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Tenth U.S. National Conference on Earthquake Engineering Frontiers of Earthquake Engineering July 21-25, 2014 10NCEE Anchorage, Alaska

UNCERTAINTY QUANTIFICATION AND SENSITIVITY ANALYSIS OF EARTHQUAKE CASUALTIES

M. Gobbato1, W.M. Yeo2 and N. Shome3

ABSTRACT

Severe earthquake events can cause significant injuries and fatalities which can then impact considerably the overall financial loss of the insurance and reinsurance companies managing large life-risk portfolios. In this context, six increasingly severe injury levels — defined by the National Council of Compensation Insurance (NCCI) and ranging from medical-only to fatal injuries — are considered in this study. An earthquake casualty vulnerability model, properly developed and validated against historical casualty data, is then used to estimate the number of injuries and their associated financial losses in each one of these six NCCI-based injury levels. Given the occurrence of an earthquake event, such a model predicts what portion (or rate) of the impacted population falls into each injury level by analyzing the location-level (i.e., building- level) exposures within the event footprint. Each casualty rate is treated as a random variable with its mean and standard deviation being generally a function of the earthquake intensity as well as the building construction class, building age and height. In this study, a fault-tree analysis (FTA) approach is used to (i) characterize the joint and the marginal probability distribution functions of the casualty rates and (ii) estimate the first two moments of these marginal distributions, conditional on the earthquake intensity. Subsequently, these results are used to provide the coefficient-of-variation (CV) versus mean casualty rate (MCR) relationships for all six injury levels. Furthermore, the influence of building age and construction class on the portfolio-level number of fatalities is also discussed in the paper. For illustration purposes, the geographic scope of this study is restricted to the United States.

1Senior modeler, model development, Risk Management Solutions Inc., 7575 Gateway Blvd., Newark, CA 94560 2Lead modeler, model development, Risk Management Solutions Inc., 7575 Gateway Blvd., Newark, CA 94560 3Senior director, model development, Risk Management Solutions Inc., 7575 Gateway Blvd., Newark, CA 94560

Gobbato M., Yeo W.M., and Shome N. Uncertainty Quantification and Sensitivity Analysis of Earthquake Casualties. Proceedings of the 10th National Conference in Earthquake Engineering, Earthquake Engineering Research Institute, Anchorage, AK, 2014. Tenth U.S. National Conference on Earthquake Engineering Frontiers of Earthquake Engineering July 21-25, 2014 10NCEE Anchorage, Alaska

Uncertainty Quantification and Sensitivity Analysis of Earthquake Casualties

M. Gobbato1, W.M. Yeo2 and N. Shome3

ABSTRACT

Severe earthquake events can cause significant injuries and fatalities which can then impact considerably the overall financial loss of the insurance and reinsurance companies managing large life-risk portfolios. In this context, six increasingly severe injury levels — defined by the National Council of Compensation Insurance (NCCI) and ranging from medical-only to fatal injuries — are considered in this study. An earthquake casualty vulnerability model, properly developed and validated against historical casualty data, is then used to estimate the number of injuries and their associated financial losses in each one of these six NCCI-based injury levels. Given the occurrence of an earthquake event, such a model predicts what portion (or rate) of the impacted population falls into each injury level by analyzing the location-level (i.e., building-level) exposures within the event footprint. Each casualty rate is treated as a random variable with its mean and standard deviation being generally a function of the earthquake intensity as well as the building construction class, building age and height. In this study, a fault-tree analysis (FTA) approach is used to (i) characterize the joint and the marginal probability distribution functions of the casualty rates and (ii) estimate the first two moments of these marginal distributions, conditional on the earthquake intensity. Subsequently, these results are used to provide the coefficient-of-variation (CV) versus mean casualty rate (MCR) relationships for all six injury levels. Furthermore, the influence of building age and construction class on the portfolio-level number of fatalities is also discussed in the paper. For illustration purposes, the geographic scope of this study is restricted to the United States.

