# Phase-Resolved Swells Across Ocean Basins in SWOT Altimetry Data: Revealing Centimeter-Scale Wave Heights Including Coastal Reflection

Fabrice Ardhuin, Beatriz Molero, Alejandro Bohé, Frédéric Nouguier, Fabrice Collard, Isabel Houghton, Andrea Hay, Benoit Legresy

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Fabrice Ardhuin, Beatriz Molero, Alejandro Bohé, Frédéric Nouguier, Fabrice Collard, et al. Phase-Resolved Swells Across Ocean Basins in SWOT Altimetry Data: Revealing Centimeter-Scale Wave Heights Including Coastal Reflection. Geophysical Research Letters, 2024, Science from the Surface Water and Ocean Topography Satellite Mission, 51 (16), pp.e2024GL109658. 10.1029/2024GL109658. hal-04676626

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# Geophysical Research Letters®

## RESEARCH LETTER

10.1029/2024GL109658

**Special Collection:**

Science from the Surface Water and Ocean Topography Satellite Mission

**Key Points:**

* Surface Water Ocean Topography (SWOT) data provide the first open ocean spatial measurements of phase-resolved swells with wavelength 500–1,050 m

* Swells with heights as low as 3 cm are well detected by SWOT, allowing tracking across oceans

* Swell reflection off the coast can be separated from incident waves

**Supporting Information:**

Supporting Information may be found in the online version of this article.

**Correspondence to:**

F. Ardhuin,
ardhuin@ifremer.fr

**Citation:**

Ardhuin, F., Molero, B., Bohé, A., Nouguier, F., Collard, F., Houghton, I., et al. (2024). Phase-resolved swells across ocean basins in SWOT altimetry data: Revealing centimeter-scale wave heights including coastal reflection. *Geophysical Research Letters*, *51*, e2024GL109658.
https://doi.org/10.1029/2024GL109658

Received 18 APR 2024

Accepted 7 AUG 2024

© 2024. The Author(s).
This is an open access article under the terms of the Creative Commons Attribution License, which permits use, distribution and reproduction in any medium, provided the original work is properly cited.

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## Phase-Resolved Swells Across Ocean Basins in SWOT Altimetry Data: Revealing Centimeter-Scale Wave Heights Including Coastal Reflection

Fabrice Ardhuin¹ ID, Beatriz Molero², Alejandro Bohé³, Frédéric Nouguier¹ ID, Fabrice Collard⁴, Isabel Houghton⁵ ID, Andrea Hay⁶,⁷ ID, and Benoit Legresy⁶,⁷,⁸ ID

¹Laboratoire d'Océanographie Physique et Spatiale (LOPS), Univ. Brest, CNRS, IRD, Ifremer, IUEM, Brest, France, ²CLS, Ramonville St Agne, France, ³CNES, Toulouse, France, ⁴ODL, Locmaria-Plouzane, France, ⁵Sofar Ocean Technologies, San Francisco, CA, USA, ⁶School of Geography, Planning, and Spatial Sciences, University of Tasmania, Hobart, TAS, Australia, ⁷Commonwealth Scientific and Industrial Research Organisation, Hobart, TAS, Australia, ⁸Integrated Marine Observing System, Hobart, TAS, Australia

**Abstract** Severe storms produce ocean waves with periods of 18–26 s, corresponding to wavelengths 500–1,055 m. These waves radiate globally as swell, generating microseisms and affecting coastal areas. Despite their significance, long waves often elude detection by existing remote sensing systems when their height is below 0.2 m. The new Surface Water Ocean Topography (SWOT) satellite offers a breakthrough by resolving these waves in global sea level measurements. Here we show that SWOT can detect 25-s waves with heights as low as 3 cm, and resolves period and direction better than in situ buoys. SWOT provides detailed maps of wave height, wavelength, and direction across ocean basins. These measurements unveil intricate spatial patterns, shedding light on wave generation in storms, currents that influence propagation, and refraction, diffraction and reflection in shallow regions. Notably, the magnitude of reflections exceeds previous expectations, illustrating SWOT's transformative impact.

**Plain Language Summary** Wind storms at sea make waves that increase in size with wind speed, and with the distance over which the high winds have been able to amplify the waves. Once generated these waves propagate as swell around the world ocean: in that stage the wave period remains constant while the wave height decay away from the source. Waves with periods longer than 18 s are relatively infrequent, but they are an important source of seismic waves and coastal impacts. However, current remote sensing techniques miss long waves under 0.2 m high. The Surface Water Ocean Topography (SWOT) satellite mission changes this, spotting 25-s waves with heights as low as 3 cm. SWOT maps wave height, wavelength, and direction worldwide, revealing the influence of winds, currents and water depth. For example, We found stronger than expected coastal reflection, which will help revise wave forecasting models and their application in seismology.

## 1. Introduction

Only the most severe wind storms are capable of generating significant ocean wave energy with periods longer than 18 s, corresponding to a deep water wavelength of 500 m. Once generated, these waves radiate away from the storm as swell, following great circles across ocean basins (Munk et al., 1963), with a possible weak effect of currents on their propagation paths (Gallet & Young, 2014). The infrequent occurrences of these long period waves are found in ocean time series of surface elevation or near-surface pressure (Darbyshire, 1950), or land-based records of microseisms that provide a global but very indirect view of these long swells (Hanafin et al., 2012; Husson et al., 2012). Satellite observations are more direct but existing techniques, real aperture radar (Hauser et al., 2021; Jackson et al., 1985) and synthetic aperture radar (SAR) (Ardhuin et al., 2015; Hasselmann et al., 1985), are less reliable at detecting long waves with heights under 0.2 m.

