The Establishment and Verification of a Velocity Doppler Transfer Model for Dual-Beam Squint Airborne SAR

Abstract

Measuring ocean currents is essential for oceanographic studies, and dual-beam squint airborne SAR measurements provide significant advantages, including flexibility, cost-effectiveness, and extensive coverage. However, substantial attitude changes in the airborne platform introduce challenges to achieving accurate ocean current measurements. Additionally, existing attitude correction methods fail to account for the off-nadir angle and squint angle errors of targets located at the edge of the beam’s ground footprint, further impacting measurement precision. To address these limitations, this paper proposes a dual-beam squint airborne velocity Doppler transfer model. The squint antenna view vector is initially defined in the aircraft-centered frame of reference and subsequently described using the flightpath frame of reference. By estimating the Doppler frequency caused by aircraft attitude changes, the velocity Doppler transfer model is established. This model is then applied to invert sea surface currents. An error analysis is conducted, and the Monte Carlo method is employed to validate the model’s accuracy. The results demonstrate that the proposed velocity Doppler transfer model effectively inverts sea surface currents with high accuracy in both velocity and direction. Compared to pre-existing methods, the proposed model shows superior performance, particularly in addressing off-nadir and squint angle errors, thereby enhancing overall measurement precision.

1. Introduction

Ocean currents play a pivotal role in the movement of seawater and exert a profound influence on weather and climate over the oceans [1,2]. Remote sensing, as an all-day, all-weather detection method, provides significant advantages in acquiring dynamic information about sea surface currents, including high efficiency and relatively low cost. Among various remote sensing instruments, synthetic aperture radar (SAR) stands out due to its ability to measure parameters such as the backscattering coefficient, Doppler centroid frequency (FDC), and others, which are highly advantageous for estimating sea surface currents. There are two primary methods used in SAR-based measurements of sea surface currents: the interferometric method and the Doppler centroid inversion method.

The interferometric method was proposed by Goldstein et al. [3,4] and Shemer et al. [5]. Then, Romeiser et al. constructed a new sea surface backscattering model named the M4S model [6,7]. The scattering mechanism assumed by this model is the Bragg scattering and specular scattering, which is used to infer the sea surface currents by taking into account the interaction between wind and current [8,9,10]. Through these studies, Romeiser et al. [6,7,9] established the connection between sea surface currents and the interferometric phase information, and the interferometric method has been fully developed. Then, many studies have analyzed the effect of the interferometric method. In 2001, Romeiser et al. [9] compared results of along-track interferometric (ATI) SAR inversion and acoustic Doppler current profiler (ADCP) and circulation model to analyze the feasibility and limitations of inverting sea surface currents by ATI. The results show that ATI can be used to invert sea surface currents in coastal zones with high resolution, and the inverted sea surface currents can be used to further invert underwater topography with high resolution. After that, Kersten et al. [11] proved that the phase error of the along-track interferometric SAR (ATI-SAR) is smaller than that of single-phase center SAR; that is, the velocity error is smaller. Moreover, for the system with same power and total antenna size, ATI method has higher sensitivity [11]. Since then, multi-aperture along-track interferometric (MA-ATI) is conducted and the concept of multi-beam interference is extended to obtain the direction of sea surface currents [11,12,13,14,15,16,17,18,19]. When ATI method is used to measure sea surface currents, the interferometric phase must be accurate enough. Therefore, the phase imbalance caused by error of antenna phase center and attitude cannot be ignored. So, the correction of attitude error and other variables is very important for sea surface current measurement.

The feasibility of the Doppler centroid inversion method was first analyzed by Charpon et al. [20] in 2005, and a simplified Doppler centroid model was also provided. They further analyzed the factors that may cause deviations in Doppler information [20]. Building on the work of Charpon et al., Johannessen et al. established the relationship between the radar imaging model (RIM) model and Doppler information and constructed the DopRIM model [21]. This model was then used to analyze the Doppler imaging of the Agulhas current in southern South Africa [22,23]. Hansen et al. employed the DopRIM model to simulate wave–current interaction in strong tidal current areas, further illustrating the modulation effect of strong surface currents on sea surface roughness and slant distance Doppler signals [24]. Charpon’s team has also been exploring the use of Doppler information to invert ocean circulation and its mesoscale–submesoscale characteristics [25,26,27,28,29,30,31,32,33]. The calculation of Doppler centroid is closely related to the attitude of the radar platform, so the correction of attitude error and other variables is very important for sea surface current measurement.

