As mentioned above, the accuracy of shipborne acoustic Doppler current measurements is affected in a dominant way by the platform motion compensation process. In the spaceborne context, the platform velocity is almost 3 orders of magnitude larger (7000 ms-1 vs. 10 ms-1 for shipborne measurements). Thus, the accuracy requirements are tremendously exacerbated. In particular, attention must be paid to the detailed effects of the antenna radiation diagram and the sea surface normalized radar cross section (NRCS) variations with space and observation azimuth. A detailed discussion of these effects is given in Appendix .

In summary, in the case of a sufficiently narrow radiation diagram, VNG can be approximated as the radar carrier velocity projected on an effective look direction. This effective look direction differs from the geometric boresight direction by an effective azimuthal mispointing δφ due to the finite antenna beamwidth combined with the variations of NRCS within the radar footprint, as well as by an effective incidence angle mispointing δθ due to radar timing or surface-tracking errors.

The beamwidth at the working incidence angle is thus a very important parameter of a radar intended for TSCV measurements. Table  summarizes the parameters of the KuROS and KaRADOC antennas. For KuROS they have been determined following the procedure detailed in Appendix . For KaRADOC, they are the result of anechoic chamber measurements (Appendix ). As discussed in Appendix , these parameters describe the antenna radiation diagrams when expressed as functions of variables, α and β, which do not coincide with azimuth and incidence angle. In the case of constant-altitude flight and near-nadir observations with the antenna looking towards azimuth φb, one can, however, obtain a Gaussian approximation of the one-way radiation diagram as 2G≃exp⁡-φ-φb22sin⁡2(θ)σα2+β0-tan⁡(θ)tan⁡(θ)σβ2, where σα=α-3dB/8log⁡(2) and σβ=β-3dB/8log⁡(2). For 12∘ observations the second term in the exponential can safely be neglected, and the effective azimuthal beamwidth can be estimated as 3φ-3dB=α-3dBsin⁡(θ). When projected on the ground, φ-3dB is thus larger than α-3dB by a factor 1/sin⁡(θ), equal to 4.8 for 12∘ measurements. Provided that the beam is not too wide, the Gaussian approximation in Eq. () of G as a function of φ can then be used with the parameter 4σφ≃α-3dB/sin⁡θ8log⁡(2).

Due to the width of the azimuthal aperture, the NRCS-weighted line-of-sight azimuth φa can differ from the boresight azimuth φb by a mispointing angle δφ. Expressions for δφ are obtained in Appendix  in the two limiting cases of slow linear and fast sinusoidal variations of the ocean surface NRCS with respect to azimuth. In the slow variation case, δφ is obtained as 5δφ=φa-φb=12σα2sin⁡2θ1σ0∂σ0∂φ. Denoting by φt the flight track azimuth and Vp the along-track flight velocity, the spurious azimuth gradient Doppler (AGD) contribution to the DV caused by the mispointing reads 6UAGD=sin⁡φb-φtVp2σα2sin⁡2θ1σ0∂σ0∂φ.

As an example, Fig.  shows the variations of the two-way antenna radiation diagram G2, its Gaussian approximation and the G2σ0̃ (see Eq. ) product as a function of azimuth at a 12∘ incidence angle for a northward-looking KuROS antenna (φb=0∘), using σ0 data from the Drift4SKIM campaign on 22 November 2018. The effect of the wind-induced azimuthal gradient of σ0 is to shift the effective radiation diagram towards the brighter upwind and downwind directions, with an apparent pointing azimuth φa. The shift induced in this case is δφ=φa-φb=-0.81∘=-15×10-3rad. For comparison (see Sect.  and Table ), the pointing accuracy required to achieve a 15 cm s-1 error on the horizontal current in the airborne configuration is 1.2 mrad.

Here, it is important to note that KuROS was not specifically designed for this experiment but primarily as a calibration and validation instrument for the CFOSAT mission, which required a broad radiation diagram. Though the analysis of the KuROS data helped uncover many interesting effects relevant to Doppler observations of the sea surface, its design was not fully appropriate to validate the inversion of the geophysical velocities, for which the pencil-beam antenna diagram of KaRADOC was better suited.

