Surface Vector Current Retrieval by Single-Station High-Frequency Surface Wave Radar Based on Ocean Dynamics in the Taiwan Strait

Abstract

In order to address the issue of limited common coverage and high cost in mapping ocean surface vector current by two (or more) high-frequency surface wave radars, this paper proposes a single-station surface wave radar vector current inversion algorithm. The feasibility of this algorithm has been validated in the Taiwan Strait. Based on the ocean dynamic characteristics of the Taiwan Strait, the algorithm utilizes the radial current obtained from a high-frequency surface wave radar (HFSWR) in Fujian Province to invert the ocean surface vector current. The surface vector current can be decomposed into three primary dynamic components: tidal currents, wind-driven currents, and geostrophic currents. Firstly, tidal current forecasting models and Ekman and Stokes theories are used to calculate the tidal and wind-driven currents in the Taiwan Strait, respectively. Subsequently, the directions of geostrophic currents in the Taiwan Strait are determined with sea surface height data, and the magnitudes of the geostrophic currents are constrained using the radial current from the single HFSWR. Finally, the three components are added together to obtain the vector current. Comparative results demonstrate that the efficacy of the algorithm has been validated through field experiments (with two HFSWRs and two drifting buoys) conducted in the southwestern of the Taiwan Strait. Further research is needed on the applicability of this algorithm to other sea areas and monitoring systems.

1. Introduction

The Taiwan Strait is an important maritime passage, and observations of surface currents in this region play a significant guiding role in various aspects, such as navigational safety, maritime search and rescue, marine fisheries, pollution monitoring, and the development of marine resources [1,2,3,4]. In recent decades, high-frequency surface wave radars (HFSWRs) have been used as over-the-horizon radars for monitoring oceanic currents. Both research teams from mainland China and Taiwan have deployed several HFSWRs to monitor surface currents in the Taiwan Strait. These include the marine observation demonstration area of the Fujian Provincial Ocean Forecasting Station, which consists of two radar stations in Dongshan and Longhai, and the Taiwan Ocean Radar Observing System (TOROS), which has 19 HFSWRs deployed along the coasts of Taiwan and Penghu Islands [5,6,7,8,9]. Their observation coverage spans nearly the entire Taiwan Strait.

According to the Doppler principle, a single HFSWR can only detect the radial component of the current. This means that at least two HFSWRs are required to obtain the vector current within their common coverage area [10,11,12,13,14]. However, this poses two challenges: it increases the demands on the layout and location selection of radar arrays, leading to higher costs, and it restricts the detection range of the vector currents to jointly covered areas. Therefore, estimating the vector currents solely based on the radial currents detected by a single HFSWR is a significantly valuable and critical area of focus. Previous studies have proposed various algorithms to address these challenges and estimate vector currents based on radial currents. These algorithms can be categorized into two types: those based on algebraic-geometric relationships and those based on fluid dynamic constraints. They provide new approaches for inverting vector currents, expanding the potential applications of HFSWRs.

The major examples include the Parameter Estimation Algorithm [15], Stream Function Method [16], Optimal Interpolation (OI) method [17], Vortex Identification Method [18], and Stream Potential Function Method [19]. The Parameter Estimation Algorithm fundamentally utilizes the least squares method to estimate the magnitude and direction parameters of vector currents. It has been previously applied in shore-based and shipboard radar systems [20,21], yielding positive results. However, the algorithm’s limitations are based on the assumption of spatially uniform ocean currents, resulting in significant errors under moderate to high sea conditions. The Stream Function (SFM) is contingent upon the assumption of horizontally non-divergent ocean dynamics, while the Stream Potential Function (SPFM) incorporates horizontally divergent potential functions, making it more suitable for realistic oceanic scenarios compared to SFM. Marmain et al. achieved vector current mapping from a single HFSWR by combining non-divergent ocean currents with the Vortex Identification Method (VIM), yet lacked experimental validation [18]. Additionally, Wang et al. proposed a method based on sparse representation and unitary transformation techniques for estimating vector currents based on the stream function and verified the algorithm through simulations, but did not conduct field experiments [22]. Furthermore, Igor used radial currents as one of the inputs, significantly improving the assimilated model output of oceanic currents [23]. Han et al. developed an inversion algorithm by combining variational adjoint assimilation methods and the POM numerical model, and achieved promising results in field experiments; however, the duration of comparative validation was limited [24].

