Nat Hazards DOI 10.1007/s11069-015-1988-z ORIGINAL PAPER
Seismic hazard estimation: directly using observations versus applying seismic hazard model T. J. Liu1 • A. Pozos-Estrada2 • R. Gomez2 • H. P. Hong1
Received: 11 February 2015 / Accepted: 17 September 2015 Ó Springer Science+Business Media Dordrecht 2015
Abstract Seismic hazard evaluation for a site can be carried out based on direct observations such as ground motion records at the site or based on an adopted seismic hazard model, including the distribution of earthquake intensity, the spatio-temporal occurrence model of earthquakes in a region, and the ground motion prediction equations. The former is adequate if sufficient records at the site are available, while the latter is capable to incorporate information from similar tectonic regions. This study compares the seismic hazard estimated using these two approaches for a recording site in Mexico City. The comparison is carried out in terms of uniform hazard spectra by considering interplate earthquakes alone, inslab earthquakes alone, and both interplate and inslab earthquakes. Up to date ground motion records obtained at the site are employed for the analysis. The results showed quantitatively the differences between the uniform hazard spectrum (UHS) obtained by the two approaches and the differences between the estimated UHS and the design spectrum, which is important for seismic risk assessment and design code making. The results are also used to identify the differences between the standard deviation of the residuals for GMPEs developed using records from a single station and that obtained considering a broad network of stations. Keywords
Ground motion prediction equation Seismic hazard Distribution fitting
& H. P. Hong
[email protected] 1
Department of Civil and Environmental Engineering, University of Western Ontario, London, ON N6A 5B9, Canada
2
Institute of Engineering, Universidad Nacional Auto´noma de Me´xico, Ciudad Universitaria, 04510 Mexico, D.F., Mexico
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Nat Hazards
1 Introduction Seismic hazard evaluation is frequently carried out using the procedure proposed by Cornell (1968) and Esteva (1968) for an adopted seismic hazard model. The seismic hazard model includes the distribution of earthquake intensity, the spatio-temporal occurrence of earthquakes in a region, and the ground motion prediction equations (GMPEs) (or attenuation relations). The distribution of the earthquake intensity is described by the magnitude-recurrence modeling; the spatio-temporal earthquake occurrence model specifies the seismic source zones in space and the temporal earthquake occurrence process. They are inferred from historical seismic events, and attention is needed to judiciously define the (boundary of) seismic source zones, which smooth the observed irregular spatial occurrence pattern of historical seismic events into a pattern that is characterized by a probabilistic model (Frankel 1995; Woo 1996; Hong et al. 2006). The ground motion records are used to develop GMPEs for ground motion measures such as peak ground acceleration (PGA) and spectral acceleration (SA) (Esteva 1968; Douglas 2011). If there are sufficient observed strong ground motion records at a site of interest, the direct use of the observations and the application of extreme value theory could be made to estimate the seismic hazard (Milne and Davenport 1969). This avoids the need to make some of the required assumptions on the seismic hazard model to evaluate the seismic hazard. Since the available ground motion records at the Ciudad Universitaria (CU) station (i.e., five stations around 19.33°N, 99.183°W) in Mexico City are rich as compared to other sites, Ordaz and Reyes (1999) exploited this richness by comparing the observed exceedance rate of SA to that calculated based on an adopted seismic hazard model for Mexican interplate earthquakes. They showed that the calculated exceedance rate is greater than that observed for moderate SA but the difference is not very large, and that in general the computed exceedance rates agree reasonably well with the data. These conclusions provide a verification of and support to the use of the seismic hazard model to evaluate the seismic hazard, at least for the CU station in Mexico City and Mexican interplate earthquakes. There are, however, indications that seismic hazard at the site is also affected by Mexican inslab earthquakes (Jaimes and Reinoso 2006; Singh et al. 2015). Additional discussion on the value of using the field observation is given by Beauval et al. (2008) for seismic hazard assessment and by Crowley et al. (2008) for seismic loss modeling. The main objective of this study is to compare the seismic hazard estimated by applying a seismic hazard model and by directly using observations at the CU station. The comparison considered the interplate as well as the inslab earthquakes, and uses ground motion records up to 2012 at the site. The analysis is carried out by considering interplate earthquakes alone, inslab earthquakes alone, and both interplate and inslab earthquakes. Observations on the statistics of the residuals of the GMPEs developed for a single site versus those developed based on a broad network of stations are also made.
2 Estimating seismic hazard by directly using observations The available ground motion records for the interplate and inslab earthquakes at the CU station in Mexico City are rich as compared to other sites. Based on a series of studies on the Mexican interplate and inslab earthquakes (Ordaz and Reyes 1999; Jaimes and Reinoso 2006; Arroyo et al. 2010; Singh et al. 2015) and the available Mexican ground motion record database, two sets of events that are recorded at CU station are summarized in
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Nat Hazards Table 1 Mexican interplate earthquakes with ground motion record available at CU station Date
M
H (km)
Lat. N.
23/08/1965
7.8
33
15.38
03/02/1968
5.6
33
16.37
02/08/1968
7.4
33
19/03/1978
6.4
29/11/1978
Long. W.
