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".
 Attachment 1 pertains to earthquake ground motions in Western North America (WNA).
 Attachment 2 pertains to earthquake ground motions in Eastern North America (ENA).
 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
May 2005 – I. M. Idriss
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:



Ln  y   f1 M,rrup  Ff3  M   HWf4 M,rrup

[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, rrup 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 f1  M,rrup  is given by.
for M  c1


f1 M,rrup  a1  a2  M  c1   a12  8.5  M   a3  a13  M  c1   Ln  R 
for M  c1
n
[1-2]


f1 M,rrup  a1  a4  M  c1   a12  8.5  M   a3  a13  M  c1   Ln  R 
n
2
 c42 in Eq. [1-2}.
Note that R  rrup
The function f3  M  is described below:
f3  M   a5
f3  M   a5 
for M  5.8
 a6  a5 
for 5.8  M  c1
c1  5.8
f3  M   a6
[1-3]
for M  c1
The function f4  M,rrup  is assumed to consist of the product of the following two functions,
namely:


 
f4 M,rrup  fHW  M  fHW rrup
Attenuation Relations
WNA
Page 1-1
[1-4]
May 2005 – I. M. Idriss
The function fHW  M  is evaluated as follows:
fHW  M   0
for M  5.5
fHW  M   M  5.5
for 5.5  M  6.5
fHW  M   1
[1-5]
for M  6.5
The function fHW  rrup  is given by:
 
fHW rrup  0
for rrup  4
 rrup  4 
fHW rrup  a9 

 4 
 
for 4  rrup  8
 
fHW rrup  a9
for 8  rrup  18
rrup  18 

fHW rrup  a9  1 

7


for 18  rrup  24
 
 
fHW rrup  0
[1-6]
for rrup  25
The standard error terms are given by the following equations:
 total  M   b5
for M  5
 total  M   b5  b6  M  5 
for 5  M  7
 total  M   b5  2b6
for M  7
[1-7]
Values of the coefficients a1 .. a6 , a9 .. a13 , c1 , c4 , c5 , 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 b5 and b6
are listed in Table 1-2 for the same periods.
Attenuation Relations
WNA
Page 1-2
May 2005 – I. M. Idriss
Table 1-1 Coefficients for the Median Spectral Ordinates Using Equations Derived by Abrahamson and Silva (1997)
Period
c4
a1
a2
a3
a4
a5
a6
a9
a10
a11
a12
0.01
0.02
0.03
0.04
0.05
0.06
0.075
0.09
0.1
0.12
0.15
0.17
0.2
0.24
0.3
0.36
0.4
0.46
0.5
0.6
0.75
0.85
1
1.5
2
3
4
5
5.60
5.60
5.60
5.60
5.60
5.60
5.58
5.54
5.50
5.39
5.27
5.19
5.10
4.97
4.80
4.62
4.52
4.38
4.30
4.12
3.90
3.81
3.70
3.55
3.50
3.50
3.50
3.50
1.640
1.640
1.690
1.780
1.870
1.940
2.037
2.100
2.160
2.272
2.407
2.430
2.406
2.293
2.114
1.955
1.860
1.717
1.615
1.428
1.160
1.020
0.828
0.260
-0.150
-0.690
-1.130
-1.460
0.512
0.512
0.512
0.512
0.512
0.512
0.512
0.512
0.512
0.512
0.512
0.512
0.512
0.512
0.512
0.512
0.512
0.512
0.512
0.512
0.512
0.512
0.512
0.512
0.512
0.512
0.512
0.512
-1.1450
-1.1450
-1.1450
-1.1450
-1.1450
-1.1450
-1.1450
-1.1450
-1.1450
-1.1450
-1.1450
-1.1350
-1.1150
-1.0790
-1.0350
-1.0052
-0.9880
-0.9652
-0.9515
-0.9218
-0.8852
-0.8648
-0.8383
-0.7721
-0.7250
-0.7250
-0.7250
-0.7250
-0.144
-0.144
-0.144
-0.144
-0.144
-0.144
-0.144
-0.144
-0.144
-0.144
-0.144
-0.144
-0.144
-0.144
-0.144
-0.144
-0.144
-0.144
-0.144
-0.144
-0.144
-0.144
-0.144
-0.144
-0.144
-0.144
-0.144
-0.144
0.610
0.610
0.610
0.610
0.610
0.610
0.610
0.610
0.610
0.610
0.610
0.610
0.610
0.610
0.610
0.610
0.610
0.592
0.581
0.557
0.528
0.512
0.490
0.438
0.400
0.400
0.400
0.400
0.260
0.260
0.260
0.260
0.260
0.260
0.260
0.260
0.260
0.260
0.260
0.260
0.260
0.232
0.198
0.170
0.154
0.132
0.119
0.091
0.057
0.038
0.013
-0.049
-0.094
-0.156
-0.200
-0.200
0.370
0.370
0.370
0.370
0.370
0.370
0.370
0.370
0.370
0.370
0.370
0.370
0.370
0.370
0.370
0.370
0.370
0.370
0.370
0.370
0.331
0.309
0.281
0.210
0.160
0.089
0.039
0.000
-0.417
-0.417
-0.470
-0.555
-0.620
-0.665
-0.628
-0.609
-0.598
-0.591
-0.577
-0.522
-0.445
-0.350
-0.219
-0.123
-0.065
0.020
0.085
0.194
0.320
0.370
0.423
0.600
0.610
0.630
0.640
0.664
-0.230
-0.230
-0.230
-0.251
-0.267
-0.280
-0.280
-0.280
-0.280
-0.280
-0.280
-0.265
-0.245
-0.223
-0.195
-0.173
-0.160
-0.136
-0.121
-0.089
-0.050
-0.028
0.000
0.040
0.040
0.040
0.040
0.040
0.0000
0.0000
0.0143
0.0245
0.0280
0.0300
0.0300
0.0300
0.0280
0.0180
0.0050
-0.0040
-0.0138
-0.0238
-0.0360
-0.0460
-0.0518
-0.0594
-0.0635
-0.0740
-0.0862
-0.0927
-0.1020
-0.1200
-0.1400
-0.1726
-0.1956
-0.2150
The coefficients a13 = 0.17, c1 = 6.4, c5 = 0.03, and n = 2 for all periods.
Attenuation Relations
WNA
Page 1-3
May 2005 – I. M. Idriss
Table 1-2 Coefficients for Standard Error Terms Using Equations Derived by Abrahamson and
Silva (1997)
Attenuation Relations
WNA
Period - sec
b5
b6
0.01
0.02
0.03
0.04
0.05
0.06
0.075
0.09
0.1
0.12
0.15
0.17
0.2
0.24
0.3
0.36
0.4
0.46
0.5
0.6
0.75
0.85
1
1.5
2
3
4
5
0.70
0.70
0.70
0.71
0.71
0.72
0.73
0.74
0.74
0.75
0.75
0.76
0.77
0.77
0.78
0.79
0.79
0.80
0.80
0.81
0.81
0.82
0.83
0.84
0.85
0.87
0.88
0.89
0.135
0.135
0.135
0.135
0.135
0.135
0.135
0.135
0.135
0.135
0.135
0.135
0.135
0.135
0.135
0.135
0.135
0.132
0.130
0.127
0.123
0.121
0.118
0.110
0.105
0.097
0.092
0.087
Page 1-4
May 2005 – I. M. Idriss
1.2 ATTENUATION RELATIONSHIPS DERIVED BY BOORE, JOYNER AND FUMAL
(1997)
The following equation was derived by Boore et al for spectral ordinates at rock sites:
V 
2
Ln  y   b1  b2  M  6   b3  M  6   b5 Ln  r   bV Ln  S 
 VA 
[1-8]
r  r jb2  h2
 b1SS

