Towards the Mitigation of Discrepancies in Sea Surface Parameters Estimated from Low- and High-Resolution Satellite Altimetry

Abstract

In this study, we present an extension to existing numerical retrackers of synthetic-aperture radar (SAR) altimetry signals. To our knowledge at the time of writing this manuscript, it offers the most consistent retrieval of geophysical parameters compared to low-resolution mode (LRM) retracking results. We achieve this by additionally estimating the standard deviation of vertical wave-particle velocities ${\sigma }_v$ and a new parameter $u_x$, linked to a residual Doppler in the returned radar echoes, which can be related to wind speed and direction. Including this new parameter into the SAR stack retracker mitigates sea surface height estimation errors by up to two centimeters for Sentinel-6MF SAR mode results. Additionally, we found a closed-form equation to describe $u_x$ as a function of eastward and northward wind variables, which allows mitigating the effects of this parameter on a SAR stack within level 1B processing and generating a lookup table to correct sea surface height estimates in SAR mode. This additionally opens up the door to estimating the wind speed and direction from SAR altimetry stacks. Additionally, we discuss how this new retracker performs with respect to different planned future baseline processor changes of Sentinel-6MF, namely F09 and F10, by attempting to imitate their level 2 processing. This is achieved by processing cycles 017 to 051 (nearly a full year) of Sentinel-6MF level 1A data on a global scale. We observe that the new retracking method is, on average, more accurate with respect to LRM. However, there is a slight increase in measurement noise due to the introduction of an additional parameter. To ensure that the results of the new retracker are not biased, we retrack using both the new method and the SINCS-OV ZSK retracker on Sentinel-6MF stack data produced in a Monte Carlo simulation. We analyze the simulation results with respect to accuracy, precision, and correlations between estimated parameters. We show that the accuracy of the new retracker is better than SINCS-OV ZSK but less precise, which could be related to higher correlation coefficients—especially with respect to the new parameter $u_x$—between estimated parameters.

1. Introduction

For several decades, nadir-looking satellite radar altimeter missions have been routinely used to monitor ocean surface heights and sea state parameters. In a process called “retracking”, parameterized models for the expected radar power returned from a randomly rough surface are fit to the averages of radar pulse echoes in order to retrieve the parameters known as sea surface height (SSH) and significant wave height (SWH), and the normalized radar cross-section, ${\sigma }_0$, from which wind speed is derived. The parameters are sensitive to an area of ocean surface called the measurement “footprint”.

In the first few decades of altimetry, only incoherent averaging of pulse echoes was used, a technique now called low-resolution mode (LRM). The parametric model for LRM is known as the Brown model [1], and the retracker for LRM is the MLE3/4 retracker [2], where MLE stands for the maximum likelihood estimator, and the number afterward denotes the number of estimated parameters. The LRM footprint is in a circle of a few km in diameter, and the diameter increases as SWH increases [3].

In the last 12 years, SAR altimeter missions have furnished radar echoes that can be coherently processed to narrow the footprint in the direction of spacecraft flights. Fully focused SAR (FF-SAR) altimetry [4] takes aperture synthesis to the maximum limit and can measure water surface heights in rivers and canals that are very narrow (a few meters) in the direction of flight [5]. However, over the open ocean, the standard practice, known as unfocused SAR (UF-SAR) or Delay/Doppler altimetry [6], synthesizes an aperture for only a few milliseconds of the flight, narrowing the footprint to about 300 m in the flight direction while leaving the across-flight dimension the same size as for LRM altimetry. This standard approach to ocean SAR altimetry is the concern of this paper; here, to distinguish it from LRM altimetry, it is simply called “SAR” altimetry. The SAMOSA2 retracker [7]—named after the project “SAR Altimetry Mode Studies and Applications over Ocean, Coastal Zones and Inland Water (SAMOSA)”—retrieves SSH, SWH, and wind speed from SAR altimeter radar echo power displayed as a one-dimensional function of the two-way travel time of the radar pulse.

Geophysical parameters retrieved from SAR altimetry can be more precise than those retrieved from LRM altimetry, but LRM and SAR retrievals may have different accuracies and biases. SAR altimetry exploits Doppler shifts arising from relative motion between the altimeter antenna and the radar scattering points on the sea surface; this may make the SAR parameter estimates sensitive to the direction of ocean surface motions caused by winds and waves. LRM gives equal weight to radar scatterers lying at all azimuths within the circular measurement area; thus, LRM parameter estimates should be independent of any angle between wind or wave propagation and the spacecraft flight direction.

The first generation of SAR altimeters, CryoSat-2 and Sentinel-3, could operate in either LRM or SAR modes, but only by one mode at a time, exclusive of the other mode. Differences in LRM and SAR estimates, if any, had to be found by comparing SAR mode estimates to observations from so-called pseudo-LRM signals, which mimic LRM, but are computed in SAR mode. In this way, it was found that the CryoSat-2 SAR mode SWH retrievals differed from pseudo-LRM by up to 20 cm. Buchhaupt [8] showed that the standard deviation of vertical wave-particle velocities ${\sigma }_v$ would blur the Doppler spectrum exploited in the SAR mode; this would lead to biased SWH estimates because one-dimensional SAR retrackers that are in use would not be able to distinguish between the effect of SWH and the effect of ${\sigma }_v$.

To improve the accuracy of SWH retrievals from SAR altimetry, Buchhaupt [8] introduced the SINCS-OV (signal model involving numerical convolutions for SAR introducing orbital velocities) retracker, which estimates the geophysical parameters and ${\sigma }_v$ by fitting a model to a two-dimensional “stack” displaying radar echo power as a function of both two-way travel time and Doppler frequency. While this approach mitigated the differences between pseudo-LRM and SAR estimates of SWH, the ${\sigma }_v$ estimates did not agree well with observations from buoys and model forecasts from the European Centre for Medium-Range Weather Forecasts (ECMWF). This issue was addressed in Buchhaupt et al. [9] by considering that the SAR altimeter may be sensitive to vertical wave motion only where wave slopes are close to radar incidence angles, making the observable ${\sigma }_v$ smaller than the actual ${\sigma }_v$ by a factor $a_v$, which depends mainly on wave steepness, $S_m$; correcting ${\sigma }_v$ estimates from SINCS-OV for the $a_v$ effect resulted in ${\sigma }_v$ values consistent with ECMWF and buoy measurements in the German Bight.

Although the ${\sigma }_v$ effect with its $a_v$ correction seemed to solve the problem of the SWH estimation, Raynal et al. [10] found SSH anomaly differences between Sentinel-3 SAR retrievals and pseudo-LRM retrievals. The geographical distribution of these differences suggested a correlation with the global pattern of the north–south wind speed.

With the launch of Sentinel-6MF, it has been possible to simultaneously make geophysical retrievals from both LRM and SAR modes, enabling a direct and global investigation of discrepancies in the two types of retrievals. We found that the differences in SSH retrieved from Sentinel-6MF by LRM and by SAR are linked to both the wind speed and the wind direction relative to the flight direction [11]; see Figure 1 and Figure 2. This effect arises through a wave Doppler-induced bias, as the wind speed determines the wave slopes and the orbital velocity of water particles [12].

This study focuses on mitigating these SSH inconsistencies. As Sentinel-6MF is the new reference mission after Jason-3, this study will discuss the results of this mission to provide an initial perspective. In future studies, other SAR altimetry missions will be considered as well.

Section 2 discusses how these velocities are introduced into the SAR stack model and Section 3 shows how this new parameter is implemented in the SAR altimetry stack model. Section 4 shows that atmospheric refraction has the same effect on a stack as horizontal velocities and provides a formulation to introduce this effect in SAR data processing. In Section 5, we discuss the implementation choices of the Sentinel-6MF processing campaign presented in this study. The same processing parameters are then used in the Monte Carlo processing presented in Section 6. These simulations are performed to ensure that the proposed retracking scheme provides bias-free estimates. Additionally, the Monte Carlo runs provide important information about the reachable accuracy, precision, and correlation between estimated parameters. In Section 7, we discuss the results of the Sentinel-6MF processing campaign, and afterward we present our conclusions.

