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Data-driven and physics-guided hindcasting of measured wave spectra

Sander Rikka

arXiv:2610.29306v1physics.ao-phcs.LG

TL;DR

Short-range prediction of measured wave spectra requires assessing spectral structure as well as wave height. This paper hindcasts twelve hours from a buoy spectrum using local environmental inputs and compares descriptor supervision with local physical guidance. The reported benefits are modest and depend on the prediction objective, while operational forecasting remains unevaluated.

  • Problem

    The paper examines how to hindcast measured wave spectra over short ranges while evaluating both spectral structure and wave-height accuracy.

    Source: §1 · p. 1, §5 · p. 12

  • Method

    A recurrent model predicts twelve hours from one measured buoy spectrum using local wind, fetch, depth, and comparisons of descriptor supervision and local physical constraints.

    Source: §1 · pp. 2–3

  • Results

    Physical guidance produces modest and mixed changes: local relaxation improves wave-height accuracy, extended relaxation improves spectral accuracy, and empirical wind-sea bounds worsen pooled results.

    Source: §5 · p. 13

  • Takeaways & Limitations

    The most accurate objective depends on the quantity of interest, and local physical guidance offers modest, objective-dependent benefits.

    Source: §5 · pp. 12–13

    Operational evaluation requires winds available at issue time and the actual availability time of the initial measured spectrum.

    Source: §4.3 · p. 12

Abstract

from arXiv · show

Short-range hindcasts of measured wave spectra are evaluated using a recurrent model initialized by one hourly buoy spectrum and supplied with local wind, fetch and depth. Measurements from three Baltic Sea stations are combined for training in 2017--2020, validation in 2021 and testing in 2022. ERA5 reanalysis supplies winds throughout the twelve-hour prediction interval. The comparison examines supervision of spectral variance, peak frequency and peak amplitude, followed by empirical wind-sea bounds and a local relaxation constraint. Variance and peak-frequency supervision gives the lowest spectral error within the data-driven comparison, whereas additional peak-amplitude supervision gives the lowest wave-height error. In the matched physical comparison, relaxation reduces height RMSE by 5.01\% to \PhPtwoHs{}~cm. Extending this constraint through hours 13--24, without spectral targets at those hours, reduces primary spectral RMSE by 2.31\% to \PhPthreeSpec{}~$\unitE$. Upper-bound constraints increase errors. Wind-condition comparisons show spectral gains from extended relaxation in increasing, decreasing and steady strong winds, with little change in steady weak winds. The results support modest, objective-dependent benefits from local physical guidance. Wave-height scores concern the retained frequency band, and operational forecasting and transfer to independent stations remain to be evaluated.

1 Introduction

The paper examines short-range hindcasts of measured wave spectra, asking how observational descriptors and local physical constraints affect prediction. Baltic Sea conditions and differences between spectrum-level and integrated wave-height evaluation motivate the comparison.

1 Introduction

  • Baltic waves are locally wind-generated, and fetch, coastline, ice, and changing winds shape their short-term evolution.The Baltic has limited ocean-swell penetration, while waves may persist after the generating wind weakens.Source: §1 · pp. 1–2
  • Wave-height accuracy alone cannot establish spectral accuracy because spectra with the same variance can differ in peak location, peak density, and shape.The paper situates its work alongside spectral models and machine-learning approaches, which differ in their targets and information constraints.Source: §1 · p. 2
  • The study compares descriptor supervision with physical guidance for short-range hindcasts of measured one-dimensional wave spectra.The comparison adds variance, peak-frequency, and peak-amplitude supervision, then tests empirical wind-sea bounds and local relaxation.Source: §1 · p. 2
  • One buoy spectrum initializes a twelve-hour prediction using local wind, directional fetch, and depth, with ERA5 winds supplied throughout.Measurements from three Baltic stations are combined; this retrospective setup motivates examining observed initial spectra alongside evolving local winds.Source: §1 · pp. 2–3

2 Data and methods

The hindcasting task uses hourly spectra from three Baltic buoy stations, with a shared retained-frequency grid and period-separated training, validation, and test windows. ERA5 winds, fetch, and station depth provide environmental inputs.

