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

Transcription

1 remote sensing Artile An Imaging Compensation Algorithm for Spaeborne High-Resolution SAR Based on a Continuous Tangent Motion Model Ze Yu 1,, Shusen Wang 1, and Zhou Li 2, *, 1 Shool of Eletronis and Information Engineering, BeiHang University, Beijing 1191, China; yz613@buaa.edu.n (Z.Y.); shusenwang@buaa.edu.n (S.W.) 2 Beijing Institute of Remote Sensing Information, Beijing 1192, China * Correspondene: lizhou1321@126.om; Tel.: These authors ontributed equally to this work. Aademi Editors: Josef Kellndorfer and Prasad S. Thenkabail Reeived: 25 November 215; Aepted: 4 Marh 216; Published: 1 Marh 216 Abstrat: This paper develops a ontinuous tangent motion model by approximating relative trajetories between spaeborne Synti Aperture Radar (SAR) and its targets during every transmitting period and every reeiving period as tangential segments. Compared with existing motion models, inluding stop-go model and ontinuous retilinear model, ontinuous tangent model is muh loser to atual relative motion. Based on new motion model, an imaging ompensation algorithm is proposed that onsiders spae-variant property of SAR ehoes more fully by introduing three spae-variant fators: time-saling fator, transmitting reeiving rate, and transmitting reeiving onstant. As a result, spaeborne SAR imaging with 1-meter resolution and wide-swath, up to 15 km ˆ 15 km, is ahieved. Simulation results indiate that novel algorithm provides a superior and more onsistent fousing quality aross whole swath ompared with existing algorithms based on stop-go or ontinuous retilinear models. Keywords: high-resolution imaging; motion model; phase ompensation; synti aperture radar 1. Introdution Motion ompensation is key to Synti Aperture Radar (SAR) or Inverse Synti Aperture Radar (ISAR) imaging [1,2]. The motion model, whih desribes relative motion between SAR and its targets, i.e., anything illuminated by antenna mainlobe, plays a ruial role in SAR imaging. Preise alulation of relative distanes aording to motion model is basis of good fousing quality. If motion model is not suffiiently aurate, performane of range migration orretion and phase ompensation based on relative distanes will diminish, and fousing quality will be affeted. The aim of this paper is to onstrut a novel motion model for spaeborne SAR and develop a ompensation algorithm, based on motion model, for use in high-resolution and wide-swath imaging. Satellites have advantages for SAR observation ompared with or platforms [3]. The relative motion of spaeborne SAR is ompliated and annot be expressed expliitly beause SAR travels along an elliptial orbit while targets rotate with earth. In this ase, an approximate motion model has to be adopted to derive orresponding algorithm for effiient and preise imaging. The stop-go model is most ommonly applied motion model for spaeborne SAR [4,5]. This model inludes two assumptions. The first is that relative trajetory during aperture time is approximated as a straight line, even if it is stritly urved. The seond is that SAR is assumed to remain stationary at a single position while transmitting one pulse and reeiving orresponding eho before moving Remote Sens. 216, 8, 223; doi:1.339/rs

2 Remote Sens. 216, 8, of 23 to next position. If SAR satellite is in an orbit with a height of 6 km, it atually takes more than 4 ˆ 1 3 s for a pulse to omplete a round trip. During this amount of time, SAR satellite travels at least 3 meters. For spaeborne SAR, stop-go model is valid when aperture time is short enough for differene between atual and approximated trajetories to be negligible, and signal band is small enough for distane error indued by atual relative motion during transmitting and reeiving to be less than one range resolution ell [5]. As is well known, aperture time and signal band are key fators in azimuth and range resolutions, respetively. Therefore, appliation of stop-go model is limited by resolution. Imaging algorithms based on stop-go model, suh as Range-Doppler (RD) [6,7] and Chirp-Saling (CS) [8,9], exhibit good performane only when resolution is worse than a threshold whih is about.3 meters for X-band. Currently, majority of spaeborne SARs operate below this threshold. However, re are times when required resolution is better than this threshold, suh as TerraSAR-X NG, whih will produe 5-meter resolution images beyond year 225 [1]. Better performane of target detetion and reognition an be ahieved based on images with better resolution [11,12]. When required resolution is higher, validity of stop-go model is redued. Prats-Iraola et al. pointed out that in order to aquire 1-meter resolution images, urvature of orbit and relative motion between transmitting and reeiving should be taken into aount [13]. Orwise, residual phase indued by stop-go model is intolerable and quality inevitably dereases. For urvature-indued imaging degradation, Prats-Iraola et al. applied range history of a referene target in swath enter to ompensate for eho phase to be pure hyperboli [13]. Conventional imaging algorithms [6 9] based on retilinear approximation were n adapted to proess ompensated ehoes. However, beause range histories orresponding to enter and or positions of swath are different, ompensation effet gradually delines as swath beomes wider. As a result, this method annot be applied to imaging of large swaths. The swath width of TerraSAR-X Staring Spotlight (ST) mode, whih has adopted this method, is less than 5.3 km [13,14]. The stop-go model was also investigated by Liu et al. who reonstruted eho model by taking relative motion during transmitting and reeiving into aount. A orresponding imaging algorithm was n produed to obtain a high fousing quality with a -m resolution and an 8 km (slant range) ˆ 4 km (azimuth) swath width [15]. However, appliability of Liu s algorithm to spaeborne SAR was lessened by a retilinear motion assumption made during derivation of algorithm. Prats-Iraola also studied this problem and analyzed mismathing phenomenon that ourred during filtering along range by referring to frequeny-modulated ontinuous-wave (FMCW) systems [16,17]. A orretion method was presented in two-dimensional frequeny domain [13]. This method is not suitable for wide-swath imaging beause spae variane is not onsidered in orretion fator. This paper will present a ompensation algorithm to ahieve high-resolution SAR imaging with a wide swath, up to 15 km, by onstruting a ontinuous tangent motion model. This novel motion model assumes that SAR moves without stopping and approximates relative trajetories during every transmitting and reeiving period as tangent lines. This assumption fully reflets properties of relative motion. The paper is strutured as follows. Setion 2 onstruts ontinuous tangent model and derives two-way distane orresponding to every sampling point for spaeborne SAR. Setion 3 presents a new eho expression that onsiders spae-variant property of SAR ehoes more fully than or existing eho expressions. A novel imaging ompensation algorithm is presented in Setion 4 and simulated results are ompared with algorithms based on existing motion models in Setion 5. Setion 6 onludes paper.

