An Imaging Compensation Algorithm for Spaceborne High-Resolution SAR Based on a Continuous Tangent Motion Model

Abstract: This paper develops a continuous tangent motion model by approximating the relative trajectories between spaceborne Synthetic Aperture Radar (SAR) and its targets during every transmitting period and every receiving period as tangential segments. Compared with existing motion models, including the stop-go model and the continuous rectilinear model, the continuous tangent model is much closer to the actual relative motion. Based on the new motion model, an imaging compensation algorithm is proposed that considers the space-variant property of SAR echoes more fully by introducing three space-variant factors: the time-scaling factor, the transmitting–receiving rate, and the transmitting–receiving constant. As a result, spaceborne SAR imaging with 0.21-meter resolution and wide-swath, up to 15 kmˆ 15 km, is achieved. Simulation results indicate that the novel algorithm provides a superior and more consistent focusing quality across the whole swath compared with existing algorithms based on the stop-go or continuous rectilinear models.


Introduction
Motion compensation is the key to Synthetic Aperture Radar (SAR) or Inverse Synthetic Aperture Radar (ISAR) imaging [1,2].The motion model, which describes the relative motion between SAR and its targets, i.e., anything illuminated by the antenna mainlobe, plays a crucial role in SAR imaging.Precise calculation of relative distances according to the motion model is the basis of good focusing quality.If the motion model is not sufficiently accurate, the performance of range migration correction and phase compensation based on relative distances will diminish, and the focusing quality will be affected.The aim of this paper is to construct a novel motion model for spaceborne SAR and develop a compensation algorithm, based on the motion model, for use in high-resolution and wide-swath imaging.
Satellites have advantages for SAR observation compared with other platforms [3].The relative motion of spaceborne SAR is complicated and cannot be expressed explicitly because SAR travels along an elliptical orbit while targets rotate with the earth.In this case, an approximate motion model has to be adopted to derive the corresponding algorithm for efficient and precise imaging.The stop-go model is the most commonly applied motion model for spaceborne SAR [4,5].This model includes two assumptions.The first is that the relative trajectory during aperture time is approximated as a straight line, even if it is strictly curved.The second is that SAR is assumed to remain stationary at a single position while transmitting one pulse and receiving the corresponding echo before moving to the next position.If the SAR satellite is in an orbit with a height of 600 km, it actually takes more than 4 ˆ10 ´3 s for a pulse to complete a round trip.During this amount of time, the SAR satellite travels at least 30 meters.For spaceborne SAR, the stop-go model is valid when the aperture time is short enough for the difference between the actual and approximated trajectories to be negligible, and the signal band is small enough for the distance error induced by the actual relative motion during transmitting and receiving to be less than one range resolution cell [5].As is well known, the aperture time and the signal band are key factors in the azimuth and range resolutions, respectively.Therefore, the application of the stop-go model is limited by resolution.Imaging algorithms based on the stop-go model, such as Range-Doppler (RD) [6,7] and Chirp-Scaling (CS) [8,9], exhibit good performance only when the resolution is worse than a threshold which is about 0.3 meters for X-band.Currently, the majority of spaceborne SARs operate below this threshold.However, there are times when the required resolution is better than this threshold, such as TerraSAR-X NG, which will produce 0.25-meter resolution images beyond the year 2025 [10].Better performance of target detection and recognition can be achieved based on images with better resolution [11,12].
When the required resolution is higher, the validity of the stop-go model is reduced.Prats-Iraola et al. pointed out that in order to acquire 0.21-meter resolution images, the curvature of the orbit and the relative motion between transmitting and receiving should be taken into account [13].Otherwise, the residual phase induced by the stop-go model is intolerable and the quality inevitably decreases.
For curvature-induced imaging degradation, Prats-Iraola et al. applied the range history of a reference target in the swath center to compensate for echo phase to be pure hyperbolic [13].Conventional imaging algorithms [6][7][8][9] based on the rectilinear approximation were then adapted to process compensated echoes.However, because range histories corresponding to the center and other positions of the swath are different, the compensation effect gradually declines as the swath becomes wider.As a result, this method cannot be applied to the imaging of large swaths.The swath width of the TerraSAR-X Staring Spotlight (ST) mode, which has adopted this method, is less than 5.3 km [13,14].
The stop-go model was also investigated by Liu et al. who reconstructed the echo model by taking the relative motion during transmitting and receiving into account.A corresponding imaging algorithm was then produced to obtain a high focusing quality with a 0.2-m resolution and an 8 km (slant range) ˆ4 km (azimuth) swath width [15].However, the applicability of Liu's algorithm to spaceborne SAR was lessened by a rectilinear motion assumption made during the derivation of the algorithm.Prats-Iraola also studied this problem and analyzed the mismatching phenomenon that occurred during filtering along the range by referring to frequency-modulated continuous-wave (FMCW) systems [16,17].A correction method was presented in the two-dimensional frequency domain [13].This method is not suitable for wide-swath imaging because space variance is not considered in the correction factor.This paper will present a compensation algorithm to achieve high-resolution SAR imaging with a wide swath, up to 15 km, by constructing a continuous tangent motion model.This novel motion model assumes that SAR moves without stopping and approximates the relative trajectories during every transmitting and receiving period as tangent lines.This assumption fully reflects the properties of relative motion.
The paper is structured as follows.Section 2 constructs the continuous tangent model and derives the two-way distance corresponding to every sampling point for spaceborne SAR.Section 3 presents a new echo expression that considers the space-variant property of SAR echoes more fully than other existing echo expressions.A novel imaging compensation algorithm is presented in Section 4 and simulated results are compared with algorithms based on existing motion models in Section 5. Section 6 concludes the paper.