2. Including Horizontal Velocities in an Analytical Description of SAR Altimetry Signals over a Random Sea Surface

In SAR altimetry, it is necessary to describe the echoes scattered off the ocean surface as a two-dimensional delay–Doppler map (DDM) (or stack) in both the time delay, with respect to the tracking range gate ($\tau$), and Doppler frequency ($f_D$) domains. In Buchhaupt et al. [13], Buchhaupt [8], and Buchhaupt et al. [14], we first introduced the representation of the DDM as a fast convolution method, which we aim to expand in this study. In the fast convolution approach, we define the backward Fourier transform (FT) from $f_D$ to the slow time, $t_s$, and the forward FT from $\tau$ to the frequency f of the SAR altimetry stack [14]. After these FTs, the DDM can be described as the multiplication of three terms [14]:

$$\widehat{\widehat{P}}\left(f,t_s+\frac{f}{s}\right)=F\widehat{\widehat{\mathit{SS}}}R\left(f,t_s\right)P\widehat{\widehat{T}}R\left(f,t_s+\frac{f}{s}\right)P\widehat{\widehat{D}}F\left(f,t_s\right)$$

(1)

where s is the chirp or sweep signal slope and $F\widehat{\widehat{\mathit{SS}}}R(f,t_s)$ is the flat sea surface response (FSSR), describing the altimeter impulse response in the slow-time/frequency domain. $P\widehat{\widehat{T}}R(f,t_s)$, the point target response (PTR), describes the radar response to a single isotropic scatterer. $P\widehat{\widehat{D}}F(f,t_s)$ introduces a rough random sea surface, which mainly causes range-smearing due to random elevations and azimuth-smearing due to random vertical wave-particle velocities. For the sake of convenience, we provide the computations of all three terms in Appendix A, Appendix B, Appendix C.

The benefit of Equation (1) compared to SAR altimetry signal representations in the $\tau$/$f_D$ domain is that the FSSR and PTR terms are constant for each DDM. Only the probability density function (PDF) term contains geophysical parameters and, therefore, needs to be evaluated in every iteration of a nonlinear optimization process. Additionally, no convolutions need to be solved and no approximation of the PTR is necessary.

To introduce vertical and horizontal velocities on the sea surface, we define—similar to [15]—a Doppler pulsation term, ${\mathsf{\Omega }}_L$, containing horizontal and vertical dynamics of a larger scattering surface element located at the along-track coordinate x and across-track coordinate y, which, according to [12], dominate the Doppler shifts at small incidence angles, such as for nadir-looking altimeters:

$${\mathsf{\Omega }}_L=\frac{2}{{\lambda }_c}\left(z_t-{\alpha }_x{c}_x\frac{x}{h_s}-{\alpha }_y{c}_y\frac{y}{h_s}\right)$$

(2)

where $z_t$ denotes the vertical wave-particle velocities, ${\lambda }_c$ denotes the carrier wavelength of the emitted signal, $h_s$ denotes the satellite altitude with respect to the reference surface, ${\alpha }_x$ denotes the along-track curvature coefficient, ${\alpha }_y$ denotes the across-track curvature coefficient, $c_x$ denotes the along-track horizontal sea surface velocities, and $c_y$ denotes the across-track horizontal sea surface velocities.

The line of sight variance of ${\mathsf{\Omega }}_L$, together with the nonlinear sea surface elevation $\eta$ and slopes ${\eta }_x$/${\eta }_y$—under the assumption that horizontal wave-particle velocity components are negligible for nadir-looking altimeters—were thoroughly discussed in [9]. Therefore, this study focuses on the mean line of sight Doppler pulsation, which can be written with $\frac{x^2}{h_s^2}\ll 1$ and $\frac{y^2}{h_s^2}\ll 1$ as follows:

$$\begin{array}{cc}\hfill {\overline{\mathsf{\Omega }}}_L& =E\left[{\mathsf{\Omega }}_L\left|\left(z_x=\frac{{\alpha }_{x}x}{h_s},z_y=\frac{{\alpha }_{y}y}{h_s}\right)\right.\right]\hfill \end{array}$$

(3a)

$$\begin{array}{cc}\hfill & \approx \frac{2}{{\lambda }_c}\underset{u_x}{\underbrace{\left(a_{vx}\frac{{\sigma }_v}{{\sigma }_x}-c_y\right)}}\frac{{\alpha }_{x}x}{h_s}\hfill \end{array}$$

(3b)

$$\begin{array}{cc}\hfill & +\frac{2}{{\lambda }_c}\underset{u_y}{\underbrace{\left(a_{vy}\frac{{\sigma }_v}{{\sigma }_y}-c_x\right)}}\frac{{\alpha }_{y}y}{h_s}\hfill \end{array}$$

(3c)

where $a_{vx}$ and $a_{vy}$ are auxiliary parameters mostly depending on the correlation vertical wave-particle velocities and along-track or across-track wave slopes. They are given as [9]:

$$\begin{array}{cc}\hfill a_{vx}& =\frac{{\rho }_{xt}-{\rho }_{yt}{\rho }_{xy}}{1-{\rho }_{xy}^2}\hfill \end{array}$$

(4a)

$$\begin{array}{cc}\hfill a_{vy}& =\frac{{\rho }_{yt}-{\rho }_{xt}{\rho }_{xy}}{1-{\rho }_{xy}^2}\hfill \end{array}$$

(4b)

where ${\rho }_{xt}$ is the correlation coefficient between along-track wave slopes and vertical wave-particle velocities, ${\rho }_{yt}$ is the correlation coefficient between across-track wave slopes and vertical wave-particle velocities, ${\rho }_{xy}$ is the correlation coefficient between along-slopes and across-track wave slopes.

In Equation (3), $u_x$ and $u_y$ behave as horizontal sea surface velocities in along- and across-track directions. Even so, they also contain vertical components of wave velocities at a given wave slope. Therefore, $u_x$ can be referred to as he mean along-track line-of-sight surface velocity and $u_y$ as the across-track mean line-of-sight surface velocity.

Including Equation (3) into the stack model can be achieved by multiplying Equation (37) from [14], where $exp\left\{2\pi i{\overline{\mathsf{\Omega }}}_L{t}_s\right\}$, and following the derivation of the stack model presented in [14]. The approach in Buchhaupt et al. [14] considers an arbitrary number of bursts $N_b$ used in the SAR processing. In this study, only one burst ($N_b=1$) is considered. The scaling of the along-track velocity of the nadir and a non-zero across-track velocity, the resulting stack model is identical to [14]. The horizontal velocity terms become:

$$\begin{array}{cc}\hfill v_x& \mapsto \left(1+\frac{u_x}{v_x}\right)v_x\hfill \end{array}$$

(5a)

$$\begin{array}{cc}\hfill v_y& \mapsto u_y\hfill \end{array}$$

(5b)

Due to the range ambiguity of nadir-looking altimeters, across-track horizontal velocities from the left and right sides of the altimeter mostly cancel each other out and are, therefore, negligible. Therefore, in the following, only $u_x$ is considered.

