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

Svenja Ehlers<EMAIL_ADDRESS>[ Marco Klein  Alexander Heinlein
Mathies Wedler  Nicolas Desmars  Norbert Hoffmann  Merten Stender  Hamburg
University of Technology, Dynamics Group, Schloßmühlendamm 30, 21073 Hamburg,
Germany  German Aerospace Center (DLR), Institute of Maritime Energy Systems,
21502 Geesthacht, Germany  Delft University of Technology, Delft Institute of
Applied Mathematics, 2628 CD Delft, Netherlands  Imperial College London,
Department of Mechanical Engineering, London SW7 2AZ, United Kingdom
Technische Universität Berlin, Cyber-Physical Systems in Mechanical
Engineering, 10623 Berlin, Germany

###### Abstract

Accurate short-term prediction of phase-resolved water wave conditions is
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 modeling assumptions that compromise real-time capability or
accuracy of the entire 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 modeling 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-based network performs better
in handling the data structure imposed by wave physics due to its global
approach to learn the mapping between input and desired output in Fourier
space.

###### keywords:

deep operator learning , Fourier neural operator , nonlinear ocean waves ,
phase-resolved surface reconstruction , X-band radar images , radar inversion

††journal: Ocean Engineering

url]www.tuhh.de/dyn

##  1 Introduction

Offshore installations and vessels are strongly impacted by the dynamics of
the surrounding ocean waves. Thus, accurate predictions of future wave
conditions are desirable for enhancing their safe and efficient operation. For
this purpose, several numerical wave prediction methods have been developed,
involving two fundamental steps: the _assimilation_ of initial wave conditions
from wave measurement data, followed by the _forecast_ of future wave
evolution. While one line of research focuses on predicting simplified phase-
averaged wave quantities based on statistical parameters, marine applications
such as wind turbine installation, helicopter landings, or controlling wave
energy converters require phase-resolved spatio-temporal wave information
$\eta(x,t)$ to identify periods of low wave conditions or enable extreme event
warning. The X-band radar is a remote sensing device that can obtain such
phase-resolved wave information. However, the radar backscatter is affected by
the geometrical mechanism of tilt and shadowing modulation, creating a
nonlinear and sparse relationship between radar measurement intensities
$\xi(x,t)$ and the actual ocean wave surface elevation $\eta(x,t)$. This makes
a reconstruction of wave information from radar information necessary in the
assimilation step, which is also referred to as radar inversion and is
graphically exemplified in Figure 1.

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 long-crested waves travelling in one spatial dimension. The radar
measurement is a snapshot acquired at time instant $t_{\mathrm{s}}$, which is
utilized for reconstructing the wave surface elevation at the same time
instant. This reconstruction serves as the initial condition for forecasting
the further wave evolution at times $t>t_{\mathrm{s}}$.

Contemporary phase-resolved wave prediction methods face a trade-off between
accuracy and real-time capability. To achieve computationally efficient
predictions, linear wave theory (LWT) is commonly employed during the forecast
step [cf. Morris et al., 1998, Naaijen and Wijaya, 2014, Hilmer and Thornhill,
2015], along with prior spectral- or texture-analysis-based reconstruction of
initial wave conditions from radar data [Borge et al., 2004, Dankert and
Rosenthal, 2004]. However, these reconstruction methods necessitate additional
calibration by wave buoys or rely on simplified assumptions concerning the
radar backscatter. Furthermore, the accuracy of the linear approach decreases
remarkably for larger temporal horizons of prediction and increasing wave
steepness [Lünser et al., 2022], necessitating a wave prediction using
nonlinear wave models, especially for capturing safety-critical events such as
rogue waves [Ducrozet et al., 2007, Kharif et al., 2009]. Comparative studies
on phase-resolved nonlinear ocean wave prediction have demonstrated that the
high-order spectral (HOS) method, introduced by West et al. [1987] and
Dommermuth and Yue [1987], provides the best forecast accuracy [Klein et al.,
2020]. While the HOS forecast step itself is also numerically efficient, the
assimilation step currently represents the weakest part in the entire process
[Köllisch et al., 2018]: the inversion of initial conditions relies on an
optimization procedure of the wave model parameters for the subsequent
forecast [Wu, 2004, Blondel-Couprie, 2009], which decreases the possible
forecast horizon of the entire prediction and hinders the real-time capability
so far [Desmars, 2020]. Even though the alternative for the HOS inversion
proposed by Köllisch et al. [2018] is able to improve the real-time
capability, this method instead assumes an unrealistic radar snapshots data
rate $\Delta t_{\mathrm{r}}$, making it not suitable for real-world
applications [Desmars, 2020].

The aforementioned shortcomings of conventional ocean wave reconstruction and
forecasting methods have motivated the exploration of alternatives based on
machine learning (ML) techniques. For instance, ML methods are able to
forecast simple phase-averaged wave quantities such as significant wave height
$H_{\mathrm{s}}$, peak period $T_{\mathrm{p}}$ or mean wave direction [cf. Deo
et al., 2001, Asma et al., 2012, James et al., 2018, Wu et al., 2020, Yevnin
and Toledo, 2022]. Recent advancements have also allowed for the more complex
task of forecasting the spatio-temporal evolution of phase-resolved wave
fields, achieved by training multilayer perceptrons (MLPs) [Desouky and
Abdelkhalik, 2019, Law et al., 2020, Duan et al., 2020a, Zhang et al., 2022],
recurrent neural networks (RNNs) [Kagemoto, 2020, Mohaghegh et al., 2021, Liu
et al., 2022], or convolutional neural networks (CNNs) [Klein et al., 2022,
Wedler et al., 2023] on synthetic or experimental one-dimensional elevation
data. However, these studies presuppose that either temporal sequences of wave
elevations can be solely measured at a single point in space by buoys
$\eta(x=x_{\mathrm{p}},t)$ or snapshots of initial wave conditions that are
available throughout the entire space domain $\eta(x,t=t_{\mathrm{s}})$. In
practice, neither of these assumptions is feasible due to the lack of
directional wave information of single-point measurements and the fact that
the acquisition of spatial snapshots using remote sensing systems such as
radars lead to sparse and unscaled observations $\xi(x,t=t_{\mathrm{s}})$,
requiring a reconstruction of wave surface elevations first.

Consequently, it would be advantageous to employ ML methods also for the
phase-resolved reconstruction of wave elevations $\eta(x,t)$ from X-band radar
data $\xi(x,t)$. However, as far as the authors are aware, this topic has not
yet been addressed. Prior studies have solely focused on reconstructing phase-
averaged statistical parameters of the prevailing sea state from radar data.
For instance, Vicen-Bueno et al. [2012] and Salcedo-Sanz et al. [2015]
improved the estimation of $H_{\mathrm{s}}$ by extracting scalar features from
sequences of radar images $\xi(x,t)$ in a preprocessing step, which in turn
were employed to train MLPs and support vector regression models. In contrast,
Yang et al. [2021] extracted features from each of the consecutive radar
images itself for improved $H_{s}$ estimation at the current time instant.
While these methods rely on handcrafted features acquired during a
preprocessing step, end-to-end approaches that automatically extract important
features from their input have also been proposed. For instance, Duan et al.
[2020b] and Chen and Huang [2022] used CNN-based methods to estimate
$H_{\mathrm{s}}$ and $T_{\mathrm{p}}$ from radar images.

Although there seems to be no relevant investigation into the ML-based
reconstruction of phase-resolved wave surfaces from sparse X-Band radar data,
we hypothesize that ML offers a valuable alternative for the radar inversion
task (Hypothesis 1), as it shares many similarities to typical inverse
problems in imaging [Bertero et al., 2022, Ongie et al., 2020] such as
inpainting [Pathak et al., 2016] and restoration [Zhang et al., 2017]. Two
neural network architectures, with network components involving either a local
or global approach of data processing, are investigated in detail for their
performance in our task. Specifically, we will adapt the U-Net proposed by
Ronneberger et al. [2015], a fully convolutional neural network that employs a
mapping approach in Euclidean space, and the Fourier neural operator (FNO)
proposed by Li et al. [2020], which is designed to learn a more global mapping
in Fourier space. Despite the success of CNN-based approaches in imaging
problems, we hypothesize that FNO models may be better suited for handling the
complex and dynamic nature of ocean waves (Hypothesis 2), since we can assume
that the wave features are already explicitly encoded in the network
structure. In contrast, the U-Net needs to learn these wave features by
aggregating information from multiple layers. Lastly, we expect that
incorporating historical context via spatio-temporal radar data will enhance
the reconstruction quality of both ML architectures (Hypothesis 3).

In general, the fast inference capabilities of trained ML models, make them
ideal for maintaining real-time capability of the entire prediction process
(Criterion 1) due to the rapid surface reconstruction without particular data
preprocessing. Beside the real-time capability, ensuring high reconstruction
accuracy is crucial to prevent initial reconstruction errors that will
accumulate and deteriorate the subsequent wave forecast. Hence, we strive for
an empirical reference value for the surface similarity parameter (SSP) error
metric [Perlin and Bustamante, 2014] of less than $\mathrm{SSP}\leq 0.10$
between ground truth and reconstructed wave surfaces (Criterion 2), which is a
commonly used error threshold in ocean wave research [Klein et al., 2020,
Lünser et al., 2022]. In addition, the proposed ML methods must be capable of
handling real-world measurement conditions of radar snapshots taken at
intervals of $\Delta t_{\mathrm{r}}=[1,\,2]\,\mathrm{s}$ (Criterion 3), a
common X-band radar revolution period [Neill and Hashemi, 2018].

To summarize, the objective of this work is to develop an ML-based approach
for phase-resolved radar inversion. This involves training of an ML models to
learn a mapping functions $\mathcal{M}$ that are able to reconstruct a spatial
wave elevation snapshot $\eta(x,t=t_{\mathrm{s}})$ from one or
$n_{\mathrm{s}}$ consecutive historical radar snapshots $\xi(x,t_{i})$, where
$t_{i}=\\{t_{\mathrm{s}}-i\Delta
t_{\mathrm{r}}\\}_{i=0,\dots,n_{\mathrm{s}}-1}$. As obtaining ground truth
wave surface elevation data for large spatial domains in real ocean conditions
is almost impractical, in Section 2, we first generate synthetic yet highly
realistic one-dimensional spatio-temporal wave surfaces $\eta(x,t)$ using the
high-order spectral method for different sea states. The corresponding X-band
radar surfaces $\xi(x,t)$ are generated using a geometric approach and
incorporate tilt- and shadowing modulations. In Section 3, two neural network
architectures are introduced, a U-Net-based and FNO-based network, which
should be investigated for their suitability for radar inversion. In Section
4, we discuss the computational results. In particular, we first compare the
wave reconstruction performance of the U-Net-based and the FNO-based models
$\mathcal{M}_{\mathrm{F},n_{\mathrm{s}}}$, each trained using either
$n_{\mathrm{s}}=1$ radar snapshot in each input or spatio-temporal input data,
meaning that multiple consecutive radar images $n_{\mathrm{s}}$ are provided.
Afterwards, the observations are generalized for the entire data set and
discussed. Finally, in Section 5, we draw conclusions based on these results
and suggest future research directions.

##  2 Data generation and preparation

This section briefly introduces the generation of long-crested nonlinear
synthetic wave data $\eta(x,t)$ using the high-order spectral method, followed
by the generation of synthetic radar data $\xi(x,t)$ that accounts for the
tilt- and shadowing modulation mechanisms. The final step involves extracting
a number of $N$ input-output $(\mathbf{x}_{i}$,$\mathbf{y}_{i}),i=1,\dots,N$
data samples from the synthetic radar and wave data, which we employ to train
the supervised ML models in the subsequent steps of this work.

###  2.1 Nonlinear synthetic wave data

To generate synthetic one-dimensional wave data, the water-wave problem can be
expressed by potential flow theory. Assuming a Newtonian fluid that is
incompressible, inviscid, and irrotational, the underlying wave model is
described by a velocity potential $\Phi(x,z,t)$ satisfying the Laplace
equation

$\nabla^{2}\Phi=\frac{\partial^{2}\Phi}{\partial
x^{2}}+\frac{\partial^{2}\Phi}{\partial z^{2}}=0$ (1)

within the fluid domain, where $z=0\,\mathrm{m}$ is the mean free surface with
$z$ pointing in upward direction. The domain is bounded by the kinematic and
dynamic boundary conditions at the free surface $\eta(x,t)$ and the bottom
boundary condition at the seabed at depth $d$

$\displaystyle\eta_{t}+\eta_{x}\Phi_{x}-\Phi_{z}\hskip 28.45274pt$
$\displaystyle\text{on }z=\eta(x,t)$
$\displaystyle\Phi_{t}+g\eta+\frac{1}{2}\left(\Phi_{xx}^{2}+\Phi_{zz}^{2}\right)\hskip
28.45274pt$ $\displaystyle\text{on }z=\eta(x,t)$ (2)
$\displaystyle\Phi_{z}=0\hskip 28.45274pt$ $\displaystyle\text{on }z=-d.$

Solving this system of equations is challenging due to the nonlinear terms in
the boundary conditions, which must be satisfied additionally at the unknown
free surface $\eta(x,t)$. Even though linear wave theory [Airy, 1849] provides
adequate approximations for certain engineering applications, capturing
realistic ocean wave effects requires modeling of nonlinear behaviour of
surface gravity waves. Thus, we employ the HOS method, as formulated by West
et al. [1987], which transforms the boundary conditions to the free surface
and expresses them as a perturbation series of nonlinear order $M$ around
$z=0$. In practice, an order of $M\leq 5$ is sufficient for capturing the
nonlinear wave effects of interest [Desmars, 2020, Lünser et al., 2022]. The
HOS simulation is linearly initialized by spatial wave surface elevation
snapshots $\eta(x,t_{\mathrm{s}}=0)$ sampled from the JONSWAP spectrum for
finite water depth [Hasselmann et al., 1973, Bouws et al., 1985]. The
corresponding initial potential is linearly approximated. Subsequently the
initial elevation and potential are propagated nonlinearly in time with the
chosen HOS order $M$. The referred JONSWAP spectrum attains its maximum at a
peak frequency $\omega_{\mathrm{p}}$, whereas the peak enhancement factor
$\gamma$ determines the energy distribution around $\omega_{\mathrm{p}}$. The
wave frequencies $\omega$ are linked to the wavenumbers $k$ by the linear
dispersion relation $\omega=\sqrt{gk\cdot\tanh{(kd)}}$. The relations
$\omega=\nicefrac{{2\pi}}{{T}}$ and $k=\nicefrac{{2\pi}}{{L}}$ allow for
substituting the peak frequency with a peak period $T_{\mathrm{p}}$, peak
wavelength $L_{\mathrm{p}}$, or peak wavenumber $k_{\mathrm{p}}$. Moreover, a
dimensionless wave steepness parameter
$\epsilon=k_{\mathrm{p}}\cdot\nicefrac{{H_{\mathrm{s}}}}{{2}}$ is defined
based on the significant wave height $H_{\mathrm{s}}$. For more details on the
HOS simulation, consider the work of Wedler et al. [2023] or Lünser et al.
[2022], for example.

