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Regularized Migration/Inversion
Henning Kuehl (Shell Canada)Mauricio Sacchi (Physics, UofA)
Juefu Wang (Physics, UofA)Carrie Youzwishen (Exxon/Mobil)
This doc -> http://www.phys.ualberta.ca/~sacchi/pims03.ppt
Email: [email protected]
Signal Analysis and Imaging Group Department of PhysicsUniversity of Alberta
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Outline
• Motivation and Goals• Migration/inversion - Evolution of ideas and
concepts• Quadratic versus non-quadratic regularization
two examples• The migration problem
RLS migration with quadratic regularzationExamplesRLS migration with non-quadratic
regularizationExamples
• Summary
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Motivation
To go beyond the resolution provided by the data (aperture and band-width) by incorporating quadratic and non-quadratic regularization terms into migration/inversion algorithms
This is not a new idea…
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Migration with Adjoint Operators [Current technology]
RLS Migration (Quadratic Regularization) [Not in production yet]
RLS Migration (Non-Quadratic Regularization) [??]
Resolution
Evolution of ideas and concepts
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Migration with Adjoint Operators
1 RLS Migration (Quadratic Regularization)
2 RLS Migration (Non-Quadratic Regularization)
Resolution
Evolution of ideas and concepts
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LS Deconvolution Sparse Spike Deconvolution
LS Radon Transforms HR Radon Transforms
Quadratic Regularization
Stable and Fast Algorithms
Low Resolution
Non-quadratic Regularization
Requires Sophisticated Optimization
Enhanced Resolution
Evolution of ideas and conceptsTwo examples
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Deconvolution
Quadratic versus non-quadratic regularization
€
W r = s
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Convolution / Cross-correlation / Deconv
sWr
srW
'~ ==Convolution
Cross-correlation
€
˜ r = W 'W r =R r
r= R−1 ˜ r
= R−1W ' s
Deconvolution
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Quadratic and Non-quadratic Regularization
€
J =||W r − s ||2 +μR(r)
R =|| r ||2, r = (W 'W + μI)−1W 's
R = Φ(r), r = (W 'W + μQ(r))−1W 's
Φ(r) = ln(1+ri
2
α 2i
∑ )
Cost
Quadratic
Non-quadratic
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€
r
W
s = W r
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€
˜ r = W 's
r = (W 'W + αI)−1W 's
r = (W 'W + μQ(r))−1W 's
Adj
LS
HR
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Radon Transform
Quadratic versus non-quadratic regularization
€
m(t − φ(x, p), p)d∫ p = d(t, x)
or
Lm = d
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LS Radon
h q
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HR Radon
qh
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Modeling / Migration / Inversion
€
Lm = d
˜ m = L'd
< m,L'd >=< d,L m >
Modeling
Migration
De-blurring problem
€
˜ m = L'Lm =K m
m= K−1 ˜ m
= K−1L'd
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Inducing a sparse solution via non-quadratic regularization appears to be a good idea (at least for the two previous examples)
Q. Is the same valid for Migration/Inversion?
Back to migration
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Inducing a sparse solution via non-quadratic regularization appears to be a good idea (at least for the two previous examples)