3. Implementing Horizontal Doppler Shifts in the SAR Altimetry Model

Since only $u_x$ needs to be considered, it can be observed that $u_x$ only reduces the along-track nadir velocity $v_x$, which in turn only appears as a scaling term of the slow-time $t_s\mapsto \left(1+\frac{u_x}{v_x}\right)t_s$. All other terms are identical to [9,14]. A slow-time scaling can be implemented during the azimuth Fourier transform via linear substitution:

$$\widehat{P}\left(f,f_D\right)=\frac{e^{-2\pi i\frac{f}{s}{f}_D}}{|1+\frac{u_x}{v_x}|}\underset{\mathbb{R}}{\int }\widehat{\widehat{P}}\left(f,t_s+\frac{f}{s}\right)exp\left\{-2\pi i\frac{f_D{t}_s}{1+\frac{u_x}{v_x}}\right\}d{t}_s$$

(6)

where $\widehat{\widehat{P}}(f,t_s)$ is identical to the results from [9,14]; moreover, a small ${\sigma }_v$ and an along-track PTR scaling error can be neglected. Additionally, the amplitude factor of $\frac{1}{|1+\frac{u_x}{v_x}|}$ leads to a small Sentinel-6MF ${\sigma }_0$ error of about 0.007 dB for $u_x=10$ m/s, which is negligible. Therefore, in the following, this amplitude factor is ignored.

The power of the DDM is then calculated—after applying the slant range correction by adding a phase shift of $2\pi \frac{c{h}_s}{{\alpha }_x}{\left(\frac{f_D}{2f_c{v}_x}\right)}^{2}f+2\pi i\frac{f_D}{s}f$ to Equation (6)—in the time-delay/Doppler frequency domain with an inverse Fourier transform:

$$\begin{array}{ccc}\hfill P\left(\tau ,f_D\right)& \approx & \underset{\mathbb{R}}{\int }exp\left\{2\pi i\frac{c{h}_s}{{\alpha }_x}{\left(\frac{f_D}{2f_c{v}_x}\right)}^{2}f\right\}\underset{\mathbb{R}}{\int }\widehat{\widehat{P}}\left(f,t_s+\frac{f}{s}\right)\hfill \\ & \times & exp\left\{-2\pi i\frac{f_D{t}_s}{1+\frac{u_x}{v_x}}+2\pi i\tau f\right\}d{t}_{s}df\hfill \end{array}$$

(7)

Simply put, the line of sight velocities cause scaling of the stack sampling in the azimuth domain.

Since many optimizing algorithms run more stably and faster when the Jacobian is provided, the partial derivative of Equation (6), with respect to $u_x$, is given as follows:

$$\begin{array}{ccc}\hfill \frac{\partial P\left(\tau ,f_D\right)}{\partial u_x}& \approx & \frac{v_x{f}_D}{{\left(v_x+u_x\right)}^2}\underset{\mathbb{R}}{\int }exp\left\{2\pi i\frac{c{h}_s}{{\alpha }_x}{\left(\frac{f_D}{2f_c{v}_x}\right)}^{2}f\right\}\hfill \\ & \times & \underset{\mathbb{R}}{\int }2\pi i{t}_s\widehat{\widehat{P}}\left(f,t_s+\frac{f}{s}\right)\hfill \\ & \times & exp\left\{-2\pi i\frac{f_D{t}_s}{1+\frac{u_x}{v_x}}+2\pi i\tau f\right\}d{t}_{s}df\hfill \end{array}$$

(8)

Within the numerical retracking—such as SINCS and SINCS-OV [8,9,14]—the Fourier transforms are calculated numerically. The inverse Fourier transform from the frequency to time delay is performed via a complex to real inverse fast Fourier transform (FFT). On the other hand, a Fourier transform from equidistant slow-time values to scaled or non-equidistant Doppler frequencies is needed and, therefore, a direct FFT approach does not work. In this study, a type 2 nonuniform FFT [16] (Section 3.1) is used as it allows some shortcuts in the implementation, leading to a higher processing speed. However, a chirp Z-transform [17] is a valid alternative as well.

4. Atmospheric Refraction

In Section 2, line-of-sight velocities are mainly introduced into the SAR altimetry DDM model. However, it is important to consider another effect causing Doppler frequency scaling, namely atmospheric refraction.

It describes how objects are seen by an observer located on the sea surface. To be more precise, atmospheric refraction describes the incidence angle difference between the geometrical incidence angles $\Delta {\theta }_i$—associated with the travel path of light through a vacuum—and the real incidence angle (due to refraction caused by variations in the air density). Nadir-looking instruments are given in radians, as in [18]:

$$\Delta {\theta }_i\approx {\left(n-1\right)}_{tpf}\frac{x}{h_s}$$

(9)

where ${\left(n-1\right)}_{tpf}$ describes the refractive modulus. The tpf subscript means that it considers the temperature, total air pressure, and water vapor pressure.

According to [19], the refractive modulus of air is given for microwave or radio-wave signals as a function of pressure $P_d$ in the atmosphere, temperature T in degrees Celsius, and water vapor pressure $P_w$ in the atmosphere:

$${\left(n-1\right)}_{tpf}=\frac{0.000288P_d}{1+0.003661T}-\frac{0.000024P_w}{1+0.003661T}+\frac{0.005099P_w}{{\left(1+0.003661T\right)}^2}$$

(10)

The water vapor pressure is given according to [20] as follows:

$$P_w=0.006028exp\left\{\frac{17.2694T}{T+237.29}\right\}$$

(11)

It is possible to interpret atmospheric refraction as an apparent horizontal velocity—a similar mean vertical velocity observed at a specific incidence angle—with a negative sign. This means that for a scatterer, the satellite appears to be slower. The apparent horizontal velocity of the sea surface can be written as

$$v_{atm}=v_x{\left(n-1\right)}_{tpf}$$

(12)

or for Sentinel-6MF at a standard atmosphere of $P_d=1$ atm, $T=15$ °C, and an assumed nadir velocity of $v_x=5940.3$ m/s of $v_{atm}=2.2$ m/s.

Figure 3 shows the probability density of $v_{atm}$ modeled for the Sentinel-6MF processing campaign, including cycles 017 to 051. It can be seen that the dry component of the atmosphere has a bigger contribution but most of the variation of $v_{atm}$ is caused by the wet component.

It is worth mentioning that it is possible to consider atmospheric refraction during the L1B processing by adjusting the Doppler frequency sampling if the local air pressure and the temperature at mean sea level are known. Alternatively, standard atmosphere conditions can be used to mitigate most of the atmospheric refraction. However, in this study, the L1B processing is not adjusted to include atmospheric refraction. On the other hand, it means we have to consider it when interpreting the $u_x$ retracking results.

5. Retracking of Sentinel-6MF Signals

In order to test and validate our findings presented in the previous sections, we perform a validation campaign that encompasses one year of global Sentinel-6MF data. Additionally, Monte Carlo runs are employed to investigate the possible performance of the retracker with respect to accuracy and precision. Although the following focuses on real data processing, the simulations attempt to adopt these steps as much as feasible.

The SAR stacks were produced using the scientific LSAR-v1.1 L1A to L2 processor—being the experimental in-house processor of the Laboratory for Satellite Altimetry (LSA)—by means of an unfocused SAR processing approach, including the so-called range walk correction [21] via a chirp Z-transform. Additionally, LSAR attempts to reduce the size of the resulting DDMs by using a non-exact beam-steering approach, which results in $O_x{N}_p$ Doppler beams with equidistant Doppler frequencies per 20-Hz surface location. $N_p$ is the number of pulses per burst and $O_x$ is the along-track oversampling factor. Since each radar cycle consists of $N_b$ bursts for each surface location and Doppler frequency, $N_b$ Doppler beams occur. Since these will not provide further spatial information about the sea surface, they are averaged to one Doppler beam, resulting in an improved signal-to-noise-ratio (SNR) by a factor of $\sqrt{N_b}$. It is important to note that the zero skewness (ZSK) transform is performed before averaging the $N_b$ Doppler beams as it only works on exponentially distributed data, and after averaging, the samples would adhere to a Gamma distribution.

