Experts in satellite data are continuing to develop methods that will increase the accu-
racy, resolution or repetition of wind/wave measurements/products in the coastal zone. A
spatial resolution of 500 m for the wind and waves seems to be reachable from SAR and/or
altimetry. They will also extend the number of variables that can be derived from space, in
particular providing surface current maps from SAR (Gommenginger et al. 2019; Moiseev
et al. 2022). Over the coming years, Turner et al. (2021) confidently anticipated the com-
bined benefits of global coverage, access to an archive of routine and repeated measure-
ments that already span several decades, and the opportunities for automated and “always
on” monitoring of present conditions will further drive a substantial uptake in use of satel-
lite data for coastal applications. Although not yet quite matching the accuracy of land-
based and ship-based sensors, the continued improvement in spatial and temporal reso-
lution of space-borne instruments promises significant potential for providing additional
information and insights to inform coastal management, with advances in algorithms
(including machine learning, e.g., Danilo and Melgani 2019; Sagawa et al. 2019) and the
development of techniques that shift our capabilities from ‘simple’ individual snapshots of
the bathymetry toward seamless three-dimensional monitoring of the complete, evolving,
coastal zone (e.g., Bergsma et al. 2021) particularly exciting in this respect (Turner et al.
2021).

Of course, issues of resilience and protection are not only relevant to the built envi-
ronment, but also to the management of the health of the coastal ecosystem underpinning
the blue economy, as well as the safety of the communities whose livelihood and leisure
largely depend on it (Rayner et al. 2019). Of fundamental importance to coastal applica-
tions are the data to support trajectory modeling in studies of transport of plankton, fish
eggs and larvae to aid stock assessment (e.g., Röhrs et al. 2014); pollutant dispersion such
as marine litter (e.g., van Sebille et al. 2020); oil spill tracking to assist clean-up operations
(e.g., Jones et al. 2016); as well as the planning of search and rescue missions (e.g., Breivik
et al. 2013), among others. Such use cases need near real-time information on surface cur-
rents and surface winds—often, in addition, waves—to properly account for the advection
and modification of particles within their leeway drift predictions. EO information is either
able to be processed and used as forcing for Lagrangian simulations directly, or assimilated
in coastal models to predict the fate of these objects. As with all remote sensing opportuni-
ties, the increasing prevalence of large constellations of satellites promises a significant
potential for providing new data.

The augmentation of wind, wave and current measurements with additional comple-
mentary space-borne information (e.g., ocean color, temperature, communications; Melet
et al. 2020)—such as that used for the monitoring of the broader non-metocean aspects
affecting water quality monitoring (e.g., Chawla et al. 2020), pollutant (e.g., Viatte et al.
2020) and oil spill detection (e.g., Fingas and Brown 2018) as well as search & rescue tar-
get identification (e.g., Kanjir et al. 2018)—are essential to realizing truly integrated (and
data-driven) approaches to effective coastal management. A cross-sectional view on the
use of space observations to monitor and study coastal areas, estuaries and deltas is pre-
sented by Laignel et al. (2023). in this special issue.

4.4 Future Prospects

In terms of both offshore industry and coastal zone management applications, it is possible
to identify a several unifying priorities for the use of EO and satellite data:
* Maintaining the existing observational timeseries. Users have highlighted that a primary benefit offered by satellite data is the long (and consistent) time series that exists, with the potential to also support the application of new techniques that make use of third-party signals not originally designed for the purpose for which they can be employed.

* Providing higher spatial / temporal resolution data. Users commonly identify a need for higher resolution data, in space and time, which represents a challenge for EO acquisition, even with the increased number of satellite altimeter and SAR missions now operating. For example, Ardhuin et al. (2019c) specified the following requirements for global to regional wave climate data: 3-hourly measurements; 25 km resolution, uncertainty in SWH of 10 cm / 5% and long-term stability in SWH of 5 cm/decade. This is difficult for along-track measuring sensors, but new missions with swath measuring capability (CFOSAT, SWOT) should offer improvements.

* Development of instrumentation/techniques to support emerging / evolving requirements, such as the provision of full wave energy spectra in engineering applications.

* Augmentation with a range of measured and modeled oceanographic data, which needs to be interoperable and translatable to derive relevant metrics to support simple multi-variate decision-making by non-experts; for example through integration with metocean information systems.

