Nat Hazards
DOI 10.1007/s11069-014-1448-1
ORIGINAL PAPER
On the use of AFOSM to estimate major earthquake
probabilities in Taiwan
J. P. Wang • H. Kuo-Chen
Received: 26 May 2014 / Accepted: 15 September 2014
Springer Science+Business Media Dordrecht 2014
Abstract Advanced first-order second-moment (AFOSM) is commonly used to obtain an
upper-bound estimate for a probabilistic analysis. This study presents a new AFOSM
application to engineering seismology, estimating major earthquake probabilities based on
fault length and slip rate, along with an earthquake empirical model subject to a model
error of 0.26 Mw. The AFOSM analysis shows that the probability could be as high as 64 %
for a major earthquake in northern Taiwan to exceed Mw 7.0, considering the length and
slip rate of the Sanchiao fault are equal to 36 km and 2 ± 1 mm per year. By contrast, the
other case study shows that for the Meishan fault in central Taiwan, the probability is
‘‘only’’ 4 % for earthquake magnitude to exceed Mw 7.0, given a shorter fault length of
14 km and a larger slip rate of 6 ± 3 mm per year.
Keywords
AFOSM Earthquake probability Taiwan
1 Introduction
The region around Taiwan is known for high seismicity because of the unique tectonic
background. In average, there are around 2,000 earthquakes above ML 3.0 occurring in the
region every year, with major events such as the ML 7.3 Chi–Chi earthquake that could
recur in decades. Under the circumstance, a variety of earthquake studies focusing on the
region were conducted and reported, including earthquake early warning (Hsiao et al.
2011), seismic hazard analysis (Cheng et al. 2007; Wang and Huang 2014), and earthquake
J. P. Wang (&)
Department of Civil and Environmental Engineering, Hong Kong University of Science and
Technology (HKUST), Hong Kong, China
e-mail: [email protected]
H. Kuo-Chen
Department of Earth Sciences, National Central University, Taiwan, Republic of China
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Nat Hazards
statistics studies (Chen et al. 2013; Wang et al. 2014), among others (e.g., Sokolov et al.
2000; Campbell et al. 2002; Huang et al. 2007).
The uncertainty or the variability of a random variable could have a significant influence
on an analysis. In order to take such factors into account, more and more probabilistic
analyses were employed for a variety of topics (e.g., Bianchini and Hewage 2012; Castaldo
et al. 2013; Cho 2014; Zhang et al. 2014). However, because the analytical solution of
probabilistic analyses is usually unavailable, the alternatives such as Monte Carlo Simulation (MCS), first-order second-moment (FOSM), advanced first-order second-moment
(AFOSM), and the total-probability algorithm were commonly employed for a probabilistic analysis. For example, Probabilistic Seismic Hazard Analysis (PSHA) is on the basis
of using the total-probability algorithm to estimate the exceedance probability of earthquake motions, given the analytical solution is difficult to develop (Kramer 1996).
Understandably, each of the probabilistic approaches has its advantages and disadvantages.
For example, although it is relatively easy to perform a MCS, the simulation is relatively timeconsuming, and the results are usually associated with some analytical presumption. Take this
study for example, if we used MCS to estimate earthquake probabilities, we would have
assumed fault slip rate followed some kind of probability distributions (e.g., normal distribution, uniform distribution, etc.), because relevant studies on the topic are not reported.
By contrast, although the FOSM computation is less demanding than MCS, the alternative solution could be quite different from the analytical solution, when the random
variables of the probabilistic analysis follow an asymmetrical distribution (Ang and Tang
2007). To improve the situation, AFOSM was then developed on the basis of FOSM
(Hasofer and Lind 1974; Ayyub and Halder 1984), aiming to obtain a more reliable
estimate regardless of the variable’s probability distribution being symmetrical or asymmetrical. Besides, the estimates of AFOSM are usually considered an upper-bound value
owing to the optimization-based computation, making it more suitable for engineering
exercises with some conservatism added to the result (e.g., Kwak and Lee 1987; Ganji and
Jowkarshorijeh 2012).
