The Doppler effect for SAR 1

The Doppler effect for SAR 1 Semyon Tsynkov 2,3 3 Department of Mathematics North Carolina State University, Raleigh, NC 2016 AFOSR Electromagnetics Contractors Meeting January 5 7, 2016, Arlington, VA 1 Work supported by AFOSR, Dr. Arje Nachman 2 In collaboration with Dr. Mikhail Gilman, NCSU S. Tsynkov NCSU) The Doppler effect for SAR AFOSR, 1/6/16, Arlington, VA 1 / 25

Plan of Presentation 1 Motivation and objectives of the study 2 Basic SAR for reference purposes) Key assumptions Data inversion Factorized GAF 3 The role of antenna motion for SAR The Lorentz transform and Doppler effect Corrected matched filter for data inversion What if the filter were not corrected? 4 The Doppler effect in slow time Doppler interpretation of azimuthal reconstruction Signal compression in slow time 5 Discussion S. Tsynkov NCSU) The Doppler effect for SAR AFOSR, 1/6/16, Arlington, VA 2 / 25

Motivation and objectives of the study Start-stop approximation and its shortcomings Synthetic array is a set of successive locations of one antenna. Target in the far field of the antenna is in the near field of the array. The antenna is assumed motionless when sending/receiving signals simplifies the analysis yet neglects important effects: Displacement of the antenna during the pulse round-trip; Doppler frequency shift. The corresponding image distortions may be substantial, even though typically v c 1. Under the start-stop approximation, there may be no physical Doppler effect, because the velocity is considered zero. Yet azimuthal reconstruction is often attributed to the Doppler effect. This is a common misconception in the literature. S. Tsynkov NCSU) The Doppler effect for SAR AFOSR, 1/6/16, Arlington, VA 3 / 25

Motivation and objectives of the study Objectives To quantify the distortions of SAR images due to the start-stop approximation: To identify imaging regimes for which the distortions may be large; To identify key parameters that control the magnitude of distortions. To introduce a minimally invasive procedure for reducing or removing those distortions: To correct the matched filter so that it accounts for antenna motion; To show that the distortions become small after the correction; To compare against the previously studied ionospheric corrections. To analyze the Doppler interpretation of azimuthal reconstruction: To distinguish between the Doppler effect in fast time i.e., physical Doppler effect) and the Doppler effect in slow time; To identify the mechanism of signal compression in slow time and the parameters that define it. S. Tsynkov NCSU) The Doppler effect for SAR AFOSR, 1/6/16, Arlington, VA 4 / 25

Basic SAR Key model assumptions Key assumptions Imaging setup, propagation, and scattering: Linear flight path; Broadside stripmap imaging; Scalar radar signals no polarization effects); Unobstructed propagation between the antenna and the target; Linearized scattering at the target via the first Born approximation; Deterministic and dispersionless targets; Start-stop approximation for signal processing. Interrogating waveforms linear chirps: Pt) = At)e iω 0t, where At) = χ τ t)e iαt2 and χ τ t) is the indicator function of the interval [ τ/2, τ/2]. ω 0 central carrier frequency, τ duration, α = B 2τ chirp rate. The instantaneous frequency varies linearly along the chirp: ωt) def = d dt ω 0t + αt 2 ) = ω 0 + 2αt = ω 0 + B τ t, B 2 ω B 2. S. Tsynkov NCSU) The Doppler effect for SAR AFOSR, 1/6/16, Arlington, VA 5 / 25

Schematic Basic SAR Key assumptions orbit 3 x H θ L SA R R z ground track R R z x 1 L 0 φ z ψ beam footprint S. Tsynkov NCSU) The Doppler effect for SAR AFOSR, 1/6/16, Arlington, VA 6 / 25 2

SAR data inversion Basic SAR Data inversion Incident field retarded potential from the antenna at x R 3 : u 0) t, z) = 1 4π Pt z x /c). z x Scattered field for monostatic imaging ν ground reflectivity): u 1) t, x) νz)p t 2 x z /c) dz. SAR data inversion: reconstruct νz) from the given u 1) t, x). The inversion is done in two stages: Application of the matched filter range reconstruction); Summation along the synthetic array azimuthal reconstruction). S. Tsynkov NCSU) The Doppler effect for SAR AFOSR, 1/6/16, Arlington, VA 7 / 25

Basic SAR SAR data inversion cont d) Data inversion Matched filter R y y x, R z z x ): I x y) = Pt 2R y /c)u 1) t, x) dt χ = dz νz) dt Pt 2R y /c)pt 2R z /c). χ } {{ } W xy,z) PSF Synthetic aperture determined by the antenna radiation pattern): Iy) = I x ny) = W x ny, z)νz)dz n n [ ] = W x ny, z) νz) dz = Wy, z)νz) dz = W ν. n Wy, z) GAF or imaging kernel) convenient for analysis. Actual processing is for the entire dataset rather than for each y. S. Tsynkov NCSU) The Doppler effect for SAR AFOSR, 1/6/16, Arlington, VA 8 / 25

