Троян В.Н. Принципы решения обратных геофизических задач
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ˆ
~
θ
1
(1)
= max
~
θ
1
g
2
(~u
k
,
~
θ
1
,
ˆ
~
θ
2
(1)
),
~
Y
k1
= ~u
T
k
−
~
f
2k
(
ˆ
~
θ
2
).
ˆ
~
θ
2
(2)
ˆ
~
θ
1
(1)
g
2
(
ˆ
~
θ
1
(n)
,
ˆ
~
θ
2
(n)
) − g
2
(
ˆ
~
θ
1
(n−1)
,
ˆ
~
θ
2
(n−1)
)
g
2
(
ˆ
~
θ
1
(n−1)
,
ˆ
~
θ
2
(n−1)
)
< δ ∼ 10
−2
÷10
−3
,
M = 3
∆g
(n)
2
∆g
(n−1)
2
> δ,
ˆ
~
θ
2
ˆ
~
θ
1
M = 3 g
3
(~u
k
,
ˆ
~
θ
1
,
ˆ
~
θ
2
,
~
θ
3
) =
X
k
~
Y
T
k3
R
−1
~
f
k3
(
~
θ
3
) −
−
1
2
X
k
~
f
T
3k
(
~
θ
3
)R
−1
~
f
3k
(
~
θ
3
)
~
Y
T
k3
= ~u
T
k
−
~
f
T
1k
(
ˆ
~
θ
1
) −
~
f
T
2k
(
ˆ
~
θ
2
)
ˆ
~
θ
3
(1)
= max
~
θ
3
g
3
(~u
k
,
ˆ
~
θ
1
,
ˆ
~
θ
2
,
~
θ
3
).
α
2
3
=
X
k
~
f
T
3k
(
ˆ
~
θ
(1)
3
)R
−1
~
f
3k
(
ˆ
~
θ
(1)
3
) ≥ α
2
0P
e
,
α
2
3
< α
2
0P
e
,
ˆ
~
θ
1
ˆ
~
θ
2
ˆ
~
θ
2
(1)
ˆ
~
θ
1
(1)
g
3
(
ˆ
~
θ
1
(n)
,
ˆ
~
θ
2
(n)
,
ˆ
~
θ
3
(n)
) − g
3
(
ˆ
~
θ
1
(n−1)
,
ˆ
~
θ
(n−1)
2
,
ˆ
~
θ
3
(n−1)
)
g
3
(
ˆ
~
θ
1
(n−1)
,
ˆ
~
θ
(n−1)
2
,
ˆ
~
θ
(n−1)
3
)
< δ,
m = 4
∆g
(n)
3
∆g
(n−1)
3
> δ,
~
θ
1
~
θ
2
~
θ
3
m m
M = m, g
m
(~u
k
,
ˆ
~
θ
1
,
ˆ
~
θ
2
, . . .
~
θ
m
) =
X
k
~
Y
T
km−1
R
−1
~
f
mk
(
~
θ
m
) −
−
1
2
X
k
~
f
T
mk
(
~
θ
m
)R
−1
~
f
mk
(
~
θ
m
)
~
Y
T
km
= ~u
T
k
−
m−1
X
µ=1
~
f
T
µk
(
ˆ
~
θ
µ
)
ˆ
~
θ
m
(1)
= max
~
θ
m
g
m
(~u
k
,
ˆ
~
θ
1
,
ˆ
~
θ
2
, . . . ,
~
θ
m
).
α
2
m
=
X
k
~
f
T
mk
(
ˆ
~
θ
(1)
m
)R
−1
~
f
mk
(
ˆ
~
θ
(1)
m
) ≥ α
2
0P
e
,
α
2
m
< α
2
0P
e
,
(m−1)
ˆ
~
θ
1
. . . ,
ˆ
~
θ
m−1
)
ˆ
~
θ
(1)
1
, . . . ,
ˆ
~
θ
(1)
m−1
,
ˆ
~
θ
(1)
m
ˆ
~
θ
(2)
1
, . . . ,
ˆ
~
θ
(2)
m−1
,
ˆ
~
θ
(2)
m
g
m
(
ˆ
~
θ
(n)
1
, . . . ,
ˆ
~
θ
(n)
m−1
,
ˆ
~
θ
(n)
m
) − g
m
(
ˆ
~
θ
(n−1)
1
, . . . ,
ˆ
~
θ
(n−1)
m−1
,
ˆ
~
θ
(n−1)
m
)
g
m
(
ˆ
~
θ
(n−1)
1
, . . . ,
ˆ
~
θ
(n−1)
m−1
,
ˆ
~
θ
(n−1)
m
)
< δ,
M = m + 1
∆g
(n)
m
∆g
(n−1)
m
> δ,
~
θ
1
. . . ,
~
θ
m−1
,
~
θ
m
M = 1, g
1
(~u
k
,
~
θ
1
) =
X
k
[~u
T
k
R
−1
~
f
1k
(
~
θ
1
) −
−
1
2
~
f
T
1k
(
~
θ
1
)R
−1
~
f
1k
(
~
θ
1
)]
ˆ
~
θ
1
= max
~
θ
1
g
1
(~u
k
,
~
θ
1
).
α
2
1
=
X
k
~
f
T
k
(
ˆ
~
θ
1
)R
−1
~
f
k
(
ˆ
~
θ
1
) ≥ α
2
0P
e
,
α
2
1
< α
2
0P
e
,
M = 2, g
2
(~u
k
,
ˆ
~
θ
1
,
~
θ
2
) =
X
k
~
Y
T
k2
R
−1
~
f
2k
(
~
θ
2
) −
−
1
2
X
k
~
f
T
2k
(
~
θ
2
)R
−1
~
f
2k
(
~
θ
2
)
~
Y
T
k2
= ~u
T
k
−
~
f
1k
(
ˆ
~
θ
1
).
