The challenge stems from the nature of wildfire emissions themselves. Unlike the steady biogenic emissions from forests or the well-documented output of industrial sources, wildfires are rare, explosive and deeply unpredictable events. The location, size and intensity of new fires can only be estimated probabilistically, and the height at which smoke is injected into the atmosphere remains highly uncertain. Because air quality forecasts depend so heavily on emissions, these uncertainties cascade through the entire dispersion calculation, producing errors that vary enormously in space and time. The 2023 Canadian fire season, which consumed more than 15 million hectares of forest, made these limitations painfully visible as smoke crossed the Atlantic and degraded air quality in European cities.

Operational air quality assimilation systems, which blend model forecasts with surface observations to produce analyses, rely on accurate estimates of background forecast errors. Yet the error formulations used in most operational systems are highly simplified, often assuming uniform statistical properties that cannot capture the anisotropic, case-dependent error structure of a wildfire plume. Ensemble forecasting, the gold standard in numerical weather prediction, offers a way around this, but it comes at a steep price. Building a meaningful ensemble for air quality requires perturbing emissions, deposition rates and boundary conditions whose error characteristics are poorly known, and even ensembles of 50 to 100 members are orders of magnitude too small to sample the enormous dimensionality of the atmospheric system, leading to sampling errors that demand additional fixes like localization and inflation.

Vogel and her colleagues propose a fundamentally different approach rooted in ideas that trace back to stochastic dynamic prediction in the 1960s. Instead of running dozens of parallel forecasts, their method, called a parametric uncertainty forecast, evolves the error statistics themselves as prognostic variables within the forecast model. The key insight is that for a Gaussian error distribution, a handful of parameters, principally the error standard deviation and its spatial correlation length, are enough to describe the leading structure of the uncertainty. By deriving explicit equations for how these parameters change under advection, diffusion and emissions, the researchers can propagate uncertainty through the model at a tiny fraction of the computational cost of an ensemble.

The theoretical core of the paper lies in deriving these prognostic equations for the three processes that matter most for near-surface smoke forecasting. Advection turns out to be the simplest case: error standard deviation is transported by exactly the same wind field as the concentration itself, so the model’s existing advection scheme can handle it without modification. Vertical diffusion is more subtle. When a spatially correlated concentration field diffuses, its errors shrink, and the researchers show that this reduction depends on the vertical error correlation length, a new parameter that must be supplied to the model. Emissions act as a source of uncertainty, increasing concentration error in proportion to the correlated emission error standard deviation, a quantity that combines how uncertain the emissions are with how those errors relate to existing concentration errors.

A particularly important theoretical result concerns the choice between error variance and error standard deviation as the prognostic variable. Although the two are mathematically equivalent, the standard deviation form yields simpler equations with fewer terms, avoids an extra spatial derivative that introduces discretization errors, and shares the same units as concentration itself, making implementation and interpretation far easier. In the diffusion equation, the variance form requires an additional gradient term that the standard deviation form does not, and idealized experiments showed the standard deviation formulation tracks a full Kalman filter more accurately. The team therefore selected standard deviation as the working variable throughout their implementation.

The researchers embedded this parametric uncertainty forecast into GEM-MACH, Canada’s operational coupled meteorology and air quality model, which runs at 10-kilometer resolution over North America with 84 vertical layers and hourly wildfire emissions from the Canadian Forest Fire Emissions Prediction System. They chose fine particulate matter, PM2.5, as the target species and added its error standard deviation as a new prognostic field. For initial and boundary conditions, they replaced the operational bounded statistical fit between concentration and error variance with a simple linear relation applied to the instantaneous forecast concentrations, ensuring that uncertainties from emissions inside the simulation period remain consistent with those entering from outside it.

To demonstrate the approach, the team applied it to the June 6, 2023 smoke episode, when large fires in central Quebec sent plumes toward populated areas while older smoke lingered over Ottawa and Montreal. The parametric forecast produced dramatically different error fields from the operational setup: large, plume-shaped uncertainty structures grew directly from the remote fire hotspots and were carried southward by the winds, while the operational error fields, derived from two-month running statistics, remained nearly flat across the smoke regions. When these error estimates were fed into the hourly surface analysis system, the differences were striking. In well-observed urban areas, the two experiments produced similar corrections, but in the sparsely monitored north, the parametric experiment generated analysis increments exceeding 80 micrograms per cubic meter in plumes located roughly 400 kilometers from the nearest observation stations.

That last result may be the most consequential. Because the analysis spreads observation information according to the background error covariance, and because the parametric forecast assigns very large error standard deviations within the plumes, even weakly correlated distant observations can now exert meaningful corrections on unobserved smoke. The researchers emphasize that these far-reaching increments arise purely from the improved variance fields, not from exaggerated error correlations, meaning the sparse observing network is used far more efficiently in a highly case-dependent and anisotropic way. The method can even produce opposite-signed corrections within a single plume when nearby observations disagree, something the operational setup cannot achieve.

The authors are careful to note that their formulation remains deliberately simplified. Emission errors are assumed proportional to local emissions, the vertical error correlation length is prescribed rather than forecast, and processes like area emissions, chemistry, deposition and wind uncertainty are excluded for now. Sensitivity tests showed the results depend on the chosen correlation length, with shorter lengths producing shorter uncertainty plumes. Still, the study marks the first implementation of parametric diffusion and emission error dynamics in a state-of-the-science atmospheric model and leads the way toward a full parametric Kalman filter for operational smoke assimilation. As fire seasons intensify and public demand for trustworthy air quality warnings grows, this efficient new mathematics may soon help forecasters tell not just where the smoke will go, but how sure they are when they say so.

Subject of Research: Parametric uncertainty forecasting for wildfire smoke data assimilation in operational air quality models

Article Title: Formulation of parametric uncertainty forecasts towards operational wildfire smoke assimilation

Article References: Vogel, A., Ménard, R., Abu, J., & Chen, J. (2026). Formulation of parametric uncertainty forecasts towards operational wildfire smoke assimilation. Nonlinear Processes in Geophysics, 33(3), 347-371. https://doi.org/10.5194/npg-33-347-2026

Image Credits: AI Generated

DOI: 10.5194/npg-33-347-2026

Keywords: wildfire smoke, air quality forecasting, data assimilation, parametric Kalman filter, uncertainty quantification, GEM-MACH, PM2.5, error covariance, ensemble forecasting, emissions uncertainty, Canada wildfires, atmospheric modeling