Statistical Inference of Spatial Random Functions

A methodology to infer the spatial distribution of the statistical moments of a stationary normal random spatial function based on a finite set of measured values is described. The uncertainty of estimation of the parameters characterizing the probability density function (pdf) is incorporated in the process of inferring and conditioning. Relationships generalizing the traditional approach of stochastic interpolation and simulation have been derived. They yield: (i) an estimate of the conditional expectation; (ii) an estimate of the conditional multivariate normal (MVN) covariance and variance; and (iii) a new quantity, the variance of the estimated variance. The method is illustrated for a two‐dimensional synthetic example with both estimated mean and variance regarded as random. A set of 60 points was generated within a square of sides equal to unity from a MVN population with zero mean, unity variance and exponential covariance with a linear integral scale of 0.3. Contour maps of the inferred expected values and variances are drawn and compared with maps of kriged values and maps of kriged variances. Contour maps of variances of estimation of the variance are given to demonstrate the uncertainty of variance estimates. The unconditional and conditional statistical moments of the space average (block value), , were also evaluated. The reduction of the uncertainty of the inferred random variable as the number of measured points increases is demonstrated with the same synthetic example.