Spaceborne SAR Semyon Tsynkov 1 1 Department of Mathematics North Carolina State University, Raleigh, NC http://www4.ncsu.edu/ stsynkov/ tsynkov@math.ncsu.edu +1-919-515-1877 Mathematical and Computational Aspects of Radar Imaging ICERM, Brown University, October 16 20, 2017 S. Tsynkov (NCSU) Spaceborne SAR ICERM, October 18, 2017 1 / 37
Collaborators and Support Collaborators: Dr. Mikhail Gilman (Research Assistant Professor, NCSU) Dr. Erick Smith (Research Mathematician, NRL) Support: AFOSR Program in Electromagnetics (Dr. Arje Nachman): Awards number FA9550-14-1-0218 and FA9550-17-1-0230 S. Tsynkov (NCSU) Spaceborne SAR ICERM, October 18, 2017 2 / 37
New research monograph (April 2017) S. Tsynkov (NCSU) Spaceborne SAR ICERM, October 18, 2017 3 / 37
What I plan to accomplish in this talk Spaceborne SAR is a vast area: Any attempt to do a broad overview will be superficial. Instead, I would like to: Present some recent findings that are unexpected/intriguing; Identify some common misconceptions in the SAR literature. Three subjects: Doppler effects and the start-stop approximation; Ionospheric turbulence; Interpretation of targets. S. Tsynkov (NCSU) Spaceborne SAR ICERM, October 18, 2017 4 / 37
Main idea of SAR Coherent overhead imaging by means of microwaves: Typically, P-band to X-band (1 meter to centimeters wavelengh). A viable supplement to aerial and space photography. To enable imaging, target must be in the near field instrument size must be very large unrealistic for actual physical antennas. Synthetic array is a set of successive locations of one antenna: Fraunhofer length of the antenna 2D 2 λ that of the array (aperture); Target in the far field of the antenna is in the near field of the array. S. Tsynkov (NCSU) Spaceborne SAR ICERM, October 18, 2017 5 / 37
Introduction: monostatic broadside stripmap SAR x 0 H θ antenna γ n L SA D R x n 3 orbit (flight track) ground track R n z n R y L θ 0 z y beam footprint 1 2 S. Tsynkov (NCSU) Spaceborne SAR ICERM, October 18, 2017 6 / 37
Conventional SAR data inversion Interrogating waveforms linear chirps: P(t) = A(t)e iω 0t, where A(t) = χ τ (t)e iαt2. ω 0 central carrier frequency, τ duration, α = B 2τ chirp rate. Incident field retarded potential from the antenna at x R 3 : u (0) (t, z) = 1 4π P(t z x /c). z x Scattered field for monostatic imaging (ν ground reflectivity that also absorbs the geometric factors): u (1) (t, x) ν(z)p (t 2 x z /c) dz. Obtained with the help of the first Born approximation. SAR data inversion: reconstruct ν(z) 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) Spaceborne SAR ICERM, October 18, 2017 7 / 37
Generalized ambiguity function (GAF) Matched filter (R y y x, R z z x ): I x (y) = P(t 2R y /c)u (1) (t, x) dt χ = dz ν(z) dt P(t 2R y /c)p(t 2R z /c). χ } {{ } W x(y,z) PSF Synthetic aperture (determined by the antenna radiation pattern): I(y) = I x n(y) = W x n(y, z)ν(z)dz n n [ ] = W x n(y, z) ν(z) dz = W(y, z)ν(z) dz = W ν. n W(y, z) GAF (or imaging kernel) convenient for analysis. Actual processing done for the entire dataset rather than for each y. S. Tsynkov (NCSU) Spaceborne SAR ICERM, October 18, 2017 8 / 37
Factorization of the GAF and resolution analysis Factorized form of the GAF: ( B ) ( W(y, z) = W(y z) τsinc c (y k0 L ) 2 z 2 ) sin θ Nsinc SA R (y 1 z 1 ). For narrow-band pulses the factorization error is small: O( B ω 0 ). W(y z) δ(y z), so the imaging system is not ideal. Resolution semi-width of the main lobe of the sinc ( ): Azimuthal: A = πr = πrc ; k 0 L SA ω 0 L SA What would it be with no phase modulation? Range: R = πc B. 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. Bτ is the compression ratio of the chirp (or TBP); must be large. 2π S. Tsynkov (NCSU) Spaceborne SAR ICERM, October 18, 2017 9 / 37
