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Many empirical hurricane economic loss models consider only wind speed and neglect storm size. These models may be inadequate in accurately predicting the losses of super-sized storms, such as Hurricane Sandy in 2012. In this study, we examined the dependences of normalized US hurricane loss on both wind speed and storm size for 73 tropical cyclones that made landfall in the US from 1988 through 2012. A multi-variate least squares regression is used to construct a hurricane loss model using both wind speed and size as predictors. Using maximum wind speed and size together captures more variance of losses than using wind speed or size alone. It is found that normalized hurricane loss (L) approximately follows a power law relation with c a b maximum wind speed (V ) and size (R), L=10 V R , with c determining an overall scaling max max factor and the exponents a and b generally ranging between 4–12 and 2–4 respectively. Both a and b tend to increase with stronger wind speed. Hurricane Sandy’s size was about three times of the average size of all hurricanes analyzed. Based on the bi-variate regression model that explains the most variance for hurricanes, Hurricane Sandy’s loss would be approximately 20 times smaller if its size were of the average size with maximum wind speed unchanged. It is important to revise conventional empirical hurricane loss models that are only dependent on maximum wind speed to include both maximum wind speed and size as predictors. Keywords: hurricane, economic loss, storm size, wind speed 1. Introduction hurricane loss model should consider all these factors, although empirical models usually include a limited number Landfalling hurricanes cause large amounts of economic of predictors for macro-scale assessment of hurricane dama- damage, injury and loss of life. In the United States, hurricane ges resulted from climate and socio-economic changes (e.g., Nordhaus 2010, Emanuel 2011, Mendelsohn et al 2012). We losses account for the largest fraction of insured losses from focus on the dependence of hurricane loss on maximum wind all natural hazards (Bevere et al 2013). It is expected that speed and storm size on an aggregate scale in this study. hurricane loss (L) would depend on maximum wind speed The dependence of hurricane loss on maximum wind (V ), storm size (R), duration, wind direction, precipitation, max speed has been examined extensively. It has been shown that storm surge, and local exposure and vulnerability factors such hurricane economic loss follows an approximate power-law as the number of housing units, composition, value and age of relationship with maximum wind speed, i.e., L = CV , with max properties, and demographic information including popula- a ranging between 3 and 9 (Nordhaus 2010, Bouwer and tion density (Vickery et al 2006, Bouwer 2011, Murphy and Botzen 2011, Emanuel 2011) and C being a scaling factor. Strobl 2010, Nordhaus 2010, Schmidt et al 2010, Mendelsohn Murnane and Elsner (2012) analyzed normalized US hurri- et al 2012, Czajkowski and Done 2014). A comprehensive cane losses from 1900 to 2011 using a linear ﬁtting between log (L) and V for the top 10%, 25%, 50%, 75%, and 90% 10 max Content from this work may be used under the terms of the of hurricane losses. This method, called quantile regression Creative Commons Attribution 3.0 licence. Any further approach, suggested an exponential relationship between distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI. normalized loss and maximum wind speed. 1748-9326/14/064019+09$33.00 1 © 2014 IOP Publishing Ltd Environ. Res. Lett. 9 (2014) 064019 A R Zhai and J H Jiang Table 1. The list of landfalling Atlantic tropical cyclones used in the study. The normalized losses are in billions of 2013 US dollars. The maximum wind speeds (V ) are in miles per hour (mph). The radii of 34 knots wind speed (R34) are in nautical miles (nm). max Loss V R34 max Rank Name Date ($Billion) (mph) (nm) 1 Katrina 8/29/2005 96.56 125 175 2 Andrew 8/24/1992 53.77 170 97.5 3 Sandy 10/29/2012 51.21 75 385 4 Wilma 10/23/2001 24.56 120 181.25 5 Ike 9/13/2008 20.32 110 168.75 6 Ivan 9/16/2004 18.39 120 175 7 Charley 8/13/2004 17.5 150 78.75 8 Hugo 9/21/1989 17 140 137.5 9 Rita 9/24/2005 11.92 115 145 10 Frances 9/5/2004 11.65 105 155 11 Jeanne 9/26/2004 8.94 120 145 12 Allison 6/5/2001 7.68 50 95 13 Floyd 9/16/1999 7.47 105 162.5 14 Irene 8/27/2011 7.33 75 182.5 15 Fran 9/5/1996 6.09 115 181.25 16 Opal 10/3/1991 5.93 115 181.25 