In this study, we select a wave domain length of $4000\,\mathrm{m}$,
discretized by $n_{x}=1024$ grid points, resulting in $\Delta
x=3.906\,\mathrm{m}$. A peak enhancement factor of $\gamma=3$ is employed to
emulate North Sea conditions. The water depth is $d=500\,\mathrm{m}$ and the
sea state parameters peak wavelength $L_{\mathrm{p}}$ and steepness $\epsilon$
are varied systematically over
$L_{\mathrm{p}}\in\\{80,90,\ldots,190,200\\}\,\mathrm{m}$ and
$\epsilon\in\\{0.01,0.02,\ldots,0.09,0.10\\}$, resulting in 130 possible
$L_{\mathrm{p}}$-$\epsilon$-combinations. For each
$L_{\mathrm{p}}$-$\epsilon$-combination, we generate four different initial
surfaces $\eta(x,t_{\mathrm{s}}=0)$ by superimposing the wave components of
the JONSWAP spectrum with random phase shifts. The subsequent wave evolution
$\eta(x,t>0)$ for $t=0,\dots,50\,\mathrm{s}$ with $\Delta
t_{\mathrm{save}}=0.1\,\mathrm{s}$ is performed considering the nonlinearities
imposed by HOS order $M=4$. As a result, we generate a total of 520 unique
spatio-temporal HOS wave data arrays, each of shape
$E_{\mathrm{HOS}}\in\mathbb{R}^{1024\times 500}$, where
$(E_{\mathrm{HOS}})_{ij}=\eta(x_{i},t_{j})$ with $x_{i}=i\cdot\Delta x$ and
$t_{j}=j\cdot\Delta t_{\mathrm{save}}$.

###  2.2 Corresponding synthetic radar data

As X-band radar systems are often pre-installed on marine structures for
object detection purposes, they also gained attention for observing ocean
surface elevations [Borge et al., 1999]. The system antenna rotates with a
device-specific revolution time $\Delta t_{\mathrm{r}}$ of between
$1-2\,\mathrm{s}$ [Neill and Hashemi, 2018] while emitting radar beams along a
range $r$, which are interacting with short-scale capillary waves distributed
on large-scale ocean surface waves by the Bragg resonance phenomenon,
resulting in backscatter to the antenna [Valenzuela, 1978]. This procedure
provides measurement data $\xi(r,t)$ as a proxy of wave surface elevations
$\eta(r,t)$, which are not directly relatable to each other due to the
different modulation mechanisms. Most influential are assumed to be tilt
modulation [Dankert and Rosenthal, 2004], shadowing modulation [Borge et al.,
2004, Wijaya et al., 2015] or a combination of both [Salcedo-Sanz et al.,
2015].

In order to generate synthetic radar snapshots for this work, the modulation
mechanisms is simulated according to Salcedo-Sanz et al. [2015] and Borge et
al. [2004], as illustrated in Figure 2. Tilt modulation refers to the
variation in radar backscatter intensity depending on the local incidence
angle $\tilde{\Theta}(r,t)$ between the unit normal vector $\mathbf{n}(r,t)$
perpendicular to the illuminated wave facet $\eta(r,t)$ and the unit normal
vector $\mathbf{u}(r,t)$ pointing towards the antenna. As the backscatter
cannot reach the antenna if the dot product $\mathbf{n}\cdot\mathbf{u}$
approaches negative values for $|\tilde{\Theta}|>\frac{\pi}{2}$, the tilt
modulation $\mathcal{T}$ is simulated by

$\mathcal{T}(r,t)=\mathbf{n}(r,t)\cdot\mathbf{u}(r,t)=\cos\tilde{\Theta}(r,t)\hskip
14.22636pt\text{if}\;\;|\tilde{\Theta}(r,t)|\leq\frac{\pi}{2}$ (3)

The shadowing modulation instead occurs when high waves located closer to the
antenna obstruct waves at greater distances. Shadowing depends on the nominal
incidence angle $\Theta(r,t)$ of a wave facet $\eta(r,t)$ with horizontal
distance $R(r)$ from the antenna at height $z_{\mathrm{a}}$ above the mean sea
level, geometrically expressed as

$\Theta(r,t)=\tan^{-1}\left[\frac{R(r)}{z_{\mathrm{a}}-\eta(r,t)}\right].$ (4)

At a specific time instance $t$, a wave facet $\eta(r,t)$ at point $r$ is
shadowed in case there is another facet $\eta^{\prime}=\eta(r^{\prime},t)$
closer to the radar $R^{\prime}=R(r^{\prime})<R(r)$ that satisfies the
condition $\Theta^{\prime}=\Theta(r^{\prime},t)\geq\Theta(r,t)$. The
shadowing-illumination mask $\mathcal{S}$ can be constructed from this
condition as follows

$\mathcal{S}(r,t)=\begin{cases}0&\hskip
14.22636pt\text{if}\;\;R(r^{\prime})<R(r)\text{ and
}\Theta(r^{\prime},t)\geq\Theta(r,t),\\\ 1&\hskip
14.22636pt\text{otherwise}.\end{cases}$ (5)

Assuming that tilt-and shadowing modulation contribute to the radar imaging
process, the image intensity is proportional to the local radar cross section,
that is $\xi(r,t)\sim\mathcal{T}(r,t)\cdot\mathcal{S}(r,t)$. As marine radars
are not calibrated, the received backscatter $\xi(r,t)$ may be normalized to a
user-depended range of intensity values.

modulation
---
tilt
---
${\color[rgb]{0.551,0.121,0.133}\mathbf{u}}$
---
${\color[rgb]{0.551,0.121,0.133}\mathbf{n}}$
---
antenna
---
$z_{a}$
---
wave $\eta$
---
${\color[rgb]{0.551,0.121,0.133}\tilde{\Theta}}$
---
shadowing
---
${\color[rgb]{0.106,0.455,0.569}\Theta}$
---
${\color[rgb]{0.106,0.455,0.569}\Theta^{\prime}}$
---
$r_{0}$
---
$r$
---
modulation
---
radar beams
---
${\color[rgb]{0.106,0.455,0.569}R}$
---
${\color[rgb]{0.106,0.455,0.569}R^{\prime}}$
---
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.

In this work, we assume an X-band radar system with an antenna installation
height of $z_{\mathrm{a}}=18\,\mathrm{m}$. Around this antenna, the system’s
dead range is estimated to be $r_{\mathrm{min}}=100\,\mathrm{m}$, and it scans
the wave surface with a spatial range resolution of $\Delta r=3.5\,\mathrm{m}$
at $n_{r}=512$ grid points. Thus, the maximum observation range is computed as
$r_{\mathrm{max}}=1892\,\mathrm{m}$. The radar revolution period is chosen
according to 1 as a snapshot each $\Delta t_{\mathrm{r}}=1.3\,\mathrm{s}$,
i.e., $n_{t}=38$ radar snapshots for $50\,\mathrm{s}$ of simulation time.
Using these definitions, we first transform the 520 wave data arrays
$E_{\mathrm{HOS}}\in\mathbb{R}^{1024\times 500}$ from their HOS grid to the
radar system’s grid, yielding $E_{\mathrm{sys}}\in\mathbb{R}^{512\times 38}$,
where $(E_{\mathrm{sys}})_{ij}=\eta(r_{i},t_{j})$ with $r_{i}=i\cdot\Delta r$
and $t_{j}=j\cdot\Delta t_{\mathrm{r}}$. To obtain highly realistic
corresponding radar observations, we model tilt modulation $\mathcal{T}(r,t)$
and shadowing modulation $\mathcal{S}(r,t)$, resulting in 520 radar data
arrays, each denoted as $Z_{\mathrm{sys}}\in\mathbb{R}^{512\times 38}$ with
$(Z_{\mathrm{sys}})_{ij}=\xi(r_{i},t_{j})$.

###  2.3 Preparation of data for machine learning

To train a supervised learning algorithm, labeled input-output data pairs are
required. As visualized in Figure 3, from each of the 520 generated radar-wave
arrays-pairs we extract six radar input snapshots $\mathbf{x}$ from
$(Z_{\mathrm{sys}})_{ij}=\xi(r_{i},t_{j})$ and wave output snapshots
$\mathbf{y}$ from $(E_{\mathrm{sys}})_{ij}=\eta(r_{i},t_{j})$ at six distinct
time instances $t_{\mathrm{s}}$ with the largest possible temporal distance.
Each output $\mathbf{y}\in\mathbb{R}^{512\times 1}$ contains a single snapshot
at time $t_{\mathrm{s}}$, while each input $\mathbf{x}\in\mathbb{R}^{512\times
n_{\mathrm{s}}}$ incorporates a number of $n_{\mathrm{s}}$ historical radar
snapshots at discrete times $\\{t_{\mathrm{s}}-i\cdot\Delta
t_{\mathrm{r}}\\}_{i=0,\dots,n_{\mathrm{s}}-1}$. A single snapshot
($n_{\mathrm{s}}=1$) at a time $t_{\mathrm{s}}$ can be used as input, however,
as we assumed in 1, larger temporal context may enhance the quality of a
network’s reconstruction $\mathbf{\hat{y}}$. The optimal value of
$n_{\mathrm{s}}$ is subject of investigation as discussed in Sections 4.1.2
and 4.2.2. In total, $N=6\cdot 520=3120$ data samples are generated, each
corresponding to a single $L_{\mathrm{p}}$-$\epsilon$-combination. The data
set takes the of shape $\mathbf{X}\in\mathbb{R}^{3120\times 512\times
n_{\mathrm{s}}}$ and $\mathbf{Y}\in\mathbb{R}^{3120\times 512\times 1}$ and is
split into $60\%$ training, $20\%$ validation, and $20\%$ test data using a
stratified data split w.r.t. the sea state parameters
$(L_{\mathrm{p}},\epsilon)$. This ensures an equal representation of each wave
characteristic in the resulting subsets, as described in detail in A.

$n_{\mathrm{s}}$
---
radar surface array
---
wave surface array
---

$n_{r}=512$
---

$n_{r}=512$
---

$n_{t}=38$
---
$N=3120$ samples
---
$n_{\mathrm{s}}$
---
$t_{\mathrm{s,1}}$
---
$t_{\mathrm{s,2}}$
---
$t_{\mathrm{s,3}}$
---
$t_{\mathrm{s,4}}$
---
$t_{\mathrm{s,5}}$
---
$t_{\mathrm{s,6}}$
---

$n_{r}=512$
---

$\mathbf{X}$
---
$\mathbf{Y}$
---
inputs
---
outputs
---
$t_{\mathrm{s,1}}$
---
$\Delta t_{\mathrm{r}}=1.3\,\mathrm{s}$
---
$1$
---
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 $\mathbf{x}$ consists $n_{\mathrm{s}}$ radar snapshots
acquired at intervals of $\Delta t_{\mathrm{r}}=1.3\,\mathrm{s}$, while each
output $\mathbf{y}$ represents a single-snapshot wave surface elevation at
time instant $t_{\mathrm{s}}$. In total $N=6\cdot 520=3120$ data samples are
generated.

##  3 Machine learning methodology

The U-Net [Ronneberger et al., 2015] and the Fourier neural operator (FNO) [Li
et al., 2020] are neural network architectures for data with grid-like
structures such as our radar and wave surface elevation snapshots. Their
fundamental difference is the inductive bias encoded by each architecture,
which refers to prior assumptions about either the solution space or the
underlying data-generating process [Mitchell, 1980, Battaglia et al., 2018].
The U-Net is a special type of CNN [LeCun et al., 1989] and imposes an
inductive bias by assuming that adjacent data points in Euclidean space are
semantically related and learns local mappings between input patches and
output features in each layer. This local information is aggregated to more
global features due to the utilization of multiple downsampling and
convolutional layers. In contrast, the FNO operates under the assumption that
the data information can be meaningfully represented in Fourier space. It
employs multiple Fourier transformations to learn a mapping between the
spectral representation of the input and desired output, directly providing a
global understanding of the underlying patterns in the data. This section
presents the U-Net- and FNO-based network architectures used in our study for
radar inversion. In addition, suitable loss and metric functions are
introduced for assessing the models performance.

###  3.1 U-Net-based network architecture

We first adopt the U-Net concept, originally developed for medical image
segmentation by Ronneberger et al. [2015], which has since been applied to a
variety of image-to-image translation and surrogate modeling problems, for
instance by Isola et al. [2016], Liu et al. [2018], Stoian et al. [2019], Wang
et al. [2020], Eichinger et al. [2022], Niekamp et al. [2023] and Stender et
al. [2023]. The mirrored image dimensions in a fully convolution autoencoder
network allow for the U-Net’s key property, that is the use of skip-
connections for concatenating the output features from the encoding path with
the inputs in the decoding path. This enables the reuse of data information of
different spatial scales that would otherwise be lost during downsampling and
assists the optimizer to find the minimum more efficiently [Li et al., 2018].

Our proposed encoder-decoder architecture is the result of a four-fold cross-
validated hyperparameter study, documented in Table 2 in the appendix. As
depicted in Figure 4, the adapted U-Net architecture, has a depth of
$n_{\mathrm{d}}=5$ consecutive encoder blocks followed by the same number of
consecutive decoder blocks with skip-connections between them. Each encoder
block is composed of a 1D convolutional layer with $n_{\mathrm{k}}=32$ kernels
of size $s_{\mathrm{k}}=5$, that are responsible for identifying specific
features in the input by shifting the smaller-sized kernels, containing the
networks trainable weights, across the larger input feature maps in a step-
wise manner. Each convolutional layer is followed by a GeLU activation
function $\sigma$ [Hendrycks and Gimpel, 2016] and an average pooling
downsampling layer of size 2. To summarize, in the encoding path each input
sample $\mathbf{x}\in\mathbb{R}^{n_{r}\times n_{\mathrm{s}}}$ is transformed
by the first convolutional layer resulting in
$v_{\mathrm{c1}}\in\mathbb{R}^{n_{r}\times n_{\mathrm{k}}}$, with $n_{r}=512$
being the number of spatial grid-points and $n_{\mathrm{s}}$ being the
historic snapshots in the radar input. Subsequently this intermediate output
is send through $\sigma$, before the pooling layer reduces the spatial
dimension to $v_{\mathrm{p1}}\in\mathbb{R}^{\frac{1}{2}n_{r}\times
n_{\mathrm{k}}}$. This process is repeated until the final encoding block’s
output being $v_{\mathrm{p5}}\in\mathbb{R}^{\frac{1}{32}n_{r}\times
n_{\mathrm{k}}}$. Next, the decoding blocks are applied, each consisting of a
convolutional layer with again $n_{\mathrm{k}}=32$ kernels of size
$s_{\mathrm{k}}=5$, followed by GeLU activation. Afterwards the feature maps’
spatial dimensions are upsampled using transpose convolutional layers with
linear activation. The resulting feature maps then are concatenated with the
output of the corresponding stage in the encoding path via skip-connections,
before the next convolution is applied. This process is repeated until the
final output $\mathbf{\hat{y}}\in\mathbb{R}^{n_{r}\times 1}$ is calculated
using a convolutional layer with a single kernel and linear activation.

