The study, carried out by Pattharaporn Thongnim of the Department of Mathematics at Burapha University in Chonburi, Thailand, rests on an extraordinary dataset: 353,644 water-level readings recorded at roughly ten-minute intervals between May 2019 and June 2026, obtained from the Marine Department of Thailand. Before any analysis could begin, the raw record had to be cleaned. Duplicated timestamps were merged, exact zeros removed, and a suspicious plateau in late 2022, during which the sensor was evidently stuck at between 5.67 and 5.69 metres, was discarded. Isolated spikes were excised with a Hampel filter, a robust statistical technique that rejects values deviating by more than 0.25 metres from a centred 70-minute running median, applied in two passes. Crucially, the threshold was derived from the distribution of the deviations themselves rather than from the residuals of a tidal model, so that genuinely unexplained observations were not thrown away at the screening stage. In total, 1,458 readings were removed, leaving 351,212 clean measurements spanning seven monsoon cycles.

The first analytical step was a classical one. A least-squares harmonic analysis, in the tradition going back to Doodson’s 1921 development of the tide-generating potential, fitted 33 tidal constituents with nodal corrections to the cleaned record. This decomposition explained 94.7 percent of the hourly variance, a remarkably good fit for a shallow tropical estuary. The relative importance of semidiurnal versus diurnal constituents, expressed as a form factor of 1.28, places Bang Pakong firmly in the mixed, mainly semidiurnal regime that characterises the Gulf of Thailand, where the semi-diurnal tide is known to propagate clockwise around the basin.

But the headline finding concerns asymmetry. In a perfectly linear tidal system, flood and ebb would take equal time. In shallow water, nonlinear effects break that symmetry: crests travel faster than troughs, the flood shortens, and the ebb lengthens. At Bang Pakong, measured directly across 7,889 individual tidal cycles, the mean rise takes 6.26 hours while the mean fall takes 8.12 hours, a duration asymmetry of 1.86 hours. That is a strong flood dominance, and it matters enormously for the estuary’s future. Following the framework established by Friedrichs and Aubrey in their 1988 synthesis of tidal distortion in shallow estuaries, flood dominance generally implies net landward sediment transport, which in turn governs whether the channel silts up or scours out, and how far salt intrudes during the dry season.

The real novelty lies in what happens when the analysis is allowed to vary through the year. By permitting annual modulation of the non-degenerate tidal constituents, the study shows that the shallow-water overtides, the harmonic constituents generated by the nonlinear distortion itself, swing seasonally two to three times more than the principal astronomical constituents. Relative seasonal variations of 22 to 37 percent were found for MS4, M4, MO3, MN4 and M6, against just 13 percent for the dominant lunar semidiurnal constituent M2. Every one of these overtides peaks between June and August, precisely when the southwest monsoon drives maximum river discharge and the mean water level sits 14.8 centimetres lower than average. The physical interpretation is compelling: lower mean depth amplifies the nonlinear shallow-water processes that generate the overtides, so the estuary’s tidal distortion is literally modulated by the monsoon.

Not every seasonal signal is straightforward. The M2 amplitude itself shows a statistically significant decline of 0.55 plus or minus 0.09 centimetres per year, a trend reproduced independently in both the wet and dry seasons and therefore not attributable to errors in the nodal corrections that account for the 18.6-year lunar cycle. Meanwhile, attempts to detect annual modulation in the K1, S2 and P1 constituents proved unidentifiable when the S1 and T2 constituents were retained in the model, a degeneracy that would yield physically meaningless amplitudes if ignored. This is a cautionary tale for anyone fitting seasonal tidal models: some apparent seasonal variability is an artefact of constituent interference rather than real physics.

The second methodological pillar of the study addresses a problem that plagues every long-term observational record: gaps. Some 6.4 percent of the hourly water-level values were missing, and rather than simply interpolating, Thongnim turned to Gaussian process regression, a Bayesian machine-learning technique that treats the unknown function as a draw from a probability distribution over functions defined by a covariance kernel. The Gaussian process reconstruction served two distinct purposes. First, it validated the duration analysis: filling the gaps recovered 11.6 percent more complete tidal cycles, and the asymmetry barely moved, settling at 1.83 hours instead of 1.86, confirming that the flood dominance is robust to missing data.

Second, and more subtly, the reconstruction became a testbed for kernel selection. Seven different covariance structures were compared using three independent criteria: closed-form leave-one-out residuals, the Akaike information criterion, and withheld-data experiments. The winner was a two-scale Matérn 3/2 kernel with a semiannual mean term, a choice that captures both short-term tidal roughness and longer seasonal structure. Perhaps the most striking methodological result is that the choice of kernel changes the predictive uncertainty far more than it changes the reconstructed values themselves. In other words, different reasonable kernels agree on what the missing data probably looked like, but disagree substantially on how confident one should be about that agreement, a distinction with real consequences for downstream uncertainty analysis.

