remote sensing Article
An Imaging Compensation Algorithm for Spaceborne High-Resolution SAR Based on a Continuous Tangent Motion Model Ze Yu 1,† , Shusen Wang 1,† and Zhou Li 2, *,† 1 2
* †
School of Electronics and Information Engineering, BeiHang University, Beijing 100191, China;
[email protected] (Z.Y.);
[email protected] (S.W.) Beijing Institute of Remote Sensing Information, Beijing 100192, China Correspondence:
[email protected]; Tel.: +86-138-1101-1901 These authors contributed equally to this work.
Academic Editors: Josef Kellndorfer and Prasad S. Thenkabail Received: 25 November 2015; Accepted: 4 March 2016; Published: 10 March 2016
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 km ˆ 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. Keywords: high-resolution imaging; motion model; phase compensation; synthetic aperture radar
1. 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
Remote Sens. 2016, 8, 223; doi:10.3390/rs8030223
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to the next position. If the SAR satellite is in an orbit with a height of 600 km, it actually takes more than 4 ˆ 10´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 [6–9] 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) ˆ 4 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.
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3 of 23 2. Continuous Tangent Model and Two-Way Distance 2. Continuous Tangent Model and Two-Way Distance The two-way distance refers to the sum of propagation distances for electromagnetic signals The two-way distance refers to the sum Distance of propagation distances for electromagnetic signals 2.from Continuous Tangent Model and SAR to the target and back to Two-Way SAR. It is applied in the imaging to implement range correction 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 for electromagnetic calculating the two-way two-way distanceThe refers to themodel sum of distances signals and The phase compensation. stop-go is propagation 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 from SAR to the target and back to SAR. It is applied in the imaging to implement range correction 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 two-way order to and phase compensation. The stop-go model is mostof commonly for calculating model, which significantly improves the accuracy two-way used distance calculationthe in order to enhance focusing quality. distance This section will construct a continuous tangent model, an update to the stop-go enhance [4,13]. focusing quality. model, which significantly improves the accuracy of two-way distance calculation in order to enhance 2.1. Continuous Tangent Model focusing quality.Tangent 2.1. Continuous Model Figure 1 demonstrates the actual relative motion between spaceborne SAR and the target. For Figure 1 demonstrates 2.1. Continuous Tangent Modelthe actual relative motion between spaceborne SAR and the target. For convenience, time is divided into two parts: azimuth slow time ttii and range fast time ττ . Here, convenience, time is dividedthe into two parts: azimuth time spaceborne and range fastand timethe .target. Here, relative motionslow between SAR t i = iFigure / f p i1= demonstrates 0,1, 2,3, ) and f p actual ( i = 0,1, is the pulse repetition frequency (PRF). As a result, the continuous t = i / f f i p p 2,3, For convenience, time is) and divided into parts: azimuth slow time ti and fastthe time τ. Here, ( is thetwo pulse repetition frequency (PRF). Asrange a result, continuous T / 2, t mthe + Tcontinuous / 2 t ma −result, t = t + τ t i ttime “ i{ f (i “ 0, 1, 2, 3, ¨ ¨ ¨ ) and f is the pulse repetition frequency (PRF). As p is expressed as t = t +pτ . The mth pulse is transmitted during t m − T / 2, t m + T / 2 , where i , where i time t tis expressed is expressed . Themth mth pulseisistransmitted transmittedduring duringrt ´ T{2, t ` T{2s, where time as as t “ ti `The τ. The T T denotes the pulse width. center ofpulse the transmitting waveform ismmarked m as A, which leaves T denotes the pulse width. The center of the transmitting waveform is marked as A, which leaves denotes the pulse width. The center of the transmitting waveform is marked as A, which leaves the the radar at ttmm , reaches the target, and is reflected to the receiver at ttnn ++ ΔΔττ . Similarly, another radar at tm ,atreaches the target, is reflected to the receiver tn ` ∆τ.atSimilarly,. another point B is the radar , reaches the and target, and is reflected to the at receiver Similarly, another t m + τ s and received at t n + Δτ + τ r . point B is transmitted at transmitted at tm ` τs and at tnreceived ` ∆τ ` at τr . t n + Δτ + τ r . m + τ s and point B is transmitted at treceived
(a) (a)
(b) (b)
Figure 1. (a)The The actualrelative relative motionbetween between spaceborneSynthetic Synthetic ApertureRadar Radar (SAR)and and the Figure Figure1.1. (a) (a) The actual actual relativemotion motion betweenspaceborne spaceborne SyntheticAperture Aperture Radar(SAR) (SAR) andthe the target; (b)The The correspondingtime time sequenceofoftransmitting transmitting andreceiving. receiving. target; target;(b) (b) Thecorresponding corresponding timesequence sequence of transmittingand and receiving.
In the stop-go model, as demonstrated in Figure 2, the relative trajectory is approximated as a In the the stop-go stop-go model, model, as as demonstrated demonstrated in in Figure Figure 2, 2, the the relative relative trajectory trajectory isis approximated approximated as as aa In straight line during the aperture time. Transmitting and receiving are assumed to occur at the same straight line line during during the the aperture aperture time. time. Transmitting Transmitting and and receiving receiving are areassumed assumed to to occur occur at atthe thesame same straight position, which indicates that the two-way distances for points A and B are the same, and the width position,which whichindicates indicatesthat thatthe thetwo-way two-waydistances distances points and B are same, width position, forfor points AA and B are thethe same, andand thethe width of of the transmitting pulse equals the width of the echo from one isolated point. of the transmitting pulse equals width of the echo from isolated point. the transmitting pulse equals the the width of the echo from oneone isolated point.
