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Impact of fault parameter uncertainties on earthquake recurrence probability examined by Monte Carlo simulation — an example in Central Taiwan
Engineering Geology 126 (2012) 67–74
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Impact of fault parameter uncertainties on earthquake recurrence probability examined by Monte Carlo simulation — an example in Central Taiwan Jui-Pin Wang a,⁎, Chii-Wen Lin b, Hamed Taheri c, Wen-Shan Chan d a
Department of Civil & Environmental Engineering, The Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong Geology Division, Central Geological Survey, Ministry of Economic Affairs, Taiwan, ROC Department of Civil & Environmental Engineering, The Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong d Dept. Geosciences, National Taiwan University, Taiwan, ROC b c
a r t i c l e
i n f o
Article history: Received 12 August 2010 Received in revised form 9 April 2011 Accepted 2 December 2011 Available online 2 January 2012 Keywords: Uncertainty Earthquake recurrence probability Monte Carlo simulation Probabilistic analysis
a b s t r a c t The region around Taiwan is well known for its unique geological setting which has resulted in a high seismicity. The Central Geological Survey, Taiwan (CGST), has launched a series of investigation on characterizing the active faults on the island. The CGST revealed that there are 33 active faults and their fault parameters are suggested in the latest publication in 2010. However, the uncertainties of return period and earthquake magnitude are not provided. The main objective of the paper is to study the impact of “unspecified” parameter uncertainties on recurrence earthquake probabilities. In addition, a numerical approach involving the use of Monte Carlo simulation was proposed to estimate the exceedance probabilities in which the parameter uncertainties were taken into account. The Meishan fault in Central Taiwan with the best-estimate return period of 162 years and earthquake magnitude of 7.1 was used as an example. The analyses verified the impact of parameter uncertainties on the exceedance probability in fault-induced earthquake recurrence. When uncertainty is high, the exceedance probability associated with large magnitudes is high. It was also observed that the probabilistic analysis in this study can generate a realistic, continuous distribution of exceedance probabilities that cannot be obtained by a deterministic approach. The best-estimate distribution of exceedance probability was established by enveloping the curves estimated by both deterministic and probabilistic approaches. Accordingly, the best-estimate exceedance probability for a 7.5-magnitude earthquake recurring in 2011–2060 induced by the Meishan fault in Central Taiwan was estimated as 11%. © 2011 Elsevier B.V. All rights reserved.
1. Introduction Taiwan is located at the Circum Pacific Belt, i.e., the collision zone of the Asian Plate and Philippine Sea Plate, leading to frequent occurrences of catastrophic earthquakes compared to other regions. The latest event with the magnitude of 6.4 stroke the southern part of Taiwan on March 4, 2010, causing more than 100 people injured and a few damages to structures and roads. In 1999, the Chi-Chi earthquake with the magnitude of 7.3 brought the disaster causing the casualty of more than 2400 people, which was unforgettable for many Taiwanese people. In earthquake engineering, a variety of approaches have been developed from seismic hazard assessment to hazard mitigation. For instance, the probabilistic seismic hazard analysis (PSHA) is the most commonly used approach to evaluate site-specific seismic hazard (Bommer and Abrahamson, 2006). Regarding to seismic hazard mitigation, the earthquake early warning system (EEWS) is one of
⁎ Corresponding author. Tel.: + 852 2358 8482; fax: + 852 2358 1534. E-mail address: [email protected] (J.-P. Wang). 0013-7952/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.enggeo.2011.12.012
the approaches that has been studied and developed (Allen and Kanamori, 2003; Horiuchi et al., 2005; Wu et al., 2007). In contrast with other earthquake prediction models, an EEWS is a more practical, effective approach in the seismic hazard mitigation on a short time scale (Kanamori et al., 1997; Allen and Kanamori, 2003). However, the PSHA and EEWS cannot be exempted from the uncertainty resulting from methodology itself to input parameters. Bommer and Abrahamson (2006), for instance, pointed out that the ground-motion variability plays an important role in a high estimate of seismic hazard in recent PSHA studies. Klugel (2007) and Wang and Zhou (2007) echoed the study and emphasized the importance of the dependency among earthquake parameters in monitoring the ground-motion variability. Certainly, this becomes one of the issues debating on the reliability of the methodology among others (Castanos and Lomnitz, 2002; Bommer, 2003). In EEWSs, Iervolino et al. (2006, 2007, 2009) has conducted a comprehensive study focusing on the uncertainty of the real-time ground shaking using a regional EEWS. Furthermore, most onsite EEWSs were established by correlating initial indications, e.g., first-three-second displacement, and ground motion shaking e.g., peak ground velocity (Wu and Kanamori, 2005). Based on available observations, the empirical
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Fig. 1. Spatial distribution of the 33 active faults on the island of Taiwan (after Lin et al., 2008, 2009).
equation can be established by a regression model. Since the two variables impossibly present a perfect correlation, uncertainty is unavoidable in onsite EEWSs. A few probability distributions have been discussed regarding their appropriateness to simulate the recurrence interval of earthquakes. The lognormal and Weibull distributions are the ones generally accepted and frequently used (Hagiwara, 1974; Rikitake, 1976, 1991; Nishenko and Buland, 1987). Abaimov et al. (2008) suggested that the Weibull distribution is the preferred distribution based on the study on the San Andreas fault. It can be used to estimate the risk of recurrence of earthquakes on the San Andreas fault and elsewhere (Abaimov et al., 2008). Regarding the probability functions
for earthquake magnitude, McGuire and Arabasz (1990) derived the probability density function based on recurrence parameters. The function is then incorporated into the PSHA framework. Gan and Tung (1983) proposed an extreme value distribution to simulate the annual largest magnitudes of world shallow earthquakes from 1904 to 1980, and the observed and derived cumulative probabilities show a good agreement. Wang et al. (2011) examined the suitability of the five commonly-used probability models in simulating the annual maximum earthquake magnitude around Taiwan since 1900. The study reveals that lognormal and Gamma distributions are more appropriate to model annual maximum earthquake magnitudes. Note that the magnitude distribution functions are either derived or
J.-P. Wang et al. / Engineering Geology 126 (2012) 67–74
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Table 1 Fault parameters of the nine active faults located in Southwestern Taiwan (after Lin et al., 2008, 2009). Sense of Dip angle/ Across-fault horizontal Fault velocity change by GPS length movement dip (mm/yr) direction (km)
Fault name
Parallel to fault
Normal to fault
Across-fault vertical velocity change Short(mm/yr) term slip rate (mm/yr) Precise GPS Geological leveling method
Chiuchiungkeng Fault Meishan fault Tachienshan Fault Muchiliao Fault Liuchia Fault
17
Reverse
20–30°/E
+ 0.8 ± 2.6
2.3 ± 3.9 − 15.3
13 + 25
>60° >60°/SE
+ 1.6 ± 3.2 − 2.6 ± 3.9
0.9 ± 3.3 − 0.8 0.5 ± 3.8 − 11.6
– 30°/E
+ 2.0 ± 3.6 + 2.0 ± 3.6
2.3 ± 2.9 – 2.3 ± 2.9 0.3
Chukou Fault Hisnhu Fault Houchiali Fault Tsaochen Fault
28 6+ 12 10
Dextral Reverse w. dextral Reverse Reverse w. sinistral Reverse Dextral Reverse Sinistral
50–60°/E >60° >35°/W >60°/N
+ 3.2 ± 4.2 11.1 ± 3.4 3.1 + 8.8 ± 3.9 1.6 ± 4.2 − 5.0 + 3.5 ± 2.8 6.2 ± 3.8 7.9 − 1.4 ± 8.6 6.3 ± 7.9 –
7 17
verified by statistical procedures based on observed regional seismicity. To the authors' best knowledge, there has not yet a probability distribution in discussing the recurrence earthquake magnitude induced by a specific fault. The Monte Carlo Simulation (MCS) is a useful numerical approach for probabilistic analysis and it is widely used in geosciences. Based on limited paleoseismic records, Parsons (2008) used the MCS to establish the distribution of recurrence parameters for the strike-slip faults in California. Refice and Capolongo (2002) evaluated the earthquake-triggered landslide hazard in Italy in a probabilistic framework, in which the uncertainties of input parameters (e.g., geological, geotechnical) were taken into account by using the MCS. The United States Nuclear Regulatory Commission (2007) published a technical reference in which the MCS is used to consider the uncertainties of soil properties in a site amplification analysis. In this paper, we reviewed the latest publications regarding the active faults in Taiwan that have been investigated and published by the Central Geological Survey, Taiwan (CGST). It is found that the fault parameters such as return period and induced-earthquake magnitude are provided but their uncertainty is not specified. Therefore, the main objective in this study is to examine and demonstrate
Long-term Recurrence Last reactivated interval slip rate time (yr) (yr) (mm/yr)
Probable maximum earthquake (M)
–
14.3–20.7 –
199
b 18,540 B.P.
