1. Introduction
There has been considerable research into quantifying hurricane risks by many different research groups over many decades. Published examples include Friedman (1972), (1975), Clark (1986), Vickery et al. (2000), Jagger and Elsner (2006), Emanuel et al. (2006), Hall and Jewson (2007), Lee et al. (2018), Bloemendaal et al. (2020), Arthur (2021), Meiler et al. (2022), and Tamaki et al. (2022). In addition, there has been considerable proprietary research and development undertaken by the insurance industry, some of which is described in submissions to the Florida Commission on Hurricane Loss Projection Methodology (FSBoA 2022). There has also been a large amount of research into understanding how hurricane characteristics might be changing because of climate change. Recent research includes Bhatia et al. (2019), Ting et al. (2019), Murakami et al. (2020), Lee et al. (2020), Emanuel (2020), and Garner et al. (2021), and recent reviews include Walsh et al. (2015) and Knutson et al. (2019, 2020). Combining these two lines of research has led to investigations into how hurricane risks might be changing because of climate change such as Gettelman et al. (2017), Levin and Murakami (2019), Trepanier (2020), and Carney et al. (2022).
Various approaches have been used to study the impact of climate change on hurricane risk. For instance, some studies have used statistical downscaling of climate models using hurricane risk models, to overcome the limitations of climate models with respect to capturing the details of hurricane behavior (Emanuel et al. 2006; Lee et al. 2018; Bloemendaal et al. 2020). The risk models are embedded in the climate model output. Different studies of this type have used different statistical downscaling approaches and may give different results. These methodologies are potentially able to capture many of the ways hurricanes may be changing due to climate change, including changes in the frequencies of storms of different intensities, genesis locations, storm tracks, storm size, forward speed, transitioning behavior, and rainfall. However, there have as yet been too few of these studies to lead to a full understanding of the robustness of the results to different model formulations.
Other studies, including much of the proprietary research into hurricane risk in the insurance industry, have used the approach of adjusting the output from existing risk models using factors derived from climate model results. Depending on the details of the risk model, this approach may be more limited, as it may not be possible to adjust all aspects of hurricane behavior. However, it is a simpler approach, and hence allows for the rapid generation of alternative views of risk and investigation of sensitivities and uncertainties. It is also easier to embed this approach into the workflows of many insurance companies. Overall, understanding hurricane risks, and how they may be changing, is challenging, and our understanding benefits from the use of multiple different approaches and perspectives.
In this study we use the approach of adjusting an existing risk model. The risk model we use is simple and is not intended to be used for detailed calculations of risk. The motivation for using a simple model is to enable us to investigate in detail the mechanisms by which hurricane frequency changes may lead to changes in loss, including understanding sensitivities to methodological assumptions, the role of uncertainty and the impacts on different loss metrics. This will hopefully lead to useful insights into how changes in losses occur, which can then help prioritize decisions being made when building more complex models that are intended to be used for more detailed calculation of risks.
The simplifications we use in our study are as follows:
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We only consider the impact of changes in the frequency of hurricanes of different intensities since frequency changes (and mean intensity changes, which can be derived from frequency changes) are the most widely studied aspect of the possible impact of climate change on hurricanes. There have been sufficiently many studies of possible changes in hurricane frequency and intensity that Knutson et al. (2020) were able to produce distributions of possible changes, where the spread in the distribution represents the different results from the different studies. We neglect the impact of all other possible changes in hurricane behavior including changes in genesis regions, tracks, forward speed, size, and transitioning, even though these changes may be important.
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We only consider the impact of wind and storm-surge hazard. In the insurance industry, hurricane wind hazard is the most important risk and is the focus of much of the current activity in that industry with respect to understanding the possible effects of climate change. The inclusion of storm-surge losses in our calculations arises from its inclusion in the historical losses that we use. The changes in loss we calculate only reflect the impact of frequency change on storm-surge damage and neglect the effect of sea level rise. We do not consider changes in rainfall and flood risk. Changes in risk due to sea level rise and changes in rainfall are likely large, and merit detailed investigation, but are out of scope of this study.
Our methodology works as follows. We use the simple hurricane risk model described in Jewson (2023a), which is constructed using publicly available data for normalized hurricane damages (Weinkle et al. 2018; Martinez 2020). This model can simulate an arbitrary number of synthetic hurricane events with relatively realistic values of landfall frequency, landfall intensity and U.S. loss. We then derive projections of changes in hurricane landfall frequencies from the results in Knutson et al. (2020) using the postprocessing methods described in Jewson (2023b). We use these projections to adjust the risk model by adjusting the landfall frequencies of hurricanes in different intensity categories. This creates a new version of the risk model that gives future projected losses, which we analyze. We perform a number of sensitivity tests to study how the changes in losses derived from the risk model change when we vary how the frequency projections and the risk model are constructed. The insights this gives into the relative importance of different factors can help inform the design of future hurricane projections and future risk models. We also study the sensitivity of the losses in the model to changes in the frequency of hurricanes of different intensities. This helps us to understand which categories of hurricanes are driving the changes in losses that we see. Finally, we consider individual scenarios from within the distribution of projection uncertainty to give an indication of the extent to which our current best estimates of the losses are uncertain and may change in the future as the science progresses.
In section 2 we describe the normalized hurricane loss data and risk model that we use. In section 3 we discuss the hurricane projections. In section 4 we discuss projected changes in the average annual loss. In section 5 we discuss projected changes in the distribution of annual losses. In section 6 we consider various sensitivity tests, and in section 7 we summarize our results and conclude.
2. Normalized hurricane losses and a simple hurricane risk model
a. Normalized hurricane losses
The risk model for U.S. hurricane losses we use is derived from the published datasets for normalized hurricane losses from Weinkle et al. (2018, hereinafter W2018) and Martinez (Martinez 2020, hereinafter M2020). The W2018 analysis is a continuation of earlier analyses by Pielke and Landsea (1998) and Pielke et al. (2008). The M2020 analysis is based on the W2018 dataset but includes additional adjustments in an attempt to better account for building cost inflation. We do not evaluate the relative merits of the W2018 and M2020 loss estimates and include results for both. Given the many challenges involved in estimating normalized historical damages, both datasets should be considered highly uncertain. For instance, the base losses used in W2018 may disagree with base losses from other sources. Also, the normalization methodologies used are only a simplification of the many processes that lead to changes in losses. For instance, they do not account for possible reductions in vulnerability, such as those that would be expected in regions where stricter building codes have come into force in recent decades. We do not attempt to quantify these uncertainties but simply note them as one of the “external” sources of uncertainty in our study. The goal of our analysis is not to produce detailed loss estimates, but rather to build a model that can be interrogated in order to deepen our understanding of what influences and determines losses and changes in losses, with a focus on frequency and frequency changes. Using both datasets allows us to evaluate whether the conclusions we draw are robust to the differences between the datasets.
