Lecture 2: Seismic Inversion and Imagingmueller/graduate_workshop/pdfs_of... · Outline 1 Subsurface Applications 2 Schematic Description of Seismic imaging 3 Travel-Time Tomography

Lecture 2: Seismic Inversion and Imaging

Youzuo Lin1

Joint work with: Lianjie Huang1

Monica Maceira2

Ellen M. Syracuse1

Carene Larmat1

1: Los Alamos National Laboratory2: Oak Ridge National Laboratory

Graduate Student Workshop on Inverse Problem, 2016

Outline

1 Subsurface Applications

2 Schematic Description of Seismic imaging

3 Travel-Time Tomography and Full-Waveform InversionTravel-Time TomographyFull-Waveform InversionRegularization TechniquesA Modified Total-Variation Regularization Scheme

4 Computation Methods

5 Application to Global Seismology

6 Application to Enhanced Geothermal SystemsReservoir Monitoring with Time-Lapse DataBrady’s EGS SiteRobustness Tests

7 Conclusions

Youzuo Lin (LANL) Seismic Inversion Inverse Problems Workshop 1 / 77

Outline







7 Conclusions


Enhanced Geothermal System

• Geothermal is a sustainable energy because it is clean andreliable, however the exploration and drilling remain expensiveand risky.

• Quantitative monitoring for enhanced geothermal systems canhelp optimize the geothermal production and the placement ofnew wells.Youzuo Lin (LANL) Seismic Inversion Inverse Problems Workshop 2 / 77


• According to US Energy Information Administration (EIA), 9western states together have the geothermal potential to provideover 20% of electricity needs

• DOE aims a ten-fold increase of US electricity production fromgeothermal reservoirs within the next 10 years



However, the reality is that• High Risk: Two to five out of every 10 geothermal wells

prospected end up dry

• Expensive to Drill: Wells cost between $2 million and $5 million,meaning geothermal investors risk losing millions on poor odds

An accurate characterization of the subsurface structure is thekey to a successful drilling schemes


Quantitative Monitoring for CO2 Storage

• Quantifying changes in CO2 reservoirs is a key element forassessing the performance of geologic carbon sequestration.

• Reliably monitoring of potential CO2 leakage through fault zonesis crucial for ensuring safe CO2 storage.


Global Seismology

• Seismic wave is currently the only effective tool that can penetratethe entire Earth

• Seismic inversion (tomography) is used to obtain the structuralinformation of the Earth


Outline







7 Conclusions


Problem Description - Data Measurement

• Data Measurement

Source

Receiver

Media & Velocity 2Media & Velocity 1

��

��

��

Reflection Data

Source

Media & Velocity 1

Receiver

Media & Velocity 2

��

��

��

Transmission Data


Problem Description - Data Usage

0 100 200 300 400 500 600 700 800−1

−0.5

0

0.5

1

1.5

2x 10

6

Time

Dis

pla

ce

me

nt

First Arrival Time

• First Arrival Time• Pros: Simplify the nonlinear problem to be linear; Efficient to solve.• Cons: Low-resolution imaging.


Problem Description - Data Usage

0 100 200 300 400 500 600 700 800−1

−0.5

0

0.5

1

1.5

2x 10

6

Time

Dis

pla

ce

me

nt

Whole Waveform

• The Whole Waveform• Pros: High-resolution imaging.• Cons: Problem stays nonlinear; Computation load is expensive.


Problem Description - Inversion

• Data Inversion• A Model to Get Started (Initial Model)

• Generate The Simulated Data (Forward Modeling)• Match The Simulated Data to Measurement (Data Matching)• Use The Difference to Update The Initial Model (Model Update)

Initial ModelYouzuo Lin (LANL) Seismic Inversion Inverse Problems Workshop 9 / 77


• Data Inversion• A Model to Get Started (Initial Model)• Generate The Simulated Data (Forward Modeling)

• Match The Simulated Data to Measurement (Data Matching)• Use The Difference to Update The Initial Model (Model Update)

Initial Model0 100 200 300 400 500 600 700 800

−1

−0.5

0

0.5

1

1.5

2x 10

6

Time

Dis

pla

ce

me

nt

Forward ModelingYouzuo Lin (LANL) Seismic Inversion Inverse Problems Workshop 9 / 77


• Data Inversion• A Model to Get Started (Initial Model)• Generate The Simulated Data (Forward Modeling)• Match The Simulated Data to Measurement (Data Matching)

• Use The Difference to Update The Initial Model (Model Update)

Initial Model0 100 200 300 400 500 600 700 800

−1

−0.5

0

0.5

1

1.5

2x 10

6

Time

Dis

pla

ce

me

nt

Measurement

Simulating Data

Data MatchingYouzuo Lin (LANL) Seismic Inversion Inverse Problems Workshop 9 / 77


• Data Inversion• A Model to Get Started (Initial Model)• Generate The Simulated Data (Forward Modeling)• Match The Simulated Data to Measurement (Data Matching)• Use The Difference to Update The Initial Model (Model Update)

Initial Model

?+

Model UpdateYouzuo Lin (LANL) Seismic Inversion Inverse Problems Workshop 9 / 77

Problem Description - Regularization & PriorInformation

• Inversion for ill-posed nonlinear problems can be much morechallenging to solve• Ill-posedness (cause of limited data coverage)• Local minima (cause of nonlinearity and non-convex optimization)

• Include prior information to constrain the inversion• To avoid the instability during the inversion of data.• To obtain more accurate reconstructions and a faster convergent

rate.• What can be prior information?

