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

Sander Rikka

physics.ao-phOct 21, 2026

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

Authors: Sander Rikka

Organizations: Department of Marine Systems, Tallinn University of Technology

Abstract

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.

Figures & tables

Station	Latitude ( ∘ N)	Longitude ( ∘ E)	Depth (m)	Train	Validation	Test
BS	61.8000	20.2333	112.2	23,844	8,515	8,633
GoF	59.9650	25.2350	49.7	22,264	6,794	6,851
NBP	59.2500	21.0000	107.0	27,685	8,553	7,695

Table 1: Buoy locations, water depths and aligned prediction-origin counts.

ID	Descriptive name	Objective
A0	Spectrum baseline	Log-spectrum reconstruction and temporal increments
A1	Variance supervision	A0 plus observed spectral variance
A2	Peak-frequency supervision	A1 plus observed soft peak frequency
A3	Peak-amplitude supervision	A2 plus observed peak spectral density
P1	Wind-sea bounds	A2 plus upper bounds on wind-sea variance and peak period
P2	Relaxation	A2 plus a local relaxation residual at hours 1–12

Table 2: Comparison objectives. All variants share the same inputs and network.

Descriptor comparison
	Spectral RMSE			Wave-height metrics
Model	$E$ ( $\mathrm{m^{2}\,Hz^{-1}}$ )	$f_{p}$ (Hz)	$A_{p}$ ( $\mathrm{m^{2}\,Hz^{-1}}$ )	RMSE (cm)	Bias (cm)	$r$	SI
A0	0.41	$4.95\times 10^{-2}$	1.60	19.54	-6.26	0.97	0.20
A1	0.41	$4.90\times 10^{-2}$	1.48	19.00	-3.98	0.97	0.19
A2	0.40	$4.80\times 10^{-2}$	1.56	19.62	-5.08	0.97	0.20
A3	0.43	$4.88\times 10^{-2}$	1.33	18.78	-4.32	0.97	0.19

Table 3: Pooled errors over leads 1–12 h. The two A2 rows are separate fits of the same supervised objective, one for each comparison. The WAM row provides regional context from Aguiar et al. (2024) , rather than a matched benchmark.

Figure 1: Spectral and wave-height RMSE by lead time: descriptor objectives (a,b) and physical constraints (c,d). The lower panels use the matched A2 reference. All curves use the same 23,179 test origins; persistence repeats the initial spectrum.

		Spectral density			Wave height
Station	Model	RMSE ( $\mathrm{m^{2}\,Hz^{-1}}$ )	RMSE/RMS	$S$ (%)	RMSE (cm)	SI	$S$ (%)
BS	A2	0.48	0.56	33.06	22.54	0.23	49.69
BS	P3	0.47	0.54	34.55	21.47	0.22	52.10
GoF	A2	0.14	0.47	37.77	13.87	0.18	50.11
GoF	P3	0.14	0.45	39.29	13.96	0.18	49.78
NBP	A2	0.45	0.47	36.65	22.30	0.19	47.74

Table 4: Station errors for the matched A2 reference and extended relaxation (P3).

S

is RMSE reduction relative to persistence.

Figure 2: Wind-conditioned normalized errors for the matched A2 reference, extended relaxation (P3) and persistence. Rows show BS, GoF and NBP. Each value pools leads 1–12 within its station and wind class. Steady denotes small net wind-speed change, subdivided by mean speed; denominators use observations within each subset.

Figure 3: NBP spectral hindcasts and supplied wind for two forcing-selected origins: 30 June 2022, 21:00 UTC (a–d), and 2 December 2022, 14:00 UTC (e–h). Spectral panels compare observations, the matched A2 reference, P3 and persistence at one, six and twelve hours. Cases are selected from wind-class medians without using prediction errors.

Figure 4: Continuation to 24 hours for the A2 reference, relaxation (P2), extended relaxation (P3) and persistence on 22,892 common test origins. The dashed line marks the end of spectral supervision. P3 alone received a physical residual at hours 13–24 during training.

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