The W2018 analysis includes two versions of normalized hurricane losses, of which we use “CL18,” whereas the M2020 analysis includes four versions, of which we use “CLa18.” We only use one dataset from each source as the differences between the two versions given in W2018, and the differences between the four versions given in M2020, are smaller than the differences between W2018 and M2020. Trends and variability in normalized damage estimates are discussed in Klotzbach et al. (2018).
These datasets give estimates of the loss that 197 U.S. landfalling historical hurricanes from 1900 to 2017 would have caused if they had occurred in 2018. The losses generally cover wind and storm-surge damage but not inland flood damage, with some exceptions (see the discussion in W2018). They take into account the 2018 distribution of economic exposure, and the losses are given in 2018 U.S. dollars. We consider 2018 values, rather than more recent values, since neither dataset has been updated beyond 2018 at this point.
Figure 1 shows the estimated losses for the 197 historical hurricanes, for the two datasets. We see that the M2020 dataset gives higher loss values for most storms. A summary of key statistics from the two datasets is given in Table 1. The average annual loss (AAL) column in this table shows how the AAL breaks down by hurricane category. Definitions of the hurricane intensity categories are given in appendix A. We see that the largest contribution to AAL is from category-4 (cat4) storms and the second largest is from category-3 (cat3) storms. Category-2 (cat2), category-1 (cat1), and category-5 (cat5) storms contribute significantly less to AAL. The dominance of cat4 storms in this respect is because cat4 storms cause larger losses individually than cat1–cat3 storms, on the one hand, and are more frequent than cat5 storms, on the other.
Normalized hurricane losses from the W2018 and M2020 datasets, on logarithmic scales, for 197 U.S. landfalling hurricanes in the period 1900–2017. Monetary values are in billions of 2018 U.S. dollars, and intensities refer to landfall.
Citation: Journal of Applied Meteorology and Climatology 62, 12; 10.1175/JAMC-D-23-0087.1
Historical hurricane loss statistics derived from the normalized hurricane losses created by W2018 and M2020 (the CL18 and CL18a versions, respectively). Monetary values are in billions of 2018 U.S. dollars ($B). Column 1 gives the hurricane intensity category at landfall. Column 2 gives the number of U.S. landfalling hurricanes during the period 1900–2017. Column 3 converts column 2 into an average number of storms per year. Columns 4, 5, and 6 are each split into two subcolumns for results derived from the W2018 and the M2020 datasets. They give the mean loss per storm, standard deviation of loss per storm, and average annual loss.
b. A hurricane risk model
The hurricane risk model we use is from Jewson (2023a). It is a model for the total U.S. loss due to hurricanes. Modeling total U.S. loss gives us good sample sizes for estimating the parameters we wish to estimate from the historical loss data for all except cat5 storms. Modeling total U.S. loss is also sufficient for us to investigate the questions we want to investigate in this study. We have not extended the risk model to represent U.S. regional losses, even though that would potentially allow more in-depth analysis, for two reasons. First, it is unclear whether the loss data would support building such a model. Dividing the loss data regionally may lead to sample sizes that would be so small that randomness in the historical losses would dominate the attempt to resolve regional signals. Second, regional variations within the United States in the changes in hurricane landfall frequencies due to climate change, perhaps due to changes in regions of hurricane genesis, the shapes of hurricane tracks, or changes in the way that hurricanes intensify during their lifetime, are currently poorly understood. They are not quantified in the Knutson et al. (2020) hurricane projections we use.
The risk model represents the frequency of hurricanes in each landfall intensity category as being Poisson distributed, with mean given by the annual number of storms per year from Table 1, column 3. The loss for events in each intensity category is modeled as being lognormally distributed, with mean and standard deviation also as given in Table 1, columns 4 and 5. Extensive validation of the fit of the distributions is given in Jewson (2023a). We use these fitted distributions to simulate 1 million years of synthetic hurricane losses, from which we derive various metrics for the baseline loss climate. We then adjust the frequencies in the risk model in various ways, as described below, and resimulate and recalculate the metrics.
Our model is based on storm intensity defined using maximum wind speed, as opposed to minimum mean sea level pressure (MMSLP). This is necessary for compatibility with the results from Knutson et al. (2020). However, there is a good argument for using MMSLP instead, as MMSLP correlates better with damage (Klotzbach et al. 2020). This suggests that in future it may be appropriate to rederive the impacts of climate change using storm intensity categories defined from MMSLP.
c. Simulation methodology
The simulations for the baseline climate use standard fixed-parameter Monte Carlo simulations for the Poisson and lognormal distributions. Fixed-parameter simulations use the same parameters to generate each year of simulation. For instance, the Poisson simulations use the same frequency in each year of the simulation. The number of hurricanes then varies from year to year as samples from the Poisson distribution. The simulations for the adjusted models make use of two techniques that extend standard Monte Carlo simulation. The first technique, from Jewson (2022a), extends fixed-parameter Monte Carlo to allow for uncertainty in the frequency parameters. This extension is required in our study because the climate projections we use include uncertainty estimates that we wish to propagate into the results. Jewson (2022a) refers to this method as stochastic-parameter simulation. It involves using different values for the frequency parameters for each year of simulation, where the different values are sampled from the frequency uncertainty distribution. This method is analogous to sampling from the parameter posterior in Bayesian statistics in order to generate a predictive distribution. Stochastic-parameter simulation uses a single simulation to achieve the same results as generating a large number of separate fixed-parameter simulations, one for each parameter setting, and then combining the results from those different simulations together.
The second simulation technique is the incremental simulation method from Jewson (2023a), which improves convergence of the estimates of changes between the original and adjusted simulations. Rather than generating the adjusted simulations from an independent Monte Carlo simulation, the incremental simulation method generates them by adding or removing events from the original simulation. This minimizes the differences between the two simulations and hence reduces the noise around estimates of change.
