_p is the external pressure coefficient, and GC_pi is a product of the gust-effect factor and internal pressure coefficient. The velocity pressure is calculated by Eq. 2: (2)qh=0.613KhKztV2[Pa] where K_h is the velocity pressure exposure coefficient, K_zt is the topographic factor, and V[m/s] is the wind speed. For the particular wind speed, the eight directions of the wind were considered following the approach from Cope (2004) and Gurley et al. (2005). We assumed the same probability of the wind from all of the directions.

As the first of the vulnerability estimation, we checked the opening failure mode for all windows, garage doors, and doors. The windows have two failure modes: one is a failure by pressure, and the other is a failure by debris impact (Gurley et al., 2005) (Eq. 3): (3)pD(V)=1−exp(−A⋅NA⋅B⋅C⋅D) where p_D(V) is the probability of impact, A is a fraction of missile objects, N_A is a total number of missile objects, B is the number of the objects that are currently in the air, C is a fraction of the unprotected windows in the wall, and D is the probability if described object destroys the window.

The failure of the openings may allow the wind to penetrate the interior of the house and produce the additional internal pressure. Internal pressure was increased if Eq. 4 was satisfied, by changing the category of the building from enclosed to partially enclosed: (4)Ao>1.10Aoi where A_o is the total area of openings in the wall that receives positive external pressure, A_oi is the sum of the areas of openings in the building envelope excluding A_o.

The second house element checked was the roof structure, which is composed of the roof cover, roof sheathing, and roof-to-wall connection. The roof cover is the main barrier preventing water penetration. Water penetration is the main reason for the internal loss—a significant part of the total loss. For internal loss calculation, the approach from Holmes (1999) was utilized. The roof sheathing and the roof cover have one failure mode, which is a failure by pull out. If the roof sheathing panel is damaged, the corresponding roof cover was assumed as damaged. The last, but the most crucial element of the roof structure, is the roof-to-wall connection. The price of roof-to-wall connection is relatively low compared with other components, but it has a crucial role in the overall capacity of the roof structure. After the failure of a particular roof-to-wall connection, the force from this connection was redistributed to the remaining roof-to-wall connections of the same roof area. After ensuring the roof-to-wall connection, the program checks the roof uplift using Eq. 5. If the roof is uplifted, it is assumed that the whole roof structure (the roof cover, the roof sheathing, and the roof-to-wall connection) is destroyed: (5)R>(W−D) where R is the sum of the capacity of the roof-to-wall connections that have not failed, W is uplift wind load, and D is the dead load. Larger dead load values have beneficial effect on the overall roof capacity.

The last step of the loss analysis is the analysis of walls. Walls have three failure modes, namely, sheathing failure, lateral failure (Cope, 2004), and foundation failure due to the shear force. The first mode is progressive whereas the second and third modes are abrupt. The second and third modes destroy the whole house structure.

After all elements were checked, the damage was assigned into the damage matrices. Based on the damage matrices and replacement costs of failed components (Means, 2015), the vulnerability at a particular wind speed was calculated.

The loss analysis was performed in collaboration with CoreLogic company, which developed a fully validated software for loss estimation. The methodology used in the regional loss analysis follows CoreLogic EQECAT (2015). Figure 2 shows the overall algorithm that was applied to the regional loss estimation. The methodology can be categorized into three modules: the hazard module, the vulnerability module, and the financial module.

The first module named as the hazard module estimates the wind speed at all property locations, defined by their latitude and longitude coordinates or street addresses. The storm database used in this research includes a combination of the historical and stochastic storms combining overall 16,000 high wind speed events that affect Florida. Based on the data from National Hurricane Center HURDAT2, we were able to calculate the storm intensity approximated from the coastline dependent smoothed hurricane landfall maximum wind speed distributions (Ho et al., 1987).

The synthetic hurricanes were calculated following the algorithm from Batts et al. (1980) and Vickery and Twisdale (1995). The central pressure difference (Eq. 6) was the main factor in determining the radius of maximum wind speeds for the high wind speed event (Krayer and Marshall, 1992): (6)Δp(t)=Δp0exp(−a⋅t) where a is a filling constant, Δp₀ is a central pressure difference at the time of landfall.

