A Statistical Approach for Characterizing the Behaviour of Roughness Parameters Measured by a Multi-Physics Instrument on Ground Surface Topographies: Four Novel Indicators

Abstract

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.

1. Introduction

As geologists study the terrain relief [1], surface topography investigators aim to understand the history of the surface [2], but physical phenomena at different scales [3,4] could be linked (scale continuity), as in [5]. Particularly in tribology, understanding physical phenomena is necessary to functionalize surface topography in order to tune the behaviour of a final product, according to its use [6,7,8,9]. These tribological phenomena occur at the interface between two topographical features in contact and moving relative to each other. This is why it is necessary to analyse the topography of these surfaces, because they play a crucial role in how these phenomena manifest, and thus they are key to explaining them. Among these tribological phenomena, there are friction [10], wear [11], which results from the progressive deterioration of surfaces caused by friction, lubrication [12], which tends to modify the friction conditions and avoid premature wear, adhesion [13], which is an attractive force which contributes to friction conditions, contact fatigue [14], and even tribo-corrosion [15]. Understanding these tribological phenomena through surfaces topographies significantly contributes to various fields, including materials engineering [5] and mechanical systems design [16,17].

There exist two types of instruments for measuring surface topographies: tactile and optical. Tactile profilometry stands as one of the most commonly utilized techniques for surface metrology in both research and industry, but this measurement technic is, little by little, being replaced by the optical profilometers. The advantages of the optical profilometers are their ability to carry out quick 3D measurements with different technologies, such as Focus Variation [18,19], Confocal Microscopy [20] and Interferometry [21,22]. The diversity of optical instruments allows us to choose the most adequate technology regarding the features of the measured surface (form, slope, brightness, smoothness, etc.). However, depending on the characteristics of the surface, measurement uncertainties can occur, especially with optical instruments. The instrument choice could be thus determined by its ability to measure a given surface landscape regarding the generated uncertainties (as defined in [23]).

Choosing an adequate instrument with its significant settings [24,25] is not the only problematic factor in metrology. It is necessary to perform a judicious image treatment [26,27] to put forward the required information, and then describe the surface topographies with roughness parameters. A question could arise: what is a good parameter and how can we determine its reliability for the carried-out measurement campaign?

1.1. State of the Art

1.1.1. Sources of Roughness Parameter Fluctuations

The roughness parameters are indicators that can describe a range of features such as characteristic heights, height distribution maps (based on the Abbott–Firestone curve), periodicity, directionality of textures, local gradients, fractal aspects, volume characteristics, and surface topography motifs. These various types of parameters can also be influenced by numerous factors, including the instrument used to perform the measurement, the inherent properties of the surface, the measurement plan (such as repositioning and the duration of acquisition), and uncertainties (such as environmental disturbances or instrument drift).

Surface/Instrument Interaction

In the field of this study, a Sensofar S neox (Sensofar^®, Terrassa, Spain) device is used. This device includes three types of instruments: Coherence Scanning Interferometry (CSI), 3D field Confocal Microscope (CM) and Focus Variation or Shape-from-Focus (FV). Regarding the state of the art, only these instruments are taken into account. In this part, the focus is on the interaction between instruments and intrinsic properties of the surface. Sometimes, no reference is found, due to the incompatibility of the measuring conditions and the instruments.

The studied surface landscape type determines the entire investigation process. Indeed, the optical and geometrical properties of a physical specimen play a main role regarding the most adequate instrument and its ability to measure, more especially the interaction between the surface and the instrument. These intrinsic properties are described in Appendix A and are faced regarding the instrument type defined before, to highlight if an instrument is commonly used regarding a given property (optical or geometrical).

Measurement Conditions

The specimen repositioning is the major source of topographical variations in metrology, whether repositioning is carried out manually or by a displacement table [28]. This is why a measurement plan including multiple measurements at a given position reproduced at multiple positions could impact the results. But the instrument settings used to perform the measurement campaign have an impact too. The magnification, the lateral resolution [29], the scanning speed [30], the type and the intensity of light source [31] are some examples of a non-exhaustive list of major instrument settings which have a crucial impact.

Environment

Instruments can face external fluctuations such as temperature, light, the hygrometry or human activities. These aspects of fluctuation depend on the place when the instrument is installed. Some preconisation is explained by the VDI/VDE 2655 [32]. These evolutions could be followed during the measuring time with sensors such as a vibrometer, a anemometer, or others, in order to try to describe more precisely the impact of the environment on the measurements.

1.2. Uncertainty Determination

1.2.1. Classical Methods

In surface metrology, uncertainty is a complex and critical issue, influenced by various factors and requiring rigorous methodological approaches for a precise evaluation. Among the methods used to quantify these uncertainties, the Guide to the Expression of Uncertainty in Measurement (GUM) [33] provides a standardized framework based on the propagation of uncertainties. The Monte Carlo method, recommended by the Supplement 1 of GUM [34], allows for the simulation and assessment of the impact of uncertainties by generating probability distributions for the measured variables, thus providing a robust estimation of combined uncertainties.

The uncertainties calculated with direct methods are obtained by comparison of the signal obtained by the instrument with the theorical signal, while the uncertainties calculated with indirect methods are obtained by subtraction. Substitution is a commonly used technique where the direct measurement is replaced by an indirect measurement of a known quantity, facilitating the evaluation of uncertainties [35,36] when direct measurement is difficult or impossible. The uncertainties are composed of many sources of fluctuation, such as the measurement noise generated by speckle fluctuations in White Light Interferometry (WLI) [37,38,39], and are calculated from intensity. It is essential to account for this noise, as well as other deviations such as flatness correction [40] or lateral resolution [29] and the Z-axis scale [41], to ensure the reliability of measurements.