Figure a shows a typical example of the azimuthal variation of σ0̃ at a 12∘ incidence angle for the Ku band. As expected for near-nadir measurements , the NRCS is largest in the downwind look direction (φ=320∘), has a secondary peak in the upwind direction and is weakest in the crosswind look directions. Figure b shows the corresponding UAGD contribution for the KuROS and KaRADOC cases using an aircraft velocity Vp=120ms-1 and the Ku-band NRCS fit for both instruments (this is a reasonable assumption for order-of-magnitude estimates). As detailed in Appendix , Eqs. () and () only apply for a narrow beam when projected on the ground, which is not a very good approximation for the KuROS case, even at a 12∘ incidence angle. As shown in Fig. , the Gaussian approximation for the antenna diagram as a function of φ gives a distribution that is too narrow and does not properly take into account the azimuthal integration, leading to an overestimation of UAGD. It is clear, however, that even the more exact Eq. () gives very large correction magnitudes, in excess of 1.2ms-1 in some azimuth ranges.

Because the azimuth gradient UAGD contribution to the observed DV is proportional to Vpσφ2, this effect is much larger (and correcting it is correspondingly more demanding in terms of antenna characterization) for KuROS than for KaRADOC or DopplerScatt thanks to their narrow azimuthal beam aperture. Another remark is that the approximate expression in Eq. (), though it gives the appropriate dependency of UAGD with respect to look azimuth, tends to overpredict its magnitude as the widening associated with the ground projection saturates for broad beams.

Although the relative variations ∂φσ0/σ0 are larger for larger incidence angles, this is more than compensated for by the 1/sin⁡2θ reduction in azimuthal diversity across the footprint. This effect can thus be neglected for much higher incidence angles .

The geophysical part of the DFS measured by a microwave radar over the ocean, using both along-track interferometry and Doppler centroid techniques, emerges from the average over the instrument field of view (FOV) of the backscatter-weighted line-of-sight projection of the surface velocity, as illustrated in Fig. .

In the well-understood case of decametric electromagnetic waves interacting with the sea surface at grazing incidence, the interaction is dominated by the Bragg coherent backscattering mechanism , in which the backscattered field reflects the properties (amplitude, phase speed) of a very finely selected component of the sea state, namely that whose wave vector is precisely equal to the so-called Ewald vector, the difference between the wave vectors of the scattered and incident electromagnetic waves. Exploiting the deviation of the phase speed of this sea state component from its theoretical value is the principle of the HF radars operationally used to measure ocean TSCV in coastal areas .

In the case of the near-nadir interaction of microwaves with the sea surface, which is the configuration considered for SKIM and used by the AirSWOT, KuROS and KaRADOC airborne instruments, this mental picture must be adapted: the Bragg scattering mechanism is not dominant, and the main contribution comes from quasi-specular reflections on those facets of the sea surface which are normal to the Ewald vector. The backscattering cross section of the sea surface and DFS in this case do not depend on the properties of a single Fourier component of the sea state but on the probability density function of the sea surface slope, which is a complex functional of its entire directional spectrum.

As discussed by , who applied it to the analysis of AirSWOT NRCS and DFS data collected during the Gulf of Mexico LASER experiment in 2016, the theoretical framework appropriate for this configuration is the Kirchhoff approximation . In this approximation, the geophysical DFS ωGD can be expressed as 7ωGD=-i∂τCC(τ)τ=0, where C(τ) is the temporal covariance function of the ensemble-averaged electromagnetic field backscattered in the direction of the radar.

Assuming Gaussian statistics for the sea surface, introducing ρ(ξ,τ)=η(x+ξ,t+τ)η(x,t), the space–time covariance function of the sea surface elevation, and QH=-4πλsin⁡(θ)eφ and Qz=4πλcos⁡(θ), the horizontal and vertical components of the Ewald vector (with λ the radar wavelength), one obtains C(τ) and ∂τC as 8C(τ)=∫eiQH⋅ξeQz2ρ(ξ,τ)-ρ(0,0)-e-Qz2ρ(0,0)dξ,9∂τC=Qz2∫∂τρ(ξ,τ)eiQH⋅ξeQz2ρ(ξ,τ)-ρ(0,0)dξ.

The clear upwind–downwind asymmetry of σ0 observed in the Drift4SKIM radar observations (see Fig. ) shows that the Gaussian assumption, which is unable to describe such skewness-related effects, is clearly questionable. It is, however, the only practical option, as going further would require prescriptions for the higher-order statistics of the sea surface, which are at present not available.