Basic SAR Factorization of the GAF Factorized GAF Factorized form of the GAF: Wy, z) W Σ y, z) W R y, z). The azimuthal factor L SA is the length of the array): W Σ y, z) = n e2ik 0R n z R n y) = e 2ik 0 Ly 2 z 2 ) R n e2ik 0y 1 z 1 ) e iφ k0 0 L ) Nsinc SA R y def 1 z 1 ) = e iφ 0 W A y, z). The range factor n = n c center of the array): W R y, z) = At 2R c y/c)at 2R c z/c)dt = χ = e iαt 2Rc y /c)2 e iαt 2Rc z /c)2 dt χ = e iα4trc y Rc z B ) )/c dt τsinc c Rc y R c z). χ L SA n RN S. Tsynkov NCSU) The Doppler effect for SAR AFOSR, 1/6/16, Arlington, VA 9 / 25

Resolution analysis Basic SAR Factorized GAF The GAF Wy, z) = Wy z) is the image of a point target. Yet Wy z) δy z), so the imaging system is not ideal. Resolution semi-width of the main lobe of the sinc ): Two point targets at this distance can be told apart. Azimuthal resolution: A = πr = πrc. k 0 L SA ω 0 L SA Range resolution: R = πc B. What would it be without phase modulation? The range resolution would be the length of the pulse τc. τc The actual range resolution is better by a factor of = Bτ R π. The quantity Bτ is the compression ratio of the chirp also TBP). 2π Compression ratio must be large a tip for waveform design. S. Tsynkov NCSU) The Doppler effect for SAR AFOSR, 1/6/16, Arlington, VA 10 / 25

Factorization error Basic SAR Factorized GAF Error due to factorized representation of the GAF: Error = Wy, z) W R y, z) W Σ y, z) W W RΣ). Denote T n = R n y R n z)/c and τ n = τ 2 T n, then W W RΣ) = n e 2iω 0T n [τ n sinc 2ατ n T n ) τ c sinc 2ατ c T c )]. The factorization error is caused by the dependence on n in the argument of the sinc ). The following estimate holds: maxw W RΣ) ) max W RΣ) π 8 B ω 0. For narrow-band pulses the factorization error is small. For wide-band pulses the factorized GAF may be inaccurate. Another tip for waveform design and resolution analysis. S. Tsynkov NCSU) The Doppler effect for SAR AFOSR, 1/6/16, Arlington, VA 11 / 25

The role of antenna motion for SAR Waves from moving sources The Lorentz transform and Doppler effect We consider a straightforward uniform motion of the antenna with the speed v yet do not employ the start-stop approximation: 1 2 u u c 2 t 2 2 z 2 2 u 1 z 2 2 u 2 z 2 3 = Pt)δz xt)) Pt)δz 1 vt)δz 2 + L)δz 3 H), β = The solution in Lorentz-transformed coordinates σ = 1 t vz ) 1 β c 2, ζ 1 = 1 ) vt + z 1, β is a retarded potential 1 v2 c 2 ): ) where ρ = ζ 21 + z 2 + L) 2 + z 3 H) 2 : u 0) σ, ζ 1, z 2, z 3 ) = 1 4πβ Pσ ρ/c)/β). ρ S. Tsynkov NCSU) The Doppler effect for SAR AFOSR, 1/6/16, Arlington, VA 12 / 25

The role of antenna motion for SAR The Lorentz transform and Doppler effect Schematic for the analysis of antenna motion x0) x xt) v γ z γ y L SA H r R y orbit flight track) 3 ground track L R z R θ 1 z 0 y beam footprint 2 S. Tsynkov NCSU) The Doppler effect for SAR AFOSR, 1/6/16, Arlington, VA 13 / 25

The role of antenna motion for SAR The Liénard-Wiechert potentials The Lorentz transform and Doppler effect Solution by Kirchhoff integral where µ = µt ) def = z xt ) + ct ): u 0) t, z) = 1 t dt δ z z ct t )) 4π R 3 t t Pt )δz xt ))dz = 1 4π = 1 4π = 1 4π t µt) δ z xt ) ct t )) t t Pt )dt z xt ) δµ ct) c z xt ) z 1 vt )v)t t ) Pt )dµ cpt ) c z xt ) z 1 vt )v µt )=ct. Equation µt ) = ct is simple for straightforward uniform motion: z 1 vt ) 2 + z 2 + L) 2 + z 3 H) 2 + ct = ct, in which case u 0) reduces to that obtained by Lorentz transform. Otherwise, it may be complex; often solvable only numerically. S. Tsynkov NCSU) The Doppler effect for SAR AFOSR, 1/6/16, Arlington, VA 14 / 25