Like other earthquake studies focusing on the high seismicity region, the key scope of this
study is to estimate major earthquake probabilities associated with two active faults in
Taiwan. Essentially, the probability estimates of this study were based on fault lengths, slip
rates, and an earthquake empirical model in the literature (Cheng et al. 2007; Anderson et al.
1996). As mentioned previously, since the probability distribution of fault slip rate is
unknown, MCS is not applicable to this study unless some assumption was made. Besides, in
order to obtain an upper-bound estimate, we therefore used AFOSM in this study, which is, to
the best of our knowledge, the first application of AFOSM to engineering seismology.
According to the AFOSM calculations, as a major earthquake is induced by the Sanchiao fault in northern Taiwan, the probability could be as high as 64 % for earthquake
magnitude to exceed Mw 7.0 (moment magnitude), given fault length of 36 km, slip rate of
2 ± 1 mm per year, and the error of the earthquake empirical model equal to ±0.26 Mw.
By contrast, the probability could be ‘‘only’’ 4 % for earthquake magnitude to exceed Mw
7.0 as far as the Meishan fault is concerned, given a shorter fault length of 14 km and a
larger slip rate of 6 ± 3 mm per year.
2 Background of the two active faults in Taiwan
The Sanchiao fault in northern Taiwan (Fig. 1a) is a normal fault with a length of around
40 km (e.g., Shyu et al. 2005; Huang et al. 2007). Recently, the evidence from seismic
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(a)
120.5
25.5
121.0
121.5
122.0
25.5
Sa
nc
hi
ao
tra
it
25.0
24.5
25.0
Taipei City
Ta
iwa
nS
Latitude ( 0 N)
fa
ul
t
Fig. 1 Locations of two active
faults in Taiwan: a the Sanchiao
fault, and b the Meishan fault
24.5
20 km
24.0
120.5
121.0
121.5
24.0
122.0
Longitude ( 0 E)
120.0
24.0
120.5
121.0
24.0
nS
iwa
Meishan fault
Ta
Latitude ( 0 N)
tra
it
(b)
23.5
23.0
120.0
Chaiyi City
20 km
120.5
23.5
23.0
121.0
Longitude ( 0 E)
survey suggested that the fault could further extend to the offshore regions in northern
Taiwan, meaning that the fault length could be longer than what we usually expected. More
importantly, because the fault is stretching along the west of Taipei City, the most
important city in Taiwan, the earthquake activity associated with the fault is therefore a
major concern to the local community.
Under the circumstance, some studies aiming to evaluate the earthquake potentials and
seismic hazards in Taipei City were reported from different perspectives. For example, by
examining the sediments along the fault line, Huang et al. (2007) considered that a few
major earthquakes could have been induced by the Sanchiao fault in the past 10,000 year
(or in Holocene), in addition to the event causing the Taipei Lake in the seventeenth
century owing to large ground subsidence at that time (Huang et al. 2007). Besides, Wang
et al. (2013a) conducted a PSHA for the city, including the recommendations of a few
earthquake time histories for earthquake-resistant design based on the seismic hazard
evaluated.
In 1906, an ML 7.1 (local magnitude) earthquake near Meishan Township in central
Taiwan was occurring, killing around 3,500 people at that time. (Owing to the location, the
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earthquake and the fault were therefore referred to as the Meishan earthquake and Meishan
fault.) Field investigations at that time found that the Meishan fault (Fig. 1b) is a strike-slip
fault with a length of around 14 km (Omori 1907). Recently, sophisticated geophysical
investigations indicated that the rupture zone in the region should be more extensive and
complex than expected (Shih et al. 2003).