Introduction

Large earthquake events can lead to a significant number of injuries and fatalities which can then impact considerably the overall financial loss of the insurance and reinsurance companies managing large life-risk portfolios [1]. In this context, the number of injuries is calculated based on six increasingly severe injury levels, as defined by the National Council of Compensation Insurance (NCCI), ranging from medical-only to fatal injuries; see Table 1. In order to estimate the number of injuries and their associated financial losses in each one of the six injury levels, a proper earthquake casualty vulnerability model must be developed, calibrated and validated against historical data. Such a model aims at providing, in probabilistic terms, the

Gobbato M., Yeo W.M., and Shome N. Uncertainty Quantification and Sensitivity Analysis of Earthquake Casualties. Proceedings of the 10th National Conference in Earthquake Engineering, Earthquake Engineering Research Institute, Anchorage, AK, 2014. Table 1. Description of the six NCCI-based injury levels considered in this study.

Injury level and rate Description

IL1 (CR1) Minor injury without any permanent impairment. Examples Medical Only include abrasions, lacerations, strains, sprains, and contusions. Injury that results in an individual’s inability to work but from IL2 (CR2) which the individual can fully recover within a reasonably short Temporary Total period of time. Examples include simple broken bones, loss of consciousness, serious strains, and sprains. A permanent injury that results in ongoing partial disability. IL3 (CR3) Examples include complicated fractures, serious joint injury, Permanent Partial – Minor concussions, or minor crush injury. A permanent injury that results in a disability level greater than IL4 (CR4) 25% (but less than total disability) with no return to work. Permanent Partial – Major Examples include massive organ injury, heart laceration, loss of limb(s), or crushed extremities. The most severe type of non-fatal injury leading to a total IL5 (CR5) disability state, where the individual is unable to work again. Permanent Total Examples include spinal cord syndrome, crush syndrome, and massive head injury. IL6 (CR6) Immediate death or fatal injury resulting in death. Fatal

distribution of the injured population impacted by the earthquake event among the six NCCI- based injury levels aforementioned (i.e., IL1, IL2, …, IL6). In other words, the model predicts what portion (or rate) of the impacted population falls into each injury level, with the logical constraint that the sum of all injury/casualty rates across the six NCCI-based injury levels must 6 be less or equal to 1 (i.e., i 1cri 1). Each injury level is associated with a casualty rate which is treated as a random variable whose mean and standard deviation are assumed to be a function of the earthquake intensity as well as the building construction class, building age and height. A choice driven by the (i) limited availability of casualty data and (ii) the lack of detailed location- level information found in common insurance portfolios. It is worth noting that, in reality, injury rates and their associated cost depend on many other factors such as building occupancy class (e.g., ware house, office, or retail store), age of the working population inside the building, etc. This study focuses on the development of a simulation-based methodology for characterizing the uncertainty in the casualty rates. More specifically, a fault-tree analysis (FTA) approach is first exploited to characterize the joint and marginal probability distribution functions (PDFs) of the casualty rates and then estimate the first two moments of these marginal distributions conditional on a given earthquake intensity. Subsequently, these results are used to provide the coefficient- of-variation (CV) versus mean-casualty rate (MCR) relationships for all injury levels. More specifically, Monte-Carlo simulations, with a large number of realizations, are used to generate random realizations of a six-component vector (collecting the six casualty rates aforementioned) across the whole spectrum of possible earthquake intensities. These data points are then used to fit and validate — through standard hypothesis testing procedures [2] — a mixed PDF model (i.e., a PDF model with a discrete as well as a continuous component). This model is chosen to (i) provide a better fit to the simulated injury rates and (ii) improve the performance and stability of the injury rate sampling process which represents a key component of the portfolio loss analyses discussed in the second part of the paper. The nodes of the event-tree are designed to account for all pertinent sources of uncertainty such as the ground shaking intensity, the time of occurrence of the seismic event, the geographical distribution of the population, the building vulnerability, and the probability of having the building occupants (as well as the people in the immediate proximity of the building) in a given injury level. Additionally, it should be pointed out that the input distributions for all different nodes in the event-tree are based either on publicly available data (see Refs. [3] and [4]) or proprietary and licensed third-party databases; however, it should be emphasized that the primary focus of this study is on the illustration of the proposed FTA methodology and not on the numerical results. The proposed approach, discussed in this paper, represents a rigorous method of characterizing the joint and marginal distributions of casualty rates without imposing any restrictive and/or a priori assumption on their form. For illustration purposes, the geographic scope of this study is restricted to the United States (U.S.), and the inputs to the event-tree simulation model are therefore chosen accordingly. Finally, in the last part of the paper, an industry portfolio with a well distributed exposure across the state of California is used to analyze the influence of building age and construction class on the number of fatalities at the portfolio-level.

