Industry: Environmental Product: ModelRisk Application: Quantifying uncertainty in air quality monitoring and regulatory compliance
A city air-quality agency runs a PM2.5 monitoring station and reports to the regulator against two benchmarks: an annual-mean standard of 9 µg/m³ and a 24-hour standard of 35 µg/m³. Plug the average emission level, the average meteorology and the average season into the spreadsheet and it returns an annual mean of 8.6 µg/m³ — comfortably under the line. On that single number the station is compliant and no action is required.
Run the same physics 50,000 times with the day-to-day variability left in, and the annual mean averages 9.9 µg/m³ with an 80% chance of exceeding the 9 µg/m³ standard in any given year. The deterministic estimate didn't just understate the risk — it landed on the wrong side of the regulatory line. The agency rebuilt the assessment in Vose Software's ModelRisk so that compliance is reported as a probability, not a point.
Ambient concentration is a product of variable terms, not a sum of fixed ones. A daily PM2.5 reading is a local emission contribution multiplied by a meteorological dispersion factor — how well the atmosphere mixes and ventilates that day — on top of a seasonal inversion pattern and a regional background. Plugging mean values into a product of skewed factors discards the right tail, and it is the right tail that breaches standards. Worse, the bad days are not independent: a single stagnant, low-mixing-height day concentrates every source at once, so exceedance days cluster instead of averaging away. A model that treats each day or each source as independent quietly deletes that clustering and reports a tail far thinner than reality.
The agency modelled daily concentration as background + emission × meteorology × season, with emission strength LogNormal, the meteorological dispersion factor a shared LogNormal drawn once per day and applied to all sources, and a seasonal shape peaking in winter. Imposing that shared meteorological factor produced a same-day correlation between two source-driven concentrations of 0.58; replacing it with an independent per-source factor collapsed the correlation to 0.14 — confirming that the clustering is the common meteorological cause, exactly the structure a sum-of-independent-sources model destroys.
background + emission × meteorology × season
Rolling the daily series up into an annual mean over 50,000 simulated years gives the headline picture.
The annual mean averages 9.88 µg/m³ (median 9.83), with a P90 of 11.22 µg/m³ and a P99 of 12.48 µg/m³. Against the 9 µg/m³ annual standard, the probability of exceedance is 80%. The deterministic point estimate of 8.64 µg/m³ sits below the standard and reports the station compliant — hiding an 80% breach risk. The mean of a product of variable factors is simply not where the regulatory decision should be made.
The 24-hour standard of 35 µg/m³ is a separate test: not the annual average but the count of individual days that spike above the limit. The natural regulatory question is "how many exceedance days will we see, and what is the chance we blow the budget?"
The station averages 1.7 exceedance days per year, with a P90 of 4 days and a P99 of 7. Reading off the curve: there is a 74.2% chance of at least one exceedance day, a 45.2% chance of more than one, and a 12.1% chance of more than three — the practical compliance trigger. Only 25.8% of years are entirely clean. A single annual-average number cannot express any of this; the exceedance-day curve is the shape an enforcement conversation actually needs.
Sensitivity ranking on the P99 exceedance-day count shows where tightening the inputs would tighten the forecast.
The dominant driver is the daily meteorological dispersion spread — the day-to-day variability in how well the atmosphere ventilates — at roughly ±3.0 days around the P99 of 7. Winter inversion amplitude and local emission strength follow. The standard level itself (35 vs a tighter 30 µg/m³) is a smaller lever than the meteorology the agency cannot control. That ranking tells the agency its monitoring dollars are better spent characterising local dispersion than refining the emission inventory.
The same engine, re-run with the local emission term scaled down, turns control options into shifted distributions rather than single promised numbers.
A 20% local-emission cut pulls the mean annual concentration to 8.43 µg/m³ and the chance of breaching the annual standard from 80% to 25%. A 40% cut combined with cleaner winter heating reaches a mean of 6.99 µg/m³ and an exceedance probability of 0.7% — the only path that brings the station to confident compliance. Presenting the two standards as full distributions let the agency cost each option against the probability of compliance it actually delivers, rather than against a point estimate that was already on the wrong side of the line.
Reporting air quality as a single average is a forecast of one number on one side of a regulatory line. Reporting it as a distribution — built in ModelRisk inside the agency's existing Excel workflow — is a forecast of how often that line is actually crossed, which is the only thing the standard was ever asking.