Estimation of a Piecewise Constant Function Using Reparameterized Level-Set Functions

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1 Estimation of a iecewise Constant Function Using Reparameterized Level-Set Functions nga Berre 1,2, Martha Lien 2,1, and Trond Mannseth 2,1 1 Department of Mathematics, University of Bergen, Johs. Brunsgt. 12, N-5008 Bergen, Norway. 2 CR - Centre for ntegrated etroleum Research, University of Bergen, Realfagbygget, Allégaten 41, N-5007 Bergen, Norway. nga.berre@mi.uib.no, Martha.Lien@cipr.uib.no, and Trond.Mannseth@cipr.uib.no Abstract. n the last decade, the use of level-set functions has gained increasing popularity in solving inverse problems involving the identification of a piecewise constant function. Normally, a fine-scale representation of the level-set functions is used, yielding a high number of degrees of freedom in the estimation. n contrast, we focus on reparameterization of the level-set functions on a coarse scale. The number of coefficients in the discretized function is then reduced, providing necessary regularization for solving ill-posed problems. A coarse representation is also advantageous to reduce the computational work in solving the estimation problem. 1 Level-Set Representation of a iecewise Constant arameter Function The identification of a piecewise constant parameter function is an inverse problem arising in various applications. Eamples are image segmentation and identification of electric or fluid conductivity in reservoirs. Three features of the function can potentially be unknown: the number of regions of different constant value, the geometry of the regions, and the constant values of the parameter function. For representing piecewise constant functions in a manner suitable for identification, the level set approach [6] has become popular as it provides a fleible tool to represent region boundaries. The first approach related to inverse problems is due to Santosa [7]. Recent reviews are given by Tai and Chan [8], Burger and Osher [2], and Dorn and Lesselier [3]. The main idea is that the boundary between two regions can be represented implicitly as the zero level set of a function the level-set function. f we consider a domain Ω

2 2 nga Berre, Martha Lien, and Trond Mannseth consisting of two regions Ω 1 and Ω 2, the level-set function is defined to have the following properties: φ() > 0 for Ω 1 ; φ() < 0 for Ω 2 ; φ() = 0 for Ω 1 Ω 2. Hence, the boundary between the regions Ω 1 and Ω 2 is incorporated as the zero level-set of φ(). A parameter function p() taking different constant values c 1 in Ω 1 and c 2 in Ω 2 can now be written as p() = c 1 H(φ()) + c 2 [1 H(φ())], where H is the Heaviside function. Commonly, the level-set functions are initialized as signed distance functions. For representation of a partitioning with more than two regions, Vese and Chan [9] provide an etension of the above idea, which enables representation of up to 2 l regions with l level-set functions. 2 Coarse-Scale Level-Set Representation Usually, level-set functions are represented with one degree of freedom for each grid cell of the computational grid. We apply an approach based on a coarse-scale representation, where each level-set function is reparameterized by a few coefficients only. This approach has several advantages: the reduced number of coefficients makes sensitivity computations less demanding; the need for regularization is diminished since the possible variations in the region boundaries are more limited; and, we can achieve convergence in a low number of iterations. The latter is particularly important in solving inverse problems requiring computationally demanding forward computations and sensitivity calculations. A prime eample is the inverse problem of fluid conductivity estimation for oil reservoirs [1, 5]. A reparameterization of the level-set function is written φ() = n a i θ i (), i=1 where {θ i } denotes the set of basis functions and {a i } denotes the set of coefficients in the discretization. A coarse representation of the level-set functions enables fast identification of coarse-scale features of the parameter function with a low number of estimated coefficients. n addition, the approach provides regularization as the boundaries are confined to certain shapes based on the chosen representation. For fine-scale solutions of inverse problems, the coarse representation can enhance convergence by serving as a preconditioner for