It is evident from the above discussion that correcting attitude errors is crucial for accurate sea surface current measurements. To calculate these errors, the SAR platform velocity Doppler transfer model (VDTM) must first be established. In 2005, Cumming & Wong developed the spaceborne VDTM [34]. While this model is effective for correcting attitude errors in spaceborne platforms, it is not applicable to airborne platforms. This limitation arises because airborne platforms do not need to account for factors such as Earth’s curvature and rotation, which are critical considerations for spaceborne systems. Moreover, airborne platforms exhibit more pronounced attitude changes compared to their spaceborne counterparts. Despite these differences, airborne platforms play a vital role in sea surface current measurement. They serve as test beds for spaceborne systems and are indispensable in the development of new technologies. Additionally, airborne platforms offer advantages in terms of flexibility and cost-effectiveness that spaceborne platforms cannot match. Therefore, it is essential to establish a model specifically designed for attitude error correction in airborne platforms. Current methods only account for the need to obtain radial components of ocean currents without addressing the requirement for attitude variation correction in acquiring ocean current vectors; however, the acquisition of ocean current vectors is of importance for oceanographic research. To obtain an ocean current vector, observations must be made from two different directions which means two beams are needed [35]. In this framework, beam 1 and beam 2 illuminate the same sea surface scene from two different directions successively, retrieving two radial components of the ocean current vector. The two radial components are then combined through vector synthesis to derive the ocean current vector. So, it is also important to establish a VDTM specifically designed for dual-beam squint airborne SAR.

For radial components of ocean current vector estimation, the ATI method and the DCA method can be used. Since the DCA method offers advantages over the ATI method in terms of easier data acquisition, greater operational flexibility, a simpler and more efficient processing workflow, and insensitivity to phase unwrapping issues. Although the spatial resolution of the ATI method is higher, the DCA method can achieve comparable ocean current measurement accuracy to that of the ATI method while not having higher requirements for radar design [35].

Motivated by these considerations, this paper aims to build the dual-beam squint airborne VDTM. This model is then utilized to invert sea surface currents, followed by a verification of the VDTM. In this model, the DCA method is used to retrieve the radial component of the ocean current vector, with a time interval of 1 ms between consecutive pulses. In this paper, Section 2 details the development of the dual-beam squint airborne VDTM and outlines the inversion process for sea surface currents. Section 3 presents the ocean current inversion result based on VDTM, compares it with the actual ocean current, and analyzes potential errors in the VDTM. Section 4 verifies the accuracy of VDTM. Finally, Section 5 offers a comprehensive summary of the entire study.

2. Methods

This section aims to establish the dual-beam squint airborne VDTM, which will be used to invert sea surface currents measured by dual-beam squint airborne SAR.

2.1. Establishment of Aircraft-Centered Frame of Reference (ACFR)

The ACFR serves as the primary reference frame, as illustrated in Figure 1. In this frame, the $X$-axis denotes the forward direction pointing toward the aircraft’s nose, the $Y$-axis points to the left side of the aircraft within its horizontal plane and is orthogonal to the $X$-axis, while the $Z$-axis represents the upward vertical direction, orthogonal to the $XOY$ plane and directed away from the aircraft’s center. This configuration establishes an orthogonal reference frame. Within the ACFR, the position of the aircraft ${\mathit{A}}_0$, is denoted as follows:

$${\mathit{A}}_0={\left[0,0,0\right]}^T,$$

(1)

where ${(·)}^T$ represents the transpose operation.

For illustrative purposes, we first consider the off-nadir angle, as shown in Figure 2. Assuming that the radar antenna is mounted on the aircraft body such that the radar boresight lies within the $YOZ$ plane, the unit boresight vector ${\mathit{U}}_{0f}$ can be defined as:

$${\mathit{U}}_{0f}={[0,sin(\gamma ),cos(\gamma \left)\right]}^T,$$

(2)

where $\gamma$ represents the off-nadir angle, defined as the angle between the $OZ$ axis and the beam direction.

