MS egusphere-2023-2755

Title: Bridging classical data assimilation and optimal transport

Authors: Bocquet et al

Summary

The authors propose a modification of 3D-Var in which instead of using a classic quadratic error function in both the observational and background error, a variant of the Wasserstein distance is used. The authors argue that this could potentially improve results as certain problems with the quadratic error function (including what is known as the ``double penalty error'') are thus avoided. The authors motivate the detailed form of the cost function and discuss possible approaches to carry out the 3D-Var minimisation in order to obtain the analysis.

General Comments

The paper discusses a very interesting and pertinent problem and proposes a solution that is certainly worth looking at further. The continued use of quadratic error functional is somewhat anachronistic given that the main motivation for using it are simplicity of analytic computations and an assumption of Gaussian errors. The first argument carries less weight in the age of supercomputers, and the second was always known to be wrong except in (important) special cases.

I cannot, however, recommend the publication of the paper in its present form and believe that the paper requires a major revision. This is due to the following major concerns (see below for a few more minor concerns)

(1) I find the first section very confusing. The authors are discussing problems related to the so-called double penalty error and the non-overlap of functions (or distributions) that appear in data assimilation. It is not clear however on what level the authors are working. More specifically, it seems first that the authors want to work on the level of probability distributions for state variables. Later however it turns out that they want to work on the level of state variables directly, yet focus on those that represent meteorological fields which essentially have the character of distributions. This, however, has to be clear from the very beginning.

(2) I understand what the authors label as the first (of two) weaknesses of classical DA which is often termed the double penalty error. This problem however is ultimately an issue resulting from a mismatch between the employed distance and the smoothness of meteorological fields. If the correct metric is selected depending on the smoothness of the meteorological fields, there is no double penalty problem. Labelling this as a problem of ``classical DA'' however inplies that classical DA only uses the mean square error which is not correct (error covariances are important part of the error functional and have a strong influence on whether the double penalty problem occurs or not).

(3) Related to the previous question, I do NOT understand what the authors label as the second weakness of classical DA. In fact, the last paragraph of Sec. 1.1 hardly makes any sense when it talks about ``overlap in space and time'' between background and observations. The material seems to draw on intuition coming from the Bayes rule but that applies to probability densities; the 3D-Var analysis is an operator-convex combination of meteorological fields which is something completely different.

(4) As far as I can see, the technique can only assimilate meteorological fields that essentially behave like distributions, which is clearly a major restriction. I believe the assumption that the fields be positive is not enough (after all, by choice of origin any meteorological field can be assumed to have nonnegative values). All the examples mentioned by the authors are extensive quantities. Is there an issue with applying the approach to an intensive quantity such as, say, the temperature?

Minor Comments

I have only a few minor comments at this point but I believe that a quite a few more might pop up once the major issues above have been clarified

(a) It is not clear how the authors deal with comparing fields that do not have the same mass. Also, it is not clear what the cumbersome result in Fig. 5 has to do with the assumption whether or not the two fields have the same mass.

(b) The concept of the entropy regularisation is not clear. It is not even clear why this renders the problem convex or at least uniquely solvable.

(c) There are other ways to measure distances between meteorological fields that avoid or alleviate the double penalty error (depending on the smoothness of the fields). Have the authors compared their Wasserstein approach with other metrics, also given that the entropy regularised Wasserstein distances are not easy to calculate and optimise?

(d) On pg 6 the authors claim that ``our problem is not subject to the curse of dimensionality''. Although this is true, the curse of dimensionality (in the sense implied by the discussion here) is not actually a concern in 3D-Var either which is what the method should be compared with. So this remark is misleading.

The paper proposed a different objective function for the data assimilation problem, which is a mixed sum between the classic cost function and the Wasserstein metrics. With assumptions on strictly convexity and linearity, the problem is convex. The paper then derives the optimization conditions for the new objective function by combining convex optimization analysis and the duality concerned about the Wasserstein metric with entropy regularization. It is further supported by several 1D and 2D data assimilation problems.

1. My biggest complaint is the paper title. I think the title does not match the content of the paper. Based on the title, it sounds like a providing a theoretical connection between classical data assimilation and OT. However, it is more like introducing an application of OT theory for data assimilation problems. I suggest the authors change the title of this paper to a more descriptive one.
2. The assumption in equation (11) is rather strong. H is linear, and all the costs are convex. That means the paper only deals with log-concave distributions, which does not apply to the multi-modal distributions that are challenging to handle. The baseline DA (17) is a strictly convex problem that can be solved easily. Even if one is worried about noise overfitting, many existing good methods exist to handle this.
3. Between (17) and (18), there are new introductions of x^b and x^o with Wasserstein metrics, turning the problem from strongly convex to a mixed problem. The motivation is not very clearly stated. The significantly increased computational cost associated with (18) has to be supported by very strong reasons. For example, what properties can we achieve by combining these two different cost functions?
4. Section 2.4 has too many details about the derivation that are standard steps in convex optimization. I suggest putting many in an appendix and only stating the main formula.
5. The numerical examples in Sections 3 & 4 are a bit too simple. Of course, 1D and 2D OT are not so costly. However, when the dimension becomes large, the extra two terms in (18) become increasingly cumbersome, and computational cost is forbidden. This work is for geoscience applications with often high-dimensional state space.

Overall, I feel the paper title is too big of a summary for the paper, and the numerical examples, on the other hand, are elementary. While I can relate the formulation from (6) to (7), which is very neat and has a clear mathematical intuition, the hybrid sum in (18) seems to be a "cocktail" of two different metrics. Further understanding is necessary even if the authors don't plan on proving any mathematical properties.