An important result of the last section was
the development of a ``layer matrix" (14)
that is,
a matrix which can be used to extrapolate waves
observed in one layer to the waves observed in the next layer.
This process may be continued indefinitely.
To see how to extrapolate from layer 1 to layer 3
substitute (14) with *k* = 1 into
(14) with *k* = 2, obtaining

(24) |

Inspection of this example
suggests the general form for a product of
*k* layer matrices

(25) |

Now let us verify that (25)
is indeed the general form.
We assume (25) is correct for *k* - 1;
then we multiply (25)
by another layer matrix and see if the product
retains the same form with *k* - 1 increased to *k*.
The product is

(26) |

By inspecting the product
we see that the scaling factor is of the same form
with *k* - 1 changed to *k*.
Also the 22 matrix element can be obtained
from the 11 element by replacing *Z* with
1/*Z* and multiplying by *Z*^{k}.
Likewise,
the 21 element is obtained from the 12 element;
thus (25) does indeed represent a general form.
The polynomials *F*(*Z*) and *G*(*Z*) of order *k*
are built up,
starting from *F _{1}*=1 and

(27) | ||

(28) |

By inspecting (27) and (28) we can see some of the
details of *F* and *G*.
From (27) we see that the lead coefficient
*f _{0}* of

(29) | ||

(30) |

It may be noted in (29) and (30)
and proved from the recurrence relations
(27) and (28) that the coefficients of *F*
contain even powers of *c* and that *G* contains
odd powers of *c*.
This means that if all *c* change sign,
*G* will change sign but *F* is unchanged.

The polynomials *F*(*Z*) and *G*(*Z*) are not independent
and a surprising energy-flux-like relationship exists
between them.
By substitution from (27) and (28)
one may directly verify that

(31) |

Since *F _{1}*(

(32) |

Equation (32) is a surprising equation because
on the left-hand side we have two spectra,
the spectrum of *f*_{t} and the spectrum of *g*_{t},
but the right-hand side is a positive,
frequency-independent constant.
Since the spectrum of *f*_{t} is thus greater than the
spectrum of *g*_{t},
we may apply the theorem of adding garbage to a minimum-phase
wavelet to deduce from (27) and from knowledge that
|*c*_{k}| < 1 that *F*_{k}(*Z*) is minimum-phase if
*F*_{k-1}(*Z*) is minimum-phase.
Since *F _{1}*(

Let a stack of layers be sandwiched in between
two halfspaces Figure 9.
An impulse is incident from below.
The backscattered wave is called *C*(*Z*)
and the transmitted wave is called *T*(*Z*).

Figure 11

Mathematically, we describe the situation with the equations

(33) |

We may solve the first of (33) for
the transmitted wave *T*(*Z*)

(34) |

and introduce the result back into the second of (33) to obtain the backscattered wave

(35) |

The mathematical fact that *F*(*Z*) is minimum-phase
corresponds to the physical fact that the *C*(*Z*) and *T*(*Z*)
have finite energy;
therefore the denominators of (34) and (35)
cannot have zeros inside the unit circle.
Since we know that the backscattered wave *C*(*Z*) contains
less energy than the incident wave by reference to
(23) we know that a positive real function is given by

(36) |

Now let us see how to reconstruct the
reflection coefficients *c*_{j} from the observed
scattered wave *C*(*Z*).
Referring to Figure 9 we have

(37) |

The first coefficient of *C*(*Z*) is *c*_{k}
[this is physically obvious but may also be seen from (30)].
Thus the layer matrix in (37) is known.
Multiplying (37) through by the inverse of the
layer matrix we will have obtained *U*_{k-1}(*Z*)
and *D*_{k-1}(*Z*).
The next reflection coefficient *c*_{k-1} is obviously *d _{0}*/

Next let us reconsider the reflection seismology geometry. We have

(38) |

From the first equation we may solve for *R*(*Z*)

(39) |

The denominator occurs so often that we give it
the name *A*(*Z*)

(40) |

*A*(*Z*), like *F*(*Z*), is minimum-phase.
The second of (38) gives the escaping wave as

simplifying with (32) we get

(41) |

Let us confirm^{}
that the definition (40) of *A*_{k}(*Z*)
is consistent with our chapter definition of the Levinson
recursion ().
First, rewrite (40) in various ways

(42) | ||

(43) | ||

(44) | ||

(45) |

(46) | ||

(47) | ||

(48) |

The positive real function is

(49) | ||

As mentioned earlier,
if the equations are interpreted in terms of acoustics,
then *Y*(*D* - *U*)/(*D* + *U*) is interpreted as vertical velocity
divided by pressure.
It is called the *admittance* which is the inverse of the impedance.

We have now completed the task of solving for the waves given the reflection coefficients. In the subsequent section we attach the inverse problems of getting the reflection coefficients from knowledge of various waves.

- In Figure 9 let , ,and .What are the polynomial ratios
*T*(*Z*) and*C*(*Z*)? - 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*' =*c*.]^{2} - 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*). - Let an impulse be incident on a stack of layers
and let a wave
*C*(*Z*) be reflected. What is the reflection coefficient at the first layer encountered? What would be the reflected wave as a function of*C*for a situation which differs from the above by the removal of the first reflector? - Consider the earth to be modeled by layers over a halfspace.
Let an impulse be incident from below (Figure 10).
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 .How is*X*related to*E*? This relation illustrates the principle of reciprocity which says source and receiver may be interchanged.**E8-3-5**Earthquake geometry.

Figure 9 - Show that ,which shows that one may autocorrelate the transmission seismogram to get the reflection seismogram.
- Refer to Figure 11. Calculate
*R*' from*R*.**E8-3-7**Stripping off the surface layer.

Figure 10

10/30/1997