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LEMDA: A Lagrangian-Eulerian Multiscale Data Assimilation Framework

Quanling Deng, Nan Chen, Samuel N. Stechmann, Jiuhua Hu

Abstract

Lagrangian trajectories are widely used as observations for recovering the underlying flow field via Lagrangian data assimilation (DA). However, the strong nonlinearity in the observational process and the high dimensionality of the problems often cause challenges in applying standard Lagrangian DA. In this paper, a Lagrangian-Eulerian multiscale DA (LEMDA) framework is developed. It starts with exploiting the Boltzmann kinetic description of the particle dynamics to derive a set of continuum equations, which characterize the statistical quantities of particle motions at fixed grids and serve as Eulerian observations. Despite the nonlinearity in the continuum equations and the processes of Lagrangian observations, the time evolutions of the posterior distribution from LEMDA can be written down using closed analytic formulae. This offers an exact and efficient way of carrying out DA, which avoids using ensemble approximations and the associated tunings. The analytically solvable properties also facilitate the derivation of an effective reduced-order Lagrangian DA scheme that further enhances computational efficiency. The Lagrangian DA within the framework has advantages when a moderate number of particles is used, while the Eulerian DA can effectively save computational costs when the number of particle observations becomes large. The Eulerian DA is also valuable when particles collide, such as using sea ice floe trajectories as observations. LEMDA naturally applies to multiscale turbulent flow fields, where the Eulerian DA recovers the large-scale structures, and the Lagrangian DA efficiently resolves the small-scale features in each grid cell via parallel computing. Numerical experiments demonstrate the skilful results of LEMDA and its two components.

LEMDA: A Lagrangian-Eulerian Multiscale Data Assimilation Framework

Abstract

Lagrangian trajectories are widely used as observations for recovering the underlying flow field via Lagrangian data assimilation (DA). However, the strong nonlinearity in the observational process and the high dimensionality of the problems often cause challenges in applying standard Lagrangian DA. In this paper, a Lagrangian-Eulerian multiscale DA (LEMDA) framework is developed. It starts with exploiting the Boltzmann kinetic description of the particle dynamics to derive a set of continuum equations, which characterize the statistical quantities of particle motions at fixed grids and serve as Eulerian observations. Despite the nonlinearity in the continuum equations and the processes of Lagrangian observations, the time evolutions of the posterior distribution from LEMDA can be written down using closed analytic formulae. This offers an exact and efficient way of carrying out DA, which avoids using ensemble approximations and the associated tunings. The analytically solvable properties also facilitate the derivation of an effective reduced-order Lagrangian DA scheme that further enhances computational efficiency. The Lagrangian DA within the framework has advantages when a moderate number of particles is used, while the Eulerian DA can effectively save computational costs when the number of particle observations becomes large. The Eulerian DA is also valuable when particles collide, such as using sea ice floe trajectories as observations. LEMDA naturally applies to multiscale turbulent flow fields, where the Eulerian DA recovers the large-scale structures, and the Lagrangian DA efficiently resolves the small-scale features in each grid cell via parallel computing. Numerical experiments demonstrate the skilful results of LEMDA and its two components.
Paper Structure (27 sections, 46 equations, 10 figures)

This paper contains 27 sections, 46 equations, 10 figures.

Table of Contents

  1. Introduction
  2. The LEMDA Framework
  3. The Lagrangian particle model
  4. The continuum model for describing Eulerian quantities
  5. Stochastic approximate forecast model for the flow field
  6. The LEMDA framework with closed analytic solution
  7. Assessing the Skill of DA
  8. Path-wise measurements for assessing the skill in the posterior mean estimate
  9. An information measurement for assessing the skill in the posterior distribution
  10. Lagrangian DA
  11. Performance of the Lagrangian DA
  12. Reduced-order Lagrangian DA
  13. Eulerian DA
  14. Eulerian DA with no particle collisions
  15. Recovery of large-scale flow feature using Eulerian DA in the presence of particle collisions
  16. ...and 12 more sections

