Computational Uncertainty

Surface measurement instruments, such as stylus and optical profilers, are used to characterize the roughness of engineering surfaces. Most of these measurement instruments are microcomputer-based systems, which contain both surface analysis software and data storage facilities. Each measurement instrument has its own unique analysis software and data format. When a surface is measured, the measurement instrument generates a data file and stores it on a local disk. The data file is then analyzed by the analysis software provided in the surface measurement instrument. When compared, different analysis software implementations can yield significant differences in output parameters. The factors affecting software performance include the choice of analysis method, the quality of software, and characteristics of data points. The diagram and equations below shows some sources of uncertainty that contribute to   measurement results by analysis software.


This diagram shows some sources of uncertainty that contribute to measurement results by analysis software. The observable data points after a surface profile measured contain both measurement errors and true surface data point. When these observable data points are inputted into the analysis software, the reported value will include true value, computational errors, and propagated data errors.


Let f : R -> R and x -> f(x). When we have an inexact input x^ and approximate function f^, total error = f^(x^) -f(x) =[f^(x^) - f(x^)] + [f(x^) - f(x)] = computational errors + propagated data errors = truncation / rounding errors + condition of the problem



Uncertainty due to propagated data errors:

This diagram shows how computational uncertainty for calculated surface parameter is calculated. A total profile, noise in z direction with standard uncertainty of 1% of Pq, and noise in x direction with standard uncertainty of 2% of spacing are inputted into the Monte Carlo Simulation. Inside Monte Carlo Simulation, it contains Gaussian filter or 2RC filter block connecting to Surface Parameter Calculation block. The modified profiles (roughness and waviness) from filter block are inputted into Surface Parameter Calculation block to calculate surface parameter. By iterating the simulation N time, an array of roughness parameter and waviness with length N can be obtained. Mean and standard deviation of roughness parameter and waviness parameter array are obtained with 95% converge.


Monte Carlo Simulation
Example:

Monte Carlo Example for Ra Parameter. Ra parameter has a mean of 0.6399 micron and standard uncertainty of 6.80 nm.
Distribution function of Ra


Uncertainty due to truncation and rounding errors:

See Section F.2.2.3 of "American National Standard for Calibration - U.S. Guide to the Expression of Uncertainty in Measurement".


References
American National Standard for Calibration - U.S. Guide to the Expression of Uncertainty in Measurement, Boulder, CO : NCSL International , c1997 (2000 printing)

M. G. Cox, M. P. Dainton, and P. M. Harris, Software Support for Metrology Best Practice Guide No. 6, Uncertainty and Statistical Modelling, National Physical Laboratory, UK, 2002


Michael Krystek, Measurement uncertainty propagation in the case of filtering in roughness measurement, Measurement Science and Technology, 12, Issue 1, 2001, p. 63-67