This paper presents practical methods for the sequential generation or simulation of a Gaussian two-dimensional random field. The specific realizations typically correspond to geospatial errors or perturbations over a horizontal plane or grid. The errors are either scalar, such as vertical errors, or multivariate, such as , , and errors. These realizations enable simulation-based performance assessment and tuning of various geospatial applications. Both homogeneous and non-homogeneous random fields are addressed. The sequential generation is very fast and compared to methods based on Cholesky decomposition of an a priori covariance matrix and Sequential Gaussian Simulation. The multi-grid point covariance matrix is also developed for all the above random fields, essential for the optimal performance of many geospatial applications ingesting data with these types of errors.
Sufficient conditions for strictly positive definite correlation functions are developed. These functions are associated with wide-sense stationary stochastic processes and provide practical models for various errors affecting tracking, fusion, and general estimation problems. In particular, the expected magnitude and temporal correlation of a stochastic error process are modeled such that the covariance matrix corresponding to a set of errors sampled (measured) at different times is positive definite (invertible) -a necessary condition for many applications. The covariance matrix is generated using the strictly positive definite correlation function and the sample times. As a related benefit, a large covariance matrix can be naturally compressed for storage and dissemination by a few parameters that define the specific correlation function and the sample times. Results are extended to wide-sense homogeneous multi-variate (vector-valued) random fields. Corresponding strictly positive definite correlation functions can statistically model fiducial (control point) errors including their inter-fiducial spatial correlations. If an estimator does not model correlations, its estimates are not optimal, its corresponding accuracy estimates (a posteriori error covariance) are unreliable, and it may diverge. Finally, results are extended to approximate error covariance matrices corresponding to non-homogeneous, multi-variate random fields (a generalization of non-stationary stochastic processes). Examples of strictly positive definite correlation functions and corresponding error covariance matrices are provided throughout the paper.
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