• Good initial guess (starting models)• Smoothness of the models• Locations of the regions of interests• Shapes of the reconstructions• etc.

• Regularization technique is a method to introduce priorinformation to the inversion.Youzuo Lin (LANL) Seismic Inversion Inverse Problems Workshop 10 / 77

Outline







7 Conclusions


Travel-Time Tomography

• The tomography is given as an over-determined linear systems:

w1GTph w1GTp

vp 0w2GTs

h 0 w2GTsvs

whLh 0 00 wpLvp 00 0 wsLvs

λhI 0 00 λpI 00 0 λsI

δhδmpδms

=

w1dTp

w2dTs

00000

,

where GTph , GTs

h , GTpvp , GTs

vs are the sensitivity matrices of the P- andS-arrivals, Lh, Lvp , and Lvs are the first-order smoothing matricesfor h, mp, and ms with weights of wh, wp, and ws; I is the identitymatrix weighted by λh, λp, and λs.


Forward Waveform Modeling

• Forward Modeling

Acoustic-wave equation in the time-domain[1

K (r)∂2

∂t2 −∇ ·(

1ρ(r) ∇

)]p(r, t) = s(t),

where ρ(r) is density, K (r) is bulk modulus, s(t) is source, andp(r, t) is pressure field.


Seismic Inversion

• Inverse Problem

Waveform Inversion

minm

{‖d− f (m)‖22

},

where ‖d− f (m)‖22 is the misfit function, d is recorded waveform data, and || · ||2 stands for the `2 norm.

• Difficulties and Solution• Ill-posedness, multiple minima, computational costly, slow

convergence• Regularization Techniques


Inverse Problem & Regularization Techniques

• General Regularization Methodology

Full-waveform inversion (FWI) with regularization

minm

{‖d− f (m)‖22 + λR(m)

},

where ‖d− f (m)‖22 is data fidelity term, R(m) is the regularization term and λ is the regularization parameter.

• Specific Regularization and Its Characteristics• Total-Variation (TV): R(m) = ‖∇m‖1 =

∑i |(δm)i |, (1-D)

Best suited for reconstructing piecewise-constant functions, computationally expensive

• Tikhonov (TK): R(m) = ‖L m‖2 =∑

i(δm)2i , (1-D)

Best suited for reconstructing smooth functions, computationally cheap

5

2

1

2

• TVstep = 5;TVsmooth = 2 + 2 + 1 = 5.

• TKstep = 52 = 25;TKsmooth = 22+22+1 = 9←.


A New Travel-Time Tomography

What does this application tell us?

• Sharp velocity contrast (total-variation might be a better choice)• Geologic data are available in the shallow layers (some kind of

a priori information)

Travel-Time Tomography with a Modified Total-Variation (MTV)Regularization (TomoMTV, Lin et. al., GJI (201) 2015):

E(m, u) = minm,u

{∥∥∥Gm− d∥∥∥2

2+ λ1 ‖m− u‖22 + λ2 ‖w ∇u‖1

}

•∥∥∥Gm− d

∥∥∥2

2is the data misfit term;

• ‖m− u‖22 and ‖w ∇u‖1 are the regularization terms;• λ1 and λ2 are the regularization parameters;• w incorporates the a priori information from geologic data.


A Closer Look at the TomoMTV

• What does ‖w ∇u‖1 do to the inversion?• Avoid the smoothing of the inversion due to the TV term

• Further encourage the inversion at the sharp velocity contrastinferred from the geologic data

wi,j =

{0 if point (i, j) is on the interface1 if point (i, j) is off the interface .

• How to pick the two regularization parameters, λ1 and λ2?• We use L-curve method to pick λ1 due to its simplicity.

• We use UPRE method to pick the λ2.


A Modified Total-Variation Regularization Scheme

FWI with Modified Total-Variation (MTV) Regularization (Lin &Huang, GJI (200) 2015):

E(m,u) = minm,u

{‖d − f (m)‖22 + λ1 ‖m− u‖22 + λ2 ‖u‖TV

},

where λ1 and λ2 are both positive regularization parameters.

• The MTV regularization term contains a new variable u and anadditional term ‖m− u‖22 compared to the conventional TVregularization term.


