Closure of the sea surface height budget with a Stokes offset

Bruce Haines; Charly de Marez; Jinbo Wang; J\"orn Callies

REVIEW 3 major objections 4 minor 4 references

Closure of the sea surface height budget with a Stokes offset

T0 review · 3 major / 4 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read A wave-following GPS buoy measures a Lagrangian sea surface height, and correcting for the Stokes offset closes the height budget.

desk verdict A parameter-free Stokes offset closes the sea surface height budget at low frequencies, but the paper's load-bearing assumption—that the moored GPS buoy follows the wave orbit like a Lagrangian particle—is asserted rather than verified. read the letter →

arxiv 2506.06956 v2 pith:ZZQWUCSD submitted 2025-06-08 physics.ao-ph

Jörn Callies , Charly de Marez , Jinbo Wang , Bruce Haines This is my paper

classification physics.ao-ph

keywords seasurfaceheightbudgetStokesoffsetGPSbuoyLagrangianmeangravitywavesSWOTcalibrationstericbottompressure

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper claims that moored GPS buoys measure the sea surface height as a Lagrangian mean, not the Eulerian mean that satellite altimeters see, and that the difference, the Stokes offset, must be included before the sea surface height budget closes. Using a California Current System mooring with a GPS buoy, bottom pressure recorder, CTDs, and atmospheric reanalysis, the authors show the budget residual equals the Stokes offset estimated independently from the measured wave spectrum via $h_l = \int (\omega^2/g) S(\omega)\,d\omega$. The match has no adjustable parameter and reduces the root-mean-square residual from 3.6 cm to 2.8 cm at periods beyond one day. If correct, GPS buoys used for SWOT calibration and as open-ocean tide gauges must apply a wave-dependent correction of up to 16 cm.

What carries the argument

The central object is the Stokes offset $h_l$, the upward shift of a wave-following instrument's time-mean height relative to the Eulerian mean sea surface height. It is computed from the measured variance spectrum $S(\omega)$ by $h_l = \int (\omega^2/g) S(\omega)\,d\omega$, using the deep-water dispersion relation $k = \omega^2/g$; this generalizes the plane-wave result $h_l = \tfrac{1}{2} k a^2$. The argument works by forming the hydrostatic sea surface height budget $h' - h'_a - h'_b = h'_s$ from GPS buoy height, bottom pressure, steric height, and atmospheric pressure, then showing that the residual equals the Stokes offset once the buoy height is treated as the Lagrangian mean $h_g = h + h_l$.

What would settle it

An instrument that measures the buoy's horizontal and vertical orbital motion relative to nearby water parcels would settle it: if the buoy's wave-following amplitude is significantly less than the surface wave orbital amplitude, the offset predicted by (7) would overestimate the true bias and the budget residual would not close after correction.

Watch

Extended reading notes

Core claim

The central claim is that the observed budget residual $h'_g - h'_a - h'_b - h'_s$ is not a failure of the sea surface height budget but the signature of measuring height with a wave-following buoy: the buoy spends more time near crests, so its 20-minute mean is higher than the Eulerian mean by the Stokes offset. The authors estimate that offset from the GPS-measured wave spectrum using the deep-water dispersion relation, $h_l = \int (\omega^2/g) S(\omega)\,d\omega$, and find that it tracks the budget residual in time, including the spikes during large-amplitude wave events. Because the estimate uses only the measured wave spectra and no fitted constants, the close match supports the interpretation that GPS buoy heights are Lagrangian measurements.

Load-bearing premise

The load-bearing premise is that the moored GPS buoy follows the wave orbit closely enough that its 20-minute average equals the Lagrangian mean surface height; line tension can pull the buoy off that orbit, especially at maximum displacement from the anchor.