Start-stop approximation and its shortcomings The antenna is assumed motionless during the time of emission and reception of a given signal: Then, it moves to the next sending/receiving location. This assumption simplifies the analysis yet neglects two 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 frequently encountered mistake in the literature. S. Tsynkov (NCSU) Spaceborne SAR ICERM, October 18, 2017 10 / 37
Schematic for the analysis of antenna motion x(0) x γ y H x(t) v γ z L SA r R y orbit (flight track) 3 ground track L R z R θ 1 z 0 y beam footprint 2 S. Tsynkov (NCSU) Spaceborne SAR ICERM, October 18, 2017 11 / 37
Waves from moving sources Analysis uses Lorentz transforms or Liénard-Wiechert potentials. Incident field 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 in the original coordinates: ( ( u (1) (t, x ) ν(z)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) Spaceborne SAR ICERM, October 18, 2017 12 / 37
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 the antenna motion: ( ( I x (y) = dzν(z) dtp t 1 + 2 v ) ( c cos γ y 1 + v ) ) c cos γ y χ ( ( P t 1 + 2 v ) c cos γ z 2R y c 2R z c A simple correction; only geometric info is required. ( 1 + v c cos γ z) ). Summation along the synthetic array: I(y) = I x n(y) = W x n(y, z)ν(z)dz n n [ ] = W x n(y, z) ν(z) dz = W(y, z)ν(z) dz = W ν. n S. Tsynkov (NCSU) Spaceborne SAR ICERM, October 18, 2017 13 / 37
The GAF in the presence of antenna motion The azimuthal factor: ( k0 L ( W A (y, z) Nsinc SA (y 1 z 1 ) + 2 L(y 2 z 2 ) v ) ). R }{{}}{{ R c } original due to Doppler The range factor: typically O(1) ( B ( W R (y, z) τsinc (y 2 z 2 ) sin θ (y c }{{} 1 z 1 ) v {( }}{ 1 + ω 0 c ) ) ). } c {{ αr 2 } original due to Doppler Cross-contamination is small; the resolution remains unaffected. Factorization error is still O( B ω 0 ): max W W R W A max W R W A B 8ω 0 π A (y 1 z 1 ) + L(y ( 2 z 2 ) v 4 + ω ) 0c. R c 2αR S. Tsynkov (NCSU) Spaceborne SAR ICERM, October 18, 2017 14 / 37
SAR with no filter correction: the real start-stop Plain non-corrected filter applied to Doppler-based propagator: I x (y) = dzν(z) χ ( dtp t 2R y c ( ( P t 1 + 2 v c cos γ z ) ) 2R z c ( 1 + v c cos γ z) ). The factors W R and W A change insignificantly. Yet they cannot be directly used for resolution analysis due to the factorization error: max W W R W A B π max W R W A 8ω 0 (y 1 z 1 ) + v ( c R 1 + ω ) 0 c. αr 2 c αr 2 The term ω 0 A may become large if, e.g., α is small. Ignoring this error may seriously underestimate image distortions: FMCW SAR may be particularly prone to deterioration. S. Tsynkov (NCSU) Spaceborne SAR ICERM, October 18, 2017 15 / 37
ViSAR project by DARPA A SAR instrument to operate in the EHF band (at 235GHz). FMCW interrogating waveforms due to hardware limitations: FMCW = Frequency Modulated Continuous Wave. S. Tsynkov (NCSU) Spaceborne SAR ICERM, October 18, 2017 16 / 37
Doppler interpretation of azimuthal reconstruction Azimuthal reconstruction is often erroneously attributed to the physical Doppler effect, which does not exist under start-stop. Can one think of a meaningful counterpart? Linear variation of the instantaneous frequency along the chirp 2αt = ω(t) ω 0 yields the range factor of the GAF: W R (y, z) = e iα4t(rc y R c z)/c dt = e 2i(ω(t) ω 0)(R c y R c z)/c dt. χ In the azimuthal factor, there is a linear variation of the local wavenumber along the array, k(n) = k 0 L SA n RN : W A (y, z) = L SA n n e2ik 0 RN (y 1 z 1 ) = n e2ik(n)(y 1 z 1 ). Can be thought of as a chirp of length L SA in azimuth. χ S. Tsynkov (NCSU) Spaceborne SAR ICERM, October 18, 2017 17 / 37
