Measurement Noise in Profile Metrology: Concepts, Characterization, and Metrological Implication

1. Introduction

Profile surface texture measurement is widely used to characterize functional surfaces in precision engineering and dimensional metrology. Despite the growing use of optical and scanning probe techniques, contact stylus instruments remain important for profile surface texture measurement in both industrial applications and high-accuracy metrology. They play a key role in calibrating surface texture standards and establishing metrological traceability.

At the nanometre scale, measurement results can be significantly affected by contributions from the measuring system and the measurement environment. Measurement noise is particularly relevant in this context, as it is superimposed on the measured profile and can influence the evaluated surface parameters. In recent years, considerable attention has focused on characterizing measurement noise in areal surface topography measurements within the ISO 25178 series. However, a standardized approach to noise characterization is currently lacking for profile surface texture measurements. The ISO 21920 series specifies profile surface texture metrology, in which separating spatial scales is an important element of the specification operator. This separation is achieved by applying appropriate filters, including those defined in the ISO 16610 series. Fundamentally, however, these filters separate signal components by spatial-frequency content and do not distinguish components by physical origin. Consequently, filtering alone cannot distinguish surface features from noise components within the same spatial-frequency range. This limitation becomes particularly important when measurement-noise amplitude approaches that of the surface signal. Internal sensor and electronic noise, thermal fluctuations, and residual mechanical and acoustic vibrations may then constitute a significant fraction of the recorded profile. For positive-definite parameters, particularly those based on the mean square of deviations, positive and negative random noise components do not cancel. Instead, their squared contributions increase the evaluated parameter value.

The EMPIR project ProbeTrace 18RPT01 [1] investigated the influence of random measurement noise and ways to reduce it numerically. One project objective was to develop numerical noise-reduction methods and software to preprocess raw roughness and form profiles, aiming to reduce the contribution of random noise to overall measurement uncertainty. The main approach was based on Noise Bias Reduction (NBR). In addition to NBR, we implemented two frequency-domain approaches: reducing Fourier components and excluding random components. Dedicated NBR software was developed, and the implemented procedures were evaluated using both simulated and experimental measurement data. Results from the ProbeTrace project showed that random noise can contribute significantly to measured profile deviations and that reducing it can improve measurement results and reduce associated measurement uncertainty.

As a follow-up to the ProbeTrace activities, the Croatian Science Foundation (HRZZ) project NanoSurf was initiated to investigate further factors limiting the accuracy of surface measurements at the nanometre scale. While ProbeTrace addressed numerical methods for reducing random noise in roughness and form measurements, NanoSurf extends this research by focusing on characterizing and reducing measurement noise in profile surface texture measurements and assessing its influence on measurement results and associated measurement uncertainty.

In this context, the study focuses on characterizing measurement noise in high-accuracy contact stylus profile measurements, with particular emphasis on its metrological interpretation and quantitative assessment methods. It also examines the applicability of noise-characterization procedures established for areal surface topography measurements to profile measurements and their use in the metrological characterization of high-accuracy stylus instruments.

2. Definition of Noise—Normative Background

In surface metrology, measurement noise is an important limiting factor, especially in high-precision applications. Controlling the measurement system’s mechanical, environmental, and electrical conditions primarily reduces its influence. This includes using stable, high-mass granite supports and passive or active vibration-isolation systems, stabilizing ambient and instrument temperatures, reducing airflow-induced disturbances, and ensuring proper electrical grounding and shielding. Signal-processing techniques also reduce unwanted signals, with S-filtering commonly applied to attenuate high-frequency components and averaging. Determining the measurement noise level is important for metrological characterization of the measuring instrument and for evaluating measurement uncertainty during calibration. Within the normative framework of surface texture metrology, several terms describe noise in the measurement process, including instrument noise, measurement noise, dynamic noise, static noise, and basic noise. Although these terms are related, their definitions and the conditions under which they are determined may differ depending on the applied standard and measurement method. Before considering these specific terms, it is helpful to introduce some general definitions related to noise in measurement.

