François Blateyron

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).

This study investigates two measurement campaigns: extended time and short time, to determine the stability of roughness measurements, focusing on the Sdr parameter. Extended-time measurements were conducted using the most sensitive instrument available to follow metrological fluctuations. The results revealed that Sdr exhibits the clearest trend and the highest dispersion among all roughness parameters, making it the most relevant indicator for tracking temporal deviations. Other parameters, such as Sa, Sq, and Sds, also emerged as potential candidates. These results were validated through a stability analysis (SI), showing that Sdr is the worst stable roughness parameter. To ensure the robustness of the findings and be closer to the real conditions, a short-time assessment was performed using a dedicated measurement plan performed on multiple instruments. The results confirmed that measurement fluctuations are instrument-dependent, but similar results are found across the same technologies (CSI(S) and CSI(B)). The short-time study included a quality inspection, a drift/stability analysis employing AR (2) models on the time series data systematically and a relevance measurement assessment using ANOVA. The study was conducted using a full-scale roughness analysis and could potentially be applied to a multiscale approach. These findings highlight the ability of Sdr to monitor metrological fluctuation during a long-time acquisition and according to a dedicated measurement plan.

With a view to improve measurements, this paper presents a statistical approach for characterizing the behaviour of roughness parameters based on measurements performed on ground surface topographies (grit #080/#120). A S neox^TM (Sensofar^®, Terrassa, Spain), equipped with three optical instrument modes (Focus Variation (FV), Coherence Scanning Interferometry (CSI), and Confocal Microscopy (CM)), is used according to a specific measurement plan, called Morphomeca Monitoring, including topography representativeness and several time-based measurements. Previously applied to the Sa parameter, the statistical approach based here solely on the Quality Index (QI) has now been extended to a multi-parameter approach. Firstly, the study focuses on detecting and explaining parameter disturbances in raw data by identifying and quantifying outliers of the parameter’s values, as a new first indicator. This allows us to draw parallels between these outliers and the surface topography, providing reflection tracks. Secondly, the statistical approach is applied to highlight disturbed parameters concerning the instrument mode used and the concerned grit level with two other indicators computed from QI, named homogeneity and number of modes. The applied method shows that a cleaning of the data containing the parameters values is necessary to remove outlier values, and a set of roughness parameters could be determined according to the assessment of the indicators. The final aim is to provide a set of parameters which best describe the measurement conditions based on monitoring data, statistical indexes, and surface topographies. It is shown that the parameters Sal, Sz and Sci are the most reliable roughness parameters, unlike Sdq and S5p, which appear as the most unstable parameters. More globally, the volume roughness parameters appear as the most stable, differing from the form parameters. This investigated point of view offers thus a complementary framework for improving measurement processes. In addition, this method aims to provide a global and more generalizable alternative than traditional methods of uncertainty calculation, based on a thorough analysis of multi-parameter and statistical indexes.

Editors of surface texture analysis software need to validate their parameter calculations, ideally in an implementation-independent way. Today, there is no universal reference method for validating parameter compliance and more specifically with regard to material ratio parameters, Rk parameters, and volume parameters that are based on the Abbott curve. Several years ago, at Digital Surf, we developed parameter validation methods that are based on mathematics. A mathematical softgauge, defined by an equation, is injected through the mathematical parameter definition, to obtain a mathematical true value of the parameter value. In a parallel path, a discrete softgauge, derived from the mathematical softgauge, is injected through an algorithm and the resulting value is compared to the approximated true value. This method makes it possible to check the accuracy of an algorithm for calculating a particular parameter. For this study, two families of mathematical softgauges were specifically designed to test material ratio parameters. More precisely, the two main goals of this study are, (1) to advocate an algorithm validation approach that is independent from any software implementation, and (2) to obtain true values for all parameters based on the Abbott curve.

