Estimation in the Presence of Outliers: The Capillary Pressure Case

Abstract

The inversion of laboratory centrifuge data to obtain capillary pressure functions in petroleum science leads to a Volterra integral equation of the first kind with a right-hand side defined by a set of discrete data. The problem is ill-posed in the sense of Hadamard [4]. The discrete data lead to a discretized equation of the form

$$ A\overrightarrow c = \overrightarrow b + \overrightarrow \varepsilon , $$

, where b^→ represents the observation vector, A is an ill-conditioned matrix derived from the forward problem, c^→ is the coefficients in a representation of the inverse capillary function, i.e., parameters to be determined, and ∈^→ is the error vector associated with b^→. If ∈^→∼N(0,σ²), and satisfies the Gauss-Markov (G-M) conditions, then an estimate, c^→ _λ, of c^→ is BLUE [9]. In the presence of outliers, the G-M conditions and/or the normality assumption can be violated.

In this paper we parameterize the capillary pressure function using B-splines and address the issue of ill-posedness by reformulating the problem as a constrained optimization task involving the determination of the spline coefficients. By the nature of the experimental procedure, we expect the G-M conditions to be satisfied. A systematic method of outlier elimination and a choice of knots is employed to ensure satisfaction of the normality assumption and thereby derive capillary pressure curves to a high degree of accuracy. A robust method for estimating the solution curve, which accommodates both outliers and influential points, namely the L ₁-norm solution, is also presented. The method is demonstrated on synthetic data.