6.5
5.8 ± 2.3 – 13.6 ± 10.7 –
17.3–22.4 – – –
162 –
A.D.1906 A.D.1999
7.1 6.7
5.6 ± 2.0 5.6 ± 2.0
10.2–22.3 11.2 ± 4.0 10.2–22.3 11.2 ± 4.0
430 430
b 100,000 B.P. 6.0 b 10,000 B.P. 6.5
– – 5–6 –
5.3–16.7 4.2–13.6 5.2–9.5 –
1314 188 936 –
b 10,000 B.P. A.D.1946 b 100,000 B.P. b 100,000 B.P.
4.7 ± 8.9
9.3 ± 8.4 9.3 ± 8.4 3.8 ± 7.2 3.2 ± 8.9 − 0.4 ± 7.6 4.1 ± 7.9
– – – –
6.8 6.1 6.3 6.2
the impact of fault parameter uncertainties on earthquake recurrence probabilities using the proposed procedure in which the MCS was used. Using the approach, an example in Central Taiwan was demonstrated. 2. Geological investigations on active faults in Taiwan A strong earthquake could bring a horrific consequence toward the society and the rate of its recurrence in Taiwan is known to be high. The CGST has thoroughly begun the investigation on active faults on the island of Taiwan and updated the result with the latest findings periodically. In the latest publication, 33 faults were reported and their spatial distribution is shown in Fig. 1 (Lin et al., 2008, 2009). Table 1 summarizes the nine faults located in Southwestern Taiwan. The fault parameters reported by the CGST include the fault length, slip characteristics, dip angle and orientation of the fault plane, slip rate, return period, etc. The approaches used in the investigation include geological studies, historical records, subsurface geophysical investigation, geodetic measurement, and others. However, it is noted that the uncertainties of return period and earthquake magnitude are not provided except their best estimates. 3. Deterministic approach estimating earthquake recurrence probability
0.4
Recurrence Probability
(100, 0.365) 0.3
The recurrence probability of earthquakes can be estimated by using the Poisson sequence that is widely accepted to simulate the occurrence of rare events. Accordingly, the probability distribution of recurrence time follows the exponential distribution. Within a time interval t, the probability distribution function of the next occurrence is:
0.2
(50, 0.203)
−λt
P ðT≤t Þ ¼ 1−e
ð1Þ
0.1
where λ is the mean occurrence rate equal to (1/r), where r is return period. If a fault-induced earthquake has not occurred until a specific year (n) since last occurrence (c), the occurrence probability within next t-year becomes:
(10, 0.044) 0.0
(1, 0.005) 0
20
40
60
80
100
Period [Years] Fig. 2. Earthquake recurrence probabilities induced by the Meishan fault estimated by a deterministic approach.
! F ðt; n; c; r Þ ¼ P n−cbT≤t þ n−cjT > n−c
¼
Pðn−cbT≤t þ n−cÞ : 1−PðT≤n−cÞ ð2Þ
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Exceedance Probability
0.006
0.06
(a) 1-Year Era (2011)
0.004
0.04
0.002
0.02
0.000
0.00
6.0
6.5
7.0
7.5
6.0
8.0
0.2
0.4
0.1
0.2
0.0
0.0
6.5
7.0
7.5
6.5
7.0
7.5
8.0
(d) 100-Year Era (2011~2111)
(c) 50-Year Era (2011~2060)
6.0
(b) 10-Year Era (2011~2020)
8.0
Magnitude
6.0
6.5
7.0
7.5
8.0
Magnitude
Fig. 3. Exceedance probabilities of earthquakes induced the Meishan fault in four eras estimated by the deterministic approach.
Combing Eqs. (1) and (2), F ðt; n; c; r Þ ¼
−n−c r
e
−tþn−c r
−e n−c e− r
:
ð3Þ
Two implications must be noted in the derivation. First, Eq. (3) computes the probability regardless of the uncertainty of the inputs. Without the consideration of the uncertainty, it is regarded as a deterministic approach. Extra attention is noted that the deterministic approach simply refers a generic description on the computation and has no revelation to the deterministic seismic hazard analysis (DSHA) or others. Second, the recurrence probability presents the fault-induced earthquake associated with a specific magnitude. Therefore, the magnitude does not appear in the derivations. According to the CGST, the best-estimate return period and last occurrence of the Meishan fault are 162 years and in 1906, respectively. Fig. 2 shows the recurrence probabilities from a 1-year era to a 100-year era after 2010. Since the best-estimate earthquake magnitude is 7.1, the probabilities in fact present that a 7.1-magnitude earthquake recurs in different eras. For instance, the probabilities are 0.5%, 4.4%, 20.3%, and 36.5% considering 1-year, 10-year, 50-year and 100-year, respectively. However, it must be noted that the recurrence probability with earthquake magnitudes greater or less than 7.1 cannot be determined. Therefore, we introduced the exceedance probabilities to consider the recurrence probability associated with specific magnitudes of interest. Define P(M ≥ m*) as exceedance probability associated with a specific magnitude m*. The recurrence probability regardless of earthquake magnitude shown in Fig. 2 can be extended to exceedance probabilities corresponding to a specific magnitude as Fig. 3 shows. Take the Meishan fault for example, when m* less than 7.1,
P(M ≥ m*) is equal to the probability associated with magnitude = 7.1. In contrast, when m* is greater than 7.1, P(M ≥ m*) drops to zero in a sudden. Therefore, the exceedance probability distribution presents a discrete pattern using the deterministic approach. 4. Probabilistic analysis 4.1. Exceedance probability by MCS As shown in Figure 3, it is unrealistic that the exceedance probability of recurrence earthquakes features a sudden change around the best-estimate magnitude. This implies that the deterministic approach cannot provide a legitimate estimation for a broad magnitude range. In addition, the uncertainty of the parameters is not considered in the deterministic computation, which leads to another unrealistic expectation that the exceedance probabilities only rely on the best estimates and are independent of parameter uncertainties. Therefore, a probabilistic approach was used and a numerical procedure was proposed in this study. The procedure involved the use of the Monte Carlo simulation to generate random fault parameters with prescribed distributions. By changing COVs, the impact of uncertainty on the exceedance probability can be assessed. In the example of the Meishan fault, because the last occurrence was reported in 1906, the date was considered as a constant. Therefore, its uncertainty was not discussed in the following probability analyses. When n trials were performed in the probabilistic analysis, two probabilities 1) recurrence probability PR and 2) magnitude exceedance probability PM ≥ m* were computed based on randomly generated return periods (ri) and magnitudes (mi). Note that PR is continuous and PM ≥ m* is discrete with its value equal to either 1 or
J.-P. Wang et al. / Engineering Geology 126 (2012) 67–74
(a) Magnitude
~ Normal; Return Period ~ Normal
0.30
0. As a result, the probabilistic analysis estimates the exceedance probability (P^ M≥m ) associated with magnitude ≥ m* as:
Exceedance Probability
n P
0.25
5%-COV 10%-COV 20%-COV
0.20
0.10 0.05
6.0
6.5
7.0
7.5
8.0
Magnitude
(b) Magnitude
~ Lognormal; Return Period ~ Lognormal
0.30
Exceedance Probability
P^ M≥m ¼ i¼1
P R P M≥m n
:
ð4Þ
4.2. Probability models and results
0.15
0.00
0.25
5%-COV 10%-COV 20%-COV
0.20 0.15 0.10 0.05 0.00 6.0
6.5
7.0
7.5
8.0
Magnitude
(c) Magnitude
~ Lognormal; Return Period ~ Weibull
0.30
Exceedance Probability
71
0.25
5%-COV 10%-COV 20%-COV
0.20