~
θ
1
ˆ
~
θ
1
ˆ
~
θ
2
= max
~
θ
2
g
2
(~u
k
,
ˆ
~
θ
1
,
~
θ
2
).
α
2
2
=
X
k
~
f
T
2k
(
ˆ
~
θ
2
)R
−1
~
f
2
k(
ˆ
~
θ
2
) ≥ α
2
0P
e
,
α
2
2
< α
2
0P
e
,
~
θ
1
M = 3, g
3
(~u
k
,
ˆ
~
θ
1
,
ˆ
~
θ
2
,
~
θ
3
) =
X
k
~
Y
T
k3
R
−1
~
f
3k
(
~
θ
3
) −
−
1
2
X
k
~
f
T
3k
(
~
θ
3
)R
−1
~
f
3k
(
~
θ
3
)
~
Y
T
k3
= ~u
T
k
−
~
f
T
3k
(
ˆ
~
θ
1
) −
~
f
T
3k
(
ˆ
~
θ
2
)
ˆ
~
θ
3
= max
~
θ
3
g
3
(~u
k
,
ˆ
~
θ
1
,
ˆ
~
θ
2
,
~
θ
3
).
α
2
3
=
X
k
~
f
T
k
(
ˆ
~
θ
3
)R
−1
~
f
k
(
ˆ
~
θ
3
) ≥ α
2
0P
e
,
α
2
3
< α
2
0P
e
,
ˆ
~
θ
1
ˆ
~
θ
2
m m
M = m, g
m
(~u
k
,
ˆ
~
θ
1
,
ˆ
~
θ
2
, . . .
~
θ
m
) =
X
k
~
Y
T
km−1
R
−1
~
f
mk
(
~
θ
m
) −
−
1
2
X
k
~
f
T
mk
(
~
θ
m
)R
−1
~
f
mk
(
~
θ
m
)
~
Y
T
km−1
= ~u
T
k
−
m−1
X
µ=1
~
f
T
µk
(
ˆ
~
θ
µ
)
ˆ
~
θ
m
= max
~
θ
m
g
m
(~u
k
,
ˆ
~
θ
1
,
ˆ
~
θ
2
, . . . ,
~
θ
m
).
α
2
m
=
X
k
~
f
T
mk
(
ˆ
~
θ
m
)R
−1
~
f
mk
(
ˆ
~
θ
m
) ≥ α
2
0P
e
,
α
2
m
< α
2
0P
e
,
(m − 1)
ˆ
~
θ
1
. . . ,
ˆ
~
θ
m−1
m
R =
σ
2
I
~
f
k
(
~
θ
m
) = kA
m
ϕ(t
i
− τ
m
− k∆xγ
m
)k
n
i=1
,
M = m, g
m
(~u
k
, A
m
, τ
m
, γ
m
) =
A
m
σ
2
X
i
ϕ(t
i
)
X
k
y(t
i
+ τ
mk
) −
−
A
2
m
2σ
2
K
X
i
ϕ
2
(t
i
),
y(t
i
+ τ
mk
) = u(t
i
+ τ
mk
) −
X
µ
′
6=m
A
µ
′
ϕ(t
i
− τ
µ
′
k
− τ
µk
).
∂g
m
∂A
m
= 0 ⇒
ˆ
A
m
=
P
i
ϕ(t
i
)
P
k
y(t
i
+ τ
mk
)
P
i
ϕ
2
(t
i
)
.
ˆ
A
m
g
m
(~u
k
, A
m
, τ
m
, γ
m
)
g
m
(τ
m
, γ
m
) =
1
σ
2
[
P
i
ϕ(t
i
)
P
k
y(t
i
+ τ
mk
)]
2
K
P
i
ϕ
2
(t
i
)
.
τ
m
γ
m
m
ˆτ
m
, ˆγ
m
= max
τ
m
,γ
m
g
m
(~u
k
,
ˆ
A
m
, τ
m
, γ
m
).
α
2
m
=
ˆ
A
2
m
σ
2
X
k
X
i
ϕ
2
(t
i
− ˆτ
m
− k∆xˆγ)) ≥ α
2
0P
e
,
M = m+1
α
2
m
< α
2
0P
e
,
ˆ
~
θ{
ˆ
A
1
, ˆτ
1
, ˆγ
1
,
ˆ
A
2
, ˆτ
2
, ˆγ
2
, . . . ,
ˆ
A
m
, ˆτ
m
, ˆγ
m
}.
m
µ
ε R = σ
2
I
f
kµ
=
I
µ
h
µ
h
2
µ
+ (x
k
− ξ
2
µ
)
= I
µ
ϕ
k
(h
µ
, ξ
µ
),
ϕ
k
(h
µ
, ξ
µ
) =
h
µ
h
2
µ
+ (x
k
− ξ
2
µ
)
.
M = m, g
m
(u
k
, I
m
, h
m
, ξ
m
) =
I
m
σ
2
X
k
y
km
ϕ(h
m
, ξ
m
) −
−
1
2
I
2
m
σ
2
X
k
ϕ
2
k
(h
m
, ξ
m
),
y
km
= u
k
−
X
µ
′
6=m
I
µ
′
ϕ
k
(h
µ
′
, ξ
µ
′
).
∂g
m
∂I
m
= 0,
ˆ
I
m
=
P
k
y
km
ϕ
k
(h
m
, ξ
m
)
P
k
ϕ
2
k
(h
m
, ξ
m
)
.