The word noise has many definitions. In common use, the word noise means any unwanted sound. In both analog and digital electronics, noise, or signal noise, is an unwanted perturbation to a wanted signal. In signal processing or computing, noise can be considered unwanted data without meaning; that is, data not used to transmit a signal but produced as an unwanted by-product of other activities [2]. From a metrological perspective, noise is generally understood as the non-deterministic component of a measurement signal that is unrelated to the measurand and contributes to measurement uncertainty [3,4,5]. Therefore, within the normative framework, noise is not classified strictly according to its deterministic or non-deterministic nature. Instead, it is categorized according to its origin, measurement method and conditions, and contribution to measurement uncertainty.

Under the earlier, withdrawn ISO 3274 [6] standard on the nominal characteristics of contact (stylus) instruments, noise was not included among the metrological characteristics. Instead, the concept of the residual profile was used, obtained by measuring an optical flat. The residual profile comprised deviations of the guide, external and internal disturbances, and deviations in profile transmission. A similar approach is retained in the current ISO 12179 [7] standard for calibrating contact (stylus) instruments, where noise is not treated as a separate metrological characteristic. DKD-R 4-2 applies the same basic measurement principle: repeated measurements on a plane glass plate. However, it treats the resulting instrument-related deviations differently. For the calibration of the vertical measuring system, DKD-R 4-2 Sheet 2 [8] defines the term basic noise, expressed as Rz₀ calculated on the measured profile. At the same time, the straightness deviation of the guidance is characterized separately by Wt₀. DKD-R 4-2 separates the instrument’s basic noise and the guidance straightness deviation into distinct quantities. In addition, DKD-R 4-2 recommends how to include basic instrument noise in the calibration uncertainty budget.

For the calibration and metrological characterization of high-accuracy laboratory contact (stylus) instruments, even when used exclusively for profile measurements, the ISO 25178 series can and should be used, although it establishes the metrological framework primarily for areal surface topography measurement.

3.1. Static and Dynamic Noise

ISO 25178-701 [9], which specifies calibration methods for contact (stylus) instruments, distinguishes static and dynamic noise and provides procedures for their assessment. Definitions of static and dynamic noise, as well as instrument and measurement noise, are given in ISO 25178-600 [10].

Static noise (N_S) is the combination of instrument noise and environmental noise on the output signal when the instrument is not scanning laterally. The static noise test verifies the instrument’s setup and performance by identifying potential noise sources. Although it is not evaluated separately in the measurement uncertainty [11], its contribution is inherently included in the dynamic noise. Since no lateral motion is involved, determining static noise does not require a calibrated material measure. However, this condition cannot be realized on all instruments, as some systems do not allow profile acquisition unless the lateral scanning axis is in motion. If the dynamic response of the measuring system is also to be characterized, a reference flat surface, such as a calibrated optical flat, can be scanned. Under these conditions, the measured signal includes noise contributions associated with the lateral motion of the scanning system, and the resulting quantity is referred to as dynamic noise N_D. By definition, dynamic noise occurs during the motion of the drive units on the output signal. It includes contributions from the mechanical and control behavior of the scanning system, such as drive-induced vibrations, guideway or bearing effects, servo-related disturbances, and other motion-dependent fluctuations. Its magnitude and spectral content may depend on operating parameters such as scanning speed and the measurement system’s dynamic response. Under a strict interpretation of noise as a non-deterministic contribution to the measurement signal, the terms static and dynamic noise should be used with some caution. In practical measurements, particularly during scanning, the experimentally determined dynamic noise may also contain repeatable or deterministic motion-dependent contributions, such as those associated with the guideway or drive system. Therefore, for the metrological characterization of an instrument, the terms instrument noise and measurement noise may better describe the relevant noise contributions than a classification based solely on static and dynamic measurement conditions. This distinction provides a more direct link between the relevant noise contributions and the instrument’s metrological characteristics, rather than the specific procedure used for noise evaluation.