As further validation, and to illustrate the output from the risk model, Fig. 2 compares the frequency of events exceeding different levels of loss from the risk model with the same from the historical data, for models fitted separately to each of the two normalized loss datasets. To compare the modeled and historical losses, we use event exceedance frequency (EEF) graphs, also known as frequency–severity graphs. Assigning the position on the vertical axis of an EEF requires the use of a plotting-position rule. To plot the ith storm, we use the Nyears/(i − 0.5) rule. This rule assigns an equal range in frequency to each storm and plots at the middle of the frequency range. Many other plotting-position rules are available: for instance, the R statistical software package lmomco contains seven different rules as options. Figure 2 also gives 5%–95% bootstrap uncertainty ranges around the historical losses, and we see that the risk models lie well within the range. We conclude that they give a reasonably good fit to the historical data in both cases. We also see that the distribution of modeled losses based on the M2020 data gives much larger values than that based on the W2018 data.
Estimated return periods of storms exceeding each level of loss, using EEF curves for historical events (jagged red curves) and 10 000 yr of simulated events (smooth black curves). Return period is calculated as 1 divided by the frequency. Horizontal gray lines show the 5%–95% bootstrapped ranges for the historical losses. (a) Results derived from W2018 historical losses; (b) results derived from M2020 historical losses.
Citation: Journal of Applied Meteorology and Climatology 62, 12; 10.1175/JAMC-D-23-0087.1
3. Hurricanes and climate change
Hurricanes cause damage in many ways: extreme winds damage structures, extreme rainfall causes damage by water ingress into damaged structures and by freshwater flooding, and extreme sea level causes damage by saltwater flooding and wave impact. Climate change is likely influencing all these aspects of hurricane damage in some way. In this article, we will only consider one aspect of the effect of climate change on hurricanes and hence damage: the effects of changing frequency of storms of different intensities. The other aspects of the effect of climate change on hurricanes and damage are at least as important but are not within the scope of this study. By considering different changes in the frequencies of different intensity storms we automatically account for changes in the mean intensity of storms, as shown by Jewson (2022b).
Many studies have considered the effects of climate change on storm frequencies and mean intensities. Recent reviews include Knutson et al. (2020, hereinafter K2020) and Sobel et al. (2021). The results in K2020 suggest that, on average over many models, weaker storms may become less frequent and stronger storms may become more frequent (where storm strength is measured using lifetime maximum intensity). However, there are large differences between the different models, and these average results are an average over a wide range. These average results should therefore be seen as being of low confidence, and both the magnitude and the direction of changes may be incorrect. This large uncertainty arises from the many limitations that climate models show with respect to simulating realistic tropical cyclone behavior. For instance, it has recently been highlighted (Sobel et al. 2023) that many climate models struggle to produce realistic simulations of the variability of east Pacific equatorial sea surface temperatures over recent decades. Many climate models show a bias toward positive sea surface temperatures, that is, positive ENSO conditions, relative to the observations. On a year-to-year basis, the ENSO phase has a larger impact on Atlantic hurricane activity than projected levels of climate change, with positive ENSO phase typically leading to fewer hurricanes. As a result, this bias in sea surface temperatures could be leading to negative biases in projections of future hurricane activity.
Changes in frequencies of hurricanes of different intensities are related to changes in mean intensity and the proportion of intense storms. The on-average projected reduction in the frequency of weak storms and the on-average projected increase in the frequency of stronger storms contribute in roughly equal measures to a projected increase in the average intensity of storms (Jewson 2022b) and to a projected increase in the proportion of storms that are more intense.
a. Postprocessing the K2020 results
Different studies have given different estimates for the impact of climate change on hurricane frequencies. As a result, using just a single study to model impacts would introduce considerable arbitrariness into the results. One way to avoid this arbitrariness is to consider distributions of results from many studies, and K2020 have compiled distributions for the frequency changes for cat0–cat5 and cat4–cat5 tropical cyclones from many different studies in this way [according to K2020, category 0 (cat0) refers to tropical cyclones of tropical storm intensity but not hurricane intensity (18–32 m s−1, or 34–63 kt)]. For a number of technical reasons, the K2020 results cannot, however, be applied directly to our risk model, but first need postprocessing, as described in Jewson (2023b). The starting point for the postprocessing is the data used by K2020 to produce Figs. 1b and 2b in that paper. The postprocessing steps applied to these data in Jewson (2023b) are as follows:
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Distributions are fit to the K2020 quantiles.
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The correlations between the distributions of changes in cat0–cat5 and cat4–cat5 storm frequencies are modeled, and the correlations are used to derive the implied distribution of changes in cat03 frequencies.
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Information about changes in the frequencies of cat0–cat3 and cat4–cat5 storms is converted into information about changes in the frequencies of storms in each category from cat0 to cat5. Part of this conversion uses linear interpolation of the frequency changes as a function of intensity. This interpolation is performed in such a way that it conserves the frequency changes for the cat0–cat3 and cat4–cat5 bins but assigns different frequency changes within those bins.
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Information about changes in storm frequencies as a function of lifetime maximum intensity is converted into information about changes in storm frequencies as a function of landfall intensity. This is required because the normalized hurricane losses, and hence our risk model, use landfall intensity.
Jewson (2023b) applied these various steps to produce distributions of landfall frequency changes for tropical cyclones in 18 global regions. The continental United States is covered by two of these regions, which correspond to the Gulf of Mexico and the U.S. East Coast. Since our risk model only considers damage for the whole of the United States, we have repeated the analysis from Jewson (2023b) but for a single region that makes up the whole of the continental United States. The size of the changes for the whole of the United States lie in between the changes given by Jewson (2023b) for the Gulf of Mexico and U.S. East Coast.
b. Frequency change versus GMST
K2020 considered climate change consistent with a 2°C increase in global mean surface temperature (GMST), and we will do the same. K2020 also assumed that the percentage change in frequency is the same whatever the start and end date of the 2°C change (T. Knutson 2020, personal communication). For example, the 2°C change could be considered to start in 1880, or it could be considered to start in 2000. This assumption, that the rate of change versus GMST is constant in time, is unlikely to be completely true in detail, but is a reasonable first approximation given the current state of understanding of the impact of climate change on hurricane frequencies. The presentation of changes in hurricane characteristics as a function of GMST changes is very convenient, as it allows different climate model results to be combined and allows users of the projections to apply the changes to any time period. We can report, anecdotally, that this assumption has greatly facilitated the application of hurricane projections to risk analyses in the insurance industry, since such analyses may vary in terms of start and end date. The mathematics and implications of this assumption, and how it allows risk models based on different historical periods to predict different future periods, are investigated in detail in Jewson (2021).
c. Step, linear, and landfall models
To help to understand which of the sequence of postprocessing steps described above has the biggest impact on calculated loss changes, we create three sets of hurricane frequency changes, two of which do not include all the steps. Understanding which parts of the postprocessing are important can help risk modelers decide which to include in their analyses.
