Surface-consistent Gabor deconvolution

Surface-consistent
Surface consistent
Gabor deconvolution
Carlos A. Montaña, Monta , Gary F. Margrave,
and Dave Henley

Outline
Why could we need SCGABOR?
Overview of Gabor deconvolution
A surface consistent Gabor
algorithm
Example
Conclusions

DIVETSCO TESTS
(From Perz et al., 2005, CSEG meeting)

Outline
Why could we need SCGABOR?
Overview of Gabor deconvolution
A surface consistent Gabor
algorithm
Example
Conclusions

t
From Wiener to Gabor
1. Constant Q theory 3. Gabor transform
W matrix Q matrix r s
τ τ
t
G
τ t
[] s ( τ , f ) ∫
2. Nonstationary convolutional model
s(
f )
∞
− 2π
ift
= s ( t ) g ( t − τ ) e dt
− ∞
∞
−2πifτ
= w(
f ) ∫α
Q(
f , τ ) r(
τ ) e dτ
−∞

Fourier of
the
wavelet
Gabor of
the trace
Nonstationary conv. conv.
Model
in the Gabor domain
Factorization of the nonstationary
convolutional model
s(
f
)
∞
−2πifτ
= w(
f ) ∫α
Q ( f , τ ) r(
τ ) e dτ
−∞
Approximated
factorization
G Q τ
[] s ( τ , f ) ≈ w(
f ) α ( τ , f ) G[]
r ( , f )
≈
Attenuation
function
Gabor of the
reflectivity

Gabor deconvolution
Time
Gabor transform of the trace
Frequency
Deconvolutional operator:
- smoothing
-phase:
using Hilbert
transform
Estimate of the
Gabor reflectivity
Wavelet removal and compensation for attenuation are simultaneous

Φ(f)
Minimum phase, linearity,
causality and Hilbert
transform.
Minimum phase
– Explosive sources
are also minimum
phase.
f
A(f)
=H(log( )
f
Futterman (1962) showed that
wave attenuation in a causal,
linear theory is always
minimum phase.

Outline
Why could we need SCGABOR?
Overview of Gabor deconvolution
A surface consistent Gabor
algorithm
Example
Conclusions

=
)
( t
σ
σ(t)
Surface Consistency
Surface Consistency
)
,
(
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( t
s
a
t
s =
σ
)
,
(
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,
(
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,
(
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,
(
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,
,
,
,
(
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,
(
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(
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,
,
,
(
ω
ω
ω
ω
ω
σ
σ
h
D
x
C
r
B
s
A
h
x
r
s
t
h
d
t
x
c
t
r
b
t
s
a
t
h
x
r
s
=
⊗
⊗
⊗
=
s
)
,
(
)
,
(
)
,
,
( t
r
b
t
s
a
t
r
s ⊗
=
σ
r
)
,
(
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,
(
)
,
(
)
,
,
,
( t
x
c
t
r
b
t
s
a
t
x
r
s ⊗
⊗
=
σ
x
)
,
(
)
,
(
)
,
(
)
,
(
)
,
,
,
,
( t
h
d
t
x
c
t
r
b
t
s
a
t
h
x
r
s ⊗
⊗
⊗
=
σ
h

Surface-consistent Surface consistent Gabor
deconvolution
G[] ( τ , f ) ≈ w(
f ) α ( τ , f ) G[]
Q ρ ( τ , f )
σ
≈
[ w ( f , s)
][ α ( f , τ , h)
][ Gρ(
f , τ , h)
][ w ( f , ) ]
G s
Q
r
σ ( f , τ , h,
r,
s)
=
r
h, r, s: midpoint, receiver and source coordinates respectively

Surface-consistent Surface consistent Gabor algorithm
For the i, j, k trace: ith midpoint, jth receiver and kth source indexes
≈
s 1 s 2 s 3 … s k .. s Ns w
Sources array
r 1 r 2 r 3 … r i .. r Nr
Receivers array
h 1 h 2 h 3 … h i .. h Nm
Midpoints array

Surface-consistent Surface consistent Gabor algorithm
For the i, j, k trace: ith midpoint, jth receiver and kth source indexes
≈
s 1 s 2 s 3 … s k .. s Ns
( w )
s
r 1 r 2 r 3 … r i .. r Nr
j
=
M j
∑
m=
1
w ( f , s )
s
M
j
m
j
( w )
r
k
N k
∑
n=
wr
f rn
=
N
1
( , )
k
ijk
k
m 1 m 2 m 3 … m i .. m Nm
Li
∑
l=
i
Ai
=
f x
L
1
α(
, τ ,
i
[ ( w ) ] . ( w )
θ
( f , τ ) = A * * *
l i
s
j
[ ]
r
k

Outline
Why could we need SCGABOR?
Overview of Gabor deconvolution
A surface consistent Gabor
algorithm
Example
Conclusions

Synthetic raw data
(Courtesy DIVESTCO)
The dataset is made up of 78 shots,
96 channels per shot
Q=40, sample rate=2ms, length=2 sec.
Station interval=34 m.
Brute stack

Synthetic raw data
(Courtesy DIVESTCO)
Q=40
⊗
s
⊗
r
V=3500
Strong attenuation Surf. Consist. wavelets Strong random noise

After single channel Gabor

After Surf. Cons. Gabor

Conclusions
A poor S/N could harm the estimation of the
minimum phase Gabor deconvolution operator,
introducing undesirables artefacts
The Surface-Consistent Surface Consistent implementation of
Gabor deconvolution allows a robust estimation
of the minimum phase deconvolution operator
in the presence of
– Strong random noise
– Strong variations of the near-surface near surface features

Acknowledgements
CREWES sponsors
POTSI sponsors
MITACS
NSERC
Mike Perz