Why does this matter beyond one Thai estuary? Monsoon-influenced shallow estuaries line the coasts of South and Southeast Asia, West Africa and tropical Australia, and they host some of the world’s most productive and most densely populated deltas. Tidal asymmetry is the engine of sediment transport in these systems, and if the asymmetry itself varies seasonally by tens of percent, then annual sediment budgets computed from a single season, or from a time-averaged harmonic analysis, may be systematically biased. The finding that shallow-water constituents respond to river discharge and mean water level also provides a sensitive diagnostic for climate-driven change: as sea level rises or monsoon rainfall patterns shift, the seasonal modulation of tidal distortion should shift with them, offering an early warning signal encoded in the tide gauge record itself.

The study also demonstrates how modern statistical machinery can extract more from classical oceanographic data. The combination of a 33-constituent harmonic fit, direct duration statistics across nearly eight thousand tidal cycles, and a carefully validated Gaussian process reconstruction represents a template that can be applied to any long water-level record with gaps. As coastal managers across monsoon Asia grapple with accelerating subsidence, changing sediment loads and intensifying floods, tools like these, which quantify not just the tide but the tide’s seasonal personality, are likely to become indispensable. The Bang Pakong estuary, it turns out, has been quietly broadcasting the signature of its monsoon all along; it simply needed the right mathematics to listen.

Subject of Research: Seasonal variation of tidal asymmetry and Gaussian process reconstruction of water levels in the Bang Pakong estuary, Thailand

Article Title: Gaussian process reconstruction and seasonal change in tidal non-linearity in a monsoon-influenced estuary

Article References: Thongnim, P. (2026). Gaussian process reconstruction and seasonal change in tidal non-linearity in a monsoon-influenced estuary. Ocean Dynamics, 76(11), Article 114. https://doi.org/10.1007/s10236-026-01869-6

Image Credits: AI Generated

DOI: 10.1007/s10236-026-01869-6

Keywords: tidal asymmetry, estuarine tides, shallow-water constituents, seasonal modulation, Gaussian process, covariance kernel, Bang Pakong, monsoon, harmonic analysis, Gulf of Thailand, flood dominance, ocean dynamics

The study, led by Somayeh Abdollahi of Ghent University together with colleagues from Flanders Hydraulics, the National Geographic Institute, and the Agency for Maritime and Coastal Services, harnessed twenty-three years of high-resolution water-level observations recorded between 2001 and 2023. Four tide gauges anchored the analysis: Nieuwpoort and Oostende on the open coast, the harbour station at Zeebrugge, and the offshore Westhinder platform, designated MP7, which stands sentinel over the shipping approaches far from land. For the first time in Belgian waters, the team systematically compared multiple analysis windows, reconstruction strategies, and tidal-constituent selection procedures, all within the widely used UTide harmonic-analysis framework developed by David Codiga at the University of Rhode Island.

Harmonic analysis is the classical engine of tidal science, tracing its lineage back to Arthur Doodson’s landmark 1921 decomposition of the tide-generating potential. The idea is elegant: the tide at any location can be represented as the sum of dozens of sinusoidal constituents, each with a precisely known astronomical frequency but an amplitude and phase that must be estimated from observations. Once those amplitudes and phases are locked in, the tide can be predicted forward or backward in time indefinitely. LAT is then defined as the lowest water level the astronomical tide alone can produce, excluding the chaotic contributions of storm surges and weather. In practice, however, the answer depends heavily on which constituents are included, how long a record is analysed, and how the reconstruction is performed.

The researchers confronted this methodological fragility head-on. They tested four period-selection approaches, labelled M1 through M4: a single long-record harmonic analysis spanning 2001 to 2019 with annual reconstructions; separate year-by-year analyses; annual analyses used to predict the following nineteen years; and a sophisticated complex averaging of annual harmonic coefficients that preserves the vector nature of tidal amplitudes and phases. Each method was applied at all four stations, and the spread in the resulting LAT estimates turned out to be far from trivial, underscoring that a single ‘correct’ LAT value is an illusion unless the analytical recipe is specified precisely.

