Towards the Application of Complex-Valued Variograms in Soil Research

Variograms are a cornerstone of spatial analysis in geostatistics, traditionally applied to real-valued variables under the intrinsic hypothesis. Many soil properties, particularly when integrating magnetic and geochemical measurements, can be expressed as complex-valued variables that capture both magnitude and phase information. In the case of magnetic susceptibility, the imaginary component reflects energy losses associated with viscous magnetization, which in soils can indicate the presence of pedogenic ferrimagnetic minerals, while its relative increase may also reveal anthropogenic magnetite contamination. This study examines the formulation and application of variograms for such complex-valued variables in the context of soil research. Two complementary definitions are considered: an intrinsic-based approach, which directly estimates the variogram from increments and is applicable under the intrinsic hypothesis, and a covariance-based approach, which requires stronger second-order stationarity. Simulated complex-valued soil property data with controlled spatial structures were used to compare the behaviour of these formulations with their real-valued counterparts. The findings indicate that complex-valued variograms preserve additional spatial information, particularly related to local phase shifts, while maintaining compatibility with conventional variographic modelling.