High-Resolution High-Squint Large-Scene Spaceborne Sliding Spotlight SAR Processing via Joint 2D Time and Frequency Domain Resampling

Abstract

A frequency domain imaging algorithm, featured as joint two-dimensional (2D) time and frequency domain resampling, used for high-resolution high-squint large-scene (HHL) spaceborne sliding spotlight synthetic aperture radar (SAR) processing is proposed in this paper. Due to the nonlinear beam rotation during HHL data acquisition, the Doppler centroid varies nonlinearly with azimuth time and traditional sub-aperture approaches and two step approach fail to remove the inertial Doppler aliasing of spaceborne sliding spotlight SAR data. In addition, curved orbit effect and long synthetic aperture time make the range histories difficult to model and introduce space-variants in both range and azimuth. In this paper, we use the azimuth deramping and 2D time-domain azimuth resampling, collectively referred to as preprocessing, to eliminate the aliasing in Doppler domain and correct the range-dependent azimuth-variants of range histories. After preprocessing, the squint sliding spotlight SAR data could be considered as equivalent broadside strip-map SAR during processing. Frequency domain focusing, mainly involves phase multiplication and resampling in 2D frequency and RD domain, is then applied to compensate for the residual space-variants and achieve the focusing of SAR data. Moreover, in order to adapt higher resolution and larger scene cases, the combination of the proposed algorithm and partitioning strategy is also discussed in this paper. Processing results of simulation data and Gaofen-3 experimental data are presented to demonstrate the feasibility of the proposed methods.

1. Introduction

With the improvements of requirements for resolution, revisiting capability, and coverage [1,2,3,4,5] of synthetic aperture radar (SAR), precise and efficient frequency domain algorithms for high-resolution high-squint large-scene (HHL) spaceborne sliding spotlight SAR processing become more and more significant in the field of SAR signal processing. The frequency domain processing [5] for traditional spaceborne sliding spotlight SAR data mainly includes two steps [6]. The first step is de-aliasing [7] in Doppler domain, which is used to remove the inertial Doppler aliasing of sliding spotlight SAR data [8,9] and the second step is focusing, which is used to focus the SAR data by frequency domain imaging algorithms. However, as the resolution of SAR data reaches decimeter level and squint angle reaches 45°, a series of challenges arise [6,10,11] and result in the traditional de-aliasing algorithms inefficient and imaging algorithms invalid.

Traditional de-aliasing methods can be broadly divided into two categories: sub-aperture approaches [12,13,14,15] and two step approach (TSA) [8,9]. However, these methods lose their effectiveness or introduce new problems in situation of HHL [10,11,16]. Azimuth deramping methods [10,11,16] are novel and efficient means used for de-aliasing. Such algorithms remove the Doppler aliasing of by phase multiplication between the raw data and the deramlping function in range frequency and azimuth time domain, thereby transforming the squint sliding spotlight SAR data into equivalent broadside strip-map SAR data. The core of the deramping algorithms lies in the construction of deramping function. The [10,11] construct the function by calculating the range history between the antenna phase center (APC) and the virtual rotation point (VRP). However, as squint angle become bigger and imaging scene become larger, the VRP may no longer exist and the concept of VRP should be extended to virtual rotation curve. The [16] assume the scanning angle of antenna beam is linear versus azimuth time and construct the deramping function as shown in [16]. However, for HHL SAR echo acquisition, the scanning angle varies nonlinearly versus azimuth time and the corresponding deramping function should be refined accordingly.

Slant range model and derivation of the corresponding spectrum are significant parts for designing spaceborne SAR imaging algorithms. Hyperbolic range model is a widely used range model for spaceborne SAR imaging [5]. However, in case of high-resolution cases, due to the curved orbit effect and long synthetic aperture time [6], this model fail to describe range histories between APC and targets accurately. In order to adapt high-resolution cases, advanced hyperbolic range model [17], equivalent acceleration range model [18], squint equivalent acceleration range model [19], and so on [20,21] had been proposed and succeeded in improving the accuracy to some extent when expressing slant range histories. However, these models become less accurate as the resolution increases further. Moreover, it is also a cumbersome problem to derive the corresponding accurate spectrum when the range model is determined. Expressing the range histories by high-order polynomial (the order of polynomial can be adjusted adaptively according to resolution) and deriving the spectrum by Principle of Stationary Phase (POSP) and Series Revision (SR) are good methods to circumvent this problem [22,23,24,25,26]. However, the expression of spectrum may become complex when the order of polynomial is high and the design for following corrections may be difficult.

Correcting space-variants, including space-variant azimuth modulation, range cell migration (RCM), and range-azimuth cross coupling (RACC) in range and azimuth direction, in time domain and focusing the SAR data in frequency domain are another important parts for spaceborne SAR imaging algorithms design. Correcting space-variants in time domain removes some of the space-variants by changing the range histories of SAR signals. Motion compensation (MOCO) [27,28], imported from airborne SAR signal processing, is a kind of time-domain space-variants correction methods that corrects the space-variants by MOCO principle [6,29,30,31,32]. However, such methods have to be combined with sub-aperture approaches or TSA and thereby the processing cannot be performed efficiently. Time domain azimuth resampling method [16,18,19,33,34] is another kind of time-domain space-variants correction methods, which corrects the azimuth-variants by performing an elaborated azimuth resampling in 2D time domain. However, most time domain azimuth resampling methods assume that the azimuth-variants are the same at different range cells and the assumption no longer holds when the range swath is large. Although method in [16] takes into account the range-dependent characteristic of azimuth-variants, the resampling function still could to be refined.

Focusing the SAR data in frequency domain eliminates residual space-variants and achieves focusing by compensating space-variant azimuth modulation, RCM, and high order RACC in frequency domain. Nonlinear chirp scaling methods [35,36,37,38,39,40] are efficient frequency domain focusing methods, but these methods cause additional high-order space-variants and image geometric distortion that are not conducive to ultra-high resolution SAR processing [33]. Frequency domain resampling methods [5,34,41,42,43], featured as frequency domain phase multiplication and subsequent frequency domain resampling, is another kind of frequency domain focusing methods derived from RMA and extended RMA [41]. These methods assume the residual space-variants vary linearly with range or azimuth position and the ignored nonlinear space-variants deteriorate the focusing quality of image edges in the case of HHL.

