Geology Reference
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or
m
b 0 ,
a k φ bb (
k )
=
(2.93)
k = 0
on using the relaxed definition (2.22) of autocorrelation for energy signals.
For p =
1,..., m ,wehave
δ
j b l j
I
∂Re a p =
p
p
k b l k a j b l j +
a k b l k δ
=
0,
(2.94)
j , k , l
δ
j b l j
i I
p
p
k b l k a j b l j a k b l k δ
∂Im a p =−
=
0.
(2.95)
j , k , l
On adding equations (2.94) and (2.95), we get
m
m
+
n
p
j b l j =
b l k b l p =
a k b l k δ
a k
0.
(2.96)
j , k , l
k
=
0
l
=
0
Again using the relaxed definition of autocorrelation (2.22), we find that
m
a k φ bb ( p
k )
=
0,
p
=
1,..., m .
(2.97)
k
=
0
In matrix form, equations (2.93) and (2.97) give the minimum error energy sys-
tem for the inverse sequence,
b 0
0
. . .
0
a 0
a 1
. . .
a m
φ bb (0)
··· φ bb (
m )
. . .
. . .
. . .
=
.
(2.98)
φ bb ( m )
··· φ bb (0)
On scaling the inverse sequence by its first term, this system takes on the same
form as the prediction error equations (2.80), giving rise to the description of this
method as predictive deconvolution.
Returning to expression (2.89), the error energy can be written
m
m
a 0 b 0 +
a j
I
=
1
a 0 b 0
a k φ bb ( j
k ).
(2.99)
j
=
0
k
=
0
From equations (2.93) and (2.97), when I
=
I min ,
m
b 0 δ
0
a k φ bb ( j
k )
=
j ,
j
=
0,..., m .
(2.100)
k = 0
 
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