# Application of Areal Topography Parameters in Surface Characterization
## 3 Assessment of the Quality of Measurement Results Obtained via an Atomic Force Microscope  
The quality of measurement results obtained using an atomic force microscope (AFM) depends on several interrelated factors, including system calibration, environmental stability, and the choice of scanning parameters. An accurate interpretation of surface characteristics requires not only high-resolution data but also confidence in the traceability and repeatability of results. Therefore, the assessment of measurement quality implies the evaluation of measurement uncertainty. Measurement uncertainty is defined as a non-negative parameter that characterizes the dispersion of quantity values attributed to a measurand [54]. According to the _Guide to the Expression of Uncertainty in Measurement_[55], measurement uncertainty is a parameter associated with the result of a measurement, characterizing the dispersion of values that could reasonably be attributed to the measurand. Uncertainty implies doubt, and in its broadest sense, measurement uncertainty reflects doubt about the validity of a measurement result. According to Measurement Systems Analysis [56], measurement uncertainty is an estimate of the range within which the true value is believed to lie. A clear understanding and proper estimation of measurement uncertainty is essential for ensuring the reliability and comparability of AFM-based results, particularly in high-precision or interlaboratory applications. Although surface topography is a well-established field in dimensional metrology, there is still limited work in the literature that explicitly recognizes the importance of measurement uncertainty estimation as a quality indicator of measurement results. In many studies, results are reported without a corresponding uncertainty evaluation, which limits their reliability and comparability, especially in high-precision applications. A significant contribution in this context is the doctoral dissertation of one of the authors, titled Estimation of Measurement Uncertainty in the Field of Atomic Force Microscopy in Dimensional Nanometrology [57], which presents research findings focused on uncertainty estimation in AFM-based surface measurements. The dissertation includes the development of mathematical models for estimating measurement uncertainty, calculations based on both Monte Carlo and Bayesian methods, and results from interlaboratory comparisons. These findings illustrate how various sources of uncertainty can significantly influence the reliability and variability of AFM measurement results. For uncertainty evaluation, the following influential input quantities were analyzed: measured value (\(h_{x}\)), scan rate (\(r\)), scanning time (\(t\)), scan length (\(l\)), nominal step height (\(d\)), thermal expansion coefficient (\(\alpha\)), temperature difference (\(\Delta\vartheta\)), repeatability (\(\delta r\)), reproducibility (\(\delta R\)), and probe-related uncertainty (_probe_). The evaluation model is given by expression (1). \[h=h_{x}+r\cdot t\cdot l+d\cdot\alpha\cdot\Delta\vartheta+\delta r+\delta R+ \delta probe \tag{1}\]  
The probability density function for the step height was obtained using 100,000 simulations and is shown in Figure 6.  
The expanded measurement uncertainty at a confidence level of \(p=95\%\) is \(U=2.9\) nm, with the corresponding interval limits \(Y_{0.025}=96.0\) nm and \(Y_{0.975}=101.8\) nm (red lines in Figure 6). This result was compared with the certified value in the standard's calibration certificate. Results are shown in Table 2. Our results are labeled as Lab A in the table. The comparison of results is performed by calculating the compatibility factor \(En\), Equation (2).