By Klees R., Haagmans R. (eds.)
This publication comprises state of the art non-stop wavelet research of 1 and extra dimensional (geophysical) signs. distinctive consciousness is given to the reconaissance of particular houses of a sign. It additionally includes an extension of normal wavelet approximation to the appliance of so-called moment iteration wavelets for effective illustration of signs at quite a few scales even at the sphere and extra complicated geometries. moreover, the publication discusses the applying of harmonic (spherical) wavelets in capability box research with emphasis at the gravity box of the Earth. Many examples are given for functional program of those instruments; to help the textual content workouts and demonstrations can be found on the net.
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Additional resources for Wavelets in the Geosciences
Example text
We note by E (r(t)) the expectation of the r a n d o m variable r. The correlation of two time points t, u is the mean of the r a n d o m function ~(t) s(u) ¢(t, u) = E A white noise s is a "function" t h a t is of zero mean, and for which different time points are completely un-correlated E (s(t)) = 0, E (s(t) s(u)) = 5(t - u). Consider the wavelet transform of such a random function. It is itself a random function, but this time over the half-plane. =E (/~+CCdtgb,a(t)s(t) f_+~dU-jb,,a,(u) s(u)) : o o) In the last equation we have used the correlation function of a white noise.
G(t) = ~ 23 cos(~t), dw(~(w) - ~ ( - w ) )e ~W* O0 = 2--~ dw~(co) sin(wt), and thus the real part is an even function whereas the imaginary part is an odd function. They are related by a Hilbert transform ~g = - H . ~g = H ~g. This transformation is essentially defined as the multiplication by sign w = w~ lwl in Fourier space H: ~'(w) ~ - i sign (w) - ~'(w) (14) Since in Fourier space H is multiplication with a bounded function it follows that H is continuous on L 2 (~). Now multiplication in Fourier space corresponds to convolution, which in the case of the Hilbert transform reads H s ( t ) = -1 lim f + ~ d s(u) ~-~0 j _ ~ t - u ( t - u) 2 + ~2 Note that this is formally a convolution with 1/(zrt).
Thus s may be recovered fl'om its wavelet transform via a wavelet synthesis. • For s progressive and g, h arbitrary we have • For s regressive and g, h arbitrary we have A4g Wgs = c~hs. • For s arbitrary and g or h progressive we have Thus we extract the positive frequencies of the signal. • For s arbitrary and g or h regressive we have Mg Wgs = c~,flI-s. Continuous Wavelet Analysis 39 Thus we extract the negative frequencies of the signal. • For s real valued and g or h progressive we obtain the analytic signal associated with s.