characterization of heterogeneous hydraulic conductivity ... · right: a probability measure...

Characterization of heterogeneous hydraulic conductivityfield via Karhunen-Loeve expansions and ameasure-theoretic computational method

Jiachuan He

University of Texas at Austin

April 15, 2016

Jiachuan He (UT Austin) Spring 2016 group meeting April 15, 2016 1 / 22

Problem

Specification of a variety of parameters, such as hydraulic conductivity, isrequired in models that have been widely used to analyze groundwatersystems and contaminant transport. However, it is practically impossibleto characterize the model parameters exhaustively due to the complexhydrogeological environment. For this reason, inverse modeling is neededto identify input parameters by incorporating observed model responses,e.g., hydraulic conductivities are derived based on observed hydraulic head.

Given uncertain hydraulic head data from observation wells, the stochasticinverse problem is solved numerically to obtain a probability measure onthe space of unknown parameters characterizing the heterogeneity of thehydraulic conductivity field.


Overview

1 Introduction

2 ParameterizationKarhunen-Loeve expansions

3 Inverse EstimationMeasure-theoretic framework

4 Illustrative exampleSynthetic problem


Groundwater flow equation

S∂h(x, t)

∂t+∇ · q(x, t) = g(x, t) (1)

q(x, t) = −K (x)∇h(x, t) (2)

subject to initial and boundary conditions

h(x, 0) = H0(x), x ∈ D (3)

h(x, t) = H(x, t), x ∈ ΓD (4)

q(x, t) · n(x) = Q(x, t), x ∈ ΓN (5)

where S is specific storage (LT−1), h is hydraulic head (L), and g issource or sink (T−1).


Karhunen-Loeve expansions

Inverse problems are often ill-posed. Inverse estimation of permeabilityfields is commonly performed by reducing the number of original unknownsof a heterogeneous hydraulic conductivity field to a smaller group ofunknowns. This makes the inverse problem better posed by reducingredundancy while capturing the most important features of the field. TheKarhunen-Loeve Expansion (KLE) is a classical option for derivinglow-dimensional parameterizations for inverse estimation applications [1].With the knowledge of the property covariance function, the KLE canprovide an accurate characterization of a complex field.

”In the theory of stochastic processes, the Karhunen-Loeve theorem(named after Kari Karhunen and Michel Loeve), also known as theKosambiKarhunenLove theorem is a representation of a stochastic processas an infinite linear combination of orthogonal functions...” (wikipedia)


Let Y (x, ω) = ln[K (x, ω)] be a random process, where K is the hydraulicconductivity, x is the position vector defined over the domain D, and ωbelongs to the space of random events Ω. Let Y (x) denote the expectedvalue of Y (x, ω) over all possible realizations of the process, and C (x1, x2)denote its covariance function. Being an autocovariance function,C (x1, x2) is bounded, symmetric, and positive definite. Thus, it has thespectral decomposition

C (x1, x2) =∞∑n=1

λnfn(x1)fn(x2) (Mercer ′sThm) (6)

where λn and fn(x) are the solutions to the homogeneous Fredholmintegral equation of the second kind:∫

DC (x1, x2)fn(x1)dx1 = λnfn(x2). (7)


The eigenfunctions are orthogonal and form a complete set. They can benormalized according to the following criterion∫

Dfn(x)fm(x) = δnm. (8)

Hence, Y (x, ω) can be written as

Y (x, ω) = Y (x) +∞∑n=1

ξn(ω)√λnfn(x), (Karhunen − LoeveThm) (9)

where ξn(ω) is a set of random variables to be determined. The KLE ofa Gaussian field has the further property that ξn(ω) are independentstandard normal random variables (Ito-Nisio Theorem). Truncating theseries in Eq (9) at the Nth term, gives

Y (x, ω) = Y (x) +N∑

n=1

ξn(ω)√λnfn(x). (10)


Inverse estimation

Y (x, ω) = Y (x) +N∑

n=1


The KLE provides a flexible and effective method for describing a spatiallydistributed hydraulic conductivity field. It represents the uncertainhydraulic conductivity field as weighted sums of predefined spatiallyvariable basis functions. The random variables ξn in basis function weightswill be quantified with a recently developed measure-theoretic framework.

The stochastic inverse estimation in this characterization of aheterogeneous hydraulic conductivity field problem is to determine aprobability measure on ξn (input parameters) such that mapping thisprobability measure through the model produces the same probabilitymeasure as the one defined on the observable hydraulic head.