FWI with Modified Total-Variation Regularization


E(m,u) = minu

{min

m

{‖d − f (m)‖22 + λ1 ‖m− u‖22

}+ λ2 ‖u‖TV

},


• The regularization parameter λ1 controls the trade-off between thedata misfit term and the Tikhonov regularization term, and λ2balances the amount of interface-preservation in FWI.


FWI with MTV Regularization (A Closer Look)


E(m,u) = minu

{min

m

{‖d − f (m)‖22 + λ1 ‖m− u‖22

}+ λ2 ‖u‖TV

},


• The inner problem is to solve for m using a conventional FWI withthe Tikhonov regularization and prior model u.

• The outer subproblem is to solve for u using a standard L2-TVminimization method to preserve the sharpness of interfaces ininversion result m.

• The interleaving of solving these two subproblems leads to aninversion that not only improves the minimization of the data misfit,but also enhances the sharpness of interfaces.


Outline







7 Conclusions


Computation Methods

We employ the Alternating Direction Method of Multipliers (ADMM) tosolve our new FWI with the modified total-variation regularization term

Alternating Direction Method of Multipliers (ADMM)

m(k) = argminm

{E1(m)}

= argminm

{‖d − f (m)‖22 + λ1

∥∥∥m− u(k−1)∥∥∥2

2

}u(k) = argmin

u{E2(u)}

= argminu

{∥∥∥m(k) − u∥∥∥2

2+ λ2 ‖u‖TV

}


Selection of the Regularization Parameter: λ1

• The subproblem of m(k) is a classical FWI with Tikhonovregularization.

• Various parameter estimation method has been developed:L-Curve, GCV, etc.

• We employ the following formula:

Selection of λ1:

λ1 =‖d − f (m)‖22

k∥∥m− u(k−1)

∥∥22

,

where k is a dimensionless number, which is approximately 10.


Selection of the Regularization Parameter: λ2

• The subproblem of u(k) is a classical L2-TV minimization.

• Surprisingly, not many effective methods in existing references.

• We employ the unbiased predictive risk estimator (UPRE):

Selection of λ2, (Lin et. al., SP (90) 2010):

λ2 = argminλ2

{1n‖rλ2‖

22 +

2σ2

ntrace(ATV,λ2)− σ

2}.


Solving for m(k): Nonlinear Conjugate Gradients

• CG direction at each iteration is used as search direction of dk .* The search direction dk needs to be a descent direction

cos θ =∇ET

k dk

||∇Ek || ||dk ||< 0.

• Line search for βk , the Armijo condition is used{E(m(k) + β(k)d(k)) ≤ E(m(k)) + c1β

(k)(d(k))T∇E(m(k))

(d(k))T∇E(m(k) + β(k)d(k)) ≥ c2(d(k))T∇E(m(k)).

• Updating scheme

m(k+1) = m(k) + β(k)d(k).


Gradients

• The gradient of the data misfit can be obtained by the adjoint-statemethod

∇m ‖d− f (m)‖22 =−2m3

∑shots

∑t

∂2−→f (k)

∂t2 ·←−r(k),

where−→f (k) is the forward propagated wavefield, and

←−r(k) is the

backward propagated residual at iteration k , which is furtherdefined as r(k) = d− f (m(k)).

• The gradient of L2-norm term can be simply derived as,

∇m

∥∥∥m− u(k−1)∥∥∥2

2= 2(m− u(i−1)).


Solving for u(k)

• The subproblem of solving u(k) is a L2 − TV denoising problem.

• Any TV-solver can be used: Iteratively Reweighted NormAlgorithm, Lagged Diffusivity Fixed Point Iteration Algorithm,Split-Bregman Method, etc.


Solving for u(k): Split-Bregman Method

• Reformulate the minimization of u(k) as an equivalent problembased on the Bregman distance:

E(u,dx ,dz) = minu,dx ,dz

∥∥∥u−m(k)∥∥∥2

2+ λ2 ‖u‖TV

+ µ∥∥∥dx −∇xu− b(k)

x

∥∥∥2

2+ µ

∥∥∥dz −∇zu− b(k)z

∥∥∥2

2.

• An alternating minimization algorithm can be employed, wheretwo subproblems need to be further minimized:

minu

∥∥∥u−m(k)∥∥∥2

2+µ∥∥∥d(k)

x −∇xu− b(k)x

∥∥∥2

2+µ∥∥∥d(k)

z −∇zu− b(k)z

∥∥∥2

2,

and

mindx ,dz

λ2 ‖u‖TV + µ∥∥∥dx −∇xu− b(k)

x

∥∥∥2

2+ µ

∥∥∥dz −∇zu− b(k)z

∥∥∥2

2.


Computational Cost Analysis

• Suppose that the size of the model is m ∈ <m×n and the datap ∈ <q×n, where m is the depth, n is the offset, and q is the timesteps. We assume there are s shots and the finite-differencecalculation employs a scheme of o(δt2, δh4).