Aiming at addressing the shortcomings of the existing algorithms above, this paper proposes a HHL spaceborne sliding spotlight SAR processing algorithm via joint 2D time and frequency domain resampling. The proposed algorithm is constituted by pre-processing in time domain and focusing in frequency domain. The pre-processing includes the azimuth deramping and 2D azimuth resampling in 2D time domain where the former is used to remove the aliasing in Doppler domain by implementing phase multiplication between raw data and an elaborated deramping function and the latter is used to circumvent the range-dependent azimuth-variants. The focusing in 2D frequency domain mainly includes range and azimuth resampling in the 2D frequency domain, residual RCMC, phase multiplication, and azimuth resampling in range-Doppler (RD) domain, which is used to correct the residual space-variants and achieve focusing. Moreover, for ultra-high resolution cases, the full-aperture processing cannot obtain consistent and well-focused results across scene and the partitioning strategy have to be applied. Therefore, the processing algorithm incorporated with partitioning strategy is also presented in this paper.

The remainder of this paper is organized as follows. Section 2 describes the azimuth deramping and 2D azimuth resampling in time domain. Section 3 introduces the signal model and derives the corresponding spectrum. Section 4 gives a detailed description about the focusing in 2D frequency domain. Section 5 combines the processing algorithm and partitioning strategy. Section 6 presents the processing results of simulation data and Gaofen-3 (GF-3) experimental data [44,45,46]. Finally, Section 7 concludes this paper.

2. Pre-Processing

The processing flows, shown as Figure 1, of the proposed algorithm could be divided into two parts: pre-processing in time domain and focusing in frequency domain. The pre-processing includes azimuth deramping and 2D azimuth resampling. After this procedure, the squint sliding spotlight SAR data could be treated as equivalent broadside strip-map SAR data.

2.1. Azimuth Deramping

The polynomial is applied to describe the relative distance between APC and target in this paper:

$$\begin{array}{cc}\hfill R_{ini}\left(\eta \right) & =r+p_1(\eta -{\eta }_c)+p_2{(\eta -{\eta }_c)}^2+\cdots \hfill \\ \hfill \eta & \in [{\eta }_{-},{\eta }_{+}]\hfill \end{array}$$

(1)

where $R_{ini}\left(\eta \right)$ represents the initial slant range, $\eta$ represents the azimuth time, ${\eta }_{-}$ and ${\eta }_{+}$ represent the start and end synthetic aperture time, ${\eta }_c$ represents the beam center crossing time, r represents the instantaneous range at ${\eta }_c$, and $p_i$ (i = 1, 2) are range model coefficients.

The change of $p_1$ in (1) with ${\eta }_c$ represents the variation of Doppler centroid with azimuth time and the removal of this variation from the SAR data completes the de-aliasing. Therefore, we model $p_1$ as a polynomial function of ${\eta }_c$:

$$p_1\left({\eta }_c\right)=p_{10}+{\displaystyle \sum _{v=1}^V}{p}_{1,a_v}\times {\eta }_c^v$$

(2)

where V is the order of the polynomial, and the $p_{10}$ and $p_{1,a_v}$ ($i=1,\cdots ,V$) are azimuth-variant coefficients.

Assuming that the corresponding range history of deramping function is $R_d\left(\eta \right)$, the range history of target lies in $(r,{\eta }_c)$, where r and ${\eta }_c$ represent the range and azimuth position respectively, after azimuth deramping could be expressed as

$$C\left(\eta \right)=r+p_1(\eta -{\eta }_c)+{\displaystyle \sum _{m=2}^M}{p}_m{(\eta -{\eta }_c)}^m-R_d\left(\eta \right)$$

(3)

Finding the derivative of function $C\left(\eta \right)$ at $\eta ={\eta }_c$:

$${\displaystyle \frac{\partial C}{\partial \eta }}{|}_{\eta ={\eta }_c}=p_1\left({\eta }_c\right)-{\displaystyle \frac{\partial R_d}{\partial \eta }}{|}_{\eta ={\eta }_c}$$

(4)

The ${\displaystyle \frac{\partial C}{\partial \eta }}{|}_{\eta ={\eta }_c}$ represents the variation of Doppler centroid versus azimuth time due to beam rotation and could be eliminated totally by setting it equals to zero. The corresponding range history could be expressed as:

$$R_d\left(\eta \right)=\int p_1\left(\eta \right)\mathbf{d}\eta =p_{10}\eta +{\displaystyle \sum _{v=1}^V}{\displaystyle \frac{p_{1,a_v}}{v+1}}\times {\eta }^{v+1}.$$

(5)

The azimuth deramping and pulse compression are implemented by the phase multiplication between SAR data and the elaborated deramping function (shown as (6)) in range frequency and azimuth time domain.

$$H_{dramp}(f_r,\eta )=exp\left[{\displaystyle \frac{+j4\pi (f_r+f_c)R_d\left(\eta \right)}{c}}+{\displaystyle \frac{j\pi f_r^2}{K_r}}\right]$$

(6)

where j is $\sqrt{-1}$, $f_r$ is the range frequency, $f_c$ is the carrier frequency, $K_r$ is frequency modulation rate of transmitted signal, and c is the speed of light.

It is worth pointing out that if $p_{1,a_1}$ and higher order coefficients in (5) equal to zero, the corresponding deramping function is the correction function used for linear range walk correction in [47] and if $p_{1,a_2}$ and higher order coefficients equal to zero, the corresponding function is the deramping function used in TSA [8,9,19].

In order to validate the effectiveness of the proposed method, a simulation with parameters shown in

Table 1. Spaceborne SAR Orbit and SAR System Parameters.

Orbit Parameter	Value
Eccentricity	0.0001
Semi-major axis	7084 km
Inclination	98.5°
Accening node	197°
Argument of perigee	90°
SAR System Parameter	Value
Carrier frequency	9.6 GHz
Signal bandwidth	600 MHz
Range sampling frequency	720 MHz
Pulse repetition frequency (PRF)	2166.4 Hz
Off-nadir angle	37°
Central squint angle	45°
Antenna beam width	0.3°
Synthetic aperture time for scene center	14.24 s
Scene center resolution (slant range × azimuth)	0.22 m × 0.34 m
Azimuth and range swath (ground range)	15 km × 20 km

is carried out. The scanning angle during data acquisition is nonlinear with respect to azimuth time as shown in Figure 2. The range histories of targets distributed in illuminated scene before and after deramping are shown in Figure 3. As shown in Figure 3b,c, the range histories of squint sliding spotlight SAR after deramping could be equivalent to range histories of broadside strip-map SAR.