$n_{\mathrm{k}}$
---
convolution ($n_{\mathrm{k}}=32,\,s_{\mathrm{k}}=5$) + GeLU
---
average pooling
---
up-convolution
---
skip-connection
---
convolution ($n_{\mathrm{k}}=1,\,s_{\mathrm{k}}=5$)
---

$n_{r}=512$
---

$\frac{1}{2}n_{r}$
---
$1$
---
$n_{\mathrm{s}}$
---

$\mathbf{x}$
---

$v_{\mathrm{p1}}$
---

$v_{\mathrm{c2}}$
---

$v_{\mathrm{p2}}$
---

$v_{\mathrm{c3}}$
---

$v_{\mathrm{c1}}$
---

$\frac{1}{4}n_{r}$
---
$\frac{1}{8}n_{r}$
---
$\frac{1}{16}n_{r}$
---
$\frac{1}{32}n_{r}$
---

$\mathbf{\hat{y}}$
---

$n_{\mathrm{k}}$
---
encoder
---
decoder
---
Figure 4: Fully convolutional encoder-decoder architecture based on the U-Net
[Ronneberger et al., 2015]. Each input $\mathbf{x}$ 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 $\mathbf{\hat{y}}$. Moreover, the outputs of the encoding
stages are transferred to the decoding path via skip-connections.

As indicated above, the U-Net architecture assumes local connections between
neighboring data points, which is accomplished through two mechanisms.
Firstly, the convolutional layers use kernels with a receptive field of
$s_{\mathrm{k}}=5$ pixels to process different local parts of the larger input
feature maps in the same manner. This is referred to as weight sharing,
causing a property called translational equivariance: each patch of the input
is processed by the same kernels. Secondly, the pooling layers induce locality
by assuming that meaningful summations of information from small local regions
in the intermediate feature maps can be made and creates a property referred
to as translational invariance [Goodfellow et al., 2016].

###  3.2 FNO-based network architecture

In the second step, we explore a neural network based on the FNO [Li et al.,
2020]. While a CNN is limited to map between finite-dimensional spaces, neural
operators are in addition capable to learn nonlinear mappings between a more
general class of function spaces. This makes the FNO well-suited for capturing
the spatio-temporal patterns that govern the dynamics of various physical
problems that obey partial differential equations, if the solutions are well
represented in Fourier space. FNO variants have been applied to e.g., fluid
dynamics [Peng et al., 2022, Li et al., 2022], simulation of multiphase flow
[Yan et al., 2022, Wen et al., 2022], weather forecasting [Pathak et al.,
2022], material modeling [Rashid et al., 2022, You et al., 2022], and image
classification [Williamson et al., 2022].

The FNO-based iterative architecture approach ($\mathbf{x}\rightarrow
v_{0}\rightarrow v_{1}\rightarrow\ldots\rightarrow\mathbf{\hat{y}}$) applied
in this work is illustrated in Figure 5, while Table 3 in the appendix
summarizes the determination of model hyperparameters by a four-fold cross-
validation. Our FNO transforms radar input data
$\mathbf{x}\in\mathbb{R}^{n_{r}\times n_{\mathrm{s}}}$ into a higher-
dimensional latent representation $v_{0}\in\mathbb{R}^{n_{r}\times
n_{\mathrm{w}}}$ of channel width $n_{\mathrm{w}}=32$, using a linear neural
network layer $P$ with $n_{\mathrm{w}}$ nodes. Subsequently, the latent
representation passes through $n_{\mathrm{f}}=3$ Fourier layers, each
consisting of two paths. In the upper path, a global convolution operator
defined in Fourier space is applied to each channel of $v_{0}$ separately
utilizing discrete Fourier transforms $F$. A linear transformation $R_{0}$ is
then applied to the lower-order Fourier modes after truncating the Fourier
series at a maximum number of $n_{\mathrm{m}}=64$ modes. Subsequently, this
scaled and filtered content is back-transformed to the spatial domain using an
inverse discrete Fourier transforms $F^{-1}$. In the lower path, a linear
transformation $W_{0}$ in the spatial domain is applied to the input $v_{0}$
to account for non-periodic boundary conditions and higher-order modes that
are neglected in the upper path of the Fourier layer. The outputs of the upper
and lower paths are added, and the sum is passed through a nonlinear GeLU
activation $\sigma$ resulting in $v_{1}\in\mathbb{R}^{n_{r}\times
n_{\mathrm{w}}}$, before entering the next Fourier layer. In summary, the
output of the $(i+1)$-th Fourier layer is defined as

$v_{i+1}=\sigma\left(F^{-1}\left(R_{i}\cdot F(v_{i})\right)+W_{i}\cdot
v_{i}\right).$ (6)

Finally, the output $v_{3}$ of the last Fourier layer is transferred to the
target output dimension $\mathbf{\hat{y}}\in\mathbb{R}^{n_{r}\times 1}$ using
another linear layer $Q$. In summary, the FNOs weights correspond to
$P\in\mathbb{R}^{n_{\mathrm{s}}\times d_{\mathrm{w}}}$,
$Q\in\mathbb{R}^{n_{\mathrm{s}}\times d_{\mathrm{w}}}$ and all
$R_{i}\in\mathbb{C}^{d_{\mathrm{w}}\times d_{\mathrm{w}}\times
d_{\mathrm{m}}}$ and $W_{i}\in\mathbb{R}^{d_{\mathrm{w}}\times
d_{\mathrm{w}}}$. As the $R_{i}$-matrices contain the main portion of the
total number of weights, most parameters are learned in the Fourier space
rather than the original data space.

$n_{r}=512$
---
$n_{\mathrm{s}}$
---
Fourier layer 1
---
Fourier layer 2
---
Fourier layer 3
---
$P$
---
$\mathbf{x}$
---
$v_{0}$
---
$v_{1}$
---
$v_{2}$
---
$\mathbf{\hat{y}}$
---
$F$
---
$F^{-1}$
---
$n_{r}$
---
$Q$
---
$n_{r}$
---
$n_{r}$
---
$n_{\mathrm{w}}$
---
$+$
---
$\sigma$
---
$W_{0}$
---
$R_{0}$
---
$v_{3}$
---
$n_{\mathrm{w}}$
---
$n_{\mathrm{w}}$
---
Figure 5: Network architecture based on the Fourier neural operator [Li et
al., 2020]. Each input $\mathbf{x}$ 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
$\mathbf{\hat{y}}$ by another neural network $Q$. More specifically, each
Fourier layer is composed of two paths added in the end. The upper one learns
a mapping in Fourier space by adapting $R_{i}$ for scaling and truncating the
Fourier Series after $n_{\mathrm{m}}$ modes, while the lower one learn a local
linear transform $W_{i}$.

As previously noted, the FNO architecture incorporates a global inductive bias
that assumes the input data exhibits approximately periodic properties and can
be effectively represented in Fourier space. Furthermore, the FNO’s design
presupposes that the Fourier spectrum of the input data is smooth, enabling
its frequency components to be represented by a limited number of low-
wavenumber Fourier coefficients, as the $R_{i}$ matrices, which are
responsible for the global mapping, truncate higher-frequency modes.

###  3.3 Training and evaluation

Both the U-Net- and FNO-based architecture are implemented using the PyTorch
library [Paszke et al., 2019]. In order to enable a fair comparison, both
networks employ the same loss function for training. To account for wave
training data of varying spatial scales, the relative L2-norm of the error is
utilized as the loss function, where $\mathbf{y}$ and
$\mathbf{\hat{y}}\in\mathbb{R}^{512\times 1}$ represent the true and predicted
wave surfaces for one training sample

$\mathcal{L}(\mathbf{y},\mathbf{\hat{y}})=\mathrm{nL2}(\mathbf{y},\mathbf{\hat{y}})=\frac{\|\mathbf{\hat{y}}-\mathbf{y}\|_{2}}{\|\mathbf{y}\|_{2}}.$
(7)

To minimize the loss, we use the Adam optimizer [Kingma and Ba, 2014] with a
learning rate of 0.001. The training is executed for 800 epochs on an NVIDIA
GeForce RTX 3050 Ti Laptop GPU. For both, the U-Net-based models
$\mathcal{M}_{\mathrm{U},n_{\mathrm{s}}}$ and FNO-based models
$\mathcal{M}_{\mathrm{F},n_{\mathrm{s}}}$, only the models with the lowest
test loss within the 800 epochs is stored for performance evaluation and
visualization.

Established machine learning metrics based on Euclidean distances treat the
deviation of two surfaces in frequency or phase as amplitude errors [Wedler et
al., 2022]. Therefore, we introduce the surface similarity parameter (SSP)
proposed by Perlin and Bustamante [2014] as an additional performance metric

$\mathrm{SSP}(\mathbf{y},\mathbf{\hat{y}})=\frac{\sqrt{\int|F_{\mathbf{y}}(k)-F_{\mathbf{\hat{y}}}(k)|^{2}dk}}{\sqrt{\int|F_{\mathbf{y}}(k)|^{2}dk}+\sqrt{\int|F_{\mathbf{\hat{y}}}(k)|^{2}dk}}\in[0,1],$
(8)

where $k$ denotes the wavenumber vector and $F_{\mathrm{y}}$ denotes the
discrete Fourier transform of a surface $\mathbf{y}$. The SSP is a normalized
error metric, with $\mathrm{SSP}=0$ indicating perfect agreement and
$\mathrm{SSP}=1$ a comparison of phase-inverted surfaces. As the SSP combines
phase-, amplitude-, and frequency errors in a single quantity and is a
normalized error metric, it is used in recent ocean wave prediction studies by
Klein et al. [2020, 2022], Wedler et al. [2022, 2023], Desmars et al. [2021,
2022] and Lünser et al. [2022].

While metrics such as the $\mathrm{nL2}$ or $\mathrm{SSP}$ evaluate the
average reconstruction quality of each
$\mathbf{\hat{y}}\in\mathbb{R}^{n_{r}\times 1}$ across the entire spatial
domain with $n_{r}=512$ grid points, it is important to consider the potential
imbalance in reconstruction error between those areas where the radar input
was either shadowed or visible. This imbalance ratio can be quantified by
$\tfrac{\mathrm{nL2}_{\mathrm{shad}}}{\mathrm{nL2}_{\mathrm{vis}}}$. Here,
$\mathrm{nL2}_{\mathrm{shad}}=\mathrm{nL2}(\mathbf{y}_{\mathrm{shad}},\mathbf{\hat{y}}_{\mathrm{shad}})$
and
$\mathrm{nL2}_{\mathrm{vis}}=\mathrm{nL2}(\mathbf{y}_{\mathrm{vis}},\mathbf{\hat{y}}_{\mathrm{vis}})$
are the nL2 norms of the output wave elevations in the shadowed or visible
areas, respectively. We separate the visible and shadowed parts using the
shadowing mask $\mathcal{S}$ introduced in Eq. (5), where
$\mathbf{y}_{\mathrm{vis}}=\mathcal{S}\cdot\mathbf{y}$ and
$\mathbf{y}_{\mathrm{shad}}=(1-\mathcal{S})\cdot\mathbf{y}$. Afterwards, all
cells with zero entries are removed from the output arrays, such that the
number of visible or invisible data points is $n_{\mathrm{vis}}$ or
$n_{\mathrm{shad}}$, respectively, and
$\mathbf{y}_{\mathrm{vis}},\,\mathbf{\hat{y}}_{\mathrm{vis}}\in\mathbb{R}^{n_{\mathrm{vis}}\times
1}$ and
$\mathbf{y}_{\mathrm{shad}},\,\mathbf{\hat{y}}_{\mathrm{shad}}\in\mathbb{R}^{n_{\mathrm{shad}}\times
1}$ satisfy $n_{\mathrm{vis}}+n_{\mathrm{shad}}=n_{r}=512$. To conclude, a
high value of the
$\tfrac{\mathrm{nL2}_{\mathrm{shad}}}{\mathrm{nL2}_{\mathrm{vis}}}$-ratio
indicates that the reconstruction in areas that were shadowed in the input is
much worse than in the visible areas. We thus not only strive for low nL2
values across the entire spatial domain, but also for low
$\tfrac{\mathrm{nL2}_{\mathrm{shad}}}{\mathrm{nL2}_{\mathrm{vis}}}$ to achieve
uniform reconstructions. We use a ratio metric only based on the Euclidean
distance based nL2 and not on the SSP, as small sections of $\mathbf{y}$ and
$\mathbf{\hat{y}}$ cannot be meaningfully considered in Fourier space.

##  4 Results

This work explores the potential of utilizing machine learning for the
reconstruction of one-dimensional ocean waves surfaces
$\eta(x,t=t_{\mathrm{s}})$ from radar measurement surfaces $\xi(x,t_{i})$ with
$t_{i}=\\{t_{\mathrm{s}}-i\Delta
t_{\mathrm{r}}\\}_{i=0,\dots,n_{\mathrm{s}}-1}$. Therefore, each radar input
sample $\mathbf{x}\in\mathbb{R}^{512\times n_{\mathrm{s}}}$, acquired
according to Section 2, is to be mapped to the desired wave surface output
$\mathbf{y}\in\mathbb{R}^{512\times 1}$ via a ML model
$\mathcal{M}:\mathbf{x}\rightarrow\mathbf{y}$. We examine the impact of the
number of historical radar snapshots $n_{\mathrm{s}}$ as well as the impact of
the inductive bias of the U-Net-based models
$\mathcal{M}_{\mathrm{U},n_{\mathrm{s}}}$ and the FNO-based models
$\mathcal{M}_{\mathrm{F},n_{\mathrm{s}}}$, proposed in Section 3. We train the
models using a total of $N_{\mathrm{train}}+N_{\mathrm{val}}=2496$ samples and
evaluate their performance using the previously excluded test set of
$N_{\mathrm{test}}=624$ samples. The results are summarized in Table 1 and are
reviewed and discussed regarding the pre-stated 1-1 and 1-1 in detail in the
subsequent subsections.