The maximum gradient wind speed (Eq. 7) and the maximum 10 min wind speed at the height of 10 m above the ocean (Eq. 8) were calculated to estimate the 10 min wind speed over the ocean at any point in the hurricane (Eq. 9): (7)Vgx=KΔp−Rmax2f where K is a constant, Δp central pressure difference. R_max is the radial distance from the center of the storm to the location where the maximum wind speeds occur, and f is a Coriolis parameter: (8)V(10,Rmax)=0.856Vgx+0.5c where c is the translation speed of the hurricane: (9)V(10,r,θ)=V(10,Rmax)⋅Vr(r)−0.5c(1−cosθ) where V_r(r) is a ratio of maximum 10 min wind speed at the radius of maximum winds to the 10 min mean at a distance r from the center of circulation (Batts et al., 1980).

The model computes wind speeds using the storm’s maximum sustained wind speed, the filling rate, the radius of maximum winds, the storm track, the translation speed, the gust factor, the storm profile (attenuation of wind speed outward from the center), and the friction caused by local terrain and man-made structures. The model used in this research addresses the loss estimates for wind effects only. Storm surge losses are outside the scope of this research. The model used in this research addresses the loss estimates for wind effects only. Storm surge losses were not included but follow the similar methodology. The examination of climate change in future hurricane activities were outside scope of this research and were not included during simulating future hurricanes. Liu (2014) developed a database of synthetic hurricanes that carry the effect of the surface temperature and annual frequency change, which can be addressed in the future study.

The second module titled as the vulnerability module provides damage estimates for each of the property assets being analyzed during the hurricane event. The vulnerability functions used in this research were delivered based on the engineering approach and were explained more thoroughly in Section “Vulnerability Estimation.”

The last module, called the financial module, calculates the loss using vulnerability functions associated with the structures. In this research, we calculated three different loss metrics. Details of these metrics will be explained in the next section.

In this research, we compare three different loss metrics using the regional loss analysis of Florida counties. The first metric is the average annual loss (AAL) (%), described by Eq. 10. AAL represents the expected loss ratio per year. The dollar loss of the region is normalized by the total replacement value of all houses in the region (only the building value without the price of the lot): (10)AAL=1M∑i=1MLi¯ where M is the number of events in the time period Δt (in the current research we assumed Δt = 113 years from 1900 to 2013); and Li¯ is the percentage event loss caused by ith event (Eq. 11): (11)Li¯=1K∑j=1KLj(Vij) where K is the number of locations in the considered region (in this research region = a county); V_ij is the maximum wind speed observed at the jth location during the ith event; and L_j(V_ij) is the percentage loss obtained from the particular vulnerability function.

The second loss metric is the average annual dollar loss (AADL) ($) characterized by Eq. 12. AADL represents the expected dollar amount loss per year, i.e., the dollar loss of all structures in the investigated region: (12)AADL=1M∑i=1MLei¯ where Lei¯ is the dollar event loss in the considered region: (13)Lei¯=1K∑j=1KLj(Vij)⋅Hj where H_j is the price of the property at the jth location (without price of the lot).

The last loss metric is the average annual population-weighted loss AAPL (people) defined in Eq. 14, which is a new measure proposed in this research. The population-weight reflects the number of people influenced by the hurricane. As an example, consider two different counties with identical AAL. With AAL alone, the two counties appear to have the same level of risks against hurricanes. However, if one county has a lot more population than the other, a measure needs to reflect this difference in population: (14)AAPL=1M∑i=1MLpi¯ where Lpi¯ is the population event loss in the considered region: (15)Lpi¯=1K∑j=1KLj(Vij)⋅Pj where P_j is the exposed population at the jth location (number of people occupying the examined structure).

To illustrate the differences among loss metrics, an example loss analysis was conducted. The first step was to construct vulnerability functions. We chose a low-rise wood residential structure with gable roof built before hurricane Andrew (built before 1992). The geometry of the investigated house was chosen in a way to represent the highest percentage of existing structures in Florida. The following dimensions have been utilized: overhang = 0.61 m (2 ft), length = 18.29 m (60 ft), width = 11.58 m (38 ft), walls height = 3.05 m (10 ft), roof height = 4.14 m (13.6 ft), roof angle = 23°, and truss spacing = 0.61 m (2 ft). This geometry was chosen based on Pinelli et al. (2003), where a significant number of houses were investigated and this structure was identified as the most representative one.

Table 1 shows the capacities of the different house elements used for the vulnerability estimation (see Vulnerability Estimation). The capacities represent the structures built before hurricane Andrew. It should be noticed that those capacities are relatively lower compared with structures built more recently. After hurricane Andrew, the building code in Florida has been changed to decrease the loss from future hurricane events. Since many houses in Florida were built before 1992, this structure still represents the majority of the wood residential structures (Pita et al., 2008).