The probability models simulating return period and earthquake magnitude need to be determined in the probability analysis using the MCS. We used a few combinations from common models (i.e., normal and lognormal) to suggested distributions to reveal and verify the expected impact. Fig. 4 shows the exceedance probabilities in 2011–2060 of the earthquake induced by the Meishan fault using (normal, normal), (lognormal, lognormal) and (Weibull, lognormal) simulating return period and magnitude, respectively. The sample size was 50,000 in the three analyses. A few uncertainty levels (5%, 10%, 20%) were assigned simultaneously to examine their impact on exceedance probabilities. Note that the best estimate level of uncertainty is unknown and the determination of its value is not the scope of the study. The results shown in Fig. 4 reveal that the proposed probability analysis generates a realistic, continuous distribution of exceedance probability. A few features in the exceedance probability were also observed in the probabilistic analysis. The exceedance probability reduced more quickly with the increase of magnitude when a low parameter uncertainty was considered. It was also observed that the three curves intersected around the best-estimate magnitude of 7.1, resulting in high exceedance probabilities associated with magnitude ≥ 7.1 when uncertainty was high. The variations in the three analyses considering different models were not conspicuous, showing that the model uncertainty did not have a significant impact on exceedance probabilities. The pattern of exceedance probability distributions can be explained by the fact that the magnitude uncertainty dominates the distributions. When magnitude uncertainty is low, random generated magnitudes concentrate near the best-estimate magnitude. This leads to the random magnitude exceedance probability PM ≥ m* equal to 1 in each MCS trail when m* is small. On the contrary, PM ≥ m* is equal to 0 when a large m* is considered. However, the random recurrence probabilities associated with random return periods are continuously distributed and the variations between MCS trials are less significant compared to PM ≥ m*. Due to the discrete nature of PM ≥ m*, the curve of exceedance probabilities estimated by the probabilistic approach features a gentle slope when a high uncertainty of parameters is considered.
0.15
4.3. Best-estimate exceedance probability distribution 0.10 0.05 0.00 6.0
6.5
7.0
7.5
8.0
Magnitude Fig. 4. Uncertainty impact on exceedance probabilities induced by the Meishan fault in 2011–2060 estimated by the probabilistic approach involved the use of Monte Carlo simulation. a) magnitude and return period both following the normal distribution, b) magnitude and return period both following the lognormal distribution, and c) magnitude and return period following the lognormal distribution and Weibull distribution, respectively.
According to the aforementioned three analyses, the impact of model uncertainty on exceedance probabilities is not significant. To ensure a more reliable estimation, the suggested probability models along with best-estimate fault parameters were used to obtain the best-estimate exceedance probability of earthquakes induced by the Meishan fault in Central Taiwan. The suggested models are the Weibull distribution and lognormal distribution for return period and earthquake magnitude, respectively. Fig. 5 shows 16 distributions of exceedance probabilities in 2010–2060 considering various combinations of magnitude and return-period uncertainties. At a given magnitude uncertainty, high return-period uncertainties result in high exceedance probabilities. Without further evidence supporting specific uncertainty levels inherited in return period and magnitude, a conservative perspective was adopted to determine the best-estimate exceedance probability.
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Exceedance Probability
0.35
(a) COV of Magnitude = 5%
0.35
0.30
0.30
0.25
0.25
0.20
0.20
0.15
0.15
COV of R.P. = 10% COV of R.P. = 20% COV of R.P. = 30% COV of R.P. = 40%
0.10 0.05
COV of R.P. = 10% COV of R.P. = 20% COV of R.P. = 30% COV of R.P. = 40%
0.10 0.05
0.00
0.00 6.0
0.35
(b) COV of Magnitude = 10%
6.5
7.0
7.5
8.0
(c) COV of Magnitude = 15%
6.0 0.35
0.30
0.30
0.25
0.25
0.20
0.20
0.15
0.15
COV of R.P. = 10% COV of R.P. = 20% COV of R.P. = 30% COV of R.P. = 40%
0.10 0.05
7.0
7.5
8.0
(d) COV of Magnitude = 20%
COV of R.P. = 10% COV of R.P. = 20% COV of R.P. = 30% COV of R.P. = 40%
0.10 0.05
0.00
6.5
0.00 6.0
6.5
7.0
7.5
8.0
6.0
6.5
Magnitude
7.0
7.5
8.0
Magnitude
Fig. 5. Distributions of exceedance probabilities induced by the Meishan fault in 2011–2060 with the considerations of magnitude and return-period uncertainties up to 20%-COV and 40%-COV, respectively.
Therefore, we used the envelope among various considerations as the best-estimate distribution of exceedance probability. Fig. 6 shows the distributions estimated by four probability analyses and by the deterministic approach along with the corresponding bestestimate distribution. Note that we considered the magnitude and return-period uncertainties up to 20%-COV and 40%-COV, respectively, to ensure a reliable and realistic estimation when specific uncertainties are not available. In such a case, the exceedance probabilities of earthquakes with magnitudes 6.5, 7.0 and 7.5 were 29%, 20% and 11%, respectively. Also note that the 11%-probability of a recurrence earthquake with magnitude ≥ 7.5 in 2011–2060 is higher than 5% estimated by using the distribution of annual maximum earthquakes for Central Taiwan (Wang et al., 2011).
distributions were still assigned for return period and earthquake magnitude, respectively, and their uncertainty levels were both set at 5%COV. The results showed that when dependency increased, the recurrence probabilities decreased in the entire range of magnitude but the differences were found insignificant. This numerical demonstration implied that 1) the dependency had little impact on earthquake exceedance probability, 2) a conservative estimate was actually rendered considering the independency between return period and earthquake magnitude. As a result, without a specific correlation that is supported, the best-estimate exceedance probability considering the independency is recommended.
0.35 0.30
The aforementioned analyses were performed based on the assumption that the return period and earthquake magnitude of the Meishan fault is statistically independent. However, a positive dependency between them could be more likely. However, there have not been specific studies available on the dependency between the return period and earthquake magnitude of the Meishan fault. In addition, it becomes another issue when a universal relationship, if any, is used without verifying its applicability on the fault of interest. Therefore, in order to examine the issue, we performed another series analysis using hypothetical correlation coefficients, in which correlated fault parameters were generated in the MCS. Fig. 7 shows the exceedance probabilities in 2011–2060 considering independent (ρ = 0) and dependent scenarios (ρ = 0.2 and ρ = 0.4). Note that in these analyses, the suggested Weibull and lognormal
Exceedance Probability
5. Discussion 5.1. Dependency of magnitude and return period
50-Year Era (2011 ~ 2060)
Best-Estimate Distribution of Exceedance Probability
0.25 0.20 0.15 0.10
COVM = 20% COVM = 15% COVM = 10%
0.05 Deterministic
0.00 6.0
6.5
7.0
7.5
COVM = 5%
8.0
Magnitude Fig. 6. Best-estimate exceedance probability induced by the Meishan fault in Central Taiwan during 2011–2060.
J.-P. Wang et al. / Engineering Geology 126 (2012) 67–74
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6. Conclusion
Exceedance Probability
0.25
ρ=0 ρ = 0.2 ρ = 0.4
0.20
0.15
0.10
0.05
0.00 6.0
6.5
7.0
7.5
8.0
Magnitude Fig. 7. Illustration of the impact of dependency between magnitude and return period on exceedance probabilities.