b1  b1RV
b
 1ALL


for reverse-slip earthquakes 
if mechanism is not specified 

for strike-slip earthquakes
[1-9]
The variable y is spectral acceleration (5% damping) in g's; M is moment magnitude, Vs is the
average shear wave velocity (in m/sec)in the upper 30 m of the profile under consideration, V A is
reference shear wave velocity (in m/sec), and r jb is the closest horizontal distance (in km) from
the site to the surface projection of the source. Values of the coefficients
b1SS , b1RV , b1ALL , b2 , b3 , b5 , bV , VA , and h (in km) are listed in Table 1-3. Also listed in
Table 1-3 are the values of the standard error terms.
Attenuation Relations
WNA
Page 1-5
May 2005 – I. M. Idriss
Table 1-3
Coefficients Derived by Boore, Fumal and Joyner (1997)
Period
b1ss
b1RV
b1ALL
b2
b3
b5
bV
VA
h
SE
0
0.10
0.11
0.12
0.13
0.14
0.15
0.16
0.17
0.18
0.19
0.20
0.22
0.24
0.26
0.28
0.30
0.32
0.34
0.36
0.38
0.40
0.42
0.44
0.46
0.48
0.50
0.55
0.60
0.65
0.70
0.75
0.80
0.85
0.90
0.95
1.00
1.10
1.20
1.30
1.40
1.50
1.60
1.70
1.80
1.90
2.00
-0.313
1.006
1.072
1.109
1.128
1.135
1.128
1.112
1.090
1.063
1.032
0.999
0.925
0.847
0.764
0.681
0.598
0.518
0.439
0.361
0.286
0.212
0.140
0.073
0.005
-0.058
-0.122
-0.268
-0.401
-0.523
-0.634
-0.737
-0.829
-0.915
-0.993
-1.066
-1.133
-1.249
-1.345
-1.428
-1.495
-1.552
-1.598
-1.634
-1.663
-1.685
-1.699
-0.117
1.087
1.164
1.215
1.246
1.261
1.264
1.257
1.242
1.222
1.198
1.170
1.104
1.033
0.958
0.881
0.803
0.725
0.648
0.570
0.495
0.423
0.352
0.282
0.217
0.151
0.087
-0.063
-0.203
-0.331
-0.452
-0.562
-0.666
-0.761
-0.848
-0.932
-1.009
-1.145
-1.265
-1.370
-1.460
-1.538
-1.608
-1.668
-1.718
-1.763
-1.801
-0.242
1.059
1.130
1.174
1.200
1.208
1.204
1.192
1.173
1.151
1.122
1.089
1.019
0.941
0.861
0.780
0.700
0.619
0.540
0.462
0.385
0.311
0.239
0.169
0.102
0.036
-0.025
-0.176
-0.314
-0.440
-0.555
-0.661
-0.760
-0.851
-0.933
-1.010
-1.080
-1.208
-1.315
-1.407
-1.483
-1.550
-1.605
-1.652
-1.689
-1.720
-1.743
0.527
0.753
0.732
0.721
0.711
0.707
0.702
0.702
0.702
0.705
0.709
0.711
0.721
0.732
0.744
0.758
0.769
0.783
0.794
0.806
0.820
0.831
0.840
0.852
0.863
0.873
0.884
0.907
0.928
0.946
0.962
0.979
0.992
1.006
1.018
1.027
1.036
1.052
1.064
1.073
1.080
1.085
1.087
1.089
1.087
1.087
1.085
0.000
-0.226
-0.230
-0.233
-0.233
-0.230
-0.228
-0.226
-0.221
-0.216
-0.212
-0.207
-0.198
-0.189
-0.180
-0.168
-0.161
-0.152
-0.143
-0.136
-0.127
-0.120
-0.113
-0.108
-0.101
-0.097
-0.090
-0.078
-0.069
-0.060
-0.053
-0.046
-0.041
-0.037
-0.035
-0.032
-0.032
-0.030
-0.032
-0.035
-0.039
-0.044
-0.051
-0.058
-0.067
-0.074
-0.085
-0.778
-0.934
-0.937
-0.939
-0.939
-0.938
-0.937
-0.935
-0.933
-0.930
-0.927
-0.924
-0.918
-0.912
-0.906
-0.899
-0.893
-0.888
-0.882
-0.877
-0.872
-0.867
-0.862
-0.858
-0.854
-0.850
-0.846
-0.837
-0.830
-0.823
-0.818
-0,813
-0.809
-0.805
-0.802
-0.800
-0.798
-0.795
-0.794
-0.793
-0.794
-0.796
-0.798
-0.801
-0.804
-0.808
-0.812
-0.371
-0.212
-0.211
-0.215
-0.221
-0.228
-0.238
-0.248
-0.258
-0.270
-0.281
-0.292
-0.315
-0.338
-0.360
-0.381
-0.401
-0.420
-0.438
-0.456
-0.472
-0.487
-0.502
-0.516
-0.529
-0.541
-0.553
-0.579
-0.602
-0.622
-0.639
-0.653
-0.666
-0.676
-0.685
-0.692
-0.698
-0.706
-0.710
-0.711
-0.709
-0.704
-0.697
-0.689
-0.679
-0.667
-0.655
1396
1112
1291
1452
1596
1718
1820
1910
1977
2037
2080
2118
2158
2178
2173
2158
2133
2104
2070
2032
1995
1954
1919
1884
1849
1816
1782
1710
1644
1592
1545
1507
1476
1452
1432
1416
1406
1396
1400
1416
1442
1479
1524
1581
1644
1714
1795
5.57
6.27
6.65
6.91
7.08
7.18
7.23
7.24
7.21
7.16
7.10
7.02
6.83
6.62
6.39
6.17
5.94
5.72
5.50
5.30
5.10
4.91
4.74
4.57
4.41
4.26
4.13
3.82
3.57
3.36
3.20
3.07
2.98
2.92
2.89
2.88
2.90
2.99
3.14
3.36
3.62
3.92
4.26
4.62
5.01
5.42
5.85
0.520
0.479
0.481
0.485
0.486
0.489
0.492
0.495
0.497
0.499
0.501
0.502
0.508
0.511
0.514
0.518
0.522
0.525
0.530
0.532
0.536
0.538
0.542
0.545
0.549
0.551
0.556
0.562
0.569
0.575
0.582
0.587
0.593
0.598
0.604
0.609
0.613
0.622
0.629
0.637
0.643
0.649
0.654
0.660
0.664
0.669
0.672
Attenuation Relations
WNA
Page 1-6
May 2005 – I. M. Idriss
1.3 ATTENUATION RELATIONSHIPS DERIVED BY CAMPBELL (1997)
The following equations were derived by Campbell for the median value of peak horizontal
acceleration ( AH in g's) and the median value of horizontal spectral ordinates ( SAH in g's) for 5%
spectral damping:
Ln  AH   3.512  0.904M
 1.328Ln