The range histories after deramping could be rewritten as

$$\begin{array}{cc}\hfill R_{dp}\left(\eta \right) & =R_{ini}\left(\eta \right)-Rd\left(\eta \right)=r+{\displaystyle \sum _{n=2}^N}{k}_n{(\eta -{\eta }_0)}^n\hfill \\ \hfill \eta & \in [{\eta }_{-},{\eta }_{+}]\hfill \end{array}$$

(7)

where ${\eta }_0$ is the zero Doppler time and $k_n$, $(n=2,\cdots ,N)$, are range history coefficients after deramping obtained by polynomial fitting.

The ${\eta }_0$ is defined as

$${\eta }_0=\underset{\eta }{arg}\left\{{\displaystyle \frac{\partial R_{dp}\left(\eta \right)}{\partial \eta }}=0\right\}$$

(8)

2.2. 2D Azimuth Resampling in 2D Time Domain

The space-variant of range model coefficient $k_2$ could be modeled as:

$$\begin{array}{cc}\hfill k_2({\eta }_0,r)& =k_{20}+{\displaystyle \sum _{u=1}^U}{k}_{2,r_u}{r}^u+{\displaystyle \sum _{v=1}^V}{k}_{2,a_v}{\eta }_0^u+{\displaystyle \sum _{x=1}^X\sum _{y=1}^Y}{k}_{2,r_x{a}_y}{r}^x{\eta }_0^y\hfill \\ \hfill & =\left(k_{20}+{\displaystyle \sum _{u=1}^U}{k}_{2,r_u}{r}^u\right)+\left(k_{2,a_1}+{\displaystyle \sum _{x=1}^X}{k}_{2,r_x{a}_1}{r}^x\right){\eta }_0\hfill \\ & +\left(k_{2,a_2}+{\displaystyle \sum _{x=1}^X}{k}_{2,r_x{a}_2}{r}^x\right){\eta }_0^2+\cdots \hfill \\ \hfill & =k_{20}\left(r\right)+{\displaystyle \sum _{v=1}^V}{k}_{2,a_v}\left(r\right){\eta }_0^v\hfill \end{array}$$

(9)

where $k_{2,r_u}$, $k_{2,a_v}$, and $k_{2,r_x{a}_y}$ are space-variant coefficients of the polynomial in (9). The U, V, X, and Y are orders of the polynomial whose value could be selected adaptively according to the parameters of SAR system and illuminated scene.

Denoting the azimuth time and zero Doppler time after azimuth resampling as $t_a$ and $t_c$ and the 2D resampling funcition as $\chi (\eta ,r)$. The second-order term in the (7) after resampling could be written as:

$$\begin{array}{cc}\hfill C_2 & =K_2{(t_a-t_c)}^2=K_2{\left\{\chi \left[\eta ,R_{dp}\left(\eta \right)\right]-\chi ({\eta }_0,r_c)\right\}}^2\hfill \\ \hfill \eta & \in \left[{\eta }_{-},{\eta }_{+}\right]\hfill \end{array}$$

(10)

where $K_2$ is the second-order coefficient after resampling, $R_{dp}\left(\eta \right)$ is the range history, and $r_c$ is the instantaneous distance between APC and target at ${\eta }_0$. In contrast to azimuth resampling proposed in [34], the azimuth resampling function proposed here is range-dependent, and the RCM of SAR signal is taken into account during the derivation of resampling function.

By extending the azimuth timescale transformation in [34] to two-dimension, the second derivative of $C_2$ versus $\eta$ at ${\eta }_0$ could be expressed as:

$$k_2={\displaystyle \frac{1}{2}}\left({\displaystyle \frac{{\partial }^2{C}_2}{\partial {\eta }^2}}\right){|}_{\eta ={\eta }_0}=K_2{\left({\displaystyle \frac{\partial \chi }{\partial \eta }}+k_1{\displaystyle \frac{\partial \chi }{\partial R_{dp}}}\right)}^2{|}_{\eta ={\eta }_0}$$

(11)

The derivation of (11) could be found in Appendix A.

Since ${\eta }_0$ is the zero Doppler time as shown in (8), the $k_1$ in (11) is zero. Therefore, we can obtain:

$$t_a=\int \sqrt{{\displaystyle \frac{k_2(\eta ,r)}{k_{20}\left(r\right)}}}\mathbf{d}\eta =\int \sqrt{1+{\displaystyle {\sum }_{v=1}^V}{\displaystyle \frac{k_{2,a_v}\left(r\right)}{k_{20}\left(r\right)}}{\eta }^v}\mathbf{d}\eta$$

(12)

Setting the value of V in (12) to 3, an example of polynomial relation between $t_a$ and $\eta$ could be expressed as

$$\begin{array}{cc}\hfill t_a & =\chi (\eta ,r)\hfill \\ \hfill & \approx \eta +{\displaystyle \frac{{\alpha }_1\left(r\right)}{4}}{\eta }^2+\left[{\displaystyle \frac{4{\alpha }_2\left(r\right)-{\alpha }_1^2\left(r\right)}{24}}\right]{\eta }^3+\left[{\displaystyle \frac{{\alpha }_1^3\left(r\right)-4{\alpha }_1\left(r\right){\alpha }_2\left(r\right)+8{\alpha }_3\left(r\right)}{64}}\right]{\eta }^4\hfill \\ \hfill {\alpha }_i\left(r\right) & ={\displaystyle \frac{k_{2,a_i}\left(r\right)}{k_{20}\left(r\right)}}i=1,2,3\hfill \end{array}$$

(13)

The azimuth resampling function is range-dependent, meaning that the SAR signal lies in different range cells in 2D time domain experiencing different resampling function due to RCM. It is difficult to obtain an analytical relation between range model coefficients before and after resampling as [16,33,34]. In this paper, we refit the range model coefficients after resampling by polynomial fitting.

The range histories after resampling could be written as (14), and $t_c$ is still defined as the azimuth time at which the instantaneous Doppler frequency is zero:

$$\begin{array}{cc}\hfill t_c & =\underset{t_a}{arg}\left\{{\displaystyle \frac{\partial R\left(t_a\right)}{\partial t_a}}=0\right\}\hfill \\ \hfill R\left(t_a\right) & =r+{\displaystyle \sum _{n=2}^N}{K}_n{(t_a-t_c)}^n{t}_a\in [t_{-},t_{+}]\hfill \end{array}$$

(14)

where N is the order of the polynomial and its value could be selected adaptively according to resolution, r is the instantaneous range at $t_c$, $t_{-}$ and $t_{+}$ is the start and end of synthetic aperture time after resampling. $K_n$, $(n=2,\cdots ,N)$, is the slant range coefficients after resampling and is obtained by polynomial fitting between resampled azimuth time and resampled range histories.