During the L1B to L2 process—usually referred to as retracking—the computation of the model DDMs is performed following [8] (Section 3.3). One difference is that the computations of $F\widehat{\widehat{\mathit{SS}}}R$ and $P\widehat{\widehat{T}}R$ start in the f/$t_s$ domain (contrary to the f/$f_D$ domain, as in [8]). Therefore, a frequency vector with $N=O_t{N}_t{N}_s$ samples with a resolution of $df=\frac{f_s}{N_t{N}_s}$ and a slow-time vector with $M=O_x{N}_x{N}_p$ samples with a distance of $d{t}_s=\frac{1}{N_x{f}_p}$ are used for the computations. $N_s$ is the number of samples in an echo, $O_t$ is the oversampling factor in the time-delay domain, $N_t$ is the receiving window-widening factor in the time-delay dimension, and $N_x$ is the widening factor in the azimuth dimension. Mission parameters are presented in

Table 1. Summary of Sentinel-6MF mission parameters used to simulate the DDMs. It is important to note that the pulse repetition frequency is not constant for Sentinel-6MF. The $f_p$ value given here is a proxy used when simulating signals.

Symbol	Description	Value
${\theta }_{3dBx}$	Along-track half-power beamwidth	1.34°
${\theta }_{3dBy}$	Across-track half-power beamwidth	1.34°
s	Negative chirp slope	9.9748 MHz/$\mathsf{\mu }$s
$f_c$	Central frequency	13.575 GHz
$f_p$	Pulse-repetition frequency	9100.2 Hz
$f_s$	Time-delay sample frequency	395 MHz
B	Usable pulse bandwidth	320 MHz
$N_s$	Number of samples per echo	128
$N_p$	Number of pulses per burst	64
$k_0$	Reference gate	40

, as well as the pulse repetition frequency, which is extracted from the L1A product.

The widening and oversampling parameters used in this study, are presented in

Table 2. Sampling and window-widening parameters used in calculating the modeled Sentinel-6MF DDM.

Symbol	Description	Value
$O_t$	Time delay oversampling factor	2
$N_t$	Time delay window-widening factor	8
$N_s$	Number of samples per echo	128
$O_x$	Azimuth oversampling factor	2
$N_x$	Azimuth window-widening factor	4
$N_b$	Number of bursts per radar cycle	7
$N_p$	Number of pulses per burst	64

.

The retracking is performed with a Levenberg–Marquardt algorithm [22]. As discussed in [8], it is necessary to retrack the whole DDM to estimate ${\sigma }_v$, as a waveform retracker is not able to distinguish between $H_s$ and ${\sigma }_v$. The same applies to $u_x$ and $t_0$.

Table 3. Summary of retrackers used in this study.

Retracker	Mode	Input	Estimated Parameter
SINC2	LRM	waveform	A, $t_0$, ${\sigma }_z$
SINCS	SAR	waveform	A, $t_0$, ${\sigma }_z$
SINCS-OV	SAR	stack	A, $t_0$, ${\sigma }_z$, ${\sigma }_v$
SINCS-OV2	SAR	stack	A, $t_0$, ${\sigma }_z$, ${\sigma }_v$, $u_x$

presents an overview of the retrackers considered in this study. The abbreviation SINC stands for the signal model involving numerical convolutions. SINC2 was the first of the developed numerical retrackers and the name incorporating the PTR follows a squared sine Cardinalis function. The S in SINCS stands for SAR. OV means orbital velocity as it introduces a parameter based on the vertical component of orbital wave motions. In this study, we introduce another parameter, which mostly depends on the mean line-of-sight motions, or in other words, vertical wave-particle velocities, at a given wave slope; thus, we decided to name the new retracker SINCS-OV2, indicating two parameters that are mainly based on orbital velocities.

where ${\sigma }_z=H_s/4$ describes the standard deviation of sea surface elevation displacements.

It is important to note that all retrackers accommodate possible negative ${\sigma }_z$ and ${\sigma }_v$ values—caused by noise—by setting ${\sigma }_z^2\mapsto {\sigma }_z\left|{\sigma }_z\right|$ and ${\sigma }_v^2\mapsto {\sigma }_v\left|{\sigma }_v\right|$. Additionally, retrackers using input signals transformed with the ZSK approach estimate the thermal noise $t_n$ as well.

If not otherwise stated, all retrackers use a constant short-wave non-linearity factor of $\mu =0.0546$ and a spectral narrowness parameter of $\nu =0.39$. These values are chosen in such a way that the underlying wave spectrum is the Joint North Sea Wave Observation Project (JONSWAP) spectrum and the resulting elevation displacement skewness is $Skew\left[\eta \right]=3\mu (1-\nu )=0.1$.

6. Evaluating the Impact of the Mean Line-of-Sight Velocities with Monte Carlo Runs

Before beginning a real data analysis, it is important to investigate whether the new retracker SINCS-OV2 is capable of estimating bias-free geophysical parameters and what precision is achievable. Here, Monte Carlo runs of Sentinel-6MF DDMs are performed to accomplish this. As this process can be very time-consuming if all parameter combinations are simulated, some restrictions need to be set first to reduce the workload.

• The observed surface is only affected by wind waves, ensuring that no currents or swell effects are considered.
• The local wave field is fully developed and unidirectional, which means that it can be described by a Pierson–Moskowitz spectrum [23].
• A standard atmosphere is assumed with $T=15$ °C and $P=1$ atm to simplify the implementation of $v_{atm}$.

Putting these restrictions into relationships with respect to $H_s$ yields the following:

$$\begin{array}{cc}{U}_{10}\hfill & =2.1375\sqrt{g{H}_s}\hfill \end{array}$$

(13a)

$$\begin{array}{cc}{\sigma }_v\hfill & =\sqrt{\mu g{\sigma }_z}\hfill \end{array}$$

(13b)

$$\begin{array}{cc}{u}_x\hfill & =\sqrt{U_{10}}cos{\phi }_w+u_{★}-v_{atm}\hfill \end{array}$$

(13c)

where $U_{10}$ describes the total wind speed ten meters above sea level. Equation (13a) results from restrictions 1 and 2, leading to a wave field described by the Pierson–Moskowitz spectrum, Equation (13b), from the definition of the short-wave non-linearity coefficient $\mu =\frac{{\sigma }_v^2}{g{\sigma }_z}$, Equation (13c) follows the findings from Section 7.2. A formulation of the friction velocity $u_{★}$ is given in Equation (16).

In this study, $H_s=\{0 \mathrm{m},1 \mathrm{m},2 \mathrm{m},4 \mathrm{m},8 \mathrm{m},12 \mathrm{m}\}$, a short-wave non-linearity coefficient of $\mu =0.0546$, a spectral narrowness coefficient of $\nu =0.425$, a mean gravity acceleration of $g=9.81$ m/s, and wind directions with respect to the satellite flight path of ${\phi }_w=\{{0.0}^{\circ },{22.5}^{\circ },\cdots {180.0}^{\circ }\}$ are used for the Monte Carlo runs (negative values for ${\phi }_w$ are not shown here as $u_x$ is symmetric with respect to ${\phi }_w$ (see Equation (13c))). For each $H_s$/${\phi }_w$ realization, $M_{sim}=10,000$ simulations are performed.

Each simulation is conducted using the following steps:

• Compute a noise-free SINCS-OV2 DDM for current $H_s$/${\phi }_w$ realization with arbitrary amplitude A and thermal noise of $t_n=\frac{A}{1000}$.
• Add exponentially distributed noise to the DDM and apply the ZSK transform [8] (Equation (7.1)) to the resulting noisy DDM. Repeat this step $N_b$ times, since in LSAR v1.1, this many Doppler beams are summed per Doppler beam. For Sentinel-6MF, $N_b=7$.
• Sum all $N_b$ DDMs and normalize the result, such that the maximum power within it equals one.
• Retrack the DDM with SINCS-OV ZSK and SINCS-OV2 ZSK.
• Repeat all steps $M_{sim}-1$ time for each $H_s$/${\phi }_w$ realization.

