Seismic Hazard Estimate from Background Seismicity in Southern California

Transcription

1 Bulletin of the Seismological Society of America, Vol. 86, No. 5, pp , October 1996 Seismic Hazard Estimate from Background Seismicity in Southern California by Tianqing Cao, Mark D. Petersen, and Michael S. Reichle Abstract We analyzed the historical seismicity in southern California to develop a rational approach for calculating the seismic hazard from background seismicity of magnitude 6.5 or smaller. The basic assumption for the approach is that future earthquakes will be clustered spatially near locations of historical mainshocks of magnitudes equal to or greater than 4. We analyzed the declustered California seismicity catalog to compute the rate of earthquakes on a grid and then smoothed these rates to account for the spatial distribution of future earthquakes. To find a suitable spatial smoothing function, we studied the distance (r) correlation for southern California earthquakes and found that they follow a 1/r ~ power-law relation, where/1 increases with magnitude. This result suggests that larger events are more clustered in space than smaller earthquakes. Assuming the seismicity follows the Gutenberg- Richter distribution, we calculated peak ground accelerations (PGA) for 1% probability of exceedance in 5 yr. PGA estimates range between.25 and.35 g across much of southern California. These ground-motion levels are generally less than half the levels of hazard that are obtained using the entire seismic source model that also includes geologic and geodetic data. We also calculated the overall uncertainty for the hazard map using a Monte Carlo method and found that the coefficient of variation is about.24.1 for much of the region. Introduction Since the 17 January 1994 Northridge earthquake, the California Department of Conservation, Division of Mines and Geology has been assessing the ground shaking hazard of Los Angeles, Ventura, and Orange counties to facilitate cost-effective distribution and prioritize expenditure of mitigation funds (Petersen et al., 1996). As part of the effort to map ground shaking hazard in southern California, this study estimates that portion of hazard associated with background seismicity. We define background seismicity as earthquakes of magnitude 6.5 or smaller and assume that they can be forecasted based on M => 4 historical seismicity. For many earthquakes larger than M 6.5, recurrence times are on the order of several hundreds or thousands of years. That prohibits us from estimating the hazard associated with M > 6.5 earthquakes using historical seismicity, which spans only about 15 yr. Instead, we use geologic and geodetic information to account for these longer-term hazards. The general method to calculate probabilistic seismic hazard by integrating all the influences of potential point, line, or areal sources is well established (Cornell, 1968; McGuire, 1976). The Working Group on California Earthquake Probabilities (WGCEP 1995) estimated background seismicity contribution using a catalog of M => 6 earthquakes. These earthquakes are called distributed earthquakes and are as- sumed to follow a modified Gutenberg-Richter magnitude distribution with an upper limiting magnitude of This method was adopted by Petersen et al. (1996) for the part of the hazard due to background seismicity. Because of the limited knowledge of seismic sources in the region, Frankel (1995a) used a simpler approach for mapping the seismic hazards in the central and eastern United States. The major features of Frankel's approach are to abandon seismotectonic zones and to use point sources (or infinitely small source zones) in seismic hazard analysis. Another difference between Frankel's approach and the WGCEP model (1995) is the seismicity smoothing function: the former uses a Gaussian function, and the latter uses a 1/r function. In this article, we follow Frankel's (1995a) approach to calculate the hazard from background seismicity for southern California. This approach is simpler for hazards calculation and is based solely on the recorded seismicity history. Most importantly, we will show that the results are not significantly different from those calculated from using seismic source zones. We also study the statistics of the earthquake spatial distributions to improve the spatial seismicity smoothing. Finally, we present a background seismicitybased hazard map and its uncertainties for southern California. 1372

