Risk and Reliability in a Natural Hazard System

M. Khaleghy Rad and S.G. Evans

Natural Disaster Systems Research Group, Department of Earth and Environmental Sciences, University of Waterloo, Ontario, Canada, N2L 3G1

Abstract

A quantitative risk assessment framework for natural hazards is essential for mitigation, risk management, and reinsurance purposes. FN-curves have been a major tool of societal risk assessment in human-caused (industrial) accidents and natural hazard events since their invention in 1967 (Farmer, 1967). In this approach, F is the annual frequency of death cumulatively plotted versus number of fatalities (N) due to an event. Since it is a process-independent approach, FN-curves are useful for the development of a unified risk assessment framework. However, to have a comparable framework with engineering-based F-N-curves featuring probability of failure versus loss (Baecher and Christian, 2003) and to have a frame-independent approach, a normalization of conventional FN-curves on both horizontal and vertical axes is needed. The common acceptable level of risk adopted for natural hazards, based on FN-curves, is the human-caused (industrial) acceptable risk. We examine the reliability of natural hazard systems and using comparative risk assessment, we show that the risk of natural hazards far exceeds the risk of human-caused (industrial) events. This finding suggests the need to define independent credible criteria for natural hazards.

Résumé

Un cadre évaluation quantitative des risques pour les dangers naturels est essentielle pour l’atténuation, de gestion des risques, et à des fins de réassurance. FN-courbes ont été un outil majeur de l’évaluation des risques dans la société d’origine humaine (industrielle) les accidents et les catastrophes naturelles depuis leur invention en 1967. Dans cette approche, F est la fréquence annuelle de la mort cumulativement tracé par rapport à nombre de décès (N) en raison d’un événement. Puisqu’il s’agit d’une approche processus indépendant, FN-courbes sont utiles pour l’élaboration d’un cadre unifié d’évaluation des risques. Cependant, pour avoir un cadre comparable à l’ingénierie à base de courbes FN avec la probabilité de défaillance par rapport à la perte (Baecher et Christian, 2003) et d’avoir une approche cadre indépendant, une normalisation des classiques de la bande-courbes sur les deux axes horizontaux et verticaux nécessaires. Le niveau de risque acceptable commune adoptée pour les dangers naturels, basée sur la bande-courbes, est l’origine humaine (industrielle) risque acceptable. Nous examinons la fiabilité des systèmes de catastrophes naturelles et l’utilisation de l’évaluation comparative des risques, nous montrons que le risque de catastrophes naturelles dépasse de loin le risque de d’origine humaine (industrielle) des événements. Ce résultat suggère la nécessité de définir des critères indépendants crédibles pour les dangers naturels.

1 INTRODUCTION

F−N diagrams have been used for industrial risk assessment since they were developed in the 1960s. F−N curves are log-log plots with F, cumulative annual frequency of N or more consequences vesus the value of consequences. Here we consider number of fatalities as consequence of natural hazards. The comparative risk assessment of dfferent accidenst/events can be shown on these plots. On the other hand there are other diagrams introduced in the field of engineering which are log-log plots of annual probability of failure versus the value of

consequences, Pf-N curve (Baecher, 2003). The

interpretation of these two plots are different sinc the first one, F-N curve, is generated based on the frequency of the consequence but the second one, P

f-N curve, is based on

the probability of failure which is not neccessarily dependant on the consequences (i.e. the calculated value of P

f is not

directly the result of the magnitude of consequences). Figure 1 compares both traditional plots. Left plot is the P

f-N curve

from Baecher (1982) and the right plot is the acceptable level of risk in Hong-Kong on F-N curve Hong-Kong-Government-Planning-Department (1994).

To compare these two plots we suggest the normalization of the F-N plot on y-axis with the total number of events (including no consequences) which indicates the meaning of probability of failure. In this study, we use the earthquake disaster fatalities in the world (1950-2010; Khaleghy-Rad and Evans, unpublished data) to apply the methods that we present for normalization. Figure 2 illustrates the normalization of F-N curve of earthquake disasters on y-axis in comparison with the overall trend of the P

f-N Baecher

Figure 1 Comparison of Pf-N (left, Baecher, 1982) and F-N (right, Hong-Kong Government Planning Department, 1994) traditional plots

Figure 2 Normalized F-N curve of earthquake disasters (1950-2010) on y-axis in comparison with Pf-N curve (Baecher, 1982) and R2P2 criterion (Health and Safey Executive, 2001)

(1982). This plot also includes the acceptable risk proposed by Health and Safety-Executive (2001), i.e., the R2P2 (Reducing Risk, Protecting People) point which is 50 fatalities with the frequency of 1 in 5000 normalized by the total number of industrial and man-caused events in UK (EM-DAT data). Figure 2 shows that 1) the acceptable criteria of risk on the F-N curve is not comparable with the equivalent criteria on P

f-N curve and 2) the earthquake

disaster risk is not accceptable according to either.

