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Sampling error mitigation through spectrum smoothing: first experiments with ensemble transform Kalman filters and Lorenz models

Bosu Choi, Yoonsang Lee

TL;DR

This work introduces a nonintrusive sampling error mitigation method that modifies the ensemble to ensure a smooth turbulent spectrum and turns out that the ensemble modification to satisfy the smooth spectrum leads to inhomogeneous localization and inflation, which apply spatially varying localization and inflation levels at different locations.

Abstract

In data assimilation, an ensemble provides a way to propagate the probability density of a system described by a nonlinear prediction model. Although a large ensemble size is required for statistical accuracy, the ensemble size is typically limited to a small number due to the computational cost of running the prediction model, which leads to a sampling error. Several methods, such as localization and inflation, exist to mitigate the sampling error, often requiring problem-dependent fine-tuning and design. This work introduces a nonintrusive sampling error mitigation method that modifies the ensemble to ensure a smooth turbulent spectrum. It turns out that the ensemble modification to satisfy the smooth spectrum leads to inhomogeneous localization and inflation, which apply spatially varying localization and inflation levels at different locations. The efficacy of the new idea is validated through a suite of stringent test regimes of the Lorenz 96 turbulent model.

Sampling error mitigation through spectrum smoothing: first experiments with ensemble transform Kalman filters and Lorenz models

TL;DR

This work introduces a nonintrusive sampling error mitigation method that modifies the ensemble to ensure a smooth turbulent spectrum and turns out that the ensemble modification to satisfy the smooth spectrum leads to inhomogeneous localization and inflation, which apply spatially varying localization and inflation levels at different locations.

Abstract

In data assimilation, an ensemble provides a way to propagate the probability density of a system described by a nonlinear prediction model. Although a large ensemble size is required for statistical accuracy, the ensemble size is typically limited to a small number due to the computational cost of running the prediction model, which leads to a sampling error. Several methods, such as localization and inflation, exist to mitigate the sampling error, often requiring problem-dependent fine-tuning and design. This work introduces a nonintrusive sampling error mitigation method that modifies the ensemble to ensure a smooth turbulent spectrum. It turns out that the ensemble modification to satisfy the smooth spectrum leads to inhomogeneous localization and inflation, which apply spatially varying localization and inflation levels at different locations. The efficacy of the new idea is validated through a suite of stringent test regimes of the Lorenz 96 turbulent model.
Paper Structure (13 sections, 1 theorem, 33 equations, 5 figures, 3 tables, 1 algorithm)

This paper contains 13 sections, 1 theorem, 33 equations, 5 figures, 3 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Sampling error in data assimilation
  3. Ensemble Transformation Kalman Filter
  4. Localization and inflation
  5. Sampling error mitigation based on smoothness constraint
  6. Turbulence Theory
  7. Spectrum smoothing
  8. Numerical experiments
  9. Test model and test regimes
  10. Test results
  11. Comparison with the classic ETKF in various turbulent regimes
  12. Inhomogeneous localization and inflation effect of spectrum smoothing
  13. Conclusion and Discussion

Key Result

Theorem 1

For the correlation function $R(x)$ of a random flow field, $u(x)$, which has translation invariant mean and correlation function and also satisfies the condition it is necessary and sufficient that it has a representation of the form where $F(\omega)$ is a spectral distribution of $u(x)$.

Figures (5)

  • Figure 1: Lorenz 96 mean power spectrum comparison while varying the ensemble size: (a) spectrum smoothing is not applied and (b) spectrum smoothing is applied
  • Figure 2: Time series of 128-dimensional Lorenz 96 with varying $F \in\{4, 8, 16\}$
  • Figure 3: Ensemble spectrum of 128-dimensional Lorenz 96 for various turbulent regimes. Each spectrum is obtained by using an ensemble of size $K=1000$ mean power spectrum comparison while varying $F$, each mean power spectrum is obtained by taking the mean from ensemble of size $K=1000$
  • Figure 4: Ratio of the prior variances between before and after the spectrum smoothing at various times. ETKF using Lorenz 96 model with $F=8$
  • Figure 5: Ratio of the covariances between adjacent components (first off-diagonal components of the prior covariance matrix) between before and after the spectrum smoothing at various times. ETKF using Lorenz 96 model with $F=8$

Theorems & Definitions (1)

  • Theorem 1: Khinchin Theorem