Lihua WANG Yanghua GAO , Peng LU , Li FAN , Yunxuan ZHOU

1 Department of Geography and Spatial Information Techniques, Center for Land and Marine Spatial Utilization and Governance Research, Ningbo University, Ningbo 315211, China

2 Institute of East China Sea, Ningbo University, Ningbo 315211, China

3 Chongqing Institute of Meteorological Sciences, Chongqing 401147, China

4 College of Geo-Exploration Science and Technology, Jilin University, Changchun 130026, China

5 State Key Laboratory of Estuarine and Coastal Research, East China Normal University, Shanghai 200062, China

Abstract Values for Doppler center frequency are calculated from the echo signal at the satellite using the Doppler centroid method and so include the predicted Doppler frequency caused by the relative motion of the satellite and the Earth, which is the main component of Doppler center frequency and must be removed to obtain the Doppler frequency anomaly for ocean current measurement. In this paper, a new Doppler frequency anomaly algorithm was proposed when measuring surface currents with synthetic aperture radar(SAR). The key of the proposed algorithm involved mean f iltering method in the range direction and linear f itting in the azimuth direction to remove the radial and the azimuthal component of predicted Doppler frequency from the Doppler center frequency, respectively. The basis is that the theoretical Doppler center frequency model of SAR exhibits an approximately linear characteristic in both the range direction and in the azimuth direction. With the help of the new algorithm for predicted Doppler frequency removal,the estimation error of Doppler frequency anomaly can be reduced by avoiding employing the theoretical antenna pattern and imperfect satellite attitude parameters in the conventional Doppler frequency method.SAR measurement results demonstrated that, compared to the conventional Doppler frequency with/without error correction method, the proposed algorithm allows for a pronounced improvement in the current measuring accuracy in comparison with the global ocean multi-observation (MOB) products. In addition,the eff ectiveness and robustness of the proposed Doppler algorithm has been demonstrated by its application in the high velocity current in the Kuroshio region.

Keyword: synthetic aperture radar (SAR); Doppler frequency; Doppler frequency anomaly; current retrieval; ocean surface currents

1 INTRODUCTION

Ocean current is a form of large-scale movement of seawater, ref lecting the transportation status of quasisteady seawater at diff erent scales (Kneller and Buckee, 2000; Chiri et al., 2019). The implementation of ocean current observations, comprehension of the characteristics and changing laws of ocean current f ields are of great signif icance for ocean transportation,ocean numerical forecasting, ocean f ishery production,and global climate change research (Chapron et al.,2005; Rouault et al., 2010; Liu et al., 2019).

Synthetic aperture radar (SAR) images can capture the spatial distribution of sea surface micro-scale waves caused by the sea surface current f ield (Goldstein et al., 1989; Chapron et al., 2005). It is currently the most economical and eff ective data source for obtaining large coverage and high spatial resolution ocean current information (Romeiser et al., 2010;Hansen et al., 2011a; Wang et al., 2014; Ren et al.,2017; Martin et al., 2018). The SAR Doppler spectrum of the sea surface scattering echo has the capability of ref lecting the dynamic modulation characteristics of the sea surface itself. The Doppler frequency shift of its line-of-sight direction is proportional to the relative velocity of the sea surface movement and the radar platform, which has gradually become an important sea surface dynamic remote sensing parameter(Chapron et al., 2005; Hansen et al., 2011a; He et al.,2020). It has been widely applied in ocean surface state parameter inversion, sea clutter suppression, and sea surface target detection and recognition, etc.,especially is signif icant for the study of sea surface currents and meso-sub-mesoscale ocean current processes (Purkis and Klemas, 2011; Klemas, 2012).

