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Uncertainty Quantificationof Velocity Models and Seismic Imaging
Gregory Ely1, Alison Malcolm2 , Oleg Poliannikov1, David P. Nicholls3
Massachusetts Institute of Technology1Memorial University of Newfoundland2University of Illinois at Chicago3
MIT Earth Resources Laboratory2017 Annual Founding Members Meeting
Seismic Inversion: Challenges
• Single image• Sensitive to initial model (non-convex, non-linear problem)• Expensive forward solves
– (finite difference, finite element)• No uncertainty quantification
– How large is the trap?
Reconstruction
2Example From:http://pysit.readthedocs.org/en/release/0.5/quick_start/marmousi.html#details-‐of-‐parallelmarmousi-‐py
Time (s)
rece
iver
inde
x
200 400 600 800 1000 1200 1400 1600 1800 2000 2200
0
0.5
1
1.5
2
2.5
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Bayesian Seismic Inversion
• Goal: Estimate distribution of velocity models given data.
• Challenges– Large dimensionality (128 X 384 pixels)
• Displaying & Interpreting error• Calculating or storing covariance
– Expensive forward solvers– Non-linear forward model
• Multimodal & non-Gaussian model distribution
• Solution: Markov-Chain Monte Carlo & fast forward model
3
Distance (m)
Dept
h (m
)
0 1000 2000 3000 4000 5000 6000 7000 8000 9000
0
500
1000
1500
2000
2500
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Velo
city
(m/s
)
1500
2000
2500
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3500
4000
4500
5000
5500
: Observed dataforward model
: Velocity model
Outline
• Bayesian model Inversion– Bayesian inverse problem– MCMC Sampling
• Fast Forward Solver– Field Expansion Method– Gradient Adaptation
• Results– Simple synthetic– Marmousi based model
4
Bayesian Model Inversion
5
: Observed data: wave equation forward model
Likelihood function
Gaussian noise
Posterior Distribution
MCMC: Adaptive Metropolis-Hastings Summary
6
• Generates samples directly from posterior (m0,m2,….,mn)– Non parametric– ’Invaraint’ to starting model– Self tuning proposal distribution
• Requirements & Drawbacks– 100,000 evaluations of Likelihood– Requires 200< parameters
• Finite difference, finite element– Too slow– Large dimensionality
Haario, H., E. Saksman, and J. Tamminen, 2001, An Adaptive Metropolis Algorithm:Bernoulli, 7, 223.
Outline
• Bayesian model Inversion– Bayesian inverse problem– MCMC Sampling
• Fast Forward Solver– Field Expansion Method– Gradient Adaptation
• Results– Simple synthetic– Marmousi based model
7
Fast Forward Solve: Field Expansion
• Reduced model space & fast– Discrete layers with perturbations (no pixels)– O(LPN2Log(P))– L: number of layers P: spatial modes N: Taylor series terms
• Acoustic Helmholtz solver (Frequency domain)– Parallelizable over shot
• Limited velocity models– Piecewise constant– Continuous layers
v1(x,y)
y=a1
y=a1+g1(x)
y=a2+g2(x)
y=a2
v2(x,y)
v3(x,y)
Region of interest
Domain size
8
(A) (B)
A. Malcolm and D. P. Nicholls, "A Field Expansions Method for Scattering by Periodic Multilayered Media," Journal of the Acoustical Society of America, Volume 129, Number 4, 1783-1793 (2011)
Field Expansion: Gradient Adaptation
• Algorithm scales as O(LPN2Log(P))– Linear in # of layers L
• Parameterize model to master layers with gradient– Low dimensionality & complex models
• Datta and Sen 2016: global velocity model estimation• Fraser and Sen 1985: synthetic seismograms• Zelt and Smith 1992: Tomography
9
2000 4000 6000 8000Depth (m)
0
500
1000
1500
2000
2500
3000
Offs
et (m
)
↑V4U ↓V5
D
↑V3U
↓V4D
↑V2U
↓V3D
↑V1U
↓V2D
↓VD1
2000 4000 6000 8000Depth (m)
0
500
1000
1500
2000
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0 2000 4000 6000 8000Depth (m)
0
500
1000
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2500
30002000
2500
3000
3500
4000
Velo
city
(m/s
)
Parameterization Interpolated Layers Velocity ModelDe
pth (m
)
Offset (m)Offset (m)Offset (m)
Field Expansion: Comparison
• Top: Marmousi Bottom: Foothills
• Field expansion: 5 master layers, 7 nodes per interface, 10 layers per gradient• Pixel Model: 46,848 parameters , Field Expansion: 41 parameters
10
1 km
A B C
2000
3000
4000
5000
Velo
city
(m/s
)
1 km1500
2000
2500
3000
3500
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4500
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Vel
ocity
(m/s
)
1 km1500
2000
2500
3000
3500
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4500
5000
5500
Vel
ocity
(m/s
)
1 km1500
2000
2500
3000
3500
4000
4500
5000
5500
Vel
ocity
(m/s
)
Original Gaussian Smoothed Field Expansion
Foothills model: Gray, S. H., and K. J. Marfurt, 1995, Migration from topography:Improving the near-surface image: Canadian Journalof Exploration Geophysics, 31, 18–24.