Continuous Tangent Model and Two-Way Distance
The two-way distance refers to the sum of propagation distances for electromagnetic signals from SAR to the target and back to SAR.It is applied in the imaging to implement range correction and phase compensation.The stop-go model is most commonly used for calculating the two-way distance [4,13].This section will construct a continuous tangent model, an update to the stop-go model, which significantly improves the accuracy of two-way distance calculation in order to enhance focusing quality.

Continuous Tangent Model
Figure 1 demonstrates the actual relative motion between spaceborne SAR and the target.For convenience, time is divided into two parts: azimuth slow time t i and range fast time τ.Here, t i " i{ f p (i " 0, 1, 2, 3, ¨¨¨) and f p is the pulse repetition frequency (PRF).As a result, the continuous time t is expressed as t " t i `τ.The mth pulse is transmitted during rt m ´T{2, t m `T{2s, where T denotes the pulse width.The center of the transmitting waveform is marked as A, which leaves the radar at t m , reaches the target, and is reflected to the receiver at t n `∆τ.Similarly, another point B is transmitted at t m `τs and received at t n `∆τ `τr .

Continuous Tangent Model and Two-Way Distance
The two-way distance refers to the sum of propagation distances for electromagnetic signals from SAR to the target and back to SAR.It is applied in the imaging to implement range correction and phase compensation.The stop-go model is most commonly used for calculating the two-way distance [4,13].This section will construct a continuous tangent model, an update to the stop-go model, which significantly improves the accuracy of two-way distance calculation in order to enhance focusing quality.In the stop-go model, as demonstrated in Figure 2, the relative trajectory is approximated as a straight line during the aperture time.Transmitting and receiving are assumed to occur at the same position, which indicates that the two-way distances for points A and B are the same, and the width of the transmitting pulse equals the width of the echo from one isolated point.In the stop-go model, as demonstrated in Figure 2, the relative trajectory is approximated as a straight line during the aperture time.Transmitting and receiving are assumed to occur at the same position, which indicates that the two-way distances for points A and B are the same, and the width of the transmitting pulse equals the width of the echo from one isolated point.

Continuous Tangent Model and Two-Way Distance
The two-way distance refers to the sum of propagation distances for electromagnetic signals from SAR to the target and back to SAR.It is applied in the imaging to implement range correction and phase compensation.The stop-go model is most commonly used for calculating the two-way distance [4,13].This section will construct a continuous tangent model, an update to the stop-go model, which significantly improves the accuracy of two-way distance calculation in order to enhance focusing quality.In the stop-go model, as demonstrated in Figure 2, the relative trajectory is approximated as a straight line during the aperture time.Transmitting and receiving are assumed to occur at the same position, which indicates that the two-way distances for points A and B are the same, and the width of the transmitting pulse equals the width of the echo from one isolated point.The continuous rectilinear model was first applied to airborne FMCW SAR imaging [16], while Liu et al. applied the model to SAR imaging [15].As demonstrated in Figure 3, the continuous rectilinear model assumes that SAR moves along a straight line without stopping.The model decomposes relative motion into two kinds.One is the relative motion between adjacent pulses, which indicates that the transmitting and corresponding receiving occur in different pulse repetition periods.The other is relative motion during one pulse, which implies that different points in the same pulse propagate along different transmitting and receiving paths, i.e., τ r ‰ τ s .