Distributing targets across the illuminated scene, calculating $K_n$, $(n=2,\cdots ,N)$, and constructing the function relation between $K_n$ and $(r,t_c)$ as shown in (15). The form of the function could be polynomial of two variables as (9) and it can be other forms. The form of the function can even be non-analytic. The referenced method in Figure 4 and Figure 5.

$$K_n=f_{it,n}(r,t_c),n=2,\cdots ,N$$

(15)

In order to validate the effectiveness of the proposed 2D azimuth resampling, a computer simulation with parameters shown in Table 1 is executed and results are shown in Figure 4 and Figure 5. As shown in these figures, after the resampling in [33,34], the azimuth-variant of $K_2$ at reference range is corrected completely while the azimuth-variant still remain at the edge of range, whereas the proposed method removes the space-variant of $K_2$ completely for the whole scene.

3. Signal Model and Derivation of Spectrum

The signal model and the corresponding spectrum are essential for design of frequency domain imaging algorithms. In this section, we derive the spectrum of range model shown as (14) by POSP and SR and decompose the spectrum by 2D taylor expansion.

3.1. Derivation of Spectrum by POSP and SR

By applying POSP [11], the spectrum of target located in $(r,t_c)$ after deramping, pulse compression, and azimuth resampling could be expressed as

$$\begin{array}{cc}\hfill s(f_r,f_a;r,t_c) & ={\displaystyle \frac{1}{\sqrt{\frac{\partial f_a}{\partial t_a}}}}exp\left[j{\mathsf{\Phi }}_{2df}(f_r,f_a;r,t_c)\right]\hfill \\ \hfill {\mathsf{\Phi }}_{2df}(f_r,f_a;r,t_c) & =-{\displaystyle \frac{4\pi (f_r+f_c)R\left(t_a\right)}{c}}-2\pi f_a{t}_a\hfill \\ \hfill t_a-t_c & =p(f_r,f_a;r,t_c)\hfill \end{array}$$

(16)

where $R\left(t_a\right)$ is shown as (14) and $f_a$ is the azimuth frequency. The reciprocal of $\sqrt{{\displaystyle \frac{\partial f_a}{\partial t_a}}}$ represents the amplitude modulation in 2D frequency domain and the expression of ${\displaystyle \frac{\partial f_a}{\partial t_a}}$ is shown as (17), ${\mathsf{\Phi }}_{2df}(f_r,f_a;r,t_c)$ is the phase modulation in 2D frequency domain, the third formula in (16) represents the mapping relationship between $t_a$ and $f_a$ obtained by POSP [5,11]. The r and $t_c$ after semicolon represent the spectrum, modulation, and mapping relation are variant versus r and $t_c$.

$${\displaystyle \frac{\partial f_a}{\partial t_a}}={\displaystyle \frac{2(f_r+f_c)}{c}}{\displaystyle \sum _{n=0}^{N-2}}(n+1)(n+2)K_{n+2}{(t_a-t_c)}^n$$

(17)

The core of obtaining the spectrum lies in the calculation of $p(f_r,f_a;r,t_c)$. By applying SR [24,25,26], the relationship could be expressed as:

$$\begin{array}{cc}\hfill t_a-t_c & ={\displaystyle \sum _{i=1}^I}{A}_i{f}_e^i\hfill \\ \hfill & ={\displaystyle \frac{f_e}{2K_2}}+{\displaystyle \frac{(-3K_3)f_e^2}{8K_2^3}}+{\displaystyle \frac{(9K_3^2-4K_2{K}_4)f_e^3}{16K_2^5}}+\cdots \hfill \\ \hfill f_e & =-{\displaystyle \frac{c{f}_a}{2(f_r+f_c)}}\hfill \end{array}$$

(18)

where $A_i$ are polynomial coefficients in (18) obtained by SR and I is the order of polynominal.

The spectrum of target located in $(r,t_c)$ could be obtained by substituting (18) into (16), the N in (14) and I could be selected adaptively according to the resolution and synthetic aperture time. It is worth mentioning that we do not intend to derive the analytic form of the spectrum to avoid the difficulty introduced by the complex form of spectrum in the case of high-resolution and high-squint. Moreover, the phase function and resampling function in frequency domain are also calculated by numerical methods.

3.2. Decomposition of Phase Modulation

The reference function multiply (RFM) is an essential step in frequency domain resampling methods. The function used for RFM is the reciprocal of the spectrum of target located at $(r_{ref},t_{c,ref})$ where $r_{ref}$ is the reference range position and $t_{c,ref}$ is the azimuth position. After RFM, the residual phase modulation could be expressed as

$$\begin{array}{cc}\hfill \mathsf{\Phi }(f_r,f_a;r,t_c) & ={\mathsf{\Phi }}_{2df}(f_r,f_a;r,t_c)-{\mathsf{\Phi }}_{2df}(f_r,f_a;r_{ref},t_{c,ref})\hfill \end{array}$$

(19)

The $\mathsf{\Phi }(f_r,f_a;r,t_c)$ could be decomposed as

$$\begin{array}{cc}\hfill \mathsf{\Phi }(f_r,f_a;r,t_c) & =\mathsf{\Psi }(f_r,f_a;r,t_c)+\psi (f_a;r,t_c)\hfill \\ \hfill \mathsf{\Psi }(f_r,f_a;r,t_c) & =\mathsf{\Phi }(f_r,f_a;r,t_c)-\mathsf{\Phi }(0,f_a;r,t_c)\hfill \\ \hfill \psi (f_a;r,t_c) & =\mathsf{\Phi }(0,f_a;r,t_c)\hfill \end{array}$$

(20)

where $\mathsf{\Psi }(f_r,f_a;r,t_c)$ represents the space-variant RCM and high-order RACC, collectively known as RACC [5], and $\psi (f_a;r,t_c)$ represents the space-variant azimuth modulation.

By expanding the $\mathsf{\Psi }(f_r,f_a;r,t_c)$ in (20) versus r and $t_c$ at $(r_{ref},t_{c,ref})$, $\mathsf{\Psi }(f_r,f_a;r,t_c)$ could be rewritten as

$$\begin{array}{cc}\hfill \mathsf{\Psi }(f_r,f_a;r,t_c) & \approx \left({\displaystyle \frac{\partial \mathsf{\Psi }}{\partial r}}{|}_{\begin{array}{c}r=r_{ref}\\ t_c=t_{c,ref}\end{array}}\right)\times (r-r_{ref})+\left({\displaystyle \frac{\partial \mathsf{\Psi }}{\partial t_c}}{|}_{\begin{array}{c}r=r_{ref}\\ t_c=t_{c,ref}\end{array}}\right)\times (t_c-t_{c,ref})\hfill \end{array}$$

(21)

where $\left({\displaystyle \frac{\partial \mathsf{\Psi }}{\partial r}}{|}_{\begin{array}{c}r=r_{ref}\\ t_c=t_{c,ref}\end{array}}\right)$ and $\left({\displaystyle \frac{\partial \mathsf{\Psi }}{\partial t_c}}{|}_{\begin{array}{c}r=r_{ref}\\ t_c=t_{c,ref}\end{array}}\right)$ are functions versus $f_r$ and $f_a$ and keep constant versus r and $t_c$.