The recently launched Surface Water Ocean Topography (SWOT) satellite mission uses a completely new type of instrument, a Ka-band Radar Interferometer (KaRIN) that consists of a pair of SARs that provides across-track interferometry from a single satellite pass, as described by Fu et al. (2024). SWOT has a unique capability of resolving kilometer-scale oscillations in global measurements of sea surface height (SSH) that can be used to map long swell parameters in unprecedented detail. Over the open ocean, SWOT provides measurements of radar
<figure><img src="image_3.png" /><figcaption>**Figure 1.** (a–c) Samples (40 km across) of the sea surface elevation (sea surface height) field relative to the reference geoid and detrended over 5 km tiles. Arrows indicate the direction of North (N), satellite flight (Vsat) and radar look direction (Look). The location of these samples, around Australia, are marked $S_1$, $S_2$, $S_3$. Background colors in panel (d) show the maximum modeled value of $H_{18}$ for the entire month of June 2023, overlaid with Surface Water Ocean Topography ground tracks (red lines, numbered from 1 to 27) and a selection of in situ buoys (pink lines in Indian Ocean and squares along tracks 15 and 19).</figcaption></figure>

backscatter ($\sigma_0$) and SSH with a 250 m pixel size across a 120 km swath (with a center gap of 20 km). Figure 1 shows three samples of the ocean surface elevation, labeled $S_1–S_3$, each 40 km across.

These samples range from an apparently simple swell case in Figure 1a to barely visible features with a shorter wavelength in Figure 1b, and crossing swells in Figure 1c. We will quantify the magnitude of these waves with $H_{18}$, the wave height for waves longer than 18 s. Using buoy or model data this is defined as 4 times the square root of the integral of the wave frequency spectrum $E(f)$ from a frequency $f_{min} = 1/30$ s to a frequency $f_{max} = 1/18$ s. For SWOT data, this is taken as 4 times the square root of the integral—restricted to swell signals—of the wavenumber spectrum $E(k_x, k_y)$ for a spatial frequency $k = \sqrt{k_x^2 + k_y^2} < 2$ cycles per kilometer (cpk), that is, wavelengths larger than 500 m.

The long waves in Figures 1a–1c are all coming from the same storm. That storm was the most powerful on the globe for the months June–August 2023 as illustrated on Figure 1d. The maximum value of $H_{18}$ exceeded 10 m, and the total significant wave height reached $H_s = 17.2$ m at 54°S, 86°E and 15:00 UTC on June 6, according to a wave model hindcast (Ardhuin & Accensi, 2024).
We expect that SWOT data will bring fresh insights on a wide range of physical processes from the generation of waves in storms to their propagation over currents and shallow topography. From March 2023 to July 2023, SWOT acquired data over the same orbit every 23.8 hr. We use that data to illustrate SWOT's wave-resolving capability, and provide the first validation of SWOT swell measurements (sample S₁), large scale evidence of coastal wave reflection (sample S₂) and island scattering (sample S₃). Reflection is locally relevant for nearshore morphodynamics, and probably dominant at global scales for the generation of microseisms with frequencies around 0.1 Hz (Ardhuin & Roland, 2012). Scattering by shallow water around small islands has not yet been explored scientifically, but traditional navigators from the Marshall Islands know this effect very well and use it for detecting islands beyond the horizon (Genz et al., 2009). Here we perform a similar detection using evidence of waves arriving from the 15 km wide Norfolk Island group located 300 km from a SWOT wave measurement. In Section 2 we detail an automatic method for retrieving swell parameters that works well for cases like sample S₁. In Section 3 we discuss samples S₂ and S₃ that we interpret as evidence for coastal reflection and scattering. Conclusions and perspectives follow in Section 4.

## 2. Interpretation and Validation of SWOT Long Wave Signals

### 2.1. SWOT Processing and Analysis

We have used the Low Resolution (LR) “Unsmoothed” SWOT product (SWOT project, 2023), from the KaRIN interferometer, with pixels posted at approximately 250 m in each direction. In LR mode, the interferogram, whose phase is utlimately converted to surface elevation, is averaged on-board using a 500 m-wide filter. This filter combines with an effective point target response, so that the SWOT data is a convolution of the true SSH. The same convolution is performed on the single-channel powers to produce σ₀ on the same grid. To first order, components of the sea level power spectral density (PSD) at any spatial frequency (kₓ, kᵧ) should be equal to the wave spectrum multiplied by a transfer function G(kₓ, kᵧ). We have chosen the x-axis oriented cross-track and the y-axis along-track. As we illustrate in this paper, the Unsmoothed KaRIN product resolves long swells, although with an amplitude heavily suppressed: as an example, G is typically 0.2 for cross-track propagating waves with kₓ = 1 cpk and kᵧ = 0 meaning that for this wavelength of 1 km, the true amplitude of waves is larger than the amplitude of the SSH field by factor 1/√0.2 = 2.2.

Also, Doppler Beam formation in the presence of random velocities along the line of sight leads to a loss of resolution in the azimuth direction giving a multiplicative term exp(-k²yλ²c) that is included in G, where λc is the same azimuthal cut-off wavelength as any other SAR imaging systems, and is a function of the mean wave period Tm0.2 and significant wave height Hs (Stopa et al., 2015). More details are given in the Supporting Information (see Figure S1 in Supporting Information S1 for G and its different components). Finally, because of the radar integration of echoes on iso-range surfaces, wave signals are generally distorted by the non-linear mapping from range to the cross-track dimension x. That “range bunching”, also called “surf board” effect (Peral et al., 2015), is strongest near storms.