Intercomparison of instruments, based on statistical methods, is necessary to evaluate the influence of each instrument on the measurement. This includes comparisons of optical profilometers, where the results from different instruments must be equivalent in accordance with filtering and lateral-resolution standards. However, it is crucial to consider the specific metrological characteristics of each instrument as defined by ISO 25178-6 [42] and ISO 25178-60X [43,44,45,46,47,48] standards, as well as to calibrate and verify these instruments to correctly assess uncertainties, particularly by using recommended good practice guides.

Due to the complexity of applying classical methods for determining uncertainties on real surfaces and estimating these uncertainties across all roughness parameters, alternative methods have been developed.

1.2.2. Unconventional Methods

(X, Y) Method

A method based on gradients and height standard deviation of surface topography pixels was developed to study measurement fluctuations [49,50]. This method quantifies uncertainties from repeated measurements at the same position and links them to surface gradients through a visual tool, B²D plots in 2D and 3D. As a result, height fluctuations (standard deviation) are correlated to gradients, and the height fluctuations depend on both abrasion level and light/sub-layer interaction for the abraded lens.

Allan Deviation

Surface uncertainty assessment using Allan deviation [51] reveals the limitations of traditional methods. Two types of Allan deviation can be calculated for surface topography series: lateral and temporal deviation, both based on Sq². The lateral deviation describes the variation in Z (X, Y) using the root mean square of Sq between each map and the mean map of a given series. In contrast, the temporal deviation accounts for multiple subsets of long-term measurement series and includes a mean filter to describe a time trend. Unlike ISO 25178-600 [35] and ISO 25178-700 [36] standards that assume stationary and randomly distributed noise, the Allan deviation shows that these assumptions do not always hold. The temporal Allan deviation indicates that averaging over longer periods may not improve mean values if drifts or spikes occur. The lateral Allan deviation reveals implicit or explicit filtering, often unnoticed in classical analyses, affecting the lateral resolution. These methods enable in-depth analysis of non-stationary noise and lateral correlations, providing a detailed understanding of uncertainties in surface measurements.

Coefficient Determination Method

Another method allowing us to follow the measurement deviation is presented in [52,53], using S neox^TM (Sensofar^®, Terrassa, Spain), and shows several important findings in real-world conditions. The method is based on the determination coefficient R² and shows the Z (X, Y) fluctuations between two measurements or a data set, allowing us to compare different configurations. This method can be used to show the time evolution of height regarding the time or the difference of height obtained with two different measurement modes. It allows us to quickly assess the repeatability, precision, and uncertainty of topographic measurements. Each height value from the maps is plotted on a 2D chart, with the height of the nth map as the X coordinate and the height of the (n + 1)th map as the Y coordinate, allowing us to compute the R² coefficient. It also determines how consistent the measurements are between different optical modes. Measurements with the same mode show very low height variations and a strong coefficient (R² > 0.99). However, measurements between different modes correlate less (R² ≈ 0.88), and display different slopes on their height–height plots. Overall, this assessment method shows the Confocal mode is the most repeatable, with the strongest correlations, and tends to measure slightly higher heights compared to the other modes. The method efficiently estimates measurement reliability and noise levels, crucial for evaluating topographic data accuracy.

Index Method

Finally, the method developed in [54] shows statistical indicators developed to discriminate two surfaces with neighbouring roughness, while considering stability, drift, and signal-to-noise ratio, in order to help improve the quality of measurements. Unlike the precedent approach, the developed indexes are computed from roughness parameters calculated on the height maps, and not directly from the height maps. By using these indexes, it is possible to determine if an instrument can effectively distinguish two similar surfaces, to evaluate the impact of the stitching process (combining measurements) on data quality [55], and thus to enhance the precision and relevance of measurements in surface topography studies.

More globally, absolute metrology refers to the precise and unbiased measurement of surfaces, aiming to provide results that are not influenced by the calibration of instruments based on standardized specimens. However, the challenge arises when the real-world surfaces differ from these standardized specimens, leading to instrument bias. Additionally, roughness parameters, typically developed for simplified surfaces, may not accurately represent real surfaces, creating a bias in the measurement results. Uncertainties in this context are also problematic, as they stem from various sources, including environmental factors and instrument limitations, making it difficult to quantify them accurately. To address these challenges and to be independent of the device set-up, relative metrology steps in, focusing on the comparison of surfaces (A/B), considering roughness parameters, scale, and process conditions. A well-structured measurement plan can help validate parameters and control uncertainties, ensuring better quality, stability, and relevance in measurements.

1.2.3. Uncertainty Methods Comparison

Table 1. Summary of uncertainty calculation methods, showing their respective strengths and weaknesses.

Uncertainty Method	Strengths	Weaknesses
GUM/ISO	Approved by the ISO committee and supported by most metrologists Established for standard surfaces Takes into account multiple sources of uncertainties	Difficult to apply to real surfaces Time effect not taken into account Could be difficult to implement for non-linear effects Time consuming
X, Y mapping	Mapping visualisation and correlation with the surface gradients	Sensitive to specimen drift Depends on the number and frequency of map acquisitions
Allan deviation (lateral and temporal)	Time effect taken into account Non-stationary noise detected (drift, spikes, modulation or regime changes)	Method more difficult to implement than ISO 25478-600/700 [35,36] Noise distribution is not clearly Gaussian, making it difficult to model
Correlation	Easy to implement Trend modelling is straightforward	Uncertainty quantification is not direct Height sensitivity to outliers’ map values
Statistical index	Allows for the clear assessment of the instruments’ performances Includes four type of indicators which are complementary Adapted to real surfaces and comparisons	Can be difficult to implement directly Statistical knowledge required Roughness parameter selection is not direct in assessment of the instrument performance

presents a comparison of the conventional and unconventional methods for estimating measurement uncertainties.