3.2. Measurement and Instrument Noise

Instrument noise N_I is defined as internal noise added to the output signal caused by the instrument if ideally placed in a noise-free environment. Instrument noise should be regarded primarily as the instrument’s intrinsic noise contribution and, consequently, as an idealized lower limit of its noise performance. In practice, instrument noise cannot be completely separated from contributions arising from the environment and measurement conditions. Nevertheless, the concept remains useful for distinguishing instrument-generated noise from other contributions. For evaluating measurement uncertainty, however, the relevant quantity is the measurement noise affecting the measurement result, rather than instrument noise considered as an isolated characteristic of the instrument. This is consistent with the GUM approach, in which measurement uncertainty is associated with the measurement result and not the instrument itself [12,13,14]. The measurement procedures and methods for determining instrument noise are specified in ISO 25178-700 [15]. To evaluate instrument noise in surface topography measurements, a flat plane (AFL) defined in ISO 25178-70 [16], such as an optical flat, is commonly used as the reference artefact [17].

Measurement noise, N_M refers to the noise added to the output signal occuring during the normal use of the instrument. It comprises instrument noise and additional contributions from environmental and operational influences, including thermal effects, mechanical vibrations, air turbulence, and other external disturbances [11,15,18]. This represents the fundamental noise concept from a metrological perspective, as it reflects the combined effect of all noise contributions present under actual measurement conditions. The measurement procedures and methods for determining measurement noise are specified in ISO 25178-700 [15]. For this purpose, a representative surface should be used, with characteristics as similar as possible to those of the surface intended to be measured in normal use. Regardless of the surface used, determine the noise by applying either the subtraction method or the averaging method. Both methods rely on repeated measurements and aim to separate the measuring system’s random noise contribution from the repeatable surface topography in the measured signal.

ISO 25178-700 [15] also introduces the concept of stabilized measurement noise, N_Mi for evaluating the temporal stability of measurement noise. Although stabilized measurement noise does not represent a distinct type or physical source of noise, determining it from repeated measurements enables assessment of whether the estimated noise level remains consistent across successive measurements or varies with time-dependent or environmental influences. Stabilized measurement noise therefore measures the stability of measurement noise under repeated measurement conditions.

Table 1 presents an overview of noise concepts, while Table 2 presents the relationships between these concepts.

3. Measurement Noise Characterization

Measurement noise can be characterized at different levels of statistical and signal complexity. The simplest description uses a scalar measure of amplitude, such as the standard deviation or root-mean-square value. Although these quantities estimate the overall noise level, they do not describe the probability distribution of the fluctuations, their spatial dependence, spectral content, or stability over position and time.

3.1. Amplitude Characterization of Noise in Surface Texture

Dynamic noise is determined using an optical flat, whereas static noise may be evaluated on an arbitrary surface, since no lateral motion occurs during the measurement. The maximum height of profile R_z, was typically used as the evaluated parameter in both tests. DKD-R 4-2 Sheet 2 [8] addresses basic noise and estimates it using the R_z0 parameter of plane glass plate. This approach therefore characterized noise primarily through a scalar maximum-height descriptor. Good practice guidance for contact stylus instruments subsequently introduced root mean square (RMS) based quantities to statistically describe noise amplitude. NPL Measurement Good Practice Guide No. 129 [19] recommends quantifying static noise in profile measurements using the RMS deviation of the evaluated profile. For filtered profile data, the parameter R_q is used, whereas P_q is applied to unfiltered profile data. For areal measurements, the corresponding parameter is S_q. The same guide also describes subtraction and averaging approaches for isolating measurement noise from the intrinsic topography and flatness contribution. This provides a direct methodological bridge between the earlier maximum-height characterization, and the later RMS based ISO approach. A more recent and generalized approach is presented in ISO 25178-700 [15]. Since profile data can be extracted from areal topography measurements, the standard also notes that most of these procedures can be adapted for profile. The default procedure for determining measurement noise is the subtraction method, in which two measurements are acquired at the same location with the shortest practical time interval and then subtracted. Ideally, this cancels the common topographic contribution, and the remaining difference predominantly represents the combined noise of the two measurements. For areal data the noise is evaluated from S_q:

$$N_M={\displaystyle \frac{\mathit{Sq}\left[z_1\left(x,y\right)-z_2\left(x,y\right)\right]}{\sqrt{2}}}$$