5.2. Impact of the date of last occurrence In the example computations, the uncertainty of the date of last occurrence was not taken into account in the probabilities analysis because it was certain and considered as a constant. However, it is not usually the case that the date of last occurrence is well known. Under such a circumstance, the analysis similar to the study can still be performed by generating additional random last-occurrence dates along with random magnitudes and return periods. The procedure to estimate the exceedance probability remains the same and the impact of parameter uncertainties can be examined in an identical manner as illustrated in this study. However, it must be noted that in an attempt to use the probabilistic analysis to estimate the exceedance probability with the consideration of the last-occurrence uncertainty, suitable probability models for the random variable need to be evaluated in advance in order to obtain a reliable recurrence probability. 5.3. Generic applications featured in the proposed methodology It must be noted that the proposed approach is generic. It can be used in any given fault as its parameters are “available” either they are reliably estimated by legitimate methods and available evidences, or they are suggested by a subjective estimation. The advantage of the probabilistic approach over the deterministic approach includes a realistic estimation in exceedance probabilities for faultinduced earthquakes by considering parameter uncertainties. The more reliable information (e.g., fault parameters, uncertainty level, correlation between parameters, probability models) is available, the more reliable exceedance probabilities can be estimated. In the example, we used the best-estimate fault parameters and models in the referred publications to demonstrate the uncertainty impact and to provide a best-estimate in exceedance probabilities. The best-estimate uncertainty level and correlations are unavailable so that they are suggested subjectively and used to ensure proper conservatism in exceedance probabilities. Note that using hypothetical values does not affect the legitimacy of the analytical framework. We agree that exceedance probabilities estimated by the approach will become more reliable and convincible when all necessary, reliable inputs are available. However, the determination of best-estimate inputs is not the scope of the study and we believe that it is difficult that the task is accomplished without extensive studies. We suggest that the analysis is re-conducted to adjust the best-estimate exceedance probabilities with new information that becomes available.
This paper presents a numerical approach to estimate earthquake recurrence probabilities and to examine the impact of fault parameter uncertainties on them. A numerical example using the Meishan fault in Central Taiwan is demonstrated. The probabilistic analysis using the Monte Carlo Simulation shows that when a high uncertainty level is considered, the exceedance probabilities are high associated with large-magnitude earthquakes. The results also show that the impact of magnitude uncertainty is predominant in exceedance probability distributions. The best-estimate distribution of exceedance probabilities is proposed using the envelope of the curves estimated by both probabilistic and deterministic approaches. Considering the magnitude uncertainty up to 20%-COV and return period uncertainty up to 40%-COV, the exceedance probability is 11% considering an earthquake with magnitude greater than 7.5 induced by the Meishan fault in Central Taiwan. Acknowledgments We appreciate Prof. Ling at Columbia University, Prof. Tang and Prof. Tung at the Hong Kong University of Science and Technology for their constructive and insightful suggestions. In addition, the valuable comments from anonymous reviewers are greatly appreciated. References Abaimov, S.G., Turcotte, D.L., Shvherbakov, R., Rundle, J.B., Yakovlev, G., Goltz, C., Newman, W.I., 2008. Earthquakes: recurrence and interoccurrence times. Pure and Applied Geophysics 165, 777–795. Allen, R.M., Kanamori, H., 2003. The potential for earthquake early warning in Southern California. Science 300, 786–789. Bommer, J., 2003. Uncertainty about the uncertainty in seismic hazard analysis. Engineering Geology 70, 165–168. Bommer, J.J., Abrahamson, N.A., 2006. Why do modern probabilistic seismic-hazard analyses often lead to increased hazard estimates. Bulletin of the Seismological Society of America 96 (6), 1967–1977. Castanos, H., Lomnitz, C., 2002. PSHA: is it science? Engineering Geology 66, 315–317. Gan, Z.J., Tung, C.C., 1983. Extreme value distribution of earthquake magnitude. Physics of the Earth and Planetary Interiors 32, 325–330. Hagiwara, Y., 1974. Probability of earthquake occurrence as obtained from a Weibull distribution analysis of crustal strain. Tectonophysics 23 (3), 313–318. Horiuchi, S., Negishi, H., Abe, K., Kamimura, A., Fujinawa, Y., 2005. An automatic processing system for broadcasting earthquake alarms. Bulletin of the Seismological Society of America 95, 708–718. Iervolino, I., Convertito, V., Giorgio, M., Manfredi, G., Zollo, A., 2006. Real-time risk analysis for hybrid earthquake early warning systems. Journal of Earthquake Engineering 10, 867–885. Iervolino, I., Giorgio, M.C., Manfredi, G., 2007. Expected loss-based alarm threshold set for earthquake early warning systems. Earthquake Engineering and Structural Dynamics 36, 1151–1168. Iervolino, I., Giorgio, M., Galasso, C., Manfredi, G., 2009. Uncertainty in early warning predictions of engineering ground motion parameters: what really matters? Geophysical Research Letters 36, 1–6. Kanamori, H., Hauksson, E., Heaton, T., 1997. Real-time seismology and earthquake hazard mitigation. Nature 390, 461–464. Klugel, J.U., 2007. Comment on Why do modern probabilistic seismic-hazard analyses often lead to increased hazard estimates by Julian J. Bommer and Norman A. Abrahamson or how not to treat uncertainties in PSHA. Bulletin of the Seismological Society of America 97, 2198–2207. Lin, C.W., Lu, S.T., Shih, T.S., Lin, W.H., Liu, Y.C., Chen, P.T., 2008. Active faults of Central Taiwan. Special Publication of Central Geological Survey 21, 148 (In Chinese with English abstract). Lin, C.W., Chen, W.S., Liu, Y.C., Chen, P.T., 2009. Active faults of Eastern and Southern Taiwan. Special Publication of Central Geological Survey 23, 178 (In Chinese with English abstract). McGuire, R.K., Arabasz, W.J., 1990. An introduction to probabilistic seismic hazard analysis. In: Ward, S.H. (Ed.), Geotechnical and Geoenvironemtnal Geophysics: Society of Exploration Geo-physicists, Vol. 1, pp. 333–353. Nishenko, S.P., Buland, R., 1987. A generic recurrence interval distribution for earthquake forecasting. Bulletin of the Seismological Society of America 77, 1382–1399. Parsons, T., 2008. Monte Carlo method for determining earthquake recurrence parameters from short paleoseismic catalogs: example calculation for California. Journal of Geophysical Research 113, B03302. doi:10.1029/2007JB004998. Refice, A., Capolongo, D., 2002. Probabilistic modeling of uncertainties in earthquake induced landslide hazard assessment. Computers & Geosciences 28, 735–749. Rikitake, T., 1976. Recurrence of great earthquakes at subduction zones. Tectonophysics 35 (4), 335–362.
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Rikitake, T., 1991. Assessment of earthquake hazard in the Tokyo area, Japan. Tectonophysics 199 (1), 121–131. U.S. Nuclear Regulatory Commission, 2007. A performance-based approach to define the site-specific earthquake ground motion. Regulatory Guide 1.208. Wang, Z., Zhou, M., 2007. Comment on Why do modern probabilistic seismic-hazard analyses often lead to increased hazard estimates by Julian J. Bommer and Norman A. Abrahamson. Bulletin of the Seismological Society of America 97 (6), 2212–2214. Wang, J.P., Chan, C.H., Wu, Y.M., 2011. The distribution of annual maximum earthquake magnitude around Taiwan and its application in estimation of catastrophic earthquake recurrence probability. Natural Hazard 59, 553–570.
Wu, Y.M., Kanamori, H., 2005. Rapid assessment of damaging potential of earthquakes in Taiwan from the beginning of P waves. Bulletin of the Seismological Society of America 95, 1181–1185. Wu, Y.M., Kanamori, H., Allen, R.M., Hauksson, E., 2007. Determination of earthquake early warning Paraameters, τc and Pd, for southern California. Geophysical Journal International 170, 711–717.