Fault-tree analysis methodology

The FTA approach depicted in Fig. 1 is used to characterize the uncertainty of earthquake-driven injuries in the Western U.S. macro zone. For demographical purposes, this macro zone is in turn conveniently divided into three sub-regions, namely WUS1 (California), WUS2 (Oregon and Washington), and WUS3 (Alaska, Arizona, Colorado, Hawaii, Idaho, Montana, Nevada, New Mexico, Utah, and Wyoming). Based on the notation used in Fig. 1, each FTA run involves the following steps:  Based on either a scalar or a vector-valued intensity measure, simulate an earthquake event (denoted as EQ in Fig. 1) of random intensity during either day- or night-time. A Bernoulli random number generator is used to establish if an event has occurred during day- or night-time [5].  Select randomly a geographical area where the event occurs. For the sake of simplicity, this area is assumed to be entirely encompassed by 8one of the three sub-regions outlined above (i.e., WUS1, WUS2, or WUS3).  Simulate, within the impacted area at the time of the event, what portion of the population is commuting (COM) and what portion is instead located either inside or in the immediate premises of each one of the five main construction classes used for this simulation exercise: wood frame, unreinforced masonry (URM), reinforced masonry (RM), reinforced concrete (RC), and steel frame buildings. For this study, a portfolio with well distributed exposure across the entire U.S. is used to model the distribution of the indoor exposed population among the five main construction classes aforementioned. On the other hand, the criteria used to distribute the outdoor and commuting population data are provided by HAZUS [4]. Lastly, the random distribution of the impacted population among multiple nodes is achieved by using the Dirichlet distribution [6].  Determine the damage state of each construction class (conditional on the simulated earthquake intensity) by (i) using a pre-defined set of structural fragility functions, for example those given in [7-9], and (ii) explicitly accounting for the uncertainty related to building age and height. For simplicity, only three possible damage states are considered in this study: collapse (C), heavy damage (HD), or neither of the above (

Layer selecting which Damage Non geographical area is Wood state Fatal Fatal impacted by the event In- door Non Fatal URM C Fatal Day Out- time WUS1 door Total rate of RM

Figure 1. Illustration of the FTA approach used to compute the MCR-CV relationship for the fatality rate in the Western U.S. macro zone (i.e., WUS1, WUS2, and WUS3). Fault-tree analysis Results

The FTA approach briefly outlined in the previous section was used to carry out one million swipes/simulations across all possible earthquake intensities and impacted regions within the western U.S. region. These analyses provided one million random realizations of the casualty T rate vector — i.e., CR cr , with CR defined as CR CR1,,,,, CR 2 CR 3 CR 4 CR 5 CR 6 — and allowed constructing the scatter plots of simulated casualty rates vs. earthquake intensity used as basis for the characterization of the joint and marginal PDFs of the six casualty rates for given/fixed earthquake intensities along the entire intensity spectrum considered in this study. In addition, research studies (see Ref. [10] for instance) show that for any given hazard level considered, the marginal PDF of CRi , f cr with i 1, ,6 , is well described by a mixed CRi i PDF model defined as

f cr p cr f cr i 1, ,6 (1) CRii i i i CR i

where pi represents the probability mass at cri 0 , cri denotes the Dirac Delta, and f cr refers to the continuous part of the distribution over the interval 01. The mean and CRi i standard deviation of the whole marginal PDF f cr are denoted as and , CRi i cri cri respectively. Similarly, the mean and standard deviation of the central portion of marginal PDF f cr are denoted as and , respectively. Using the simulation results from the CRi i cri cri proposed FTA, it is possible to derive the functional relationship between and for all casualty rates (i.e., ). The results obtained are summarized in Fig. 2 and, as expected,

~ ~ ~

Figure 2. Relationship between the probability mass pi and the mean value ( ) of the central part of the PDF for i 1, ,6 . it was found that p is a strictly decreasing function of with p 01. In other i cri i cri words, when the mean value of the marginal PDF f cr is equal to zero, the probability mass CRi i has its peak and it then gradually decreases as increases. Additionally, it should also be noted that all six vs. curves reported in Fig. 2 are very similar to one another; however, the same types of curves would have been substantially different if plotted against rather than cri .

