Троян В.Н. Принципы решения обратных геофизических задач
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u
D
1
=
1
w
n
t
1
t
2
. . . t
n
h
z r
1
z
r
1
1
ih
z r
2
z
r
2
1
i
. . .
h
z r
n
z
r
n
1
i
u
D
n+1
u
D
1
=
1
w
n
z
n
F
n
(1/z) z
n
G
n
(1/z)
G
n
(z) F
n
(z)
u
D
n+1
F
n
(z) =
1
t
1
t
2
. . . t
n
(1 + F
1
z + F
2
z
2
+ . . . + F
n−1
z
n−1
)
G
n
(z) =
1
t
1
t
2
. . . t
n
(r
n
+ G
1
z + G
2
z
2
+ . . . + G
n−1
z
n−1
)
n
(n + 1)
u
D
1
u
D
n+2
⇒
1
w
n
z
n
F
n
(1/z) z
n
G
n
(1/z)
G
n
(z) F
n
(z)
×
×
1
wt
n+1
z zr
n+1
r
n+1
1
=
1
w
n+1
t
n+1
×
×
z
n
F
n
(1/z)z + z
n
G
n
(1/z)r
n+1
z
n
F
n
(1/z)zr
n+1
+ z
n
G
n
(1/z)
G
n
(z)z + F
n
(z)r
n+1
G
n
(z)zr
n+1
+ F
n
(z)
=
1
w
n+1
z
n+1
F
n+1
(1/z) z
n+1
G
n+1
(1/z)
G
n+1
(z) F
n+1
(z)
F
n+1
(z) = (F
n
(z) + r
n+1
zG
n
(z))t
−1
n+1
G
n+1
(z) = (r
n+1
F
n
(z)+zG
n
(z))t
−1
n+1
R(z)
u
D
1
=
R(z)
1 + R(z)
,
u
D
n+1
=
0
T ( z)
T ( z)
R(z)
1 + R(z)
=
1
w
n
z
n
F (1/z) z
n
G(1/z)
G(z) F (z)
0
T ( z)
R(z)
1
=
1
w
n
z
n
F (1/z) z
n
G(1/z)
G(z) − z
n
F (1/z) F (z) − z
n
G(1/z)
0
T ( z)
M( z) = F (z) − G(1/z)z
n
T ( z) =
w
n
M( z)
; R(z) =
z
n
G(1/z)
M( z)
M( z) = M
0
+ M
1
z + . . . + M
n
z
n
M
0
=
1
t
1
t
2
. . . t
n
; M
n
= −
r
n
t
1
t
2
. . . t
n
M( z) = exp(inω∆t)/T (z)
|z| ≤ 1
M( z) 1/M(z)
ω 1/M(z)
|z| ≤ 1
[1 + R(z) + R(1/z)]M(1/z) =
= M(1/z) + R(z)M(1/z) + R(1/z)M(1/z) =
= M(1/z) +
z
n
G(1/z)
M( z)
M(1/z) +
z
−n
G(z)
M(1/z)
M(1/z) =
= F (z)M(1/z)/M(z) + z
−n
G(z) =
= [F (z)M(1/z) + z
−n
G(z)M(z)]/M(z)
1
=
= [F (z)F (1/z) − z
−n
F (z)G(z) + z
−n
F (z)G(z) −
− z
n
z
−n
G(z)G(1/z)]/M(z) =
= [F (z)F (1/z) − G(z)G(1/z)]/M(z)
det
1
w
n
z
n
F (1/z) z
n
G(1/z)
G(z) F (z)
= F (z)F (1/z) − G(z)G(1/z)
det
1
wt
1
z zr
1
r
1
1
1
wt
2
z zr
2
r
2
1
. . .
1
wt
n
z zr
n
r
n
1
=
=
n
Y
i=1
det
1
wt
i
z zr
i
r
i
1
=
n
Y
i=1
z(1 − r
i
)(1 + r
i
)
w
2
t
2
i
=
n
Y
i=1
1 − r
i
t
i
M(z) = F (z) − z
n
G(1/z) M(1/z) = F(1/z) − z
−n
G(z)
z = w
2
; (1 + r
i
) = t
i
[1 + R(z) + R(1/z)]M(1/z) = (
n
Y
i=1
(1 − r
i
)/t
i
)/M(z)
1/M(z)
t < 0
z
R(t) = R
1
δ(t−2∆t)+R
2
δ(t−4∆t)+. . .+R
n
δ(t−2n∆t)+. . .
R(z) = R
1
z + R
2
z
2
+ . . . + R
n
z
n
+ . . .
(1 + R
1
z + . . . + R
n
z
n
+ . . . + R
1
z
−1
+ . . . + R
n
z
−n
+ . . .)×
×(M
0
+ M
1
z
−1
+ . . . + M
n
z
−n
)
M
0
R
1
+ M
1
+ M
2
R
1
+ . . . + M
n
R
n−1
= 0
M
1
/M
0
+ M
2
R
1
/M
0
+ . . . + M
n
R
n−1
/M
0
= −R
1
(z
−1
)
M
0
R
2
+ M
1
R
1
+ M
2
+ . . . + M
n
R
n−2
= 0
M
1
R
1
/M
0
+ M
2
/M
0
+ . . . + M
n
R
n−2
/M
0
= −R
2
(z
−2
)
1 R
1
R
2
. . . R
n−1
R
1
1 R
1
. . . R
n−2
R
2
R
1
1 . . . R
n−3
. . . . . . .
