Method for deghosting seismic data acquired by a marine seismic source and receiver assembly
In a method for deghosting seismic data acquired by a marine seismic source and receiver assembly effects of seismic reflections by the water surface, known as ghost signals, are removed by a deghosting algorithm, which transforms input seismic data with the surface ghost reflections into source- and receiver-deghosted seismic data using a sparse-inversion technique both for hydrophone and/or geophone recordings, which technique includes equation (26), thereby considerably improving usuable bandwidth and giving rise to a significant imaging uplift.
1. A method comprising:
deghosting seismic data of a subsurface area acquired by a marine seismic source and receiver assembly, wherein effects of seismic reflections by the water surface, known as ghost signals, are removed by a deghosting algorithm which transforms input seismic data with the surface ghost reflections into source-deghosted and receiver-deghosted seismic data, thereby generating an output of deghosted primaries, using a sparse-inversion technique, wherein the deghosting algorithm comprises a minimization scheme based on the formula
V (a) (ω)= D (a) (ω)− G (r) (ω) X (ω) G (s) (ω) W (ω),∀ω
where the matrices V (a) (ω) denote residual terms such that V (a) rs (ω) is the residual at frequency ω of the signal from source s at receiver r, the matrices D (a) (ω) denote acquired data with or without multiples, the matrices X(ω) denote ghost-free data, the matrices W(ω) denote the wavelet information, and the matrices G (s,r) (ω) denote ghost functions, the latter being defined as
G (s) ( x s′ ,x s ,ω)=∫ dk ∥ e ik ∥ (x s −x s′ ) γ (s) ( k ⊥ ,z s ),
G (r) ( x r ,x r′ ,ω)=∫ dk ∥ e ik ∥ (x r′ −x r ) γ (r) ( k ⊥ ,z r ),
γ (r) ( k ⊥ ,z r )=2 i sin( k ⊥ z r ),
γ (s) ( k ⊥ ,z s )=2 i sin( k ⊥ z s )/(2 i k ⊥ z s ),
(ω/ c ) 2 =( k ∥ ) 2 +( k ⊥ ) 2 ,
with c the p-wave velocity of the subsurface top layer and z s,r the depth of source (s) and receiver (r); and
generating an image of the output after subsequent reverse-time migration.
2. The method of claim 1 , wherein the deghosting algorithm comprises a minimization scheme based on the formula
V (p) (ω)= D (p) (ω)− {tilde over (G)} (r) (ω) G (r) (ω) X (ω) G (s) (ω) {tilde over (G)} (s) (ω) W (ω),∀ω
where the matrices V (p) (ω) denote preconditioned residual terms defined as
V (p) (ω)= {tilde over (G)} (r) (ω) V (a) (ω) {tilde over (G)} (s) (ω),∀ω,
the matrices D (p) (ω) denote preconditioned data terms defined as
D (p) (ω)= {tilde over (G)} (r) (ω) D (a) (ω) {tilde over (G)} (s) (ω),∀ω,
and the matrices {tilde over (G)} (s,r) (ω) denote preconditioning functions, defined as
G
~
(
s
)
(
x
s
′
,
x
s
,
ω
)
=
∫
dk
∥
e
i
k
∥
(
x
s
-
x
s
′
)
γ
~
(
s
)
(
k
⊥
,
z
s
)
,
G
~
(
r
)
(
x
r
,
x
r
′
,
ω
)
=
∫
dk
∥
e
i
k
∥
(
x
r
′
-
x
r
)
γ
~
(
r
)
(
k
⊥
,
z
r
)
,
γ
~
(
r
)
=
γ
(
r
)
γ
(
r
)
2
+
ϵ
,
γ
~
(
s
)
=
γ
(
s
)
γ
(
s
)
2
+
ϵ
,
with ϵ<<1.
3. The method of claim 2 , wherein the deghosting algorithm comprises a minimization scheme based on the formula
V (a) (ω)= D (a) (ω)− G (r) (ω) X 0 (ω) G (s) (ω) W (ω)+ G (r) (ω) X 0 (ω)[ G (r) (ω)] −1 D (a) (ω),∀ω
where the matrices V (a) (ω) denote residual terms such that V (a) rs (ω) is the residual at frequency ω of the signal from source s at receiver r, the matrices D (a) (ω) denote acquired data with multiples, the matrices X 0 (ω) denote ghost-free and surface-multiple free data, the matrices W(ω) denote the wavelet information, and the matrices G (s,r) (ω) denote ghost functions, the latter being defined as
G (s) ( x s′ ,x s ,ω)=∫ dk ∥ e ik ∥ (x s −x s′ ) γ (s) ( k ⊥ ,z s ),
G (r) ( x r ,x r′ ,ω)=∫ dk ∥ e ik ∥ (x r′ −x r ) γ (r) ( k ⊥ ,z r ),
γ (r) ( k ⊥ ,z r )=2 i sin( k ⊥ z r ),
γ (s) ( k ⊥ ,z s )=2 i sin( k ⊥ z s )/(2 i k ⊥ z s ),
(ω/ c ) 2 =( k ∥ ) 2 +( k ⊥ ) 2 ,
with c the p-wave velocity of the subsurface top layer and z s,r the depth of source (s) and receiver (r).
4. The method of claim 3 , wherein the deghosting algorithm comprises a minimization scheme based on the formula
V (a) (ω)= D (a) (ω)− {tilde over (G)} (r) (ω) G (r) (ω) X 0 (ω) G (s) (ω) {tilde over (G)} (s) (ω) W (ω)+ {tilde over (G)} (r) (ω) G (r) (ω) X 0 (ω)[ {tilde over (G)} (r) (ω) G (r) (ω)] −1 D (p) (ω),∀ω
where the matrices V (p) (ω) denote preconditioned residual terms defined as
V (p) (ω)= {tilde over (G)} (r) (ω) V (a) (ω) {tilde over (G)} (s) (ω),∀ω,
the matrices D (p) (ω) denote preconditioned data terms defined as
D (p) (ω)= {tilde over (G)} (r) (ω) D (a) (ω) {tilde over (G)} (s) (ω),∀ω,
and the matrices {tilde over (G)} (s,r) (ω) denote preconditioning functions, defined as
G
~
(
s
)
(
x
s
′
,
x
s
,
ω
)
=
∫
dk
∥
e
i
k
∥
(
x
s
-
x
s
′
)
γ
~
(
s
)
(
k
⊥
,
z
s
)
,
G
~
(
r
)
(
x
r
,
x
r
′
,
ω
)
=
∫
dk
∥
e
i
k
∥
(
x
r
′
-
x
r
)
γ
~
(
r
)
(
k
⊥
,
z
r
)
,
γ
~
(
r
)
=
γ
(
r
)
γ
(
r
)
2
+
ϵ
,
γ
~
(
s
)
=
γ
(
s
)
γ
(
s
)
2
+
ϵ
,
with ϵ<<1.
5. The method of claim 4 , wherein the residual is approximated by
V (p) (ω)= D (p) (ω)− {tilde over (G)} (r) (ω) G (r) (ω) X 0 (ω) G (s) (ω) {tilde over (G)} (s) (ω) W (ω)+ X 0 (ω) D (p) (ω),∀ω
by making the assumption that
∫ dk x e ik x (x−x′) {tilde over (γ)} (i) ( k z ,z i )γ (i) ( k z ,z i )˜δ( x−x′ ) for i=s,r,
of which the validity stems from the fact that γ (i) {tilde over (γ)} (i) ≈1 away from the ghost notches.
6. The method of claim 1 , wherein the marine seismic source and a receiver assembly comprises fixed-depth hydrophones.