Table 1: Reconstruction results averaged across the entire test set evaluated with different metrics for the investigated U-Net-based models $\mathcal{M}_{\mathrm{U},n_{\mathrm{s}}}$ and FNO-based models $\mathcal{M}_{\mathrm{F},n_{\mathrm{s}}}$ trained with either one or multiple radar snapshots $n_{\mathrm{s}}$ in each samples’ input. model | mean errors on test set
---|---
name | architecture | $n_{\mathrm{s}}$ | epochs | investigated in | $\mathrm{nL2}$ | $\frac{\mathrm{nL2}_{\mathrm{shad}}}{\mathrm{nL2}_{\mathrm{vis}}}$ | $\mathrm{SSP}$
$\mathcal{M}_{\mathrm{U},1}$ | U-Net-based | 1 | 150 |  Sec. 4.1.1 | 0.329 | 2.679 | 0.171
$\mathcal{M}_{\mathrm{U},10}$ | U-Net-based | 10 | 592 |  Sec. 4.1.2 | 0.123 | 1.755 | 0.061
$\mathcal{M}_{\mathrm{F},1}$ | FNO-based | 1 | 721 |  Sec. 4.2.1 | 0.242 | 1.886 | 0.123
$\mathcal{M}_{\mathrm{F},9}$ | FNO-based | 9 | 776 |  Sec. 4.2.2 | 0.153 | 1.381 | 0.077

###  4.1 Performance of the U-Net-based model

In the first step of our investigation, we examine the ability of U-Net-based
models $\mathcal{M}_{\mathrm{U},n_{\mathrm{s}}}$ to reconstruct wave surfaces
using single snapshot radar input data $\mathbf{x}$ for training. In the
second step, we utilize the same architecture to determine the best number of
historical snapshots $n_{\mathrm{s}}$ required in each radar input to achieve
the best reconstruction performance. To evaluate the effectiveness of the
trained models, we also visually compare the predicted wave elevations
$\mathbf{\hat{y}}$ of selected test samples with their corresponding true
elevations $\mathbf{y}$.

####  4.1.1 U-Net using single-snapshot radar data

Mapping of single snapshot radar data refers to mapping radar snapshots
$\mathbf{x}\in\mathbb{R}^{512\times 1}$ to wave snapshots
$\mathbf{y}\in\mathbb{R}^{512\times 1}$ recorded at the same time instant
$t_{\mathrm{s}}$. According to Table 1, the trained U-Net-based model
$\mathcal{M}_{\mathrm{U},1}$ achieves a performance given by a mean loss value
of $\mathrm{nL2}=0.329$ across all test set samples after 150 epochs of
training. Afterwards, the model tends to overfit the training data, as shown
in the loss curve in the appendix Figure 15(a). The observed error corresponds
to a mean value of $\mathrm{SSP}=0.171$ on test set, which fails to satisfy
the predefined 1 of reconstruction errors below $\mathrm{SSP\leq 0.1}$.

To identify the origin of reconstruction errors, we employed model
$\mathcal{M}_{\mathrm{U},1}$ to predict reconstructions $\mathbf{\hat{y}}$ of
individual samples from the test set. Despite the stratified data split
ensuring an equal distribution of sea state parameter combinations
$(L_{\mathrm{p}},\epsilon)$ in the training and test set, the errors are
unevenly distributed across individual samples, as illustrated in Figure 6.
The sample in Figure 6(a) corresponds to a peak wavelength
$L_{\mathrm{p}}=180\,\mathrm{m}$ and small amplitudes caused by a small
steepness of $\epsilon=0.01$. It exhibits a minor impact of the shadowing
modulation mechanism only affecting $9.4\%$ of the total radar-illuminated
surface $\mathbf{x}$ in the top panel. The corresponding predicted surface
reconstruction $\mathbf{\hat{y}}$ in the bottom panel closely approximates the
true wave elevation $\mathbf{y}$, as evidenced by the sample-specific error of
$\mathrm{nL2}=0.152$ or $\mathrm{SSP}=0.076$. In contrast, the sample in
Figure 6(b) with the same $L_{\mathrm{p}}=180\,\mathrm{m}$ but increased
$\epsilon=0.10$ shows $71.5\%$ of the spatial $r$-domain being affected from
shadowing modulation causing zero-valued intensities. This results in a high
reconstruction error of $\mathrm{nL2}=0.541$ or $\mathrm{SSP}=0.311$.
Particularly the shadowed areas seem to contribute to the poor reconstruction,
as their error is 2.96 times higher than in the visible areas, indicated by
$\frac{\mathrm{nL2}_{\mathrm{shad}}}{\mathrm{nL2}_{\mathrm{vis}}}$.

(a)

(b)

Figure 6: Two samples from the test set described by same wavelength
$L_{\mathrm{p}}=180\,\mathrm{m}$, but different wave steepness $\epsilon$,
reconstructed by the U-Net-based architecture $\mathcal{M}_{\mathrm{U},1}$.
(a) Small $\epsilon$ cause minor impact of the shadowing modulation in the
radar input and allow accurate predictions. (b) Larger $\epsilon$ create more
extensive shadowed areas and cause higher prediction errors.

####  4.1.2 U-Net using spatio-temporal radar data

To improve the reconstruction quality of the U-Net-based architecture,
especially for high wave steepness, we took inspiration from classical
spectral-analysis- and optimization-based approaches [cf. Borge et al., 2004,
Wu, 2004]. These approaches use spatio-temporal radar data by considering
temporal sequences of $n_{\mathrm{s}}$ historical radar snapshots for
reconstruction. Thus, we use multiple historical radar snapshots
$n_{\mathrm{s}}$ that satisfy 1 with $\Delta t_{\mathrm{r}}=1.3\,\mathrm{s}$
for each input sample $\mathbf{x}\in\mathbb{R}^{512\times n_{\mathrm{s}}}$,
while the output remains a single snapshot $\mathbf{y}\in\mathbb{R}^{512\times
1}$ at the respective last time instant $t_{\mathrm{s}}$. We conducted 14
additional training runs of the same architecture to determine the best number
of input snapshots $n_{\mathrm{s}}$. The boxplot in Figure 7 shows that the
model’s mean performance across the entire test set significantly improves up
to an value of $n_{\mathrm{s}}=10$, confirming 1 as the reconstruction quality
improves by incorporating multiple radar snapshots in the input. Moreover, the
individual samples’ error values become less scattered around the mean value.

The model $\mathcal{M}_{\mathrm{U},10}$ determined by the boxplot analysis
achieves a final mean performance of $\mathrm{nL2}=0.123$ or
$\mathrm{SSP}=0.061$ set, as shown in Table 1, now satisfying 1 of
$\mathrm{SSP}\leq 0.10$ and thus confirms 1. In addition, it yields a lower
ratio of
$\tfrac{\mathrm{nL2}_{\mathrm{shad}}}{\mathrm{nL2}_{\mathrm{vis}}}=1.755$
compared to $2.679$ for model $\mathcal{M}_{\mathrm{U},1}$, indicating a more
balanced reconstruction between shadowed and visible areas. Moreover, the
model does not exhibit early overfitting anymore, achieving the best
performance after 592 epochs, shown in Figure 15(b) in the appendix.

Figure 7: Boxplot depicting the error distribution on test set, depending on
the number of historical radar snapshots $n_{\mathrm{s}}$ provided to train
U-Net-based architectures $\mathcal{M}_{\mathrm{U},n_{\mathrm{s}}}$. The best
model performance is achieved for $n_{\mathrm{s}}=10$.

Figure 8 further confirms the improvement of the reconstruction by
$\mathcal{M}_{\mathrm{U},10}$ for the same two test set samples reconstructed
by $\mathcal{M}_{\mathrm{U},1}$ in Figure 6. The top panels display the most
recent radar snapshot $t_{\mathrm{s}}$ in the darkest shading and preceding
snapshots in increasingly lighter shades. The sample with small
$\epsilon=0.01$ in Figure 8(a) experiences only a slight reduction in
reconstruction error, while the sample with $\epsilon=0.10$ in Figure 8(b)
exhibits a substantial reduction around one-third of the previous nL2 or SSP
value. The improved performance seems mainly attributable to the enhanced
reconstruction of shadowed areas.

(a)

(b)

Figure 8: Two samples from the test set described by same wavelength
$L_{\mathrm{p}}=180\,\mathrm{m}$, but different wave steepness $\epsilon$,
reconstructed by the U-Net based architecture trained with $n_{\mathrm{s}}=10$
historical snapshots in the radar input $\mathcal{M}_{\mathrm{U},10}$.
Compared to $\mathcal{M}_{\mathrm{U},1}$, a strong reconstruction improvement
is observed, especially for the sample with high $\epsilon=0.10$ in (b).

###  4.2 Performance of the FNO-based model

The U-Net-based model $\mathcal{M}_{\mathrm{U},10}$ already supported 1 and 1
by demonstrating the potential to reconstruct wave surface elevations from
radar data in general and improving reconstruction quality by including
additional historical radar data. However, we also hypothesized that the FNO-
based architecture may outperform CNN-based methods, such as the U-Net, due to
its global inductive bias (1), which may be beneficial for the wave data
structure. To investigate this, we train FNO-based models
$\mathcal{M}_{\mathrm{F},{n_{\mathrm{s}}}}$ with $n_{\mathrm{s}}=1$ radar
snapshot in each input $\mathbf{x}$ and subsequently determine the number
$n_{\mathrm{s}}$ to achieve the best performance. We again visually compare
true and predicted wave elevations $\mathbf{y}$ and $\mathbf{\hat{y}}$.

####  4.2.1 FNO using single-snapshot radar data

The FNO-based model $\mathcal{M}_{\mathrm{F},1}$ trained with of
$n_{\mathrm{s}}=1$ snapshot in each input, attains its best performance
$\mathrm{nL2}=0.240$ after 721 training epochs, as shown Table 1 and also
demonstrated in the loss curve in Figure 16(a) in the appendix. Although the
corresponding $\mathrm{SSP}=0.123$ does not attain the 1, it still presents a
notable improvement compared to the $\mathrm{SSP}$ value of 0.171 previously
obtained by the U-Net-based model $\mathcal{M}_{\mathrm{U},1}$. Moreover,
$\mathcal{M}_{\mathrm{F},1}$ not only reduces the error across the entire
spatial $r$-domain in terms of nL2 or SSP, but also reconstructs the waves
more uniformly between shadowed and visible areas compared to
$\mathcal{M}_{\mathrm{U},1}$. This is evident by the decrease in the mean
$\tfrac{\mathrm{nL2}\mathrm{shad}}{\mathrm{nL2}\mathrm{vis}}$-ratio from
$2.679$ to $1.886$.

This improved wave reconstruction can be illustrated by comparing the
reconstructions of the same two test set samples generated by
$\mathcal{M}_{\mathrm{F},1}$ in Figure 9 to $\mathcal{M}_{\mathrm{U},1}$ in
Figure 6. As depicted in Figure 9(a), the overall nL2 or SSP metrics of this
sample are only slightly improved, but the ratio
$\tfrac{\mathrm{nL2}_{\mathrm{shad}}}{\mathrm{nL2}_{\mathrm{vis}}}$ achieved
by $\mathcal{M}_{\mathrm{F},1}$ is substantially smaller than observed for
$\mathcal{M}_{\mathrm{U},1}$. This observations are even more pronounced for
the sample with high $\epsilon$ in Figure 9(b). The
$\mathcal{M}_{\mathrm{F},1}$ reduces the error over the entire range in terms
of nL2 or SSP by almost half and also produces a more uniform reconstruction
between shadowed and visible areas.

(a)

(b)

Figure 9: Two samples from the test set described by same wavelength
$L_{\mathrm{p}}=180\,\mathrm{m}$, but different wave steepness $\epsilon$
reconstructed by the FNO-based architecture $\mathcal{M}_{\mathrm{F},1}$. The
$\mathcal{M}_{\mathrm{F},1}$ outperforms the $\mathcal{M}_{\mathrm{U},1}$ in
reconstructing the shadowed areas, especially noticeable for the sample with
large $\epsilon=0.10$ in (b).

####  4.2.2 FNO using spatio-temporal radar data

Although the FNO-based model $\mathcal{M}_{\mathrm{F},1}$ outperforms the
U-Net-based model $\mathcal{M}_{\mathrm{U},1}$, it does not achieve the
desired reconstruction quality of $\mathrm{SSP}\leq 0.10$ (1). To enhance the
model performance, we analyze the effect of including multiple historical
snapshots in the input $\mathbf{x}\in\mathbb{R}^{512\times n_{\mathrm{s}}}$.
Again, 14 additional training runs were conducted, each with an increasing
number of $n_{\mathrm{s}}$. The results, depicted in Figure 10, demonstrate an
initial improvement in performance for the models
$\mathcal{M}_{\mathrm{F},n_{\mathrm{s}}}$, which is slightly less notable than
that observed for the U-Net-based models
$\mathcal{M}_{\mathrm{U},n_{\mathrm{s}}}$ in Figure 7 before. The FNO-based
models achieve their best performance for $n_{\mathrm{s}}=9$ input snapshots,
beyond which the mean error slightly increases.

Figure 10: Boxplot depicting the error distribution on test set, depending on
the number of historical radar snapshots $n_{\mathrm{s}}$ provided to train
FNO-based architectures $\mathcal{M}_{\mathrm{F},n_{\mathrm{s}}}$. The best
model performance is achieved for $n_{\mathrm{s}}=9$.

According to Table 1, the model $\mathcal{M}_{\mathrm{F},9}$ attains a mean
performance of $\mathrm{nL2}=0.153$ on the test set, after 776 training
epochs, as depicted by the loss curve in the appendix Figure 16(b). This error
value corresponds to $\mathrm{SSP}=0.076$, fulfilling the 1 of a
$\mathrm{SSP\leq 0.10}$ . However, in comparison to the U-Net-based model
$\mathcal{M}_{\mathrm{U},10}$, which achieved a final mean value of
$\mathrm{SSP}=0.061$, the performance measured across the entire $r$-domain of
$\mathcal{M}_{\mathrm{F},9}$ is slightly inferior, even though in the single-
snapshot case $\mathcal{M}_{\mathrm{F},1}$ outperformed
$\mathcal{M}_{\mathrm{U},1}$. Nevertheless, compared to all investigated
models, $\mathcal{M}_{\mathrm{F},1}$ on average achieves the best best
reconstruction uniformity between shadowed and visible areas indicated by a
mean $\tfrac{\mathrm{nL2_{\mathrm{shad}}}}{\mathrm{nL2}_{\mathrm{vis}}}=1.381$
on test set.

Figure 11 shows the reconstruction of the same two samples from the test set
using $\mathcal{M}_{\mathrm{F},9}$. Compared to $\mathcal{M}_{\mathrm{F},1}$
in Figure 9 both samples experience an almost similar increase in
reconstruction quality measured in terms of the individual SSP and nL2 errors.
In addition, this values are is comparable to that achieved by
$\mathcal{M}_{\mathrm{U},10}$ in Figure 8. However, for the sample with small
$\epsilon=0.01$ in Figure 11(a), $\mathcal{M}_{\mathrm{F},9}$ generates a more
balanced reconstruction than $\mathcal{M}_{\mathrm{U},10}$, as reflected by
the reduction of
$\tfrac{\mathrm{nL2_{\mathrm{shad}}}}{\mathrm{nL2}_{\mathrm{vis}}}$ from 2.201
to 1.665 for this individual sample. For the higher-steepness sample in Figure
11(b), the increase of reconstruction uniformity is less significant but still
present.

