Hydrology deals with water movement through landscapes — rainfall becomes runoff, infiltrates soils, recharges aquifers, and flows through river networks. The models that describe these processes are complex, nonlinear, and uncertain. Bayesian methods have become central to modern hydrology because they provide a coherent framework for calibrating these models against observed data while fully quantifying the uncertainty in predictions that inform flood warnings, dam operations, and water resource planning.
Bayesian Calibration of Rainfall-Runoff Models
Rainfall-runoff models transform precipitation and evapotranspiration inputs into streamflow predictions through conceptual representations of soil moisture storage, groundwater, and channel routing. These models have numerous parameters — infiltration capacity, soil depth, recession coefficients — that must be calibrated against observed streamflow. Bayesian calibration replaces point estimates with full posterior distributions over the parameter space.
With Heteroscedastic Error Model Qₜ = Mₜ(θ) + σₜ · εₜ, εₜ ~ N(0,1)
σₜ = a + b · Mₜ(θ)ᶜ [variance scales with flow]
The DREAM algorithm (DiffeRential Evolution Adaptive Metropolis), developed specifically for hydrological applications, efficiently samples the posterior in high-dimensional, multimodal parameter spaces that are common in watershed models. The posterior reveals parameter equifinality — the phenomenon where multiple parameter combinations produce equally good fits — which Bayesian methods quantify rather than ignore.
Keith Beven's concept of equifinality — that many parameter sets are equally acceptable — troubled hydrologists for decades. Bayesian inference reframes equifinality as a feature: the posterior distribution naturally assigns probability to all parameter combinations consistent with the data. Rather than searching for a single "best" parameter set, Bayesian prediction marginalizes over the entire posterior, producing uncertainty bounds that reflect the true identifiability of the model.
Flood Frequency Analysis
Estimating the magnitude of rare floods — the 100-year or 500-year event — is essential for dam design, floodplain mapping, and insurance. Bayesian flood frequency analysis fits extreme value distributions (generalized extreme value, log-Pearson III) to annual maximum flows while incorporating regional information through hierarchical priors. The resulting posterior predictive distribution of flood quantiles captures uncertainty from the short record length, the choice of distribution, and spatial variability.
μᵢ ~ N(Xᵢβ, τ²) [regional regression on catchment attributes]
P(Q₁₀₀ | data) = ∫ P(Q₁₀₀ | μ, σ, ξ) · π(μ, σ, ξ | data) dμdσdξ
Bayesian methods also enable the incorporation of historical and paleoflood data — records of extreme floods from sediment deposits, tree-ring scars, or historical documents — as censored or imprecise observations, dramatically extending the effective record length for rare event estimation.
Water Quality and Contaminant Transport
Bayesian methods calibrate groundwater flow and contaminant transport models, which are used to predict the spread of pollutant plumes, design remediation systems, and assess drinking water safety. The high dimensionality of spatially distributed hydraulic conductivity fields requires advanced techniques such as Bayesian geostatistics and dimension reduction through Karhunen-Loeve expansion of the prior.
"All models are wrong, but in hydrology, we must also ask: how wrong, where, and what are the consequences? Bayesian methods force us to answer these questions honestly." — Jasper Vrugt, developer of the DREAM algorithm
Climate Change and Nonstationarity
Traditional flood frequency analysis assumes stationarity — that the statistical properties of floods do not change over time. Under climate change, this assumption is increasingly untenable. Bayesian nonstationary models allow distribution parameters to vary as functions of time, climate indices, or land-use change, with the posterior revealing whether observed trends are statistically credible or within the range of natural variability. Bayesian model comparison via Bayes factors tests whether nonstationary models are warranted by the data, providing a principled approach to a contentious question.