Method of analyzing seismic data
Post-migration common image gathers (CIGs) are generated in a dip angle domain from measured seismic data. From a CIG, a hybrid Radon model is determined, including a reflection model related to concave features in the CIG and a diffraction model related to linear features in the CIG. The reflection model is transformed with a reflection Radon operator applied along inversion trajectories restricted around apices of the concave features to obtain reflection data. The diffraction model is transformed with a diffraction Radon operator to obtain diffraction data. The reflection and diffraction data at different horizontal positions can then be combined and summed to generate a migrated image of the subsurface.
1. A method for analyzing seismic data, comprising:
generating a post-migration common image gather in a dip angle domain from measured seismic data;
determining a hybrid Radon model associated with the common image gather, including a reflection model related to concave features in the common image gather and a diffraction model related to linear features in the common image gather;
transforming the reflection model with a reflection Radon operator applied along inversion trajectories restricted around apices of the concave features to obtain reflection data; and
transforming the diffraction model with a diffraction Radon operator to obtain diffraction data, so as to eliminate a major part of the noise from the data while keeping the information contained in the diffraction components of a dip angle common image gather.
2. The method as claimed in claim 1 , further comprising adding the reflection data and diffraction data obtained by transforming the reflection and diffraction models with the Radon operators, to obtain a processed common image gather.
3. The method as claimed in claim 1 , wherein determining the hybrid Radon model associated with the common image gather comprises minimizing an objective function:
F ( m d ,m r )=∥ L d ·m d +L r ·m r −d∥ 2 +ε d ∥W d ·m d ∥ 2 +ε r ∥W r ·m r ∥ 2
where m d and m r are data vectors of the diffraction model and of the reflection, model, respectively, L d and L r are the diffraction and reflection Radon operators, respectively, d represents a data vector of the common image gather, W d and W r are model space weights, ε d and ε r are measures of sparseness for diffraction and reflection, respectively, and ∥·∥ 2 is an L 2 measure.
4. The method as claimed in claim 1 , wherein transforming the reflection model m r (ζ, c, z′) with the reflection Radon operator comprises computing the reflection data D r (α, z) as:
D
r
(
α
,
z
)
=
∑
c
∑
ζ
m
r
(
ζ
,
c
,
z
′
=
z
-
c
·
(
α
-
ζ
)
2
)
where α and z are respectively dip angle and depth coordinates in the common image gather, c, ζ and z′ are respectively parabola curvature, parabola apex shift and depth coordinates in the reflection model, the sum on the parabola apex shift coordinate ζ being restricted to values of ζ around the dip angle coordinate α.
5. The method as claimed in claim 2 , further comprising obtaining processed common image gathers for different horizontal positions and summing, the data in the processed common image gathers to generate a migrated image of the subsurface.
6. The method as claimed in claim 4 , wherein, when computing the reflection data D r (α, z) for a dip angle coordinate α and a depth coordinate z, the sum on the parabola apex shift coordinate ζ is restricted, for a given, value of the parabola curvature coordinate c, to values of ζ different from α by less than about
±
λ
4
c
,
where λ is an average wavelength of seismic waves used to measure the seismic data.
7. A system for analyzing seismic data, comprising a computer resource configured to perform steps of:
generating a post-migration common image gather in a dip angle domain from measured seismic data;
determining a hybrid Radon model associated with the common image gather, including a reflection model related to concave features in the common image gather and a diffraction model related to linear features in the common image gather;
transforming the reflection model with a reflection Radon operator applied along inversion trajectories restricted around apices of the concave features to obtain reflection data; and
transforming the diffraction model with a diffraction Radon operator to obtain diffraction data; said system eliminates a major part of the noise from the data while keeping the information contained in a diffraction component of the dip angle common image gather.
8. The system as claimed in claim 7 , wherein the computer resource is further configured to add the reflection data and diffraction data obtained by transforming the reflection and diffraction models with the Radon operators, to obtain a processed common image gather.
9. The system as claimed in claim 7 , wherein determining the hybrid Radon model associated with the common image gather comprises minimizing an objective function:
F ( m d ,m r )=∥ L d ·m d +L r ·m r −d∥ 2 +ε d ∥W d ·m d ∥ 2 +ε r ∥W r ·m r ∥ 2
where m d and m r are data vectors of the diffraction model and of the reflection model, respectively, L d and L r are the diffraction and reflection Radon operators, respectively, d represents a data vector of the common image gather, W d and W r are model space weights, ε d and ε r are measures of sparseness for diffraction and reflection, respectively, and ∥·∥ 2 is an L 2 measure.
