Árbók VFÍ/TFÍ - 01.01.1997, Blaðsíða 230

228 Árbók VFÍ/TFÍ 1995/96
in a suitable interval of time, a frequency analysis of wave periods is performed by counting
the number of cycles in the record with the same period “n(T)”. This number plotted as a
function of the period, represented by the broken line in Fig. 3, strongly reflects the shape of
the corresponding amplitude spectrum. It is therefore possible to obtain the probability distri-
bution of wave periods as
FT(x)=P[T^x]=ljA(T)dT (l|)
o
where D is a scaling factor such that
large x
— |A(T)dT = l (12)
o
An earthquake motion process is now put together as follows:
N(t)
X(t) = w(t -Tj, Ai, Tj, Oj) (13)
i-0
where Xj are the Poisson arrival times with intensity v(t), A| a representative random ampli-
tude, Tj the random wave periods, distributed as shown above, and <f>, are random phase angl-
es, uniformly distributed between 0 and 2n.
A suitable shape function for an earthquake motion process has been suggested to be of
the form (see Solnes, [21], [22]).
0 *rr
w(t-x,T,0) = A(T)exp[-f(T,Q)(t-x)]sin[Í2-(t-x)+$>] (14)
where A(T) is the amplitude to be drawn from an appropriate Fourier amplitude spectrum,
and f(T,Q) is some function of the wave period and the attenuation factor Q of the earthquake
waves or the quality factor, (see Knopoff, [14], Aki and Richards, [1], and Joyner and Boore,
[13]). In earlier studies of the shape function, the attenuation term was assumed to be the same
as for body waves in solids, that is, exp[-7tt/TQ], [21]. This results in a very harsh compres-
sion of the high frequency components in the thus defined surface motion, which does not
hold for real measured records. Therefore, it is suggested that the square root of the period T
be used instead. Thus, the attenuation term becomes exp[-7tt/(Q T)]. This modification seems
to impart a reasonable frequency contenl to the signal.
The amplitude spectrum selected for the generation of artificial earthquakes can be based
on average smoothed measurements of earthquake spectral amplitudes, theoretical earthquake
spectra or otherwise available spectra, representative of the ground displacement, the ground
velocity or the surface acceleration according to the mode desired for the site under study,