2.1 Measured spectra and hindcasting task

  • One measured spectrum at each hourly origin initializes predictions for the next twelve hours, using a single model trained jointly on three buoy stations.The stations are in the Northern Baltic Proper, Bothnian Sea, and Gulf of Finland.Source: §2.1 · p. 3, Table 1 · p. 3
  • Spectra are interpolated onto 52 frequencies from 0.085 to 0.595 Hz, and retained-band variance and significant wave height can differ from full-spectrum values.The interpolation does not explicitly preserve integrated variance; the same grid is used for observations, predictions, and persistence.Source: §2.1 · p. 3
  • Training targets cover 2017–2020, validation targets 2021, and test targets 2022; successive hourly origins overlap and are not independent events.All three stations contribute to each period, and target windows must remain within their assigned period.Source: §2.1 · p. 3

2.2 Wind, fetch and depth

  • ERA5 hourly winds are taken from the nearest 0.25° grid point and supplied at the origin and throughout the prediction interval.Fetch is measured along eight coastline rays, and depth comes from the nearest Baltic bathymetry-grid coordinate.Source: §2.2 · pp. 3–4

2.3 Model and comparison design

All objectives share a recurrent encoder–decoder model and differ in the observation-descriptor or physical terms added to the baseline loss. The comparisons test how supervision choices shape the hindcasts.

2.3 Model and comparison design

  • Seven objectives share the same inputs and network, while physical variants add empirical wind-sea bounds or a local relaxation ODE rather than a spatial wave-action PDE.The physical terms encode reduced local assumptions, not a complete source balance.Source: Table 2 · p. 4, §2.3 · p. 4
  • The shared encoder–decoder LSTM predicts spectral increments autoregressively, feeding its own prediction into the next hour during training and evaluation.The encoder receives the initial spectrum, wind, and fetch; the decoder also receives station depth.Source: §2.3 · p. 4
  • The baseline loss combines reconstruction and increment losses over twelve prediction hours, with larger weights on the earliest hourly changes.The increment weights are 1.25, 1.15, and 1.05 for the first three adjacent-hour pairs, then 1 thereafter.Source: §2.3 · p. 4
  • Descriptor objectives add measured variance, differentiable peak-frequency, or peak-amplitude supervision to the baseline objective.Peak amplitude uses linear density, whereas variance and peak-frequency supervision use logarithmic or normalized quantities.Source: §2.3 · pp. 4–5

2.4 Local physical constraints

Physical guidance uses local equilibrium wind-sea relations and applies penalties only under selected conditions. The tested terms include upper bounds and relaxation toward equilibrium, with one variant continuing beyond the hours with observed spectral targets.

2.4.1 Equilibrium wind sea

  • Young–Verhagen growth relations estimate equilibrium variance and peak frequency from hourly mean wind and fetch, positive depth, and a dimensionless growth parameter.A JONSWAP shape projects equilibrium variance onto represented wind-sea frequency bins.Source: §2.4.1 · p. 5

2.4.2 Applicability and calibration

  • Physical penalties apply only to eligible transitions meeting wind-speed, bin-count, wind-change, wind-turn, observed-variance, and valid-input conditions.Calibration uses eligible first transitions; stationwise tolerances set upper bounds, while a residual-based selection gives a relaxation time of 12 hours.Source: §2.4.2 · p. 5

2.4.3 Bounds and relaxation

  • P1 penalizes excess wind-sea variance and soft peak period, whereas P2 and P3 constrain a local relaxation equation toward equilibrium.For P3, the model continues to hour 24 although observed-spectrum targets stop at hour 12; validation remains a twelve-hour task.Source: §2.4.3 · p. 6

2.5 Training and evaluation

Training uses a fixed architecture and optimization protocol, with physical comparisons evaluated against a separately fitted A2 reference. Performance is assessed over leads 1–12 and, for continuation, on a common subset with complete 24-hour observations.