3 Remote Sens. 216, 8, of 24 Remote Sens. 216, 8, of 24 Remote Sens. 216, 8, of Continuous Tangent Model and Two-Way Distane 2. Continuous Tangent Model and Two-Way Distane The two-way distane refers to sum of propagation distanes for eletromagneti signals 2. Continuous The two-way Tangent distane Model refers andto Two-Way sum Distane of propagation distanes for eletromagneti signals from SAR to target and bak to SAR. It is applied in imaging to implement range orretion from SAR to target and bak to SAR. It is applied in imaging to implement range orretion and The phase two-way ompensation. distanethe refers stop-go model sum of is propagation most ommonly distanes used for eletromagneti alulating two-way and phase ompensation. The stop-go model is most ommonly used for alulating two-way signals from distane SAR[4,13]. to This target setion and bak will toonstrut SAR. It is applied ontinuous tangent imaging model, to implement an update range to orretion stop-go distane [4,13]. This setion will onstrut a ontinuous tangent model, an update to stop-go and model, phase whih ompensation. signifiantly The improves stop-go model auray is mostof ommonly two-way used distane for alulating alulation in two-way order to model, whih signifiantly improves auray of two-way distane alulation in order to distane enhane [4,13]. fousing This quality. enhane fousing quality. setion will onstrut a ontinuous tangent model, an update to stop-go model, whih signifiantly improves auray of two-way distane alulation in order to enhane fousing 2.1. Continuous quality. Tangent Model 2.1. Continuous Tangent Model Figure demonstrates atual relative motion between spaeborne SAR and target. For 2.1. Continuous Figure 1 demonstrates Tangent Model atual relative motion between spaeborne SAR and target. onveniene, time is divided into two parts: azimuth slow time t i and range fast time τ For. Here, onveniene, Figure 1 time demonstrates is divided into atual two parts: relative azimuth motionslow between time t t spaeborne SAR and target. i = i / fp ( i =,1,2,3, f i and range fast time τ. Here, ) and p For ti = onveniene, i / fp is pulse repetition frequeny (PRF). As result, ontinuous ( i =,1,2,3, time is divided f ) and p into is two pulse parts: repetition azimuth frequeny slow time (PRF). t i and As range a result, fast time ontinuous τ. Here, ttime i i{ f t p (i is expressed, 1, 2, 3, as ) and t= ti f+ p τis. The pulse mth pulse repetition is transmitted frequeny during (PRF). As tm a result, T/2, tm + T ontinuous /2, where time time denotes t t is expressed is expressed pulse as as t= ti + τ width. t t i `The τ. The. The enter mth mth of pulse pulse transmitting is is transmitted transmitted waveform during during tm T/2, tm+ T/2 rt is m marked T{2, t m as `A, T{2s, whih where, where leaves T T denotes denotes pulse pulse width. width. The The enter enter of of transmitting transmitting waveform waveform is marked is marked as A, as whih A, whih leaves leaves radar at t radar at t m, reahes m, reahes target, and is refleted to reeiver at t target, and is refleted to reeiver at t n ` τ. Similarly, n +Δ τ. Similarly, anor radar at t m, reahes target, and is refleted to reeiver at t anor point B is transmitted point is transmitted at t m ` τ s and at t m + τ s reeivedand at t n reeived ` τ ` τat t r. n +Δ τ + τ n +Δ τ. Similarly, anor r. point B is transmitted at t m + τ s and reeived at t n +Δ τ + τ r. (a) (b) (a) (b) Figure Figure 1. (a) The atual relative motion between spaeborne Synti Aperture Radar (SAR) and Figure (a) (a) The The atual atual relative relative motion motion between between spaeborne spaeborne Synti Synti Aperture Aperture Radar Radar (SAR) (SAR) and and target; target; (b) The orresponding time sequene of transmitting and reeiving. target; (b) (b) The The orresponding orresponding time time sequene sequene of of transmitting transmitting and and reeiving. reeiving. In stop-go model, as demonstrated in Figure 2, relative trajetory is approximated as In stop-go model, as demonstrated in Figure 2, relative trajetory is approximated as a straight line during aperture time. Transmitting and reeiving are assumed to our at same straight line during aperture time. Transmitting and reeiving are assumed to our at same position, whih indiates that two-way distanes for points and are same, and width position, whih indiates that two-way distanes for for points A A and and B are B are same, same, and and width width of of transmitting pulse equals width of eho from one isolated point. of transmitting pulse pulse equals equals width width of of eho eho from from one one isolated isolated point. point. Figure 2. The stop-go model. Figure The stop-go model.

4 Remote Sens. 216, 8, of of Remote Sens. 216, 8, of 24 The ontinuous retilinear model was first applied to airborne FMCW SAR imaging [16], while The The ontinuous retilinear model model was was first first applied applied to to airborne FMCW FMCW SAR SAR imaging [16], [16], while while Liu et al. applied model to SAR imaging [15]. As demonstrated in Figure 3, ontinuous Liu Liu et et al. al. applied applied model model to SAR to SAR imaging imaging [15]. As [15]. demonstrated As demonstrated Figurein 3, Figure ontinuous 3, ontinuous retilinear model assumes that SAR moves along a straight line without stopping. The retilinear model model retilinear assumes model thatassumes SAR moves that along SAR a moves straight along line without a straight stopping. line The without model stopping. deomposes The relative model deomposes relative motion into two kinds. One is relative motion between adjaent pulses, motion deomposes into two relative kinds. motion One isinto relative two kinds. motion One between is relative adjaent motion pulses, between whih indiates adjaent that pulses, whih indiates that transmitting and orresponding reeiving our in different pulse transmitting whih indiates and orresponding that transmitting reeiving our and inorresponding different pulse repetition reeiving periods. our The in different or is relative pulse repetition periods. The or is relative motion during one pulse, whih implies that different motion repetition during periods. one pulse, The whih or implies is relative that motion differentduring points inone pulse, same pulse whih propagate implies along that different points in same pulse propagate along different transmitting and reeiving paths, i.e., τ r τ s transmitting points in and same reeiving pulse propagate paths, i.e., along τ r τdifferent s. transmitting and reeiving paths, i.e., τ r τ. s. Figure 3. The ontinuous retilinear model. Figure 3. The ontinuous retilinear model. The ontinuous retilinear model does not aount for urvature of orbit and The ontinuous retilinear model does not aount for urvature of orbit and approximated The ontinuous trajetory retilinear is still model a straight does not aount line [15]. foras urvature aperture of time orbitbeomes and approximated longer, this approximated trajetory is still a straight line [15]. As aperture time beomes longer, this trajetory approximation is still awill straight indue line inreasingly [15]. As aperture unaeptable time beomes errors at longer, edge this of approximation synti will aperture. indue approximation will indue inreasingly unaeptable errors at edge of synti aperture. inreasingly The proposed unaeptable ontinuous errors tangent at model edgeis ofdemonstrated synti aperture. in Figure The 4. It proposed inludes ontinuous advantages tangent of The proposed ontinuous tangent model is demonstrated in Figure 4. It inludes advantages of model ontinuous is demonstrated retilinear in Figure model 4. and It inludes furrmore advantages approximates of relative ontinuous trajetories retilinear during model every and ontinuous retilinear model and furrmore approximates relative trajetories during every furrmore transmitting approximates period and every relative reeiving trajetories period during as tangential every transmitting segments, period respetively. and every The reeiving two-way transmitting period and every reeiving period as tangential segments, respetively. The two-way period distane asfor tangential every sampling segments, point respetively. an n The be alulated two-way distane more aurately, for every sampling espeially point at an edge nof be a distane for every sampling point an n be alulated more aurately, espeially at edge of a alulated synti aperture, more aurately, beause espeially this novel at motion edge of model a synti is muh aperture, loser beause to this atual novel relative motion motion model synti aperture, beause this novel motion model is muh loser to atual relative motion is ompared muh loser with toexisting atual models. relative motion ompared with existing models. ompared with existing models. Figure 4. The ontinuous tangent model. Figure 4. The ontinuous tangent model. We denote distane vetors during mth transmitting period and nth reeiving period as We denote distane vetors during mth transmitting period and nth reeiving period as s( t m, τ ) t, τ ) and R r( t n, τ ) and R t τ ), respetively. In ontinuous tangent model, R s( t m, τ ) R s pt m, τq and R r pt n, τq, respetively., respetively. In ontinuous In ontinuous tangent model, tangent R s pt m, τq model, and R r pt R n s, ( τ t `τ ) τqand and an be R r R expressed ( t n, Δ τ + τ) r( t Δ τ + τ) as an be expressed as an be expressed as R s pt m, τq «R s pt m q ` V m τ τ P T ( ) T τ, Rs tm, τ T, T T τ R s tm, τ Rs( tm) + Vmτ 2, T j (1) 2 Rs( tm) + Vmτ 2 2 (1) 2 2 (1) T R r pt n, τ ` τq «R r pt n, τq ` V r τ τ P T2 τ, 1 τ T j (2) 1f T τ T Δτ, p 2 1 Δτ T (2) Rr( tn, Δ τ + τ) Rr( tn, Δ τ) + Vrτ τ 2 Δτ, f p Δτ 2 where V (2) Rr tn, Δ τ + τ Rr( tn, Δ τ) + Vrτ 2 f p 2 m and V r are relative veloity vetors at t m and t n ` τ, respetively. where Vm where V and Vr m and V are relative veloity vetors at t m r are relative veloity vetors at t and t m and t n +Δ τ n +Δ τ, respetively., respetively.