Continuous Tangent Model
Remote Sens. 2016, 8, 223 4 of 24 The continuous rectilinear model was first applied to airborne FMCW SAR imaging [16], while Liu et al. applied the model to SAR imaging [15].As demonstrated in Figure 3, the continuous rectilinear model assumes that SAR moves along a straight line without stopping.The model decomposes relative motion into two kinds.One is the relative motion between adjacent pulses, which indicates that the transmitting and corresponding receiving occur in different pulse repetition periods.The other is relative motion during one pulse, which implies that different points in the same pulse propagate along different transmitting and receiving paths, i.e., r s τ τ ≠ .The continuous rectilinear model does not account for the curvature of the orbit and the approximated trajectory is still a straight line [15].As the aperture time becomes longer, this approximation will induce increasingly unacceptable errors at the edge of the synthetic aperture.The proposed continuous tangent model is demonstrated in Figure 4.It includes the advantages of the continuous rectilinear model and furthermore approximates relative trajectories during every transmitting period and every receiving period as tangential segments, respectively.The two-way distance for every sampling point can then be calculated more accurately, especially at the edge of a synthetic aperture, because this novel motion model is much closer to the actual relative motion compared with existing models.
( ) where m V and r V are relative velocity vectors at m t and n t τ + Δ , respectively.The continuous rectilinear model does not account for the curvature of the orbit and the approximated trajectory is still a straight line [15].As the aperture time becomes longer, this approximation will induce increasingly unacceptable errors at the edge of the synthetic aperture.The proposed continuous tangent model is demonstrated in Figure 4.It includes the advantages of the continuous rectilinear model and furthermore approximates relative trajectories during every transmitting period and every receiving period as tangential segments, respectively.The two-way distance for every sampling point can then be calculated more accurately, especially at the edge of a synthetic aperture, because this novel motion model is much closer to the actual relative motion compared with existing models.The continuous rectilinear model was first applied to airborne FMCW SAR imaging [16], while Liu et al. applied the model to SAR imaging [15].As demonstrated in Figure 3, the continuous rectilinear model assumes that SAR moves along a straight line without stopping.The model decomposes relative motion into two kinds.One is the relative motion between adjacent pulses, which indicates that the transmitting and corresponding receiving occur in different pulse repetition periods.The other is relative motion during one pulse, which implies that different points in the same pulse propagate along different transmitting and receiving paths, i.e., r s τ τ ≠ .The continuous rectilinear model does not account for the curvature of the orbit and the approximated trajectory is still a straight line [15].As the aperture time becomes longer, this approximation will induce increasingly unacceptable errors at the edge of the synthetic aperture.The proposed continuous tangent model is demonstrated in Figure 4.It includes the advantages of the continuous rectilinear model and furthermore approximates relative trajectories during every transmitting period and every receiving period as tangential segments, respectively.The two-way distance for every sampling point can then be calculated more accurately, especially at the edge of a synthetic aperture, because this novel motion model is much closer to the actual relative motion compared with existing models.
( ) where m V and r V are relative velocity vectors at m t and n t τ + Δ , respectively.We denote distance vectors during the mth transmitting period and the nth receiving period as R s pt m , τq and R r pt n , τq, respectively.In the continuous tangent model, R s pt m , τq and R r pt n , ∆τ `τq can be expressed as where V m and V r are relative velocity vectors at t m and t n `∆τ, respectively.