The occurrence of ρ as the argument of an exponential in these integrals renders further analytical progress difficult see, however,. Approximate expressions can, however, be obtained by performing a Taylor expansion of ρ in the neighborhood of the origin. This results in a Gaussian approximation of the integrand. The integrals can be readily evaluated, yielding (denoting by “⋅” the usual matrix product) 10ωGD≃-QHT⋅∇ξξρ-1⋅∂τ∇ξρ. The derivatives of ρ are taken at ξ=0 and τ=0, and they can be expressed as moments of the directional sea state spectrum Sd(k) as 11∂τ∇ξρ=msv,∇ξξρ=-Mss, where, in the notation of , msv stands for the mean slope velocity and Mss for the mean square slope matrix, 12msv=mssxtmssyt,Mss=mssxxmssxymssyxmssyy, with 13mssxαyβtγ=2∫R2kxαkyβωγSd(k)dk. The surface current enters through its effect on the dispersion relation of surface waves ω(k). In the presence of a vertically homogeneous current U a detailed discussion of the effect of shear can be found in, 14ω(k)=k⋅U+ω0(|k|), where 15ω0(|k|)=g|k|1+|k|2/κM2 is the dispersion relation of gravity–capillary waves in deep water, with k the wave vector and κM=363.2radm-1 the wavenumber corresponding to the gravity–capillary regime transition. Introducing this expression in Eq. (), and defining msv0 as the spectral moment obtained using the dispersion relation of Eq. () in Eq. (), one obtains the approximate DFS as 16ωGD=QHT⋅Mss-1⋅msv0+U and the corresponding VGD as 17VGD=-sin⁡(θ)eφ⋅Mss-1⋅msv0+U. While clearly oversimplified (it is, for instance, independent of the electromagnetic wavelength, which is known to have a significant influence on σ0), this expression has a definite pedagogical interest, as it allows one to distinguish a number of interesting features.

The raw velocity projection VGD accessible to Doppler radar instruments is composed of a “genuine” current Doppler (CD) component VCD, equal to the projection of the TSCV along the radar line of sight, plus a wave Doppler (WD) component VWD induced by the natural motion of the sea surface.

Figure  gives orders of magnitude for the natural range of variability of the different factors thus isolated. Figure a shows the variability of the Stokes drift velocity estimated following and using wind and directional wave measurements collected from 2010 to 2017 at Ocean Station Papa. Even though US is highly correlated with the wind speed with a Pearson's linear correlation coefficient of 0.85, a strong dependence on the long-wavelength part of the spectrum, for which Hs is a proxy, definitely has to be accounted for.

Figure b, taken from , instead shows the dependence on wind speed and Hs of Ku- and Ka-band effective mean squared slope mssshape retrieved from the GPM (Global Precipitation Mission) satellite measurements. The variability is even more strongly dominated by the dependence on wind speed, the variability due to the long-wavelength part of the spectrum being much smaller. These measurements very clearly show the filtering effect of the electromagnetic wavelength and are a clear warning that Eq. (), suggestive though it is, should be considered with caution.

Finally, Fig. c and d show the magnitude of the horizontal UWD component as a function of wind speed and Hs, estimated by numerically evaluating the integrals of Eqs. () and () for C(τ) and ∂τC using the numerical tools of on the basis of long-wavelength spectra extracted from global runs of the WAVEWATCH III model , completed in the high-wavenumber range by spectral tails. The shading in the background represents the histogram of the different (wind speed, MWD) pairs. As could be hoped for, the opposing influences of the wind speed on msv0 and Mss tend to counteract each other, greatly reducing the range of variability of MWD with wind. This effect appears stronger in the Ka rather than the Ku band, possibly due to the saturation of the Ku-band mssshape at high winds. These figures show a strong remaining impact of the long-wavelength waves, which clearly must be accounted for. As wind speed and significant height are highly correlated variables, the frequently encountered situations fall in a quite narrow interval MWD≃C0, with C0≃2.6 and C0≃2.2 m s-1 in the Ku and Ka band, respectively. In other words, most of the variability of UWD is controlled by the directionality effect, and the magnitude MWD is a weakly varying function of the wind, the wave age and the presence of swell see also.

A final remark is that, though these general patterns can probably be assumed to be robust, the precise numerical values depend on the parametric spectral shapes which have been used to fill the high-wavenumber range of the spectra. Changing, for instance, the high-wavenumber azimuthal spreading functions, which are for the moment not very well constrained observationally, has different impacts on the msv0 and Mss terms and can thus be expected to marginally change the numbers.

