Analytical relationship between Gaussian and transformed-Gaussian spatially distributed fields

A transformation of a Gaussian random field results in a change of the spatial dependence. In simple terms, the range and the shape of a variogram are different between an underlying Gaussian and a transformed Gaussian field. There is a relationship between the underlying and the transformed spatial dependence. This paper shows how this relationship can be expressed analytically. Rather than generating multidimensional random fields and conducting spatial statistical analysis, we develop an accurate and efficient approach based on a unique mapping of the correlation coefficients of the original multi-Gaussian fields to the transformed correlation coefficients to evaluate the spatial correlation of transformed non-Gaussian random fields for any type of geostatistical parameterization. Such a mapping can also yield accurate estimation of the spatial correlation of the underlying Gaussian field given the spatial correlation of the transformed field. Results indicate that (1) a nonlinear transformation of spatially distributed fields usually changes the spatial dependence structure; and (2) the relationship between the dependence structures of the underlying and the transformed field can be expressed analytically, and it is sufficient to do this in one dimension. We use the developed approach to investigate the change of correlations of connected random fields generated by the absolute-value transformation. Results show (1) the correlation lengths of the underlying Gaussian fields are 1.67 and 2.64 times of those of transformed non-Gaussian fields for Gaussian and exponential covariance models, respectively; (2) the anisotropic ratio does not change; and (3) the anticorrelation in hole-effect correlation models disappear. Citation: Gong, R., C. P. Haslauer, Y. Chen, and J. Luo (2013), Analytical relationship between Gaussian and transformed-Gaussian spatially distributed fields, Water Resour. Res., 49, 1735-1740, doi:10.1002/wrcr.20143.