Two-Way Distance
The two-way distance for point A in Figure 4 is where c is the speed of light.The difference between |R s pt m q| and |R r pt n , ∆τq| can be derived as where ∆V m and ∆r m are the transmitting-receiving rate and the transmitting-receiving constant, respectively.They can be expressed as where V a denotes the average relative velocity vector during the period rt 0 , t m´1 s.
The two-way distance for point B is where τ r and τ s satisfy (see Appendix B) and where θ r is the angle between V r and R r pt n , ∆τq and θ s is the angle between V m and R s pt m q.
Because the time difference between transmitting and receiving is very small, |V r | « |V m |, θ r « θ s , and k 1 « k 2 .The last term in Equation ( 7) can then be simplified as c ¨rτ r ´τs s " m t τ r (11) where |V m | and cosθ s vary with azimuth slow time and the looking angle.As demonstrated in Figure 5, for a fixed looking angle, |V m | cosθ s varies linearly with the azimuth time.Therefore, m t can be approximated as: where k m is the time-scaling factor.According to Equations ( 7), ( 11) and ( 13), the two-way distance ( ) R t τ for the point received at n t τ + can be expressed as where

Echo Expression of Spaceborne SAR Based on the Continuous Tangent Model
The SAR echo expression is the foundation for deriving an imaging algorithm.Because the amplitude weighting is usually not taken into account in the derivation of SAR imaging algorithms [16,18], the weighting factor of the antenna pattern is ignored.The echo from an isolated point can be expressed as  According to Equations ( 7), ( 11) and ( 13), the two-way distance R w pt n , τq for the point received at t n `τ can be expressed as where τ " τ `pn ´mq { f p .

Echo Expression of Spaceborne SAR Based on the Continuous Tangent Model
The SAR echo expression is the foundation for deriving an imaging algorithm.Because the amplitude weighting is usually not taken into account in the derivation of SAR imaging algorithms [16,18], the weighting factor of the antenna pattern is ignored.The echo from an isolated point can be expressed as where σ is the scattering cross-section of the target, a p τq is the rectangular amplitude modulation, R w pt i , τq is the two-way distance, and λ is the wavelength.In this paper, we use a chirp signal as the transmitting signal, i.e., ϕ p τq " ´πb τ2 , where b is the linear frequency modulation rate.
By substituting Equation ( 14) into Equation ( 15), SAR echo received at t m `τ can be expressed as where According to spaceborne SAR system parameters designed for 0.2-meter resolution, ϕ 1 pt m q and ϕ 2 `tm 2 ˘will not exceed 1 ˝, which hardly affects the focusing quality.Therefore, Equation ( 16) can be simplified as If constants are omitted, the Fourier transformation of Equation ( 19) along the range time can be expressed as where f τ denotes frequency in the range domain.
For comparison, echo expressions based on the stop-go model and the continuous rectilinear model are listed as Equations ( 21) [4] and (22) [15], respectively: where V denotes the flight speed.By comparing Equations ( 20)-( 22), differences between these three echo expressions can be summarized as follows: (1) Relative to Equation (21), Equation (20) has three new phase terms: the third, fourth, and fifth terms.If these terms are not compensated for, they will have varying effects on focusing quality.The third phase will lead to asymmetric distortion of side lobes.The fourth phase results in the offset of the Doppler centroid and causes further offset of imaging results along the azimuth.The fifth term represents the linear variation with f τ , which means that an offset of the range migration is produced.Therefore, the azimuthal matching filter cannot be designed according to the right range gate, which causes azimuthal defocusing.
(2) Compared with Equation ( 22), Equation ( 20) more fully emphasizes the space-variant properties of spaceborne SAR echoes by introducing three factors: ∆V m , ∆r m and k m .These factors are different for different positions in the swath, which can be observed from Equations ( 5), (6), and (13).Therefore, the new echo expression, Equation (20), can describe echoes much more precisely for a wider swath.This can also be proven by the simulation results in Section 5, which demonstrate that the imaging algorithm based on Equation (20) achieves better focusing quality than that based on Equation ( 22) at the azimuth edge of the swath, although both algorithms have nearly the same performance at the swath center.Since k m , ∆V m , and ∆r m are important, their space-variant properties and influence on focusing quality will be analyzed in the following.