The first set of frequency changes comes from the “step” model from Jewson (2023b), which involves fitting distributions to the K2020 results, modeling correlations between changes in cat0–cat5 and cat4–cat5 and converting changes in cat0–cat5 and cat4–cat5 into changes in cat0–cat3 and cat4–cat5 (i.e., steps 1 and 2 from section 3a). It does not use interpolation of frequency changes as a function of intensity, or conversion to landfall. It is called the “step” model since the lack of interpolation means that there is a large step between the frequency changes for cat3 and cat4 storms, which might be considered implausible. The second set of frequency changes comes from the “linear” model, which, relative to the step model, additionally includes linear interpolation of the frequency changes versus intensity, but also does not include conversion to landfall (i.e., uses steps 1–3 from section 3a). The frequency interpolation removes the large step between the frequency changes for cat3 and cat4 storms. The landfall model is derived from the linear model by applying landfall adjustments (i.e., uses steps 1–4 from section 3a). This is the complete version of the model.
d. Frequency changes
The means and standard deviations of hurricane frequency changes by category from the step, linear, and landfall models, corresponding to the 2°C scenario, are given in Table 2 and illustrated in Fig. 3d. The frequency changes are presented as multiplicative changes and are modeled by lognormal distributions to capture uncertainty. The lognormal shape captures the possible range of multiplicative changes from zero to infinity, avoids the possibility of negative frequencies, and also captures the positive skew of the distributions of change given in K2020. We give values for the means of the frequency change distributions, as a measure of central tendency, while K2020 gave values for the medians of the distributions. Since the distributions from K2020 are positively skewed, especially for cat45, the means of the distributions are higher than the medians.
Means and standard deviations of the distributions of hurricane frequency adjustments we apply for a 2°C climate change scenario for the three models described in the text: step, linear, and landfall. A value of 1 is no change, and 1.1 is a 10% increase in frequency. The distributions of frequency change are lognormal, with the given mean and standard deviations. The distributions for different storm categories are perfectly correlated.
U.S. landfalling hurricane frequencies, changes in frequencies, losses, changes in losses, and category-by-category contribution to average annual loss: (a) frequencies by storm landfall category derived from W2018, (b) the mean loss per event, by category, from the W2018 and M2020 datasets, (c) the AAL by category from the same datasets, (d) distributions of frequency changes by category from Table 2, for the step, linear, and landfall models described in the text (the bar heights show the mean changes, and the three horizontal lines show the 25%, 50% and 75% quantiles), (e) the changes in AAL derived from combining the baseline AAL in (c) with the frequency changes in (d), (f) the changes in AAL but now as a percentage of the total baseline AAL (total across all categories, separately for the two datasets), and the contribution of each category of event to AAL, for the baseline and climate change scenarios, showing the same data but on (g) linear and (h) log scales (in both cases, the gray lines give constant contribution to AAL, and the highest contribution to AAL is for points in the top-right-hand corner).
Citation: Journal of Applied Meteorology and Climatology 62, 12; 10.1175/JAMC-D-23-0087.1
The step model shows constant changes within the cat0–cat3 and cat4–cat5 bins. On average, it suggests that the frequency of cat0–cat3 storms decreases versus GMST and the frequency of cat4–cat5 storms increases versus GMST. The linear model interpolates the step model frequency changes, leading to slightly lower mean frequency changes for cat0 and slightly higher mean frequency changes for cat5. Cat3 storms switch from decreasing mean frequency in the step model to increasing mean frequency in the linear model. The landfall model shows larger frequency increases for all categories of storms than the linear model. This is a straightforward consequence of the fact that most storms make landfall at a lower intensity than the lifetime maximum intensity (and the K2020 results are in terms of lifetime maximum intensity). This effect is discussed in detail in Jewson (2022c, 2023b).
Table 2 also gives changes in cat0–cat5 and cat1–cat5 frequencies from the step, linear, and landfall models, although we do not use these in our analysis. Notably, the mean frequency change of cat1–cat5 storms switches from negative in the K2020 results, the step model and the linear model, to positive in the landfall model, because of the landfall adjustment postprocessing step. In all three models, the uncertainty of the frequency changes (as given by the standard deviations in Table 2) is large relative to the size of the mean frequency change. This uncertainty is mostly due to the range of results from the different studies collated by K2020, and in the landfall model is also partly due to additional uncertainty introduced in the conversion to landfall.
To completely specify our statistical model, we must also specify the correlations between the distributions of change for storms rated at different categories. The analysis in Jewson (2022b) showed that only rank correlations of 1 are consistent with the K2020 results, and so that is what we use. In section 7, we will test the sensitivity to this correlation to help risk modelers to understand whether this correlation is important to include in their modeling.
e. Comparison with the historical landfall record
Statistically significant increases in landfalling hurricane frequencies of sizes similar to those shown in Table 2 have not been detected in the historical U.S. landfalling hurricane record. However, this should not be taken to imply that the changes are not occurring. The changes in Table 2 are small relative to the size of interannual variability in landfalling hurricane numbers and would not be expected to be detectable because of the signal-to-noise ratio. We provide a statistical analysis that demonstrates this in appendix B. This analysis shows that even trends in cat1–cat5 landfall frequencies as large as the mean plus two standard deviations from the landfall model would not be expected to be detectable in the historical record. The lack of detectable trends in the observations therefore neither validates nor invalidates the existence of a trend but is simply an absence of evidence.
In the absence of detectable observed trends in hurricane numbers, we rely on climate models to quantify the possible trends. This puts our estimates of changes in hurricane frequency on a less strong evidential footing than estimates of changes in temperature or sea level, since changes in temperature and sea level are clearly evident in observations. Nevertheless, it is important not to neglect the implications about hurricane frequency trends that are emerging from climate models, given the large impact such trends may have long before they become observationally detectable.
4. Average annual loss changes
We now combine the hurricane risk models described in section 2 and the hurricane frequency changes due to climate change described in section 3 to estimate the impact of the changes in frequency on loss. In this section, we will consider the impact on AAL while in the next section we will consider the impact on distributions of annual loss.