17 Isabel 9/18/2003 4.76 105 237.5 18 Gustav 9/1/2008 4.53 105 180 19 Bob 8/19/1991 3.46 105 112.5 20 Georges 9/28/1998 2.85 105 121.25 21 Dennis 7/10/2005 2.66 120 153.75 22 Isaac 8/28/2012 2.41 80 152.5 23 Andrew 8/25/1992 2.24 115 125 24 Gordon 11/15/1990 1.64 50 120 25 Irene 10/14/1995 1.33 80 90 26 Lili 10/2/1998 1.27 90 142.5 27 Bonnie 8/26/1998 1.26 110 156.25 28 Georges 9/25/1998 1.19 105 112.5 29 Dolly 7/23/2008 1.11 85 110 30 Alberto 7/3/1994 1.03 65 60 31 Erin 8/2/1995 0.69 85 87.5 32 Erin 8/3/1995 0.69 100 87.5 33 Fay 8/19/2008 0.59 65 65 34 Ernesto 8/29/2006 0.56 45 31.25 35 Bertha 7/12/1996 0.51 105 143.75 36 Josephine 10/6/1992 0.5 70 50 37 Isidore 9/26/2002 0.49 65 235 38 Cindy 7/5/2005 0.38 75 37.5 39 Gabrielle 9/14/2001 0.35 70 87.5 40 Marco 10/10/1986 0.27 35 87.5 41 Hermine 9/6/2010 0.26 70 37.5 42 Dennis 9/4/1999 0.26 70 97.5 43 Claudette 7/15/2003 0.25 90 96.25 44 Gilbert 9/16/1988 0.25 70 237.5 45 Chantal 8/1/1989 0.24 80 100 46 Danny 7/19/1997 0.18 80 45 47 Hanna 9/6/2008 0.17 70 106.25 48 Jerry 10/14/1985 0.17 85 87.5 49 Gaston 8/29/2004 0.17 75 47.5 50 Earl 9/2/1998 0.14 80 83.75 51 Mitch 11/4/1994 0.14 65 143.75 52 Charley 8/14/2004 0.12 80 63.75 53 Bret 8/22/1999 0.1 115 67.5 54 Charley 8/22/1998 0.087 45 106.25 55 Ophelia 9/15/2005 0.083 75 97.5 56 Bill 6/30/2003 0.071 60 91.25 57 Humberto 9/13/2007 0.054 90 37.5 58 Arlene 6/20/1993 0.047 40 31.25 2 Environ. Res. Lett. 9 (2014) 064019 A R Zhai and J H Jiang Table 1. (Continued. ) Loss V R34 max Rank Name Date ($Billion) (mph) (nm) 59 Barry 8/5/2001 0.046 70 60 60 Hanna 9/14/2002 0.03 60 62.5 61 Harvey 9/21/1999 0.025 60 43.75 62 Earl 9/2/2010 0.019 70 170 63 Gordon 9/17/2000 0.017 65 118.75 64 Keith 11/22/1984 0.015 65 162.5 65 Alex 6/30/2010 0.011 70 137.5 66 Beryl 8/9/1988 0.008 50 62.5 67 Florence 9/9/1988 0.007 80 75 68 Kyle 10/10/1998 0.007 40 10 69 Fay 9/7/2002 0.007 60 42.5 70 Alex 8/3/2004 0.005 80 63.75 71 Chris 8/28/1988 0.003 45 60 72 Allison 6/5/1995 0.003 70 87.5 73 Gustav 9/10/2002 0.0001 65 106.25 While V is found to explain the largest variance of (V /V ) , for quick estimates of hurricane damage. max max max0 TM hurricane damage (Nordhaus 2006, Schmidt et al 2010, Although both HHI and CHI take into account storm size, Jain 2010, Murphy and Strobl 2010), the fact that many low the exact dependence of loss on size varies. Thus, it is imperative to examine historical data to quantify the rela- intensity storms caused substantial losses suggests that it is tionship between hurricane loss and size on an aggregate necessary to consider factors beyond maximum wind speed in level. estimating hurricane losses. For example, Hurricane Sandy Emanuel (2005) provided a theoretical basis for a pos- (2012) was a category 1 hurricane with a V of 75 miles per max sible relation between hurricane loss and size. He expressed hour (mph) when it made landfall on 29 October 2012. It the total power dissipation (PD) of a tropical cyclone (TC) as caused widespread damage in New Jersey and New York, the integral of the cubic of maximum wind speed over the size with a total loss around 51.2 billion when normalized to 2013 () R of a storm through its lifetime (τ) US dollars (USD) according to the ICAT damage estimates (http://www.icatdamageestimator.com). On the other hand, τ R Hurricane Andrew (1992) had a V of 170 mph (category 5) PD =2d πρ C V rrtd, (1) max ∫∫ at landfall and its total normalized loss was similar to the Sandy’s, about 53.7 billion in 2013 USD (table 1). Obviously, where C is the surface drag coefﬁcient, ρ is the surface air factors other than maximum wind speed contribute to the density, |V| is the magnitude of surface wind speed, r is the similar losses between Hurricane Andrew (1992) and Sandy radius of the storm, and the integral is from storm center to (2012). In the top 30 most costly tropical cyclones (TCs) the outer storm limit and over the lifetime () τ of the storm. whose normalized losses are greater than one billion (table 1), Since the economic loss of a hurricane is driven by PD, 9 (30%) storms have V less than 95 mph, the upper limit of max equation (1) shows that loss would increase with the square of category 1 Hurricane on the Safﬁr–Simpson hurricane wind average storm size. Powell and Reinhold (2007) proposed to scale. use the integrated kinetic energy (IKE) over the volume of a Kantha (2006) pointed out that storm size is an important TC as an indicator of storm destructive potential. In their parameter that must be incorporated into the index for