Estimation of a Piecewise Constant Function
Using Reparameterized Level-Set Functions
Inga Berre1,2 , Martha Lien2,1 , and Trond Mannseth2,1
1
2
Department of Mathematics, University of Bergen, Johs. Brunsgt. 12, N-5008
Bergen, Norway.
CIPR - Centre for Integrated Petroleum Research, University of Bergen,
Realfagbygget, Allégaten 41, N-5007 Bergen, Norway.
[email protected], [email protected], and
[email protected]
Abstract. In 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. In 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 Piecewise Constant
Parameter Function
The identification of a piecewise constant parameter function is an inverse
problem arising in various applications. Examples 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 flexible 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. If we consider a domain Ω
2
Inga Berre, Martha Lien, and Trond Mannseth
consisting of two regions Ω1 and Ω2 , the level-set function is defined to have
the following properties:
φ(x) > 0 for x ∈ Ω1 ;
φ(x) < 0 for x ∈ Ω2 ;
φ(x) = 0 for x ∈ ∂Ω1 ∩ ∂Ω2 .
Hence, the boundary between the regions Ω1 and Ω2 is incorporated as the
zero level-set of φ(x). A parameter function p(x) taking different constant
values c1 in Ω1 and c2 in Ω2 can now be written as
p(x) = c1 H(φ(x)) + c2 [1 − H(φ(x))],
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 extension of the
above idea, which enables representation of up to 2l 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 example is the inverse problem of fluid conductivity
estimation for oil reservoirs [1, 5].
A reparameterization of the level-set function is written
φ(x) =
n
X
ai θi (x),
i=1
where {θi } denotes the set of basis functions and {ai } 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. In 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
Parameter 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. In 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. In 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. In 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(x) 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(x), where the jumps are determined by the spatial support of the
basis functions for φ(x), 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 θ1 = [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 θ1 and θ2 being the
standard linear basis on [0, 1].
As a simple example of a continuous reparameterization of φ(x) 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
Dr = [0, 1] × [0, 1] is given by
φ(x1 , x2 ) = a1 (x1 − 1)(x2 − 1) − a2 (x1 − 1)x2 − a3 x1 (x2 − 1) + a4 x1 x2 .
By increasing the resolution of the quadrilateral grid, the boundaries can have
more complex 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
Inga Berre, Martha Lien, and Trond Mannseth
φ
φ
a1
a1
0
a2
1
x
p
c1
c2
0
0
a2
1
x
p
c1
c2
1
x
0
1
x
Fig. 1. Coarse representation of φ with two coefficients, a1 and a2 , by characteristic
(top left) and linear (top right) basis functions. Optimizing with respect to a1 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 (I)
and four production (P ) 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(x),
and C denotes the (diagonal) covariance matrix 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(x) 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.
Parameter estimation using level-set functions
I
I
I
I
4.7
30
I
I
5
4.7
30
4.6
P
25
P
4.5
4.6
P
25
P
4.5
4.4
20
4.3
I
I
I
4.4
20
15
4.1
4.3
I
4.2
I
I
15
4.1
4
10
P
4
10
P
3.8
I
I
5
10
15
I
20
P
3.9
5
25
4.2
3.7
P
3.9
3.8
5
I
30
I
5
10
I
15
20
25
3.7
30
Fig. 2. Reference fluid conductivity and final estimation result.
1.5
1
0.5
0
−0.5
−1
−1.5
−2
−2.5
−3
30
20
10
5
10
15
20
25
30
Fig. 3. Level-set function at convergence together with the surface z = 0.
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