hydraulic inverse modeling using total-variation ...mads.lanl.gov/presentations/lin hydraulic...

Hydraulic Inverse Modeling UsingTotal-Variation Regularization with

Relaxed Variable-Splitting

Youzuo Lin, Velimir Vesselinov, Daniel O’Malley, & Brendt Wohlberg

Los Alamos National LaboratoryLos Alamos, NM 87545

March 15, 2017

Outline

1 Introduction

2 Hydrogeologic Inverse ModelingForward Problem - Groundwater Flow EquationInverse Problem - Least-Squares Problem

3 Regularization Theory

4 Total Variation Regularization with Relaxed Variable-SplittingScheme

5 Computation Methods

6 Numerical Results

7 Conclusions

Youzuo Lin et al. Inverse Modeling March 15, 2017 1 / 28

Outline

1 Introduction

2 Hydrogeologic Inverse ModelingForward Problem - Groundwater Flow EquationInverse Problem - Least-Squares Problem

3 Regularization Theory

4 Total Variation Regularization with Relaxed Variable-SplittingScheme

5 Computation Methods

6 Numerical Results

7 Conclusions

Youzuo Lin et al. Inverse Modeling March 15, 2017 2 / 28

Introduction

• Inverse modeling seeks model parameters given a set ofobserved-state variables.

• For many practical hydrogeological problems, because the datacoverage is limited, the inversion can be ill-posed and unstable.

• To stabilize the inversion, regularization techniques can beemployed to eliminate the ill-posedness.

• The most commonly used type of regularization include Tikhonovand Total-Variation (TV).

• However, Tikhonov regularization tends to yield smoothedinversion results, and conventional TV regularization can becomputationally unstable and yield unwanted artifacts.

• We have developed a novel hydraulic inverse modeling methodusing a TV regularization with relaxed variable-splitting scheme topreserve sharp interfaces and improve the accuracy of inversion.

Youzuo Lin et al. Inverse Modeling March 15, 2017 2 / 28

Outline

1 Introduction

2 Hydrogeologic Inverse ModelingForward Problem - Groundwater Flow EquationInverse Problem - Least-Squares Problem

3 Regularization Theory

4 Total Variation Regularization with Relaxed Variable-SplittingScheme

5 Computation Methods

6 Numerical Results

7 Conclusions

Youzuo Lin et al. Inverse Modeling March 15, 2017 3 / 28

Hydrogeologic Inverse Modeling - Illustration

• Input: Measured values (hydraulic heads) at N observation wells.

• Output: Model parameter values (conductivity or transmissivity)at every grid node of the model.

Youzuo Lin et al. Inverse Modeling March 15, 2017 3 / 28

Forward Problem

The forward problem of hydrogeologic inverse modeling is governed bythe groundwater flow equation,

Groundwater Flow Equation

∇ · (T∇h) = gg(x , y) = 0∂h∂x

∣∣∣∣a,y

=∂h∂x

∣∣∣∣b,y

= 0

h(x , c) = 0,h(x ,d) = 1

where h is the hydraulic head, T is the transmissivity and g is asource/sink (here, set to zero).

Youzuo Lin et al. Inverse Modeling March 15, 2017 4 / 28

Forward Problem

Using the operator, the forward modeling problem of the hydrogeologicinverse modeling can be simplified as,

Groundwater Flow Equation - Operator Form

h = f (T),

where f (·) is the forward operator mapping from the model parameterspace to the measurement space.

Youzuo Lin et al. Inverse Modeling March 15, 2017 5 / 28

Inverse Problem

Correspondingly, the problem of model calibration can be posed as adamped least-squares problem,

Hydrogeologic Inverse Modeling

m = arg minm

{‖d− f (m)‖22

},

where d represents a recorded hydraulic head dataset, m is thecalibrated model parameter, ‖d− f (m)‖22 measures the data misfit,|| · ||2 stands for the L2 norm.

Youzuo Lin et al. Inverse Modeling March 15, 2017 6 / 28

Outline

1 Introduction

2 Hydrogeologic Inverse ModelingForward Problem - Groundwater Flow EquationInverse Problem - Least-Squares Problem

3 Regularization Theory

4 Total Variation Regularization with Relaxed Variable-SplittingScheme

5 Computation Methods

6 Numerical Results

7 Conclusions

Youzuo Lin et al. Inverse Modeling March 15, 2017 7 / 28

Regularization Theory

• Inverse modeling with general regularization term can be posedas,

Hydrogeologic Inverse Modeling with Regularization

m̂ = arg minm

{‖d− f (m)‖22 + λR(m)

},

where R(m) is a general regularization term and the parameter λis a parameter controlling the amount of regularization in theinversion.

Youzuo Lin et al. Inverse Modeling March 15, 2017 7 / 28

Inverse Problem & Regularization Techniques

• General Regularization Methodology

Inverse Modeling 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←.