Considering the Bragg scattering, specular scattering and wave breaking, the backscatter coeffi cient model (Kudryavtsev et al., 2003) and the two-dimensional radar imaging model (RIM)(Kudryavtsev et al., 2005) were constructed.Analyzing the two models, we f ind that the backscattering coeffi cient model is more suitable for analyzing the ocean surface background f low f ield of mesoscale and small-scale motion. For the study of large-scale sea surface f low f ield, Doppler frequency information needs to be introduced. Chapron et al.(2005) f irst elaborated theoretically the feasibility of Doppler signal inversion of ocean currents, and analyzed the relationship between Doppler frequency shift signal and backscatter coeffi cient. Based on the theoretical framework of the RIM model, the Doppler information was introduced to construct the Doppler RIM (DopRIM) model (Johannessen et al., 2008),which can be used to analyze the interaction among wind, waves, and currents under the diff erent sea surface condition. Utilizing DopRIM model to simulate the wave-current interaction in the strong tidal current area, the research results conf irmed the modulation of the strong surface current on the sea surface roughness and slant distance Doppler frequency signal (Johannessen et al., 2008). By considering the diff erences of radar polarization and incidence angle, DopRIM can separate non-Bragg scattering components from the Doppler center frequency. The research of Doppler frequency information has greatly promoted the measurement of sea surface dynamic processes. The diff erence in radial velocity monitored by high frequency (HF)radar and SAR in the strong tidal current area found that the SAR inversion results are aff ected by the wind f ield due to the interaction of wind waves and surface currents, which restricts the accuracy of SAR ocean current inversion (Danilo et al., 2007; Han et al., 2017). The C-band Doppler shift (CDOP) model was proposed to calculate the wind-induced Doppler frequency shift (Mouche et al., 2012), which can achieve to remove the contribution of ocean surface wind to the Doppler shift from the SAR Doppler center frequency. The SAR ocean current inversion method based on the Doppler frequency shift theory mainly removes the Doppler frequency components caused by the motion of the non-sea surface current from the Doppler center frequency, then construct Doppler centroid frequency anomaly algorithm of ocean current to derive the SAR radial f low f ield. The error sources of SAR ocean current inversion based on Doppler frequency shift theory have been pointed out and the error correction method has been proposed.This method has been widely used in the monitoring of Agulhas Flow (Rouault et al., 2010), Norwegian Atlantic Slope Current (Hansen et al., 2011b), strong tidal currents off the coast of Normandy (Chapron et al., 2005), and Changjiang (Yangtze) Coastal Current(Wang et al., 2014).

It is worthwhile to note that, for SAR ocean current monitoring based on the Doppler frequency shift method, one of the most critical parameters in the inversion processes is the original Doppler frequency anomalyfDca(Wang et al., 2014; He et al., 2020), its value is generally obtained by calculating the diff erence between the measured Doppler frequencyfDcand the predicted Doppler frequencyfDp, wherefDcis the echo frequency of the radar beam center, andfDpdenotes the Doppler frequency caused by the relative motion between the Earth and the satellite. The calculation accuracy offDpis one of the main factors aff ecting the accuracy of SAR ocean current inversion(Hansen et al., 2011a). Usually, thefDpcalculation is implemented with the help of the pre-compiled C language library CFI software released by European Space Agency (ESA), using timing information and orbit state vectors (Hansen et al., 2011a; Wang et al.,2014). That is to say,fDpis calculated under ideal operating conditions from the attitude control of the satellite inertial navigation system. However, this calculation requires accurate SAR orbit timing, sensor position, and attitude parameters, etc. There is a certain deviation in the acquisition of these actual parameters.The theoretical antenna pattern rather than the actual azimuth antenna pattern was utilized within the SAR processor, which leads to inaccurate electronic pointing, in turn, increases physical mispointing owing to the imperfect known satellite attitude parameters(Hansen et al., 2011a). This induced an off set of varying strongly with the viewing angle in the range direction. Specif ically, a 0.01° pointing error in yaw(rotation about the vertical axis) would give an error infDpof about 14 Hz atγ=20° and about 27 Hz atγ=40°(Nilsson and Tildesley, 1995; Hansen et al., 2011a).