Date
M
H (km)
Lat. N.
Long. W.
96.12
19/09/1985
8.0
15
18.08
102.94
99.40
21/09/1985
7.6
15
18.02
101.48
16.07
98.10
30/04/1986
7.0
20
18.02
103.06
36
17.03
99.76
25/04/1989
6.9
19
16.60
99.40
7.8
19
16.00
96.69
31/05/1990
6.1
16
17.11
100.89
14/03/1979
7.6
28
17.49
101.26
24/10/1993
6.7
19
16.54
98.98
25/10/1981
7.3
14
17.88
102.15
10/12/1994
6.6
20
18.02
101.56
07/06/1982
6.9
18
16.17
98.36
14/09/1995
7.3
22
16.31
98.88
07/06/1982
7.0
24
16.26
98.51
20/03/2012
7.4
16
16.25
98.52
Table 1 for interplate earthquakes and in Table 2 for inslab earthquakes. In some cases, records at several recording sites within CU are obtained for the same seismic event. In such a case, only the record from the ‘‘backyard’’ station at CU is considered; this is consistent with the extreme value analysis approach since each record must represent a single event. Also, events with records at CU station whose associated ground motion measures are small are neglected. Table 1 shows a list of 18 interplate earthquakes with focal depth, H, ranging from 14 to 36 km and moment magnitude, M, varying from 5.6 to 8.0. The epicentral distance to the CU station ranges from about 280 to 470 km. Table 2 shows a list of 18 recorded inslab events with H ranging from 35 to 70 km and M varying from 5.2 to 7.4. The epicentral distance to the CU station ranges from 140 to 440 km. The events shown in Tables 1 and 2 represent major Mexican interplate and inslab earthquakes in recent history. The 1973 Orizaba (inslab) earthquake is not included in Table 2 since it was not recorded at the CU station (Singh et al. 2015). The annual occurrence rate of interplate earthquake calculated based on the events shown in Table 1 is 0.3651 (/year), while that of inslab earthquake calculated based on Table 2 (including Orizaba earthquake) is 0.3796 (/year). The records at the CU station as well as source parameters for the events shown in Tables 1 and 2 are obtained from the Mexican strong ground motion database and the Department of Seismic Instrumentation, Institute of Engineering, UNAM. The records are processed by applying a baseline correction (Boore 2005) and a high-pass filter with corner
Table 2 Mexican inslab earthquakes with ground motion record available at CU station Date
M
H (km)
Lat. N.
Long. W.
Date
M
H (km)
Lat. N.
Long. W.
06/07/1964
6.7
55
18.03
100.77
03/04/1997
5.2
52
18.51
98.10
07/06/1976
6.4
57
17.09
100.85
22/05/1997
6.5
54
18.37
101.82
24/10/1980
7.0
70
18.03
98.29
20/04/1998
5.9
60
18.35
101.19
05/08/1993
5.2
54
17.43
98.337
15/06/1999
6.9
61
18.13
97.54
23/02/1994
5.8
75
17.75
97.27
21/06/1999
6.3
53
18.15
101.72
06/05/1994
5.2
57
18.39
97.98
30/09/1999
7.4
47
16.03
96.96
23/05/1994
6.2
50
18.02
100.57
21/07/2000
5.9
50
18.11
98.97
10/12/1994
6.4
50
17.98
101.52
22/05/2009
5.7
45
18.13
98.44
11/01/1997
7.1
35
18.37
101.82
11/12/2011
6.5
58
17.84
99.98
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Nat Hazards
1
2
3
40
20
0 0
4
1
M = 7.8
30
20
10
1
2
3
60
40
20
1
M=7
30
20
10
3
50
100 80 60 40 20 2
3
30
20
10
1
2
3
M = 6.1 15
10
5
1
2
3
10
1
4
7/06/1982 M = 6.9
40
20
0 0
4
1
2
3
4
Tn (s) 21/09/1985 M = 7.6
30
20
10
1
2
3
30/04/1986 M=7 15
10
5
0 0
4
1
2
3
4
Tn (s) 20
24/10/1993 M = 6.7 15
10
5
1
2
3
4
10/12/1994 M = 6.6 15
10
5
0 0
1
2
3
4
Tn (s)
80
14/09/1995
Spectral acceleration (Gal)
Spectral acceleration (Gal)
3
Tn (s)
50
3
20
Tn (s)
M = 7.3
40
30
20
10
0 0
2
40
0 0
4
2
60
Tn (s) 31/05/1990
0 0
4
1
80 M = 7.3
20
Tn (s)
5
Tn (s)
20
0 0
4
Spectral acceleration (Gal)
M = 6.9
10
Tn (s)
M=8
1
M = 6.4
0 0
4
30
0 0
4
20
25/04/1989
3
25/10/1981 40
Tn (s)
Spectral acceleration (Gal)
Spectral acceleration (Gal)
3
19/09/1985 120
0 0
4
2
50
Tn (s)
0 0
2
Spectral acceleration (Gal)
Spectral acceleration (Gal)
Spectral acceleration (Gal)