hur- paper, the IKE is a quadratic function of maximum wind ricane hazard in addition to maximum wind speed and storm speed and several size measures. Whether the theoretical translation speed (S). He proposed a Hurricane Hazard Index models are applicable to actual hurricane losses needs to be 2 3 (HHI) in the form of (R/R ) (V /V ) (S /S), in which the 0 max max0 0 tested with real data. subscript ‘0’ indicates corresponding reference values. Storm The availability of storm size information from the size not only determines the impacted area, but also affects National Hurricane Center (NHC) Extended Best Track wind duration and wind directional change in the census (EBT) database (Demuth et al 2006) provides us an oppor- tracts, both factors being important drivers of losses on local tunity to determine the dependence of hurricane loss on size scale (Czajkowski and Done 2014, Powell et al 1995, from observations. In this study, we aim to quantify the Jain 2010). Holland et al (2010) also showed that wind speed, relationship between hurricane loss and the hurricane max- size and translation speed all contribute to the offshore energy imum wind speed and size using historical data and create an industry losses. Based on the work of Kantha (2006), Dr empirical model for hurricane loss using both maximum wind Steve Smith of Willis Re, created the CME Hurricane speed and size as predictors. The estimated hurricane losses TM TM 3 Index (CHI ), i.e., CHI = (V /V ) + (3/2)(R/R ) by the bivariate regression model are compared with those max max0 0 3 Environ. Res. Lett. 9 (2014) 064019 A R Zhai and J H Jiang from the simple regression models using maximum wind For storm size, ﬁve metrics are available in the EBT speed or size alone. In particular, the relative roles of max- database, dating back to 1988 (Demuth et al 2006). R34, R50 imum wind speed and size in determining Hurricane Sandy’s and R64 represent the radii of a storm where wind speeds at loss are analyzed. 10 m height above the surface are 34, 50 and 64 knots, The structure of the paper is as follows. Section 2 respectively. R represents the radius of maximum wind max describes the data for hurricane loss, maximum wind speed speed. R is the radius of the outmost closed isobar, i.e., the out and size, as well as the analysis method. Sections 3 and 4 outer limit of a storm. The size data are given at four quad- show the relationship between loss and maximum wind rants for each storm at 6-hourly interval. Averages of radii at speed, and between loss and size, respectively. Section 5 the four quadrants are used in this study, although different presents the bi-variate regression results, the sensitivity, weights for each quadrant may be explored in the future. The uncertainty and statistical signiﬁcance of the regression data at the closest 6-hourly interval prior to landfall are used. coefﬁcients. Section 6 discusses the importance of the size for While R50, R64 and R are highly correlated with R34 with out predicting Hurricane Sandy’s loss, and conclusions are given correlation coefﬁcients close to 0.8, R and R34 are only max in section 7. weakly correlated with a correlation coefﬁcient of 0.13 for all available size data since 1988. The correlation between nor- malized hurricane loss and R is found to be less than 0.1. max 2. Data and approach Therefore, only R34 is used as a size metric for the regression models for loss. A total of 73 TCs that made landfall in the The US hurricane loss data are downloaded from the ICAT US between 1988 and 2012 form the basis of this analysis. Damage Estimator website (http://www.icatdamageestimator. Table 1 lists the 73 cases with storm name, date of landfall, com/viewdata). ICAT is an insurance company that provides normalized loss, maximum wind speed and R34 in descend- catastrophe insurance coverage to business and homeowners ing order of loss values. in the US. The losses are normalized to 2013 USD, taking To quantify the relationship of hurricane loss with into account of inﬂation, wealth and population differences maximum wind speed and size, we use the multi-variate least- between the years that landfalling hurricanes occurred (Pielke squares regression analysis tool. The best-ﬁt regression line is et al 2008) and the reference year (2013 in our study). The identiﬁed by minimizing the sum of the squares of the vertical loss data include only direct losses associated with a hurri- deviations from each data point to the line. The regression cane’s impact and do not