Scaling laws of mean and uncertainty in 3D surface texture measurements

Abstract Taylor’s law, originally formulated in ecology, describes a universal scaling relationship between variance σ p and mean μ p with σ p = e α μ p β . This study investigates the validity of this law in surface metrology as a new method of uncertainty estimation by analyzing areal surface parameters from ISO 25178-2. The amplitude, spatial, slope, curvature, fractal and directional families of parameters are especially analyzed. These parameters are calculated on 3600 topographies measured by interferometry at the same location of a #120 ground TA6V surface for 6 days. By using a two-level bootstrap combined with multi-scale filtering (band-pass, high-pass, and low-pass), we systematically tested the robustness of the law. Results show that intercepts α evolve consistently with filtering strategies, slopes β remain statistically indistinguishable from unity at the 95% confidence level, and goodness-of-fit coefficients (R 2 ) confirm the validity of the power-law across scales. Among parameter families, spatial descriptors exhibit the highest statistical stability, while amplitude and directional parameters provide complementary information: amplitude parameters exhibit moderate dispersion while directional parameters reflect anisotropic surface complexity. These findings demonstrate that Taylor’s law reliably governs surface texture parameter variability and can be used to establish predictive uncertainty laws from a single measurement. This opens perspectives for both industrial applications (process monitoring, quality control, surface engineering, etc) and academic one (non-invasive diagnostics, metrology, historical materials, etc).