Árbók VFÍ/TFÍ - 01.01.1997, Blaðsíða 231

Simulation of stochastic 229
[21]. In lack of measurements or if no provisions for a suitable amplitude spectra are
available, a way of constructing a theoretical Fourier amplitude spectrum for a specific site
can be based on the work of Hanks and McGuire, [10], and Boore, [5], which use the Brune
spectrum [6], [7] to explain the high frequency portion of the motion. Such a spectrum can
have the form
| A(io,r,M0) | = |X(m,r,M0)| = C-r(r,r0)-Q(m,M0)-P(W)-HQ(u),r)-|HB(co)| (15)
where A(œ,r,M0) is the modulus of an amplitude spectrum of the horizontal surface accelera-
tion of an earthquake, r is the distance in km from the site to the hypocentre, Mn is the seismic
moment of the earthquake, C is a scaling factor, T is a geometric spreading function, Q is the
source spectral model, P is an amplification factor, HQ is an attenuation factor and H„ is a low-
pass filter. For a full discussion and a detailed description of the different factors and para-
meters of Eq. (15), the reader is referred to Bessason, [3].
All the main ingredients for the earthquake shape functions Eq. (14) are thus in place. The
main statistical parameters of the process are easily obtained. The mean value of the random
process, Eq. (13), is given by (cf. Eq. (3))
i
E[X(t)]=|E[w(t-x,T,0)]v(t)dr
0
and the second moment (s<t) by
s
E[X(t)X(s)] = JE [w(t -x, T, $ )w(s -x, T, $)] v(x)dx
o
The expectation of the earthquake shape functions is given by
E [w(t-x,T, (Þ)] = f —~x) dx f—dz /"exp[- ‘n'(t ^]
J0 D Jo2ít { Qfi
•sin(—(t-y) +z) v(y)dy
x
(16)
(17)
(18)
2n 2n
Since j sinzdz = I coszdz = 0, E[w(t-x,T,<Þ] = 0 and the mean value E[X(t)| (see Eq. (16)) is
0 0
zero. The second moment can be obtained in the same rnanner. It is completely determined
when the two functions A(x) and V(y) have been obtained, i.e.