consider indirect damage and tool yields R as explained variance and p-value for statistical longer-term macroeconomic effects (Pielke et al 2008). They signiﬁcance of each ﬁt. The explained variance indicates how are based on the historical economic damage compiled by much variance of the predictand (y) can be accounted for by Landsea (1991) from the monthly weather review annual the regression model using the predictor(s). The higher R summaries and more recently from the storm summary corresponds to a better ﬁtting in terms of capturing the var- archive at the NHC website (NHC 2006). They are total iations of a predictand. The p-value is the probability of the losses, including both wind and ﬂooding damages, roughly ﬁtting coefﬁcients for each predictor being zero. In other following a doubling of the insured losses, although adjust- words, it is the chance of the dependence of the predictand on ments are made on a storm-by-storm basis (Pielke et al 2008). a predictor being purely random. To reject the null hypothesis The uncertainties of the normalized loss data come from both that the dependence is random at a 95% statistical signiﬁcance the original hurricane damage estimate and the normalization level, the p-value should be less than 0.05. The smaller the procedure. Although Downton and Pielke (2005) found that p-values, the more statically signiﬁcant it is that the ﬁtting the individual damage estimates between states and the fed- coefﬁcients are nonzero. eral government over different time periods could vary by as We conduct regression analyses for losses expressed as a much as 40% for a large-impact storm with a loss of more function of maximum wind speed, a function of size, and a than $500 million, no systematic biases in damage estimate function of both wind speed and size. Sensitivity of the ﬁt- are found (Downton et al 2005). The assumptions in the tings to storm intensity (i.e., maximum wind speed) is normalization methodology, such as the growth rate of examined. The uncertainty and statistical signiﬁcance of the national wealth, could have a sizeable impact on the nor- ﬁtting results are quantiﬁed using the bootstrapping resam- malized losses (Pielke and Landsea 1998) especially for the pling method (Efron 1979). The details are described in years before 1940. We use the loss data starting from 1988, section 5. when storm size data became available. The exposure and vulnerability characteristics are more similar in the past 25 years than over a longer period and therefore the normal- 3. The relationship between loss and maximum wind ization is prone to less error. Details about the normalized loss speed data can be found in Pielke et al (2008). The maximum wind speeds at landfall for each storm are Figure 1 is a scatter plot between losses and maximum wind also provided by ICAT. We have veriﬁed that the maximum speeds for the 73 cases. Both quantities are expressed in wind speeds at ICAT are consistent with the NHC data at the logarithms of base 10. There is an approximate linear relation closest 6-hourly point prior to landfall, rounded up at a between loss and wind speed in logarithmic scale, suggesting 5 mph interval. a power-law relationship between L and V . A least-squares max 4 Environ. Res. Lett. 9 (2014) 064019 A R Zhai and J H Jiang Figure 1. The scatter plot of loss versus maximum wind speed for the Figure 2. The scatter plot of loss versus R34 for the 73 tropical 73 tropical cyclone cases. Both loss and wind speed are shown in cyclone cases. Both loss and R34 are shown in logarithm of base 10. logarithm of base 10. the Caribbean. On the basis of individual storms, the corre- −1.39 5.27 2 linear ﬁt yields L = 10 V , which gives an R of 0.39. max lation between V and R34 is much weaker. max The economic loss model thus explains 39% of the variance Following the approximate power-law relations shown in −9 of the loss with a p-value of 3.32 × 10 for statistical sig- the preceding sections, a general form of the loss model is niﬁcance level of 95%. This small p-value suggests that it is assumed to be statistically signiﬁcant at 95% level to reject a null hypothesis ca b LV = 10 (R34) , (2) that the coefﬁcient for V equals zero. The correlation max max between the logarithms of L and V is 0.63. The calculated max where a and b represent the power-law dependences of losses root mean square (rms) for the least-squares ﬁt residuals of on maximum wind speed and size, respectively. They are log (L) is 0.93. The rms accounts for how