Figure 2. The stop-go model. Figure2.2. The Thestop-go stop-gomodel. model. Figure
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The continuous rectilinear model was first applied to airborne FMCW SAR imaging [16], while The continuous continuous rectilinear model model was was first first applied applied to airborne airborne FMCW FMCW SAR SAR imaging [16], [16], while while The Liu et al. applied therectilinear model to SAR imaging [15]. As to demonstrated in Figureimaging 3, the continuous Liu et al. applied the model to SAR imaging [15]. As demonstrated in Figure 3, the continuous Liu et al. applied model tothat SARSAR imaging [15].along As demonstrated in Figure 3, thestopping. continuous rectilinear rectilinear modeltheassumes moves a straight line without The model rectilinear model assumes that SAR moves along a straight line without stopping. The model model assumes that SAR moves along line without stopping. The model decomposes decomposes relative motion into twoa straight kinds. One is the relative motion between adjacent relative pulses, decomposes relative motion into two kinds. One is the relative motion between adjacent pulses, motion into two kinds. Onetransmitting is the relativeand motion between adjacent pulses, which the which indicates that the corresponding receiving occur inindicates differentthat pulse which indicates that the transmitting andincorresponding receivingperiods. occur The in different pulse transmitting and corresponding receiving occur different pulse repetition other is relative repetition periods. The other is relative motion during one pulse, which implies that different repetition periods. The which other implies is relative one which implies along that different motion during one pulse, that motion differentduring points in thepulse, same pulse propagate different points in the same pulse propagate along different transmitting and receiving paths, i.e., ττ r ≠≠ττ s . r s. points in theand same pulse propagate transmitting and receiving paths, i.e., transmitting receiving paths, i.e.,along τr ‰ τdifferent . s
Figure 3. The continuous rectilinear model. Figure 3. 3. The The continuous continuous rectilinear rectilinear model. model. Figure
The continuous rectilinear model does not account for the curvature of the orbit and the The continuous rectilinear model does not account for the curvature of the orbit and the approximated trajectory is still a straight line [15]. aperture longer, this The continuous rectilinear model does not account forAs the the curvature of thetime orbitbecomes and the approximated 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. trajectory is still a straight line [15]. As the aperture time becomes longer, this approximation will induce approximation will induce increasingly unacceptable errors at the edge of the synthetic aperture. The proposed continuouserrors tangent model in FigureThe 4. Itproposed includescontinuous the advantages of increasingly unacceptable at the edgeisofdemonstrated the synthetic aperture. tangent The proposed continuous tangent model is demonstrated in Figure 4. It includes the advantages of the continuous rectilinear model4.and furthermore approximates trajectories during every model is demonstrated in Figure It includes the advantages of therelative continuous rectilinear model and the continuous rectilinear model and furthermore approximates relative trajectories during every transmitting approximates period and every receiving period as tangential segments, period respectively. The receiving two-way furthermore relative trajectories during every transmitting and every transmitting period and every receiving period as tangential segments, respectively. The two-way distance every sampling can thenThe be calculated more accurately, especiallypoint at the edge period asfor tangential segments,point respectively. two-way distance for every sampling can thenofbea distance for every sampling point can then be calculated more accurately, especially at the edge of a synthetic aperture, because this novel motion is much closerbecause to the this actual relative calculated more accurately, especially at the edge ofmodel a synthetic aperture, novel motionmotion model synthetic aperture, because this novel motion model is much closer to the actual relative motion compared withtoexisting models. is much closer the actual relative motion compared with existing models. compared with existing models.
Figure 4. The model. Figure 4. The continuous continuous tangent tangent model. Figure 4. The continuous tangent model.
We denote distance vectors during the mth transmitting period and the nth receiving period as We Wedenote denotedistance distance vectors vectors during during the the mth mth transmitting transmitting period period and and the the nth nth receiving receiving period period as as R s (t m ,τ ) and R r (t n ,τ ) , respectively. In the continuous tangent model, R s (t m ,τ ) and RRs pt , τq and R pt , τq, respectively. In the continuous tangent model, R pt , τq and R pt , ∆τ ` τq can s m model, r Rn s (t m ,τ ) and s (tmm ,τ ) andr nR r (t n ,τ ) , respectively. In the continuous tangent R rexpressed (t n , Δτ + τ ) as be can be expressed as „ R r (t n , Δτ + τ ) can be expressed as T T Rs ptm , τq « Rs ptm q ` Vm τ τ P ´T T , (1) τ ∈ − T2, T2 (1) R s ( t m ,τ ) ≈ R s (t m ) + Vmτ „τ ∈ − 2 , 2 (1) R s ( t m ,τ ) ≈ R s (t m ) + Vmτ T 2 2 1 T Rr ptn , ∆τ ` τq « Rr ptn , ∆τq ` Vr τ τ P ´ ∆τ, ´ ∆τ ´ (2) T 1f T 2 2 τ ∈ T − Δτ , 1 p− Δτ − T (2) R r ( t n , Δτ + τ ) ≈ R r (t n , Δτ ) + Vrτ τ ∈ 2 − Δτ , f p − Δτ − 2 (2) where Vm and Vr are relative 2 fp R r ( t nvelocity , Δτ + τ ) ≈vectors R r (t n , Δat τ )t+mVand 2∆τ, respectively. rτ tn ` m and V r are relative velocity vectors at t m and t n + Δτ , respectively. where V where V m and V r are relative velocity vectors at t m and t n + Δτ , respectively.
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2.2. Two-Way Distance The two-way distance for point A in Figure 4 is “ ‰ R A ptm ; tn ` ∆τq “ |Rs ptm q| ` |Rr ptn , ∆τq| “ c ¨ pn ´ mq { f p ` ∆τ
(3)
where c is the speed of light. The difference between |Rs ptm q| and |Rr ptn , ∆τq| can be derived as (see Appendix A) |Rr ptn , ∆τq| ´ |Rs ptm q| “ 2∆Vm tm ` 2∆rm (4) where ∆Vm and ∆rm are the transmitting–receiving rate and the transmitting–receiving constant, respectively. They can be expressed as V a ¨ Vm c
(5)
Rs pt0 q ¨ Vm c
(6)
∆Vm “ ∆rm “
where Va denotes the average relative velocity vector during the period rt0 , tm´1 s. The two-way distance for point B is RB ptm ` τs ; tn ` ∆τ ` τr q “ |Rs ptm , τs q| ` |Rr ptn , ∆τ ` τr q| “ R A ptm q ` c ¨ rτr ´ τs s “ 2 |Rs ptm q| ` 2∆Vm tm ` 2∆rm ` c ¨ rτr ´ τs s
(7)
where τr and τs satisfy (see Appendix B) τs “
pc ´ k1 q τr pc ` k2 q
(8)
and k1 “ Vr ¨ Rr ptn , ∆τq{ |Rr ptn , ∆τq| “ |Vr | ¨ cosθr
(9)
k2 “ Vm ¨ Rs ptm q{ |Rs ptm q| “ |Vm | ¨ cosθs
(10)
where θr is the angle between Vr and Rr ptn , ∆τq and θs is the angle between Vm and Rs ptm q. Because the time difference between transmitting and receiving is very small, |Vr | « |Vm |, θr « θs , and k1 « k2 . The last term in Equation (7) can then be simplified as c ¨ rτr ´ τs s “ mt τr
(11)
mt “ c ¨ pk1 ` k2 q{ pc ` k2 q « 2 |Vm | cosθs
(12)
where |Vm | and cosθs vary with azimuth slow time and the looking angle. As demonstrated in Figure 5, for a fixed looking angle, |Vm | cosθs varies linearly with the azimuth time. Therefore, mt can be approximated as: mt “ 2km tm (13) where km is the time-scaling factor.