Equally consequential was the choice of tidal constituents. The team compared three selection strategies: the default constituent set offered by UTide, a fixed set traditionally used by the Dutch hydrographic service, and a novel iterative procedure guided by the spectrum of the analysis residuals. In the residual-spectrum-guided approach, the researchers analysed the tide, examined the frequency content of what remained unexplained, and added constituents corresponding to prominent residual peaks, repeating the process until the spectrum was clean. This data-driven strategy, implemented entirely with UTide routines, emerged as the preferred configuration, capable of capturing shallow-water constituents that generic or borrowed constituent sets overlook in the complicated hydrodynamics of the southern North Sea.

The preferred configuration, based on the 2001–2019 input window, yielded deterministic LAT values of −0.648 metres at Nieuwpoort, −0.499 metres at Oostende, −0.251 metres at Zeebrugge, and −0.300 metres at the offshore Westhinder station, all expressed relative to the Belgian national height datum TAW/DNG. The pronounced spatial variation, with LAT nearly half a metre deeper at Nieuwpoort than at Zeebrugge, reflects the complex interplay of coastal geometry, bathymetry, and shallow-water tidal dynamics along the Belgian coast. No single conversion grid, the researchers caution, can faithfully represent such station-specific behaviour without careful local calibration.

To quantify how confident one can be in these numbers, the team propagated uncertainty in the harmonic parameters using 1000 Gaussian Monte Carlo realisations, perturbing the estimated amplitudes and phases according to their statistical uncertainties and recomputing the LAT minimum for each realisation. The resulting conditional standard deviations were remarkably tight, ranging from 0.0031 to 0.0034 metres, suggesting that once the harmonic model and constituent set are fixed, the LAT estimate is numerically robust. The authors are careful, however, to frame this spread as conditional on the chosen model rather than a complete uncertainty budget; it captures parameter uncertainty but not the full spectrum of errors that could affect an operational datum.

The Monte Carlo experiments also delivered a reassuring message about computational practicality. Comparing ensembles of 500 and 1000 realisations across the four period-selection methods produced maximum absolute differences of only 0.007 metres in the Monte Carlo mean, 0.009 metres in the conditional standard deviation, and 0.020 metres in one percentile-interval bound. For the preferred residual-spectrum-guided configuration, those differences shrank further to 0.003, 0.004, and 0.006 metres respectively. In other words, halving the ensemble size barely moves the answer, a finding that will ease the computational burden for hydrographic offices considering similar analyses elsewhere.

Throughout the study, the researchers treated existing operational products, including station values published by the Agency for Maritime and Coastal Services and the official LAT conversion grid, as consistency benchmarks rather than independent ground truth. This distinction matters: the operational values were not used to validate the new estimates in an absolute sense, but to check whether the updated analysis remains compatible with the datums that Belgian mariners and coastal engineers rely on today. The deterministic LAT, defined as the minimum of the unperturbed harmonic reconstruction, and the Monte Carlo mean, the average of minima across perturbed realisations, were deliberately reported separately, since the two quantities answer subtly different questions.

The broader significance of the work extends beyond Belgium’s short coastline. The 18.61-year nodal cycle, in which the lunar orbit’s precession modulates tidal amplitudes by up to tens of centimetres, means that short analysis windows can bias LAT estimates substantially, a phenomenon documented in tide-gauge records worldwide. By demonstrating the value of a nineteen-year input window, roughly one full nodal cycle, and by providing a reproducible framework for constituent selection and uncertainty propagation, the Belgian team has offered a template that other hydrographic services can adapt. As sea levels rise and extreme water levels evolve, the low end of the tidal envelope deserves the same scrutiny as the high end, and this study shows exactly how that scrutiny should be applied. The tide-gauge data underpinning the analysis remain freely available through the Meetnet Vlaamse Banken portal, inviting researchers everywhere to test, refine, and extend the approach.

Subject of Research: Assessment of Lowest Astronomical Tide at Belgian North Sea tide gauges using long-term harmonic analysis and residual-spectrum-guided constituent selection

Article Title: Assessment of lowest astronomical tide at belgian north sea tide gauges using long-term harmonic analysis and residual-spectrum-guided constituent selection

Article References: Abdollahi, S., Vanlede, J., Verbeurgt, J., Verstraeten, J., De Kuyper, A., Gurdebeke, P., & De Wulf, A. (2026). Assessment of lowest astronomical tide at belgian north sea tide gauges using long-term harmonic analysis and residual-spectrum-guided constituent selection. Ocean Dynamics, 76(9), Article 96. https://doi.org/10.1007/s10236-026-01850-3

Image Credits: AI Generated

DOI: 10.1007/s10236-026-01850-3

Keywords: Lowest Astronomical Tide, Belgian North Sea, harmonic analysis, tide gauges, tidal constituents, UTide, Monte Carlo uncertainty, vertical datum, ocean dynamics, hydrography, coastal sciences, residual spectrum