Simulation of Multivariate Gaussian Fields Conditioned by Realizations of the Fields and Their Derivatives

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Abstract
This paper investigates conditional simulation technique of multivariate Gaussian random fields by stochastic interpolation technique. For the first time in the literature a situation is studied when the random fields are conditioned not only by a set of realizations of the fields, but also by a set of realizations of their derivatives. The kriging estimate of multivariate Gaussian field is proposed, which takes into account both the random field as well as its derivative. Special conditions are imposed on the kriging estimate to determine the kriging weights. Basic formulation for simulation of conditioned multivariate random fields is established. As a particular case of uncorrelated components of multivariate field without realizations of the derivative of the random field, the present formulation includes that of univariate field given by Hoshiya. Examples of a univariate field and a three component field are elucidated and some numerical results are discussed. It is concluded that the information on the derivatives may significantly alter the results of the conditional simulation.