Next, the squint angle is taken into account, as illustrated in Figure 3. The specific pointing angles of the fore and aft beams are defined by the view vectors ${\mathit{U}}_{0{\mathrm{f}}_{\mathrm{s}1}}$ and ${\mathit{U}}_{0{\mathrm{f}}_{\mathrm{s}2}}$ as follows:

$$\begin{gathered} {\mathit{U}}_{0f_{s1}}={\mathit{tan}\left({\displaystyle \frac{\pi }{2}}-\zeta \right)\times \mathit{U}}_{0f}+{[\begin{array}{ccc}1& 0& 0\end{array}]}^{\prime }, \\ {\mathit{U}}_{0f_{s2}}={\mathit{tan}\left({\displaystyle \frac{\pi }{2}}+\zeta \right)\times \mathit{U}}_{0f}+{[\begin{array}{ccc}-1& 0& 0\end{array}]}^{\prime }, \end{gathered}$$

(3)

where $\zeta$ represents the squint angle, defined as the angle between the side-looking radar antenna and the squint antenna. For consistency with other parameters, ${\mathit{U}}_{0{\mathrm{f}}_{\mathrm{s}1}}$ and ${\mathit{U}}_{0{\mathrm{f}}_{\mathrm{s}2}}$ need to be normalized into unit view vectors ${\mathit{U}}_{0\mathrm{f}{\mathrm{u}}_{\mathrm{s}1}}$ and ${\mathit{U}}_{0\mathrm{f}{\mathrm{u}}_{\mathrm{s}2}}$ respectively. The off-nadir angle together with the squint angle, determines the three-dimensional pointing direction of the radar antenna. So, the off-nadir angle is critical for determining the radial velocity components of ocean current and subsequently deriving the ocean current vector.

2.2. Establishment of Flightpath Frame of Reference (FFR)

The aircraft attitude is typically described using the FFR, as illustrated in Figure 4. In this reference frame the $X$-axis aligns with the north direction, the $Y$-axis aligns with the west direction, and the $Z$-axis extends downward, orthogonal to the horizontal plane from the aircraft’s center. The origin of the FFR corresponds to the projection of the aircraft’s center onto the ground.

To define variables in the FFR, ACFR should be converted to FFR, it is necessary to apply a rotational transformation to the ACFR. Specifically, the first step involves rotating ACFR clockwise around the positive $X$-axis by an angle $\phi$, which is referred to as the roll angle (as illustrated in Figure 5). This transformation is achieved through a series of well-defined mathematical operations as follows:

$${\mathit{T}}_x=\left[\begin{array}{ccc}1& 0& 0\\ 0& cos\left(\phi \right)& -sin\left(\phi \right)\\ 0& sin\left(\phi \right)& cos\left(\phi \right)\end{array}\right].$$

(4)

Next, the ACFR is rotated clockwise around the positive $Y$-axis by an angle $\theta$, which is referred to as the pitch angle (as depicted in Figure 6). This rotation is implemented through a transformation matrix or a set of mathematical operations as follows:

$${\mathit{T}}_y=\left[\begin{array}{ccc}cos\left(\theta \right)& 0& sin\left(\theta \right)\\ 0& 1& 0\\ -sin\left(\theta \right)& 0& cos\left(\theta \right)\end{array}\right].$$

(5)

Finally, the ACFR is rotated clockwise around the positive $Z$-axis by an angle $\psi$, which is referred to as the yaw angle (as illustrated in Figure 7). This rotation is accomplished through a specific transformation matrix or mathematical operation as follows:

$${\mathit{T}}_z=\left[\begin{array}{ccc}cos\left(\psi \right)& -sin\left(\psi \right)& 0\\ sin\left(\psi \right)& cos\left(\psi \right)& 0\\ 0& 0& 1\end{array}\right].$$

(6)

