EFFECTS OF RUPTURE DIRECTIVITY ON PROBABILISTIC SEISMIC HAZARD ANALYSIS Norman A. Abrahamson ABSTRACT For long period structures such as bridges that are near faults with high activity rates, it can be important to explicitly include directivity effects in the attenuation relations for either probabilistic or deterministic analyses. There are two rupture directivity effects. The first effect is a change in strength of shaking of the average horizontal component of motion, and the second effect is the systematic differences in the strength of shaking on the two horizontal components oriented perpendicular and parallel to the strike of the fault. The new San Francisco Oakland Bay Bridge is used as an example to show the effects of including rupture directivity in probabilistic seismic hazard analysis. For this example, including both effects results in about a 30% increase in the T=3 second spectral acceleration for a 500 year return period as compared to traditional analyses. Introduction Rupture directivity effects can lead to large long period pulses in the ground motion. Recently, models have been developed to quantify the directivity effect (e.g. Somerville et al, 997). With these models of the rupture directivity effect, directivity can be included in either deterministic or probabilistic seismic hazard analyses. This paper demonstrates the effect of rupture directivity on probabilistic seismic hazard analyses. Attenuation Relations and Rupture Directivity For design of long-period structures such as bridges, characterization of long-period motion is essential. Attenuation relations commonly used in California do not explicitly include rupture directivity effects but they can adjusted to account for near-fault directivity effects using the Somerville et al. (997) fault-rupture directivity model. The Somerville et al. (997) model comprises two period-dependent scaling factors that may be applied to horizontal attenuation relationship. One of the factors accounts for the change in shaking intensity in the average horizontal component of motion due to near-fault rupture directivity effects (higher ground motions for rupture toward the site and lower ground motions for rupture away from the site). The second factor reflects the directional nature of the shaking intensity using two ratios: fault normal (FN) and fault parallel (FP) versus the average (FA) Pacific Gas and Electric Company, 245 Market Street, Mail Code N4C, San Francisco, CA 9405
component ratios. The fault normal component is taken as the major principal axis resulting in an FN/FA ratio larger than and the fault parallel component is taken as the minor principal axis with an FP/FA ratio smaller than. The two scaling factors depend on whether fault rupture is in the forward or backward direction, and also the length of fault rupturing toward the site. There are several aspects of the empirical model for the average horizontal component scale factors developed by Somerville et al that needed to be modified to make the model applicable to a probabilistic hazard analysis. As published, the directivity model is independent of distance. The data set used in the analysis includes recordings at distances of 0 to 50 km. A distance dependent taper function was applied to the model that reduces the effect to zero for distances greater than 60km. T d (r) = for r < 30 km - (r-30)/30 for 30 km < r < 60 km () 0 for r > 60 km As published, the model is applicable to magnitudes greater than 6.5. A magnitude taper was applied that reduces the effect to zero for magnitudes less than 6.0. T m (m) = for m = 6.5 - (m-6.5)/0.5 for 6 = m < 6.5 (2) 0 for m < 6 The empirical model uses two directivity parameters: x and θ where x is defined as the fraction of the fault length that ruptures toward the site and θ is the angle between the fault strike and epicentral azimuth (see Somerville et al for details). The worst case is x= and θ=0. The empirical model uses a form that increases a constant rate as x increases from 0 to. There is little empirical data with x cos(θ) values greater than 0.6, and the extrapolation of the model to larger x cos(θ) values is not well constrained. Based on an evaluation of empirical recordings and numerical simulations, the form of the directivity function is modified to reach a maximum at x cos(θ) =0.4. The slope is greater than the Somerville model, but it flattens out at a lower level. The T=3 second value is used to guide the adjustment of the model at all periods. The resulting model is given by y Dir (x,θ,τ) = C (T) +.88 C 2 (T) x cos(θ) for x cos(θ) = 0.4 (3) C (T) + 0.75 C 2 (T) for x cos(θ) > 0.4 where C (T) + C 2 (T) are from Somerville et al. (997), and are listed in Table.