This paper presents a radar vector current retrieval algorithm based on ocean dynamic characteristics in the Taiwan Strait. Previous research suggests that the currents in the Taiwan Strait consist of three main components: tidal currents, geostrophic currents, and wind-driven currents, which can be written as below,

$$\left\{\begin{array}{c}\stackrel{\rightharpoonup }{V}={\stackrel{\rightharpoonup }{V}}_t+{\stackrel{\rightharpoonup }{V}}_w+{\stackrel{\rightharpoonup }{V}}_g\hfill \\ u=u_t+u_w+u_g\hfill \\ v=v_t+v_w+v_g\hfill \end{array}\right.,$$

(1)

where $\stackrel{\rightharpoonup }{V}$ is the total vector current, ${\stackrel{\rightharpoonup }{V}}_t$ is the tidal current, ${\stackrel{\rightharpoonup }{V}}_w$ is the wind-driven current and ${\stackrel{\rightharpoonup }{V}}_g$ is the geostrophic current. Tidal currents are generated by the tide-generating forces of the sun and moon [25,26,27,28], while geostrophic currents result from the balance between the horizontal pressure gradient force and the Coriolis force, influenced by sea surface heights (SSH) [25]. Wind-driven currents are caused by friction between the sea surface wind and the sea surface. Theoretically, sea surface vector currents can be directly calculated using dynamic theory, eliminating the need for remote sensing methods. However, ideal assumptions and other factors in dynamic theory may lead to variations in actual sea surface currents. Therefore, we propose combining dynamic theory with single-station HFSWR observations to invert the surface vector currents and verify the effectiveness of the algorithm using the vector currents measured by two HFSWRs and drifters in the field experiment.

This paper is organized as follows. Section 2 presents the dataset and methodology utilized to conduct the research. Section 3 presents the results. A discussion is provided in Section 4, followed by the conclusions in Section 5.

2. Data and Methods

2.1. HFSWR Experimental

Filed observations were conducted using two HFSWRs deployed along the coast of Dongshan and Longhai in Fujian Province from 22 August 2021 to 27 August 2021 (UTC+8). The HFSWRs operated at the 8.75 MHz frequency, with an output of the ocean surface currents at a spatial resolution of 5 km every 10 min. The radial currents derived in this study represent the projected vector currents toward the Dongshan station. Specific parameters of HFSWRs can be found in the literature [29].

Within the combined coverage area of these two radars, drifters were deployed from vessels to validate the effectiveness of the proposed algorithm in this paper. The drifters used were LTWDB01 long endurance surface drifters (as shown in Figure 1b), capable of floating on the ocean surface for extended periods to measure the surface ocean current speed and direction. These buoys weigh no more than 5 kg and utilize the BeiDou II satellite positioning system, providing an accuracy of 10 m and a data update rate of 5 min. Figure 1c depicts the trajectories of the two drifters. It is evident that the deployment positions of the two drifters differed, and both buoys followed trajectories oriented from the southwest to the northeast. The timestamps for the drifter data can be found in

Table 1. Information of drifters.

Drifters	Start Time	End Time	Positioning Accuracy
1	17:16, 22 August 2021	16:25, 26 August 2021	BeiDou II, <10 m
2	10:57, 22 August 2021	02:23, 27 August 2021	BeiDou II, <10 m

(UTC+8). Additionally, Figure 1a shows the wind rose during the experimental period.

There is no moored buoy in this experiment, and there is no ADCP data available for verification in this sea area. Our proposed algorithm needs to be verified by comparing it with HFSWR current products and drifter currents. In this study, Dongshan radar station measurements were used to constrain and adjust the speed of the geostrophic current.

Prior to employing the radial current to constrain and adjust the geostrophic current, an assessment is conducted on the HF radar observation currents each time, assigning a credibility value C. We utilize a scale of 0 to 1 to denote the credibility (C) of the ocean currents at each spatial point at each time, where 0 signifies low confidence, and 1 signifies high confidence.

The HFSWR utilizes the first-order spectrum in the echo range Doppler spectrum to retrieve ocean surface currents. The signal-to-noise ratio (SNR) in the first-order spectrum serves as a critical indicator of radar signal quality. A high SNR implies that the signal strength significantly exceeds the background noise, directly impacting the extraction of velocity information using Doppler frequency shifts. Moreover, in scenarios with elevated noise levels, the signal may be engulfed by noise, rendering the effective detection of Doppler frequency shifts unfeasible.

The performance of the Multiple Signal Classification (MUSIC) algorithm in estimating the Direction-of-Arrival (DOA) is crucial. By applying spatial filtering and processing to the signals, the MUSIC algorithm can more precisely determine the location of ocean surface currents, thereby facilitating the extraction of directional information of the ocean currents.

