Turning Bands in Functional Geostatistics: A Simulation Tool for Spatially Correlated Curves

Abstract

Functional geostatistics enables the spatial prediction of curves, profiles, or time series indexed over space. It assumes that each sampling location is associated with a function rather than a scalar variable. Although a substantial body of literature exists on curve prediction (functional geostatistics and related methods), comparatively fewer contributions have focused on simulation methods capable of generating spatially correlated functional data under controllable dependence structures. Such methods are essential for validating methodological developments, comparing predictors, and quantifying uncertainty. In this work, we propose an extension of the turning bands simulation method to the context of functional geostatistics (TBFG). Curves are represented using a functional basis (e.g., Fourier or spline functions), and the corresponding coefficient vectors are modeled as spatially correlated Gaussian random fields. The turning bands simulation is performed in the coefficient space, after which the curves are reconstructed using the selected functional basis. In the implementation, the one-dimensional processes simulated along the bands employ auxiliary covariance functions whose angular averages reproduce the desired two-dimensional covariance structure. A key feature of TBFG is the inclusion of an explicit spatial range parameter, which allows direct control over the strength of spatial dependence among the simulated curves. Spatial dependence is assessed through the trace-variogram, an extension of the classical variogram defined in the functional space (L2). The simulations show that the range of the tracevariogram increases as the imposed spatial range parameter becomes larger, providing a clear and interpretable diagnostic of the strength of spatial dependence. An appendix includes R code that allows the results to be fully reproduced.