5 Remote Sens. 216, 8, of Two-Way Distane The two-way distane for point A in Figure 4 is R A pt m ; t n ` τq R s pt m q ` R r pt n, τq pn mq { f p ` τ (3) where is speed of light. The differene between R s pt m q and R r pt n, τq an be derived as (see Appendix A) R r pt n, τq R s pt m q 2 V m t m ` 2 r m (4) where V m and r m are transmitting reeiving rate and transmitting reeiving onstant, respetively. They an be expressed as V m V a V m r m R spt q V m where V a denotes average relative veloity vetor during period rt, t m 1 s. The two-way distane for point B is R B pt m ` τ s ; t n ` τ ` τ r q R s pt m, τ s q ` R r pt n, τ ` τ r q R A pt m q ` rτ r τ s s 2 R s pt m q ` 2 V m t m ` 2 r m ` rτ r τ s s (5) (6) (7) where τ r and τ s satisfy (see Appendix B) τ s p k 1q p ` k 2 q τ r (8) and k 1 V r R r pt n, τq{ R r pt n, τq V r osθ r (9) k 2 V m R s pt m q{ R s pt m q V m osθ s (1) where θ r is angle between V r and R r pt n, τq and θ s is angle between V m and R s pt m q. Beause time differene between transmitting and reeiving is very small, V r «V m, θ r «θ s, and k 1 «k 2. The last term in Equation (7) an n be simplified as rτ r τ s s m t τ r (11) where m t pk 1 ` k 2 q{ p ` k 2 q «2 V m osθ s (12) V m and osθ s vary with azimuth slow time and looking angle. As demonstrated in Figure 5, for a fixed looking angle, V m osθ s varies linearly with azimuth time. Therefore, m t an be approximated as: m t 2k m t m (13) where k m is time-saling fator.

6 Remote Sens. 216, 8, of 23 Remote Sens. 216, 8, of 24 Figure Variation of of V m V osθ m os s with θ s with azimuth azimuth time and time and looking angle looking by applying angle by parameters applying in Table 1. Three lines are presented under looking angles 15, 35 and 55 degrees, whih represent parameters in Table 1. Three lines are presented under looking angles 15, 35 and 55 degrees, whih low, medium and high looking angles, respetively. For a looking angle, V m osθ s varies linearly with represent low, medium and high looking angles, respetively. For a looking angle, V m osθ s azimuth time. varies linearly with azimuth time. Table 1. Simulation parameters. Table 1. Simulation parameters. Parameters Value Value Orbital inlination Orbital height 68 km Orbital height 68 km Eentriity Elevation Eentriity angle 35.1 Elevation Squint angle BeamSquint rotation angle veloity 2±6.21 /s Beam rotation Bandwidth veloity 1.2 GHz /s Wavelength.3 m Bandwidth 1. GHz Wavelength.3 m Aording to Equations (7), (11) and (13), two-way distane R w pt n, τq R for w( t ) Aording to Equations (7), (11) and (13), two-way distane n, τ point reeived at t n ` τ an be expressed as for point reeived at t n + τ an be expressed as R w pt n, τq Rw( tn, τ 2 R s pt m q ` 2 V ) m t m ` 2 r m ` 2k m t m τ ` pn mq { fp r2 R s pt m q ` 2 V m t m ` 2 r m s { ( = 2 Rs( tm) + 2Δ Vmtm + 2Δ rm + 2 kmtm{ τ + ( n m) / fp 2 Rs( tm) + 2Δ Vmtm + 2Δrm «2 R s pt m q ` 2 V m t m ` 2 r m ` 2k m t m ˆτ 2 R j (14) s pt m q } (14) 2 R s( t m) 2 R s( tm) + 2Δ Vmtm + 2Δ rm + 2kmtm τ where ˆτ τ ` pn mq { f p. τ τ + ( ) where 3. Eho = Expression n m / of fp Spaeborne SAR Based on Continuous Tangent Model. The SAR eho expression is foundation for deriving an imaging algorithm. Beause 3. amplitude Eho Expression weightingof is Spaeborne usually not taken SAR into Based aount on in Continuous derivation Tangent of SAR imaging Model algorithms [16,18], weighting fator of antenna pattern is ignored. The eho from an isolated point an be expressed as The SAR eho expression is foundation for deriving an imaging algorithm. Beause amplitude weighting is usually not taken j into " aount in derivation j* of SAR imaging algorithms [16,18], Sweighting pt i, ˆτq σ fator a ˆτ Rw pt i, ˆτq of antenna exp pattern jϕ ˆτ is Rw pt i, ˆτq ignored. The eho exp from j 2π j λan R wisolated pt i, ˆτq point (15) an be expressed as where σ is sattering ross-setion of target, a p ˆτq is retangular amplitude modulation, Rw( ti, τ ) Rw( ti, τ R w pt i, ˆτq is two-way ) 2π St ( i, distane, τ) = σ a and λ is τ wavelength. exp jϕ In this τ paper, we use a exp j Rw( ti, hirp signal as τ transmitting signal, i.e., ϕ p ˆτq πb ) ˆτ 2, where λ (15) b is linear frequeny modulation rate.

7 Remote Sens. 216, 8, of 23 where By substituting Equation (14) into Equation (15), SAR eho reeived at t m ` ˆτ an be expressed as S pt m, ˆτq ˆ σ a 1 2k mt m # exp jπb exp " j 4π λ ϕ 1 pt m q 4πbk mt m 2 ˆ ˆτ 2 R s pt m q ˆ 1 2k mt m ˆ ϕ 2 t m 2 4πbk mt m 2 2 ˆτ 2 R s pt m q 2 j p V mt m ` r m q 2 p V mt m ` r m q j 2 ` jϕ 1 pt m q ` jϕ 2 `tm 2 + R s pt m q ` k m ˆτt m ` V m t m ` r m j* ˆτ 2 R ˆ s pt m q ˆτ 2 R j s pt m q 2 r m ˆ ˆτ 2 R ˆ s pt m q k m ˆτ 2 R j s pt m q ` 2 V m Aording to spaeborne SAR system parameters designed for -meter resolution, ϕ 1 pt m q and ϕ 2 `tm 2 will not exeed 1, whih hardly affets fousing quality. Therefore, Equation (16) an be simplified as S pt m, ˆτq ˆ «σ a 1 2k mt m # exp jπb exp " j 4π λ ˆ ˆτ 2 R s pt m q ˆ 1 2k mt m ˆτ 2 R s pt m q 2 j p V mt m ` r m q 2 p V mt m ` r m q j 2 + R s pt m q ` k m ˆτt m ` V m t m ` r m j* If onstants are omitted, Fourier transformation of Equation (19) along range time an be expressed as S pt m, f τ q ˆ 1 exp j4π λ ` fτ exp jπ f τ 2 exp exp b j4π ˆ 1 λ ` fτ j R s pt m q ˆ4km t m f τ jπ ` 4k2 mt 2 j m λb λ 2 b j V m t m exp j4π ˆ j fτ r m where f τ denotes frequeny in range domain. For omparison, eho expressions based on stop-go model and ontinuous retilinear model are listed as Equations (21) [4] and (22) [15], respetively: S a pt i, f τ q ˆ j 1 S s pt i, f τ q exp j4π λ ` fτ R s pt m q exp jπ f τ 2 b exp «exp j4π jπ ˆ j 1 λ ` fτ R s pt m q exp 4V 2 t m f τ λb R s pt m q ` 4V 4 t 2 m λ 2 b R s pt m q 2 jπ f τ 2 b ff exp (16) (17) (18) (19) (2) (21) j (22) j 4πV2 λ t m where V denotes flight speed. By omparing Equations (2) (22), differenes between se three eho expressions an be summarized as follows:

8 Remote Sens. 216, 8, of 23 (1) Relative to Equation (21), Equation (2) has three new phase terms: third, fourth, and fifth terms. If se terms are not ompensated for, y will have varying effets on fousing quality. The third phase will lead to asymmetri distortion of side lobes. The fourth phase results in offset of Doppler entroid and auses furr offset of imaging results along azimuth. The fifth term represents linear variation with f τ, whih means that an offset of range migration is produed. Therefore, azimuthal mathing filter annot be designed aording to right range gate, whih auses azimuthal defousing. (2) Compared with Equation (22), Equation (2) more fully emphasizes spae-variant properties of spaeborne SAR ehoes by introduing three fators: V m, r m and k m. These fators are different for different positions in swath, whih an be observed from Equations (5), (6), and (13). Therefore, new eho expression, Equation (2), an desribe ehoes muh more preisely for a wider swath. This an also be proven by simulation results in Setion 5, whih demonstrate that imaging algorithm based on Equation (2) ahieves better fousing quality than that based on Equation (22) at azimuth edge of swath, although both algorithms have nearly same performane at swath enter. Sine k m, V m, and r m are important, ir spae-variant properties and influene on fousing quality will be analyzed in following Time-Saling Fator Suppose k m,p and k m, denote time-saling fators orresponding to a ertain target P T in swath and swath enter, respetively. When eho from P T is proessed, k m,p should be adopted during imaging. If k m,p is replaed by k m,, aording to Equation (2) residual phase after ompensation is ψ km ` fη, f τ ˆ4km,p π λb ˆ f τ t k,p ` km,p λ t k,p 2 ˆ4km, π λb ˆ f τ t k, ` km, λ t k, 2 (23) where t k,p «R p λr p f η sinϕ p osϕ p b V p V p 4V p2 λ 2 2 f η t k, «R λr f η sinϕ osϕ b V V 4V 2 λ 2 2 f η (24) (25) and f η denotes azimuthal frequeny. V p, ϕ p, and R p are equivalent veloity, equivalent squint angle, and referene range orresponding to P T, respetively, and V, ϕ, and R are those orresponding to swath enter. The equivalent veloity V e and equivalent squint angle ϕ e an be alulated aording to d ˆ λr f r λ 2 fd V e ` (26) 2 ϕ e aros 2 ˆ λ f d 2V e where f d is Doppler entroid frequeny, f r is Doppler frequeny rate, and R is referene range defined as slant range from SAR to target at entral moment of aperture time. By applying parameters in Table 1, residual phases at swath edges are analyzed and shown in Figure 6, representing maximums of ψ km ` fη, f τ along range and azimuth, respetively. Figure 6a demonstrates that residual phase is less than.1 at range edge, whih is 7.5 km away from swath enter. Figure 6b shows that residual phase is less than 1 at azimuth edge, whih is also 7.5 km away from swath enter. Sine both residual phases are far less than π/4 [19], influene of residual phases indued by time-saling fator on imaging performane are negligible, implying that it is not neessary to update time-saling fator and k m, an be applied to imaging of whole swath. (27)

9 away from swath enter. Figure 6b shows that residual phase is less than 1 at azimuth edge, whih is also 7.5 km away from swath enter. Sine both residual phases are far less than π/4 [19], influene of residual phases indued by time-saling fator on imaging performane are negligible, implying that it is not neessary to update time-saling fator and k Remote m, Sens. 216, 8, of 23 an be applied to imaging of whole swath. (a) (b) Figure Figure Phase Phase errors errors indued indued by by time-saling time-saling fator fator k k m. The distribution of residual phases m. The distribution of residual phases in in two-dimensional two-dimensional frequeny frequeny domain domain at at range range edge edge (a) and (a) and azimuth azimuth edge edge (b). (b) Transmitting Reeiving Transmitting Reeiving Rate Rate V ΔVm m ΔV m If If transmitting reeiving rate V m,, whih, orresponds to to swath enter, is is applied to to imaging for for whole swath, n n residual phase is is ˆ ` 1 fτ Δ ψ ΔV ( f, f ) 4 m η τ = π 1 ψ Vm fη, f τ 4π + p( ΔVm, p ΔVm λ ) λ ` fτ t k,p ` Vm,p V m, (28) ΔV mp where, where V m,p is is transmitting reeiving rate rate orresponding to ato ertain a ertain position position in in swath. ` swath. Aording to Table 1, Figure 7 gives residual phases ψ Δψ Vm V ( fη f,, f f m η τ ) Aording to Table 1, Figure 7 gives residual phases Δ τ at range edge and azimuth edge, respetively, whih are all 7.5 km away from swath enter. at As range demonstrated edge and in Figure azimuth 7a, edge, residual respetively, phase is whih less than are all 1.5, 7.5 km whih away will from not affet swath enter. fousing As quality demonstrated in Figure edge. 7a, Although residual residual phase phase is less inthan Figure 1.5, 7bwhih reahes will 11 not ataffet azimuth fousing edge, quality variation at isrange linearedge. with Although azimuth frequeny, residual whih phase means in Figure only7b an offset reahes along 11 azimuth at azimuth will be produed. edge, This variation offset is is less linear thanwith.14 m. azimuth The above frequeny, analysis indiates whih means that only residual an offset phase along aused azimuth by applying will be V produed. This offset is less than.14 m. The above analysis indiates that residual phase m, instead of V m,p only leads to geometri distortion at swath edge, and does not degrade fousing ΔV aused quality. by applying Therefore, m, V ΔV, instead of mp m, will be adoptedonly in imaging leads to asgeometri transmitting reeiving distortion at rate swath foredge, Remote Sens. 216, 8, of 24 whole swath. ΔV and does not degrade fousing quality. Therefore, m, will be adopted in imaging as transmitting reeiving rate for whole swath. (a) (b) Figure 7. Phase errors indued by transmitting reeiving rate Δ Vm. The distribution of Figure 7. Phase errors indued by transmitting reeiving rate V m. The distribution of residual phases residual inphases two-dimensional in two-dimensional frequeny domain frequeny at domain rangeat edge (a) range andedge azimuth (a) and edge azimuth (b). edge (b) Transmitting Reeiving Constant Δrm Aording to Equation (6), variation of Δrm in whole swath is demonstrated in Figure 8. Δr m does not hange along range diretion and varies linearly with azimuth. The maximum of Δrm is less than two meters at azimuth edge and minimum is zero at azimuth enter. Therefore, aording to Equation (2), an offset of range migration will be Δr

