A Bayesian Hierarchical Model to Simulate Temporal Variability in Urban Stormwater Event Mean Concentrations

Event Mean Concentration (EMC) is widely used to estimate stormwater pollutant loads due to its simplicity and low data requirements. However, conventional deterministic approaches typically use a single representative EMC for a catchment to estimate loads, thereby neglecting temporal variability across seasons and storm events, and potentially biasing event-level load estimates. To address this limitation, we present a Bayesian hierarchical linear mixed model that explicitly quantifies seasonal and inter-event variability in EMCs by estimating full posterior distributions, rather than a single deterministic EMC value, while retaining the operational simplicity of the traditional EMC approach. The model decomposes EMC into three hierarchical components: a global fixed effect, a seasonal random effect, and an event-level random effect. This structure enables EMC variability to be partitioned across multiple temporal scales and propagated into predictive uncertainty. The approach is demonstrated using soluble reactive phosphorus (SRP) data from a mixed urban catchment in London, Canada, comprising 18 monitored storm events across summer and fall seasons. A suite of models is developed to systematically evaluate methodological choices, including models with increasing levels of EMC variability representation (from global-only to full hierarchical structures) and models considering alternative land-use representations (lumped versus distributed). Results indicate that the full hierarchical model consistently outperforms simplified structures that exclude key variability components, as evaluated using leave-one-out cross-validation (LOO). Models that explicitly represent distinct land-use types demonstrate improved predictive performance compared to lumped representations; however, spatial disaggregation increases marginal variance, reflecting additional uncertainty. For the full hierarchical, land-use-distributed model, seasonal-level effects account for the largest share of marginal variability (median 63%), indicating that EMC variability for SRP manifests mainly as seasonal changes. Overall, these findings demonstrate that a single representative EMC is insufficient to characterize intrinsic temporal variability. By explicitly propagating uncertainty across hierarchical levels, the proposed Bayesian framework improves the reliability of stormwater load predictions and provides a more robust basis for management decisions.