• The cost of solving for m(k):

COST1 ≈ k1 · (l + 3) · O(s · m · n · q) + (l + 5) · O(m · n),

where l is the number of trials in the line search for β(k) and k1 isthe total iteration steps.

• The cost of solving for u(k):

COST2 ≈ 18 · O(m · n).

• Therefore, the cost of solving for u(k) is trivial compared to thecost of solving for m(k).


Outline







7 Conclusions


Results - True Model

19.2˚

19.4˚

1 km

19.2˚

19.4˚

3 km

-155.4˚ -155.2˚ -155˚ -154.8˚

19.2˚

19.4˚

5 km

7 km

9 km

-155.4˚ -155.2˚ -155˚ -154.8˚

11 km

4 5 6 7

Vp, km/sYouzuo Lin (LANL) Seismic Inversion Inverse Problems Workshop 28 / 77

Results - Tikhonov Inversion

19.2˚

19.4˚

1 km

19.2˚

19.4˚

3 km

-155.4˚ -155.2˚ -155˚ -154.8˚

19.2˚

19.4˚

5 km

7 km

9 km

-155.4˚ -155.2˚ -155˚ -154.8˚

11 km

4 5 6 7


Results - Total Variation Inversion

19.2˚

19.4˚

1 km

19.2˚

19.4˚

3 km

-155.4˚ -155.2˚ -155˚ -154.8˚

19.2˚

19.4˚

5 km

7 km

9 km

-155.4˚ -155.2˚ -155˚ -154.8˚

11 km

4 5 6 7


Numerical Results - Edge-Guided Inversion

19.2˚

19.4˚

1 km

19.2˚

19.4˚

3 km

-155.4˚ -155.2˚ -155˚ -154.8˚

19.2˚

19.4˚

5 km

7 km

9 km

-155.4˚ -155.2˚ -155˚ -154.8˚

11 km

4 5 6 7


Results - Tikhonov Inversion Difference

19.2˚

19.4˚

1 km

19.2˚

19.4˚

3 km

-155.4˚ -155.2˚ -155˚ -154.8˚

19.2˚

19.4˚

5 km

7 km

9 km

-155.4˚ -155.2˚ -155˚ -154.8˚

11 km

-0.2 -0.1 0.0 0.1 0.2


Results - Total Variation Inversion Difference

19.2˚

19.4˚

1 km

19.2˚

19.4˚

3 km

-155.4˚ -155.2˚ -155˚ -154.8˚

19.2˚

19.4˚

5 km

7 km

9 km

-155.4˚ -155.2˚ -155˚ -154.8˚

11 km

-0.2 -0.1 0.0 0.1 0.2


Results - Edge-Guided Inversion Difference

19.2˚

19.4˚

1 km

19.2˚

19.4˚

3 km

-155.4˚ -155.2˚ -155˚ -154.8˚

19.2˚

19.4˚

5 km

7 km

9 km

-155.4˚ -155.2˚ -155˚ -154.8˚

11 km

-0.2 -0.1 0.0 0.1 0.2


Results - Robustness Tests on the Sparse Data

• Randomly eliminate 50% of the earthquake events and stations(Left: Tikhonov, Middle: TV, Right: Edge-Guided TV)

19.2

˚

19.4

˚

1 k

m

19.2

˚

19.4

˚

3 k

m

19.2

˚

19.4

˚

5 k

m

19.2

˚

19.4

˚

7 k

m

19.2

˚

19.4

˚

9 k

m

-155.4

˚-1

55.2

˚-1

55˚

-154.8

˚

19.2

˚

19.4

˚

11 k

m

1 k

m

3 k

m

5 k

m

7 k

m

9 k

m

-155.4

˚-1

55.2

˚-1

55˚

-154.8

˚

11 k

m

19.2

˚

19.4

˚

1 k

m

19.2

˚

19.4

˚

3 k

m

19.2

˚

19.4

˚

5 k

m

19.2

˚

19.4

˚

7 k

m

19.2

˚

19.4

˚

9 k

m

-155.4

˚-1

55.2

˚-1

55˚

-154.8

˚19.2

˚

19.4

˚

11

km

45

67

Vp, km

/s


Results - Robustness Tests on the Sparse Data

• Randomly eliminate 50% of the earthquake events and stations(Left: Tikhonov, Middle: TV, Right: Edge-Guided TV)

