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.

Variation Due to Site Response

1a

Basement Rock

Accelerometer2a

1 2a a≠

Applications of SVGM Studies

Mathematical Representation of Ground Motion Variability

Coherency

Phase variation between two recordings (j and k) is estimated by coherency

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∆

Correlation Studies

1f 2f

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

Amplitude variability only

Phase variability only

Phase and amplitude variability

State-of-Knowledge : Coherency• Abrahamson and others – mean γ

LSST Model can be used over most level sites

State-of-Knowledge : Amplitude Variability

•Abrahamson and others:

State-of-Knowledge : Correlation

Sa

T

Correlation Studies available for Sa Baker and Cornell (2006)

No work on correlation of coherency or amplitdue variability

What is Needed to Advance State-of-Knowledge

1. Dense Arrays2. Variable site conditions3. Array spacing of 1 to 100 m4. Stable source of funding to maintain

array and archive the data