1.3. Objectives

The primary objective of this paper is to develop a methodology based on a statistical index (here the Quality Index) for identifying reliable roughness parameters coming from a given surface topography (surface #080/#120) and the instruments used (S neox^TM Sensofar^®, Terrassa, Spain), including Focus Variation, Confocal and Interferometry profilometers). This involves evaluating parameters both through direct raw data analysis and a developed statistical index (Quality Index). An important aspect of this study is to identify the best and worst parameters which highlight measurement problems, and to study the worst parameters to know if they can be valuable indicators.

To achieve this, four novel indicators were developed based on the Quality Index (QI): the percentage of outliers of the roughness parameter values, the number of modes, and the homogeneity of QI. The percentage of outliers is calculated from the raw data to identify values that deviate from the main population. Subsequently, the QI is applied to each roughness parameter, grit level, and measurement mode. This process allows for the computation of two additional indicators based on the QI distribution: the number of modes in the QI distribution and the homogeneity which reflects the ability to fit the QI distribution using unimodal and polymorphic modelling. The goal is to assess whether measurement conditions (such as measurement plans and instrument modes) or the level of surface grinding influence the quality index through a comparative assessment of these indicators.

This methodology aims to identify reliable parameters and explain discrepancies through topographical analysis. It includes comparing the raw data approach with statistical index analysis to decide if the outliers of the roughness parameter values should be removed from the database, and validating the results with statistical indexes. Finally, we will discuss how this new methodology could replace or complement existing uncertainty calculation methods, which are often limited to specific cases, thus providing a more generalized approach.

2. Materials and Methods

2.1. Surface Description

The TA6V material was selected for its ability to maintain surface conditions. Two TA6V rods, of 10 mm thickness and 30 mm diameter, were cut using a machine with a diamond cutting disc, and pre-ground with a Silicon Carbide (SiC) grinding paper P#320 to remove cutting scratches and residual stresses. They were then ground with SiC papers from grit P#80 to grit P#1200 for 2 min each, at 300 rpm, with a normal force of 30 N. Finally, one specimen was ground with P#80 (called #080) and the other with P#120 (called #120) for 15 min, at 300 rpm, with a normal force of 30 N, using water lubrication. The surface mechanisms are described in Appendix B.

2.2. Measurement Description

2.2.1. Instrument Settings

The S neox^TM (Sensofar^®, Terrassa, Spain) 3D optical profilometer was chosen to carry out the measurement campaign, supporting Focus Variation (FV), Confocal Microscopy (CM), and Coherence Scanning Interferometry (CSI). The FV and CM modes used a green LED light source and a 20× EPI Nikon lens, while the CSI mode employed a white LED light source and a 20× DI Mirau lens. The system featured a 1.2-megapixel camera with a pixel size of 0.69 µm, offering Z-scaling [56] of 8 nm for continuous CM and FV modes, and 1 nm for CSI mode. The vertical scan range was configured to 128 µm for FV and CM modes, and 30 µm for CSI mode.

2.2.2. Measurement Strategy

A dedicated measurement plan was used to perform measurements [54]. The measurement process was automated. This measurement plan included a multi-measurement strategy: 30 zones per specimen (30 repetitions) and 10 iterations at each given location. The zones were randomly determined at the beginning of the measurement process, then set for the entirety of the study and for the three instrument modes. Each measurement lasted between 1 and 3 s. Each measurement presented raw data without any post-processing. The measurements were of the elementary surfaces, avoiding stitching, and thus ensuring detailed topographies sufficient for statistically robust roughness-parameter calculations.

Figure 1 shows the measurement order according to the two grit levels (in green), the three instrument modes (in red), the repetitions (in blue) and the iterations (in black). For instance, during 1 iteration step, 30 repetitions are performed, and 60 grit alternations are made.

Figure 2 allows for the illustration of the measurement process according to the repetitions, the iterations and the alternance between the grit levels. It is specified that the grit level alternance is made at each measurement. The repetitions are incremented at each position on a given specimen, as shown in Figure 2. After measuring all measurement repetitions, i.e., all positions on the surface, a new measurement cycle is performed, corresponding to the iteration step. This measurement process is carried out for all instrument modes without removing the specimen, to measure the same area and to have comparative measurements.

2.3. Data Post-Processing

2.3.1. Topography Processing

The surface post-processing, performed with MountainsMap^®, is simplified to facilitate direct comparison of measurements, without the need for operator intervention to clean the surface. Figure 3 illustrates the difference in roughness parameter calculation between a raw surface and a surface where a second-order removal form was applied. Removing the form significantly influences the results, highlighting the necessity of this step in our case. No S-filter or C-filter was applied to the surface topographies, to ensure that the comparison remains unaffected by image processing.

2.3.2. Index Computation

To simplify the analysis and avoid working directly on the raw data of roughness parameters, a series of statistical indexes was developed [54]. In this publication, only the Quality Index (QI) is considered for the evaluation of the measurements. QI measures the quality of the measurement for each roughness parameter, defined as a signal-to-noise ratio between the roughness-parameter-inter-position (RP-inter-position) and the roughness-parameter-intra-position (RP-intra-position). For the computed values, a ratio is determined between RP-inter-position and RP-intra-position to create histograms of QI values.

RP-intra-position represents the measurement variability within the iterations performed at each position (30 repetitions per surface) on the two ground surfaces. Each series has its own standard deviation, according to a given roughness parameter. RP-inter-position represents the topographical variability between repetitions (30 positions per surface) with 10 iterations (10 measurements at the same position) on the two ground surfaces. Each set of repetitions has its own roughness-parameter standard deviation.