Q. In the same valid for Migration/Inversion?
A. Not so fast… First, we need to define what we are inverting for…
Back to migration
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Migration as an inverse problem
€
W L m = d
˜ m = L'W 'd
d = d(s,g, t)
m = m(x,z)
m = m(x,z, p)
Forward
Adjoint
Data
“Image” or
Angle Dependent Reflectivity
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• There is No general agreement about what type of regularization should be used when inverting for m=m(x,z)… Geology is too complicated…
• For angle dependent images, m(x,z,p), we attempt to impose horizontal smoothness along the “redundant” variable in the CIGs:
€
J =||W (L m − d) ||2 +μ || Dpm(x,z, p) ||2
Migration as an inverse problem
Kuehl, 2002, PhD Thesis UofA
url: cm-gw.phys.ualberta.ca~/sacchi/saig/index.html
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Quadratic Regularization Migration Algorithm
Features:DSR/AVP Forward/Adjoint Operator (Prucha et. al, 1999)
PSPI/Split Step
Optimization via PCG
2D/3D (Common Azimuth)
MPI/Open MP
€
J =||W ( L m − d) ||2 +μ || Dpm(x,z, p) ||2
Kuehl and Sacchi, Geophysics, January 03 [Amplitudes in 2D]
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2D - Synthetic data example
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Marmousi model (From H. Kuehl Thesis, UofA, 02)
0
1
2
3
Depth (km)
3 4 5 6 7 8 Horizontal distance (km)
130018002300280033003800430048005300
m/s
3
0
1
2
Depth (km)
3 4 5 6 7 8 Horizontal distance (km)
0.4
0.6
0.8
1.0
x10 4
kg/m^3
AVA target
Velocity
Density
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De
pth
[ km
]
0
0.5
1.0
1.5
2.0
2.5
7000 7500 8000CMP position [m]
0
0.5
1.0
1.5
2.0
2.5
Dep
th [
km]
0 200 400 600 800Ray parameter [mu s/m]
Migrated AVA
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De
pth
[ km
]
0
0.5
1.0
1.5
2.0
2.5
7000 7500 8000CMP position [m]
0
0.5
1.0
1.5
2.0
2.5
De
pth
[ km
]
0 200 400 600 800Ray parameter [mu s/m]
Regularized Migrated AVA
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Migration RLSM
0 10 20 30 40 5000.020.040.060.08
0.10.120.140.160.18
0.2
Angle of incidence (deg)
Ref
lect
ion
coef
ficie
nt
0 10 20 30 40 5000.020.040.060.08
0.10.120.140.160.18
0.2
Angle of incidence (deg)
Ref
lect
ion
coef
ficie
nt
0.5
0.7
0.9
1.1
Dep
t h [ k
m]
0 200 400 600 800Ray parameter [mu s/m]
0.5
0.7
0.9
1.1
Dep
th [
k m]
0 200 400 600 800Ray parameter [mu s/m]
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Incomplete CMP (30 %) Complete CMP
0
0.5
1.0
1.5
2.0
2.5
Tim
e [ s
ec]
200 400 600 800 1000 1200Offset [m]
0
0.5
1.0
1.5
2.0
2.5
Tim
e [s
ec]
200 400 600 800 1000 1200Offset [m]
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Migration RLSM
0.5
0.7
0.9
1.1
Dep
th [ k
m]
0 200 400 600 800Ray parameter [mu s/m]
0.5
0.7
0.9
1.1
Dep
th [
k m]
0 200 400 600 800Ray parameter [mu s/m]
0 10 20 30 40 5000.020.040.060.08
0.10.120.140.160.18
0.2
Angle of incidence (deg)
Ref
lect
ion
coef
ficie
nt
0 10 20 30 40 5000.020.040.060.08
0.10.120.140.160.18
0.2
Angle of incidence (deg)
Ref
lect
ion
coef
ficie
nt
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Constant ray parameter image (incomplete)
0
0.5
1.0
1.5
2.0
2.5
De
pth
[k
m]
3000 4000 5000 6000 7000 8000 9000
CMP position [m]
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Constant ray parameter image (RLSM)
0
0.5
1.0
1.5
2.0
2.5
De
pth
[k
m]
3000 4000 5000 6000 7000 8000 9000
CMP position [m]
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3D - Synthetic data example(J Wang)
Parameters for the 3D synthetic data
x-CDPs : 40Y-CDPs : 301Nominal Offset: 50dx=5 mdy=10 mdh=10 m
90% traces are randomly removed to simulate a sparse3D data.
