Inversion of heterogeneous parabolic‐type equations using the pilot points method

Abstract

SUMMARYThe inverse problem (also referred to as parameter estimation) consists of evaluating the medium properties ruling the behaviour of a given equation from direct measurements of those properties and of the dependent state variables. The problem becomes ill-posed when the properties vary spatially in an unknown manner, which is often the case when modelling natural processes. A possibility to ÿght this problem consists of performing stochastic conditional simulations. That is, instead of seeking a single solution (conditional estimation), one obtains an ensemble of ÿelds, all of which honour the small scale variability (high frequency uctuations) and direct measurements. The high frequency component of the ÿeld is di erent from one simulation to another, but a ÿxed component for all of them. Measurements of the dependent state variables are honoured by framing simulation as an inverse problem, where both model ÿt and parameter plausibility are maximized with respect to the coe cients of the basis functions (pilot point values). These coe cients (model parameters) are used for parameterizing the large scale variability patterns. The pilot points method, which is often used in hydrogeology, uses the kriging weights as basis functions. The performance of the method (both its variants of conditional estimation=simulation) is tested on a synthetic example using a parabolic-type equation. Results show that including the plausibility term improves the identiÿcation of the spatial variability of the unknown ÿeld function and that the weight assigned to the plausibility term does lead to optimal results both for conditional estimation and for stochastic simulations.

Regularized pilot points method for reproducing the effect of small scale variability: Application to simulations of contaminant transport

“…These weights were always 1.0, both for the term of state variables and for the plausibility term. This finding confirms the results of Alcolea et al (2006b).…”

Stochastic Inverse Methods to Identify non-Gaussian Model Parameters in Heterogeneous Aquifers

“…It was soon realized that the concept of the SCM could be implemented in the original PiPM, and it has been applied multiple times since then (e.g., Alcolea et al, 2006Alcolea et al, , 2008Lavenue and De Marsily, 2001). Gómez-Hernández et al (2003) reviewed stochastic conditional inverse modeling showing the strengths of SCM.…”

“…The update process for the PiPM is illustrated in " Figure 1" of RamaRao et al (1995) and in " Figure 1" of Alcolea et al (2006). For the sake of completeness and comparison, we graphically show an sketch of the update algorithm for all the three methods (Figure 2.1).…”

“…In this respect, Neuman (1973) proposed the multiple-objective algorithm in which the model parameters are constrained not only by minimization of the reproduction error but also by a physical plausibility criterion. A similar strategy is used by Carrera and Neuman (1986a) and Medina and Carrera (1996) in the MLM, in which prior information is combined into the likelihood functions, or by Alcolea et al (2006) in the PiPM, in which a regularization term is added to the objective function.…”

“…Each realization represents a possible case of the unknown reality. Flow and transport are modeled in each realization and all model predictions are collected to characterize uncertainty and to define an optimal prediction (e.g., pilot point method (Alcolea et al, 2006;RamaRao et al, 1995), self-calibration and gradual deformation ).…”