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Impact of fault parameter uncertainties on earthquake recurrence probability examined by Monte Carlo simulation — an example in Central Taiwan Jui-Pin Wang a,⁎, Chii-Wen Lin b, Hamed Taheri c, Wen-Shan Chan d a
Department of Civil & Environmental Engineering, The Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong Geology Division, Central Geological Survey, Ministry of Economic Affairs, Taiwan, ROC Department of Civil & Environmental Engineering, The Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong d Dept. Geosciences, National Taiwan University, Taiwan, ROC b c
a r t i c l e
i n f o
Article history: Received 12 August 2010 Received in revised form 9 April 2011 Accepted 2 December 2011 Available online 2 January 2012 Keywords: Uncertainty Earthquake recurrence probability Monte Carlo simulation Probabilistic analysis
a b s t r a c t The region around Taiwan is well known for its unique geological setting which has resulted in a high seismicity. The Central Geological Survey, Taiwan (CGST), has launched a series of investigation on characterizing the active faults on the island. The CGST revealed that there are 33 active faults and their fault parameters are suggested in the latest publication in 2010. However, the uncertainties of return period and earthquake magnitude are not provided. The main objective of the paper is to study the impact of “unspecified” parameter uncertainties on recurrence earthquake probabilities. In addition, a numerical approach involving the use of Monte Carlo simulation was proposed to estimate the exceedance probabilities in which the parameter uncertainties were taken into account. The Meishan fault in Central Taiwan with the best-estimate return period of 162 years and earthquake magnitude of 7.1 was used as an example. The analyses verified the impact of parameter uncertainties on the exceedance probability in fault-induced earthquake recurrence. When uncertainty is high, the exceedance probability associated with large magnitudes is high. It was also observed that the probabilistic analysis in this study can generate a realistic, continuous distribution of exceedance probabilities that cannot be obtained by a deterministic approach. The best-estimate distribution of exceedance probability was established by enveloping the curves estimated by both deterministic and probabilistic approaches. Accordingly, the best-estimate exceedance probability for a 7.5-magnitude earthquake recurring in 2011–2060 induced by the Meishan fault in Central Taiwan was estimated as 11%. © 2011 Elsevier B.V. All rights reserved.
1. Introduction Taiwan is located at the Circum Pacific Belt, i.e., the collision zone of the Asian Plate and Philippine Sea Plate, leading to frequent occurrences of catastrophic earthquakes compared to other regions. The latest event with the magnitude of 6.4 stroke the southern part of Taiwan on March 4, 2010, causing more than 100 people injured and a few damages to structures and roads. In 1999, the Chi-Chi earthquake with the magnitude of 7.3 brought the disaster causing the casualty of more than 2400 people, which was unforgettable for many Taiwanese people. In earthquake engineering, a variety of approaches have been developed from seismic hazard assessment to hazard mitigation. For instance, the probabilistic seismic hazard analysis (PSHA) is the most commonly used approach to evaluate site-specific seismic hazard (Bommer and Abrahamson, 2006). Regarding to seismic hazard mitigation, the earthquake early warning system (EEWS) is one of
⁎ Corresponding author. Tel.: + 852 2358 8482; fax: + 852 2358 1534. E-mail address: [email protected] (J.-P. Wang). 0013-7952/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.enggeo.2011.12.012
the approaches that has been studied and developed (Allen and Kanamori, 2003; Horiuchi et al., 2005; Wu et al., 2007). In contrast with other earthquake prediction models, an EEWS is a more practical, effective approach in the seismic hazard mitigation on a short time scale (Kanamori et al., 1997; Allen and Kanamori, 2003). However, the PSHA and EEWS cannot be exempted from the uncertainty resulting from methodology itself to input parameters. Bommer and Abrahamson (2006), for instance, pointed out that the ground-motion variability plays an important role in a high estimate of seismic hazard in recent PSHA studies. Klugel (2007) and Wang and Zhou (2007) echoed the study and emphasized the importance of the dependency among earthquake parameters in monitoring the ground-motion variability. Certainly, this becomes one of the issues debating on the reliability of the methodology among others (Castanos and Lomnitz, 2002; Bommer, 2003). In EEWSs, Iervolino et al. (2006, 2007, 2009) has conducted a comprehensive study focusing on the uncertainty of the real-time ground shaking using a regional EEWS. Furthermore, most onsite EEWSs were established by correlating initial indications, e.g., first-three-second displacement, and ground motion shaking e.g., peak ground velocity (Wu and Kanamori, 2005). Based on available observations, the empirical
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Fig. 1. Spatial distribution of the 33 active faults on the island of Taiwan (after Lin et al., 2008, 2009).
equation can be established by a regression model. Since the two variables impossibly present a perfect correlation, uncertainty is unavoidable in onsite EEWSs. A few probability distributions have been discussed regarding their appropriateness to simulate the recurrence interval of earthquakes. The lognormal and Weibull distributions are the ones generally accepted and frequently used (Hagiwara, 1974; Rikitake, 1976, 1991; Nishenko and Buland, 1987). Abaimov et al. (2008) suggested that the Weibull distribution is the preferred distribution based on the study on the San Andreas fault. It can be used to estimate the risk of recurrence of earthquakes on the San Andreas fault and elsewhere (Abaimov et al., 2008). Regarding the probability functions
for earthquake magnitude, McGuire and Arabasz (1990) derived the probability density function based on recurrence parameters. The function is then incorporated into the PSHA framework. Gan and Tung (1983) proposed an extreme value distribution to simulate the annual largest magnitudes of world shallow earthquakes from 1904 to 1980, and the observed and derived cumulative probabilities show a good agreement. Wang et al. (2011) examined the suitability of the five commonly-used probability models in simulating the annual maximum earthquake magnitude around Taiwan since 1900. The study reveals that lognormal and Gamma distributions are more appropriate to model annual maximum earthquake magnitudes. Note that the magnitude distribution functions are either derived or
J.-P. Wang et al. / Engineering Geology 126 (2012) 67–74
69
Table 1 Fault parameters of the nine active faults located in Southwestern Taiwan (after Lin et al., 2008, 2009). Sense of Dip angle/ Across-fault horizontal Fault velocity change by GPS length movement dip (mm/yr) direction (km)
Fault name
Parallel to fault
Normal to fault
Across-fault vertical velocity change Short(mm/yr) term slip rate (mm/yr) Precise GPS Geological leveling method
Chiuchiungkeng Fault Meishan fault Tachienshan Fault Muchiliao Fault Liuchia Fault
17
Reverse
20–30°/E
+ 0.8 ± 2.6
2.3 ± 3.9 − 15.3
13 + 25
>60° >60°/SE
+ 1.6 ± 3.2 − 2.6 ± 3.9
0.9 ± 3.3 − 0.8 0.5 ± 3.8 − 11.6
– 30°/E
+ 2.0 ± 3.6 + 2.0 ± 3.6
2.3 ± 2.9 – 2.3 ± 2.9 0.3
Chukou Fault Hisnhu Fault Houchiali Fault Tsaochen Fault
28 6+ 12 10
Dextral Reverse w. dextral Reverse Reverse w. sinistral Reverse Dextral Reverse Sinistral
50–60°/E >60° >35°/W >60°/N
+ 3.2 ± 4.2 11.1 ± 3.4 3.1 + 8.8 ± 3.9 1.6 ± 4.2 − 5.0 + 3.5 ± 2.8 6.2 ± 3.8 7.9 − 1.4 ± 8.6 6.3 ± 7.9 –
7 17
verified by statistical procedures based on observed regional seismicity. To the authors' best knowledge, there has not yet a probability distribution in discussing the recurrence earthquake magnitude induced by a specific fault. The Monte Carlo Simulation (MCS) is a useful numerical approach for probabilistic analysis and it is widely used in geosciences. Based on limited paleoseismic records, Parsons (2008) used the MCS to establish the distribution of recurrence parameters for the strike-slip faults in California. Refice and Capolongo (2002) evaluated the earthquake-triggered landslide hazard in Italy in a probabilistic framework, in which the uncertainties of input parameters (e.g., geological, geotechnical) were taken into account by using the MCS. The United States Nuclear Regulatory Commission (2007) published a technical reference in which the MCS is used to consider the uncertainties of soil properties in a site amplification analysis. In this paper, we reviewed the latest publications regarding the active faults in Taiwan that have been investigated and published by the Central Geological Survey, Taiwan (CGST). It is found that the fault parameters such as return period and induced-earthquake magnitude are provided but their uncertainty is not specified. Therefore, the main objective in this study is to examine and demonstrate
Long-term Recurrence Last reactivated interval slip rate time (yr) (yr) (mm/yr)
Probable maximum earthquake (M)
–
14.3–20.7 –
199
b 18,540 B.P.
