Quantifying internal variability in large-ensemble climate data with the Wasserstein distance

Technical Summary: Quantifying Internal Variability in Large-Ensemble Climate Data with the Wasserstein Distance

Problem Statement
Quantifying the relative contributions of forced and internal variability is essential for detecting climate change signals and assessing projection uncertainty. Single-Model Initial-condition Large Ensembles (SMILEs) are the standard tool for this, where the ensemble mean represents forced variability and the residual represents internal variability. Existing metrics for comparing these components rely primarily on variance (e.g., $\gamma_{var}$) or information-theoretic quantities like Shannon entropy and mutual information (e.g., $\gamma_{info}$). However, variance-based metrics fail to adequately capture non-Gaussian distributions, which are common in climate variables (e.g., skewed or heavy-tailed precipitation). Conversely, information-theoretic metrics, while robust to non-Gaussianity, require data binning into histograms, introducing dependence on arbitrary parameters (bin width and range) and increasing computational complexity. There is a need for a metric that reflects the entire distribution shape, handles non-Gaussianity and outliers, and requires no additional parameters.

Methodology
The authors propose a new metric, $\gamma_{W1}$, based on the 1-Wasserstein distance (optimal transport cost) on the one-dimensional real line ($\mathbb{R}^1$).

1. Definitions:

   • Let $x_{n,t}$ be the climate variable for ensemble member $n$ at time $t$.
   • The ensemble mean $m_t$ is defined as forced variability.
   • The residual $\eta_{n,t} = x_{n,t} - m_t$ is defined as internal variability.
   • The total mean $m$ is the time average of $m_t$.

2. Metric Construction:

   • The magnitude of internal variability ($w_{int}$) is calculated as the 1-Wasserstein distance between the distribution of all samples ($x_{n,t}$) and the distribution of the ensemble mean ($m_t$).
   • The magnitude of forced variability ($w_{ext}$) is the 1-Wasserstein distance between the distribution of the ensemble mean ($m_t$) and the total mean ($m$).
   • To compute these distances, the authors sort the $N_{ens} \times N_T$ values of $x_{n,t}$ and the replicated values of $m_t$ in ascending order. The distance is the average absolute difference between corresponding quantiles.
   • The final metric is the ratio:
     $$\gamma_{W1} := \frac{w_{int}}{w_{int} + w_{ext}}$$
   • This ratio ranges from 0 to 1, where higher values indicate a larger relative contribution of internal variability.

3. Computational Advantages:

   • The method requires only sorting and differencing quantiles.
   • It involves no additional parameters (unlike histogram binning).
   • It applies to any distribution shape, including non-Gaussian variables.

Validation and Results
The study validates $\gamma_{W1}$ using synthetic climate data (Gaussian, uniform, and lognormal distributions) and applies it to the Community Earth System Model Large Ensemble (CESM-LE) for 2 m air temperature and total precipitation under historical (20C) and RCP8.5 forcing scenarios.

• Synthetic Data Analysis:

  • Robustness: Unlike $\gamma_{var}$ (sensitive to outliers) and $\gamma_{info}$ (sensitive to outliers and binning choices), $\gamma_{W1}$ provides stable estimates regardless of distribution shape or the presence of outliers.
  • Ensemble Size: The metric's stability depends on ensemble size ($N_{ens}$). Estimates become reliable when $N_{ens} \approx 40$ or more. Below this threshold, sampling error in the ensemble mean affects the metric.
  • Comparison: $\gamma_{W1}$ is computationally more efficient than $\gamma_{info}$ (which requires searching for optimal bin numbers, $O(NB)$) but slightly less efficient than $\gamma_{var}$ ($O(N)$ vs. $O(N \log N)$ for sorting).

• CESM-LE Application:

  • Temperature: All metrics indicated that the relative contribution of internal variability decreases as radiative forcing increases (from 20C to RCP8.5). However, $\gamma_{W1}$ showed the clearest signal of this response, with the largest amplitude in the difference maps.
  • Precipitation: Precipitation data often exhibit heavy-tailed, non-Gaussian distributions. Under both forcing scenarios, $\gamma_{var}$ saturated near 1 globally, and $\gamma_{info}$ showed minimal change, failing to detect the response to increased forcing. In contrast, $\gamma_{W1}$ did not saturate, revealing clear spatial structures (e.g., reduced internal variability at high latitudes and in tropical oceans) and successfully detecting the forced response.

Key Contributions

1. Novel Metric: Introduction of $\gamma_{W1}$, a parameter-free metric based on optimal transport that quantifies the relative magnitude of internal variability.
2. Distribution Agnosticism: The metric captures the entire distribution shape, making it suitable for non-Gaussian and heavy-tailed climate variables where variance-based methods fail.
3. Robustness: Demonstrated stability against outliers and independence from arbitrary parameters (like bin width), provided the ensemble size is sufficient ($N_{ens} \gtrsim 40$).
4. Empirical Validation: Application to CESM-LE demonstrates that $\gamma_{W1}$ is more effective than existing metrics at detecting changes in internal variability relative to forced variability, particularly for precipitation.

Significance and Claims
The paper claims that the 1-Wasserstein distance provides a "simple and useful tool" for analyzing large-ensemble datasets. The authors position $\gamma_{W1}$ as a superior alternative to variance-based metrics for non-Gaussian data and a more robust, parameter-free alternative to information-theoretic metrics. The study suggests that $\gamma_{W1}$ offers a clearer signal of the forced response in climate projections, especially for variables with complex distributional properties. The authors note that while the metric is effective, a detailed analysis of the specific causes of internal and forced variability using this new tool remains a subject for future work. They also outline potential extensions, such as using Wasserstein barycenters for ensemble means and applying the metric to multi-model ensembles to address model uncertainty.