# 5 Conclusions

In this paper we have presented how satellites provide essential observations of wind, waves and surface currents in three domains: (1) monitoring and modelling the atmospheric and oceanic conditions at a global scale, (2) to contribute to research studies on the coupled atmosphere/ocean system, and on evaluation of the impact of climate change, (3) to support applications dedicated to users of the open and coastal ocean. With more than 30 years of satellite missions defined to measure wind, waves and currents at the ocean surface, most space-borne sensors and inversion methods can be considered as mature. However, progress is still expected in several domains thanks to interesting advances in instrumental concepts. For example, a better characterization of extreme events like tropical cyclones will be possible in the future thanks to the choice of polarimetry diversity on future wind scatterometers (with cross-polarization the most sensitive to high wind) and to a more systematic use of low-frequency microwave radiometer measurements. A better characterization of the air/sea exchanges will also be possible with missions which will simultaneously provide measurements of wind, wave surface currents (projects WacM-Odysea in the USA, Rodriguez et al. 2019), Seastar in Europe, Gommenginger et al. 2019) and eventually combined with other important parameters of the air/sea exchanges like the sea-surface temperature (Harmony project in Europe, López-Dekker et al. 2019). In parallel, the development of new inversion methods applied to the most recent altimeter or synthetic aperture radar missions, will allow better spatial resolution of sea-state parameters, which will benefit the characterization of non-homogenous conditions (coastal zones including, e.g., coral reef islands, fronts, eddies, marginal ice zone, etc.) and the associated need to protect the fragile environments under the pressure of climate change. It must also be recalled here that new perspectives are sometimes opened by new techniques applied to remotely-sensed observations not originally designed for the purpose for which they were defined. For example, the GNSS constellation of satellites primarily used for navigation,
are generating signals that are reflected off the sea surface (radio occultation and scatterometry) can be related to sea state. Multi-band optical images acquired quasi-instantaneously, such as on the Sentinel-2 mission, also offer promises opportunities for providing more systematic details on wave evolution and/or bathymetry. Lidar measurements defined for sea-ice monitoring can provide interesting by-products on waves. Finally, the development of new services based on a wide-open access to a long archive of satellite products (e.g., the Copernicus Marine Service, see Le Traon et al. 2019) will continue to facilitate the use of these observations for a wide range of applications either in research or oriented toward the blue economy.

## Appendix 1: List of sensors and satellite missions mentioned in this paper

In the Table of this Appendix, we provide the list of sensors and satellite missions mentioned in the paper and include the name of agencies responsible of the sensors, as well as the period of measurements.

More details can be found in the data base maintained by the CEOS (Committee on Earth Observation Satellites) (http://database.eohandbook.com/).

