The usual way to combine uncertainties assumes the measurement model behaves like a straight line over the small range where the errors live, so that each input's uncertainty can be scaled by a slope and added in quadrature. For many measurements that assumption is fine, but for strongly nonlinear models it can misstate the result. A Monte Carlo uncertainty budget takes a different route: instead of linearising the model, it simulates the measurement thousands of times with randomly drawn inputs and looks at how the results spread out. This guide explains what the Monte Carlo method is, why it suits highly nonlinear flow models where the analytical law of propagation struggles, and how it fits alongside the traditional approach.
Monte Carlo Uncertainty in one line: A Monte Carlo uncertainty budget evaluates measurement uncertainty numerically by drawing many random samples from the probability distribution of each input, running each set of samples through the measurement model, and building up the distribution of the result from those many outcomes. From that simulated distribution it reads off the standard uncertainty and a coverage interval directly, without relying on the analytical law of propagation of uncertainty. It is the method described in GUM Supplement 1 and is especially useful for strongly nonlinear models, such as orifice flow metering, where the linear analytical approach can be inaccurate.
The traditional analytical method combines uncertainties by treating the measurement model as approximately linear around the operating point: it takes the slope of the model with respect to each input, scales each input's standard uncertainty by that slope, and combines the results in quadrature. This is the law of propagation of uncertainty, and it is really propagating slopes and standard deviations, not full distributions. It works well when the model is close to linear over the range the input errors span, which is often the case.
The Monte Carlo method throws out the linearisation and works with the whole distributions instead. For each input you specify not just a standard uncertainty but a probability distribution - normal, rectangular, triangular, or whatever fits how that input is known. The computer then draws a random value from each input's distribution, feeds that complete set of values through the exact measurement model, and records the result. Doing this a large number of times produces a large sample of possible results, and the spread of that sample is the distribution of the measurement itself.
From the simulated distribution of results you read the answers off directly. The standard deviation of the simulated results is the combined standard uncertainty, and a coverage interval is obtained by finding the range that contains the desired proportion of the simulated outcomes. Crucially, none of this required assuming the model was linear or assuming the result would be normally distributed; the shape of the output distribution emerges from the simulation, whatever it turns out to be. That is the essential difference: the analytical method propagates summary statistics through a linearised model, while Monte Carlo propagates the full distributions through the exact model.
The place this really earns its keep is a strongly nonlinear measurement, and orifice flow metering is the standard example. In a differential-pressure meter the flow rate goes with the square root of the differential pressure, and the full calculation folds in density, expansion, and coefficient terms that make the overall model distinctly nonlinear. When a model curves like this, the slope used by the analytical method is only exactly right at the single point where it is evaluated, and it drifts as you move away from that point. If the input uncertainties are large enough to reach into regions where the slope has changed, the linear approximation starts to misstate the result.
Two things go wrong with the linear approach in these cases. First, the combined standard uncertainty itself can be off, because a single slope cannot represent a curved response over a wide input range. Second, and often more importantly, the result's distribution may not be symmetric or normal, so the usual step of multiplying by a coverage factor and quoting a symmetric interval can misrepresent the real coverage. A square-root response, for instance, skews the output distribution, and a symmetric interval centred on the estimate no longer contains the intended proportion of outcomes on each side.
Monte Carlo handles both problems naturally because it never linearises and never assumes a shape for the answer. It runs the actual nonlinear flow equation for every sampled set of inputs, so the curvature is fully respected, and it reads the coverage interval from the simulated distribution, so any skew or asymmetry is captured as it really is. This is why GUM Supplement 1 positions the Monte Carlo method as the tool to reach for when the conditions that justify the analytical law of propagation are in doubt, and why it is increasingly used to check or replace the traditional budget for demanding flow measurements.
The Monte Carlo method is not a rejection of the analytical budget so much as a more general tool that includes the simpler case. Where a model really is close to linear and the inputs are well behaved, the two approaches agree closely, and the analytical method has the advantages of being fast, transparent, and easy to document line by line, showing exactly how each input contributes. Many measurements are handled perfectly well by the analytical budget, and there is no need to simulate what a slope and a quadrature sum already capture correctly.
The practical way the two fit together is often to use the analytical budget as the working method and to use Monte Carlo as a validation, running the simulation to confirm that the linearised result is trustworthy for a given measurement, or to correct it where the model turns out to be too nonlinear for the linear law. When the two disagree, that disagreement is itself informative, because it flags that curvature or a non-normal output is significant enough to matter. This makes Monte Carlo valuable even in shops that quote analytical budgets, as the check that tells them when the simpler method is safe.
For flow and process measurement carried in flow computers and SCADA systems, the connection is that the same inputs the analytical budget uses - differential pressure, pressure, temperature, and the coefficient and geometry terms - are exactly the inputs a Monte Carlo simulation samples. A cloud SCADA platform such as Merobix that logs those inputs and their operating ranges provides the real-world context that makes a Monte Carlo study meaningful, since the input distributions should reflect how the measurement actually operates, not just nominal conditions. The method itself is an offline computation rather than something the SCADA runs live, but the data the SCADA collects is what grounds the input distributions and lets an engineer confirm that a metering point's stated uncertainty holds up under the nonlinear reality of how it runs.
It works by drawing many random samples from the probability distribution of each input, running each complete set of samples through the exact measurement model, and collecting the results. Repeating this a large number of times builds up the distribution of the measurement result, from which the standard uncertainty and a coverage interval are read directly. It never linearises the model or assumes the result is normally distributed.
The standard analytical method linearises the model, using a single slope for each input and combining in quadrature, which is only accurate when the model is close to linear over the range the input errors span. For strongly nonlinear models like orifice flow metering, where flow goes with the square root of differential pressure, that assumption can misstate the uncertainty and produce a skewed output distribution. Monte Carlo runs the exact nonlinear model and captures the true distribution, so it is more reliable in those cases.
GUM Supplement 1 is the document that describes propagating distributions by a Monte Carlo method as an alternative to the analytical law of propagation of uncertainty in the main GUM. It sets out how to represent each input by a probability distribution, sample from them, propagate the samples through the model, and derive the result's distribution and coverage interval. It is the reference practitioners cite when using Monte Carlo for uncertainty evaluation.
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