The position of the aircraft ${\mathit{A}}_1$ and two view vectors of the squint antennas ${\mathit{U}}_{1_{\mathrm{s}1}}$, ${\mathit{U}}_{1_{\mathrm{s}2}}$ in FFR can be expressed as follows:

$$\begin{gathered} {\mathit{A}}_1=\left[\begin{array}{ccc}1& 0& 0\\ 0& cos\left(\phi \right)& -sin\left(\phi \right)\\ 0& sin\left(\phi \right)& cos\left(\phi \right)\end{array}\right]\left[\begin{array}{ccc}cos\left(\theta \right)& 0& sin\left(\theta \right)\\ 0& 1& 0\\ -sin\left(\theta \right)& 0& cos\left(\theta \right)\end{array}\right]\left[\begin{array}{ccc}cos\left(\psi \right)& -sin\left(\psi \right)& 0\\ sin\left(\psi \right)& cos\left(\psi \right)& 0\\ 0& 0& 1\end{array}\right]{\left[0,0,0\right]}^T+{\left[\begin{array}{ccc}0& 0& -h\end{array}\right]}^{\prime }, \\ {\mathit{U}}_{1_{s1}}=\left[\begin{array}{ccc}1& 0& 0\\ 0& cos\left(\phi \right)& -sin\left(\phi \right)\\ 0& sin\left(\phi \right)& cos\left(\phi \right)\end{array}\right]\left[\begin{array}{ccc}cos\left(\theta \right)& 0& sin\left(\theta \right)\\ 0& 1& 0\\ -sin\left(\theta \right)& 0& cos\left(\theta \right)\end{array}\right]\left[\begin{array}{ccc}cos\left(\psi \right)& -sin\left(\psi \right)& 0\\ sin\left(\psi \right)& cos\left(\psi \right)& 0\\ 0& 0& 1\end{array}\right]{\mathit{U}}_{0f{u}_{s1}}, \\ {\mathit{U}}_{1_{s2}}=\left[\begin{array}{ccc}1& 0& 0\\ 0& cos\left(\phi \right)& -sin\left(\phi \right)\\ 0& sin\left(\phi \right)& cos\left(\phi \right)\end{array}\right]\left[\begin{array}{ccc}cos\left(\theta \right)& 0& sin\left(\theta \right)\\ 0& 1& 0\\ -sin\left(\theta \right)& 0& cos\left(\theta \right)\end{array}\right]\left[\begin{array}{ccc}cos\left(\psi \right)& -sin\left(\psi \right)& 0\\ sin\left(\psi \right)& cos\left(\psi \right)& 0\\ 0& 0& 1\end{array}\right]{\mathit{U}}_{0f{u}_{s2}}, \end{gathered}$$

(7)

where $h$ is the height of the aircraft.

2.3. Estimation of FDC

Having detailed the aircraft position and the two view vectors of the squint antennas in FFR, it is necessary to estimate the FDC resulting from the aircraft attitude in order to construct the VDTM. The relative velocities of the aircraft, denoted as $V_{{\mathrm{r}\mathrm{e}\mathrm{l}}_{\mathrm{s}1}}$ and $V_{{\mathrm{r}\mathrm{e}\mathrm{l}}_{\mathrm{s}2}}$, are derived from the following:

$$\begin{gathered} V_{{rel}_{s1}}={\mathit{V}}_1\cdot {\mathit{U}}_{1_{s1}}-{\mathit{V}}_q\cdot {\mathit{U}}_{1_{s1}}, \\ {V}_{{rel}_{\mathrm{s}2}}={\mathit{V}}_1\cdot {\mathit{U}}_{1_{s2}}-{\mathit{V}}_q\cdot {\mathit{U}}_{1_{s2}}. \end{gathered}$$

(8)

Here, ${\mathit{V}}_{\mathrm{q}}$ denotes the velocity of the target and is set to 0 to emphasize the impact of the aircraft’s attitude, while ${\mathit{V}}_1$ represents the velocity of the aircraft. The FDC resulting from the aircraft’s attitude can be determined from the following:

$$\begin{gathered} f_{a1}={\displaystyle \frac{-2V_{{rel}_{s1}}}{\lambda }}, \\ {f}_{a2}={\displaystyle \frac{-2V_{{rel}_{s2}}}{\lambda }}. \end{gathered}$$

(9)

Here, $f_{\mathrm{a}1}$ and $f_{\mathrm{a}2}$ represent the FDC of the two squint beams caused by the aircraft’s attitude, while $\lambda$ denotes the radar wavelength. At this point, the VDTM has already been established.