10. The system as claimed in claim 7 , wherein transforming the reflection model m r (ζ, c, z′) with the reflection Radon operator comprises computing the reflection data D r (α, z) as:
D
r
(
α
,
z
)
=
∑
c
∑
ζ
m
r
(
ζ
,
c
,
z
′
=
z
-
c
·
(
α
-
ζ
)
2
)
where α and z are respectively dip angle and depth coordinates in the common image gather, c, ζ and z′ are respectively parabola curvature, parabola apex shift and depth coordinates in the reflection model, the sum on the parabola apex shift coordinate ζ being restricted to values of ζ around the dip angle coordinate α.
11. The system as claimed in claim 8 , wherein the computer resource is further configured to obtain processed common image gathers for different horizontal positions and to sum the data in the processed common image gathers to generate a migrated image of the subsurface.
12. The system as claimed in claim 10 , wherein, when computing the reflection data D r (α, z) for a dip angle coordinate α and a depth coordinate z, the sum on the parabola apex shift coordinate ζ is restricted, for a given value of the parabola curvature coordinate c, to values of ζ different from α by less than about
±
λ
4
c
,
where λ is an average wavelength of seismic waves used to measure the seismic data.
13. A non-volatile computer readable medium having stored thereon a computer program product for a system for analyzing seismic data, wherein the program product comprises instructions to have said system performing steps of:
generating a post-migration common image gather in a dip angle domain from measured seismic data;
determining a hybrid Radon model associated with the common image gather, including a reflection model related to concave features in the common image gather and a diffraction model related to linear features in the common image gather;
transforming the reflection model with a reflection Radon operator applied along inversion trajectories restricted around apices of the concave features to obtain reflection data; and
transforming the diffraction model with a diffraction Radon operator to obtain diffraction data, wherein instructions of the program product eliminates a major part of the noise from the data while keeping the information contained in the diffraction components of a dip angle common image gather.
14. The non-volatile computer readable medium as claimed in claim 13 , wherein the program product further comprises instructions to have said system adding the reflection data and diffraction data obtained by transforming the reflection and diffraction models with the Radon operators, to obtain a processed common image gather.
15. The non-volatile computer readable medium as claimed in claim 13 , wherein determining the hybrid Radon model associated with the common image gather comprises minimizing an objective function:
F ( m d ,m r )=∥ L d ·m d +L r ·m r −d∥ 2 +ε d ∥W d ·m d ∥ 2 +ε r ∥W r ·m r ∥ 2
where m d and m r are data vectors of the diffraction model and of the reflection model, respectively, L d and L r are the diffraction and reflection Radon operators, respectively, d represents a data vector of the common image gather, W d and W r are model space weights, ε d and ε r are measures of sparseness for diffraction and reflection, respectively, and ∥·∥ 2 is an L 2 measure.
16. The non-volatile computer readable medium as claimed in claim 13 , wherein transforming the reflection model m r (ζ, c, z′) with the reflection Radon operator comprises computing the reflection data D r (α, z) as:
D
r
(
α
,
z
)
=
∑
c
∑
ζ
m
r
(
ζ
,
c
,
z
′
=
z
-
c
·
(
α
-
ζ
)
2
)
where α and z are respectively dip angle and depth coordinates in the common image gather, c, ζ and z′ are respectively parabola curvature, parabola apex shift and depth coordinates in the reflection model, the sum on the parabola apex shift coordinate ζ being restricted to values of ζ around the dip angle coordinate α.
17. The non-volatile computer readable medium as claimed in claim 14 , wherein the program product further comprises instructions to have said system obtaining processed common image gathers for different horizontal positions and summing the data in the processed common image gathers to generate a migrated image of the subsurface.
18. The non-volatile computer readable medium as claimed in claim 16 , wherein, when computing the reflection data D r (α, z) for a dip angle coordinate α and a depth coordinate z, the sum on the parabola apex shift coordinate ζ is restricted, for a given value of the parabola curvature coordinate c, to values of ζ different from α by less than about
±
λ
4
c
,
where λ is an average wavelength of seismic waves used to measure the seismic data.