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Fundamentals of Seismic Wave Propagation

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and equation (6.3.3), we can reduce it to
(WTI2)A = p3(WTI2).
Post-multiplying this by the matrix W, pre-multiplying equation (6.3.3) by matrix
WTI2, and subtracting, we find that
WTI2W = K, (6.3.24)
say, must be diagonal. The eigenvector columns of matrix W are
w(0) = wE
(
gˆE
−pkc3k gˆE
)
, (6.3.25)
where wE is an arbitrary normalization (no summation over E). Again, for future
use, it is useful to define the polarization (cf. equation (6.3.6))
g = wE gˆE . (6.3.26)
From result (6.3.24), we find that the diagonal elements of matrix K are
KE = −(2ρV3w2)E , (6.3.27)
where equation (5.3.20) defines the component of the ray velocity, V3. Thus the
required inverse matrix in definition (6.3.22) can be computed simple as
W−1 = K−1WTI2. (6.3.28)
If wE = 1/
√±2ρV3,
w(0) = 1√±2ρV3
(
gˆE
−pkc3k gˆE
)
, (6.3.29)
with the positive sign for E = 1 to 3, and the negative sign for E = 4 to 6, then
K = K−1 = I3. For numerical purposes, especially for evanescent waves, it is sim-
pler to take wE = 1, when K is defined by equation (6.3.27). It is important to
remember that the normalization wE affects the numerical values of the reflec-
tion/transmission coefficients, but not, of course, the resultant amplitude of the
field variables. The reflection/transmission coefficients are with respect to the ba-
sis vectors, (6.3.25) w(0), and changes in one are compensated by changes in the
6.3 Reflection/transmission coefficients 213
other so the appropriate products remain independent of the normalization factor,
wE .
For future use it is convenient to introduce a notation for the sub-matrices in the
eigenmatrix W. Thus we write
W =
(
W´ W`
)
=
(
W11 W12
W21 W22
)
, (6.3.30)
where the 6 × 3 sub-matrices W´ and W` contain the up and down-going eigenvec-
tors, respectively, and Wi j are 3 × 3 sub-matrices. We define a symplectic trans-
form of the velocity-traction vector, w (6.1.2)
w‡ = −wTI2 = −
(
tTn vT
) (6.3.31)
(this is a different symplectic transform from that used for the dynamic ray sys-
tem, (5.2.32)). Then with the normalization wE = 1/
√±2ρV3, the inverse matrix
(6.3.24) is
W−1 = I3WTI2 =
(−WT21 −WT11
WT22 W
T
12
)
=
(
W´‡
−W`‡
)
. (6.3.32)
For eigenvectors wi of the matrix A, we have the orthonormal relationship
w
‡
i w j = ±δi j , (6.3.33)
where the sign depends on the propagation direction, i.e. positive for i = 1 to 3,
and negative for i = 4 to 6, with our ordering convention. For propagating rays,
this normalization is connected with the energy flux in the nˆ direction. For evanes-
cent rays, this connection breaks down, but the normalization is still useful. An
orthonormality relationship like result (6.3.33) was appreciated by Herrera (1964)
and Alsop (1968), but without the connection to the symplectic symmetry of the
differential system. Biot (1957) discussed energy flux results. It is important to re-
member that the orthonormality (6.3.33) applies to rays with the same slowness,
p⊥, parallel to the surface used to define the traction, tn . For different ray types,
these will be propagating in different directions. It does not apply to different rays
propagating in the same direction or with the same total slowness.
Thus the inverse matrix W−1 in equation (6.3.22) is known without inverting
any matrix, so
Q = I3 WT1 I2 W2, (6.3.34)
and the coefficients (6.3.23) can be calculated by inverting only one 3 × 3 matrix.