The first and second term on the right-hand of the approximate equal sign represent the range-variant and azimuth-variant RACC including range-variant and azimuth-variant RCM and high-order RACC respectively. All of them could be corrected by resampling in 2D frequency domain.

By expanding $\psi (f_a;r,t_c)$ in (20) versus $t_c$ at $t_{c,ref}$, $\psi (f_a;r,t_c)$ could be rewritten as

$$\begin{array}{cc}\hfill \psi (f_a;r,t_c) & \approx \psi (f_a;r,t_{c,ref})+\left({\displaystyle \frac{\partial \psi }{\partial t_c}}{|}_{t_c=t_{c,ref}}\right)\times (t_c-t_{c,ref})\hfill \end{array}$$

(22)

The first term on the right-hand of the approximate equal sign represents the azimuth-invariant part of azimuth modulation which could be compensated by phase multiplication in RD domain. The second term represents the azimuth-variant part of azimuth modulation which could be compensated by azimuth resampling in RD domain.

4. Focusing by Resampling in 2D Frequency Domain

The focusing in frequency domain, which could also be regarded as compensation of the residual space-variant after preprocessing, corresponds to the focusing part in Figure 1. The processing flow of focusing includes RFM, range resampling in 2D frequency domain, azimuth resampling in 2D frequency domain, residual RCMC, azimuth matched filtering, and azimuth resampling in RD domain.

4.1. RFM

The RFM is used to compensating the amplitude modulation and phase modulation by a reference target. Assuming the reference target is located at $(r_{ref},t_{c,ref})$ after pre-processing in slant range plane. The matched filter could be defined as the reciprocal of spectrum of reference target

$$\begin{array}{c}{H}_{ref}(f_r,f_a) =\sqrt{{\displaystyle \frac{\partial f_a}{\partial t_a}}}\times exp\left[-j{\mathsf{\Phi }}_{2df}(f_r,f_a;r_{ref},t_{c,ref})\right]\end{array}$$

(23)

There $H_{ref}(f_r,f_a)$ could be calculated by (16)–(18), and (23). $K_n$ used in (18), $(n=2,\cdots ,N)$, for the reference point is obtained by space-variant functions constructed in (15)

$$\begin{array}{cc}\hfill K_n & =f_{it,n}(r_{ref},t_{c,ref})\hfill \\ \hfill n & =2,\cdots ,N\hfill \end{array}$$

(24)

The phase modulation of target located at $(r,t_c)$ after RFM could be expressed as (19).

4.2. Range Resampling in 2D Frequency Domain

The range resampling in 2D frequency domain, which can also be regarded as the modified Stolt interpolation [5], is used to correct the range-variant RCM and high-order RACC. The range resampling is implemented by range interpolation in 2D frequency domain. The resampling function could be expressed as

$$\left[{\displaystyle \frac{\partial \mathsf{\Psi }(f_r,f_a;r,t_{c,ref})}{\partial r}}{|}_{r=r_{ref}}\right]\Rightarrow -{\displaystyle \frac{4\pi f_r^{\prime }}{c}}$$

(25)

where $f_r^{\prime }$ is the range frequency after range resampling in 2D frequency domain.

By arranging (25) further and we can obtain

$$\begin{array}{cc}\hfill f_r^{\prime }& =X(f_r,f_a)=\left(-{\displaystyle \frac{c}{4\pi }}\right)·\left[{\displaystyle \frac{\partial \mathsf{\Psi }(f_r,f_a;r,t_{c,ref})}{\partial r}}{|}_{r=r_{ref}}\right]\hfill \end{array}$$

(26)

The inverse function of resampling function (26) could be expressed by

$$f_r=X^{-1}(f_r^{\prime },f_a)$$

(27)

It should be noted that the resampling function $X(f_r,f_a)$ and its inverse function $X^{-1}({f_r}^{\prime },f_a)$ are all calculated by numerical method.

Here we exhibit an example of the calculation of $X(f_r,f_a)$ by least square method (LSM). Distributing targets in slant range plane after pre-processing at $(r_i,t_{c,ref})$, $(i=1,\cdots ,P_r)$, as shown in Figure 6a and the illuminated scene should be included in the scope spanned by these targets.

Calculating RACC $\mathsf{\Psi }(f_r,f_a;r_i,t_{c,ref})$, $(i=1,\cdots ,P_r)$, for targets in Figure 6a by (16)–(20). The range model coefficients used for these formulas are calculated by using $K_n(r,t_c)$ constructed in (15):

$$\begin{array}{cc}\hfill K_n(r_i,t_{c,ref})& =f_{it,n}(r_i,t_{c,ref})\hfill \\ \hfill n& =2,\cdots ,N\hfill \\ \hfill i& =1,\cdots ,P_r\hfill \end{array}$$

(28)

where $P_r$ is the number of distributed targets in Figure 6.

For a certain azimuth frequency $f_a$, we can construct the observation matrix $\mathsf{\Theta }\left(f_a\right)$ and coefficient matrix A as:

$$\mathsf{\Theta }\left(f_a\right)=\left[\begin{array}{c}\mathsf{\Psi }(f_r,f_a;r_1,t_{c,ref})\\ ⋮\\ \mathsf{\Psi }(f_r,f_a;r_i,t_{c,ref})\\ ⋮\\ \mathsf{\Psi }(f_r,f_a;r_{P_r},t_{c,ref})\end{array}\right]A=\left[\begin{array}{c}{r}_1-r_{ref}\\ ⋮\\ r_{P_r}-r_{ref}\end{array}\right]$$

(29)

where the size of $\mathsf{\Theta }\left(f_a\right)$ is $P_r\times N_r$ and the size of A is $P_r\times 2$. The $N_r$ is the length of $f_r$ in discrete domain.The least square solution could be expressed as:

$$\xi ={\left(A^{T}A\right)}^{-1}{A}^T\mathsf{\Theta }\left(f_a\right)$$

(30)

where ${[·]}^{-1}$ represents matrix inverse operator, and ${[·]}^T$ represents matrix transpose operator.