Outline
• Bayesian model Inversion– Bayesian inverse problem– MCMC Sampling
• Fast Forward Solver– Field Expansion Method– Gradient Adaptation
• Results– Simple synthetic– Marmousi based model
11
Simple Model
• Velocity: (1500, 2500, 3000) m/s• 5Hz, Single source, 256 receivers, Gaussian noise: SNR 1.2• MCMC Run: 100,000 iterations
– Migrate reflector for each velocity model
12
(A) (B)
Simple Model: Realizations from MCMC
• MCMC Run: 100,000 iterations, discard first 50,000• Showing every 100th sample• Layers compensate thickness for velocity
13
Simple Model: Visualizing Error
• Uncertainty increases with depth• Error bars (standard deviation) misrepresent uncertainty
– Stable shape– Depth more uncertain
14
1 km
A B C
D E F
1400
1600
1800
2000
2200
2400
2600
2800
3000
3200
Velo
city
(m/s
)
-1000 -500 0 500 1000Offset (m)
0
500
1000
1500
2000
2500
3000
Dep
th (m
)
2644 +/- 191 m/s
2050 +/- 186 m/s
1489 +/- 45 m/s
Models from chainError Representation
Simple Model: Quantities of Interest
• Relative height more stable than depth ( 165 +/- 15m, 2200 +/- 135m)• Focusing on quantities of interests simplifies analysis
– Reduces dimensionality– Easy to interpret– Visualize non-Gaussian easily
15
0 200 400 600 800Relative Height (m)
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2C
ount
s×104 Anticline Height
1800 2000 2200 2400 2600Depth (m)
0
500
1000
1500
2000
2500
3000
3500
4000
Cou
nts
Reflector depth
Marmousi Example
• 3Hz single shot, 256 receivers, SNR 0.75, 41 velocity model parameters• MCMC Run: 100,000 iterations, discard first 50,000
• Focus on stability of images (reflectors of interest)
16
1 km
A B C
2000
3000
4000
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Velo
city
(m/s
)
1 km
A B C
2000
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Velo
city
(m/s
)
1 km
A B C
2000
3000
4000
5000
Velo
city
(m/s
)
Original
Smoothed
Field Expansion
Posterior Distribution: Velocity Models
• 50,000 models of 100,000 discarded, 6 models randomly selected• Shallow velocity structure better constrained than deeper• Too complex to show error bars
17
1 km
A B C
D E F
2000
2500
3000
3500
4000
4500
5000
Velo
city
(m/s
)
Posterior Distribution: Migrated Images
• Generated reflectivity model à travel times• Zero offset migrate travel times with each velocity model
• Upper structure more stable than lower• Red anticline shape poorly constrained• Salt flank/sill a trap? (D,F)
18
A B C
D E F
1 km
Posterior Distribution: Quantities of Interest
• Ran MCMC 50 times à 2.5 million posterior samples• Deeper red anticline area, poorly constrained
