Territory-wide predictions of mortality in landslide disasters

Transcription

1 Territory-wide predictions of mortality in landslide disasters M. Pacheco University of the State of Rio de Janeiro, Brazil Abstract This paper is based on the outcomes of previous landslide failures and related mortality in a given region, providing input values to estimate credible ranges of mortality in future catastrophic events. Specific probability functions to model the distribution of mortality in landslides are presented. The proposed distributions are developed theoretically enforcing the principle of maximum entropy and the theory of statistics of extremes, leading however to very simple formulae for practical applications. The proposed distributions are compared to historical landslide mortality in Petrópolis (State of Rio de Janeiro, Brazil), Recife (State of Pernambuco, ortheast Brazil), Hong Kong, Rio de Janeiro, Canada, and Italy with encouraging results. Keywords: landslides, mortality, principle of maximum entropy, statistics of extremes, PLL (potential loss of life), F- curves. 1 Introduction As with other natural disasters, it is nearly impossible to avoid entirely losses of life in landslides. This problem is particularly severe in heavily populated urban environments, where civil defense and public safety bodies should enforce effective strategies for risk control. The most effective strategies include real time alert warnings to the population at risk and other actions that may eventually lead to the prompt evacuation of the endangered communities. To accomplish this, it is essential to estimate the size of the vulnerable population and the expectation of mortality resulting from landslide activity. Specific probability distributions to model the distribution of mortality in landslides are presented, allowing predicting ranges of expected mortality and extreme fatality in future catastrophic events The distributions are based on the Carnot Principle

2 210 Risk Analysis IV of Maximum Entropy, where the term entropy (widely used in Thermodynamics) is used in the sense of uncertainty, due to the similarity of concepts between Thermodynamics and Mathematical Statistics. Accordingly, a distribution of maximum uncertainty (or maximum entropy) is used to take into account that: a- the uncertainty associated to predictions of mortality in landslides is intrinsically very high; b- previous statistics of mortality in landslides are often scarce and inaccurate; c- previous statistics generally do not make distinction between different sliding and/or triggering mechanisms, making it difficult to correlate mortality to natural causes, anthropogenic action, continuous population growth, or to specific geotechnical, geologic, and geomorphologic features, thus leading to increasing and uncontrolled uncertainty. Finally, the theory of statistics of extremes is used to predict extreme events, taking a distribution of maximum entropy as a parent distribution to determine the resulting distribution of extremes. 2 Theoretical background: the principle of maximum entropy The principle of maximum entropy is used to determine the distribution of mortality in landslides in single slopes, in cases where the domain of mortality is bounded by 0 max [9][10][11][13]. The method of Lagrange multipliers is used to maximize the entropy h of a continuous probability density function f X (x), a X b, by the following set of equations : b f X ( x) dx = 1 (1) a ϕ ( α, β) = α( X b) + β( X a) + 2X ( a + b) = 0 (2) h ϕ + λ = 0 α α (3) h ϕ + λ = 0 (4) β β b h = f ( x)ln f ( x dx (5) a X X ) The equations above are solved for f X (x) given by the beta distribution [7][13], where α and β are the parameters of the distribution. The constraint ϕ(α,β) represents the known mean value X of the beta distribution. It is shown that the condition of maximum entropy is satisfied when α=0 and β>0 for skewed-right (reverse-j shaped) beta distributions, or β=0 and α>0 for skewedleft (J-shaped) distributions. For symmetrical distributions, α=β=0 (rectangular distribution). The reverse-j shaped beta distribution (α=0 and β>0) expressed in cumulative descending form is:

3 Risk Analysis IV 211 β + 1 max F d ( ) = (6) max max 2E[ ] max β = E[ ] < (7) E[ ] 2 F d () represents the probability of loss of or more lives in the event of a landslide. Eqn (6) is helpful in cases where the maximum anticipated mortality max is reasonably well estimated, such as in the cases of buildings in the path of potential landslides, where max is the maximum building occupancy. E[] is the mean value of the distribution or the expectation of mortality. The applicability of eqn (6) to model an unbounded distribution of mortality is enhanced by letting max and E[]=PLL, when eqn (6) approaches the exponential distribution, which is expressed in cumulative descending form as: F d ( ) = exp (8) PLL PLL is the Potential Loss of Life representing the historical mean mortality over a given period of observation T, expressed as: T PLL = f i i = (9) T i where f i is the frequency of occurrence of i or more deaths. Eqn (9) is equivalent to the mean value of a discrete random variable and therefore the Potential Loss of Life represents the mean yearly mortality over a given period of observation T. Accordingly, T is the total mortality recorded during the period of observation T. The theory of statistics of extremes is used for predictions of extreme events. Accordingly, the descending cumulative function of extremes derived from the exponential distribution is given as [3]: F dex ( ) = 1 exp n PLL se (10) F dex () is the probability of loss of or more lives in a future catastrophic (extreme) event, whereas F d () is the probability of loss of or more lives in any event, eqns (6) and (8). The parameter n s is the sample size, i.e., the number of past landslides causing 0 fatalities. The modal value u n of the distribution of extremes is called the characteristic value [3]:

4 212 Risk Analysis IV u n = PLL ln( n s ) (11) The characteristic value u n denotes a particular point in eqn (10) representing the most likely extreme event. Predictions of extreme mortality are dependent on the sample size n s in eqns (10) and (11), i.e., the higher the sample size, the higher the probability of predicting an extreme value. The parameter n s is usually unknown since past history data usually do not account for the many previous landslides with zero fatalities. Predictions can be made taking n s >500 [11]. The landslide inventory of Petrópolis, a mountain City with intense landslide activity in the State of Rio de Janeiro, Brazil, includes n s =1190 landslides where as many as 1050 significant events took place with no loss of life. Among those 1190 landslides, only 140 fatal events produced 535 deaths from 1933 to 1988 [12], hence PLL=535/55=9.73, eqn (9). If the value n s =1000 is plugged into eqn (11), the characteristic value u n can be estimated with good approximation as: u n 7PLL (12) Another point of practical interest in eqn (10) is the Maximum Extreme Mortality MEM, defined as the expected mortality corresponding to a probability of exceedance of one to one million. MEM is estimated from eqn (10) taking n s =1000 as: MEM 21PLL (13) 3 Landslides with mean mortality approximately constant When the yearly mean mortality PLL in a given region is approximately constant throughout the period of observation T, the historical data points plot reasonably close to the entropy distribution given by eqn (8). This is shown in the examples below. Historical data points plotting close to eqn (8) are not regarded as extreme events, even when the mortality in the most tragic events is high. In contrast, extreme events will plot near the distribution of extremes, eqn (10). 3.1 Landslides in Petrópolis (Southeast Brazil) Petrópolis is a tourist mountain City in the State of Rio de Janeiro, Brazil, with intense landslide activity. The F- historical data points corresponding to landslides in Petrópolis (open squares) are compared to eqn (8) (black diamonds, PLL=9.73) in Figure 1, with good agreement. This indicates that those disasters are not regarded as extreme events, including the worst event that claimed 40 lives [12]. Eqn (10) gives the solid line representing the distribution of extreme mortality in Figure 1, with no previous fatal events fitting this curve. The distribution of extremes contains the open triangle denoting the most likely extreme mortality u n =69, eqn (12). Accordingly, a future extreme event in Petrópolis may produce a mortality close to u n =69 lives (open triangle in Figure 1). This indicates that

5 Risk Analysis IV 213 the local civil defence should be prepared to face a future extreme event claiming about 70 lives. Eqn (10) contains the open diamond denoting the Maximum Extreme Mortality MEM=204, eqn (13). Since this point represents an extreme mortality corresponding to a probability of exceedance of one to one million, it can be used for practical purposes to indicate a limiting mortality that is unlikely to be exceeded. F ( or more deaths/year) ,E+00 1,E-02 1,E-04 1,E-06 1,E-08 PLL=9.73 Petrópolis Un=69 MEM=205 Figure 1: Landslides in Petrópolis [12]. 3.2 Landslides in Recife (ortheast Brazil) The F- historical data points corresponding to landslides in the metropolitan area of Recife (open squares), State of Pernambuco, Brazil, are shown in Figure 2 [4], with very good agreement with eqn (8), PLL=7.6. The worst disaster claiming =42 lives (open circle) plots to the right of eqn (8) and approaches the distribution of extremes, eqn (10). F ( or more deaths/year) ,E+00 1,E-02 1,E-04 1,E-06 1,E-08 PLL=7.6 Recife =42 Un=53 MEM=160 Figure 2: Landslides in Recife [4].