Time-Scaling Factor
Suppose k m,p and k m,c denote time-scaling factors corresponding to a certain target P T in the swath and the swath center, respectively.When the echo from P T is processed, k m,p should be adopted during imaging.If k m,p is replaced by k m,c , according to Equation ( 20) the residual phase after compensation is where and f η denotes the azimuthal frequency.V p , ϕ p , and R p are the equivalent velocity, the equivalent squint angle, and the reference range corresponding to P T , respectively, and V c , ϕ c , and R c are those corresponding to the swath center.The equivalent velocity V e and the equivalent squint angle ϕ e can be calculated according to where f d is the Doppler centroid frequency, f r is the Doppler frequency rate, and R 0 is the reference range defined as the slant range from SAR to the target at the central moment of the aperture time.
By applying the parameters in Table 1, the residual phases at the swath edges are analyzed and shown in Figure 6, representing maximums of ∆ψ k m `fη , f τ ˘along range and azimuth, respectively.Figure 6a demonstrates that the residual phase is less than 0.1 ˝at the range edge, which is 7.5 km away from the swath center.Figure 6b shows that the residual phase is less than 1 ˝at the azimuth edge, which is also 7.5 km away from the swath center.Since both residual phases are far less than π/4 [19], the influence of the residual phases induced by the time-scaling factor on imaging performance are negligible, implying that it is not necessary to update the time-scaling factor and k m,c can be applied to imaging of the whole swath.
away from the swath center.Figure 6b shows that the residual phase is less than 1° at the azimuth edge, which is also 7.5 km away from the swath center.Since both residual phases are far less than π/4 [19], the influence of the residual phases induced by the time-scaling factor on imaging performance are negligible, implying that it is not necessary to update the time-scaling factor and  , which corresponds to the swath center, is applied to imaging for the whole swath, then the residual phase is

Transmitting-Receiving Rate
where is the transmitting-receiving rate corresponding to a certain position in the swath.
According to Table 1, Figure 7 gives the residual phases ( ) at the range edge and the azimuth edge, respectively, which are all 7.5 km away from the swath center.As demonstrated in Figure 7a, the residual phase is less than 1.5°, which will not affect the focusing quality at the range edge.Although the residual phase in Figure 7b reaches 110° at the azimuth edge, the variation is linear with the azimuth frequency, which means only an offset along azimuth will be produced.This offset is less than 0.14 m.The above analysis indicates that the residual phase caused by applying will be adopted in imaging as the transmitting-receiving rate for the whole swath.

Transmitting-Receiving Rate ∆V m
If the transmitting-receiving rate ∆V m,c , which corresponds to the swath center, is applied to imaging for the whole swath, then the residual phase is where ∆V m,p is the transmitting-receiving rate corresponding to a certain position in the swath.According to Table 1, Figure 7 gives the residual phases ∆ψ ∆V m `fη , f τ ˘at the range edge and the azimuth edge, respectively, which are all 7.5 km away from the swath center.As demonstrated in Figure 7a, the residual phase is less than 1.5 ˝, which will not affect the focusing quality at the range edge.Although the residual phase in Figure 7b reaches 110 ˝at the azimuth edge, the variation is linear with the azimuth frequency, which means only an offset along azimuth will be produced.This offset is less than 0.14 m.The above analysis indicates that the residual phase caused by applying ∆V m,c instead of ∆V m,p only leads to geometric distortion at the swath edge, and does not degrade the focusing quality.Therefore, ∆V m,c will be adopted in imaging as the transmitting-receiving rate for the whole swath.is less than two meters at the azimuth edge and the minimum is zero at the azimuth center.Therefore, according to Equation ( 20), an offset of the range migration will be r Δ

Transmitting-Receiving Constant ∆r m
According to Equation ( 6), the variation of ∆r m in the whole swath is demonstrated in Figure 8. ∆r m does not change along the range direction and varies linearly with the azimuth.The maximum of ∆r m is less than two meters at the azimuth edge and the minimum is zero at the azimuth center.Therefore, according to Equation ( 20), an offset of the range migration will be produced by ∆r m along the range direction and this offset varies with the azimuth., is adopted in imaging for the whole swath, the same azimuth matching filter will be applied to process echoes for Because of ∆r m , the azimuth matching filter designed for the range gate R p `∆r m should be applied to process echoes from targets like P T whose reference range equals R p .However, if the transmitting-receiving constant corresponding to the swath center, i.e., ∆r m " 0, is adopted in imaging for the whole swath, the same azimuth matching filter will be applied to process echoes for targets whose reference range equals R p `∆r m .This leads to a quadratic azimuthal phase error, which can be expressed as where ρ a denotes the azimuthal resolution.
According to Equation (29), the phase error ∆ψ ∆r m will lead to more azimuthal mismatching and defocusing as the target is much farther away from the swath center along azimuth, because ∆ψ ∆r m increases with ∆r m .Therefore, different ∆r m must be adopted for different echo segments along the azimuth.

