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Structurally informed data assimilation in two dimensions

Tongtong Li, Anne Gelb, Yoonsang Lee

TL;DR

This paper develops a structurally informed data assimilation framework using ensemble transform Kalman filtering (ETKF), which introduces gradient-based weighting matrices constructed from finite difference statistics of the forecast ensemble, thereby allowing the assimilation process to dynamically adjust the influence of observations and prior estimates according to local roughness.

Abstract

Accurate data assimilation (DA) for systems with piecewise-smooth or discontinuous state variables remains a significant challenge, as conventional covariance-based ensemble Kalman filter approaches often fail to effectively balance observations and model information near sharp features. In this paper we develop a structurally informed DA framework using ensemble transform Kalman filtering (ETKF). Our approach introduces gradient-based weighting matrices constructed from finite difference statistics of the forecast ensemble, thereby allowing the assimilation process to dynamically adjust the influence of observations and prior estimates according to local roughness. The design is intentionally flexible so that it can be suitably refined for sparse data environments. Numerical experiments demonstrate that our new structurally informed data assimilation framework consistently yields greater accuracy when compared to more conventional approaches.

Structurally informed data assimilation in two dimensions

TL;DR

This paper develops a structurally informed data assimilation framework using ensemble transform Kalman filtering (ETKF), which introduces gradient-based weighting matrices constructed from finite difference statistics of the forecast ensemble, thereby allowing the assimilation process to dynamically adjust the influence of observations and prior estimates according to local roughness.

Abstract

Accurate data assimilation (DA) for systems with piecewise-smooth or discontinuous state variables remains a significant challenge, as conventional covariance-based ensemble Kalman filter approaches often fail to effectively balance observations and model information near sharp features. In this paper we develop a structurally informed DA framework using ensemble transform Kalman filtering (ETKF). Our approach introduces gradient-based weighting matrices constructed from finite difference statistics of the forecast ensemble, thereby allowing the assimilation process to dynamically adjust the influence of observations and prior estimates according to local roughness. The design is intentionally flexible so that it can be suitably refined for sparse data environments. Numerical experiments demonstrate that our new structurally informed data assimilation framework consistently yields greater accuracy when compared to more conventional approaches.

Paper Structure

This paper contains 18 sections, 47 equations, 8 figures, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Problem set up
  4. Ensemble transform Kalman filter (ETKF)
  5. Inflation and localization
  6. Structurally informed prior design
  7. Covariance-based weighting matrix
  8. Gradient-based weighting matrix
  9. The construction of $\widehat{S}$
  10. Scaling and localization
  11. Directional structure-based correlation refinement
  12. Numerical examples
  13. Evaluation metrics
  14. Linear advection equation with dense observations
  15. Burgers' equation with dense observations
  16. ...and 3 more sections

Figures (8)

  • Figure 1: Gradient statistics \ref{['eq: x gradient']} for (top-left) $x$ and (top-right) $y$; (bottom-left) aggregate gradient statistic \ref{['eq: joint gradient']}; and (bottom-right) variance $\widehat{V}_{i,j}$\ref{['eq: variance']} at time $t =2$. Here $(\vartheta,\varphi)=(1,1)$ and $\widetilde{\beta} = 10^{-3}$ in \ref{['eq:betat']} is used to determine $\beta$ in \ref{['eq: Sd']}.
  • Figure 2: Numerical solutions for the posterior mean ${\mathbf{m}}$ for $W_C^D$ with $\alpha = 4, 6$ and $W_S^D$ with $(\vartheta,\varphi) = (1/2,1), (1/2,2), (1,1), (1,2), (2,1)$, and $(2,2)$ respectively (left to right).
  • Figure 3: Posterior mean solution ${\mathbf{m}}$\ref{['eq:mean_update']} at time $t=2$ obtained by $W_C^D$ in \ref{['eq: Cd']} (left), and $W_S^D$ in \ref{['eq: Sd']} with $(\vartheta,\varphi)=$ (1/2, 1) (middle left), (1,1) (middle right), and (2,1) (right).
  • Figure 4: Numerical solutions for the posterior mean ${\mathbf{m}}$ (top) and the corresponding pointwise error $err$ (bottom) for $W_C^D$ and $W_S^D$ with $(\vartheta, \varphi)=$ (1/2, 1), (1,1) , and (2,1) (from left to right) respectively.
  • Figure 5: Relative errors $err_{l_1}$\ref{['eq:l1err']} (left) and $err_{l_2}$\ref{['eq:l2err']} (middle), along with the complementary pattern correlation $1-Pcorr$\ref{['eq:pcorr']} (right), obtained for the posterior mean using $W_C^D$ and $W_S^D$ with momentum parameters $\vartheta = 1/2, 1, 2$.
  • ...and 3 more figures

Theorems & Definitions (8)

  • Remark 2.1
  • Remark 3.1
  • Remark 3.2
  • Remark 3.3
  • Remark 3.4
  • Remark 3.5
  • Remark 3.6
  • Remark 4.1