6 214 Risk Analysis IV The characteristic value u n =53 is a reasonable estimate of the most tragic event in Recife (=42 lives), whereas it seems very unlikely the outcome of a future tragic event claiming more than MEM=160 lives. The civil defence in Recife enforces a successful alert warning to the low-income population when severe weather conditions are forecast. This program is considered one of the most effective in Brazil, as it brings the mean yearly mortality in the metropolitan area of Recife among the lowest compared to other over populated areas with intense landslide activity in Brazil. 3.3 Landslides in Hong Kong before 1978 Figure 3 shows the historical F- data points representing the distribution of mortality in Hong Kong before 1978 [5], also with good agreement with eqn (8), except for the extreme event that claimed 72 lives in F ( or more deaths/year) 1.E+00 1.E-02 1.E-04 1.E-06 1.E-08 PLL=10.2 HK (pre 1978) 1972 disasters Un=71 MEM=204 Figure 3: Landslides in Hong Kong before 1978 [5]. The open triangle in Figure 3 indicates that this tragic event compares very well with u n =71 predicted by eqn (12), indicating the ability of eqns (10) and (11) to predict extreme events. 3.4 Landslides in Hong Kong after 1978 Figure 4 shows the historical F- data points representing the distribution of mortality in Hong Kong after 1978 [5], also with very good agreement with eqn (8). The historical mortality PLL=10.2 in Hong Kong before 1978 (Figure 3) dropped to PLL=2.1 after 1978 (Figure 4), as a result of the effective governmental actions towards landslide control enforced by the Geotechnical Engineering Office of Hong Kong. According to eqn (12), a future extreme event in Hong Kong today is expected to claim u n =15 lives, in contrast with u n =71 lives before In addition, the maximum extreme mortality dropped from

7 Risk Analysis IV 215 MEM=204 to MEM=15, indicating that the occurrence of an extreme event claiming more than 15 lives in Hong Kong today is very unlikely. F ( or more deaths/year) E+00 1.E-02 1.E-04 1.E-06 1.E-08 PLL=2.1 HK (after 1978) Un=15 MEM=45 Figure 4: Landslides in Hong Kong after 1978 [5]. F ( or more deaths/year) ,E+00 1,E-02 1,E-04 1,E-06 1,E-08 PLL=15.14 Un= disaster 1988 disaster Figure 5: Landslides in Rio de Janeiro. 3.5 Landslides in Rio de Janeiro The local landslide inventory of the City of Rio de Janeiro comprises 530 deaths in 593 landslides from 1962 to 1996 (T=35 years) [1][2]. These figures include the worst disaster of 1966, where 114 people were killed during the simultaneous collapse of 2 buildings and 1 house in the same landslide, yielding PLL=530/35=15.14, eqn (9). The second worst disaster claimed 27 lives in 1988, during the collapse of an elderly care clinic. The landslide inventory presents the yearly distribution of mortality and as such does not allow the development of the historical F- data points. However, the existing data allow

