Stochastic description of the seismic source

The description of the seismic source follows an approach proposed by Boore (1983). Boore’s method starts with the generation of a gaussian white noise. A window is applied to the noise which has the form

w
a is set to 1 since the scaling is done in the frequency domain, H(t) is the Heaviside function, and b is given by the relation

b

and

formula C=b/(? Td)

where

td

It reaches its maximum at the fraction e of the duration Td. A commonly used value for is e = 0.2. At the time t=Td the window amplitude has decreased to the threshold (for example h= 0.05). The windowed noise is filtered in the frequency domain with a bandpass filter given by

form

where

C* constant for geometrical spreading and radiation pattern

M0 seismic moment

S(f, f0) f2/(1+(f2/f0)2)
P(f, fmax) (1+(f/fmax)2q)-1/2

q parameter of the steepness of the high frequency decay (here q=4).

 

The filter is zerophase and therefore has an acausal response. For engineering seismology the acausal part of the signal is negligible. The spectra of the windowed random sequences are scaled with respect to their flat part between f0 and fmax, to a value of M04p2f02/C.

The corner frequency f0 is obtained from the source radius r0 in terms of Brune’s (1970, 1971) model, i e.,

whereas fmax has to be specified according the ideas of the user. Often a value fmax = 20 Hz is recommended.