<table><thead><tr><th>Platform Acronym</th><th>Instruments mentioned in the text</th><th>instrument agency, country</th><th>Start/end Dates</th></tr></thead><tbody><tr><td>ADEOS-I<br>ADEOS-II<br>(Advanced Earth Observing Satellites)</td><td>NSCAT (NASA Scatterometers)<br>Quikscat</td><td>USA</td><td>1996–1997<br>2002–2003</td></tr><tr><td>Aqua</td><td>AMSR-E (Advanced Microwave Scanning Radiometer EOS)</td><td>NASA (USA) and JAXA (Japan)</td><td>In operation since 2002</td></tr><tr><td>CFOSAT<br>(China France Oceanography SATellite)</td><td>SWIM (Surface Waves Investigation and Monitoring)<br>SCAT (Scatterometer)</td><td>CNES (France) and CNSA (China)</td><td>In operation since 2018</td></tr><tr><td>Coriolis</td><td>Windsat</td><td>DoD (USA)</td><td>2003–2020</td></tr><tr><td>Cryosat-2</td><td>SIRAL (SAR Interferometer Radar Altimeter)</td><td>ESA (Europe)</td><td>In operation since 2010</td></tr><tr><td>CYGNSS</td><td>DDMI (Delay Doppler Mapping Instrument)</td><td>NASA &amp; NOAA (USA)</td><td>In operation since 2016</td></tr><tr><td>DMSP, (Defense Meteorological Satellite)-satellite series F</td><td>SSM/I Special Sensor Microwave Imager on F-8 to F-15<br>SSM/IS Special Sensor Microwave-Imager / Sounder on F16–F19</td><td>NOAA and DoD (USA)</td><td>Successive satellites in operation since 1987</td></tr><tr><td>Envisat</td><td>ASAR (Advanced SAR) Radar altimeter</td><td>ESA</td><td>2002–2012</td></tr><tr><td>ERS-1/2<br>(European Remote Sensing Satellites -1/2)</td><td>SAR, altimeter</td><td>ESA</td><td>1991–2000</td></tr></tbody></table>
<table><thead><tr><th>Platform</th><th>Acronym</th><th>Instruments mentioned in the text</th><th>instrument agency, country</th><th>Start/end Dates</th></tr></thead><tbody><tr><td>FY3-E, FY3-H, FY3-I</td><td>WindRAD (Wind Radar)</td><td></td><td>CNSA (China)</td><td>In operation since 2021</td></tr><tr><td>GCOM (Global Change Observation Mission-Water)</td><td>AMSR2 (Advanced Microwave Scanning Radiometer 2)</td><td></td><td>JAXA (Japan)</td><td>In operation since 2012</td></tr><tr><td>GFO (Geosat Follow-on)</td><td>Radar altimeter</td><td></td><td>NASA (USA)</td><td>1998–2008</td></tr><tr><td>GPM (Global Precipitation mission)</td><td>GMI (GMP Microwave imager)</td><td></td><td>NASA (USA)</td><td>In operation since 2014</td></tr><tr><td>Harmony</td><td>Passive Synthetic Aperture Radar</td><td></td><td>ESA (Europe)</td><td>Launch expected in 2029</td></tr><tr><td>HY2</td><td>ALT (Radar Altimeter) SCAT (Scatterometer)</td><td></td><td>CNSA (China)</td><td>In operation since 2021</td></tr><tr><td>HydroGNSS</td><td>GNSS Reflectometer</td><td></td><td>ESA (Europe), UKSA (UK)</td><td>Launch expected in 2024</td></tr><tr><td>Icesat-2 (Ice, Cloud, and land Elevation Satellite 2)</td><td>ATLAS (Advanced Topographic Laser Altimeter System)</td><td></td><td>NASA (USA)</td><td>In operation since 2018</td></tr><tr><td>ISS (International space station)</td><td>RapidScat</td><td></td><td>NASA (USA)</td><td>2014–2016</td></tr><tr><td>Jason 1 to 3</td><td>Altimeters POSEIDON-2 (Jason-1), POSEIDON-3 (Jason-2), POSEIDON-3B (Jason-3)</td><td></td><td>CNES (France)</td><td>Jason 1: 2011–2013<br>Jason 2: 2008–2019<br>Jason 3: in operation since 2016</td></tr><tr><td>Metop A-B-C</td><td>ASCAT (Advanced Scatterometer)</td><td></td><td>EUMETSAT, ESA (Europe)</td><td>Metop-A: 2006–2021<br>Metop-B: In operation since 2012<br>Metop-C: In operation since 2018</td></tr><tr><td>Oceansat-2<br>Oceansat-3</td><td>Scatterometer</td><td></td><td>ISRO (India)</td><td>In operation since 2009<br>In operation since 2022</td></tr><tr><td>QuikSCAT</td><td>Seawinds scatterometer</td><td></td><td>NASA (USA)</td><td>1999–2018</td></tr><tr><td>RADARSAT-1<br>RADARSAT-2</td><td>Synthetic Aperture Radar</td><td></td><td>CSA (Canada)</td><td>1994–2013<br>In operation since 2007</td></tr><tr><td>SARAL</td><td>AltiKa (Ka-band Altimeter)</td><td></td><td>CNES (France)</td><td>In operation, since 2013</td></tr><tr><td>Scatsat-1</td><td>Scatsat-1</td><td></td><td>ISRO (India)</td><td>2016–2017</td></tr><tr><td>Seasat</td><td>SAR, altimeter, Scatterometers<br>Radiometer SMMR (Scanning Multi-Channel Microwave Radiometer)</td><td></td><td>NASA (USA)</td><td>1978–1978</td></tr><tr><td>Seastar</td><td>SAR (squinted along-track interferometry)</td><td></td><td>ESA (Europe)</td><td>Under study for an Earth Explorer satellite (Phase 0)</td></tr><tr><td>Sentinel-1 (A-B-C-D...)</td><td>SAR</td><td></td><td>ESA (Europe)</td><td>Series started in 2014</td></tr><tr><td>Sentinel-2 (A-B-C-D...)</td><td>Optical multi-spectral Instrument</td><td></td><td>ESA (Europe)</td><td>Series started in 2015</td></tr><tr><td>Sentinel-3 (1-B-C-D...)</td><td>SAR radar altimeter (SRAL)</td><td></td><td>ESA (Europe)</td><td>Series started in 2016</td></tr></tbody></table>
<table><thead><tr><th>Platform Acronym</th><th>Instruments mentioned in the text</th><th>instrument agency, country</th><th>Start/end Dates</th></tr></thead><tbody><tr><td>Sentinel-6 Michael Freilich (A-B)</td><td>Poseidon-4 SAR Radar Altimeter</td><td>ESA (Europe) & CNES (France)</td><td>Operational since 2020</td></tr><tr><td>SKIM</td><td>Doppler near-nadir incidence scatterometer</td><td>ESA (Europe)</td><td>Abandoned (proposed as an Earth Explorer satellite)</td></tr><tr><td>SMAP Soil Moisture Active Passive</td><td>L-Band Radiometer</td><td>NASA (USA) & CSA (Canada)</td><td>Operational since 2015</td></tr><tr><td>SMOS Soil Moisture and Ocean Salinity</td><td>Microwave Imaging Radiometer using Aperture Synthesis (MIRAS)</td><td>ESA (Europe)</td><td>Operational since 2009</td></tr><tr><td>SWOT Surface Water Ocean Topography</td><td>Ka-band Radar Interferometer (KaRIN)—Interferometric altimeter</td><td>NASA (USA) and CNES (France)</td><td>Launch planned in 2023</td></tr><tr><td>Tandem-X</td><td>SAR</td><td>DLR (Germany)</td><td>In operation since 2010</td></tr><tr><td>TerraSAR-X</td><td>SAR</td><td>DLR (Germany)</td><td>In operation since 2007</td></tr><tr><td>Topex</td><td>Altimeters (Topex and Poseidon-1)</td><td>NASA (USA) and CNES (France)</td><td>1992–2006</td></tr></tbody></table>