– Non-Gaussian distribution – Significant samples near zero area
19
2200 2300 2400 2500 2600 2700 2800 2900 3000Depth (m)
0
0.5
1
1.5
2
Cou
nts
×105 Red Depth: 2657 +/- 131 m
1700 1800 1900 2000 2100 2200 2300 2400 25000
0.5
1
1.5
2
2.5
3
Cou
nts
×105 Blue Depth: 2126 +/- 109 m
0 1 2 3 4 5 6 7 8×104
0
0.5
1
1.5
2
2.5
3 ×105 Blue Area: 2.192e+04 +/- 8225 m2
0 1 2 3 4 5 6 7 8Depth (m) ×104
0
2
4
6
8
10
12 ×104 Red Area: 3.001e+04 +/- 16718 m2
A B C
D E F
1 km
Area (m2)
Blue
Red
Summary
• Presented framework for uncertainty quantification– Reduced dimensionality of forward solver– Adapted field expansion to model gradients– Combined field expansion with adaptive metropolis hasting algorithm
• Demonstrated algorithm on synthetic models– Simple & Marmousi velocity models– Showed methods to visualize error– Focused on quantities of interest
• Robustness of algorithm– Estimates of uncertainty from single MCMC run– Handles incorrect parameterization– Not truly invariant to starting models, tolerates 20% gradient perturbations
20
Initial Model Choice: Linear Gradient
• Re-ran MCMC inversion with linear gradient initial model 50 times– Linear Gradient: 1500 m/s – 4300 m/s
• Distributions for quantities of interest within 1 STD – Cross sectional area matches well
21
2200 2300 2400 2500 2600 2700 2800 2900 3000Depth (m)
0
0.5
1
1.5
2
Cou
nts
×105 Red Depth: 2592 +/- 147 m
1700 1800 1900 2000 2100 2200 2300 2400 25000
0.5
1
1.5
2
2.5
3
Cou
nts
×105 Blue Depth: 2050 +/- 143 m
0 1 2 3 4 5 6 7 8×104
0
0.5
1
1.5
2
2.5
3 ×105 Blue Area: 2.438e+04 +/- 9102 m2
0 1 2 3 4 5 6 7 8Depth (m) ×104
0
2
4
6
8
10
12 ×104 Red Area: 2.579e+04 +/- 17041 m2
True initial
Linear Model
Blue
Red
Initial Model Choice: Incorrect Linear Gradient
• Ran 20 chains for 8 linear gradients– Slowest: 1500-2900 m/s Fastest: 1500-5400 m/s
• Poor convergence with 25% gradient perturbation– MCMC didn’t find global minima– Quantities of interest biased
• Below 20% perturbation consistent quantities of interest
22
3000 3500 4000 4500 5000 55000
0.5
1
1.5
2
2.5
Cro
ss S
ectio
nal A
rea
(m2 ) ×104 Deep Anticline
3000 3500 4000 4500 5000 55000
2000
4000
6000
8000
10000 Shallow Anticline
3000 3500 4000 4500 5000 5500Bottom Velocity (m/s)
-535
-530
-525
-520
-515
-510
-505
-500 L
og L
ikel
ihoo
dProposal Liklihood
3000 3500 4000 4500 5000 5500Bottom Velocity (m/s)
0
50
100
150
200
250
Dep
th (m
)
3000 3500 4000 4500 5000 5500Bottom Velocity (m/s)
0
50
100
150
200
250
Initial Model Choice: Incorrect Parameterization
• 3 – 5 layers, 3 – 11 nodes• 10 trials per layer node combination
• Number of nodes significant impact.• Number of layers lesser impact.
