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

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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

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:  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

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Bayesian Model Inversion

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:  Observed  data:  wave  equation  forward  model

Likelihood  function

Gaussian  noise

Posterior  Distribution

MCMC: Adaptive Metropolis-Hastings Summary

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• 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

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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

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(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

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↑V4U ↓V5

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↓VD1

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Parameterization Interpolated  Layers Velocity  ModelDe

pth  (m

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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

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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

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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

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(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

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Simple Model: Visualizing Error

• Uncertainty increases with depth• Error bars (standard deviation) misrepresent uncertainty

– Stable shape– Depth more uncertain

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2644 +/- 191 m/s

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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

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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)

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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

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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)

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D E F

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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

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×105 Red Depth: 2657 +/- 131 m

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3 ×105 Blue Area: 2.192e+04 +/- 8225 m2

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12 ×104 Red Area: 3.001e+04 +/- 16718 m2

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Area  (m2)

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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

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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

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×105 Red Depth: 2592 +/- 147 m

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3 ×105 Blue Area: 2.438e+04 +/- 9102 m2

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12 ×104 Red Area: 2.579e+04 +/- 17041 m2

True  initial

Linear  Model

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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

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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.

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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

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anticline 1 area standard deviation

True ModelGradient

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Blue  area  standard  deviation

Red  area  standard  deviation

Adaptive Metropolis-Hastings: Outline

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• 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

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Seismic Inversion: Overview

• Marmousi acoustic velocity model• 6 finite difference shots

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ReceiverSource

Outline

• Bayesian model Inversion– Bayesian inverse problem– MCMC Sampling

• Fast Forward Solver– Field Expansion Method– Gradient Adaptation

• Results– Simple synthetic– Marmousi based model

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Seismic Inversion: Overview

• Record gathers• Estimate velocity model

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Time  (s)

Receiver  Index

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Seismic Inversion: Overview

• Need accurate initial model • Local optimization

Measured

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Residual

Wave  Equation:Finite  Difference

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Extra Slides

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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

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Absorption2468

101214

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e (s)

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Offset (m)-2000 -1000 0 1000 2000

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Removing Wood’s Anomalies

• 3 layer velocity model (1500, 2500, 3500 m/s)• Solution: top layer absorptive

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Depth (m)

No AbsorptionVe

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Wood’s Anomalies: Convexity & Cost Function

• 3 layer velocity model (Top Layer, depth: 500 m, Velocity 1100 m/s)• Anomalies create local minima

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Top  layer

Top  layer

Finite Difference0

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Comparison to Existing Methods

• 3 layer model sinusoidal perturbation• Seismic Unix finite difference solver

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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

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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

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Bayesian Model Inversion

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:  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

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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

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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

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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

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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

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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)

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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