(1)

ISO 25178-700 [15] also requires reporting the relevant acquisition time and any spatial or temporal filtering that can alter the evaluated noise bandwidth. Alternatively, it specifies an averaging method based on at least three repeated measurements acquired at the same location. The individual topographies are averaged to obtain a mean topography, which is then subtracted from each individual measurement. The resulting residual topographies predominantly represent the non-repeatable component of the measurement. Their RMS values can be expressed as:

$$N_M=\sqrt{{\displaystyle \frac{{\sum }_{i=1}^{n}S{q}^2\left(z_i\left(x,y\right)-z_m\left(x,y\right)\right)}{n-1}}},z_m\left(x,y\right)={\displaystyle \frac{1}{n}}{\sum }_{i=1}^n{z}_i\left(x,y\right)$$

(2)

ISO 25178-700 [15] further allows stabilized measurement noise to be evaluated from repeated noise measurements, with the standard deviation of the individual noise values providing information on noise stability. For profile measurements, the stabilized-noise procedure based on the subtraction method can be applied directly, since it is formulated in terms of repeated individual noise values, whereas the expression associated with the stabilized averaging method can be adapted to profile measurements by replacing the areal parameter S_q with the corresponding profile parameter R_q.

An overview of noise concepts, the corresponding reference documents, and the parameters used for amplitude characterization is presented in Table 3.

3.2. Autocorrelation-Based Characterization

The autocorrelation function (ACF) was initially used to characterize the spatial structure of measured surface profiles, whereas more recent studies have extended its use to identify and suppress measurement noise. In 1981, Bennett and Dancy [22] used correlation-based statistical analysis to determine the parameters of smooth optical surfaces. Recent studies further demonstrated the applicability of ACF-based methods for surface topography analysis, particularly for identifying measurement noise and assessing the roughness characteristics of machined surfaces [23,24,25]. In addition, correlation-based approaches have been incorporated into traceable roughness measurement procedures, supporting the reliable characterization of surface texture in practical metrological applications [26].

In addition to amplitude-based parameters, the spatial structure of surface data can be characterized using the autocorrelation function, as defined in ISO 25178-2 [20] and ISO 21920-2 [21]. For areal and profile surface texture, respectively. In both cases, the ACF describes the degree of correlation between the measured surface or profile and a spatially shifted version of the same data.

Amplitude-based quantities such as S_q, R_q and N_M quantify the magnitude of measurement noise but do not provide information on its spatial content. Consequently, measurement systems exhibiting similar RMS noise amplitudes may nevertheless show substantially different spatial noise characteristics. When applied to residual data obtained using the subtraction or averaging procedures described above, the ACF provides complementary information on the spatial dependence of the non-repeatable measurement component.

For profile measurements, autocorrelation function describes the correlation between a measurement profile and the same profile spatially shifted by a distance t_x. When applied to a residual noise profile, the ACF shape can distinguish different forms of spatial noise behavior. The ACF rapidly approaches zero for non-zero spatial lags, indicating weak spatial correlation and suggesting that the measurement noise is approximately white. For a finite data set, however, the estimated sample autocorrelations typically fluctuate around zero rather than being exactly zero. Following the statistical criterion used by NIST [27], approximate 95% confidence limits for the autocorrelation coefficients under the white noise hypothesis are given by $\pm {\displaystyle \frac{1.96}{\sqrt{N}}}$, where N is the number of observations. In contrast, a slow decrease of the ACF indicates spatial correlation, while oscillations or repeated peaks indicate periodic components in the residual signal, potentially related to vibration, motion dynamics, or other periodic disturbances. The ACF complements RMS based amplitude approach.

3.3. Power Spectral Density

The power spectral density (PSD) of a surface is a mathematical tool that decomposes a surface into contributions from different spatial frequencies (wavevectors). Mathematically, the PSD is the Fourier transform of the autocorrelation function of the signal, which contains only the power (not the phase) across a range of wavevectors [28]. Equivalently, by virtue of the convolution theorem, the PSD can be obtained directly from the surface signal as the squared magnitude of its Fourier transform [29].