10 (a) (b) Figure 7. Phase errors indued by transmitting reeiving rate Δ Vm. The distribution of residual phases in two-dimensional frequeny domain at range edge (a) and azimuth edge (b). Remote Sens. 216, 8, of 23 Δr 3.3. Transmitting Reeiving Constant m 3.3. Transmitting Reeiving Aording to Equation Constant (6), variation r m of Δrm in whole swath is demonstrated in Figure 8. Δr m Aording does not tohange Equation along (6), variation range diretion of r m inand whole varies swath linearly is demonstrated with azimuth. in Figure The 8. r maximum m does not of hange Δrm is along less than range two meters diretion at and varies azimuth linearly edge with and azimuth. minimum The is zero maximum at of azimuth r m is less enter. thantherefore, two meters aording at azimuth to Equation edge and (2), an minimum offset of is zero range at migration azimuthwill enter. be Therefore, produed aording by Δrm along to Equation range (2), diretion an offset and of this range offset migration varies with will be azimuth. produed by r m along range diretion and this offset varies with azimuth. Figure The variation of of rδ m rmin in whole whole swath. swath. It is It obvious is that that r m Δ does rm does not hange not hange along along range range diretion, diretion, and varies and varies linearly linearly with with azimuth. azimuth. The absolute The absolute maximum maximum of r m isof less Δ rthan m is two less meters than two at meters azimuth at edge, azimuth and edge, minimum and minimum is zero at is zero swath at enter. swath enter. Beause Beause of of r Δr m,, azimuth p +Δr azimuth mathing mathing filter filter designed designed for for range range gate gate R p ` r m should should be be applied to proess ehoes from targets like P T whose referene range equals R p. However, if p applied to proess ehoes from targets like PT transmitting reeiving onstant orresponding towhose swath referene enter, i.e., range r m equals, is adopted. However, in imaging if for transmitting reeiving whole swath, onstant same azimuth orresponding mathingto filter will swath be applied enter, toi.e., proess Δ r m = ehoes, is adopted for targets in whose imaging referene for range whole equals swath, R p `same r m. azimuth This leads mathing to a quadrati filter will azimuthal be applied phase to error, proess whih ehoes an for be expressed as ψ rm πλ r m 8ρ 2 a sin 2 (29) ϕ where ρ a denotes azimuthal resolution. Aording to Equation (29), phase error ψ rm will lead to more azimuthal mismathing and defousing as target is muh farr away from swath enter along azimuth, beause ψ rm inreases with r m. Therefore, different r m must be adopted for different eho segments along azimuth. 4. Imaging Compensation Algorithm As we all know, existing spaeborne SAR imaging proessors are designed based on stop-go motion model. With onstrution of ontinuous tangent motion model, we have updated SAR eho expression, and analyzed new harateristis of updated eho expression. In order to satisfy existing SAR proessors, we ompensated different parts between updated eho expression and that based on stop-go model. Aording to eho expression (Equation (19)) and spae-variant analysis of three key fators, k m, V m, and r m, a ompensation algorithm with better fousing performane is proposed as follows.

11 Remote Sens. 216, 8, of 23 (1) The first step is to divide ehoes into several segments along azimuth. Every eho segment orresponds to a blok of swath. This is required beause r m is spae-variant and imaging for whole swath annot be implemented one. Orwise, defousing will our at azimuth edge of swath, whih was analyzed in Setion 3. The ephemeris data that are olleted by spaeborne instruments and transmitted to ground station with SAR ehoes ontain information on satellite position vetors. By applying ephemeris data, relative distane vetors and relative veloity vetors an be aquired. The distribution of r m aross whole swath an n be alulated aording to Equation (6). It is applied to determine segment strategy, whih must guarantee that quadrati phase error indued by r m at edge of every swath blok satisfies requirements of fousing quality. For lth eho segment, transmitting reeiving rate V m,l and transmitting reeiving onstant r m,l orresponding to enter of lth swath blok an be alulated aording to Equations (5) and (6). The time-saling fator k m,l an be estimated by a linear fitting of m t aording to Equation (13). (2) The azimuthal deramping [2,21] and Fourier transform along range are n implemented on lth eho segment suessively to aquire S E pt i, f τ q, whih has same form as Equation (2). (3) The first ompensation fator 1 pt i, f τ q is designed as follows ˆ j 1 1 pt i, f τ q exp j4π λ ` fτ V m,l t i exp j 4π f j τ r m,l After multiplying S E pt i, f τ q by 1 pt i, f τ q, a Fourier transform along azimuth is performed to aquire S F ` fη, f τ, whih is S F ` fη, f τ ˆ exp jπ f τ 2 ˆ R osϕ exp j2π f η b dv exp j 4π p f ` f τ q R sinϕ 2 f 2 η 1 $ 4V 2 p f ` f τ q 2 & ˆ4km,l exp % j π Rosϕ λr f η sinϕ b λb V V 4V 2 λ 2 2 f» fi η, f τ ` km,l Rosϕ λr f η sinϕ b fl ` j k m,lπλ 2 R 2. λ V V 4V 2 λ 2 2 f η V 4 sin 2 ϕ f η 3 - (3) (31) (4) The seond ompensation fator 2 ` fη, f τ is designed as 2 ` fη, f τ $ & exp % j» f τ ` km,l λ ˆ4km,l π λb Rosϕ V Rosϕ λr f η sinϕ b V V 4V 2 λ 2 2 f fi η, λr f η sinϕ b fl j k m,lπλ 2 R 2. V 4V 2 λ 2 2 f η V 4 sin 2 ϕ f η 3 - (32) ` ` After multiplying S F fη, f τ with 2 fη, f τ, azimuth equals an inverse Fourier transform along ` ` S K pt i, f τ q IFFT fη SF fη, f τ 2 fη, f τ ˆ j 1 exp j4π λ ` fτ R s pt i q exp jπ f τ 2 b (33)

12 Remote Sens. 216, 8, of 23 (5) S K pt i, f τ q has same form as Equation (21), whih is eho expression based on stop-go model. Therefore, sliding spotlight imaging algorithms based on stop-go model an be applied to proess S K pt i, f τ q and ahieve a high-resolution image for lth eho segment. Here, deramping ωk algorithm is adopted [13,18]. Figure 9 shows ompensation flow for every eho segment. After eah eho segment has been imaged, results are stid toger along azimuth to form a final omplete image of Remote Sens. 216, 8, of 24 whole swath. Figure Figure 9. An9. imaging An imaging ompensation algorithm based on ontinuous tangent model. model. 5. Simulation and Validation 5. Simulation and Validation 5.1. Simulation Method and Parameters 5.1. Simulation Method and Parameters SAR eho simulation methods are typially based on stop-go model. A new method is SAR proposed eho here simulation to simulate methods ontinuous are typially motion based of on satellite. stop-go The key model. to this A method new method is to is proposed aquire herean toaurate simulatetransmitting ontinuous distane motion aording of satellite. to reeiving The key distane to this method for every is toeho aquire an aurate sampling transmitting point. distane aording to reeiving distane for every eho sampling point. As demonstrated As demonstrated in in Figure Figure 1, 1, reeiving momentδδ i i for for ith ithsampling sampling point point during during tn, t n ` t 1{ n, t f n + p 1 an fp be expressed as an be expressed as δ i i δt i n = `tnτ + g τ` i g + i i =,1,2, 2, (34) (34) F s s where g where τ τ g is sampling is sampling start start time time and and F s isf s is sampling sampling rate. rate. In order to alulate transmitting moment orresponding to δ In order to alulate transmitting moment orresponding to i, iterations are performed as, iterations are performed demonstrated as demonstrated in Figure in Figure 1. Detailed 1. Detailed steps steps are as are follows. as follows.

13 Remote Sens. 216, 8, of 23 Remote Sens. 216, 8, of 24 Figure 1. A workflow diagram of simulation method. The ore of flow hart is to aurately Figure 1. A workflow diagram of simulation method. The ore of flow hart is to aurately estimate transmitting moment orresponding to every sampling moment within synti estimate transmitting moment orresponding to every sampling moment within synti aperture time. Afterwards, transmitting and reeiving distanes an be alulated and aperture time. Afterwards, transmitting and reeiving distanes an be alulated and substituted substituted into eho expression to ahieve simulated ehoes. into eho expression to ahieve simulated ehoes. (1) The initial estimation of transmitting moment orresponding to δ i is assumed to be (1) The initial estimation of transmitting moment orresponding to δ i is assumed to be ζ = δ R δ (35) R( δ ) where i ( ) i, i 2 i ζ i, δ i 2R pδ i q{ (35) is relative distane between SAR and target at moment δ i. R (2) If Equation (36) an be satisfied, transmitting distane is onsidered to be ( ζ, ) ik where R pδ i q is relative distane between SAR and target at moment δ i., (2) If Equation (36) an be satisfied, ζ whih denotes relative distane at ik, transmitting ( k =,1,2, distane is onsidered to be R `ζ ). Orwise, Step (3) is performed. i,k, whih denotes relative distane at ζ i,k (k, 1, 2, ). Orwise, Step (3) is performed. R( ζ ik, ) + R( δ i) 15 ( δi ζ i, k) < 1 (36) R `ζ i,k ` R pδi q `δ ˇ i ζ ă i,k ˇˇˇˇˇ 1 15 (36) ζ (3) Let ik, + 1 = ζ ik, Δ ζ. Δζ is (3) Let ζ i,k`1 ζ i,k ζ. ζ is ( R( ζ ik, ) + R( δi) ) ( δi ζ ik, ) Δ ζ = (37) `ζi,k 2 `R ` R pδi q { `δ i ζ i,k (4) Let k = k + 1, and return ζ to Step (2). (37) 2 By performing iteration method for every sampling point, transmitting and reeiving distanes (4) Let k an k be ` 1, preisely and return alulated to Step and (2). substituted into Equation (15) to form simulated ehoes. By performing iteration method for every sampling point, transmitting and reeiving distanes Simulation an be preisely parameters alulated are listed in and Table substituted 2. The simulated into Equation swath overing (15) toan form area of 15 simulated km 15 km ehoes. is Simulation demonstrated parameters in Figure are 11, listed where inthree Table point 2. The targets, simulated P1, P2, and swath P3, overing are arranged an area at ofenter, 15 km ˆ 15 km is demonstrated range edge, and in Figure azimuth 11, where edge, three respetively. point targets, P 1, P 2, and P 3, are arranged at enter, range edge, and azimuth edge, respetively.

