Controlling Ensemble Variance in Diffusion Models: An Application for Reanalyses Downscaling

TL;DR

This work addresses how to control ensemble variance in diffusion-based downscaling of reanalysis data. By linking DDIM diffusion steps to the variance of reverse-diffusion ensembles, the authors propose an ERA5-to-CERRA downscaling model that calibrates global and spatial variance to a reference (CERRA-EDA) while also applying the approach to the Arctic CARRA-EAST dataset. The study demonstrates that global variance increases with the number of diffusion steps and that spatial variance can be tuned by selecting an optimal step count (e.g., $N\approx8$ for spatial alignment and $N\approx12$ for global variance matching), with EMOS-based baselines providing a reference calibration. The results show diffusion-based ensembles can achieve high-resolution, uncertainty-aware downscaling efficiently, offering a practical tool for climate analyses and operational applications, including regions lacking ensemble data.

Abstract

In recent years, diffusion models have emerged as powerful tools for generating ensemble members in meteorology. In this work, we demonstrate how a Denoising Diffusion Implicit Model (DDIM) can effectively control ensemble variance by varying the number of diffusion steps. Introducing a theoretical framework, we relate diffusion steps to the variance expressed by the reverse diffusion process. Focusing on reanalysis downscaling, we propose an ensemble diffusion model for the full ERA5-to-CERRA domain, generating variance-calibrated ensemble members for wind speed at full spatial and temporal resolution. Our method aligns global mean variance with a reference ensemble dataset and ensures spatial variance is distributed in accordance with observed meteorological variability. Additionally, we address the lack of ensemble information in the CARRA dataset, showcasing the utility of our approach for efficient, high-resolution ensemble generation.

Figures (9)

• Figure 1: Comparison of CERRA and CARRA-EAST domains.
• Figure 2: Reverse diffusion signal and noise ratios with cosine scheduling, including the reduced signal rate.
• Figure 3: Plotting the global mean variance $\mu_V$ of ensemble diffusion for different number of steps against CERRA-EDA.
• Figure 4: Comparing the spatial variance of CERRA-EDA with ensemble diffusion at different diffusion steps. The comparison relative to the testing year 2016 and is performed in a 3 monthly manner to highlight seasonal variations.
• Figure 5: Evolution of the mean variance discrepancy (MVD) scores with the number of diffusion steps. The plot visualizes the values reported in Table \ref{['tab:MVD_between_variance_images']} for the yearly and three-monthly periods (JFM, AMJ, JAS, OND) during 2016. Lower values indicate a closer match between the spatial variance distributions of Ensemble Diffusion and CERRA-EDA. The yearly curve (orange) highlights the global trend, while the seasonal curves illustrate the differing optimal step counts across the annual cycle.

Theorems & Definitions (12)

• Proposition 3.1
• proof
• Theorem 3.2
• proof
• Theorem 3.3
• proof : Sketch of the proof
• proof
• Proposition 2.1
• proof
• Corollary 2.2