the Creative Commons Attribution 4.0 License.
the Creative Commons Attribution 4.0 License.
| 28 Nov 2024
Alignment 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 underline physical processes.
Status: final response (author comments only)
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AC1: 'Comment on egusphere-2024-3623', Yicun Zhen, 02 Dec 2024
Feel free to question and comment! Thanks.
Citation: https://doi.org/10.5194/egusphere-2024-3623-AC1 -
RC1: 'Comment on egusphere-2024-3623', Anonymous Referee #1, 17 Feb 2025
General comments:
In this manuscript, Zhen and co-authors propose an physics-obeying framework to estimate alignment fields to solve displacement errors in geophysical data assimilation. The idea of working from the manifold of physically-allowable fields is interesting.
However, I must regrettably recommend rejecting this manuscript because the manuscript seems unsuitable for NPG. >90% of DA practitioners (the manuscript's targeted audience) will not understand this manuscript due to the sheer amount of mathematical background required. The typical DA scientist knows some linear algebra (up to eigenvectors and linear spaces), some basic multivariate calculus (think Calculus III at Zhen's alma mater in Penn State), some statistics and some probability theory. Considering that the lead author has published with DA practitioners like the famous late Fuqing Zhang (Zhen and Zhang 2014, Monthly Weather Review), they should be aware of the general level of mathematical expertise that DA practitioners have. Considering the effort needed to rewrite this manuscript for a DA audience, I recommend re-submitting this manuscript to an applied mathematics journal or a computer science journal.
If the authors did successfully rewrite their manuscript for a DA audience (no more than Calculus III level!), the following comments should then be considered.
Major comments:
1) Please consider more recent advances in the field alignment literature. In particular, I recommend reviewing the following works and related references. Please compare and contrast your method with methods like these (e.g., the physics-based penalty terms used to prevent grossly unphysical alignment fields)
Stratman, D. R., and C. K. Potvin, 2022: Testing the Feature Alignment Technique (FAT) in an Ensemble-Based Data Assimilation and Forecast System with Multiple-Storm Scenarios. Mon. Wea. Rev., 150, 2033–2054, https://doi.org/10.1175/MWR-D-21-0289.1.
Ying, Y., 2019: A Multiscale Alignment Method for Ensemble Filtering with Displacement Errors. Mon. Wea. Rev., 147, 4553–4565, https://doi.org/10.1175/MWR-D-19-0170.1.
2) The experiment presented in the manuscript (Section 3.2) is inadequate because only a single time-step DA outcome is shown. Please perform some cycled OSSEs that demonstrate that your framework results in root-mean-square-errors that are consistently better than a standard DA method like the EnKF. Even perfect model OSSEs will do.3) To convince your audience that your framework is superior to other alignment estimation approaches, please perform cycled DA OSSEs with an alternative approach. In my experience, the FAT-type method is not difficult to implement and try.
4) Please provide estimates for the computational cost/complexity of your framework, and whether your framework is embarrassingly parallel. A key difficulty in practical DA is the high dimensionality (>10^7) of the state space. If your framework's computational cost/complexity does not scale linearly with dimensionality or is inherently serial (i.e., impossible to parallelize), it cannot be used for practical DA.
Minor comments:
1) Your in-text reference citations are incorrectly formatted. I believe this is an issue with your citation command in LaTeX or your chosen template.
2) Please display your algorithms using a text Table, not a print screen.
3) Please check to see if you have defined every symbol you used. For example, the epsilon symbol on used in Line 21 is not defined and sv in (2) is not defined.
4) It is unclear to me how (1) leads to (2). Please derive (2) an/or provide a reference that connects (1) and (2).
5) L95: Please define the Hodge star operator.
6) Fig 3 & 4: Please swap the second and third rows.
Citation: https://doi.org/10.5194/egusphere-2024-3623-RC1 - CC1: 'Reply on RC1', Yicun Zhen, 19 Feb 2025
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RC2: 'Comment on egusphere-2024-3623', Anonymous Referee #2, 22 Feb 2025
Summary
The authors develop a method to nudge model estimates of state variables to match observed states. Using knowledge of the dynamics of the physical system, this method constrains the possible nudging functions to the set of diffeomorphisms on the state space. It is shown that this method preserves physically relevant details in the nudged system, such as conservation laws. The method is compared to other data assimilation strategies.
