| 12 Nov 2025
Quantifying the minimum ensemble size for asymptotic accuracy of the ensemble Kalman filter using the degrees of instability
Abstract. The ensemble Kalman filter (EnKF) is widely used for state estimation in chaotic dynamical systems, including the atmosphere and ocean. However, the required ensemble size for accurate state estimation remains unclear. In this study, we define filter accuracy based on its time-asymptotic performance relative to the observation noise. We then investigate the minimum ensemble size, m*, required to achieve this accuracy, linking it to the degrees of instability in the chaotic dynamics. Since the well-defined characteristic numbers of dynamical systems called the Lyapunov exponents (LEs) quantify the timeasymptotic exponential growth or decay rates of infinitesimal perturbations, we define the degrees of instability N+ by the number of positive LEs. In the EnKF, capturing such instabilities with limited ensemble is crucial for achieving long-term filter accuracy. Therefore, we propose an ensemble spin-up and downsizing method within data assimilation cycles. Numerical experiments applying the EnKF to the Lorenz 96 model show that the minimum ensemble size required for filter accuracy is estimated by m* = N+ +1. This study provides a practical estimate for the minimum ensemble size based on a priori information about the target dynamics, along with a method to achieve long-term accuracy.
Competing interests: Some authors are members of the editorial board of journal NPG.
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In this manuscript, two goals are introduced: 1) the manuscript tries to demonstrate that the EnKF can asymptotically converge to a below observation errors as long as the ensemble member is more than the dimension of the unstable subspace. 2) the manuscript proposes an ensemble downsizing approach, which is supposed to accelerate the convergence of the ensemble Kalman filter (EnKF) with smaller ensemble size. However, I find this manuscript can be benefited from a more in-depth analysis, and therefore recommend for at least a major revision.
1. In L61 - 62, the manuscripts claims that it addresses the time-asymptotic accuracy of the ETKF instead of the time-averaged analysis error within a finite-time interval. However, I find it difficult to understand the differences between them. In the referenced literature, the time-average is typically calculated after a spin-up/burn-in time when the filters are converged, which is very similar to time-asymptotic accuracy. In fact, the results in Fig.2 are in general consistent with previous studies, e.g., with an observation error of 1, in Fig. 3 of Bocquet, M.: Ensemble Kalman filtering without the intrinsic need for inflation, Nonlin. Processes Geophys., 18, 735–750, https://doi.org/10.5194/npg-18-735-2011, 2011.
However, Fig.2 is still an interesting figure. The filter accuracy, as defined by Eq.(17), is violated for large observation errors. This accuracy only starts to appear from ensemble size larger than the unstable subspace (m = 14). These results show clear importance of the accuracy definition of Eq. (17) and the observation noises. Could the authors provide references and explanations of Eq.(17)? It would be also of interests to have an explanation for the sensitivity to observation noises, which is absent from the manuscript. One possible explanation might be the observation noises affecting the linearity which is briefly demonstrate in the Appendix of in Marc Bocquet & Alberto Carrassi (2017) Four-dimensional ensemble variational data assimilation and the unstable subspace, Tellus A: Dynamic Meteorology and Oceanography, 69:1, 1304504, DOI: 10.1080/16000870.2017.1304504. If the linearity is the cause, would smaller observation interval allow for larger observation noises? Nevertheless, these results seem to suggest that to get accurate analysis, an ensemble size of 14 requires very stringent condition.Â
2. The manuscript claims that the ensembled downsizing method can "mitigate the slow convergence of uncertainty". However, this is not clearly shown in available Figures. In the comparison between N_spinup = 0 and N_spinup = 720 in Fig.3, it is not clear that N_spinup=720 converges faster. Moreover, the figure presents m=15 which is different from m=14 as claimed in the conclusion section. However, this figure indeed shows only a smaller inflation factor is needed. However, it is not clear if this effect is related to the unstable-neutral subspace. Is it possible that the initial ensemble size mitigates the sampling errors? I wonder if it would be useful to investigate the alignment of error covariance matrix and Lyapunov vectors over time for the ensemble downsizing method.
3. The motivation of F=16 is not very clear to me. Is it just used to validate the results for F=8?
Minor comments:
L19: "For continuous-time non-autonomous dynamical system, ..., at least one of the LEs is zero". With this condition, this is still not completely correct for certain dissipative systems, I think.
L28: (ETKF, Bishop et al., 2001). If LaTex is used, try \citep[ETKF,][]{ref}. The same follows for cases in L140, L146.
L77, L117, L175: No bracket for the reference.
L106: Consistency for v_j(t) or v_j.
L112: At least one exponent is zero
L124: For Nx \in N number of variables, ...
L128: used in data assimilation algorithms.
L165: What does this mean for "taking over the observation noises and initial ensemble"? Is the expectation computed over different observation noises?
Figure 2 and Figure 6: in both x- and y-axis only 10^{-} is shown. caption: black line indicates the order of r^2
Figure 3, 4: if spin-up time is 0, it is perhaps better to avoid 41 -> 14 in the legend.