To synthesize vector currents in the common coverage area, at least two radar stations positioned at different locations are required. The Geometric Dilution of Precision (GDOP) is also a vital parameter, as it directly influences the accuracy of radar measurements. A lower GDOP value indicates that the radar system can provide more precise information regarding the current speed and direction.

With reference to the method outlined in [30,31], the credibility of the ocean currents, denoted as (C), can be represented by the following formula:

$$\left\{\begin{array}{c}d{v}_1={\displaystyle \frac{1}{v}}\left[{\displaystyle \frac{v_1-v_{2}cos\left({\theta }_1-{\theta }_2\right)}{si{n}^2\left({\theta }_1-{\theta }_2\right)}}\right]\Delta v_1\\ d{v}_2={\displaystyle \frac{1}{v}}\left[{\displaystyle \frac{v_1-v_{2}cos\left({\theta }_1-{\theta }_2\right)}{si{n}^2\left({\theta }_1-{\theta }_2\right)}}\right]\Delta v_1\\ d{\theta }_1={\displaystyle \frac{1}{v}}\left[{\displaystyle \frac{v_1{v}_2}{sin\left({\theta }_1-{\theta }_2\right)}}-{\displaystyle \frac{v_1^2+v_2^2-2v_1{v}_{2}cos\left({\theta }_1-{\theta }_2\right)}{si{n}^3\left({\theta }_1-{\theta }_2\right)}}cos\left({\theta }_1-{\theta }_2\right)\right]\Delta {\theta }_1\\ d{\theta }_2={\displaystyle \frac{1}{v}}\left[{\displaystyle \frac{v_1{v}_2}{sin\left({\theta }_1-{\theta }_2\right)}}+{\displaystyle \frac{v_1^2+v_2^2-2v_1{v}_{2}cos\left({\theta }_1-{\theta }_2\right)}{si{n}^3\left({\theta }_1-{\theta }_2\right)}}cos\left({\theta }_1-{\theta }_2\right)\right]\Delta {\theta }_2\\ C=d{v}_1+d{v}_2+d{\theta }_1+d{\theta }_2\end{array}\right.$$

(2)

where, $d{v}_1$ and $d{v}_2$ are the errors of radial current velocities measured by two radar sites respectively, and $d{\theta }_1$ and $d{\theta }_2$ are the errors of DOA. Radial current errors $d{v}_1$ and $d{v}_2$ are characterized by the first-order spectral SNR corresponding to the single radar site radial current.

In Figure 1c, the color overlay illustrates the spatial distribution of the mean credibility during the experimental period. It can be observed that the central region of the radar footprint exhibits higher credibility, exceeding 0.7, while the edge areas display lower credibility, dropping below 0.4.

2.2. Tidal Current

Seawater experiences a regular horizontal motion caused by tidal forces, known as tidal currents. The current creates an elliptical pattern when connecting the endpoints of the tidal current vector over one tidal cycle. The pattern is used to represent the rotational direction and magnitude of the tidal current. In coastal areas, tidal currents are constrained by the coastline, resulting in very small semi-minor magnitudes of the tidal ellipses. However, in more distant regions from the coastline, surface currents are influenced by the Coriolis force and predominantly exhibit rotary currents. Surface tidal currents dominate most of the region of the Taiwan Strait [25]. During high tide, offshore tidal currents flow into the central Taiwan Strait, with a northeastward current in the seas off the southern coast of Fujian and a northward deviation within the central and southern parts of the strait. Conversely, during low tide, the tidal currents in the Taiwan Strait exhibit an outflow pattern, with a southwestward current in the seas off the southern coast of Fujian and a southward deviation within the central and southern parts of the strait.

It is generally believed that the tidal current is the superposition of cosine oscillations of various tidal components. Therefore, by expanding the tidal force according to frequency, various tidal components can be obtained. Based on the periodicity of these component tides during their propagation and coupled with observational data (such as tidal gauge stations, buoys, and satellite data), tidal currents can be obtained through numerical simulation methods that consider the impact of actual topography. The tidal inversion and prediction system OTIS (The Oregon State University Tidal Inversion System), developed by Oregon State University, assimilates coastal tidal gauge data along with satellite remote sensing data (TPXO8.0-atlas), enabling the output of tides and tidal currents in the northern South China Sea and East China Sea [32] with a spatial resolution of 1/30° in the Taiwan Strait region. The output includes the tidal ellipse parameters of eight major tidal constituents, such as M2, S2, O1, K1, N2, K2, P1, Q1. OTIS has been validated through multiple comparisons, including satellite remote sensing, high-frequency radar, and moored buoys, all of which have produced positive results since its introduction in 2004 [33,34,35].