The spatial-frequency content of the signal can be characterized using the power spectral density. For areal surface texture, ISO 25178-2 [20] defines the areal PSD, while for profile measurements, ISO 21920-2 [21] defines the PSD. In both standards, PSD describes the distribution of signal power in the spatial-frequency domain. PSD characterization of measurement noise is particularly important when applying filtering or noise-bias removal. Complementing ACF-based analysis, Podulka et al. used PSD to examine the spectral characteristics of high-frequency measurement noise and to distinguish noise-related components from the measured surface topography. These approaches were applied to surfaces produced by different processing routes, including laser texturing, fatigue bending, milling, burnishing, grinding, and turning, demonstrating the applicability of PSD-based noise characterization across a broad range of surface conditions [30,31,32].

Filtering can reduce measured noise, but it may also alter the measured surface texture because it removes or reduces some spatial-frequency components. ISO 25178-700 [15] therefore requires filters that influence the spatial frequencies over which noise is determined to be reported. Knowledge of the noise spectrum is also important for noise-bias removal. Spectral analysis can be used to identify noise-related Fourier components and support spectral noise-bias correction or reduction.

Table 4 summarizes the methods used to characterize and analyse measurement noise, along with their purpose, analytical outcome, and relevant standards.

4. Mathematical and Statistical Characterization of Noise

Measurement noise is generally regarded as a non-deterministic or stochastic component of the measurement signal. However, unwanted contributions in a measurement system may include both deterministic and stochastic components. In roundness measurement, for example, spindle error may include a repeatable component associated with the spindle’s angular position, along with a random component arising from mechanical, thermal, or operational variations. The repeatable component is therefore treated as a deterministic error contribution, whereas the random component contributes to the overall measurement noise [3].

A simple statistical characterization of the measured profile may be expressed with an additive model:—z_m(x) = z(x) + e_r(x) + n(x) (3)

where z_m(x) denotes the measured profile, z(x) the true profile, e_r(x) the repeatable and therefore deterministic error component, and n(x) the random component—measurement noise.

Once the repeatable error component has been identified and subsequently removed (by averaging or by subtraction), the measurement model can be simplified to:

z_m(x) = z(x) + n(x) (4)

If n(x) is regarded as a random process, its first and second order statistical behaviour can be described by its mean, variance, and autocorrelation function.

$${\mu }_n\left(x\right)=E\left[n\left(x\right)\right];{\sigma }_n^2\left(x\right)=E\left[{\left(n\left(x\right)-{\mu }_n\left(x\right)\right)}^2\right];R_{\mathit{nn}}\left(x,x+\tau \right)=E\left[n\left(x\right)n\left(x+\tau \right)\right]$$

(5)

where τ represents a temporal or spatial lag, depending on whether x denotes time or position. In an idealized measurement model, the noise is often assumed to have zero mean: μ_n(x) = 0.

A random process is called nonstationary if its mean μ_n(x) or autocorrelation function R_nn(x, x + τ) changes with the absolute position (time) x [33,34,35]. Nonstationarity may arise from thermal drift, changes in vibration conditions, gradual changes in contact conditions, variations in gain or speed, or other effects whose intensity changes during the measurement. In this context, a further distinction can be made between short-term and long-term noise, depending on the characteristic temporal or spatial scale over which the fluctuations occur. Short-term noise is associated with rapidly varying fluctuations occurring over relatively small scales. In contrast, long-term noise reflects slower variations that become apparent over longer measurement intervals or larger spatial scales [36].

In contrast, a process is called weakly stationary, or wide-sense stationary (WSS), if its first- and second-order statistical properties do not depend on the absolute time or position at which the process is observed. In this case, the mean is constant, μ_n(x) = μ_n and the autocorrelation function depends only on the lag τ: R_nn(x, x + τ) = R_nn( τ). For measurement-noise analysis, WSS is often the most practical form of stationarity to consider. Strict stationarity is a stronger condition. It requires all joint probability distributions of the process to remain unchanged under an arbitrary common shift in time or space. In practical metrology, however, strict stationarity can rarely be demonstrated directly from limited measurement data. Ergodicity is practically important because it allows the statistical properties of a process to be estimated from a single sufficiently long realization. For an ergodic process, the temporal or spatial average of one realization converges to the corresponding ensemble expectation. Ergodicity is a fundamental assumption in surface roughness characterization [13], allowing spatial statistics from a single profile to represent an ensemble of statistically similar surfaces. The same concept underpins statistical modeling of noise in profile measurements, where noise is treated as a random process and quantified using parameters such as variance, standard deviation, and RMS. Only stationary random processes can be ergodic.

