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

SUMMARY The 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 toght this problem consists of performing stochastic conditional simulations. That is, instead of seeking a single solution (conditional estimation), one obtains an ensemble ofelds, all of which honour the small scale variability (high frequencyuctuations) and direct measurements. The high frequency component of the �eld is dierent from one simulation to another, but axed component for all of them. Measurements of the dependent state variables are honoured by framing simulation as an inverse problem, where both modelt and parameter plausibility are maximized with respect to the coecients of the basis functions (pilot point values). These coecients (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 identication of the spatial variability of the unknown �eld function and that the weight assigned to the plausibility term does lead to optimal results both for conditional estimation and for stochastic simulations. Copyright ? 2006 John Wiley & Sons, Ltd.