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University of British ColumbiaSLIM
Ning Tu, Xiang Li and Felix J. Herrmann
Controlling linearization errors in sparse seismic inversion with rerandomization
Migration:linearized modelling
Real data:approximately
linear
Motivation
LS : minimize�m
krF[m0,q]�m� dk2,
rF
m0
q
d
�m : model perturbation: linearized modelling operator: background model: vectorized source wavefields: vectorized residual wavefields
Linearized inversion
: Curvelet transform: tolerance for noise/modelling error�
C
BPDN : minimize kxk1subject to krF[m0,q]C
⇤x� dk2 �
Herrmann and Li, 2012
Fast inversion with sparsity promotion
: sparsity level
LASSO : minimize krF[m0,q]C⇤x� dk2
subject to kxk1 ⌧
⌧
van den Berg and Friedlander, 2008
Alternative formulation
•For each LASSO subproblem, we draw:‣new randomized source aggregates‣new frequency subsets
•Leads to faster convergence in terms of model error.•Also leads to high robustness to linearization errors.
Herrmann and Li, 2012Tu and Herrmann, 2012
Rerandomization
Example: faster convergence
•model grid spacing: 5 meters•150 sources/receivers, 15m spacing•122 frequencies in 0-‐60Hz range• linearized modelling data•using 2 simultaneous sources, 15 frequencies for fast inversion
•running for 305 iterations, simulation cost ~ 1 RTM with all sources and frequencies
Tu and Herrmann, 2012
True perturbation
Baseline image: Inversion with all sources and frequencies
Fast inversion with subsampling, no randomization[120X speed-up]
Fast inversion with subsampling, with randomization[120X speed-up]
0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6x 104
0.4
0.5
0.6
0.7
0.8
0.9
Number of PDE solves
Rel
ativ
e m
odel
erro
r
no redrawingwith redrawing
Model error decrease
d = rF[m0,q]�m
d = F[m,q]� F[m0,q]
Example: robustness to linearization errors
Using a simple two-‐layer model, we compare two datasets and the inversion results:• linearized data:
• forward modelling data:
Lateral distance (m)
Dep
th (m
)
0 200 400
0
100
200
300
400Lateral distance (m)
Dep
th (m
)
0 200 400
0
100
200
300
400
True model, background model and model perturbation
Lateral distance (m)
Dep
th (m
)
0 200 400
0
100
200
300
400
0.15 0.2 0.25!15
!10
!5
0
5
10
15
Time (s)
Am
plit
ude blue: linearized data
green: forward modelling datared: difference
Comparison of traces from the two datasets
Inversion with all the data
•21 sequential sources•61 frequencies •100 maximal # iterations
Lateral distance (m)
De
pth
(m
)
0 200 400
0
100
200
300
400
Inversion of linearized data
Lateral distance (m)
De
pth
(m
)
0 200 400
0
100
200
300
400
Inversion of forward modelling data
Lateral distance (m)
De
pth
(m
)
0 200 400
0
100
200
300
400
with small run maximal #iterations
with “true”optimization stops prematurely
� �
Fast inversion with subsampling
•5 simultaneous sources•15 random frequencies •~ 16X subsampling•100 maximal # iterations
Lateral distance (m)
De
pth
(m
)
0 200 400
0
100
200
300
400
Inversion of linearized data
no rerandomization run maximal # iterations
Lateral distance (m)
De
pth
(m
)
0 200 400
0
100
200
300
400
Inversion of forward modelling data
Lateral distance (m)
De
pth
(m
)
0 200 400
0
100
200
300
400
with rerandomization run maximal # iterations
no rerandomization run maximal # iterations
•a 2D slice of the SEG/EAGE salt model•smooth background model, including smooth salt boundaries•3.9km deep, 15.7km wide•80 ft grid spacing•5Hz Ricker wavelet, 8s recording•323 sources with 160 ft spacing at 80ft depth•using linearized data and forward-‐modelling data•using 15 frequencies, 15 simultaneous sources for fast inversion•running for 100 iterations, simulation cost ~ 1.45X RTM with all sources and frequencies
Case study
True model
Lateral distance (m)
Dep
th (m
)
0 5000 10000 15000
0100020003000
Smooth background model
Lateral distance (m)
Dep
th (m
)
0 5000 10000 15000
0100020003000
0 0.5 1 1.5 2 2.5!2
!1
0
1
Time (s)
Am
plit
ud
e
blue: linearized datagreen: forward modelling datared: difference; top salt event at ~2s
Comparison of traces from the two datasets
Lateral distance (m)
De
pth
(m
)
0 5000 10000 15000
0
1000
2000
3000
True model perturbation
Lateral distance (m)
De
pth
(m
)
0 5000 10000 15000
0
1000
2000
3000
Fast inversion of linearized data, no rerandomization
~1.45X the simulation cost of a single RTM with all data
Fast inversion of linearized data, with rerandomization
Lateral distance (m)
Dep
th (m
)
0 5000 10000 15000
0100020003000
~1.45X the simulation cost of a single RTM with all data
Lateral distance (m)
De
pth
(m
)
0 5000 10000 15000
0
1000
2000
3000
Fast inversion of linearized data, no rerandomization
~1.45X the simulation cost of a single RTM with all data
Lateral distance (m)
De
pth
(m
)
0 5000 10000 15000
0
1000
2000
3000
~1.45X the simulation cost of a single RTM with all data
Fast inversion of forward modelling data, no rerandomization
Lateral distance (m)
De
pth
(m
)
0 5000 10000 15000
0
1000
2000
3000
~1.45X the simulation cost of a single RTM with all data
Fast inversion of forward modelling data, with rerandomization
Lateral distance (m)
De
pth
(m
)
0 5000 10000 15000
0
1000
2000
3000
RTM of forward modelling data, with all sources and frequencies
Lateral distance (m)
De
pth
(m
)
0 5000 10000 15000
0
1000
2000
3000
Lateral distance (m)
Depth
(m
)
0 5000 10000 15000
0
1000
2000
3000
RTMLateral distance (m)
De
pth
(m
)
0 5000 10000 15000
0
1000
2000
3000
True perturbationFast inversion
Details[red arrow: true reflector; yellow arrow: artifacts]
•The linearization error is more an issue for dimensionality reduced system than the full system.
•Simply allowing a tolerance in the inversion does not address the issue.
•Rerandomization can: i) lead to faster convergence; ii) increase robustness to linearization errors.
•We can potentially better resolve fine sub-‐salt features with sparse inversion in a computationally efficient way.
Conclusions
Acknowledgements
This work was in part financially supported by the Natural Sciences and Engineering Research Council of Canada Discovery Grant (22R81254) and the Collaborative Research and Development Grant DNOISE II (375142-‐08). This research was carried out as part of the SINBAD II project with support from the following organizations: BG Group, BGP, BP, CGG, Chevron, ConocoPhillips, ION, Petrobras, PGS, Total SA, WesternGeco, and Woodside.
Thank Eric Verschuur for sharing the salt dome model, and the authors of the SEG/EAGE salt model.Thank all SLIM group members for beneficial discussions.Thank you for your attention!