It is easy the to get that the size of $\xi$ is $2\times N_r$. The least square solution of $X(f_r,f_a)$ at certain azimuth frequency $f_a$ could be expressed as:

$$X(f_r,f_a)=\left(-{\displaystyle \frac{c}{4\pi }}\right)\times \xi$$

(31)

The resampling function $X(f_r,f_a)$ could be obtained by repeating the calculation from (29) to (31) for each azimuth frequency $f_a$. After the calculation of $X(f_r,f_a)$, the inverse function $f_r=X^{-1}({f_r}^{\prime },f_a)$ could be obtained by interpolation. It should be noted that LMS is not the only method to obtain $X(f_r,f_a)$ and the calculation could also be achieved by other numerical methods.

The RACC of target at $(r,t_c)$ after range resampling could be obtained by substituting (27) into $\mathsf{\Psi }(f_r,f_a;r,t_c)$:

$${\mathsf{\Psi }}_r(f_r^{\prime },f_a;r,t_c)=\mathsf{\Psi }\left[X^{-1}({f_r}^{\prime },f_a),f_a;r,t_c\right]$$

(32)

4.3. Azimuth Resampling in 2D Frequency Domain

Azimuth resampling in 2D frequency domain, which can be regarded as the extension of Stolt interpolation in azimuth, is used to correct the azimuth-variant RCM and high-order RACC. The azimuth resampling is implemented by azimuth interpolation in 2D frequency domain. The resampling function could be expressed as:

$$\left[{\displaystyle \frac{\partial {\mathsf{\Psi }}_r(f_r^{\prime },f_a;r_{ref},t_c)}{\partial t_c}}{|}_{t_c=t_{c,ref}}\right]\Rightarrow -2\pi f_a^{\prime }$$

(33)

where $f_a^{\prime }$ is the azimuth resampling function after azimuth resampling in 2D frequency domain.

By arranging (33) further and we can obtain

$$\begin{array}{c}\hfill f_a^{\prime }=Y(f_r^{\prime },f_a)\left(-{\displaystyle \frac{1}{2\pi }}\right)·\left[{\displaystyle \frac{\partial {\mathsf{\Psi }}_r(f_r^{\prime },f_a;r_{ref},t_c)}{\partial t_c}}{|}_{t_c=t_{c,ref}}\right]\end{array}$$

(34)

The inverse function of resampling function (34) could be expressed by:

$$f_a=Y^{-1}(f_r^{\prime },f_a^{\prime })$$

(35)

The resampling function $Y(f_r^{\prime },f_a)$ could also be obtained by distributing targets along the azimuth at reference range as shown in Figure 6b and repeating the calculation like (28) to (31) for each range frequency $f_r^{\prime }$. The inverse function $Y^{-1}(f_r^{\prime },f_a^{\prime })$ is also calculated by interpolation.

The phase modulation ${\mathsf{\Phi }}_a(f_r^{\prime },f_a^{\prime };r,t_c)$ of target at $(r,t_c)$ after range and azimuth resampling in 2D frequency domain could be obtained by substituting $f_a=Y^{-1}(f_r^{\prime },f_a^{\prime })$ and $f_r=X^{-1}(f_r^{\prime },f_a)$ into (19) and (20). The expression of ${\mathsf{\Phi }}_a(f_r^{\prime },f_a^{\prime };r,t_c)$ is shown as (36). It should be noted that the $X^{-1}\left[f_r^{\prime },Y^{-1}(f_r^{\prime },f_a^{\prime })\right]$ in (36) also should be calculated by numerical methods.

$$\begin{array}{cc}\hfill {\mathsf{\Phi }}_a(f_r^{\prime },f_a^{\prime };r,t_c)& ={\mathsf{\Psi }}_r\left[f_r^{\prime },Y^{-1}(f_r^{\prime },f_a^{\prime });r,t_c\right]+{\mathsf{\Psi }}_{az}\left[Y^{-1}(f_r^{\prime },f_a^{\prime });r,t_c\right]\hfill \\ \hfill & ={\mathsf{\Phi }}_{2df}\left\{X^{-1}\left[{f_r}^{\prime },Y^{-1}(f_r^{\prime },f_a^{\prime })\right],Y^{-1}(f_r^{\prime },f_a^{\prime });r,t_c\right\}\hfill \\ & -{\mathsf{\Phi }}_{2df}\left\{X^{-1}\left[{f_r}^{\prime },Y^{-1}(f_r^{\prime },f_a^{\prime })\right],Y^{-1}(f_r^{\prime },f_a^{\prime });r_{ref},t_{c,ref}\right\}\hfill \end{array}$$

(36)

In order to validate the effectiveness of the proposed azimuth resampling, computer simulation with orbit parameters shown in Table 1 and SAR system parameters shown in

Table 2. Spaceborne SAR Orbit and SAR System Parameters.

Parameter	Value
Carrier frequency	9.6 GHz
Signal bandwidth	1200 MHz
Range sampling frequency	1440 MHz
PRF	2166.4 Hz
Off-nadir angle	37°
Central squint angle	45°
Antenna beam width	0.3°
Synthetic aperture time for scene center	28.77 s
Scene center resolution (slant range × azimuth)	0.11 m × 0.17 m
Azimuth and range swath (ground range)	15 km × 20 km

was carried out. Nine targets are distributed on the surface of the earth as shown in Figure 7a and their distribution after deramping in slant range plane are shown in Figure 7b. SAR echo of targets $P_4$, $P_5$, and $P_6$ (along the azimuth) are processed by method in Figure 1. The results with azimuth resampling and without azimuth resampling are shown in Figure 8. It could be found that the curved RCMs in Figure 8a,c becomes straight (as shown in Figure 8d,f) after the azimuth resampling which verifies the effectiveness of the proposed method.

The high-order RACC could be ignored after 2D resampling, thereby ${\mathsf{\Phi }}_a(f_r^{\prime },f_a^{\prime };r,t_c)$ could be expanded as

$$\begin{array}{cc}\hfill {\mathsf{\Phi }}_a(f_r^{\prime },f_a^{\prime };r,t_c)& \approx \varphi (f_a^{\prime };r,t_c)+{\varphi }_{rcm}(f_a^{\prime };r)\times f_r^{\prime }\hfill \\ \hfill \varphi (f_a^{\prime };r,t_c)& ={\mathsf{\Phi }}_a(0,f_a^{\prime };r,t_c)\hfill \\ \hfill {\varphi }_{rcm}(f_a^{\prime };r)& ={\displaystyle \frac{\partial {\mathsf{\Phi }}_a\left(f_r^{\prime },f_a^{\prime };r,t_{c,ref}\right)}{\partial f_r^{\prime }}}{|}_{f_r^{\prime }=0}\hfill \end{array}$$

(37)

where $\varphi (f_a^{\prime };r,t_c)$ is the residual azimuth modulation and is variant in both range and azimuth, and ${\varphi }_{rcm}(f_a^{\prime };r)$ is the residual RCM and is only variant in range.