2 F−N CURVES AND NATURAL HAZARDS

Although the annual frequency of fatalities due to natural hazard events have been one of the bases of developing the acceptable risk criteria on FN-curves, these criteria appear far too conservative for natural hazard risk assessment. The comparion of acceptable level of risk in UK and Netherlands (as examples of the acceptable risk criteria) with the F-N curve of earthquake disasters in the world is shown in Fig. 3. Fig. 3 shows that the earthquake disasters of the world are around 2 orders of magnitude above the limits of acceptable level of risk.

Figure 3 F-N curve of earthquake disasters (1950-2010) with respect to the acceptable level of risk in UK and Netherlands

2.1 NORMALIZATION OF F−N CURVES

Normalization of F-N curve has been previously suggested in the literature, e.g. Prugh (1992), Nishenko and Barton (1994), Hungr and Wong (2007) and Horn et al. (2008) The normalization method that Prugh (1992) and Hungr and Wong (2007), introduced is associated with the vertical axis of the curve, i.e., cumulative frequency of more than specific number of fatalities, N, per year. Since the unit of the vertical

axis is [cumulative] number of events (with more than N deaths) per year, if any normalization is needed, it should be in the same unit, number of events, while in the mentioned literature, the vertical axis is normalized by the number of people exposed by the event. On the other hand, the horizontal axis has the unit of number of fatalities which is should be normalized by another number, exposed population. This method of normalization is also suggested by Nishenko and Barton (1994) and Horn et al. (2008). Here, we suggest normalization of F-N curves both on the horizontal axis, division of the number of fatalities by the number of exposed population, ExP, and vertical axis, division of number of more than N fatalities per year by the total number of events including no fatalities, e.g. all earthquake events with the same condition as the events that passed the criteria of M>5.5, in the same period of time. This approach will help comparison of risk of different events with different exposed population. For the purpose of this initial study, we take the total population of the country in the natural disaster event year to be an index of the exposed population of that country subject to the earthquake disaster event in question. Since the urban population density growth is a function of the area underlay of the population (Clark (1951); Makse and Stanley (1995)), the normalization of x-axis by the exposed population will include the normalization in area indirectly. Illustration of the proposed normalization is shown in Figure 4. Figure 4 shows the comparison of the normalization of the frequency-magnitude of the fatality of global earthquake disasters with the acceptable risk criterion proposed by Health and Safety-Executive (2001), R2P2 that estimated for the world population, 5000 fatalities at the frequency of 1 in 5000, normalized by about 6 billion population on the x-axis and by about 7800 total number of industrial and man-caused events in the world on the y-axis.

Figure 4 also represents a new assessment of frequency-magnitude of fatalities, since the number of fatalities is normalized by the number of exposed population. In this assessment, the same number of fatalities in two events that have two different exposures (ExP), have different frequencies; e.g. earthquakes in Morocco, 1960, and Iran, 1968, both had 15,000 fatalities, but the ExP of Iran 1968 earthquake is almost twice as large as ExP of Morocco 1960 earthquake, therefore, Iran 1968 earthquake is more frequent than Morocco 1960 in the normalized magnitude-frequency curve. This means that the Morocco 1960 earthquake had more risk to its people than Iran 1960 earthquake.

3 RELIABILITY OF NATURAL HAZARD

Reliability is the ability of a system or a component to perform its required functions under stated conditions for a specified period of time. Hence, the value of reilability determines how effective the system is in any condition. A system is fully reliable if its reliability is 1 and it is totally unreliable if its reliability is 0. In the context of natural hazards, a reliable system requires a system with no losses. We consider an end of the world scenario in which the population of the world vanishes due to an asteroid impact, for example, as an earth protection system having a reliability of 0.

3.1 CONCEPTUAL MODEL

The definition of risk according to Aven (2011) in ISO documents is the effect of uncertainty on objectives. In other words it includes two main components: 1) indetermination (probability of events) and 2) consequences (damages or deaths) (Proske, 2008). The conceptual, mathematical and engineering definitions of risk (to life) always carry these two components. The conceptual definition of risk, Eq. [1, is based on common knowledge of risk which includes the two components mentioned above. Here Hazard is defined as a risk source event where consequences relate to harm (Aven, 2011). The conventional definition of risk, Eq. 2. (Birkmannn, 2006), on the other hand, introduces two parameters of Vulnerability and Exposure which are defined as the effect of uncertainty on objectives at occurrence of an event, and the presence of an object being subject to an event or risk source, respectively (Aven, 2011). In Eq. [3 we introduce a new parameter in the definition of risk, 1/Resistance , which is defined as the level of tolerance at the occurrence of an event.

Risk = Hazard×Consequenses [1]

=Hazard×Vulnerability(×Exposure) [2]

=Hazard×{ 1

Resistance×Exposure} [3]

Reliability, Rel, of any system is defined as the probability of survival of the system over a period of time. In the natural disaster context, this means the probability of no losses due to an event. Since time in the context of natural disasters

systems can be accomodated in the Resistance of the system and since higher resistance corresponds to less losses, one minus the probability of exceeding of a certain Resistance is our refined definition of Reliability for application of natural disaster risk analysis purposes, Eq. [6.

R(t) =1 −F(t) [4]

F(t) =1− t0f(x)dx [5]

Rel(r0) =

∞

rof(r)dr [6]

where r here is Resistance in “Risk” equation, [3, and r

0 is an acceptable defined resistance.