Based on the report of the Central Geological Survey Taiwan, the return period of the
Meishan earthquake was considered only around 160 years, which implies that the event
could recur around year 2070 on a deterministic basis, and that the recurrence earthquake
probability in next 50 years could be around 30 % (Wang et al. 2012). To mitigate the high
earthquake risk around the area, new intrumentations were installed to monitor the
earthquake activity along the fault, and to study the characteristics of the earthquake
motions for engineering purposes (Wu et al. 2009). In addition, based on the best-estimate
earthquake magnitude and return period, Wang et al. (2013b) evaluated seismic hazard
associated with the Meishan fault, offering a more tangible reference for developing a
proper earthquake-resistant design for the region.
3 Overview of an earthquake empirical model
Like in other fields, a few earthquake empirical models were developed and used in
engineering seismology. A well-known example is in the study of Wells and Coppersmith
(1994), proposing a few empirical relationships between earthquake magnitude and fault
length, rupture area, etc.
In addition to those parameters, fault slip rate was also considered a plausible indicator
to earthquake magnitude (Kanamori and Allen 1986; Scholz et al. 1986). As a result,
Anderson et al. (1996) conducted a new regression study on historical earthquake data and
proposed an empirical relationship to estimate earthquake magnitude with fault length and
slip rate combined:
Mw ¼ 5:12 þ 1:16 log L 0:2 log S þ e
ð1Þ
where L is fault length in km, S is slip rate in mm per year, and e is the error of the
regression model owing to the randomness in the earthquake samples. According to the
regression study, the standard deviation of e in this empirical relationship is equal to 0.26
Mw. Besides, e is a random variable following the normal distribution with mean = 0,
based on the fundamentals of regression analysis (Ang and Tang 2007).
The application of this empirical model can be demonstrated with an example as follows: Given an active fault with L = 10 km and S = 1 mm per year, this empirical model
would suggest that earthquake magnitude should have a mean value and standard deviation
equal to Mw 6.28 and 0.26, and that the variable should follow the normal distribution.
4 Fault length and slip rate from the literature
As mentioned previously, a variety of earthquake studies focusing on the region around
Taiwan were conducted. In the seismic hazard study of Cheng et al. (2007), two hazard
maps were proposed with PSHA, with those input data such as fault length and slip rate
also summarized in the report. For example, the ‘‘deterministic’’ fault lengths of the
Sanchiao fault and Meishan fault were considered 36 km and 14 km, respectively. By
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contrast, the ‘‘probabilistic’’ estimates on slip rate were 2 ± 1 mm per year for the Sanchiao fault, and 6 ± 3 mm per year for the Meishan fault, with standard deviations also
reported to account for the uncertainty.
5 Performance function of the probabilistic analysis
Given that the best-estimate fault length is a ‘‘deterministic’’ estimate without its standard
deviation reported, the length was first substituted into Eq. 1 to develop the performance
function of this study. Taking the Sanchiao fault with L = 36 km for example, the performance function becomes:
Mw ¼ 6:93 0:2 log S þ e ;
for Sanchiao fault
ð2Þ
As a result, the performance function is governed by two random variable S and e, and
their mean values and standard deviations are available in the literature. But as mentioned
previously, because the probability distribution of fault slip rate S is unknown, we cannot
apply MCS to this study without making an assumption on it, which is one of the reasons
we used AFOSM in this study instead, aiming to make the result more transparent without
the assumption.
6 Overview of AFOSM
AFOSM starts with the Taylor expansion on the performance function. From the name of
the methodology, it is understood that the expansions up to the first order are only retained
in an AFOSM analysis. Take Z = f(X, Y) for example, the performance function can be
approximated as follows:
Z ¼ f ða; bÞ þ ðX aÞfX ða; bÞ þ ðY bÞfY ða; bÞ
ð3Þ
of
of
and oy
, respectively.
where a and b are the expansion points for X and Y; fx and fy denote ox
With a detailed derivation given in the ‘‘Appendix’’, the mean value of Z (denoted as lZ)
can be written as:
lZ ¼ f ða; bÞ þ fx ða; bÞ ðlX aÞ þ fy ða; bÞ ðlY bÞ
ð4Þ
where lX and lY denote the mean values of X and Y, respectively. Similarly, the standard
deviation of Z (denoted as rZ) can be written as follows:
2
ð5Þ
r2Z ¼ ½fx ða; bÞ2 r2X þ fy ða; bÞ r2Y
Note that Eq. 5 is on a condition that X and Y are independent of each other, or an
additional term accounting for their correlation will be added.
