A Robust Variational Framework for Cyclogeostrophic Ocean Surface Current Retrieval

Bertrand, Vadim; Le Sommer, Julien; Vianna Zaia De Almeida, Victor; Samson, Adeline; Cosme, Emmanuel

doi:https://doi.org/10.5194/egusphere-2025-4172

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https://doi.org/10.5194/egusphere-2025-4172

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04 Sep 2025

| 04 Sep 2025

Status: this preprint is open for discussion and under review for Ocean Science (OS).

A Robust Variational Framework for Cyclogeostrophic Ocean Surface Current Retrieval

Vadim Bertrand, Julien Le Sommer, Victor Vianna Zaia De Almeida, Adeline Samson, and Emmanuel Cosme

Abstract. Estimations of surface currents at submesoscales (1–50 km) are crucial for operational applications and environmental monitoring, yet accurately deriving them from satellite observations remains a challenge. While the geostrophic approximation has long been used to infer ocean surface currents from Sea Surface Height (SSH), it neglects nonlinear advection, which can become significant at submesoscales. To address this limitation, we present a robust and efficient variational method for inverting the cyclogeostrophic balance equation, implemented in the open-source Python library jaxparrow. Unlike the traditional iterative approaches, our method reformulates the inversion as an optimization problem, providing stable estimates even in regions where a cyclogeostrophic solution may not exist. Using both DUACS and the high-resolution NeurOST SSH products, we demonstrate that cyclogeostrophic corrections become increasingly relevant at finer spatial scales. Validation against drifter-derived velocities shows that our approach consistently improves current estimates in energetic regions, reducing errors by up to 20 % compared to geostrophy alone in energetic regions of the global ocean. These results support the systematic inclusion of cyclogeostrophic inversion in the analysis of high-resolution SSH fields.

Received: 27 Aug 2025 – Discussion started: 04 Sep 2025

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Vadim Bertrand, Julien Le Sommer, Victor Vianna Zaia De Almeida, Adeline Samson, and Emmanuel Cosme

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'Comment on egusphere-2025-4172', Anonymous Referee #1, 13 Oct 2025 reply
While balanced motions dominate at scales consistent with or larger than the ocean mesoscale, in cases with substantial curvature (e.g. due to eddies), cyclogeostrophic effects can be key contributors to overall oceanic motions. As the authors note, this topic has been explored extensively over the past ~80 years both in meteorological and oceanographic literature. In this study, the authors develop and text an open source utility to compute the cyclogeostrophic correction for gridded altimeter data. The results show that a variational approach is more successful than a more commonly used iterative approach, and that in high-energy regions, both improve on the more limited geostrophic correction. The software utility and discussion will be of interest to readers. However, the statistical diagnostics have some inconsistencies. Revisions will be needed prior to publication.
A critical concern stems from the fact that equation (8) defines the misfit epsilon as the square root of the squared misfit of the two components. This is a positive quantity that should have a Rayleigh distribution. However, the subsequent statistics treat it as a Gaussian quantity in order to determine the standard error of the mean in equation (10). This is not automatically true. More attention needs to be given to the metrics used to assess performance.

Because the analysis is based on a Rayleigh distribution, the distinction between the mean (epsilon) and the standard deviation (sigma) is not very meaningful. If the mode of the Rayleigh distribution is defined as sigma (using the notation from Wikipedia---apologies this is a different definition of sigma than the standard deviation), then the mean is sigma sqrt(pi/2), and the standard deviation is sigma sqrt((4-pi)/2). This means that Figure 4 and Figure C2 present redundant information. Moreover, because the mean and the standard deviation convey the same information, it’s not clear how to interpret the information in Figures 4 and C2.

The manuscript uses surface drifters as a reference for assessing whether the cyclogeostrophic velocity provides a more realistic assessment of total velocity, but the plotted results do not provide compelling evidence. Figure 4a shows that cyclogeostrophic velocities have greater spread relative to drifters in regions of high kinetic energy, but this does not show whether the means have converged. Figure C2a shows that there is on average a non-zero difference from drifters and that the mean speed difference is larger in regions of high eddy energy, which seems unsurprising. Figure 4b shows that for NeurOST the spread of the corrected velocities is smaller in eddy-intense regions such as the Kuroshio Extension, while Figure 4c shows that with DUACS data the spread is slightly larger in eddy-intense regions. This does not formally tell us whether the cyclogeostrophic correction brings the altimeter-derived currents into better agreement with drifter velocities. Figures C2b and C2c should contain information showing a reduction in misfit with the cyclogeostrophic correction, but they are not discussed in the text. Figure 5 shows misfit but does not indicate a statistically significant improvement from the cyclogeostrophic correction. The manuscript needs to provide a clear metric showing that the variational method works.

