Correcting Spatial Variance of RCM for GEO SAR Imaging Based on Time-Frequency Scaling

Geosynchronous synthetic aperture radar (GEO SAR) operates at an altitude ~36,000 km [1]. Compared with a low-Earth orbit (LEO) SAR, greater coverage can be achieved by GEO SAR because of its much higher orbit [2]. Furthermore, GEO SAR can guarantee observation of the same location every 24 h with the same incidence angle, which cannot be realized by LEO SAR [3]. Given its special characteristics, GEO SAR has attracted much attention [4]. It has become part of a global earthquake satellite system to monitor the global seismic state, as proposed by NASA and JPL in 2003 [5]. Another GEO SAR system called Geosynchronous Earth Monitoring by Interferometry and Imaging (GEMINI) was put forward in 2012 to acquire Earth surface data through GEO SAR interferometry [6]. For GEO SAR systems, imaging is always a major problem, the key to which is correction of the spatial variance of range cell migration (RCM).

RCM determines the distribution of echoes in the time and frequency domains. The spatial variance of RCM causes the spectrums corresponding to targets at different locations to be different. Therefore, in order to accomplish precise focusing in the frequency domain, the spatial variance must be corrected to eliminate the difference of RCM between any target in the swath and the reference point, usually the swath center. For LEO SAR with zero-Doppler steering, RCM is spatially variant along the range direction only, which can be processed by algorithms, such as range Doppler (RD) [7], chirp scaling (CS) [8] and wavenumber domain (WD) [9]. For high-squint LEO SAR, RCM varies along both range and azimuth slightly, and can be corrected by azimuth frequency scaling and block processing [10]. Compared with LEO SAR, higher altitude makes the synthetic aperture time of GEO SAR increase by over 100 times, which causes the relative trajectories between SAR and targets to become much more curved. As a result, for GEO SAR, the spatial variance of RCM is not only present in both range and azimuth directions but also much greater and more complicated. Moreover, the better the spatial resolution is, the much more serious the spatial variance of RCM is. How to correct the two-dimensional spatial variance of RCM is the specific imaging issue to realize high-resolution GEO SAR imaging. To address this problem, a precise range model has to be constructed to analyze RCM. Furthermore, an imaging algorithm based on the model can be presented to correct spatial variance and focus echo data.

According to this basic thought, in 2011, Bao et al. proposed a polynomial model to approximate RCM [11]. The following year they proposed a modified chirp scaling algorithm to correct linear spatial variance along the range direction [12]. After applying this algorithm, the quadratic residual RCM persists after correction. For this defect, Hu et al. forwarded a modified non-linear chirp scaling (NCS) algorithm based on correction of the quadratic residual RCM, but this still neglected spatial variance along the azimuth [13]. To correct the first-order azimuth variance, three different techniques were developed. Sun et al. adopted azimuth scaling combined with chirp scaling [14]. Hu et al. proposed a wavenumber-domain imaging algorithm based on modified Stolt interpolation [15]. Ding et al. constructed a fourth-order space-variant slant range model, applied the quadratic factor compensation in the two-dimensional time domain to reduce the variance of the azimuth phase, and adopted NCS to accomplish imaging. Simulation results demonstrated that this algorithm could enlarge the azimuth size of a well-focused image with a moderate resolution [16]. However, these techniques cannot correct higher-order azimuth variance contained in the quadratic and cubic phases, which is also crucial for imaging quality. In 2015, Li et al. proposed a fifth-order slant range model, and corrected quadratic azimuth spatial variance by exploiting azimuth time scaling. This technique focused on azimuth processing, while the range variance was not considered adequately [17].

The goal of this paper is to develop an algorithm that can correct high-order spatial variance of RCM along the range and azimuth directions, and perform GEO SAR imaging with a resolution of 2 m in both the ground cross-range and range directions. The most important innovation of this algorithm is the implementation of time-frequency scaling to correct linear and quadratic azimuth variance.

This paper is structured as follows: Section 2 constructs an echo model based on approximation of the range history, analyzes spatial variance in the time and frequency domain, and proposes explicit expressions for phases in terms of spatial variables. To correct the spatial variance, Section 3 presents the basic methodology. A GEO SAR imaging algorithm is advanced in Section 4, which is composed of five steps, i.e., an initial azimuth scaling, RCM correction (RCMC) & range compression, second azimuth scaling, azimuth focusing, and geometric correction. In Section 5, simulation results are addressed to verify the validity of the proposed algorithm. Section 6 concludes the paper.

For SAR, after demodulation, the echo corresponding to an isolated point target can be represented by: where $\eta$ and $\tau$ denote the slow time along the azimuth and the fast time along the range, respectively. The constant $\sigma$ is the backscattering coefficient of the target, $c_0$ is light speed, and $\lambda$ is wavelength. $a\left(\tau \right)$ represents the transmitted signal. Here, the chirp signal is the transmitted signal, which implies $a\left(\tau \right)=rect\left[\tau /T_p\right]\exp \left[j\pi K_r{\tau }^2\right]$. $K_r$ is the linear frequency modulation rate, $T_p$ is pulse duration, and $rect[\cdot ]$ denotes the rectangular envelope. ${\eta }_c$ is the beam crossing time and $T_s$ is the synthetic aperture time.