19.2

˚

19.4

˚

1 k

m

19.2

˚

19.4

˚

3 k

m

19.2

˚

19.4

˚

5 k

m

19.2

˚

19.4

˚

7 k

m

19.2

˚

19.4

˚

9 k

m

-155.4

˚-1

55.2

˚-1

55˚

-154.8

˚

19.2

˚

19.4

˚

11 k

m

1 k

m

3 k

m

5 k

m

7 k

m

9 k

m

-155.4

˚-1

55.2

˚-1

55˚

-154.8

˚

11 k

m

19.2

˚

19.4

˚

1 k

m

19.2

˚

19.4

˚

3 k

m

19.2

˚

19.4

˚

5 k

m

19.2

˚

19.4

˚

7 k

m

19.2

˚

19.4

˚

9 k

m

-155.4

˚-1

55.2

˚-1

55˚

-154.8

˚19.2

˚

19.4

˚

11

km

-0.2

-0.1

0.0

0.1

0.2

dV

p, km

/s


Outline







7 Conclusions


Reservoir Monitoring with Time-Lapse Data

• Conventional Inversions

δmconv =

f−1(dtime 2)−f−1(dtime 1),

where dtime 1 and dtime 2are data collected at twodifferent times.

Image

data

Inversion

Baseline

Image

Inversion

Image

dataTime−lapse

Time−lapse

Difference

Baseline

Figure: Conventional InversionYouzuo Lin (LANL) Seismic Inversion Inverse Problems Workshop 37 / 77


• Double-Difference FWIThe method jointlyinverts time-lapse datafor reservoir changes.

δmDDFWI =

f−1(dsim time 2)−f−1(dsim time 1),

wheredsim time 1 = f (mbaseline)anddsim time 2 = dsim time 1 +(dtime 2 − dtime 1)

Modeling

Inversion

Baseline

Image

Inversion

DifferenceImage

Baseline Time−lapse

Data Data

Simulated

Time−lapse Data

ImageTime−lapse

SimulatedBaseline Data

Difference

Initial Model

Figure: Double-Difference FWI



• Double-Difference FWI V.S. Conventional Inversions• Computational Cost: There are 2 FWIs involved in both

approaches. → Similar cost to the conventional inversions

• Noise and Artifacts: Noises common to data are removed by datadifferentiation. → Less noise and artifacts generated comparingto the conventional inversions

• Stability: The inversion of the simulated time-lapse data focuses toimage the changes of physical properties. → Easier to applydifferent priory information to stabilize the inversion

• The baseline model is the KEY for a successfulimplementation of double-difference FWI


Double-Difference FWI with MTV Regularization

Double-Difference FWI with MTV Regularization (Lin & Huang,GJI(203) 2015):

mbaseline = m1 = argminm1

{‖dtime1 − f (m1)‖22 + ‖m1‖MTV

},

mtime-lapse = m2 = argminm2

{‖dsim time2 − f (m2)‖22 + ‖δm‖MTV

},

where δm = m2 −m1 and ‖m‖MTV = λ1 ‖m− u‖22 + λ2‖u‖TV.

• Our FWI with the MTV regularization improves inversion of thebaseline dataset to obtain an accurate baseline model.

• we focus our inversion on the regions where time-lapse changesoccur using the regularization term ‖δm‖MTV to constrain thetime-lapse model differences.


Reservoir Monitoring for EGS

0

0.5

1.0

1.5

De

pth

(km

)

0 0.5 1.0 1.5 2.0Horizontal Distance (km)

2000

2500

3000

3500

4000

4500

5000

Ve

locity

(m/s

)

Baseline Vp Model

0

0.5

1.0

1.5

De

pth

(km

)


2000

2500

3000

3500

4000

4500

5000

Ve

locity

(m/s

)

Baseline Vs Model

• Models are built based on the Brady’s enhanced geothermalsystem (EGS) field (Lin & Huang, SGW 2012)



0

0.5

1.0

1.5

De

pth

(km

)


2000

2500

3000

3500

4000

4500

5000

Ve

locity

(m/s

)

Vp Model After Stimulation

0

0.5

1.0

1.5

De

pth

(km

)


2000

2500

3000

3500

4000

4500

5000

Ve

locity

(m/s

)

Vs Model After Stimulation

• Models are built based on the Brady’s enhanced geothermalsystem (EGS) field

• Stimulation leads to decreases in Vp and Vs



0

0.5

1.0

1.5

De

pth

(km

)


-250

-200

-150

-100

-50

0

50

100

150

200

250

Ve

locity

Pe

rturb

atio

n (m

/s)

Time-lapse Difference of Vp

0

0.5

1.0

1.5

De

pth

(km

)


-250

-200

-150

-100

-50

0

50

100

150

200

250

Ve

locity

Pe

rturb

atio

n (m

/s)

Time-lapse Difference of Vs

• Monitoring regions with time-lapse difference of Vp abd Vs



0

0.5

1.0

1.5

De

pth

(km

)


2000

2500

3000

3500

4000

4500

5000

Ve

locity

(m/s

)

Vp Initial Model

0

0.5

1.0

1.5

De

pth

(km

)


2000

2500

3000

3500

4000

4500

5000

Ve

locity

(m/s

)

Vs Initial Model

• The initial Vp and Vs models for inversion.• 96 sources and 500 receivers on the top surfaces of the models.• Ricker’s wavelet with a center frequency of 25 Hz is used as the

source function.