8 216 Risk Analysis IV to infer that the mean yearly mortality in Rio is nearly constant and therefore the distribution of landslide mortality can be estimated with reasonable accuracy by eqn (8), as shown in Figure 5. The second worst disaster (open dot) plots near the entropy distribution (eqn 8) and as such is not regarded as an extreme event. The 1966 disaster (black dot), however, plots to the right of the entropy distribution and therefore is interpreted as an extreme event. According to eqn (12), the most likely extreme mortality u n =106 compares well with =114 fatalities recorded in the 1966 disaster. 4 Landslides with variable mean mortality The results presented in session 3 indicate that the entropy distribution of mean µ =PLL provides a good match with the historical data points only in cases where the mean yearly mortality is approximately constant throughout the period of observation T. Otherwise, a distinct interpretation of eqn (8) is needed, as in the case of landslides in Canada and Italy, shown next. 4.1 Landslides in Canada It was shown in session 3 that the extreme events plot to the right of eqn (8). When the yearly mean mortality is variable, the periods of most intense landslide activity produce a mean mortality higher than PLL and the corresponding F- data points approach the distribution of extremes, eqn (10). The extreme events plot continuously further to the right as PLL increases, as shown in Figure 6 [6]. When the mean mortality is not constant the characteristic value u n and the Maximum Extreme Mortality MEM can still be estimated by eqs (12) and (13), replacing however PLL by PMMM, the Past Maximum Mean Mortality. PMMM represents the historical mean mortality corresponding to the most tragic period. The loss of life in Canadian landslides was subdivided into selected smaller periods, according to distinct levels of observed landslide activity [6]. The corresponding results are listed in Table 1. It is seen that the criterion used in Table 1 provided local PLL values that ranged between 0.3 and 9.7, with a global mean PLL=3.52. Table 1: Mean mortality in Canada. Period Loss of Life PLL (years) (local mean)

9 Risk Analysis IV 217 PMMM is determined from eqn (10) as: max PMMM = ln 1/ T (14) ( ) The value max relates to the mortality reported in the worst landslide disaster within the period of observation T. Once the past maximum mean mortality is known, the desired extreme values u n and MEM are determined as: 7 u n = max (15) ln 1/ ( T ) 21 MEM = max (16) ln 1/ ( T ) The Frank Slide disaster (the worst in Canada, claiming 70 lives in 1903) is simulated taking the second most tragic Canadian landslide as input ( max =56 deaths, Britannia Creek, British Columbia, 1921). For T=155 years and max =56, eqn (15) yields u n =77, comparing well with =70 fatalities at the Frank slide. Taking max =70 (Frank Slide) yields PMMM=13.8. This value is compared to the upper mean mortality (PLL=9.7) corresponding to the period between 1889 and 1929 in Table 1. However, eqn (14) represents a rational criterion to estimate the past maximum mean mortality regardless of arbitrary selections of time intervals, as in Table 1. The distribution of mortality corresponding to PMMM=13.8 is shown by black squares in Figure 6, comparing very well with the high-mortality events in Canada. In contrast, the curve PLL=3.5 (black diamonds) models adequately the low-mortality events. Therefore, the region between both curves contains all distributions whose mean yearly mortality ranges between 3.5 PLL F ( or more deaths/year) 1.E+00 1.E-02 1.E-04 1.E-06 1.E PLL=3.5 PMMM=13.8 Canada Un=97 MEM=291 Figure 6: Landslides in Canada [6].

10 218 Risk Analysis IV The distribution of extremes shown in Figure 6, obtained for PMMM=13.8, leads to u n =97. This is clearly a somewhat pessimistic prediction to future extreme events in Canada today, considering that the mean mortality there was significantly lowered in the past 20 years. For PLL=3.52, eqn (12) produces u n =25, a more reasonable prediction. However, this value may still be interpreted as an over prediction considering that the mean mortality in Canada the last years is about PLL=0,57 (last line in Table 1). 4.2 Landslides in Italy The historical F- data points corresponding to the distribution of mortality in Italy (open squares, PLL=59.4, T=100 years, [8]) are shown in Figure 7. They compare well to eqn (8) up to 100. However, the points corresponding to the 3 most tragic (extreme) events of the 20 th Century depart from eqn (8): Vajont (1917 deaths, 1963); Monte Antelao (341 deaths, 1925) and Stava (269 deaths, 1985), shown by the last two open squares (Stava and Monte Antelao) and the black dot (Vajont) in Figure 7. To predict the mortality in the worst disaster (Vajont), the second worst mortality (Monte Antelao, max =341) is taken as input in eqns (15) and (16), yielding u n =610 and MEM=1830, respectively. The characteristic value u n falls between the 2 worst disasters of the 20 th Century and compares well to =600 deaths reported in the worst disaster of the 19 th Century (Roccamontepiano, 1765 not shown in Figure 7). Although u n =610 is nearly one third the death toll reported at Vajont, it should be taken into account that the mortality there was greatly due to the water wave and subsequent outburst flooding following a major rockslide. evertheless, the Maximum Extreme Mortality MEM=1830 compares well with the total mortality reported in Vajont. F ( or more deaths/year) ,E+00 1,E-02 1,E-04 1,E-06 1,E-08 PLL=59.4 Italy Vajont Un=416 MEM=1247 Figure 7: Landslides in Italy [8].