Editorial extensions

If this is right

GPS buoy heights must be corrected by the Stokes offset before they are compared with SWOT or other altimetry, because the two instruments measure different mean heights.
The empirical sea state bias correction previously applied to these GPS data is likely dominated by the Stokes offset; its linearized coefficient, about 0.021, is close to the 0.018 inferred earlier.
Including the offset reduces the root-mean-square budget residual from 3.6 cm to 2.8 cm and brings the residual variance at periods longer than one day down to a small fraction of the surface height signal.
The budget still does not close at tidal periods and at periods shorter than about 6 hours, which the authors attribute to remaining measurement errors and possibly non-hydrostatic internal waves.
Drifting-buoy sea level measurements, such as those proposed for global sea level monitoring, would also require a Stokes offset correction to the extent that the buoys follow wave orbits.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

If GPS buoys are Lagrangian and altimeters are Eulerian, then any comparison of their trends inherits the trend in wave climate; the expected 21st-century increase in wave height would act as a time-varying bias in buoy-based sea level estimates.
A direct test of the perfect-following assumption would be to compare the buoy's horizontal orbital motion with that of a nearby wave-following drifter or acoustic tracker; an attenuated buoy orbit would predict a smaller offset than (7) gives.
The same offset logic should apply to any wave-following platform, including surface drifters and possibly wave gliders, so the closure demonstrated here provides a template for correcting those measurements.
Because the offset depends on the second spectral moment rather than the third, GPS sampling at about 1 Hz may be adequate; an independent wave measurement at a colocated buoy or lidar could verify the spectral estimate.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

This is a useful paper, and the central claim mostly lands. Using a moored GPS buoy, a bottom pressure recorder, and CTD profiles from one California Current mooring, the authors show that a standard Stokes offset formula, with no fitted parameters, explains the residual peaks in the sea surface height budget. The interpretation that the GPS buoy measures a Lagrangian rather than Eulerian surface height, and that the previously empirical sea-state bias is mostly Stokes drift, is a genuinely new application of known theory. The paper is transparent: data are public, processing is clear, and limitations are discussed more openly than in most papers.

What I like: Eq. (7) is standard, the offset is estimated from measured wave spectra with no tuning, and the match in Fig. 3c between the residual and the Stokes offset at the big wave events is impressive, especially at periods longer than a day. The RMS residual reduction from 3.6 to 2.8 cm is modest but real. The paper does not oversell—it explicitly notes that residual errors persist at tidal and sub-6-hour periods.

The soft spot is exactly the one the stress-test note flagged: the mooring buoy is assumed to follow the wave orbit in both vertical and horizontal directions, so the 20-minute mean height is a Lagrangian mean. Line tension will suppress horizontal motion, and the paper admits this but gives no quantitative bound. Since the Stokes offset is computed from the same GPS record whose residual it explains, the agreement is not an independent confirmation. I don't think this is fatal—the low-frequency closure is a genuinely nontrivial check, and a parameter-free match carries some weight—but the central claim rests on an instrument-kinematics assumption that is asserted rather than verified.

A minor point: the high-frequency failure to close is acknowledged but not analyzed, and the remaining tidal peaks suggest unresolved issues in the pressure or steric terms.

Who benefits: oceanographers working on SWOT cal/val, GPS buoy development, and anyone using mooring-based sea level budgets. It is a solid methodological contribution, not a transformative discovery.

Yes, this deserves a serious referee. I would recommend sending it to review, and I would suggest the authors add a quantitative estimate of the line-tension effect (they record line tension on the buoy) or at least bound how well the buoy follows horizontal orbits. That would harden the central claim considerably.

Referee Report

3 major / 4 minor

Summary. The paper argues that a moored GPS buoy measuring sea surface height behaves approximately as a Lagrangian particle, so its 20-minute mean height differs from the Eulerian mean sea surface height by a Stokes offset. Using a mooring in the California Current System, the authors estimate this offset from the measured wave spectrum via Eq. (7), subtract it from the residual of the sea surface height budget (Eq. 5), and find that the RMS residual drops from 3.6 cm to 2.8 cm, with clear improvement at periods longer than one day. They identify the Stokes offset as the likely source of the previously reported empirical sea-state bias and discuss implications for SWOT calibration and drifting-buoy sea level measurements.