To calculate resistance in the Risk equation, Eq. [3 for any natural hazard system, we need to have the value of Hazard, the history of fatalities, Risk, here called Loss because of the context of past events, and the Exposure value. As the value of Resistance is calculated from this equation, we can estimate the probability distribution of it and calculate the parameters associated with the best fit to the distribution. By having the probability distribution function, generating several random variables with the parameters of our dataset is straightforward. Cumulative probability or probability of exceedance of any value from this distribution is the next step of the calculations and finally we can have Rel(r

0), Eq. [6, for any

r0. The values of resistance in our datasets can be

assessed after this from the generated Reliability function.

3.2 EARTHQUAKE EXAMPLE

The risk of earthquake disasters is considered in this example. We gathered the data of global earthquakes with more than 1000 fatalities from 1973 to 2010 (Khaleghy-Rad

and Evans, unpublished data). The reliability of the global earthquake disaster system is calculated in this section. In our dataset, we obtained the value of Exposed Population, here population of the world as a proxy for exposure, the number of fatalities due to earthquake disasters, summed up for each year. Furthermore, each year, the probability of magnitude 5.5 and more is our proxy of “Hazard”. To calculate the probability of hazard, the probability of

Figure 5 Log-Normal distribution as the best fit to the earthquake disasters (with more than 1000 fatalities) of 1973-2010 (based on Khaleghy-Rad and Evans, unpublished data).

exceeding magnitude of 5.5 each year is calculated based on NEIC catalog of earthquakes. The distribution of earthquake occurrence in each year is best fitted to the Poisson distribution.

The next step for calculation of Resistance was to use

Eq. 3 for each year. The distribution of “Resistance” is best fitted to the Log-Normal distribution in Fig. 5.

Fig. 5 shows that Log-Normal distribution is a very good fit to the dirsibution of Resistance values. Now, by using these distribution parameters, we applied Monte-Carlo simulation to generate thousands of resistance data and generate a reliability curve for resistance values (Fig.6). Fig. 6 shows that as the resistance goes up the system is more reliable. As an example we estimated the resistance of the world toward earthquake disasters in year 2011, the estimated value has the reliability of around 0.15, which is not suggestive of a reliable system.

4 SUMMARY AND DISCUSSION

In this paper, natural disasters are considered as disaster systems and we compare their risk and reliability with industrial systems. As we demonstrated in Fig. 3, the societal risk of earthquake disasters are far above the acceptable level of risk imposed for industrial accidents. We suggest that it is too conservative for natural hazards. We interpret the reason of a low acceptable level of risk with respect to natural hazard F-N curves is that the reliability of

industrial systems is higher than natural hazard systems which is the result of controllable hazard in the industrial context in the equation of risk (Eq. 3). Therefore, developing a reasonable and independant acceptable risk level for

natural systems is essential for rational natural hazard risk assessment. The method of reliability assessment of natural systems can be applicable to any natural hazard, such as landslides and hurricanes. In the continuation of this research we apply the reliability analysis for rockfalls on the Sea-to-Sky Highway of British Columbia.

REFERENCES

Aven, T. (2011). On the new iso guide on risk management terminology. Reliability Engineering and System Safety, In Press, Corrected Proof.

Baecher, G. B. (1982). Statistical methods in site characterization. Updating subsurface samplings of soils and rocks and their in-situ testing, pages 463–492. Engineering Foundation, Santa Barbara, Calif.

Baecher, G. B., J. T. C. (2003). Reliability and statistics in geotechnical engineering. John Wiley & Sons Ltd., The Atrium, Southern Gate, Chichester, West Sussex PO19 *SQ, England.

Birkmannn, J. (2006). Measuring Vulnerability to Natural Hazards. United Nations University Press.

Clark, C. (1951). Urban population densities. J. Roy. Stat. Soc., 114:490-496.

Hong Kong Government Planning Department (1994) Potentially hazardous installations. Hong Kong

planning standards and guidelines, Chap. 11.

Figure 6 Reliability of Resistance of earthquake disasters

Health and Safety Executive, S. (2001). Reducing Risk, Protecting People. HSE Books.

Horn, M. E. T., Fulto, N., and Westcott, M. (2008). Measures of societal risk and their potential use in civil aviation. Risk Analysis, 28(6):1711–1726.

Hungr, O. and Wong, H. N. (2007). Landslide risk acceptability criteria: Are f-n plots objective? Geotechnical News, 25(4):47–50.

Makse, H.A., S. H. and Stanley, H. (1995). Modeling urban growth patterns. Nature, 377:608–612.

Nishenko, S. and Barton, C. (1994). Scaling laws for natural disaster fatalities. In Rundle, J., Turcotte, D., and Klein, W., editors, Reduction and Predictability of Natural Disasters, pages 3–17. Addison Wesley.

Proske, D. (2008). Catalogue of Risks Natural, Technical, Social and Health Risks. Springer, Berlin.

Prugh, R. (1992). Improve f/n graph presentation and criteria. J. Loss Prev. Process Indust., 5:239–247.