4.4. Residual RCMC

Since the range-variant is severe in case of HHL, the expanding of phase modulation versus range position r in (21) may be not accurate and the residual range-variant RACC, especially residual RCM in (37), may result in impulse response function (IRF) distortion at range edge. In order to circumvent this problem, we execute residual RCMC in RD domain.

The residual RCM for target located at $(r,t_{c,ref})$ could be expressed as

$$\begin{array}{cc}\hfill RCM(f_a^{\prime };r)& =\left(-{\displaystyle \frac{c}{4\pi }}\right)\times {\varphi }_{rcm}(f_a^{\prime };r)\hfill \\ \hfill & =\left(-{\displaystyle \frac{c}{4\pi }}\right)\times {\displaystyle \frac{\partial {\mathsf{\Phi }}_a\left(f_r^{\prime },f_a^{\prime };r,t_{c,ref}\right)}{\partial f_r^{\prime }}}{|}_{f_r^{\prime }=0}\hfill \end{array}$$

(38)

It is worth mentioning that the residual RCMC can only correct the residual RCM of target at $t_{c,ref}$ because it is performed in RD domain.

The residual RCM used for residual RCMC could be expressed by

$$U_r(f_a^{\prime };r)=RCM(f_a^{\prime };r)-RCM(0;r)$$

(39)

The $U_r(f_a^{\prime };r)$ could be obtained by numerical method. Distributing targets as shown in Figure 6a and calculating $U_r(f_a^{\prime };r)$ for each targets. The size of $U_r(f_a^{\prime };r_i)$, $(i=1,\cdots ,P_r)$, is much smaller than the size of SAR data, thus a 2D interpolation should be carried out to make the size of $U_r(f_a^{\prime };r_i)$ same as SAR data. The execution of residual RCMC is the same as the RCMC in RDA [5].

In order to validate the effectiveness of the residual RCMC, SAR echo of targets $P_2$, $P_5$, and $P_8$ (along the range) in Figure 7 are processed by proposed method. The results with residual RCMC and without residual RCMC are shown in Figure 9. It could be found that the curved RCMs in Figure 9a,c becomes straight (as shown in Figure 9d,f) after the residual RCMC which verifies the effectiveness of the proposed method.

4.5. Azimuth Matched Filtering

The residual azimuth modulation after 2D resampling could also be expanded as (22)

$$\begin{array}{cc}\hfill \varphi (f_a^{\prime };r,t_c)& \approx \varphi (f_a^{\prime };r,t_{c,ref})+\left({\displaystyle \frac{\partial \varphi }{\partial t_c}}{|}_{t_c=t_{c,ref}}\right)\times (t_c-t_{c,ref})\hfill \end{array}$$

(40)

The azimuth matched filter could be constructed by (22) as

$$H_{az}(f_a^{\prime };r)=exp\left[-j\varphi (f_a^{\prime };r,t_{c,ref})\right]$$

(41)

The calculation of $\varphi (f_a^{\prime };r,t_{c,ref})$ is the same as the calculation of residual RCM $U_r(f_a^{\prime };r)$.

4.6. Azimuth Resampling in RD Domain

The azimuth resampling in RD domain is used for correcting the azimuth-variant modulation which is implemented by azimuth interpolation in RD domain. The resampling function in RD domain could be expressed by

$$\left[{\displaystyle \frac{\partial \varphi (f_a^{\prime };r,t_c)}{\partial t_c}}{|}_{t_c=t_{c,ref}}\right]\Rightarrow -2\pi f_a^{″}$$

(42)

where $f_a^{″}$ is the azimuth frequency after azimuth resampling in RD domain.

By arranging the function further, the resampling function could be expressed as

$$\begin{array}{cc}\hfill f_a^{″}& =Z(f_a^{\prime };r)=\left(-{\displaystyle \frac{1}{2\pi }}\right)·\left[{\displaystyle \frac{\partial \varphi (f_a^{\prime };r,t_c)}{\partial t_c}}{|}_{t_c=t_{c,ref}}\right]\hfill \end{array}$$

(43)

The resampling function $Z(f_a^{\prime };r)$ is different in different range cell, which means the range-dependent characteristic of azimuth-variant azimuth modulation is also considered during RD domain resampling.

The resampling function $Z(f_a^{\prime };r)$ could also be obtained by LSM and interpolation. Distributing targets across the scene as shown in Figure 6c. For each range position $r_i$, $(i=1,\cdots ,P_r)$, the resampling function $Z(f_a^{\prime };r_i)$ could be obtained by LSM like shown by (28)–(31) and the observation matrix and coefficient matrix could be expressed as

$$\begin{array}{c}\hfill \mathsf{\Theta }\left(f_a\right)=\left[\begin{array}{c}\varphi (f_a^{\prime };r_i,t_{c,1})-\varphi (f_a^{\prime };r_i,t_{c,ref})\\ ⋮\\ \varphi (f_a^{\prime };r_i,t_{c,j})-\varphi (f_a^{\prime };r_i,t_{c,ref})\\ ⋮\\ \varphi (f_a^{\prime };r_i,t_{c,P_a})-\varphi (f_a^{\prime };r_i,t_{c,ref})\end{array}\right] A=\left[\begin{array}{c}{t}_{c,1}-t_{c,ref}\\ ⋮\\ t_{c,P_a}-t_{c,ref}\end{array}\right]\end{array}$$

(44)

After obtaining the resampling function for each i, the $Z(f_a^{\prime };r_i)$ should be interpolated to the same size as SAR data.

5. Combination of Proposed Method and Partitioning Strategy

When the resolution and observed scene are improved further, the processing flow in Figure 1 may still not be able to obtain the consistent imaging results for the whole scene. Inspired by method proposed in [11], we combine the proposed method in Figure 1 and partitioning strategy to adapt higher resolution cases in this section. The processing flow is shown as Figure 10 which includes pre-processing, image partitioning, focusing, and images combination. It should be noted that the focusing in Figure 10 is the same as the focusing in Figure 1. The core of this strategy lies in partitioning.