The overall changes in AAL due to the frequency changes given in Table 2 are $2.2B (billion) (14.4%) and $3.6B (14.0%), for the models based on the W2018 and M2020 datasets, respectively. We see that the absolute differences in the changes between the two datasets are much larger than the percentage changes, which is a useful result, since it is the percentage changes that are of the most interest to users of climate change information. The absolute changes are different because the baseline AALs are different, with much larger values for the M2020 dataset (see Table 1).
We can better understand the AAL changes by decomposing the change by category and by investigating how the AAL varies with GMST in our model, as follows.
a. AAL change by category
The terms in the middle and on the right-hand side of Eq. (1) are shown in Fig. 3. Figure 3e shows the AAL change by category, Fig. 3d shows the expected frequency change, and Fig. 3b shows the mean loss by event by category. For both normalized loss datasets, we see from Figs. 3e and 3f that the largest changes in AAL are for cat4 hurricanes. This is because cat4 hurricanes have a large loss per event, are somewhat frequent, and the cat4 frequency changes are large in percentage terms. The second largest contribution to AAL change comes from cat3 storms, followed by cat5, cat2, and cat1 storms. Cat5 storms contribute more to the AAL increase than cat1 or cat2 storms, even though they contribute less to the baseline AAL, because the frequency increases for cat5 are larger than the frequency increases for cat1 or cat2.
The contributions of storms in each category to the AAL are illustrated in Figs. 3g and 3h. These figures show the mean loss per storm and the mean frequency in each category, with and without climate change adjustments. The lines show lines of constant AAL. Figure 2h uses logarithmic scales on the axes so that the lines of constant AAL become straight.
If we divide Eq. (1) by the total AAL to give percentage changes, we can see that the percentage changes are determined by the expected frequency changes, which are the same for the two datasets, and by the relativities between the mean losses per event, which are similar for the two datasets (see Fig. 3b). This explains why the percentage changes in AAL are similar for the two datasets. We note that the change in AAL due to climate change that we have derived is considerably smaller than the difference between the AALs estimated from the W2018 and M2020 datasets, which is $8.9B.
b. AAL change with GMST
Change in AAL vs GMST, as based on our calculations for AAL change, combined with a model for interpolation of hurricane frequencies vs GMST (solid black line). Approximations are based on just the linear term (dashed red line), the linear term and the first quadratic term (dotted blue line), and the linear term and both quadratic terms (dot–dashed purple line).
Citation: Journal of Applied Meteorology and Climatology 62, 12; 10.1175/JAMC-D-23-0087.1
5. Loss distribution changes
We now consider the impact of hurricane frequency changes due to climate change on two annual loss distributions derived from our risk model: the total annual loss distribution (TALD; i.e., the distribution of the total loss per year from all hurricanes) and the maximum annual loss (MALD; i.e., the distribution of the maximum loss per year, from the most damaging hurricane of each year). We follow risk modeling convention and present these distributions as exceedance probabilities (EPs), that is, the probability of exceeding a certain value, which is one minus the cumulative distribution function. In some cases, we also present probabilities as return periods, defined as one over the EP. Results for the TALD are given in the text, and the corresponding results for the MALD are given in the online supplemental material.
Impact of climate change
Figures 5a and 5b show baseline values and changes for the TALD. For W2018 data (Fig. 5a), the TALD at 100-yr return period is roughly $170B, while for the M2020 data (Fig. 5b), it is roughly $300B.
Total annual loss (TAL) distributions, and changes in TAL distributions due to climate change, based on risk models derived from the W2018 and M2020 normalized hurricane losses: Return periods, defined as 1 divided by the probability of exceeding a given loss, for (a) W2018 and (b) M2020, showing the TAL distribution for a baseline climate and for a climate in which the frequencies of hurricane events have been adjusted for climate change. Also shown are (c) the differences between (a) and (b), represented as differences in probabilities for a given loss, and (d) the same differences, but now represented as differences in loss for a given probability.
Citation: Journal of Applied Meteorology and Climatology 62, 12; 10.1175/JAMC-D-23-0087.1
When we consider changes in the TAL and MAL distributions, we can either consider the change in the probability for a given level of loss (the change along a vertical line in Fig. 5a), or, conversely, the change in the loss for a given level of probability (the change along a horizontal line in Fig. 5a). We will investigate both and refer to them as the probability changes and the loss changes.
The TALD probability changes at fixed loss (Fig. 5c) show increases in the probability of losses due to climate change. This is because the frequency change distributions we have applied mostly give frequency increases (see Fig. 3d). The increases in probability get larger as the loss increases. This is because higher losses are caused by events rated at higher categories, and we are applying larger frequency increases to the higher-category events. The probability changes for the W2018 data are higher than for the M2020 data, for a given loss. This is because, for a given loss, the W2018 losses are determined by higher-category events, to which we are applying larger frequency increases.
The TALD loss changes at fixed probability (Fig. 5d) show similar results for W2018 and M2020 datasets. This is because losses at a given probability are determined by more or less the same categories of events in the two models derived from the two datasets. The losses then change according to the changes in the frequencies of those categories of events. The change in loss at fixed probability is therefore a somewhat more robust measure of the percentage impact of climate change than the change in probability at fixed loss, as it is less sensitive to the absolute level of loss in the underlying model.
Percentage changes in loss at short return periods in the TAL and MAL distributions (Fig. 5d and the online supplemental material) arise as ratios of small or even zero losses and can take very large values, from −100% to positive infinity. In our case they tend to −100% as the return period reduces. Figure 6a shows a close-up of the baseline and climate change versions of the TALD from Fig. 5 at very short return periods to illustrate this. We see from the vertical sections of the curves at zero loss that the probability of having no losses in a year increases slightly because of climate change, and this leads to changes of −100% for a small range of return periods. The probability of having no losses in a year increases because the distributions of frequency change include some scenarios in which overall frequencies decrease. Further analysis (not illustrated) of the behavior of the TALD and MALD at short return periods has shown that the behavior is sensitive to the correlations between the frequency changes for the different categories of storms. We are using rank correlations of 1, following the analysis given in Jewson (2022b). Switching to a correlation of 0, as a sensitivity test, leads to the percentage changes in TALD and MALD diverging to infinity at short return periods.
