Троян В.Н. Принципы решения обратных геофизических задач
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ˆ
θ
MLS
M[
ˆ
~
θ] = (ψ
T
ψ)
−1
ψ
T
M[~u] =
= (ψ
T
ψ)
−1
ψ
T
M[ψ
~
θ + ε] =
~
θ,
M[ε] = 0
ˆ
θ
MLS
R
ss
(
ˆ
~
θ) ≤ R
ss
(
~
θ
∗
) s = 1, . . . , S,
~
θ
∗
~u
ˆ
~u = ψ
ˆ
~
θ
ˆσ
2
ε
⇒ M[(~u − ψ
ˆ
~
θ)
T
(~u − ψ
ˆ
~
θ)] = σ
2
ε
(n − S) ⇒
⇒ ˆσ
2
ε
= (~u − ψ
ˆ
~
θ)
T
(~u − ψ
ˆ
~
θ)/(n − S) =
ˆ
~ε
T
ˆ
~ε/(n − S).
τ
s
~u = ψ
~
A + ~ε,
ψ = ||ϕ(t
i
− τ
s
)||
n×S
~ε
ˆ
~
A = (ψ
T
ψ)
−1
ψ
T
~u
ˆ
~
A
R
ˆ
A
= ˆσ
2
ε
(ψ
T
ψ)
−1
,
ˆσ
2
ε
=
(~u − ψ
ˆ
~
A)
T
(~u − ψ
ˆ
~
A)
n − S
~u = ψ
~
θ + ~ε
ψ ψ
is
= x
s
i
i ψ
ψ
i
= kϕ
0
(x
i
), ϕ
1
(x
i
), . . . , ϕ
S−1
(x
i
)k.
n
X
i=1
ϕ
s
(x
i
)ϕ
s
′
(x
i
) =
n
P
i=1
ϕ
2
s
(x
i
)
s = s
′
0
s 6= s
′
ψ
T
ψ
ψ
T
ψ =
n
P
i=1
ϕ
2
0
(x
i
) 0 . . . 0
0
n
P
i=1
ϕ
2
1
(x
i
) . . . 0
. . . . . . . . . . . .
0 0 . . .
n
P
i=1
ϕ
2
S−1
(x
i
)
.
ϕ
0
(x
i
) = 1, ϕ
1
(x
i
) = x
i
+ b
0
ϕ
0
(x
i
).
b
0
ϕ
0
(x
i
)
b
0
⇒
n
X
i=1
ϕ
1
(x
i
)ϕ
0
(x
i
) =
n
X
i=1
x
i
ϕ
0
(x
i
) + b
0
n
X
i=1
ϕ
2
0
(x
i
).
n
X
i=1
ϕ
1
(x
i
)ϕ
0
(x
i
) = 0,
b
0
= −
n
P
i=1
x
i
ϕ
0
(x
i
)
n
P
i=1
ϕ
2
0
(x
i
)
,
ϕ
1
(x
i
)
ϕ
1
(x
i
) = x
i
−
n
P
i=1
x
i
ϕ
0
(x
i
)
n
P
i=1
ϕ
2
0
(x
i
)
ϕ
0
(x
i
).
ϕ
2
(x
i
)
ϕ
2
(x
i
) = x
2
i
+ b
1
ϕ
1
(x
i
) + b
0
ϕ
0
(x
i
).
b
0
b
1
n
X
i=1
ϕ
2
(x
i
)ϕ
0
(x
i
) = 0, b
0
= −
n
P
i=1
x
2
i
ϕ
0
(x
i
)
n
P
i=1
ϕ
2
0
(x
i
)
,
n
X
i=1
ϕ
2
(x
i
)ϕ
1
(x
i
) = 0, b
1
= −
n
P
i=1
x
2
i
ϕ
1
(x
i
)
n
P
i=1
ϕ
2
1
(x
i
)
.
b
0
b
1
ϕ
2
(x
i
)
ϕ
2
(x
i
) = x
2
i
−
n
P
i=1
x
2
i
ϕ
1
(x
i
)
n
P
i=1
ϕ
2
1
(x
i
)
ϕ
1
(x
i
) −
n
P
i=1
x
2
i
ϕ
0
(x
i
)
n
P
i=1
ϕ
2
0
(x
i
)
ϕ
0
(x
i
).