Table. Model Coefficients for the Modified Somerville et al. (997) Directivity Effects for the Average Horizontal Component Period (sec) C C2 0.60 0.000 0.000 0.75-0.084 0.85.00-0.92 0.423.50-0.344 0.759 2.00-0.452 0.998 3.00-0.605.333 4.00-0.73.57 5.00-0.797.757 Finally, including the directivity effect should results in a reduction of the standard deviation of the attenuation relation. Based on an evaluation of the empirical data, at T=3 seconds, there is a reduction of the standard deviation of about 0.05 natural log units due to adding the directivity term into the ground motion model. The period dependence of the reduction was approximated by the period dependence of the slope of the directivity effect. To account for the reduction in the standard deviation due to including the directivity effect as part of the model, the standard deviations for the published attenuation relations were modified for use in the hazard analysis using the following relation: σ dir (M,T) = σ (M,T) - 0.05 C 2 (T)/.333 (4) where C 2 (T) is given in Table and σ(m,,t) is the standard deviation from the published attenuation relation (without directivity effects). The final modified Somerville model for the average horizontal component for strike-slip faults is given by ln Sa dir (M,r,x,θ,Τ) = ln Sa(M,r) + y Dir (x,θ,τ) T d (r) T m (m) (5) where Sa(M,r) is an empirical attenuation relation without directivity. This modified model is shown in Figure a. The FN/Avg ratios are shown in Figure b for the modified Somerville et al. (997) model. Incorporating Rupture Directivity Into Probabilistic Seismic Hazard Analysis It is straightforward to include rupture directivity into a probabilistic hazard analysis. The main change is that the location of the hypocenter on the rupture area needs to be included as an additional source of (aleatory) variability (Figure 2). In a standard hazard calculation, the hazard is given by: ν i (A > z) = N i (M min ) W = 0 RA =0 Ex= 0 Ey M max i m = M min f mi (m) f Wi (m,w ) f RAi (m, RA) f Exi (x) f Eyi (m,x) P(A > z m,r i (x,y,ra,w))dw dradx dy dm (6)
where N i (M min ) is the rate of earthquakes with magnitude greater than M min from the i th source; m is magnitude; M maxi is the maximum magnitude (for the i th source); fm(m), fw(m,w), fra(m,ra), fex, and fey are probability density functions for the earthquake magnitude, rupture width, rupture area, location of the center of rupture along strike and location of the center of rupture down dip, respectively; and P(A>z m,r,ε) is the probability that the ground motion exceeds the test level z for a given magnitude and distance. Including directivity effects for strike-slip faults, the hazard is given by ν i (A > z) = N i (M min ) W = 0 RA =0 Ex= 0 Eyhx= 0 m = M min M max i f mi (m) f Wi (m,w) f RAi (m,ra) f Exi (x) f Eyi (m, x) f hx (h x ) P(A > z m,r, X,θ)dW dradx dy dh x dm (7) where r, X, and θ are computed from the rupture location and the hypocenter location. In this calculation, we need to define and additional probability density function for the location of the hypocenter on the rupture, f hx (h x ). Here, I have assumed uniform distributions for the hypocenter location (e.g. no preferred locations for the nucleation of the rupture). It could be argued that the effect on directivity on the average horizontal component is already included in the standard deviation of the ground motion of the published attenuation relations. There are two reasons why the published standard deviations do not adequately account for directivity effects. First, the standard deviation of most attenuation relations is averaged over all distances. As a result, the standard deviation of the near fault ground motion is underestimated by the average standard deviation over all distances. Second, the size of the directivity effect can vary significantly for different locations that are the same distance away from the fault. That is, a particular site-fault geometry may be more likely to experience forward directivity effects than other sites. Example Calculation As an example, the probabilistic seismic hazard is computed with and without directivity. As discussed above, we can consider the effects of directivity on the average horizontal component or on the individual components oriented parallel and perpendicular to the strike of the fault. The base case is using the attenuation relations as published. These are for the geometric mean of the two horizontal components. This case is label as without directivity in Figure 3. Including just the effect of directivity on the average horizontal component increases the hazard at return periods greater than 500 years. The hazard with directivity can be deaggregated in terms of the rupture directivity parameter, Xcos(θ). The deaggregation at the 500 year return period is shown in Figure 4. This figure shows that most of the long period ground motion hazard is from forward directivity cases.