2.4. Inversion of Sea Surface Current

After establishing the VDTM, the sea surface current can be inverted using this model. First, the FDC of the target located at the center of the two squint beams, denoted as $f_1$ and $f_2$, is obtained by measurement through the average cross-correlation coefficient (ACCC) [34].

$$\begin{gathered} f_1={\displaystyle \frac{\angle \left\{\sum _i{s_1}^{*}(i·∆t)·s_1\left(\right(i+1)·∆t)\right\}}{2\pi }}·F_{prf}, \\ {f}_2={\displaystyle \frac{\angle \left\{\sum _i{s_2}^{*}(i·∆t)·s_2\left(\right(i+1)·∆t)\right\}}{2\pi }}·F_{prf}. \end{gathered}$$

(10)

Here, $\mathrm{\angle }\left\{·\right\}$ represents the phase of a complex number, $∆t$ is the azimuth signal sampling interval, $F_{\mathrm{prf}}$ is the pulse repetition frequency, $s$ denotes the azimuth signal, and ${\left(·\right)}^{\mathrm{*}}$ indicates the complex conjugate. The relationship between the FDC caused by the sea surface current and the aircraft attitude can be expressed as follows:

$$\begin{gathered} f_1=f_{c1}+f_{a1}, \\ {f}_2=f_{c2}+f_{a2}. \end{gathered}$$

(11)

Here, $f_{\mathrm{c}1}$ and $f_{\mathrm{c}2}$ represent the FDC of the two squint beams induced by the sea surface current, while $f_{\mathrm{a}1}$ and $f_{\mathrm{a}2}$ represent the FDC of the two squint beams caused by the aircraft attitude. To obtain $f_{\mathrm{c}1}$ and $f_{\mathrm{c}2}$, $f_{\mathrm{a}1}$ and $f_{\mathrm{a}2}$ are calculated using VDTM. The velocities of the sea surface current, when projected along the two radar view vectors, can then be derived from the following:

$$\begin{gathered} V_{c1}={\displaystyle \frac{-f_{c1}\ast \lambda }{2}}, \\ {V}_{c2}={\displaystyle \frac{-f_{c2}\ast \lambda }{2}}. \end{gathered}$$

(12)

Here, $V_{\mathrm{c}1}$ and $V_{\mathrm{c}2}$ represent the velocity components of the sea surface current projected along the two radar view vectors. Finally, the sea surface current vector can be determined using the following:

$$\begin{gathered} {\mathit{V}}_c·{\mathit{U}}_{1_{s1}}=V_{c1}, \\ {\mathit{V}}_c·{\mathit{U}}_{1_{s2}}=V_{c2}. \end{gathered}$$

(13)

It can be expanded as follows:

$$\begin{gathered} {\mathit{V}}_c\left(1\right){\mathit{U}}_{1_{s1}}\left(1\right)+{\mathit{V}}_c\left(2\right){\mathit{U}}_{1_{s1}}\left(2\right)=V_{c1}, \\ {\mathit{V}}_c\left(1\right){\mathit{U}}_{1_{s2}}\left(1\right)+{\mathit{V}}_c\left(2\right){\mathit{U}}_{1_{s2}}\left(2\right)=V_{c2}. \end{gathered}$$

(14)

$$\begin{gathered} {\mathit{V}}_c(y)={\displaystyle \frac{{\mathit{U}}_{1_{s1}}\left(1\right)\ast V_{c2}-{\mathit{U}}_{1_{s2}}\left(1\right)\ast V_{c1}}{{{\mathit{U}}_{1_{s1}}\left(1\right)\ast \mathit{U}}_{1_{s2}}\left(2\right)-{{\mathit{U}}_{1_{s1}}\left(2\right)\ast \mathit{U}}_{1_{s2}}\left(1\right)}}, \\ {\mathit{V}}_c\left(x\right)={\displaystyle \frac{V_{c1}-{\mathit{V}}_c\left(y\right)\ast {\mathit{U}}_{1_{s1}}\left(2\right)}{{\mathit{U}}_{1_{s1}}\left(1\right)}}. \end{gathered}$$

(15)

Here, ${\mathit{V}}_{\mathrm{c}}\left(y\right)$ represents the velocity component of the sea surface current along the $Y$-axis of the FFR, and ${\mathit{V}}_{\mathrm{c}}\left(x\right)$ represents the velocity component of the sea surface current along the $X$-axis of the FFR. They denote the coordinates of the ocean current vector in the FFR. Thus, the sea surface current has now been successfully inverted. The above steps and modules together constitute VDTM.