accurate the model termed ‘elasticity’ in Nordhaus (2010) and can be obtained as is when estimating the actual loss. A low rms means the the regression slopes from a double-logarithmic (for pre- model’s estimated values are close to the actual values while a dictand and predictors) least-squares regression. The constant high rms means the model’s estimated values are far off from c is a scaling factor that can be obtained using the regression the actual values. Therefore, a low rms is preferred. Here, a y-intercept. It approximately represents the impacts of factors rms of 0.93 for log (L) suggests that the ﬁtting errors for other than V and R34, including the underlying exposure max losses are on average within a factor of 10. and vulnerability characteristics. This multiplicative relation for maximum wind speed and size, equation (2), is partly based on the theoretical consideration that the destructiveness 4. The relationship between loss and storm size of a storm is proportional to its total energy such as the IKE (Powell and Reinhold 2007) or the PD in equation (1) Figure 2 shows the relationship between loss and storm size, (Emanuel 2005). It also partly resembles the HHI proposed by represented by R34. Their logarithms exhibit an approxi- Kantha (2006). On the other hand, an additive relation for mately linear relation, but with more scatter than the coun- maximum wind speed and size is also possible such as the terpart for loss and wind speed. The least-squares ﬁt yields CHI or the discrete equations for the IKE calculations (Powell 3.94 2.36 L = 10 (R34) . This linear ﬁt captures only 26% of the and Reinhold 2007). As the function forms do not need to be variance of the loss, with the corresponding correlation of unique, we focus on the multiplicative function form as in −6 0.51 and a p-value of 5.04 × 10 for statistical signiﬁcance equation (2) in this paper. The additive function for V and max level of 95%. The rms for the least-squares ﬁt residue of R34 is explored but not shown here because they generally log (L) is 1.03, somewhat larger than that for the regression 10 explain similar or even less variance than equation (2). using V . max As in Nordhaus (2010), we ﬁnd the ﬁtting coefﬁcients are somewhat different for subsets of the data grouped by max- imum wind speed, shown in table 2. For all 73 TC cases 5. Dependence of loss on maximum wind speed (V ⩾ 35 mph), a is 4.18, b is 1.25, and c is −1.83. For max and size category 1 or higher hurricanes (V ⩾ 75 mph), a is 4.98, b max is 2.66, and c is −6.22. For major hurricanes of category 3 or Using multi-variate linear regression, a loss model using both higher (V ⩾ 110 mph), a and b increase to 11.97 and 4.44, max maximum wind speed and size as predictors can be obtained. respectively, and c is −24.62. When V ⩾ 120 mph, a and b max The correlation between V and R34 is about 0.34 for the 73 slightly decrease to 9.97 and 3.52, respectively; however, the max TCs (0.16 for the 43 hurricanes with V ⩾ 75 mph), indi- sample size is very small (only 8) and the results are not max cating that they could serve as two nearly ‘independent’ statistically signiﬁcant at these extremely high wind speeds variables for the prediction of losses. Quiring et al (2011) (p-value > 0.05, thus not shown in table 2). Figure 3 shows the showed that annual averages of V and R34 over the general increasing trend of ‘elasticity’ for wind speed and size max Atlantic Basin from 1988 to 2008 are correlated at 0.55 with with storm intensity. The higher elasticity on wind speed for even higher correlations up to 0.81 in sub-basins such as over stronger storms is consistent with the previous studies 5 Environ. Res. Lett. 9 (2014) 064019 A R Zhai and J H Jiang probability of b being 2.66 or greater for the all storm cases is only 0.004, suggesting that the two means of the exponent b for all storms and hurricanes are statistically different at 99.6% level. The increased power-law dependence of loss on size for hurricanes suggests that it is particularly important to consider the impact of size on loss for high-intensity storms, which are generally associated with greater losses than weaker tropical storms. For lower intensity tropical storms, the underlying exposure and vulnerability characteristics may mask the dependence of loss on size. Figure 3. Variations of bi-variate regression coefﬁcients, a for Furthermore, the explained variances by the bi-variate maximum wind speed and b for size, with increasing threshold regressions are