6.5
5.8 ± 2.3 – 13.6 ± 10.7 –
17.3–22.4 – – –
162 –
A.D.1906 A.D.1999
7.1 6.7
5.6 ± 2.0 5.6 ± 2.0
10.2–22.3 11.2 ± 4.0 10.2–22.3 11.2 ± 4.0
430 430
b 100,000 B.P. 6.0 b 10,000 B.P. 6.5
– – 5–6 –
5.3–16.7 4.2–13.6 5.2–9.5 –
1314 188 936 –
b 10,000 B.P. A.D.1946 b 100,000 B.P. b 100,000 B.P.
4.7 ± 8.9
9.3 ± 8.4 9.3 ± 8.4 3.8 ± 7.2 3.2 ± 8.9 − 0.4 ± 7.6 4.1 ± 7.9
– – – –
6.8 6.1 6.3 6.2
the impact of fault parameter uncertainties on earthquake recurrence probabilities using the proposed procedure in which the MCS was used. Using the approach, an example in Central Taiwan was demonstrated. 2. Geological investigations on active faults in Taiwan A strong earthquake could bring a horrific consequence toward the society and the rate of its recurrence in Taiwan is known to be high. The CGST has thoroughly begun the investigation on active faults on the island of Taiwan and updated the result with the latest findings periodically. In the latest publication, 33 faults were reported and their spatial distribution is shown in Fig. 1 (Lin et al., 2008, 2009). Table 1 summarizes the nine faults located in Southwestern Taiwan. The fault parameters reported by the CGST include the fault length, slip characteristics, dip angle and orientation of the fault plane, slip rate, return period, etc. The approaches used in the investigation include geological studies, historical records, subsurface geophysical investigation, geodetic measurement, and others. However, it is noted that the uncertainties of return period and earthquake magnitude are not provided except their best estimates. 3. Deterministic approach estimating earthquake recurrence probability
0.4
Recurrence Probability
(100, 0.365) 0.3
The recurrence probability of earthquakes can be estimated by using the Poisson sequence that is widely accepted to simulate the occurrence of rare events. Accordingly, the probability distribution of recurrence time follows the exponential distribution. Within a time interval t, the probability distribution function of the next occurrence is:
0.2
(50, 0.203)
−λt
P ðT≤t Þ ¼ 1−e
ð1Þ
0.1
where λ is the mean occurrence rate equal to (1/r), where r is return period. If a fault-induced earthquake has not occurred until a specific year (n) since last occurrence (c), the occurrence probability within next t-year becomes:
(10, 0.044) 0.0
(1, 0.005) 0
20
40
60
80
100
Period [Years] Fig. 2. Earthquake recurrence probabilities induced by the Meishan fault estimated by a deterministic approach.
! F ðt; n; c; r Þ ¼ P n−cbT≤t þ n−cjT > n−c
¼
Pðn−cbT≤t þ n−cÞ : 1−PðT≤n−cÞ ð2Þ
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Exceedance Probability
0.006
0.06
(a) 1-Year Era (2011)
0.004
0.04
0.002
0.02
0.000
0.00
6.0
6.5
7.0
7.5
6.0
8.0
0.2
0.4
0.1
0.2
0.0
0.0
6.5
7.0
7.5
6.5
7.0
7.5
8.0
(d) 100-Year Era (2011~2111)
(c) 50-Year Era (2011~2060)
6.0
(b) 10-Year Era (2011~2020)
8.0
Magnitude
6.0
6.5
7.0
7.5
8.0
Magnitude
Fig. 3. Exceedance probabilities of earthquakes induced the Meishan fault in four eras estimated by the deterministic approach.
Combing Eqs. (1) and (2), F ðt; n; c; r Þ ¼
−n−c r
e
−tþn−c r
−e n−c e− r
:
ð3Þ
Two implications must be noted in the derivation. First, Eq. (3) computes the probability regardless of the uncertainty of the inputs. Without the consideration of the uncertainty, it is regarded as a deterministic approach. Extra attention is noted that the deterministic approach simply refers a generic description on the computation and has no revelation to the deterministic seismic hazard analysis (DSHA) or others. Second, the recurrence probability presents the fault-induced earthquake associated with a specific magnitude. Therefore, the magnitude does not appear in the derivations. According to the CGST, the best-estimate return period and last occurrence of the Meishan fault are 162 years and in 1906, respectively. Fig. 2 shows the recurrence probabilities from a 1-year era to a 100-year era after 2010. Since the best-estimate earthquake magnitude is 7.1, the probabilities in fact present that a 7.1-magnitude earthquake recurs in different eras. For instance, the probabilities are 0.5%, 4.4%, 20.3%, and 36.5% considering 1-year, 10-year, 50-year and 100-year, respectively. However, it must be noted that the recurrence probability with earthquake magnitudes greater or less than 7.1 cannot be determined. Therefore, we introduced the exceedance probabilities to consider the recurrence probability associated with specific magnitudes of interest. Define P(M ≥ m*) as exceedance probability associated with a specific magnitude m*. The recurrence probability regardless of earthquake magnitude shown in Fig. 2 can be extended to exceedance probabilities corresponding to a specific magnitude as Fig. 3 shows. Take the Meishan fault for example, when m* less than 7.1,
P(M ≥ m*) is equal to the probability associated with magnitude = 7.1. In contrast, when m* is greater than 7.1, P(M ≥ m*) drops to zero in a sudden. Therefore, the exceedance probability distribution presents a discrete pattern using the deterministic approach. 4. Probabilistic analysis 4.1. Exceedance probability by MCS As shown in Figure 3, it is unrealistic that the exceedance probability of recurrence earthquakes features a sudden change around the best-estimate magnitude. This implies that the deterministic approach cannot provide a legitimate estimation for a broad magnitude range. In addition, the uncertainty of the parameters is not considered in the deterministic computation, which leads to another unrealistic expectation that the exceedance probabilities only rely on the best estimates and are independent of parameter uncertainties. Therefore, a probabilistic approach was used and a numerical procedure was proposed in this study. The procedure involved the use of the Monte Carlo simulation to generate random fault parameters with prescribed distributions. By changing COVs, the impact of uncertainty on the exceedance probability can be assessed. In the example of the Meishan fault, because the last occurrence was reported in 1906, the date was considered as a constant. Therefore, its uncertainty was not discussed in the following probability analyses. When n trials were performed in the probabilistic analysis, two probabilities 1) recurrence probability PR and 2) magnitude exceedance probability PM ≥ m* were computed based on randomly generated return periods (ri) and magnitudes (mi). Note that PR is continuous and PM ≥ m* is discrete with its value equal to either 1 or
J.-P. Wang et al. / Engineering Geology 126 (2012) 67–74
(a) Magnitude
~ Normal; Return Period ~ Normal
0.30
0. As a result, the probabilistic analysis estimates the exceedance probability (P^ M≥m ) associated with magnitude ≥ m* as:
Exceedance Probability
n P
0.25
5%-COV 10%-COV 20%-COV
0.20
0.10 0.05
6.0
6.5
7.0
7.5
8.0
Magnitude
(b) Magnitude
~ Lognormal; Return Period ~ Lognormal
0.30
Exceedance Probability
P^ M≥m ¼ i¼1
P R P M≥m n
:
ð4Þ
4.2. Probability models and results
0.15
0.00
0.25
5%-COV 10%-COV 20%-COV
0.20 0.15 0.10 0.05 0.00 6.0
6.5
7.0
7.5
8.0
Magnitude
(c) Magnitude
~ Lognormal; Return Period ~ Weibull
0.30
Exceedance Probability
71
0.25
5%-COV 10%-COV 20%-COV
0.20