It should be noted here that two radial components of the ocean current vector are not obtained at the same time. When beam 1 illuminates a scene, we use VDTM to obtain the radial component along the direction of beam 1 of this scene. As the radar platform moves, when beam 2 illuminates the same scene, we use VDTM to obtain the radial component along the direction of beam 2 of the same scene. Due to the high speed of the radar platform, the ocean current vector remains constant between two illuminations though the dynamic sea surface changes. Finally, the ocean current vector can be derived through vector synthesis.

Although the sea surface current has been successfully retrieved, measurement errors are inevitable under real-world conditions. Therefore, suppressing these errors is critical for improving accuracy. Instrument measurement errors primarily include attitude angle measurement errors, look angle errors, and off-nadir angle errors, among others. These errors can be effectively mitigated through a two-step process.

In the first step, error reduction is achieved by segmenting the original data along the azimuth and range directions. For each segment, the off-nadir angle and look angle are determined individually. In the second step, the VDTM is applied to a stationary target with its FDC set to 0, enabling the derivation of more precise off-nadir angles and look angles. This approach enhances the overall accuracy of the measurements.

3. Results

After successfully inverting the sea surface current and suppressing some measurement errors, the error of the sea surface current inverted by VDTM can now be evaluated. The evaluation process utilized Matlab (R2024b).

Sea surface current errors primarily consist of FDC measurement errors influenced by additive and multiplicative noise, as well as Position and Orientation System (POS) measurement errors. These errors can be simulated using a statistically independent normal distribution. However, the range of measurement errors varies for different variables. Therefore, it is necessary to adjust the mathematical expectation $\mu$ and variance ${\sigma }^2$ of the normal distribution to ensure its limits align with the measurement errors of the respective variables. Here, $\mu$ is set to 0, and ${\sigma }^2$ is set to 0.5 for the normal distribution to ensure that the root mean square error (RMSE) of the POS’ measurement errors for platform speed remains within the range of −0.5 m/s to 0.5 m/s. Additionally, $\mu$ is set to 0, and ${\sigma }^2$ is set to 0.01 for the normal distribution to constrain the RMSE of measurement errors for aircraft attitude within −0.01° to 0.01°, a tolerance achievable by modern POS such as the Applanix POS MV [36].

The measurement errors can be expressed as follows:

$$\begin{gathered} E_v~N(0,0.5), \\ {E}_a~N(0,0.01). \end{gathered}$$

(16)

Here, $E_{\mathrm{v}}$ represents the instrumental measurement errors of velocity, $E_{\mathrm{a}}$ represents the instrumental measurement errors of aircraft attitude, and $N$ denotes the normal distribution. The parameters utilized in the simulation are presented in

Table 1. Simulation parameters.

Parameters	Value
Ocean current	$\mathrm{Direction:\; 45}°$, Velocity: 1.42 m/s
RMSE of roll measured by POS	≤0.01°
RMSE of pitch measured by POS	≤0.01°
RMSE of yaw measured by POS	≤0.01°
RMSE of platform speed measured by POS	≤0.5 m/s

.

Furthermore, to achieve more accurate error analysis of VDTM, an aerodynamic simulation model was developed for a Boeing 747 flying at an altitude of 2000 m and a speed of 150 m/s. This model integrates principles from aerodynamics, control theory, and computational modeling. The primary objective of this model is to enable more precise error analysis of the VDTM by providing a robust framework for testing and validating control systems under various flight conditions. This model is based on the work referenced from [37].

