Maximizing the statistical diversity of an ensemble of bred vectors by using the geometric norm

Abstract: We show that the choice of the norm has a great impact on the construction of ensembles of bred vectors. The geometric norm maximizes (in comparison with other norms like the Euclidean one) the statistical diversity of the ensemble while, at the same time, enhances the growth rate of the bred vector and its projection on the linearly most unstable direction, i.e. the Lyapunov vector. The geometric norm is also optimal in providing the least fluctuating ensemble dimension among all the spectrum of q-norms studied. We exemplify our results with numerical integrations of a toy model of the atmosphere (the Lorenz-96 model), but our findings are expected to be generic for spatially extended chaotic systems.

• The paper demonstrates that using the geometric norm (q=0) maximizes ensemble diversity, yielding higher ensemble dimensions and reduced temporal fluctuations.
• It shows that geometric-norm bred vectors maintain significant alignment with the dominant Lyapunov vector while achieving faster exponential growth rates.
• The study implies that applying geometric norms in operational forecasting could improve data assimilation and prevent ensemble collapse.

Maximizing Ensemble Diversity of Bred Vectors with the Geometric Norm

Introduction

This paper rigorously examines how the selection of norm impacts the diversity and dynamical properties of bred vector (BV) ensembles in spatially extended chaotic systems, exemplified by the Lorenz-96 model. While established forecasting centers often employ Euclidean-type norms for bred vector rescaling, the authors identify and characterize profound distinctions in ensemble characteristics when alternative norms—specifically, the geometric (0-norm)—are adopted. The central focus is on the statistical diversity, Lyapunov vector (LV) alignment, and growth rates of bred vector ensembles as induced by the choice of norm.

Impact of Norm Choice on Bred Vector Ensembles

The construction of BVs involves periodic rescaling of finite perturbations to a fixed amplitude using a prescribed norm. Traditionally, the Euclidean norm has been predominant, yet the paper demonstrates that the inherent properties of the resulting ensemble are fundamentally altered by the specific choice of norm. The study systematically considers $q$-norms, where $q=2$ yields the standard energy-type (Euclidean) norm and $q=0$ results in the geometric (logarithmic) norm, with higher and lower $q$ values interpolating between these cases.

A crucial finding is that the geometric norm ($q\to 0$) achieves maximal ensemble diversity, quantified by the ensemble dimension, in direct comparison to norms with $q>0$. The geometric norm produces ensembles that possess a higher degree of linear independence among their members and facilitate greater statistical diversity, enhancing the ensemble's capability to span the most unstable directions in the system's phase space.

Numerical Analysis in the Lorenz-96 Model

Using the Lorenz-96 system with standard parameters for strong spatiotemporal chaos, the paper conducts detailed numerical integrations to evaluate key metrics across different norms:

• Ensemble Dimension: As a measure of effective ensemble diversity, the geometric norm consistently produces higher average ensemble dimensions than other $q$-norms for equivalent perturbation amplitudes. Importantly, the transition to loss of diversity (collapse to a single direction) is less abrupt for the geometric norm.
• Fluctuations of Ensemble Dimension: The geometric norm yields the smallest relative temporal fluctuations in ensemble dimension over all tested norms, indicating more robust and stable sampling of phase space directions.
• Alignment with Lead Lyapunov Vector: For small but finite perturbation amplitudes, ensembles constructed with the geometric norm are able to maintain significant projection on the dominant Lyapunov vector while preserving diversity. In contrast, Euclidean and other $q>0$ norms rapidly sacrifice diversity for improved alignment.
• Growth Rates: Logarithmic BVs (geometric norm) realize the fastest exponential growth rates, approaching the leading Lyapunov exponent more efficiently while maintaining ensemble diversity.

Theoretical and Practical Implications

The results decisively challenge the previously held notion that the choice of norm for BV construction is largely inconsequential. The geometric norm is empirically superior in promoting both diversity and dynamical relevance (projection onto the leading LV) in ensemble forecasting strategies. This is of substantial importance for operational data assimilation and ensemble prediction systems, suggesting that geometric-norm BVs could offer significant gains in forecast skill and reliability, particularly by mitigating ensemble collapse and under-dispersion.

An open avenue highlighted by the authors involves extending the analysis to practical application in operational numerical weather prediction models, which are high-dimensional and feature inhomogeneous dynamics. Additionally, integrating geometric-norm-based BVs within ensemble Kalman filtering frameworks represents a promising direction, with the potential for improved filter performance due to enhanced ensemble diversity and dynamical adaptability.

Conclusion

The comprehensive analysis elucidates that the geometric norm ($q=0$) is optimal for constructing ensembles of bred vectors within spatially extended chaotic systems, as assessed by ensemble diversity, stability, projection onto the main LV, and growth rates. These findings advocate for a reevaluation of norm selection in operational ensemble generation procedures, and provide a robust foundation for future developments in data assimilation and ensemble-based numerical forecasting methodology.