Based on the FTA simulation results, it was also found that the marginal PDFs f cr CRi i in Eq. 1 (for i 1, ,6 ), conditional on any given earthquake intensity within the range of earthquake intensities considered in the analyses, are very well represented by a Beta distribution [12]. As an illustration, Fig. 3 shows the proposed fitting with a Beta PDF for the simulated fatality rates (i.e., cr6) at a given severe hazard level; i.e., towards the upper extreme of the range of earthquake intensities considered. As can be seen in both Fig. 3(a) and Fig. 3(b), the simulated data is very well fitted by the Beta PDF and the Kolmogorov-Smirnov test [2] confirmed the goodness of fit. Furthermore, as shown in Fig. 4, it was observed that the marginal PDF can always be fitted by a strictly decreasing Beta PDF at all intensity levels. It can in fact be noticed from Fig. 4 that the coefficient of variation (CV) of the random casualty rates,

CRi , viewed as a continuous function of and defined as CV (for ), is cri cr i cr i always lower than the theoretical upper bound beyond which the Beta PDF is no longer a strictly decreasing function in the interval [0 , 1]. This theoretical upper bound is denoted by the dashed red lines in the six sub-plots in Fig. 4 and its analytical expression is provided in Eq. 2:

1 cri CVcr i 1, ,6 (2) i 2 2 crii cr

(a) (b)

Figure 3. Illustration of Beta PDF fitting for the simulated number of fatalities at a given hazard level: (a) histogram from simulated number of fatalities with corresponding Beta PDF fitting for f cr ; (b) comparison between empirical vs. analytical CDFs for CR6. CR6 6 Upper bound for strictly decreasing Beta PDF

~ ~ ~

Figure 3. Relationship between the coefficient of variation ( CV ) and the mean value ( ) of cri cri the central part of the PDF f cr for i 1, ,6 . CRi i

Portfolio-level Analysis Results

An industrial portfolio with a well distributed exposure across the state of California is used in this section to analyze the influence of the time of occurrence of an earthquake as well as the influences of building age and building construction class on the number of casualties (and in particular fatalities) at the portfolio-level. Three historical earthquake events are used to investigate the first effect mentioned above: the 1906 San Francisco earthquake, the 1994 Northridge earthquake, and the 1868 Hayward earthquake. Concurrently, four different times of the day for the occurrence of each event were considered: 8:00 AM, 12:00 PM, 2:00 PM, and 6:00 PM. Each combination of earthquake event and time of occurrence is then used to estimate the mean portfolio-level loss in terms of mean number of fatalities and mean number of overall injuries; the results are shown in Fig. 5. Using 8:00AM as reference time, it is estimated that the number of casualties and financial losses caused by earthquake events striking at 12:00 PM and 2.00 PM are about 45% and 65% higher, respectively. On the other hand, an earthquake event occurring at 6.00 PM would cause only 30% to 35% of the number of casualties and financial losses produced by the same seismic event striking at 8:00 AM. It should be noted that the relative differences mentioned above are solely driven by the temporal variation of the modeled working population exposed to the event and not by changes in the simulated injury rates. Additionally, as an illustration, Fig 6 shows the geographical distribution of the number of fatalities (i.e., the quantity represented by the leftmost blue bar in Fig. 5(a)) caused by the 1906 San Francisco earthquake event assumed to strike at 8:00 AM. It is worth noting that in Fig 6 the estimated number of fatalities at the building-level is aggregated at the zip-code level for illustration purposes.