2
RSEIS
 0.149 exp  0.647M  
2

 1.125  0.112Ln  RSEIS   0.0957 M  F
[1-10]
 0.440  0.171Ln  RSEIS   SSR
 0.405  0.222Ln  RSEIS   SHR
Ln  SAH   Ln  AH   c1  c2 tanh c3  M  4.7  
  c4  c5 M  RSEIS  0.5c6 SSR  c6SHR
[1-11]
 c7 tanh  c8 D  1  SHR   fSA  D 
In these equations, M is moment magnitude, and the source-to-site distance, RSEIS , is the shortest
distance between the recording site and the assumed zone of seismogenic rupture on the fault.
Campbell indicates, based on the work of Marone and Scholz (1988), that the upper 2 to 4 km of
the fault zone is typically non-seismogenic. The style of faulting variable, F , is equal to zero for
strike slip faulting and is equal to unity for all other style of faulting. The parameters SSR and
SHR define the local site conditions as follows:
for soft rock sites; and
SSR  1 & SHR  0
SSR  0
&
SHR  1
for hard rock sites.
The parameter D is depth to basement rock below the site. The function fSA  D  is given by:
For D  1 km
fSA  D   0
[1-12]
For D  1
fSA  D   c6 1  SHR  1  D   0.5c6 1  D  SSR
The values of the coefficients c1 ......c8 are provided in Table 1-4.
Campbell derived relationships for the standard error of estimate of Ln  AH  as a function of
mean predicted value of Ln  AH  and mean earthquake magnitude. Thus:
Attenuation Relations
WNA
Page 1-7
May 2005 – I. M. Idriss
For AH  0.068g
[1-13]
SE  0.55
For 0.068g  AH  0.21g
SE  0.173  0.14Ln  AH 
For AH  0.21g
SE  0.39
For M  7.4
SE  0.889  0.0691M
[1-14]
For M  7.4
SE  0.38
Campbell suggests that Eq. 1-13 is more statistically robust than Eq. 1-14.
Table 1-4
Coefficients Derived by Campbell (1997)
Period
c1
c2
c3
c4
c5
c6
c7
c8
0.05
0.075
0.1
0.15
0.2
0.3
0.5
0.75
1
1.5
2
3
4
0.05
0.27
0.48
0.72
0.79
0.77
-0.28
-1.08
-1.79
-2.65
-3.28
-4.07
-4.26
0
0
0
0
0
0
0.74
1.23
1.59
1.98
2.23
2.39
2.03
0
0
0
0
0
0
0.66
0.66
0.66
0.66
0.66
0.66
0.66
-0.0011
-0.0024
-0.0024
-0.0010
0.0011
0.0035
0.0068
0.0077
0.0085
0.0094
0.0100
0.0108
0.0112
0.000055
0.000095
0.000007
-0.00027
-0.00053
-0.00072
-0.001
-0.001
-0.001
-0.001
-0.001
-0.001
-0.001
0.20
0.22
0.14
-0.02
-0.18
-0.40
-0.42
-0.44
-0.38
-0.32
-0.36
-0.22
-0.30
0
0
0
0
0
0
0.25
0.37
0.57
0.72
0.83
0.86
1.05
0
0
0
0
0
0
0.62
0.62
0.62
0.62
0.62
0.62
0.62
Attenuation Relations
WNA
Page 1-8
May 2005 – I. M. Idriss
1.4 ATTENUATION RELATIONSHIPS DERIVED BY IDRISS (2002)
The following equation was derived by Idriss for spectral ordinates at rock sites:
Ln  y   1  2 M    1  2 M  Ln  R  10    F
[1-15]
y is the median spectral acceleration in g (5% damping), or peak ground acceleration (pga), in
g's, M is moment magnitude, R is the closest distance to the rupture plane in km, F is the fault
type (1 for reverse and reverse/oblique, and 0 otherwise), and 1 , 2 , 1 ,  2 are coefficients,
whose values are listed in Table 1-5a, 1-5b; and 1-5c. Table 5a provides values of these
coefficients for M  6 . Values for M  6 to M  6.5 are listed in Table 5b, and those for
M  6.5 in table 5c.
Standard error terms are obtained using the following expressions:
 max

