Strong Motion Arrays for Study of Ground Motion Variability
Jonathan Stewart, Bob Nigbor and Timothy Ancheta
Collaborators: Norman Abrahamson, Jonathan Bray, John Osteraas, Brian McDonald, Akshay Gupta, Melanie Walling
Causes of Variable Ground Motions
Deterministic Stochastic
1Abrahamson, N.A, “Seismic Ground Motion and Fault Displacement Evaluation, San Diego –Coronado Bay Bridge, Caltrans Contract No. 59N771” Figure G.1, May, 1994.
Coherency
Phase variation between two recordings (j and k) is estimated by coherency
( ) ( ),
( ) ( )jk
jj
jkkk
SS S
ωγ ξ ω
ω ω=
×
Coherency
Phase variation between two recordings (j and k) is estimated by coherency
( ) ( ),
( ) ( )jk
jj
jkkk
SS S
ωγ ξ ω
ω ω=
×
( ) ( ) ( ), , exp ,jk jk jk
iγ ξ ω γ ξ ω θ ξ ω⎡ ⎤= ⎣ ⎦
Coherency
Phase variation between two recordings (j and k) is estimated by coherency
( ) ( ),
( ) ( )jk
jj
jkkk
SS S
ωγ ξ ω
ω ω=
×
( , )γ ξ ω
( ) ( ) ( ), , exp ,jk jk jk
iγ ξ ω γ ξ ω θ ξ ω⎡ ⎤= ⎣ ⎦
Lagged coherency -describes stochastic component
Coherency
Phase variation between two recordings (j and k) is estimated by coherency
( ) ( ),
( ) ( )jk
jj
jkkk
SS S
ωγ ξ ω
ω ω=
×
( , )γ ξ ω ( , ) /r cθ ξ ω ωξ=
( ) ( ) ( ), , exp ,jk jk jk
iγ ξ ω γ ξ ω θ ξ ω⎡ ⎤= ⎣ ⎦
Wave passage phase shift due to different arrival times at two recordings
Amplitude Variability
• Quantified by the standard error of the Fourier Amplitude spectrum
• The amplitude variation is given by:
– is normally distributed– Mean = 0– due the differencing
( ) ( ) ( )ln lnijk ij ikA A Aω ω ω∆ = −
( )2 ,A Ajσ σ ω ξ∆ =
A∆
Effect of |γ|, ∆Amp, and ρ, on Ground Motion Variation
Use Strain as a metric of Ground Motion Variation
( ) ( ) ( )1 2t tt
rε
∆ −∆=
Random components are uncorrelated
1 2
r
State-of-Knowledge : Coherency• Abrahamson and others – mean γ
LSST Model can be used over most level sites
State-of-Knowledge : Correlation
Sa
T
Correlation Studies available for Sa Baker and Cornell (2006)
No work on correlation of coherency or amplitdue variability