Considering the errors on the different terms to be independent, developing Eq. () allows one to derive the error variance of Doppler radar measurements of the TSCV as 18VarδUCD=UCDtan⁡θ2Var(δθ)+VarδVLOSsin⁡2θ+VarδVNGsin⁡2θ+VarδUWD. As a first step, four contributions to the uncertainty on UCD can thus be isolated, with different origins.

A first part corresponds merely to the error caused by imperfect knowledge of the projection angle between the TSCV and the line of sight. Its order of magnitude is controlled by the TSCV, and it is thus negligible with respect to similar terms that involve the platform velocity.

The Drift4SKIM experiment differs from previous airborne Doppler radar campaigns in two important respects: in order to observe the effect of wave development on the geophysical Doppler velocity UGD, it was performed in a midlatitude, eastern basin oceanic environment open to offshore swells. Also, given the campaign objectives of demonstrating the sensitivity of airborne radar Doppler measurements to the geophysical contributions of currents and waves, it comprised an extensive in situ component designed to have commonly accepted reference measurements for these parameters.

Fieldwork was performed in two areas (denoted by square boxes in Fig. ) named the “offshore” area, centered on the Trèfle buoy (see below), and the “Keller Race” area to the north of the island of Ushant. Both locations are in the range of coverage of a two-site WERA high-frequency radar system, operated by Service Hydrographique et Oceanographique de la Marine (Shom) and already used for several studies, in particular related to wave–current interactions .

Keller Race is an area with very strong horizontal gradients of the current . Although it is easy to show a strong effect of the current on the measured DFS, the spatial variability of the sea state is difficult to measure in situ, introducing uncertainties when combining UCD+UWD in a forward model or using UWD estimates when retrieving UCD from the measured UGD. The offshore area, on the other hand, was chosen for its spatial uniformity, being located far enough from the islands and with a near-uniform depth of 110 m. Only airborne data acquired over the offshore area are presented in this paper.

The week around spring tides in November 2018 was selected in order to allow for a wide range of current speeds (Fig. a).

The KuROS and KaRADOC radars were installed on an ATR-42 plane operated by the French institutional scientific flight facility, SAFIRE, which is equipped with an AIRINS™ GNSS-FOG INS providing position, pitch, roll and heading information with stated tolerances of a few centimeters, 0.005, 0.005 and 0.01∘, respectively.

Ground truth measurements comprised two permanent operational systems: the HF radar system mentioned previously, with an expected depth of measurement around 1 m , and the Pierres Noires (WMO no. 62069) wave-measuring buoy. Dedicated instrumentation was also deployed for the campaign.

The Trèfle buoy was moored at 5∘15′ W, 48∘15′ N at the center of the offshore area. This buoy monitored the surface current and provided directional wave spectra (Fig. ).

A wide range of geophysical conditions were encountered during the 1-week-long campaign. Four flights were performed over the offshore area on 21 November from 13:50 to 15:50 UTC, on 22 November from 12:15 to 15:00 UTC, on 24 November from 11:20 to 13:20 UTC, and finally on 26 November from 09:40 to 11:00 UTC. In this paper, we focus on data acquired on 22 and 24 November as the geophysical conditions were interesting and complementary (see below), and data were acquired with the largest azimuth diversity on these two days.

The 22 November flight took place at the end of a steady southeasterly wind episode (13 m s-1 from 140∘). The 24 November flight, in contrast, took place during a steady weak southwesterly wind period (5 m s-1 from 225∘) (Fig. b).

The wave height during the campaign was dominated by the presence of two swell systems from North Atlantic remote storms. The swell height decreased from 2.5 m on 21 November to 0.9 m on 24 November, with a peak frequency increasing from 0.07 to 0.1 Hz and a mean direction gradually veering from northwest to west. This swell has a small contribution to the Stokes drift of the order of 10 % of the wind-sea contribution on 22 November.

The main environmental conditions at the time of these star-pattern flights are summarized in Table .

KuROS is a Ku-band (13.5 GHz) pulse-pair Doppler radar with a dual antennae system and azimuthal scanning possibility, which was developed in the framework of the CFOSAT prelaunch studies. Of the two antennas, the low-incidence (LI) antenna is nominally centered on a 14∘ incidence angle, while the medium-incidence (MI) antenna is nominally centered on a 40∘ incidence angle. Only the LI antenna, which was the more relevant for SKIM, was used during the campaign. This antenna uses a horizontal transmit–receive polarization (HH) polarization. A comprehensive description of the system can be found in . A new antenna was used for the Drift4SKIM campaign, with characteristics given in Table .

