Árbók VFÍ/TFÍ - 01.01.1997, Page 228

226 Árbók VFÍ/TFÍ 1995/96
X(t) =X) Yj'lf('cj)w(t,xi)
i
The first two moments are now found to be
t
E [X(t)] = v0jiY J v)/(x)w(t,t)dx
o
and
(5)
(6)
>S[x(t)X(s)] = v0(oy+i4) | i]/2(x)w(t,t)w(s,x)dt (7)
0
An interesting case arises when the mean value |aY = 0- Then the mean value E[X(t)] is also
zero and the second moment is given by
S
[X(t)X(s)] = E[X(t)X(s)] = oY j'v0i}f2(t)w(t,t)w(s,t)dt (8)
0
Consider a non-homogeneous compound Poisson process with a zero mean with the second
cumulant given by Eq. (4), that is,
s
K,[X(t)X(s)] =E[X(t)X(s)] = oY |w(t ,t)w(s ,t) v(t)dt (9)
0
since jjY = 0. Up to their second moments, the non-homogeneous compound Poisson process
with the intensity function
v(t) = v04(t) (10)
and the amplitude-modulated homogeneous compound Poisson process, Eq. (5), are equival-
ent. Their higher order moments, of course, need not be equal. This interesting result, pointed
out by Shinosuka et al. [19], provides a simple method for generating numerically sample
functions of' non-homogeneous processes, which will be utilised in the following.
Earthquake Wave Forms
The observed or recorded earthquake motion at a certain location („building site“) may be
considered to be a superposition of random wave forms with random amplitudes and random
phases arriving in chaos according to a non-homogeneous compound Poisson process with an
intensity function v(t). It is plausible to assume that the intensity function will be similar to
the form shown in Fig. 2. 2, that is, during the initial phase, a few incoming waves are