Methods of enhancing separation of primary reflection signals and noise in seismic data using radon transformations
View Patent ↗Improved methods of processing seismic data which comprise amplitude data assembled in the offset-time domain in which primary reflection signals and noise overlap are provided for. The methods include the step of enhancing the separation between primary reflection signals and coherent noise by transforming the assembled data from the offset-time domain to the time-slowness domain. More specifically, the assembled amplitude data are transformed from the offset-time domain to the time-slowness domain using a Radon transformation according to an index j of the slowness set and a sampling variable Δp; wherein j = p max - p min + 1 μ sec / m Δ p , Δp is from about 0.5 to about 4.0 μsec/m, p max is a predetermined maximum slowness, and p min is a predetermined minimum slowness. Alternately, an offset weighting factor x n is applied to the assembled amplitude data, wherein 0<n<1, and the amplitude data are transformed with a Radon transformation. The assembled amplitude data also may be transformed with a Radon transformation applied within defined slowness limits p min and p max , where p min is a predetermined minimum slowness and p max is a predetermined maximum slowness, with a hyperbolic Radon transformation, or, as applied to data which have been uncorrected for normal moveout, with a hyperbolic or parabolic Radon transformation. Thus, the primary reflection signals and coherent noise transform into different regions of the time-slowness domain to enhance the separation therebetween.
1. A method of processing seismic data to remove unwanted noise from meaningful reflection signals indicative of subsurface formations, wherein said seismic data comprise amplitude data recorded over time at a number of seismic receivers in an area of interest, said amplitude data are assembled into common geometry gathers in an offset-time domain, and said amplitude data comprise primary reflection signals and coherent noise that overlap in an offset-time domain; said method comprising the step of enhancing the separation between primary reflection signals and coherent noise by transforming said data from the offset-time domain to the time-slowness domain, wherein:
(a) an offset weighting factor x n is applied to said amplitude data, wherein x is offset and 0<n<1; and
(b) said offset weighted amplitude data are transformed from the offset-time domain to the time-slowness domain using a Radon transformation;
(c) whereby said primary reflection signals and said coherent noise transform into different regions of the time-slowness domain to enhance the separation therebetween.
2. The method of claim 1 , wherein said assembled data are uncorrected for normal moveout.
3. The method of claim 1 , wherein n in said offset weighting factor x n is approximately 0.5.
4. The method of claim 3 , wherein said assembled data are uncorrected for normal moveout.
5. The method of claim 1 , wherein said assembled amplitude data are transformed using a hyperbolic Radon transformations.
6. The method of claim 5 , wherein said assembled data are uncorrected for normal moveout.
7. The method of claim 1 , wherein said offset weighting factor x n is applied and said amplitude data are transformed with a continuous Radon transformation defined as follows:
u
(
p
,
τ
)
=
∫
-
∞
∞
ⅆ
x
∫
-
∞
∞
ⅆ
t
d
(
x
,
t
)
x
n
δ
[
f
(
t
,
x
,
τ
,
p
)
]
or a discrete Radon transformation approximating said continuous Radon transformation,
where
u(p,τ)=transform coefficient at slowness p and zero-offset time τ
d(x,t)=measured seismogram at offset x and two-way time t
x n =offset weighting factor (0<n<1)
δ=Dirac delta function
ƒ(t,x,τ,p)=forward transform function.
8. The method of claim 7 , wherein said assembled data are uncorrected for normal moveout.
9. The method of claim 7 , wherein said forward transform function, ƒ(t,x,τ,p), is selected from the transform functions for linear slant stack, parabolic, and hyperbolic kinematic travel time trajectories, which functions are defined as follows:
(a) transform function for linear slant stack:
ƒ( t,x,τ,p )= t−τ−px
(b) transform function for parabolic:
ƒ( t,x,τ,p )= t−τ−px 2
(c) transform function for hyperbolic:
ƒ( t,x,τ,p )= t −√{square root over (τ 2 +p 2 x 2 )}.
10. The method of claim 9 , wherein said assembled data are uncorrected for normal moveout.
11. The method of claim 1 , wherein said offset weighting factor x n is applied and said amplitude data are transformed with a continuous hyperbolic Radon transformation defined as follows:
u
(
p
,
τ
)
=
∫
-
∞
∞
ⅆ
x
x
n
d
(
x
,
τ
2
+
p
2
x
2
)
or a discrete hyperbolic Radon transformation approximating said continuous hyperbolic Radon transformation, where
u(p,τ)=transform coefficient at slowness p and zero-offset time τ
x n =offset weighting factor (0<n<1).
12. The method of claim 11 , wherein said assembled data are uncorrected for normal moveout.
13. The method of claim 1 , wherein said offset weighting factor x n is applied and said amplitude data are transformed with a discrete Radon transformation defined as follows:
u
(
p
,
τ
)
=
∑
l
=
-
L
L
∑
k
=
1
N
d
(
x
l
,
t
k
)
x
l
n
δ
[
f
(
t
k
,
x
l
,
τ
,
p
)
]
Δ
x
l
Δ
t
k
where
u(p,τ)=transform coefficient at slowness p and zero-offset time τ
d(x l ,t k )=measured seismogram at offset x l and two-way time t k
x l n =offset weighting factor at offset x l (0<n<1)
δ=Dirac delta function
ƒ(t k ,x l ,τ,p)=forward transform function at t k and x l
Δ
x
l
=
x
l
+
1
-
x
l
-
1
2
for
l
=
0
,
±
1
,
…
,
±
(
L
-
1
)
Δx L =x L −x L−1
Δx −L =x −L+1 −x −L
Δ
t
k
=
t
k
+
1
-
t
k
-
1
2
for
k
=
2
,
…
,
N
-
1
Δt 1 =t 2 −t 1
Δt N =t N −t N−1 .
14. The method of claim 13 , wherein said assembled data are uncorrected for normal moveout.
15. The method of claim 13 , wherein said forward transform function, ƒ( t k ,x l ,τ,p), is selected from the transform functions for linear slant stack, parabolic, and hyperbolic kinematic travel time trajectories, which functions are defined as follows:
(a) transform function for linear slant stack:
ƒ( t k ,x l ,τ,p )= t k −τ−px l
(b) transform function for parabolic:
ƒ( t k ,x l ,τ,p )= t k −τ−px l 2
(c) transform function for hyperbolic:
ƒ( t k ,x l ,τ,p )= t k −√{square root over (τ 2 +p 2 x l 2 )}.
16. The method of claim 15 , wherein said assembled data are uncorrected for normal moveout.
17. The method of claim 1 , wherein said offset weighting factor x n is applied and said amplitude data are transformed with a discrete hyperbolic Radon transformation defined as follows:
u
(
p
,
τ
)
=
∑
l
=
-
L
L
x
l
n
d
(
x
l
,
τ
2
+
p
2
x
l
2
)
Δ
x
l
where
u(p,τ)=transform coefficient at slowness p and zero-offset time τ
x l n =offset weighting factor at offset x l (0<n<1)
Δ
x
l
=
x
l
+
1
-
x
l
-
1
2
for
l
=
0
,
±
1
,
…
,
±
(
L
-
1
)
Δx L =x L −x L−1
Δx −L =x −L+1 −x −L .
18. The method of claim 17 , wherein said assembled data are uncorrected for normal moveout.