For each $H_s$/${\phi }_w$ realization, the mean differences, standard deviations, and correlation coefficients between the estimated parameters depending on SWH and the wind direction are calculated. Figure 4 shows the range of biases. It can be observed that SINCS-OV ZSK, given in the left plot of Figure 4, observes the SWH and wind-direction dependent bias, e.g., for $H_s=1$ m varying from $-2$ mm to 9 mm. For zero SWH, a 3 mm bias can be observed, as caused by atmospheric refraction. These differences are caused by $u_x$—being proportional to SWH due to the restriction set in this study—leading to an azimuth scaling, which is compensated by the retracker with the range of biases. On the other hand, SINCS-OV2 ZSK is bias-free, which is a good result, as the SNR and the distribution of each sample of the DDM might lead to retracking biases.

Similar observations can be made for SWH, as given in Figure 5. SINCS-OV ZSK returns biased SWHs, varying from $-0.4$ cm to $3.2$ cm at $H_s=1$ m or $-2.8$ cm to $4.5$ cm at $H_s=12$ m. These biases are not as crucial since most SWH requirements demand an accuracy of about fifteen centimeters, but since the range estimates do not fulfill these, it would still be necessary to use SINCS-OV2 ZSK, which allows bias-free SWH estimates, except for the zero SWH, where a $-9$ mm bias is still present.

For ${\sigma }_v$ given in Figure 6, similar conclusions can be drawn for SINCS-OV ZSK, showing, besides a sign change, very similar $H_s$/${\phi }_w$ behavior. Since no accuracy requirements for altimetry results exist (the parameter is very new), no statement about the significance can be made. SINCS-OV2 ZSK allows an almost bias-free ${\sigma }_v$ retrieval.

Since SINCS-OV ZSK does not estimate $u_x$, the values given in Figure 7 show $u_x$. SINCS-OV2 ZSK—as shown on the right plot of Figure 7—is able to estimate this new parameter bias-free.

The standard deviations presented in

Table 4. Sentinel-6MF standard deviations for estimated parameters retrieved from 10,000 Monte Carlo runs. The first value denotes the SINCS-OV ZSK standard deviation and the second one denotes the SINCS-OV2 ZSK value.

SWH [m]	Range Std [cm]	SWH Std [cm]	${\mathit{\sigma }}_{\mathit{v}}$ Std [cm/s]	${\mathit{u}}_{\mathit{x}}$ Std [m/s]
0	0.8/1.3	8.6/9.8	16.8/19.5	-/2.8
1	1.1/ 1.4	2.6/3.3	6.5/7.5	-/3.6
2	1.3/1.8	2.8/3.7	6.5/7.5	-/4.2
4	1.7/2.4	3.5/5.1	7.0/8.75	-/5.4
8	2.3/3.3	5.4/8.0	9.5/11.7	-/7.2
12	2.8/4.1	7.5/9.8	12.5/14.8	-/8.7

are important as they give the precision of retrieved parameters at different SWH realizations. Wind-direction dependencies were not observable and are, therefore, not shown in Table 4. The first value denotes the SINCS-OV ZSK standard deviation and the second value denotes the SINCS-OV2 ZSK standard deviation. It is observable that the retrieved precision from SINCS-OV2 ZSK is worse compared to SINCS-OV ZSK, which is no surprise since it estimates an additional parameter, $u_x$. This parameter is estimated with a low SNR since standard deviations are bigger than the expected values at ${\phi }_w=0^{\circ }$, as given in Figure 7, degrading the precision of other parameters.

Finally,

Table 5. Sentinel-6MF correlation coefficients for estimated parameters retrieved from 10,000 Monte Carlo runs at an SWH of two meters. The first value denotes the SINCS-OV ZSK correlation coefficients and the second one denotes the SINCS-OV2 ZSK values.

Corr(${\mathit{X}}_{\mathit{i}}$,${\mathit{X}}_{\mathit{j}}$)	Range	SWH	${\mathit{\sigma }}_{\mathit{v}}$	${\mathit{u}}_{\mathit{x}}$
Range	$+1.00$/$+1.00$	$+0.60$/$+0.70$	$+0.18$/$-0.34$	$+0.00$/$-0.67$
SWH	$+0.60$/$+0.70$	$+1.00$/$+1.00$	$-0.26$/$-0.52$	$+0.00$/$-0.53$
${\mathbf{\sigma }}_{\mathbf{v}}$	$+0.18$/$-0.34$	$-0.26$/$-0.52$	$+1.00$/$+1.00$	$+0.00$/$+0.68$
${\mathbf{u}}_{\mathbf{x}}$	$+0.00$/$-0.67$	$+0.00$/$-0.53$	$+0.00$/$+0.68$	$+1.00$/$+1.00$

presents the correlation coefficients of retrieved parameters. For different SWH realizations, the values are slightly different, but for the sake of readability, only values at an SWH of two meters are presented. It can be seen that $u_x$ is moderately to strongly correlated with other parameters; with respect to ${\sigma }_v$, the correlation, 0.68, is high. Additionally, the correlation coefficients of all parameters increase significantly if SINCS-OV2 ZSK is used as a retracker. The impact of this effect (and how it might be mitigated) shall be left as an open question for further study.

7. Global Sentinel-6MF Data Investigation of LRM/SAR Inconsistencies

In this section, we present a comparison of LRM and SAR geophysical parameters retrieved from a global one-year processing campaign of Sentinel-6MF L1A data. In this study, cycles 017 to 051 were processed, covering a time span from 25 April 2021 to 7 April 2022. All L1A files used in this study are from processing baseline F006.

The processing was performed using our in-house experimental L1A to L2 processor, denoted as LSAR. As a reference dataset, we chose ERA5 data retrieved from the Copernicus Climate Change Service [24].

Table 6. Summary of the used ERA5 parameters. The long name here is identical to how the parameters are listed on the Copernicus Climate Data Store. The abbreviation denotes the designation of the parameter in the ERA5 network common data form (NetCDF) file. The symbol provides the usage of the parameter in this study.

Long Name	Abbr.	Symbol
Mean zero-crossing wave period	mp2	$T_{02}$
Significant height of combined wind waves and swell	swh	$H_s$
10 m eastward wind component	u10	$u_{10}$
10 m northward wind component	v10	$v_{10}$
Mean sea level pressure	msl	P
2 m temperature	t2m	T

gives the parameters which were downloaded and used in this study:

The total wind speed $U_{10}$, the wind direction relative to the satellite heading ${\phi }_w$, the standard deviation of vertical wave-particle velocities ${\sigma }_v$, and the mean line-of-sight velocity corrected for atmospheric refraction $u_x$ can be calculated from the parameters of Table 6 by:

$$\begin{array}{cc}\hfill U_{10}& =\sqrt{u_{10}^2+v_{10}^2}\hfill \end{array}$$

(14a)

$$\begin{array}{cc}\hfill {\phi }_w& =arctan2\left(u_{10},v_{10}\right)-{\phi }_s\hfill \end{array}$$

(14b)

$$\begin{array}{cc}\hfill {\sigma }_v& =\frac{\pi }{2}\frac{H_s}{T_{02}}\hfill \end{array}$$

(14c)

$$\begin{array}{cc}\hfill u_x& =\sqrt{U_{10}}cos{\theta }_w+u_{★}-v_{atm}\hfill \end{array}$$

(14d)

where ${\phi }_s$ is the course angle of the satellite.

We retrieved the ERA5 data at intervals of six hours.

Table 7. Summary of datasets created for this study: one year of global Sentinel-6MF L1A data was processed for each dataset.

Abbr.	LRM	SAR	Note
3P	SINC2 STD	SINCS STD	As S6-MF baseline F09
3P+OV	SINC2 ZSK	SINCS-OV ZSK	As S6-MF baseline F10
3P+OV2	SINC2 ZSK	SINCS-OV2 ZSK	Estimates $u_x$ in SAR

describes the datasets and the corresponding retracker processed for this study:

In the following subsections, we present how different processing approaches behave with respect to LRM/SAR discrepancies. Parameters of interest are sea level anomaly (SLA), SWH, ${\sigma }_v$, and $u_x$.

