The Stochastic Inverse Problem in Groundwater Hydrology: A Kalman Filter Approach

Size: px

Start display at page:

Download "The Stochastic Inverse Problem in Groundwater Hydrology: A Kalman Filter Approach"

Marybeth Paul
5 years ago
Views:

1 The Stochastic Inverse Problem in Groundwater Hydrology: A Kalman Filter Approach Franz Konecny Institute of Mathematics and Applied Statistics C/muerszZ?/ o/ Agrzcw/Zwm/ ^czences, Fzermo /W J#0 Ore^or Men(fe/-^m^e gg, Fzenna, /l^^rm konecny@mail.boku.ac.at Abstract In this paper we are concerned with the identification of the hydraulic conductivity field on the basis of noisy head observations. An augmented Kalman smoother is applied to the stochastic flow equation. The implementation of the smoother requires the solution of a Riccati type partial differential equation for the smoother covariance and a stochastic partial differential equation to obtain the optimal estimate of the augmented state. We discussfinitedimensional approximations of the smoothing problem and propose an EM algorithm for the estimation of the conductivity parameters. 1 Introduction Groundwater models are used as tools for decision making in water resources management. The accuracy of prediction depends of the reliability of the parameter estimates. In this paper we are mainly concerned with the identification of a hydraulic conductivity field from noisy head data. The Kalman filter (or Kalman-Bucyfilter)is the optimal estimator of the state of a linear system at the current time, given noisy measurements. By augmentation of the state space, we are able not only to estimate the hydraulic heads as the state of the flow equation, but also the conductivity field. In several papers (reviewed in Ginn [4] and Me Laughlin and Townley [6]) extended Kalmanfilteringwas

2 298 Computer Methods in Water Resources XII used for Bayes estimation of the parameters in the context of the deterministic inverse problem. Kalman smoothing refers to estimating the state of a linear system, given noisy measurements of a fixed time interval. In [8], Riedel was concerned with the evolution of the temperature, where the diffusivity was an unobservable time dependent randomfield.he pointed out that inverse problems are generally analyzed off-line and therefore the Kalman smoother is more appropriate then the Kalman filter. In this paper, we adopt this approach and combine it with the EM algorithm. This algorithm was introduced by Dempster et al. [3] as a general framework for handling incomplete data problems. In our problem we use the EM algorithm for the estimation of unknown system parameters. We are unaware of any usage of the EM algorithm in the context of distributed parameter systems. 2 The mathematical model We consider a groundwater flow through a porous, heterogenous, isotropic medium. The equation governing the flow is = V (#V6) 4- *(f, z) + f (f, z), zed, (1) where h hydraulic head, S specific storage coefficient, K hydraulic conductivity, < = deterministic source term, (= stochastic forcing term and D flow region. The equation may be adapted to ID, 2D or 3D-flow, subject to initial and boundary conditions: h(t,x) = hi(t,x), O I xedd,0<t<tf (2) (Z,x) = &2(<,x), xe(9d,0 <Z <*/, where bo,/%i,&2 are given functions. The log-conductivity y(x) = log/i(x) is assumed to be a homogeneous and isotropic Gaussian randomfield,which can be written as Y(x) = ^y+^(x), (3) where //y = E[y(x)] and y(x) is the fluctuation part with E[y(x)] = 0. In the literature, the following covariance function is often used to represent the spatial variability of hydraulic conductivity: (4)

3 Computer Methods in Water Resources XII 299 0y is the variance of Y and ly the correlation length. We assume the stochastic forcing term (*, x) representing the modeling error is a distributed Gaussian white noise process with the following properties: )] = 0 (5) Then the hydraulic heads ft(*,x) are also random. We define ft(*,x) = E[h(t,x)] and h(t,x) = h(t,x) -h(t,x). Neclecting nonlinear terms Dagan [1] obtained the following equation with ko = e^ subject to the same initial and boundary conditions as equation (1). The linearized evolution equation for the head fluctuations is r\ T ^ a7 with the homogenous boundary condition and the initial condition A(Z,x) = 0, xe<9d, (8) ^(0,x) = ((x), (9) where ((%) also is Gaussian noise with E[((x)] = 0 and the covariance function PO(X,X') =E[((x)((x')]. An important task is the estimation of the unknown values of /iy,0y and ly. Usually only few measured values of the conductivity are available, which cannot provide reliable estimates of the unknown parameters. Therefore estimation has to be based on the measurements of the heads. For this task we augment the state space by adjunction of the auxiliar equation Tt = (io) with the initial condition /(0,x) = y(x). (11)