Most of the research focuses on the correction of Doppler shift anomalies from the SAR range and azimuth directions (Johannessen et al., 2008; Hansen et al., 2011a). The azimuth direction correction is mainly based on the linear relationship between the azimuthal gradients of backscatter variation in a single Doppler frequency grid and the gradient change offDcaover the land cover. The range direction correction relies on the reference SAR image with enough land cover, from adjacent orbits/acquisition time (±3 days) of SAR main image for ocean current monitoring. It is mainly realized by constructing a polynomial function of the Doppler frequency anomalyfDcaerror caused by the altitude angle of the SAR reference image. The factors considered in this correction method are very comprehensive (Hansen et al., 2011a; Wang et al., 2014; He et al., 2020), but the actual processing process is complicated, and there is an over-correction of Doppler center frequency anomaly (He et al., 2020). In addition, not all ocean currents (such as Kuroshio mainstream, Equatorial Current, and West Wind Drift) are adjacent to the suffi cient land cover areas. Therefore, we proposed to start from the Doppler center frequency model of the SAR echo signal and analyze the characteristics of the Doppler center frequency itself to further clarify the variation law of the Doppler center frequency in the azimuth and range directions. Based on this change rule, the inf luence of the predicted Doppler shift could be removed from the azimuthal and radial directions,respectively. This method can eff ectively avoid thefDpcalculation that depends on the precise SAR orbital time, position, sensor attitude parameters, and radar pointing angle, etc., which will improve the calculation accuracy offDca, and then obtaining high precision sea surface radial current f ield.

The structure of this paper is as follows. In Section 2, the theoretical Doppler center frequency model in the range and azimuth directions and its change characteristics are introduced. Section 3 gives the algorithm construction to remove thefDpfrom the two directions; in Section 4 we apply the proposed algorithm for ocean surface radial current retrieval from SAR. The eff ectiveness and robustness of the proposed algorithm in the current measuring are assessed in Section 5 through the comparison with the global ocean multi-observation (MOB) products.Section 6 presents our conclusions.

2 THEORETICAL MODELS OF DOPPLER CENTER FREQUENCY

The Doppler center frequency estimation of the SAR echo signal is a key technology for SAR imaging(Wong and Cumming, 1996; Toporkov and Brown,2002). Its spatial resolution is usually on the kilometer level, which is much lower than the spatial resolution of SAR backscatter intensity image of typical 10-100 m (Fig.1). The SAR echo signal contains the Doppler frequency shift information generated by the relative movement of the sensor and the sea surface state. The research on the characteristics of the Doppler center frequency of the SAR echo signal is helpful for the SAR monitoring of the dynamic sea surface. Starting from the existing theoretical model of Doppler center frequencyfDc, according to the geometric characteristics of satellite imaging, the characteristics offDcin the range and azimuth directions are discussed, respectively.

2.1 Theoretical model

The Doppler center frequencyfDcis mainly composed of two parts, the predicted Doppler frequencyfDp, and the Doppler frequency anomalyfDcaproduced by complex sea surface conditions. The relationship can be expressed as:

wheref*Ris the radial component of the corresponding Doppler frequency,f*AZis the azimuthal component of the corresponding Doppler frequency.

Fig.1 The backscatter of the SAR scene over Changjiang (Yangtze) River coastal area on Jan. 31, 2005 (a); the Doppler centroid grid (points) has superimposed (b)

2.1.1 Doppler center frequency in the range direction

The hyperbolic form of the range equation is a commonly used range model in spaceborne SAR imaging algorithms. In this case, the satellite’s actual motion trajectory is assumed to be local uniform linear motion, and the Earth is assumed to be locally f lat and not rotating. Since the synthetic aperture time of the low Earth orbit (LEO) SAR is relatively short,generally less than 2 s, the accuracy of the hyperbolic range model can meet the demand when the spatial resolution requirement is not very high. Without loss of generality, the geometry of spaceborne SAR is shown in the Fig.2.

Using the parameters def ined in the Fig.2, the expression of Doppler center frequency is (Raney,1986; Madsen, 1989; Cumming and Wong, 2005)

Fig.2 The geometry of spaceborne SAR

Assuming thatR0is the slant distance at a certain instantaneous point,φshown in Eq.6 is expanded into a f irst-order Taylor series, which is given by:wheref0represents the Doppler center frequency expanded by the off -nadir angle at a certain instantaneous point, which can be approximated as a constant. Therefore, the Doppler center frequencyfDccan be approximated as a linear relationship with the slant range distance.