14 Remote Sens. 216, 8, of 23 Remote Sens. 216, 8, of 24 Remote Sens. 216, 8, of 24 Table Table Simulation Simulation parameters. parameters. Table Parameters Parameters 2. Simulation parameters. Value Value Orbital Parameters inlination 98.6 Value Orbital inlination 98.6 Orbital Orbital height inlination height km km Eentriity Orbital Eentriity height km Elevation Elevation Eentriity angle angle Elevation Wavelength angle.3 35 m Beam Beam width Wavelength width (Azimuth) m Beam Squint width angle angle (Azimuth) ± Beam Beam Squint rotation angle veloity 2 /s 2 /s ±6.21 Signal Signal bandwidth bandwidth 1. 1.GHz Beam rotation veloity 2 /s Pulse Pulse width 4 4 μs µs Signal bandwidth 1. GHz Pulse repetition frequeny 4 Hz Pulse repetition Pulse width frequeny 4 μs Hz Pulse repetition frequeny 4 Hz Figure Figure 11. The 11. The simulated swath: PP1 1 is is at swath enter. P2 Pand 2 and P3 Pare 3 are at at range range edge edge and and azimuth azimuth Figure edge, 11. edge, respetively, The simulated whih swath: are are all P1. P1 all is at 7.5 kmswath away enter. fromp2 P 1 and. P3 are at range edge and azimuth edge, respetively, whih are all 7.5 km away from P Simulation Results 5.2. Simulation Results 5.2. Simulation Figures Results demonstrate imaging results of P1, P2, and P3, where profiles represent Figures fousing quality demonstrate along range imaging and azimuth, results and of ontour P 1, P 2, and maps P denote 3, where two-dimensional profiles represent Figures demonstrate imaging results of P1, P2, and P3, where profiles represent fousing quality. In along every figure, range re andare azimuth, three kinds andof ontour results. The maps first denote result is ahieved two-dimensional fousing quality along range and azimuth, and ontour maps denote two-dimensional using fousing fousing Deramping quality. quality. In ωk every In algorithm, every figure, figure, whih re re are is are based three three on kinds kinds of stop-go of results. results. model The The [18]. first first result The result seond is is ahieved ahieved is ahieved using using Deramping aording Deramping ωkto algorithm, ωk algorithm algorithm, whih presented is whih basedis in on based [15], whih stop-go on is stop-go based model on model [18]. The ontinuous [18]. seond The seond retilinear is ahieved is ahieved model. aording to The aording algorithm third is to ahieved presented algorithm using [15], presented novel whih imaging in is [15], based ompensation whih onis based ontinuous method presented ontinuous retilinear this retilinear model. paper. The model. third is ahieved The third usingis ahieved novelusing imaging novel ompensation imaging ompensation method presented method presented in this paper. in this paper. (a) (a) Figure 12. Cont. Figure 12. Cont. Figure 12. Cont. (b) (b)

15 Remote Sens. 216, 8, of 23 Remote Sens. 216, 8, of () (e) (d) (f) (g) (i) (h) Figure 12. Imaging results of P1, point at swath enter. (a ) The range profile, azimuth Figure 12. Imaging results of P profile and ontour map 1, point at swath enter. (a ) The range profile, azimuth ahieved using Deramping ωk algorithm based on stop-go profile model; and (d f) ontour The range map profile, ahieved using azimuth Deramping profile and ωk algorithm ontour map based ahieved on stop-go using model; (d f) algorithm The range based profile, on ontinuous azimuth profile retilinear andmodel; ontour (g i) The map range ahieved profile, using azimuth algorithm profile and based on ontinuous ontour map retilinear ahieved using model; (g i) novel The imaging rangeompensation profile, azimuth method presented profile and in this paper. ontour map ahieved using novel imaging ompensation method presented in this paper.

16 Remote Sens. 216, 8, of 23 Remote Sens. 216, 8, of (a) () (e) (b) (d) (f) (g) (h) Figure 13. Cont. Figure 13. Cont.

17 Remote Sens. 216, 8, of 23 Remote Sens. 216, 8, of 24 Remote Sens. 216, 8, of (i) (i) Figure 13. Imaging results of P2, point at range edge. Images (a i) are in aordane with Figure 13. Imaging results of of P2P, 2 edge.images Images (a i) in aordane Figure 13. Imaging results,12. point pointat at range range edge. (a i) areare in aordane with with Figure Figure 12. Figure 12. (a) (a) () () (d) (d) -.9 (b) (b) - (e) (e) Figure 14. Cont (f) Figure 14. Cont. Figure 14. Cont (f).5

18 Remote Sens. 216, 8, of 23 Remote Sens. 216, 8, of (g) (i) (h) Figure 14. Imaging results of P3, point at azimuth edge. Images (a i) are in aordane with Figure 12. Figure 14. Imaging results of P 3, point at azimuth edge. Images (a i) are in aordane with Figure 12. By applying resolution, peak to side lobe ratio (PSLR), and integrated side lobe ratio (ISLR) as indiators to evaluate Figures 12 14, evaluation results along range and azimuth are By presented applying Tables resolution, 3 and 4, peak respetively. to side lobe ratio (PSLR), and integrated side lobe ratio (ISLR) as indiators to evaluate Figures evaluation results along range and azimuth are presented in Tables 3 and 4 respetively. Table 3. Evaluation of imaging quality along range. Motion Model Target Resolution (m) PSLR (db) ISLR (db) Table 3. EvaluationP1 of imaging quality.1444 along range Stop-go model P Motion Model Target P3 Resolution.1444 (m) PSLR (db) ISLR (db) P P Continuous retilinear model P2 Stop-go model P P P Continuous tangent model Continuous retilinear model P P P2 P P3 P P Continuous tangent model Table 4. Evaluation P of imaging quality along azimuth Motion Model Target P 3 Resolution.1342 (m) PSLR (db) ISLR 9.8 (db) P Stop-go model Table 4. EvaluationP2 of imaging quality 35 along azimuth P Motion Model Target P1 Resolution 156 (m) PSLR 13.7 (db) ISLR 1.39 (db) Continuous retilinear model P P P Stop-go model P P1 P Continuous tangent model P P P Continuous retilinear model P P Continuous tangent model P P P