2 Seismic Hazard Estimate from Background Seismicity in Southern California 1373 Model and Hazard Calculation Our model uses the historical seismicity with magnitudes from 4 to 6.5 to obtain the a-value distribution over the study area between 121 W to 115 W and 32 N to 36 N. Part of the background seismicity with magnitude smaller than 4 is not used because the catalog is not complete, and they rarely cause any hazards. The a-value is a constant defined as the seismicity rate in the Gutenberg-Richter magnitude-frequency relation log N = a - bm, where N is the number of events with magnitude equal to or greater than M and b is a constant for the frequency ratio between different magnitude events. The study area is divided into grid cells of.124 by.1 (-11 by 11 km) in size. A spatial smoothing function is applied to the original a-value map. The basic assumption behind this model is that future moderate background earthquakes will be located near the historical events of M => 4 (Kagan and Jackson, 1994; Frankel, 1995a). In southern California, the catalog should be complete down to magnitude 4 since 1932 (WGCEP, 1995), which provides more than 6 yr of seismicity data. The rate for M -- 6 earthquakes within this period is very close to the period from 1852 to 1932 (Table 4 of WGCEP, 1995), although the pre-188 catalog is probably incomplete (Toppozada, 1995). Another assumption is the use of point sources to represent the seismic events that occur along finite faults. From the Gutenberg-Richter relation, the a-value map is mostly determined by earthquakes of about magnitude 4, which usually have a source dimension much smaller than the map grid size used in the calculations. Thus the point source assumption is justified. For hazard calculations, we choose a low cutoff magnitude of Mo = 5, which is based on the observation that earthquakes with magnitude less than 5 are generally unlikely to cause significant damage. The upper limit Mx = 6.5 is a relatively arbitrary choice in the definition of background seismicity; however, we will show later that this parameter affects the hazard calculations the least. Our model is determined fully by the historical seismicity recurrence process, the a-value smoothing function, and the groundmotion attenuation relation. The hazard can be calculated following the standard methodology (Cornell, 1968) to obtain annual probabilities of exceeding specified ground motions at a site (Frankel, 1995a). The steps leading to the final hazard results include (1) bin the earthquakes with magnitude equal to or greater than 4 to each grid cell i (.1 in latitude by.124 in longitude) to get ni, the total number of earthquakes for cell i; (2) convert the cumulative n i to its incremental value according to Herrman (1977), which is the number of events from M to M + AM in cell i; (3) use the b-value to convert the incremental number of events for M -> 4 to the number for M => ; (4) smooth and normalize the incremental values and denote as/~i; (5) bin the/~i values for all the cells within the area defined by the same distance Dk to the site plus a small distance increment, and denote this partial sum of ai from an annulus centered at the site as Ark; and (6) calculate the annual rate 2(u > u) of ground motion u exceeding u according to the following sum over distance and magnitude: ~(u > Uo) = Z Z l O[l g(nk/~) -- bm'] p( u > UolDk, M I), k(o<dtc<dx) I(Mo<--MI<--Mx) where T is the time in years of the earthquake catalog used and P(u > uold k, Ml) is the conditional probability that u will exceed Uo when an earthquake of magnitude MI occurs at a distance D k from the site. This conditional probability is dependent on the attenuation relation and its uncertainties. We use the formula and parameters of Boore et al. (1993, 1994) for the random component of peak ground acceleration on soil sites of type C (alluvium). The factor before P(u > uoldk, Mr) is the annual rate of earthquakes in the distance bin k and magnitude bin l (Frankel, 1995a). In this term, we replaced b(ml - Mo) in Frankel (1995a) by bmb The summations over k and l cover all the distance and magnitude contributions to the site; k is bounded by an arbitrary distance Dx, which is large enough for an event of magnitude Mx at this distance to have no effect on the calculation (5 km is used in this article); and 1 is bounded by M (= 5.) and Mx ( = 6.5). The importance of b-value to hazard assessments is easily seen from equation (1). The number Ark is relatively certain despite its Poissonian fluctuation and dependency on smoothing function; but a low b-value will overestimate the hazard, and a high b-value will underestimate the hazard. Ideally, the data would allow us to obtain a b-value distribution over the grid