SE   1  0.12M

 min
for M  5


for 5  M  7¼ 

for M  7¼

[1-16]
The values of  , 1 ,  min , and  max are listed in Table 1-6.
Attenuation Relations
WNA
Page 1-9
May 2005 – I. M. Idriss
Table 1-5a Coefficients for the Median Spectral Ordinates Using Equations
Derived by Idriss (2002) for M  6
Period - sec
1
2
1
2
0.01
0.03
0.04
0.05
0.06
0.07
0.075
0.08
0.09
0.1
0.11
0.12
0.13
0.14
0.15
0.16
0.17
0.18
0.19
0.2
0.22
0.24
0.25
0.26
0.28
0.3
0.32
0.34
0.35
0.36
0.38
0.4
0.45
0.5
0.55
0.6
0.7
0.8
0.9
1
1.5
2
3
4
5
2.5030
2.5030
2.9873
3.2201
3.2988
3.2935
3.2702
3.2381
3.1522
3.0467
2.9308
2.8093
2.6859
2.5579
2.4301
2.3026
2.1785
2.0543
1.9324
1.8129
1.5794
1.3575
1.2490
1.1435
0.9381
0.7437
0.5553
0.3755
0.2883
0.2049
0.0362
-0.1223
-0.4985
-0.8415
-1.1581
-1.7051
-1.9821
-2.4510
-2.8715
-3.2511
-4.7813
-5.9481
-7.7976
-9.3398
-10.7364
M 6
0.1337
0.1337
0.0290
0.0099
0.0187
0.0378
0.0489
0.0606
0.0845
0.1083
0.1316
0.1541
0.1758
0.1966
0.2166
0.2357
0.2541
0.2718
0.2888
0.3051
0.3360
0.3646
0.3782
0.3913
0.4161
0.4394
0.4612
0.4816
0.4914
0.5008
0.5190
0.5361
0.5749
0.6091
0.6393
0.7087
0.7127
0.7514
0.7847
0.8139
0.9288
1.0246
1.2121
1.4047
1.5973
2.8008
2.8008
2.7850
2.7802
2.7784
2.7777
2.7774
2.7773
2.7770
2.7767
2.7763
2.7759
2.7754
2.7748
2.7741
2.7733
2.7724
2.7714
2.7704
2.7693
2.7668
2.7641
2.7626
2.7611
2.7580
2.7548
2.7514
2.7480
2.7462
2.7445
2.7410
2.7374
2.7285
2.7197
2.7112
2.7030
2.6878
2.6742
2.6624
2.6522
2.6206
2.6097
2.6086
2.6012
2.5703
-0.1970
-0.1970
-0.2081
-0.2092
-0.2083
-0.2072
-0.2068
-0.2065
-0.2061
-0.2060
-0.2061
-0.2063
-0.2066
-0.2070
-0.2074
-0.2079
-0.2083
-0.2088
-0.2092
-0.2096
-0.2105
-0.2112
-0.2116
-0.2119
-0.2126
-0.2132
-0.2137
-0.2143
-0.2145
-0.2147
-0.2152
-0.2156
-0.2167
-0.2176
-0.2185
-0.2194
-0.2211
-0.2228
-0.2244
-0.2259
-0.2326
-0.2368
-0.2385
-0.2336
-0.2250
Attenuation Relations
WNA
Page 1-10
May 2005 – I. M. Idriss
Table 1-5b Coefficients for the Median Spectral Ordinates Using Equations
Derived by Idriss (2002) for M  6 to M  6.5
Period - sec
1
2
1
2
0.01
0.03
0.04
0.05
0.06
0.07
0.075
0.08
0.09
0.1
0.11
0.12
0.13
0.14
0.15
0.16
0.17
0.18
0.19
0.2
0.22
0.24
0.25
0.26
0.28
0.3
0.32
0.34
0.35
0.36
0.38
0.4
0.45
0.5
0.55
0.6
0.7
0.8
0.9
1
1.5
2
3
4
5
4.3387
4.3387
3.9748
3.9125
3.8984
3.8852
3.8748
3.8613
3.8249
3.7774
3.7206
3.6562
3.5860
3.5111
3.4327
3.3515
3.2683
3.1837
3.0980
3.0117
2.8382
2.6648
2.5786
2.4929
2.3231
2.1559
1.9916
1.8306
1.7512
1.6728
1.5183
1.3671
1.0036
0.6598
0.3347
0.0271
-0.5407
-1.0522
-1.5147
-1.9343
-3.5364
-4.5538
-5.5133
-5.5624
-5.0154
M  6 to M  6.5
-0.1754
-0.1754
-0.1244
-0.0972
-0.0777
-0.0613
-0.0536
-0.0462
-0.0319
-0.0181
-0.0046
0.0086
0.0215
0.0341
0.0464
0.0586
0.0704
0.0821
0.0934
0.1046
0.1263
0.1471
0.1572
0.1671
0.1863
0.2049
0.2228
0.2400
0.2484
0.2567
0.2728
0.2883
0.3251
0.3591
0.3906
0.4200
0.4729
0.5193
0.5603
0.5966
0.7255
0.7945
0.8254
0.7672
0.6513
3.2564
3.2564
3.0378
2.9689
2.9481
2.9448
2.9458
2.9477
2.9527
2.9578
2.9623
2.9659
2.9685
2.9703
2.9712
2.9713
2.9708
2.9697
2.9681
2.9660
2.9608
2.9544
2.9509
2.9472
2.9393
2.9310
2.9224
2.9135
2.9091
2.9045
2.8955
2.8864
2.8639
2.8419
2.8206
2.8002
2.7624
2.7283
2.6980
2.6712
2.5803
2.5443
2.5790
2.7072
2.8979
-0.2739
-0.2739
-0.2468
-0.2381
-0.2355
-0.2352
-0.2354
-0.2358
-0.2367
-0.2376
-0.2385
-0.2393
-0.2401
-0.2407
-0.2412
-0.2416
-0.2420
-0.2423
-0.2425
-0.2426
-0.2428
-0.2428
-0.2428
-0.2427
-0.2425
-0.2423
-0.2419
-0.2416
-0.2414
-0.2412
-0.2407
-0.2403
-0.2391
-0.2379
-0.2367
-0.2356
-0.2334
-0.2315
-0.2298
-0.2284
-0.2246
-0.2252
-0.2354
-0.2537
-0.2773
Attenuation Relations
WNA
Page 1-11
May 2005 – I. M. Idriss
Table 1-5c Coefficients for the Median Spectral Ordinates Using Equations
Derived by Idriss (2002) for M  6.5
Period - sec
1
2
1
2
0.01
0.03
0.04
0.05
0.06
0.07
0.075
0.08
0.09
0.1
0.11
0.12
0.13
0.14
0.15
0.16
0.17
0.18
0.19
0.2
0.22
0.24
0.25
0.26
0.28
0.3
0.32
0.34
0.35
0.36
0.38
0.4
0.45
0.5
0.55
0.6
0.7
0.8
0.9
1
1.5
2
3
4
5
6.5668
6.5668
6.1747
6.2734
6.4228
6.5418
6.5828
6.6162
6.6541
6.6594
6.6436
6.6084
6.5639
6.5085
6.4448
6.3778
6.3077
6.2366
6.1623
6.0872
5.9380
5.7915
5.7213
5.6485
5.5097
5.3744
5.2428
5.1167
5.0563
4.9957
4.8752
4.7604
4.4900
4.2369
4.0027
3.7826
3.3750
3.0078
2.6734
2.3648
1.1109
0.1818
-1.1016
-1.9306
-2.5042
M  6.5
-0.5164
-0.5164
-0.4717
-0.4675
-0.4696
-0.4704
-0.4698
-0.4685
-0.4642
-0.4580
-0.4504
-0.4419
-0.4328
-0.4233
-0.4137
-0.4040
-0.3945
-0.3850
-0.3758
-0.3668
-0.3494
-0.3331
-0.3253
-0.3178
-0.3035
-0.2900
-0.2774
-0.2656
-0.2600
-0.2545
-0.2441
-0.2342
-0.2119
-0.1922
-0.1747
-0.1589
-0.1314
-0.1078
-0.0870
-0.0683
0.0068
0.0649
0.1532
0.2153
0.2579
3.2606
3.2606
3.0156
2.9671
2.9677
2.9791
2.9850
2.9904
2.9989
3.0044
3.0071
3.0077
3.0067
3.0044
3.0012
2.9974
2.9931
2.9885
2.9836
2.9786
2.9684
2.9580
2.9529
2.9477
2.9376
2.9276
2.9178
2.9082
2.9034
2.8987
2.8894
2.8803
2.8581
2.8367
2.8160
2.7960
2.7580
2.7227
2.6901
2.6603
2.5501
2.4928
2.4711
2.4953
2.5107
-0.2740
-0.2740
-0.2461
-0.2400
-0.2396
-0.2406
-0.2413
-0.2419
-0.2429
-0.2437
-0.2442
-0.2446
-0.2448
-0.2448
-0.2448
-0.2447
-0.2446
-0.2444
-0.2442
-0.2440
-0.2436
-0.2431
-0.2428
-0.2426
-0.2421
-0.2417
-0.2412
-0.2408
-0.2405
-0.2403
-0.2399
-0.2395
-0.2384
-0.2374
-0.2363
-0.2353
-0.2333
-0.2314
-0.2295
-0.2278
-0.2211
-0.2176
-0.2168
-0.2190
-0.2199
Attenuation Relations
WNA
Page 1-12
May 2005 – I. M. Idriss
Table 1-6 Style of Faulting and Standard Error Coefficients for Spectral Ordinates
Using Equations Derived by Idriss (2002)
Period - sec