The probability models simulating return period and earthquake magnitude need to be determined in the probability analysis using the MCS. We used a few combinations from common models (i.e., normal and lognormal) to suggested distributions to reveal and verify the expected impact. Fig. 4 shows the exceedance probabilities in 2011–2060 of the earthquake induced by the Meishan fault using (normal, normal), (lognormal, lognormal) and (Weibull, lognormal) simulating return period and magnitude, respectively. The sample size was 50,000 in the three analyses. A few uncertainty levels (5%, 10%, 20%) were assigned simultaneously to examine their impact on exceedance probabilities. Note that the best estimate level of uncertainty is unknown and the determination of its value is not the scope of the study. The results shown in Fig. 4 reveal that the proposed probability analysis generates a realistic, continuous distribution of exceedance probability. A few features in the exceedance probability were also observed in the probabilistic analysis. The exceedance probability reduced more quickly with the increase of magnitude when a low parameter uncertainty was considered. It was also observed that the three curves intersected around the best-estimate magnitude of 7.1, resulting in high exceedance probabilities associated with magnitude ≥ 7.1 when uncertainty was high. The variations in the three analyses considering different models were not conspicuous, showing that the model uncertainty did not have a significant impact on exceedance probabilities. The pattern of exceedance probability distributions can be explained by the fact that the magnitude uncertainty dominates the distributions. When magnitude uncertainty is low, random generated magnitudes concentrate near the best-estimate magnitude. This leads to the random magnitude exceedance probability PM ≥ m* equal to 1 in each MCS trail when m* is small. On the contrary, PM ≥ m* is equal to 0 when a large m* is considered. However, the random recurrence probabilities associated with random return periods are continuously distributed and the variations between MCS trials are less significant compared to PM ≥ m*. Due to the discrete nature of PM ≥ m*, the curve of exceedance probabilities estimated by the probabilistic approach features a gentle slope when a high uncertainty of parameters is considered.
0.15
4.3. Best-estimate exceedance probability distribution 0.10 0.05 0.00 6.0
6.5
7.0
7.5
8.0
Magnitude Fig. 4. Uncertainty impact on exceedance probabilities induced by the Meishan fault in 2011–2060 estimated by the probabilistic approach involved the use of Monte Carlo simulation. a) magnitude and return period both following the normal distribution, b) magnitude and return period both following the lognormal distribution, and c) magnitude and return period following the lognormal distribution and Weibull distribution, respectively.
According to the aforementioned three analyses, the impact of model uncertainty on exceedance probabilities is not significant. To ensure a more reliable estimation, the suggested probability models along with best-estimate fault parameters were used to obtain the best-estimate exceedance probability of earthquakes induced by the Meishan fault in Central Taiwan. The suggested models are the Weibull distribution and lognormal distribution for return period and earthquake magnitude, respectively. Fig. 5 shows 16 distributions of exceedance probabilities in 2010–2060 considering various combinations of magnitude and return-period uncertainties. At a given magnitude uncertainty, high return-period uncertainties result in high exceedance probabilities. Without further evidence supporting specific uncertainty levels inherited in return period and magnitude, a conservative perspective was adopted to determine the best-estimate exceedance probability.
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Exceedance Probability
0.35
(a) COV of Magnitude = 5%
0.35
0.30
0.30
0.25
0.25
0.20
0.20
0.15
0.15
COV of R.P. = 10% COV of R.P. = 20% COV of R.P. = 30% COV of R.P. = 40%
0.10 0.05
COV of R.P. = 10% COV of R.P. = 20% COV of R.P. = 30% COV of R.P. = 40%
0.10 0.05
0.00
0.00 6.0
0.35
(b) COV of Magnitude = 10%
6.5
7.0
7.5
8.0
(c) COV of Magnitude = 15%
6.0 0.35
0.30
0.30
0.25
0.25
0.20
0.20
0.15
0.15
COV of R.P. = 10% COV of R.P. = 20% COV of R.P. = 30% COV of R.P. = 40%
0.10 0.05
7.0
7.5
8.0
(d) COV of Magnitude = 20%
COV of R.P. = 10% COV of R.P. = 20% COV of R.P. = 30% COV of R.P. = 40%
0.10 0.05
0.00
6.5
0.00 6.0
6.5
7.0
7.5
8.0
6.0
6.5
Magnitude
7.0
7.5
8.0
Magnitude
Fig. 5. Distributions of exceedance probabilities induced by the Meishan fault in 2011–2060 with the considerations of magnitude and return-period uncertainties up to 20%-COV and 40%-COV, respectively.
Therefore, we used the envelope among various considerations as the best-estimate distribution of exceedance probability. Fig. 6 shows the distributions estimated by four probability analyses and by the deterministic approach along with the corresponding bestestimate distribution. Note that we considered the magnitude and return-period uncertainties up to 20%-COV and 40%-COV, respectively, to ensure a reliable and realistic estimation when specific uncertainties are not available. In such a case, the exceedance probabilities of earthquakes with magnitudes 6.5, 7.0 and 7.5 were 29%, 20% and 11%, respectively. Also note that the 11%-probability of a recurrence earthquake with magnitude ≥ 7.5 in 2011–2060 is higher than 5% estimated by using the distribution of annual maximum earthquakes for Central Taiwan (Wang et al., 2011).
distributions were still assigned for return period and earthquake magnitude, respectively, and their uncertainty levels were both set at 5%COV. The results showed that when dependency increased, the recurrence probabilities decreased in the entire range of magnitude but the differences were found insignificant. This numerical demonstration implied that 1) the dependency had little impact on earthquake exceedance probability, 2) a conservative estimate was actually rendered considering the independency between return period and earthquake magnitude. As a result, without a specific correlation that is supported, the best-estimate exceedance probability considering the independency is recommended.
0.35 0.30
The aforementioned analyses were performed based on the assumption that the return period and earthquake magnitude of the Meishan fault is statistically independent. However, a positive dependency between them could be more likely. However, there have not been specific studies available on the dependency between the return period and earthquake magnitude of the Meishan fault. In addition, it becomes another issue when a universal relationship, if any, is used without verifying its applicability on the fault of interest. Therefore, in order to examine the issue, we performed another series analysis using hypothetical correlation coefficients, in which correlated fault parameters were generated in the MCS. Fig. 7 shows the exceedance probabilities in 2011–2060 considering independent (ρ = 0) and dependent scenarios (ρ = 0.2 and ρ = 0.4). Note that in these analyses, the suggested Weibull and lognormal
Exceedance Probability
5. Discussion 5.1. Dependency of magnitude and return period
50-Year Era (2011 ~ 2060)
Best-Estimate Distribution of Exceedance Probability
0.25 0.20 0.15 0.10
COVM = 20% COVM = 15% COVM = 10%
0.05 Deterministic
0.00 6.0
6.5
7.0
7.5
COVM = 5%
8.0
Magnitude Fig. 6. Best-estimate exceedance probability induced by the Meishan fault in Central Taiwan during 2011–2060.
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6. Conclusion
Exceedance Probability
0.25
ρ=0 ρ = 0.2 ρ = 0.4
0.20
0.15
0.10
0.05
0.00 6.0
6.5
7.0
7.5
8.0
Magnitude Fig. 7. Illustration of the impact of dependency between magnitude and return period on exceedance probabilities.