ϕ
s
(x
i
)
ϕ
s
(x
i
) = x
s
i
−
n
P
i=1
x
s
i
ϕ
s−1
(x
i
)
n
P
i=1
ϕ
2
s−1
(x
i
)
ϕ
s−1
(x
i
)−. . .−
n
P
i=1
x
s
i
ϕ
0
(x
i
)
n
P
i=1
ϕ
2
0
(x
i
)
ϕ
0
(x
i
),
ϕ
s
(x
i
) = x
s
i
+ b
s−1
ϕ
s−1
(x
i
) + . . . + b
0
ϕ
0
(x
i
).
b
q
q < s
n
X
i=1
x
s
i
ϕ
q
(x
i
) + b
q
n
X
i=1
ϕ
2
q
(x
i
) = 0, b
q
= −
n
P
i=1
x
s
i
ϕ
q
(x
i
)
n
P
i=1
ϕ
2
q
(x
i
)
.
ˆ
θ
s
ˆ
θ
s
=
n
X
j=1
ϕ
sj
u
j
, s = 0, 1, 2, . . . , S − 1,
ˆ
~
θ = Φ~u,
Φ
Φ =
ϕ
0
(x
1
) ϕ
0
(x
2
) . . . ϕ
0
(x
n
)
ϕ
1
(x
1
) ϕ
1
(x
2
) . . . ϕ
1
(x
n
)
. . . . . . . . . . . .
ϕ
S−1
(x
1
) ϕ
S−1
(x
2
) . . . ϕ
S−1
(x
n
).
n
X
i=1
ϕ
2
si
= 1.
R
ε
= σ
2
ε
I
R
θ
= σ
2
ε
I.
ˆσ
2
ε
=
1
n − S
(~u − ψ
ˆ
~
θ)
T
(~u − ψ
ˆ
~
θ).
~u = ψ
~
θ + ~ε
A
~
θ =
~
V ,
A [K × S]
V [K × 1]
˜
~
θ
~
θ
˜
~
θ
T
=
ˆ
~
θ
T
+ (
~
V
T
−
ˆ
~
θ
T
A
T
)(A(ψ
T
ψ)
−1
A
T
)
−1
A(ψ
T
ψ)
−1
R
˜
~
θ
˜
~
θ
R
˜
~
θ
= R
ˆ
~
θ
− σ
2
ε
(ψ
T
ψ)
−1
A
T
(A(ψ
T
ψ)
−1
A
T
)
−1
A(ψ
T
ψ)
−1
R
ˆ
~
θ
= ˆσ
2
ε
(ψ
T
ψ)
−1
, ˆσ
2
ε
=
1
n − S − K
(~u − ψ
˜
~
θ)
T
(~u − ψ
˜
~
θ).
~u
(
~
V )
~u
~u
f(~x,
~
θ)
~
θ
L(~u,
~
θ)
L(~u,
~
θ) = f(~x,
~
θ).
~
θ
L(~u,
~
θ) ln L(~u,
~
θ)
ˆ
~
θ
MLM
= arg max ln L(u
1
, . . . , u
n
;
~
θ).
ln L θ
s
∂ ln L(u
1
, . . . , u
n
;
~
θ)
∂θ
s
= 0.
ˆ
~
θ
MLM
ˆ
~
θ
MLM
M[
ˆ
~
θ
n
]
n→∞
= θ
ˆ
~
θ
MLM
√
n(
ˆ
~
θ
n
−
~
θ)
n→∞
∈ N(0, [I
(F )
(θ)]
−1
)
I
(F )
(θ)
ˆ
~
θ
MLM
σ
2
ˆ
θ
n
n→∞
= [I
(F )
(θ
n
)]
−1
R(
ˆ
~
θ)
n→∞
= [I
(F )
(
ˆ
~
θ
n
)]
−1
.
~u =
~
f(
~
θ) + ~ε, ~u = (u
1
, . . . , u
n
), ε ∈ N(0, R
ε
).
l(~u,
~
θ) = −
n
2
ln 2π −
1
2
ln R
ε
−
1
2
(~u−
~
f(
~
θ))
T
R
−1
ε
(~u−
~
f(
~
θ)),
l
1
(~u,
~
θ) = −
1
2
(~u −
~
f(
~
θ))
T
R
−1
ε
(~u −
~
f(
~
θ)).
R
ε
= σ
2
ε
I,
l
1
(~u,
~
θ) = −
1
2σ
2
ε
(~u −
~
f(
~
θ))
T
(~u −
~
f(
~
θ)).
(~u −
~
f(
~
θ))
T
(~u −
~
f(
~
θ)),
ε
(~u −
~
f(
~
θ))
T
R
−1
ε
∂
~
f
∂θ
s
= 0, s = 1, 2, . . . , S.
~u = ψ
~
θ + ~ε, ε ∈ N(0, R
ε
)