Uncertainties in a probabilistic model for seismic hazard analysis in Japan

Uncertainties in a probabilistic model for seismic hazard analysis in Japan T. Annaka* and H. Yashiro* * Tokyo Electric Power Services Co., Ltd., Japan ** The Tokio Marine and Fire Insurance Co., Ltd., Japan Abstract We developed a logic tree model for evaluating uncertainty in seismic hazard for Japan. The uncertainties in an attenuation model and a seismic source model with temporal dependence of large earthquake occurrence were represented by either discrete branches or continuous probability distributions associated with each node in the logic tree. The nodes for the attenuation model were distance measure, site correction coefficient, and standard error and those for the seismic source model were magnitude distribution, maximum magnitude, mean and variability of recurrence interval, seismic source location and geometry, and division of seismic source zones. Uncertainty in the hazard was evaluated using a Monte Carlo approach and displayed by 5-, 16-, 50-, 84- and 95-percentile hazard curves. 1 Introduction We proposed a probabilistic model for seismic hazard analysis in Japan (Annaka and Yashiro [1,2]). The model consists of a seismic source model with temporal dependence of large earthquake occurrence and an attenuation model developed based on the records obtained by the JMA-87 type strong motion accelerometer network of the Japan Meteorological Agency (JMA). Although we can obtain a 'best-estimated' hazard curve from the model, it is also very important to evaluate the uncertainty in the hazard curve for a practical judgment. The objective of the present study is to develop a logic tree model for evaluating uncertainty in seismic hazard for Japan. The logic tree approach has been used for assessing the uncertainty in seismic hazard (e.g. Pacific Gas and Electric Company [3]). Logic trees provide a useful

370 tool for representing the uncertainties in the various model parameters. Although discrete branches with discrete probability distribution are usually defined for each node in the logic tree, the continuous probability distribution of the model parameter is also used for the node (e.g. Cramer et al. [4]). In the present study we used both discrete and continuous distributions of the uncertain quantities. A Monte Carlo approach of randomly sampling the logic tree can be used to simulate a distribution of seismic hazard curves. 2 Uncertainties in attenuation model Nodes in logic tree for the attenuation model are shown in Figure 1. Distance measure, site correction coefficient, and standard error are defined as the nodes. P denotes the probability of each branch. A shortest distance (SD) and an equivalent hypocentral distance (EHD) from a site to a finite fault are considered to be useful for taking account of the effect of fault size in the near-fault area. Annaka et al. [5] proposed attenuation equations for peak ground motions and 5 % damped acceleration response spectra based on both SD and EHD using the same data set. The attenuation equations are derived from the records of the JMA-87 type accelerometers obtained by JMA during the period from August 1988 to March 1996. The attenuation equation using SD for peak ground acceleration (PGA) is as follows: log A = 0.606M + 0.00459/7^-2.136log(/? + 0.334exp(0.653)) +1.730 (1) where A is the mean PGA of two horizontal components, M is JMA magnitude, Hc=H for// < 100km and //<-=! 00 for 100km<//<200km, H is the depth of the center of a fault plane of an earthquake in km, and R is the shortest distance in km to the fault plane. Common logarithms are used. The constants, 0.334 and 0.663, are determined by the constraint that a peak ground acceleration becomes independent on magnitude when R reaches 0 km. The constant of 1.730 was determined so that the attenuation equation can be applied to a site whose subsurface S wave velocity is about 400 m/s. The attenuation equation using EHD is as follows: log A = 0.573M + 0.00429//C - log X - 0.00207X - 0.243 (2) where X is the equivalent hypocentral distance in km and the others are the same Distance Measure of Attenuation Equation Site Correction Coefficient Standard Error of Attenuation Equation Shortest Distance Constant P=0.5 Normal Distribution / / P=0.5 Equivalent +- (Continuous) \ ^Hypocentral Distance \ Magnitude Dependent P=0.5 P=0.5 Figure 1: Nodes in logic tree for attenuation model.