7.1. Post-Processing of Retracked Sentinel-6MF Data

Since geophysical parameters retrieved from radar altimetry data tend to be noisy, it is necessary to perform an outlier detection before the validation itself. In this study, this step is conducted on 20-Hz data.

The first step is a general threshold-based outlier detection where all values outside a defined range are replaced with not-a-number (NaN). This is conducted for all parameters, meaning that, e.g., if an SWH value is above 20 m, then all other parameters at the same surface location and epoch are set to NaN as well.

Table 8. Summary of hard outlier thresholds.

Parameter	From	To
${\sigma }_0$	0 dB	∞ dB
$SLA$	$-3$ m	3 m
$H_s$	$-2$ m	20 m
${\sigma }_v$	$-1$ m/s	3 m/s
$u_x$	$-60$ m/s	60 m/s

shows the valid ranges of each parameter. The values in Table 8 are selected based on the 1-Hz histograms—derived without applying an outlier detection—of each parameter, and its signal-to-noise ratio. For example, for $u_x$, we decided that values outside of the $\pm 15$ m/s range should not be feasible based on its 1-Hz histogram. Additionally, we assumed a signal-to-noise ratio of about one, observing 20-Hz standard deviations. We combine this with three sigma criteria as outlier detection criteria give $\pm 15\mathrm{m}/\mathrm{s}\pm 3\times 15\mathrm{m}/\mathrm{s}\mapsto \pm 60\mathrm{m}/\mathrm{s}$. A similar approach was used for all other parameters besides ${\sigma }_0$ for which only negative values are considered as outliers.

The second step is a dynamic approach based on a moving median filter, which is applied to $H_s=4{\sigma }_z$ and ${\sigma }_v$. First, a twenty-second-long moving median filter is applied to $H_s$ (further called smoothed $H_s$) and afterward the same filter is applied to the absolute difference between $H_s$ and smoothed $H_s$ (called smoothed absolute deviation). An outlier is detected if $H_s$ deviates by more than six times the smoothed absolute deviation from the smoothed $H_s$. The same is conducted for ${\sigma }_v$ but its absolute value is tested against six times the smoothed absolute deviation. This approach is based on a median absolute deviation-based outlier detection [25], but it uses a moving median filter and tests for four times the standard deviation. Notably, six times the median absolute deviation approximately equates to four times the standard deviation for a normally distributed parameter.

Finally, the 1-Hz data compression is straightforward, performed by averaging twenty corresponding 20-Hz data values. NaN values are ignored. If only ten values or less remain after the outlier detection, the 1-Hz value is set to NaN.

7.2. Discussion of $u_x$ Results with Respect to ERA5

The new parameter from this study, $u_x$, which can only be estimated with the SINCS-OV2 ZSK retracker, does not have any requirements yet, but

Table 9. The 1-Hz $u_x$ noise for SAR Sentinel-6MF in meters per second.

Retracker	${\mathit{H}}_{\mathit{s}}=1$ m	${\mathit{H}}_{\mathit{s}}=2$ m	${\mathit{H}}_{\mathit{s}}=5$ m	${\mathit{H}}_{\mathit{s}}=8$ m
SINCS-OV2 ZSK	1.4	1.6	2.4	2.9

shows that overall precision of this parameter is low with standard deviation values of more than one meter per second whereas mean values vary in most cases between $\pm 5$ m/s.

From Section 2 and Section 4, it can be concluded that $u_x$ depends mainly on line-of-sight velocities caused by wave slope/vertical-velocity correlations, mean horizontal velocities, and atmospheric refraction. Nonlinear effects were not modeled in this study. In the following, we attempt to define a wind-speed-dependent formulation of $u_x$. To achieve this, we binned estimated 1-Hz $u_x$ values from SINCS-OV2 ZSK retracking for different ERA5 wind speeds and directions with respect to the satellite track. For wind-speed $U_{10}$, central values from $0.5$ m/s to $15.5$ m/s with a step size of 1 m/s were chosen. For wind direction, values ranged from $-{175}^{\circ }$ to ${175}^{\circ }$, with a step size of ${10}^{\circ }$. An $u_x$ value is then assigned to the nearest corresponding wind speed and wind direction. From our observations, as $u_x$ closely follows a normal distribution, we compute the median for each wind-speed/wind-direction realization. For each wind-speed realization, we fit the resulting curve of median $u_x$ values with the following function

$${\tilde{u}}_x(U_{10},{\phi }_w)=A_{u_x}\left(U_{10}\right)cos\left({\phi }_w\right)+b_{u_x}\left(U_{10}\right)$$

(15)

where $A_{u_x}$ and $b_{u_x}$ are estimated parameters that describe the amplitude of the directional term and a mean offset.

In the following, we present closed-form solutions for the parameters $A_{u_x}$ and $b_{u_x}$ and attempt to find well-established sea state parameters, e.g., in this study, the friction velocity $u_{★}$ for $b_{u_x}$. However, we do not claim that $b_{u_x}$ equals $u_{★}$. We only observe that both parameters show very similar wind-speed behaviors.

Figure 8 provides the $A_{u_x}$ (left plot) and $b_{u_x}$ (right plot) estimates for the given ERA5 wind speeds. The left plot shows that the directional amplitude $A_{u_x}$ can be described—with an error of about 10%—by the square root of the wind speed $U_{10}$. The corresponding geophysical parameter is $a_{vx}\frac{{\sigma }_v}{{\sigma }_x}$, approximately describing the ratio between the slope/velocity covariance and the along-track wave slope variance [9]. The right plot provides the offset parameter $b_{u_x}$ with respect to the ERA5 wind speed $U_{10}$. The red curve denotes the friction velocity computed with $U_{10}$, as described in Edson et al. [26]; Figure 10 shows good agreement with $b_{u_x}$.

A detailed formulation of the computation of $u_{★}$ from the wind wave spectra at high wind speeds is given in Takagaki et al. [27]. In this study, we used a simpler formulation derived in Edson et al. [26], which describes $u_{★}$ as a linear function of $U_{10}$ for three different wind speed cases. However, we fit the formula given by Edson et al. [26] with a hyperbola:

$$u_{★}\approx \sqrt{{\left(0.2923\mathrm{m}/\mathrm{s}\right)}^2+{\left(0.062U_{10}\right)}^2}-0.2923\mathrm{m}/\mathrm{s}$$

(16)

This representation leads to a good match with $b_{u_x}$.

In Figure 9, we present a map of smoothed $u_x$ estimates derived from ascending and descending passes from cycles 017 to 051. On the top and bottom plots, the geographical features of $u_x$ are well observable, especially in the northwest of South America and the west of Africa. Additionally, it can be seen that $u_x$ results from ascending and descending passes, with different signs at latitudes between $\pm {45}^{\circ }$. Outside these latitudes, the signs are mostly identical due to similar course angles in these regions.

Another interesting piece of information is the variation of $u_x$ in different regions. To present this—as shown in Figure 10—we estimated the standard deviation of $u_x$ over cycles to describe the variation over time. Since $u_x$ follows a normal distribution, here, the standard deviation is estimated as $1.4826MAD\left(u_x\right)$ to make the map more robust with respect to extreme events or outliers. Interestingly, the variation does not differ much between ascending and descending passes. Additionally, the standard deviations at latitudes between $\pm {30}^{\circ }$ are low in areas where $u_x$ is in terms of a big amount. Outside these latitudes, the standard deviations are high in areas with large means, $u_x$.

Figure 11 shows the difference between the mean $u_x$ as shown in Figure 9 and the mean $u_x$ calculated with ERA5 data. It can be seen that the differences vary between $\pm 1$ m/s and the relative error is usually within 10%. However, we cannot state if the remaining errors are within ERA5 wind speed data or if it is necessary to include currents and swell to improve the match. Nevertheless, the results are quite promising and model $u_x$ seems to match the measured $u_x$ well enough to be used as an input parameter for a lookup table (LUT).