It is understood that the probability Pr (Z [ z*) is a function of mean value and standard
deviation of Z, and because both are a function of expansion points a and b, Pr (Z [ z*)
becomes a function of a and b as well, or expressed as follows:
PrðZ [ z Þ ¼ gðlZ ; rZ Þ ¼ hða; bÞ
ð6Þ
The next step of AFOSM is to search for the maximum value of Pr (Z [ z*) by varying the
two expansion points. In other words, AFOSM involves optimization in the computation,
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making the estimate usually more conservative than those from MCS, FOSM, etc. More
importantly, the optimization-based AFOSM can provide a reliable estimate even though
the variables of the probabilistic analysis are asymmetrical, a key improvement over
FOSM (Ayyub and Halder 1984).
Understandably, AFOSM is a generic framework, and it is applicable to any different
performance function f. However, the formulation of f will determine how to carry out the
optimization in a more effective manner. For example, when the performance function is in
a form of Z = nX ? m log Y (n and m are constants) like those in this study, the calculation
of Pr (Z [ z*) will be independent of the expansion point for X, because lZ and rZ are the
same regardless of the expansion point (Ayyub and Halder 1984). Realizing the nature of
the AFOSM computation, we can facilitate the optimization by only varying the expansion
point for Y, which is the expansion point for fault slip rate in this study.
As mentioned previously, given a regression model Y = f(Xis) ? e, Y is considered
following the normal distribution based on the fundamental of regression analysis. As a
result, on the use of the regression model shown in Eq. 1, Mw in this study is considered a
random variable following the normal distribution. Therefore, from Eq. 6, the governing
equation of this study in the estimating of magnitude exceedance probability Pr (M [ m*)
can be expressed as follows:
m lM
¼ hðbÞ
ð7Þ
PrðM [ m Þ ¼ 1 PrðM m Þ ¼ 1 U
rM
where U is the cumulative density function of a standard normal variate (i.e., mean = 0
and standard deviation = 1), and b is the expansion point for fault slip rate. As explained
previously, owing to the unique form of the performance function of this study, the
calculation of Pr (M [ m*) is only governed by the expansion point of fault slip rate, so
that the AFOSM calculation of this study is to search for the maximum value of
Pr (M [ m*) by trying different b values.
7 Major earthquake probability and the Sanchiao fault
Given fault slip rate S = 2 ± 1 mm per year and model error e = 0 ± 2.6 Mw in the
performance function (i.e., Eq. 2), Fig. 2 shows the relationship between exceedance
probability and earthquake magnitude for the Sanchiao fault in northern Taiwan.
Accordingly, the probability could be as high as 64 % for a major earthquake to exceed Mw
7.0, based on the fault length, slip rate, and the earthquake empirical model. By contrast,
the probability was reduced to 6 % for a major earthquake to exceed Mw 7.5, based on the
same input data with the AFOSM analysis.
Figure 3 shows the expansion points for fault slip rate that can maximize Pr (M [ m*)
in the AFOSM calculations. It demonstrates that in the six AFOSM calculations the
expansion point was decreasing almost linearly with the increase in earthquake magnitude.