(a)

(b)

Figure 11: Two samples from the test set described by same wavelength
$L_{\mathrm{p}}=180\,\mathrm{m}$, but different wave steepness $\epsilon$,
reconstructed by the FNO-based architecture trained with $n_{\mathrm{s}}=9$
historical snapshots in the radar input $\mathcal{M}_{\mathrm{F},9}$. Compared
to $\mathcal{M}_{\mathrm{F},1}$ a reconstruction improvement is visible for
both samples. Moreover, for this two samples the reconstruction quality on the
entire $r$-domain is almost equivalent to the results of
$\mathcal{M}_{\mathrm{U},10}$, but especially for the small steepness sample
in (a) the error ratio between shadowed and visible areas is remarkably
smaller using $\mathcal{M}_{\mathrm{F},9}$, which indicates the potential of a
more uniform reconstruction.

###  4.3 Comparative discussion

The aforementioned visual observations have been limited to the examination of
only two samples from the test set, both described by peak wavelength
$L_{\mathrm{p}}=180\,\mathrm{m}$ and either steepness $\epsilon=0.01$ or
$\epsilon=0.10$. To avoid any possible incidental observations, the
generalization of the error values needs to be examined. This can be achieved
by plotting individual error values against their combination of peak
wavelength $L_{\mathrm{p}}$ and steepness $\epsilon$ for all 624 test set
samples reconstructed using the U-Net based models
$\mathcal{M}_{\mathrm{U},1}$ and $\mathcal{M}_{\mathrm{U},10}$ or FNO-based
models $\mathcal{M}_{\mathrm{F},1}$ and $\mathcal{M}_{\mathrm{F},9}$.

####  4.3.1 Discussion of overall reconstruction quality

Figure 12 illustrates the reconstruction error as the mean nL2 value across
4-5 test samples available for each specific
$L_{\mathrm{p}}$-$\epsilon$-combination. Additionally, red dots in the cell
centers indicate the combinations that achieved a mean $\mathrm{SSP}\leq 0.10$
(1).

(a)

(b)

(c)

(d)

Figure 12: Error surfaces generalizing the previous observations for the four
investigated models $\mathcal{M}$ depending on the
$L_{\mathrm{p}}$-$\epsilon$-combination of the samples from the test set. Red
dots indicate parameter combinations that meet the 1 of reconstruction errors
$\mathrm{SSP}\leq 0.10$. The upper subplots illustrate the result of (a) the
U-Net based model and (b) the FNO-based model, both trained with only one
radar snapshot ($n_{\mathrm{s}}=1$) in each input. The same architectures were
trained with multiple historic radar snapshots in each input, as demonstrated
in the lower subplots. Specifically, (c) shows the U-Net-based model trained
with $n_{\mathrm{s}}=10$ and (d) the FNO-based model trained with
$n_{\mathrm{s}}=9$.

Figure 12(a) confirms the findings presented in Section 4.1.1 for the U-Net-
based model $\mathcal{M}_{\mathrm{U},1}$ trained with one radar snapshot in
each input ($n_{\mathrm{s}}=1$). The reconstruction errors increase with
increasing steepness $\epsilon$ and thus with increasing wave height. Moreover
we now observe that this effect occurs almost independent of the peak
wavelength $L_{\mathrm{p}}$. For samples described by $\epsilon>0.02$,
$\mathcal{M}_{\mathrm{U},1}$ fails to meet the 1, which is attributable to the
geometrical radar imaging problem demonstrated in Figure 2, showing that the
increase in wave height, caused by increased $\epsilon$, results in more and
larger shadowed areas. Figure 13 demonstrates that the occurrence of shadowing
mainly increases with increasing $\epsilon$ and is less influenced by
$L_{\mathrm{p}}$. While $\epsilon=0.01$ on average only causes around $10\%$,
$\epsilon=0.10$ instead causes approximately $70-75\%$ of each input
$\mathbf{x}$ being affected by shadowing modulation. This results in zero-
valued intensities that complicate the radar inversion task. Understanding the
challenges faced by model $\mathcal{M}_{\mathrm{U},1}$ in reconstructing
shadowed areas, requires revisiting the U-Net’s local mode of operation,
outlined in Section 3.1, and the exemplary radar input depicted in the upper
panel of Figure 6(b). Due to shadowing, numerous local areas exhibit zero-
intensities covering up to approximately $200\,\mathrm{m}$, especially for
greater distances from the radar system. However, the kernels in the first
convolutional layer with a kernel size of $s_{\mathrm{k}}=5$ only cover a
domain of $s_{\mathrm{k}}\cdot\Delta r=17.5\,\mathrm{m}$ while being shifted
across the input feature map in a step-wise manner. While the U-Net’s
translational equivariance property is useful for translating radar
intensities to wave surface elevation regardless of their spatial location, it
thus also causes kernels to be shifted across large areas with zero-input
only, which cannot be processed in a meaningful way. Although the pooling
layers subsequently reduce the dimension of feature maps, resulting in an
increased ratio of kernel size to feature size, the problem of radar inversion
can be assumed to be based on the mapping of individual pixel values, known as
low-level features. These features are learned in the early layers of a CNN-
based network [Zeiler and Fergus, 2014]. Accordingly, the initial stages of
the U-Net-based architecture are more important for our task than for its
original purpose of image segmentation [Ronneberger et al., 2015] that is
based on mid- to high-level features extracted in the later layers. We
therefore face problems applying $\mathcal{M}_{\mathrm{U},1}$ for
reconstruction, as important kernels in the early layers receive a significant
amount of sparse, not valuable content. Although increasing the kernel size
$s_{\mathrm{k}}$ is a theoretically possible solution, doing so would
compromise the U-Net’s local key property. Moreover, when processing 2D
surfaces with 2D convolutional kernels in future research, it would result in
a quadratic increase in the number of weights, leading to computational
issues.

Figure 13: Graphs visualizing the average proportion of each input
$\mathbf{x}$ being affected by shadowing modulation in dependency of the
samples wave steepness values $\epsilon=0.01-0.10$ for the shortest, one
medium and the longest peak wavelength $L_{\mathrm{p}}$ occurring in the test
set.

For this reason, the approach of providing $n_{\mathrm{s}}=10$ consecutive
radar snapshots governed according to 1 for the training of U-Net based model
$\mathcal{M}_{\mathrm{U},10}$ in Section 4.1.2, more effectively accounts for
the sparsity in the input data. The upper panel of Figure 8(b) demonstrated
the presence of input information across the majority of the $r$-domain. The
wave surfaces undergo shape variations while traveling towards the radar due
to differing phase velocities of their components caused by dispersion. This
results in a different part of the radar surface being shadowed or visible at
each time step and seem to allows to capture more information about the wave
on average, as the reconstruction quality significantly improves compared to
$\mathcal{M}_{\mathrm{U},1}$. Therefore we infer that the spatial and temporal
shifts of the additional radar intensities acquired at
$t_{i}=\sum_{i=0}^{n_{\mathrm{s}}-1}t_{\mathrm{s}}-i\Delta t_{\mathrm{r}}$ can
be compensated successfully. This may be attributed to the fact that each
kernel applied to the input has its own channel for each snapshot, allowing
for separate processing to counterbalance the shift first, followed by the
addition of results to one feature map utilized as part of the input for the
next layer. The improved reconstruction observed for
$\mathcal{M}_{\mathrm{U},10}$ is further supported by its performance
generalization shown in Figure 12(c). Compared to Figure 12(a), the mean nL2
error is substantially smaller and reconstruction errors are more evenly
distributed across the $L_{\mathrm{p}}$-$\epsilon$-space, resulting in a
satisfactory SSP value (1) for almost all samples. Although there is still a
slight increase in the error for cells with higher $L_{\mathrm{p}}$ and
$\epsilon$, the proposed model $\mathcal{M}_{\mathrm{U},10}$ can accurately
reconstruct samples with varying wave characteristics and degrees of
shadowing, thus supporting 1 and 1 of this work.

Motivated by the inherent patterns in wave data and the successful application
of the Fourier neural operator (FNO) to systems exhibiting certain periodic
properties, we conducted a comparative analysis of the global inductive bias
of this network architecture with the local inductive bias of the CNN-based
U-Net. As discussed in Section 4.2.1, our observations indicate that the FNO-
based model $\mathcal{M}_{\mathrm{F},1}$, trained with only one snapshot
($n_{\mathrm{s}}=1$) outperforms the U-Net-based $\mathcal{M}_{\mathrm{U},1}$
in reconstructing shadowed areas in the input, as evidenced, i.e., by
comparing the reconstruction in Figure 9(b) to 6(b). This observation is
generalizable to the entire test data set, as shown in Figure 12(b). Although
errors in the FNO error surface still increase with higher steepness
$\epsilon$ and, consequently, with an increase in the percentage of shadowing
according to Figure 13, the increase is much less severe than that obtained by
$\mathcal{M}_{\mathrm{U},1}$ shown in Figure 12(a).

The improved ability of the FNO-based model $\mathcal{M}_{\mathrm{F},1}$ in
reconstructing shadowed areas from a single-snapshot input can be attributed
to its mode of operation outlined in Section 3.2. Although the latent
representation $v_{0}$ in Figure 5 is usually not explicitly known, we can
infer that the layer $P$ with $n_{\mathrm{s}}=1$ input nodes and
$n_{\mathrm{w}}$ output nodes only performs linear transformations to each
radar input $\mathrm{x}$. As the radar inputs exhibit kinks at the transitions
from visible to shadowed areas, consequently $v_{0}$ will have similar
characteristics along the range direction. These transitions result in peaks
for specific wavenumbers $k$ in the spectrum $F(k)$. However, the desired wave
outputs $\mathbf{y}$ of the training data posses smooth periodic properties,
without peaks at the kink-related wavenumbers in $F_{\mathbf{y}}(k)$. Since
the $R_{i}$ matrices in the Fourier layers scale the radar inputs spectrum to
the wave outputs spectrum, they learn small coefficients for the corresponding
entries to reduce the peaks. Therefore, the FNO’s global inductive bias,
combined with the data structure of wave surfaces, can efficiently correct
sparse, shadowed regions in spectral space, resolving the issue of
insufficient local information for reconstruction that arises with the U-Net-
based model $\mathcal{M}_{\mathrm{U},1}$ in Euclidean space. Thus it can be
also stated that the FNO explicitly hard-encodes prior knowledge about
physical wave properties through its network structure and thus can be assumed
to be a physics-guided design of architecture [cf. Willard et al., 2022, Wang
and Yu, 2023] for our problem.

Despite the better performance of the FNO-based model
$\mathcal{M}_{\mathrm{F},1}$ compared to $\mathcal{M}_{\mathrm{U},1}$ in
reconstructing shadowed radar inputs that already supports 1, Figure 12(b)
still reveals that most of the test set samples fail to meet the 1 of
$\mathrm{SSP}\leq 0.10$. However, this issue was resolved by training a FNO-
based model $\mathcal{M}_{\mathrm{F},9}$ with $n_{\mathrm{s}}=9$ historical
radar snapshots in each input. This was demonstrated for the two test set
examples in Figure 11 and is generalized in Figure 12(d). We observe from that
Figure, that the slightly higher mean error across the entire test set of
$\mathcal{M}_{\mathrm{F},9}$ compared to $\mathcal{M}_{\mathrm{U},10}$, is
primarily caused by samples with low steepness $\epsilon$ or short wavelengths
$L_{\mathrm{p}}$. It is worth noting, that the observed minimal increase in
errors for short wavelengths cannot be attributed to a truncation at an
insufficient number of Fourier series modes $n_{\mathrm{m}}$ in the Fourier
layers. In this work, $n_{\mathrm{m}}$ is determined as 64 and the spectral
representation is discretized by $\Delta k=\tfrac{2\pi}{n_{r}\cdot\Delta
r}=0.00351\,\mathrm{m}^{-1}$. The highest peak wavenumber of
$k_{\mathrm{p}}=0.0785\,\mathrm{m}^{-1}$ in our data set is reached for
samples with $L_{\mathrm{p}}=80\,\mathrm{m}$. The spectral density around
$k_{\mathrm{p}}$ has decayed almost completely at
$k_{\mathrm{filt}}=n_{\mathrm{m}}\cdot\Delta k=0.2264\,\mathrm{m}^{-1}$, such
that no important wave components are filtered out, as is visualized in the
Figure 17 in the appendix. Therefore, the small unequal tendency in error
distribution achieved by $\mathcal{M}_{\mathrm{F},9}$ in Figure 12(d) for
samples described by different $L_{\mathrm{p}}$-$\epsilon$, is likely caused
by other factors than by an unsuitable network hyperparameter
$n_{\mathrm{m}}$. Moreover, we observed in the loss curve shown in Figure
16(b) in the appendix that further training for more than 800 epochs could
potentially improve the model’s performance, whereas the best performance on
the test set for $\mathcal{M}_{\mathrm{U},10}$ seems to have been already
reached, as the model begins to overfit the training, as depicted in Figure
15(a).

####  4.3.2 Discussion of reconstruction uniformity

So far the generalization of the reconstruction quality has been evaluated
based on nL2 or SSP values across the entire $r$-domain only. However, Table 1
indicates that the FNO-based model $\mathcal{M}_{\mathrm{F},9}$ achieves a
more uniform reconstruction between shadowed and visible areas. This is
demonstrated by the mean ratio of
$\tfrac{\mathrm{nL2}_{\mathrm{shad}}}{\mathrm{nL2}_{\mathrm{vis}}}=1.381$
across all samples in the test set, while the U-Net-based model
$\mathcal{M}_{\mathrm{U},10}$ still struggles with reconstructing shadowed
areas as inferred by its
$\tfrac{\mathrm{nL2}_{\mathrm{shad}}}{\mathrm{nL2}_{\mathrm{vis}}}=1.755$. For
the $\tfrac{\mathrm{nL2}_{\mathrm{shad}}}{\mathrm{nL2}_{\mathrm{vis}}}$-ratio,
the error distribution is displayed in Figure 14 for each test set sample,
based on their $L_{\mathrm{p}}$-$\epsilon$-combination. The model
$\mathcal{M}_{\mathrm{U},10}$ generates an error surface shown in Figure 14(a)
that exhibits broadly varying levels of uniformity in the reconstruction, even
for neighboring $L_{\mathrm{p}}$-$\epsilon$-combinations. In some cases, the
reconstruction errors in shadowed areas exceeds those in visible areas by more
than 2.5 times. This undesired effect is much less pronounced for the FNO-
based model $\mathcal{M}_{\mathrm{F},9}$, as a comparison with Figure 14(b)
reveals.