cells with smaller variance. Unfortunately, there are not sufficient data to calculate statistically significant b-values over areas in about a 1-km dimension in southern California for magnitudes 4 or greater. Therefore, we use a constant b-value for all of southern California. Figure 1 shows the b-value calculations using Weichert's (198) formulation, which is a maximum likelihood estimate but allows unequal observation periods for different magnitudes. The calculated b-values for southern California for the full catalog and the declustered catalog are.968 and.933, respectively. The value for the full catalog is consistent with the value given by Hileman et al. (1973). The declustering procedure reduces the b-value and the rate of occurrence slightly by removing mainly aftershocks. a-value Smoothing Functions To show the significance of the spatial smoothing functions on the hazard calculation, we compare two such functions: the l/r (WGCEP, 1995) and the Gaussian (Frankel, 1995a) distribution functions. Both functions represent the probability, out to some maximum distance r m ( = 25 km), for a future earthquake to occur at a distance r from an earlier (1)

3 1374 T. Cao, M. D. Petersen, and M. S. Reichle rr -~ <._> E 1-1 b =.933 +] /-.23 \\ 1"2 ] [] 1.3 I..,... t..., Magnitude.8.6 "~.4-.2 (l/r) ~aussian Figure 1. Calculated b-values from declustered (squares) and complete (triangles) catalogs for southern California with least-squares fitting to each data set. earthquake. In the Gaussian function, the probability of future earthquakes is proportional to exp( - r2/c2), where c is a characteristic correlation distance. For the 1/r function, a characteristic distance does not exist. Figure 2 is a schematic diagram showing the relation between these two smoothing functions. Also, we compare two other extreme smoothing functions: the uniform distribution function and the delta function. The uniform distribution function will smooth out any spatial inhomogeneities, and the delta function preserves any inhomogeneities. The area under each curve has been normalized. Figure 2 shows that the Gaussian function has more weight between 1 to 7 km than the 1/r function, and beyond 7 km the weighing relation is reversed. Within 1 km, Figure 2 shows a higher probability for the 1/r function, but actually it does not produce high a-value peaks since we chose a grid-size dimension of about 11 km and since high peaks are averaged out over each grid. In order to compare the differences of the above two functions on spatial a-value smoothing, we apply them to the southem California seismicity distribution from 1932 to 1994 with magnitude equal to or greater than 4. The seismicity catalog is compiled by Petersen (written comm.) in local magnitude ML. However, the various magnitude measurements (M L, M w, and Ms, for example) are roughly the same for magnitude below 6.5 (Bullen and Bolt, 1985; Reiter, 199). Therefore, for those events having magnitudes equal to or greater than 6 in the Petersen catalog, we replace the ML magnitudes by the Mw magnitudes listed in Ellsworth. I t -1 O O Distance r (km) Figure 2. A schematic diagram for different smoothing functions (distance correlation density functions). The four curves indicate the uniform, Gaussian, power-law, and delta function distributions. (199) and Hauksson et al. (1994). With these alterations, the magnitudes in the Petersen catalog approximate moment magnitude. Next we perform the declustering procedure to remove aftershocks according to Reasenberg (1985), so that the Poissonian assumption applies (i.e., independence among events). Finally, we smooth the a-value map using l/r and Gaussian functions. The smoothed a-value maps obtained by using these two smoothing functions are shown in Figures 3a and 3b. Because the a-value is incremental, we contour the number of events within (-hak//2, Z~JI///2) at magnitude for each grid cell. The software for equal interval contouring in this article is provided by Wessel and Smith (1991). The total number of events within the map area is kept the same for Figures 3a and 3b through normalization. The b-value used is.93 (see Fig. 1). The smoothed a-value maps show significant differences between the two smoothing functions (Figs. 3a and 3b). In the high seismicity areas, smoothing with a Gaussian function (Fig. 3a) produces higher a-value areas than the 1/r function. In the lower seismicity areas, the 1/r function (Fig. 3b) gives slightly higher a-values. These differences can be explained by the different nature of the two smoothing functions described above and shown in Figure 2.