In this study, OTIS is adopted to obtain the tidal currents in the Taiwan Strait and support the synthesis of surface vector currents. It is noted that the authors performed spatial interpolation on OTIS to maintain consistency between its spatial latitude and longitude grid points and the radial currents detected by the DS high-frequency radar.

2.3. Wind-Driven Current

The annual average wind speed in the Taiwan Strait ranges from 8.0 to 8.5 m/s. Higher wind speeds are generally experienced in the central and southwestern regions, while the northeastern sea areas have slightly weaker winds [36]. From mid-May to August, the prevailing wind is the southwesterly monsoon, with wind directions mainly from the south to the southwest across most sea areas. The distribution of average wind speeds is greater in the north and south and smaller in the central area, ranging from 6.0 to 9.0 m/s. The average wind speed in the central part of the Taiwan Strait, Kinmen, and Penghu ranges from 8.0 to 9.0 m/s. During the period from October to April of the following year, the prevailing wind in the Taiwan Strait is the northeasterly monsoon, with wind directions mainly from the north to the northeast. This is accompanied by strong winds that reach their peak from December to January of the following year. In January, the frequency of northeasterly winds in the Taiwan Strait can reach up to 60% to 78.8%, and the average wind speed during the northeasterly monsoon period is 9.0 to 13.0 m/s [37].

The wind-driven current ${\stackrel{\rightharpoonup }{V}}_w$ consists of Ekman and Stokes components, which can be computed based on Ekman theory [38,39] and Stokes theory [40], respectively. Ekman components ${\stackrel{\rightharpoonup }{U}}_E$ can be represented as:

$$\left\{\begin{array}{c}{\displaystyle \frac{\partial }{\partial z}}\left(A_z{\displaystyle \frac{\partial {\stackrel{\rightharpoonup }{U}}_E}{\partial z}}\right)=if{\stackrel{\rightharpoonup }{U}}_E\hfill \\ {\stackrel{\rightharpoonup }{U}}_E\left|z=h=0\right.\hfill \\ A_z{\displaystyle \frac{\partial {\stackrel{\rightharpoonup }{U}}_E}{\partial z}}\left|z=0={\displaystyle \frac{-\stackrel{\rightharpoonup }{\tau }}{{\rho }_w}}\right.\hfill \end{array}\right.,$$

(3)

where, $z$ represents the water depth, f is the Coriolis parameter, $A_z$ is the eddy viscosity [41]. The following two formulas are the boundary conditions. Where $h$ is the total water depth, ${\rho }_w$ is the density of seawater. $-\stackrel{\rightharpoonup }{\tau }={\rho }_a{C}_D{U}_{10}^2$ represents wind stress. ${\rho }_a$ is the air density. $C_D$ is the drag coefficient. $U_{10}$ is wind speed at a height of 10 m above ocean surface. According to Wu et al. [42,43], $C_D=\left(0.8+0.065U_{10}\right)\times {10}^{-3}$ is suitable for both high and low wind speeds. According to boundary conditions, it is possible to solve ${\stackrel{\rightharpoonup }{U}}_E$ for different water depths:

$${\stackrel{\rightharpoonup }{U}}_E={\displaystyle \frac{\stackrel{\rightharpoonup }{\tau }{e}^{-i\pi /4}}{{\rho }_w\sqrt{f{A}_z}}}{\displaystyle \frac{\mathit{sinh}[\left(1+i\right)\left(h-z\right)\sqrt{f/\left(2A_z\right)}]}{\mathit{cosh}[\left(1+i\right)h\sqrt{f/\left(2A_z\right)}]}}$$

(4)

Due to the nonlinearity of surface gravity waves, Stokes drift ${\stackrel{\rightharpoonup }{U}}_S\left(z\right)$ occurs in the wave propagation direction. ${\stackrel{\rightharpoonup }{U}}_S\left(z\right)$ can be calculated from the non-directional ocean wave spectrum $F\left(\omega \right)$ [44,45]:

$${\stackrel{\rightharpoonup }{U}}_S\left(z\right)={\displaystyle \frac{2}{g}}{{\displaystyle \int }}_0^{\infty }{\omega }^3{e}^{2\widehat{k}z}F\left(\omega \right)d\omega$$

