### 2.69 Tong and Katayama (1988)

• Ground-motion model is:

where A is in gal, T is predominant period of site, α = 0.509, β = 2.32, γ = 0.039 and δ = 2.33 (σ not given).

• Correlation coefficient between magnitude and distance is 0.84, so magnitude and distance cannot be considered independent, so attenuation rate, β, is difficult to find.
• First step fit log A = -βi log(Δ + 10) + δi to each earthquake. Define reliability parameter, ψi = NiRi2, where Ni is degrees of freedom for i earthquake and Ri is correlation coefficient. Plot ψi against βi and find attenuation rate scattered, between -6 and 9, for ψi < 1 (Group B) and for ψ1 > 1 attenuation rate converges (Group U).
• Group B includes earthquakes with focal depths > 388km, earthquakes with small magnitudes and records from distances 100km, earthquakes with records from great distances where spread of distances is small, earthquakes recorded by only 3 stations and earthquakes with abnormal records. Exclude these records.
• Apply multiple regression on Group U to find α, β, γ and δ simultaneously. Also fix β = ψiβi ψi and find α, γ and δ. Find different coefficients but similar correlation coefficient. Conclude due to strong correlation between M and Δ so many regression planes exist with same correlation coefficient.
• Perform Principal Component Analysis (PCA) on log A, M, log(Δ + 10), T and log A∕A and find that equation found by fixing β is not affected by ill-effect of correlation between M and Δ.
• Omit T from regression and find little effect in estimation.