2.1.2 Doppler center frequency in the azimuth direction

Low Earth satellites equipped with SAR sensors generally operate in near-circular orbits below 1 000 km. The speed V and acceleration A of the satellite can be considered constant in a local area, that is, | R|and V are constant near the instantaneous deployment point. Therefore, the azimuth echo signal of the spaceborne SAR is approximately linear frequency modulated signal, andfDchas an approximate linear relationship with time in the azimuth direction.

At present, the orbital height of the spaceborne SAR in orbit is less than 1 000 km, which is the LEO SAR. When the imaging resolution is not high or the scene is not large, e.g. SEASAT satellite SAR, ERS-1/-2 satellite SAR, and ENVISAT satellite ASAR wide swath, alternating polarization or image mode,the hyperbolic range model is adequate over the duration of the target exposure time, which is typically on the order of a second. Under this circumstance, the Doppler center frequencyfDcchanges approximately linearly with the slant range in the range direction and with the azimuth time in the azimuth direction.Previous studies have also conf irmed the approximate linear relationship. The study of Madsen (1989)pointed that, by theory, the Doppler centroid is very close to a linear function of range. And then, by f itting the estimated Doppler centroids to a linear function of range and calculating the root mean square deviation of the observations relative to the f it, SEASAT SAR was employed to discuss the diff erent Doppler center algorithms. Wang (2005) described the technical process of calculating the Doppler center frequency with correlation Doppler estimator (CDE) method,and calculated the Doppler center frequency of the ERS-1 SAR image by using CDE. The calculation results conf irmed that there is an approximate linear relationship between the Doppler center frequency and the slant distance. Yu (2006) exploited the approximate linear relation of the Doppler center frequency against the slant range. Based on this linear relationship, a new method was proposed to resolve SAR pulse repetition frequency ambiguity. Zhao et al.(2013) studied the practicality of the hyperbolic range model in low-orbit SAR. Taking TerraSAR-X as an example, the maximum synthetic aperture time that the hyperbolic range model can be used for is 4.4 s,and the highest resolution is 0.36 m.

2.2 Change characteristics of Doppler center frequency

Fig.3 The f Dc from the SAR scene on Jan. 31, 2005 (a) and the Doppler change characteristics of the three lines along the range direction (b)

The metadata of the ENVISAT ASAR wide swath mode data product includes parameter f iles and datasets, such as the geolocation grid(GEOLOCATION_GRID_ADS), Doppler centroid parameter dataset (DOP_CENTROID_COEFFS_ADS), and antenna elevation pattern dataset(ANTENNA_ELEV_PATTERN_ADS). The data record of the Doppler centroid parameter dataset takes the receiving time of the SAR echo signal in the range direction as thexcoordinate, and the receiving time in the azimuth direction as theycoordinate,forming a coordinate system with the unit of time interval. The time coordinate in the range direction takes the time interval of nanoseconds as the unit, and the time interval from the start time to the end time is gridded into 100 columns. The time coordinate in the azimuth direction is based on the movement time of the satellite in the azimuth direction, also takes the time interval of nanoseconds as the unit. The grid value is based on the length of azimuth time between the start time and the end time.

A heterogeneous Changjiang River coastal SAR scene acquired on Jan. 31, 2005, including both land and water area, is shown in Fig.1a. From the Doppler centroid parameter dataset, the baseband Doppler center frequencyfDccan be extracted and is shown in Fig.3a.

2.2.1 The range direction of Doppler center frequency

From thefDcgrid of the SAR image on Jan. 31,2005, three section lines in Fig.3a along the range direction are randomly selected to demonstrate the linear change characteristics of the Doppler center frequency. The freedom degrees of the three section lines on SAR image aren-2=100-2=98. The linear f itting degrees are 0.920 9, 0.916 7, and 0.942 2,respectively. These all pass the signif icance test with a conf idence level of 0.001 (freedom degrees=100,critical value=0.321 1), which verif ies that the change of Doppler center frequency in the range direction is approximately linear.

Fig.4 The f Dc from the SAR scene on Jan. 31, 2005 (a) and the Doppler change characteristics of the three lines along the azimuth direction (b)

2.2.2 The azimuth direction of Doppler center frequency

Similarly, three section lines in Fig.4a along the azimuth direction are selected from the Doppler grid image on Jan. 31, 2005. The interval of each section line is approximate 20 grid widths in the range direction. The freedom degrees of the three lines aren-2=52-2=50, the linear f itting degrees are 0.976 6,0.972 5, and 0.734 3, respectively. These all pass the signif icance test with a conf idence level of 0.001(freedom degrees=50, critical value=0.443 3), which show that the change of Doppler center frequency in the azimuth direction is approximately linear.