2.5 Training and evaluation

  • Training uses batch size 256, an initial learning rate of 1.758 × 10−3 and weight decay 0.05, with early stopping and checkpoints selected by 2021 reconstruction loss.The architecture and optimization are fixed; physical comparisons use a separately fitted A2 reference.Source: §2.5 · p. 6
  • Primary evaluation pools spectral RMSE across origins, leads and frequencies, and assesses peak frequency, peak amplitude and Hs over leads 1–12.The aggregate includes 278,148 overlapping origin–lead pairs; wave-height bias is prediction minus observation, and SI is RMSE/Hs.Source: §2.5 · p. 6
  • Continuation models are evaluated without refitting on 22,892 origins with complete 24-hour observations, scoring hours 1–12 and 13–24 separately.Normalized spectral error uses the subset’s observed spectral-density RMS; normalized height error uses mean observed Hs.Source: §2.5 · p. 6

2.6 Wind-condition analysis

Wind-condition comparisons classify each twelve-hour test interval by net wind-speed change and mean speed. Errors and normalization denominators are calculated separately within each station and wind class.

2.6 Wind-condition analysis

  • Intervals with net wind-speed gradient above 0.1 m s−1 h−1 are increasing, below −0.1 are decreasing, and the remaining small-trend intervals are divided by mean speed.The low, middle and high speed thresholds are the training 20th and 80th percentiles, 4.4 and 9.7 m s−1; small net trend does not imply constant wind.Source: §2.6 · p. 7
  • Wind classes are assigned from ERA5 wind speed over each twelve-hour test interval.The supplied passage begins describing this classification but does not provide the full calculation.Source: §2.6 · p. 7
  • Errors and their normalization denominators are calculated within each station and wind class.Two NBP examples are selected near class medians of initial Hs, mean wind speed and net gradient, without using prediction errors.Source: §2.6 · p. 7

3 Results

The preferred objective depends on whether spectral error, wave height or peak descriptors are prioritized, and physical guidance has mixed, station- and wind-dependent effects. Extended relaxation provides a modest spectral benefit beyond twelve hours, with a height tradeoff.

3.1 Spectral and wave-height accuracy

  • A2 gives the lowest spectral error (0.40 m2 Hz−1) and peak-frequency error (4.80 × 10−2 Hz), while A3 gives the lowest height and amplitude errors (18.78 cm and 1.33 m2 Hz−1).All learned models improve pooled spectral, wave-height, peak-frequency and peak-amplitude RMSE relative to persistence.Source: §3.1 · p. 7, Table 3 · p. 7
  • 5.01%: twelve-hour relaxation (P2) reduces height RMSE to 19.26 cm; extended relaxation (P3) lowers spectral RMSE to 0.39 m2 Hz−1 and reduces spectral and height errors by 2.31% and 3.04%.Wind-sea bounds (P1) instead increase spectral and height RMSE by 8.44% and 4.89%; the matched A2 reference retains the lowest peak-frequency error.Source: §3.1 · p. 7, Figure 1 · p. 8

3.2 Station differences

  • P3 reduces both spectral and height errors at BS and NBP, but at GoF spectral error decreases slightly while height error increases; both models retain positive persistence skill at all stations.The lower absolute spectral error at GoF partly reflects lower observed spectral density, while its normalized error is close to NBP’s.Source: §3.2 · p. 8, Table 4 · p. 9

3.3 Dependence on wind conditions

  • P3 lowers pooled spectral error by 2.79% in increasing winds, 1.59% in decreasing winds and 2.80% in high-speed small-trend conditions, but changes it by less than 0.1% in low-speed small-trend conditions.P1 increases spectral error by 20.70% during increasing winds; low-speed small-trend height error increases slightly under P3.Source: §3.3 · pp. 8, 10, Figure 2 · p. 9

3.4 Continuation beyond twelve hours

  • At hours 13–24, P3’s spectral RMSE is 0.45 m2 Hz−1, 0.76% below A2 and 6.30% below P2, while its height error is 1.46% higher than P2’s.P3 alone received a physical residual during hours 13–24, beyond the end of spectral supervision.Source: §3.4 · p. 10, Figure 4 · p. 11

4 Discussion

Physical guidance has objective- and wind-dependent effects, while assumptions and retrospective inputs constrain interpretation. The band-limited, location-specific evaluation leaves operational use and transfer to be tested.