## Appendix 2: Principles of wave estimation from SAR images

Effects and properties that contribute to the formation and/or the degradation of satellite SAR ocean wave images are summarized here-below:

* *Tilt modulation* variation in backscatter due to the local modification of the relative incidence angle by long wave slopes. The longer waves locally modify the exact plane of incidence, producing a local change in cross section (following the geometrical characteristics of the instrument).

* *Hydrodynamic modulation* variation in backscatter due to the non-uniform distribution of scatterers (i.e., short waves) along a long wave profile. Existing parameterizations of this phenomenon are still subject of on-going active research efforts and usually follow weakly nonlinear theory solutions (Alpers et al. 1981; Alpers and Hasselmann 1978; Hasselmann and Hasselmann 1991). Because the radar cross section at medium incidence (typically 10–25°) is not very sensitive to the short waves, the hydrodynamic effect on the signal modulation can be generally neglected for observations at these incidence angles.

* *Constructive azimuth velocity bunching* deterministic misregistrations in the azimuth direction, associated with long wave orbital motions, leading to an apparent constructive redistribution of the backscatter intensity along the azimuth direction.

* *Destructive azimuth velocity bunching* random misregistrations in the azimuth direction, associated with long wave orbital motions, leading to possible significant degradation in the azimuth resolution, leading to distortions of the resulting SAR ocean image spectra (nonlinear relationship between SAR image spectra and ocean wave spectra).

* *Range elevation bunching* misregistrations associated with long wave slope variations. When the angle of incidence is smaller than the slope of the waves (i.e., for incidence
angles of a few degrees, depending on sea-state) the iso-range plane of the radar signal may cut the rough surface at multiple points, like a long surfboard in choppy seas. This induces misregistrations which are proportional to the surface elevation and depend on the nominal instrument incidence angle. However, it is acknowledged that this range bunching effect can be neglected for observations at incidence angles larger than about 10° (Jackson 1981).

* Acceleration smearing smearing in the SAR image associated with relative scatterer velocity changes during the SAR integration time.

* Azimuth smearing reduced along-track resolution associated with both the unresolved random scatter motions and the limited scatter lifetime during the SAR integration time (finite scene coherence time).

* Speckle multiplicative noise in the SAR images due to the coherent processing. Its statistical characteristics will depend upon the chosen look summation technique. The inter-correlation (cross-spectral) technique will help to almost entirely remove the speckle distribution in the spectral domain.