7/06/1982
40
1
Tn (s) 140
2
10
50
M = 7.6
0 0
4
50
1
20
19/03/1978 15
Tn (s) 14/03/1979
Tn (s)
40
30
0 0
4
Spectral acceleration (Gal)
29/11/1978 40
0 0
3
80
50
0 0
2
M = 5.6
Tn (s)
Spectral acceleration (Gal)
Spectral acceleration (Gal)
Tn (s)
Spectral acceleration (Gal)
60
20
3/02/1968 40
Spectral acceleration (Gal)
10
M = 7.4
Spectral acceleration (Gal)
20
2/08/1968
Spectral acceleration (Gal)
M = 7.8
30
0 0
50
80 23/08/1965
40
Spectral acceleration (Gal)
50
Spectral acceleration (Gal)
Spectral acceleration (Gal)
frequency of 0.05 Hz for events with M [ 6.5 and 0.1 Hz for the remaining events. The spectral acceleration (SA) for the natural vibration period Tn up to 4.0 s is calculated. These values are shown in Figs. 1 and 2. It can be observed from Fig. 1 that there is a significant peak for the vibration period Tn near 2 s for several interplate records. In general, the plots give the impression that the ordinate of the SA is significant for Tn greater than about 0.25 s and that the value of the SA for several records does not decrease
1
2
Tn (s)
3
4
20/03/2012 M = 7.4 60
40
20
0 0
1
2
3
4
Tn (s)
Fig. 1 Spectral acceleration for Mexican interplate events recorded at the CU station (damping ratio n = 5 %). The solid line shows the geometric mean, while the dashed and dotted lines represent the values of SA along the recording orientations
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Nat Hazards
10
3
10
0 0
4
1
Tn (s)
Spectral acceleration (Gal)
Spectral acceleration (Gal)
23/02/1994 M = 5.8
6
4
2
0 0
1
2
3
Spectral acceleration (Gal)
Spectral acceleration (Gal)
11/01/1997 M = 7.1
10
5
06/05/1994 M = 5.2
6
4
2
1
3
Spectral acceleration (Gal)
Spectral acceleration (Gal)
15/06/1999 M = 6.9
30
20
10 1
2
3
20
1
3
M = 5.2
2
1
2
3
15
10
5
2
M = 6.2
10
5.0
1
3
2
3
3
4
10/12/1994 M = 6.4 15
10
5
0 0
4
1
2
3
4
Tn (s) 22/05/1997 M = 6.5
8
6
4
2
1
2
3
20/04/1998 M = 5.9
8
6
4
2
0 0
4
1
2
3
4
Tn (s) 50 30/09/1999 M = 7.4
40
30
20
10 0
4
2
10
1
2
3
4
Tn (s)
Tn (s) 50
21/07/2000 M = 5.9
40
30
20
10
0 0
1
2
3
4
Tn (s)
100
22/05/2009
Spectral acceleration (Gal)
Spectral acceleration (Gal)
1
Tn (s)
50
M = 6.3
1
2
Tn (s) 21/06/1999
Tn (s)
4
20 23/05/1994
0 0
4
Tn (s)
20
6
0 0
4
10
4
M = 5.7
40
30
20
10
0 0
3
M = 5.2
Tn (s)
6
0 0
4
2
15
0 0
4
03/04/1997 8
0 0
4
50
0
2
10
Tn (s)
40
40
05/08/1993 8
Tn (s)
Tn (s)
15
2
60
20
8
0 0
4
20
1
4
10
Tn (s)
0 0
3
M = 7.0
Tn (s)
10
8
2
10 24/10/1980
0 0
Spectral acceleration (Gal)
2
20
Spectral acceleration (Gal)
1
30
Spectral acceleration (Gal)
0 0
M = 6.4
Spectral acceleration (Gal)
20
07/06/1976 40
Spectral acceleration (Gal)
30
80
50
Spectral acceleration (Gal)
M = 6.7
Spectral acceleration (Gal)
06/07/1964 40
Spectral acceleration (Gal)
Spectral acceleration (Gal)
Spectral acceleration (Gal)
50
1
2
Tn (s)
3
4
11/12/2011 M = 6.5
80
60
40
20
0 0
1
2
3
4
Tn (s)
Fig. 2 Spectral acceleration for Mexican inslab events recorded at the CU station (damping ratio n = 5 %). The solid line shows the geometric mean, while the dashed and dotted lines represent the values of SA along the recording orientations
very quickly as Tn increases beyond 2 s. Except for one record, the maximum of the SA for Tn up to 4 s for each of the considered interplate earthquake records is greater than 10 Gal. However, unlike the case for the interplate earthquake records, the SA values for some of the inslab earthquake records are very pronounced for Tn near 0.25 s. This is consistent with findings from previous studies (Garcı´a et al. 2005; Garcı´a 2006). The overriding impression is that the SA value decreases quickly as Tn increases beyond 1 s. The maximum of the SA for several considered records is less than 10 Gal, while the maximum SA for all the considered inslab earthquake records is about 90 Gal.
