2023-04-20
Uncertainties in Measurements
When it comes to gas concentration measurements, it is a matter of complex systems with many individual sources of uncertainty.
A single source can perhaps be capable of reporting the measurand within a single percent from the actual value or so, sometimes even better. However, taking all factors into account, the total uncertainty is usually an order of magnitude higher, also when the instrument performs at its best. So what is the measurement uncertainty in practice?
The Probability Approach
Uncertainties are not calculated by imagining what would be the worst possible error for each source of uncertainty, and then adding all worst-cases to form a total uncertainty. The probability of that happening is far too low – it is unlikely that an individual error is maximised, and even more unlikely that all potential errors are maximised and drive the resulting total error in the same direction.
Instead, uncertainty calculations are based on probabilities with distribution functions and standard deviations for each source of uncertainty. The total uncertainty is usually expressed based on a 95 % confidence interval. This reduces the effects of individual sources and acknowledges that there is no such thing as an absolutely true uncertainty, but just a reasonable estimation of what the uncertainty might be.
A Certain Example of Uncertainty
Looking at gas concentration monitors to be approved under European requirements, the uncertainty sources to be considered and the requirements on total uncertainty are stated in EN 15267-3 and other EN standards – google “QAL1” and thou shall find.
Among the sources are repeatability, reproducibility, cross-sensitivity, lack-of-fit, temperature and pressure sensitivity, long-term drift, and uncertainty in calibration gas. In each case, both a distribution function and an individual uncertainty (standard deviation) must be estimated. Then, the combined (total) uncertainty is calculated as the “square root of the sum of the uncertainties squared”. Finally, the “total expanded uncertainty” is calculated as two times the combined uncertainty, to express the 95-percentile of the error at the nominal value.
The Percentile Notation
This can be stated as a percentage requirement: with 95 % confidence, the value reported by the instrument must be within ±XX % of the actual level, usually an emission or air quality limit. Alternatively expressed, 95 % of a large number of measurement results at the level of the limit value must be within ±XX % of the true value, i.e. the limit.
Permissible Uncertainties
“XX” above varies between different types of gases. This is because some measurement techniques have lower built-in uncertainties than others, and to be fair, the requirements must relate to the techniques practically applicable to the detection of a specific gas. By example, to pass QAL1 tests, the relative total expanded uncertainty must be lower than 15 % for SO2 and NOX, but only lower than 30 % for HCl and HF.
The percentages stipulated by the QAL1 requirements tell how bad it may get without disqualifying the instrument type. An actual, individual instrument is normally better than this (95 % of all readings are within the ±XX % uncertainty) but the measurement errors are still roughly in the range of 10 % or so, many a bit better but some also a bit worse.
As Good as it Gets
A ten percent measurement error may sound terrible. However, that is sometimes as good as it gets with today’s technology readily available for practical use. The acceptance of uncertainties in the range of 10 % is confirmed by for example the European Union Industrial Emissions Directive (the IED) which effectively gives a discount on the measurements before any comparison is made against emission limits. The discount process is called “validation”, and simply means subtracting a certain percentage of the emission limit value from the reported levels.
By example, the percentage is 10 % for CO, 20 % for SO2 and NOX, and 30 % for dust. This compensates for the risk of a correctly operating measurement system (within the uncertainty stipulated by EN 15267-3) reporting “on the high side” and thereby potentially claiming exceedances of emission limits that are false. Thanks to the discount, a reported exceedance is most likely also a true exceedance (with 95 % confidence), all uncertainties considered.