To conclude this subsection, it can be stated that with $u_x$, a new geophysical parameter can be estimated from SAR stacks. Due to $u_x$ mainly depending on the wind speed and direction, it opens up the possibility of estimating directional winds from SAR altimetry.

7.3. Discussion of SAR ${\sigma }_v$ Results

Due to the fact that LRM is unaffected by vertical wave-particle velocities, it is not possible to estimate ${\sigma }_v$ by retracking LRM waveforms. Therefore, no comparison between SAR and LRM ${\sigma }_v$ could be performed. As this parameter is quite new, we could not find accuracy or noise requirements for it. However, we still provide the 1-Hz standard deviation of SAR ${\sigma }_v$ in

Table 10. The 1-Hz ${\sigma }_v$ noise for SAR Sentinel-6MF in centimeters per second.

Retracker	${\mathit{H}}_{\mathit{s}}=1$ m	${\mathit{H}}_{\mathit{s}}=2$ m	${\mathit{H}}_{\mathit{s}}=5$ m	${\mathit{H}}_{\mathit{s}}=8$ m
SINCS-OV ZSK	8.5	9.2	12.3	17.2
SINCS-OV2 ZSK	13.2	14.2	20.4	28.6

. The main observation from Table 10 is the worse precision of SINCS-OV2 ZSK compared to SINCS-OV ZSK, which is hinted at in Table 4 and Table 5, indicating that $u_x$/${\sigma }_v$ correlations in the retracking significantly reduce the precision of ${\sigma }_v$.

As no differences between ascending and descending passes could be observed for the mean ${\sigma }_v$ or the standard deviation of ${\sigma }_v$, Figure 12 shows only ascending passes. The upper plot presents the mean overall cycles and the lower plot presents the standard deviation. Both show results retrieved with the SINCS-OV2 ZSK SAR retracker. Comparing the upper plot of Figure 12 with Figure 9, no similarities can be seen. This is probably caused by the fact that $u_x$ depends on the wind direction and ${\sigma }_v$ does not.

On the other hand, when comparing ${\sigma }_v$ from SINCS-OV2 ZSK and ${\sigma }_v$ from SINCS-OV ZSK, as shown in Figure 13, a clear resemblance with Figure 9 showing $u_x$ is observable. This leads to the conclusion that ${\sigma }_v$ estimates are affected by $u_x$ as well, but the magnitude is less than ten centimeters and only a few percentage points of ${\sigma }_v$.

Overall, it can be stated that the impact of $u_x$ on ${\sigma }_v$ is a few percentage points and, therefore, is rather small.

7.4. Discussion of LRM/SAR SLA Discrepancies

SLA is the most important parameter in radar altimetry for ocean applications as it provides crucial information about the sea level rise and local events, such as storms/hurricanes and currents. Therefore, mission requirements regarding these parameters are rather strict, requiring range estimate accuracies in the centimeter range.

Table 11. The 1-Hz SLA noise for LRM Sentinel-6MF in centimeters.

Retracker	${\mathit{H}}_{\mathit{s}}=1$ m	${\mathit{H}}_{\mathit{s}}=2$ m	${\mathit{H}}_{\mathit{s}}=5$ m	${\mathit{H}}_{\mathit{s}}=8$ m
Requirement	1.20	1.50	2.40	3.20
SINC2 STD	1.25	1.40	1.82	2.02
SINC2 ZSK	1.25	1.35	1.70	1.95

presents the range noise upper limits for LRM and the achieved 1-Hz standard deviations of SLA measurements. The value for $H_s=2$ m is a mission requirement. It can be observed that SINCS2 STD and SINC2 ZSK both reach the upper limits and the mission requirement for SWH values above two meters. For the one-meter case, they are $1.25$ cm above the $1.2$ cm requirement. Additionally, we note that SINC2 ZSK has slightly better precision than SINC2 STD. Additionally, it is worth mentioning that there is a goal for $1.0$ cm at $H_s=2$ m, which SINC2 ZSK makes progress toward.

For the SAR mode, the SLA standard deviations are presented in

Table 12. The 1-Hz SLA noise for SAR Sentinel-6MF in centimeters.

Retracker	${\mathit{H}}_{\mathit{s}}=1$ m	${\mathit{H}}_{\mathit{s}}=2$ m	${\mathit{H}}_{\mathit{s}}=5$ m	${\mathit{H}}_{\mathit{s}}=8$ m
Requirement	0.70	0.80	1.30	2.00
SINCS STD	0.45	0.60	1.22	2.22
SINCS-OV ZSK	0.45	0.56	1.11	1.86
SINCS-OV2 ZSK	0.56	0.70	1.25	2.02

. SINCS STD, which will be implemented in Sentinel-6MF baseline F09, does not fulfill the eight-meter SWH requirements of $2.00$ cm as it is 10% bigger. On the other hand, SINCS-OV ZSK performs better at high sea states and it reaches the $H_s=8$ m requirement with a value of $1.86$ cm. SINCS-OV2 ZSK—estimating additionally $u_x$—performs worse for small to medium sea states, but almost reaches the high sea state’s upper noise limit by exceeding the value by 1%.

However, the accuracy of an estimated parameter, which is more important than the precision discussed before, is usually described by the root-mean-square error (RMSE). In the following, we will focus on mean differences between LRM and SAR with respect to ERA5 SWH and $u_x$.

Table 13. Regression of SAR minus LRM 1-Hz SLA values with respect to $u_x$. The slope is given in cm/(m/s). Offset, median, and STDD in centimeters.

Abbr.	Slope	Offset	Median	STDD	NP
3P	$-0.277$	$0.63$	$1.11$	$1.93$	$16,474,989$
3P+OV	$-0.253$	$0.15$	$0.57$	$1.70$	$16,674,069$
3P+OV2	$0.007$	$0.14$	$0.12$	$1.68$	$16,571,956$

provides the main statistics of this section for all three datasets. The slope and offset denote the linear regression parameter, median denotes the median of the SAR/LRM SLA differences, STDD denotes the standard deviation of differences (STDD), and NP denotes the number of points. The 3P dataset representing SINCS STD for SAR and SINC2 STD for LRM shows the highest dependency on SLA differences with $u_x$, with a regression slope of $-0.277$ cm/(m/s), the highest offset ($6.3$ mm), median ($11.1$ mm), and STDD ($19.3$ mm). Surprisingly, the 3P+OV dataset—using ZSK transforms and a vertical velocity tracker—shows better performance with a significantly lower offset ($1.5$ mm), median ($5.7$ mm), and STDD ($17.0$ mm), but the $u_x$ dependency has a regression slope of $-0.253$ cm/(m/s) that is almost unchanged compared to the 3P dataset. By far, the best agreement between SAR and LRM is reached with the 3P+OV2 dataset, which estimates additionally $u_x$ in SAR, with an insignificant slope of $0.007$ cm/(m/s), an offset of 1.4mm, a median of differences of $1.2$ mm, and an STDD of 16.8 mm. Therefore, SINCS-OV2 ZSK eliminates the $u_x$ dependency.

The following figures provide a visualization of Table 13.

As indicated by Figure 14, the mean differences between the SAR and LRM SSH estimates—retrieved from the SINCS STD SAR waveform retracker and SINC2 STD LRM retracker—look very similar compared to Figure 9, showing mean $u_x$ values. The mean differences are over two centimeters, which does not fulfill the mission requirements.

The same can be stated about Figure 15 showing the standard deviations over all cycles as it looks very similar to Figure 10, where the $u_x$ standard deviation is presented. Figure 15 shows that the variations in mean SSH differences between SINCS STD and SINC2 STD vary between two and ten millimeters. As in Figure 10, at latitudes between $\pm {45}^{\circ }$, the standard deviations are low when the mean differences are big, and outside this region, the standard deviations become bigger if the mean increases.

Figure 16 shows the mean differences over all cycles between SAR and LRM for the 3P+OV2 dataset, which estimates $u_x$ as well. It can be observed that no obvious SSH inconsistencies exist between SAR and LRM besides a bias of $1.4$ mm, as shown in Table 13. Ascending and descending passes show the same behavior as no differences can be observed.