(5)

where, g is the acceleration due to gravity, $\omega$ is the angular frequency. $\widehat{k}$ is the unit vector in the direction of wave propagation $\theta$. In this paper, JONSWAP wave spectrum is used to calculate ${\stackrel{\rightharpoonup }{U}}_S\left(z\right)$ [46]:

$$\begin{gathered} F{(k)}_{Jonswap}={\displaystyle \frac{0.0081}{2k^4}}\mathit{exp}[{\displaystyle \frac{-5}{4}}({\displaystyle \frac{k_m}{k}}{)}^2]\cdot {\gamma }^{\mathit{exp}[{\displaystyle \frac{-1}{2{\sigma }^2{k}_m}}\left(\sqrt{k}-\sqrt{k_m}{)}^2\right]}, \\ {k}_m={\displaystyle \frac{22g^{8/3}}{U_{19.5}{L}^{2/3}}}, \sigma =\left\{\begin{array}{c}0.07, k\le k_m\\ 0.09, k>k_m\end{array}\right., \lambda =3.3 \end{gathered}$$

(6)

where $k_m$ is the peak wave number, $U_{19.5}$ is the sea surface wind speed at 19.5 m height, $L=100 \mathrm{km}$ is the fetch length, σ is a parameter that controls the shape of the non-directional ocean wave spectrum.

The surface wind data required for calculating wind-driven currents are the sea level 10 m wind U/V components from the ECMWF reanalysis data ERA5, with a temporal resolution of 1 h and a spatial resolution of 0.25°. The sea surface wind speed $U_{19.5}$ at the height of 19.5 m required in Equation (6) can be calculated based on the relationship between wind and height [47]. The wind-driven currents are interpolated to match the spatial and temporal characteristics of the radial current at the DS Station. Distance-weighted interpolation is employed for spatial alignment, while linear interpolation is used for temporal alignment.

2.4. Geostrophic Current

Geostrophic currents, which result from the balance between the horizontal pressure gradient force and Coriolis force, are present not only in open ocean regions but also in nearshore areas, where they contribute to local circulation along with tidal currents. These currents play a significant role in upper ocean mixing and vertical transport [48,49]. A study by Li and Liao found a correlation between sea surface height and winter monsoons, concluding that the persistent influence of monsoons causes sea surface height to rise in the northern Taiwan Bank, similar to how strong winds lift water masses [50,51].

In the southwest region of the Taiwan Strait, offshore and alongshore currents flow eastward or northeastward when the pressure gradient force is balanced with the Coriolis force. According to Oey’s work, the geostrophic balance causes an eastward alongshore flow during the monsoon relaxation period, which then turns north after crossing the Penghu Channel [52]. Wang utilized vector currents obtained from high-frequency radars to verify the timing and extent of stable geostrophic currents. Wang concluded that the geostrophic balance dominates the residual current in the across-strait direction, while in the alongshore direction, both the geostrophic balance and wind stress explain the residual current in this region [53].

Daily datasets were extracted from the Global Ocean Physics Reanalysis Products GLOBAL-REANALYSIS-PHY-001-030 (GLORYS12V1) provided by the Copernicus Marine Environment Monitoring Service (CMEMS, http://copernicus.eu, accessed on 10 September 2023). The GLORYS12V1 product is a CMEMS global ocean eddy-resolving (1/12° horizontal resolution and 50 vertical levels) reanalysis covering the altimetry era of 1993–2024. The calculation of geostrophic currents based on the sea surface height gradient is a well-established theoretical method, demonstrated by Equation (7),

$$\left\{\begin{array}{c}-fv=-g{\displaystyle \frac{\partial \zeta }{\partial x}}\\ fu=-g{\displaystyle \frac{\partial \zeta }{\partial y}}\end{array}\right.,$$

(7)

where $\zeta$ represents the SSH, x, and y are the axes of the Cartesian coordinate system. u and v represent the geostrophic currents calculated using Equation (7) in the equatorial and meridional directions. These values establish the directions of the geostrophic currents within the radar footprint. The spatial distribution of geostrophic currents calculated from sea surface height at UTC 12:00 22 August 2021 is shown in Figure 2, where the majority of the geostrophic currents flow eastward or northeastward. Particularly, the geostrophic current exceeding 30 cm/s south of the DS radar station flows eastward along the southern edge of the Taiwan Bank. Upon nearing the Penghu Islands, it shifts its direction to the northeast and intensifies to 60 cm/s, resembling the formation of a weak cyclonic feature.

