chapter  8
40 Pages

## Conditionally Simulated Ground Motions

The purpose of interpolation and conditional simulation of spatially variable ground motions is similar: Given a set of seismic time histories recorded at an array of sensors, evaluate the time histories at target locations where data are not available. The simplest approach is deterministic interpolation: For example, Thra´insson et al.  interpolated seismic time histories by considering that the Fourier transform of the motion at a target location for each discrete frequency can be expressed as the weighted average of the Fourier transforms of the data recorded at a set of stations. The weights in their approach were the inverse squared distance of the target location to each recording station, and the phases were also modiﬁed to account for a deterministic wave passage effect. A more reﬁned scheme was reported by Abrahamson , who ﬁrst used a frequency-wavenumber (F-K) analysis, as described in Section 4.1, to decompose the recorded waveﬁeld to its signal components and noise, and expressed the time history at the location of each recording station, rm , as:

U (rm, ω) = q∑

j=1 A j (ω) exp[i κ j (ω) · rm + θ j (ω)] + η(rm, ω) (8.1)

where q is the number of plane-wave signals identiﬁed from the F-K analysis, A j (ω), κ j (ω) and θ j (ω) are the amplitude, wavenumber and phase, respectively, of the

j-th signal, and η(rm, ω) is the noise at the station. Abrahamson  then estimated the interpolated time series at any location r by recombining the signal and noise components as:

U (r , ω) = q∑

j=1 A j (ω) exp[i κ j (ω) · r + θ j (ω)] + μη(r , ω) exp(iφη(r , ω)) (8.2)

with μη(r , ω) being the weighted mean amplitude of the noise, η(rm, ω), at the recording stations. The phase, φη(r , ω) in Eq. 8.2, was described by a normal random variable with mean value equal to the phase of the weighted sum of the noise and variance evaluated from the phases of the noise at the recording stations. The weights, in this approach, were functions of the coherency of the noise.