Additionally, four new indicators based on the Quality Index Probability Density Function (QI PDF) have been developed to assess the reliability of the parameters under different measurement modes used and grit level measured. These indicators include the following: the mean value of QI (called Mean_Q, Equation (1)), the percentage of outliers in the raw dataset (referred to as the %-Out Index), the homogeneity of QI, which is the ability of the QI PDF to fit a unimodal polymorphic distribution (referred to as the Homo_Q), and the number of modes present in the QI PDF (referred to as the NBmode Index).

$$Mean\_Q(RP,M,G)={\displaystyle \frac{{\sum }_{i=1}^n{QI}_i(RP,M,G)}{n}}$$

(1)

• RP: roughness parameter
• M: instrument mode
• G: grit level
• n: number of QI values

Homo_Q is an indicator used to assess the quality of the fitting of a chosen statistical model in relation to the QI PDF. First, QI PDF is fitted using a unimodal and polymorphic distribution, specifically the Johnson SU model [57]. The Johnson SU distribution is used here, as this is characterized by four shape parameters ($\gamma ,\delta , \sigma \xi$) (Equation (2)), which offer considerable flexibility for modelling a wide range of data distributions. The shape parameter $\gamma$ adjusts the curvature of the distribution, affecting its skewness and overall shape. The location parameter $\delta$ shifts the distribution along the value axis, positioning its central tendency, such as the mode or median. The scale parameter $\sigma$ controls the dispersion of the distribution, influencing its spread and standard deviation. Lastly, the location parameter $\xi$ affects the distribution symmetry and its central location, influencing its skewness and kurtosis. By convention, $\delta >0$ and $\sigma >0$.

$$Z=\gamma +\delta {\mathit{sinh}}^{-1}\left({\displaystyle \frac{X-\xi }{\sigma }}\right)$$

(2)

Secondly, a ${\chi }^2$ calculation is made between the data set and the Johnson SU fitting, to determine the ‘quality’ of the statistical model adjustment. The number of histogram classes for which the histograms are calculated is different. Therefore, this factor has to be taken into account for comparison purposes. Finally, thanks to this degree of freedom, we can build the Homo_Q indicator according to Equation (3).

$$Homo\_Q(RP,M,G)={\displaystyle \frac{{\chi }^2(RP,M,G)}{Df(RP,M,G)}}$$

(3)

• $Df=k-1-p$ (Degree of Freedom)
• $k$: number of histogram classes
• $p$: number of Johnson SU model parameters

The NBmode represents the number of modes in the probability density function of QI by utilizing a procedure that counts the modes after selecting an appropriate bandwidth. A histogram is generated with an overlaid kernel density estimate, where the bandwidth is carefully chosen to ensure an accurate representation of the distribution.

In summary, the optimal configuration of indicators includes a higher value of Mean_Q, a lower value of Homo_Q, and a lower value of NBmode for a reliable parameter. It is noted that a low %-Out is preferred.

2.3.3. Roughness Parameter Ranking Method

Since a global view combining all new indicators cannot be analysed directly, a ranking strategy is adopted to identify the most reliable roughness parameters. The approach involves creating a primary criterion called severity, based on the Mean_Q, Homo_Q, and NBmode indicators. The percentage of outliers is excluded from the severity criterion, as data have already been cleaned (outliers are removed from data).

The severity rate is determined by the ranking position of each roughness parameter across different measurement configurations. These ranking positions are derived from the charts presented in Appendix A. A function defined and called $Rk$ is used to rank the parameters from 1st to 50th (corresponding to the 50 roughness parameters calculated for the study). If two parameters have the same value for a given indicator, they share the same rank, and the subsequent parameter does not receive the next rank but skips to the following position. The severity rate, expressed as a percentage, is calculated using Equation (4).

$$Sev={\displaystyle \frac{\left({Rk}^{-1}\left(Mean\_Q\right)\times Rk\left(Homo\_Q\right)\times Rk\left(NBmode\right)\right)}{{NP}^3}}\times 100$$

(4)

• Sev: severity rate of a given roughness parameter for a specific measurement mode and grit level;
• Rk: ranking position of the roughness parameter (increasing order);
• Rk⁻¹: ranking position of the roughness parameter (decreasing order);
• Mean_Q: mean value of QI for a given roughness parameter in a specific measurement mode and grit level;
• Homo_Q: homogeneity value of QI for a given roughness parameter in a specific measurement mode and grit level;
• NBmode: number of modes of QI PDF for a given roughness parameter in a specific measurement mode and grit level;
• NP: total number of parameters (in this case, 50).

2.3.4. Summary of the Methodology

Figure 4 presents the methodology adopted to analyse the results obtained from QI. Firstly, the roughness parameters are plotted as timestamp visualization, parameter by parameter, for each measurement mode and grit level. At this step, three aspects are investigated: the stability through the iteration series, the topographic representativeness of the parameter regarding the measured zones, and the global way of the data populations. For the stability and the representativeness, the analysis is conducted directly through the monitoring, and hypotheses can be made about the reliability of the parameter a priori. In parallel, the QI values and its PDF are computed, allowing us to obtain the mean value of QI, the homogeneity of QI and the number of PDF modes.

Sometimes, roughness parameters are more prone to have outliers in their results. For this reason, it is necessary to clean the data in each case. An algorithm is used to remove these outliers, to make all the cases studied comparable. The outliers are removed from the dataset according to the MCB method as explained in Appendix C. A topographical inspection could be also performed to determine the origin of the outliers if a difference between the a priori and a posteriori result is noted, in order to try to identify the physical causes. The aim is to link the parameter behaviour with the measurement fluctuations.

