Characterization of heterogeneous hydraulic conductivity field via Karhunen-Loève expansions and a measure-theoretic computational method

Transcription

1 Characterization of heterogeneous hydraulic conductivity field via Karhunen-Loève expansions and a measure-theoretic computational method Jiachuan He University of Texas at Austin April 15, 2016 Jiachuan He (UT Austin) Spring 2016 group meeting April 15, / 22

2 Problem Specification of a variety of parameters, such as hydraulic conductivity, is required in models that have been widely used to analyze groundwater systems and contaminant transport. However, it is practically impossible to characterize the model parameters exhaustively due to the complex hydrogeological environment. For this reason, inverse modeling is needed to 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 stochastic inverse problem is solved numerically to obtain a probability measure on the space of unknown parameters characterizing the heterogeneity of the hydraulic conductivity field. Jiachuan He (UT Austin) Spring 2016 group meeting April 15, / 22

3 Overview 1 Introduction 2 Parameterization Karhunen-Loève expansions 3 Inverse Estimation Measure-theoretic framework 4 Illustrative example Synthetic problem Jiachuan He (UT Austin) Spring 2016 group meeting April 15, / 22

4 Groundwater flow equation h(x, t) S + q(x, t) = g(x, t) t (1) q(x, t) = K(x) h(x, t) (2) subject to initial and boundary conditions h(x, 0) = H 0 (x), x D (3) h(x, t) = H(x, t), x Γ D (4) q(x, t) n(x) = Q(x, t), x Γ N (5) where S is specific storage (LT 1 ), h is hydraulic head (L), and g is source or sink (T 1 ). Jiachuan He (UT Austin) Spring 2016 group meeting April 15, / 22

5 Karhunen-Loève expansions Inverse problems are often ill-posed. Inverse estimation of permeability fields is commonly performed by reducing the number of original unknowns of a heterogeneous hydraulic conductivity field to a smaller group of unknowns. This makes the inverse problem better posed by reducing redundancy while capturing the most important features of the field. The Karhunen-Loeve Expansion (KLE) is a classical option for deriving low-dimensional parameterizations for inverse estimation applications [1]. With the knowledge of the property covariance function, the KLE can provide an accurate characterization of a complex field. In the theory of stochastic processes, the Karhunen-Loève theorem (named after Kari Karhunen and Michel Loève), also known as the KosambiKarhunenLove theorem is a representation of a stochastic process as an infinite linear combination of orthogonal functions... (wikipedia) Jiachuan He (UT Austin) Spring 2016 group meeting April 15, / 22

6 Let Y (x, ω) = ln[k(x, ω)] be a random process, where K is the hydraulic conductivity, x is the position vector defined over the domain D, and ω belongs to the space of random events Ω. Let Ȳ (x) denote the expected value of Y (x, ω) over all possible realizations of the process, and C(x 1, x 2 ) denote its covariance function. Being an autocovariance function, C(x 1, x 2 ) is bounded, symmetric, and positive definite. Thus, it has the spectral decomposition C(x 1, x 2 ) = λ n f n (x 1 )f n (x 2 ) (Mercer sthm) (6) n=1 where λ n and f n (x) are the solutions to the homogeneous Fredholm integral equation of the second kind: C(x 1, x 2 )f n (x 1 )dx 1 = λ n f n (x 2 ). (7) D Jiachuan He (UT Austin) Spring 2016 group meeting April 15, / 22

7 The eigenfunctions are orthogonal and form a complete set. They can be normalized according to the following criterion f n (x)f m (x) = δ nm. (8) Hence, Y (x, ω) can be written as Y (x, ω) = Ȳ (x) + D ξ n (ω) λ n f n (x), (Karhunen LoeveThm) (9) n=1 where {ξ n (ω)} is a set of random variables to be determined. The KLE of a Gaussian field has the further property that ξ n (ω) are independent standard normal random variables (Itô-Nisio Theorem). Truncating the series in Eq (9) at the N th term, gives N Y (x, ω) = Ȳ (x) + ξ n (ω) λ n f n (x). (10) n=1 Jiachuan He (UT Austin) Spring 2016 group meeting April 15, / 22

8 Inverse estimation Y (x, ω) = Ȳ (x) + N ξ n (ω) λ n f n (x). (11) n=1 The KLE provides a flexible and effective method for describing a spatially distributed hydraulic conductivity field. It represents the uncertain hydraulic conductivity field as weighted sums of predefined spatially variable basis functions. The random variables ξ n in basis function weights will be quantified with a recently developed measure-theoretic framework. The stochastic inverse estimation in this characterization of a heterogeneous hydraulic conductivity field problem is to determine a probability measure on ξ n (input parameters) such that mapping this probability measure through the model produces the same probability measure as the one defined on the observable hydraulic head. Jiachuan He (UT Austin) Spring 2016 group meeting April 15, / 22

9 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 L as a transverse parameterization. Right: A probability measure described as a density on D maps uniquely to a probability density on L. Figures adopted from [2] P L (A) = A ρ L dµ L = Q L (A) ρ D dµ D = P D (Q L (A)) (12) Jiachuan He (UT Austin) Spring 2016 group meeting April 15, / 22

10 Inverse problem To calculate the probability of more arbitrary events in Λ, we make use of the Disintegration Theorem and a standard ansatz to compute a probability measure P Λ in terms of an approximate density ρ Λ that is consistent with P D. By saying P Λ is consistent with P D, we mean that if B is any event in D (so Q 1 (B) is a generalized contour event in Λ), then ρ Λ (λ)dµ Λ = P Λ (Q 1 (B)) = P D (B) = ρ D dµ D. (13) Q 1 (B) B Jiachuan He (UT Austin) Spring 2016 group meeting April 15, / 22