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|Vm V | cosθ the azimuth time and the and looking by applying θ s with Figure 5.5.Variation Variationof of the azimuth time the angle looking angle byparameters applying s with m cos in Table 1. Three lines are presented under looking angles 15, 35 and 55 degrees, which represent the parameters in Table 1. Three lines are presented under looking angles 15, 35 and 55 degrees, which low, medium and high looking angles, respectively. For a looking angle, |Vm | cosθs varies linearly with represent the low, medium and high looking angles, respectively. For a looking angle, Vm cosθ s the azimuth time. varies linearly with the azimuth time. Table 1. Simulation parameters. Table 1. Simulation parameters. Parameters Value Parameters Value ˝ Orbitalinclination inclination 98.06 Orbital 98.06° Orbital height 680 km Orbital height 680 km Eccentricity 0.001 Eccentricity ˝ Elevation angle 350.001 Elevation angle 35˝ ° Squint angle ˘6.21 ˝ /s Beam rotation velocity 0.42±6.21 Squint angle ° Bandwidth 1.00.42 GHz°/s Beam rotation velocity Wavelength 0.03 m Bandwidth 1.0 GHz Wavelength 0.03 m According to Equations (7), (11) and (13), the two-way distance Rw ptn , τqRfor tthe received at ( ,τ )point to Equations (7), (11) and (13), the two-way distance w n for the point tn ` τAccording can be expressed as received at t n + τ can be expressed as Rw ptn , τq R wq| (t n ,τ2∆V ) m tm ` 2∆rm ` 2km tm τ ` pn ´ mq { f p ´ r2 |Rs ptm q| ` 2∆Vm tm ` 2∆rm s {c( “ 2 |Rs pt m ` „ (14) n −s pt mm = 2 R s ( t m ) + 2ΔV mt m + 2Δrm + 2 k mt m τ +2 (|R ) /q|f p − 2 R s ( t m ) + 2ΔV mt m + 2Δrm c 2∆Vm tm ` 2∆rm ` 2k m tm τˆ ´ « 2 |Rs ptm q| ` (14) c 2 t R ( ) s m ≈ 2 R s ( t m ) + 2ΔV mt m + 2Δrm + 2 k mt m τ − c where τˆ “ τ ` pn ´ mq { f p .
{
}
3. Echoτ =Expression τ + ( n − m ) / of f p Spaceborne SAR Based on the Continuous Tangent Model . where The SAR echo expression is the foundation for deriving an imaging algorithm. Because the amplitude weightingof is Spaceborne usually not taken account in the derivationTangent of SAR imaging 3. Echo Expression SARinto Based on the Continuous Modelalgorithms [16,18], the weighting factor of the antenna pattern is ignored. The echo from an isolated point can be expressed as The SAR echo expression is the foundation for deriving an imaging algorithm. Because the „ into"account „ * „ SAR imaging algorithms amplitude weighting is usually in of ˆ ˆ Rwnot pti ,taken τq Rwthe pti ,derivation τq 2π ˆ ˆ ˆ ˆ point (15) S pt , τq “ σ ¨ a τ ´ ¨ exp jϕ τ ´ ¨ exp ´j Rw pti , τq i [16,18], the weighting factor of thec antenna pattern is ignored. can c The echo fromλan isolated be expressed as ˆ is the rectangular amplitude modulation, where σ is the scattering cross-section of the target, a pτq R and τ is R t i In ,τ this w t i ,λ w ˆ is the two-way Rw pti , τq distance, the wavelength. paper, signal as the π use achirp 2we ⋅ exp jϕ τ − ⋅ exp S t i ,τ = σ ⋅ a τ − R w t i ,τ −j (15) 2 ˆ ˆ transmitting signal, i.e., ϕ pτq “ ´πbcτ , where b is the linear rate. c frequency λ modulation
( )
( )
( )
( )
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By substituting Equation (14) into Equation (15), SAR echo received at tm ` τˆ can be expressed as ˙ˆ ˙ 2 |Rs ptm q| 2 2k m tm τˆ ´ ´ p∆Vm tm ` ∆rm q c c c + # 2 „ ` 2˘ 2 |Rs ptm q| 2 ´ p∆Vm tm ` ∆rm q ` jϕ1 ptm q ` jϕ2 tm ¨ exp ´jπb τˆ ´ c c " „ˆ ˙ * 4π 2k m tm ˆ m ` ∆Vm tm ` ∆rm |Rs ptm q| ` k m τt ¨ exp ´j 1´ λ c „ˆ
ˆ S ptm , τq
where
“ σ¨a
1´
˙ „ ˆ ˙ ˆ 2 |Rs ptm q| 4πbk m tm 2 |Rs ptm q| ¨ τˆ ´ ϕ1 ptm q “ ¨ c τˆ ´ ´ 2∆rm c c c2 ˆ ˙„ ˆ ˙ ´ ¯ 2 |Rs ptm q| 2 |Rs ptm q| 4πbk m tm 2 ˆ ˆ τ ´ k ` 2∆V τ ´ ϕ2 t m 2 “ ´ m m c c c2
(16)
(17) (18)
According to spaceborne SAR system parameters designed for 0.2-meter resolution, ϕ1 ptm q and ˘ ϕ2 tm 2 will not exceed 1˝ , which hardly affects the focusing quality. Therefore, Equation (16) can be simplified as `
ˆ S ptm , τq
„ˆ ˙ˆ ˙ 2 2k m tm 2 |Rs ptm q| ´ p∆Vm tm ` ∆rm q 1´ τˆ ´ c c c # „ 2 + 2 |Rs ptm q| 2 ¨ exp ´jπb τˆ ´ ´ p∆Vm tm ` ∆rm q c c " „ˆ ˙ * 4π 2k m tm ˆ m ` ∆Vm tm ` ∆rm |Rs ptm q| ` k m τt ¨ exp ´j 1´ λ c « σ¨a
(19)
If constants are omitted, the Fourier transformation of Equation (19) along the range time can be expressed as ˙ 1 fτ ` |Rs ptm q| “ exp ´j4π c ˜ ¸λ ˙ „ ˆ fτ 2 4k m tm f τ 4k2m t2m ¨ exp jπ ` ¨ exp jπ b λb λ2 b „ ˆ ˙ „ ˆ ˙ 1 fτ fτ ¨ exp ´j4π ` ∆Vm tm ¨ exp ´j4π ∆rm λ c c „
S ptm , f τ q
ˆ
(20)
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: „
ˆ
Ss pti , f τ q “ exp ´j4π
1 fτ ` λ c
˙
˜ ¸ fτ 2 |Rs ptm q| ¨ exp jπ b
˜ ¸ fτ 2 |Rs ptm q| ¨ exp jπ Sa pti , f τ q “ exp ´j4π b « ˜ ¸ff „ 2 4 2 4V tm f τ 4V tm 4πV 2 ¨ exp jπ ` ¨ exp ´j t m λb |Rs ptm q| λ2 b |Rs ptm q|2 cλ „
ˆ
1 fτ ` λ c
(21)
˙
(22)
where V denotes the flight speed. By comparing Equations (20)–(22), differences between these three echo expressions can be summarized as follows:
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(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: ∆Vm , ∆rm 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 , ∆Vm , and ∆rm are important, their space-variant properties and influence on focusing quality will be analyzed in the following. 3.1. Time-Scaling Factor Suppose k m,p and k m,c denote time-scaling factors corresponding to a certain target PT in the swath and the swath center, respectively. When the echo from PT 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 ˆ ∆ψkm f η , f τ “ `
˘
4k m,p π λb
˙ ˆ ˙ ˙ ˆ ˙ ˆ k m,p 4k m,c π k m,c 2 2 ´ ¨ f τ tk,p ` t ¨ f τ tk,c ` t λ k,p λb λ k,c
where
(23)
tk,p «
Rp λR p f η sinϕ p cosϕ p ´ b Vp Vp 4Vp 2 ´ λ2 f η 2
(24)
tk,c «
λRc f η sinϕc Rc cosϕc ´ b Vc Vc 4Vc 2 ´ λ2 f η 2
(25)
and f η denotes the azimuthal frequency. Vp , ϕ p , and R p are the equivalent velocity, the equivalent squint angle, and the reference range corresponding to PT , respectively, and Vc , ϕc , and Rc are those corresponding to the swath center. The equivalent velocity Ve and the equivalent squint angle ϕe can be calculated according to d ˆ ˙ λR0 f r λ fd 2 ` Ve “ (26) 2 2 ˆ ˙ λ fd ϕe “ arccos ´ (27) 2Ve where f d is the Doppler centroid frequency, f r is the Doppler frequency rate, and R0 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 ∆ψkm f η , 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 k
m ,c Sens. Remote 8, 223 can2016, be applied
to imaging of the whole swath.