In Equation (1), $R_{actual}\left(\eta \right)$ is the equivalent slant range between the spaceborne SAR and the target, which is the average of the one-way slant range when transmitting a pulse and that when receiving the echo. Usually $R_{actual}\left(\eta \right)$ can be approximated by a polynomial: where $r_n$ is the nth-order coefficient, and $N_r$ denotes the order. All SAR imaging algorithms are derived based on an appropriate slant range model, like Equation (2). The higher the order, the better the fit, which leads to better imaging performance. However, a high order increases the complexity of imaging algorithms. Therefore, determining an $N_r$ that balances imaging quality and complexity of the algorithm is a major challenge.

Usually the aperture time of LEO SAR is so short that Equation (2) with $N_r=2$ is adequate to approximate the slant range, which indicates that the phase error induced by the approximation is too small to affect imaging quality [18]. For GEO SAR, orbital altitude increases the aperture time over one hundred times. For example, if the ground resolution is 2 m, the aperture times for LEO and GEO SARs are about 6 s and 750 s, respectively. Thus, to describe the much more complicated relative motion with the longer aperture time, the order of Equation (2) must be redetermined by evaluating the impact of different $N_r$ on imaging quality, which can be represented by the resolution, peak side-lobe level ratio (PSLR), and integral side-lobe level ratio (ISLR) [18].

The following matching filtering is adopted: where $\otimes$ represents convolution and $N_r$ may equal 4, 5 or ∞. When $N_r$ approaches ∞, Equation (3) indicates ideal filtering, which can achieve optimal imaging quality per the theory of matching filtering [18].

Evaluation results of applying Equation (3) and parameters in Table 1 are shown in Figure 1. Azimuthal resolution, PSLR, and ISLR achieved with $N_r=5$ are as nearly identical to results of the ideal filtering during the entire orbital period. However, when $N_r=4$, results are much worse. Therefore, the fifth-order approximation to $R_{actual}(\eta )$ is adequate to acquire optimum imaging quality.

By combining Equations (1) and (2), the echo model can be expressed as:

According to Equation (2), RCM is determined by slant range coefficients (i.e., $r_n$). In order to correct the spatial variance of RCM, this paper adopts time-frequency scaling to modify $r_n$.

To introduce this idea, a one-dimensional signal $s\left(t\right)$ is assumed to be: where $T$ is the signal duration time. $t_c$ is the center time and $d_n$ is the nth-order time-domain phase coefficient. According to Equation (6), $d_n$ can be assumed quadratically variant with $t_c$. The aim of the time-frequency scaling method is to remove the spatial variance in $d_n$, which indicates that $d_n$ doesn’t depend on $t_c$ after scaling.

By applying series reversion method [21], the corresponding frequency-domain spectrum of Equation (14) can be attained as: $D_n$ is the frequency-domain spectrum phase coefficient and can be obtained by applying the method in Appendix B. In order to avoid time-domain aliasing, $D_1$ will not be modified by scaling.

Modification of frequency-domain spectrum phase coefficients leads to modification of time-domain phase coefficients. As a result, the following frequency scaling function can be used:

After multiplication by Equation (16), Equation (15) becomes:

By applying series reversion method to Equation (17), the following expression can be obtained: where:

$d_n^{\prime }$ is the reconstructed nth-order time-domain phase coefficient. Like $d_n$, $d_n^{\prime }$ is also spatially variant with $t_c$. According to Equation (6), spatial variance is up to the second order along both the range and the azimuth. As a result, $d_n^{\prime }$ can be assumed a second-order function of $t_c$, i.e., where $d_{n,ref}^{\prime }$ corresponds to the swath center. Then, the spatial difference between $d_n^{\prime }$ and $d_{n,ref}^{\prime }$ is:

In order to eliminate the first-order spatial variance of $d_n^{\prime }$, the time scaling function to be multiplied with Equation (18) is designed as:

By multiplying Equations (18) and (22), the linear component of $\Delta d_n^{\prime }$ will be removed. Therefore, for correcting the linear spatial variance of all $d_n^{\prime }$ ($n\ge 2$), the complete time scaling function is:

After multiplying Equation (18) with Equation (23), the signal becomes: where:

The new time-domain phase coefficient is:

Similar to $d_n^{\prime }$, $d_n^{″}$ also varies with $t_c$. By applying the series reversion method to Equation (24), the spectrum is: where $B_m$ can be acquired by the method in Appendix B.

Because the time scaling has removed the linear spatial variance, $B_m$ satisfies:

In order to remove the quadratic spatial variance in $B_m$ ($m=2,3$), the following equation should be satisfied by assigning the appropriate value of $Z_n$:

According to Equation (13), ${\varphi }_k$ ($k>3$) is spatially invariant in GEO SAR imaging. And after time-frequency scaling, the linear and quadratic spatial variance has been removed from $B_m$ ($m=2,3$), as shown in Equations (28) and (29). Therefore, focusing for the whole swath can be accomplished in the frequency domain. Although the time-frequency scaling is only applied to correct the azimuth variance in this section, it can also be applied for the range variance correction, as shown in Section 4.

Based on the basic idea in Section 3, a GEO SAR imaging algorithm composed of five steps is proposed. The first is to eliminate linear azimuth variance of ${\varphi }_k$ through the first azimuth time scaling. The second is to achieve RCM correction and range compression. The third is to correct residual azimuth variance by the second azimuth time-frequency scaling. The fourth and final steps are to accomplish azimuth focusing and correct geometric distortion.