Doppler effect in slow time The instantaneous wavenumber k(n) can be transformed as k(n) = 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 k(n) 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) Spaceborne SAR ICERM, October 18, 2017 18 / 37
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 (azimuthal compression ratio) 1? Because 2L2 SA λ 0 is the Fraunhofer distance of the synthetic array. S. Tsynkov (NCSU) Spaceborne SAR ICERM, October 18, 2017 19 / 37
Ionospheric distortions of SAR images EM waves in the ionosphere are subject to temporal dispersion: vgr < c group delay; vph > c phase advance; The duration τ of the chirp and its rate α change. Mismatch between the received signal and the matched filter. Can be reduced by adjusting the filter real-time TEC/gradients needed can be obtained with the help of dual carrier probing: Correction is more involved than in the case of Doppler; Correction is efficient if the moments don t vary over the image. However, the ionosphere is a turbulent medium. Synthetic aperture may be comparable to the scale of turbulence. Parameters of the medium will fluctuate from one pulse to another: Using a single correction may still leave room for mismatches. One needs to quantify the image distortions due to turbulence: How can one marry the deterministic and random errors? S. Tsynkov (NCSU) Spaceborne SAR ICERM, October 18, 2017 20 / 37
Monostatic broadside stripmap SAR x 0 H θ antenna γ n L SA D R x n 3 orbit (flight track) ground track R n z n R y L θ 0 z y beam footprint 1 2 S. Tsynkov (NCSU) Spaceborne SAR ICERM, October 18, 2017 21 / 37
Recalling the math of transionospheric SAR Convolution representation of the image (requires linearity): I(y) = ν(z)w(y, z)dz. The imaging kernel (or GAF generalized ambiguity function): W(y, z) = n e 2iω 0T n ph τ/2 τ/2 e 4i αn T n grt dt, where T n ph, gr = T ph, gr(x n, y, ω 0 ) T ph, gr (x n, z, ω 0 ) and T ph, gr (x, z, ω 0 ) = Factorization: R z W(y, z) τnsinc 0 1 v ph, gr (s) ds R z 0 1 ( 1 1 4πe 2 c 2 m e ω0 2 ) N e (s) ds = ( B(y2 z 2 ) sin θ ) ( ω0 (y 1 z 1 )L ) sinc SA. v gr R v ph R z v ph, gr. S. Tsynkov (NCSU) Spaceborne SAR ICERM, October 18, 2017 22 / 37
In the presence of turbulence The electron number density: N e = N e + µ(x), µ = 0. The travel times become random: T ph, gr (x, z, ω 0 ) = R z 1 4πe 2 v ph, gr 2c m e ω0 2 R z 0 µ(x(s))ds Accordingly, the GAF also becomes random (stochastic): R z v ph, gr ϕ 2c. W (y, z) = n τ/2 e 2iω 0T ph n e 4iαT n gr t dt, τ/2 where T ph, n gr = Rn y T ph, gr (x n, z, ω 0 ) = Rn y R n z ± ϕ n v ph, gr v ph,gr 2c. S. Tsynkov (NCSU) Spaceborne SAR ICERM, October 18, 2017 23 / 37
Statistics of propagation Correlation function of the medium (turbulent fluctuations): V(x, x ) def = µ(x )µ(x ) = µ 2 V r (r) = M 2 N e 2 V r (r), where V r (r) V r ( x x ) decays rapidly, e.g., V r (r) = e r/r 0. Other short-range correlation functions include Gaussian and Kolmogorov-Obukhov. Correlation radius of the medium (outer scale of turbulence): r 0 def = 1 V r (0) 0 V r (r)dr. Variance of the eikonal quantifies the magnitude of phase fluctuations: ϕ 2 ( 4πe 2 ) 2M Rz 2 = m e ω0 2 N e (h(s)) 2 ds 2 V r (r) dr. 0 0 Covariance of the eikonal is of central importance: ϕ m ϕ n. S. Tsynkov (NCSU) Spaceborne SAR ICERM, October 18, 2017 24 / 37
Covariance of the eikonal (phase path) Affected by statistics of the medium and propagation geometry: ( 4πe 2 ) 2RM 2 ϕ m ϕ n = m e ω0 2 a 1 0 du N e (h(ur)) 2 dsv r ( u 2 x m x n 2 + s 2). A common misconception: not accounting for the ionopause. ionosphere ionopause b r 0 c d z e f pr R r 0 r be R With no ionopause, ϕ m ϕ n decays slowly even if the correlation function of the medium is short-range. With the ionopause, ϕ m ϕ n is also short-range (like V r ) and r ϕ r 0. Why is ϕ m ϕ n important? S. Tsynkov (NCSU) Spaceborne SAR ICERM, October 18, 2017 25 / 37