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(a)
(b)
Figure6.6.Phase Phaseerrors errorsinduced induced the time-scaling factor distribution of residual the residual phases m . The Figure byby the time-scaling factor k m .kThe distribution of the phases in in two-dimensional the two-dimensional frequency domain atrange the range (a) and the azimuth the frequency domain at the edgeedge (a) and the azimuth edgeedge (b). (b).
3.2.Transmitting–Receiving Transmitting–ReceivingRate Rate∆VΔV m 3.2. m ΔV m,,cwhich corresponds to the swath center, is applied to IfIfthe m,c , which corresponds to the swath center, is applied to thetransmitting–receiving transmitting–receivingrate rate∆V imaging for the whole swath, then the residual imaging for the whole swath, then the residualphase phaseisis
` ˘ 1 f τf τ + ) = 4π1 λ` ∆ψ∆VΔmψ Δf ηV, f(τf η ,“f τ4π λ cc ˆ
m
` ˘ V mm,p V m ,c ) k ,p ( Δ , p −´Δ∆V tt k,p ∆V m,c
˙
(28) (28)
ΔV , p the transmitting–receiving rate corresponding to a certain position in the swath. where is the transmitting–receiving rate corresponding to a certain where∆Vm,pmis ` ˘position in the swath. According to Table 1, Figure 7 gives the residual phases ∆ψ ∆VΔmV (f ηf η, ,f τf τ )at the range edge and Δψ According Table 1, Figure 7 gives the7.5 residual phases at the edge and the azimuth edge,torespectively, which are all km away from the swath center. As range demonstrated 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 in Figure the residual phase phase is less in than 1.5°,7bwhich will the focusing quality at the ˝ ataffect range edge.7a, Although the residual Figure reaches 110not the azimuth edge, the variation range edge. Although the residual phase in Figure 7b reaches 110° at the azimuth edge, the is linear with the azimuth frequency, which means only an offset along azimuth will be produced. variation is linear with the azimuth frequency, which means only an offset along azimuth will This offset is less than 0.14 m. The above analysis indicates that the residual phase caused by applyingbe produced. offsetonly is less 0.14 m. The aboveatanalysis indicates that thenot residual ∆V insteadThis of ∆V leadsthan to geometric distortion the swath edge, and does degradephase the m
m,c
m,p
ΔV m , p in imaging as the transmitting–receiving rate for the ΔV m ,c∆V will be adopted focusing Therefore, instead of only leads to geometric distortion at the swath edge, caused quality. by applying m,c Remote Sens. 2016, 8, 223 10 of 24 whole swath. and does not degrade the focusing quality. Therefore, ΔV m,c will be adopted in imaging as the transmitting–receiving rate for the whole swath.
(a)
(b)
Figure 7. Phase errors induced by the transmitting–receiving rate ΔV . The distribution of the Figure 7. Phase errors induced by the transmitting–receiving rate ∆Vm . Themdistribution of the residual residual in the two-dimensional frequency domain the (a) range (a) andedge the azimuth phases in phases the two-dimensional frequency domain at the rangeatedge and edge the azimuth (b). edge (b).
3.3. Transmitting–Receiving Constant Δrm According to Equation (6), the variation of Δrm in the whole swath is demonstrated in Figure 8. Δrm does not change along the range direction and varies linearly with the azimuth. The maximum of Δrm 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 Δr
(a)
(b)
Figure 7. Phase errors induced by the transmitting–receiving rate ΔV m . The distribution of the residual phases in the two-dimensional frequency domain at the range edge (a) and the azimuth edge (b). Remote Sens. 2016, 8, 223
3.3. Transmitting–Receiving Constant Δrm
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According to EquationConstant (6), the variation of Δrm in the whole swath is demonstrated in Figure 8. 3.3. Transmitting–Receiving ∆rm Δrm does not change along the range direction and varies linearly with the azimuth. The According to Equation (6), the variation of ∆rm in the whole swath is demonstrated in Figure 8. Δrm isalong ∆r does not the range direction linearly maximum maximum of change less than two meters at and the varies azimuth edge with and the azimuth. minimumThe is zero at the m
of ∆rm is less thanTherefore, two meters at the azimuth edge and zero at the azimuthwill center. azimuth center. according to Equation (20),the anminimum offset of is the range migration be Therefore, according to Equation (20), an offset of the range migration will be produced by ∆r along Δ r m m along the range direction and this offset varies with the azimuth. produced by the range direction and this offset varies with the azimuth.