Figures (10)

  • Figure 2.1: Overview of the LEMDA framework. Panel (a): the Lagrangian observations and Lagrangian DA (see Section \ref{['Subsec:Lagrangian_Model']}). Note the strong nonlinearity in the observational process of the Lagrangian DA. Panel (b): the Eulerian description of particle statistics, which can be derived from the Boltzmann description of the particle system (see Section \ref{['Subsec:Eulerian_Model']}). Panel (c): applying the Eulerian DA to the cases where the observed tracers are the ice floes with collisions (see Section \ref{['Subsec:Collisions']}). Panel (d): applying the Lagrangian DA to each grid cell to recover refined features that are missed by the large-scale Eulerian DA in the multiscale DA framework. Panel (e): the development of cheap stochastic surrogate models for the underlying flow field, which is a crucial part of allowing the analytic solvers of both the Eulerian and the Lagrangian DAs (see Section \ref{['Subsec:Approximate_Models']}). Panel (f): rigorous derivation of reduced-order DA schemes for Lagrangian DA (see Section \ref{['Subsec:Reduced_LDA']}). The closed analytic solutions for the posterior distributions of LEMDA are shown in Section \ref{['Subsec:Closed_Formulae']}.
  • Figure 4.1: Lagrangian DA with parameters in \ref{['Parameters_LaDA_model1']} and \ref{['Parameters_LaDA_model2']}. Panels (a)--(c) show the RMSE \ref{['Skill_Scores_RMSE']}, pattern correlation \ref{['Skill_Scores_Corr']} and uncertainty reduction \ref{['Signal_Dispersion']} as a function of the tracer numbers $L$ (x-axis) and drag coefficient $\beta$ (different curves). Panel (d) shows the real part of the posterior estimate of the mode $(-4,-4)$ with different $L$ and $\beta$. Panel (e) shows the flow field at $t=8$ with different $L$ and $\beta$. The last subpanel includes the truth. Note that in this and the subsequent figures, all flow fields are shown in the non-dimensional units. The velocity with an amplitude of $0.5$ corresponds to $0.05$m/s.
  • Figure 4.2: Lagrangian DA and its reduced-order form, where $\beta=1$ is adopted. Panel (a) compares the skill scores of the full and reduced-order DA solutions in terms of the normalized RMSE and the pattern correlation of mode $(-4,-4)$. Panel (b) shows the diagonal entry of the posterior variance of mode $(-4,-4)$ resulting from the full DA scheme averaged over the time interval $[1,10]$ with that given by the predetermined value in \ref{['root_Rk_SI']}. Panel (c) shows the posterior covariance from the filtering solution at the last time instant $t=10$ with $L=20$ tracers. Panel (d) shows the posterior covariance when the reduced-order DA scheme is applied. Panels (e)--(f) demonstrate the posterior mean time series and the posterior uncertainty for mode $(-4,-4)$. The posterior uncertainty is given by the two standard deviations from the mean value.
  • Figure 5.1: Eulerian DA of the flow field using without particle collisions. The observations of the momentums are computed based on $L = 2000$ observed particle and the Eulerian observations are taken at a $9\times 9$ mesh grid. Panels (a)--(b) compare the filtering results with the truth for two Fourier modes $(1,1)$ and $(4,4)$. Panel (c) shows the comparison of the truth and the recovered velocity field in physical space at $t=1,3$ and $5$. Panel (d) shows the skill scores of the velocity field in physical space.
  • Figure 5.2: Skill scores of Eulerian DA with different number of particles $L$ and grid size $N_x=N_y$. Panels (a)--(b) show the RMSE and pattern correlation as a function of time using different $L$, where the mesh size $N_x=N_y=9$ is fixed. Panel (c) shows the skill scores average over time as a function of $L$. Panel (d) shows the skill scores as a function of $N_x=N_y$, where $L=8000$ is used.
  • ...and 5 more figures