General Comments
As written, I think this paper would be more at home in an applied mathematics journal. The general readership of a geophysical journal will struggle with the level of mathematical abstraction here. Many geophysical scientists do know and use some differential geometry, but the majority that I have seen approach it from a background of physics (think index notation) rather than mathematics (index-free notation). I think representing at least a few of the foundational equations here in index notation or providing a reference that makes such a connection would be helpful. For example the discussion surrounding the definition of Lie derivatives in eqs. (4) - (11) is very terse and abstract, and the notation used is quite different from the notation of physicists in, e.g. general relativity. This section might benefit from a simple figure showing \Phi effecting “flow down the integral curves” defined in (9). See for example Figures B2 and B3 in Sean Carroll’s Spacetime and Geometry. You could just point the reader to such a reference.
There is a lot of (justified) technical detail here, but I think the paper would be considerably easier to understand if each example were brought explicitly back to the simplest problem set up in Figure 1. What are S, \Omega, and T in each example? What physical constraint are you applying to constrain T? I sometimes found it difficult to see the forest for the trees. I think examples 2.2.1 and 2.2.2 do a good job of this, but I’m still not totally sure how \Phi relates to T#.
It would be helpful to have more explicit figure captions.
Specific Comments
Since the notation is pretty abstract, it would be helpful to make more explicit use of the simple example presented in Figure 1. What is \Omega in this case? Is it simply the xy plane? In general would it be a more abstract state space? Is the field S in this example just the color assigned to each point? Are S1 and S2 the estimate and the observation? What is the optimal T that results in a rightward shift of the bright spot? Is it just a vector field that points everywhere rightward?
I don’t understand the difference between T and T#. If T#S mean “the map T applied to the field S,” what does T# mean on its own — is this a “sharp” in the sense of the musical isomorphisms between tangent and cotangent bundles on a manifold? The paragraph from lines 43-53 seems to make use of a distinction between T and T#, but I don’t understand what that is.
What exactly is the operation T#S? It seems like it is usually either S composed with T or its inverse, but Line 280 seems to imply that this is only an assumption. When is this assumption invalid? When do you compose with T and when with its inverse?
Ex. 2.1.1 — (16) and (17) are inscrutable to me; could you include more description in the surrounding text? Does the asterisk denote the Hodge star? Some other operation? If it’s a Hodge star it should be typeset in the same way as on Line 95 when it is introduced. I’m also not sure why the inverse of T is used.
Exs 2.2.1 and 2.2.2: I think you should be really explicit about what the T (or T#?) ultimately is here. My understanding is that it is an approximation of \Phi — is that correct?
Line 228: If M is the configuration space shouldn’t a state vector S_i be an element of the tangent bundle of M rather than of M itself?
Line 315: “But the second part is merely for mathematical reasons.” Should be more explicit.
It would be helpful to have more explanation of what exactly to look for in figures 3-7. For example Line 317 states that “…the difference between the morphed h_2 field and the target h_1 field is much smaller than between the morphed \omega_2 and the target \omega_1.” This isn’t terribly obvious from just looking at the figures — is there a statistical analysis that justifies this statement, or should it be clear just from examining the figure? Why do you suspect that this difference is attributable to the positiveness of the h field?
The sentence beginning on Line 328 “The space of ensemble members… to become a curved manifold.” Is confusing, I would reword it.
Technical Corrections
Line 9: “underline” should be “underlying”
Line 36: “the most transforms” should be “transforms fastest” or “transforms most quickly”
Line 56: “underline” should be “underlying”
Line 94: “writes” should be “is written as” or just “is”
Line 123: “an unique” should be “a unique”
Line 150: “translates on” should be “translates to” or “becomes”
Line 159: “writes” should probably be “can be written as” or something similar to that.
Line 204: “enable to also transport” should be “also enable us to transport” or “also enable transport of” or something similar.
Line 245: “can be thought as” should be “can be thought of as”
Line 250: “idealistic” should be “idealized” or “idealised.” Spelling difference here is simply a US/UK convention.
Line 282: “ensue” should be “ensure”
Line 312: “Expected” should be “As expected”
Line 316: “maintain” should be “maintains”
Line 348: “underline” should be “underlying”
Citation: https://doi.org/10.5194/egusphere-2024-3623-RC2 -
CC5: 'Reply on RC2', Yicun Zhen, 27 Feb 2025
Dear anonymous reviewer,
Thank you for your patient review. For the communication of purpose, this is my response to your specific questions. Hope it will help you and the other readers. Formal response will be provided in case it is needed.
Best wishes,
Yicun Zhen
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CC5: 'Reply on RC2', Yicun Zhen, 27 Feb 2025
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