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Error analysis of the projected PO method with additive inflation for the partially observed Lorenz 96 model

Kota Takeda

Abstract

We consider the filtering problem with the partially observed Lorenz 96 model. Although the accuracy of the 3DVar filter in this problem has been established, the theoretical guarantee for the ensemble Kalman filter (EnKF) remains limited due to the analytical difficulty of handling non-symmetric matrices that emerge in the partial observation setting. This study establishes uniform-in-time error bounds of a stochastic variant of the EnKF, known as the perturbed observation (PO) method. By utilizing additive covariance inflation, we successfully obtain the bounds both with and without projecting the background covariance onto the observation space. Our analysis with the projection complements existing results for the deterministic variant of the EnKF, while our approach without the projection offers an extended mathematical framework to handle the non-symmetric matrix products directly. A numerical example validates the theoretical findings and shows comparable accuracies between the two settings.

Error analysis of the projected PO method with additive inflation for the partially observed Lorenz 96 model

Abstract

We consider the filtering problem with the partially observed Lorenz 96 model. Although the accuracy of the 3DVar filter in this problem has been established, the theoretical guarantee for the ensemble Kalman filter (EnKF) remains limited due to the analytical difficulty of handling non-symmetric matrices that emerge in the partial observation setting. This study establishes uniform-in-time error bounds of a stochastic variant of the EnKF, known as the perturbed observation (PO) method. By utilizing additive covariance inflation, we successfully obtain the bounds both with and without projecting the background covariance onto the observation space. Our analysis with the projection complements existing results for the deterministic variant of the EnKF, while our approach without the projection offers an extended mathematical framework to handle the non-symmetric matrix products directly. A numerical example validates the theoretical findings and shows comparable accuracies between the two settings.

Paper Structure

This paper contains 11 sections, 10 theorems, 83 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Nonlinear data assimilation problem and algorithm
  3. Notations
  4. State space model
  5. The ensemble Kalman filter
  6. Error analysis
  7. Error bound with the covariance projection
  8. Error bound without the covariance projection
  9. Numerical example
  10. Summary
  11. Matrix calculations

Key Result

Lemma 2.2

\newlabellem:l96_absorbing0 Let $\rho = \sqrt{2J}|F|$ and $B(\rho) \coloneq \{\bm{u} \in \mathbb{R}^J \mid |\bm{u}| \le \rho\}$. Then, for any $\bm{u}_0 \in \mathbb{R}^J$, there exists $T = T(|\bm{u}_0|) \ge 0$ such that $\bm{u}(t) \in B(\rho)$ for the solution $\bm{u}(t)$ to eq:l96_dissipative wi

Figures (1)

  • Figure 1: Plot of the Mean Squared Error $\frac{1}{m} \sum_{k=1}^m \mathbb{E} [\|\bm{\delta}^{(k)}_n\|^2]$ vs. observation time step with a log-scale y-axis. The covariance inflation methods are the additive inflation \ref{['eq:inflated_covariance']} (add) and the projected additive inflation \ref{['eq:inflated_projected_covariance']} (proj) with $\alpha = 0.0, 0.5, 2.0$. Each sample path is indicated in low opacity.

Theorems & Definitions (22)

  • Definition 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Definition 2.4
  • Definition 2.5
  • Theorem 3.1: Error bound of the projected PO method
  • Lemma 3.2
  • Proof 1
  • Lemma 3.3
  • Proof 2: proof of \ref{['lem:analysis_bound']}
  • ...and 12 more