214 Rays at an interface
6.3.2.2 The reciprocity of coefficients
Finally we need to prove the reciprocity of the reflection/transmission coefficients
TTT (6.3.23). In the reciprocal rays, the slownesses in the plane of the interface are
reversed and the matrix A becomes
A′ = A(−pν). (6.3.35)
The eigen-solution becomes
A′W′ = −W′pn, (6.3.36)
where the change of sign occurs as the slowness surface has point symmetry. The
revised eigenvectors, W′, are related by
W′ = −I3W, (6.3.37)
i.e. the traction components change sign. Importantly the propagation directions
of the columns of W′ are reversed, so equation (6.3.18) becomes
W′1
(
I 0
TTT ′11 TTT ′12
)
= W′2
(TTT ′21 TTT ′22
0 I
)
. (6.3.38)
Taking the transpose of this equation (6.3.38) and multiplying by I1 times equation
(6.3.18), we obtain(
I TTT ′ T11
0 TTT ′ T12
)
W′ T1 I1W1
(TTT 11 TTT 12
I 0
)
=
(TTT ′ T21 0
TTT ′ T22 I
)
W′ T2 I1W2
(
0 I
TTT 21 TTT 22
)
.
(6.3.39)
Using definition (6.3.37), it is straightforward to simplify this as
W′ TI1W = −WTI T3I1W = WTI2W = K =
(
K´ 0
0 K`
)
, (6.3.40)
where we have expanded the matrix (6.3.24) into diagonal 3 × 3 sub-matrices for
the positive and negative travelling waves. Then expanding equation (6.3.39), it is
seen to be equivalent to
TTT ′ =
(
K`−11 0
0 −K´−12
)
TTT T
(
−K´1 0
0 K`2
)
, (6.3.41)
which is the reciprocity result for reflection/transmission coefficients. If the eigen-
vectors are normalized so K = I3, then this simplifies to TTT ′ = TTT T, or
TTT (−p1 , −p2) = TTT T(p1 , p2). (6.3.42)
6.3 Reflection/transmission coefficients 215
This equation describes the fact that if the source and receiver are interchanged,
requiring the swapping of subscript indices on Ti j and the reversal of the slowness
components parallel to the interface, then the reflection/transmission coefficients
are equal (with suitable normalization of the eigenvectors), i.e. they satisfy reci-
procity. This result is far from trivial and does not correspond to just reversing
time. Apart from the reversal of the source and receiver rays, the other generated
rays are completely different in the reciprocal experiments. The reciprocal result
(6.3.42) does depend on the polarizations being defined in a consistent manner.
Changes in sign of the polarization, permitted by the eigen-equation (6.3.14), will
result in changes in the sign of coefficients and reciprocity will only be satisfied
for the combination of coefficient times polarization. Equation (6.3.42) will con-
tain sign mismatches.
6.3.2.3 Energy flux conservation
The orthonormality relation (6.3.33), which resulted in the simple reciprocity re-
lationship (6.3.42), is connected with the energy flux of the eigenvectors when the
waves are propagating, not evanescent. It leads to another relationship between the
coefficients.
Expanding the continuity equation (6.3.18), the first row is
W´1TTT 11 + W`1 = w = W`2TTT 21. (6.3.43)
Multiplying both sides by the transform (6.3.31), we obtain
−
(
TTT TW´T1 + W`T
)
I2
(
W´1TTT 11 + W`1
)
= w‡ w = −TTT T21W`T2I2W`2TTT 21. (6.3.44)
Expanding, and using the orthonormality (6.3.33), we have
TTT T11TTT 11 + TTT T21TTT 21 = I. (6.3.45)
The second row of equation (6.3.18) leads to the similar result
TTT T22TTT 22 + TTT T12TTT 12 = I. (6.3.46)
Scalar examples of results (6.3.45) and (6.3.46) have already been noted, results
(6.3.9) and (6.3.10). The simple result and proof break down at a fluid–solid inter-
face – equation (6.3.43) no longer applies as the tangential displacement is discon-
tinuous and tangential traction is zero in w (see Section 6.5).
Physically, these results, (6.3.45) and (6.3.46), express the conservation of en-
ergy flux across the interface when the waves are propagating, and the coefficients
are real. The results remain true when any waves are evanescent, and coefficients
complex, but no longer express conservation of energy flux.
216 Rays at an interface
6.3.3 Isotropic coefficients
In isotropic media, we follow exactly the same procedure as in anisotropic media
(Section 6.3.2) but significant simplifications are possible, and explicit expressions
can be obtained for the coefficients. These were first obtained by Knott (1899) and
later by Zoeppritz (1919) and are known by both names although usually the latter.
Their

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