Further analysis and sensitivity tests related to the results shown in Fig. 5: (a) a close-up of Fig. 5a for short return periods and small losses, (b) the impact on Fig. 5c of using two alternative sets of frequency changes to calculate the impact of climate change, based on “step” and “linear” models, as discussed in the text, (c) the impact of using just the median and the mean of the frequency change distribution rather than the full distribution, and the impact of setting correlations between frequency changes to zero, and (d) the impact of using bootstrap models for historical losses by category instead of fitted lognormal distributions.
Citation: Journal of Applied Meteorology and Climatology 62, 12; 10.1175/JAMC-D-23-0087.1
The changes in absolute loss increase as return period increases (because the losses are more influenced by higher-category events, to which we are applying larger frequency increases). However, Fig. 5d shows percentage change, not absolute change, and shows decreasing percentage increases as return period increases. The percentages decrease because the numerator in the calculation of the percentage (the change in loss) increases less quickly than the denominator (the baseline loss).
6. Sensitivity and uncertainty
a. Sensitivity to methodological choices
Figures 6b–d look at a number of sensitivities of the loss results to methodological decisions. Figure 6b looks at the changes in the TALD probabilities for the step, linear, and complete models. We see that moving from the step model to the linear model has a material impact on the changes in probability and increases the changes. This increase is mainly because the linear model tends to increase the frequency change of cat3 storms. Moving from the linear model to the complete model causes an even larger increase in the changes in probability. This is because the complete model accounts for the fact that many landfalling cat3 and cat4 storms would previously have been higher intensity and applies higher-frequency adjustments accordingly. The conclusion from Fig. 6b is that it is essential to consider how intensity changes are interpolated and that it is also essential to include adjustments for the difference between lifetime maximum intensity and landfall intensity.
Figure 6c looks at the changes in the TALD probabilities using the complete model, as used to generate Figs. 5 and 6, and the same, but with three simplifications: using the median frequency changes only, using the mean frequencies changes only, and using the whole distribution of frequency changes but with correlations between changes of different category events set to zero. One of the motivations for performing these sensitivity tests is that many modelers in the insurance industry have been using the K2020 results, but only using the median, since the median, but not the mean, was provided by K2020 as a measure of central tendency. Using just the median or just the mean, or setting the correlations to zero, are not mathematically correct but might be justified if they give a good approximation to the results from the complete model.
We see that using the median frequencies gives the lowest changes in probability. This is because the distributions of frequency change are positively skewed, and the median is a low value relative to the mean and relative to much of the distribution. Using the mean frequency changes gives higher results than using the medians because the means of the frequency distributions are higher than the medians. The TALD based on mean frequency crosses the TALD for the complete model and gives lower results at long return periods. This difference can be understood as follows. In the complete model, using the whole distribution of frequency changes via the stochastic-parameter simulation approach incorporates changes in the frequency that are lower than the change in the mean in some years of simulation, and higher than the change in the mean in others. The inclusion of realizations with higher changes in frequency than the mean creates more extreme loss years, and so leads to the TALD for the complete model showing larger probability increases at the high loss levels.
Figure 6c also shows results for the version of the model with zero correlation between the frequency changes for different categories. In the complete model, the inclusion of correlations between the frequency changes leads to more extreme years, because there is a greater chance of having multiple events from different intensity categories. This leads to the TALD for the complete model showing larger probability increases than the zero-correlation model at high loss levels.
The overall conclusion from Fig. 6c is that correct propagation of the uncertainty around the frequency changes and correct representation of the correlations between the frequency changes are essential because they have a large impact on loss changes. Based on these results, we would advocate that risk modelers should aim to propagate the entire distribution of frequency uncertainty through their risk models, using the stochastic-parameter simulation method. Propagating the distribution of uncertainty is only slightly more difficult than using the mean or median to adjust a risk model.
Figure 6d compares the TALD changes from the lognormal loss model with those from a model that replaces the lognormal distribution with bootstrapping of the historical losses within each category. The bootstrap loss changes are much more noisy because they lack the smoothing effect of fitting the lognormal distribution. This is a test of the lognormal model in the sense that if the bootstrap loss changes were completely different then that would imply that the lognormal may not be appropriate in some way. There is, however, reasonably good agreement between the two.
b. Sensitivity by category
We saw in section 4 above that the change in the AAL due to climate change in our model is mostly caused by the increase in the frequency of cat4 storms. We now consider the question of which categories of storms are causing the changes in the TALD shown in Fig. 5. We address this question by first investigating the sensitivity of the TALD to changes in the frequencies of storms of different categories. We do this by increasing the frequency of each category of storm separately by 10% and looking at the impact on the TALD in each case.
Figure 7a, for the TALD, shows that increasing the frequency of cat1 and cat2 storms by 10% only has an effect at the very lowest loss levels. Increasing the frequency of cat3 storms by 10% has a bigger effect than increasing the frequency of other categories of storms up to losses of around $8B. From losses of $8B upward, increasing the frequency of cat4 storms has the largest effect. Increasing cat5 frequencies has a relatively small effect, even at losses of $150B. The dominance of cat4 storms above $8B is due to the large mean loss per event for cat4 storms (Fig. 3b), combined with cat4 storms being somewhat frequent in the baseline model. Because they are somewhat frequent, increasing the frequency by a given percentage adds a material number of new storms. The lower impact of changing the frequency of cat3 storms is because cat3 storms individually have lower losses than cat4 storms. The lower impact of changing the frequency of cat5 storms is because cat5 storms are so infrequent that increasing the frequency by a fixed percentage does not add many new storms. If we were to change the frequencies of storms by a fixed number of storms, instead of by a fixed percentage, these results would be completely different, and cat5 would dominate.
(a) The change of the baseline TALD (from Fig. 5a) resulting from separate 10% changes in the frequencies of each category of storm. It also shows the sum of the changes, and the change from applying 10% frequency changes to all categories at once. (b) The impact of the frequency changes given in Table 2, applied separately by category, and together, and the sum of the results from applying them separately. All changes are shown as changes in probability for a given loss. Also shown are the impacts of climate change on the TALD, as shown in Fig. 5 (black line), and the same but calculated using individual quantiles from the internal distribution of frequency changes, as labeled, for (c) changes in probabilities for a given loss and (d) changes in losses for a given probability or return period.