Number of fatalities (x1000) Overall number of injuries (x1000) 1.2 14 (a) SF 1906 (b) SF 1906 1.0 Northridge 1994 12 Northridge 1994 Hayward 1868 Hayward 1868 10 0.8 8 0.6 6 0.4 4 0.2 2 0.0 0 8:00 AM 12:00 PM 2:00 PM 6:00 PM 8:00 AM 12:00 PM 2:00 PM 6:00 PM

Figure 5. Effect of the time of occurrence of an earthquake on the number of injuries at the global portfolio-level: (a) number of fatalities, (b) overall number of injuries derived as the sum of injuries across all six NCCI-based injury levels defined in Table 1.

Number of fatalities in each zip code

0 >10

Site-adjusted PGA

Figure 6. Zip-code level resolution of the simulated number of fatalities produced by the 7.8 Mw 1906 San Francisco earthquake footprint analyzed against the chosen industrial portfolio at 8:00 AM on a weekday.

Finally, in order to evaluate the effect of building age and building construction class on the mean annual number of fatalities at the global portfolio level, a stochastic event dataset is ran against some properly modified versions of the 2011 RMS WC California-IED exposure. The results reported in Fig. 7(a) highlight the effect of building age and are obtained by manually changing the year of construction of each building part of the portfolio being analyzed. A total of five analyses are therefore needed to create the bar plot shown in Fig. 7(a) and, as it can be inferred from the figure, it is estimated that the 2011 RMS California-IED composed entirely of pre-1940 buildings leads to financial losses about five times larger than those experienced by the same portfolio entirely formed by post-2000 constructions. The effect of the building construction type is even more dramatic: as shown in Fig. 7(b), the 2011 RMS California-IED composed entirely of unreinforced masonry building produces financial losses about three hundred times larger than the corresponding losses that would be experienced by the same portfolio composed entirely of wood-frame buildings.

5 1000 Results are normalized w.r.t. the number Results are normalized w.r.t. the of fatalities obtained from the analysis of number of fatalities obtained from 4 the selected industrial portfolio entirely the analysis of the selected portfolio composed ofpost-2000 buildings. 100 entirely composed of wood frame 3 buildings. (a) (b) 2 10

Normalized Normalized number offatalities Normalized Normalized number offatalities 0 1 Prior 1940 1940-1970 1970-1990 1990-2000 post 2000 URM RM RC Steel Wood Year of construction Construction type

Figure 7. Effects of building characteristics on the mean annual number of fatalities at the global portfolio-level: (a) Effect of the building age normalized with respect to (w.r.t.) post-2000 buildings, (b) effect of the building construction type normalized w.r.t. wood frame buildings.

Conclusions

This study focused on the development of a simulation-based methodology for characterizing the uncertainty in the number of injuries caused by earthquakes. A fault-tree analysis (FTA) approach was presented and the results obtained from the Monte Carlo simulations were used to characterize the joint and marginal probability distribution functions (PDFs) of the casualty rates conditional on a given earthquake intensity, the mean casualty rate vulnerability curves (i.e., the curves providing the mean casualty rates as a function of the earthquake intensity. The nodes of the event-tree were designed to account for all pertinent sources of uncertainty in the estimation of casualty rates such as the ground shaking intensity, the time of occurrence of the seismic event, the geographical distribution of the population at the time of the event, the building vulnerability, and the probability of having the building occupants (as well as the people in the immediate proximity of the building) in a given injury level. The results indicate that a mixed PDF model — i.e., a PDF model with a discrete as well as a continuous component — fits the simulated casualty rate data well. Finally, the portfolio-level analyses performed in this study provided (i) the number of expected injuries for a well distributed life-risk portfolio in the state of California using three historical earthquake records and (ii) analyzed the influence of the time of occurrence of an earthquake as well as the influences of building age and building construction class on the estimated number of casualties.

Disclaimer

The views and opinions expressed in this article are those of the authors and do not necessarily reflect the position of Risk Management Solutions Inc. (RMS). Examples of analysis performed within this article are only examples. They should not be utilized in any real world analytics as they are based only on limited information. Assumptions made within the analyses performed for this study are not reflective of the position of RMS.

References

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