Risk Analysis II 371 as eqn (1). The EHD is proposed by Ohno et al. [6] and interpreted as a distance from a virtual point source that provides the same energy to the site as does a finite-size fault. The comparison of contour maps for PGA between the two equations is shown in Figure 2. The magnitude is assumed as 7.9. The fault model obtained for the 1923 Kanto earthquake by Matsu'ura et al. [7] is used. The PGA by EHD is relatively higher than that by SD in the area above the fault plane. Site correction coefficient (SCC) for PGA is defined as the natural logarithms of the geometrical average ratio of observation to calculation by a reference empirical attenuation equation. The uncertainty in SCC is represented by a normal distribution. If no observed records are obtained for a specified site, the SCC must be estimated from such site condition as velocity structure and geomorphological land classification. In this case the standard deviation (S.D.) of the estimated SCC might be about 0.3 at best. If observed records are obtained, the SCC can be estimated fairly precisely. In this case the S.D. might be reduced to about 0.1-0.15. Two branch models for the node of the standard error of attenuation equation are shown in Figure 3. The left and right represent constant (C) and magnitude dependent (MD) models, respectively. Youngs et al. [8] suggest a significant dependence of the standard error on magnitude: the standard error decreases with increasing magnitude. The uncertainty in the standard error is represented by a normal distribution. The values of a, and a 2 in Figure 3 were ^assumed as 0.1 and 0.05, respectively. An example of percentile hazard curves for Tokyo is shown in Figure 4(a). Uncertainty in the hazard only due to the uncertainties in the attenuation model was evaluated using a Monte Carlo approach. The 'best-estimated' seismic source model proposed by the previous papers [1,2] was used. Total 200 curves were simulated and uncertainty was displayed by 5-, 16-, 50-, 84- and 95- percentile hazard curves. The S.D. of 0.15 is used for the SCC. The comparison of 50-percentile hazard curves for four combinations of distance measure and standard error is shown in Figure 4(b). The difference between SD and EHD 139 E 140 E 139 E 140 E Figure 2: Comparison of contour maps for PGA distance measures. (cm/s^) between two

372 Risk Analysis II 0. 7 0.7 0.6 0. 6 o. 5 C/D 0.4. 5 0. 4 0.3 4. 0 5.0 6.0 7.0 Magnitude 0.3 8.0 4.0 5.0 6.0 7.0 Magnitude 8.0 Figure 3: Constant (left) and magnitude dependent (right) models for standard error of attenuation equation. 10 15 510' o Q. (a) ^ 10 g!10' 10^ CL84 ).5 ).16 0.05 io- _EHD±C ID EHD4MD 10-10-5L- 10' PGA (cm/si \ PGA (cm/s') Figure 4: (a) Percentile hazard curves for Tokyo due to the uncertainties in attenuation model, (b) Comparison of median hazard curves for four combinations of distance measure and standard error. increases with increasing PGA. 3 Uncertainties in seismic source model The islands of Japan lie mainly on the Eurasia and Okhotsk plates. The Pacific and Philippine Sea plates subduct beneath the islands towards the west-northwest and northwest, respectively. The seismic activity in and around Japan is controlled by the interaction of the four plates. For representing the seismic activity we used two types of seismic sources: (1) fault source generating large characteristic earthquakes, and (2) background seismic source generating small and moderate earthquakes.

373 3.1 Fault sources Nodes in logic tree for magnitude-frequency distribution for fault source is shown in Figure 5. A characteristic earthquake model proposed by Youngs and Coppersmith [9] was used for fault source. Magnitude distribution, central magnitude, and recurrence interval of characteristic earthquakes are defined as the nodes. Two types of magnitude distribution, uniform and truncated normal distributions, are used as the branches. The bandwidth of magnitude range is fixed to be 0.5. The uncertainty in central magnitude is represented by a truncated normal distribution. The S.D. of 0.1 is used for both truncated normal distributions and the distribution of magnitude is truncated at ± 0.25 from the center. The uncertainties in the mean and variability of recurrence intervals are represented by a normal distribution. The assumed standard deviations of natural logarithm of mean recurrence interval for the fault sources are listed in Table 1. Table 1. Assumed log standard deviation of mean recurrence intervals Source Identification Earthquake Data Used Data for Estimating Mean Historical Data <?1n 0.15 Active Fault Data Trenching Survey 0.15 Degree of Activity 030 The values are different according to the used data for estimating mean recurrence interval. The mean and standard deviation of the variability of recurrence intervals were taken as 0.30 and 0.05, respectively. The configuration of a model case and the obtained percentile hazard curves for Tokyo due to the uncertainties in magnitude-frequency distribution for fault source are displayed in Figures 6(a) and 6(b), respectively. The mean of central magnitude is 7.9 and the median of mean recurrence interval is 200 years. The attenuation equation (1) was used. The distributions of the expected number of earthquakes for the Nankaido earthquake during the coming 20 and 50 years due to the uncertainties in the mean and variability of recurrence intervals are shown in Figure 7. The median of the mean recurrence intervals and the age of the most recent earthquake for Magnitude Distribution Central Magnitude Recurrence Interval Uniform Distribution (Mean and Variability) Truncated Normal Distribution Truncated Normal *~ Distribution +- (Continuous) Distribution (Continuous) P=0.5 Figure 5: Nodes in logic tree for magnitude-frequency distribution for fault source.