noticeably higher when maximum wind speed maximum wind speed, assuming that loss (L) follows the function c a b is higher. For example, the value of R increases from 45% form L=10 V (R34) . max for all storms with V ⩾ 35 mph to 69% for hurricanes with max V ⩾ 75 mph, as shown in table 2, column 3. The smaller max sample size for hurricanes (43) than for all storms (73) may (Nordhaus 2010, Murnane and Elsner 2012) despite that contribute somewhat to the larger R for hurricane cases as fewer but more recent samples are examined here. less variability is present in the fewer samples. However, our To test the statistical signiﬁcance of the larger exponents bootstrapping resamples suggest that the sample size is not a and b for hurricanes (V ⩾ 75 mph) than those for all max the dominant factor for the greater explained variance for tropical storms (V ⩾ 35 mph), we use the bootstrapping max higher intensity TCs. resampling (with replacement) method (Efron 1979)to Figure 5 shows the distributions of R for the 10 000 identify the distributions of the ﬁtting exponents and then times of bootstrapping drawn from the 73 all storm cases and compute the probability of the two means being from the the 43 hurricane cases. For all storms, only 43 samples are same sample population. Similar tests can be conducted for all drawn randomly each time. Thus, the sample size for bi- subsamples. However, we only focus on comparing the ﬁt- variate regressions is kept at 43 for both bootstrapping pro- tings for all tropical storms (V ⩾ 35 mph) and for all hur- max cedures. The blue (from all storms) and red (from hurricanes) ricanes (V ⩾ 75 mph) because of the physical importance max histograms of R yield sample means of 0.46 and 0.71, of these two distinct groups and for compactness. respectively. The probably of R being 0.71 and higher from First, 43 samples are drawn randomly from the 73 storms the regressions of all storms is only 0.002, suggesting the and a bi-variate regression to the 43 losses using V and max means for the two sample populations are statistically dif- R34 as predictors is conducted. This procedure is repeated ferent at 99.8% level. Therefore, the larger explained variance 10 000 times by bootstrapping with replacement, resulting in by V and R34 for hurricane losses than for the losses of all max a distribution of the ﬁtting exponents a and b, the blue his- storms indicates that wind speed and size play a greater role in tograms in ﬁgure 4. Similarly, 43 bootstrapping resamples determining the losses for higher intensity TCs. Other factors (with repetitions) of the hurricane cases (V ⩾ 75 mph) are max such as storm path, wind direction, duration, and local performed and the corresponding distributions for exponents exposure and vulnerability characteristics are less important a and b are shown in red histograms in ﬁgure 4. The mean for high intensity TCs, but they may have a comparable role values of a and b for all storms are 4.24 and 1.27, respec- as maximum wind speed and size in driving the losses when tively, very close to the results from direct bi-variate regres- storm intensity is relatively weak. sion of the cases (table 2, the second row, columns 4 and 5). Table 2 also lists the regression coefﬁcients and The standard deviations of a and b for all storms based on the explained variances if only wind speed or size is used for the bootstrapping resamples are 0.98 and 0.5, respectively. In the least-squares ﬁt for each subset of samples (columns 7–10 in cases of hurricanes only, the means of a and b become 5.04 table 2). Using two predictors consistently captures more and 2.66, respectively, nearly identical to the ﬁtting coefﬁ- variance of losses than using either wind speed or size alone cients in table 2 (the sixth row, column 4 and 5). The cor- in any subsets of samples. The statistical signiﬁcance of responding standard deviations for a and b are 1.2 and 0.58, higher explained variance by two predictors (V and R34) respectively. max than by one predictor (V ) is tested using the 10 000 rea- For the exponent a, the two sample populations for max lizations of bootstrapping drawn from the 43 hurricanes V ⩾ 35 mph and V ⩾ 75 mph have quite some over- max max (ﬁgure 6). The histogram of the explained variance by uni- lapping. The probability of a being 5.04 or greater for the all variate regression using V only (blue in ﬁgure 6) is com- storm cases is about 21% based on the blue histogram max pared with the explained variance by bi-variate regression (ﬁgure 