Why does understanding the uncertainty of a measurement matter? A particularly egregious example would be filling up your 15-gallon gasoline tank with the meter (and subsequently the price) reading 42 gallons. This example would be sure to catch the attention (and incite the anger of) the consumer quickly. However, in other more critical situations, such as ensuring the shutdown of a nuclear reactor during a seismic event, vibration measurement levels need not only report values close to the actual physical event, but the uncertainty of the measurement should be known. This allows factors of safety to be built into systems with regard to uncertainty so that the outcome of an event (seismic or otherwise) can produce a repeatable response. Essentially, a measurement by itself is rendered useless without stating the bounds of uncertainty, since all measurements are guaranteed to have some level of error. Accurate interpretation of the measurement results requires consideration of the uncertainty in the measurement, such that the measurement becomes more of a range than a number.
So what is uncertainty? At a fundamental level, uncertainty can be defined as “…doubt about the validity of the result of a measurement” [GUM]. Uncertainty can encompass anything from the susceptibility of a measurement to temperature changes to the inaccuracy associated with quantization error in data acquisition. In order to convey the analysis of measurement uncertainty, an uncertainty budget is used to summarize and clearly present these sources. A powerful tool in assisting you on your way to generating your own uncertainty budget is BIPM’s Guide to the Expression of Uncertainty in Measurement (or GUM for short). This guide gives a broad overview of guidelines for evaluating and calculating the uncertainty of measurement.
Contributors to measurement uncertainty can be divided into two subgroups: Type A and Type B. Type A data refers to the characterization of a random uncertainty contribution – often through empirical gathering and statistical analysis. An important notion to grasp with Type A data gathering is that once completed, the dataset will encompass all random influences on the measurement process. Using vibration metrology as an example, this contribution can be characterized at different frequency ranges through the repeated calibration of a test accelerometer. This means that many uncertainty components can be characterized by this testing such as: Cable and connector influences, operator technique, acquisition noise, and mounting torque variation. Data can even be taken over a longer period (months) to allow for determination of the influence of environmental factors on the measurement.
Type B uncertainty evaluation encompasses all contributors that can be determined through known specifications or characterized from the results of data that produces a systematic (non-random) error. Drawing from our vibration measurement uncertainty illustration, some examples of Type B uncertainty components include: calibration uncertainty of the reference accelerometer (for back-to-back comparison calibrations), drift of the reference accelerometer (whose contribution will be modified by a chosen calibration interval), and transverse and rocking motion of the vibration shaker.
In addition to BIPM’s GUM, applicable ISO standards (such as 16063-21 for secondary vibration calibration by comparison) can include common sources of uncertainty for the described calibration method. Annex A of 16063-21, for example, gives a near comprehensive list of possible contributors. Descriptions for calculating method-specific contributors (such as transverse and rocking motion), are also described in detail. Depending upon the specific equipment used for measurement, or the specific algorithms used for data acquisition and analysis, the uncertainty contributions from each of these may be more or less valid. For transverse and rocking motion specifically, a key performance specification of the modern calibration shaker, understanding and reporting the measurement uncertainties due to this contributor is much more significant when using a traditional flexure-based shaker as compared to a modern air bearing calibration shaker (see "The Effect of High Transverse Inputs on Accelerometer Calibration" by Rick Bono and Eric Seller for more information). Using The Modal Shop’s uncertainty budgets as an example, in the 920 Hz-2 kHz range, our mainstream accelerometer calibration using a modern air bearing shaker adds a relative uncertainty contribution of 0.34% (k=2). By comparison, our uncertainty budget when using a flexure-based shaker, necessary for larger payloads such as very heavy seismic accelerometers in the exact same frequency range, adds a relative uncertainty contribution of 1.66% (k=2) - nearly 5x more. In practice, there are many laboratories currently using flexure-based shakers which exhibit transverse motion characteristics beyond the recommendations of table 2 in section 4.4 of ISO standard 16063-21. This does not invalidate the labs’ capability to perform proper vibration metrology measurements, as long as the contribution to measurement uncertainty results is properly stated. To reiterate from last month’s article, it is the responsibility of the laboratory to make sure that the reported values of expanded uncertainty are credible.