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Alignments of Geophysical Fields: a differential geometry perspective

Yicun Zhen, Valentin Resseguier, Bertrand Chapron

Abstract

To estimate the displacements of physical state variables, the physics principles that govern the state variables must be considered. Technically, for a certain class of state variables, each state variable is associated to a tensor field. Ways displacement maps act on different state variables will then differ according to their associated different tensor field definitions. Displacement procedures can then explicitly ensure the conservation of certain physical quantities (total mass, total vorticity, total kinetic energy, etc.), and a differential-geometry-based optimisation formulated. Morphing with the correct physics, it is reasonable to apply the estimated displacement map to unobserved state variables, as long as the displacement maps are strongly correlated. This leads to a new nudging strategy using all-available observations to infer displacements of both observed and unobserved state variables. Using the proposed nudging method before applying ensemble data assimilation, numerical results show improved preservation of the intrinsic structure of underlying physical processes.

Alignments of Geophysical Fields: a differential geometry perspective

Abstract

To estimate the displacements of physical state variables, the physics principles that govern the state variables must be considered. Technically, for a certain class of state variables, each state variable is associated to a tensor field. Ways displacement maps act on different state variables will then differ according to their associated different tensor field definitions. Displacement procedures can then explicitly ensure the conservation of certain physical quantities (total mass, total vorticity, total kinetic energy, etc.), and a differential-geometry-based optimisation formulated. Morphing with the correct physics, it is reasonable to apply the estimated displacement map to unobserved state variables, as long as the displacement maps are strongly correlated. This leads to a new nudging strategy using all-available observations to infer displacements of both observed and unobserved state variables. Using the proposed nudging method before applying ensemble data assimilation, numerical results show improved preservation of the intrinsic structure of underlying physical processes.
Paper Structure (33 sections, 2 theorems, 91 equations, 9 figures)

This paper contains 33 sections, 2 theorems, 91 equations, 9 figures.

Table of Contents

  1. introduction
  2. A differential geometry formulation of the optimisation problem
  3. A detailed formulation of $T^\#$ for physical fields
  4. A physical interpretation of the tensor field definitions
  5. Comparison with OF algorithms
  6. Comparison with other methods
  7. About the boundary condition of \ref{['eq: optical flow for tensor fields, compact with boundary']}
  8. The case for localized observations
  9. A displacement-based formulation of the nudging method in continuous time and continuous space
  10. Instantaneous nudging based on displacement error and its application in data assimilation
  11. Methodology
  12. Numerical results
  13. Conclusion
  14. Differential forms, pull-back map $T^*$, Hodge star operator and Lie derivatives in Euclidean spaces for practitioners in data assimilation
  15. Vector field
  16. ...and 18 more sections

Key Result

Theorem 2.1

For smooth tensor fields and vector fields $\theta_1,\theta_2,$ and ${\bf v}$, the optimisation problems eq: optical flow for tensor fields, compact no boundary, v2 and eq: optical flow for tensor fields, compact with boundary, v2 are always uniquely solvable. And the solution is smooth.

Figures (9)

  • Figure 1: Suppose that the difference between the model estimate $x^b$ and observed state $y^o$ only differ by the position, then the most natural nudging strategy is to gradually moves $x^b$ rightward.
  • Figure 2: Suppose that the original field $S$ is a rotating wind field and the given displacement map $T$ is clockwise rotation by $90^\circ$. Then the direct composition of $S$ and $T$ results in a wind field of completely different feature.
  • Figure 3: Comparison of the target fields (the first row), the original fields (the third row), and the morphed fields (the second row). This is the result of one of the ensemble members. The displacement map is calculated based on the $\omega$-field and the $h$-field. Then the displacement map is applied to the buoyancy field $\Theta$ and the velocity field ${\bf v}$. For this ensemble member, it is not hard to see that both the buoyancy and the velocity are partially aligned with the truth. However, the results for another ensemble member is not so optimistic. See Fig.\ref{['fig: xfmember2']}.
  • Figure 4: Comparison of the target fields (the first row), the original fields (the third row), and the morphed fields (the second row). This is the result for another ensemble member. The displacement map is calculated based on the $\omega$-field and the $h$-field. Then the displacement map is applied to the buoyancy field $\Theta$ and the velocity field ${\bf v}$. For this ensemble member, the initial discrepancy between $\Theta_2$ and $\Theta_1$ is larger than that in Fig.\ref{['fig: xfmember1']}. And the result is not as good as the result for the ensemble member shown in Fig.\ref{['fig: xfmember1']}.
  • Figure 5: The target field (first row), the original field (third row), and the field morphed by $T^\#S = S\circ T^{-1}$ for $S = h,\omega, \Theta, {\bf v}$ (second row). Since $T^{\#}S = S\circ T$ is not physically consistent, a singularity has been generated in the morphed $\omega_2$-field.
  • ...and 4 more figures

Theorems & Definitions (7)

  • Definition 2.0.1
  • Theorem 2.1
  • Example 2.1.1
  • Example 2.1.2
  • Example 2.2.1
  • Example 2.2.2
  • Theorem 2.2