To address the essential problem in surface metrology of establishing functional correlations spatial, frequencies in topographic measurements are progressively decomposed into a large number of narrow bands. Bandpass filters and commercially available software are used. These bands can be analyzed with conventional surface texture parameters, like average roughness, Sa, or other parameters, for detailed, multiscale topographic characterizations. Earlier kinds of multiscale characterization, like relative area, required specialized software performing multiple triangular tiling exercises. Multiscale regression analyses can test strengths of functional correlations over a range of scales. Here, friction coefficients are regressed against standard surface texture parameters over the range of scales available in a measurement. Correlation strengths trend with the scales of the bandpass filters. Using bandpass frequency, i.e., wavelength or scale, decompositions, the R² at 25 μm, exceeds 0.9 for Sa compared with an R² of only 0.2 using the broader band of conventional roughness filtering. These improved, scale-specific functional correlations can facilitate scientific understandings and specifications of topographies in product and process design and in designs of quality assurance systems.

A new method is proposed to investigate the measurement variation of instruments and their ability to discriminate two TA6V surfaces obtained by grinding with SiC grit papers (80 and 120). Three optical measurement technologies included into a single apparatus, namely Confocal Microscopy (CM), Focus Variation (FV) and Coherence Scanning Interferometry (CSI), were used to measure both ground surfaces according to a specific experimental protocol (MorphoMeca Monitoring) based on measurement iterations and repetitions. From the measurements, four indices were built from the Sa roughness parameter to assess the Quality (ratio of the topographical variability over the measurement noise), the Drift (related to surface fluctuations) and the Stability (estimation of the temporal drift using a second order autoregressive model) of the three measurement technologies (modes), as well as their Relevance (computed from an analysis of variance) to discriminate ground surfaces with close roughness. For these investigated surfaces, it was especially highlighted that the Quality Index is four times higher for the CSI mode than for the FV and CM modes. All the used measurement modes experienced small drifting (Drift Index) during the measurements. The CSI mode was slightly more sensitive to drift than the FV and CM modes. The Stability Index indicated that there was no memory effect between two acquisitions for each measurement mode, i.e., the temporal drift did not depend on the measurement mode. Moreover, the three measurement modes are able to discriminate the two ground surfaces with the same accuracy through the Sa parameter. However, the Sa values may vary depending on the measurement modes, and it was finally shown that the Sa parameter is minimized by 0.03 μm for the FV mode compared with the CSI mode, due to its smoothing effect.

Stitching methods allow one to measure a wider surface without the loss of resolution. The observation of small details with a better topographical representation is thus possible. However, it is not excluded that stitching methods generate some errors or aberrations on topography reconstruction. A device including confocal microscopy (CM), focus variation (FV), and coherence scanning interferometry (CSI) instrument modes was used to chronologically follow the drifts and the repositioning errors on stitching topographies. According to a complex measurement plan, a wide measurement campaign was performed on TA6V specimens that were ground with two neighboring SiC FEPA grit papers (P#80 and P#120). Thanks to four indicators (quality, drift, stability, and relevance indexes), no measurement drift in the system was found, indicating controlled stitching and repositioning processes for interferometry, confocal microscopy, and focus variation. Measurements show commendable stability, with interferometric microscopy being the most robust, followed by confocal microscopy, and then focus variation. Despite variations, robustness remains constant for each grinding grit, minimizing interpretation biases. A bootstrap analysis reveals time-dependent robustness for confocal microscopy, which is potentially linked to human presence. Despite Sa value discrepancies, all three metrologies consistently discriminate between grinding grits, highlighting the reliability of the proposed methodology.

Topographic maps are composed of pixels associated with coordinates (x, y, z) on a surface. Each pixel location (x, y) is linked with fluctuations in a measured height sample (z). Fluctuations here are uncertainties in heights estimated from multiple topographic measurements at the same position. Height samples (z) are measured at individual locations (x, y) in topographic measurements and compared with gradients on topographies. Here, gradients are slopes on a surface calculated at the scale of the sampling interval from inclination angles of vectors that are normal to triangular facets formed by adjacent height samples (z = z(x, y)). Similarities between maps of gradients logs and height fluctuations are apparent. This shows that the fluctuations are exponentially dependent on local surface gradients. The highest fluctuations correspond to tool/material interactions for turned surfaces and to regions of maximum plastic deformation for sandblasted surfaces. Finally, for abraded, heterogeneous, multilayer surfaces, fluctuations are dependent on both abrasion and light/sub-layer interactions. It appears that the natures of irregular surface topographies govern fluctuation regimes, and that regions which are indicative of surface functionality, or integrity, can have the highest fluctuations.