4(a)). In other words, the mean values of the two sample populations for V ⩾ 35 mph and V ⩾ 75 mph are using V and R34 together (red in ﬁgure 6). The two sample max max max statistically different at about 80% level. On the other hand, populations of R are statistically different at 98.5% level, as the two sample populations of the exponent b for all storms the probability of R being 0.71 from the uni-variate regres- and hurricanes are well separated (ﬁgure 4(b)). The sions is merely 0.015. Thus, we are conﬁdent that using both 6 Environ. Res. Lett. 9 (2014) 064019 A R Zhai and J H Jiang c a b Table 2. Regression results using maximum wind speed and/or size as predictors for loss, following the function form L=10 V (R34) . See max text for details. R is the explained variance of loss by a regression model. 2 2 Threshold Sample R a (V R b (R34 max V size R ab c (V only) only, b=0) (R34 only) only, a=0) max max ⩾ 35 73 0.45 4.19 1.25 −1.83 0.39 5.27 0.26 2.36 ⩾ 60 64 0.58 6.78 1.43 −7.31 0.52 7.77 0.23 2.57 ⩾ 65 60 0.55 6.92 1.44 −7.62 0.48 7.69 0.18 2.32 ⩾ 70 53 0.62 6.29 1.82 −7.11 0.49 7.60 0.31 2.75 ⩾ 75 43 0.69 4.98 2.66 −6.22 0.40 7.11 0.51 3.36 ⩾ 80 38 0.75 6.53 2.61 −9.30 0.57 9.01 0.51 3.92 ⩾ 85 30 0.75 6.82 2.48 −9.64 0.50 8.07 0.41 3.10 ⩾ 90 27 0.74 7.80 2.59 −11.90 0.44 8.42 0.37 2.85 ⩾ 100 24 0.64 8.82 3.13 −15.17 0.30 6.73 0.16 2.09 ⩾ 110 15 0.75 11.97 4.44 −24.62 0.23 6.54 0.16 2.17 ⩾ 115 13 0.80 12.11 4.34 −24.72 0.25 6.92 0.20 2.31 V and R34 as predictors captures signiﬁcantly more vari- Clearly, the enormous size of Hurricane Sandy plays a pre- max abilities of the losses than using V alone. dominant role in its economic loss. max 6. The importance of the size on Hurricane 7. Conclusions Sandy’s loss The US normalized hurricane losses are found to have an Hurricane Sandy (2012) is the largest Atlantic hurricane on approximate power-law relation with maximum wind speed record in terms of size. At its peak (20 h before landfall), and size, indicated by the radius of tropical-storm force Sandy’s tropical storm-force winds (wind speed greater than winds. The power-law order for maximum wind speed ranges 34 knots) spanned 1100 miles, about 1/5 of the area of the from between 4 and 12, while the power-law order for size is entire United States. At landfall, it covered almost 900 miles approximately between 2 and 4. The high elasticity on wind across with R34 being 385 nautical miles (nm). However, speed is consistent with previous studies (Bouwer and Bot- Sandy’s V is only 75 mph at landfall. Out of the top ten zen 2011, Howard et al 1972, Nordhaus 2010). This study, max most expensive storms in table 1, Sandy is the only category 1 for the ﬁrst time, presents a quantitative relationship between hurricane at landfall; all other storms have loss and size using historical data. V ⩾ 105 mph (category 2 and higher). Out of the 43 hur- The dependence on the storm size is consistent with the max ricanes analyzed, Sandy’s V is 74% of the average hurri- expectation that the potential destructiveness of a storm is max cane intensity (101 mph) and its R34 is about three times of proportional to the area of the tropical-storm force winds the average hurricane size (127 nm). (Emanuel 2005, Kantha 2006). The exact elasticity (the To estimate the role of its enormous size in determining power-law order) is sensitive to the storm intensity—stronger Sandy’s loss, we use the bi-variate regression model for the storms have higher order power-law dependence on wind 43 hurricanes, speed and size than the weaker storms, suggesting that it is especially important to take into account storm size when −6.22 4.98 2.66 LV = 10 (R34) , (3) estimating losses for high-intensity hurricanes. max Storm size by itself does not account for a large fraction because this model explains the largest variance of losses of the variance of hurricane losses. However, using wind among all models applicable to Sandy (the second to sixth speed and size together explains much more variance of rows in table 2). Considering that the standard deviation of losses than using the wind speed alone. Based on this study, the ﬁtting coefﬁcient b is 0.58 based on the bootstrapping conventional empirical models based on only maximum wind sample population (ﬁgure 4(b)), b may vary between 2.08 and speed for hurricane loss should be revised to include both 3.24 considering 1σ errors. Therefore, given that Sandy’s size wind speed and size as predictors. (R34) is about three times of the average storm, its