Citation: Journal of Applied Meteorology and Climatology 62, 12; 10.1175/JAMC-D-23-0087.1
We sum the changes in probability from our 10% frequency adjustments and compare the sum with the change in probability from adjusting all categories of storms by 10% at once. Figure 7a shows that these two are roughly equal. Although the impacts of changes in the frequencies of different categories of storms do not in general combine in this linear way, for small changes it seems that it is a good approximation, which makes interpretation of small frequency changes much easier. The reason it is close to linear is that the probability of a 10% increase in the frequency of storms in two different categories leading to extra storms in both categories in the same year is very low.
c. Climate change impact by category
We now investigate the changes in the TALD due to the climate change adjustments but applied separately by storm category (Fig. 7b). The impacts of the changes due to climate change are a combination of the size of the frequency changes being applied combined with the sensitivities shown in Fig. 7a. Cat4 is now slightly more dominant over the lower categories because the TALD not only has a high sensitivity to cat4 storms, but also because we are applying larger frequency changes to cat4 storms. Cat5 changes are now relatively more important because we are applying larger frequency changes to cat5 than to the lower categories. For the impacts of climate change in Fig. 7b, the sum of the individual impacts due to frequency changes in each category of storm separately is not a particularly good approximation to the impact of changing frequencies in all categories at once. Further analysis (not illustrated) has demonstrated that this is due to the correlation of 1 between frequency changes in the climate change adjustments. Setting the correlation to zero (artificially) gives linear behavior. A correlation of 1 does not give linear behavior because it increases the chance that there will be extra storms in multiple storm categories at once due to climate change.
The conclusion from Figs. 7a and 7b is that for the ranges of losses and return periods that are typically considered most important for risk management (10-yr return period losses to 100-yr return period losses) projected changes in cat4 storms dominate the changes in probabilities at fixed loss. The same is true for changes in losses at fixed probabilities (not illustrated). Changes in weaker storms are less important because their losses are smaller, and changes in cat5 storms are less important because they are so infrequent. This suggests that particular attention should be applied to determining the possible changes in cat4 storms, to the extent that is possible.
d. The impact of uncertainty
There are many uncertainties in the estimation of future hurricane damages. In our study, some aspects of uncertainty are external to our analysis, and not quantified, while others are internal, incorporated into the analysis, and propagated through the calculations. In this section we deconstruct the impacts of the internal uncertainty related to the uncertainty around the input hurricane frequencies we are using. This uncertainty is given in Table 2. This uncertainty mostly arises because of the range of results from the different climate model results collected in K2020 and given by the ranges in the figures in that paper (their Figs. 1b and 2b). It also partly arises as a result of the uncertainty around the conversion of frequencies as a function of lifetime maximum intensity to frequencies as a function of landfall intensity.
The results shown in Figs. 5 and 6 integrate over this internal uncertainty in the frequency changes. In other words, the uncertainty is incorporated into the distributions, and makes them wider than they would otherwise be if they were based on fixed estimates of future hurricane frequencies. This incorporation of uncertainty is the correct mathematical approach for propagating internal uncertainty and for determining the best estimate of how the distributions of loss will change. “Best estimate” in this sense means best within the context of the model: the model still has flaws and uncertainties, and fixing these would give better estimates. In addition to the best estimate, we can also consider individual uncertainty scenarios from points on the distribution of frequency uncertainty. This is useful because it can give us an indication of how much our results are influenced by somewhat arbitrary choices in methodology, such as the choice of studies considered by K2020. Points on the distribution of uncertainty roughly correspond to results from the individual climate model studies that were included in K2020, although the correspondence is not exact, since the uncertainty also includes the landfall conversion uncertainty. This uncertainty analysis is also useful because it can give an indication of how much our results may change over time. If different uncertainty scenarios give very different results, that would suggest that the results may be unstable and may change rapidly as new studies become available. These uncertainty results do not incorporate sensitivity related to the choice of historical loss dataset, as that has been addressed by presenting results from both the W2018 and M2020 datasets.
Figures 7c and 7d show changes in the TALD probabilities and losses for the complete model (as shown in Figs. 5 and 6), as well as results from using frequency changes based on the 10th, 25th, 50th, 75th, and 90th percentiles from the distribution of frequency changes used as input for that model. We see that these different scenarios give very different loss impacts. This is a consequence of the wide distributions of frequency uncertainty, as shown in Fig. 3d. Considering the changes in the probabilities (Fig. 7c): for the highest losses, and the 90th percentile, the changes in the probabilities are greater than a 100% increase. Considering the changes in the losses (Fig. 7d), we see that for the 50th, 75th, and 90th percentiles the loss changes increase at short return periods. This is because the percentile loss estimates are based on fixed frequency changes with no uncertainty, and it is the uncertainty that leads to the complete model results converging to −100% at short return periods. For the 100-yr return period (RP) in the TALD, the 90th percentile gives losses increasing by more than 30%. This corresponds to a large absolute increase in loss, since this is a 30% increase on what is already a large loss.
The conclusion from Figs. 7c and 7d is that the best estimate results shown in Figs. 5 and 6 are an average over a wide range of possibilities. They are likely quite heavily affected by some of the somewhat arbitrary decisions that were made in the modeling chain and are therefore not particularly robust to modeling decisions. They may change significantly over the next few years as new studies become available that reduce the uncertainty around possible hurricane frequency changes.
7. Conclusions
We have considered some of the possible impacts of climate change on U.S. hurricane economic loss, using two versions of a simple hurricane risk model based on two datasets for normalized historical losses. We have focused on the effect of changing frequencies of hurricanes of different intensities on wind and storm-surge damage and have ignored the effects of changing rainfall, sea level rise, and exposure. We have also ignored possible changes in storm tracks, genesis regions, storm size, forward speed, and transitioning characteristics. All of these may be important for both the overall losses and the regional losses. However, changes in genesis regions, tracks, size, speed, and transitioning characteristics have been less widely studied, and it would be difficult to form a consensus distribution for how they might change. Changes in rainfall and sea level due to climate change are extremely important but are outside the scope of this study. We have based our frequency changes on the results from the K2020 metastudy, postprocessed to create changes in frequencies of landfalling storms using the methods from Jewson (2023b). These frequency changes provide estimates of both a mean change and a standard deviation that quantifies uncertainty, including uncertainties estimated from the range of studies considered by K2020.
Considering loss changes corresponding to a 2°C increase in GMST, we have been able to draw a number of conclusions about changes in average annual loss (AAL), changes in loss distributions, the sensitivities of our results to methodology choices, the impact of events of different categories, and the impact of uncertainty.