Youzuo Lin et al. Inverse Modeling March 15, 2017 8 / 28

Outline

1 Introduction

2 Hydrogeologic Inverse ModelingForward Problem - Groundwater Flow EquationInverse Problem - Least-Squares Problem

3 Regularization Theory

4 Total Variation Regularization with Relaxed Variable-SplittingScheme

5 Computation Methods

6 Numerical Results

7 Conclusions

Youzuo Lin et al. Inverse Modeling March 15, 2017 9 / 28

Total Variation Regularization with RelaxedVariable-Splitting Scheme

The misfit function of hydraulic inverse modeling using total-variationregularization with relaxed variable-splitting is:

A New Misfit Function of Hydraulic Inverse Modeling

E(m,u) = minm,u

{‖d− f (m)‖22 + λ1 ‖m− u‖22 + λ2 ‖∇u‖1

},

• ‖d− f (m)‖22 is the data misfit term;

• ‖m− u‖22 and ‖∇u‖1 are the regularization terms;• λ1 and λ2 are the regularization parameters;

Youzuo Lin et al. Inverse Modeling March 15, 2017 9 / 28

Total-Variation Regularization with RelaxedVariable-Splitting

A New Misfit Function of Hydraulic Inverse Modeling

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 regularization terms contain a new variable u and anadditional term ‖m− u‖22 compared to the conventional TVregularization term.

Youzuo Lin et al. Inverse Modeling March 15, 2017 10 / 28

Total-Variation Regularization with RelaxedVariable-Splitting

A New Misfit Function of Hydraulic Inverse Modeling

E(m,u) = minu

{min

m

{‖d− f (m)‖22 + λ1 ‖m− u‖22

}+ λ2 ‖u‖TV

},

where λ1 and λ2 are both positive regularization parameters.

• 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 inversemodeling.

Youzuo Lin et al. Inverse Modeling March 15, 2017 11 / 28

A New Misfit Function of Hydraulic InverseModeling (A Closer Look)

A New Misfit Function of Hydraulic Inverse Modeling:

E(m,u) = minu

{min

m

{‖d− f (m)‖22 + λ1 ‖m− u‖22

}+ λ2 ‖u‖TV

},

where λ1 and λ2 are both positive regularization parameters.

• The inner problem is to solve for m using a conventional inversemodeling with the 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.

Youzuo Lin et al. Inverse Modeling March 15, 2017 12 / 28

Outline

1 Introduction

2 Hydrogeologic Inverse ModelingForward Problem - Groundwater Flow EquationInverse Problem - Least-Squares Problem

3 Regularization Theory

4 Total Variation Regularization with Relaxed Variable-SplittingScheme

5 Computation Methods

6 Numerical Results

7 Conclusions

Youzuo Lin et al. Inverse Modeling March 15, 2017 13 / 28

Computation Methods

We employ the Alternating Direction Method of Multipliers (ADMM) tosolve our new hydraulic inverse modeling

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

}

Youzuo Lin et al. Inverse Modeling March 15, 2017 13 / 28

Selection of the Regularization Parameter: λ1

• The subproblem of m(k) is a classical inverse modeling withTikhonov regularization.

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

Youzuo Lin et al. Inverse Modeling March 15, 2017 14 / 28

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}.

Youzuo Lin et al. Inverse Modeling March 15, 2017 15 / 28

Outline

1 Introduction

2 Hydrogeologic Inverse ModelingForward Problem - Groundwater Flow EquationInverse Problem - Least-Squares Problem

3 Regularization Theory

4 Total Variation Regularization with Relaxed Variable-SplittingScheme

5 Computation Methods

6 Numerical Results

7 Conclusions

Youzuo Lin et al. Inverse Modeling March 15, 2017 16 / 28

Problem Setup and Model Discretization

• The reference problem is steady-state groundwater flow on thesquare domain, [0,1]× [0,1], with fixed hydraulic head at y = 0and y = 1, zero flux boundaries at x = 0 and x = 1, and zerorecharge.

• We run the tests on a Linux desktop with 32 cores of 2.0 GHz IntelXeon E5-2650 CPU, and 16.0 GB memory.

• The groundwater flow equation is solved using the finite differencemethod on a uniform grid. The parameter grids are composed ofhorizontal and vertical transmissivity nodes (as are illustrated infigure below).

Youzuo Lin et al. Inverse Modeling March 15, 2017 16 / 28

Model Calibration in Hydrology - True Model

5 10 15 20 25 30

5

10

15

20

25

30-2

-1

0

1

2

3

4

5

6

7

8

True Model• The plus (“+”) are the hydraulic-head observation points (wells).• A horizontal profile indicated by the red dotted line will be used to

compare the results.