Figure thethe whole swath. It is that ∆rmΔdoes not change along the rm does Figure 8.8.The Thevariation variationof of∆rΔmrmin in whole swath. It obvious is obvious that not change along range direction, and varies linearly with the azimuth. The absolute maximum of ∆rm isofless two Δrthan the range direction, and varies linearly with the azimuth. The absolute maximum is less m meters at the azimuth and edge, the minimum is zero at the swath center. than two meters at theedge, azimuth and the minimum is zero at the swath center. R + Δr m should be Because Δrmm, ,the Because of of ∆r theazimuth azimuthmatching matchingfilter filterdesigned designed for for the the range range gate gate R p p` ∆rm should be applied to process echoes from targets like PT whose reference range equals RRp . However, if the p T whose reference range equals . However, if the applied to process echoes fromcorresponding targets like Pto transmitting–receiving constant the swath center, i.e., ∆rm “ 0, is adopted in imaging Δrm =echoes 0 , is adopted for the whole swath, theconstant same azimuth matching to filter be applied process for targets in transmitting–receiving corresponding thewill swath center, toi.e., whose reference ∆rm . azimuth This leadsmatching to a quadratic azimuthal phasetoerror, which can for be imaging for the range wholeequals swath,Rthe filter will be applied process echoes p `same expressed as πλ∆rm ∆ψ∆rm “ (29) 8ρ a 2 sin2 ϕ
where ρ a denotes the azimuthal resolution. According to Equation (29), the phase error ∆ψ∆rm will lead to more azimuthal mismatching and defocusing as the target is much farther away from the swath center along azimuth, because ∆ψ∆rm increases with ∆rm . Therefore, different ∆rm must be adopted for different echo segments along the azimuth. 4. 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 , ∆Vm , and ∆rm , a compensation algorithm with better focusing performance is proposed as follows.
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(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 ∆rm 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 ∆rm 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 ∆rm at the edge of every swath block satisfies the requirements of the focusing quality. For the lth echo segment, the transmitting–receiving rate ∆Vm,l and the transmitting–receiving constant ∆rm,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 mt 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 SE pti , f τ q, which has the same form as Equation (20). (3) The first compensation factor ∆1 pti , f τ q is designed as follows „ ˆ ˙ „ fτ 4π f τ 1 ` ∆Vm,l ti ¨ exp j ∆rm,l ∆1 pti , f τ q “ exp j4π λ c c
(30)
After multiplying SE pti , f τ q by ∆1 pti , f τ q, a Fourier transform along the azimuth is performed to ` ˘ acquire S F f η , f τ , which is ˙ ˆ ˙ f τ2 R0 cosϕ “ exp jπ ¨ exp ´j2π f η dV ¸ ˜ b c2 f η2 4π p f 0 ` f τ q R0 sinϕ ¨ exp ´j 1´ c 4V 2 p f 0 ` f τ q2˛ $ ¨ & ˆ 4k π ˙ λR0 f η sinϕ R0 cosϕ m,l ‚ ¨ exp j ¨˝ ´ b % λb V V 4V 2 ´ λ2 f η 2 , » ¨ ˛fi 2 2 λR0 f η sinϕ k πλ R0 3 . k m,l ˝ R0 cosϕ ‚fl ` j m,l ¨ – fτ ` ´ b fη λ V cV 4 sin2 ϕ V 4V 2 ´ λ2 f 2 ˆ
`
SF f η , f τ
˘
(31)
η
` ˘ (4) The second compensation factor ∆2 f η , f τ is designed as ∆2 f η , f τ `
˘
$ ¨ ˛ & ˆ 4k π ˙ λR f sinϕ R cosϕ 0 η m,l 0 ‚ “ exp ´j ¨˝ ´ b % λb V 2 2 2 V 4V ´ λ f η , » ¨ ˛fi 2 2 λR0 f η sinϕ k πλ R0 3 . k m,l ˝ R0 cosϕ ‚fl ´ j m,l ¨ – fτ ` ´ b fη λ V cV 4 sin2 ϕ V 4V 2 ´ λ2 f 2
(32)
η
` ˘ ` ˘ After multiplying S F f η , f τ with ∆2 f η , f τ , an inverse Fourier transform along the azimuth equals SK pti , f τ q
“ ` ˘ ` ˘‰ “ IFFT f η S F f η , f τ ¨ ∆2 f η , f τ ˜ ¸ „ ˆ ˙ 1 fτ fτ 2 “ exp ´j4π ` |Rs pti q| ¨ exp jπ λ c b
(33)
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(5) SK pti , 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 SK pti , 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 Sens. 2016, 8, 223 13 of 24 wholeRemote swath.
Figure 9. imaging An imaging compensationalgorithm algorithm based based on tangent model. Figure 9. An compensation onthe thecontinuous continuous tangent model.
5. Simulation and Validation
5. Simulation and Validation
5.1. Simulation Method and Parameters
5.1. Simulation Method and Parameters
SAR echo simulation methods are typically based on the stop-go model. A new method is
SAR echo here simulation methods are typically based onsatellite. the stop-go model. new method proposed to simulate the continuous motion of the The key to thisAmethod is to is proposed here to simulate the continuous motion of the satellite. The key to this method is to acquire acquire an accurate transmitting distance according to the receiving distance for every echo an accurate transmitting distance according to the receiving distance for every echo sampling point. sampling point. As demonstrated in Figure moment δδi i for for the theithithsampling sampling point during As ‰demonstrated in Figure1,1,the thereceiving receiving moment point during “ tn , tn ` 1{ f can be expressed as t n , t n +p 1 f p
can be expressed as
ii δi “δtin=`t nτ+gτ` i i“= 0, 0,1,1,2,2, ¨ ¨ ¨ g + FFs s
(34) (34)
τ g sampling start time and Fs is Fthe wherewhere τg is the rate. is the sampling start time and s issampling the sampling rate. In order to calculate the transmitting moment corresponding to δδii, iterations are performed as , iterations are performed In order to calculate the transmitting moment corresponding to demonstrated in Figure 10. Detailed stepssteps are as as demonstrated in Figure 10. Detailed arefollows. as follows.
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Figure 10. A workflow diagram of the simulation method. The core of the flow chart is to accurately
Figure 10. 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 estimate the transmitting moment corresponding to every sampling moment within the synthetic aperture time. Afterwards, the transmitting and receiving distances can be calculated and aperture time. Afterwards, the transmitting and receiving distances can be calculated and substituted substituted into the echo expression to achieve simulated echoes. into the echo expression to achieve simulated echoes.
(1) The initial estimation of the transmitting moment corresponding to δ i is assumed to be
(1) The initial estimation of the transmitting moment corresponding to δi is assumed to be ζ i ,0 = δ i − 2R (δ i ) c
where
R (δ i )
ζ
(35)
(35)
“ δ ´ 2R pδ q{c
i,0 i i is the relative distance between SAR and the target at the moment δ i .
where R(2) pδi qIfisEquation the relative betweenthe SAR and the target at the δi .to be R (ζ i ,k ) , (36) distance can be satisfied, transmitting distance is moment considered ` ˘ (2) If Equation (36) can be satisfied, the transmitting distance is considered to be R ζ i,k , which which denotes the relative distance at ζ i ,k ( k = 0,1, 2, ). Otherwise, Step (3) is performed. denotes the relative distance at ζ i,k (k “ 0, 1, 2, ¨ ¨ ¨ ). Otherwise, Step (3) is performed.