For the cases shown in Figure 1, we expect that range bunching can be neglected and we estimate the wave spectrum E from the SWOT SSH PSD Es as

$$E(k_x, k_y) = E_s(k_x, k_y)/G. \qquad (1)$$

Our processing starts with computing the SSH PSD from the largest possible continuous square region in the SWOT swath, using 40 km by 40 km samples (as in Figure 1), in the middle of each left and right sub-swath, using a two-dimensional Welch (1967) method with tiles of 5 by 5 km and 50% overlap. This provides coarse spectra at low wavenumbers (as in Figure 2), but very accurate spectral densities and SSH-σ₀ cross spectra, with 128 degrees of freedom. The only delicate operation is to define a “swell mask” where Equation 1 is valid, avoiding parts of the spectrum with strong non-linear contributions around k = 0. For automatic processing, we rely on the expected coherence between the SSH and σ₀, as shown in Figures 2g–2i.

We identify as swell those components that have a σ₀-SSH coherence higher than 0.05, and we dilate that region of the spectrum by one pixel to facilitate comparison with model output that may have slightly shifted peaks. This works well for the easy cases such as sample S₁.
<figure><img src="image_4.png" /><figcaption>**Figure 2.** Top: power spectral density $E_S(k_x, k_y)$, second line: after filter correction $E(k_x, k_y) = E_S(k_x, k_y)/G$, third line: coherence from the $\sigma_0$-SSH cross spectra, bottom line: associated phase. The black polygons show the edge of the “swell mask” using an automatic method: waves in the mask are in the direction from which they are propagating. Colored polygons in panels (b–d) are alternative swell masks defined a priori.</figcaption></figure>

Finally, we restrict our swell mask to directions from which the waves arrive. For this we assume that for any wavelength and direction there are no waves in the opposite direction, and we use the phase of the SSH-$\sigma_0$ cross-spectrum to determine the direction. For waves propagating in the cross-track direction, $\sigma_0$ is maximum on the slope of the wave facing the radar, hence a cross-spectrum phase near 90° (red) for $k_x > 0$ and -90° (blue) for $k_x < 0$. Doppler processing on KaRIN modifies the maps of SSH and $\sigma_0$: compared to a true geographical mapping, the velocity bunching effect displaces the ocean surface along the satellite track, which produces oscillations of $\sigma_0$. These $\sigma_0$ oscillations are out of phase with the SSH for waves that propagate in the direction of the
satellite flight (Collard et al., 2022), hence a darker tone of red or blue in the cross-spectrum phase (Figures 2j and 2l). When waves propagate in the direction opposite to the satellite motion, SSH and $σ_0$ are in phase, giving a lighter tone of red or blue, as is the case in Figure 2k. This expected phase relationship indicates that the dominant swell direction was from the South in sample $S_1$ and $S_3$ (darker color, satellite flies to the North) and from the North for the magenta component in sample $S_2$ (lighter color). For waves propagating exactly in the cross-track direction, this method cannot separate between waves from the left or from the right (red box in Figure 2c).

## 2.2. Validating SWOT Resolved Wave Heights, Wavelengths and Directions

Because few buoys are within or near the SWOT swath, we use a two-step validation. To validate SWOT we use a numerical wave model simulation, that provides a complete spatial coverage (Ardhuin & Accensi, 2024). This model uses the configuration T707GQM described in Alday and Ardhuin (2023), and has been extensively validated with satellite altimeter and in situ buoy data. The spatial resolution is 0.5 by 0.5°, with a 15° angular resolution and exponentially spaced discrete frequencies, with a relative increment of 1.1 from one frequency to the next. Two features of this model make it unique: it includes coastal reflections even if very crudely, and it uses accurate nonlinear four-wave interactions (Gagnaire-Renou et al., 2010; Lavrenov, 2001).

As models can have large scale biases on swell parameters, first we use buoy data to validate this numerical wave model. Away from islands and major currents, swell fields are expected to vary smoothly over large scales (Delpey et al., 2010). A few buoys, even hundreds of kilometers from a SWOT swath, measure a model bias on $H_{18}$ that is relevant to estimate the bias in that SWOT swath. The Southern Ocean Flux Station (SOFS) was positioned in the right swath of SWOT track 19, at 47.15°S 141.3°E. It is equipped with a dual-frequency geodetic quality Global Navigation Satellite System (GNSS) receiver, providing accurate wave measurements (Hay et al., 2023). The low frequencies of the heave spectrum have a white noise level around 0.01 m²/Hz, corresponding to a noise floor of 10 cm for low frequency swell heights. Spotter buoys (Raghukumar et al., 2019) from South-East Indian ocean provide a good sampling close to the storm (Houghton et al., 2024). These buoys use the less noisy GPS velocities instead of vertical positions, and thanks to their free drift and in spite of a lower quality GPS receiver compared to the SOFS mooring, they also achieve a noise floor of the order of 10 cm for $H_{18}$. Spotter buoys data are processed with a high pass 4th order Butterworth filter that removes spurious energy below the critical filter frequency of 0.033 Hz. This has a less than 0.1 dB impact for wavelengths shorter than 2 km and can thus be neglected. Additional data corresponding to sample $S_4$ (Figures S2–S3 in Supporting Information S1), were obtained from the moored Coastal Data Information Program station 166 located at 49.9 N 145.2 W (Thomson et al., 2013).

Data transmission constrains the spectral resolution of the Spotter buoys to 0.01 Hz. The model output was interpolated on the same Spotter frequency grid and $H_{18}$ was estimated from the sum of the wave spectrum over the lowest 3 frequency bands, corresponding to a cut-off frequency of 0.0537 Hz (a period of 18.6 s). The peak of the swell event happens on June 6 for the region East of 90°E and South of 50°, and at later times, up to June 10 at the location of sample A, as the swell propagates to the East and North. Figure 4a shows that the model is generally consistent with buoy data when the local maximum value of $H_{18}$ occurs. One exception is to the North-West of the model 1.0 m contour, where the model underestimates $H_{18}$ by up to a factor 2. This result is specific to our use of accurate nonlinear wave parameterization (Lavrenov, 2001) and probably reveals deficiencies on the parameterization of other wave evolution processes in storms (Alday & Ardhuin, 2023). This topic is beyond the scope of the present paper. To the South of 40°S, the model slightly overestimates the maximum values of $H_{18}$.