 max
1
 min
0.01
0.03
0.04
0.05
0.06
0.07
0.075
0.08
0.09
0.1
0.11
0.12
0.13
0.14
0.15
0.16
0.17
0.18
0.19
0.2
0.22
0.24
0.25
0.26
0.28
0.3
0.32
0.34
0.35
0.36
0.38
0.4
0.45
0.5
0.55
0.6
0.7
0.8
0.9
1
1.5
2
3
4
5
0.320
0.320
0.320
0.320
0.320
0.320
0.320
0.320
0.320
0.320
0.324
0.327
0.330
0.332
0.335
0.337
0.339
0.341
0.343
0.345
0.348
0.352
0.353
0.355
0.357
0.360
0.360
0.360
0.360
0.360
0.360
0.360
0.360
0.360
0.350
0.340
0.322
0.307
0.294
0.282
0.236
0.204
0.158
0.125
0.100
0.720
0.720
0.720
0.720
0.730
0.739
0.743
0.747
0.753
0.760
0.765
0.770
0.775
0.779
0.783
0.787
0.791
0.794
0.797
0.800
0.806
0.811
0.814
0.816
0.820
0.825
0.829
0.832
0.834
0.836
0.839
0.842
0.849
0.856
0.862
0.867
0.877
0.885
0.893
0.900
0.900
0.900
0.900
0.900
0.900
1.320
1.320
1.320
1.320
1.330
1.339
1.343
1.347
1.353
1.360
1.365
1.370
1.375
1.379
1.383
1.387
1.391
1.394
1.397
1.400
1.406
1.411
1.414
1.416
1.420
1.425
1.429
1.432
1.434
1.436
1.439
1.442
1.449
1.456
1.462
1.467
1.477
1.485
1.493
1.500
1.500
1.500
1.500
1.500
1.500
0.450
0.450
0.450
0.450
0.460
0.469
0.473
0.477
0.483
0.490
0.495
0.500
0.505
0.509
0.513
0.517
0.521
0.524
0.527
0.530
0.536
0.541
0.544
0.546
0.550
0.555
0.559
0.562
0.564
0.566
0.569
0.572
0.579
0.586
0.592
0.597
0.607
0.615
0.623
0.630
0.630
0.630
0.630
0.630
0.630
Attenuation Relations
WNA
Page 1-13
May 2005 – I. M. Idriss
1.5 ATTENUATION RELATIONSHIPS DERIVED BY SADIGH, CHANG, EGAN,
MAKDISI, AND YOUNGS (1997)
The following equation was derived by Sadigh et al for spectral ordinates at rock sites:
Ln  y   C1  C2 M  C3  8.5  M 