11 Risk Analysis IV Conclusions A bounded probability distribution following the principle of maximum entropy was developed for risk assessments of individual slopes. This distribution approaches the exponential distribution when the upper bound approaches infinity, allowing predictions of mortality in multiple slopes. The exponential distribution models the distribution of mortality in multiple slopes when PLL is nearly constant. The probability function of extremes derived from the exponential distribution enhances predictions of extreme events in multiple slopes. Two particular points of the distribution of extremes are the characteristic value u n (representing the most likely extreme mortality), and the Maximum Extreme Mortality MEM (probability of exceedance of one to one million). When the mean mortality is not constant, PMMM (past maximum mean mortality) should replace PLL to predict extreme events. Predictions of extreme mortality in Petrópolis, Hong Kong, Canada, Rio de Janeiro, and Italy provided encouraging results. Acknowledgments The financial support provided by the Brazilian Agency CPq and by the program PROCIECIA of the University of the State of Rio de Janeiro is greatly acknowledged. References [1] Amaral, C Landslides in Rio de Janeiro: Inventory, geological features and risk management (in Portuguese), doctoral thesis, Pontific Catholic University, Rio de Janeiro. [2] Amaral, C. & Palmeiro, C Local landslide inventory of Rio de Janeiro: State-of-the-art and access, II Pan-American symposium of landslides Vol. I, Rio de Janeiro, pp [3] Ang, A. & Tang, W Probability concepts in engineering planning and design Vol II decision, risk and reliability, John Wiley, ew York. [4] Bandeira, A Risk map of erosion and landslides in the County of Camaragibe, Pernambuco, Brazil, M.Sc. dissertation, Federal University of Pernambuco. [5] DV Det orske Veritas Quantitative landslip risk assessment of pre GCO man-made slopes and retaining walls risk assessment report for GEO, Hong Kong. [6] Evans, S Fatal landslides and landslide risk in Canada, proceedings of the international workshop on landslide risk assessment, Honolulu, Hawaii, Balkema Publishers. [7] Geraldo, F.C Principle of maximum entropy: fundamentals and application in geotechnics, M.Sc. dissertation (in Portuguese), COPPE, Federal University of Rio de Janeiro.

12 220 Risk Analysis IV [8] Guzzetti, F Landslide fatalities and the evaluation of landslide risk in Italy, Engineering Geology, Vol. 58, o. 2, pp [9] Harr, M.E Reliability-based design in civil engineering, McGraw- Hill Inc., U.S.A. [10] Pacheco, M.P Frequency distribution of mortality in cut and fill slopes, Soils and Rocks, The Latin American geotechnical journal, Vol. 24, o. 1, pp , ABMS, São Paulo. [11] Pacheco, M.P Modeling uncertainty in landslide risk assessments, international conference on probabilistics in geotechnics, technical and economic risk estimation, Graz, Austria, pp [12] Pinhel, A.S Probabilistic modeling of mortality in landslides, M.Sc. dissertation (in Portuguese), Polytechnic Institute, University of the State of Rio de Janeiro. [13] Santa Maria, P.E.L., Santa Maria, F.C.M. & Pacheco, M.P The principle of maximum entropy and its applicability to a case of foundations (in Portuguese), Soils and Rocks, the Latin-American journal of geotechnics, Vol. 19, o. 2, pp