Attenuation Relations Page 1 May 2005 – I. M. Idriss ATTENUATION RELATIONSHIPS The enclosed three Attachments summarize a number of attenuation relationships currently available to estimate earthquake ground motions in Western North America (WNA), Eastern North America (ENA), and for earthquakes occurring on subduction zones. These attenuation relationships provide estimates of earthquake ground motions at "rock sites". ¾ Attachment 1 pertains to earthquake ground motions in Western North America (WNA). ¾ Attachment 2 pertains to earthquake ground motions in Eastern North America (ENA). ¾ Attachment 3 pertains to earthquake ground motions generated by earthquakes occurring in subduction zones. Note: This document was prepared for distribution at the Refresher Course on "Seismic Analysis and Retrofitting of Lifeline Buildings in Delhi, India", held at the Auditorium of the Delhi Secretariat Building on May 26, 2005.Attenuation Relations Page 1-1 May 2005 – I. M. Idriss WNA ATTACHMENT 1 ATTENUATION RELATIONSHIPS FOR MOTIONS IN WESTERN NORTH AMERICA (WNA) The attenuation relationships derived by Abrahamson and Silva (1997), by Boore, Joyner and Fumal (1997), by Campbell (1997), by Idriss (2002), and by Sadigh et al (1997), are summarized in this Attachment. 1.1 ATTENUATION RELATIONSHIPS DERIVED BY ABRAHAMSON & SILVA (1997) The functional form adopted by Abrahamson and Silva for spectral ordinates at rock sites is the following: ()()()()1rup 3 4rupLn y f M,r Ff M HWf M,r=++ [1-1] in which, y is the median spectral acceleration in g (5% damping), or peak ground acceleration (pga), in g's, M is moment magnitude, rupr is the closest distance to the rupture plane in km, F is the fault type (1 for reverse, 0.5 for reverse/oblique, and 0 otherwise), and HW is a dummy variable for sites located on the hanging wall (1 for sites over the hanging wall, 0 otherwise). The function ()1rupfM,r is given by. for 1Mc≤ ()()()()()n1rup12 112 313 1fM,r a aMc a 8.5M a a Mc LnR⎡⎤=+ − + − + + −⎣⎦ for 1Mc≥ ()()()()()n1rup14 112 313 1fM,r a aMc a 8.5M a a Mc LnR⎡⎤=+ − + − + + −⎣⎦ [1-2] Note that 22rup 4Rr c=+ in Eq. [1-2}. The function ()3fM is described below: ()35fM a= for M5.8≤ ()()65351aafM ac5.8−=+− for 15.8 M c<< ()36fM a= for 1Mc≥ [1-3] The function ()4rupfM,r is assumed to consist of the product of the following two functions, namely: ()()()4 rup HW HW rupfM,r f Mf r= [1-4]Attenuation Relations Page 1-2 May 2005 – I. M. Idriss WNA The function ()HWfM is evaluated as follows: ()HWfM0= for M5.5≤ ()HWfMM5.5=− for 5.5 M 6.5<< ()HWfM1= for M6.5≥ [1-5] The function ()HW rupfr is given by: ()HW rupfr 0= for rupr4< ()rupHW rup 9r4fr a4−⎛⎞=⎜⎟⎝⎠ for rup4r 8<< ()HW rup 9fr a= for rup8r 18<< ()rupHW rup 9r18fr a17−⎛⎞=−⎜⎟⎝⎠ for rup18 r 24<< ()HW rupfr 0= for rupr25< [1-6] The standard error terms are given by the following equations: ()total 5Mbσ= for M5≤ ()()total 5 6MbbM5σ=− − for 5M7<< ()total 5 6Mb2bσ=− for M7≥ [1-7] Values of the coefficients 16913145a .. a , a .. a , c , c , c , and n are listed in Table 1-1 for periods ranging from T = 0.01 sec (representing zpa) to T = 5 sec. Values of the coefficients 5b and 6b are listed in Table 1-2 for the same periods.Attenuation Relations Page 1-3 May 2005 – I. M. Idriss WNA Table 1-1 Coefficients for the Median Spectral Ordinates Using Equations Derived by Abrahamson and Silva (1997) Period 4c 1a 2a 3a 4a 5a 6a 9a 10a 11a 12a 0.01 5.60 1.640 0.512 -1.1450 -0.144 0.610 0.260 0.370 -0.417 -0.230 0.0000 0.02 5.60 1.640 0.512 -1.1450 -0.144 0.610 0.260 0.370 -0.417 -0.230 0.0000 0.03 5.60 1.690 0.512 -1.1450 -0.144 0.610 0.260 0.370 -0.470 -0.230 0.0143 0.04 5.60 1.780 0.512 -1.1450 -0.144 0.610 0.260 0.370 -0.555 -0.251 0.0245 0.05 5.60 1.870 0.512 -1.1450 -0.144 0.610 0.260 0.370 -0.620 -0.267 0.0280 0.06 5.60 1.940 0.512 -1.1450 -0.144 0.610 0.260 0.370 -0.665 -0.280 0.0300 0.075 5.58 2.037 0.512 -1.1450 -0.144 0.610 0.260 0.370 -0.628 -0.280 0.0300 0.09 5.54 2.100 0.512 -1.1450 -0.144 0.610 0.260 0.370 -0.609 -0.280 0.0300 0.1 5.50 2.160 0.512 -1.1450 -0.144 0.610 0.260 0.370 -0.598 -0.280 0.0280 0.12 5.39 2.272 0.512 -1.1450 -0.144 0.610 0.260 0.370 -0.591 -0.280 0.0180 0.15 5.27 2.407 0.512 -1.1450 -0.144 0.610 0.260 0.370 -0.577 -0.280 0.0050 0.17 5.19 2.430 0.512 -1.1350 -0.144 0.610 0.260 0.370 -0.522 -0.265 -0.0040 0.2 5.10 2.406 0.512 -1.1150 -0.144 0.610 0.260 0.370 -0.445 -0.245 -0.0138 0.24 4.97 2.293 0.512 -1.0790 -0.144 0.610 0.232 0.370 -0.350 -0.223 -0.0238 0.3 4.80 2.114 0.512 -1.0350 -0.144 0.610 0.198 0.370 -0.219 -0.195 -0.0360 0.36 4.62 1.955 0.512 -1.0052 -0.144 0.610 0.170 0.370 -0.123 -0.173 -0.0460 0.4 4.52 1.860 0.512 -0.9880 -0.144 0.610 0.154 0.370 -0.065 -0.160 -0.0518 0.46 4.38 1.717 0.512 -0.9652 -0.144 0.592 0.132 0.370 0.020 -0.136 -0.0594 0.5 4.30 1.615 0.512 -0.9515 -0.144 0.581 0.119 0.370 0.085 -0.121 -0.0635 0.6 4.12 1.428 0.512 -0.9218 -0.144 0.557 0.091 0.370 0.194 -0.089 -0.0740 0.75 3.90 1.160 0.512 -0.8852 -0.144 0.528 0.057 0.331 0.320 -0.050 -0.0862 0.85 3.81 1.020 0.512 -0.8648 -0.144 0.512 0.038 0.309 0.370 -0.028 -0.0927 1 3.70 0.828 0.512 -0.8383 -0.144 0.490 0.013 0.281 0.423 0.000 -0.1020 1.5 3.55 0.260 0.512 -0.7721 -0.144 0.438 -0.049 0.210 0.600 0.040 -0.1200 2 3.50 -0.150 0.512 -0.7250 -0.144 0.400 -0.094 0.160 0.610 0.040 -0.1400 3 3.50 -0.690 0.512 -0.7250 -0.144 0.400 -0.156 0.089 0.630 0.040 -0.1726 4 3.50 -1.130 0.512 -0.7250 -0.144 0.400 -0.200 0.039 0.640 0.040 -0.1956 5 3.50 -1.460 0.512 -0.7250 -0.144 0.400 -0.200 0.000 0.664 0.040 -0.2150 The coefficients 13a = 0.17, 1c = 6.4, 5c = 0.03, and n = 2 for all periods.Attenuation Relations Page 1-4 May 2005 – I. M. Idriss WNA Table 1-2 Coefficients for Standard Error Terms Using Equations Derived by Abrahamson and Silva (1997) Period - sec 5b 6b 0.01 0.70 0.135 0.02 0.70 0.135 0.03 0.70 0.135 0.04 0.71 0.135 0.05 0.71 0.135 0.06 0.72 0.135 0.075 0.73 0.135 0.09 0.74 0.135 0.1 0.74 0.135 0.12 0.75 0.135 0.15 0.75 0.135 0.17 0.76 0.135 0.2 0.77 0.135 0.24 0.77