Getting reflection coefficients from the waves

The remainder of chapter 8 in FGDP shows how the algebraic techniques developed here can be solved in reverse. Given various waves such as the reflection data R(Z) or the earthquake data X(Z) we may deduce the reflection sequence $(c_1, c_2, c_3, \cdots)$.This appears to be the solution to the practical problem. It turns out however, for various complicated reasons, that this does not seem to be a methodology in widespread use in the seismic exploration business.

EXERCISES:

1. In Figure 9 let $c_1 = {1 \over 2}$, $c_2 = -{1 \over 2}$,and $c_3 = {1 \over 3}$.What are the polynomial ratios T(Z) and C(Z)?
2. For a simple interface, we had the simple relations t = 1 + c, t' = 1 + c', and c = -c'. What sort of analogous relations can you find for the generalized interface of Figure 9? [For example, show 1 - T(Z)T'(1/Z) = C(Z)C(1/Z) which is analogous to 1 - tt' = c².]
3. Show that T(Z) and T'(Z) are the same waveforms within a scale factor. Deduce that many different stacks of layers may have the same T(Z).
4. Consider the earth to be modeled by layers over a halfspace. Let an impulse be incident from below (Figure 8). Given F(Z) and G(Z), elements of the product of the layer matrices, solve for X and for P. Check your answer by showing that $P(Z) \bar{P}(1/Z) = 1$.How is X related to E? This relation illustrates the principle of reciprocity which says source and receiver may be interchanged.
5. Show that $1 + R(1/Z) + R(Z) = (\mbox{scale factor}) X(Z) X(1/Z)$,which shows that one may autocorrelate the transmission seismogram to get the reflection seismogram.
6. Refer to Figure 10. Calculate R' from R.
    

   E8-3-7 Figure 10 Stripping off the surface layer.