Based on the results of the ‘Fujian Coastal Zone Survey’ conducted by the Oceanographic Division of Fujian Provincial Science Committee from 1960 to 1964 and the ‘Comprehensive Survey of the Western Taiwan Strait’ carried out from 1984 to 1985 [48], a persistent northeastward geostrophic current has been identified in the Taiwan Strait [49,50,51]. However, it is important to note that the magnitude of this geostrophic current is not constant, a phenomenon that has also been confirmed by numerous scholars. Assuming that the dynamic theoretical calculations of tidal currents and wind-driven currents are accurate, the determination of the magnitude of the geostrophic current is the only requirement for inferring sea surface vector currents.

In this study, the directions of geostrophic currents existing in the footprint of the HFSWRs are retrieved from SSH. Then, we set the initial geostrophic current as 20 cm/s and introduce a scaling factor $\eta$. We determine the magnitude of the geostrophic current using an iterative method. The value of $\eta$ needs to be determined based on the radial current ${\stackrel{\rightharpoonup }{V}}_{E\_ssw}$. If the value of $\eta$ is correct, the projection of ${\stackrel{\rightharpoonup }{V}}_t$ + ${\stackrel{\rightharpoonup }{V}}_w$ + $\eta \cdot {\stackrel{\rightharpoonup }{V}}_g$ in the radial direction at the DS Station, denoted as ${\stackrel{\rightharpoonup }{V}}_{E\_proj}$, will have the minimum error with respect to ${\stackrel{\rightharpoonup }{V}}_{E\_ssw}$. In this study, the $\eta$ value is determined using radial currents with credibility greater than 0.5 within the detection range. The iterative process is illustrated in Figure 3. Here, ${\chi }^2$ represents the cost function:

$${\chi }^2={\displaystyle \frac{1}{C}}(\sum {[{\stackrel{\rightharpoonup }{V}}_{E\_ssw}-{\stackrel{\rightharpoonup }{V}}_{E\_proj}]}^2)$$

(8)

where ${\stackrel{\rightharpoonup }{V}}_{E\_ssw}$ represents the radial current at the DS radar Station, ${\stackrel{\rightharpoonup }{V}}_{E\_proj}$ denotes the projection of the sea current vector in the radial current direction, and c denotes the total number of grid points where the credibility is equal to or greater than 0.5. We apply this iterative algorithm to determine the value of η at every moment, and the value η is constant for all radial current positions at one moment. Setting $\eta$ as 3 initially, the range for the geostrophic current search is between 0~60 cm/s. By employing the iterative algorithm, we calculate the $\eta$ value that minimizes the error ${\chi }^2$, which represents the specific $\eta$ at that moment.

The temporal and spatial resolutions of the three current components are shown in

Table 2. The temporal and spatial resolutions of the data used in this study.

Current Component	Temporal Resolution	Spatial Resolution	Data Source
Tidal currents	1 h	1/30°	OTIS (TPXO8.0-atlas)
Wind-driven currents	1 h	1/4°	ECMWF (ERA5)
Geostrophic currents	24 h	1/12°	CMEMS (GLORYS12V1)
Radial currents	10 min	5 km	HF radar

, as well as those of the HF radar radial currents. To match the radial currents, the three current components are interpolated to match the spatial characteristics of the radial current at the DS radar station. Distance-weighted interpolation is employed for spatial alignment, while linear interpolation is used for temporal alignment.

3. Results

Based on the methods and data introduced in Section 2, we conducted an inversion of the surface vector currents in the southwestern Taiwan Strait. The calculated wind-driven currents, tidal currents, and geostrophic currents at 14:20 on 22 August 2021 (UTC+8) are shown in Figure 4. Wind-driven currents flowed toward the northeast in most regions, and large wind-driven currents exceeding 60 cm/s were found in the northern region, while small ones varied in the central and southern regions. The tidal currents in the Taiwan Bank were relatively strong, with the maximum velocities exceeding 100 cm/s. Following this are the tidal currents in the Xiamen-Penghu Depression located north of the Taiwan Bank, where the tidal velocities were approximately 40 cm/s. The weakest tidal currents are found south of the Taiwan Bank, with speeds of less than 10 cm/s.

The spatial distribution of the ocean surface current credibility at this moment is depicted in Figure 5. For clarity, Figure 5 displays only regions with credibility equal to or greater than 0.5, which covers most of the radar footprint. Areas with credibility below 0.5 primarily correspond to ocean regions farther away from the radar station (e.g., east of the Taiwan Bank).