 C7 Ln rrup  2

2.5
 C4 Ln  rrup  exp C5  C6 M  
[1-17]
y is the median spectral acceleration in g (5% damping), or peak ground acceleration (pga), in
g's, M is moment magnitude, rrup is the closest distance to the rupture plane in km, and C1 ...C7
are coefficients. The values of the standard error terms are listed in Table 1-7. The values of the
coefficients C1 ...C7 are provided in Table 1-8.
Table 1-7 Coefficients for Standard Error Terms Using
Equations Derived by Sadigh et al (1997)
Period - sec
zpa
0.07
0.10
0.20
0.30
0.40
0.50
0.75
≥1.00
Attenuation Relations
WNA
Standard Error
Term
1.39  0.14M
1.40  0.14M
1.41  0.14M
1.43  0.14M
1.45  0.14M
1.48  0.14M
1.50  0.14M
1.52  0.14M
1.53  0.14M
Page 1-14
Minimum Value for
M  7.21
0.38
0.39
0.40
0.42
0.44
0.47
0.49
0.51
0.52
May 2005 – I. M. Idriss
Table 1-8 Coefficients for the Median Spectral Ordinates Using Equations
Derived by Sadigh et al (1997)
Period
C1
C2
zpa
0.03
0.07
0.1
0.2
0.3
0.4
0.5
0.75
1
1.5
2
3
4
-0.624
-0.624
0.110
0.275
0.153
-0.057
-0.298
-0.588
-1.208
-1.705
-2.407
-2.945
-3.700
-4.230
1
1
1
1
1
1
1
1
1
1
1
1
1
1
zpa
0.03
0.07
0.1
0.2
0.3
0.4
0.5
0.75
1
1.5
2
3
4
-1.237
-1.237
-0.540
-0.375
-0.497
-0.707
-0.948
-1.238
-1.858
-2.355
-3.057
-3.595
-4.350
-4.880
1.1
1.1
1.1
1.1
1.1
1.1
1.1
1.1
1.1
1.1
1.1
1.1
1.1
1.1
C3
C4
M  6.5
0.000
-2.100
0.000
-2.100
0.006
-2.128
0.006
-2.148
-0.004
-2.080
-0.017
-2.028
-0.028
-1.990
-0.040
-1.945
-0.050
-1.865
-0.055
-1.800
-0.065
-1.725
-0.070
-1.670
-0.080
-1.610
-0.100
-1.570
M  6.5
0.000
-2.100
0.000
-2.100
0.006
-2.128
0.006
-2.148
-0.004
-2.080
-0.017
-2.028
-0.028
-1.990
-0.040
-1.945
-0.050
-1.865
-0.055
-1.800
-0.065
-1.725
-0.070
-1.670
-0.080
-1.610
-0.100
-1.570
C5
C6
C7
1.29649
1.29649
1.29649
1.29649
1.29649
1.29649
1.29649
1.29649
1.29649
1.29649
1.29649
1.29649
1.29649
1.29649
0.250
0.250
0.250
0.250
0.250
0.250
0.250
0.250
0.250
0.250
0.250
0.250
0.250
0.250
0.000
0.000
-0.082
-0.041
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
-0.48451
-0.48451
-0.48451
-0.48451
-0.48451
-0.48451
-0.48451
-0.48451
-0.48451
-0.48451
-0.48451
-0.48451
-0.48451
-0.48451
0.524
0.524
0.524
0.524
0.524
0.524
0.524
0.524
0.524
0.524
0.524
0.524
0.524
0.524
0.000
0.000
-0.082
-0.041
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
Note that the above coefficients are applicable to ground motions generated by a strike slip event.
Sadigh et al suggest that the calculated spectral ordinates be multiplied by a factor of 1.2 for
reverse / thrust events.
Attenuation Relations
WNA
Page 1-15
May 2005 – I. M. Idriss
ATTACHMENT 2
ATTENUATION RELATIONSHIPS FOR MOTIONS
IN EASTERN NORTH AMERICA (ENA)
The attenuation relationships derived by Toro et al (1997) and those by Atkinson & Boore (1997)
are summarized in this Attachment.
2.1 ATTENUATION RELATIONSHIPS BY TORO, ABRAHAMSON AND SCHNEIDER
(1997)
The functional form adopted by Toro et al is the following:
Ln  y   C1  C2  M  6   C3  M  6   C4 Ln  RM 
2
[2-1]
 R  
 C5  C4  max Ln  M  ,0   C6 RM
  100  
[2-2]
RM  R 2jb  C72
in which, y is the median spectral acceleration in g (5% damping) or peak ground acceleration
(pga) in g's, M is either Lg magnitude  mbLg  or moment magnitude M , C1 through C7 are
coefficients that depend on the frequency and on the region under consideration, and R jb is the
closest horizontal distance to the rupture surface if this surface is projected to the ground surface
(i.e., the Joyner-Boore distance) in km.
Two regions were considered by Toro et al (1997) in deriving attenuation relationships for