* System transfer function point target response characteristics that lead to varying sensitivity over the spectral domain.

The tilt ($T_m^{\text{tilt}}$), the hydrodynamic ($T^{\text{hydr}}$) and the range elevation bunching ($T^{\text{rb}}$) are the dominant processes contributing to the Real Aperture Modulation (RAR) function $T_m^{\text{RAR}}$:

$$T_m^{\text{RAR}} = T_m^{\text{tilt}} + T^{\text{hydr}} + T^{\text{rb}} \quad (15)$$

Applying the Bragg approximation for the short waves on a two-scale scattering model, based on a geometric optical solution for a perfect conducting surface: the linear backscatter RAR modulation transfer function (MTF) can be separated into two parts, one geometrical MTF caused by surface tilting and one MTF caused by hydrodynamic interactions. Under this assumption, the right-hand side terms of (Eq. 15) can be expressed as a function of radar parameters (incidence, polarization), surface parameters (wavenumber of the tilting waves), and relaxation rate of the Bragg waves (Alpers et al. 1981; Alpers and Hasselmann 1978; Hasselmann and Hasselmann 1991).

An alternative semi-parametric model of the RAR MTF has been proposed to retrieve ocean wave spectra from SAR images (see Johnsen et al. (2022)). It makes use of an empirical GMF function (the so-called CMOD GMF) to relate the $\sigma_m$ observations to wind speed (Stoffelen et al. 1997; 2017a). This technique is sought to avoid uncertainty associated with the use of a pure-Bragg scattering model, and offers an efficient alternative for C-band SAR data. In this model, the RAR MTF amplitude is proportional to the local derivative of relative cross section along the long wave phase, and explicitly dependent on the wind field through the CMOD function:

$$T_m^{\text{RAR}}(\underline{k}) = \frac{1}{\sigma_m} \frac{\partial \sigma_m}{\partial \theta} \cdot \left\{ ik_x + \frac{k_x^2}{|\underline{k}|} \tan \theta \right\} \quad (16)$$

where $k_x$ is the radar wavenumber vector, $k_x$ its component along the range-direction, $\theta$ is the radar beam incidence, and $\sigma_m$ is the CMOD radar cross-section model function.

To complete the SAR imaging mechanism, the induced Doppler shifts, mainly associated with the longer wave orbital velocity field, must be considered as a very crucial mapping process. This is the velocity bunching mechanism. The motion effect strongly influences the SAR imaging mechanism by affecting the complex backscatter signals.
Doppler shifts associated with the longer wave orbital velocities can also potentially contribute to an additional constructive modulation mechanism (Alpers et al. 1981). In such a case, the so-called velocity bunching mechanism may help to image swell wave components in the along-track direction. This constructive mechanism is especially efficient under low wind conditions (relatively deterministic phase component redistribution). The velocity bunching effect is controlled by the range-to-velocity ratio and the relative radial component of the scatterers’ velocity. The corresponding azimuth position shift of the scatterer is given as (Alpers and Rufenach 1981):

$$ \xi_y = \frac{R}{V} U_r \qquad (17) $$

where R is the range distance from satellite to target, V is the satellite velocity, $U_r$ is the orbital velocity of the target projected into the radar look direction. The velocity bunching mechanism is fully controlled by the non-dimensional azimuth wavenumber, $k_y \equiv k_y \rho_{\xi_y \xi_y} (0)^{1/2}$, where $\rho_{\xi_y \xi_y}$ is the covariance function of the azimuth shift $\xi_y$, and $k_y$ is the wavenumber component in azimuth direction (Krogstad 1992). In the case of small to moderate $k_y$, the imaging is constructive and the velocity bunching MTF can be written as (Alpers et al. 1981):

$$ T^{\text{vel}}(\underline{k}) = \frac{R}{V} \omega_{|\underline{k}|} \left\{ \frac{k_x}{|\underline{k}|} \sin \theta + i \cos \theta \right\} \qquad (18) $$

