- Ground-motion model is (case II):
where y is in gal, C5 = 0.231 and C6 = 0.626, for rock C1 = 0.894, C2 = 0.563, C4 = 1.523 and σ = 0.220
and for soil C1 = 1.135, C2 = 0.462, C4 = 1.322 and σ = 0.243 (these coefficients are from regression
assuming M and R are without error).
- Use two site categories:
- Supplement western USA data in large magnitude range with 25 records from 2 foreign earthquakes with
magnitudes 7.2 and 7.3.
- Note that there are uncertainties associated with magnitude and distance and these should be considered
in derivation of attenuation relations.
- Develop method, based on weighted consistent least-square regression, which minimizes residual error of
all random variables not just residuals between predicted and measured ground motion. Method considers
ground motion, magnitude and distance to be random variables and also enables inverse of attenuation
equation to be used directly.
- Note prediction for R > 100km may be incorrect due to lack of anelastic attenuation term.
- Use both horizontal components to maintain their actual randomness.
- Note most data from moderate magnitude earthquakes and from intermediate distances therefore result
possibly unreliable outside this range.
- Use weighted analysis so region of data space with many records are not overemphasized. Use M-R
subdivisions of data space: for magnitude M < 5.5, 5.5 ≤ M ≤ 5.9, 6.0 ≤ M ≤ 6.4, 6.5 ≤ M ≤ 6.9,
7.0 ≤ M ≤ 7.5 and M > 7.5 and for distance R < 3, 3 ≤ R ≤ 9.9, 10 ≤ R ≤ 29.9, 30 ≤ R ≤ 59.9,
60 ≤ R ≤ 99.9, 100 ≤ R ≤ 300 and R > 300km. Assign equal weight to each subdivision, and any data
point in subdivision i containing ni data has weight 1∕ni and then normalise.
- To find C5 and C6 use 316 records from 7 earthquakes (5.6 ≤ M ≤ 7.2) to fit log Y = ∑
C4 log[r + ∑
i=1mR0,iEi], where Ei = 1 for ith earthquake and 0 otherwise. Then fit R0 = C5 exp(C6M)
- Also try equations: log y = C1+C2M-C4 log[R+C5] (case I) and log y = C1+C2M-C3M2-C4 log[R+
C5 exp(C6M)] (case III) for M ≤ Mc, where impose condition C3 = (C2 -C4C6∕ln10)∕(2Mc) so ground
motion is completely saturated at M = Mc (assume Mc = 8.0).
- Find equations for rock and soil separately and for both combined.