The radar transmits a frequency-modulated pulse (chirp) with a 100 MHz bandwidth, achieving a 1.5 m range resolution and an effective ground-projected resolution of approximately 7 m (at 12∘). The one-way 3 dB footprint in azimuth is 580 m wide at 12∘ and 3000 m of flight altitude. The pulse repetition frequency (PRF = 1 / PRI) depends on the altitude and is 23 kHz when the aircraft flies at 3000 m. The ambiguity of the Doppler velocity measurement (see Sect.  in the Appendix) is about 126 m s-1, which is much larger than expected from the measurements (below aircraft speed of 120 m s-1). In order to reduce the thermal noise contribution, the range-resolved pulse-pair signal is coherently averaged in the instrument over 1 ms, corresponding to 22 pulse pairs per instrument sample. For the purpose of this article, this was further coherently averaged per blocks of 15 samples.

As discussed in Appendix , accuracy requirements for observation geometry are much less stringent for cross-track than for up-track and down-track Doppler velocity observations. The Doppler velocity data discussed in this article were all collected with the KuROS antenna in the port-looking orientation. This configuration also ensures an overlap with the KaRADOC footprint.

The Ka-band RADar for Ocean Current (KaRADOC) airborne radar sensor was developed for the Drift4SKIM campaign. KaRADOC is derived from the Still WAter Low Incidence Scattering (SWALIS) instrument, developed for the measurement of the NRCS of inland water surfaces in the Ka band. Further details on the system are given in Appendix .

KaRADOC was mounted under the ATR-42 aircraft in a port-looking configuration. The two-way 3 dB footprint from 3000 m of altitude over a flat sea surface is an ellipse with diameters 45 and 60 m in the cross-track and along-track directions, respectively. The antenna is of the slotted-waveguide type and allows steering of the beam in elevation (incidence angle) by varying the working frequency. Data were acquired at different incidence angles from 6 to 14∘, corresponding to a range of frequencies from 32.5 to 38.2 GHz. This article focuses on the observations collected at θ=12∘ at 33.7 GHz.

KaRADOC does not implement a range-resolution scheme: the transmitted pulses last several microseconds, and the whole FOV is illuminated simultaneously. The demodulated return signal is sampled at 15 MHz and archived. It is essentially constant while the electromagnetic wave is actually interacting with the sea surface. The useful signal segment is selected and its average is computed in order to reduce the thermal noise contribution, yielding one complex amplitude for each pulse. Several hundred pulses are sent at 4 kHz PRF for each burst of measurements, with a burst repetition frequency of the order of 5 to 10 Hz, depending on the number of incidence angles in the scanning sequence. These parameters were varied during the acquisitions. Though they have a strong impact on NRCS and DFS estimate quality, we have found the low-pass-filtered DFS signal to be robust.

The pulse-pair complex signal is averaged for each burst in order to reduce the effect of coherent speckle. One complex pulse-pair sample is thus obtained per burst. Even at the lowest burst repeat frequency of 5 Hz, the plane moves by less than a third of the FOV along-track extension between bursts.

The impact of the acquisition parameters on the KaRADOC measurement normalization is not yet fully understood, and the NRCS measurements could not be exploited in the scope of this study. The noise-filtered DFS measurements are, however, not affected by these normalization changes and are valid.

The KuROS NRCS–DFS imagery reveals a host of interesting features, modulations and dependencies. An in-depth analysis of all these processes is clearly out of the scope of this paper and will be the subject of forthcoming contributions from the Drift4SKIM team. This section thus only provides a cursory description of a few segments of σ0 and DV data collected on 22 November 2018 when the wind speed was approximately 11 m s-1, which are displayed in Fig. .

A first remark is that the NRCS is smooth, with a typical modulation depth of 1 dB after removing its mean trend as a function of incidence angle (Fig. a). This smoothness is in part due to the large footprint, but it also shows that the radiometric quality of the data and the coherent averaging performed are sufficient to control the thermal noise. Speckle noise is, however, still present, with different statistics depending on the radar look direction and the variable considered (not shown). The cross-track observation geometry leads to the best speckle noise reduction for the NRCS but to the worst-case speckle noise statistics for the DV.

