## 4 Instrument Noise  
From the observations above, it would appear prudent to avoid quoting the vertical resolution of instrument, except as a qualitative idea. This leaves us without a performance parameter related to the minimum detectable height difference. A reasonable alternative is to explicitly quantify the measurement noise--an idea supported by established norms (VIM 4.14 [7]). For the purpose of specifying instrument performance, ISO 25178 documents recognize instrument noise as the internal noise added to the output signal by the metrology system when it is placed in an ideal environment with minimal disturbances [6; 9]. A basic test for noise in areal surface topography maps consists of repeated measurements on a polished, flat part. The surface topography repeatability (STR) tells us how close we can expect the indicated height value for a specific sample point (or camera pixel) to repeat if we measure it over and over again without changing the conditions of measurement [9]. The value is a standard deviation or noise-equivalent \(S_{d}\), and is readily computed from statistics over a full image of surface points. \begin{table}
\begin{tabular}{c c c} \hline
**Instrument** & **Specification** & **Value (nm)** \\ \hline A & Height resolution & 0.001 \\ B & Repeatability of surface RMS (Z) & 0.003 \\ C & RMS repeatability (RMS\(\sigma\)) & \textless{}0.01 \\ D & Vertical resolution & 0.01 \\ E & RMS repeatability of surface accuracy & 0.01 \\ F & RMS repeatability & \textless{}0.02 \\ G & Noise floor & 0.05 \\ H & Vertical resolution & \textless{}0.1 \\ I & Vertical resolution & 0.1 \\ J & RMS repeatability & 0.3 \\ K & Vertical resolution & 1 \\ \hline \end{tabular}
\end{table} 
Table 1: Published specifications related to the concept of vertical resolution for a selection of commercially manufactured 3D interference microscopes. A simple STR test is to measure a smooth, flat part twice, subtract the difference of the two resulting topography maps, and calculate the standard deviation \(S_{q}\) of this difference map [6; 9; 14]. The STR, denoted here as \(N_{M}\), is  
\[N_{M}=S_{q}/\sqrt{2} \tag{1}\]  
Several methods use multiple measurements to provide a robust estimate of STR [9; 15; 16]. Common practice in areal measurement specifications for decades has been to quote a vertical resolution or its equivalent without any indication of how long it would take to achieve it. This is contrary to good engineering practice in specifying sensor noise [17; 18]. Excluding the measurement time in a specification can lead to expectations that do not correspond to realistic instrument use. Conversely, once we have a noise level expressed with respect to a time bandwidth, we can often adjust the data acquisition time or number of averages to reach a desired measurement precision. To illustrate the importance of including data acquisition time in an STR evaluation, Figure 2 shows the use of time averaging with sinusoidal phase shifting interferometry [19]. A 100 Hz, 1024 \(\times\) 1024 pixel camera allows for five averages per second, resulting in a measurement noise level of 0.072 nm/Hz. Another approach to reducing noise is to more densely sample the signal, which can trade a longer acquisition time for lower noise levels without averaging individual measurements [20]. For instruments that scan over an adjustable range of surface heights, the data acquisition rate is not easily expressed as a time bandwidth in Hz. For these systems, a meaningful expression of STR would be a dimensional value together with a height-scanning rate and the corresponding camera format, as in STR = 0.1 nm at 10 \(\upmu\)m/s for 1 k \(\times\) 1 k pixels. Another important factor in evaluating the STR as a measure of performance relates to spatial filtering. The application of a denoising or high spatial frequency S filter can improve the STR at the expense of reduced lateral sampling.