Khan et al. (2026) Advanced Bayesian spatio-temporal frameworks for predicting precipitation at ungauged sites and times

Identification

• Journal: Theoretical and Applied Climatology
• Year: 2026
• Date: 2026-04-10
• Authors: Muhammad Asif Khan, Rangjian Qiu, Muhammad Zubair, Shah Fahd, Zeeshan Zafar
• DOI: 10.1007/s00704-026-06205-y

Research Groups

• State Key Laboratory of Water Resources Engineering and Management, Wuhan University, Wuhan, China
• Key Laboratory of Geographic Information Science, Ministry of Education, East China Normal University, Shanghai, China
• School of Geographic Sciences, East China Normal University, Shanghai, China
• College of Urban and Environmental Sciences, Northwest University, Xi’an, China

Short Summary

This study developed and evaluated advanced Bayesian spatio-temporal frameworks, specifically Gaussian Process (GP) and Auto-Regressive (AR) models, to predict monthly precipitation at ungauged sites and times in Pakistan's Indus Basin. The AR model, combined with a square root transformation, demonstrated superior temporal forecasting accuracy, while both models provided reliable spatial predictions.

Objective

• To quantify spatial dependence in monthly precipitation and predict precipitation at ungauged sites using a Matérn-kernel Gaussian Process within a hierarchical Bayesian framework.
• To improve time-wise forecasting by incorporating autoregressive dependence and compare against the GP-only alternative.
• To evaluate variance-stabilizing transformations and pre-specify an objective selection criterion (PMCC) to mitigate skewness and heteroscedasticity.
• To deliver probabilistic prediction maps and station-level forecasts with uncertainty quantification for decision support (agriculture, hydrology, and disaster risk management).

Study Configuration

• Spatial Scale: Upper and middle Indus Basin, Pakistan, covering an area with elevations ranging from 139 meters to 8,069 meters above sea level. Data from 14 rain gauge stations.
• Temporal Scale: Monthly precipitation data collected over 14 years (2010–2023).

Methodology and Data

• Models used: Bayesian spatio-temporal framework, Hierarchical Nugget Effect Model (HNEM), Gaussian Process (GP) modeling (with Matérn covariance function), Auto-Regressive (AR) modeling. Markov Chain Monte Carlo (MCMC) method (Gibbs sampling, Metropolis-Hastings Algorithm) for parameter estimation.
• Data sources: Rainfall, temperature, humidity, and wind speed datasets collected from 14 regulatory monitors (rain gauges) provided by the Pakistan Meteorological Department (PMD).

Main Results

• The Auto-Regressive (AR) model significantly improved temporal forecast accuracy, achieving a mean absolute error (MAE) of 6.31 millimeters (temporal MAE of 6.42 millimeters with square root transformation).
• The Gaussian Process (GP) model performed better in spatial prediction, achieving a MAE of 24.33 millimeters (spatial MAE of 24.83 millimeters with square root transformation). The AR model also showed comparable spatial MAE (25.19 millimeters).
• The model's predictive performance significantly improved when the Matérn covariance function was combined with a square root transformation of the data, which minimized the Predictive Model Choice Criterion (PMCC) score (AR: 10,345.1; GP: 13,162.7).
• Relative humidity showed a strong positive correlation with precipitation (GP: 0.1227, AR: 0.089), while wind speed had a significant negative effect (GP: -0.1503, AR: -0.0734).
• Spatial variance (ση²) exceeded the nugget effect (σε²) in both models (e.g., GP: 7.358 vs. 0.1034), indicating that structured spatio-temporal processes are the main drivers of precipitation variability.
• Forecasted monthly precipitation maps for 2023 showed the AR model exhibiting greater smoothness and coherence in spatial generalization compared to the GP model.
• Precipitation levels are anticipated to increase during summer months (March to September), with minor reductions in winter, but no statistically significant annual trend was found.

Contributions

• Developed an end-to-end, reproducible Bayesian spatio-temporal workflow for monthly precipitation prediction in the topographically complex upper–middle Indus Basin.
• Provided a transparent separation of spatial versus temporal predictive skill, reported with dedicated holdout protocols and task-specific metrics.
• Detailed operational Bayesian kriging (posterior predictive sampling and back-transformation to millimeters) for direct replication and deployment.
• Consolidated reporting of priors and MCMC settings to improve reproducibility.
• First Bayesian model-based spatio-temporal analysis of precipitation for the Indus Basin, facilitating uncertainty quantification and probabilistic forecasting.

Funding

• National Natural Science Foundation of China (52322904)

Citation

@article{Khan2026Advanced,
  author = {Khan, Muhammad Asif and Qiu, Rangjian and Zubair, Muhammad and Fahd, Shah and Zafar, Zeeshan},
  title = {Advanced Bayesian spatio-temporal frameworks for predicting precipitation at ungauged sites and times},
  journal = {Theoretical and Applied Climatology},
  year = {2026},
  doi = {10.1007/s00704-026-06205-y},
  url = {https://doi.org/10.1007/s00704-026-06205-y}
}