4 Seismic Hazard Estimate from Background Seismicity in Southern California 1375 (a) 36 1 " 35 ~ 34"N 33"N 32 N 121 W 12 W 119"W 118 W 11TW 116 W 115*W (b) 36o,,, 35"1' 34"h 33 N 32 N 121 W 12 W 119 W 118 W 117 W 116 W 115 W Figure 3. Maps of smoothed a-values. The values of 1 ~ in incremental magnitude AM =.1 for each grid (--.1 by.124 ) are contoured. Some of the faults in this area and all the earthquakes from 1932 to 1994 with magnitudes equal to or greater than 4 (solid triangles) are plotted for reference. The smoothing functions (correlation density functions) are (a) Ganssian and (b) 1/r distributed. a-value Smoothing Function for Southern California Because the purpose of smoothing is to better predict the locations of future earthquakes, we analyze the number of pairs of events N(r) as a function of distance r to determine a smoothing function appropriate for southern California. We call this function a distance correlation density function C(r), and it is defined as follows: C(r) = d[n(r)/r2]/dr. (2) distance r follows a power law N(r) -- r, as shown by many studies (e.g., Kagan, 1994; Mandelbrot, 1983), then we have: C(r) = l/r ~, (3) where/t = 3 - D, and a constant factor has been dropped from (2) to (3) because we are only interested in fitting the above form (3). Notice that because dn(r)/dr = Dr - 1, we can rewrite (2) as If we assume N(r), the number of pairs within epicentral pair C(r) = (dn(r)/dr)/r 2, (4)

5 1376 T. Cao, M. D. Petersen, and M. S. Reichle i i i where a constant factor has again been dropped from (2) to (4). With (3) and (4), we can simply bin the pairs within pair 8 distances r to r + dr and then normalize it by r 2 to get C(r). m(4.-4.5) Then kt is determined by curve fitting according to (3). In practice, we take the logarithm of (3) to make/2 a linear ~>'8- function of the data. A straight line is fit to the data using "~,- Q subroutine FIT from Press et al. (1992). a The distance correlation density function defined above ~ 4ois very similar to the correlation integral function defined by "~ Volant and Grasso (1994). The only difference is that the i.. correlation integral function is cumulative, and the correla- o O 2 tion density function is incremental. The incremental version is preferred to show the probability of occurrence of future events as a function of distance. It is a probability density function for pair distance distribution without normalization. 1' 2' llrv,...l For southern California earthquake data, such a function Distance r (kin) is shown in Figure 4. When the data are fitted by a function 1/r ~, we get/2 = 1.2 for all the events with magnitudes 4 (b) to 4.5 (Fig. 4a), which is close to the 1/r function of the 12 WGCEP (1995). When the magnitude range is raised from 4. to 4.5 to 4.5 to 5., the estimate of/2 increases to m(4.5-5.) (Fig. 4b). This means that the larger events are more spatially concentrated than the smaller ones. Hence we propose 1/r ~ as the a-value spatial smoothing function with/2 linearly n depending on magnitude (/2 = e + fm, e and fare constants ~- -.5 and.4). Because the number of earthquakes decreases rapidly with increasing magnitude (b =.93), this modification does not introduce significant change from the ~ l/r function on the a-value smoothing. However, it does reflect the spatial concentration of large events, which is supported by observations (e.g., Volant and Grasso, 1994). (a) to I" 8 O 6..~ o 4 O 2 Background Seismicity Hazard Maps Figure 5a is the calculated ground-motion map for 1% probability of exceedance in 5 yr for southern California from background seismicity with magnitude from 5. to 6.5. The a-value smoothing function used is 1/r ~ with/2 a linear function of magnitude. This map predicts that the ground motion due to background seismicity for most areas in southern California ranges between.2 and.35 g. Two higherhazard areas are the Big Bend of the San Andreas fault near Tejon Pass and the southern section of the San Andreas fault system. The seismic activity associated with these higherhazard areas are on the Pleito-White Wolf fault system, and the San Gorgonio bend, especially between the Mission Creek fault and the Banning fault (Hill et al., 199). The ground-motion estimates from background seismicity are high but still less than half of the levels of hazards that are obtained for the entire seismic source model that includes geologic and geodetic data (e.g., Petersen et al., 1996). We also calculated a ground-motion map using the simpler 1/r smoothing function (Fig. 5b). The resulting hazard map appears very similar to Figure 5a. The changes are small. A close comparison of the two maps shows that the