- Ground-motion model is:
where A is in g, a1 = 0.432(0.072), a2 = 0.110(0.012), a3 = -0.0947(0.0101), a4 = 6.35(3.24),
σ1 = 0.0351(0.0096) (inter-event) and σ2 = 0.0759(0.0042) (intra-event), where numbers in brackets are
the standard errors of the coefficients.
- Use exploratory data analysis (EDA) and alternating conditional expectations (ACE) techniques.
- Firstly sought to determine functions θ(A), ϕ(M) and ψ(d) so that θ(A)ϕ(M)+ψ(d), i.e. an approximately
additive relationship. Prefer additivity because of linearity, ease of interpolation and interpretation and
departures from fit are more easily detected.
- Use ACE procedure to find model. For set of data, with response yi and predictors wi and xi find functions
to minimize: ∑
i=1n[θ(yi) - ϕ(wi) - ψ(xi)]2 subject to ∑
ϕ(wi) = 0, ∑
ψ(xi) = 0, ∑
θ(yi) = 0 and
θ(yi)2 = n. Search amongst unrestricted curves or unrestricted monotonic curves. Use EDA to select
specific functional forms from the estimates of θ, ϕ and ψ at each data point.
- Do not use weighting because does not seem reasonable from statistical or seismological points of view.
- Do not want any individual earthquake, e.g. one with many records, overly influencing results.
- Note that because each earthquake has its own source characteristics its records are intercorrelated.
Therefore use ‘random effects model’ which accounts for perculiarities of individual earthquakes and
correlation between records from same event.
- On physical grounds, restrict θ, ϕ and ψ to be monotonic and find optimal transformation of magnitude
is approximately linear, optimal transformation of distance is logarithmic and cube root is optimal for
- Note that need correlations between coefficients, which are provided, to attach uncertainties to estimated
- Provide method of linearization to give 95% confidence interval for acceleration estimates.
- Also provide a graphical procedure for estimating accelerations that does not rely on an assumed functional
- Examine residual plots (not shown) and found a candidate for an outlying observation (the record from
the Hollister 1974 earthquake of 0.011g at 17.0km).
- Find that assumption of normality after transformation seems reasonable.