0

0.5

1.0

1.5

De

pth

(km

)


2000

2500

3000

3500

4000

4500

5000

Ve

locity

(m/s

)

Vp Model

0

0.5

1.0

1.5

De

pth

(km

)


2000

2500

3000

3500

4000

4500

5000

Ve

locity

(m/s

)

Vs Model

FWI without Regularization• Interfaces of the reconstruction are degraded.



0

0.5

1.0

1.5

De

pth

(km

)


2000

2500

3000

3500

4000

4500

5000

Ve

locity

(m/s

)

Vp Model

0

0.5

1.0

1.5

De

pth

(km

)


2000

2500

3000

3500

4000

4500

5000

Ve

locity

(m/s

)

Vs Model

FWI with TV Regularization• Lots of noise and artifacts are generated.



0

0.5

1.0

1.5

De

pth

(km

)


2000

2500

3000

3500

4000

4500

5000

Ve

locity

(m/s

)

Vp Model

0

0.5

1.0

1.5

De

pth

(km

)


2000

2500

3000

3500

4000

4500

5000

Ve

locity

(m/s

)

Vs Model

FWI with MTV Regularization• Interfaces of the reconstruction are well preserved.• Noise and artifacts are eliminated.



0

0.5

1.0

1.5

De

pth

(km

)

2000 3000 4000 5000Velocity (m/s)

Vp Model

0

0.5

1.0

1.5

De

pth

(km

)

2000 3000 4000 5000Velocity (m/s)

Vs Model

FWI without Regularization• Reconstructed values are still off the true values.



0

0.5

1.0

1.5

De

pth

(km

)

2000 3000 4000 5000Velocity (m/s)

Vp Model

0

0.5

1.0

1.5

De

pth

(km

)

2000 3000 4000 5000Velocity (m/s)

Vs Model

FWI with TV Regularization• Reconstructed values are highly oscillated.



0

0.5

1.0

1.5

De

pth

(km

)

2000 3000 4000 5000Velocity (m/s)

Vp Model

0

0.5

1.0

1.5

De

pth

(km

)

2000 3000 4000 5000Velocity (m/s)

Vs Model

FWI with MTV Regularization• Reconstructed values are very close to the true values.



0

0.5

1.0

1.5

De

pth

(km

)


-250

-200

-150

-100

-50

0

50

100

150

200

250

Ve

locity

Pe

rturb

atio

n (m

/s)

Vp Model

0

0.5

1.0

1.5

De

pth

(km

)


-250

-200

-150

-100

-50

0

50

100

150

200

250

Ve

locity

Pe

rturb

atio

n (m

/s)

Vs Model

Conventional Inversions

• Hard to identify the location of the reservoir.• Artifacts are significant.



0

0.5

1.0

1.5

De

pth

(km

)


-250

-200

-150

-100

-50

0

50

100

150

200

250

Ve

locity

Pe

rturb

atio

n (m

/s)

Vp Model

0

0.5

1.0

1.5

De

pth

(km

)


-250

-200

-150

-100

-50

0

50

100

150

200

250

Ve

locity

Pe

rturb

atio

n (m

/s)

Vs Model

Double-Difference FWI without Regularization

• Results are significantly improved.• The results still contain some artifacts.



0

0.5

1.0

1.5

De

pth

(km

)


-250

-200

-150

-100

-50

0

50

100

150

200

250

Ve

locity

Pe

rturb

atio

n (m

/s)

Vp Model

0

0.5

1.0

1.5

De

pth

(km

)


-250

-200

-150

-100

-50

0

50

100

150

200

250

Ve

locity

Pe

rturb

atio

n (m

/s)

Vs Model


• The monitoring regions are very easy to visualize.• Background noise and artifacts are significantly reduced.



0

0.5

1.0

1.5

De

pth

(km

)

-300 -200 -100 0 100Velocity Perturbation (m/s)

Vp Model

0

0.5

1.0

1.5

De

pth

(km

)


Vs Model

Conventional Inversions

• The magnitudes of the inversion artifacts are almost the samelevel to those of the inversion results.



0

0.5

1.0

1.5

De

pth

(km

)


Vp Model

0

0.5

1.0

1.5

De

pth

(km

)


Vs Model

Double-Difference FWI without Regularization

• The inversion results have been improved.• The artifacts above and below the reservoir are still strong.



0

0.5

1.0

1.5

De

pth

(km

)

-300 -200 -100 0 100 200Velocity Perturbation (m/s)

Vp Model

0

0.5

1.0

1.5

De

pth

(km

)

-300 -200 -100 0 100 200Velocity Perturbation (m/s)

Vs Model


• Reconstructed values are very close to its true values.• The profiles are much less oscillated than all the other results.