The role of antenna motion for SAR The Lorentz transform and Doppler effect Waves from moving sources cont d) Incident field back in the original coordinates where r = x z ): u 0) t, z) 1 P t r c) 1 + v c cos γ )) z. 4π r Linear Doppler frequency shift can be obtained by taking t. The scattered field by the first Born approximation: u 1) t, x ) νz)p t r c x z ) 1 + v z) ) c c cos γ dz. Next steps another Lorentz transform, because the antenna is moving, and transformation back to the original coordinates: u 1) t, x ) νz)p t 1 + 2 v ) c cos γ z 2r 1 + v z) ) c c cos γ dz. The receiving location x is not the same as the emitting locationx: Two different factors multiplying the time t and the retarded time 2r c. S. Tsynkov NCSU) The Doppler effect for SAR AFOSR, 1/6/16, Arlington, VA 15 / 25

The role of antenna motion for SAR Corrected matched filter for data inversion Data inversion in the presence of antenna motion Not a fully relativistic treatment; the terms O v2 ) are neglected. c 2 Matched filter that accounts for antenna motion: I x y) = dzνz) χ dtp t 1 + 2 v c cos γ y P t 1 + 2 v ) c cos γ z ) 2R y c 2R z c 1 + v ) ) c cos γ y 1 + v c cos γ z) ), where the interior integral χ dt... is the PSF W xy, z). A potentially much simpler correction than that for the ionosphere. No parameters to be determined; only geometric data are needed. Summation along the synthetic array: W x ny, z)νz)dz Iy) = I x ny) = n n [ = n ] W x ny, z) νz) dz = Wy, z)νz) dz = W ν. S. Tsynkov NCSU) The Doppler effect for SAR AFOSR, 1/6/16, Arlington, VA 16 / 25

The role of antenna motion for SAR Corrected matched filter for data inversion The GAF in the presence of antenna motion The azimuthal factor: W Σ y, z) = n e iω 0T n κ n y +κn z ), where and T n = Rn y 1 v ) c c cos γn y Rn z 1 v ) c c cos γn z κ n y = 1 + 2 v c cos γn y and κ n z = 1 + 2 v c cos γn z. Reduces to the case of the start-stop approximation if v = 0. After evaluating the sum: W Σ y, z) e iφ k0 0 L Nsinc SA y 1 z 1 ) + 2 Ly 2 z 2 ) v ) ). R }{{}}{{ R c } original due to Doppler S. Tsynkov NCSU) The Doppler effect for SAR AFOSR, 1/6/16, Arlington, VA 17 / 25

The role of antenna motion for SAR Corrected matched filter for data inversion The GAF in the presence of antenna motion cont) The range factor: W R y, z) = e 2iαTn tκ n y )2 +κ n z )2 )+iω 0 tκ n y κn z ) dt χ B τsinc y 2 z 2 ) sin θ c } {{ } original O1) y 1 z 1 ) v { }}{ 1 + ω 0 c ) ) ) } c {{ αr 2 } due to Doppler Reduces to the case of the start-stop approximation if v = 0. In many cases, the quantity ω 0 c happens to be O1). αr 2 Otherwise, there may be substantial image distortions. The cross-contamination terms for W Σ and W R are small, O v c ). As such, both range and azimuthal resolution stay unaffected. Estimate for the factorization error where A = πrc ω 0 L SA ): maxw W RΣ) ) B π max W RΣ) 8ω 0 y 1 z 1 ) + Ly 2 z 2 ) v 4 + ω ) 0c. R c 2αR A S. Tsynkov NCSU) The Doppler effect for SAR AFOSR, 1/6/16, Arlington, VA 18 / 25

The role of antenna motion for SAR What if the filter were not corrected? Performance of SAR with no filter correction Plain non-corrected filter applied to Doppler-based propagator: I x y) = dzνz) dtp t 2R ) y χ c P t 1 + 2 v ) c cos γ z 2R z 1 + v z) ) c c cos γ. The azimuthal factor: W Σ y, z) e iφ 0 Nsinc k0 L SA y 1 z 1 ) + R v )). R c Displacement of the entire image in azimuth by R v c not large. The range factor: B W R y, z) τsinc y 2 z 2 ) sin θ y 1 z 1 v 1 + ω 0 c )) ). c 2 c αr 2 An insignificant coefficient of 1 2 in front of the Doppler term. S. Tsynkov NCSU) The Doppler effect for SAR AFOSR, 1/6/16, Arlington, VA 19 / 25