(a)

(b)

Figure 14: Error surfaces depicting the ratio
$\tfrac{\mathrm{nL2}_{\mathrm{shad}}}{\mathrm{nL2}_{\mathrm{vis}}}$ between
the reconstruction quality achieved on shadowed and visible areas depending on
the specific $L_{\mathrm{p}}$-$\epsilon$-combination of the samples from the
test set. The individual cell entries display the mean ratio across the 4-5
samples available for each specific parameter combination. The uniformity of
the reconstructions achieved by the U-Net-based model
$\mathcal{M}_{\mathrm{U},10}$ in (a) thus is compared to the one achieved by
the FNO-based model $\mathcal{M}_{\mathrm{F},9}$ in (b).

####  4.3.3 Final comparison

For a final evaluation, either the general reconstruction quality nL2 on the
entire $r$-domain can be chosen as the main performance criterion, which, in
our case, would argue for the selection of the U-Net-based model
$\mathcal{M}_{\mathrm{U},10}$, or instead the uniformity of the reconstruction
indicated by
$\tfrac{\mathrm{nL2}_{\mathrm{shad}}}{\mathrm{nL2}_{\mathrm{vis}}}$, which
would argue for the FNO-based model $\mathcal{M}_{\mathrm{F},9}$. This
decision should be made based on the application case. If the ML-reconstructed
wave surface is intended to be used as an initial condition for subsequent
forecasting with the HOS method, we would expect a more uniform reconstruction
to represent a more physical result, and consequently the FNO-based
reconstruction to be less likely affect the entire prediction in a negative
way. Moreover, we observed that the FNO-based models global approach would
allow for a reasonably more meaningful reconstruction of shadowed areas even
with fewer historical radar snapshots $n_{\mathrm{s}}$. This is not
necessarily the case using the U-Net-based models.

Besides, the trained FNO-based model $\mathcal{M}_{\mathrm{F},9}$ in this work
allows for much faster inference speed than the U-Net-based
$\mathcal{M}_{\mathrm{U},10}$, even though $\mathcal{M}_{\mathrm{F},9}$ is
constructed as a custom implementation and contains more weights compared to
$\mathcal{M}_{\mathrm{U},10}$, which uses standard layers from the PyTorch
library that are in addition probably optimized. More specifically, using the
hardware specifications outlined in Section 3.3, our
$\mathcal{M}_{\mathrm{F},9}$ is able to generate predictions for a new input
sample in an average time of $1.9\cdot 10^{-3}\,\mathrm{s}$, which is
approximately 20 times faster than the average time required by
$\mathcal{M}_{\mathrm{U},10}$, which is $3.7\cdot 10^{-4}\,\mathrm{s}$ for the
same task.

##  5 Conclusion

This work introduces a novel machine learning-based approach for the phase-
resolved reconstruction of ocean wave surface elevations from sparse radar
measurements. To evaluate the performance of our approach, we generate
synthetic nonlinear wave surface data for a wide range of sea states and
corresponding radar surface data by incorporating both tilt- and shadowing
modulation mechanisms. Two neural network architectures, based on the U-Net
and the Fourier neural operator, are trained, both provided with varying
amounts of spatio-temporal radar surface measurement input.

Our results and discussion indicate that both models are capable of producing
high-quality wave surface reconstructions with average errors below
$\mathrm{SSP}\leq 0.10$ when trained with a sufficient amount of
$n_{\mathrm{s}}=10$ or $9$ consecutive radar snapshots. Furthermore, both
models generalize well across different sea states. On average, the U-Net-
based model achieves slightly smaller errors across the entire spatial domain
of each reconstructed wave sample, while the FNO-based model produces a more
uniform wave reconstruction between areas that were shadowed and visible in
the corresponding radar input. This observation is further confirmed by the
edge case of instantaneous inversion, i.e., if the networks are trained with
only a single radar snapshot in each input. The weakness in the reconstruction
of shadowing-affected areas of the U-Net-based model can be attributed to the
local operation of the network architecture, where its small convolutional
kernels do not receive processable information when shifted across shadowed
input areas with zero-intensities only. The problem can be circumvented by
using the FNO-based network, which learns a global mapping between radar input
and wave output in the Fourier space. Its network structure already encodes
prior physical knowledge about the periodic data structure apparent in ocean
waves and is therefore possibly better suited for our use case.

Our findings suggest that the FNO-based network may provide additional
advantages for example concerning smaller training data sets, noisy input
radar data, or the reconstruction of two- dimensional ocean wave surfaces,
which require the additional reconstruction of wave direction components.
Future investigations may explore these potentials.

##  Appendix A Influence of neural network hyperparameters

To mitigate the high cost of obtaining a larger data set, a four-fold cross-
validation approach with an independent test set was utilized for finding the
network hyperparameters, as recommended for example by Raschka [2018]. The
data set of $N=3120$ samples was divided into a fixed and independent test set
comprising $20\%$ or $N_{\mathrm{test}}=624$ samples, with the remaining 2496
samples partitioned into four equal-sized parts based on the governing sea
state parameters $(L_{\mathrm{p}},\epsilon)$ using a stratified data split
technique to ensure equal representation of each wave characteristic in the
resulting subsets. During each cross-validation step, one part with
$N_{\mathrm{val}}=624$ samples was used as the validation set, and the
remaining three parts with $N_{\mathrm{train}}=1872$ samples constituted the
training set.

Tables 2 and 3 present the results of the four-fold cross-validation
hyperparameter studies for the U-Net- and FNO-based architectures. For both
network types, the same fixed test set was excluded from this investigation.
The metrics
($\mathrm{nL2},\,\frac{\mathrm{nL2}_{\mathrm{shad}}}{\mathrm{nL2}_{\mathrm{vis}}},\,\mathrm{SSP}$)
and the number of epochs necessary to attain the best performance represent
average values across all four folds.

Table 2: Results of the hyperparameter study for the U-Net-based architecture. Each investigated architecture is characterized by a depth $n_{\mathrm{d}}$, a kernel size $s_{\mathrm{k}}$ and an approach for the number of convolutional kernels $n_{\mathrm{k}}$ in each layer. The approaches for $n_{\mathrm{k}}$ explored doubling the number of kernels per layer with increasing encoder depth (and in reverse halving in the decoder) and another keeping $n_{\mathrm{k}}$ the same in all convolutional layers. The performance, measured by the mean nL2 value on validation set displays only slight variations among architectures, indicating that the U-Net is not highly sensitive to these changes. Nevertheless, we select the best model according to the validation nL2, that highlighted in blue, as our $\mathcal{M}_{\mathrm{U},1}$. U-Net hyperparameters |  | $\mathrm{nL2}$ | $\frac{\mathrm{nL2}_{\mathrm{shad}}}{\mathrm{nL2}_{\mathrm{vis}}}$ | SSP
---|---|---|---|---
depth $n_{\mathrm{d}}$ |  kernels each layer $n_{\mathrm{k}}$ |  k. size $s_{\mathrm{k}}$ | #weights | epochs | train | val | train | val | train | val
3 | $[8,16,32]$ | 3 | 12,401 | 785 | 0.392 | 0.395 | 2.124 | 2.119 | 0.208 | 0.210
5 | 18,305 | 791 | 0.360 | 0.367 | 2.512 | 2.526 | 0.189 | 0.193
4 | $[8,16,32,64]$ | 3 | 48,433 | 645 | 0.352 | 0.371 | 2.475 | 2.527 | 0.185 | 0.195
5 | 72,769 | 189 | 0.347 | 0.361 | 2.536 | 2.575 | 0.182 | 0.189
5 | $[8,16,32,64,128]$ | 3 | 192,177 | 278 | 0.342 | 0.360 | 2.601 | 2.644 | 0.179 | 0.189
5 | 290,241 | 154 | 0.303 | 0.344 | 2.276 | 2.438 | 0.158 | 0.179
6 | $[8,16,32,64,128,256]$ | 3 | 766,385 | 223 | 0.336 | 0.356 | 2.557 | 2.601 | 0.176 | 0.186
5 | 1,159,361 | 78 | 0.326 | 0.356 | 2.365 | 2.452 | 0.170 | 0.186
3 | $[16,32,64]$ | 3 | 49,121 | 779 | 0.388 | 0.392 | 2.149 | 2.144 | 0.206 | 0.208
5 | 72,705 | 675 | 0.353 | 0.366 | 2.543 | 2.582 | 0.185 | 0.193
4 | $[16,32,64,128]$ | 3 | 192,865 | 310 | 0.356 | 0.371 | 2.439 | 2.479 | 0.187 | 0.195
5 | 290,177 | 99 | 0.349 | 0.364 | 2.520 | 2.551 | 0.183 | 0.191
5 | $[16,32,64,128,256]$ | 3 | 767,073 | 140 | 0.343 | 0.365 | 2.490 | 2.532 | 0.181 | 0.192
5 | 1,159,297 | 76 | 0.298 | 0.342 | 2.364 | 2.476 | 0.155 | 0.178
6 | $[16,32,64,128,256,512]$ | 3 | 3,062,369 | 225 | 0.338 | 0.358 | 2.568 | 2.611 | 0.177 | 0.188
5 | 4,634,241 | 124 | 0.346 | 0.366 | 2.419 | 2.487 | 0.183 | 0.194
3 | $[32,32,32]$ | 3 | 38465 | 778 | 0.388 | 0.392 | 2.135 | 2.137 | 0.206 | 0.208
5 | 52865 | 724 | 0.351 | 0.363 | 2.581 | 2.611 | 0.185 | 0.191
4 | $[32,32,32,32]$ | 3 | 49,825 | 569 | 0.350 | 0.364 | 2.555 | 2.583 | 0.184 | 0.192
5 | 70,369 | 199 | 0.345 | 0.358 | 2.631 | 2.663 | 0.181 | 0.188
5 | $[32,32,32,32,32]$ | 3 | 61,185 | 224 | 0.344 | 0.357 | 2.654 | 2.663 | 0.180 | 0.187
5 | 87,873 | 183 | 0.310 | 0.341 | 2.536 | 2.659 | 0.161 | 0.177
6 | $[32,32,32,32,32,32]$ | 3 | 72,545 | 395 | 0.350 | 0.362 | 2.548 | 2.562 | 0.184 | 0.191
5 | 105,377 | 217 | 0.333 | 0.350 | 2.693 | 2.741 | 0.174 | 0.183
3 | $[64,64,64]$ | 3 | 152,705 | 739 | 0.383 | 0.389 | 2.186 | 2.189 | 0.203 | 0.206
5 | 210,177 | 394 | 0.351 | 0.365 | 2.539 | 2.568 | 0.185 | 0.192
4 | $[64,64,64,64]$ | 3 | 197,953 | 333 | 0.351 | 0.367 | 2.482 | 2.524 | 0.184 | 0.193
5 | 280,001 | 120 | 0.343 | 0.359 | 2.611 | 2.646 | 0.179 | 0.187
5 | $[64,64,64,64,64]$ | 3 | 243,201 | 191 | 0.342 | 0.356 | 2.687 | 2.694 | 0.179 | 0.186
5 | 349,825 | 182 | 0.316 | 0.348 | 2.482 | 2.603 | 0.165 | 0.182
6 | $[64,64,64,64,64,64]$ | 3 | 288,449 | 255 | 0.345 | 0.362 | 2.532 | 2.568 | 0.181 | 0.190
5 | 419,649 | 138 | 0.323 | 0.349 | 2.598 | 2.684 | 0.168 | 0.182
Table 3: Results of the hyperparameter study for the FNO-based architecture. Each investigated architecture is characterized by a number of Fourier layers $n_{\mathrm{f}}$, a width of the latent representation $n_{\mathrm{w}}$ and a number of modes $n_{\mathrm{m}}$ for truncation of the layers’ Fourier series. Our observations reveal that the performance on the validation set does not show relevant improvement for $n_{\mathrm{w}}>32$, and that $n_{\mathrm{f}}=3$ lead to slightly better performance than the larger number of layers. The performance, measured by the mean nL2 value on validation set, displays a little more distinct variation among architectures than the U-Net before. Nevertheless, neighbouring parameter combination of the FNO result in almost the same performance, suggesting that the FNO, is also not overly sensitive to hyperparameter changes. We select the best model according to the smallest mean validation sets nL2 highlighted in blue as our $\mathcal{M}_{\mathrm{F},1}$. FNO hyperparameters |  | $\mathrm{nL2}$ | $\frac{\mathrm{nL2}_{\mathrm{shad}}}{\mathrm{nL2}_{\mathrm{vis}}}$ | SSP
---|---|---|---|---
layers $n_{\mathrm{f}}$ |  modes $n_{\mathrm{m}}$ |  width $n_{\mathrm{w}}$ | #weights | epochs | train | val | train | val | train | val
3 | 32 | 16 | 52,321 | 793 | 0.316 | 0.334 | 1.380 | 1.437 | 0.163 | 0.173
32 | 204,225 | 790 | 0.262 | 0.313 | 1.338 | 1.520 | 0.134 | 0.160
48 | 455,969 | 540 | 0.216 | 0.296 | 1.371 | 1.685 | 0.110 | 0.150
64 | 807,553 | 500 | 0.214 | 0.296 | 1.362 | 1.681 | 0.109 | 0.151
3 | 40 | 16 | 64,609 | 791 | 0.284 | 0.306 | 1.469 | 1.547 | 0.145 | 0.157
32 | 253,377 | 703 | 0.229 | 0.290 | 1.402 | 1.654 | 0.116 | 0.148
48 | 566,561 | 480 | 0.212 | 0.291 | 1.380 | 1.701 | 0.108 | 0.148
64 | 1,004,161 | 395 | 0.207 | 0.288 | 1.390 | 1.719 | 0.105 | 0.146
3 | 48 | 16 | 76,897 | 788 | 0.272 | 0.296 | 1.481 | 1.573 | 0.139 | 0.152
32 | 302,529 | 605 | 0.219 | 0.279 | 1.419 | 1.675 | 0.111 | 0.142
48 | 677,153 | 342 | 0.216 | 0.285 | 1.373 | 1.661 | 0.110 | 0.145
64 | 1,200,769 | 258 | 0.214 | 0.286 | 1.360 | 1.642 | 0.109 | 0.146
3 | 56 | 16 | 89,185 | 787 | 0.243 | 0.270 | 1.602 | 1.704 | 0.124 | 0.137
32 | 351,681 | 714 | 0.202 | 0.265 | 1.499 | 1.817 | 0.102 | 0.135
48 | 787,745 | 397 | 0.208 | 0.265 | 1.511 | 1.767 | 0.105 | 0.134
64 | 1,397,377 | 307 | 0.204 | 0.267 | 1.475 | 1.753 | 0.104 | 0.136
3 | 64 | 16 | 101,473 | 798 | 0.237 | 0.265 | 1.644 | 1.767 | 0.102 | 0.135
32 | 400,833 | 534 | 0.199 | 0.256 | 1.561 | 1.837 | 0.101 | 0.130
48 | 898,337 | 349 | 0.199 | 0.257 | 1.560 | 1.844 | 0.101 | 0.130
64 | 1,593,985 | 276 | 0.190 | 0.261 | 1.501 | 1.822 | 0.096 | 0.133
3 | 72 | 16 | 113,761 | 742 | 0.234 | 0.267 | 1.588 | 1.750 | 0.119 | 0.136
32 | 449,985 | 600 | 0.197 | 0.257 | 1.575 | 1.887 | 0.100 | 0.131
48 | 1,008,929 | 367 | 0.189 | 0.258 | 1.568 | 1.889 | 0.096 | 0.132
64 | 1,790,593 | 244 | 0.187 | 0.258 | 1.523 | 1.851 | 0.095 | 0.131
4 | 32 | 16 | 68,977 | 789 | 0.281 | 0.316 | 1.386 | 1.520 | 0.144 | 0.162
32 | 270,817 | 520 | 0.220 | 0.296 | 1.355 | 1.668 | 0.112 | 0.151
48 | 605,777 | 394 | 0.190 | 0.289 | 1.358 | 1.760 | 0.097 | 0.147
64 | 1,073,857 | 233 | 0.185 | 0.292 | 1.303 | 1.711 | 0.094 | 0.148
4 | 40 | 16 | 85,361 | 787 | 0.250 | 0.291 | 1.469 | 1.658 | 0.128 | 0.148
32 | 336,353 | 608 | 0.193 | 0.281 | 1.397 | 1.817 | 0.098 | 0.143
48 | 753,233 | 331 | 0.188 | 0.283 | 1.360 | 1.771 | 0.095 | 0.144
64 | 1,336,001 | 167 | 0.206 | 0.293 | 1.314 | 1.638 | 0.105 | 0.149
4 | 48 | 16 | 101,745 | 720 | 0.245 | 0.289 | 1.408 | 1.612 | 0.125 | 0.148
32 | 401,889 | 327 | 0.212 | 0.282 | 1.365 | 1.662 | 0.103 | 0.143
48 | 900,689 | 185 | 0.205 | 0.282 | 1.355 | 1.676 | 0.104 | 0.144
64 | 1,598,145 | 431 | 0.144 | 0.282 | 1.324 | 1.785 | 0.073 | 0.142
4 | 56 | 16 | 118,129 | 703 | 0.216 | 0.265 | 1.533 | 1.767 | 0.110 | 0.135
32 | 467,425 | 518 | 0.172 | 0.269 | 1.420 | 1.873 | 0.087 | 0.136
48 | 1,048,145 | 285 | 0.171 | 0.269 | 1.406 | 1.834 | 0.087 | 0.137
64 | 1,860,289 | 156 | 0.177 | 0.269 | 1.385 | 1.778 | 0.089 | 0.136
4 | 64 | 16 | 134,513 | 784 | 0.204 | 0.258 | 1.555 | 1.843 | 0.103 | 0.131
32 | 532,961 | 270 | 0.186 | 0.258 | 1.497 | 1.846 | 0.095 | 0.131
48 | 1,195,601 | 149 | 0.187 | 0.260 | 1.476 | 1.831 | 0.094 | 0.132
64 | 2,122,433 | 114 | 0.176 | 0.259 | 1.450 | 1.826 | 0.089 | 0.131
4 | 72 | 16 | 150,897 | 740 | 0.189 | 0.256 | 1.528 | 1.903 | 0.095 | 0.130
32 | 598,497 | 262 | 0.180 | 0.257 | 1.496 | 1.903 | 0.091 | 0.130
48 | 1,343,057 | 181 | 0.168 | 0.257 | 1.454 | 1.879 | 0.085 | 0.130
64 | 2,384,577 | 121 | 0.169 | 0.259 | 1.406 | 1.817 | 0.086 | 0.132