Eastern North America (ENA), namely the Mid-Continent and the Gulf Crustal Regions. The
extent of each region is depicted in Fig. 2-1. The coefficients C1 ... C7 for various frequencies
and for zpa and applicable to each region are listed in Table 2-1 for equations using moment
magnitude and for equations using Lg magnitude.
2.2 ATTENUATION RELATIONSHIPS BY ATKINSON AND BOORE (1997)
The following functional form was used by Atkinson & Bore (1997) to calculate the median
spectral acceleration
Ln  y   C1  C2  M  6   C3  M  6   Ln  R   C4 R
2
[2-3]
in which, y is the median spectral acceleration in g (5% damping) or peak ground acceleration
(pga) in g's, M is moment magnitude C1 through C4 are coefficients that depend on frequency,
and R is the hypocentral distance in km. The coefficients C1 ... C4 for various frequencies and
for pga are listed in Table 2-2.
Attenuation Relations
ENA
Page 2-1
May 2005 – I. M. Idriss
Table 2-1 Coefficients of Attenuation Equations Derived by Toro et al (1997)
Frequency
(Hz)
0.5
1
2.5
5
10
25
35
pga
0.5
1
2.5
5
10
25
35
pga
0.5
1
2.5
5
10
25
35
pga
0.5
1
2.5
5
10
25
35
pga
C1
C2
C3
Coefficient
C4
C5
C6
Mid-Continent Region – Equations Using Moment Magnitude
-0.74
1.86
-0.31
0.92
0.46
0.0017
0.09
1.42
-0.20
0.90
0.49
0.0023
1.07
1.05
-0.10
0.93
0.56
0.0033
1.73
0.84
0
0.98
0.66
0.0042
2.37
0.81
0
1.10
1.02
0.0040
3.68
0.80
0
1.46
1.77
0.0013
4.00
0.79
0
1.57
1.83
0.0008
2.20
0.81
0
1.27
1.16
0.0021
Mid-Continent Region – Equations Using Lg Magnitude
-0.97
2.52
-0.47
0.93
0.60
0.0012
-0.12
2.05
-0.34
0.90
0.59
0.0019
0.90
1.70
-0.26
0.94
0.65
0.0030
1.60
1.24
0
0.98
0.74
0.0039
2.36
1.23
0
1.12
1.05
0.0043
3.54
1.19
0
1.46
1.84
0.0010
3.87
1.19
0
1.58
1.90
0.0005
2.07
1.20
0
1.28
1.23
0.0018
Gulf Region – Equations Using Moment Magnitude
-0.81
-1.60
-0.26
0.74
0.71
0.0025
0.24
-0.60
-0.15
0.79
0.82
0.0034
1.64
0.80
-0.08
0.99
1.27
0.0036
3.10
2.26
0
1.34
1.95
0.0017
5.08
4.25
0
1.87
2.52
0.0002
5.19
4.35
0
1.96
1.96
0.0004
4.81
3.97
0
1.89
1.80
0.0008
2.91
2.07
0
1.49
1.61
0.0014
Gulf Region – Equations Using Lg Magnitude
-1.01
2.38
-0.42
0.75
0.83
0.0032
0.06
1.97
-0.32
0.80
0.92
0.0030
1.49
1.74
-0.26
1.00
1.36
0.0032
3.00
1.31
0
1.35
2.03
0.0014
4.65
1.30
0
1.78
2.41
0.0000
5.08
1.29
0
1.97
2.04
0.0000
4.68
1.30
0
1.89
1.88
0.0005
2.80
1.31
0
1.49
1.68
0.0017
Attenuation Relations
ENA
Page 2-2
C7
6.9
6.8
7.1
7.5
8.3
10.5
11.1
9.3
7.0
6.8
7.2
7.5
8.5
10.5
11.1
9.3
6.6
7.2
8.9
11.4
14.1
12.9
11.9
10.9
6.8
7.3
9.0
11.4
13.8
12.9
11.9
10.9
May 2005 – I. M. Idriss
Table 2-2 Coefficients of Attenuation Equations Derived by Atkinson & Boore (1997)
Frequency
(Hz)
Coefficient
C1
C2
C3
C4
0.5
0.8
1.0
1.3
2.0
3.2
5.0
7.9
10
13
20
pga
-1.660
-0.900
-0.508
-0.094
0.620
1.265
1.749
2.140
2.301
2.463
2.762
1.841
1.460
1.462
1.428
1.391
1.267
1.094
0.963
0.864
0.829
0.797
0.755
0.686
-0.039
-0.071
-0.094
-0.118
-0.147
-0.165
-0.148
-0.129
-0.121
-0.113
-0.110
-0.123
0
0
0
0
0
0.00024
0.00105
0.00207
0.00279
0.00352
0.00520
0.00311
Fig. 2-1 Regions Considered by Toro et al (1997) in Deriving Attenuation
Relationships for Eastern North America (ENA)
Attenuation Relations
ENA
Page 2-3
May 2005 – I. M. Idriss
ATTACHMENT 3
ATTENUATION RELATIONSHIPS FOR SUBDUCTION EARTHQUAKES
The attenuation relationships derived by Youngs et al (1997) are summarized in this Attachment.
3.1 ATTENUATION RELATIONSHIPS DERIVED BY YOUNGS, CHIOU, SILVA AND
HUMPHREY (1997)
The following attenuation relationship was derived by Youngs et al (1997):

