Unilateral ARMA processes on a square net by the herringbone method

Because multidimensional ARMA processes have great potential for the simulation of geological parameters such as aquifer permeability, it was important to resolve which of two proposed alternative methods should be used for determining the two-dimensional weighting parameter, , for a unilateral ARMA (1, 0) process on a square net. Practical simulations demonstrates that the correct formulation is: =₁₀/(1+ ₁₀ ² where _r,s is the correlation between lattice points at lagsr and s. When the simulations are performed with correlations of 0.8 or more a residual bias was detected which was found to be caused by a difference in the variance between the one- and two-dimensional models. This can be rectified by modifying the two- dimensional model as follows: z_ij=(z_{i–1, j} + z_{i, j–1}) + a_ij where ²=1/(1 + ₁₀ ² ).