Given this, we obtain the total retrieval currents by adding these three kinds of currents. In Figure 6, the blue arrows represent the vector currents (HF) measured by HFSWRs, while the red arrows depict the retrieval results (RE). The mean root mean square errors were calculated in the northward and eastward directions according to the method in Hatayama’s study [54], defined as

$$\left\{\begin{array}{c}RMS{E}_u=\sqrt{{\displaystyle \frac{1}{m}}{\displaystyle \sum }_1^m{\left[U_{RE}\left(m\right)-U_{HF}\left(m\right)\right]}^2}\\ RMS{E}_v=\sqrt{{\displaystyle \frac{1}{m}}{\displaystyle \sum }_1^m{\left[V_{RE}\left(m\right)-V_{HF}\left(m\right)\right]}^2}\end{array}\right.$$

(9)

to evaluate the differences. Here U and V are the eastward and northward components of the vector current, and m is the total number of effective currents within the radar footprint.

In the central sea areas covered by radar where HF generally could demonstrate higher accuracy [8], RE shows good agreement with HF, with root mean square errors (RMSE) of 11.5 cm/s in the eastward direction (U component) and 18.2 cm/s in the northward direction (V component). The relative errors of the U and V components are 19.05% and 21.17%, respectively. However, in the marginal zones, the agreement between RE and HF is not as strong as that in the central areas.

Another example of three components of currents is shown in Figure 7. The direction of the wind-driven currents predominantly aligns with the northwesterly direction, with magnitudes that are generally modest, not exceeding a velocity of 30 cm/s. In comparison to the previous example, the tidal currents near the Taiwan Bank and the Penghu Islands are less intense, with maximum values reaching up to 70 cm/s. Within the Xiamen-Penghu Depression, tidal current velocities range between 40 and 50 cm/s.

Similar to Figure 5, Figure 8 also displays only the regions with credibility greater than or equal to 0.5. This area is slightly smaller than that shown in Figure 5, particularly near the DS radar station. The variability in credibility is influenced by numerous factors, including SNR at different times, radio interference, and the operational status of the radar equipment. Subsequently, we compare the ocean currents of the RE and HF within this area.

Figure 9 displays the current pattern at 09:00 on 26 August 2021, showing a southwest-to-northeast direction in a large portion of the sea area. Similar to the previous case, the agreement between RE and HF in the marginal zones was relatively poor. Within the common radar coverage areas, the RMSE of the U component is 11.3 cm/s, and the RMSE of the V component is 17.3 cm/s. The relative errors of the U and V components are 15.62% and 18.62%, respectively.

In order to validate the algorithm, we compared RE and HF with vector current measurements obtained from drifters. Before comparison, spatiotemporal matching was conducted due to the mobility of the drifters. The nearest neighbor principle was used to determine the spatial and temporal closest ocean current data for comparison. The comparisons of U and V components are illustrated in Figure 10. It is evident that both the U and V components exhibit strong periodicity, confirming the dominance of tidal currents in the Taiwan Strait, a region known for strong tidal influence. Furthermore, in Figure 10b,d, both the HF and RE exhibit a closer alignment with the drifters in the V direction. However, the alignment appears less pronounced in the U direction. This could be attributed to significant disturbances in the U direction during the experimental period, as indicated by abrupt changes in the U component, as shown in Figure 10c.

Table 3. RMSE of the comparison, which listed before and after “/” correspond to Drifter 1 and 2.

	RMSE_U (cm/s)	RMSE_V (cm/s)
HF	11.9/16.5	13.1/22.2
RE	12.8/18.5	13.9/24.1

presents the RMSE of the HF and RE in comparison with the two drifters. The RMSE values listed before and after “/” correspond to drifters 1 and 2, respectively. From Table 3, it is observed that the RMSE of the V component exceeds that of the U component, and the RMSE obtained from the comparison with Drifter2 is higher than that of Drifter1. The accuracy of HF slightly surpasses that of RE, with a maximum difference of not more than 4 cm/s.

Figure 11 presents scatter plots comparing the HF, RE, and drifters in the U and V components, with the color of the scatter points indicating the quantity of samples. Figure 11a–f depict the comparison results with Drifter1, while Figure 11g–l depict the comparison results with Drifter2. The solid black line in each plot represents $y=x$, with the correlation coefficient (cc) also indicated in each subplot. Their p-values are extremely small, approaching zero, suggesting that the positive correlation between the two is not a random occurrence but should be considered statistically significant. From Figure 11, it is evident that the results for the V component are more concentrated along the black line, resulting in higher correlation coefficients for the V component compared to the U component. When comparing with the drifters, it is observed that, except for the RE correlation coefficients being higher than HF in Figure 11a,b, the remaining plots show higher correlation coefficients between HF and the drifter results. Additionally, the correlation coefficients between RE and HF are higher than those between RE and drifters, indicating that the variations in RE and HF are more closely aligned, given that RE is obtained through the inversion of radial flow results.

