ABSTRACT
Suppose that the random process in question is composed of two types of phenomena. One is random and very localized in time and space, while the other occurs in the form of propagating waves. We want to separate the propagating waves from the localized phenomena. Assume that we have N time series that represent measurements made along a line at positions xi, i=1, …, N. The measurements are represented by
where
µ2(? ) is the spectral density of w, and is a phase that depends on ? . We assume that w1 and w2 have the same spectral density. The terms ni(t) are random processes that are uncorrelated with each other or with wi. Let
for i=1, 2, …N; while, by hypothesis,
Remember that I call ni(t) noiselike components because each one is uncorrelated with anything else; they are not like noise in the sense of being only a nuisance that interferes with measurement of something else that is really important. In particular, do not confuse them with measurement errors; they are assumed to be real phenomena that are important in their own right. The propagating part of the signal is wi. While ni might have periodic components, we assume that these periodic
components are not correlated between different ni. Also, it is not necessary to assume that ni is d-correlated in t .