Time series in Figures 3b–3e reveal more details. At buoy 010,379 (panel b), the agreement is generally good on June 8, followed by a model underestimation on June 9 and 10, except for the model with coastal reflection after 06:00 UTC on June 10, with model and buoy giving values around 20 cm. Without SWOT data, we would have discarded this as a coincidence. Directional buoy parameters are too noisy for such low wave heights and long period and do not provide a reliable estimate of the mean wave direction. A similar 20 cm reflected component is predicted by the model at the location of sample $S_2$, consistent with buoy energy levels, and the model even predicts up to 40 cm along SWOT tracks 8 and 19 near sample $S_1$.

We now compare SWOT data with buoy and model values. The modeled spectra are transformed to be comparable to the SWOT SSH spectra shown in Figure 2: Each spectrum is (a) regridded in wavenumber space to a resolution 3 times finer than in Figure 2 and for a larger domain from -4 to 4 cpm in both directions, (b) made centrally symmetric, (c) multiplied by G, (d) folded to the interval [-2, -1] cpm to reproduce aliasing, (e) block-averaged to the same resolution as in Figure 2 (0.2 cpm along the x direction) to produce a filtered spectrum $E_s(k_x)$,
<figure><img src="image_5.png" /><figcaption>**Figure 3.** (a) Map showing the maximum value of *H*<sub>18</sub> for June 6 to 12 June 2023 from the Spotter buoys (dots and written values from 0.7 to 4.9 m), and from the wave model (contours). Thick contours show the model maximum between June 1 and June 12. Locations of Surface Water Ocean Topography (SWOT) sample data are indicated by *S*<sub>1</sub>–*S*<sub>6</sub> (b–e) time series of *H*<sub>18</sub> at Spotter buoys with identification number 010,379, 010,351 and 010,253, and Southern Ocean Flux Station buoy, and co-located model values, these buoys are closest to SWOT data samples, respectively. Light blue lines “NOREF” corresponds to the same model without coastal reflection.</figcaption></figure>

$k_y$), and (f) divided by $G$ to give an estimate of $E(k_x, k_y)$. From these SWOT and model spectra, $H_{18}$ is computed from twice the spectrum in the swell mask, to account for double-sidedness. The swell direction $\theta_{18}$ is taken as the direction of the energy-weighted mean wavenumber vector, and the wavelength $L_{18}$ as the ratio of zeroth and first moment of the two-dimensional masked wavenumber spectrum (Ardhuin, 2024). We note that the azimuth cut-off filter associated with velocity bunching, which is part of the transfer function $G$, requires some estimation of the root mean square orbital velocity $w_{rms}$, including waves not resolved by SWOT. We use the modeled $w_{rms}$, which was validated to be within 10% of buoy-derived values.

We start with a simple example: sample $S_1$ of the SWOT track number 19 around the SOFS mooring, for cycle 546 on June 9, 2 UTC, as shown in Figure 1a. The 1.1 m value for $H_{18}$ is consistent with buoy and model data. Figure 4a gives a broader context (Movies S1 and S2 for more details). Sample $S_1$ has a mean wavelength of 760 m, and this value increases along the SWOT track toward the Australian coast, where it exceeds 1,000 m in deep water, corresponding to a wave period $T = 25.3$ s. As SWOT flies to the North, it catches up with the swell forerunners that have traveled faster and further. This is consistent with linear wave dispersion that gives a deep water group speed $C_g = gT/(4\pi)$ (Munk, 1947). Our SWOT-derived estimate of $H_{18}$ closely follows the model value decreasing from 1.0 m to under 10 cm. To the South, we note that between latitudes 46 and 53°S, the mean direction $\theta_{18}$ veers by $\pm$ 6°. Since the left and right swath give two independent measurements at locations separated by 60 km, their correlation shows that these fluctuations in wave direction are not random but associated with features of the wave field. The local minimum of $H_{18}$ from SWOT, at 48°S, is consistent with refraction over a current jet located upwave of the measurement location, similar to cases in Drake Passage discussed by Ardhuin et al. (2017). The maximum rotation of the wave direction occurs at locations consistent with that upwave current and the local current maximum (Villas Bôas & Young, 2020). Currents were not taken into account in our model simulation.

South of 50°S, the model overestimates the SWOT $H_{18}$ values, which can be due (in part) to a model bias in that region, or a bias in our estimate of the $G$ transfer function for wavelengths under 700 m and in the presence of a strong windsea. In sample $S_1$ this wind sea, unresolved in the LR SWOT data, gives a total wave height of 5.1 m as measured by the buoy.
<figure><img src="image_6.png" /><figcaption>**Figure 4.** Examples of (a) swell height $H_{18}$, (b) swell wavelength $L_{18}$, (c) swell direction $\theta_{18}$ and (d) azimuthal cut-off $\lambda_c$ along Surface Water Ocean Topography track 19, South of Australia on June 9th, around 2:10 UTC. Same quantities in panels (e-h) for track 21, South of Australia on June 11th.</figcaption></figure>

3. Coastal Reflection and Scattering: From Big to Small Islands

Guza and O'Reilly (2001) performed the only study that found evidence of coastal reflection in wave heights measured in deep water at a location sheltered from the offshore direct wave propagation. Otherwise reflected waves have been incredibly challenging to observe. In general, large scale shoreline reflection coefficients for the wave energy are very poorly known and poorly constrained by microseism measurements, giving energy reflection coefficients values $0.02 < R^2 < 0.1$ (Stutzmann et al., 2012). Our numerical simulation with a constant $R^2 = 0.1$ was thus expected to overestimate reflection.