19 Remote Sens. 216, 8, of 23 Aording to imaging profiles and evaluation results, first imaging algorithm, whih is based Remote Sens. 216, 8, of 24 on stop-go model, exhibits worst performane. Along range and azimuth, main lobe orresponding Aording to every to imaging targetprofiles is broadened, and evaluation whihresults, redues first resolution. imaging algorithm, Although whih PSLR is based and ISLR are better on for stop-go this method model, exhibits than or worst two performane. methods, Along it omes range at and ost azimuth, of a loss main resolution. lobe Furrmore, orresponding this method to every brings target is asymmetri broadened, phenomena whih redues to azimuth resolution. sidealthough lobes, whih PSLR indiates and ISLR that quadrati are better and high-order for this method phase than errors or are not two ompensated methods, it omes for ompletely at ost in of imaging. a loss in resolution. The Furrmore, seond algorithm, this method whih brings is asymmetri based on phenomena ontinuous to azimuth retilinear side lobes, model, whih anindiates ahievethat thinner main lobes. quadrati Compared and high-order with phase first errors method, are not ompensated range resolutions for ompletely are improved in imaging. by about 6.35% 7.6%. The seond algorithm, whih is based on ontinuous retilinear model, an ahieve thinner The azimuth resolutions are improved by about 6.26% 7.59%. However, for P 3, whih is at azimuth main lobes. Compared with first method, range resolutions are improved by about edge of swath, azimuth resolution, PSLR, and ISLR deay obviously ompared with swath 6.35% 7.6%. The azimuth resolutions are improved by about 6.26% 7.59%. However, for P3, whih enter, is as at demonstrated azimuth edge in Figure of 14 swath, and Table azimuth 4. resolution, PSLR, and ISLR deay obviously The ompared algorithm with based swath on enter, ontinuous as demonstrated tangent in model Figure preserves 14 and Table 4. fousing performane with seondthe method algorithm at based swathon enter ontinuous and range tangent edge, model andpreserves furr improves fousing fousing performane quality at with azimuth seond edge. method The azimuth at swath resolution, enter and PSLR, and range ISLR edge, orresponding and furr improves to P 3 are improved fousing by 3.56%, quality 1.79 db, at and 1.78 azimuth db, respetively. edge. The azimuth As a result, resolution, fousing PSLR, qualities and ISLR for Porresponding 1, P 2, and P 3 are to P3 nearly are same, improved whih indiate by 3.56%, that 1.79 imaging db, and ohereny 1.78 db, respetively. for whole As a swath result, isfousing good. qualities for P1, P2, and P3 To furr are nearly demonstrate same, whih performane indiate that imaging of proposed ohereny algorithm, for whole a two-dimensional swath is good. example To furr demonstrate performane of proposed algorithm, a two-dimensional is provided based on an airborne SAR image (Courtesy of Sandia National Laboratories, Airborne example is provided based on an airborne SAR image (Courtesy of Sandia National Laboratories, ISR). The resolution and size of this image are.1 m and 66 ˆ 66 m, respetively. It is deployed Airborne ISR). The resolution and size of this image are.1 m and 66 m 66 m, respetively. It is at deployed top left orner at top of left orner simulated of swath, simulated andswath, simulated and simulated eho aneho be ahieved. an be ahieved. By applying By algorithms applying based algorithms on stop-go based on model, stop-go ontinuous model, retilinear ontinuous model, retilinear and model, ontinuous and tangent model, ontinuous differenttangent imaging model, results different are aquired imaging (Figure results are 15). aquired (Figure 15). (a) (b) () (d) Figure 15. Cont. Figure 15. Cont.

20 Remote Sens. 216, 8, of 23 Remote Sens. 216, 8, of 24 (e) (f) (g) (h) Figure 15. Comparison of imaging results: (a) real SAR image; (b) enlarged view of seletable Figure 15. Comparison of imaging results: (a) real SAR image; (b) enlarged view of seletable portion in (a); () stop-go model image; (d) enlarged view of seletable portion in (); (e) portion in (a); () stop-go model image; (d) enlarged view of seletable portion in (); (e) ontinuous ontinuous retilinear model image; (f) enlarged view of seletable portion in (e); (g) ontinuous retilinear tangent model model image; image; (f) (h) enlarged enlarged view viewof ofseletable portion portion in (g). in (e); (g) ontinuous tangent model image; (h) enlarged view of seletable portion in (g). There are three red retangles in every enlarged view of seletable portion. As shown in There bottom are retangle three red of Figure retangles 15b, re in every are three enlarged sattering viewenters. of However, seletableonly portion. two enters As shown are in reognized in Figure 15d, while three enters an be distinguished in both Figure 15f,h. This bottom retangle of Figure 15b, re are three sattering enters. However, only two enters are indiates that algorithm based on stop-go model has worst fousing quality. In top reognized in Figure 15d, while three enters an be distinguished in both Figure 15f,h. This indiates retangle of Figure 15b, only edge of one sattering enter exists. However, something more, that aused algorithm by side based lobes, appears on stop-go at same model positions has of Figure worst 15f,h. fousing This is quality. an isolate In spot top in retangle top of Figure retangle 15b, of only Figure 15h. edgethe of spot one in sattering top retangle enter exists. of Figure However, 15f is onneted something with more, sattering aused by side lobes, enter appears beause at algorithm same positions based on of Figure ontinuous 15f,h. retilinear This is an model isolate has spot worse inpslr top and retangle ISLR of Figure than 15h. proposed The spotalgorithm in top based retangle on of ontinuous Figure 15f tangent is onneted model. The withdifferene sattering between enter beause middle algorithm retangles in based Figure on15f,h ontinuous also demonstrates retilinear this point. model As has a result, worse PSLR algorithm and ISLR proposed than proposed in this algorithm paper has based better on fousing ontinuous performane tangent than or model. two The algorithms. differene between middle retangles in Figure 15f,h also demonstrates this point. As a result, algorithm proposed in this paper has 5.3. Computational Load better fousing performane than or two algorithms. Computational load is a key element restriting appliation of an algorithm. As 5.3. Computational demonstrated Load in Figure 9, after first and seond ompensations, ehoes an be suessfully foused using urrent algorithms based on stop-go model. Therefore, analysis of Computational load is a key element restriting appliation of an algorithm. As demonstrated omputational load fouses on first and seond ompensations. Although hirp saling in Figure algorithm 9, after (CSA) first [8,9] and annot seond ahieve ompensations, 1 m resolution ehoes and 15 ankm beswath suessfully width for foused spaeborne usingsar, urrent algorithms is worth basedomparing on stop-go CSA and model. se Therefore, ompensations analysis from ofaspet omputational of omputational load fouses load on first and beause seond CSA ompensations. is reognized as an Although effiient algorithm hirpand saling has been algorithm widely applied. (CSA) [8,9] annot ahieve 1 m resolution and 15 km swath width for spaeborne SAR, it is worth omparing CSA and se ompensations from aspet of omputational load beause CSA is reognized as an effiient algorithm and has been widely applied.

