Steepest descent method for representing spatially correlated uncertainty in GIS

All spatial data in a geographic information system (GIS) intrinsically contain uncertainty. Simulations could be used for many GIS applications in order to estimate confidence ranges of certain analyses and project worst-case scenarios. For those applications, generation of Gaussian random fields is essential to simulate the uncertainty effects, because errors in spatial data are assumed dependent upon the Gaussian distribution. Gaussian fields with no spatial dependency could be assumed because of their simple concept and easy computation, but the reality is that spatial errors have a spatially correlated nature. For this reason, the intensive matrix computation for generating spatially autocorrelated Gaussian random fields requires the solution of a large, sparse linear system: X = ρ WX + ε. There has been substantial development of direct and iterative methods for solving a large, sparse linear system. In this research, those methods are presented and compared in terms of computation complexity for the particular system. The writer presents the steepest descent method as the best possible method with linear complexity.