Sample $S_2$ is not the most obvious case of reflected waves in SWOT data, but it is easy to analyze quantitatively due to the absence of incident waves at the same time (they can be seen on the previous day). Unfortunately the directional parameters from the buoy are noisy and the directional spread, around 60° (Figure S6 in Supporting Information S1), could be interpreted as a nearly isotropic directional spectrum or can be biased high due to noise (O'Reilly et al., 1996). In contrast, the SWOT cross-spectrum phase with values under 45° in Figure 2k for the swell peak, clearly indicates a dominant wave direction from the North.

For sample $S_2$, our automatic processing gives SWOT wave heights that are biased high because, once corrected for the transfer function G, the swell peak blends with the noise on the edge of the spectral domain at ($k_x$ = 0, $k_y$ = 2) (Figure 2e). As a result we have used a fixed “alternative mask”, the magenta box in Figure 2b.
It is noteworthy that $L_{18}$ clearly increases from 570 m at 41°S to 670 m at 52°S, consistent with a dispersive propagation of waves from North to South. Overall, the buoy and SWOT data are consistent with the model reflected components coming from the region around Cape Leeuwin. Our model parameter choice $R^2 = 0.1$ is surprisingly accurate for this region and those waves.

It is possible that the opposite trend of $L_{18}$ at latitudes 57–54°S, increasing from South to North, is actually due to reflections off small icebergs that are very numerous around 62°, and appear as strong bright/dark dipoles in maps of $\sigma_0$ (Movie S2). For other regions, such as the Great Australian Bight, the model overestimation of $H_{18}$ at buoy 010,253 on June 10 (Figure 3d), and the lower SWOT values on track 19 (not shown) suggest that the reflection coefficient should be smaller, probably associated to fewer cliffs and less steep beaches along the shoreline encountered by these reflected waves (Ardhuin & Roland, 2012).

The presence of waves is much easier to detect for the longest swell components and when the noise level is lower, such as in sample $S_3$, acquired North of New Zealand on track 17. In that case, the main peak (within the black polygon on Figure 2c) is associated with a height $H_{18} = 0.30$ m for both model and SWOT. A weaker component coming from East or West is barely visible, and is highlighted by the red square in Figure 2f. That cross-track component is found consistently along the SWOT track from 30.7 to 28.7°S. Outside of that area, the SSH spectral density in the red square in Figure 2c) is 18–20 dB (60–100 m$^4$), giving a background noise wave height of 2 cm. This value rises to only 4 cm in sample $S_3$: subtracting the background noise gives a wave height corrected to $\sqrt{(4^2 - 2^2)} \approx 3$ cm.

Where are these 3 cm waves coming from? Although we cannot lift the 180° direction ambiguity from sample $S_3$ alone, the rotation of the mean direction between neighboring samples is consistent with a localized source to the West of the SWOT track, with waves veering from an azimuth of 280° to 260° over a distance of 100 km along the track (Movie S4). Assuming a circular wave front, this rotation would correspond to a radius of 270 km. At this latitude, the SWOT track number 4 lies 300 km to the West of track 17, providing data 13 hr after sample $S_3$. Indeed, along track 4, from 29.4 to 28.4°S, SWOT covers part of the shelf of Norforlk Island, which is much wider (100 km) than the island itself (15 km). SWOT swells have a much broader directional spectrum at the edge of the shelf (see sample $S_7$ Figures S2–S3 in Supporting Information S1 and Movie S4). Therefore the crossing sea pattern in $S_3$ can only be reasonably explained by refraction over the shelf surrounding Norfolk island, an effect not yet considered in global ocean wave models.

The wider variety of wave patterns in both SSH and $\sigma_0$ fields, and amazing uninterrupted coverage provided by SWOT is more fully visualized in Movies S1–S4, and Figure S2 in Supporting Information S1. Sample $S_4$ taken on June 17 near station PAPA, 17,000 km away from the storm, is shown in Supporting Figure S2 in Supporting Information S1.

## 4. Conclusions and Perspectives

As demonstrated by Fu et al. (2024), SWOT resolution and accuracy is providing an order of magnitude improvement on the measurement of “ocean topography”, compared to nadir satellite altimeters. That ocean topography contains the large scale component of the surface elevation that is related to currents by the geostrophic balance. However, SWOT data actually contains many other components of ocean dynamics, in both maps of surface elevation and radar backscatter. We have shown that SWOT data contains useful and accurate information on swell height, wavelength and direction, for the infrequent but important cases with wavelengths in the range 500–1,050 m. The swell detection level varies with wavelength and other factors: in calm areas a swell height of 3 cm and 1 km wavelength was detected, far beyond the reach of existing remote sensing techniques using radar or optical imagery, and even beyond the reach of standard surface oceanographic buoys. A detailed validation will require instruments and processing more precise than standard wave buoys. Other instruments such as underwater pressure recorders are sensitive enough but usually mounted on the bottom and can only measure swells in shallow enough areas that are not necessarily representative of open ocean conditions (Snodgrass et al., 1966).

The main limitation of the global SWOT data is the 250 m pixel size. Computing a surface elevation spectrum on board, which could have given access to shorter waves before data is averaged, is not feasible on SWOT but may be possible on future similar satellites. This additional information would be most useful for mapping shorter
swells and refining the surface height error model. That possibility will be investigated using SWOT High Resolution data, available in many coastal regions and few offshore locations. Besides, a further analysis of the performance of SWOT for swell mapping will benefit from ongoing work to define a model for the SWOT “background noise”. This noise contains many effects including a nonlinear transformation of the wave spectrum (Peral et al., 2015), and it may have a level comparable to the swell signal studied here, as shown in Figures S2 and S3 in Supporting Information S1 with samples S<sub>5</sub> and S<sub>6</sub>.