Imaging Compensation Algorithm
As we all know, existing spaceborne SAR imaging processors are designed based on the stop-go motion model.With the construction of the continuous tangent motion model, we have updated the SAR echo expression, and analyzed the new characteristics of the updated echo expression.In order to satisfy the existing SAR processors, we compensated the different parts between the updated echo expression and that based on the stop-go model.According to the echo expression (Equation ( 19)) and the space-variant analysis of three key factors, k m , ∆V m , and ∆r m , a compensation algorithm with better focusing performance is proposed as follows.
(1) The first step is to divide echoes into several segments along azimuth.Every echo segment corresponds to a block of the swath.This is required because ∆r m is space-variant and imaging for the whole swath cannot be implemented once.Otherwise, defocusing will occur at the azimuth edge of the swath, which was analyzed in Section 3.
The ephemeris data that are collected by spaceborne instruments and transmitted to the ground station with SAR echoes contain information on satellite position vectors.By applying the ephemeris data, the relative distance vectors and relative velocity vectors can be acquired.The distribution of ∆r m across the whole swath can then be calculated according to Equation (6).It is applied to determine the segment strategy, which must guarantee that the quadratic phase error induced by ∆r m at the edge of every swath block satisfies the requirements of the focusing quality.
For the lth echo segment, the transmitting-receiving rate ∆V m,l and the transmitting-receiving constant ∆r m,l corresponding to the center of the lth swath block can be calculated according to Equations ( 5) and ( 6).The time-scaling factor k m,l can be estimated by a linear fitting of m t according to Equation ( 13).
(2) The azimuthal deramping [20,21] and Fourier transform along the range are then implemented on the lth echo segment successively to acquire S E pt i , f τ q, which has the same form as Equation ( 20).
(3) The first compensation factor ∆ 1 pt i , f τ q is designed as follows After multiplying S E pt i , f τ q by ∆ 1 pt i , f τ q, a Fourier transform along the azimuth is performed to acquire S F `fη , f τ ˘, which is (4) The second compensation factor ∆ 2 `fη , f τ ˘is designed as - After multiplying S F `fη , f τ ˘with ∆ 2 `fη , f τ ˘, an inverse Fourier transform along the azimuth equals (5) S K pt i , f τ q has the same form as Equation ( 21), which is the echo expression based on the stop-go model.Therefore, sliding spotlight imaging algorithms based on the stop-go model can be applied to process S K pt i , f τ q and achieve a high-resolution image for the lth echo segment.Here, the deramping ωk algorithm is adopted [13,18].
Figure 9 shows the compensation flow for every echo segment.After each echo segment has been imaged, results are stitched together along the azimuth to form a final complete image of the whole swath.

Simulation Method and Parameters
SAR echo simulation methods are typically based on the stop-go model.A new method is proposed here to simulate the continuous motion of the satellite.The key to this method is to acquire an accurate transmitting distance according to the receiving distance for every echo sampling point.
As demonstrated in Figure 1, the receiving moment i δ for the ith sampling point during , 1 where g τ is the sampling start time and s F is the sampling rate.
In order to calculate the transmitting moment corresponding to i δ , iterations are performed as demonstrated in Figure 10.Detailed steps are as follows.

Simulation Method and Parameters
SAR echo simulation methods are typically based on the stop-go model.A new method is proposed here to simulate the continuous motion of the satellite.The key to this method is to acquire an accurate transmitting distance according to the receiving distance for every echo sampling point.
As demonstrated in Figure 1, the receiving moment δ i for the ith sampling point during " t n , t n `1{ f p ‰ can be expressed as where τ g is the sampling start time and F s is the sampling rate.
In order to calculate the transmitting moment corresponding to δ i , iterations are performed as demonstrated in Figure 10.Detailed steps are as follows.(1) The initial estimation of the transmitting moment corresponding to i δ is assumed to be ( ) where ( ) i R δ is the relative distance between SAR and the target at the moment i δ .
(2) If Equation (36) can be satisfied, the transmitting distance is considered to be ( ) , and return to Step (2).
By performing the iteration method for every sampling point, the transmitting and receiving distances can be precisely calculated and substituted into Equation (15) to form the simulated echoes.
Simulation parameters are listed in Table 2.The simulated swath covering an area of 15 km × 15 km is demonstrated in Figure 11, where three point targets, P1, P2, and P3, are arranged at the center, the range edge, and the azimuth edge, respectively.(1) The initial estimation of the transmitting moment corresponding to δ i is assumed to be where R pδ i q is the relative distance between SAR and the target at the moment δ i .
´`δ i ´ζi,k ˘ˇˇˇˇă 10 ´15 (36) (4) Let k " k `1, and return to Step (2).By performing the iteration method for every sampling point, the transmitting and receiving distances can be precisely calculated and substituted into Equation (15) to form the simulated echoes.
Simulation parameters are listed in Table 2.The simulated swath covering an area of 15 km ˆ15 km is demonstrated in Figure 11, where three point targets, P 1 , P 2 , and P 3 , are arranged at the center, the range edge, and the azimuth edge, respectively.