loss would For Hurricane Sandy, its enormous size contributes pre- 2.66 be approximately 20 (≈3.0 ) times the loss of a storm with dominantly to the economic loss. Out of the 43 hurricanes that the same maximum wind speed (75 mph) and the average were examined, Sandy’s size was ∼3 times of the average 0.58 hurricane size (127 mph), with about a factor of 2 (≈3 ) hurricane size, corresponding to about 20 times greater eco- uncertainty for the ratio, i.e., 10–40 times. On the other hand, nomic loss than that by an average sized hurricane at the same Sandy’s relatively weak maximum wind speed would make maximum wind speed. Note that the uncertainty of the ratio its loss ∼20% of a storm with the average intensity (101 mph) 20 times is approximately a factor of two (i.e., 10–40 times). and Sandy’s size (385 nm). The 1σ uncertainty of the expo- The huge loss by Hurricane Sandy is clearly a demonstration nent a causes the ratio to vary between 15% and 30%. of the impact of storm size on hurricane damage. 7 Environ. Res. Lett. 9 (2014) 064019 A R Zhai and J H Jiang Figure 5. The histograms of the explained variance (R ) by the bi- variate regressions from 10 000 bootstrapping sample sets. The blue ones are from the random resamples of 43 storms from the 73 cases with V ⩾ 35 mph, while the red ones are from the bootstrapping max (with replacement) 43 hurricanes with V ⩾ 75 mph. The mean and max standard deviation of R for each group are marked. Figure 4. The histograms of bi-variate regression exponents a (top) and b (bottom) from 10 000 bootstrapping sample sets. The blue ones are from the random resamples of 43 storms from the 73 cases with V ⩾ 35 mph, while the red ones are from the bootstrapping max (with replacement) 43 hurricanes with V ⩾ 75 mph. The mean and max standard deviation of a and b for each group are marked. Figure 6. The histograms of the explained variance (R ) from 10 000 As many other factors could contribute to hurricane bootstrapping sample sets from the 43 hurricanes with V ⩾ 75 mph. The blue ones are based on the uni-regression using losses, continued work is needed to incorporate the impacts of max only V as a predictor. The red ones are based on the bi-variate max storm translation speed, wind duration, wind direction, pre- regression using both V and R34 as predictors. The mean and max cipitation rate, and rainfall amount in the loss model. In standard deviation of R for each group are marked. addition, the relatively short duration of the data and uncer- tainties in the normalized hurricane damage, maximum wind speed and size at landfall could affect the accuracy of the provide useful guidance for developing more comprehensive regression results. The varying elasticities with maximum loss models for hurricane damage research, insurance needs, wind speed could indicate more complicated nonlinear rela- and hazard preparations. tionships than the simple linear regression models represent. It is also desirable to test the generality of our regression models by independent datasets. Nevertheless, the simple Acknowledgements regression model using maximum wind speed and size as predictors provides the ﬁrst-order estimate of hurricane eco- We thank Drs Lixin Zeng, Hui Su and Chengxing Zhai for nomic damages. The quantitative dependences reported here helpful discussions and detailed comments on the manuscript. 8 Environ. Res. Lett. 9 (2014) 064019 A R Zhai and J H Jiang We thank Dr Longtao Wu and Dr Lee Poulsen for help with Jain V 2010 The role of wind duration in damage estimation AIR Currents www.air-worldwide.com/Publications/AIR-Currents/ processing the Best Track data. We are grateful to two 2010/The-Role-of-Wind-Duration-in-Damage-Estimation anonymous reviewers for constructive comments. ARZ Kantha L 2006 Time to replace the Safﬁr-Simpson hurricane scale? thanks the support from La Cañada High School, especially EOS 87 3 Ms Patricia Compeau. JHJ performs the work at the Jet Landsea C W 1991 West African Monsoonal Rainfall and Intense Propulsion Laboratory, California Institute of Technology, Hurricane Associations Paper 484 (Fort Collins, CO: Colorado State University, Department of Atmospheric under contract with NASA. Science) Mendelsohn R, Emanuel K, Chonabayashi S and Bakkensen L 2012 The impact of climate change on global tropical storm damages Nat. Clim. 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Environmental Research Letters – IOP Publishing
Published: May 1, 2014
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