8 Major earthquake probability and the Meishan fault
The same analysis was applied to another case study to estimate major earthquake probabilities associated with the Meishan fault in central Taiwan. Given fault length L = 14 km
and slip rate S = 6 ± 3 mm per year, Fig. 4 shows the relationship between exceedance
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0.7
Exceedance probability, Pr(M w > m*)
Fig. 2 Earthquake exceedance
probability associated with the
Sanchiao fault in northern
Taiwan based on the AFOSM
analysis
Performance function:
Mw = 5.12 + 1.16logL - 0.2logS + ε
0.6
0.5
Given:
L = 36 km
S = 2 +/- 1 mm per year
ε = +/- 0.26
0.4
0.3
0.2
0.1
0.0
7.0
7.1
7.2
7.3
7.4
7.5
m*
Performance function:
Mw = 5.12 + 1.16logL - 0.2logS + ε
11.0
Expansion points for slip rate S
Fig. 3 Expansion points for slip
rate S in the six AFOSM
computations for the case of the
Sanchiao fault
Given:
L = 36 km
S = 2 +/- 1 mm per year
ε = +/- 0.26
10.5
10.0
9.5
9.0
7.0
7.1
7.2
7.3
7.4
7.5
Mw
probabilities and major earthquakes for this case study with AFOSM. Accordingly, the
probability is ‘‘only’’ about 4 % for a major earthquake to exceed Mw 7.0 with a shorter fault
length and a larger slip rate. On the other hand, the AFOSM computations show that the
probability would converge to 2 % as earthquake magnitude increased, which is a common
feature in AFOSM governed by the optimization-based computation.
Figure 5 shows the expansion points for fault slip rate in the six AFOSM calculations
for the case of the Meishan fault. Unlike the previous case (i.e., Figure 3), the curve
appears a sudden change in the expansion points. To examine the cause, we reviewed the
whole optimization process in those calculations, and the observations were summarized as
follows. Take the calculation of Pr (Mw [ 7.0) for example, Fig. 6 shows each expansion
point and corresponding exceedance probability, indicating that the optimal expansion
point appeared around S = 26, although there was another ‘‘local’’ maximum point
appearing around S = 0.15. By contrast, Fig. 7 shows the AFOSM calculation of
Pr (Mw [ 7.1), demonstrating that the point S = 0.15, rather than S = 26, is associated
with the ultimate maximum probability in this case. As a result, the verification provided
support to the robustness of the AFOSM calculations shown in Figs. 4 and 5, owing to the
nature of the optimization-based analysis.
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Exceedance probability, Pr(M w > m*)
Nat Hazards
0.036
0.034
Performance function:
Mw = 5.12 + 1.16logL - 0.2logS + ε
0.032
Given:
L = 14 km
S = 6 +/- 3 mm per year
ε = +/- 0.26
0.030
0.028
0.026
0.024
0.022
7.0
7.1
7.2
7.3
7.4
7.5
m*
Fig. 4 Earthquake exceedance probability associated with the Meishan fault in central Taiwan based on the
AFOSM analysis
Expansion points for slip rate S
30
25
Performance function:
Mw = 5.12 + 1.16logL - 0.2logS + ε
20
Given:
L = 14 km
S = 6 +/- 3 mm per year
ε = +/- 0.26
15
10
5
0
7.0
7.1
7.2
7.3
7.4
7.5
Mw
Fig. 5 Expansion points for slip rate S in the six AFOSM computations for the case of the Meishan fault
9 Discussions
With fault length, slip rate, and the earthquake empirical model, the AFOSM study shows
that the Sanchiao fault in northern Taiwan should pose a higher earthquake risk than the
Meishan fault in central Taiwan, in terms of the magnitude of the recurring earthquake.
Understandably, the result of the study can be further integrated with other information
(e.g., earthquake return period and population at risk) to further quantify earthquake risk,
considering earthquake recurrence probability in a given period of time, the location of
faults (earthquake consequence), etc. For example, although earthquake magnitude and
population at risk associated with the Sanchiao fault in northern Taiwan are larger than
those of the Meishan fault in central Taiwan, the earthquake risk of the Sanchiao fault is
not necessarily higher than the Meishan fault, considering that the Meishan earthquake
should have a much shorter return period than the Sanchiao earthquake (Wang et al. 2012).