(3) Let
ζ i ,k +1 = ζ i , k − Δζ
ˇ `R (ζ ˘i ,k ) + R (δ i ) − (δ i − ζ i ,k ) < 10 ˇ −15 ˇR ζ ` ˘ˇˇ ˇ i,k `c R pδi q ´ δi ´ ζ i,k ˇ ă 10´15 ˇ ˇ ˇ c
(36)
(36)
. Δζ is
(3) Let ζ i,k`1 “ ζ i,k ´ ∆ζ. ∆ζ is
Δζ =
( R (ζ i , k ) + R (δ i ) )
c − (δ i − ζ i , k )
` ` ˘ ` ˘ 2 ˘ R ζ i,k ` R pδi q {c ´ δi ´ ζ i,k ∆ζto“Step (2). (4) Let k = k + 1 , and return 2
(37)
(37)
By performing the iteration method for every sampling point, the transmitting and receiving (4) Let k can “ k be ` 1, and return to Stepand (2).substituted into Equation (15) to form the simulated distances precisely calculated By performing the iteration method for every sampling point, the transmitting and receiving echoes. parameters are listed in Table 2. The simulated swath covering area the of 15simulated km × 15 kmechoes. distancesSimulation can be precisely calculated and substituted into Equation (15) toanform isSimulation demonstrated in Figure 11, where three point targets, P 1 , P 2 , and P 3 , are arranged at the parameters are listed in Table 2. The simulated swath covering an area ofcenter, 15 kmthe ˆ 15 km range edge, and the azimuth edge, respectively. is demonstrated in Figure 11, where three point targets, P , P , and P , are arranged at the center, the range 1
edge, and the azimuth edge, respectively.
2
3
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Table parameters. Table2.2.Simulation Simulation parameters. Table 2. Simulation parameters. Parameters Value Parameters Value Orbital inclination 98.06° Parameters Value Orbital inclination 98.06˝ Orbital height 680 km Orbital inclination 98.06° Orbital height 680 km Eccentricity 0.001 Orbital height 680 km Eccentricity 0.001 Elevation angle 35° Eccentricity 0.001 Elevation angle 35˝ Wavelength 0.03 Wavelength 0.03 m Elevation angle 35°m ˝ Beam width (Azimuth) 0.305° Beam width (Azimuth) 0.305 Wavelength 0.03 m ˝ Squint angle ˘6.21 ±6.21° BeamSquint width angle (Azimuth) 0.305° ˝ /s Beam rotation velocity 0.42 BeamSquint rotation velocity 0.42°/s angle ±6.21° Signal bandwidth 1.0GHz GHz Signal bandwidth 1.0 Beam rotation velocity 0.42°/s Pulse width 40 µs Pulse width 40GHz μs Signal bandwidth 1.0 Pulse repetition frequency 4000 Hz Pulse repetition frequency 4000 Hz Pulse width 40 μs
Pulse repetition frequency
4000 Hz
Figure 11. The simulated swath:PP at the the swath swath center. P3 P are at the range edgeedge and the Figure 11. The simulated swath: center.P2Pand at the range and the 1 1isisat 2 and 3 are azimuth edge, which 7.5 km away from azimuth edge, respectively, are away fromP . P3 are at the range edge and the Figure 11. Therespectively, simulatedwhich swath: Pare 1 all isall at7.5 thekm swath center. PP12.1and azimuth edge, respectively, which are all 7.5 km away from P1.
5.2. Simulation Results
5.2. Simulation Results
5.2. Simulation Results Figures 12–14 demonstrate the imaging results of P1, P2, and P3, where the profiles represent
Figures 12–14 demonstrate imaging resultsand of contour P1 , P2 , and Pdenote profiles represent 3 , where the focusing along the the range and azimuth, maps thethe two-dimensional 1, P2, and P3, where the profiles represent Figures quality 12–14 demonstrate the imaging results of P the focusing quality In along the range andare azimuth, and contour maps denote theachieved two-dimensional focusing quality. there three kinds results.maps The first result using the focusing qualityevery alongfigure, the range and azimuth, andof contour denote theistwo-dimensional focusing quality. In ωk every figure, there are three kinds The first is achieved achieved using the the Deramping which is are based on the of stop-go model The second is achieved focusing quality. Inalgorithm, every figure, there three kinds ofresults. results. The[18]. first result result is using Deramping ωktoalgorithm, which iswhich basedis thewhich stop-go modelon [18]. is achieved according according theωk algorithm presented inon [15], based the The continuous rectilinear model. the Deramping algorithm, based on theisstop-go model [18].second The second is achieved third istoachieved using novel imaging compensation method in this paper.The to theThe algorithm presented inthe [15], which based onisthe continuous rectilinear model. third is according the algorithm presented in is [15], which based on thepresented continuous rectilinear model. achieved using novelusing imaging compensation method presented in this paper. The third is the achieved the novel imaging compensation method presented in this paper.
(a) (a)
(b) Figure 12. Cont. Figure 12. Cont. Figure 12. Cont.
(b)
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(i) Figure 12. Imaging results of P1, the point at the swath center. (a–c) The range profile, the azimuth
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 profile and the contour map profile, achieved using the Deramping based on the stop-go model; model; (d–f) The range the azimuth profile and ωk thealgorithm contour map achieved using the (d–f)algorithm 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 andbased on the continuous rectilinear model; rangecompensation profile, the azimuth profile and thepaper. contour map the contour map achieved using the(g–i) novelThe imaging method presented in this achieved using the novel imaging compensation method presented in this paper.
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Figure 13. Cont.
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Figure 13. Imaging results of P2, the point at the range edge. Images (a–i) are in accordance with
Figure 13. Imaging results of of P2P, 2the edge.Images Images (a–i) in accordance Figure 13. Imaging results ,12. thepoint pointat at the the range range edge. (a–i) areare in accordance with with Figure Figure 12. Figure 12.
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(i) Figure 14. Imaging results of P3, the point at the azimuth edge. Images (a–i) are in accordance with Figure 12.
Figure 14. Imaging results of P3 , the point at the azimuth edge. Images (a–i) are in accordance with Figure 12. 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 By applying resolution, to side lobe ratio (PSLR), and integrated side lobe ratio (ISLR) as presented in Tables 3 and 4,peak respectively.
