Identification of Conditional Stochastic Gaussian Field

A general formulation is presented based on the maximum likelihood method to identify the best estimator of a stochastic Gaussian field when the observation is made at discrete spatial points. The uncertainty of the estimator in terms of the uncertainty of the stochastic field and in the observation noise are discussed, and the updating of the estimator's mechanism by observation is clarified. Two methodologies, that is, a simple kriging method and an extended Kalman filtering procedure are derived as special solutions of the general formulation. The method developed here may be applied to updating a general stochastic finite-element method (FEM) system by observation, where the system consists of a stochastic field of elastic moduli or loading forces or both, and the observation may be for elastic moduli and/or displacements at finite spatial points. A numerical example is carried out with a beam on a continuous elastic foundation, and the efficiency and practical applicability of the method are demonstrated.