##  Appendix B Loss curves

After determining appropriate hyperparameters for the U-Net-based and FNO-
based models in A, the train and validation data from the four-fold cross-
validation were merged. This combined data set was then used to train the
models $\mathcal{M}_{\mathrm{U},n_{\mathrm{s}}}$ and
$\mathcal{M}_{\mathrm{F},n_{\mathrm{s}}}$, with one radar snapshot in each
samples input ($n_{\mathrm{s}}=1$) or either $n_{\mathrm{s}}=9$ or
$n_{\mathrm{s}}=10$ historical radar snapshots in each input. The performance
evaluation of these models was conducted on the previously excluded test set
of $N_{\mathrm{test}}=624$ samples. The loss curves depicted in Figures
15(a)-16(b) illustrate the model performance and the impact of different
values for $n_{\mathrm{s}}$ throughout the training epochs. Deviation between
the train and test loss curves indicates overfitting, characterized by
excessive adaptation to the training data, resulting in poor generalization to
new samples. Consequently, the best models $\mathcal{M}$ were selected based
on the lowest test loss within the 800 training epochs.

(a)

(b)

Figure 15: Loss curves for training of the U-Net-based model. Subfigure (a)
depicts the loss of model $\mathcal{M}_{\mathrm{U},1}$ trained with one one
snapshot $n_{\mathrm{s}}=1$ in the radar input, where the best performance
$\mathrm{nL2}=0.329$ on test set for model evaluation is reached after 150
epochs. Afterwards the model would tend to overfit the training data.
Subfigure (b) depicts model $\mathcal{M}_{\mathrm{U},10}$ trained with
$n_{\mathrm{s}}=10$ instead, which strongly increases performance, resulting
in $\mathrm{nL2}=0.123$ after 592 epochs of training.

(a)

(b)

Figure 16: Loss curves for training of the FNO-based model. Subfigure (a)
depicts the loss of model $\mathcal{M}_{\mathrm{F},1}$ trained with one one
snapshot $n_{\mathrm{s}}=1$ in the radar input, where the best performance
$\mathrm{nL2}=0.242$ on test set for model evaluation is reached after 721
epochs. Compared to the U-Net based model $\mathcal{M}_{\mathrm{U},1}$,
$\mathcal{M}_{\mathrm{F},1}$ does not seem to be susceptible to overfitting .
Subfigure (b) depicts model $\mathcal{M}_{\mathrm{F},9}$ trained with
$n_{\mathrm{s}}=9$ instead, which increases performance, resulting in
$\mathrm{nL2}=0.153$ after 776 epochs of training. It can be expected that
training beyond 800 epochs would further slightly increase the best
performance on test set.

##  Appendix C Visualization of spectral representation

During the investigations on the FNO models (see Figure 5), a concern arose
regarding the chosen number of Fourier series modes $n_{\mathrm{m}}=64$ in the
$R_{i}$-matrices, which might lead to the omission of significant frequency
components in the wave data. To address this concern, we visualized the
JONSWAP spectra employed to initialize the HOS wave simulation for a specific
steepness value $\epsilon$ (since different $\epsilon=0.08$ just scale the
amplitude of spectral density) and all peak wavelengths
$L_{\mathrm{p}}\in\\{80,90,\ldots,190,200\\}\,\mathrm{m}$, each corresponding
to a specific $\omega_{\mathrm{p}}$ and $k_{\mathrm{p}}$. Based on the
findings depicted and explained in Figure 17, we conclude that this assumption
is invalid.

Figure 17: JONSWAP spectra used in the data generation for one exemplary
steepness value $\epsilon$, but varying peak wavelengths
$L_{\mathrm{p}}=80-200\,\mathrm{m}$. The shortest peak wavelength of
$L_{\mathrm{p}}=80\,\mathrm{m}$ corresponds to the highest peak wavenumber of
$k_{\mathrm{p}}=0.079\,\mathrm{m}^{-1}$. The filtering wavenumber of
$k_{\mathrm{filt}}=n_{\mathrm{m}}\cdot\Delta k=64\cdot
0.00351\,\mathrm{m}^{-1}=0.2246\,\mathrm{m}^{-1}$, which is indicated by the
dotted red line and defined by the Fourier layers in this work, consequently
does not truncate important wave components in our data set-up.

## Acknowledgements

This work was supported by the Deutsche Forschungsgesellschaft (DFG - German
Research Foundation) [project number 277972093: Excitability of Ocean Rogue
Waves]

## Declaration of interests

The authors declare that they have no known competing financial interests or
personal relationships that could have appeared to influence the work reported
in this paper.

## Declaration of Generative AI and AI- assisted technologies in the writing
process

The manuscript was completely written by the authors. Once the manuscript was
completed, the authors used ChatGPT in order to improve its grammar and
readability. After using this tool, the authors reviewed and edited the
content as needed and take full responsibility for the content of the
publication.

