Structure Function Sampling Uncertainty: A Gaussian-Increment Reference Framework

Andrey Y. Shcherbina 

Structure functions (SFs) are widely used to characterize scale-dependent variability in geophysical fields, but their sampling uncertainty is difficult to quantify because pairwise increments are generally dependent.
Dependence arises from both shared observations and spatial covariance of the underlying field, so pair counts alone do not determine uncertainty. The sampling variance is governed by covariance among increment powers, termed structure-function contributions (SFCs).
For the second-order SF, a Gaussian-increment closure expresses SFC covariance in terms of the SF itself, allowing fractional sampling variance to be calculated from a specified second-order model and sampling geometry. For regular one-dimensional sampling with a finite integral SFC correlation scale, the well-resolved, long-record limit gives approximately one effective degree of freedom per correlation scale sampled. The dependence of this scale on separation yields simple uncertainty scalings for finite-scale and scale-free models. In particular, for spectra near $k^{-2}$, the fractional standard error scales as the inverse square root of the number of separation-length intervals sampled. Sampling variance can be used to construct approximate confidence intervals, while cross-scale covariance allows uncertainty to be propagated to derived quantities such as scaling exponents.
Departures from the Gaussian reference enter through an additional fourth-order cumulant contribution that is more difficult to quantify. The Gaussian result thus provides a tractable baseline that explicitly accounts for sampling geometry and spatial dependence.