The same can be stated about the standard deviations of the 3P+OV2 dataset shown in Figure 17 since no significant regional characteristics are observable. In coastal regions or typical sea ice regions, the standard deviations are, in most cases, about $2.6$ mm. This is an important result as it indicates that no spatial behavior of SSH discrepancies between SAR and LRM exist when applying the same processing steps, as in the 3P+OV2 dataset.

7.5. Discussion of LRM/SAR SWH Discrepancies

Near-real-time wave heights are critical parameters for end users for navigation and safety. However, requirements are based on user requirements and an estimate of a combination of noise and systematic errors. The goal is to reduce systematic errors to 10 cm for SWH between 0.5 and 8 m. For Sentinel-6MF, an accuracy requirement of 15 cm ± 5% SWH is defined. With a current state-of-the-art SAR retracker, this requirement is not possible to achieve due to vertical wave-particle velocities affecting SAR SWH estimates. However, in LRM, these requirements are achievable due to the incoherent processing vertical velocities not impacting LRM, as shown in

Table 14. The 1-Hz SWH noise for LRM Sentinel-6MF in centimeters.

Retracker	${\mathit{H}}_{\mathit{s}}=1$ m	${\mathit{H}}_{\mathit{s}}=2$ m	${\mathit{H}}_{\mathit{s}}=5$ m	${\mathit{H}}_{\mathit{s}}=8$ m
SINC2 STD	9.0	9.2	11.2	13.2
SINC2 ZSK	4.9	4.9	6.0	7.3

. It is worth mentioning that the ZSK transform significantly improves the precision of SWH estimates by halving standard deviations.

Table 15. The 1-Hz SWH noise for SAR Sentinel-6MF in centimeters.

Retracker	${\mathit{H}}_{\mathit{s}}=1$ m	${\mathit{H}}_{\mathit{s}}=2$ m	${\mathit{H}}_{\mathit{s}}=5$ m	${\mathit{H}}_{\mathit{s}}=8$ m
SINCS STD	3.8	4.2	8.1	14.2
SINCS-OV ZSK	2.2	2.9	6.1	11.0
SINCS-OV2 ZSK	2.7	3.2	6.3	11.2

denotes the standard deviations of 1-Hz SWH SAR estimates. Again, these values are not meaningful for SINCS STD as they do not consider ${\sigma }_v$ causing large SWH biases. Even so, SINCS-OV ZSK and SINCS-OV2 ZSK estimate additional wave velocity parameters, and the 1-Hz standard deviations are significantly lower for SINCS STD.

Table 16. Regression of SAR minus LRM 1-Hz SWH values with respect to $u_x$. The slope is given in cm/(m/s). Offset, median, and STDD in centimeters.

#	Slope	Offset	Median	STDD	NP
3P	2.345	35.2	30.9	15.2	16,858,661
3P+OV	0.568	−1.4	−2.4	5.3	16,797,917
3P+OV2	−0.166	−1.5	−1.1	5.4	16,693,252

shows the statistics of the linear regression of SAR/LRM 1-Hz SWH differences with respect to $u_x$. It is important to note that least square method-based linear regressions assume normally distributed variables, which is not the case for SWH, which approximately follows a Rayleigh distribution. Therefore, the results presented in Table 16 should not be over-interpreted. However, they provide insight into potential $u_x$ dependencies.

The SINCS retracker used in the 3P dataset in Table 16 does not consider vertical velocities and, therefore, SINCS STD has bad agreement with SINC2 STD. On the other hand, the 3P+OV dataset—including ${\sigma }_v$ as a free parameter—already provides very good agreement with an offset of $-1.4$ cm, median differences of $-2.4$ cm, and STDD of $5.3$ cm. The regression slope of $0.568$ cm/(m/s) points toward a small $u_x$ dependency. However, the resulting biases are always below the accuracy requirement of 15 cm ± 5% SWH. The 3P+OV2 dataset performs similar to the 3P+OV dataset. The only noticeable differences are the lower median of differences of $-1.1$ cm and the small $u_x$ dependency causing a regression slope of $-0.166$ cm/(m/s).

Figure 18 visualizes the mean SWH differences over all cycles between SAR and LRM, or in other words, from the 3P dataset. It can be observed that ascending and descending passes behave similarly, which is no surprise as these differences are mostly caused by vertical wave-particle velocities. Therefore, Figure 18 shows very similar behavior to Figure 12, given the average ${\sigma }_v$ values. A more detailed discussion about the effect of ${\sigma }_v$ on SWH is given for CryoSat-2 in Buchhaupt [8] and Buchhaupt et al. [9]. Overall, for Sentinel-6MF, the discrepancies in SWH between the SAR waveform retracker and LRM waveform retracker exceed the accuracy requirements of 15 cm at two-meter SWH, as well as the goal of 10 cm at two-meter SWH.

On the other hand, for the 3P+OV dataset, considering vertical wave-particle velocities by estimating ${\sigma }_v$, the mean SWH differences between SAR and LRM, as given in Figure 19, become much smaller compared to the 3P dataset. The remaining differences show the same behavior as $u_x$, visualized in Figure 9, and indicated in Table 16.

Introducing the Doppler frequency scaling—by introducing $u_x$ as an estimated parameter, such as in the 3P+OV2 dataset—further reduces the SWH differences between SAR and LRM averaged over all cycles. As shown in Figure 20, ascending and descending passes behave similarly, and for latitudes between $\pm {30}^{\circ }$, the SWH is very small. However, a latitude dependency is observable, which increases with the absolute latitude. It is still not clear what causes this issue, but overall, SWH derived from SAR and LRM has very good consistency in the 3P+OV2 dataset, which includes ${\sigma }_v$ and $u_x$.

8. Conclusions

In this study, we introduce a new SAR stack retracker called SINCS-OV2 ZSK, which provides SSH and SWH estimates with very good agreement, with respect to LRM results. It estimates a novel parameter $u_x$, which acts like a horizontal along-track sea surface motion and leads to a Doppler frequency scaling. Moreover, we show that the Doppler frequency scaling is caused by:

• Line-of-sight wave motion: This is mostly caused by vertical velocities observed at a given incidence angle of the electromagnetic wave. This depends on the wind speed and wind direction. Usually between $\pm 5$ m/s.
• Atmospheric refraction: This is caused by different refractive indices at the satellite position and the observed sea surface. This depends on the dry air pressure at sea level and water vapor content at sea level, usually between $-2.5$ m/s and $-1.7$ m/s.
• Swell and currents: This was not investigated in this study, which focused on wind waves. We plan to consider swell waves and currents in a later study.

We performed Monte Carlo runs to verify that the newly presented retracker approach provides bias-free estimates and showed that not considering $u_x$ in the retracking leads to significant SSH biases up to several centimeters. On the other hand, SWH and the standard deviation of vertical wave-particle velocities ${\sigma }_v$ are less affected by biases in the order of a few percentage points. By introducing $u_x$ as a free parameter within a SAR stack retracker—such as SINCS-OV2 ZSK—we managed to eliminate almost all of the discrepancies between LRM and SAR altimetry retracking results.

Additionally, we find a formulation to describe $u_x$ as a function of eastward and northward wind components from the ERA5 model. This is important as this formulation can be used to build lookup tables to consider SSH biases, e.g., for Copernicus Sentinel-6 data provided by EUMETSAT, which does not consider $u_x$ during the retracking.

Of course, if it is possible to estimate $u_x$ from eastward and northward wind components, it should be possible to retrieve estimates of eastward and northward wind components from $u_x$ retracking results via SAR altimetry. Therefore, the SINCS-OV2 ZSK retracker opens the door to obtaining directional information about wind fields.

In further studies, we plan to investigate other SAR altimetry missions, such as Sentinel-3, to verify the results found here. Additionally, we plan to investigate the effects caused by swell waves and currents.