Deaggregation on the directivity can guide the selection of time histories in terms of selecting forward directicity time histories vs neutral or backward directivity time histories. Conclusions Rupture directivity has not been included in most probabilistic seismic hazard calculations. To accurately estimate the hazard from long period ground motions, directivity should be directly included in the hazard analysis as one of the important sources of variability of the long period ground motion. I believe that including rupture directivity will soon become the standard of practice for computing the hazard for long period ground motions near active faults. References Somerville, P. G., N. F. Smith, R. W. Graves. and N. A. Abrahamson (997). Modification of empirical strong ground motion attenuation relations to include the amplitude and duration effects of rupture directivity, Seism. Res. Let, Vol. 68, 99-222. QGa Directivity Model 0.8 0.6 0.4 0.2 0-0.2-0.4-0.6-0.8 - Modified Somerville et al. (997) directivity model Ln(FN/Avg) 0.05 a. 0 0 0. 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0 0 20 30 40 50 60 70 80 90 Xcos(Theta) Theta Figure. Modified Somerville et al. (997) rupture directivity model. 0.3 0.25 0.2 0.5 0. b. Somerville et al. (997) FN/Average Horizontal Model
QGa 0 Depth (km) 5 0 5 0 0 20 30 40 50 60 70 80 90 00 Distance (km) Figure 2. Example of the variability in hypocenter location over the rupture surface for a single rupture location. The heavy line defines the rupture area of an earthquake scenario, the circle indicates the center of the rupture, and the stars indicate the range of hypocenters. QGa Ave Horizontal without Directivity Annual Probability of Exceedance 0. 0.0 0.00 Fault Parallel Component Average Horizontal Component Fault Normal Component 0.000 0 0.05 0. 0.5 0.2 0.25 0.3 0.35 0.4 0.45 0.5 Spectral Acceleration (g) Figure 3. Effects of directivity on the hazard for T=3 seconds spectral acceleration for the new San Francisco Bay Bridge. Figure 4. Deaggregation of the hazard for a spectral period of 3.0 seconds at the 500 year return period.
KCampbell@eqecat.com To "Gilles Bureau" <gbureau@geiconsultants.com> 03/03/20050:2 AM ee john_barneich@geopentech.com, kcampbell@absconsulting.com, Nancy_ Collins@urscorp.com, bee Subject Re: Correction for fault normal I fault parallel effects Gilles, Note that there is an error in the strike-slip equation that Norm gives in his paper (personall corom., 2003). The equation (in different format) should be: (Embedded image moved to file: pic03902.jpg) Ken Kenneth W. Campbell, PhD Vice President EQECAT Inc (503) 533-4359 (503) 533-4360 (fax) kcampbell@eqecat.com http://www.eqecat.com "Gilles Bureau" <gbureau@geiconsu Itants.com> To <kcampbell@absconsulting.com>, 03/03/2005 08:55 <KCampbell@eqecat.com>, AM <john_barneich@geopentech.com>, <Phalkun_Tan@geopentech.com>, <yoshi_moriwaki@geopentech.com>, <Nancy_Collins@URSCorp.com> cc <Paul_Somerville@URSCorp.com> Subject Re: Correction for fault normal / fault parallel effects To all: Thank you for the paper and for clarifying this situation. I have deeply appreciated the fast response from Paul and Ken, after learning one was on Australia and the other in Italy.
Best regards, Gilles Gilles Bureau, P.E., G.E. Senior Consultant GEl Consultants, Inc. 220 Broadway, Suite 32 Oakland, CA 9462-307 50-835-9838 x 05 50-835-9842 (fax) gbureau@geiconsultants.com»> <Nancy_Collins@URSCorp.com> 3//2005 5:07:8 PM»> Gentlemen: Attached is the Abrahamson paper of which Paul speaks below (See attached file: Abrahamson_6lCSZ.pdf) Nancy F. Collins, Project Scientist URS Corporation 566 EI Dorado Street, Suite 200 Pasadena, CA 90 (626) 449-7650; FAX (626) 449-3536 Forwarded by Nancy Collins/Pasadena/URSCorp on 03/0/2005 04:52 PM Paul Somerville To: KCampbell@eqecat.com, robert_graves@urscorp.com, 03/0/2005 0:54 nancy_collins@urscorp.com, yoshi_moriwaki@geopentech.com, PM john_barneich@geopentech.com cc: "Gilles Bureau" <gbureau@geiconsultants.com>, kcampbell@absconsulting.com, naa2@pge.com, paul_somerville@urscorp.com fault normal/fault parallel Subject: effects Re: Correction for Gilles et al. Ken has correctly described the situation. My assistant Nancy Collins is going to fax you the paper by Norm Abrahamson that describes the tapering of the model for small magnitude and large distance. Paul