123
Nat Hazards
To estimate the seismic hazard in terms of return period values, and to provide a probabilistic characterization of the annual maximum SA, three cases are considered: Case 1: the interplate earthquakes alone; Case 2: the inslab earthquakes alone; and Case 3: both the interplate and inslab earthquakes. For the probabilistic analysis, it is considered that the earthquake occurrence (for interplate or for inslab earthquakes) can be modeled as a Poisson process and that the SA of the seismic events, y, is distributed according to the generalized Pareto distribution (GPD) or the lognormal distribution. The GPD is often used for extreme value analysis based on the method of peak over threshold. The lognormal distribution is considered because the logarithm of the SA is commonly considered as a normal variate (i.e., the residuals of the GMPE are considered to be normally distributed). If y follows the GPD, the cumulative distribution function of y, F(y), is given by (Coles 2001), FðyÞ ¼ 1 ð1 kðy uÞ=aÞ1=k ;
ð1Þ
where a is the scale parameter and k is the shape parameter, and the distribution is for y over a specified threshold, u. Note that if k [ 0, u B y B u ? a/k. If y is lognormally distributed, F(y) is given by, FðyÞ ¼ U
^ ln y ln my =rln y ;
ð2Þ ^
where U() is the standard normal distribution function, and my and rln y are model parameters. The probability that the annual maximum of the SA is less than or equal to y, FA(y), is given by (Davison and Smith 1990; Coles 2001), FA ðyÞ ¼ expðcA ð1 FðyÞÞÞ;
ð3Þ
where cA is the annual occurrence rate. The T-year return period value of SA, yT, can be estimated by solving FA ðyT Þ ¼ 1 1=T. First, consider applying Eqs. (1) and (3) to Case 1 and Case 2. The maximum likelihood method is used to fit Eq. (1) considering u = 0, a small positive u value, or only using records whose maximum SA (for all vibration periods) is greater than 10 Gal. It is observed that in many cases, the fitted Eq. (1) is associated with a small upper bound, especially for inslab earthquakes. This resulted in very narrow differences between the estimated uniform hazard spectrum or spectra (UHS) for T ranging from 125 to 2475 years. Since such narrow differences are not observed from other studies (Ordaz and Reyes 1999; Jaimes and Reinoso 2006; Garcı´a-Soto et al. 2012), the use of GPD [Eq. (1)] is not considered further. By considering that SA is lognormally distributed [Eq. (3)], the distribution fitting is carried out using the method of maximum likelihood. Samples of the fitted distributions are illustrated in Fig. 3a, b. The estimated 125-, 475-, and 2475-year return period values of SA by using the fitted distribution and Eq. (3) are shown in Fig. 3c, d and are compared with the design spectrum given in NTCDS (2004). It can be observed that the estimated UHS by considering each earthquake type alone is not very smooth and that the increase in UHS is substantial for an increased return period. Moreover, there is an increase in the 2475-year UHS for inslab earthquakes as Tn increases (for Tn between 1.5 and about 2.3 s).
123
Nat Hazards
(b)
(a) SA at 0.2 s SA at 0.2 s fitted SA at 1.0 s SA at 1.0 s fitted SA at 3.0 s SA at 3.0 s fitted
Probability
0.95 0.9 0.75
SA at 0.2 s SA at 0.2 s fitted SA at 1.0 s SA at 1.0 s fitted SA at 3.0 s SA at 3.0 s fitted
0.995 0.99 0.95 0.9 0.75
Probability
0.995 0.99
0.5 0.25
0.5 0.25
0.1 0.05
0.1 0.05
0.01 0.005 10
0.01 0.005
Interplate 0
10
1
10
2
Inslab
10-2
10-1
Spectral acceleration (Gal)
(c)500
300 200 100 0 0
0.5
1
1.5
2
Tn (s)
2.5
3
102
3.5
NTCDS (2004) T = 125 years T = 475 years T = 2475 years
Inslab
Spectral acceleration (Gal)
Spectral acceleration (Gal)
400
101
(d) 500
NTCDS (2004) T = 125 years T = 475 years T = 2475 years
Interplate
100
Spectral acceleration (Gal)
4
400 300 200 100 0 0
0.5
1
1.5
2
2.5
3
3.5
4
Tn (s)
Fig. 3 Fitted distribution for SA and estimated UHS using Eqs. (2) and (3): a fitted distribution for SA using records from interplate earthquakes shown in Table 1, b fitted distribution for SA using records from inslab earthquakes shown in Table 2, c UHS for interplate earthquakes, and d UHS for inslab earthquakes
This increasing trend, which differs from the trends for the design spectrum and the SA shown in Fig. 2, is caused by inadequate fitting or because of the inclusions of the records associated with relatively small maximum SA, which can be appreciated in Fig. 3b. To improve the fitting, one could ignore the records associated with relatively small maximum SA. For example, by considering the 12 records whose maximum SA (for all vibration periods) is greater than 10 Gal (i.e., ignoring events 05/08/1993, 23/02/1994, 06/05/1994, 03/04/1997, 22/05/1997, and 20/04/1998 shown in Table 2, whose response spectra are shown in Fig. 2), and carrying out the distribution fitting, the resulting fitted distribution for SA, F(y), is shown in Fig. 4a. Comparison of Figs. 3b and 4a indicates that the fitting is improved in the upper tail of the distribution, especially for Tn = 0.2 and 1 s. Note that the fitting for Tn = 3 s can be further improved by ignoring the records that lead to small SA. However, this is not pursued as it will be seen that the seismic hazard for return period ranging from 125 to 2475 years for the CU station is dominated by the interplate earthquakes for 1.5 s \ Tn \ 3.5 s. The estimated UHS using the fitted distribution depicted in Fig. 4a is shown in Fig. 4b, indicating a decreasing trend of UHS for an increased Tn if Tn is greater than about 1.5 s, except for Tn around 3.5 s. This shows the importance in selecting appropriate records and the probabilistic model in modeling SA to develop the UHS.