The construction of this aerodynamic simulation model can be divided into two phases: the initial phase focuses on developing linear dynamic models, while the subsequent phase advances to nonlinear simulations. In the first phase, the longitudinal state space model was formulated using the derivation outlined in Etkin and Teichmann [38]. For the longitudinal system, the normal state vector is used with the addition of the $Z_E$ to use as feedback for the altitude command. The control inputs for the longitudinal system are the elevator and throttle. The system matrix was calculated using the parameters provided for the Boeing 747 aircraft. The system matrix of the altitude control system was set as follows:

$$A_A=\left[\begin{array}{ccccc}-0.0069& 0.0139& 0& -9.81& 0\\ -0.0905& -0.3149& 235.89& 0& 0\\ 0.0004& -0.0034& -0.4281& 0& 0\\ 0& 0& 1& 0& 0\\ 0& 1& 0& -235.89& 0\end{array}\right].$$

(17)

The control matrix of the altitude control system was set as follows:

$$B_A=\left[\begin{array}{cc}-0.000057& 2.943\\ -5.4714& 0\\ -1.159& 0\\ 0& 0\\ 0& 0\end{array}\right].$$

(18)

The state space model is completed with the C and D matrices.

$$C_A=\left[\begin{array}{ccccc}1& 0& 0& 0& 0\\ 0& 1& 0& 0& 0\\ 0& 0& 1& 0& 0\\ 0& 0& 0& 1& 0\\ 0& 0& 0& 0& 1\end{array}\right],$$

(19)

$$D_A=\left[\begin{array}{ccccc}0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0\end{array}\right].$$

(20)

Similarly, the heading control system was designed for performing steady banked turns. The system matrix of the heading control system was set as follows:

$$A_H=\left[\begin{array}{ccccc}-0.0558& 0& -235.9& 9.81& 0\\ -0.0127& -0.4349& 0.4142& 0& 0\\ 0.0036& -0.0061& -0.1458& 0& 0\\ 0& 1& 0& 0& 0\\ 0& 0& 1& 0& 0\end{array}\right].$$

(21)

The control matrix of the heading control system was set as follows:

$$B_H=\left[\begin{array}{cc}0& 1.7188\\ -0.1433& 0.1146\\ 0.0038& -0.4859\\ 0& 0\\ 0& 0\end{array}\right].$$

(22)

The same C and D matrices are used as before to complete the state space model. In the second phase, nonlinear dynamics were incorporated into the existing linearized models. The nonlinear model is very similar to the linearized control system, except the nonlinear dynamics are substituted in place of the linearized state space models. Previously, the longitudinal and lateral dynamics were uncoupled, and the state space models were calculated separately. The two systems are now combined utilizing a coupled nonlinear dynamics subsystem with four inputs and all twelve outputs. The aerodynamic simulation model based on the Boeing 747 is shown in Figure 8.

This model consists of the following four submodules: the Aircraft Control Input Module (Autopilot in Figure 8, indicated in red), the Control Parameter Transmission Module (Cable and actuator dynamics in Figure 8, indicated in dark blue), the Aircraft Aerodynamics Module (DT-B747 in Figure 8, indicated in light blue), and the Noise Module (Noise in Figure 8). The Aircraft Control Input Module is used to simulate the input of aircraft flight control instructions. The Control Parameter Transmission Module is used to simulate the signals that are transmitted through the aircraft cables after the input of the Aircraft Control Input Module. The Aircraft Aerodynamics Module is used to simulate the actual attitude changes in the aircraft after the control signals are combined with the aircraft’s aerodynamic appearance. The Noise Module is used to simulate various noises throughout the entire simulation process.

Finally, the attitude and velocity data obtained from the POS, which include measurement errors $E_{\mathrm{v}}$ (velocity error) and $E_{\mathrm{a}}$ (attitude error), are simulated and input into the VDTM to calculate the FDC and ocean current vectors with associated measurement errors. Figure 9 shows the comparison between the calculated and actual ocean current vectors.

Analysis of the figure reveals a directional discrepancy of merely 2.3 degrees and a velocity difference in only 0.01 m/s between our calculated ocean current vectors and the real values. These results suggest that the VDTM method is effective in minimizing measurement errors induced by variations in platform attitude and velocity.

4. Discussion

To more precisely assess the performance of VDTM, the Monte Carlo method is utilized. This mathematical technique is widely used to simulate random phenomena by generating a large number of random samples to approximate the behavior of complex systems. In this study, the Monte Carlo random experiment is conducted 10,000 times to account for uncertainties in the measurements. The resulting velocities and directions of the sea surface currents simulated through the Monte Carlo random experiment are illustrated in Figure 10. This approach provides a robust statistical basis for evaluating the accuracy and reliability of the VDTM under realistic conditions.

