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Validating Climate Models with Spherical Convolutional Wasserstein Distance

Robert C. Garrett, Trevor Harris, Bo Li, Zhuo Wang

TL;DR

The paper tackles the challenge of validating global climate models while preserving spatial structure by introducing the Spherical Convolutional Wasserstein Distance (SCWD), built on a functional sliced Wasserstein framework applied to $L^2(\mathbb{S}^2)$ climate fields. SCWD uses a radial kernel to generate local slices on the sphere and then integrates univariate Wasserstein distances across the globe, yielding a regionally aware similarity score between model outputs and observational/reanalysis data. The approach is formalized via the Functional Sliced WD (FSW) and its spherical specialization, with a concrete implementation based on the Wendland kernel and practical considerations for CMIP5/CMIP6 comparisons of daily near-surface temperature and precipitation against ERA5/NCEP and GPCP. Results show that CMIP6 models provide modest improvements over CMIP5, particularly in precipitation, and SCWD offers interpretable spatial diagnostics that help identify where models excel or struggle, informing model development and selection for climate projection reliability.

Abstract

The validation of global climate models is crucial to ensure the accuracy and efficacy of model output. We introduce the spherical convolutional Wasserstein distance to more comprehensively measure differences between climate models and reanalysis data. This new similarity measure accounts for spatial variability using convolutional projections and quantifies local differences in the distribution of climate variables. We apply this method to evaluate the historical model outputs of the Coupled Model Intercomparison Project (CMIP) members by comparing them to observational and reanalysis data products. Additionally, we investigate the progression from CMIP phase 5 to phase 6 and find modest improvements in the phase 6 models regarding their ability to produce realistic climatologies.

Validating Climate Models with Spherical Convolutional Wasserstein Distance

TL;DR

The paper tackles the challenge of validating global climate models while preserving spatial structure by introducing the Spherical Convolutional Wasserstein Distance (SCWD), built on a functional sliced Wasserstein framework applied to climate fields. SCWD uses a radial kernel to generate local slices on the sphere and then integrates univariate Wasserstein distances across the globe, yielding a regionally aware similarity score between model outputs and observational/reanalysis data. The approach is formalized via the Functional Sliced WD (FSW) and its spherical specialization, with a concrete implementation based on the Wendland kernel and practical considerations for CMIP5/CMIP6 comparisons of daily near-surface temperature and precipitation against ERA5/NCEP and GPCP. Results show that CMIP6 models provide modest improvements over CMIP5, particularly in precipitation, and SCWD offers interpretable spatial diagnostics that help identify where models excel or struggle, informing model development and selection for climate projection reliability.

Abstract

The validation of global climate models is crucial to ensure the accuracy and efficacy of model output. We introduce the spherical convolutional Wasserstein distance to more comprehensively measure differences between climate models and reanalysis data. This new similarity measure accounts for spatial variability using convolutional projections and quantifies local differences in the distribution of climate variables. We apply this method to evaluate the historical model outputs of the Coupled Model Intercomparison Project (CMIP) members by comparing them to observational and reanalysis data products. Additionally, we investigate the progression from CMIP phase 5 to phase 6 and find modest improvements in the phase 6 models regarding their ability to produce realistic climatologies.
Paper Structure (27 sections, 1 theorem, 11 equations, 5 figures, 3 tables, 1 algorithm)

This paper contains 27 sections, 1 theorem, 11 equations, 5 figures, 3 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Climate Model Validation
  3. Previous approaches
  4. Our proposal
  5. Preliminaries
  6. Comparing Distributions of Functions
  7. Sliced WD
  8. Generalized Sliced WD
  9. Convolution Sliced WD
  10. Methods
  11. Functional Sliced Wasserstein Distance
  12. Spherical Convolutional Wasserstein Distance
  13. Choice of Kernel
  14. Climate Model Validation
  15. Evaluation of CMIP6 Models
  16. ...and 12 more sections

Key Result

Theorem 3.3

For all compact subsets $\mathcal{S}\subset\mathbb{R}^n$, $FSW_r$ is a pseudometric on $\mathcal{P}(\mathcal{F_S})$ and maintains the $r$-convexity property of the ordinary $W_r$ metric.

Figures (5)

  • Figure 1: Diagram representing the calculation of SCWD between two distributions of daily mean surface temperature fields from ERA5 and a CMIP6 model. For each day, radial projections are computed through kernel convolutions, represented here at three different locations. The resulting projections, called slices, summarise the local climate conditions in each dataset. The slices for each day are viewed as a sample from the marginal distribution at each location, represented here as histograms. SCWD is calculated as a global mean over the univariate WD between each pair of local distributions.
  • Figure 2: Ranking CMIP6 model outputs using SCWD. Each model output is represented by a point on the scatter plot and models from the same group share the color and shape. The x-axis and y-axis values represent each model's SCWD to the ERA5 surface temperature and GPCP total precipitation fields, respectively. The NCEP reanalysis is included as a blank triangle with dashed lines representing the SCWD to ERA5 and GPCP. The SCWD from the ERA5 total precipitation field to GPCP is represented as a solid line.
  • Figure 3: Top: Map of local Wasserstein distances from ERA5 to two CMIP6 2m surface temperature outputs: AWI-CM-1-1-MR and GISS-E2-2-G. Bottom: Map of local Wasserstein distances from GPCP to two CMIP6 total precipitation outputs: NorESM2-MM and BCC-ESM1. Color fill at each location is determined by the WD between the local distributions obtained from the convolution slicer in Definition \ref{['def:scwd']}. The color scale is shared for both maps and continental boundaries are included in black to aid spatial comparisons.
  • Figure 4: Boxplots of SCWD from the CMIP5 and CMIP6 model outputs to the ERA5 Reanalysis for 2m surface temperature. Each point represents the SCWD value for one climate model output to ERA5, with CMIP5 and CMIP6 separated into two boxplots for comparison. A dashed line is included to represent the SCWD from the NCEP Reanalysis to ERA5.
  • Figure 5: Boxplots of SCWD from the CMIP5 and CMIP6 model outputs to the GPCP observational dataset for total precipitation. Each point represents the SCWD value for one climate model output to GPCP, with CMIP5 and CMIP6 separated into two boxplots for comparison. Dotted and dashed lines are included to represent the SCWD from ERA5 and NCEP to GPCP, respectively.

Theorems & Definitions (5)

  • Definition 3.1: Convolution Slicer
  • Definition 3.2: Functional Sliced WD
  • Theorem 3.3
  • Definition 3.4: Spherical Convolutional WD
  • proof