## References

  * Airy [1849] Airy, G., 1849.  Tides and Waves.  Encyclopædia metropolitana, John Joseph Griffin and Company.
  * Asma et al. [2012] Asma, S., Sezer, A., Ozdemir, O., 2012.  Mlr and ann models of significant wave height on the west coast of india.  Computers and Geosciences 49, 231–237.  doi:10.1016/j.cageo.2012.05.032.
  * Battaglia et al. [2018] Battaglia, P.W., Hamrick, J.B., Bapst, V., Sanchez-Gonzalez, A., Zambaldi, V.F., Malinowski, M., Tacchetti, A., Raposo, D., Santoro, A., Faulkner, R., 2018\.  Relational inductive biases, deep learning, and graph networks.  arXiv arXiv:1806.01261.
  * Bertero et al. [2022] Bertero, M., Boccacci, P., de Mol, C., 2022.  Introduction to Inverse Problems in Imaging.  CRC Press.
  * Blondel-Couprie [2009] Blondel-Couprie, E., 2009.  Reconstruction et prévision déterministe de houle à partir de données mesurées.  Ph.D. thesis. Ecole Centrale de Nantes (ECN).
  * Borge et al. [1999] Borge, J.C.N., Reichert, K., Dittmer, J., 1999.  Use of nautical radar as a wave monitoring instrument.  Coastal Engineering 37, 331–342.  doi:10.1016/S0378-3839(99)00032-0.
  * Borge et al. [2004] Borge, J.N., RodrÍguez, G.R., Hessner, K., González, P.I., 2004\.  Inversion of marine radar images for surface wave analysis.  Journal of Atmospheric and Oceanic Technology 21, 1291 – 1300.  doi:10.1175/1520-0426(2004)021<1291:IOMRIF>2.0.CO;2.
  * Bouws et al. [1985] Bouws, E., Günther, H., Rosenthal, W., Vincent, C.L., 1985\.  Similarity of the wind wave spectrum in finite depth water: 1. spectral form.  Journal of Geophysical Research 90, 975–986.
  * Chen and Huang [2022] Chen, X., Huang, W., 2022.  Spatial–temporal convolutional gated recurrent unit network for significant wave height estimation from shipborne marine radar data.  IEEE Transactions on Geoscience and Remote Sensing 60, 1–11.  doi:10.1109/TGRS.2021.3074075.
  * Dankert and Rosenthal [2004] Dankert, H., Rosenthal, W., 2004\.  Ocean surface determination from x-band radar-image sequences.  Journal of Geophysical Research: Oceans 109\.  doi:10.1029/2003JC002130.
  * Deo et al. [2001] Deo, M., Jha, A., Chaphekar, A., Ravikant, K., 2001\.  Neural networks for wave forecasting.  Ocean Engineering 28, 889–898.  doi:10.1016/S0029-8018(00)00027-5.
  * Desmars [2020] Desmars, N., 2020.  Real-time reconstruction and prediction of ocean wave fields from remote optical measurements.  Theses. École centrale de Nantes.
  * Desmars et al. [2021] Desmars, N., Hartmann, M., Behrendt, J., Klein, M., Hoffmann, N., 2021.  Reconstruction of Ocean Surfaces From Randomly Distributed Measurements Using a Grid-Based Method.  International Conference on Offshore Mechanics and Arctic Engineering Volume 6: Ocean Engineering.  doi:10.1115/OMAE2021-62409.
  * Desmars et al. [2022] Desmars, N., Hartmann, M., Behrendt, J., Klein, M., Hoffmann, N., 2022.  Nonlinear Reconstruction and Prediction of Regular Waves.  International Conference on Offshore Mechanics and Arctic Engineering Volume 5B: Ocean Engineering; Honoring Symposium for Professor Günther F. Clauss on Hydrodynamics and Ocean Engineering.  doi:10.1115/OMAE2022-78988.
  * Desouky and Abdelkhalik [2019] Desouky, M.A., Abdelkhalik, O., 2019\.  Wave prediction using wave rider position measurements and narx network in wave energy conversion.  Applied Ocean Research 82, 10–21.  doi:10.1016/j.apor.2018.10.016.
  * Dommermuth and Yue [1987] Dommermuth, D.G., Yue, D.K.P., 1987\.  A high-order spectral method for the study of nonlinear gravity waves.  Journal of Fluid Mechanics 184, 267–288.  doi:10.1017/S002211208700288X.
  * Duan et al. [2020a] Duan, W., Ma, X., Huang, L., Liu, Y., Duan, S., 2020a.  Phase-resolved wave prediction model for long-crest waves based on machine learning.  Computer Methods in Applied Mechanics and Engineering 372, 113350.  doi:10.1016/j.cma.2020.113350.
  * Duan et al. [2020b] Duan, W., Yang, K., Huang, L., Ma, X., 2020b.  Numerical investigations on wave remote sensing from synthetic x-band radar sea clutter images by using deep convolutional neural networks.  Remote Sensing 12.  doi:10.3390/rs12071117.
  * Ducrozet et al. [2007] Ducrozet, G., Bonnefoy, F., Le Touzé, D., Ferrant, P., 2007\.  3-d hos simulations of extreme waves in open seas.  Natural Hazards and Earth System Sciences 7, 109–122.  doi:10.5194/nhess-7-109-2007.
  * Eichinger et al. [2022] Eichinger, M., Heinlein, A., Klawonn, A., 2022.  Surrogate convolutional neural network models for steady computational fluid dynamics simulations.  Electronic Transactions on Numerical Analysis 56, 235–255.
  * Goodfellow et al. [2016] Goodfellow, I., Bengio, Y., Courville, A., 2016.  Deep Learning.  MIT Press.  http://www.deeplearningbook.org.
  * Hasselmann et al. [1973] Hasselmann, K., Barnett, T., Bouws, E., Carlson, H., Cartwright, D., Enke, K., Ewing, J., Gienapp, H., Hasselmann, D., Kruseman, P., Meerburg, A., Mller, P., Olbers, D., Richter, K., Sell, W., Walden., H., 1973\.  Measurements of wind-wave growth and swell decay during the joint north sea wave project (jonswap).  Ergänzung zur Deutschen Hydrographischen Zeitschrift, Reihe A (8) 12, 1–95.
  * Hendrycks and Gimpel [2016] Hendrycks, D., Gimpel, K., 2016\.  Bridging nonlinearities and stochastic regularizers with gaussian error linear units.  CoRR abs/1606.08415.  arXiv:1606.08415.
  * Hilmer and Thornhill [2015] Hilmer, T., Thornhill, E., 2015\.  Observations of predictive skill for real-time deterministic sea waves from the wamos ii.  OCEANS 2015 - MTS/IEEE Washington , 1–7.
  * Isola et al. [2016] Isola, P., Zhu, J., Zhou, T., Efros, A.A., 2016.  Image-to-image translation with conditional adversarial networks.  arXiv arXiv:1611.07004.
  * James et al. [2018] James, S.C., Zhang, Y., O’Donncha, F., 2018.  A machine learning framework to forecast wave conditions.  Coastal Engineering 137, 1–10.  doi:10.1016/j.coastaleng.2018.03.004.
  * Kagemoto [2020] Kagemoto, H., 2020.  Forecasting a water-surface wave train with artificial intelligence- a case study.  Ocean Engineering 207, 107380\.  doi:10.1016/j.oceaneng.2020.107380.
  * Kharif et al. [2009] Kharif, C., Slunyaev, A., Pelinovsky, E., 2009.  Rogue Waves in the Ocean.  1 ed., Springer Berlin Heidelberg.  doi:10.1007/978-3-540-88419-4_5.
  * Kingma and Ba [2014] Kingma, D.P., Ba, J., 2014.  Adam: A method for stochastic optimization.  doi:10.48550/ARXIV.1412.6980.
  * Klein et al. [2020] Klein, M., Dudek, M., Clauss, G.F., Ehlers, S., Behrendt, J., Hoffmann, N., Onorato, M., 2020.  On the deterministic prediction of water waves.  Fluids 5.  doi:10.3390/fluids5010009.
  * Klein et al. [2022] Klein, M., Stender, M., Wedler, M., Ehlers, S., Hartmann, M., Desmars, N., Pick, M.A., Seifried, R., Hoffmann, N., 2022.  Application of Machine Learning for the Generation of Tailored Wave Sequences.  International Conference on Offshore Mechanics and Arctic Engineering Volume 5B: Ocean Engineering; Honoring Symposium for Professor Günther F. Clauss on Hydrodynamics and Ocean Engineering.  doi:10.1115/OMAE2022-78601.
  * Köllisch et al. [2018] Köllisch, N., Behrendt, J., Klein, M., Hoffmann, N., 2018\.  Nonlinear real time prediction of ocean surface waves.  Ocean Engineering 157, 387–400.  doi:10.1016/j.oceaneng.2018.03.048.
  * Law et al. [2020] Law, Y., Santo, H., Lim, K., Chan, E., 2020.  Deterministic wave prediction for unidirectional sea-states in real-time using artificial neural network.  Ocean Engineering 195, 106722\.  doi:10.1016/j.oceaneng.2019.106722.
  * LeCun et al. [1989] LeCun, Y., et al., 1989.  Generalization and network design strategies.  Connectionism in perspective 19, 18\.
  * Li et al. [2018] Li, H., Xu, Z., Taylor, G., Studer, C., Goldstein, T., 2018\.  Visualizing the loss landscape of neural nets.  Advances in neural information processing systems 31\.
  * Li et al. [2020] Li, Z., Kovachki, N.B., Azizzadenesheli, K., Liu, B., Bhattacharya, K., Stuart, A.M., Anandkumar, A., 2020.  Fourier neural operator for parametric partial differential equations.  arXiv arXiv:2010.08895.
  * Li et al. [2022] Li, Z., Peng, W., Yuan, Z., Wang, J., 2022.  Fourier neural operator approach to large eddy simulation of three-dimensional turbulence.  Theoretical and Applied Mechanics Letters 12\.
  * Liu et al. [2018] Liu, D., Wen, B., Liu, X., Wang, Z., Huang, T.S., 2018\.  When image denoising meets high-level vision tasks: A deep learning approach, in: Proceedings of the 27th International Joint Conference on Artificial Intelligence, AAAI Press. p. 842–848.
  * Liu et al. [2022] Liu, Y., Zhang, X., Chen, G., Dong, Q., Guo, X., Tian, X., Lu, W., Peng, T., 2022.  Deterministic wave prediction model for irregular long-crested waves with recurrent neural network.  Journal of Ocean Engineering and Science doi:10.1016/j.joes.2022.08.002.
  * Lünser et al. [2022] Lünser, H., Hartmann, M., Desmars, N., Behrendt, J., Hoffmann, N., Klein, M., 2022\.  The influence of characteristic sea state parameters on the accuracy of irregular wave field simulations of different complexity.  Fluids 7.  doi:10.3390/fluids7070243.
  * Mitchell [1980] Mitchell, T.M., 1980.  The need for biases in learning generalizations.  Citeseer.
  * Mohaghegh et al. [2021] Mohaghegh, F., Murthy, J., Alam, M.R., 2021.  Rapid phase-resolved prediction of nonlinear dispersive waves using machine learning.  Applied Ocean Research 117, 102920\.  doi:10.1016/j.apor.2021.102920.
  * Morris et al. [1998] Morris, E., Zienkiewicz, H., Belmont, M., 1998.  Short term forecasting of the sea surface shape.  International shipbuilding progress 45, 383–400.
  * Naaijen and Wijaya [2014] Naaijen, P., Wijaya, A., 2014\.  Phase resolved wave prediction from synthetic radar images, in: International Conference on Offshore Mechanics and Arctic Engineering, American Society of Mechanical Engineers. p. V08AT06A045.
  * Neill and Hashemi [2018] Neill, S.P., Hashemi, M.R., 2018\.  Chapter 7 - in situ and remote methods for resource characterization, in: Neill, S.P., Hashemi, M.R. (Eds.), Fundamentals of Ocean Renewable Energy. Academic Press. E-Business Solutions, pp. 157–191.  doi:10.1016/B978-0-12-810448-4.00007-0.
  * Niekamp et al. [2023] Niekamp, R., Niemann, J., Schröder, J., 2023.  A surrogate model for the prediction of permeabilities and flow through porous media: a machine learning approach based on stochastic Brownian motion.  Computational Mechanics 71, 563–581.  doi:10.1007/s00466-022-02250-2.
  * Ongie et al. [2020] Ongie, G., Jalal, A., Metzler, C.A., Baraniuk, R.G., Dimakis, A.G., Willett, R., 2020\.  Deep learning techniques for inverse problems in imaging.  IEEE Journal on Selected Areas in Information Theory 1, 39–56.
  * Paszke et al. [2019] Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., Desmaison, A., Köpf, A., Yang, E.Z., DeVito, Z., Raison, M., Tejani, A., Chilamkurthy, S., Steiner, B., Fang, L., Bai, J., Chintala, S., 2019.  Pytorch: An imperative style, high-performance deep learning library.  arXiv arXiv:1912.01703.
  * Pathak et al. [2016] Pathak, D., Krahenbuhl, P., Donahue, J., Darrell, T., Efros, A.A., 2016.  Context encoders: Feature learning by inpainting, in: 2016 IEEE Conference on Computer Vision and Pattern Recognition (CVPR), IEEE Computer Society, Los Alamitos, CA, USA. pp. 2536–2544.  doi:10.1109/CVPR.2016.278.
  * Pathak et al. [2022] Pathak, J., Subramanian, S., Harrington, P., Raja, S., Chattopadhyay, A., Mardani, M., Kurth, T., Hall, D., Li, Z., Azizzadenesheli, K., et al., 2022.  Fourcastnet: A global data-driven high-resolution weather model using adaptive fourier neural operators.  arXiv preprint arXiv:2202.11214.
  * Peng et al. [2022] Peng, W., Yuan, Z., Wang, J., 2022.  Attention-enhanced neural network models for turbulence simulation.  Physics of Fluids 34, 025111\.  doi:10.1063/5.0079302.
  * Perlin and Bustamante [2014] Perlin, M., Bustamante, M., 2014\.  A robust quantitative comparison criterion of two signals based on the sobolev norm of their difference.  Journal of Engineering Mathematics 101, 115–124.
  * Raschka [2018] Raschka, S., 2018.  Model evaluation, model selection, and algorithm selection in machine learning.  arXiv arXiv:1811.12808.
  * Rashid et al. [2022] Rashid, M.M., Pittie, T., Chakraborty, S., Krishnan, N.A., 2022\.  Learning the stress-strain fields in digital composites using fourier neural operator.  iScience 25, 105452\.  doi:10.1016/j.isci.2022.105452.
  * Ronneberger et al. [2015] Ronneberger, O., Fischer, P., Brox, T., 2015.  U-net: Convolutional networks for biomedical image segmentation, in: Medical Image Computing and Computer-Assisted Intervention – MICCAI 2015. Springer International Publishing, pp. 234–241.  doi:10.1007/978-3-319-24574-4_28.
  * Salcedo-Sanz et al. [2015] Salcedo-Sanz, S., Nieto Borge, J., Carro-Calvo, L., Cuadra, L., Hessner, K., Alexandre, E., 2015\.  Significant wave height estimation using svr algorithms and shadowing information from simulated and real measured x-band radar images of the sea surface.  Ocean Engineering 101, 244–253.  doi:10.1016/j.oceaneng.2015.04.041.
  * Stender et al. [2023] Stender, M., Ohlsen, J., Geisler, H., Chabchoub, A., Hoffmann, N., Schlaefer, A., 2023\.  $\hbox{U}^{p}$-net: a generic deep learning-based time stepper for parameterized spatio-temporal dynamics.  Computational Mechanics doi:10.1007/s00466-023-02295-x.
  * Stoian et al. [2019] Stoian, A., Poulain, V., Inglada, J., Poughon, V., Derksen, D., 2019.  Land cover maps production with high resolution satellite image time series and convolutional neural networks: Adaptations and limits for operational systems.  Remote Sensing 11.
  * Valenzuela [1978] Valenzuela, G.R., 1978.  Theories for the interaction of electromagnetic and oceanic waves — a review.  Boundary-Layer Meteorology 13, 61–85.  doi:10.1007/BF00913863.
  * Vicen-Bueno et al. [2012] Vicen-Bueno, R., Lido-Muela, C., Nieto-Borge, J.C., 2012.  Estimate of significant wave height from non-coherent marine radar images by multilayer perceptrons.  EURASIP Journal on Applied Signal Processing 2012\.  doi:10.1186/1687-6180-2012-84.
  * Wang et al. [2020] Wang, F., Eljarrat, A., Müller, J., Henninen, T.R., Erni, R., Koch, C.T., 2020\.  Multi-resolution convolutional neural networks for inverse problems.  Scientific reports 10, 1–11.
  * Wang and Yu [2023] Wang, R., Yu, R., 2023.  Physics-guided deep learning for dynamical systems: A survey.  arXiv:2107.01272.
  * Wedler et al. [2022] Wedler, M., Stender, M., Klein, M., Ehlers, S., Hoffmann, N., 2022.  Surface similarity parameter: A new machine learning loss metric for oscillatory spatio-temporal data.  Neural Networks 156, 123–134.  doi:10.1016/j.neunet.2022.09.023.
  * Wedler et al. [2023] Wedler, M., Stender, M., Klein, M., Hoffmann, N., 2023\.  Machine learning simulation of one-dimensional deterministic water wave propagation.  Journal of Ocean Engineering and Science doi:10.2139/ssrn.4349150.
  * Wen et al. [2022] Wen, G., Li, Z., Azizzadenesheli, K., Anandkumar, A., Benson, S.M., 2022.  U-fno—an enhanced fourier neural operator-based deep-learning model for multiphase flow.  Advances in Water Resources 163, 104180\.  doi:10.1016/j.advwatres.2022.104180.
  * West et al. [1987] West, B.J., Brueckner, K.A., Janda, R.S., Milder, D.M., Milton, R.L., 1987.  A new numerical method for surface hydrodynamics.  Journal of Geophysical Research: Oceans 92, 11803–11824.  doi:10.1029/JC092iC11p11803.
  * Wijaya et al. [2015] Wijaya, A., Naaijen, P., Andonowati, van Groesen, E., 2015\.  Reconstruction and future prediction of the sea surface from radar observations.  Ocean Engineering 106, 261–270.  doi:10.1016/j.oceaneng.2015.07.009.
  * Willard et al. [2022] Willard, J., Jia, X., Xu, S., Steinbach, M., Kumar, V., 2022\.  Integrating scientific knowledge with machine learning for engineering and environmental systems.  arXiv:2003.04919.
  * Williamson et al. [2022] Williamson, J., Brigido, H., Ladeira, M., Souza, J.C.F., 2022\.  Fourier neural operator for image classification, in: 2022 17th Iberian Conference on Information Systems and Technologies (CISTI), pp. 1–6.  doi:10.23919/CISTI54924.2022.9820128.
  * Wu [2004] Wu, G., 2004.  Direct simulation and deterministic prediction of large-scale nonlinear ocean wave-field.  Phd. thesis. Massachusetts Institute of Technology. Department of Ocean Engineering.
  * Wu et al. [2020] Wu, M., Stefanakos, C., Gao, Z., 2020.  Multi-step-ahead forecasting of wave conditions based on a physics-based machine learning (pbml) model for marine operations.  Journal of Marine Science and Engineering 8\.  doi:10.3390/jmse8120992.
  * Yan et al. [2022] Yan, B., Chen, B., Robert Harp, D., Jia, W., Pawar, R.J., 2022\.  A robust deep learning workflow to predict multiphase flow behavior during geological co2 sequestration injection and post-injection periods.  Journal of Hydrology 607, 127542\.  doi:10.1016/j.jhydrol.2022.127542.
  * Yang et al. [2021] Yang, Y., Gao, A.F., Castellanos, J.C., Ross, Z.E., Azizzadenesheli, K., Clayton, R.W., 2021\.  Seismic wave propagation and inversion with neural operators.  The Seismic Record 1, 126–134.
  * Yevnin and Toledo [2022] Yevnin, Y., Toledo, Y., 2022\.  A deep learning model for improved wind and consequent wave forecasts.  Journal of Physical Oceanography 52, 2531 – 2537.  doi:10.1175/JPO-D-21-0280.1.
  * You et al. [2022] You, H., Zhang, Q., Ross, C.J., Lee, C.H., Yu, Y., 2022\.  Learning deep implicit fourier neural operators (ifnos) with applications to heterogeneous material modeling.  Computer Methods in Applied Mechanics and Engineering 398, 115296.  doi:10.1016/j.cma.2022.115296.
  * Zeiler and Fergus [2014] Zeiler, M.D., Fergus, R., 2014\.  Visualizing and understanding convolutional networks, in: Fleet, D., Pajdla, Tomasand Schiele, B., Tuytelaars, T. (Eds.), Computer Vision – ECCV 2014, Springer International Publishing, Cham. pp. 818–83.
  * Zhang et al. [2022] Zhang, J., Zhao, X., Jin, S., Greaves, D., 2022.  Phase-resolved real-time ocean wave prediction with quantified uncertainty based on variational bayesian machine learning.  Applied Energy 324, 119711\.  doi:10.1016/j.apenergy.2022.119711.
  * Zhang et al. [2017] Zhang, K., Zuo, W., Gu, S., Zhang, L., 2017.  Learning deep cnn denoiser prior for image restoration, in: Proceedings of the IEEE conference on computer vision and pattern recognition, pp. 3929–3938.