Dip-based corrections for data reconstruction in three-dimensional surface-related multiple prediction
A best fitting trace in seismic data is determined for a desired trace to be reconstructed. A dip-based correction is calculated per trace and per sample for differences in azimuth, common midpoint coordinates, and offset between the best fitting trace and the desired trace. The dip-based correction is applied to the best fitting trace to reconstruct the desired trace for 3D surface-related multiple prediction.
1. A method for marine seismic data reconstruction for 3D surface-related multiple prediction using a programmable computer that is programmed to perform the following:
determining a best fitting trace in the seismic data for a desired trace to be reconstructed;
calculating a dip-based correction per sample for differences in azimuth, common midpoint coordinates, and offset between the best fitting trace and the desired trace; and
applying the dip-based correction to the best fitting trace to reconstruct the desired trace for 3D surface-related multiple prediction.
2. The method of claim 1 , wherein the calculating a dip-based correction comprises:
calculating a dip-based correction for the best fitting trace and the corresponding desired trace, based upon half-offset, azimuth, arrival time, slowness, common midpoint coordinates, dip, and dip direction.
3. The method of claim 2 , wherein the calculating a dip-based correction for azimuth differences comprises applying the following equation:
Δ
t
ϕ
=
-
(
h
′
)
2
2
t
p
2
sin
(
ϕ
b
-
ϕ
d
)
sin
(
ϕ
b
+
ϕ
d
-
2
p
θ
)
,
,
with
p
=
(
ⅆ
t
ⅆ
x
)
2
+
(
ⅆ
t
ⅆ
y
)
2
and
p
θ
=
cos
-
1
(
ⅆ
t
ⅆ
x
p
)
,
where h′ is the half-offset, t is the arrival time, p is slowness, θ is dip direction, φ is source-receiver azimuth and subscripts b and d denote best fitting and desired traces, respectively.
4. The method of claim 2 , wherein the calculating a dip-based correction for offset differences comprises applying the following equation:
Δ
t
h
=
2
h
d
′
Δ
h
′
t
p
2
cos
2
(
ϕ
d
-
p
θ
)
,
where h′ is the half-offset, Δh′=h′ b −h′ d is the difference in half-offsets, t is the arrival time, p is slowness, θ is dip direction, φ is source-receiver azimuth and subscripts b and d denote best fitting and desired traces, respectively.
5. The method of claim 2 , wherein the calculating a dip-based correction for common midpoint location differences comprises applying the following equation:
Δ t cmp =|p |(Δ x cmp cos( p θ )+Δ y cmp sin( p θ )),
where p is slowness, θ is dip direction, Δx cmp =x b cmp −x d cmp is the difference in the x coordinate between common midpoint locations, Δy cmp =y b cmp −y d cmp is the difference in the y coordinate between common midpoint locations and subscripts b and d denote best fitting and desired traces, respectively.
6. The method of claim 1 , wherein the determining a best fitting trace further comprising:
obtaining seismic data in a (x CMP , y CMP , h, φ, t) format;
determining a predominant dip in x and y directions for common midpoint coordinates x CMP and y CMP and arrival time t in each trace in the seismic data;
applying a filter to the dips to smooth discontinuities;
selecting a set of desired traces to reconstruct; and
determining a best fitting trace from the seismic data for each desired trace in the selected set of traces.
7. The method of claim 6 , wherein the applied filter is a median filter.
8. The method of claim 6 , wherein the determining a best fitting trace comprises minimizing a difference Φ in a weighted sum of differences in common midpoint x and y positions x CMP and y CMP , offset h, and azimuth φ between the best fitting and desired traces.
9. The method of claim 8 , wherein the minimizing a difference Φ comprises applying the following equation:
Φ=α| x b CMP −x d CMP |+α|y b CMP −y d CMP |+β|h b −h d |+ε|φ b −φ d |,
where α, β and ε are user-defined weights, and subscripts b and d denote best fitting and desired traces, respectively.
10. The method of claim 8 , wherein the minimizing a difference Φ comprises applying the following equation:
Φ=α| x b CMP −x d CMP |+α|y b CMP −y d CMP |+β|h b −h d |
where α and β are user-defined weights, and subscripts b and d denote best fitting and desired traces, respectively.
11. The method of claim 1 , wherein the calculating a dip-based correction comprises: calculating a dip-based correction for the best fitting trace and the corresponding desired trace, based upon half-offset, azimuth, arrival time, velocity, common midpoint coordinates, dip, and dip direction.
12. The method of claim 11 , wherein the calculating a dip-based correction for azimuth differences comprises applying the following equation:
Δ
t
ϕ
=
-
2
h
′
2
v
2
t
d
sin
2
(
ψ
)
sin
(
ϕ
b
-
ϕ
d
)
sin
(
ϕ
b
+
ϕ
d
-
2
θ
)
,
where h′ is half-offset, v is velocity of the medium, t is arrival time, ψ is dip angle, φ is source-receiver azimuth, θ is dip direction and subscripts b and d denote best fitting and desired traces, respectively.
13. The method of claim 11 , wherein the calculating a dip-based correction for offset differences comprises applying the following equation:
Δ
t
h
=
2
v
2
t
d
2
h
d
′
Δ
h
′
sin
2
(
ψ
)
cos
2
(
ϕ
d
-
θ
)
,
where v is velocity of the medium, t is arrival time, h′ is half-offset, Δh′=h′ b −h′ d is the difference in half-offsets, ψ is dip angle, φ is source-receiver azimuth, and θ is dip direction subscripts b and d denote best fitting and desired traces, respectively.
14. The method of claim 11 , wherein the calculating a dip-based correction for midpoint differences comprises applying the following equation:
Δ
t
cmp
=
2
sin
(
ψ
)
v
(
Δ
x
cmp
cos
(
θ
)
+
Δ
y
cmp
sin
(
θ
)
)
,
where v is velocity, ψ is dip angle, Δx cmp =x b cmp −x d cmp is the difference in the x coordinate between common midpoint locations, Δy cmp =y b cmp −y d cmp is the difference in the y coordinate between common midpoint locations, θ is dip direction and subscripts b and d denote best fitting and desired traces, respectively.
15. The method of claim 11 , wherein the calculating a dip-based correction for midpoint differences comprises applying the following equation:
Δ
t
cmp
=
4
sin
(
ψ
)
v
2
t
d
v
2
t
d
2
4
+
h
′2
sin
2
(
ψ
)
cos
2
(
ϕ
d
-
θ
)
(
Δ
x
CMP
cos
(
θ
)
+
Δ
yCMP
sin
(
θ
)
)
,
where ψ is dip angle, v is velocity, t is arrival time, h′ is half-offset, φ is source-receiver azimuth, Δx cmp =x b cmp −x d cmp is the difference in the x coordinate between common midpoint locations, Δy cmp =y b cmp −y d cmp is the difference in the y coordinate between common midpoint locations, θ is dip direction and subscripts b and d denote best fitting and desired traces, respectively.