Equation (11) is based on the fraction of standard deviation explained by the correction. A more standard metric would look at fraction of variance (using a squared quantity). The manuscript should provide more clarity about the choice of statistical metrics and the robustness of the metrics reported in the paper.

The authors cite a few studies but do not revisit the analytic gradient wind solution, discussed nicely by Penven et al (2014) and by Knox and Ohmann (2006, https://doi.org/10.1016/j.cageo.2005.09.009). To me this seems like a disappointing gap. Of note, the gradient wind approach allows an analytic solution, which is imperfect but presumably could serve as a first guess for the iterative or variational approaches. Readers who want to implement a robust algorithm will probably want to understand the performance of the variational approach in the context of the gradient wind solution and to understand whether the gradient wind solution is useful as an initial guess for other solution strategies.

Lines 40 and 44. “convective force”. By convention, in oceanography convection is buoyancy driven, and horizontal motions are advective. They are not strictly a force, since the advective terms are intrinsic to the Navier Stokes equations and not imposed externally. This usage in oceanography contrasts with some fluid mechanics literature, which can distinguish between convection (i.e. advection) vs natural convection (motion driven by buoyancy gradients). To ensure that the manuscript is accessible to readers with an oceanographic background, the term “convective force” should be replaced with “advective terms”.

Line 44. “Coriolis force”. Coriolis is usually described as a “pseudo force”. It would be more accurate to say “Coriolis term”.

Lines 80-81. “A typical case of divergence is when the cyclogeostrophic equation has no solution”. It would be useful to clarify why the equation sometimes has no solution.

Line 140. “to drogued SVP-type drifters”. Drifters are notorious for losing their drogues without being properly flagged as missing drogues. It would be helpful to remind readers which version of the data you are using and to specify how well you think the drogue losses are flagged.

Line 145. “Due to the use of Arakawa C-grids”. How is the C-grid being used? Gridded data products are not intrinsically on the C-grid, so this point requires clarification.

Equation (8). In calculating epsilon, what is the spatial separation allowed for the interpolation? Are there constraints on satellite overpass time or distance from ground track?

Equations (10-11). Is N the same for M1 and M2? If it different, the normalized difference in equation (11) could depend largely on the number of samples, which is not really the intent of this metric, I believe. On the other hand, if N is the same in both cases, then this is really a comparison of rms error in the two cases. That leads to a question of whether the variance would be more appropriate as a metric.

Line 213. “anticyclonic (cyclonic)”. Is this redundant with the previous sentence? Classic warm core rings are anticyclonic and are north of the Gulf Stream. Cold core rings are cyclonic and are south of the Gulf Stream.

Line 216. “introduces artifacts in the most dynamic parts of the jet and eddies.” More explanation would help. What accounts for the artifacts? Does the variational method implicitly impose a smoothness parameter by minimizing the global misfit?

Line 237. “NeurOST-based cyclogeostrophy clearly reduces standard error”. Given the size of the standard errors plotted in Fig. 5, does a reduction in standard error show that the results are better or just that they are slightly more consistent, although not at a level that could be judged to be statistically significant?

Lines 240-241. “we find that at the highest EKE percentiles, cyclogeostrophy reduces reconstruction uncertainty by nearly 10 % upon geostrophy when employing the variational method.” Changes shown in Figure 5 do not appear statistically significant and would not pass a Student T-test. This does not appear to be a robust measure of the success of the algorithm, or even of the relevance of the cyclogeostrophic correction. That’s not to say that it’s not useful, but the presentation will need to be reworked.

Figure 5. Reiterating my previous point, the results in the figure suggest no statistically significant improvement by using cyclogeostrophy compared with geostrophy, and no visible differences except in high energy areas, and then only for the NeurOST product. As discussed in the text, DUACS doesn't benefit from correction and is not as good as NeurOST. However, the standard error is so large that differences are not formally detectable, even after screening for high energy areas

Line 286. “can be readily integrated as a modular component into such DA systems.” It's not clear why this would be needed in an adjoint-based data assimilation system, since the model and adjoint should account for the distinctions between geostrophic and total velocity terms.

Appendix A. “We choose snapshots of the Alboran Sea as it features two large and persistent gyres subject to cyclogeostrophy.” In Figure A1, there are clear distinctions between the iterative and variational approaches, but are there skill metrics to quantify these differences? What specific aspects of this account for the distinctions between the variational and iterative approaches?

Appendix B. The SWOT inversion is interesting, but there isn’t much context, nor is there a set of in situ measurements to use to assess the skill. What science results emerge the from the SWOT demonstration? Is the SWOT discussion needed in this manuscript?