123
Nat Hazards
(a)
0.75
500
Spectral acceleration (Gal)
0.95 0.9
Probability
(b)
SA at 0.2 s SA at 0.2 s fitted SA at 1.0 s SA at 1.0 s fitted SA at 3.0 s SA at 3.0 s fitted
0.995 0.99
0.5 0.25 0.1 0.05 0.01 0.005
NTCDS (2004) T = 125 years T = 475 years T = 2475 years
Inslab 12 records 400
300
200
100
12 Inslab records -2
10
-1
10
0
10
1
10
2
10
0 0
0.5
1
1.5
Spectral acceleration (Gal)
2
Tn (s)
2.5
3
3.5
4
Fig. 4 Samples of fitted distribution for SA and estimated UHS considering 12 inslab records listed in Table 2: a fitted distribution, b estimated UHS 400 Interplate + Inslab
Spectral acceleration (Gal)
Fig. 5 Estimated UHS considering both the interplate and inslab earthquakes
NTCDS (2004) T = 125 years T = 475 years T = 2475 years
300
200
100
0
0
1
2
3
4
Tn (s)
For Case 3, it is considered that the occurrence of interplate earthquakes is independent of that of inslab earthquakes. In such a case, yT by considering both interplate and inslab earthquakes for large T (small probability of exceedance) can be obtained by solving 1 1=T ¼ ðFA ðyT ÞÞInt ðFA ðyT ÞÞIns
ð4Þ
where the subscripts Int and Ins denote the probability distribution of the annual maximum SA due to interplate earthquakes alone and due to inslab earthquakes alone, respectively. By using the distribution function leading to results shown in Figs. 3c and 4b, and solving Eq. (4), the estimated UHS for Case 3 is presented in Fig. 5 and compared with the design spectrum given in NTCDS (2004). It can be observed from this plot that the ordinates of the design spectrum envelop the UHS for T = 125 years; for T = 475 years, they follow closely to the UHS for Tn \ 0.25 and are lower than the UHS for Tn [ 1.75 s. For Tn within 0.3 and 1.5 s, the ordinates of the design spectrum are greater than those of the UHS for T = 125 and 475 years and are approximately equal to those of the UHS for T = 2475 years. This shows that the recommended design spectrum corresponds to different return period values of SA depending on the natural vibration period.
123
Nat Hazards
3 Estimating seismic hazard by applying seismic hazard model Rather than estimating the seismic hazard directly using observations, the estimation of seismic hazard is most often carried out using a seismic hazard model defined by the distribution of the intensity of earthquakes, the spatio-temporal occurrence of earthquakes in a region, and the ground motion prediction equations (Cornell 1968; Esteva 1968). The occurrence of earthquakes that affect Mexico City is often modeled as a homogeneous Poisson process (Esteva 1968), although non-Poissonian modeling was also investigated by Singh et al. (1983) and Hong and Rosenblueth (1988). Given the earthquake occurrence, the probability distribution of the earthquake magnitude could be modeled using exponential or truncated exponential distributions or modeled with a distribution including the quasi-periodic occurrence of characteristic earthquakes, especially for Mexican subduction earthquakes (Singh et al. 1983). The seismic hazard model adopted in this study for evaluating the seismic hazard for Mexico City is shown in Fig. 6. The spatial distribution of earthquake occurrence is divided into three groups according to earthquake types and magnitude-recurrence relations k(M). The source zones with their corresponding magnitude-recurrence relations for the first two groups are close to the Mexican Pacific coastal region: the first group is defined by four polygons, while the second is defined by 14 polygons (Ordaz and Reyes 1999). They are for interplate earthquakes with 4.5 B M \ 7 and M C 7, respectively. The third group is used to deal with inslab earthquakes. The source zone for this group is represented by two large polygons (Jaimes and Reinoso 2006). Although the slope of the interface between the continental plate and the oceanic plate for the region of interest has been investigated (Pardo and Sua´rez 1995), there is no statistical model for spatial varying earthquake depth. The mean and standard deviation of the focal depth for the events shown in Tables 1 and 2 are 22.2 and 7.0 km for the interplate earthquakes and 54.6 and 8.93 km for the inslab earthquakes. As a parametric investigation, four focal depth cases shown in Table 3 are considered. This table only shows the depth for inslab earthquakes since, as will be seen, the adopted GMPE for interplate earthquakes is not a function of focal depth. Case A shown in Table 3 is based on the mean focal depth for the historical events shown in Tables 1 and 2, respectively; Case B and Case C are based on the mean minus and plus one standard deviation for the historical events. Case D considers that the focal depth can be modeled as a symmetric random variable with mean and standard deviation equal to the sample mean and standard deviation, and that the point estimate method (Rosenblueth 1975; Hong 1998) can be used to assign the weights (i.e., probabilities) and the discrete values of focal depth based on the moments alone. Note that in addition to the source zones shown in Fig. 6, there are local and continental sources. These sources are not considered since they do not contribute significantly to the seismic hazard for Mexico City (Rosenblueth et al. 1989; Arroyo et al. 2010). Consider the listed events and records shown in Tables 1 and 2, the functional form of the GMPEs for SA is: log10 y ¼ c1 þ c2 ðM 6Þ þ c3 ðM 6Þ2 þ c4 log10 R þ c5 R þ e
