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Wiener Deconvolution Theoretical Basis The Wiener Deconvolution is a technique used to obtain the phase velocity dispersion curve and the attenuation coefficients by a two stations method from two pre processed traces instrument corrected registered in two different stations located at the same great circle that the epicenter Hwang and Mitchell 1986 For it the earth

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Wiener Deconvolution Theoretical Basis, In frequency domain the above mentioned convolution can be written as. g t f t h t G F H,G F H G F H,amplitude spectrum phase spectrum. F input signal spectrum H media response spectrum,G w output signal spectrum Green Function. Then the frequency domain response of the inter station media H can be. written as,Wiener Deconvolution Theoretical Basis, A problem arises from the spectral holes present in the input signal spectrum F. These points are small or zero amplitudes that can exit for some values of the. frequency At these points the spectral ratio preformed to compute H can. be infinite or a big meaningless number for these frequencies A method to avoid. this problem is the computation of the Green Function by means of a spectral ratio. of the cross correlation divided by the auto correlation. y t g t f t Y G F G F,x t f t f t X F F F F X, and the amplitude and phase spectra of H are computed by.

Y G F H Y X G F,Wiener Deconvolution Theoretical Basis. The above mentioned process is called Wiener deconvolution The goal of. this process is the determination of the Green function in frequency domain. This function H can be used to compute the inter station phase velocity c. and the attenuation coefficients by means of,ft 0 H f N. where f is the frequency in Hz 2 f N is an integer number Bath 1974. t0 is the origin time of the Green function in time domain is the inter. station distance 1 is the epicentral distance for the station near to the. epicenter and 2 is the epicentral distance for the station far to the epicenter. These distances are measured over the great circle defined by the stations and. the epicenter,Wiener Deconvolution Theoretical Basis. The inter station group velocity Ug and the quality factor Q can be. obtained from c and by means of the well known relationships Ben. Menahem and Singh 1981,U g c c dT U gT, where T is the period T 1 f The inter station group velocity also. can be computed applying the MFT to the Green function expressed. on time domain Hwang and Mitchell 1986,Wiener Deconvolution An Example.

The Wiener filtering as an example has been applied to the traces shown above. which has been instrumentally corrected These traces have been recorded at the. stations S1 more near to the epicenter E and S2 more far. Wiener Deconvolution An Example, The cross correlation of the traces S1 and S2 and the auto correlation of the trace. S1 are performed The auto correlation can be windowed to remove noise and. other undesirable perturbations which can produce spectral holes in the auto. correlation spectrum and the Green function on frequency domain. Wiener Deconvolution An Example, The phase velocity a attenuation coefficients c group velocity b and quality. factor d can be calculated through the Green function in frequency domain as. it has been explained in the previous slides,Wiener Deconvolution References. Bath M 1974 Spectral Analysis in Geophysics Elsevier Amsterdam. Ben Menahem A and Singh S J 1981 Seismic Waves and Sources. Springer Verlag Berlin, Hwang H J and Mitchell B J 1986 Interstation surface wave analysis. by frequency domain Wiener deconvolution and modal. isolation Bull of the Seism Soc of America 76 847 864. Wiener Deconvolution Web Page,http airy ual es www Wiener htm.

Approximation Theory, Given a Hilbert space H with an inner product we de ne. d 1 2 jj 1 2 jj 1 2 1 2,Given a subspace S H with a basis 1 2 n. b 2 S can be written as,b 2 S so that jj,Problem Given 2 H nd b jj is a minimum. Solution Choose so that,b v 0 for every v 2 S,Wiener Filter. f x y original picture,g x y actual picture,fb x y estimate to f x y.

Assumption the noise is additive So,g x y h x x y y f x y dxdy n x y. We assume h x x y y is known,Assumption,1 E n x y 0. E f x y f x y Rf f x y x y,E n x y n x y Rnn x y x y. Given g x y nd fb x y so that,2 E f x y fb x y is a minimum. To get a meaningful solution we need to assume that fb x y is a linear. function of g x y,fb x y m x x y y g x y dxdy, We need to calculate m x x y y as a function of g and h.

We wish that,E fb x y m x x y y g x y dxdy be a minimum. Using the orthogonality theorem we have,E fb x y x ye 0. m x x y y g x y dxdy g e e ye,Interchanging the order of the integrals. E fb x y g e,x ye x ye dxdy,E m x x y y g x y g e,or using the cross correlation. Rf g x y x E fb x y g e,e ye E g x y g e,Rgg x y x x ye.

Rf g x y x e ye m x x y y dxdy, This is still very di cult so we assume that all the statistics are homogeneous. and invariant So,Rf g x y x e y ye,Rgg x y x Rgg x xe y ye. Rnn x y x Rnn x xe y ye,m x x y y m x x y y,Rf g x e y. x ye Rgg x e y,x ye m x x y y dxdy,x ye m x y,Fourier transform yields. Sf g u v Sgg u v H u v,but from we have, g x y h x x y y f x y dxdy n x y h x x y y f x y dxdy n x y.

and in Fourier space,Wiener Khinchine Theorem, Theorem 1 The Fourier Transform of the spatial autocorrelation function is. equal to the spectral density jF u v j2, Proof The spatial autocorrelation function is de ned by. Rf f e e y ye f x y dxdy, Muliply both sides by the kernel of the Fourier transform and integrate. R x ye x ye e,Rf f e i e,e y ye f x y e,f x x dxdyde. R x ye f s1 s2 f x y e i s1 x u s2 x v,dxdyds1 ds2.

1 R i s1 u s2 v,f s1 s2 e ds1 ds2 f x y ei xu yv dxdy. F u v F u v jF u v j2, De nition A vector has a Gaussian or normal distribution if its joint. probability density function has the form,p x C p e 2 x x. where C is a nxn symmetric positive de nite matrix. De nition A vector has a Poisson distribution if its joint probability density. function has the form,p x xi is a nonnegative integer. where C is a nxn symmetric positive de nite matrix. cov X diag 1 2 n,Summary Wiener Filter, The Wiener filter is the MSE optimal stationary linear.

filter for images degraded by additive noise and blurring. Calculation of the Wiener filter requires the assumption. that the signal and noise processes are second order. stationary in the random process sense, Wiener filters are often applied in the frequency domain. Given a degraded image x n m one takes the Discrete. Fourier Transform DFT to obtain X u v The original. image spectrum is estimated by taking the product of.

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