Propagation of variance across spatially averaged Gaussian and generalized sub-Gaussian random fields

Spatial heterogeneity of hydrogeological attributes has traditionally been described using Gaussian models. However, the generalized sub-Gaussian (GSG) model captures heavy-tailed frequency distributions and leptokurtic scaling of increment distributions across a wide range of hydrogeological variables. The statistical characteristics defining spatial heterogeneity of properties, such as permeability, depend on observation scale. In this study, we investigate the scale dependency of the GSG model parameters and examine how uncertainty propagates through random fields across a range of observation scales, defined by varying spatial averaging lengths. We establish theoretical expressions describing how the variance of a GSG random field varies with spatial averaging. These formulations enable estimation of heterogeneity at a particular scale, resulting from averaging at a reference scale. Our results reveal that heterogeneity in GSG fields is highest at the finest scale and diminishes as averaging volume increases. The resulting multi-scale variance model allows efficient estimation of residual heterogeneity at larger scales and is valuable in developing computationally efficient coarse-grained hydrogeological models. To validate our approach, we compare the analytic formulae with Monte Carlo (MC) estimates from synthetic data and a multi-scale experimental dataset comprising over 2000 air permeability measurements collected at various scales on the surface of a Massilon sandstone sample. In all cases, we find excellent agreement (largest error of less than 5%), suggesting that the approach is robust.