Measurement noise can also be characterized according to its spectral properties. In this context, white and coloured noise are distinguished [37,38]. Ideal white noise has a constant power spectral density and uncorrelated samples. In real measurement systems, this assumption is valid only within the frequency range relevant to the measurement system under consideration. If the power spectral density depends on frequency, the noise is generally described as coloured noise. Filtering can also change the noise’s spectral characteristics. White noise may become colored after filtering.

Depending on the dominant frequency range, measurement noise may also be discussed in terms of low-frequency and high-frequency components. Low-frequency noise is often associated with slow variations, such as amplifier drift, or with (1/f) type behavior [39]. High-frequency noise, on the other hand, may originate from mechanical vibrations and related environmental influences, as well as from internal electrical sources [40]. This distinction is particularly relevant when the dominant noise mechanisms occur in different frequency ranges.

Independent of its spectral properties, noise can also be classified by its amplitude distribution, for which a Gaussian distribution is frequently assumed. When noise has both an approximately constant power spectral density and a Gaussian amplitude distribution, it is called white Gaussian noise (WGN), one of the most widely used idealized noise models.

Measurement noise can introduce a systematic positive bias into surface parameters that are based on squared profile deviations, even when the noise itself has approximately zero mean E[z(x)] = 0. For an additive measurement model in which the true profile and measurement noise are statistically uncorrelated, the measured RMS roughness can be expressed as:

$$E\left[z_m{\left(x\right)}^2\right]=E\left[{z\left(x\right)}^2\right]+E\left[{n\left(x\right)}^2\right]+\underset{0}{\underbrace{2E\left[z\left(x\right)n\left(x\right)\right]}}$$

(6)

Relationship for the RMS roughness parameter can be written as: ${\mathit{Rq}}_m^2={\mathit{Rq}}^2+{\mathit{Rq}}_n^2$. This means that measurement noise introduces a positive bias into the evaluated Rq² parameter, causing the measured ${\mathit{Rq}}_m^2$value to be systematically higher than the value associated with the underlying profile alone [18,41].

5. Review of Recent Studies on Measurement Noise

Earlier work on specification standards was focused almost entirely on surface profile measurement, whereas the systematic development of traceability and characterization of areal surface texture intensified only during the 1990s. Leach et al. [42] state that systematic consideration of traceability and areal surface texture characterization began in 1996, followed by further development through European research and standardization projects that contributed to establishing a structured framework for areal surface texture metrology. The first international standards of the ISO 25178 series for areal surface metrology were published in 2010, establishing a formal framework for areal surface characterization. Subsequent revisions and extensions further refined the definition of areal parameters, metrological characteristics, measurement methods, and requirements for contact and non-contact measuring instruments. Parallel with the development of standardization, research in surface texture metrology also shifted toward areal measurement and 3D surface characterization. Accordingly, areal surface metrology increasingly addresses measurement noise both as a metrological characteristic and as a contributor to measurement uncertainty.

In the uncertainty budget, measurement noise is often considered together with residual flatness/profile, since both components can be estimated from the same set of repeated measurements and, with an appropriate measurement strategy, their individual contributions can subsequently be separated. Haitjema [43] treated residual flatness and measurement noise as a combined contribution to the measurement uncertainty of the Ra, RSm, and Sa parameters obtained using a mechanical stylus surface-measuring instrument. In addition, he proposed a repeated-measurement approach that enables separation of measurement noise, specimen-related variation, and residual flatness. In the study by Gao et al. [44], the metrological characteristics of optical areal surface measuring instruments were investigated for techniques including coherence scanning interferometry (CSI), confocal microscopy (CM), and focus variation (FV). Measurement noise was treated as a separate contribution to measurement uncertainty. It was determined using the subtraction method, while repeatability was additionally included as a Type A uncertainty contribution based on repeated measurements. Since repeatability reflects random variation between repeated measurements, it may inherently include a component associated with measurement noise. Leach et al. [18,45] treated measurement noise and flatness deviation as separate contributions in the measurement uncertainty budget in a case study based on CSI. They also introduced a distinct noise-bias contribution representing the systematic increase in the evaluated Sq value caused by measurement noise.

