**Introduction to the problems**

What is powers spectrum

Seismic signal, for example displacement x(t), is a typical example of random signal. It cannot, therefore, be treated by usual spectrum and the so called "power spectral density" (PSD) should be used instead.

Suppose x(t) is displacement, then the PSD p(f) is defined as

where w=2pi f, and f is frequency. As you see, if the x(t) whould be measured in micrometers, then the PSD dimension would be micron**2/Hz.An important feature of PSD is that the variance of the signal determined from x(t):

is exactly equal (in case average of x=0) to the variance determined from the PSD:

The latter feature suggests that the PSD (or simply power spectrum) can be used to find rms motion in a particular frequency band.

Note that often the PSD is defined so that the integral from 0 to +infinity gives the variance, so there is a factor of 2 in PSD definition.

For example, this method allows to see how the motion in various frequency bands change in time [VJ95].