ð5Þ
for the interplate earthquakes (Reyes et al. 2002) and
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Fig. 6 Source zones [after Ordaz and Reyes (1999) and Jaimes and Reinoso (2006)]: a interplate earthquakes with 4.5 B M \ 7; b interplate earthquakes with M C 7; and c inslab earthquakes. The coordinates of the polygons inferred from the references can be found in Garcı´a-Soto et al. (2012). k(M) shown in the figure denotes the magnitude-recurrence relation
log10 y ¼ c1 þ c2 M þ c3 R c4 log10 R þ c5 H þ e
ð6Þ
for the inslab earthquakes (Garcı´a et al. 2005), where y is in cm/s2 (or Gal), ci (i = 1–5) are regression coefficients, and e is a zero mean error term with standard deviation r. For
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Nat Hazards Table 3 Focal depth cases considered for sensitivity analysis
Case
Focal depth for inslab earthquakes (km)
A
54.6
B
45.7
C
63.5
D
(39.2, 54.6, 70.1) with probabilities (0.167, 0.666, 0.167)
Eq. (5), R (km) is the closest distance to the fault surface, and c4 = 0.5. For Eq. (6), qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi R ¼ R2cld þ D20 , Rcld (km) is the closest distance to the fault plane for events with M [ 6.5, or the hypocentral distance for the rest, D0 ¼ 0:0075 100:507M is a near-source saturation term (Atkinson and Boore 2003).
(a)
(b)
4
0.5
Based on records of CU station Based on a broad network of stations
3 0.4
1 0
Sigma
Regression coefficient
2
-1
0.3
-2
c
-3
c
-4
c ×1000
4
c2×10
1
0.2
5
c3×10
-5 0
1
3
2
0.1
4
0
1
T (s) n
0.98 0.95 0.90
200
Probability
0.75 0.50 0.25 0.10 0.05 0.02 0.01 -1.5
3
4
(d) SA at 0.2 s SA at 1.0 s SA at 3.0 s Spectral acceleration (Gal)
(c) 0.99
2
Tn (s)
150
100
50
0 -1
-0.5
0
Residual
0.5
1
1.5
R = 50 km R = 100 km R = 200 km R = 300 km
0
1
2
3
4
T
n
Fig. 7 Ground motion prediction equation for the CU station considering the interplate earthquakes: a regression coefficients, b sigma, c residuals presented on normal probability paper (for SA at 0.2, 1.0, and 3.0 s), d illustration of predicted SA (M7, H = 22 km)
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(a)
(b)
4
c
3
c
1
0.5
c3×100
2
0.4
1 0
Sigma
Regression coefficient
2
-1
0.3
-2 0.2
-3
c4
-4 -5
0
Based on records of CU station Based on a broad network of stations
c ×100 5
1
2
3
0.1
4
0
1
Tn (s) 0.99 0.98 0.95
Probability
0.90
200
0.75 0.50 0.25 0.10 0.05 0.02 0.01 -1.5
3
4
(d) SA at 0.2 s SA at 1.0 s SA at 3.0 s Spectral acceleration (Gal)
(c)
2
Tn (s)
150
R
cld
R
cld
R
cld
= 50 km = 100 km = 200 km
Rcld = 300 km 100
50
0 -1
-0.5
0
Residual
0.5
1
1.5
0
1
2
T
3
4
n
Fig. 8 Ground motion prediction equation for the CU station considering the inslab earthquakes: a regression coefficients, b sigma, c residuals presented on normal probability paper (for SA at 0.2, 1.0 and 3.0 s), d illustration of predicted SA (M6.5, H = 55 km)
By carrying out regression analysis (Joyner and Boore 1993), the calculated regression coefficients and the standard deviation of e for a random orientation, r (i.e., sigma), for the CU station are shown in Figs. 7a, b, and 8a, b. For comparison purposes, Figs. 7 and 8 also include the sigma for the GMPEs developed based on a broad network of stations in the Pacific coastal region that are applicable for Mexican interplate earthquakes and for Mexican inslab earthquakes (Hong et al. 2009; Pozos-Estrada et al. 2014). The former is developed using 413 records from 40 interplate events recorded up to 2004 (Garcı´a 2006), while the latter is obtained using 275 records from 16 inslab events recorded up to 2004 (Garcı´a et al. 2005). The comparison presented in Fig. 7b shows that the site sigma obtained in this study is about 0.2–0.3 for interplate earthquakes, which is about 60 % of that for regional ground motion relations. This observation is in agreement with the observation made by Atkinson (2006) by considering that the seismic activity is from a single source. The plots of the residuals on normal
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(a)
(b)
4
c1
3
c2
Based on records of CU station Based on a broad network of stations 0.4
1 0
Sigma
Regression coefficient
2
-1
0.3
-2 0.2
-3
c
-4 -5
0
(c) 0.99 0.98 0.95
c5×100
4
1
2
Tn (s)
3
(d)
SA at 0.2 s SA at 1.0 s SA at 3.0 s
Spectral acceleration (Gal)
0.50 0.25 0.10 0.05 0.02 0.01
-1
-0.5
0
1
2
3
Tn (s)
150
R
cld
4
= 50 km
Rcld = 100 km
0.75
-1.5
0.1
4
0.90
Probability
0.5
c3×100
0 0.5 Residual
1
1.5
R
cld
= 200 km
Rcld = 300 km
100
50
0
0
1
2 Tn
3
4
