By Serge A. Shapiro

ISBN-10: 3540650067

ISBN-13: 9783540650065

This e-book treats a number of generalizations of the classical O'Doherty-Anstey formulation with a purpose to describe stratigraphic filtering results. those are the consequences that may be saw while elastic and electromagnetic waves propagate via multilayered buildings. Our objective used to be to regard this subject in a complete demeanour and current compact leads to a didactically easy approach, emphasizing the physics of the wave-propagation phenomena. we don't declare mathematical pressure in all our derivations, besides 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 that are necessary for varied seismological and non-seismological functions.

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**Extra resources for Elastic Waves in Random Media: Fundamentals of Seismic Stratigraphic Filtering**

**Sample text**

1994). If D = 1, there will exist only a single mode in the wavefield. A single-mode propagation is described later in Chapters 3 and 4 (see also introduction into Chapter 5). In this case the Fiirstenberg's theorem states the existence of the localization length for the wavefield. The localization length is proportional to 1/71. Interestingly, this theorem describes not the attenuating but rather the exponentially growing solutions of wave equations. A simultaneous consideration of this theorem with the Virtser's theorem shows that there exist an exponentially decreasing solution with a Lyapunov characteristic exponent of exactly the same absolute value.

Of the Earth and Planet. 153, 1997, Shapiro and Treitel, with kind permission from Elsevier Science - NL) Usually, realistic reflection coefficients have a finite correlation. 4 Generalized Primary and Coda 35 of the reflection-coefficient series. Because of the assumed ergodicity the quantity ~c(k) is equal to the ensemble-averaged autocorrelation function. 34) i=0 where Vat[z] is the variance of the quantity x. 35) 1)20~(l). i=0 k:O The average under the double sum is a fourth statistical moment of the reflectioncoefficient series.

5 Appendix: Approximation of the Polynomial A,, 41 Using this formula to express P~-I in terms of P~-2 and combining the resulting recursive equation for P~'_ 1 with the above one we obtain: P" = P ~'_ , + z(P'_l - g ~' _ 2 ) c , ~ / c , ~ - i + zc,~-lCn. _, exp (zc,~_lcn). 57) Now we make the two important assumptions: o" ---- ~2 << 1. 58) and rl--I E'~-I - E . - 2 E,~-2 J<< 1. 59) Later, we consider these assumptions in more details. 57) by the first two terms of their Taylor expansions: E n = E,~_I(1 + z Cn En- l - En- 2 Cn - 1 En - 2 + O(r12))(1 + z c n - l c n + O(c'2)).

### Elastic Waves in Random Media: Fundamentals of Seismic Stratigraphic Filtering by Serge A. Shapiro

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