Significance. If the central premise is correct, this is an important practical and conceptual result: it cautions that GPS buoy height records must be corrected for a wave-induced Lagrangian offset before being used as Eulerian sea surface height references. The paper's strengths are its parameter-free estimate (no fitted constants in Eq. (7)), the standard derivation of the budget, the use of publicly available data, and a clear comparison of spectra in Fig. 5. The maximum offset of 16 cm and the alignment of residual peaks with large-wave events give the mechanism prima facie plausibility. However, the analysis rests on an unverified kinematic assumption about the mooring, the residual improvement is modest, and the offset is estimated from the same record used to form the residual, so the quantitative closure claim is not yet established at the level the abstract implies.

major comments (3)

[§2, Eq. (7); §3 Discussion] The load-bearing assumption that the moored GPS buoy time average equals the Lagrangian mean surface height is asserted rather than independently verified. The paper itself states that "the line tension will cause deviations from the wave orbit, especially when the buoy is displaced farthest from the anchor" and only says "This does not appear to change the leading-order behavior but should be evaluated in more detail in future work." Because Eq. (7) relies on the buoy sampling wave crests preferentially through horizontal orbital motion, any suppression of horizontal wave-following by line tension directly biases the inferred Stokes offset relative to the actual difference between the buoy mean and the Eulerian mean. The authors should provide a quantitative bound on this error, for example from the recorded line tension, or an independent test using a colocated Eulerian reference, before claiming that the budget closes.
[§2, RMS residual and Fig. 5] The reported reduction from 3.6 cm to 2.8 cm RMS residual is modest, and no formal uncertainty budget is presented to support the statement that the budget closes "to within understood uncertainties." The residual remains large relative to the signal at tidal and sub-6-hour periods, and the paper notes that Wang et al. (2022) achieved a 2 cm residual with an empirical sea-state-bias correction. The authors should provide estimates of the uncertainty in each budget term (GPS height, barometric correction, bottom pressure, steric height) and demonstrate statistically that the post-Stokes residual is consistent with zero over the periods where closure is claimed.
[§2, Fig. 3c; §3 Discussion] The Stokes offset is computed from the same GPS buoy height record that defines h'_g in the budget residual, so the comparison in Fig. 3c is not fully independent. The paper acknowledges in the Discussion that other sea-state-dependent errors in the GPS solution "cannot be dismissed." Such errors could affect both the mean height and the wave spectrum and produce a spurious correlation between the residual and h'_l. An independent estimate of the Stokes offset, for example from a wave model, a nearby wave buoy, or an alternative GPS processing strategy, is needed to confirm that the residual reduction is specifically due to the Stokes offset rather than to a correlated measurement error.

minor comments (4)

[§2, Eq. (7)] The frequency variable is not defined in the text. Eq. (7) uses the deep-water dispersion relation k = ω^2/g, which implies ω is angular frequency, while Fig. 2 labels the axis in cycles per second. Please define ω and S(ω) explicitly and state the conversion used.
[Fig. 1] The caption says the anchor location was inferred by fitting a circle to the measured GPS surface positions, but the fitting procedure and its uncertainty are not described. This matters for assessing the mooring geometry and the line-tension argument in the Discussion.
[Fig. 2 caption] The phrase "calculated from the first mode of this spectrum" is ambiguous; please specify whether this is the spectral peak frequency, the first spectral moment, or another definition.
[§2, first paragraph after Eq. (5)] The notation h'_g is used before it is formally defined; please introduce the notation for the GPS-measured height and its barometric correction at the point where the budget residual is first written.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the Stokes offset is an independent spectral prediction, not a fitted input.