3 ALGORITHM CONSTRUCTION FOR PREDICTED DOPPLER FREQUENCY REMOVAL

For SAR imaged on the ocean surface, the component in the range directionfDcRoffDcis composed of two parts. The f irst is the range component of the Doppler center frequencyfDcaRgenerated by the sea surface motion. The second is the Doppler frequency component in the range directionfDpRinduced by the relative motion of the satellite and the Earth. The conceptual decomposition can be expressed as

Therefore, the Doppler frequencyfDpAZalong the azimuth direction andfDpRalong the range direction produced by the relative motion of the satellite and the Earth can be removed from the Doppler center frequency, respectively. Then, the Doppler frequency anomalyfDcacan be obtained, which is mainly caused by sea surface movement, land inf luence, wind,nonlinear hydrodynamic eff ects, etc. The f lowchart of removingfDpby the proposed algorithm is schematically shown in Fig.5.

3.1 Removal of the predicted Doppler frequency in the range direction

The mean f iltering method is used in the range direction, that is, the average value of all grids in the azimuth direction is calculated for each time gird along the range direction of the Doppler center frequency. This average value isfDpR.

Fig.5 Block diagram of the proposed algorithm for f Dp removal

3.2 Removal of the predicted Doppler frequency in the azimuth direction

Figure 4 demonstrates that the change of Doppler center frequency in the azimuth direction is approximately linear. Therefore, there is a linear relationship between the time scale grid in the azimuth direction and the Doppler frequency shift, which can be expressed as

where the constantsAandBcan be found by a linear f itting. Sinceidenotes the range indices in the set {1,2, ???, 100}, 100 linear f itting equations can be obtained. That is, the linear f itting method is used to construct the linear functionFat each range index.Also, the corresponding correlation coeffi cientR2can be obtained. The f itting line is determined the best f itting line whenR2is the maximum value.

4 NEW ALGORITHMS FOR CURRENT RETRIEVAL

Due to the complexity of the hydrodynamic interaction among actual ocean waves, there are many factors that aff ect the Doppler spectrum of backscattering from the sea surface. At present, it is still diffi cult to clarify the mechanism of action of various factors. However, the main factors and their mechanisms of action can explain the Doppler frequency shift characteristics of sea surface backscatter under diff erent observation conditions. In addition to the impact of ocean currents, these factors also include gravity wave motion, wind drift, and nonlinear hydrodynamic eff ects among waves.

4.1 Separation of the Doppler components caused by non-ocean current

The estimation errorsferr, such as the azimuthal variation of normalized radar backscatter cross section (NRCS) and strong discrete targets, that contaminate the geophysical Doppler frequency anomaly information, must be eliminated f irst. This error correction is carried out by f inding the linear relationship between the azimuthal gradients of backscatter variation in a single Doppler frequency grid and gradients of original Doppler centroid anomaly along azimuth over land cover (Chapron et al., 2005; Johannessen et al., 2008; Hansen et al.,2011a; Mouche et al., 2012; Wang et al., 2014).

Fig.6 The Doppler frequency estimated from SAR scene on Jan. 31, 2005

Fig.7 The geophysical Doppler frequency anomaly f g on Jan. 31, 2005

Wave simulation based on various empirical sea spectrum models usually produces gravity waves composed of a large-scale fundamental wave and a small-scale resonance capillary wave. This smallscale wave is superimposed on the large-scale wave.Small-scale harmonic components produce Bragg scattering, which modulate the sea surface radar echo.Ignoring the higher-order terms related to sea surface tension and sea water density, the Bragg dispersion frequency shift can be calculated by the following formula (Toporkov and Brown, 2000):

where g0is the acceleration of gravity (m/s2);kdenotes the radar incident wave number (1/m), andθis the incident angle.

Based on a three-layer neural network, an empirical geophysical model function related the parameters of incidence angle, wind speed, and wind direction to C-band Doppler shift is derived (Mouche et al., 2012),which can be used to calculate the Doppler frequency shift from wind induced ocean waves,fw.