374 Risk Analvsis II IS^w 7T~^k. (b) \ / _ : 139 E 140 E (M:7.9±0.2, Tr=200 years).._ n.95 E U.84 0 ". 1 I AO. o.05 0' \\ i - - 1 0* 1 PGA (cm/s?) Figure 6: (a) Model configuration of fault source and site, (b) Percentile hazard curves for Tokyo due to the uncertainties in magnitudefrequency distribution for fault source. 40 30 Next 20 years (2000-201 9 A.D.) -i 20 10 o nf niicflii u.iiiiir 0. 0 0. 1 0. 2 0. 3 0. 4 Expected Number of Eathquakes o 0. 0 0. 2 0. 4 0. 6 0. 8 1.0 Expected Number of Eathquakes Figure 1: Distribution of expected number of earthquakes for the Nankaido earthquake due to the uncertainties in recurrence intervals the Nankaido earthquake are 130 years and 1946 A.D., respectively. The uncertainties in fault source location and geometry are modeled as shown in Figure 8(a). The uncertainties in the location of a reference point on the fault source along strike, dip and vertical directions and dip angle are represented by a normal distribution. The percentile hazard curves for Tokyo due to the uncertainties in fault source location and geometry are shown in Figure 8(b). The used fault source is the same as the case of Figure 6(a). The standard deviations were assumed as follows: 0-5= 10km, crn=10km, o\=10km and 0-4=5. The attenuation equation (1) was used. In this case we assumed that the standard deviations are independent each other, but it becomes necessary to take account of correlations if fault source is restricted in the interface of two plates.

Risk Analysis II 375 (a) IT 15 o_ CD (b) "810" MO" E o.95 0.84 0.5 - ~071"6~ E 0.05 PGA (cm/s*) 103 Figure 8: (a) Parameter for representing the uncertainties in fault source location and geometry, (b) Percentile hazard curves for Tokyo due to the uncertainties in source location and geometry. 3.2 Background seismic sources Nodes of logic tree for magnitude-frequency distribution for background seismic source is shown in Figure 9. A truncated Gutenberg-Richter recurrence model was used for background seismic sources. Annual frequency of earthquakes with magnitude greater than or equal to 5.0, b-value of the Gutenberg-Richter relationship, and maximum magnitude are defined as the nodes. An example of the simulation on the uncertainties in magnitude-frequency frequency is shown in Figure 10. Figure 10(a) shows the annual frequency of earthquakes at intervals of 0.5 magnitude. The solid circles and the error bars denote observed data and ±1 S.D., respectively. The error was estimated from the number of earthquakes using the equations given by Wei chert [10]. Weichert shows ±1 S.D. confidence intervals for an estimated average of a Poisson variable. Figure 10(b) shows the 200 sample curves generated based on Figure 10(a). We determined the a- and b- values of the Gutenberg-Richter relationship for each sample curve. Figure 10(c) shows the relation between the obtained a- and b- values. The correlation of the two values is very high. On the contrary the annual frequency of earthquakes with magnitude greater than or equal to 5.0 is almost independent on the b-value as shown in Figure 10(d). Annual Frequency of Earthquakes (M>5.0) Normal Distribution (Continuous) b-value of Gutenberg- Richter Relationship Normal Distribution (Continuous) Maximum Magnitude Truncated Normal Distribution (Continuous) Figure 9: Nodes in logic tree for magnitude-frequency distribution for background seismic source.

376 Risk Analvsis II _g s JD :=10 (a) 8" i 6.! 6. 0 QC 6 5. 5 5.0 6.0 7.0 Magnitude (c) 8. o 5. 0 6. 0 7. 0 \\ Magnitude (M.GE. O> Ol rthq O) O far (d) 5. 0 al Frequency Ol Ol O 01 4., 3.7 0.8 ^n 0.9 1.0 1.15 0.7 0. 8 0. 9 1.0 1.1 b-value of G-R b-value of G-R Figure 10: Example of simulation on the uncertainties in magnitudefrequency distribution for background seismic source. Based on the above result we selected the annual frequency of earthquakes (M>5.0) and the b-value as the nodes. The uncertainty in the two parameters is represented by a normal distribution. The uncertainty in maximum magnitude is represented by a truncated normal distribution. The percentile hazard curves due to the uncertainties in magnitude-frequency distribution are shown in Figure 11. The geometry of the seismic source and the site is the same as the case of Figure 6(a). The means of the annual frequency of earthquakes (M=>5.0) and b-value were determined from the observed data of Figure 10(a). The S.D, of the annual frequency of earthquakes (M>5.0) can be estimated from the number of earthquakes occurring in a specified period using the equations given by Wei chert [10] and that of b-value is assumed as 0.05. The center of maximum magnitude is 7.7 and the S.D. of 0.1 is used. The distribution of magnitude is truncated at ± 0.25 from the center. The attenuation equation (2) was used. Background seismic sources were continuously defined along the upper planes of the four plates in and near Japan. The division was made based on the seismotectonic characteristics by subjective judgments. As an alternative branch

