June 2024 Efficient and effective calibration of numerical model outputs using hierarchical dynamic models
Yewen Chen, Xiaohui Chang, Bohai Zhang, Hui Huang
Author Affiliations +
Ann. Appl. Stat. 18(2): 1064-1089 (June 2024). DOI: 10.1214/23-AOAS1823


Numerical air quality models, such as the Community Multiscale Air Quality (CMAQ) system, play a critical role in characterizing pollution levels at fine spatial and temporal scales. The model outputs, however, tend to systematically over- or underestimate the real pollutant concentrations. In this study we propose a Bayesian hierarchical dynamic model to calibrate large-scale grid-level CMAQ model outputs using data from other sources, especially point-level observations from sparsely located monitoring stations. In our model a stochastic integro-differential equation (IDE) is implemented to account for space-time interactions of air pollutants. To better approximate the spatial pattern of pollutants, we employ nonregular meshes to discretize IDEs. A spatial partitioning procedure is embedded to improve the scalability of the approach for very large meshes. An algorithm based on variational Bayes and ensemble Kalman smoother is developed to accelerate the parameter estimation and calibration procedure. We apply the proposed approach to calibrate CMAQ outputs for China’s Beijing–Tianjin–Hebei region. In contrast to existing methods, the proposed approach captures space-time interactions, produces more accurate calibration results, and operates at a higher computational efficiency. A reanalysis dataset is also adopted to demonstrate the effectiveness and efficiency of our approach to large spatial data.

Funding Statement

This research is supported by the National Natural Science Foundation of China (Grant No. 12231017, No. 12292984, and No. 12161016), the MOE Project of Key Research Institute of Humanities and Social Sciences (Grant No. 22JJD910001), and the Guangdong Provincial Key Laboratory of Interdisciplinary Research and Application for Data Science (project code 2022B1212010006).


We thank the anonymous reviewers and the Associate Editor for their invaluable comments. We are particularly grateful to the Editor, Professor Karen Kafadar (former Editor-in-Chief), for her detailed comments and insightful suggestions. Their input significantly improved the quality of this article.

The first and the second authors are equal contributors.


Download Citation

Yewen Chen. Xiaohui Chang. Bohai Zhang. Hui Huang. "Efficient and effective calibration of numerical model outputs using hierarchical dynamic models." Ann. Appl. Stat. 18 (2) 1064 - 1089, June 2024. https://doi.org/10.1214/23-AOAS1823


Received: 1 February 2023; Revised: 1 August 2023; Published: June 2024
First available in Project Euclid: 5 April 2024

Digital Object Identifier: 10.1214/23-AOAS1823

Keywords: Calibration , hierarchical dynamic models , numerical model outputs , space-partitioning-based ensemble Kalman smoother , stochastic integro-differential equations , variational Bayes

Rights: Copyright © 2024 Institute of Mathematical Statistics


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Vol.18 • No. 2 • June 2024
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