Fig.8 The radial Doppler velocities derived from the SAR scene on Jan. 31, 2005

Combing the Eqs.21, 22, andfw, the geophysical contribution of ocean current to the Doppler frequency anomaly,fg(Fig.7), can be obtained from the following equation:

fg=fDca-ferr-fB-fw. (23)

4.2 Result of current retrieval

Fig.9 The Doppler centroid anomaly

Figure 8 shows the radial Doppler velocities derived from the SAR scene in the Changjiang River coastal zone on Jan. 31, 2005.Vdis interpreted to be the radial component of the true surface current. Areas of the Changjiang River coastal current whereVd＜0 correspond to yellow, red, and black, showing that surface velocities are directed towards the satellite f light projection line, that is the Changjiang River coastal current meandering to the east/southeast,generally parallel to the shoreline. Due to the narrow coastal shallow water area and the large seabed slope,the range of coastal currents is relatively narrow. The velocity is relatively uniform of 0.1-0.3 m/s, which is mainly formed by the runoff of the Changjiang River and Qiantang River into the sea. The observed velocity f ields demonstrate that the strongest is found in the Hangzhou Bay. Velocities amounting to 0.2-0.3 m/s, are mainly aff ected by the interaction of the runoff and the ocean tide. At approximately (32.0°N,123.0°E), there develops a small-scale vortex. On the south side of the vortex, from (31.5°N, 122.7°E) to the southeast to (31.2°N, 123.7°E), whereVd＞0 is shown in green, corresponding to a westerly or southwesterly f low. It has a velocity of about 0.10 m/s.

The gray regions are f lagged invalid because of land or 10-km buff er zone of land.

5 ASSESSMENT OF THE PROPOSED ALGORITHM

To show the improvements in current-measuring performance off ered by the proposed predicted Doppler frequency removal algorithm depicted in Section 3 over other existing methods, we have compared three current-retrieval methods:

(i) the Doppler frequency method without error correction;

(ii) the conventional Doppler method with error correction (Hansen et al., 2011a; Wang et al., 2014);

(iii) the Doppler frequency anomaly method that incorporates the proposed predicted Doppler shift removal algorithm as shown in Fig.5 in this paper.

The method i of Doppler frequency anomaly works by directly subtract thefDpfromfDc. And thefDpis implemented with C language library CFI software by calling four function functions (Wang et al., 2014),includingpl_emjd(time format conversion),pp_init_attitude_f ile(satellite attitude f ile initialization),pp_get_attitude_aocs(rotation angle, pitch angle, and yaw angle calculation), andpp_target(fDpand the corresponding location parameters calculation). As for the method ii, the reference SAR images with suffi cient land cover from adjacent acquisition time(±3 days) of SAR main image are selected to implement the errors correction. After implementing each of the three aforementioned methods on SAR image, the line-of-sight Doppler velocities are retrieved.

5.1 Comparison of Doppler centroid frequencies

Using software CFI to eliminate the Doppler frequencyfDpgenerated by the relative motion of the satellite and the Earth (Method i and ii), there exists the relationship between the variation along range direction and the decreased/increased frequencies aligned in the azimuth direction, which are demonstrated in Fig.9a. This phenomenon is most clear in the f irst subswath, where thefDcais about 40-70 Hz between range pixel numbers 1 and 8, azimuth pixel numbers 0 and 20. At range pixel numbers 9 and 20, azimuth pixel numbers 0 and 20, thefDcamainly reduced to about 0-20 Hz. The phenomenon at the second subswath is also apparent, while the degree of signif icance is lower than that at the f irst subswath.Electronic mis-pointing and inaccurate satellite orbitand attitude parameters (yaw, pitch, and roll) are the main reasons for the deviation. These parameters f irst bias the predicted Doppler frequencyfDp(Chapron et al., 2005; Hansen et al., 2011a), and then transmitted tofDca. However, the proposed method in this study can eff ectively avoid the dependence on the satellite orbit and attitude parameters, thereby reducing the estimation error offDca(Fig.6c).