The purpose of range partitioning is to reduce the overlap between subblocks in the subsequent 2D further partitioning to improve the efficiency of the post-processing. The Figure 11 is the coarsely imaging result after pre-processing and RFM (the simulation parameters is shown as Table 1). As shown in Figure 11, since the 2D azimuth resampling in time domain eliminates the azimuth-variant property of $K_2$, the aperture length of targets keep constant in azimuth but vary dramatically in range. In order to avoid the resolution loss of targets located in edge of subblocks, the overlap between subblocks needs to be at least greater than the aperture length after RFM thus processing cannot be carried out efficiently in case of large range swath. More detail about blocking could be found in [11].

As shown in Figure 10, each partitioning is accompanied by a RFM whose matched filter is calculated according to range model coefficients obtained by (15). It should be noted that the matched filter used in previous RFM should be compensated before executing the new RFM.

6. Verification of Prosposed Method

This section presents processing results of simulation data and GF-3 experimental data to verify the proposed algorithms in Figure 1 and Figure 10.

6.1. Verification of Full-Aperture Processing Scheme

In order to verify the effectiveness of the proposed algorithm, the simulated spaceborne sliding spotlight SAR echo data after azimuth deramping with parameters in Table 1 were processed by algorithm proposed in [34] and the proposed algorithm shown as Figure 1 separately. It should be mentioned that the algorithm in [34] is modified to accommodate long synthetic aperture time. The geometry of the simulated scene is shown as Figure 12. The processing results of targets $P_a$, $P_e$, and $P_i$ by proposed algorithm and [34] are shown in Figure 13 and Figure 14 respectively. The quantitative results, including peak sidelobe ratio (PSLR), integration sidelobe ratio (ISLR), and impulse response width (IRW), are shown in

Table 3. Point Target Results Analysis of the Referenced Method and Proposed Method.

Method	Direction	${\mathit{P}}_{\mathit{a}}$			${\mathit{P}}_{\mathit{e}}$			${\mathit{P}}_{\mathit{i}}$
		PSLR	ISLR	IRW	PSLR	ISLR	IRW	PSLR	ISLR	IRW
Referenced Method	Rg	−14.20	−11.16	0.23	−13.25	−10.04	0.22	−14.08	−11.13	0.22
	Az	−12.21	−9.72	0.35	−13.30	−10.22	0.35	−12.14	−9.35	0.36
Proposed Method	Rg	−13.20	−10.07	0.22	−13.26	−10.04	0.22	−13.23	−10.22	0.22
	Az	−12.61	−9.99	0.34	−13.30	−10.23	0.35	−12.85	−10.06	0.34

.

As shown in Figure 13 and Figure 14, and Table 3, compared to the proposed method, the IRFs of target $P_a$ and $P_i$ obtained by the [34] suffer from profile distortion, high sidelobe, and worse focusing parameters. The comparison results verify the effectiveness of 2D azimuth resampling in time domain and residual RCMC. Since the azimuth-variant RCM is slight in the case of Table 1, the imaging results of $P_d$ and $P_f$ which could be used to verify the effectiveness of 2D frequency domain azimuth resampling are not presented in this subsection, but its effectiveness had already been verified by Figure 8.

6.2. Verification of Partitioning Processing Scheme

In order to verify the effectiveness of the partitioning strategy, the simulated spaceborne sliding spotlight SAR echo data with parameters in Table 2 were processed by algorithm in Figure 10. The geometry of the simulated scene is shown as Figure 7. The processing results of targets $P_1$, $P_5$, and $P_9$ are shown in Figure 15. The quantitative results are shown in

Table 4. Point Target Results Analysis of Proposed Algorithm.

Dir	${\mathit{P}}_1$			${\mathit{P}}_5$			${\mathit{P}}_9$
	PSLR	ISLR	IRW	PSLR	ISLR	IRW	PSLR	ISLR	IRW
Rg	−13.24	−10.04	0.11	−13.25	−10.04	0.11	−13.24	−10.07	0.11
Az	−13.35	−10.64	0.18	−13.39	−10.66	0.17	−13.37	−10.67	0.16

. It should be mentioned that after pre-processing the SAR data were divided into 8 blocks in range and each range block was divided into 16 subblocks in azimuth.

As shown in Figure 15, the interpolated contours and azimuth profiles of $P_1$, $P_5$, and $P_9$ are similar to the ideal sinc function. In Table 4, the focusing parameters are ideal. These results demonstrate the effectiveness of the proposed method shown in Figure 10.

6.3. Verification by GF-3 Experimental Data

To further demonstrate the effectiveness of the proposed method, the experimental data operating in sliding spotlight mode acquired by GF-3 was processing by proposed method in Figure 10. The SAR system parameters is shown as

Table 5. Spaceborne SAR System Parameters of GF-3.

Parameter	Value
Carrier frequency	5.4 GHz
Signal bandwidth	240 MHz
Range sampling frequency	266.67 MHz
PRF	3686.9 Hz
Central squint angle	0°
Antenna beam width	0.436°
Scene center resolution (slant range × azimuth)	0.55 m × 0.72 m
Azimuth and range swath (ground range)	20 km × 10 km

. Actually, for GF-3 SAR data, conventional TSA with RMA considering range-variant equivalent velocity could obtain good imaging results. The purpose of using GF-3 SAR data in this subsection is to validate that the partitioning processing scheme in Figure 10 does not introduce paired echo and discontinuities between image blocks.

The overview of imaging results are shown in Figure 16. Obviously, there is no apparent defocusing and artifacts in the overview. The white solid lines in Figure 16 are boundary of blocks and the close-ups of subimages between blocks are shown in Figure 17. As shown in Figure 17a–c, there are no artifacts or discontinuities between subimages. The imaging results of GF-3 further validate the effectiveness of proposed algorithm.

7. Conclusions

In this paper, a frequency domain imaging algorithm for HHL spaceborne sliding spotlight SAR processing is proposed. The algorithm is constituted by pre-processing in time domain and focusing in frequency domain. The pre-processing includes azimuth deramping and 2D azimuth resampling in time domain which is used to remove the Doppler aliasing introduced by beam rotation and correct range-dependent azimuth-variants of range histories. The focusing in frequency domain mainly includes phase multiplication and resampling in 2D frequency and RD domain which is used to correct residual space-variant azimuth modulation, RCM, and high-order RACC and focus the SAR data. Moreover, in order to adapt higher resolution and larger scene cases, the combination of the proposed algorithm and partitioning strategy is also presented in this paper. Processing results of simulation data and GF-3 experimental data are presented to demonstrate the feasibility of the proposed algorithm. The proposed algorithm have great potential in HHL SAR data processing.