O Distance r (kin) Figure 4. Correlation density functions from the southem California catalog fitted by a power-law function 1/r ~. (a) is for earthquakes with magnitudes from 4. to 4.5; (b) is for earthquakes with magnitudes from 4.5 to 5.. t is magnitude dependent. size of the areas with ground-motion reaching.35 g are slightly smaller for 1/r smoothing (Fig. 5b) than in Figure 5a. This difference is due to the contributions from largermagnitude events that have higher # values in the powerlaw smoothing function. Generally, the new smoothing function proposed in this study will slightly raise the weight of large event contributions to the ground-motion calculation compared to the 1/r smoothing function. In order to study the influence of source zones (e.g., WGCEP, 1995) on the hazard, the smoothed a-values are averaged over each source zone of the WGCEP (1995), and then the hazards are calculated. Figure 6 is the source zones map from the WGCEP (1995), and Figure 7 is the calculated hazard map with a-values averaged over each zone. There are some differences between the two hazard maps with and

6 Seismic Hazard Estimate from Background Seismicity in Southern California 1377 (a) 36 t ' 35 1k 34 N 33 N 32*N 121 W 12 W 119 W 118 W 117 W 116 W 115 W (b) 36"1" 35 ~ 34 1k 33 N 32 N 121 W 12 W 119 W 118 W 117 W 116 W 115 W Figure 5. Calculated ground-motion map for peak accelerations with 1% probability of exceedance in 5 yr. (a) Power-law smoothing function 1/r ~ is used for southern California earthquakes from 1932 to 1994 with M = 5. and M x = 6.5. Some of the faults and all the earthquakes with magnitudes equal to or greater than 5. are also plotted for reference. The unit for contouring is in g. (b) The llr smoothing function is used. without source zones (Figs. 7 and 5a). In Figure 7, those areas with higher ground-motion levels reaching.3 to.35 g are smaller, and the areas with lower ground motion (.2 to.25 g) are larger compared with Figure 5a. Another significant difference is that the ground-motion contours are influenced by the source zone shapes, especially by the big zones (e.g., 52C, 53C, 56C, 57C, 58C, 6C, 61C, 62C, 63C, and 64C). However, the ground-motion levels are generally the same using either seismic source zones or point sources. The primary differences between the hazard maps (Figs. 7 and 5a) are due to the geometry of the zones. Uncertainties The uncertainties involved in the above hazard map calculations can be divided into two types, the random (aleatory) and the model (epistemic) uncertainties (Reiter, 199). In the above calculations, only one attenuation relation is used, so that the a-values, which follow a Poisson process for a declustered catalog, the b-values, and the attenuation relation all have only random uncertainties. In contrast, the choice of the high- and low-cutoff magnitudes, M o and M x, introduces model uncertainties. Before calculating the overall uncertainties for the haz-

7 1378 T. Cao, M. D. Petersen, and M. S. Reichle Figure 6. Seismic source zones from the WGCEP (1995). 36 N 35 N 34 N 33"N - 32 N t 121"W 12"W 119 W 118"W 117"W 116 W 115"W Figure 7. Calculated ground-motion map based on the seismic source zones of Figure 6. The a-values are averaged over each zone before the hazard calculations. ard map (Fig. 5a), we first test the sensitivities for parameters b, Mo, and Mx on the hazard. Assuming these parameters follow normal distributions with standard deviations.29,.5, and.5, respectively, we have b = , Mo = , Mx = 6.5 _+.5. Figures 8a, 8b, and 8c show the effect of assuming variability in the selected b-value, low-cutoff, and high-cutoff magnitudes on annual probability of exceedance versus peak acceleration at an arbitrarily chosen location of Los Angeles (34.5 N, W). We see that the high-cutoff magni- (5) tude has the least effect on the hazard calculations (Fig. 8c), which may explain why the SCEC model (WGCEP, 1995) does not produce very high levels of ground motions when using a high-cutoff magnitude 8.22 for the distributed earthquakes. The calculated hazard is not a linear function of Mx. This is due to the attenuation relation, for which the peak values follow a lognormal distribution. Increasing the highcutoff magnitude has about half of the effect on peak acceleration as decreasing the high-cutoff magnitude by the same amount (Fig. 8c). The b-value change has only a moderate effect on calculated hazards, and the relation is nearly linear (Fig. 8a). The calculations are most sensitive to the lowcutoff magnitude. This effect is nonlinear (Fig. 8b). We choose 5. as the low-cutoff magnitude, which is considered a conservative lower bound (Reiter, 199).

