Efficient internal multiple prediction methods
Methods for processing seismic data are described. The method includes: obtaining seismic data; solving a series of partial differential wave equations, wherein a first partial differential wave equation describes propagation of a seismic wave going from a first reflector to a second reflector, wherein a second partial differential wave equation describes propagation of a seismic wave going from a second reflector to a third reflector, and wherein a third partial differential wave equation describes propagation of a seismic wave going from a third reflector to a seismic receiver, wherein outputting predicted internal multiples for further imaging or attenuation.
1. A method for seismic data processing comprising:
obtaining seismic data for a subterranean structure, the seismic data including contribution from internal multiples;
solving a series of partial differential wave equations each representing a reflected portion of predicted internal multiples of the seismic data, wherein a first partial differential wave equation describes propagation of a seismic wave going from a first reflector to a second reflector, wherein a second partial differential wave equation describes propagation of the seismic wave going from the second reflector to a third reflector, and wherein a third partial differential wave equation describes propagation of the seismic wave going from the third reflector to a seismic receiver, the third partial differential wave equation providing the predicted internal multiples;
outputting the predicted internal multiples for attenuation; and
removing the predicted internal multiples from an image of the subterranean structure.
2. The method of claim 1 , wherein the series of partial differential wave equations are solved sequentially.
3. The method of claim 1 , wherein at least one of the series of partial differential wave equations describes 2-D or 3-D propagation of the seismic wave.
4. The method of claim 1 , wherein the first partial differential equation is
(
∂
∂
z
+
iq
)
p
^
1
(
k
;
ω
;
z
;
k
s
)
=
∫
b
^
1
(
k
;
-
k
′
→
;
z
)
p
^
0
(
k
′
→
;
ω
;
z
;
k
s
)
d
k
′
→
,
wherein z is depth between z 1 and z 2 , {circumflex over (p)} 0 is Fourier transform of p 0 which is Fourier conjugate to field point going from source to the reflections above depth z 1 , {circumflex over (p)} 1 is Fourier transform of p 1 which is Fourier conjugate to field point going from z 1 to z 2 , {circumflex over (b)} 1 is Fourier transform of b 1 (b 1 is first term of inverse scattering series), {right arrow over (k)}′ is wave vector directed towards observation point, and k s is Fourier conjugate of x s , ω is radial frequency, k s is Fourier conjugate of x s which is source coordinates, and
q
=
ω
c
1
-
c
2
k
x
2
ω
2
wherein c is velocity and k x is wavenumber.
5. The method of claim 1 , wherein the second partial differential equation is
(
∂
∂
z
-
iq
)
p
^
2
(
k
;
ω
;
z
;
k
s
)
=
-
∫
b
^
1
(
k
;
-
k
′
→
;
z
)
p
^
1
(
k
′
→
;
ω
;
z
;
k
s
)
d
k
′
→
,
wherein {circumflex over (p)} 2 is Fourier transform of p 2 which is Fourier conjugate to field point going from z 2 to z 3 .
6. The method of claim 1 , wherein the third partial differential equation is
(
∂
∂
z
+
iq
g
)
p
^
3
(
k
g
;
ω
;
z
;
k
s
)
=
∫
b
^
1
(
k
g
;
-
k
′
→
;
z
)
p
^
2
(
k
′
→
;
ω
;
z
;
k
s
)
d
k
′
→
,
wherein {circumflex over (p)} 2 is Fourier transform of p 3 which is Fourier conjugate to field point going from z 3 to the seismic receiver, q g is depth wavenumbers, and k g is Fourier conjugate of x g which is receiver coordinates.
7. The method of claim 1 , wherein the seismic data is 2-D or 3-D.
8. A method for seismic data processing comprising:
obtaining seismic data for a subterranean structure, the seismic data including contribution from internal multiples;
solving a series of partial differential wave equations each representing a reflected portion of predicted internal multiples of the seismic data, wherein a first partial differential wave equation describes propagation of a seismic wave going from a first reflector to a second reflector, wherein a second partial differential wave equation describes propagation of the seismic wave going from the second reflector to a third reflector, and wherein a third partial differential wave equation describes propagation of the seismic wave going from the third reflector to a seismic receiver, the third partial differential wave equation providing the predicted internal multiples; and
outputting the predicted internal multiples.