SWOT data is opening a new era of measurements for kilometer-long waves allowing many new applications. For example, SWOT is uniquely capable of mapping swell fields reflected or scattered from islands of all sizes, a phenomenon that is still poorly quantified and modeled. That capability will lead to a clear improvement of models for microseism sources, which are limited by our knowledge of ocean wave reflection at coasts or icebergs. These microseism models are now used to map the Earth's mantle (Kato & Nishida, 2023; Nishida & Takagi, 2016; Tomasetto et al., 2024), and may also be used to probe stratospheric winds (De Carlo et al., 2021; Vorobeva et al., 2024). Conversely, by providing a calibration point for recent swell events, SWOT data can now be used to further constrain historical records of ocean storms based on microseism records (Aster et al., 2023).

## Data Availability Statement

The SWOT data used in the study are version 2.0 of Level-2 KaRIn Low Rate SSH Unsmoothed (SWOT project, 2023). The SWOT L2LR_SSH data product is produced and made freely available by the joint SWOT (NASA/JPL and CNES) project. L2LR_SSH product quality is not final and will be affected by some evolutions as the SWOT project team makes progress on science data processing algorithms and instrument calibrations. SWOT project (2023). The buoy data used for model validation (Houghton et al., 2024), and the model output (Ardhuin & Accensi, 2024) are available from SEANOE under CC-BY v4 licence. The swellSWOT tool box used for present analyses is also available via SEANOE under CC-BY v4 licence (Ardhuin, 2024).

## Acknowledgments

We are very grateful for suggestions by T. Farrar, B. Chapron, T. Postec, and two anonymous reviewers, and technical support from M. Accensi. This research was made possible by the dedication of the entire SWOT mission crew and the support of U.S. and French taxpayers. SOFS data were sourced from Australia's Integrated Marine Observing System (IMOS)—IMOS is enabled by the National Collaborative Research Infrastructure Strategy (NCRIS).

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## References From the Supporting Information

Almar, R., Bergsma, E. W. J., Thoumyre, G., Baba, M. W., Cesbron, G., Daly, C., et al. (2021). Global satellite-based coastal bathymetry from waves. *Remote Sensing*, **13**(22), 4628. https://doi.org/10.3390/rs13224628

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# Supporting Information for "Phase-resolved swells across ocean basins in SWOT altimetry data: revealing centimeter-scale wave heights including coastal reflection"

DOI: 10.1002/

Fabrice Ardhuin¹, Beatriz Molero², Alejandro Bohé³, Frédéric Nouguier¹,

Fabrice Collard⁴, Isabel Houghton⁵, Andrea Hay⁶,⁷, Benoit Legresy⁷,⁸,⁶

¹Univ. Brest, CNRS, IRD, Ifremer, Laboratoire d'Océanographie Physique et Spatiale (LOPS), IUEM, Brest, France

²CLS, Ramonville St Agne, France

³CNES, Toulouse, France

⁴ODL, Locmaria-Plouzane, France

⁵Sofar Ocean Technologies, San Francisco, CA, USA

⁶School of Geography, Planning, and Spatial Sciences, University of Tasmania, Hobart, Tasmania, Australia

⁷Commonwealth Scientific and Industrial Research Organisation, Hobart, Tasmania, Australia

⁸Integrated Marine Observing System, Hobart, Tasmania, Australia

## Contents of the Supplementary information

1. Introduction
2. Figure S1: filters on the sea surface elevation corresponding to SWOT and Spotter buoy processing.

3. Figures S2 and S3: similar to Figs. 1 and 2, for different SWOT samples.

4. Discussion of SWOT filtering effects that apply to swell measurements.

5. Figure S4: illustration of aliasing in SWOT data (sample $S_5$).

6. Discussion of spectral processing choices.

7. Figure S5: alternative SWOT spectral processing, effect of spectral resolution.

8. Figure S6: Spotter buoy spectral parameters at the location of sample $S_2$: no clear evidence of reflection.

9. Figure S7: Comparison of CFOSAT measurements with SWOT along track 17.

**Additional Supporting Information (Files uploaded separately)**

Movies S1–S4 provide context for samples $S_1$, $S_2$, and $S_3$. These movies are also available as part of a broader Youtube playlist "The Ocean wears Denim: "Rosemary" storm. We follow wind-generated waves in SWOT sea level in June of 2023, across the Indian and Pacific Oceans" available at

https://www.youtube.com/ playlist?list=PLm1sPhvTQhOxMjglmUjAyB28bSM45C2eQ

The movies are based on SWOT Level-2 KaRIn Low Rate SSH Unsmoothed, (v2.0). doi: 10.24400/527896/A01-2023.016. For context we used, the global Total (COPERNICUS-GLOBCURRENT), Ekman and Geostrophic currents at the Surface and 15m. E.U. Copernicus Marine Service Information (CMEMS). Marine Data Store (MDS). DOI: 10.48670/mds-00327 (Accessed on 29-03-2024), and numerical model output from WAVE-
WATCH III - LOPS (Alday & Ardhuin, 2023), including GQM quasi-exact 4-wave inter-
actions (Ardhuin & Accensi, 2024).

S1: SWOT observation of swell, from Antarctica to Australia, June 9th 2023 2:10 UTC: track 19, right side only. This video shows swell resolved in SWOT SSH LR data (in blue) and radar backscatter (grey shades) as it flies along track 019 from Antarctica to Australia, on June 9, 2023. Note the dark patches in radar backscatter (grey shades) are associated to rain cells. Swells are generally longer further away from the storm because long waves travel faster. At the end of the video, swells get much shorter after y=2700 km as we arrive is shallower water near the Australian coast.