8 Seismic Hazard Estimate from Background Seismicity in Southern California 1379 (a) 1 "2 (b) 1 "2,...,,,,,, c "O (1) O X W (D 1 a b= c -(3 O X 1"3 W M =5"5 _Q _ t~ E E < O 1-4 n m c (- < Peak Acceleration (g) 1 1-~ 1-1 Peak Acceleration (g) 1 o (c) 1 "2 c " x W 1.3 Mx=6"O '7 C c < Peak Acceleration (g) Figure 8. Sensitivity analysis for parameters b, Mo, and Mx. We see that (a) b is moderately sensitive, (b) Mo is most sensitive, and (c) M~ is least sensitive among these three parameters.

9 138 T. Cao, M. D. Petersen, and M. S. Reichle Since the a-value changes from place to place, a sensitivity study cannot be performed similar to b, M, and Mx. Instead, the a-value sensitivity is incorporated directly in the overall uncertainty analysis. There are two end-member methods of modeling a-value temporal fluctuations (Poissonian) over a certain area. The first model assumes that all of the a-values in southern California fluctuate up or down together or coherently; the second model assumes that a- values for each grid cell (about.1 by.1 in size) fluctuate independently. From the seismic history of southern California, we know that the former model overestimates the spatial coherent fluctuation, and the latter model underestimates the spatial coherent fluctuation. According to the a- value smoothing function in Figure 4, the probability of spatial coherent seismicity fluctuation beyond 1 km is very low. Therefore, we assume that 1 by 1 is a reasonable size for coherent a-value spatial fluctuation. Then, we follow Weichert's (198) method to calculate a-value standard deviation for each grid cell in a 1 by 1 area. If the total number of events in a 1 by 1 area is K, the one standard deviation is ]~7 for a Poissonian process. For a grid cell i in this area, the total number of events in the cell is hi, and the one standard deviation for this cell is fk divided by the total number of cells nc in this area or fk/nc. This method is also used by Frankel (1995b). Since all the parameters that will be included in the uncertainty analysis are assumed to follow continuous normal distributions, we adopt a Monte Carlo simulation (Bernreuter et al., 1989) to calctilate the overall uncertainties. A recent application of this method to estimate uncertainty for seismic hazards is by Cramer et al. (1995). Their study shows that a minimum of 1 simulations is needed to obtain stable estimates of uncertainty; 12 simulations are used in this study. The simulated ground-motion results show that the coefficient of variation, which is the ratio of the standard deviation to the mean value, is for most of southern California. The overall uncertainty is up to +_ 5% or about.15 g (two standard deviations). Discussion and Conclusions In this study, we have compared different seismicity smoothing functions and have studied their effect on seismicity patterns. We have studied the statistics of spatial seismicity distributions and found that a power-law smoothing function 1/r ~ seems a natural modification to the l/r function. We also found that the value of t is magnitude dependent. The value of/z may change due to the data and numerical method used to fit the data; its magnitude dependence is strongly suggested by the seismicity data itself. However, a power-law smoothing function may not be suitable for areas such as the eastern United States, where the seismicity patterns are very different from southern California. In southern California, the seismicity is well recorded and has much shorter return periods than the eastern United States. We need less smoothing in such high seismicity areas (Fig. 1) because nature is doing its own smoothing. In California, a power-law smoothing is sufficient, while a Gaussian function will provide too much smoothing. On the other hand, the sparse distribution of events with magnitude 4 or greater in the eastern United States suggests that more smoothing is needed and that a Gaussian function may be a better choice (Frankel, 1995a). Our comparison of the hazard maps obtained using background seismicity, with and without seismic source zones, shows that in certain areas, especially in areas with very big zones, the zones may introduce undesirable distortions to the hazard contours. Our preliminary suggestion is that for areas with well-recorded seismicity patterns, seismic source zones may not be necessary in calculating the background seismicity contribution to seismic hazards. Without the source zone, the calculation is much simpler, the calculated hazards follow recorded seismicity, and the results are similar to those obtained by introducing zones. Again, this suggestion may not apply to the eastern United States. This study indicates that we can use a simple approach to calculate seismic hazard due to background seismicity in southern California, in which a power-law seismicity smoothing function and point sources are used without introducing seismic source zones. These results are particularly important for mapping ground motion in California in areas that are not close to known active faults. 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