23
2 4 6 8 10 12# of nodes
-512
-511
-510
-509
-508
-507
-506
-505
-504
-503
Log
Like
lihoo
d
3 layers4 layers5 layers
2 4 6 8 10 12# of nodes
2
2.2
2.4
2.6
2.8
3
3.2
Area
m2
×104 Shallow Mean
2 4 6 8 10 12# of nodes
3000
4000
5000
6000
7000
8000
Area
m2
Shallow Standard Deviation
2 4 6 8 10 12# of nodes
1
2
3
4
5
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Area
m2
×104 Deep Mean
2 4 6 8 10 12# of nodes
0.6
0.8
1
1.2
1.4
1.6
Area
m2
×104 Deep Standard Deviation
proposal likelihood
Convergence Results
• Dashed lines standard deviation from all 2.5 million samples (50 runs)• Good estimate of area standard deviation achieved within 50,000 sample (1 run)
– Both true and initial starting models
24
0.5 1 1.5 2 2.5samples ×105
0
2000
4000
6000
8000
10000
area
m2
anticline 1 area standard deviation
True ModelGradient
0.5 1 1.5 2 2.5samples ×105
0
1
2
3
4
5
area
m2
×104 anticline 2 area standard deviation
Blue area standard deviation
Red area standard deviation
Adaptive Metropolis-Hastings: Outline
25
• Select initial model m0
• Evaluate L0 = L(m0)
• For i = 1:100,000–
– With probability L(m*)/L(mi-1)• mi =m*
– Else• mi =mi-1
Haario, H., E. Saksman, and J. Tamminen, 2001, An Adaptive Metropolis Algorithm:Bernoulli, 7, 223.
Outline
• Motivation
• Bayesian model Inversion– Bayesian inverse problem– MCMC Sampling
• Field expansion method
• Results– Simple synthetic– Marmousi based model
• Validation– Inaccurate starting models– Convergence
26
Distance (m)
Dep
th (m
)
0 1000 2000 3000 4000 5000 6000 7000 8000 9000
0
500
1000
1500
2000
2500
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Velo
city
(m/s
)
1500
2000
2500
3000
3500
4000
4500
5000
5500
Seismic Inversion: Overview
• Marmousi acoustic velocity model• 6 finite difference shots
27
ReceiverSource
Outline
• Bayesian model Inversion– Bayesian inverse problem– MCMC Sampling
• Fast Forward Solver– Field Expansion Method– Gradient Adaptation
• Results– Simple synthetic– Marmousi based model
28
Time (s)
rece
iver
inde
x
200 400 600 800 1000 1200 1400 1600 1800 2000 2200
0
0.5
1
1.5
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2.5
3
Seismic Inversion: Overview
• Record gathers• Estimate velocity model
29
Time (s)
Receiver Index
Distance (m)
Dep
th (m
)
0 1000 2000 3000 4000 5000 6000 7000 8000 9000
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500
1000
1500
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rece
iver
inde
x
200 400 600 800 1000 1200 1400 1600 1800 2000 2200
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Seismic Inversion: Overview
• Need accurate initial model • Local optimization
Measured
Distance (m)
Dep
th (m
)
0 1000 2000 3000 4000 5000 6000 7000 8000 9000
0
500
1000
1500
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Estimate
Time (s)
rece
iver
inde
x
200 400 600 800 1000 1200 1400 1600 1800 2000 2200
0
0.5
1
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3
Residual
Wave Equation:Finite Difference
30
Extra Slides
31
Field Expansion: Wood’s Anomalies
• Diffraction grating• Resonant mode• Energy trapped in top layer• Cannot be removed by large domain size
• Solution: Absorption
v1(x,y)
y=a1
y=a1+g1(x)
y=a2+g2(x)
y=a2
v2(x,y)
v3(x,y)
Region of interest
Domain size
32Malcolm, Alison, and David P. Nicholls. "A field expansions method for scattering by periodic multilayered media.”The Journal of the Acoustical Society of America 129.4 (2011): 1783-‐1793.