S2: SWOT observation of swell, from Antarctica to Australia, June 10 2023 03:00 YTC: track 021, right side only. This video shows swell resolved in SWOT SSH LR data (in blue) and radar backscatter (grey shades) data, as the satellite flies along track 019 from Antarctica to Australia, on June 10, 2023. The video starts at the edge of the sea ice and goes to the open ocean. Many "small" icebergs are visible as "pimples" in the backscatter data, e.g. around y = 280 km. There is a bigger one at y=640 km. Note the dark patches (e.g. at y=1070 km) in radar backscatter are associated to rain cells. Swells are generally longer further away from the storm because long waves travel faster. Here the swell get shorter as we get closer to the coast of Australia indicating that these O(20 cm) high swells with wavelengths between 500 and 700 m have been reflected from the shoreline. The incident swells can be seen on the previous day. The direction of propagation of the swell is confirmed by the cross-spectral phase between sigma0 and SSH: due to velocity bunching, the phase is shifted from 90 and 270° (tilt modulation) towards 0 if the waves
travel in the direction opposite to the flight direction of the satellite. If the phase had
been shifted toward 180°, then the waves would have been dominated by propagation in
the same direction as the satellite. This swell is thus coming from storm Rosemary and
reflected off Australia around Cape Leeuwin.

S3: This video shows swell resolved in SWOT SSH (in blue) and radar backscatter (grey shades) data, as the satellite flies along track 017 over the Pacific ocean, west and North of Aotearoa New Zealand, on June 11, 2023, between 0 and 1:00 UTC, as part of cycle 548. The video captures waves reflected from land and/or scattered from shallow water areas. Note the ring-shaped waves from the East around y = 2500 km and from the West around y = 2900 km. In that second case the wave height of these waves from the West is only 3 cm. They come from Norfolk Island, 300 km to the West. Just like traditional polynesian voyagers could detect land over the horizon thanks to the small waves from the side, SWOT detects very small amplitude swells that reveal land and underwater topography.

S4: New Caledonia to NZ, June 11th 2023: track 4, left side only. Note the waves refracted / reflected off the New Caledonia lagoon around 23°S and Norfolk island shelf around 29°S.
# 1. Introduction

This Supporting information provides further context for the analysis of the 4 samples discussed in the main text, illustrates of other possible swell signatures with another 4 samples, and help visualize the breadth and detail of SWOT data coverage. Comparison with existing CFOSAT and Sentinel 1 observations of swell illustrate the ground breaking nature of the SWOT measurements. We also discuss some of the technical choices and trade offs in our spectral analysis.

# 2. Instrument filters

Estimating wave parameters from SWOT data requires to invert the filter that is applied to the true surface elevation field. Here we include the theoretical expressions for the on board processor (OBP) cross-track and along-track filters. In the along-track dimension we also include the point target response (PTR, together with the antenna pattern) and the effect of velocity bunching (Hasselmann & Hasselmann, 1991). SWOT interferometric phases are computed separately for 9 different beams obtained using Doppler Beam Sharpening, each with its own PTR (Styles & Dubois, 2023). The theoretical PTR for each of the 9 beams is well approximated by a sinc function. Their combination uses weights that vary along the orbit following the noise levels for each beam. A typical weight distribution is used here, as given in the python library lib_filters_obp.py (Ardhuin, 2024). The combination is not very different from taking the center beam only, but it provides a slightly stronger attenuation of short wave components: as a result, using the center beam only produces a slight underestimation of the wave energy.
3. More samples, including a discussion of aliasing effects

Due to the sampling, signals with wavelengths shorter than ~500 m get aliased to longer
wavelengths with spatial frequencies $k_{xa} = k_x \pm 1/0.250$ and $k_{ya} \approx k_y \pm 1/0.235$. While the
$G$ factor decreases steeply beyond $k=2$ cpk, the wave spectrum often sharply increases
towards higher wavenumbers and the aliased signals can be significant, particularly along
the cross-track dimension. One such example is shown in Supporting Figure S4.

Also, Doppler Beam formation in the presence of random velocities along the line of sight leads to a loss of resolution in the azimuth direction giving an additional filter included in the spectral transfer function $G$.

Sample $S_4$ shows remote swells at a location for which we have independent buoy data for validation (Thomson et al., 2013). For sample $S_4$, the CDIP buoy gives $H_{18} = 15 \pm 2$ cm in the 3 hours around the SWOT pass (near 23:20 UTC), with a dominant direction from the south, and the SWOT estimate for $H_{18}$ is 15 cm.

$S_5$ and $S_6$ are closer to the storm and are also located where buoy data is available (Houghton & Ardhuin, 2024). $S_5$ is located along track 010 on June 8, around 19 UTC and very near Spotter buoy number 010379 at 41 S. In that case, possible aliasing and locally steep waves (contributing to both velocity bunching and range bunching) make the general inversion of the wave spectrum problematic. Here the SWOT estimates of $H_{18}$ are underestimated by a factor 2 compared to the model, while the model is overestimating the buoy data by no more than 20%. We expect that the difference is caused by an underestimation of both the along track filtering (i.e. the theoretical PTR may be biased),
and across-track filtering (we neglected range bunching which clearly redistributes the surface elevation variance to other cross-track wavenumbers).
4. Spectral analysis: choices and trade-offs

Several details can be modified in the spectral analysis method, with different possible
choices depending on the application.

In the present study, our main goal was to separate "noise" from "signal" as clearly as
possible so that 1) the signal is clearly visible in the figures (not cluttered by fluctuations
of spectral estimates) 2) to automatically estimate wave parameters. With these goals in
mind, we have used the largest possible area, while avoiding the edges of the subswath
where SSH errors are larger, and we have thus settled for 40 by 40 km samples. Given
the large scale homogeneity of the swell fields, we could have used even larger along-
track distances for averaging spectra. At the same time we expect some possible effect of
currents and islands, so that the averaging should be limited. We have not tested how
small the sample analyzed can be. Looking at wave transformation in shallow water, it
may be possible to infer water depth by looking at only a few wavelengths (Almar et al.,
2021).

