By Serge A. Shapiro
This ebook treats numerous generalizations of the classical O'Doherty-Anstey formulation which will describe stratigraphic filtering results. those are the results that may be saw while elastic and electromagnetic waves propagate via multilayered buildings. Our target was once to regard this subject in a complete demeanour and current compact leads to a didactically basic method, emphasizing the physics of the wave-propagation phenomena. we don't declare mathematical tension in all our derivations, in spite of the fact that, we're happy to have bought very simple descriptions of scattering, transmission and mirrored image of wavefields in acoustic, elastic, and poroelastic media which are important for varied seismological and non-seismological functions.
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Additional info for Elastic Waves in Random Media: Fundamentals of Seismic Stratigraphic Filtering
19) n = z" + j=l Substituting the expression for Bnn into that for An, we obtain a closed-form equation for the polynomial An : n An(z) = 1 + Z j-1 Z CjcmzJ-mAm-l(Z)' j = l rn=0 where we defined A-1(z) = 1. 21) /4=0 i2-- E i3--1 I Ci2Zil--i2 E 12=2N-2 i2N-2--1 i3--1 ia=2N-3 ci3 E Ci4Zi3-i4 ... i4=2N-4 i2N-x--1 . . i'~N-x=l i2N=0 where N = Int((n + 1)/2) and Int(x) denotes the integer part of x. 18), we obtain exact explicit formulas for the z-transforms of the reflectivity and the transmissivity.
In the next section these formulas are explicitly written in terms of the correlation functions and the variances 52 4 Oblique Incidence of Scalar Waves of the medium fluctuations. In fact these will then be the final results of the derivation. In the derivation above we have not used any relationship between L and the small parameter e. 15), which describe arbitrary high-order approximations of the vertical phase increment and attenuation coefficient for arbitrary L. If, following Burridge et al.
A ~ ,. . . . . . . ~. . . . . . . . . . ~,. 1 0 I t I I I I 10 20 30 40 50 60 70 time delay (in tau) Fig. 6. The transient transmissivity for the reflection-coefficient series shown in Figure 5. 46), respectively. (Reprinted from Phys. of the Earth and Planet. 3 1 I t I 1 I exact stat. approx. -...... >. 1 0 50 I I 1 I t I 100 150 200 250 300 350 4 0 0 time d e l a y (in tau) Fig. 7. The same as in Figure 6, however, on another time scale. 46), respectively. (Reprinted from Phys.