Beyond its role in measurement uncertainty evaluation, detailed characterization of measurement noise is also essential for selecting appropriate noise-reduction strategies and minimizing its influence on the measured surface topography. Vanrusselt et al. [46] made an important contribution to the standardization and characterization of measurement noise by conducting an extensive international comparison of twelve optical surface topography measuring instruments across six European research laboratories. The study evaluated the practical applicability of the metrological definitions and verification procedures specified in ISO 25178-600 [10] and ISO 25178-700 [15] for FV, CM, and CSI. The authors showed that the subtraction method can estimate measurement noise from only two repeated measurements. However, substantial differences in noise magnitude were observed between individual instruments, including instruments based on the same measurement principle. Importantly, the study also showed that the conventional RMS repeatability exhibited little correlation with the ISO-defined measurement noise and, in some cases, underestimated the noise magnitude by more than two orders of magnitude. These findings highlight the importance of standardized noise-characterization procedures and reporting measurement conditions such as measurement speed, filtering, and effective lateral resolution.

For a more detailed characterization of measurement noise, Vanrusselt and Haitjema [47] applied temporal and lateral Allan deviation to measurements obtained with CSI, confocal microscopy, and focus variation instruments. The temporal Allan deviation identified non-stationary noise effects such as drift and spikes during repeated measurements, whereas the lateral Allan deviation revealed spatial correlations and the influence of implicit or explicit filtering, showing that measurement noise cannot always be assumed to be purely stationary and randomly distributed. Gomez et al. [48] showed that topography averaging directly reduces spatial height noise by taking the arithmetic mean of N consecutive topography maps, effectively suppressing both electronic and vibrational fluctuations regardless of surface tilt.

In most of these studies, researchers characterized measurement or instrument noise using reference surfaces, most commonly optical flats or roughness standards. In contrast, Podulka [49,50,51] conducted a comprehensive series of studies characterizing high-frequency measurement noise directly on real engineering surfaces, such as plateau-honed cylinder liners and turned, ground, milled, laser-textured, ceramic, and composite surfaces. Recognizing that complex functional topography features (e.g., burnished oil pockets, valleys, and scratches) can distort power spectral density and autocorrelation assessments, these studies demonstrated that high-frequency noise is often more reliably detected through 2D profile analysis of “out-of-feature” (free-of-dimple). To minimize the impact of noise without distorting essential topography features or treatment traces, Podulka systematically evaluated and compared various filtering strategies, including Fast Fourier Transform filtering (FFTF) [49,50], wavelet-based noise extraction procedures (WNEP) [51], and modified spline filters (e.g., open-profile spline filters) [40]. These works established multivariate evaluation protocols combining PSD, ACF, Texture Direction (TD) graphs, and the assessment of highly noise-sensitive areal parameters. Characterizing noise across a range of real engineering surfaces provided the basis for developing noise-reduction procedures targeting specific frequency components. Dominant high-frequency contributions were identified and attenuated using Fast Fourier Transform filtering in combination with Frequency Spectrum-Based Techniques (FSBT), thereby reducing unwanted noise while preserving the functionally relevant features of the measured surface. Haitjema and Morel proposed a novel approach aimed at eliminating noise-induced bias rather than suppressing selected frequency components for both profile [41,52] and surface [53] measurements. The approach assumes that the power of the measured profile is the sum of the true surface contribution and an independent noise contribution, while averaging n repeated measurements reduces the noise power proportionally to 1/n without affecting the true profile component. By exploiting this relationship, the noise-free Rq² or Sq² value can be obtained by extrapolation to 1/n=0 and the same principle can subsequently be applied to individual Fourier components to reconstruct a bias-reduced profile. Haitjema and Morel [54] demonstrated that the conventional averaging approach for noise reduction can be replaced by a noise-bias removal method that achieves comparable or improved results with substantially fewer repeated measurements. To extend the noise-bias reduction method to areal surface measurements, Vanrusselt and Haitjema [55] replaced the Fourier decomposition used for profiles with Legendre polynomial decomposition, which better represents smooth 2.5D surface forms.