Simulation Results
Figures 12-14 demonstrate the imaging results of P1, P2, and P3, where the profiles represent the focusing quality along the range and azimuth, and contour maps denote the two-dimensional focusing quality.In every figure, there are three kinds of results.The first result is achieved using the Deramping ωk algorithm, which is based on the stop-go model [18].The second is achieved according to the algorithm presented in [15], which is based on the continuous rectilinear model.The third is achieved using the novel imaging compensation method presented in this paper.

Simulation Results
Figures 12-14 demonstrate the imaging results of P 1 , P 2 , and P 3 , where the profiles represent the focusing quality along the range and azimuth, and contour maps denote the two-dimensional focusing quality.In every figure, there are three kinds of results.The first result is achieved using the Deramping ωk algorithm, which is based on the stop-go model [18].The second is achieved according to the algorithm presented in [15], which is based on the continuous rectilinear model.The third is achieved using the novel imaging compensation method presented in this paper.

Simulation Results
Figures 12-14 demonstrate the imaging results of P1, P2, and P3, where the profiles represent the focusing quality along the range and azimuth, and contour maps denote the two-dimensional focusing quality.In every figure, there are three kinds of results.The first result is achieved using the Deramping ωk algorithm, which is based on the stop-go model [18].The second is achieved according to the algorithm presented in [15], which is based on the continuous rectilinear model.The third is achieved using the novel imaging compensation method presented in this paper.By applying resolution, peak to side lobe ratio (PSLR), and integrated side lobe ratio (ISLR) as indicators to evaluate Figures 12-14, the evaluation results along the range and azimuth are presented in Tables 3 and 4, respectively.By applying resolution, peak to side lobe ratio (PSLR), and integrated side lobe ratio (ISLR) as indicators to evaluate Figures 12-14 the evaluation results along the range and azimuth are presented in Tables 3 and 4 respectively.

Figure 1 TFigure 1 .
Figure 1 demonstrates the actual relative motion between spaceborne SAR and the target.For convenience, time is divided into two parts: azimuth slow time i t and range fast time τ .Here, / i p t i f = ( 0,1, 2,3, i =

Figure 1 .
Figure 1.(a) The actual relative motion between spaceborne Synthetic Aperture Radar (SAR) and the target; (b) The corresponding time sequence of transmitting and receiving.

Figure 1 TFigure 1 .
Figure 1 demonstrates the actual relative motion between spaceborne SAR and the target.For convenience, time is divided into two parts: azimuth slow time i t and range fast time τ .Here, / i p t i f = ( 0,1, 2,3, i =

Figure 2 .
Figure 2. The stop-go model.Figure 2. The stop-go model.

Figure 2 .
Figure 2. The stop-go model.Figure 2. The stop-go model.

Figure 5 .
Figure 5. Variation of |V m | cosθ s with the azimuth time and the looking angle by applying parameters in Table 1.Three lines are presented under looking angles 15, 35 and 55 degrees, which represent the low, medium and high looking angles, respectively.For a looking angle, |V m | cosθ s varies linearly with the azimuth time.

Figure 6 .
Figure 6.Phase errors induced by the time-scaling factor m k .The distribution of the residual phases in the two-dimensional frequency domain at the range edge (a) and the azimuth edge (b).
geometric distortion at the swath edge, and does not degrade the focusing quality.Therefore,

Figure 6 .
Figure 6.Phase errors induced by the time-scaling factor k m .The distribution of the residual phases in the two-dimensional frequency domain at the range edge (a) and the azimuth edge (b).