The role of antenna motion for SAR What if the filter were not corrected? Factorization error for non-corrected filter Estimate for the factorization error: maxw W RΣ) ) B π max W RΣ) 8ω 0 y 1 z 1 ) + v c R 1 + ω ) 0 c. αr 2 A Unlike in the case of a corrected filter, for typical imaging regimes the terms on the RHS are of the same order of magnitude. However, those terms are controlled by independent parameters. The term with ω 0 c may become large if, e.g., α is small. Then, B 8ω 0 αr 2 π ω 0 c v A 2αR c R = 1 8 ω 0τ v L SA c R = 1 4 τ ω max, where L ω max = ω SA v 0 2R c is the maximum absolute variation of the Doppler frequency shift over the length of the synthetic array. Ignoring this error may seriously underestimate image distortions. S. Tsynkov NCSU) The Doppler effect for SAR AFOSR, 1/6/16, Arlington, VA 20 / 25

The Doppler effect in slow time Doppler interpretation of azimuthal reconstruction Doppler interpretation of azimuthal reconstruction Start-stop approximation no physical Doppler effect. Is there an analogue of the Doppler effect that still plays a role? The range factor: W R y, z) = χ e iα4trc y R c z)/c dt = χ e 2iωt) ω 0)R c y R c z)/c dt, where ωt) ω 0 = 2αt, linear variation of instantaneous frequency. The azimuthal factor: W A y, z) = L SA n n e2ik 0 RN y 1 z 1 ) = n e2ikn)y 1 z 1 ), L where kn) = k SA n 0 represents a linear variation of the local RN wavenumber along the synthetic array. This linear variation can be though of as a chirp of length L SA in the azimuthal direction. S. Tsynkov NCSU) The Doppler effect for SAR AFOSR, 1/6/16, Arlington, VA 21 / 25

The Doppler effect in slow time The Doppler effect in slow time Doppler interpretation of azimuthal reconstruction The instantaneous wavenumber kn) can be transformed as kn) = k 0 cos γn y + γ n z 2 This is similar to the physical Doppler effect: v ω ω 0 = ω 0 cos γ. c def = k 0 cos γ n. Hence, the linear variation of kn) can be attributed to a Doppler effect in slow time n, and we can write: W A y, z) = n e 2ik 0 cos γ n y 1 z 1 ) = n e 2iω 0 cos γ n y 1 z 1 )/c. There is no velocity factor in the slow time Doppler effect, because everything can be thought of as taking place simultaneously. The geometric factor cos γ n is basically the same as that from the physical Doppler effect or the Doppler effect in fast time). S. Tsynkov NCSU) The Doppler effect for SAR AFOSR, 1/6/16, Arlington, VA 22 / 25

The Doppler effect in slow time Signal compression in slow time Signal compression in the azimuthal direction Compression ratio or TBP) of the actual chirp is the ratio of its length to the range resolution: τc = Bτ R π 1. It quantifies the improvement due to phase modulation. For the chirp in the azimuthal direction we have: L SA A = 2L2 SA λ 0 1 R 1, where λ 0 = 2πc ω 0 is the central carrier wavelength. Why is this quantity 1? Because 2L2 SA distance of the synthetic array. λ 0 is the Fraunhofer S. Tsynkov NCSU) The Doppler effect for SAR AFOSR, 1/6/16, Arlington, VA 23 / 25

Discussion Discussion The start-stop approximation neglects two phenomena: displacement of the antenna and the Doppler frequency shift. Both can be accounted for by correcting the matched filter. If the matched filter is corrected for antenna motion, then the effect of the latter on the image can be disregarded. Otherwise, the image may be subject to distortions, sometimes substantial, even though v c 1 for example, CWFM SAR). The physical Doppler effect under the start-stop approximation does not exist. Yet the azimuthal reconstruction can be interpreted with the help of the Doppler effect in slow time. The latter manifests itself as a linear variation of the instantaneous wavenumber along the synthetic array, i.e., a chirp-like behavior. The corresponding compression ratio is the ratio of the Fraunhofer distance of the array to the distance from the antenna to the target. S. Tsynkov NCSU) The Doppler effect for SAR AFOSR, 1/6/16, Arlington, VA 24 / 25

Discussion Discussion cont d) Potential image distortions due to the start-stop approximation should be kept in mind for designing the interrogating waveforms. A thorough numerical analysis of the resulting algorithm with filter correction may need to be conducted. Beyond the first Born approximation scattering at the target is actually due to Bragg resonances. Hence, the Doppler frequency shift may affect the observable quantity in SAR imaging. S. Tsynkov NCSU) The Doppler effect for SAR AFOSR, 1/6/16, Arlington, VA 25 / 25