Ln  y   0.2418  1.414M  C1  C2 10  M   C3 Ln Rrup  CM
3

[3-1]
 0.00607H  0.3846Zr
CM  1.7818exp  0.554M 
[3-2]
Standard error term, SE  C4  C5 M
[3-3]
y is spectral acceleration in g's, M is moment magnitude, Rrup is closest distance from site to
rupture surface in km, H is depth in km, and Z r = 0 for interface events (such as the Cascadia
Zone) and Z r = 1 for intra-slab events (such as the Juan de Fuca).
Note that for magnitudes greater than 8, the SE term for M = 8 is to be used.
The values of the coefficients C1 ... C5 are listed in Table 3-1.
Table 3-1 Coefficients of Attenuation Equations Derived by Young et al (1997)
Period - sec
C1
C2
C3
C4
C5
0.01
0.075
0.1
0.2
0.3
0.4
0.5
0.75
1
1.5
2
3
0
1.275
1.188
0.722
0.246
-0.115
-0.400
-1.149
-1.736
-2.634
-3.328
-4.511
0
0
-0.0011
-0.0027
-0.0036
-0.0043
-0.0048
-0.0057
-0.0064
-0.0073
-0.0080
-0.0089
-2.552
-2.707
-2.655
-2.528
-2.454
-2.401
-2.360
-2.286
-2.234
-2.160
-2.107
-2.033
1.45
1.45
1.45
1.45
1.45
1.45
1.45
1.45
1.45
1.50
1.55
1.65
-0.1
-0.1
-0.1
-0.1
-0.1
-0.1
-0.1
-0.1
-0.1
-0.1
-0.1
-0.1
Attenuation Relations
Subduction
Page 3-1
May 2005 – I. M. Idriss