A breakthrough in understanding SAR ocean wave imaging was achieved with the paper by Hasselmann and Hasselmann (1991) introducing a closed form expression relating the underlying 2D ocean wave spectra to the 2D SAR image spectra, taking into account both real aperture and velocity bunching mechanisms. This closed form expression was denoted “the ocean-to-SAR spectral transform.” This derivation was later elegantly reformulated by Krogstad (1992), and used to study the nonlinearity of the SAR wave imaging process. Finally, the next breakthrough was achieved when the formalism of Krogstad (1992) was extended by Engen and Johnsen (1995) to provide a closed form of the ocean-to-SAR cross-spectral transform. This work, which takes into account the entire wave phase velocity field is also of fundamental importance in the SAR measurements, as earlier shown on airborne SAR imagery by Vachon and Raney (1991). Engen and Johnsen (1995) defined the cross-spectra between two intensity-detected looks as:

$$ P_{\text{SAR}}(\underline{k}, \tau) = \frac{1}{\langle I^{(1)}(\underline{k}, \frac{\tau}{2}) \cdot I^{(2)}(\underline{k}, -\frac{\tau}{2}) \rangle} \langle I^{(1)}(\underline{k}, \frac{\tau}{2}) \cdot I^{(2)}(\underline{k}, -\frac{\tau}{2}) \rangle \qquad (19) $$

where $I^{(1)}(\underline{k}, \frac{\tau}{2})$ and $I^{(2)}(\underline{k}, -\frac{\tau}{2})$ are the Fourier transforms of the two intensity detected look images, and $\tau$ is the look separation time. Even for the short integration time ($\tau = 0.8$s) of standard spaceborne SAR stripmap modes, a significant phase shift ($\omega_{|\underline{k}|} \cdot \tau$) is achieved for the long wavelengths of the spectra.

The ocean-to-SAR cross-spectra formalism enables, on the one hand, the elimination of the speckle noise contribution (Engen and Johnsen 1995) and, on the other hand, the removal of the 180° directional ambiguity of the imaged harmonic wave components. Such a crucial breakthrough concerning the potential use of cross-spectral analysis between SAR individual looks is now commonly adopted to improve the analysis of
SAR ocean wave data. The methodology shows that the exact propagation direction of swell systems is achieved for 90% of the measurements (as long as the signal-to-noise ratio is sufficiently high). Such a methodology has been fully validated, was applied to the Envisat ASAR Wave Mode (WM) and now also to the Copernicus Sentinel-1 WV mode imagnettes.

The general expression of the ocean-to-SAR cross-spectral transform can be written as (Engen and Johnsen 1995):

$$P(\underline{k}, \tau) \approx \frac{1}{(2\pi)^2} \int d\underline{x} e^{-ik \cdot \underline{x}} e^{\frac{k_y^2}{2} (\rho_{\xi_y\xi_y}(\underline{x}, \tau) - \rho_{\xi_y\xi_y}(0,0))} \{1 + \rho_{II}(\underline{x}, \tau) - \delta(\underline{x})\} \quad (20)$$

where $\rho_{II}$ is the covariance function of the RAR image, and $\tau$ is the inter look-separation time.

From (20), a simplified quasi-linear version can be derived as:

$$P^{\text{qlin}}(\underline{k}, \tau) = \frac{1}{2} e^{-\frac{k_y^2}{2} \rho_{\xi_y\xi_y}(0,0)} \left\{ \left|T(\underline{k})\right|^2 e^{-i\omega_{|\underline{k}|} \tau} S(\underline{k}) + \left|T(-\underline{k})\right|^2 e^{+i\omega_{|\underline{k}|} \tau} S(-\underline{k}) \right\} \quad (21)$$

where $T(\underline{k})$ is the total SAR MTF written as:

$$T(\underline{k}) = ik_x T^{\text{rb}}(\underline{k}) + ik_y T^{\text{vel}}(\underline{k}) + T^{\text{tilt}}(\underline{k}) + T^{\text{hydr}}(\underline{k}) \quad (21)$$

The nonlinearity of the velocity bunching is manifested as an exponential azimuth cutoff (the exponential term of Eq. (21)), causing a strong filtering of the azimuth wave components. The azimuth cutoff is related to the azimuth Doppler shift covariance, which can be written in terms of the underlying wave spectra $S(\underline{k})$ as:

$$\rho_{\epsilon_x, \epsilon_y}(0,0) = \frac{1}{2\pi} \int dk |T^{\text{vel}}(\underline{k})|^2 S(\underline{k}) \quad (23)$$

As indicated by Eqs. (17) and (18), the azimuth cutoff is governed by the satellite R/V ratio and the sea state (or alternatively the wind speed).

