Estimating Motions at a Site
Attenuation Relationships for Shallow Crustal Earthquakes - Western North America

These relationships are for shallow crustal earthquakes in active tectonic regions such as those found in California, Nevada, Taiwan and Turkey. These regions have by far the largest number of recorded ground motions, and thus the attenuation relationships are the most developed. It is worth noting however, that there can be large variability in the gound motion recordings from a single earthquake, as the 1999 Ji-Ji Taiwan earthquake showed. Thus it is impossible to say that this region produces so-and-so type of ground motion in anything more than a very general sense.

Several types of relationships are available:

General relations: Abrahamson & Silva, Boore, Joyner & Fumal, Campbell, Sadigh et al.
Extensional tectonic regimes: Spudich et al. (SEA96)
Modifications for fault rupture directivity: Somerville et al.


Abrahamson and Silva

This relationship is used to calculate horizontal and vertical spectral accelerations (including the peak ground acceleration) and is valid for moment magnitudes 4 to 8 and distances 0 to 100 km. The distance rrup is defined as the closest distance from the site to the rupture plane.
Spectral accelertation is given as a function of moment magnitude, distance to the rupture plane, fault type, soil conditions, and whether the site is on the hanging wall:

lnPHA = f1(M, rrup) + Ff3(M) + HWf4(M, rrup) + Sf5(PGArock)

where f1(M, rrup) is the basic attenuation form for rock sites, and the other three terms are modifications for fault type, hanging wall effects, and soil conditions, respectively.

Basic rock attenuation model:

The basic rock attenuation model is given by:
M<= c4
1(M, rrup) = a1 + a2(M-c1) + a12(8.5-M)n + [a3 + a13(M-c1)]lnR

for M> c4
1(M, rrup) = a1 + a4(M-c1) + a12(8.5-M)n + [a3 + a13(M-c1)]lnR

where R = [rrup2 + c42]1/2

Style-of-faulting factor:

The style-of-faulting factor accounts for the differences between ground motions generated by strike-slip and reverse faults. The dummy variable F, the multiplier for the style-of-faulting factor, is 1 for reverse, 0.5 for reverse/oblique, and 0 otherwise. The style-of-faulting factor is given by:
f3(M) = a5 for M<=5.8
  a5 + (a6-a5)/(c1-5.8)      5.8<M<c1
  a6      M>=c1

Hanging wall effect:

The hanging wall effect factor accounts for the differences between ground motions on the hanging wall and the foot wall of dipping faults. The dummy variable HW, the multiplier for the hanging wall effects factor, is 1 for sites over the hanging wall, and 0 otherwise. The hanging wall effect factor is given by:
4(M, rrup)
= fHW(M)fHW(rrup)

fHW(M) = 0 for M<=5.5
  M-5.5      5.5<M<6.5
  1      M>=6.5

fHW(rrup) = 0 for rrup<4
  a9(rrup-4)/4      4<rrup<8
  a6      8<rrup<18
  a9(1-(rrup-18)/7)      18<rrup<24
  0      rrup>25

Soil conditions:

The effects of nonlinear soil response are accounted for by the inclusion of the site effects term. The dummy variable S, the multiplier for the site effects term, is 1 for deep soil, and 0 for rock or shallow soil. The site effects factor is given by:
= a10 + a11ln(PGArock+c5)

where PGArock is the expected peak ground acceleration on rock (ie. the attenuation relationship for PGA with the S term set to zero).

Standard Deviation:
stotal(M) = b5 for M<=5.0
  b5 - b6(M-5)      5.0<M<7.0
  b5-2b6      M>=7.0

Links to coefficient tables can be found in Appendix A.

Interactive Example


Boore, Joyner and Fumal

This relationship is used to calculate horizontal peak ground acceleration and spectral accelerations and is valid for moment magnitudes 5.5 to 7.5 and distances 0 to 80 km. The distance rjb is defined as the closest horizontal distance from the site to a point on the earth's surface that lies directly above the rupture.
Peak horizontal acceleration is given as a function of magnitude, soil conditions and surface distance between the site and the fault:

lnPHA or Sa = b1 + b2(M-6) + b3(M-6)2 + b5lnr + bVlnVS/VA

r = [rjb2 + h2]1/2

Here d is the closest surface projection of the fault to the site in km. This relationship is only valid for magnitudes 5.5 < Mw < 7.5 and distances up to 80 km. Links to coefficient tables can be found in Appendix A.

Recommended values of average shear-wave velocity Vs:

VS recommended (m/sec)
NEHRP VS Range (m/sec)
NEHRP class A
> 1500
NEHRP class B
760 to 1500
NEHRP class C
360 to 760
NEHRP class D
180 to 360
NEHRP class E
< 180

The recommended Vs values shown above are used in the interactive example. If you wish to use a different value of VS, a Custom option is provided in the example.

Interactive Example


Other Relationships for Western North America

Modifications for Fault Rupture Directivity

Somerville et al. have developed attenuation relationships to add the effects of fault rupture directivity to ground motion parameters developed using other relationships, particularly the Abrahamson and Silva relationship. A response spectrum and/or PGA value can be converted into the spectrum or PGA value for either the fault normal or the fault parallel component. Use the following interactive example to see the effect of fault rupture directivity on spectra. See also the notes on this topic in Elastic Response.

Interactive Example