Over our large (40 km) samples, one could use high spectral resolution. The main
benefit of our choice of a coarse spectrum is the low uncertainty of the spectrum and
cross-spectrum of SSH and σ₀ which we used for automatic swell detection. In practice
this has little impact on wave parameters as illustrated for a piece of cycle 547 track 8,
from Java to Northwest Australia, using the left swath (Fig. S4). The few outliers around
11°S are caused by swell mask errors in the presence of other features (probably internal
waves). The coarse spectral grid has another drawback: as the filter response G strongly
:

varies beyond $k = 1$ cpk, the wave height estimate may have larger errors when the waves are shorter, which is what we are seeing close to the Australian coast.

Using a coarser grid for masking and a finer grid for the filter correction gives slightly less noisy results. In the end our choice of a very coarse spectrum, is mostly motivated by visualization: the mask is easier to see and understand on its original grid.

## 5. Evidence of reflection in buoy data

In the case of sample $S_2$, the first moment directional parameters from Spotter buoy 010351 are shown in Fig. S6. This buoy measurements could indicate either a low signal to noise ratio (which generally increases the directional spread) or the presence of waves in many direction, that can be caused by reflection.

## 6. Data from other remote sensing satellites: CFOSAT

The SWIM instrument on CFOSAT measures the spectrum of waves propagating in the direction of the radar beam line of sight from the modulation of the radar backscatter. For wavelengths shorter than 500 m, the modulation is well described by the tilt effect (Jackson et al., 1985). Here we have used an empirical modulation transfer function that strongly deviates from the theoretical tilt effect for wavelengths longer than 500 m.
<figure><img src="image_7.png" /></figure>

**Figure S1.** SWOT across-track filter and components of the along-track spectral filter G. The filter used in Spotter buoy processing is also shown for reference. The point target response (PTR) filter includes the antenna pattern. When the 9 beams are taken into account, the PTR is taken to be the weighted PTR of each beam using median beam weights from one SWOT pass as defined in the python library lib_filters_obp.py (Ardhuin, 2024). The filter associated to velocity bunching uses wave orbital velocities estimated from a wave model (Ardhuin & Accensi, 2024).
<figure><img src="image_8.png" /></figure>

**Figure S2.** (a,b,c,d) Samples (40 by 40 km) of the sea surface elevation (SSH) field relative the reference geoid and detrended over 5 km tiles. These samples were re-oriented to have the up direction close to north. Exact directions are indicated by the arrows: North (N), satellite flight (Vsat) and radar look direction (Look). (e,f,g,h) show the normalized radar cross section σ₀ which is related to the surface mean square slope and atmospheric attenuation (black areas here are areas with rain)
<figure><img src="image_9.png" /></figure>

**Figure S3.** Spectra from model and SWOT for samples S₄ to S₇. From top to bottom: modeled spectrum cropped at ±4 cpk; modeled spectrum transformed to SWOT filtering and sampling; SWOT SSH spectrum; SWOT SSH spectrum after correction for filter response *E*<sub>S</sub>(*k*<sub>x</sub>, *k*<sub>y</sub>)/*G*; coherence from the σ<sub>0</sub>-SSH cross spectra, bottom line: associated phase. The black polygons show the edge of the "swell mask" using an automatic method.

August 3, 2024, 4:21am
<figure><img src="image_10.png" /></figure>

**Figure S4.** Evidence of aliasing in SWOT SSH data: the spectrum of surface elevation is compared to a modeled spectrum that is first filtered and then aliased. This is sample $S_5$ from track 010 (left swath) cycle on 2023/06/07 around 41.2 S, see fig. S2.b and S3.b for further details.
<figure><img src="image_11.png" /></figure>

**Figure S5.** Influence of Welch tile size on the estimated wave parameters, illustrated with data along cycle 547 track 8, from Java to Northwest Australia, using the left swath. *L*₁ is the size of the tiles for the wave spectrum, *L*₂ is the size of the tiles for the swell mask (which is then interpolated on the finer spectral grid if *L*₂ is less than *L*₁).

<figure><img src="image_12.png" /></figure>

**Figure S6.** Spotter buoy number 010351 and model data corresponding to sample S₂, along track 21, cycle 547.
<figure><img src="image_13.png" /></figure>

**Figure S7.** Comparison of swell measurements by the 10° incidence beam of the SWIM instrument on CFOSAT (Hauser et al., 2021) on SWOT track 17, cycle 548 and 550. The SWIM data is L2S pre-validated V1.0 data using a modulation transfer function based on an empirical fit to model data (note that in the L2 CNES product the swell longer than 500 m is absent). (a) show the shifted positions of SWIM and SWOT measurements, west of New Zealand. SWIM acquisition occurred at 19 UTC, 5 hours ahead of SWOT. Swell partition wave height and wavelength. (b) Same information for SWOT cycle 550, with similar time shift. In order to compare the measurements taking into account swell propagation (the energy of 600 m waves travels 180 km in 5 hours) and location of measurements, the latitudes of SWIM data was shifted north by 3 degrees in (a) and 6 degrees in (b).

August 3, 2024, 4:21am
References

Alday, M., & Ardhuin, F. (2023). On consistent parameterizations for both dominant wind-waves and spectral tail directionality. *J. Geophys. Res.*, 128, e2022JC019581. doi: 10.1029/2022JC019581

Almar, R., Bergsma, E. W. J., Thoumyre, G., Baba, M. W., Cesbron, G., Daly, C., ... Lifermann, A. (2021). Global satellite-based coastal bathymetry from waves. *Remote Sensing*, 13(22), 4628. doi: 10.3390/rs13224628

Ardhuin, F. (2024). *Tools for the analysis of swells in swot satellite data - version 1* [Dataset]. SEANOE. doi: 10.17882/99836

Ardhuin, F., & Accensi, M. (2024). *Numerical wave model output for june 2023, co-located with swot and buoy data for the analysis of swells from storm "Rosemary"* [Dataset]. SEANOE. doi: 10.17882/99783

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