Subsequent to the development of the ocean-to-SAR spectral transforms, different inversion algorithms were developed (Engen et al. 1994; Hasselmann et al. 1996; Engen et al. 2000; Mastenbroek and de Valk 2000; Schulz-Stellenfleth 2005; Collard et al. 2005; Shao et al. 2015). Such retrieval algorithms generally attempt to reconstruct the a priori ocean wave spectrum by minimizing the difference between the theoretical SAR image spectra of Eq. (19) and the satellite observed SAR image spectra of Eq. (22). However, as a consequence of the exact derivative of the nonlinear transform being too cumbersome to compute, most of these inversion schemes ignore the complete nonlinear mapping or represent it through a stochastic model (Schulz-Stellenfleth 2005).

Alternatively, it is possible to derive the wave spectra directly from the quasi-linear transform (see Eq. (21)). This requires that the azimuth cutoff be well estimated and that the nonlinearity of the SAR image spectra is removed from the observed SAR image cross-spectra. This nonlinearity is driven by the wind field and can be predicted simply by taking the difference between the full and the quasi-linear transform, $P^{\text{lin}} = P - P^{\text{qlin}}$. This is the approach implemented in both the ASAR Envisat Wave Mode Level 2 and the Sentinel-1 WV Level 2 processors, where the $P^{\text{lin}}$ is provided in a look-up table as a function of wind speed, wind direction, and wave age.
**Funding** J. Knaff’s funding is provided by National Oceanic and Atmospheric Administration (NOAA). The views, opinions, and findings contained in this report are those of the authors and should not be construed as an official National Oceanic and Atmospheric Administration or U.S. Government position, policy, or decision. The Office of Naval Research (ONR) has provided funding for C. Sampson through Program Element 0603207N.

## Declarations

**Conflict of interest** The authors do not have financial nor non-financial conflict of interest to declare.

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## Authors and Affiliations

Danièle Hauser<fnref n="1" />☐, Saleh Abdalla<fnref n="2" />, Fabrice Ardhuin<fnref n="3" />, Jean-Raymond Bidlot<fnref n="2" />,

Mark Bourassa<fnref n="4" />, David Cotton<fnref n="5" />, Christine Gommenginger<fnref n="6" />, Hayley Evers-King<fnref n="7" />,

Harald Johnsen<fnref n="8" />, John Knaff<fnref n="9" />, Samantha Lavender<fnref n="10" />, Alexis Mouche<fnref n="3" />,

Nicolas Reul<fnref n="3" />, Charles Sampson<fnref n="11" />, Edward C.C Steele<fnref n="7" />, Ad Stoffelen<fnref n="12" />

<figure><img src="image_31.png" /></figure>

Danièle Hauser
daniele.hauser@latmos.ipsl.fr

<footnote n="1">¹ LATMOS, CNRS; University of Versailles Saint-Quentin; Sorbonne University, 11 Boulevard d'Alembert, 78280, Guyancourt, France</footnote>

<footnote n="2">² European Centre for Medium-Range Weather Forecasts, RG2 9AX Reading, UK</footnote>

<footnote n="3">³ Present Address: LOPS, Université de Bretagne Occidentale; CNRS; Ifremer; IRD, Technopôle Brest Iroise, 29280 Plouzané, France</footnote>

<footnote n="4">⁴ Centre for Ocean-Atmosphere Prediction Studies, Florida State University, FL 32306 Tallahassee, Florida, USA</footnote>

<footnote n="5">⁵ Satellite Oceanographic Consultants Limited, SK22 4QF New Mills, UK</footnote>

<footnote n="6">⁶ National Oceanography Center, SO14 3ZH Southampton, UK</footnote>

<footnote n="7">⁷ Met Office, EX1 3PB Exeter, Devon, UK</footnote>

<footnote n="8">⁸ NORCE Norwegian Research Center AS, Postboks 6434, 929 Tromsø, Norway</footnote>

<footnote n="9">⁹ Center for Satellite Applications and Research, NOAA, CO 80523-1375 Fort Collins, USA</footnote>

<footnote n="10">¹⁰ Pixalytics Ltd, PL6 8BX Plymouth, UK</footnote>

<footnote n="11">¹¹ Naval Research Laboratory, CA 93943 Monterey, California, USA</footnote>

<footnote n="12">¹² KNMI, 3731 GA De Bilt, The Netherlands</footnote>