Risk Analvsis II 377 10 fio" CO " 10-3 I " 10-" I 10-5 0.95 0.84 0.5 0.16 0.05 PGA (cm/sf) Figure 11: Percendle hazard curves for Tokyo due to the uncertainties in magnitude-frequency distribution for background seismic source. 138 E 139 E 140 E PGA (cm/si Figure 12: (a) An automatic method for dividing background seismic source zones, (b) Comparison of seismic hazard curves for Niigata among three different zone divisions. of division we selected an automatic method that needs few subjective judgments. Figure 12(a) explains the automatic method for dividing background seismic source zones. The epicenters of shallow earthquakes with magnitude greater than or equal to 5.0 during the period from 1885 to 1999 are displayed as example. The region is mechanically divided by an interval of 0.5 degree. The annual frequency for each region is determined from the historical data and then spatially smoothed. For example the annual frequency of the broken line box is given by the average of Aj, A 2, A 3, and A 4. The comparison of seismic hazard curves at Niigata is shown in Figure 12(b). Model 0 (P=0.5) denotes original division based on the seismotectonic characteristics. Model 1 (P=0.25) and Model 2 (P=0.25) corresponds to the solid line and broken line divisions in

378 Figure 12(a), respectively. Each zone for Model 2 moves to 0.25 degree in both directions from that for Model 1. 4 Conclusions We developed a logic tree model for evaluating uncertainty in seismic hazard for Japan. Uncertainty in the hazard was evaluated using a Monte Carlo approach and displayed by 5-, 16-, 50-, 84- and 95-percentile hazard curves. Although we separately demonstrated the effects of the uncertainties in the various model parameters on uncertainty in seismic hazard, total uncertainty in seismic hazard can be obtained by combining them all. References [1] Annaka, T. & Yashiro, H., A seismic source model with temporal dependence of large earthquake occurrence for probabilistic seismic hazard analysis in Japan, Risk Analysis, WITPRESS, Boston, pp. 233-242, 1998. [2] Annaka, T. & Yashiro, H., Temporal dependence of seismic hazard in Japan, Proceedings of the 12th World Conference on Earthquake Engineering, PaperID-0316,2000. [3] Pacific Gas and Electric Company. Final Report of the Diablo Canyon Long- Term Seismic Program, U.S. Nuclear Regulatory Commission docket nos., 50-275 and 50-323,1988. [4] Cramer, C. H., Petersen, M. D. & Reichle, M. S., A Monte Carlo approach in estimating uncertainty for a seismic hazard assessment of Los Angels, Ventura and Orange counties, California, Bull. Seism. Soc. Am., 86, pp. 1681-1691,1996. [5] Annaka, T., Yamazaki, F. & Katahira, R, A proposal of an attenuation model for peak ground motions and 5 % damped acceleration response spectra based on the JMA-87 type strong motion accelerograms, Proceeding of the 24^ JSCE Earthquake Engineering Symposium, pp. 161-164, 1997. [6] Ohno, S., Ohta, T., Ikeura, T. & Takemura, M., Revision of attenuation formula considering the effect of fault size to evaluate strong motion spectra in near field, Tectonophys., 218, pp. 69-81, 1993. [7] Matsu'ura, M., Iwasaki, T., Suzuki, Y. & Sato, R., Statical and dynamical study on the faulting mechanism of the 1923 Kanto earthquake, J. Phys. Earth, 28, pp. 119-143, 1980. [8] Youngs, R. R., Abrahamson, N., Makdisi, F. I. & Sadigh, K., Magnitudedependent variance of peak ground acceleration, Bull Seism. Soc. Am., 85, pp. 1161-1176, 1995. [9] Youngs, R. R. & Coppersmith, K. J., Implications of fault slip rates and earthquake recurrence models to probabilistic seismic hazard estimates, Bull Seism. Soc. Am., 75, pp. 939-964, 1985. [10] Wei chert, D. H., Estimation of the earthquake recurrence parameters for unequal observation periods for different magnitudes, Bull. Seism. Soc. Am., 70, pp. 1337-1346,1980.