A new analysis on timestamp charts can be also conduced, identical to the previous one, but without outliers, to determine firstly if the cleaning process is correct, and in order to have an idea if the parameter is reliable a posteriori through the iteration series or the topographic representativeness. In parallel, QI is observed again, also depending on the heterogeneity of the new QI values compared to a normal distribution and the number of modes in the new QI PDF. The goal is to determine if a good quality is obtained a posteriori by comparison of QI charts.

The complete set of results from the various indicators is organized according to the measuring instruments and grinding grades, to establish a ranking of the roughness parameters based on the indicator values. For each combination of instrument, grinding grade, and roughness parameter, the values of Mean_Q, Homo_Q, and NBmode are ranked. This ranking helps to identify the best roughness parameter in each scenario. This ranking is made according to a severity rate described in Equation (4).

After completing a ranking for each scenario, it is necessary to summarize all the rankings. A severity rate threshold is set at 5%, allowing us to count the number of times a roughness parameter falls below this limit. This will provide an overview of the most reliable and least stable parameters in our study.

3. Results and Discussion

3.1. Novel Indicators Applied on the Sa Parameter

In our previous article [54], four indexes were used to evaluate the performance of the S neox^TM (Sensofar^®, Terrassa, Spain) instrument with the Sa parameter concerning the two grit levels and the different measurement modes. Figure 5 presents the results of QI and the new indicators described above. This makes it possible to know if the Sa roughness parameter is reliable.

As explained in the previous article, QI is better for the CSI mode and QI is equivalent for the FV and CM modes. It can also be noted that no significant difference in QI is seen between two grit levels for a given instrument mode. Regarding Figure 5b, the Sa values seem to form a homogeneous population for the grit #080 (circle markers) and grit #120 (triangular markers) for each instrument mode.

This first approach allows us to compare the measurement modes and the grits, but other information is needed to choose the best roughness parameter. Sa has, globally, a low number of outliers (Figure 5c). This confirms that the populations in Figure 5b are homogeneous. The mean values of QI are higher for the CSI mode, as was said in a previous analysis [54], but it is also highlighted that QI is slightly higher for FV than CM. The PDF form plotted in Figure 5a gives information about the QI values distribution, i.e., the beauty of the histograms. Homo_Q is better for the FV mode (lowest Homo_Q values for the two grit levels) than the others. This means that the unimodal and polymorphic model is more capable of fitting the QI PDF of the FV mode. Moreover, FV has the most ‘beautiful’ histogram of QI. It can be noted that Homo_Q is higher for the grit #120 than the grit #080 for the CM and CSI modes. This means that the grit #080 has more ‘beautiful’ histograms than the grit #120, unlike the FV mode which has similar Homo_Q values, probably due to the smoothing effect during measurement.

The CSI and CM modes have the same number of PDF modes (one mode for #080 and three modes for #120). The one and only mode for the grit #080 confirms that the histograms are more ‘beautiful’ for this grit, as shown with a lower value of Homo_Q. The FV mode has the same number of modes for the two grits. The number of modes is therefore used to create a two-level PDF discrimination for QI.

By taking into account all these results, and in particular the FV results, Sa cannot be the only roughness parameter to be analysed. This is why a lot of roughness parameters from ISO and EUR [58] standards are studied in this paper, the aim being to find the most reliable parameters to describe the measurements. The number of outliers in the raw data is low for Sa, but it is not the case for the other parameters. As an example, the outlier’s effect for the Sp parameter is presented in Figure 6.

The Sp parameter is presented here, as this parameter is very sensitive to map fluctuations, and data clean-up could be needed. The average percentage of outliers for the grit #080 is 16.89% (+2.78%/−4.89%) and 11.55% (+4.48%/−6.55%) for the grit #120. It can be noted in Figure 6 that the range of the Sp values has the same order of magnitude for the grit #080 and #120, both before and after removing outliers. The difference in the data population between the two grits is therefore more easily established, and the Sp values for the grit #080 are more dispersed than the grit #120, due to a higher relief in topography. It can therefore be questioned how the removal of these outliers influences the QI PDF.

3.2. Example of Cases of QI Regarding the Homogeneity of PDF, the Number of PDF Modes, and the Percentage of Outliers with and Without Outliers

As introduced earlier, three indicators are built based on the QI. These characterize the QI to determine the reliability of roughness parameters regarding their measurement representativeness and repeatability.

Figure 7 presents various cases of indicator performance with no removed data (with outliers), including the best Mean_Q and the worst Homo_Q (Figure 7a), the worst Mean_Q (Figure 7b), the highest NBmode (Figure 7c), the best Homo_Q (Figure 7d) and the lowest (Figure 7e) and highest (Figure 7f) %-Out.

Table 2. Values of the indicators for each performance case with outliers illustrated in Figure 7.

Observation		Instrument Mode	Roughness Parameter	Grit
Indicator	Case			#080	#120
Mean_Q	Best	CSI	Sal	532.99	480.48
Homo_Q	Worst			150.13	250.72
Mean_Q	Worst	FV	Sds	8.80	6.71
NBmode	Highest	CM	Smr	2	4
Homo_Q	Best	FV	Vmc	1.43	0.91
%-Out (%)	Lowest	CM	Sal	0	0
%-Out (%)	Highest	FV	Sp	21.37	14.33

presents the values of the indicators associated with Figure 7.