I, Paul Somerville To: KCampbell@eqecat.com, robert-.9raves@urscorp.com, :td::~t- nancy-collins@urscorp.com, yoshi_moriwaki@geopentech.com. -, -', 03/0/20050 :54 PM john_barneich@geopentech.com cc: "Gilles Bureau" <gbureau@geiconsultants.com>, kcampbell@absconsulting.com, naa2@pge.com, paulsomerville@urscorp.com Subject: Re: Correction for fault normal I fault parallel effects Gilles et al. Ken has correctly described the situation. I+/s.o t/-v -j> -PM I~ UlVI JCUA.I /i-j- 79b- 9/9/ My assistant Nancy Collins is going to fax you the paper by Norm Abrahamson that describes the tapering of the model for small magnitude and large distance. Paul Paul G. Somerville URS Corporation 566 EI Dorado Street Pasadena, CA 90-2560 Voice: (626) 449-7650; FAX: (626) 449-3536 email: PauLSomerville@urscorp.com -KCampbell@eqecat.com wrote: ---- To: "Gilles Bureau" <gbureau@geiconsultants.com> From: KCampbell@eqecat.com Date: 03/02005 :42AM cc: kcampbell@absconsulting.com, naa2@pge.com, paulsomerville@urscorp.com Subject: Re: Correction for fault normal/fault parallel effects Gilles, I was in Italy and unable to respond to your question before now. I have carefully reviewed the 997 SRL paper and I believe that I have made a mistake in representing the directivity model in my CRC Handbook (and two other chapters that I have written). Even though they refer to the FN to FP factor in many places throughout the text, including the equations themselves, the authors state that the coefficients listed in the tables are for the FN to AV (average) components. I believe that I was confused by the ambiguity in their statements and I apologize for the error. Therefore, I believe that my Equation (5.06) should not have the /2 in the first two terms, Furthermore, the last term (0) should also apply when M > 6, according to the authors. I also neglected to put that constraint in as well. Furthermore, since the FN and FP relationships are not numerically constrained at M > 6 and Rrup > 50, I would suggest that a magnitude and distance taper similar to the one Norm applied to the spatial directivity term (my f) should also be applied to my f2 term, except that the taper should be applied between M = 5.5 and 6.5, centered on 6.0. Paul and/or Norm, can you please confirm this and respond to my taper suggestion? Ken
Kenneth W. Campbell, PhD Vice President EQECAT Inc (503) 5334359 (503) 5334360 (fax) kcampbell@eqecat.com http://www.egecat.com "Gilles Bureau" <gbureau@geiconsu Itants.com> To <kcampbell@absconsulting.com>, 02/24/20052:52 <naa2@pge.com>, PM <paulsomerville@urscorp.com> cc Subject Correction for fault normal fault parallel effects Gentlemen: Greetings. I am facing an issue with the use of the correction factor for fault normal (FN) I fault parallel effects (FP) for ground motion estimates based on your attenuation equations. Paul and Norm, in their 997, paper talk on page 24 (Seismological Research Letters Vol. 68, No.) about fitting the strike-normal to strike-parallel ratio y (y being defined as the natural logarithm of the strike-parallel to strike normal ratio at a given period), with an equation of the form: y =cos 2? [ci + c2 In(Rrup + ) + c3(mw 6)] However, elsewhere in the text and in Table 6 and 7 of the 977 reference, the c coefficients are defined for the strike-normal to average ratio. In Chapter 5 of the CRe Handbook (2003), Ken uses a similar equation for FN (and specifically refers to Norm's 2000 model) as follows: f2(rrup, Mw,?) =% (cos 2?) [c3 + c4ln(rrup + ) + c5 (Mw - 6)] [5.06] The c3, c4 and c5 are defined in Table 5.25 (Campbell, 2003) and are the
very same values as c, c2 and c3 in the aforementioned Table 7. Hence, a factor % (half the fault-normal effects) has been introduced in Equation 5.06. Equation [5.06] also provides an equal in amplitude but opposite sign correction (-/2 etc.) for FP effects when compared with the average estimates. My question is: How where the c coefficients developed? Do they relate FN to FP, or FN to average? Is a 0.5 factor correctly applied by Ken, or does it result from the language ambiguity regarding y in the 977 paper? Thank you all very much for clarifying how to correctly apply the FN correction. Your help will be greatly appreciated. Best regards to all three of you, Gilles. Gilles Bureau, P.E., G.E. Senior Consultant GEl Consultants, Inc. 220 Broadway, Suite 32 Oakland, CA 9462-307 50-835-9838 x 05 50-835-9842 (fax) gbureau@geiconsultants.com