Inverse problem

Figure : Illustrations of the inverse problem for a general two-to-one map Left:The set-valued inverse of a single output value. Middle: The representation of Las a transverse parameterization. Right: A probability measure described as adensity on D maps uniquely to a probability density on L. Figures adopted from[2]

PL(A) =

∫AρLdµL =

∫QL(A)

ρDdµD = PD(QL(A)) (12)


Inverse problem

To calculate the probability of more arbitrary events in Λ, we make use ofthe Disintegration Theorem and a standard ansatz to compute aprobability measure PΛ in terms of an approximate density ρΛ that isconsistent with PD. By saying PΛ is consistent with PD, we mean that ifB is any event in D (so Q−1(B) is a generalized contour event in Λ), then∫

Q−1(B)ρΛ(λ)dµΛ = PΛ(Q−1(B)) = PD(B) =

∫BρDdµD. (13)


Numerical Approximation of Inverse Probability

Suppose the probability density function ρD associated with PD is known.Let DkMk=1 be a partition of D, and let pk be the probability of Dk . LetA be an event in Λ. We thus have the approximation∫

Q(A)ρDdµD ≈

∑Dk⊂Q(A)

pk . (14)

Let λ(j)Nj=1 be a collection of N points in Λ, which implicitly defines an

n-dimensional Voronoi tessellation VjNj=1 associated with the points. For

example, given λ ∈ Λ, λ ∈ Vk if λ(k) is the closest λ(j) to λ. Using theimplicitly defined Voronoi tessellation and the partitioning of D, we cancalculate the probability pΛ,j associated with the implicitly defined Voronoicell Vj . Then we can approximate the probability of any event A ⊂ Λ usinga counting measure

PΛ(A) ≈ PΛ,N(A) =∑λ(j)∈A

pΛ,j . (15)


Numerical Approximation of Inverse Probability

Algorithm:

Choose points λ(j)Nj=1 ∈ Λ.

Denote the associated Voronoi tessellation VjNj=1 ⊂ Λ.

Evaluate Qj = Q(λ(j)) for all λ(j), j = 1, ..,N.

Choose a partitioning of D, DkMk=1 ⊂ D.

Compute pk ≈∫DkρDdµD, for k = 1, ...,M.

Let Ck = j |Qj ∈ Dk for k = 1, ...,M.

Let Oj = k |Qj ∈ Dk, for j = 1, ..,N.

Let Vj be the approximate measure of Vj , i.e. Vj ≈∫Vj dµ(Vj) for

j = 1, ..,N.

Set pΛ,j = (Vj/∑

i∈COjVi )pD,Oj

, j = 1, ..,N.


Illustrative example

Figure : The domain with observation wells shown as red dots


Forward model

∇ · q(x) = 0 (16)

q(x) = −K (x)∇h(x) (17)

subject to boundary conditions

h(x) = H(x), x ∈ ΓD (18)

q(x) · n(x) = Q(x), x ∈ ΓN (19)


Special types of covariance functions

C (x1, y1) = σ2exp(− |x1−y1|η1

)Eigenvalues and eigenfunctions of a covariance function can be solvedfrom the following Fredholm equation:∫

DC (x1, x2)fn(x1)dx1 = λnfn(x2) (20)

λn and fn(x) can be found analytically:

λn =2ησ2

η2w2n + 1

(21)

fn(x) =1√

(η2w2n + 1)L/2 + η

[ηwncos(wnx) + sin(wnx)] (22)

where wn are positive roots of the characteristic equation(η2w2 − 1)sin(wL) = 2ηwcos(wL).


Separable covariance function

For problems in multidimension, if the covariance function is separable, for

example, C (x, y) = C (x1, x2; y1, y2) = σ2exp[− |x1−y1|

η1− |x2−y2|

η2

](Gaussian random field with the exponential covariance function) for arectangular domain D = (x1, x2) : 0 6 x1 6 L1, 0 6 x2 6 L2Eq.(20) can be solved independently for x1 and x2 directions to obtain

eigenvalues λ(1)n and λ

(2)n , and eigenfunctions f

(1)n (x1) and f

(2)n (x2).

λn = λ(1)n λ

(2)n (23)

fn(x) = fn(x1, x2) = f(1)n (x1)f

(2)n (x2) (24)


Figure : Examples of eigenfunctions fn. Figures adopted from [6]


Y (x, ω) = Y (x) +7∑

n=1


Figure : The domain and ”true” lnK


0.00000 0.00005 0.00010 0.00015 0.00020 0.00025 0.00030 0.00035pred_QoI_ref

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0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6pred_QoI_ref

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Figure : 1D pdf of predicted hydraulic heads at blue dots


Figure : Log hydraulic conductivity fields generated using samples with thehighest probability densities


References

Ghanem, Roger G., and Pol D. Spanos. Stochastic finite elements: a spectralapproach. Courier Corporation, 2003.

Breidt, J., T. Butler, and D. Estep. ”A measure-theoretic computational methodfor inverse sensitivity problems I: Method and analysis.” SIAM journal on numericalanalysis 49.5 (2011): 1836-1859.

Butler, T., D. Estep, and J. Sandelin. ”A computational measure theoreticapproach to inverse sensitivity problems II: a posteriori error analysis.” SIAMjournal on numerical analysis 50.1 (2012): 22-45.

Butler, T., et al. ”A measure-theoretic computational method for inverse sensitivityproblems III: Multiple quantities of interest.” SIAM/ASA Journal on UncertaintyQuantification 2.1 (2014): 174-202.

Butler, Troy, et al. ”Solving stochastic inverse problems using sigma-algebras oncontour maps.” arXiv preprint arXiv:1407.3851 (2014).

Zhang, Dongxiao, and Zhiming Lu. ”An efficient, high-order perturbation approachfor flow in random porous media via KarhunenLoeve and polynomial expansions.”Journal of Computational Physics 194.2 (2004): 773-794.


Thank You


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