The KuROS data clearly show a modulation in both NRCS and UGD associated with the northwesterly swell observed by the Trèfle buoy with a peak frequency of 0.07 Hz, corresponding to a wavelength L=320 m (Fig. ). This is particularly visible on the north–south-oriented flight tracks numbered 5 and 6 in Fig. b (see also Fig. f–g for a zoom-in on track 6). The apparent swell crest direction (dashed lines in Fig. b) differs from the true direction due to the scanning distortion effect , as the swell propagates during the measurements at a phase speed of 22 m s-1, while the aircraft moves at 120 m s-1.

The shorter waves measured by the Trèfle buoy (Fig. ) occupy a wide range of directions from a narrow wind-sea peak from the south at 0.16 Hz (L=60 m) to a broad directional distribution at 0.22 Hz (L=30 m), with a mean direction of 130∘ and a half-width (spread) of 45∘, hence covering directions from 85 to 175∘. These shorter components are present in the data from flight tracks 5 and 6 in the form of very narrow stripes with orientations shown by short dashed lines in Fig. b (see also Fig. f–g for a zoom-in on track 6). The “long-crested” appearance of the short waves in (d) and (e) is an artifact due to the wavefront-matching observation geometry , with all other directions averaged out by the large azimuth width of the radar beam. If purely geophysical, the phase relationship between the DV and NRCS modulations is expected to give the wave propagation direction. For flight track 6 in (f) and (g), the long swell propagates towards the radar, and the brighter slopes (white) correspond to eastward velocities toward the radar (blue). This will be discussed in further detail below. Finally, (h) and (i) exhibit chevron patterns with crests facing both northeast and northwest. Whereas the waves from the southwest are expected to be much longer than those from the southeast, this is not apparent in the KuROS data.

In this section we discuss the dependence of the Ku-band NRCS as a function of azimuth and incidence angle for the 22 and 24 November cases. The fixed-antenna and rotating-antenna data are presented. In order to reduce the dispersion introduced by the short-scale modulating processes discussed above, the data were averaged per 1∘ incidence angle and azimuth bins. As mentioned before, full tracks are straight and relatively long (12 km), and they view a mainly homogeneous ocean region. For the fixed-antenna observations, azimuthal diversity is obtained by performing tracks in different flight directions, forming a “star” pattern.

The variations of the Ku-band σ0 are shown for 22 November in Fig. . These measurements show the expected modulation of 0.8 to 0.9 dB with azimuth, with a downwind–crosswind contrast that increases with the incidence angle. This contrast is larger for the higher winds on 22 November. The upwind–downwind asymmetry is expected from the behavior of the surface slope probability density function . The exception are the σ0 values for the flight tracks with a fixed antenna around the azimuths 90 and 270 (Fig. a), which have anomalous normalized values between 1 and 1.3 instead of expected values much closer to 1. We have no explanation for this anomaly, which is genuine. No such anomaly was found for the rotating-antenna data collected later on the same day (Fig. b).

Discarding these azimuth ranges (shaded in gray in Fig. c), the data could be well fitted with a functional form a0+a1cos⁡(φ-φσ,1)+a2cos⁡[2(φ-φσ,2)]. As explained in Sect. , measuring this azimuthal variation is critical for the interpretation of the mean Doppler velocity due to the spurious azimuth gradient contribution. As expected, the fitted directions φσ,1 and φσ,2 are very close to the wind direction, except for the lowest incidence angles for which the contrast is less than 0.05 dB.

On 24 November, the σ0 azimuthal contrast was much weaker (Fig. ) due to the much lower wind speed and was actually not aligned with the wind direction when the measurements were performed.

We now quantitatively discuss the measured Doppler velocity signal in order to assess the agreement of our theory of the wave-induced contribution UWD with the measurements. This section is focused on the KaRADOC data, which are easier to interpret than the KuROS data due to the narrower radar beam of the instrument.

We present in Fig.  the low-pass-filtered UGD estimates retrieved from the 12∘ incidence angle KaRADOC data collected on 22 November between 12:13 and 12:59 UTC.

This representation is misleading, as much of the observed variability is in fact due to the effect of the flight track orientation. For instance, the largest contrast can be observed between the northeastward- and southwestward-directed flight tracks, even though to a first approximation a mere change in observation direction has occurred.