6. Conclusions

Measurement noise is an important factor in high-accuracy surface texture metrology, particularly when its magnitude approaches the scale of the surface features being measured. Under such conditions, noise can affect the measured profile, influence evaluated surface texture parameters, and contribute to measurement uncertainty.

This review shows that the terminology and approaches used to describe noise have evolved considerably in surface texture metrology. Earlier profile-based frameworks relied on concepts such as the residual profile and basic noise, whereas the ISO 25178 framework introduced a more explicit distinction between static, dynamic, instrument, and measurement noise. Although these concepts are related, they refer to different measurement conditions and have different metrological interpretations. In particular, instrument noise describes the measuring system’s intrinsic noise contribution under idealized conditions, whereas measurement noise represents the noise present during normal use of the instrument. From a measurement-uncertainty perspective, the latter is therefore particularly important because it is directly associated with the measurement result.

The reviewed approaches also demonstrate a transition from relatively simple amplitude-based descriptions towards a more comprehensive characterization of measurement noise. Maximum-height parameters provide a scalar indication of noise magnitude but are particularly sensitive to individual peaks. RMS-based quantities provide a statistical measure of noise amplitude and form the basis of the measurement-noise procedures established in ISO 25178-700. The subtraction and averaging methods specified in this framework use repeated measurements to separate the non-repeatable component from the repeatable surface topography. These procedures therefore provide a more direct way to estimate the noise affecting the measurement than approaches based solely on a single measured profile.

Amplitude characterization alone, however, cannot fully describe measurement-noise behavior. Measurement systems with similar RMS noise levels may exhibit substantially different spatial or spectral characteristics. Autocorrelation function and power spectral density analyses provide complementary information on spatial correlation, periodic components, and frequency content, while repeated noise determinations can provide information on temporal stability and possible non-stationary behavior. A combination of amplitude, spatial, spectral, and stability-related characterization therefore provides a more complete description of measurement noise and can support both the identification of dominant noise contributions and the selection of appropriate noise-reduction strategies.

A central consideration arising from this review is that the comprehensive measurement-noise framework currently available for areal surface topography is not inherently restricted to areal data. ISO 25178-700 itself recognizes that many of its procedures can be adapted to profile measurements. In particular, the principles of subtraction and averaging transfer directly to repeated profile measurements by replacing the areal RMS parameter with the corresponding profile parameter. The same residual profiles can subsequently be examined using ACF and PSD analyses, which are already defined for profile surface texture within ISO 21920-2. This provides a consistent route for characterizing profile measurement noise by its amplitude, spatial correlation, and spectral content.

Such an extension is particularly relevant to high-accuracy contact stylus instruments used for calibration and traceability. It would allow measurement noise to be considered separately from repeatable contributions associated with the instrument datum and the measured surface, rather than combining these effects into a single residual quantity. Measurement noise could then be introduced as a dedicated contribution to the measurement uncertainty budget, while instrument noise could remain a complementary characteristic describing the measuring system’s intrinsic noise performance. This distinction would provide a clearer connection between instrument characterization, actual measurement conditions, and uncertainty associated with the measurement result.

Further work should therefore focus on experimentally validating these adapted procedures for profile measurements. Particular attention should be given to how measurement bandwidth, environmental conditions, surface characteristics, and the number and timing of repeated measurements affect the resulting noise estimate. Such investigations could support the development of practical guidance for measurement-noise characterization in profile surface texture metrology and contribute to a more consistent treatment of measurement noise in the calibration and uncertainty evaluation of high-accuracy contact stylus instruments.