Троян В.Н. Принципы решения обратных геофизических задач
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~
θ
u(t
i
) =
m
X
µ=1
A
µ
ϕ(t
i
− τ
µ
) + ε(t
i
),
ϕ(t) A
µ
τ
µ
µ
~
θ = kA
µ
, τ
µ
k
µ=1
.
A
µ
τ
µ
u(ω) =
m
X
µ=1
A
µ
Φ(ω) exp{−iωτ
µ
} + E(ω)
N(0, 0.1)
u(ω) =
∞
Z
−∞
u(t) exp{−iωt}dt,
Φ(ω) =
∞
Z
−∞
ϕ(t) exp{−iωt}dt,
E(ω) =
∞
Z
−∞
ε(t) exp{−iωt}dt.
Z
Z = e
−iω∆t
u(Z) =
m
X
µ=1
A
µ
Φ(Z)Z
τ
µ
/∆t
+ E(Z).
u(x
k
) =
M(2h
2
− x
2
k
)
(x
2
k
+ h
2
)
5/2
+ ε(x
k
),
M h
h = 1 M = 1
N(0, σ)
h = 1 M = 1 σ = 0.1
u(x
k
, y
l
, z
m
, t
i
) =
S
X
j=0
θ
j
ψ
j
(x
k
, y
l
, z
m
, t
i
) + ε(x
k
, y
l
, z
m
, t
i
).
u
u(x
k
, z
m
) =
S
X
j=0
θ
j
ψ
j
(x
k
, z
m
) + ε(x
k
, z
m
),
θ
j
j
u(x
k
, t
i
) =
S
X
j=0
θ
j
ψ
j
(x
k
, t
i
) + ε(x
k
, t
i
),
θ
j
ψ
j
(x
k
, t
i
)
u(x
k
) =
S
X
j=0
θ
j
ψ
j
(x
k
) + ε(x
k
).
u(t
i
) =
S
X
j=0
θ
j
ψ
j
(t
i
) + ε(t
i
).
~u = ψ
~
θ + ~ε,
~u =
u
1
u
k
, ~ε =
ε
1
ε
k
, ψ =
ψ
11
. . . ψ
1S
. . . . . . . . .
ψ
K1
. . . ψ
KS
.
ψ
u(x
k
, y
l
, z
m
, t
i
)
ν = 1, 2, . . . , N
=
f
1
(x
k
, y
l
, z
m
, t
i
) + ε(x
k
, y
l
, z
m
, t
i
),
f
2
(x
k
, y
l
, z
m
, t
i
) + ε(x
k
, y
l
, z
m
, t
i
, )
′
. . . . . .
f
N
(x
k
, y
l
, z
m
, t
i
) + ε(x
k
, y
l
, z
m
, t
i
)
′
f
ν
ν = 1, 2, . . . , N
u(t
i
) =
Aϕ(t
i
− τ) + ε(t
i
, )
ε(t
i
),
A τ
u(t
i
) =
A
1
ϕ(t
i
− τ
1
) + ε(t
i
)
A
2
ϕ(t
i
− τ
2
) + ε(t
i
)
A
1
τ
1
A
2
τ
2
u(t
i
) =
A
0
ϕ
0
(t
i
− τ
0
) + ε(t
i
)
A
1
ϕ
1
(t
i
− τ
1
) + A
2
ϕ
2
(t
i
− τ
2
) + ε(t
i
).
(A
0
, τ
0
)
(A
1
, τ
1
) (A
2
, τ
2
)
u(ω) =
A
0
Φ
0
(ω − ω
0
) + E(ω)
A
1
Φ
1
(ω − ω
1
) + A
2
Φ
2
(ω − ω
2
) + E(ω).
u(x
k
, y
l
, z
m
, t
i
)
ν = 1, 2, . . . , N
=
f
1
(x
k
, y
l
, z
m
, t
i
,
~
θ
1
) + ε(x
k
, y
l
, z
m
, t
i
),
or f
2
(x
k
, y
l
, z
m
, t
i
,
~
θ
2
) + ε(x
k
, y
l
, z
m
, t
i
),
. . . . . .
f
N
(x
k
, y
l
, z
m
, t
i
,
~
θ
N
) + ε(x
k
, y
l
, z
m
, t
i
),
N
f
ν
~
θ
ν
~ε ∈ N(~ε
0
, R).
~ε
p(~ε) = (2π)
−n/2
|R|
−1/2
exp{−
1
2
(~ε −~ε
0
)
T
R
−1
(~ε − ~ε
0
)},
R
ii
′
= M[(ε
i
− ε
0i
)(ε
i
′
− ε
0i
′
)]
r
ii
′
=
R
ii
′
√
R
ii
√
R
i
′
i
′
.
r =
1 r
1
r
2
. . . r
n
. . . . . . . . . . . . . . .
r
n
r
n−1
r
n−2
. . . 1
.