R
n−1
R
n−2
R
n−3
. . . 1
M
1
/M
0
M
2
/M
0
M
3
/M
0
. . .
M
n
/M
0
=
−R
1
−R
2
−R
3
. . .
−R
n
M
n
/M
0
= −r
n
, n = 1
v
i+1
ρ
i+1
=
1 − r
i
1 + r
i
ρ
i
v
i
ρ(x)
∂
2
ϕ
∂t
2
=
∂
∂x
[E(x)
∂ϕ
∂x
]; dx = v(x)dτ; τ =
x
Z
0
dx
v(x)
τ
v(x) = [E(x)/ρ(x)]
1/2
∂
2
ϕ(τ, t)
∂τ
2
−
∂
2
ϕ(τ, t)
∂t
2
= −
d ln[ρv]
dτ
∂ϕ(τ, t)
∂τ
∂
2
ϕ
∂t
2
=
1
ρ(x)v(x)
∂
∂τ
[
v
2
(x)ρ(x)
v(x)
∂ϕ
∂τ
] =
=
1
ρ(x)v(x)
[
∂
∂τ
[v(x)ρ(x)]
∂ϕ
∂τ
+ v(x)ρ(x)
∂
2
ϕ
∂τ
2
] =
=
∂
2
ϕ
∂τ
2
+
d
dτ
ln[v(x)ρ(x)]
∂ϕ
∂τ
ψ : ψ = [ρv]
1/2
ϕ
∂
2
ψ
∂τ
2
−
∂
2
ψ
∂t
2
= q(τ)ψ; q(τ) =
1
ξ
d
2
ξ
dτ
2
ψ(t, τ) = exp(−iωt)ψ(τ); ξ = (ρv)
1/2
∂
2
ψ(τ)
∂τ
2
− q(τ)ψ(τ) + ω
2
ψ(τ) = 0
d
2
ψ
dτ
2
+ [ω
2
− q(τ)]ψ = 0, L = [
d
2
dτ
2
+ (ω
2
− q(τ))]
ψ
(ϕ = (1/ξ)ψ)
1
ξ
∂
2
ψ
∂t
2
=
∂
2
∂τ
2
1
ξ
ψ
+
2
ξ
∂ξ
∂τ
∂
∂τ
1
ξ
ψ
2
ξ
∂ξ
∂τ
∂
∂τ
1
ξ
ψ
= −
2
ξ
3
∂ξ
∂τ
2
ψ +
2
ξ
2
∂ξ
∂τ
∂ψ
∂τ
∂
2
∂τ
2
1
ξ
ψ
=
∂
∂τ
−
1
ξ
2
∂ξ
∂τ
ψ +
1
ξ
∂ψ
∂τ
=
=
2
ξ
3
∂ξ
∂τ
2
ψ −
1
ξ
2
∂
2
ξ
∂τ
2
ψ −
1
ξ
2
∂ξ
∂τ
∂ψ
∂τ
+
1
ξ
∂
2
ψ
∂τ
2
−
1
ξ
2
∂ξ
∂τ
∂ψ
∂τ
1
ξ
∂
2
ψ
∂t
2
=
2
ξ
3
∂ξ
∂τ
2
ψ −
1
ξ
2
∂
2
ξ
∂τ
2
ψ −
1
ξ
2
∂ξ
∂τ
∂ψ
∂τ
+
+
1
ξ
∂
2
ψ
∂τ
2
−
1
ξ
2
∂ξ
∂τ
∂ψ
∂τ
+
2
ξ
2
∂ξ
∂τ
∂ψ
∂τ
−
2
ξ
3
∂ξ
∂τ
2
ψ
−
∂
2
ψ
∂t
2
+
∂
2
ψ
∂τ
2
−
1
ξ
∂
2
ξ
∂τ
2
ψ = 0; q(τ) =
1
ξ
∂
2
ξ
∂τ
2
∂
2
ψ
∂τ
2
+ [ω
2
− q(τ)]ψ = 0; ψ(t, τ) = exp(−iωt)ψ(τ)
q(τ) < ω
2
q(τ) > ω
2
q(τ) = ω
2
→ τ
S
11
= S
22
; |S
11
|
2
+ |S
12
|
2
= 1; |S
22
|
2
+ |S
21
|
2
= 1
S =
S
11
S
12
S
21
S
22
SS
∗
= S
∗
S = I; S
11
S
∗
21
+ S
12
S
∗
22
= 0
f
1
(τ, ω)
f
2
(τ, ω)
lim
τ→∞
f
1
(τ, ω) = exp(iωτ); lim
τ→−∞
f
2
(τ, ω) = exp(−iωτ)
S
11
(ω)f
1
(τ, ω) = f
2
(τ, −ω) + S
12
(ω)f
2
(τ, ω)
f
1
f
2
R
∞
−∞
(1 + |τ|)q(τ)dτ < ∞
S
11
(ω)
C(x, τ)
C(x, τ) = −R(τ + x) −
x
Z
−τ
C(x, η)R(τ + η)dη, τ < x
q(τ) = 2
dC(τ, τ)
dτ
; R(t) =
1
2π
∞
Z
−∞
S
12
(ω)e
−iωt
dω
