Method for deghosting marine seismic streamer data with irregular receiver positions
Seismic data are obtained for each seismic source activation in a marine streamer and for each frequency, after being transformed to a spectral domain. An iterative conjugate gradient scheme, using a physically-based preconditioner, is applied to the transformed seismic data, to provide a least squares solution to a normal set of equations for a deghosting system of equations. The solution is inverse-transformed back to a space-time domain to provide deghosted seismic data.
1. A method for deghosting seismic data recorded from marine streamers with irregular receiver positions, comprising:
obtaining seismic data for a seismic source activation and for a frequency after being transformed to a spectral domain;
applying an iterative conjugate gradient scheme, using a preconditioner constructed according to a physical problem of deghosting seismic data from marine streamers, to the transformed seismic data, to provide a least squares solution to a normal set of equations for a deghosting system of equations; and
inverse-transforming the solution back to a space-time domain to provide deghosted seismic data.
2. The method of claim 1 , wherein the obtaining seismic data comprises:
obtaining seismic data recorded with a marine streamer;
obtaining recorded seismic data for a seismic source activation;
transforming the recorded seismic data for a seismic source activation from the space-time domain to the space-frequency domain;
obtaining transformed seismic data for a frequency; and
transforming the transformed seismic data for a frequency from the space-frequency domain to the spectral domain.
3. The method of claim 2 , wherein the transforming the recorded seismic data comprises:
transforming the scattered acoustic wavefield from the space-time (x 1 R ,x 2 R ,x 3 R ,t) domain to the space-frequency (x 1 R ,x 2 R ,x 3 R ,s) domain using a Fourier transform, so:
p sct =p sct ( x 1,q R ,x 2,r R ,x 3 R ;s )
where
s=jω=j 2 πf,
and p sct is the scattered acoustic wavefield in the space-frequency domain, x 1,q R is an in-line receiver coordinate, q is an in-line receiver number, x 2,r R is a cross-line receiver coordinate, r is a cross-line receiver number, x 3 R is receiver depth as a function of x 1,q R and x 2,r R , s is a complex Laplace transform parameter, j is imaginary unit √{square root over (−1)}, ω is circular frequency, and f is frequency.
4. The method of claim 3 , wherein the transforming the transformed seismic data comprises:
transforming the scattered wave field p sct from the space-frequency domain (x 1 R ,x 2 R ,x 3 R ,s) to the spectral domain (sα n ,sβ m ,x 3 R ,s) by:
P
n
,
m
sct
=
Δ
x
1
R
Δ
x
2
R
∑
q
=
-
1
2
N
+
1
1
2
N
∑
r
=
-
1
2
M
+
1
1
2
M
exp
(
j
s
α
n
x
1
,
q
R
+
j
s
β
m
x
2
,
r
R
)
p
sct
,
where
:
s
α
n
=
n
Δ
(
s
α
)
,
s
β
m
=
m
Δ
(
s
β
)
,
and
:
Δ
(
s
α
)
=
2
π
N
Δ
x
1
R
,
Δ
(
s
β
)
=
2
π
M
Δ
x
2
R
,
and P n,m sct is the scattered acoustic wavefield in the spectral domain, n is an in-line spectral number, m is a cross-line spectral number, Δx 1 R is in-line receiver sampling distance, Δx 2 R is cross-line receiver sampling distance, jsα n is an in-line spectral Fourier parameter, Δ(sα) is in-line spectral sampling distance, jsβ m is a cross-line spectral Fourier parameter, Δ(sβ) is cross-line spectral sampling distance, N is total number of in-line receivers, and M is total number of cross-line receivers.
5. The method of claim 4 , further comprising:
determining a system of equations for deghosting the transformed seismic data for a frequency;
determining a normal set of equations for the deghosting system of equations; and
applying the constructed preconditioner to the normal set of equations.
6. The method of claim 5 , wherein the determining a system of equations for deghosting comprises applying the following equation:
∑
n
=
-
1
2
N
+
1
1
2
N
∑
m
=
-
1
2
M
+
1
1
2
M
K
q
,
r
;
n
,
m
P
n
,
m
dgh
=
p
q
,
r
sct
,
q
=
-
1
2
N
+
1
,
…
,
1
2
N
,
r
=
-
1
2
M
+
1
,
…
,
1
2
M
,
in which the system matrix K q,r;n,m is given by
K
q
,
r
;
n
,
m
=
Δ
(
s
α
)
Δ
(
s
β
)
4
π
2
∑
n
=
-
1
2
N
+
1
1
2
N
∑
m
=
-
1
2
M
+
1
1
2
M
exp
(
-
j
s
α
n
x
1
,
q
R
-
j
s
β
m
x
2
,
r
R
)
2
sinh
(
s
Γ
n
,
m
x
3
R
)
.