1 2 3 4 5 6 7 8 9 10 11 120

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Iterations

Da

ta M

isfi

t

Conventional DD AEWI

DD AEWI with Prior

DD AEWI with MTV

Data Misfit

1 2 3 4 5 6 7 8 9 10 11 120

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Iterations

Mo

de

l M

isfi

t

Conventional DD AEWI

DD AEWI with Prior

DD AEWI with MTV

Model MisfitConvergence Plots

• Reference methods: conventional DD-FWI (in blue), the methoddeveloped in Zhang & Huang (2014) (in green)

• Our method converges fastest for both data misfit and model misfit


Robustness Tests

• FWI can be very easily trapped in the local minima.• Using noisy measurements;

• Using inaccurate initial guess far away from the true model;

• Total-variation regularization can create those “stair-casing”artifacts• Smooth changes at the monitoring regions


Robustness Test - Noisy MeasurementReceiver number

Tim

e (

s)

50 100 150 200 250 300 350 400 450 500

0.25

0.5

0.75

1

1.25

1.5

Common-Shot Gather

0.25 0.5 0.75 1 1.25 1.5 0.25−2

−1.5

−1

−0.5

0

0.5

1

1.5

2x 10

−5

Time (s)

Am

plit

ud

e

Clean

Noisy

Seismogram at Recv. Num. 250

Noisy Measurement (Baseline)

• Baseline measurement• 20 dB of white noise is added


Robustness Test - Noisy MeasurementReceiver number

Tim

e (

s)

50 100 150 200 250 300 350 400 450 500

0.25

0.5

0.75

1

1.25

1.5

Common-Shot Gather

0.25 0.5 0.75 1 1.25 1.5 0.25−2

−1.5

−1

−0.5

0

0.5

1

1.5

2x 10

−5

Time (s)

Am

plit

ud

e

Clean

Noisy

Seismogram at Recv. Num. 250

Noisy Measurement (Repeat-survey)

• Repeat-survey measurement• 20 dB of different white noise is added


Robustness Test - Noisy Measurement

0

0.5

1.0

1.5

De

pth

(km

)


2000

2500

3000

3500

4000

4500

5000

Ve

locity

(m/s

)

Vp Model

0

0.5

1.0

1.5

De

pth

(km

)


2000

2500

3000

3500

4000

4500

5000

Ve

locity

(m/s

)

Vs Model

Noisy Measurement

• Some additional artifacts in the deep layers of the velocityinversion results.


Robustness Test - Noisy Measurement

0

0.5

1.0

1.5

De

pth

(km

)


-250

-200

-150

-100

-50

0

50

100

150

200

250

Ve

locity

Pe

rturb

atio

n (m

/s)

Vp Model

0

0.5

1.0

1.5

De

pth

(km

)


-250

-200

-150

-100

-50

0

50

100

150

200

250

Ve

locity

Pe

rturb

atio

n (m

/s)

Vs Model

Noisy Measurement

• Those artifacts do not effect the accuracy of the invertedtime-lapse changes in the monitoring region


Robustness Test - Initial Guess

0

0.5

1.0

1.5

De

pth

(km

)


2000

2500

3000

3500

4000

4500

5000

Ve

locity

(m/s

)

Vp Initial Model

0

0.5

1.0

1.5

De

pth

(km

)


2000

2500

3000

3500

4000

4500

5000

Ve

locity

(m/s

)

Vs Initial Model

Initial Guess

• Smoothed by 3 wavelength



0

0.5

1.0

1.5

De

pth

(km

)


2000

2500

3000

3500

4000

4500

5000

Ve

locity

(m/s

)

Vp Initial Model

0

0.5

1.0

1.5

De

pth

(km

)


2000

2500

3000

3500

4000

4500

5000

Ve

locity

(m/s

)

Vs Initial Model

Initial Guess




0

0.5

1.0

1.5

De

pth

(km

)


2000

2500

3000

3500

4000

4500

5000

Ve

locity

(m/s

)

Vp Initial Model

0

0.5

1.0

1.5

De

pth

(km

)


2000

2500

3000

3500

4000

4500

5000

Ve

locity

(m/s

)

Vs Initial Model

Initial Guess




0

0.5

1.0

1.5

De

pth

(km

)


2000

2500

3000

3500

4000

4500

5000

Ve

locity

(m/s

)

Vp Model

0

0.5

1.0

1.5

De

pth

(km

)


2000

2500

3000

3500

4000

4500

5000

Ve

locity

(m/s

)

Vs Model

Inversion Results (Baseline)• Using initial guess smoothed by 3 WL



0

0.5

1.0

1.5

De

pth

(km

)


2000

2500

3000

3500

4000

4500

5000

Ve

locity

(m/s

)

Vp Model

0

0.5

1.0

1.5

De

pth

(km

)


2000

2500

3000

3500

4000

4500

5000

Ve

locity

(m/s

)