Youzuo Lin et al. Inverse Modeling March 15, 2017 17 / 28

Inverse Modeling Result - 2D Inversion

kest

5 10 15 20 25 30

5

10

15

20

25

30-2

-1

0

1

2

3

4

5

6

7

8

2D Inversion• Inversion result using inverse modeling with conventional TV

regularization

Youzuo Lin et al. Inverse Modeling March 15, 2017 18 / 28

Inverse Modeling Result - 2D Inversion

kest

5 10 15 20 25 30

5

10

15

20

25

30-2

-1

0

1

2

3

4

5

6

7

8

2D Inversion• Inversion result using inverse modeling with our new TV

regularization

Youzuo Lin et al. Inverse Modeling March 15, 2017 19 / 28

Inverse Modeling Result - 1D Profile

0 5 10 15 20 25 30

-2

-1

0

1

2

3

4

5

6

7

8k field comparison

kref

kest

1D Horizontal Profile• Profile of the inversion (red) v.s. the true value (blue)• Inverse modeling with conventional TV method

Youzuo Lin et al. Inverse Modeling March 15, 2017 20 / 28

Inverse Modeling Result - 1D Profile

0 5 10 15 20 25 30

-2

-1

0

1

2

3

4

5

6

7

8k field comparison

kref

kest

1D Horizontal Profile• Profile of the inversion (red) v.s. the true value (blue)• Inverse modeling with our new TV method• The interface is much better preserved

Youzuo Lin et al. Inverse Modeling March 15, 2017 21 / 28

Model Calibration in Hydrology - True ModelTrue Model

5 10 15 20 25 30 35 40 45 50

5

10

15

20

25

30

35

40

45

50

True Model• The dimension of the true model is 50× 50.• Two low-permeable geologic facies are included in the true model

representing: sand (green) and clay (red). The background ishighly permeable (gravel; blue). The permeability within all thethree facies is assumed to be uniform.

Youzuo Lin et al. Inverse Modeling March 15, 2017 22 / 28

Inverse Modeling Result - 2D Inversion

kest

Reconstruction

5 10 15 20 25 30 35 40 45 50

5

10

15

20

25

30

35

40

45

50

-6.5

-6

-5.5

-5

-4.5

-4

-3.5

-3

2D Inversion• Inversion result using inverse modeling with conventional TV

regularization

Youzuo Lin et al. Inverse Modeling March 15, 2017 23 / 28

Inverse Modeling Result - 2D Inversion

kest

Reconstruction

5 10 15 20 25 30 35 40 45 50

5

10

15

20

25

30

35

40

45

50

-6.5

-6

-5.5

-5

-4.5

-4

-3.5

-3

2D Inversion• Inversion result using inverse modeling with our new TV

regularization

Youzuo Lin et al. Inverse Modeling March 15, 2017 24 / 28

Inverse Modeling Result - 1D Profile

0 5 10 15 20 25 30 35 40 45 50

-6

-5.5

-5

-4.5

-4

-3.5

-3

-2.5

-2k field comparison (Profile 1)

kref

kest

Location 1

0 5 10 15 20 25 30 35 40 45 50

-7.5

-7

-6.5

-6

-5.5

-5

-4.5

-4

-3.5

-3

-2.5

-2k field comparison (Profile 2)

kref

kest

Location 2• Profile of the inversion (red) v.s. the true value (blue)• Inverse modeling with conventional TV method

Youzuo Lin et al. Inverse Modeling March 15, 2017 25 / 28

Inverse Modeling Result - 1D Profile

0 5 10 15 20 25 30 35 40 45 50

-6

-5.5

-5

-4.5

-4

-3.5

-3

-2.5

-2k field comparison (Profile 1)

kref

kest

Location 1

0 5 10 15 20 25 30 35 40 45 50

-7.5

-7

-6.5

-6

-5.5

-5

-4.5

-4

-3.5

-3

-2.5

-2k field comparison (Profile 2)

kref

kest

Location 2• Profile of the inversion (red) v.s. the true value (blue)• Inverse modeling with our new TV method• The interface is much better preserved

Youzuo Lin et al. Inverse Modeling March 15, 2017 26 / 28

Outline

1 Introduction

2 Hydrogeologic Inverse ModelingForward Problem - Groundwater Flow EquationInverse Problem - Least-Squares Problem

3 Regularization Theory

4 Total Variation Regularization with Relaxed Variable-SplittingScheme

5 Computation Methods

6 Numerical Results

7 Conclusions

Youzuo Lin et al. Inverse Modeling March 15, 2017 27 / 28

Conclusions

• We have developed a hydraulic inverse modeling usingtotal-variation regularization with relaxed variable-splitting.

• Our numerical examples using synthetic data show that our newmethods not only preserve sharp interfaces between facies withcontrasting permeabilities, but also significantly improve theaccuracy of the inversion. Therefore, our method has greatpotential in characterizing the subsurface heterogeneity problems.

• We implement our new inverse modeling method using Julia in theMADS computational framework (http://madsjulia.lanl.gov/), whichcan be downloaded at https://github.com/madsjulia/Mads.jl.

Youzuo Lin et al. Inverse Modeling March 15, 2017 27 / 28

Acknowledgement

• This research is supported by LANL Environmental ProgramsProjects.

• In addition, Velimir V. Vesselinov was supported by the DiaMonDproject (An Integrated Multifaceted Approach to Mathematics atthe Interfaces of Data, Models, and Decisions, U.S. Department ofEnergy Office of Science, Grant #11145687).

Youzuo Lin et al. Inverse Modeling March 15, 2017 28 / 28