3 arameter estimation using level-set functions 3 more fine-scale updates. Another strategy is to apply a sequential estimation, where the detail in the representation of the level-set functions is successively refined. The number of degrees of freedom in the estimation is then gradually increased, and we can achieve estimates more in correspondence with the information content of the data. When it comes to the choice of basis in the representation of the levelset functions, various alternatives are possible. However, once a set of basis functions is chosen, the reparameterized functions are restricted with respect to which shapes they can take. n the following, we discuss two different choices: a piecewise constant representation and a continuous representation. A piecewise constant representation of the level-set functions [1, 5] is well suited in combination with adaptive multiscale estimation [4], which provides a rough identification of the number and geometry of the regions of different constant value for the parameter function. n Berre et al. [1], reparameterization by characteristic basis functions in combination with a narrow-band approach is proposed. The representation of the region boundaries is gradually refined during estimation. n Lien et al. [5], a sequential approach is developed, where the number of regions of constant parameter value is sought found as part of the estimation. Through successive estimations with increasing resolution in the representation of p() identification of rather general parameter functions can be achieved. The choice of characteristic basis functions yields piecewise constant levelset functions. This leads to updates of the boundaries between the constant states of p(), where the jumps are determined by the spatial support of the basis functions for φ(), regardless of the resolution of the computational grid for the forward problem. The two plots to the left in Figure 1 illustrate this situation in 1-D with n = 2, supp θ 1 = [0, 1/2), and supp θ 2 = [1/2, 1]. A continuous representation of the level-set functions enables more gradual updates of the region boundaries, depending on the resolution of the computational grid for the forward problem. Hence, the number of possible updates of the region boundaries is greatly enhanced. The two plots to the right in Figure 1 illustrate this situation in 1-D with n = 2 and θ 1 and θ 2 being the standard linear basis on [0, 1]. As a simple eample of a continuous reparameterization of φ() with a low number of coefficients in 2-D, we consider a bilinear basis. The computational domain is partitioned into a coarse quadrilateral grid of rectangular elements. A bilinear reparameterization of a level-set function on a reference element D r = [0, 1] [0, 1] is given by φ( 1, 2 ) = a 1 ( 1 1)( 2 1) a 2 ( 1 1) 2 a 3 1 ( 2 1) + a By increasing the resolution of the quadrilateral grid, the boundaries can have more comple shapes. An illustration of the potential of applying a coarse-scale reparameterization of the level-set functions is given below; see Figures 2 and 3. We consider the inverse problem of fluid conductivity estimation for an oil reservoir based

4 4 nga Berre, Martha Lien, and Trond Mannseth φ φ a 1 a 1 a 2 a 2 p c 1 c 2 p c 1 c 2 Fig. 1. Coarse representation of φ with two coefficients, a 1 and a 2, by characteristic (top left) and linear (top right) basis functions. Optimizing with respect to a 1 cause φ to change sign in some area. The resulting updates of the parameter function are shown in the lower figures. The initial states are drawn by solid lines and the final states by dashed lines. on a level-set representation with bilinear basis functions. The forward problem describes horizontal, two-phase, immiscible, and incompressible fluid flow in porous media. The available data for the inversion consist of time series of pressure data d logged in the injection wells. There are nine injection () and four production () wells; hence, the data are very sparsely distributed. The data are obtained from a forward simulation with the reference fluid conductivity, where normally distributed errors are added to the calculated pressures. To illustrate the regularizing effect of the coarse level-set representation, we minimize a weighted least squares objective function with no additional regularizing terms: J(p) = [m(p) d] T C 1 [m(p) d]. Here m(p) denote the calculated pressures given a parameter function p(), and C denotes the (diagonal) covariance matri of the measurement errors in the data. The reference fluid conductivity describes a channel with areas of low conductivity on both sides where the coarse-scale features are contaminated with fine-scale conductivity variation; see Figure 2 (left). The initial guess for p() was a constant value for the whole reservoir. The estimation was started with a coarse level-set representation on a grid of only one element before we continued with a quadrilateral grid of four elements. With our coarse level-set approach, we are able to recover the main trends in the conductivity distribution. The data are not reconciled by the coarse estimate though the objective function is greatly reduced. Hence, in this case, the coarse-scale estimate can serve as a good starting point for more fine-scale estimations.

5 arameter estimation using level-set functions Fig. 2. Reference fluid conductivity and final estimation result Fig. 3. Level-set function at convergence together with the surface z = 0. References 1.. Berre, M. Lien, and T. Mannseth. A level-set corrector to an adaptive multiscale permeability prediction. Accepted for publication in Comput. Geosci. 2. M. Burger and S. Osher. A survey on level set methods for inverse problems and optimal design. Euro. Jnl of Applied Mathematics, 16(2):263 1, April O. Dorn and D. Lesselier. Level set methods for inverse scattering. nverse roblems, 22:67 131, A.-A. Grimstad, T. Mannseth, G. Nævdal, and H. Urkedal. Adaptive multiscale permeability estimation. Comput. Geosci., 7(1):1 25, M. Lien,. Berre, and T. Mannseth. Combined adaptive multiscale and level-set parameter estimation. Multiscale Model. Simul., 4(4): , S. Osher and J. A. Sethian. Fronts propagating with curvature-dependent speed: algorithms based on Hamilton-Jacobi formulations. J. Comput. hys., 79(1):12 49, F. Santosa. A level-set approach for inverse problems involving obstacles. ESAM Contrôle Optim. Calc. Var., 1:17 33, 1995/ X.-C. Tai and Tony F. Chan. A survey on multiple level set methods with applications for identifying piecewise constant functions. nt. J. Numer. Anal. Model., 1(1):25 47, L. A. Vese and T. F. Chan. A multiphase level set framework for image segmentation via the Mumford and Shah model. nt. J. of Comput. Vision, 02.