In general, no major difference in Mean_Q is observed between the two grits for each case, indicating that the parameters are consistent across grit levels. It is important to note that the parameter is selected based on a global ranking across all instrument modes. It appears that some QI PDF are clearly not smooth and contain multiple modes, more especially in Figure 7a,c,e,f. The irregular shapes are highlighted by the number of modes, the homogeneity, and the percentage of outliers. In the study, the Sal parameter is highlighted three times as a specific parameter: it is ranked as one of the best Mean_Q values for the CSI mode in Figure 7a, and one of the lowest %-Out values for the CM mode (Figure 7e), but Sal is also highlighted as the worst parameter for Homo_Q for the CSI mode. The Sds parameter for the FV mode has one of the worst Mean_Q values but a low number of PDF modes. Additionally, the Smr parameter shows a high number of PDF modes for the grit #120 and for the CM mode, while the Sp parameter has one of the highest numbers of outliers.

Given the percentage of outlier points for certain roughness parameters, it may be necessary to clean the data before calculating the Quality Index and the new indicators. However, this cleaning is not intended to make the ‘poor parameters’ reliable, but rather to prevent values that deviate too much from the parameter data from being taken into account (graphs presenting parameter values versus time). This allows us to obtain valuable indicators for roughness parameters having a few unstable points, which disturb the QI results.

Similar to Figure 7, Figure 8 presents the same cases of indicator performance after outlier removal, allowing for a comparison of the indicator values with (Table 2) and without outliers (

Table 3. Values of the indicators for each performance case without outliers illustrated in Figure 8.

Observation		Instrument Mode	Parameter	Grit
Indicator	Case			#080	#120
Mean_Q	Best	CSI	Sal	531.26	472.80
Homo_Q	Worst			96.29	311.17
Mean_Q	Worst	FV	Sds	8.87	6.73
NBmode	Highest	CM	Smr	2	5
Homo_Q	Best	FV	Vmc	2.51	1.94
%-Out (%)	Lowest	CM	Sal	0	0
%-Out (%)	Highest	FV	Sp	0	0

). Overall, it is evident that the amplitudes of QI PDF remain consistent for the charts with removed outliers, except in the case of a higher percentage of outliers. This indicates that the outlier removal method does not significantly affect the Mean_Q, Homo_Q, or the number of mode indicators. It is also reassuring that the shapes of the QI distributions are generally similar. The outlier percentage is remembered, and is the same as before (Table 2).

For example, a notable difference in QI PDF is observed in the case of the Sp parameter, due to the significant number of removed data: a high percentage of outliers is computed (21% for grit #080 and 14% for grit #120). Appendix D shows the indicators without outliers for Sp and the FV mode (Figure A3 and Figure A4). Thus, the outlier removal method only impacts QI PDF for parameters with a higher percentage of outliers, validating the method application. It is also worth noting that the Mean_Q values are sometimes slightly higher in the cleaned QI results, because the data cleaning algorithm reduces the topographical representation (roughness parameter inter-standard deviation) without modifying the measurement fluctuation (roughness parameter intra-standard deviation). This results in an increase in QI (signal-to-noise ratio).

3.3. Ranking of Roughness Parameters

To conclude, while these indicators effectively describe the behaviour of roughness parameters, a case-by-case analysis for all parameters is impractical. This raises the important question of how a multi-parameter analysis can be conducted based on these indexes, emphasizing the need for a more integrated approach.

Figure 9 gives an example of the ranking detailed in Appendix E, showing the severity values in percentage for the three instrument modes and the two grit levels. It appears that the severity rates are generally higher for the grit #080 for the CM and CSI modes, likely due to the greater texture. This indicates that the roughness parameters have generally a low quality and a poor PDF for the grit #080 compared to that for the grit #120. However, this is not the case for the FV mode, where higher values are observed for the grit #120. As previously shown, the FV mode behaves differently from the other two modes when comparing grits.

A threshold of 5% is chosen to distinguish the reliable roughness parameter and unstable parameters. A 5% threshold value is used as a convention, representing the level at which the severity rate of a roughness parameter is considered unstable based on the indicators Homo_Q, Mean_Q, and NBmode. These indicators are calculated from the Quality Index (QI) probability density function, to determine whether the QI values can be considered for further analysis. In each case, approximately half of the roughness parameters are below 5% showing the threshold is correctly chosen, but only some parameters are common for all grits and instrument modes.

Figure 10 illustrates how often a parameter falls below the 5% severity rate, as shown in Appendix E. The six scenarios are analysed based on the measurement conditions. The occurrence values range from 0 to 6, with 0 indicating the worst parameters (those that consistently exceed 5% severity) and 6 indicating the best parameters (those that consistently stay below 5% severity). For example, Sa is higher than 5% of severity for the cases FV mode/grit level #080, CM/#120 and CSI/#120. The occurrence value is thus 3. In addition, it can be noted that each occurrence is represented in Figure 10, showing the 5% threshold is correctly chosen.

Notably, the best roughness parameters are Sal (spatial parameter), Sz (height parameter), and Sci (functional indice) with an occurrence of six (six times below 5%). Additionally, Vm, Vmp, Vvv, Svk, Sk, Vv, Sdc, Smr2, Sxp, Svi, Sha, Shv, and Ssk can also be considered reliable parameters, as they appear in five out of six cases. It appears that the least reliable roughness parameters are clearly Sdq and S5p, because in all cases they always have more than 5% of severity rate. It could be considered that the Smr, Sv, FLTv, Sku, FLTy parameters are unstable too.