Figure 3 describes the vulnerability curve of the house built before hurricane Andrew along with the 0.05 and 0.95 quantiles. Furthermore, Figure 3 includes the vulnerability curve from Hazus (HAZUS MHMR, 2003) as a comparison that is fully validated based on historical loss data.

Different loss metrics reveal various aspects of expected loss of the region. The AAL reflects the percentage average loss of structures in the region. This metric depends mainly on two characteristics; the wind probability in the region and the vulnerability of the investigated house structure. The AAL is useful for insurers (or future homeowners) who analyze the expected loss of a certain type of house at a location. On the other hand, the AADL is the sum of the dollar loss of all properties in the region, and therefore will be useful for both insurers and counties or state officials. Comparing with the AAL, the AADL reflects the price of the house and the number of a house structures in the investigated location.

Those two measures are useful and address many relevant issues but in some cases, they may be misleading. The AAL does not reflect the actual price of the house and the number of structures in the region. For example, a county with the highest AAL can produce relatively low AADL: e.g., a county with a small number of structures and low value of properties but exposed to high wind probability. Similarly, the AADL may produce misleading information as well: e.g., a county with the high number of expensive structures that have low AAL. Furthermore, the value of AADL is hard to compare with different states or counties due to the difference in the price of the house and average dollar household income. Moreover, both the AAL and the AADL do not directly reflect the population. These two measures do not address the number of people that are going to be affected by the high wind event.

The AAPL will reveal population aspect of hazard that cannot be directly described by the AAL and the AADL. Unlike the AADL that depends on the price of the house, we can impartially compare AAPL among different states or even among different counties. To demonstrate the need of the new metric proposed in this research, we clustered the loss data into three groups. The groups were chosen based on design wind speed maps from ASCE 7 (2010) (occupancy category II). Groups A, B, and C represent the following wind speed ranges: 120–130, 140–150, and 160–180 mph, respectively.

According to Pita et al. (2009), we were able to calculate the percentage of the investigated structures (low-rise wood residential structure with gable roof built before hurricane Andrew) among all of the residential structures. Based on actual building stock data (U.S. Census Bureau, 2015b), the number of houses, the price of houses (without the price of the lot), and the number of occupants were calculated for each of the investigated regions.

Each mark of Figure 4 represents one county in Florida. It plots the AAL along with the population that occupy the investigated type of structure (wood-frame house built before 1992), for certain county. We were not able to specify the trend function for AAL that would satisfy the two initial constrains. Firstly, for a population size of 0, then AAL is 0. Secondly, the trend line should explain at least 75% of the data around the mean (coefficient of determination >0.75). It is obvious from Figure 4 that the AAL does not reflect the population aspect.

The motivation for the measure proposed in this research was quantification of the hurricane impact on the population. The AAPL not only quantifies the impact on the population but is independent of the price differences of houses between different regions. The AAPL is an attempt to include the best characteristics of the two previous measures, which is descriptive of the loss from particular structure and impact of the loss on the region (subjective in AADP). Figure 5 defines the relationship between the AAPL and the population. The trend among points is more visible compared with Figure 4 and drawn by black dashed line. The main criteria in choosing the trend functions was the smallest coefficient of determination of the functions that satisfy the initial constrains used earlier for Figure 4. We have chosen the linear functions for Figures 5A,C and power function for Figure 5B. The coefficients of determinations exceeded 0.91, 0.78, and 0.97.

Figure 6 characterizes the differences between the loss measures among the counties in Florida. The three measures show different distribution of losses among counties. Figure 6A (AAL) is useful for insurers (and homeowners) that analyze a certain type of house in a chosen area. The map quantifies possible risk of that type of structure. Figure 6B (AADL) is convenient for insurers or government officials to assess the regions with the highest dollar loss, which may need special attention. Figure 6C (AAPL) describes the measure proposed in this research that assesses the impact of the hurricane on the population. The advantage of the AAPL is a quantification of the hurricane loss on the population rather than on dollar value. Since the exact price approximation is difficult and often impractical, we had to approximate the dollar value for many regions, raising the uncertainties of the dollar loss estimation AADL.

Table 2 complements Figure 6 and shows the counties with the highest loss calculated by three different metrics (AAL, AADL, and AAPL), in descending order. All loss measures rank the counties differently. In particular, Collier and Indian River counties appeared in AAL but not included in other metrics. Monroe county ranked the highest in AAL, but ninth in AADL, and was not included in AAPL.