5.2. Impact of the date of last occurrence In the example computations, the uncertainty of the date of last occurrence was not taken into account in the probabilities analysis because it was certain and considered as a constant. However, it is not usually the case that the date of last occurrence is well known. Under such a circumstance, the analysis similar to the study can still be performed by generating additional random last-occurrence dates along with random magnitudes and return periods. The procedure to estimate the exceedance probability remains the same and the impact of parameter uncertainties can be examined in an identical manner as illustrated in this study. However, it must be noted that in an attempt to use the probabilistic analysis to estimate the exceedance probability with the consideration of the last-occurrence uncertainty, suitable probability models for the random variable need to be evaluated in advance in order to obtain a reliable recurrence probability. 5.3. Generic applications featured in the proposed methodology It must be noted that the proposed approach is generic. It can be used in any given fault as its parameters are “available” either they are reliably estimated by legitimate methods and available evidences, or they are suggested by a subjective estimation. The advantage of the probabilistic approach over the deterministic approach includes a realistic estimation in exceedance probabilities for faultinduced earthquakes by considering parameter uncertainties. The more reliable information (e.g., fault parameters, uncertainty level, correlation between parameters, probability models) is available, the more reliable exceedance probabilities can be estimated. In the example, we used the best-estimate fault parameters and models in the referred publications to demonstrate the uncertainty impact and to provide a best-estimate in exceedance probabilities. The best-estimate uncertainty level and correlations are unavailable so that they are suggested subjectively and used to ensure proper conservatism in exceedance probabilities. Note that using hypothetical values does not affect the legitimacy of the analytical framework. We agree that exceedance probabilities estimated by the approach will become more reliable and convincible when all necessary, reliable inputs are available. However, the determination of best-estimate inputs is not the scope of the study and we believe that it is difficult that the task is accomplished without extensive studies. We suggest that the analysis is re-conducted to adjust the best-estimate exceedance probabilities with new information that becomes available.
This paper presents a numerical approach to estimate earthquake recurrence probabilities and to examine the impact of fault parameter uncertainties on them. A numerical example using the Meishan fault in Central Taiwan is demonstrated. The probabilistic analysis using the Monte Carlo Simulation shows that when a high uncertainty level is considered, the exceedance probabilities are high associated with large-magnitude earthquakes. The results also show that the impact of magnitude uncertainty is predominant in exceedance probability distributions. The best-estimate distribution of exceedance probabilities is proposed using the envelope of the curves estimated by both probabilistic and deterministic approaches. Considering the magnitude uncertainty up to 20%-COV and return period uncertainty up to 40%-COV, the exceedance probability is 11% considering an earthquake with magnitude greater than 7.5 induced by the Meishan fault in Central Taiwan. Acknowledgments We appreciate Prof. Ling at Columbia University, Prof. Tang and Prof. Tung at the Hong Kong University of Science and Technology for their constructive and insightful suggestions. In addition, the valuable comments from anonymous reviewers are greatly appreciated. References Abaimov, S.G., Turcotte, D.L., Shvherbakov, R., Rundle, J.B., Yakovlev, G., Goltz, C., Newman, W.I., 2008. Earthquakes: recurrence and interoccurrence times. Pure and Applied Geophysics 165, 777–795. Allen, R.M., Kanamori, H., 2003. The potential for earthquake early warning in Southern California. Science 300, 786–789. Bommer, J., 2003. Uncertainty about the uncertainty in seismic hazard analysis. Engineering Geology 70, 165–168. Bommer, J.J., Abrahamson, N.A., 2006. Why do modern probabilistic seismic-hazard analyses often lead to increased hazard estimates. Bulletin of the Seismological Society of America 96 (6), 1967–1977. Castanos, H., Lomnitz, C., 2002. PSHA: is it science? Engineering Geology 66, 315–317. Gan, Z.J., Tung, C.C., 1983. Extreme value distribution of earthquake magnitude. Physics of the Earth and Planetary Interiors 32, 325–330. Hagiwara, Y., 1974. Probability of earthquake occurrence as obtained from a Weibull distribution analysis of crustal strain. Tectonophysics 23 (3), 313–318. Horiuchi, S., Negishi, H., Abe, K., Kamimura, A., Fujinawa, Y., 2005. An automatic processing system for broadcasting earthquake alarms. Bulletin of the Seismological Society of America 95, 708–718. Iervolino, I., Convertito, V., Giorgio, M., Manfredi, G., Zollo, A., 2006. Real-time risk analysis for hybrid earthquake early warning systems. Journal of Earthquake Engineering 10, 867–885. Iervolino, I., Giorgio, M.C., Manfredi, G., 2007. Expected loss-based alarm threshold set for earthquake early warning systems. Earthquake Engineering and Structural Dynamics 36, 1151–1168. Iervolino, I., Giorgio, M., Galasso, C., Manfredi, G., 2009. Uncertainty in early warning predictions of engineering ground motion parameters: what really matters? Geophysical Research Letters 36, 1–6. Kanamori, H., Hauksson, E., Heaton, T., 1997. Real-time seismology and earthquake hazard mitigation. Nature 390, 461–464. Klugel, J.U., 2007. Comment on Why do modern probabilistic seismic-hazard analyses often lead to increased hazard estimates by Julian J. Bommer and Norman A. Abrahamson or how not to treat uncertainties in PSHA. Bulletin of the Seismological Society of America 97, 2198–2207. Lin, C.W., Lu, S.T., Shih, T.S., Lin, W.H., Liu, Y.C., Chen, P.T., 2008. Active faults of Central Taiwan. Special Publication of Central Geological Survey 21, 148 (In Chinese with English abstract). Lin, C.W., Chen, W.S., Liu, Y.C., Chen, P.T., 2009. Active faults of Eastern and Southern Taiwan. Special Publication of Central Geological Survey 23, 178 (In Chinese with English abstract). McGuire, R.K., Arabasz, W.J., 1990. An introduction to probabilistic seismic hazard analysis. In: Ward, S.H. (Ed.), Geotechnical and Geoenvironemtnal Geophysics: Society of Exploration Geo-physicists, Vol. 1, pp. 333–353. Nishenko, S.P., Buland, R., 1987. A generic recurrence interval distribution for earthquake forecasting. Bulletin of the Seismological Society of America 77, 1382–1399. Parsons, T., 2008. Monte Carlo method for determining earthquake recurrence parameters from short paleoseismic catalogs: example calculation for California. Journal of Geophysical Research 113, B03302. doi:10.1029/2007JB004998. Refice, A., Capolongo, D., 2002. Probabilistic modeling of uncertainties in earthquake induced landslide hazard assessment. Computers & Geosciences 28, 735–749. Rikitake, T., 1976. Recurrence of great earthquakes at subduction zones. Tectonophysics 35 (4), 335–362.
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J.-P. Wang et al. / Engineering Geology 126 (2012) 67–74
Rikitake, T., 1991. Assessment of earthquake hazard in the Tokyo area, Japan. Tectonophysics 199 (1), 121–131. U.S. Nuclear Regulatory Commission, 2007. A performance-based approach to define the site-specific earthquake ground motion. Regulatory Guide 1.208. Wang, Z., Zhou, M., 2007. Comment on Why do modern probabilistic seismic-hazard analyses often lead to increased hazard estimates by Julian J. Bommer and Norman A. Abrahamson. Bulletin of the Seismological Society of America 97 (6), 2212–2214. Wang, J.P., Chan, C.H., Wu, Y.M., 2011. The distribution of annual maximum earthquake magnitude around Taiwan and its application in estimation of catastrophic earthquake recurrence probability. Natural Hazard 59, 553–570.
Wu, Y.M., Kanamori, H., 2005. Rapid assessment of damaging potential of earthquakes in Taiwan from the beginning of P waves. Bulletin of the Seismological Society of America 95, 1181–1185. Wu, Y.M., Kanamori, H., Allen, R.M., Hauksson, E., 2007. Determination of earthquake early warning Paraameters, τc and Pd, for southern California. Geophysical Journal International 170, 711–717.