4.1 What physical guidance contributes

  • The controlled comparison finds a tradeoff: variance and peak-frequency supervision favors spectral accuracy, while peak-amplitude supervision favors height and amplitude.Local relaxation improves height, and extending it beyond labelled hours improves spectral accuracy; empirical upper bounds perform worse overall.Source: §4.1 · p. 11
  • Equilibrium bounds rely on local depth approximating fetch bathymetry and forcing persisting long enough for equilibrium, assumptions not established by instantaneous fetch and endpoint winds.The fixed JONSWAP projection and frequency cutoff also cannot resolve crossing directional systems.Source: §4.1 · p. 11
  • Calibration does not validate the estimated equilibrium; the 12-hour relaxation time is residual-selected, and its closure alone does not outperform hourly persistence on eligible training transitions.The effects of calibration rules, station masks, and loss weights were not isolated, so the results support this formulation rather than universal parameter choices.Source: §4.1 · p. 11

4.2 Wind dependence and regional context

  • P3 improves spectral accuracy in increasing, decreasing, and steady strong winds, but has very little effect in steady weak winds.These diagnostic classes differ from stepwise applicability masks, and the descriptive comparison cannot establish the physical penalty as the cause.Source: §4.2 · p. 12
  • GoF’s lower absolute errors partly reflect its lower spectral energy scale; WAM’s reported 19.00 cm height RMSE cannot be directly ranked against the present models’ differing evaluation setups.The present height RMSEs are 19.26 cm for P2, 19.66 cm for P3, and 18.78 cm for A3; differences in forcing, frequency support, and evaluation data prevent matched ranking.Source: §4.2 · p. 12

4.3 Scope and further application

  • Operational evaluation requires winds available at issue time and the actual availability time of the initial measured spectrum.A matched numerical-model comparison and a stronger statistical baseline are also needed to establish practical advantages.Source: §4.3 · p. 12
  • The retained frequency band captures a median 96.95% of measured variance across 121,800 eligible station-hours, with a median Hs reduction of 1.27 cm.Larger fractional losses occur in weak seas and some energetic NBP cases.Source: §4.3 · p. 12
  • Directional information and spatial propagation are absent, and overlapping origins mean small differences are not statistically established; adapting other spectrum sources and testing independent locations remain necessary.The physical comparisons are exploratory because 2022 informed earlier development.Source: §4.3 · p. 12

5 Conclusion

Measured spectra support twelve-hour hindcasts, but the best objective depends on the quantity evaluated, and physical guidance has modest, mixed effects. Operational forecasting and transfer remain to be evaluated.

5 Conclusion

  • Variance and peak-frequency supervision favors spectral accuracy, while peak-amplitude supervision favors wave height and peak density.All tested objectives improve spectral and wave-height errors relative to persistence.Source: §5 · pp. 12–13
  • Local relaxation improves wave-height accuracy, extending relaxation beyond labelled hours improves spectral accuracy, and empirical wind-sea bounds worsen pooled results.Wind-condition comparisons show that these effects vary with forcing and wave-energy scale.Source: §5 · p. 13
  • Retrospective forcing, band-limited spectra, and evaluation at training locations constrain interpretation; operational winds, independent locations, and matched numerical-model comparisons remain needed.The method uses local, reanalysis-conditioned data-driven and physics-guided learning.Source: §5 · p. 13

Data availability

Buoy spectra are available through the Finnish Meteorological Institute, subject to access permission; code is available from the corresponding author upon reasonable request.

Data availability

  • The buoy spectra were provided by FMI through project EAG249 and cannot be redistributed by the authors; access must be arranged with FMI.Source: Data availability · p. 13
  • ERA5 forcing is from the Copernicus Climate Data Store product cited in C3S (2018), and bathymetry is from the Baltic Sea Bathymetry Database version 0.9.3.Source: Data availability · p. 13
  • The preprocessing, training, and evaluation code is available from the corresponding author upon reasonable request.Source: Data availability · p. 13

Author contributions and competing interests

Sander Rikka is credited with conceptualization, modelling, manuscript writing, and review; no competing interests are declared.

Author contributions and competing interests

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