No Absorption
-2000 -1000 0 1000 20000
0.51
1.52
2.5
Freq
uenc
y (Hz
)
Absorption2468
101214
2468
101214
Tim
e (s)
Offset (m)-2000 -1000 0 1000 2000
Offset (m)-2000 -1000 0 1000 2000
-2000 -1000 0 1000 20000
0.51
1.52
2.5
Removing Wood’s Anomalies
• 3 layer velocity model (1500, 2500, 3500 m/s)• Solution: top layer absorptive
33
Depth (m)
No AbsorptionVe
locit
y (m
/s)
300 400 500 600 700
900
950
1000
1050
1100
1150
1200
1250
1300
Absorption
300 400 500 600 700
900
950
1000
1050
1100
1150
1200
1250
1300
Depth (m)
Wood’s Anomalies: Convexity & Cost Function
• 3 layer velocity model (Top Layer, depth: 500 m, Velocity 1100 m/s)• Anomalies create local minima
34
Top layer
Top layer
Finite Difference0
0.5
1
1.5
2
2.5
3
Field Expansion0
0.5
1
1.5
2
2.5
X Distance (m)
Dept
h (m
)
Velocity model
−2000 0 2000
0
100
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400
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800
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1000 1100
1200
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Tim
e (s)
Velo
city (
m/s)
15001500 0Offset (m)
Comparison to Existing Methods
• 3 layer model sinusoidal perturbation• Seismic Unix finite difference solver
35
Comparison to Existing Methods: Frequency domain
• 3 Flat layer model, 3Hz (-3000,3000 m)• Pysit 4th order finite difference solver
Finite Difference Field Expansion
Receiver Index Receiver Index
Amplitu
de
RealImaginary
36
Field Expansion: Finite Domain
• Finite domain size– Energy bleeds across domains
• Solution: Increase domain size
v1(x,y)
y=a1
y=a1+g1(x)
y=a2+g2(x)
y=a2
v2(x,y)
v3(x,y)
Region of interest
Domain size
37
Bayesian Model Inversion
38
: forward model
Likelihood function
Posterior Distribution
: Observed data : Gaussian Noise
Field Expansion: Finite Domain
v1(x,y)
y=a1
y=a1+g1(x)
y=a2+g2(x)
y=a2
v2(x,y)
v3(x,y)
Region of interest
Domain size
• Finite domain size– Energy bleeds across domains
• Solution: Increase domain size
39
Fast Forward Solve: Field Expansion
• Discrete layers with perturbations (no pixels)– Reduced Model space & Fast
• Acoustic Helmholtz solver (Frequency domain)– Parallelizable over shot
• Diffraction theory: Spatial frequency domain & repeating domain size
v1(x,y)
y=a1
y=a1+g1(x)
y=a2+g2(x)
y=a2
v2(x,y)
v3(x,y)
Region of interest
Domain size
40
Field Expansion: Algorithm
• O(LPN2Log(P))
v1(x,y)
y=a1
y=a1+g1(x)
y=a2+g2(x)
y=a2
v2(x,y)
v3(x,y)
Region of interest
Domain size
41
Field Expansion: Flat Layered
• f frequency• Vm displacement (scattered field)• Cm layer velocity
• Vm Unknown
v1(x,y)
y=a1
y=a2
v2(x,y)
v3(x,y)
Region of interest
Domain size, d
Helmholtz Equation
Expression for field
Boundary Conditions
System of equations
42
Field Expansion: Flat Layered
• Ap penta-diagonal, O(L)• Solve for each p, O(PL)• Plug back into field expression O(P L Log(P))
v1(x,y)
y=a1
y=a2
v2(x,y)
v3(x,y)
Region of interest
Domain size, d
System of equations
Apply Point Source
43
Field Expansion: Perturbed Layers
• Solve for N = 0 à flat layer case• Solve N = 1,2,3,4, etc.
Flat Field
v1(x,y)
y=a1
y=a1+g1(x)
y=a2+g2(x)
y=a2
v2(x,y)
v3(x,y)
Region of interest
Domain size
Perturbed Field
Recursion Relation
-1)
44
Uncertainty Quantification of Velocity Models and Seismic Imaging
Gregory Ely1, Alison Malcolm2 , Oleg Poliannikov1, David P. Nicholls3
Massachusetts Institute of Technology1
Memorial University of Newfoundland2
University of Illinois at Chicago3
45
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