Table 1 The variation of Doppler frequency anomaly over land

In addition, large deviations also appear in the overlapping and transitional areas of diff erent subswaths, which are mainly manifested as vertical strips along the azimuth direction, shown in the black elliptical area in Fig.9a. Moreover, a largefDcavariation is presented over land where the Doppler shift is expected to be zero. Especially at the boundary of land and sea, the variation is more obvious. As far as the spatial resolution is concerned, the Doppler centroid resolution (8 km in the azimuth direction and 3.5-9 km in the range direction) is much coarser than the NRCS image (75 m×75 m). Thus, much of this variability is due to NRCS gradients along the azimuth direction, which is within the estimation area of the Doppler centroid pixels (Chapron et al., 2005; Wang et al., 2014). Using the three aforementioned methods,the results all have estimation deviations of Doppler frequency anomaly caused by the sharp change of NRCS gradients in the azimuthal direction. However,the improvement in the velocity-measuring performance off ered by the proposed algorithm can be explained by looking at the Doppler frequency anomaly over land given in Table 1. After the proposed algorithm was implemented, the Doppler frequency anomaly over land decreases from 24.50 to -2.99,suggesting that the uncertainty of Doppler frequency anomaly is signif icantly reduced. After the error correction in the azimuth and range directions, mean Doppler frequency anomaly over land is reduced to 10.80 Hz, standard deviation (STD) reduced to 19.82 Hz by using the method ii (Fig.9b). As for the proposed method in this study, the mean and STD values are -2.69 and 15.18 Hz, respectively.

Fig.10 The radial Doppler velocities derived from the SAR scene on Jan. 31, 2005 with the corresponding MOB products superimposed as arrows

Therefore, according to the above comparative analysis, we can derive that the proposed method can eff ectively avoid thefDcaerror caused by employing the inaccurate satellite orbit and attitude parameters.Also, the method helps to reduce the deviation at the overlapping and transitional regions in the diff erent subswaths. In addition, the average Doppler frequency anomaly obtained by the proposed method over the land area is only -2.99 Hz. The new method has a strong inhibitory eff ect on thefDcaestimation error caused by the dramatic change of NRCS gradients in the azimuthal direction.

5.2 Comparison of current results

To validate the sea surface f low characteristic obtained using the proposed method, we projected the MOB products (Rio et al., 2014) onto the SAR radial current direction. The SAR surface radial currents were compared with the MOB products (Fig.10). For the Changjiang River coastal current, the SAR f low direction is consistent with the MOB products, and is towards the east/southeast along the coastline. The f low velocities of both results are concentrated in 0.1-0.3 m/s. At approximately (32.0°N, 123.0°E), a current with velocity of 0.1 m/s turns around at(31.5°N, 123.5°E) and gradually turns to the west at about (31.2°N, 123.2°E) and turns to north at about(31.2°N, 122.8°E), and then to east at about (32.2°N,123.0°E), form a small-scale vortex in the MOB products, which coincides with the current seen in the SAR-derived results.

Fig.11 Error analysis of the radial velocities: comparing MOB products with SAR-derived data using method i (a), method ii (b), and the proposed method (c)

To quantify the velocity-measuring performances of the three methods, the error statistical results found by comparing the radial current velocities derived from the SAR scenes using the three methods with the MOB products are shown in Fig.11.

As demonstrated in Fig.11, the velocity measuring quality improves pronouncedly after the implementation of the proposed algorithm, with the mean velocity bias reduced from 0.34 m/s down to 0.029 m/s relative to that achieved with the conventional Doppler method without error correction. By contrast, the performance improvement off ered by the method ii with error correction is not so signif icant, with the mean velocity bias reduced from 0.34 to 0.23 m/s. The reason for this is that there is still a deviation in the Doppler centroid frequency estimation caused by the sharp change of the NRCS gradient in the azimuth direction even after the error correction (Chapron et al., 2005; Hansen et al.,2011a; He et al., 2020).

We also see that, as compared to the conventional Doppler method with/without errors correction, there is also an improvement in terms of the STD of the measured velocities after applying the proposed algorithm. The STD achieved by the proposed algorithm is as low as 0.14 m/s, in contrast to 0.22 and 0.19 m/s achieved with the conventional Doppler method without/with error correction, respectively.