4 300 Computer Methods in Water Resources XII We rewrite eqs. (6) and (9) using the augmented state vector u = (hjf where u = ((, 0)^ and ^ = A(t,x)u(t,x) + ^(t,x), (12) ot b Let us assume that the head measurements are taken at fixed points x*,...,x^ E D, the measurement times are fast relative to the characteristic groundwater flow evolution time and are well modeled by a continuous measurement process. We define a m- vector hm(t) = Then, the measurement process is given by o where rj(t) is the measurement noise, a m-dimensional Gaussian white noise stochastically independent of (t,x) and ("(x). 3 Estimation theory We use the theory of optimalfilteringfor distributed parameter systems (d.p.s.) to our groundwater estimation problem. In our setting we have a linear state equation and Gaussian system- and measurement noise. Therefore we are in the realm of the Kalman filter. We follow the formal approach to optimalfilteringof Omatu and Seinfeld [7,ch. 10]. The Kalman-Filter u(t) is given by the filter equation (14) where F(t,x)=pM(t,x)R-*(t), The error-covariance function P(t,x,x') satisfies the p.d.e.(14). ot A(t, x)p(t, x, x') + A(t, x')p(t, x, x'} ), (15) P(0,x,x') = PQ(X,X').

5 Computer Methods in Water Resources XII 301 For the details of the implementation we refer to [7, p. 319]. Since we are concerned with an off-line estimation problem, we shall use the fixed interval Kalman smoother. It is an optimal estimate u(t\tf) of the state vector u(tf), based on the measurement data ZQ' = (z(f),0 <t< if). For the equations representing the estimate u(t\tf) and its variance P(J fy,x,x') we refer to Omatu and Seinfeld [7]- 4 Numerical Implementation Since the state space of a d.p.s. is an infinite-dimensional function space a finite-dimensional approximation is required. Following Seinfeld and Koda [9] we approximate the d.p.s. by a lumped parameter system at the very beginning of the problem. Then we use the conventional estimation theory forfinite-dimensionalsystems. A Galerkin approximation to (6) and (9) is constructed as follows: TV where (<^(x)} is a set of basis functions, satisfying the boundary conditions (7). The random field t/(x) may be approximated by a truncated Karhunen-Loeve expansion (cf. Wong [10]) N 77 = 1 We use the inner product notation of functions J U The expansions coefficients Hn(t) and Fn(t) are determined by the Galerkin equations (Seinfeld and Koda [9]) and / -V ^ ' ' A4 ^ '

6 302 Computer Methods in Water Resources XII This a linear system of ordinary differential equations. We set U = (H, F)T and define N x N matrices as Hence (17-18) can be written as (20) To (19) the conventional Kalman filter and Kalman smoother may be applied. If the domain is subdivided into triangular elements and & is taken as a shaping function, the procedure is the finite element method ) & Time step dt = Fig.l: Covariance function of the Kalman filter (dotted) and Kalman smoother 5 Numerical Example In order to test the estimation procedures under considerations we performed a numerical experiment. We consider a ID - flow with the domain D = [0,1], the source term 050, the boundary condition Ar(Z,0) = Ar(Z,l) = 0 and the initial condition Ao(z) = cos(trz).