full rationale

The paper's central claim is that the residual h'_g - h'_a - h'_b - h'_s should equal the Stokes offset h'_l estimated from Eq. (7), and that subtracting h'_l closes the sea surface height budget. This is not circular. Eq. (7) computes h_l from the variance spectrum S(omega) of the 1-Hz GPS buoy height, a second-moment statistic of the wave field, while the budget residual is formed from 20-minute means, a first-moment statistic. The Stokes offset is not defined as the residual, and no adjustable parameter is fitted; the comparison is therefore a genuine consistency check of the Lagrangian-mean interpretation. The only shared input is that both sides are derived from the same GPS buoy record, which weakens independence but does not reduce the comparison to an identity. The citation of Wang et al. (2022) for the mooring design and for the empirical 0.018 eta sea-state bias is a self-citation, but it is not load-bearing: the closure test stands on Eq. (7) and the hydrostatic budget, and the Discussion's comparison between a linearized Stokes offset coefficient (0.021) and the earlier empirical coefficient (0.018) is a secondary consistency note rather than the basis of the central claim. The acknowledged limitation that mooring line tension may prevent perfect wave-following is a falsifiability and uncertainty concern about the instrument, not a circularity in the derivation. Overall, the paper's derivation chain is self-contained and the budget-closure check is a non-tautological spectral prediction.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The central calculation rests on known hydrostatic balance and Stokes drift theory, plus one paper-specific kinematic assumption (the moored GPS buoy behaves as a Lagrangian wave follower). No new entities are postulated and no free parameters are fitted; data exclusions and linear detrending are transparent processing choices.

assumptions (5)

domain assumption Hydrostatic balance (Eq. 1) applies to the time-averaged large-scale flow.
Used to derive the sea surface height budget in Section 2; standard for scales larger than surface gravity waves.
domain assumption Deep-water dispersion relation k = omega^2/g for surface gravity waves.
Used in Eq. (7) to convert the frequency spectrum to the wavenumber integral for the Stokes offset; appropriate for open-ocean deep-water waves.
ad hoc to paper GPS buoy time average approximates the Lagrangian mean surface height because the buoy follows wave orbits.
Central assumption; acknowledged in the Discussion as approximate, with line tension causing deviations from the wave orbit.
domain assumption The variance spectrum of GPS buoy height accurately represents the surface elevation variance spectrum.
The Stokes offset in Eq. (7) is computed from the GPS-measured spectrum; noise and measurement resolution introduce uncertainty.
domain assumption ERA5 sea level pressure accurately represents atmospheric surface pressure at the mooring.
The barometric correction h_a uses ERA5 reanalysis; errors in pressure contribute to the residual.

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Cite this review

Pith. "Pith review of Closure of the sea surface height budget with a Stokes offset." pith.science (2026). https://pith.science/paper/ZZQWUCSD

@misc{pith2026250606956,
  author       = {Pith},
  title        = {Pith review of: Closure of the sea surface height budget with a Stokes offset},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ZZQWUCSD}},
  note         = {Machine review of arXiv:2506.06956}
}

read the original abstract

The sea surface height budget, obtained by integrating hydrostatic balance over the water column, relates sea surface height variations to variations of the seafloor pressure, density in the water column, and atmospheric surface pressure. This budget is crucial for calibrating and interpreting satellite altimetry measurements. It only holds once non-hydrostatic surface gravity waves are averaged out, however, which complicates an observational closure of the budget. Using data from the California Current System, this study demonstrates that the budget closes to within understood uncertainties if GPS buoy measurements of surface height are interpreted as Lagrangian measurements. The buoy largely follows wave motion and spends slightly more time near wave crests than troughs. The associated Stokes offset, which reaches a maximum of 16 cm in these observations, must be accounted for in the Eulerian sea surface height budget.

Figures

Figures reproduced from arXiv: 2506.06956 by the authors.

