Antoniadou et al. (2026) A Bayesian spatial framework for modeling sub-hourly to daily extreme precipitation in Denmark using SPDE with INLA

Identification

• Journal: Journal of Hydrology
• Year: 2026
• Date: 2026-04-01
• Authors: Nafsika Antoniadou, Jonas Wied Pedersen, Anders Stockmarr, Hjalte Jomo Danielsen Sørup, Torben Schmith, Per Mikkelsen
• DOI: 10.1016/j.jhydrol.2026.135428

Research Groups

• Department of Environmental and Resource Engineering, Technical University of Denmark
• Department of Applied Mathematics and Computer Science, Technical University of Denmark
• Danish Meteorological Institute

Short Summary

This study introduces a new Bayesian spatial framework for modeling sub-hourly to daily extreme precipitation in Denmark, generating spatially continuous return level maps with associated uncertainties. The two-stage model, utilizing Negative Binomial and Generalized Pareto distributions with latent spatial random effects, captures spatial variation in extreme event frequency and magnitude, showing improved performance over the existing national model for shorter durations.

Objective

• To develop a new spatial Bayesian hierarchical model for sub-daily extreme precipitation in Denmark, incorporating latent Gaussian fields to explicitly account for spatial dependence and allow stations to "borrow strength" from one another.
• To compare the performance of this new model against the latest version of the national regional model (NRM32), highlighting differences in predictive accuracy and spatial representation of extreme rainfall.

Study Configuration

• Spatial Scale: Denmark, covering a domain of approximately 400 km x 300 km, with predictions on a 10 km x 10 km grid. The model was trained using data from 229 rain gauge stations.
• Temporal Scale: Training data spans 1979 to 2019. Independent test data covers 2020 to 2024. Rainfall data aggregated to four durations: 30 minutes, 60 minutes, 360 minutes, and 1440 minutes.

Methodology and Data

• Models used:
  • Bayesian generalized additive modeling framework with a two-stage procedure:
    • Stage 1 (NB): Exceedance frequencies modeled with a Negative Binomial distribution.
    • Stage 2 (GP): Exceedance magnitudes modeled with a Generalized Pareto distribution.
  • Latent spatial random effects modeled as Gaussian random fields (GRF) using the Stochastic Partial Differential Equation (SPDE) approach.
  • Bayesian inference performed using Integrated Nested Laplace Approximation (INLA).
• Data sources:
  • Rain gauge data from two types of stations across Denmark: tipping bucket gauges (SVK stations) and weighting rain gauges (DMI stations).
  • Covariate information from the Climate Grid Denmark (CGD) dataset (10 km x 10 km gridded daily precipitation product for 1989–2010), providing Mean Annual Precipitation (MAP) and Mean Extreme Daily Precipitation (MEDP).

Main Results

• Mean Annual Precipitation (MAP) was a significant positive predictor for the frequency of extreme precipitation events across all durations (30 min to 1440 min), with its effect becoming more pronounced for longer durations.
• Mean Extreme Daily Precipitation (MEDP) was a significant positive predictor for event intensity at longer durations (360 min and 1440 min), but not for shorter durations (30 min and 60 min).
• The posterior mean of the Generalized Pareto shape parameter (γ) ranged from 0.08 (30 min) to 0.14 (360 min), then slightly decreased to 0.11 (1440 min), indicating heavier tails for intermediate durations.
• Estimated spatial ranges (ρ) for the Gaussian Random Field (GRF) in the Generalized Pareto model varied from 72 km (60 min) to 152 km (360 min).
• Sensitivity analyses with synthetic gauge networks showed that broad spatial coverage of an area is more important than station density for predictive skill, especially for the SPDE-based model.
• For longer durations (360 min and 1440 min), the new two-stage model showed good agreement with the existing National Regional Model (NRM32) in terms of 10-year return levels and their spatial patterns.
• For shorter durations (30 min and 60 min), notable differences emerged between the two-stage model and NRM32 in both spatial variability and uncertainty, with the two-stage model generally showing more local variability and lower, more spatially variable uncertainty.
• The two-stage model achieved a lower mean absolute error (MAE) on independent test data (2020-2024) for rainfall frequencies (0.79 vs. 1.84 for NRM32) and mean exceedances (2014.02 vs. 2018.42 for NRM32).

Contributions

• Introduces a novel two-stage Bayesian spatial hierarchical model for sub-daily extreme precipitation in Denmark, providing spatially continuous return level maps with associated uncertainties.
• Explicitly accounts for spatial dependence through latent Gaussian fields, allowing stations to "borrow strength" and improving parameter estimation and prediction at unobserved locations.
• Utilizes a Negative Binomial distribution for exceedance frequencies, effectively addressing overdispersion observed in data compared to the Poisson assumption in previous models.
• Offers a flexible alternative to the current national standard model (NRM32), demonstrating improved predictive accuracy and a more realistic representation of spatial variability and uncertainty, particularly for shorter durations.
• Provides insights into optimal rain gauge network design, highlighting the critical importance of broad spatial coverage over high station density for extreme rainfall modeling using the INLA-SPDE framework.

Funding

• Danish State through the National Center for Climate Research (NCKF)

Citation

@article{Antoniadou2026Bayesian,
  author = {Antoniadou, Nafsika and Pedersen, Jonas Wied and Stockmarr, Anders and Sørup, Hjalte Jomo Danielsen and Schmith, Torben and Mikkelsen, Per},
  title = {A Bayesian spatial framework for modeling sub-hourly to daily extreme precipitation in Denmark using SPDE with INLA},
  journal = {Journal of Hydrology},
  year = {2026},
  doi = {10.1016/j.jhydrol.2026.135428},
  url = {https://doi.org/10.1016/j.jhydrol.2026.135428}
}