Figure 7 .
Figure 7. Phase errors induced by the transmitting-receiving rate According to Equation (6), the variation of m r Δ in the whole swath is demonstrated in Figure8.m r Δ does not change along the range direction and varies linearly with the azimuth.The maximum of m r Δ

Figure 7 .
Figure 7. Phase errors induced by the transmitting-receiving rate ∆V m .The distribution of the residual phases in the two-dimensional frequency domain at the range edge (a) and the azimuth edge (b).

Figure 7 .
Figure 7. Phase errors induced by the transmitting-receiving rate swath is demonstrated in Figure8.m r Δ does not change along the range direction and varies linearly with the azimuth.The maximum of m r Δ is less than two meters at the azimuth edge and the minimum is zero at the azimuth center.Therefore, according to Equation (20), an offset of the range migration will be produced by m r Δ along the range direction and this offset varies with the azimuth.

Figure 8 .
Figure 8.The variation of

Figure 8 .
Figure 8.The variation of ∆r m in the whole swath.It is obvious that ∆r m does not change along the range direction, and varies linearly with the azimuth.The absolute maximum of ∆r m is less than two meters at the azimuth edge, and the minimum is zero at the swath center.

Figure 9 .
Figure 9.An imaging compensation algorithm based on the continuous tangent model.

Figure 9 .
Figure 9.An imaging compensation algorithm based on the continuous tangent model.

Figure 10 .
Figure10.A workflow diagram of the simulation method.The core of the flow chart is to accurately estimate the transmitting moment corresponding to every sampling moment within the synthetic aperture time.Afterwards, the transmitting and receiving distances can be calculated and substituted into the echo expression to achieve simulated echoes.

Figure 10 .
Figure10.A workflow diagram of the simulation method.The core of the flow chart is to accurately estimate the transmitting moment corresponding to every sampling moment within the synthetic aperture time.Afterwards, the transmitting and receiving distances can be calculated and substituted into the echo expression to achieve simulated echoes.

Figure 11 .
Figure 11.The simulated swath: P1 is at the swath center.P2 and P3 are at the range edge and the azimuth edge, respectively, which are all 7.5 km away from P1.

Figure 11 .
Figure 11.The simulated swath: P 1 is at the swath center.P 2 and P 3 are at the range edge and the azimuth edge, respectively, which are all 7.5 km away from P 1 .

Figure 11 .
Figure 11.The simulated swath: P1 is at the swath center.P2 and P3 are at the range edge and the azimuth edge, respectively, which are all 7.5 km away from P1.

Figure 12 .Figure 12 .
Figure 12.Imaging results of P1, the point at the swath center.(a-c) The range profile, the azimuth profile and the contour map achieved using the Deramping ωk algorithm based on the stop-go model;(d-f) The range profile, the azimuth profile and the contour map achieved using the algorithm based on the continuous rectilinear model; (g-i) The range profile, the azimuth profile and the contour map achieved using the novel imaging compensation method presented in this paper.

Figure 12 .
Figure 12.Imaging results of P 1 , the point at the swath center.(a-c) The range profile, the azimuth profile and the contour map achieved using the Deramping ωk algorithm based on the stop-go model;(d-f) The range profile, the azimuth profile and the contour map achieved using the algorithm based on the continuous rectilinear model; (g-i) The range profile, the azimuth profile and the contour map achieved using the novel imaging compensation method presented in this paper.

Figure 13 . 12 .Figure 13 .
Figure 13.Imaging results of P 2 , the point at the range edge.Images (a-i) are in accordance with Figure 12.

Figure 14 .
Figure 14.Imaging results of P3, the point at the azimuth edge.Images (a-i) are in accordance with Figure 12.

Figure 14 .
Figure 14.Imaging results of P 3 , the point at the azimuth edge.Images (a-i) are in accordance with Figure 12.
. Three lines are presented under looking angles 15, 35 and 55 degrees, which represent the low, medium and high looking angles, respectively.For a looking angle, |V m | cosθ s varies linearly with the azimuth time.

Table 3 .
Evaluation of imaging quality along range.

Table 4 .
Evaluation of imaging quality along azimuth.

Table 3 .
Evaluation of imaging quality along range.

Table 4 .
Evaluation of imaging quality along azimuth.