Vs Model




0

0.5

1.0

1.5

De

pth

(km

)


2000

2500

3000

3500

4000

4500

5000

Ve

locity

(m/s

)

Vp Model

0

0.5

1.0

1.5

De

pth

(km

)


2000

2500

3000

3500

4000

4500

5000

Ve

locity

(m/s

)

Vs Model




0

0.5

1.0

1.5

De

pth

(km

)


-250

-200

-150

-100

-50

0

50

100

150

200

250

Ve

locity

Pe

rturb

atio

n (m

/s)

Vp Model

0

0.5

1.0

1.5

De

pth

(km

)


-250

-200

-150

-100

-50

0

50

100

150

200

250

Ve

locity

Pe

rturb

atio

n (m

/s)

Vs Model

Inversion Results (Time-Lapse)• Using initial guess smoothed by 3 WL



0

0.5

1.0

1.5

De

pth

(km

)


-250

-200

-150

-100

-50

0

50

100

150

200

250

Ve

locity

Pe

rturb

atio

n (m

/s)

Vp Model

0

0.5

1.0

1.5

De

pth

(km

)


-250

-200

-150

-100

-50

0

50

100

150

200

250

Ve

locity

Pe

rturb

atio

n (m

/s)

Vs Model




0

0.5

1.0

1.5

De

pth

(km

)


-250

-200

-150

-100

-50

0

50

100

150

200

250

Ve

locity

Pe

rturb

atio

n (m

/s)

Vp Model

0

0.5

1.0

1.5

De

pth

(km

)


-250

-200

-150

-100

-50

0

50

100

150

200

250

Ve

locity

Pe

rturb

atio

n (m

/s)

Vs Model



Robustness Test - Smoothly Distributed Changes

0

0.5

1.0

1.5

De

pth

(km

)


-250

-200

-150

-100

-50

0

50

100

150

200

250

Ve

locity

Pe

rturb

atio

n (m

/s)

Vp Model

0

0.5

1.0

1.5

De

pth

(km

)


-250

-200

-150

-100

-50

0

50

100

150

200

250

Ve

locity

Pe

rturb

atio

n (m

/s)

Vs Model

Smoothly Distributed Changes

• For some practical applications, a spatial region with time-lapsechanges can be smoothly distributed other than piece-wiseconstant.



0

0.5

1.0

1.5

De

pth

(km

)


-250

-200

-150

-100

-50

0

50

100

150

200

250

Ve

locity

Pe

rturb

atio

n (m

/s)

Vp Model

0

0.5

1.0

1.5

De

pth

(km

)


-250

-200

-150

-100

-50

0

50

100

150

200

250

Ve

locity

Pe

rturb

atio

n (m

/s)

Vs Model


• Our inversion results preserve sharp edges and the smoothlydistributed changes.



0

0.5

1.0

1.5

De

pth

(km

)


Vp Model

0

0.5

1.0

1.5

De

pth

(km

)


Vs Model


• Our inversion results preserve sharp edges and the smoothlydistributed changes.


Outline







7 Conclusions


Conclusions

• We have developed new travel-time tomography and full-waveforminversion method using a modified total-variation regularizationscheme.

• Our new seismic inversion methods not only preserve sharpinterfaces in inversion results, but also greatly reduce inversionartifacts.

• Our new methods employs the modified total-variationregularization to improve the inversion accuracy and enhance therobustness of our methods for noisy data to some extent.


Related References

• “Lin et. al., GJI (201) 2015”: Youzuo Lin, Ellen M. Syracuse,Monica Maceira, Haijiang Zhang and Carene Larmat,“Double-difference traveltime tomography withedge-preserving regularization and a priori interfaces,”Geophysical Journal International, 201 (2): 574-594, 2015.

• “Lin & Huang,GJI (203) 2015”: Youzuo Lin and Lianjie Huang,“Quantifying Subsurface Geophysical Properties Changes UsingDouble-difference Seismic-Waveform Inversion with a ModifiedTotal-Variation Regularization Scheme,” Geophysical JournalInternational, 203 (3): 2125-2149, 2015.

• “Lin & Huang, GJI (200) 2015”: Youzuo Lin and Lianjie Huang,“Acoustic- and Elastic-Waveform Inversion Using a ModifiedTotal-Variation Regularization Scheme,” Geophysical JournalInternational, 200 (1): 489-502, 2015.


Acknowledgement

• The research was supported by (1). the Geothermal TechnologiesProgram of the U.S. Department of Energy, and (2). LANL,Laboratory Directed Research and Development (LDRD) program.

• We thank Dr. John Queen of Hi-Q Geophysical Inc. for providingus with a velocity model of Brady’s EGS field.


Documents

Lecture 2: Seismic Inversion and Imagingmueller/graduate_workshop/pdfs_of... · Outline 1 Subsurface Applications 2 Schematic Description of Seismic imaging 3 Travel-Time Tomography