indicators to evaluate Figures 12–14 the evaluation results along the range and azimuth are presented in Tables 3 and 4 respectively.Table 3. Evaluation of imaging quality along range. Motion Model
Target
Resolution (m)
PSLR (dB)
Table 3. EvaluationP1of imaging quality 0.1444 along range.−16.49 Stop-go model Motion Model Continuous rectilinear Stop-go model model
Continuous tangent model Continuous rectilinear model
P2 P3 Target P1 P1 P2 P2 P3 P3 P1 P1 P2 P2 P P3 3
0.1433 0.1444 Resolution (m) 0.1342 0.1444 0.1342 0.1433 0.1342 0.1444 0.1342 0.1342 0.1342 0.1342 0.1342 0.1342
−16.10 PSLR−15.42 (dB) −13.25 ´16.49 −13.26 ´16.10 −13.05 ´15.42 −13.25 ´13.25 −13.25 ´13.26 −13.13 ´13.05
P1 0.1342 ´13.25 Table 4. Evaluation Continuous tangent model P2 of imaging quality 0.1342 along azimuth. ´13.25 P3 0.1342 (m) ´13.13 Motion Model Target Resolution PSLR (dB)
0.2333 −15.64 P1 Stop-go model 2 imaging quality 0.2350 along azimuth. −15.29 Table 4. EvaluationPof 0.2420 −11.98 P3 0.2156(m) −13.07 P1 Motion Model Target Resolution PSLR (dB) P2 0.2203 −13.11 Continuous rectilinear model P1 0.2333 ´15.64 0.2250 −11.53 P3 Stop-go model P2 0.2350 ´15.29 1 0.2156 −13.08 P P3 0.2420 ´11.98 P2 0.2203 −13.11 Continuous tangent model P1 0.2156 ´13.07 P3 0.2170 −13.32 P2 0.2203 ´13.11 Continuous rectilinear model P3 0.2250 ´11.53 Continuous tangent model
P1 P2 P3
0.2156 0.2203 0.2170
´13.08 ´13.11 ´13.32
ISLR (dB) −13.54 −13.10 ISLR−12.51 (dB) −9.96 ´13.54 −9.96 ´13.10 −9.77 ´12.51 −9.96 ´9.96 −9.95 ´9.96 −9.80 ´9.77 ´9.96 ´9.95 ´9.80 ISLR (dB)
−13.35 −13.05 −9.79 −10.39 ISLR (dB) −10.40 ´13.35 −8.63 ´13.05 −10.39 ´9.79 −10.41 ´10.39 −10.41 ´10.40 ´8.63 ´10.39 ´10.41 ´10.41
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According to imaging profiles and evaluation results, the first imaging algorithm, which is based Remote Sens. 2016, 8, 223 20 of 24 on the stop-go model, exhibits the worst performance. Along the range and azimuth, the main lobe corresponding to every target profiles is broadened, which results, reduces resolution. Although PSLR and ISLR According to imaging and evaluation thethe first imaging algorithm, which is based are better this method than the methods, it comes at the cost of a loss in resolution. on thefor stop-go model, exhibits theother worst two performance. Along the range and azimuth, the main lobe corresponding to everybrings target is broadened, phenomena which reducestothe resolution. PSLRindicates and ISLR that Furthermore, this method asymmetric azimuth sideAlthough lobes, which are better for this method than the other two methods, it comes at the cost in of imaging. a loss in resolution. quadratic and high-order phase errors are not compensated for completely Furthermore, this method brings asymmetric phenomena to azimuth side lobes, which The second algorithm, which is based on the continuous rectilinear model, canindicates achievethat thinner quadratic and high-order phase errors are not compensated for completely in imaging. main lobes. Compared with the first method, the range resolutions are improved by about 6.35%–7.06%. The second algorithm, which is based on the continuous rectilinear model, can achieve thinner The azimuth resolutions are improved by about 6.26%–7.59%. However, for P3 , which is at the azimuth main lobes. Compared with the first method, the range resolutions are improved by about edge of the swath, the azimuth resolution, PSLR, and ISLR decay obviously compared with the swath 6.35%–7.06%. The azimuth resolutions are improved by about 6.26%–7.59%. However, for P3, which center,isas in Figure and Table 4. atdemonstrated the azimuth edge of the14 swath, the azimuth resolution, PSLR, and ISLR decay obviously The algorithm based on the continuous tangent the4.focusing performance with compared with the swath center, as demonstrated inmodel Figure preserves 14 and Table the secondThe method at the swath center and the range edge, and further the focusing quality algorithm based on the continuous tangent model preservesimproves the focusing performance the second at the swath center and theand range edge, and further improves theimproved focusing by at thewith azimuth edge.method The azimuth resolution, PSLR, ISLR corresponding to P3 are at and the 1.78 azimuth edge. The azimuth resolution, PSLR, and ISLR to Pnearly 3 are the 3.56%,quality 1.79 dB, dB, respectively. As a result, focusing qualities for Pcorresponding , P , and P are 1 2 3 improved by 3.56%, 1.79 dB, and 1.78 dB, respectively. As a result, focusing qualities for P 1, P2, and same, which indicate that imaging coherency for the whole swath is good. P3 further are nearly the same, which indicate that imaging coherency algorithm, for the whole swath is good. To demonstrate the performance of the proposed a two-dimensional example To further demonstrate the performance of the 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 0.1 m and 66 m ˆ 66 m, respectively. It is deployed Airborne ISR). The resolution and size of this image are 0.1 m and 66 m × 66 m, respectively. It is at thedeployed top left corner of the simulated andswath, the simulated echo canecho be achieved. By applying at the top left corner of theswath, simulated and the simulated can be achieved. By algorithms based on the stop-go model, continuous model, and the model, continuous applying algorithms based on the the stop-go model, rectilinear the continuous rectilinear and tangent the model, different tangent imaging results are acquired 15).acquired (Figure 15). continuous model, different imaging(Figure results are
(a)
(b)
(c)
(d) Figure 15. Cont.
Figure 15. Cont.
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(e)
(f)
(g)
(h)
Figure 15. Comparison of imaging results: (a) the real SAR image; (b) enlarged view of selectable
Figure 15. Comparison of imaging results: (a) the real SAR image; (b) enlarged view of selectable portion in (a); (c) stop-go model image; (d) enlarged view of selectable portion in (c); (e) the portion in (a); (c) stop-go model image; (d) enlarged view of selectable portion in (c); (e) the continuous continuous rectilinear model image; (f) enlarged view of selectable portion in (e); (g) continuous rectilinear model viewofofselectable selectable portion in (e); (g) continuous tangent model tangent modelimage; image;(f) (h)enlarged enlarged view portion in (g). image; (h) enlarged view of selectable portion in (g).