All these results demonstrate the eff ectiveness of the proposed algorithm. We therefore state that SAR ocean current velocities retrieved the proposed method in the paper have a high accuracy and have the potential to fully capture the current f low dynamics.

Fig.12 The SAR backscatter image acquired over the Kuroshio area on Jan. 31, 2008

5.3 Application of proposed method in high velocity current measurement

To demonstrate the eff ectiveness and robustness of the proposed Doppler algorithm depicted in Fig.5 in terms of current measurement with diff erent velocity,we have conducted current retrieval from a SAR scene acquired over the Kuroshio area that corresponds to high current velocity retrieval in a homogeneous open sea. The corresponding SAR scene is shown in Fig.12. The Kuroshio main stream is a fast-f lowing, warm, and highly saline ocean current and is an important component of the North Pacif ic subtropical circulation system. It enters the East China Sea through the passage between TaiwanIsland and Iriomote Island. Aff ected by steep continental slope, the Kuroshio amplitude, velocity,and f low vary at diff erent place (Guo et al., 2006;Geng et al., 2018; Qiu et al., 2020). The velocity is generally 0.51-1.02 m/s, and the maximum is 1.02-2.57 m/s (Ambe et al., 2004; Deng et al., 2015;Zhuang et al., 2020).

The Doppler centroid frequencyfDcdetermined from a descending SAR scene on Jan. 31, 2008 is presented in Fig.13a. Employing the proposed algorithm in the paper, Fig.13b shows the correspondingVdretrieval results. Since these are derived from a descending SAR image, areas of the Kuroshio withVd＜0 are cyan, purple, and magenta,corresponding to a surface current directed towards the satellite f light projection line, that is the main f low of the Kuroshio is meandering to the east/northeast along the Okinawa Trough. Also, a counter current withVd＞0 meanders found at approximately (26.2°N,126.7°E) and (25.8°N, 126.0°E), corresponding to a westerly or southwesterly f low in this descending image. Therefore, the SAR currents can obtain the roughly current direction ref lected by the radial information. Both the SAR currents and the MOB products capture the intensity of surface currents in the Kuroshio region and the two data sets are in good agreement (Fig.13b). As for the Kuroshio mainstream,the SAR f low direction is consistent with the MOB products, and is towards the east/northeast. The central f low velocity of the Kuroshio is high and the radial characteristics are clear.

6 CONCLUSION

In this study, we developed a new algorithm for eliminating the predicted Doppler frequency used for surface current retrieval with synthetic aperture radar.The basis behind the proposed algorithm lies in a comprehensive analysis of the theoretical model of the Doppler center frequency for the SAR. The linear characteristic of the Doppler center frequency in both the range direction and azimuth direction have been demonstrated.

The proposed algorithm includes two steps. In the f irst step, the mean f iltering method is used to obtain the component of the predicted Doppler frequency in the range direction, and then to discard from the Doppler center frequency. In the second step, we search the best f itting line according to the corresponding linear correlation coeffi cient between the time scale grid in the azimuth direction and the obtained Doppler frequency after the implementation of the f irst step. Based on the best f itting line, the component of the predicted Doppler frequency in the azimuth direction can be calculated and removed.

As compared with the conventional Doppler frequency method without/with error correction, the proposed algorithm allows for a pronounced improvement in the current measuring accuracy through the comparison with the global ocean multiobservation (MOB) products. Also, the eff ectiveness and robustness of the proposed Doppler algorithm has been demonstrated by its application in the high velocity current in the Kuroshio region.

7 DATA AVAILABILITY STATEMENT

The SAR images used in this study are distributed by ESA, and are available from https://earth.esa.int/eogateway/catalog/envisat-asar-ws-medium-resolutionl1-asa_wsm_1p-. The global ocean multi-observation(MOB) products, MULTIOBS_GLO_PHY_REP_015_004, are distributed by CMEMS, and could be downloaded from http://marine.copernicus.eu.

8 ACKNOWLEDGMENT

The authors would like to acknowledge the data made available by the European Space Agency and the Copernicus Marine Service for the global ocean multi observation products. We wish to thank the editor and anonymous reviewers for their helpful comments and suggestions.