More precisely, the focus is on the best and the worst ranked parameters, to make the link between the parameter definition and the obtained results (physical causes):

• The Sal parameter (Texture Aspect Ratio of the Surface) represents the autocorrelation length of the map, expressed in wavelength. A higher Sal value indicates a greater presence of long wavelengths within the map. The Sal parameter is a key metric used to evaluate the anisotropy of a surface, meaning the directional properties of the patterns or textures present. It is especially useful for surfaces with regular, oriented textures, such as those produced by grinding or lapping processes, which often result in surfaces with distinct striations. For striated surfaces, such as those produced by grinding or polishing, Sal is directly related to the width and spacing of the striations. A surface with wide, well-aligned striations will have a low Sal, indicating strong directionality. Conversely, if the striations are irregular or vary significantly in spacing, Sal may increase, reflecting a loss of clear directional alignment. A high Mean_Q value for Sal indicates low dispersion of the Sal values across the iteration series, highlighting broad topographic characterization. For the manufactured surfaces (#080 and #120), this suggests that the long wavelengths (wide cutting scratches on the surface) are dependent on the different measured areas, as evidenced by the wide variation in the Sal values. This shows that Sal is a good parameter for distinguishing between different measured areas, particularly regarding the noise observed in the iteration series. However, it is important to note that while variations in wavelength are averaged when calculating Sal, small variations within an iteration series may have minimal impact on the Sal values. Additionally, Homo_Q of Sal is not particularly strong, indicating that the QI ratio, which includes both intra-position and inter-position standard deviations of Sal, may not be stable across different measured areas or throughout the iteration series (i.e., zone-dependent or time-dependent). One of the main limitations of Sal in this context is its sensitivity to the size of the striations. If certain striations are significantly larger or more irregular than the rest, they can disproportionately influence the Sal value, even if they are few in number. This creates a high statistical variability, making it difficult to obtain a representative measure of the surface’s anisotropy. To address this variability, a highly effective approach is to increase the measurement area using a technique called stitching. This involves combining multiple local measurements of a surface into a single, larger image. Stitching allows for the measurement of a greater portion of the surface, thus incorporating more striations into the analysis. By increasing the number of measured striations, the influence of rare, large, or irregular striations is minimized, leading to more reliable and representative Sal measurements.
• The Sz parameter represents the maximum height of the surface, defined as the difference between the highest peak and the deepest valley. The Sz ranking is reliable, characterized by low Homo_Q and NBmode values, which suggest a well-defined histogram and a satisfactory Mean_Q value. Due to differences in topographic features between measured areas, a high dispersion in Sz can be observed. When taking multiple measurements at different locations on a randomly textured surface, such as a ground surface, the Sz parameter can exhibit significant variability, which is closely related to the extreme value theory. Since Sz is based on the highest peaks and lowest valleys of the surface, its value is highly sensitive to local irregularities, such as isolated large peaks or deep pits. On a surface with random striations and irregular features, different measurement locations may capture different extreme values, leading to a wide range of Sz results. This variability arises because each measurement could include a rare or extreme feature that disproportionately impacts the Sz value, despite the overall surface texture being relatively homogeneous. According to extreme value theory, which models the behaviour of maxima or minima in random systems, such extreme events are expected to occur infrequently, but can significantly influence the outcome when they do. This results in Sz being a less reliable indicator of the overall surface roughness, as it is heavily influenced by outliers rather than reflecting the typical texture of the surface. To address this variability, larger measurement areas using stitching techniques can be employed to capture a more representative sample of the surface, averaging out extreme features. Extreme value theory can also be applied to better understand and model the behaviour of these extreme surface features, particularly when they are critical to the application [59]. However, the ratio between topographic representativeness and noise in the iteration series is sufficient to consider Sz as a qualitative parameter, particularly given the robustness of the QI for every measured area and iteration series, as reflected in the well-formed histogram.
• The Sci parameter is a roughness index that indicates core fluid retention, calculated as the ratio of void volume in the core zone (from 5% to 80%) to the RMS deviation (Sq). Mean_Q of Sci is generally good, outperforming Sz, due to high deviations between different measured areas, which is linked to variations in the surface topographies. NBmode of Sci, while higher than Sz and Sal, is still acceptable. This could be attributed to the presence of high peaks on the surface (groove pile-up) or third core inclusions, as the calculation of Sci is highly sensitive due to its 5% threshold, leading to multiple modes in QI PDF. Nevertheless, Homo_Q remains favourable for this parameter, indicating that the NBmode values are closely aligned or nearly merged.
• The Sdq parameter represents the RMS slope of the surfaces, and appears to be unreliable in terms of ranking. However, it is noteworthy because it indicates that slope is a highly sensitive feature of the surface [49]. Despite this, Sdq performs well in terms of QI, showing that the ratio of deviation between topographical representativeness and noise in the iteration series is generally good. However, the high NBmode and Homo_Q values suggest a local instability in Sdq QI values, both intra- and inter-position. This instability likely stems from the physical limitations of the instruments in measuring slope. When the slope limit is reached, some pixels may be measured inconsistently, leading to variable pixel quality [60]. The Sdq parameter is crucial for assessing surface roughness, but it is highly sensitive to the sampling interval used in measurements. This sensitivity arises because the Sdq calculation involves the derivative of the surface profile, and any noise present in the data can significantly affect the accuracy of this derivative. A larger sampling interval can amplify noise during derivative calculations, as it may exaggerate random fluctuations in the data, leading to less accurate Sdq values. Conversely, a smaller sampling interval can enhance precision by capturing more detailed surface features, but it increases the data volume and may also magnify the effects of measurement noise. To mitigate these issues, several approaches can be employed. Data filtering techniques, such as the Gaussian filter [61], can smooth the data before derivative calculation, reducing noise while preserving essential surface characteristics. Choosing an optimal sampling interval is essential to balance detail and noise effects. Advanced interpolation methods, like B-spline interpolation, can offer more robust performance against noise compared to Lagrange interpolation by providing a smoother approximation of the surface profile. Our sensitivity analysis can help us to understand how variations in the sampling interval impact Sdq calculations, allowing for adjustments in measurement and processing methods, to ensure reliability. By addressing our indexes and implementing appropriate solutions, the accuracy and reliability of Sdq measurements can be significantly improved.
• The S5p parameter belongs to the segmentation-based family. The Wolf pruning algorithm is used to eliminate insignificant motifs by merging smaller ones into larger ones. This parameter is sensitive to the calculation method, representing the average height of the five peaks with the highest global peak height within the defined area. As a result, the area computation is sensitive to small variations in surface iteration series, as evidenced by a poor ranking of Mean_Q for this parameter. Furthermore, the high NBmode and Homo_Q values indicate instability in the inter- and intra-position standard deviations of S5p values. The variability in segmentation caused by noise means that the significant peaks identified may differ from one measurement to another, complicating the consistency and accuracy of the results. To mitigate this issue, several approaches can be employed. Data filtering techniques, such as filters [62], can smooth out the noise before segmentation, helping to preserve essential surface characteristics while reducing random fluctuations. Performing repeated measurements at the same location and obtained a mean map can also help to lessen the impact of noise and provide a more stable assessment of surface quality [50]. Advanced pruning methods, like the Wolf pruning threshold, can be used to refine the results by focusing on the most significant data points and minimizing the influence of noise. By addressing the challenges of noise and segmentation variability through these strategies, it is possible to achieve more reliable and accurate evaluations of surface quality.