Fig. 9 Ground motion prediction equation for the CU station considering 12 records for the inslab earthquakes (each with maximum SA greater than 10 Gal): a regression coefficients, b sigma, c residuals presented on normal probability paper (for SA at 0.2, 1.0, and 3.0 s), d illustration of predicted SA (M6.5, H = 55 km)
probability paper, shown in Fig. 7c, indicate that they could be modeled as normal variate. An illustration of the predicted SA by using the developed GMPE is shown in Fig. 7d. Unlike the case for interplate earthquakes, the single-station sigma for the inslab earthquakes (see Fig. 8b) increases significantly as the natural vibration period, Tn, increases. It is greater than that for the GMPEs developed based on a broad network of stations for Tn greater than 0.5 s. This ‘‘apparent paradox’’ can be explained based on the plot of residuals shown in Fig. 8c. It indicates that e is not normally distributed, and the GMPEs developed for the CU station by using all the records for the inslab earthquakes listed in Table 2 are inappropriate. To overcome this, regression analysis is carried out again but considering only those records whose maximum value of the SA for Tn up to 4 s is greater than 10 Gal (i.e., the 12 records mentioned earlier). The obtained regression coefficients and r for the 12 records are shown in Fig. 9.
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Comparison of the r values shown in Figs. 8 and 9 indicates that the use of these 12 records has effectively reduced the single-station sigma. Inspection of the plot of e shown in Fig. 9c indicates that it follows normal distribution better than that shown in Fig. 8c. Note that by using the developed GMPEs, the predicted SA shown in Fig. 8d is greater than those shown in Fig. 9d, even though the former considers records with small SA values. Since the use of the lower bound slightly greater than 10 Gal does not reduce r value significantly, and a further increase in the lower bound reduces the number of records that can be used to develop the GMPEs, the use of other lower bounds is not considered. To estimate the seismic hazard at the CU station in the following, the GMPEs shown in Eqs. (5) and (6) with coefficients presented in Figs. 7 and 9 are used. Note that c5 for Eq. (6) is positive for Tn \ 3.0 s, implying that an increase in focal depth leads to an increase in SA. This observation is in agreement with that reported in Garcı´a et al. (2005) and in Hong et al. (2009). For the numerical evaluation, simple Monte Carlo technique with the steps described in Hong et al. (2006) is followed by generating a synthetic earthquake catalog of 200,000 years. Furthermore, similar to the case in the previous section, the analysis is carried out by considering the interplate earthquake alone, the inslab earthquake alone, and both the interplate and inslab earthquakes for return periods of 125, 475, and 2475 years. First, consider Case A for the focal depths shown in Table 3. The obtained uniform hazard spectra based on the interplate earthquakes alone, the inslab earthquakes alone, and both the interplate and inslab earthquakes are shown in Fig. 10. For comparison purposes,
(a) 400 300 Observation
250
(b) 400
NTCDS (2004) T = 125 years T = 475 years T = 2475 years T = 125 years T = 475 years T =2475 years
Seismic hazard model
200
Inslab
Spectral acceleration (Gal)
Spectral acceleration (Gal)
Interplate
350
150 100 50 0
0
0.5
1
1.5
2
2.5
3
3.5
350 300
Observation
250 200
NTCDS (2004) T = 125 years T = 475 years T = 2475 years T = 125 years T = 475 years T = 2475 years
150 100 50 0
4
Seismic hazard model
0
0.5
1
1.5
2
2.5
3
3.5
4
Tn (s)
Tn (s)
Spectral acceleration (Gal)
(c) 400 Interplate + Inslab
350
Seismic hazard model
300 Observation
250 200
NTCDS (2004) T = 125 years T = 475 years T = 2475 years T = 125 years T = 475 years T = 2475 years
150 100 50 0
0
0.5
1
1.5
2
2.5
3
3.5
4
Tn (s)
Fig. 10 Estimated seismic hazard based on adopted seismic hazard model and comparison with that obtained directly using observations: a for interplate earthquakes alone, b for inslab earthquakes alone, c for both the interplate and inslab earthquakes. Results based on seismic hazard model are shown in black lines, while the results based on observations are shown in gray lines
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the UHS developed based on direct observations that have been shown in Figs. 3c, 4b, and 5 are also included in Fig. 10. Comparison of the results shown in Fig. 10a indicates that the estimated seismic hazard by considering the interplate earthquakes alone is relatively consistent, whether the analysis is carried out by directly using the observations or by applying the adopted seismic hazard model. However, Fig. 10b shows that there are significant differences in the estimated hazards due to inslab earthquake alone by using these two approaches for Tn less than about 0.5 s. The discrepancy can be explained by noting that: 1.