There are three red rectangles in every enlarged view of the selectable portion. As shown in the bottom rectangle of Figure 15b, there are three scattering two centers are in There are three red rectangles in every enlarged viewcenters. of the However, selectableonly portion. As shown recognized in Figure 15d, while three centers can be distinguished in both Figure 15f,h. This the bottom rectangle of Figure 15b, there are three scattering centers. However, only two centers are indicates that the algorithm based on the stop-go model has the worst focusing quality. In the top recognized in Figure 15d, while three centers can be distinguished in both Figure 15f,h. This indicates rectangle of Figure 15b, only the edge of one scattering center exists. However, something more, that the algorithm on the stop-go model has the worst15f,h. focusing quality. In spot the top rectangle caused by side based lobes, appears at the same positions of Figure This is an isolate in the top of Figure 15b, of only the 15h. edgeThe of spot one in scattering center exists. However, something caused by rectangle Figure the top rectangle of Figure 15f is connected with more, the scattering side lobes, at the same positions of Figure 15f,h. This is an isolate the top centerappears because the algorithm based on the continuous rectilinear model has spot worseinPSLR and rectangle ISLR of Figure The spotalgorithm in the top rectangle Figure 15ftangent is connected withdifference the scattering center than 15h. the proposed based on theofcontinuous model. The between middle rectangles in Figure 15f,h also demonstrates this point. As a result, the algorithm proposed because the algorithm based on the continuous rectilinear model has worse PSLR and ISLR than the in this paper hasbased betteron focusing performance than other twoThe algorithms. proposed algorithm the continuous tangent model. difference between middle rectangles
in Figure 15f,h also demonstrates this point. As a result, the algorithm proposed in this paper has 5.3. Computational Load better focusing performance than other two algorithms. Computational load is a key element restricting the application of an algorithm. As demonstrated Load in Figure 9, after the first and second compensations, echoes can be successfully 5.3. Computational focused using current algorithms based on the stop-go model. Therefore, the analysis of
Computational load is a key element restricting the application of an algorithm. As demonstrated computational load focuses on the first and second compensations. Although the chirp scaling in Figure 9, after the second compensations, echoes cankm beswath successfully focused using current algorithm (CSA)first [8,9]and cannot achieve 0.21 m resolution and 15 width for spaceborne SAR, algorithms based on the stop-go model. Therefore, the analysis of computational load focuses on the it is worth comparing CSA and these compensations from the aspect of the computational load first and second the chirp scaling algorithm (CSA) [8,9] cannot achieve because CSAcompensations. is recognized as anAlthough efficient algorithm and has been widely applied. 0.21 m resolution and 15 km swath width for spaceborne SAR, it is worth comparing CSA and these compensations from the aspect of the computational load because CSA is recognized as an efficient algorithm and has been widely applied.
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Computational load is evaluated according to the complex multiplication and addition in the algorithm. Multiplication of two complex numbers and addition of two real numbers require 6 FLOPs (floating point operations) and 1 FLOP, respectively. An FFT or IFFT of length N requires 5Nlog2 pNq FLOPs [4]. Suppose sampling numbers along the azimuth and range are Nazi and Nrng , respectively. The computational loads of CSA, and the first and second compensations in the proposed algorithm, are, respectively, ` ˘ NCSA “ 10Nrng Nazi log2 pNazi q ` 10Nrng Nazi log2 Nrng ` 18Nrng Nazi ` ˘ NPA “ 15Nrng Nazi log2 pNazi q ` 10Nrng Nazi log2 Nrng ` 24Nrng Nazi
(38)
In order to achieve 0.21 m resolution and a swath of 15 km ˆ 15 km, Nazi and Nrng should be at least 118,650 and 150,000, respectively. According to Equation (38), the computational load of the first and second compensations is 1.25 times that of CSA, which indicates that the compensations are efficient. 6. Conclusions This paper developed a continuous tangent model to describe the relative motion between a spaceborne SAR and its targets. An imaging compensation algorithm was proposed based on the new motion model. Compared with the algorithm based on the stop-go model, the novel algorithm improved the range resolutions and the azimuth resolutions by about 6.35%–7.06% and 6.26%–10.33%, respectively. Compared with the algorithm based on the continuous rectilinear model, the novel algorithm further improved the azimuth resolution, PSLR, and ISLR at the azimuth edge by 3.56%, 1.79 dB, and 1.78 dB, respectively. Simulation results indicated that the presented algorithm provided a superior and more consistent focusing quality across the whole swath. The novel algorithm is also applicable to the updating of existing spaceborne SAR imaging processors that are designed based on the stop-go model. A pre-processing module is the only addition necessary to implement the first and second compensations, as demonstrated in Figure 9. After being processed by this pre-processing module, echoes can be successfully focused using existing processors to produce high-quality image products. The input parameters for the new algorithm are the same as the existing algorithms, which indicates that it is not necessary to change the input interface in the updating. However, as the resolution is improved, a higher requirement for the accuracy of the motion model will be put forward. Therefore, under the condition of better resolution, the continuous tangent model should be further verified as well as the corresponding imaging compensation algorithm. Author Contributions: All authors contributed equally to this paper. Conflicts of Interest: The authors declare no conflict of interest.
Appendix A As a result of the moving distance between transmitting and receiving locations for spaceborne SAR being less than 100 m, which is very small relative to the orbital height, the following formula holds: “ ‰ Rr ptn , ∆τq ´ Rs ptm q « Vm ¨ ∆τ ` pn ´ mq { f p (A1) The term |Rr ptn , ∆τq| ´ |Rs ptm q| then satisfies |Rr ptn , ∆τq| ´ |Rs ptm q|
ˇ “ ‰ˇ « ˇRs ptm q ` Vm ¨ ∆τ ` pn ´ mq { f p ˇ ´ |Rs ptm q| ‰ Rs ptm q ¨ Vm “ « ∆τ ` pn ´ mq { f p |Rs ptm q|
(A2)
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With the equation “ ‰ |Rs ptm q| ` |Rr ptn , ∆τq| “ c ¨ ∆τ ` pn ´ mq { f p
(A3)
the following formula can be acquired: 2 ¨ Rs ptm q ¨ Vm c
(A4)
“ Rs pt0 q ` Va ¨ tm
(A5)
|Rr ptn , ∆τq| ´ |Rs ptm q| « where
mř ´1
Rs ptm q “ Rs pt0 q `
Vi
i “0
Va “
fp mÿ ´1
Vi {m
(A6)
i “0
and Vi represents the relative velocity vectors at ti . Therefore, |Rr ptn , ∆τq| ´ |Rs ptm q| can be approximated as follows: |Rr ptn , ∆τq| ´ |Rs ptm q| «
2 ¨ V a ¨ Vm 2 ¨ Rs pt0 q ¨ Vm tm ` c c
(A7)
Appendix B According to Figure 1, the following expression can be derived: #
“ ‰ |Rs ptm q| ` |Rr ptn , ∆τq| “ c ¨ ∆τ ` pn ´ mq { f p “ ‰ |Rs ptm , τs q| ` |Rr ptn , ∆τ ` τr q| “ c ¨ ∆τ ` τr ´ τs ` pn ´ mq { f p
(B1)
Therefore, |Rs ptm , τs q| ` |Rr ptn , ∆τ ` τr q| ´ |Rs ptm q| ´ |Rr ptn , ∆τq| “ c ¨ pτr ´ τs q
(B2)
which can also be expressed as |Rs ptm q ` Vm τs | ` |Rr ptn , ∆τq ` Vr τr | ´ |Rs ptm q| ´ |Rr ptn , ∆τq| “ c ¨ pτr ´ τs q
(B3)
By solving Equation (B3), the relation between τs and τr can be shown to satisfy Rr ptn , ∆τq ¨ Vr |Rr ptn , ∆τq| τ Rs ptm q ¨ Vm r c` |Rs ptm q|
c´ τs “
(B4)
References 1.
2.
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