According to Figure 10, the accurate velocity of the sea surface current is 1.41 m/s and the accurate direction of the sea surface current is 45$°$. The statistical variables are calculated as follows:

$$\begin{gathered} E_v={\displaystyle \frac{A_j-A_f}{N}}, \\ {R}_v=\sqrt{{\displaystyle \frac{1}{N}}{\sum }_{i=1}^N{\left(A_j-A_f\right)}^2}. \end{gathered}$$

(23)

Here, $E_{\mathrm{v}}$ represents the bias of the variable in the Monte Carlo random experiment, $R_{\mathrm{v}}$ denotes the RMSE of the variable in the Monte Carlo random experiment, $A_j$ is the value of the variable in the Monte Carlo random experiment with serial number $j$, $A_{\mathrm{f}}$ is the accurate value of the variable, and $N$ is the total number of trials in the Monte Carlo random experiment. For the proposed method, the bias of velocity and direction in the Monte Carlo random experiment is 0 m/s and −0.31°, respectively, while the RMSE of velocity and direction is 0.02 m/s and 3.68°, respectively. These results demonstrate the high accuracy and reliability of the proposed method in estimating sea surface current parameters.

To further highlight the superiority of our proposed method, we conducted a comparative analysis with the well-established approach proposed by Cumming and Wong [34]. Simulations were performed using their method [34], and Monte Carlo experiments were carried out under identical parameter settings (e.g., attitude angles, platform velocity) and error distributions as those used in our study. In contrast to our method, the approach by Cumming and Wong [34] does not account for off-nadir angle and squint angle errors for targets located at the edges of the beam’s ground footprint. This oversight has led to diminished precision in measuring the velocities of these edge-positioning targets. Furthermore, their methodology accounts only for variations in the three attitude angles while neglecting the critical angle between the platform’s velocity vector and its attitude orientation—an omission that may be inconsequential in spaceborne scenarios but proves significant for airborne platforms. This limitation restricts its capability to comprehensively correct velocity measurement errors, especially in scenarios where the velocity vector experiences significant changes. The results obtained using the method by Cumming and Wong [34] are presented in Figure 11, which clearly illustrates the performance differences between the two approaches.

The bias of velocity and direction in the Monte Carlo random experiment using the pre-existing method is 0.1 m/s and 26.2°, respectively, while the RMSE of velocity and direction is 0.1 m/s and 26.22°. In comparison, our proposed method demonstrates superior performance, achieving lower bias and RMSE values in both velocity and direction. This clearly indicates that our approach outperforms previous methods in terms of accuracy and reliability for estimating sea surface current parameters.

Figure 12 provides a more intuitive illustration that the ocean current vectors calculated based on VDTM are closer to the actual ocean current vectors compared to those computed by existing methods.

5. Conclusions

The correction of attitude errors plays a crucial role in the accurate measurement of sea surface currents. While there have been significant research efforts focused on attitude error correction for spaceborne platforms, the attitude changes in airborne platforms are far more pronounced. This is particularly true for dual-beam squint airborne platforms, where the dynamics of the platform introduce additional complexities not encountered in spaceborne systems. As a result, pre-existing attitude error correction methods designed for spaceborne platforms are not well-suited for dual-beam squint airborne platforms.

To address this challenge, this paper establishes a dual-beam squint airborne VDTM. Using this model, the sea surface current is inverted, providing a novel approach to account for the unique characteristics of airborne platforms. Following the development of the model, an extensive error analysis was conducted, along with 10,000 Monte Carlo random experiments, to validate the performance of the dual-beam squint airborne VDTM.

The results demonstrate that the average velocity and direction obtained from the Monte Carlo random experiments using the proposed method are significantly closer to the accurate velocity and direction compared to those achieved using pre-existing methods. This indicates that the VDTM outperforms previous approaches in terms of both bias and RMSE for both velocity and direction measurements. Overall, these findings highlight the effectiveness and superiority of the proposed VDTM for attitude error correction in dual-beam squint airborne platforms.