11 Numerical Approximation of Inverse Probability Suppose the probability density function ρ D associated with P D is known. Let {D k } M k=1 be a partition of D, and let p k be the probability of D k. Let A be an event in Λ. We thus have the approximation ρ D dµ D p k. (14) Q(A) D k Q(A) Let {λ (j) } N j=1 be a collection of N points in Λ, which implicitly defines an n-dimensional Voronoi tessellation {V j } N j=1 associated with the points. For example, given λ Λ, λ V k if λ (k) is the closest λ (j) to λ. Using the implicitly defined Voronoi tessellation and the partitioning of D, we can calculate the probability p Λ,j associated with the implicitly defined Voronoi cell V j. Then we can approximate the probability of any event A Λ using a counting measure P Λ (A) P Λ,N (A) = p Λ,j. (15) λ (j) A Jiachuan He (UT Austin) Spring 2016 group meeting April 15, / 22

12 Numerical Approximation of Inverse Probability Algorithm: Choose points {λ (j) } N j=1 Λ. Denote the associated Voronoi tessellation {V j } N j=1 Λ. Evaluate Q j = Q(λ (j) ) for all λ (j), j = 1,.., N. Choose a partitioning of D, {D k } M k=1 D. Compute p k D k ρ D dµ D, for k = 1,..., M. Let C k = {j Q j D k } for k = 1,..., M. Let O j = {k Q j D k }, for j = 1,.., N. Let V j be the approximate measure of V j, i.e. V j V j dµ(v j ) for j = 1,.., N. Set p Λ,j = (V j / i C Oj V i )p D,Oj, j = 1,.., N. Jiachuan He (UT Austin) Spring 2016 group meeting April 15, / 22

13 Illustrative example Figure : The domain with observation wells shown as red dots Jiachuan He (UT Austin) Spring 2016 group meeting April 15, / 22

14 Forward model q(x) = 0 (16) q(x) = K(x) h(x) (17) subject to boundary conditions h(x) = H(x), x Γ D (18) q(x) n(x) = Q(x), x Γ N (19) Jiachuan He (UT Austin) Spring 2016 group meeting April 15, / 22

15 Special types of covariance functions C(x 1, y 1 ) = σ 2 exp( x 1 y 1 η 1 ) Eigenvalues and eigenfunctions of a covariance function can be solved from the following Fredholm equation: C(x 1, x 2 )f n (x 1 )dx 1 = λ n f n (x 2 ) (20) λ n and f n (x) can be found analytically: D λ n = 2ησ2 η 2 w 2 n + 1 (21) f n (x) = 1 (η 2 w 2 n + 1)L/2 + η [ηw ncos(w n x) + sin(w n x)] (22) where w n are positive roots of the characteristic equation (η 2 w 2 1)sin(wL) = 2ηwcos(wL). Jiachuan He (UT Austin) Spring 2016 group meeting April 15, / 22

16 Separable covariance function For problems in multidimension, if the covariance [ function is separable, ] for example, C(x, y) = C(x 1, x 2 ; y 1, y 2 ) = σ 2 exp x 1 y 1 η 1 x 2 y 2 η 2 (Gaussian random field with the exponential covariance function) for a rectangular domain D = {(x 1, x 2 ) : 0 x 1 L 1, 0 x 2 L 2 } Eq.(20) can be solved independently for x 1 and x 2 directions to obtain eigenvalues λ (1) n and λ (2) n, and eigenfunctions f (1) n (x 1 ) and f (2) n (x 2 ). λ n = λ (1) n λ (2) n (23) f n (x) = f n (x 1, x 2 ) = f (1) n (x 1 )f (2) n (x 2 ) (24) Jiachuan He (UT Austin) Spring 2016 group meeting April 15, / 22

17 Figure : Examples of eigenfunctions f n. Figures adopted from [6] Jiachuan He (UT Austin) Spring 2016 group meeting April 15, / 22

18 7 Y (x, ω) = Ȳ (x) + ξ n (ω) λ n f n (x). (25) n=1 Figure : The domain and true lnk Jiachuan He (UT Austin) Spring 2016 group meeting April 15, / 22

19 ρ ρ ρ ρ pred_qoi_ref pred_qoi_ref pred_qoi_ref pred_qoi_ref Figure : 1D pdf of predicted hydraulic heads at blue dots Jiachuan He (UT Austin) Spring 2016 group meeting April 15, / 22

20 Figure : Log hydraulic conductivity fields generated using samples with the highest probability densities Jiachuan He (UT Austin) Spring 2016 group meeting April 15, / 22

21 References Ghanem, Roger G., and Pol D. Spanos. Stochastic finite elements: a spectral approach. Courier Corporation, Breidt, J., T. Butler, and D. Estep. A measure-theoretic computational method for inverse sensitivity problems I: Method and analysis. SIAM journal on numerical analysis 49.5 (2011): Butler, T., D. Estep, and J. Sandelin. A computational measure theoretic approach to inverse sensitivity problems II: a posteriori error analysis. SIAM journal on numerical analysis 50.1 (2012): Butler, T., et al. A measure-theoretic computational method for inverse sensitivity problems III: Multiple quantities of interest. SIAM/ASA Journal on Uncertainty Quantification 2.1 (2014): Butler, Troy, et al. Solving stochastic inverse problems using sigma-algebras on contour maps. arxiv preprint arxiv: (2014). Zhang, Dongxiao, and Zhiming Lu. An efficient, high-order perturbation approach for flow in random porous media via KarhunenLoeve and polynomial expansions. Journal of Computational Physics (2004): Jiachuan He (UT Austin) Spring 2016 group meeting April 15, / 22

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