9. The method of claim 8 , wherein the series of partial differential wave equations are solved sequentially.
10. The method of claim 8 , wherein at least one of the series of partial differential wave equations describes 2-D or 3-D propagation of the seismic wave.
11. The method of claim 8 , wherein the first partial differential equation is
(
∂
∂
z
+
iq
)
p
^
1
(
k
;
ω
;
z
;
k
s
)
=
∫
b
^
1
(
k
;
-
k
′
→
;
z
)
p
^
0
(
k
′
→
;
ω
;
z
;
k
s
)
d
k
′
→
,
wherein z is depth between z 1 and z 2 , {circumflex over (p)} 0 is Fourier transform of p 0 which is Fourier conjugate to field point going from source to reflection at depth z 1 , {circumflex over (p)} 1 is Fourier transform of p 1 which is Fourier conjugate to field point going from z 1 to z 2 , {circumflex over (b)} 1 is Fourier transform of b 1 (b 1 is first term of inverse scattering series), {right arrow over (k)}′ is wave vector directed towards observation point, and k s is Fourier conjugate of x s , ω is radial frequency, k s is Fourier conjugate of x s which is source coordinates, and
q
=
ω
c
1
-
c
2
k
x
2
ω
2
wherein c is velocity and k x is wavenumber.
12. The method of claim 8 , wherein the second partial differential equation is
(
∂
∂
z
-
iq
)
p
^
2
(
k
;
ω
;
z
;
k
s
)
=
-
∫
b
^
1
(
k
;
-
k
′
→
;
z
)
p
^
1
(
k
′
→
;
ω
;
z
;
k
s
)
d
k
′
→
,
wherein {circumflex over (p)} 2 is Fourier transform of p 2 which is Fourier conjugate to field point going from z 2 to z 3 .
13. The method of claim 8 , wherein the third partial differential equation is
(
∂
∂
z
+
iq
g
)
p
^
3
(
k
g
;
ω
;
z
;
k
s
)
=
∫
b
^
1
(
k
g
;
-
k
′
→
;
z
)
p
^
2
(
k
′
→
;
ω
;
z
;
k
s
)
d
k
′
→
,
wherein {circumflex over (p)} 3 is Fourier transform of p 3 which is Fourier conjugate to field point going from z 3 to the seismic receiver, q g is depth wavenumbers, and k g is Fourier conjugate of x g which is receiver coordinates.
14. The method of claim 8 , wherein the seismic data is 2-D or 3-D.
15. A system for seismic data processing comprising:
one or more seismic sources generating at least one seismic wave propagating into a subterranean structure;
a plurality of reflectors reflecting the at least one seismic wave, the plurality of reflectors including a first reflector, a second reflector, and a third reflector;
one or more seismic receivers capturing seismic data for the subterranean structure following reflection of the at least one seismic wave from the plurality of reflectors, the seismic data including contribution from internal multiples; and
a computer system generating predicted internal multiples by solving a series of partial differential wave equations each representing a reflected portion of the predicted internal multiples of the seismic data, the series of partial differential wave equations including a first partial differential wave equation, a second partial differential wave equation, and a third partial differential wave equation, the first partial differential wave equation describing propagation of the at least one seismic wave from the first reflector to the second reflector, the second partial differential wave equation describing propagation of the at least one seismic wave from the second reflector to the third reflector, the third partial differential wave equation describing propagation of the at least one seismic wave from the third reflector to the one or more seismic receivers.
16. The system of claim 15 , wherein the series of partial differential wave equations are solved sequentially.
17. The system of claim 15 , wherein at least one of the series of partial differential wave equations describes 2-D or 3-D propagation of the at least one seismic wave.
18. The system of claim 15 , wherein the seismic data is 2-D or 3-D.
19. The system of claim 15 , wherein the computer system removes the predicted internal multiples from a seismic image of the subterranean structure.
20. The system of claim 15 , wherein the computing system outputs the predicted internal multiples.