4. Discussion

The dynamic theory of tides is mature, and OTIS is widely utilized and substantiated by scholars; thus, we do not alter the tidal current output through iteration. The contribution of wind-driven currents is much smaller than that of tidal currents in the Taiwan Strait during the experimental period, and the theory for wind-driven currents is also mature. Wind datasets exhibit good spatial continuity and high temporal resolution. Consequently, the calculated results for wind-driven currents are relatively accurate. As a result, we also do not alter the wind-driven currents through iteration. Although the dynamic theory of geostrophic current is relatively mature, high-time-resolution sea surface height data are difficult to obtain, and the contribution of geostrophic currents is larger than that of wind-driven currents sometimes in the Taiwan Strait [37]. Therefore, using the radial current as a constraint, we determine the magnitude of the geostrophic currents through an iterative process. In a specific area of the Taiwan Strait, this method of determining geostrophic currents is effective, as most of the geostrophic currents in this area flow east or northeast all year and are not very complex. However, in areas under complex dynamic conditions, more effective methods for determining geostrophic currents are necessary.

The comparison reveals a high correlation between HF, RE, and drifter currents, particularly for the V component, where the correlation between HF and RE exceeds 0.9. Additionally, the lower precision of RE compared to HF is foreseeable, as the precision of RE depends on the detection accuracy of the radial current. Nevertheless, the difference in precision between RE and HF is not significant, with variations of up to 4 cm/s and 1 cm/s for maximum and minimum values, respectively, indicating the excellent performance of the proposed algorithm. This algorithm not only effectively inverses the temporal and spatial variations of surface vector currents but also achieves high accuracy.

Both the HF and RE exhibit clearer periodic characteristics and higher correlations in the V component. However, the U component generally shows periodicity but with a relatively lower correlation. This could be due to the performance differences between the two HFSWRs, as the DS radar station detects the radial component in the U direction for most sea areas. Drifters mainly drift on the ocean surface without the use of drogues; they are predominantly influenced by wind (Stokes drift) and tides. This factor may have contributed to the relatively large errors observed in areas with strong geostrophic currents, such as the eastern region of the DS radar station. Overall, the algorithm performs well in the Taiwan Strait, but its applicability to other ocean areas requires validation using experimental data from these areas. Additionally, the algorithm, developed based on ocean dynamics, may produce significant errors during extreme phenomena, such as typhoons, when classic drift theory-based wind-driven currents may become inaccurate and the contribution of wind-driven currents will significantly increase.

For the same sea area, not only HFSWRs but also X-band radars and satellite-based sensors [55] are capable of detecting the radial components of surface vector currents. Exploring the feasibility of this algorithm in various detection systems holds great potential for future research.

The characteristics of surface currents in the Taiwan Strait exhibit seasonal variations; however, the three primary constituent components remain unchanged. We believe that this method is also applicable to winter monsoon conditions. The fundamental principle of this method lies in reconstructing vector ocean currents through the utilization of ocean dynamics theory in combination with observations. In contrast to the prevailing direction of summer monsoons, winter monsoons are typically dominated by northeasterly winds. However, other influencing factors on the final inversion of ocean currents remain similar. We believe that both the Ekman theory and Stokes theory remain applicable under winter monsoon conditions, rendering this method suitable for such conditions. Furthermore, as indicated in the Introduction, the calculation methods for these components take into account real-time variations in tides, winds, and radial currents, resulting in the algorithm’s applicability across different seasons without the need for additional adjustments.

5. Conclusions

The Taiwan Strait is characterized by stable ocean currents and hydrodynamic features, including seasonal strong monsoons, powerful tidal waves, perennial northeastward geostrophic currents, and distinctive shallow bank topography. Based on an understanding of the oceanographic conditions in the Taiwan Strait, this study proposes an algorithm for computing the two principal components of surface vector currents: tidal currents and wind-driven currents. The algorithm utilizes radial current observations from a single-station HFSWR to determine the magnitude of the geostrophic currents, thus retrieving the surface vector current.

This algorithm not only reduces costs but also provides accurate surface vector currents comparable to those obtained by dual-site high-frequency radar observations, demonstrating its practical applicability. Although its efficacy has been validated in the context of the Taiwan Strait, additional research is imperative to evaluate its suitability across diverse oceanic regions.