Another representation of the same data is proposed in Fig. . In this figure the UGD data are represented as blue lines shifted to the right of the plane ground track (in black) by an amount proportional to the instantaneous low-pass-filtered UGD value. This representation removes the trivial effect of observation direction changes and allows subtler effects to be better appreciated. For instance, noise-free observations of a constant vector UGD would appear as straight lines parallel to the flight tracks, all crossing at the tip of the vector. Deviations from this behavior, such as can be observed in Fig. , are indicative of measurement noise, geophysical variability or geophysical phenomena not accounted for by our theory.

For 22 November, 16 flight tracks are available collected from 12:13 to 12:59 UTC, and for 24 November 17 tracks were collected from 11:27 to 13:13 UTC.

Overall, the assumption of a constant vector is good to within 0.3 m s-1. It is particularly striking that the three horizontal lines in Fig. a are almost perfectly aligned, corresponding to two flight tracks looking into azimuth 0∘ and one flight into azimuth 180∘. On 22 November, the largest dispersion is for the 315 and 135∘ azimuths for which a total of four tracks are available with very different values that are, however, consistent along each track.

Using the average values from the different tracks, we compare the measured Doppler velocity to the forward model given by Eq. (), with UWD estimated from the in situ wave buoy data using the tools discussed in Sect. . The method combines the buoy spectrum up to 0.35 Hz and adds a high-frequency tail based on the spectrum, then computes numerically the integrals of Eqs. () and () to obtain the DFS estimate. The TSCV contribution, UCD, is taken to be the drift velocity of the nearest CARTHE drifter, which is uniform to within 3 cm s-1 in the offshore area (interactive animations of all deployments and trajectories can be found at https://odl.bzh/eVRHv1TE, last access: 25 October 2020).

Figure  shows the measured mean Doppler velocity and standard deviation for each track (the standard deviation is representative of the order of magnitude of the short-scale modulations due to waves, not of the error bar for the mean DV). On 22 November (Fig. a), the current vector accounts for less than half of the observed magnitude of UGD, and it is interesting that the maximum Doppler velocity is from azimuth 147∘, between the wind direction (128∘) and the up-current direction (183∘). The directions of the modeled and measured UGD are within 5∘ of each other.

Compared to the relatively high wind condition on 22 November, it is interesting to discuss the results for 24 November (Fig. b), with a wind speed of 5.5 m s-1 instead of 11 m s-1. The amplitude of the Doppler velocity is not much reduced, in spite of more than halved current and Stokes drift. This is consistent with the expected near-constant value C0 of the wave Doppler velocity magnitude, and this is the main result of the present paper.

The process leading to the estimation of the constant C0 is, however, dependent on a number of assumptions: the directional wave spectrum must be evaluated from the buoy data, then matched to a parametric spectral shape before the necessary numerical integrations can be performed. The spectral shape we have used depends on the wind speed and direction, but also on a wave age parameter, Ω, equal to 0.84 for equilibrium seas.

Table  summarizes a subset of the extensive tests we have performed to check the sensitivities of this process. It is clear from this table that drastically changing the wind speed, as occurred between the two days, affects the magnitude of the computed UWD more at the Ku band than the Ka band, but not in a catastrophic way, and that the wave age parameter Ω can also be varied over its meaningful range quite freely. We have also checked that the transition frequency at which the spectral tail is matched to the observational data is not a very sensitive parameter, provided it is taken low enough for the buoy data to be of good quality where they are kept.

Extracting directional wave spectra from buoy data, however, is a quite an intricate and subjective step. Several methods have been developed over the years to this end, each with pros and cons seefor a review. Two of the best-established methods are the maximum entropy method (MEM) and the maximum likelihood method (MLM). The MEM is a parametric method which assumes a specific form of the directional spreading function. In each frequency band, the parameters of the spreading function are chosen such that the first moments of the azimuthal Fourier spectrum match the buoy-derived ones. The MLM is a nonparametric method akin to the Capon beamformer. In terms of directional moments measured by buoys, the MEM estimates provide spectra that exactly fit the measured moments, while the MLM produces spectra that have directional spreads larger than those obtained directly from the measured moments. However, it is not clear how they compare on other properties of the spectrum that may be relevant to the mean slope velocity. Comparing results obtained with these two methods was thus a convenient way to test the sensitivity of UWD to the sea state directional spread.