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Ultimate maximum point, the AFOSM solution
Fig. 6 Result of optimization in
the AFOSM calculation of
Pr (Mw [ 7.0) for the case of the
Meishan fault
0.035
Pr(M w > 7.0)
0.030
0.025
A local maximum point
0.0232
0.020
0.0230
0.015
0.0228
0.010
0.0
0
20
0.1
0.2
40
0.3
60
80
100
Expansion point for S
Fig. 7 Result of optimization in
the AFOSM calculation of
Pr (Mw [ 7.1) for the case of the
Meishan fault
Ultimate maximum point,
the AFOSM solution
0.025
0.0228
0.0224
Pr(Mw > 7.1)
0.020
0.0220
A local maximum point
0.015
0.0
0.1
0.2
0.3
0.010
0.005
0
20
40
60
80
100
Expansion point for S
As mentioned previously, because there is no study providing scientific references
regarding the probability distribution of fault slip rate, MCS is not applicable to this study
without making an assumption on that. In addition, another reason of using AFOSM in this
study is to obtain an upper-bound probability estimate, with some more conservatism
added to the result to compensate some uncertainty in the characterizing of the input data,
especially the mean and standard deviation of fault slip rate.
10 Conclusions
The Sanchiao fault in northern Taiwan and the Meishan fault in central Taiwan could be in
high earthquake risk. For the seismic safety of the study areas, this study presents a novel
probabilistic study using AFOSM to estimate major earthquake probabilities based on fault
length and slip rate, along with an earthquake empirical model.
Given the Sanchiao fault with length = 36 km and slip rate = 2 ± 1 mm per year, the
probability could be as high as 64 % for a major earthquake to exceed Mw 7.0 when recurring.
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By contrast, the other analysis shows that for the Meishan fault in central Taiwan, the
probability is ‘‘only’’ about 4 % for a major earthquake to exceed Mw 7.0, given a shorter fault
length = 14 km and a larger slip rate = 6 ± 3 mm per year. Like many other earthquake
studies focusing on the high seismicity region, the probability estimates of this study from a
different perspective can offer a new reference to earthquake risk around the study areas, for
developing proper measures to assure the seismic safety of the local community.
Appendix
Derivations on mean value and standard deviation of Z = f(X, Y).
For a function Z = f(X, Y), its Taylor expansion up to the first-order terms can be
expressed as follows:
Z ¼ f ða; bÞ þ ðX aÞfX ða; bÞ þ ðY bÞfY ða; bÞ
ðA:1Þ
where constants a and b are the expansion points for X and Y, respectively. As a result, the
mean value of Z, denoted as E[Z] or lZ, can be approximated as follows:
E½Z ¼ lZ ¼ E½f ða; bÞ þ ðX aÞfX ða; bÞ þ ðY bÞfY ða; bÞ
¼ E½f ða; bÞ þ E½ðX aÞfX ða; bÞ þ E½ðY bÞfY ða; bÞ
¼ f ða; bÞ þ fX ða; bÞE½X a þ fY ða; bÞE½Y b
¼ f ða; bÞ þ fX ða; bÞðlX aÞ þ fY ða; bÞðlY bÞ
ðA:2Þ
of
of
As mentioned previously, fx and fy denote ox
and oy
, respectively. Understandably, this
derivation is on the basis that the mean value of a constant [e.g., f(a, b), fX(a, b), …] is the
value itself (Ang and Tang 2007).
On the other hand, the variance of Z, denoted as V[Z] or r2Z (r is standard deviation), can
be approximated as follows:
V½Z ¼ r2Z ¼ V½f ða; bÞ þ ðX aÞfX ða; bÞ þ ðY bÞfY ða; bÞ
¼ V½ðX aÞfX ða; bÞ þ ðY bÞfY ða; bÞ
¼ ðfX ða; bÞÞ2 V½X a þ ðfY ða; bÞÞ2 V½Y b
2
ðA:3Þ
2
¼ ðfX ða; bÞÞ V½X þ ðfY ða; bÞÞ V½Y
¼ ðfX ða; bÞÞ2 r2X þ ðfY ða; bÞÞ2 r2Y
Similarly, the derivation is on the basis that the variance of a constant is equal to zero, and
X and Y are two variables independent of each other.
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