**Figure 2.** Example surface height signal from the GPS buoy dominated by surface gravity [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗

**Figure 3.** Closure of the sea surface height budget by the Stokes offset. (a) The height [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗

Figures from the paper (2 more)

**Figure 4.** Figure 4: Wave properties and the Stokes offset. (a) Significant wave height [PITH_FULL_IMAGE:figures/full_fig_p009_4.png]

**Figure 5.** Figure 5: Variance spectra of the residual of the sea surface height budget with and with [PITH_FULL_IMAGE:figures/full_fig_p010_5.png]

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Reference graph

Works this paper leans on

4 extracted references · 4 canonical work pages

[1]

(2014) Waves and Mean Flows

Bühler, O. (2014) Waves and Mean Flows . 2nd ed. Cambridge University Press. 360 pp. Casas-Prat, M. et al. (2024) Wind-Wave Climate Changes and Their Impacts. Nature Re- views Earth & Environment 5 (1), 23–42. Cazenave, A., W . Llovel (2010) Contemporary Sea Level Rise. Annual Review of Marine Science 2, 145–173. Chelton, D. B., M. G. Schlax, R. M. Samels...

work page 2014
[2]

Egbert, G

Root. Egbert, G. D., R. D. Ray (2000) Significant Dissipation of Tidal Energy in the Deep Ocean Inferred from Satellite Altimeter Data. Nature 405 (6788), 775–778. 12 Elipot, S. (2020) Measuring Global Mean Sea Level Changes with Surface Drifting Buoys. Geophysical Research Letters 47 (21), e2020GL091078. Fu, L.-L. et al. (2024) The Surface Water and Ocea...

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[3]

Haines, B., S

Root. Haines, B., S. Desai, C. Meinig, S. Stalin (2017) CAL VAL of the SWOT SSH Spectrum: Moored GPS Buoy Approach . Version

work page 2017
[4]

Inverted Barom- eter

Root. Hersbach, H. et al. (2020) The ERA5 Global Reanalysis. Quarterly Journal of the Royal Meteorological Society 146 (730), 1999–2049. Kenyon, K. E. (1969) Stokes Drift for Random Gravity Waves. Journal of Geophysical Research (1896-1977) 74 (28), 6991–6994. Le Provost, C., A. F . Bennett, D. E. Cartwright (1995) Ocean Tides for and from TOPEX/POSEIDON....

work page 2020

Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.

[1] [1]

(2014) Waves and Mean Flows

Bühler, O. (2014) Waves and Mean Flows . 2nd ed. Cambridge University Press. 360 pp. Casas-Prat, M. et al. (2024) Wind-Wave Climate Changes and Their Impacts. Nature Re- views Earth & Environment 5 (1), 23–42. Cazenave, A., W . Llovel (2010) Contemporary Sea Level Rise. Annual Review of Marine Science 2, 145–173. Chelton, D. B., M. G. Schlax, R. M. Samels...

work page 2014

[2] [2]

Egbert, G

Root. Egbert, G. D., R. D. Ray (2000) Significant Dissipation of Tidal Energy in the Deep Ocean Inferred from Satellite Altimeter Data. Nature 405 (6788), 775–778. 12 Elipot, S. (2020) Measuring Global Mean Sea Level Changes with Surface Drifting Buoys. Geophysical Research Letters 47 (21), e2020GL091078. Fu, L.-L. et al. (2024) The Surface Water and Ocea...

work page 2000

[3] [3]

Haines, B., S

Root. Haines, B., S. Desai, C. Meinig, S. Stalin (2017) CAL VAL of the SWOT SSH Spectrum: Moored GPS Buoy Approach . Version

work page 2017

[4] [4]

Inverted Barom- eter

Root. Hersbach, H. et al. (2020) The ERA5 Global Reanalysis. Quarterly Journal of the Royal Meteorological Society 146 (730), 1999–2049. Kenyon, K. E. (1969) Stokes Drift for Random Gravity Waves. Journal of Geophysical Research (1896-1977) 74 (28), 6991–6994. Le Provost, C., A. F . Bennett, D. E. Cartwright (1995) Ocean Tides for and from TOPEX/POSEIDON....

work page 2020