With regard to AAL, we have found that AAL in our normalized hurricane loss data, and hence in our risk model, is dominated by cat4 storms, with a large contribution also from cat3 storms. We find that the AAL increases by 14%. The changes in AAL due to climate change are also dominated by cat4 storms, followed by cat3 storms. The absolute values of change are very different for the two loss models we have used, but the percentage values of change are reasonably similar, suggesting that the percentage changes in AAL are reasonably robust to changes in the underlying risk model.
With regard to the distribution for total annual loss, we have found that the changes in probability of a given loss are sensitive to the underlying risk model, while changes in loss for a given probability are somewhat less sensitive. The largest percentage changes in loss appear at a return period of 4 yr, and 100-yr return period loss changes (which are ∼12%) are larger than 10-yr return period loss changes (which are ∼17%) in absolute terms but smaller in percentage terms. Changes in the distribution of maximum annual loss are slightly lower than changes in the distribution of total annual loss.
With regard to sensitivities to methodological choices, we have found that the modeling step in which we apply intensity interpolation to the K2020 frequency changes makes a large difference to the final changes in loss. The step in which we apply appropriate adjustments to convert frequency as a function of lifetime maximum intensity to frequency as a function of landfall intensity then makes an even larger difference to changes in the loss. We have investigated whether using just the mean or median of the distribution of frequency changes from K2020 would give a good approximation to using the whole distribution, and have found that it would not, and it would underestimate changes, especially for the median. We have also shown that ignoring correlation between frequency changes would underestimate loss changes.
With regard to the impact of events of different categories on the changes in the loss distributions, we find that in the very short return period range, changes in the distribution of loss are dominated by changes in cat3 frequencies. For longer return periods, including the 10-yr to 100-yr return period range, they are dominated by changes in cat4 frequencies. Cat4 frequency changes dominate both because of the large sensitivity of losses to changes in the cat4 frequency, and because of the large changes in the cat4 frequency due to climate change. Cat5 frequency changes are not as important as cat4 frequency changes, even at the longest return periods we consider, because cat5 events are so much less frequent.
With regard to uncertainty, our best estimate results are based on averaging loss changes over the distribution of frequency uncertainty. This gives moderate changes in loss under climate change. However, we find that picking quantiles from the distribution of uncertainty leads to very different losses. This is because the frequency uncertainty distribution is wide. Higher quantiles from this distribution lead to large changes in loss. This large uncertainty suggests that our best estimate results are not particularly robust and may show significant changes over the next few years if we update them as future climate modeling studies become available. This wide uncertainty suggests our changes in average annual loss are not inconsistent with those of Gettelman et al. (2017), who found decreases in loss due to climate change, based on simulations from a single climate model.
All our results are highly uncertain, because of uncertainties in the normalized hurricane loss data, uncertainty in the frequency changes we are applying, and approximations in our risk model. Nevertheless, our results hopefully provide some useful insights into possible changes in U.S. hurricane losses due to frequency changes.
Acknowledgments.
Many thanks are given to the consortium of companies that have funded this research, to Roger Pielke Jr. and Andrew Martinez for their help in using their datasets, and to the anonymous reviewers who made constructive suggestions that have improved the article. The author owns Lambda Climate Research, a think-tank set up to do and publish climate research of direct relevance to society.
Data availability statement.
The datasets used in this study are available at W2018 and Martinez (2020).
APPENDIX A
The Saffir–Simpson Wind Speed Scale
The Saffir–Simpson hurricane wind speed scale defines the intensity categories of hurricanes on the basis of maximum wind speed as follows: category 1 = 64–82 kt (1 kt ≈ 0.51 m s−1), category 2 = 83–95 kt, category 3 = 96–112 kt, category 4 = 113–136 kt, and category 5 = 137 kt or higher.
APPENDIX B
Detecting Trends in Landfalling Hurricane Numbers
We consider the question of whether we would expect to be able to detect the sizes of trends in landfalling hurricane numbers given in Table 2 in the observed landfall record. We use a simple statistical model for this purpose. This model is not intended to be entirely realistic but serves the purpose of helping us to understand the extent to which trends might be detectable. We model the number of hurricanes per year from 1900 to 2017 as a Poisson distribution with mean that increases linearly year on year. The mean value of the mean is set equal to the observed mean number of cat1–cat5 storms, which is 197/118 = 1.67 storms per year. The time trend in the mean, which we refer to as the internal trend, is set equal to a trend derived from the implied trend from the landfall model results given in Table 2. Global mean surface temperatures increased by 0.85°C during the period from 1900 to 2017, and so the trend from Table 2 is adjusted so that it applies to a 0.85°C increase. We consider three cases: the internal trend set to the mean of the implied climate change trend, the internal trend set to the mean of the implied trend plus one standard deviation and the internal trend set to the mean of the implied trend plus two standard deviations. The latter case represents the extreme upper end of the range of trends that are implied by the climate model results. We then generate 10 000 sets of 118 yr of simulated hurricane landfall numbers for each of these three models. The three internal trends, and three examples of simulated hurricane landfall numbers, are shown in Fig. B1. For each of the 30 000 sets of 118 yr of simulated hurricane numbers, we then calculate the trend in the simulated data and test whether it is significantly different from zero. Individual examples from the set of 30 000 calculated trends are shown in Fig. B1. For the case in which the internal trend is given by the mean trend from Table 2, a significant trend is detected, at a 95% confidence level, in only 471 of the 10 000 cases. For the more extreme case in which the internal trend is given by the mean trend plus one standard deviation from Table 2, a significant trend is detected in only 1208 of the 10 000 cases. For the most extreme case we test, in which the internal trend is given by the mean trend plus two standard deviations, a significant trend is detected in only 3267 of the 10 000 cases. We conclude that the trends in Table 2 are so small, relative to the interannual variability of hurricane numbers, that one would not expect them to be detectable in the historical record.
Examples of series of simulated hurricane numbers (black lines), the trends used to create the simulations (red lines), and trends fitted to the results of the simulations (blue lines) for simulations that are based on (a) a trend equal to the mean trend from the landfall model, (b) a trend equal to the mean plus 1 standard deviation from the landfall model, and (c) a trend equal to the mean plus 2 standard deviations from the landfall model.
Citation: Journal of Applied Meteorology and Climatology 62, 12; 10.1175/JAMC-D-23-0087.1
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