7. The method of claim 6 , wherein the determining a normal set of equations comprises applying the following equation:
∑
n
=
-
1
2
N
+
1
1
2
N
∑
m
=
-
1
2
M
+
1
1
2
M
A
n
′
,
m
′
;
n
,
m
P
n
,
m
dgh
=
B
n
′
,
m
′
,
n
′
=
-
1
2
N
+
1
,
…
,
1
2
N
,
m
′
=
-
1
2
M
+
1
,
…
,
1
2
M
,
where, the matrix A n′,m;n,m is given by
A
n
′
,
m
′
;
n
,
m
=
∑
q
=
-
1
2
N
+
1
1
2
N
∑
r
=
-
1
2
M
+
1
1
2
M
K
q
,
r
;
n
′
,
m
′
_
K
q
,
r
;
n
,
m
,
the known vector B n′,m′ is given by
B
n
′
,
m
′
=
∑
q
=
-
1
2
N
+
1
1
2
N
∑
r
=
-
1
2
M
+
1
1
2
M
K
q
,
r
;
n
′
,
m
′
_
p
q
,
r
sct
,
and n and n′ are in-line spectral numbers, m and m′ are cross-line spectral numbers, and the overline designates complex conjugation.
8. The method of claim 6 , wherein the preconditioner comprises an inverse of the system matrix for a horizontal streamer.
9. The method of claim 8 , wherein the applying a preconditioner comprises applying the following equation:
P
~
n
,
m
=
[
A
n
,
m
;
n
,
m
]
-
1
2
=
(
∑
q
=
-
1
2
N
+
1
1
2
N
∑
r
=
-
1
2
M
+
1
1
2
M
K
q
,
r
;
n
,
m
2
)
-
1
2
.
10. The method of claim 1 , wherein the applying an iterative conjugate gradient scheme comprises:
selecting an initial estimate for iteration counter k=0, as follows:
P n dgh(0) =0, r q (0) =p q sct ,
calculating estimates for iteration counter k=1, as follows:
g
n
(
1
)
=
P
~
n
∑
q
K
q
,
n
_
r
q
(
0
)
,
v
n
(
1
)
=
P
~
n
g
n
(
1
)
,
K
v
n
(
1
)
=
∑
n
K
q
,
n
v
n
(
1
)
,
α
(
1
)
=
∑
n
g
n
(
1
)
2
∑
q
Kv
n
(
1
)
2
,
P
n
dgh
(
1
)
=
P
n
dgh
(
0
)
+
α
(
1
)
v
n
(
1
)
,
r
q
(
1
)
=
r
q
(
0
)
-
α
(
1
)
Kv
n
(
1
)
,
performing the following until an error criterion is met:
increasing the iteration counter by k=k+1; and
calculating estimates for iteration counter k, using previous estimates calculated for k−1, as follows:
g
n
(
k
)
=
P
~
n
∑
q
K
q
,
n
_
r
q
(
k
-
1
)
,
v
n
(
k
)
=
P
~
n
g
n
(
k
)
+
∑
n
g
n
(
k
)
2
∑
n
g
n
(
k
-
1
)
2
v
n
(
k
)
,
K
v
n
(
k
)
=
∑
n
K
q
,
n
v
n
(
k
)
,
α
(
k
)
=
∑
n
g
n
(
k
)
2
∑
q
Kv
n
(
k
)
2
,
P
n
dgh
(
k
)
=
P
n
dgh
(
k
-
1
)
+
α
(
k
)
v
n
(
k
)
,
r
q
(
k
)
=
r
q
(
k
-
1
)
-
α
(
k
)
Kv
n
(
k
)
.
11. The method of claim 5 , wherein the error criterion is given by:
∑
q
r
q
(
k
)
2
∑
q
p
q
sct
2
<
10
-
3
.
12. The method of claim 4 , wherein the inverse-transforming the solution comprises applying the following equation:
p
dgh
=
Δ
(
s
α
)
Δ
(
s
β
)
4
π
2
∑
n
=
-
1
2
N
+
1
1
2
N
∑
m
=
-
1
2
M
+
1
1
2
M
exp
(
-
j
s
α
n
x
1
,
q
R
-
j
s
β
m
x
2
,
r
R
)
P
n
,
m
dgh
.