Appendix C. As noted above, the mean misfit in Appendix C presents some fundamental challenges. In addition, the wording in the figure caption is unclear. What is meant by spatial binning? I think it would be sufficient to say “but showing the mean misfit instead of the standard error of the misfit”. Ideally all four panels should be discussed, although as noted above, the interpretation is not clear. The mean misfit in panel (a) is fairly large and by definition always positive. In panel (b) the cyclogeostrophic approach shows a decrease in speed bias in western boundary currents and an increase in speed bias near the equator. In panel (c) DUACS shows an increase in speed bias in western boundary currents. Panel (d) shows better results with NeurOST relative to DUACS geostrophy, but that's presumably mostly a reflection of differences between NeurOST and DUACS.

I found the package name, jaxparrow, to be a massive distraction. As the authors are probably aware, Jack Sparrow is the name of the pirate protagonist in the Disney film, Pirates of the Caribbean. The name Jack Sparrow is also associated with the 16th to 17th century pirate Jack Ward. There has been quite a bit of historical scholarship on pirates in recent years. I’m not sure if the authors if this paper intended to invoke the Disney film or the historical antecedents. Regardless, the name was a distraction for me, and it left me asking a broad range of questions that have nothing to do with the content of the manuscript. Does the name of the software package glorify a Disney film at a moment in time when people have been asking if they should boycott Disney? Would Disney protest usage of the name, citing concerns about trademark or copyright? (I don’t think they should, given the spelling change and given the historical origins of the name that long pre-date the Disney corporation, but Disney has been a notoriously fierce defender of its trademarks.) Does the name glorify pirates, who in the modern world have been antagonists to ocean observation? All of these questions, I leave to others (and the lawyers) to untangle. I merely remind the authors that their cute choice of name comes at the cost of pulling reader attention away from the core concepts.

Minor points
Line 10, and elsewhere in the text. No space between number and % sign. ‘20 %’ -> ‘20%’, etc.

Line 33. “upper spectrum”. This term is unclear. Should it be “high-wavenumber portion of the spectrum”?

Line 53. “This has been confirmed by several authors later” . Reword for clarity: “This was confirmed subsequently by several authors (e.g. ....)”

Line 124. “As” -> “Following”. Build the references into the sentence, separating with commas, and using “and” before the last. The semi-colons are unclear within a non-parenthetical set of references.

Line 126. “DUACS” -> “the DUACS”

Line 149, Line 236, and elsewhere. “ms^{-1}”. Clarify: milliseconds or meters per second? For meters per second, I would use a space: “m s^{-1}”. Interesting, the color bar label in Figure 4(a) is unambiguous.

Line 150. “7-days moving average” -> “7-day moving average”. (The term “7-day” is used as an adjective, so it does not take an “s".)

Line 198. “location” -> “locations”

Line 211. “zoom” is a verb but not a noun. Maybe “along with an enlargement of the Gulf Stream region”, or “along with a set of regional maps showing the Gulf Stream region”

Lines 212-214. “northern (southern)” etc. Opposites in parentheses do not produce lucid, readable text. In this case the contrasts are particularly unclear. Do the northern and southern branches both produce corrections of +/- 0.2 m/s? If so, then why are the parenthetical comments needed? If not, then I think this sentence and the next one should be revised for clarity.

Figure 3 caption. “Neu-rOST”. The syllable break seems odd. Should it be “Neur-OST”?

Line 241. “upon geostrophy”. Not clear. Do you mean “relative to geostrophy”

Line 254. “5---20”. The typeset dash looks like an “em-dash” but should be an “en-dash”.

Line 275. “currents evaluation” -> “current evaluation”

Line 302. “from eNATL60-BLB002” -> “from the eNATL60-BLB002”

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Citation: https://doi.org/10.5194/egusphere-2025-4172-RC1
RC2: 'Comment on egusphere-2025-4172', Anonymous Referee #2, 14 Oct 2025 reply

Please see attached PDF.

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Citation: https://doi.org/10.5194/egusphere-2025-4172-RC2

Vadim Bertrand, Julien Le Sommer, Victor Vianna Zaia De Almeida, Adeline Samson, and Emmanuel Cosme

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Short summary

Understanding ocean surface currents is essential for navigation and environmental monitoring. Traditionally, currents are estimated from satellite sea surface height using a simplified balance that neglects inertial accelerations. We developed a new open-source method that goes beyond this approximation, providing more accurate currents estimates. Global evaluation against drifter observations shows error reductions of up to 20 %, especially in energetic regions.


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