21 Remote Sens. 216, 8, of 23 Computational load is evaluated aording to omplex multipliation and addition in algorithm. Multipliation of two omplex numbers and addition of two real numbers require 6 FLOPs (floating point operations) and 1 FLOP, respetively. An FFT or IFFT of length N requires 5Nlog 2 pnq FLOPs [4]. Suppose sampling numbers along azimuth and range are N azi and N rng, respetively. The omputational loads of CSA, and first and seond ompensations in proposed algorithm, are, respetively, N CSA 1N rng N azi log 2 pn azi q ` 1N rng N azi log 2 `Nrng ` 18Nrng N azi N PA 15N rng N azi log 2 pn azi q ` 1N rng N azi log 2 `Nrng ` 24Nrng N azi (38) In order to ahieve 1 m resolution and a swath of 15 km ˆ 15 km, N azi and N rng should be at least 118,65 and 15,, respetively. Aording to Equation (38), omputational load of first and seond ompensations is 1.25 times that of CSA, whih indiates that ompensations are effiient. 6. Conlusions This paper developed a ontinuous tangent model to desribe relative motion between a spaeborne SAR and its targets. An imaging ompensation algorithm was proposed based on new motion model. Compared with algorithm based on stop-go model, novel algorithm improved range resolutions and azimuth resolutions by about 6.35% 7.6% and 6.26% 1.33%, respetively. Compared with algorithm based on ontinuous retilinear model, novel algorithm furr improved azimuth resolution, PSLR, and ISLR at azimuth edge by 3.56%, 1.79 db, and 1.78 db, respetively. Simulation results indiated that presented algorithm provided a superior and more onsistent fousing quality aross whole swath. The novel algorithm is also appliable to updating of existing spaeborne SAR imaging proessors that are designed based on stop-go model. A pre-proessing module is only addition neessary to implement first and seond ompensations, as demonstrated in Figure 9. After being proessed by this pre-proessing module, ehoes an be suessfully foused using existing proessors to produe high-quality image produts. The input parameters for new algorithm are same as existing algorithms, whih indiates that it is not neessary to hange input interfae in updating. However, as resolution is improved, a higher requirement for auray of motion model will be put forward. Therefore, under ondition of better resolution, ontinuous tangent model should be furr verified as well as orresponding imaging ompensation algorithm. Author Contributions: All authors ontributed equally to this paper. Conflits of Interest: The authors delare no onflit of interest. Appendix A As a result of moving distane between transmitting and reeiving loations for spaeborne SAR being less than 1 m, whih is very small relative to orbital height, following formula holds: R r pt n, τq R s pt m q «V m τ ` pn mq { f p (A1) The term R r pt n, τq R s pt m q n satisfies R r pt n, τq R s pt m q «ˇˇRs pt m q ` V m τ ` pn mq { f p ˇˇ Rs pt m q «R s pt m q V m (A2) τ ` pn mq { fp R s pt m q

22 Remote Sens. 216, 8, of 23 With equation R s pt m q ` R r pt n, τq τ ` pn mq { f p (A3) following formula an be aquired: R r pt n, τq R s pt m q «2 R spt m q V m (A4) where R s pt m q R s pt q ` m 1 ř V i i f p R s pt q ` V a t m (A5) V a m 1 ÿ i V i {m and V i represents relative veloity vetors at t i. Therefore, R r pt n, τq R s pt m q an be approximated as follows: R r pt n, τq R s pt m q «2 V a V m t m ` 2 R spt q V m (A6) (A7) Appendix B Aording to Figure 1, following expression an be derived: Therefore, # whih an also be expressed as R s pt m q ` R r pt n, τq τ ` pn mq { f p R s pt m, τ s q ` R r pt n, τ ` τ r q τ ` τ r τ s ` pn mq { f p (B1) R s pt m, τ s q ` R r pt n, τ ` τ r q R s pt m q R r pt n, τq pτ r τ s q (B2) R s pt m q ` V m τ s ` R r pt n, τq ` V r τ r R s pt m q R r pt n, τq pτ r τ s q (B3) By solving Equation (B3), relation between τ s and τ r an be shown to satisfy τ s Rrpt n, τq V r R r pt n, τq ` Rspt m q V m R s pt m q τ r (B4) Referenes 1. Sadjadi, F. New omparative experiments in range migration mitigation methods using polarimetri inverse synti aperture radar signatures of small boats. In Proeedings of 214 IEEE Radar Conferene, Cininnati, OH, USA, May 214; pp Sadjadi, F.A. New Experiments in Inverse synti aperture radar image exploitation for maritime surveillane. In Proeedings of SPIE Defense+ Seurity, 214 International Soiety for Optis and Photonis, Baltimore, MD, USA, 5 9 May 214; pp. 999:1 999:9. 3. Wang, W.Q.; Jiang, D. Integrated wireless sensor systems via near-spae and satellite platforms: A review. IEEE Sens. J. 214, 14, [CrossRef]

23 Remote Sens. 216, 8, of Cumming, I.G.; Wong, F.H. Digital Proessing of Synti Aperture Radar Data: Algorithms and Implementation; Arteh House: Norwood, MA, USA, Curlander, J.; MDonough, R. Synti Aperture Radar: Systems and Signal Proessing; John Wiley & Sons: New York, NY, USA, Lim, B.G.; Woo, J.C.; Lee, H.Y.; Kim, Y.S. A Modified subpulse SAR proessing proedure based on range-doppler algorithm for synti wideband waveforms. Sensors 28, 8, [CrossRef] 7. Bamler, R.; Breit, H.; Steinbreher, U.; Just, D. Algorithms for X-SAR proessing. In Proeedings of International Geosiene and Remote Sensing Symposium, Tokyo, Japan, August 1993; pp Wang, W.Q.; Shao, H. Azimuth-variant signal proessing in high-altitude platform passive SAR with spaeborne/airborne transmitter. Remote Sens. 213, 5, [CrossRef] 9. Davidson, G.W.; Cumming, I.; Ito, M.R. A hirp saling approah for proessing squint mode SAR data. IEEE Trans. Aerosp. Eletron. Syst. 1996, 32, [CrossRef] 1. Janoth, J.; Gantert, S.; Shrage, T.; Kaptein, A. From TerraSAR-X towards TerraSAR next generation. In Proeedings of 1th European Conferene on Synti Aperture Radar, Berlin, Germany, 3 5 June 214; pp Novak, L.M.; Benitz, G.R.; Owirka, G.J.; Bessette, L.A. ATR performane using enhaned resolution SAR, aerospae/defense sensing and ontrols. In Proeedings of International Soiety for Optis and Photonis, Orlando, FL, USA, 1 June 1996; pp Novak, L.M.; Owirka, G.J.; Netishen, C.M. Performane of a high-resolution polarimetri SAR automati target reognition system. Linoln Lab. J. 1993, 6, Prats-Iraola, P.; Sheiber, R.; Rodriguez-Cassola, M.; Mittermayer, J.; Wollstadt, S.; De Zan, F.; Brautigam, B.; Shwerdt, M.; Reigber, A.; Moreira, A. On proessing of very high resolution spaeborne SAR data. IEEE Trans. Geosi. Remote Sen. 214, 52, [CrossRef] 14. Mittermayer, J.; Wollstadt, S.; Prats-Iraola, P.; Sheiber, R. The TerraSAR-X staring spotlight mode onept. IEEE Trans. Geosi. Remote Sen. 214, 52, [CrossRef] 15. Liu, Y.; Xing, M.; Sun, G.; Lv, X.; Bao, Z.; Hong, W.; Wu, Y. Eho model analyses and imaging algorithm for high-resolution SAR on high-speed platform. IEEE Trans. Geosi. Remote Sens. 212, 5, [CrossRef] 16. Meta, A.; Hoogeboom, P.; Ligthart, L.P. Signal proessing for FMCW SAR. IEEE Trans. Geosi. Remote Sens. 27, 45, [CrossRef] 17. Ribalta, A. Time-domain reonstrution algorithms for FMCW-SAR. IEEE Geosi. Remote Sens. Lett. 211, 8, [CrossRef] 18. Teng, L.; Wu, J.; Huang, Y.; Yang, J. A deramping based Omega-k algorithm for wide sene spotlight SAR. In Proeedings of International Conferene on Systems and Informatis, Yantai, China, 18 2 May 212; pp Carrara, W.G.; Goodman, R.S.; Majewski, R.M. Spotlight Synti Aperture Radar: Signal Proessing Algorithms; Arteh House: Norwood, MA, USA, Mittermayer, J.; Moreira, A.; Loffeld, O. Spotlight SAR data proessing using frequeny saling algorithm. IEEE Trans. Geosi. Remote Sens. 1999, 37, [CrossRef] 21. Wang, G.D. A deramp hirp saling algorithm for proessing spaeborne spotlight SAR data. In Proeedings of ICMMT 4th International Conferene on Mirowave and Millimeter Wave Tehnology, Beijing, China, August 24; pp by authors; liensee MDPI, Basel, Switzerland. This artile is an open aess artile distributed under terms and onditions of Creative Commons by Attribution (CC-BY) liense (