Machine learning for phase-resolved reconstruction of nonlinear ocean wave surface elevations from sparse remote sensing data

TL;DR

This study tackles phase-resolved reconstruction of ocean surface elevations from sparse X-band radar data by training two neural architectures, a U-Net and a Fourier neural operator (FNO), on synthetic nonlinear waves generated via the high-order spectral method. The data pipeline couples 1D nonlinear wave fields with tilt and shadowing radar modulations, creating 3120 input-output samples across varied sea states, and trains models to map $n_s$ historical radar snapshots to a single surface snapshot at $t_s$. Across comprehensive experiments, the U-Net benefits strongly from using $n_s=10$ histories (achieving $\mathrm{nL2}=0.123$, $\mathrm{SSP}=0.061$), while the FNO with $n_s=9$ histories attains the best uniformity in shadowed regions ($\mathrm{nL2}=0.153$, $\mathrm{SSP}=0.076$, $\tfrac{\mathrm{nL2}_{\mathrm{shad}}}{\mathrm{nL2}_{\mathrm{vis}}}=1.381$) and offers substantially faster inference. Both models generalize well to new sea states, but the study notes limitations due to synthetic data and suggests extensions to 2D surfaces and physics-informed learning for real-world radar use. The work demonstrates that ML-based radar inversion can meet practical accuracy targets and enable real-time phase-resolved wave prediction when guided by appropriate historical context and architecture choice.

Abstract

Accurate short-term predictions of phase-resolved water wave conditions are crucial for decision-making in ocean engineering. However, the initialization of remote-sensing-based wave prediction models first requires a reconstruction of wave surfaces from sparse measurements like radar. Existing reconstruction methods either rely on computationally intensive optimization procedures or simplistic modelling assumptions that compromise the real-time capability or accuracy of the subsequent prediction process. We therefore address these issues by proposing a novel approach for phase-resolved wave surface reconstruction using neural networks based on the U-Net and Fourier neural operator (FNO) architectures. Our approach utilizes synthetic yet highly realistic training data on uniform one-dimensional grids, that is generated by the high-order spectral method for wave simulation and a geometric radar modelling approach. The investigation reveals that both models deliver accurate wave reconstruction results and show good generalization for different sea states when trained with spatio-temporal radar data containing multiple historic radar snapshots in each input. Notably, the FNO demonstrates superior performance in handling the data structure imposed by wave physics due to its global approach to learn the mapping between input and output in Fourier space.

Figures (17)

• Figure 1: Graphical illustration of the phase-resolved reconstruction task of ocean wave surfaces $\eta$ from sparse radar intensity surfaces $\xi$ for the case of waves travelling in one spatial dimension. The radar measurement (left panel) is a snapshot acquired at time instant $t_\mathrm{s}$ and is considered as sparse due to reoccurring areas with zero intensity caused by the geometrical shadowing modulation. This radar snapshot is used for reconstructing the wave surface elevation at the same time instant $t_\mathrm{s}$ (right panel).
• Figure 2: Geometric display of tilt- and shadowing modulation. Tilt modulation $\mathcal{T}(r,t)$ is characterized by the local incidence angle $\Tilde{\Theta}$ between surface normal vector $\mathbf{n}$ and antenna vector $\mathbf{u}$, while shadowing modulation $\mathcal{S}(r,t)$ of a wave facet occurs if another wave closer to the radar systems obstructs the radar beams.
• Figure 3: Schematic representation of the ML training sample extraction process. The left-hand side illustrates one of the raw radar and wave surface simulations ($Z_\mathrm{sys}, E_\mathrm{sys} \in \mathbb{R}^{512 \times 38}$), which are utilized to extract input-output samples shown on the right-hand side. Each input $\mathbf{x}_i$ consists $n_\mathrm{s}$ radar snapshots acquired at intervals of $\Delta t_\mathrm{r}=1.3 \, \mathrm{s}$, while each output $\mathbf{y}_i$ represents a single-snapshot wave surface elevation at time instant $t_\mathrm{s}$. In total $N=6 \cdot 520=3120$ data samples are generated.
• Figure 4: Fully convolutional encoder-decoder architecture based on the U-Net Ronneberger2015. Each input $\mathbf{x}_i$ is processed by $n_\mathrm{d}=5$ alternating convolutional-, activation- and average pooling layers in the encoding path. The decoding path contains convolutional-, activation- and transpose convolutional layers for a gradual upsampling to calculate the output $\mathbf{\hat{y}}_i$. Moreover, the outputs of the encoding stages are transferred to the decoding path via skip-connections.
• Figure 5: Network architecture based on the Fourier neural operator Li2020. Each input $\mathbf{x}_i$ is lifted to a higher dimensional representation $v_0$ of channel width $d_\mathrm{w}$ by a neural network $P$. Afterwards, $n_\mathrm{f}=3$ Fourier layers are applied to each channel. Finally, $v_3$ is transferred back to the target dimension of the output $\mathbf{\hat{y}}_i$ by another neural network $Q$. More specifically, each Fourier layer is composed of two paths. The upper one learns a mapping in Fourier space by adapting $R_j$ for scaling and truncating the Fourier Series after $n_\mathrm{m}$ modes, while the lower one learns a local linear transform $W_j$.