7 Computer Methods in Water Resources XII 303 Then we have A(Z,z) = e-(^/*)^cos(7rz). The variances of f(z,z) and rn(t) are q = 0.5 and r = 0.1, respectively. We use a Galerkin approximation of the order TV = 3 with ^(x) = y/2sm(rnrx) and ipn(x) = Uncosvnx + sinunx, where the u^ are the positive solutions of the transcendental equation tanu = 2w/(w^ - 1). (For details concerning the Karhunen Loeve basis see Konecny [5]). Since {</> } and {ipn} are orthonormal function systems, we have d = Ci - 7, the identity matrix of order 3. Furthermore we choose 3 measurement points %I = 0.25, z% = 0.5, a^ = We obtained approximations of the optimal filter and smoother by application of the standard schemes (cf. Riedel [8],(2.4-5) and (2.9-12)) on the lumped system (19). Fig. 1 shows the error covariance functions in the measurement point x\ = 0.5 of the Kalman filter and the Kalman smoother. Fig. 2 shows a typical estimate of the log-conductivity Y(x) obtained by Kalman smoothing together with its nominal 95%-confidence band "1 0.9 "0 0, x Fig. 2: Smoothed log-conductivity with 95%-confidence bands 6 The EM Algorithm For the calibration of a groundwater model the conductivity parameters are to be estimated from the head measurements. So we are faced with a parameter estimation problem with partial observations, which can be approached by a combination of the EM algorithm and Kalman smoothing (cf. Dembo and Zeitouni [2]). An interesting feature of this iterative algorithm is that it generates a maximizing sequence of estimates in the sense that the corresponding sequence

8 304 Computer Methods in Water Resources XII of likelihood function values is monotone increasing. The procedure in general involves iterations of Kalman smoothing with known parameters and then a deterministic maximization. For a detailed description of the algorithm we refer to our working paper [5]. 7 Conclusion In this paper, we have presented an application of the augmented Kalman smoother to the stochastic inverse problem in groundwater hydrology. It has been applied to a linearized form of the flow equation augmented by a stochastic forcing term representing the model error. We have outlined the estimation of the conductivity parameters by an extension of the EM algorithm. References [1] Dagan, G.: Flow and Transport in Pourous Formations. New York: Springer-Verlag, [2] Dembo, A. and Zeitouni, O.: Parameter estimation of partially observed continuous time stochastic processes via the EM algorithm. Stochastic Processes and Applications 23 (1), , [3] Dempster, A. P., Laird, N. M. and Rubin, D. B.: Maximum Likelihood from Incomplete Data via the EM Algorithm. J. R. Statist. Soc. B 39, 1-38, [4] Ginn, T. R. and Cushman, J. H.: Inverse methods for subsurface flow: A critical review of stochastic techniques. Stochastic Hydro!. ##W. 4, 1-26, [5] Konecny, F.: The Stochastic Inverse Problem in Groundwater Hydrology: A Kalmanfilter-smootherApproach. Working paper, Institute of Mathematics and Applied Statistics, University of Agricultural Sciences, Vienna, [6] McLaughlin, D. and Town ley, L. R.: A reassessment of the groundwater inverse problem. Water Resour. Res. 32(5), , [7] Omatu, S. and J.H. Seinfeld: Distributed Parameter Systems, Theory and Applications. Oxford: Clarendon Press, [8] Riedel, K. S.: Optimal Estimation of Dynamically Evolving Diffusivities. Journal of Computational Physics 115, 1-11, [9] Seinfeld, J. H. and Koda, M.: Numerical Implementation of Distributed Parameter Filters with Application to Problems in Air Pollution. In: Distributed Parameter Systems: Modelling and Identification. Lecture Notes in Control and Information Sciences, vol. 1. New York: Springer-Verlag, [10] Wong, E.: Stochastic Processes in Information and Dynamic Systems. New York: McGraw-Hill, 1971.

State-space Model. Eduardo Rossi University of Pavia. November Rossi State-space Model Financial Econometrics / 49

State-space Model. Eduardo Rossi University of Pavia. November Rossi State-space Model Financial Econometrics / 49 State-space Model Eduardo Rossi University of Pavia November 2013 Rossi State-space Model Financial Econometrics - 2013 1 / 49 Outline 1 Introduction 2 The Kalman filter 3 Forecast errors 4 State smoothing