An analysis of the roughness parameter family reveals that the functions and related parameters (highlighted by a red box) are generally reliable, particularly the volume parameters. However, it also indicates that the form parameters are globally unstable (in yellow). As for the height parameters, they exhibit either reliability or instability, likely due to the calculation method which directly depends on the heights of the maps, and can cause some instabilities in results. No further comments can be made on the other roughness parameter families, as the parameters within them are uniformly represented in the ranking. Finally, the severity rate can describe as a Quality indicator of a Quality index, allowing us to validate whether a Quality index computed on roughness parameter values is good or not. This answers the following question: if a set of parameters is known for its physical relevance, which is the most reliable parameter?

4. Conclusions

Previously, four indexes were developed to characterize the measurements performed with a Sensofar S neox™ (Sensofar^®, Terrassa, Spain): quality index, stability index, drift index and relevance index. Each index was calculated on the Sa values computed from topographies obtained according to a specific measurement plan, named Morphomeca Monitoring. In this paper, only the quality aspect is studied through the Quality Index (QI) and four new indicators are created based on this index: the mean QI value (Mean_Q), the QI homogeneity (Homo_Q), the number of modes in the QI probability density function (NBmode) and the percentage of outliers of the parameter values (%-Out). These indicators were firstly studied for the Sa parameter, then for 50 roughness parameters (ISO and EUR standards).

If only the Sa parameter is considered, the analysis of QI across different instrument modes and grit levels shows that the CSI mode consistently has the highest Mean_Q, while the FV and CM modes perform similarly. The FV mode is notable for its superior histogram homogeneity (Homo_Q) and more consistent QI distributions, which are less influenced by grit levels. However, the FV mode tends to smooth the surface topography, which can introduce bias in the conclusions. For example, no significant differences are observed between grit #080 and grit #120 across any instrument modes. This limitation highlights the need for a multi-parameter approach to fully capture surface characteristics.

Additionally, the data of the Sa parameter have relatively few outlier values, which is why an outlier removal method was applied in this study. This method is necessary to harmonize parameters with one another. Generally, removing outliers does not significantly affect key indicators like Mean_Q or Homo_Q, except in cases with a high percentage of outliers, such as the Sp parameter.

To evaluate the roughness parameters across different instrument modes and grit levels, a ranking system was established using severity rates. A 5% threshold was set to identify reliable parameters. Through this process, Sal, Sz, and Sci emerged as the most reliable parameters, while others like Sdq and S5p were consistently unstable. The analysis also showed that function-related parameters, especially those related to volume, tended to be more reliable, whereas form parameters were generally unstable.

The proposed method adopts an overview of a large set of roughness parameters, allowing us to investigate a complementary point of view regarding the surface properties and features. The method takes into account the fact that the measurements could be performed with different instruments and different levels of surface generation intensity. However, it can be noted that just because a parameter is reliable, it does not necessarily have physical significance in the study of the surface generation process.

All of the approach is based on the QI but, as shown in [54], others indexes such as the Drift Index (DI), Stability Index (SI) and Relevance Index (RI) exist, and could be included in the study. A future work including all indexes is scheduled to find the criteria to rank roughness parameters by following a global method allowing us to determine the best instrument mode to measure a specified manufactured surface.

In the bibliography, a lack of knowledge is identified regarding the reliability of the roughness parameters with regard to a specific measurement condition (instruments and measurement plan) and intrinsic properties of a surface. These findings underscore the complexity of selecting appropriate roughness parameters and the importance of using a comprehensive, integrated approach to achieve accurate and reliable surface measurements.

To conclude, this study proposes a novel method based on the statistical analysis of roughness parameters and a severity ranking system to highlight the most reliable parameters computed from surface topographies obtained using a set of instruments and a specific measurement strategy. This method can be applied to all surface types and for other roughness parameters, even if only one instrument is analysed, although the ranking step will show only two values (1 for parameters below 5% and 0 for parameters above 5%). When only a single instrument is studied, the results become more specific, and are not meant for comparative analysis.

As a future perspective, a complementary study could be conducted on other surface conditions, such as sandblasting, femto-laser, or oxidation, to identify the most reliable parameters, regardless of surface type. This would help determine if certain parameters are sensitive enough to be attributed to specific surface types. In such cases, a particular parameter could be linked to a specific device/surface combination to monitor measurement quality. It can also be noted that the influence of instruments settings is not taken into account in the field of this study. Another work can be conducted to show the influence of magnification, acquisition velocity or other relevant settings on the roughness parameter reliability, the quality index or the others indexes developed in [54].