As Table  shows, using one technique or the other to estimate the resolved part of the wave directional spectrum does induce significant differences in the simulated UWD values, showing that the azimuthal width of the spectrum, which is currently not very well constrained observationally, is a sensitive factor. The values obtained using the broader MLM spectrum are consistently smaller than those obtained using the MEM spectrum. A broader azimuthal distribution only redistributes the weight between the different Mss components but reduces the contributions composing the msv0 vector.

The results obtained using both methods are shown in Fig.  as light green and dark green lines. It appears that the MLM processing of the buoy data gives the best fit to the radar UGD, showing that the directional spread of the sea state should not be taken too low. The possibility that the directional distribution of the spectrum could be slightly too narrow for intermediate wavelengths of 2–10 m was, for instance, discussed in specific cases by . It is, however, not yet clear if it is specific to the very young wind seas they observed, although it could also explain some properties of L-band backscatter . The MLM was used to process the Trèfle data in the rest of this study.

Conversely, Fig.  illustrates the use of the DV data for the retrieval of the surface current vector by subtraction of UWD from the fitted UGD. The norm of the difference between the in situ measured and remotely sensed UCD vectors is less than 20 cm s-1 on both days, which is significant but quite satisfying at such an early stage of the technique, especially taking into account the fact that geophysical variability due to time variations of wind and tidal current occurred over the several hours of the flight.

Due to the much broader radiation diagram of the KuROS antenna, analyzing the Ku-band data requires significantly more effort, as the UAGD spurious velocity contribution due to the azimuthal variation of σ0 across the FOV discussed in Sect.  must be compensated for. The DV measurements, corrected for the UNG platform motion contribution but not for the UAGD contribution, are represented in Fig.  as red dots, while the green line represents the projection along the line-of-sight azimuth of the sum of the TSCV and the MLM-derived UWD vectors. The difference is clearly very large, reaching 2ms-1 in places on 22 November, and smaller on 24 November as the azimuthal modulation of the radar NRCS is much weaker.

Introducing the fits to the Ku-band NRCS data discussed in Sect.  in Eq. () allows one to produce the corrected data represented by the magenta dots, which are in much better agreement with the green line (though a constant offset is apparent in the 24 November data, which is rejected by the cosine-fit procedure). Figure  summarizes the UCD retrieval operation in vector form (the magenta arrow represents only the first azimuthal harmonic component of the UAGD correction). The norm of the difference between the in situ (in gray) and remotely sensed (in blue) estimates of the TSCV is of the order of 0.5ms-1 on 22 November and of the order of 0.2ms-1 on 24 November. Again, these numbers, though admittedly not small, can be considered encouraging given the number of very large corrections applied to the data and the fact that the instrument had definitely not been designed for this purpose.

The range-resolution scheme implemented in KuROS makes it a very interesting instrument for the analysis of DFS and NRCS modulations. In particular, , with a different antenna (slightly narrower beam), have attempted to use the cross-spectrum of the DFS and NRCS to resolve the 180∘ ambiguity in the wave propagation direction. In the SKIM context, analyzing the contribution of the resolved scales to the correlation between σ0 and the DFS could permit the development of empirical methods to estimate the unresolved part and provide estimates of UWD.

In practice, with the antenna used for the Drift4SKIM flights, another contribution to the DFS modulations is also caused by the gradients of σ0 and the speed of the aircraft, just like the mean spurious UAGD velocity. Brighter areas in the field of view tend to strongly influence the DFS signal towards positive values if they are located to the front of the aircraft and negative values if they are located aft of the aircraft.

As a test of this, simulations were performed with the Radar Sensing Satellite Simulator , which are illustrated in Fig. . The amplitude of the spurious modulations is enhanced by 70 % when the antenna diagram is made 50 % wider in azimuth. With typical variations of σ0 up to 1 dB over scales of the order of 1 km (e.g., Fig. ), the variation of σ0 with azimuth φ is roughly proportional to 1/sin⁡θ, giving a UAGD that does not vary much with θ, of the order of 1.5 m s-1. This spurious velocity is larger than the 0.5 m s-1 significant orbital velocity of the swell. As a result the phase relation between DFS and σ0 can change sign as a function of azimuth due to the combination of two imaging mechanisms with comparable magnitudes and possibly opposite signs.

This effect will be weaker for shorter (wind-sea) components as soon as the wavelength and crest length become much shorter than the KuROS footprint Ly, as given by Eq. (): for a given σ0 contrast, the gradient increases linearly as the scale L is reduced, but the UAGD for a given gradient is reduced exponentially in -Ly/L.