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IncompleteData 4 adjacent CDPs
Reconstructeddata (after 12Iterations)
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A B
C
A: Iteration 1 B: Iteration 3 C: Iteration 12
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At y=950 m, x=130 m
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Misfit Curve
Convergence – Misfit vs. CG iterations
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A complete data B. Remove 90% traces C. least-squares
Stacked images comparison (x-line 30)
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Field data example
ERSKINE (WBC) orthogonal 3-D sparse land data set
dx=50.29m
dy=33.5m
€
J =||W ( L m − d) ||2 +μ || Dpm(x,z, p) ||2
Comment on Importance of W
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Iteration 1 Iteration 3 Iteration 7
CIG at x-line #10, in-line #71
€
J =||W ( L m − d) ||2 +μ || Dpm(x,z, p) ||2
Cost Optimization
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Stacked image, in-line #71
Migration RLS Migration
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Stacked image, in-line #71
Migration RLS Migration
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Image and Common Image Gather (detail)
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Image and Common Image Gather (detail)
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Stacked Image (x-line #10) comparison
Migration LS Migration
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Non-quadratic regularization applied to imaging
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€
d ≈ s + n
Rs ≈ 0
J =|| d − s ||2 +α || Ds ||2, Ds = ∂xs
J =|| d − s ||2 +α Φ(Ds)
Quadratic Smoothing
Non-Quadratic Smoothing
Non-quadratic regularization
But first, a little about smoothing:
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Quadratic regularization ->Linear filters
Non-quadratic -> Non-linear filters
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Segmentation/Non-linear Smoothing
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Segmentation/Non-linear Smoothing
1/Variance at edge position
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2D Segmentation/Non-linear Smoothing
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GRT Inversion - VSP(Carrie Youzwishen, MSc 2001)
* * ** **
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Dx Dz m
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• Quadratic constraint (Dp) to smooth along p (or h)
• Non-quadratic constraint to force vertical sparseness
€
J =||W L m − d ||2 +μ || Dh m(x,z,h) ||2 +η Φ(m)
HR Migration (Current Direction)
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Example: we compare migrated images m(x,z,h) for the following 3 imaging methods:
€
˜ m = L'W 'd
min{ J =||W L m − d ||2 +μ || m(x,z,h) ||2 }
min{J =||W L m − d ||2 +μ || Dh m(x,z,h) ||2 +η Φ(m) }
Adj
LS
HR
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True
Z(m)
X(m)
Synthetic Model
Acoustic, Linearized, Constant V, Variable Density
m(x,z)
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Data
Shots
Pre-stack data
d(s,g,t)
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Adj
Common offset images
m(x,z,h)
Z(m)
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LS
Common offset images
m(x,z,h)
Z(m)
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HR
Common offset images
m(x,z,h)
Z(m)
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Adj
Z(m)
X(m)
Stacked CIGs
m(x,z)
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LS
Z(m)
X(m)
Stacked CIGs
m(x,z)
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HR
Z(m)
X(m)
Stacked CIGs
m(x,z)
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CIG # 10 CIG # 11 CIG # 12
Z(m)
0m Offset 380m
Adj
LS
HR
CIGs
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Data
HR Prediction
LS PredictionT(s)
Shots
d(s,g,t)
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Conclusions• Imaging/Inversion with the addition of quadratic and non-
quadratic constraints could lead to a new class of imaging algorithms where the resolution of the inverted image can be enhanced beyond the limits imposed by the data (aperture and band-width).
• This is not a completely new idea. Exploration geophysicists have been using similar concepts to invert post-stack data (sparse spike inversion) and to design Radon operators.
• Finally, it is important to stress that any regularization strategy capable of enhancing the resolution of seismic images must be applied in the CIG domain. Continuity along the CIG horizontal variable (offset, angle, ray parameter) in conjunction with sparseness in depth, appears to be reasonable choice.
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Acknowledgments
• EnCana• Geo-X• Veritas • Schlumberger foundation• NSERC• AERI
• 3D Real data set was provided by Dr Cheadle