2.
In the analysis by directly using the observations, the records for inslab earthquakes are associated with a minimum epicentral distance (to CU station) greater than 140 km. Therefore, the effect of any potential future inslab earthquake with an epicentral distance less than 140 km, which may significantly increase the seismic hazard, cannot be taken into account. In the analysis based on the seismic hazard model, the CU station is very close to the seismic source zone (see Fig. 6) and it is considered that the developed GMPEs (Fig. 9) can be extrapolated and are applicable for epicentral distances smaller than 140 km. Consequently, a significant increase in seismic hazard is obtained by this approach as compared to that based on directly using observations.
However, validation of one or the other approach can only be carried out with the acquisition of additional observations. In general, the mentioned differences affect also the estimated UHS due to both the interplate and inslab earthquakes. Figure 10c shows that the ordinates of the UHS for T = 125 years are below the design spectrum. The ordinates of the UHS for T = 475 years exceed the design spectrum for Tn \ 0.25 s and Tn [ 1.75 s. The ordinates of the UHS for T = 2475 years practically envelop the design spectrum, but for Tn within 0.3–1.5 s they are approximately equal to those of the design spectrum. These observations are consistent with those by directly using the observations, even though applying the adopted seismic hazard model yields higher UHS, as shown in Fig. 10c. To investigate the effect of the focal depth models (see Case A to Case D shown in Table 3) on the estimated UHS, the analysis carried out for Fig. 10 (i.e., for Case A) is repeated by considering Case B to Case D. The results obtained indicate that the assumption on the focal depth does affect the estimated UHS. Using Case A as the benchmark and for Tn \ 1.0 s, the decrease of H from the mean to mean minus one standard deviation (i.e., Case B) resulted in 5 % decrease, on average, in the estimated UHS, and the increase of H from the mean to mean plus one standard deviation (i.e., Case C) resulted in 5 % increase, on average, in the estimated UHS. This is simply the consequence of c5 for Eq. (6) being positive, as mentioned earlier. However, the consideration of the uncertainty in H (i.e., Case D) leads to an estimated UHS almost identical to that of Case A.
4 Summary of observations A site-specific seismic hazard assessment for the Ciudad Universitaria (CU) station in Mexico City is carried out by directly using observations and by applying an adopted seismic hazard model. Several main observations from the analysis results are summarized below:
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1.
2.
3.
4.
Directly using the observations (i.e., ground motion records) and applying an adopted seismic hazard model for interplate earthquakes could lead to similar UHS. The consistency can be attributed to the fact that the records are representative for a range of earthquake magnitude and spatio-temporal occurrence that are considered in the seismic hazard model and are important for seismic hazard evaluation. The consistency can be appreciated by comparing the UHS obtained using these two approaches for interplate earthquakes (Fig. 3c). Unlike the case for the interplate earthquakes, the UHS for inslab earthquakes obtained by directly using the observations and by applying the seismic hazard model differ. This is attributed to the fact that the seismic source model does not reflect the spatial distribution of the inslab events associated with the considered records. Although the investigation of the preference of the former or the latter cannot be easily selected, it does point to the need for scrutinizing basic assumptions in seismic hazard assessment. Furthermore, analysis results also show the importance of records selection in developing the GMPEs and in assigning probabilistic model for SA conditioned on the earthquake occurrence. The single-station sigma for the GMPEs considering the interplate earthquakes is about 60 % of that obtained by using a broad network of stations. The percentage agrees with the value recommended by Atkinson (2006) based on small or moderate California earthquakes. This emphasizes that the uncertainty in the GMPEs developed for a single site can be reduced as compared to that developed based on a broad network of stations. However, unlike the case for interplate earthquakes, the singlestation sigma for inslab earthquakes and large Tn is comparable to that obtained by using a broad network of stations. We attribute this to the small sample size and samples with small SA. The seismic hazard analysis results show that the recommended design spectrum in NTCDS (2004) does not correspond to a uniform hazard spectrum. The design spectrum for Tn \ 0.25 s or Tn [ 1.75 s follows approximately the UHS for T = 475 years. For Tn within 0.3 and 1.5 s, the ordinates of the design spectrum are greater than those of the UHS for T = 125 and 475 years and are approximately equal to those of the UHS for T = 2475 years.
Acknowledgments Financial support received from the National Science and Engineering Research Council of Canada, the National Council on Science and Technology (CONACYT) of Mexico, and the University of Western Ontario is gratefully acknowledged. We thank the Coordinacio´n de Instrumentacio´n Sı´smica from the Institute of Engineering from UNAM for providing some of the records used in this work. We are grateful to Dr. D. Garcı´a for his constructive comments and suggestions, which have improved the quality and readability of the manuscript.
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