4.3 Reflection of a Plane Wave from a Layer with a Slowly Varying Thickness
75
∇ A 00 ∇τ 1re f = 0, 2i∇ A i j ∇τ 1re f + +A i−1, j−1 = 0,
(4.31)
∇ B
+
00 ∇τ 2elap = 0, 2i∇ B
+
i j ∇τ 2elap + +B
+
i−1, j−1 = 0,
(4.32)
∇ B
−
00 ∇τ 3re f = 0, 2i∇ B
−
i j ∇τ 3re f + +B
−
i−1, j−1 = 0,
(4.33)
∇C
+
00 ∇τ 3elap = 0, 2i∇C
+
i j ∇τ 3elap + +C
+
i−1, j−1 = 0,
(4.34)
∇C
−
00 ∇τ 4re f = 0, 2i∇C
−
i j ∇τ 4re f + +C
−
i−1, j−1 = 0,
(4.35)
∇ D
+
00 ∇τ 4elap = 0, 2i∇ D
+
i j ∇τ 4elap + +D
+
i−1, j−1 = 0,
(4.36)
∇ D
−
00 ∇τ 5re f = 0, 2i∇ D
−
i j ∇τ 5re f + +D
−
i−1, j−1 = 0,
(4.37)
∇ E 00 ∇τ 5elap = 0, 2i∇ E i j ∇τ 5elap + +E i−1, j−1 = 0.
(4.38)
By solving (4.27)–(4.38) with regard to (4.22)–(4.26), we obtain the eikonals for the
reflected and transmitted fields and the ray amplitudes A, B, C, D, and E (as the
first approximation).
τ 1re f = k 2x ξ 1 + k 2y ξ 2 + k 2z ξ 3 , k
2
2x + k
2
2y + k
2
2z = k
2 n
2
2 ,
(4.39)
τ 2elap = k 2x ξ 1 + k 2y ξ 2 − k
2z ξ 3 , k
2
2x + k
2
2y + k
2
2z = k
2 n
2
2 ,
(4.40)
τ 3re f = k 3x ξ 1 + k 3y ξ 2 + k
3z ξ 3 ,
(4.41)
τ 3elap = k 3x ξ 1 + k 3y ξ 2 − k
3z ξ 3 , k
2
3x + k
2
3y + k
2
3z = k
2 n
2
3 ,
(4.42)
τ 4re f = k 4x ξ 1 + k 4y ξ 2 + k
4z ξ 3 ,
(4.43)
τ 4elap = k 4x ξ 1 + k 4y ξ 2 − k
4z ξ 3 , k
2
4x + k
2
4y + k
2
4z = k
2 n
2
4 ,
(4.44)
τ 5re f = k 5x ξ 1 + k 5y ξ 2 + k
5z ξ 3 , k
2
5x + k
2
5y + k
2
5z = k
2 n
2
5 ,
(4.45)
τ 5elap = k 5x ξ 1 + k 5y ξ 2 − k 5z ξ 3 , k
2
5x + k
2
5y + k
2
5z = k
2 n
2
5 ,
(4.46)
where
k j x = kn j sin(θ ) sin(φ), k j y = kn j sin(θ ) cos(φ), k jz = kn j cos(θ ), j = 2, 5
A(ξ 1 , ξ 2 , ξ 3 , ε x , ε y ) = A
00 (t 0 ) + ε x
A
10 (t 0 ) + ξ 3 A 0000 (t 0 )
+
+ε y
A
01 (t 0 ) + ξ 3 A 0000 (t 0 )
+