Stochastic GAF Factorization: ( W B(y2 z 2 ) sin θ ) (y, z) τsinc v gr n e 2iω 0T n ph For narrow-band signals, the factorization error is small: O( B ω 0 ). The effect of turbulence on the imaging in range can be shown to be negligibly small compared to its effect on imaging in azimuth. What remains is the sum of random variables over the array. Why is it important to know how rapidly the random phase decorrelates along the array? Random phases are normal due to the central limit theorem. Terms in the sum are log-normal: Uncorrelated Independent. Statistical dependence or independence of the constituent terms directly affect the moments of the stochastic GAF. S. Tsynkov (NCSU) Spaceborne SAR ICERM, October 18, 2017 26 / 37
Errors due to randomness The mean of the stochastic GAF reduces to the deterministic GAF (subject to extinction e π ϕ2 /λ 2 0): The deterministic errors include plain SAR, ionosphere, Doppler... The errors due to randomness are superimposed on the above: When r ϕ L SA and ϕ 2 λ 0 (small-scale turbulence with small fluctuations), they are estimated by variance of the stochastic GAF: σw 2 = 2 ω 0 A c N ϕ 2 L SA /r ϕ σ 2 W A quantifies the difference between stochastic and deterministic GAF and accounts for the variation within the statistical ensemble. Yet mechanically adding the two types of errors may be ill-advised: In reality, there is a single image rather than an ensemble; For rϕ L SA, randomness manifests itself within a single image; For rϕ L SA, random phases ϕ n within the array are identical: σ 2 is large, yet it basically becomes irrelevant. W A S. Tsynkov (NCSU) Spaceborne SAR ICERM, October 18, 2017 27 / 37
Errors due to randomness (cont d) The case r ϕ L SA is very similar to fully deterministic: Image distortions due to randomness can be characterized directly in terms of the azimuthal shift and blurring (expected values), NOT as the difference between the stochastic and deterministic GAF; Blurring is significant only for much larger fluctuations than the shift: One can have a shifted yet otherwise decent (low blurring) image even for ϕ 2 λ 0. On the other hand, in the case r ϕ L SA, ϕ 2 λ 0 (smallscale turbulence with large fluctuations), the image is completely destroyed [Garnier & Solna, 2013]. No continuous transition (yet) between the small-scale case and large-scale case. Correction of image distortions due to turbulence is a major issue: Current analysis is aimed only at quantification of distortions. S. Tsynkov (NCSU) Spaceborne SAR ICERM, October 18, 2017 28 / 37
Why do we need new models for radar targets? The image I(y) is a convolution-type integral: I(y) = ν(z)w(y, z)dz, where ν characterizes the target, and W the imaging system. The reflectivity ν(z) is either phenomenological or physics-based. Scattering must be linear with respect to the target properties ν: ν(z) is proportional to the variation of the local refractive index; Assumption: Weak scattering the first Born approximation. Scattering occurs only at the surface of the target; dz = dz 1 dz 2. Why? microwaves do not penetrate under the surface. Weak scattering is inconsistent with no-penetration conjecture. A flat uniform target won t backscatter... What is the actual observable quantity in SAR? How is it related to the local properties of the scatterer? S. Tsynkov (NCSU) Spaceborne SAR ICERM, October 18, 2017 29 / 37
Scattering about a dielectric half-space Scattering off a material half-space with a planar interface: { n 2 1, z 3 > 0, (z) = ε(z) = ε (0) + ε (1) (z 1, z 2 ), z 3 < 0, where ε (1) ε (0) = const. The background permittivity ε (0) can be large, so the scattering is not necessarily weak. The method of perturbations: where u (1) u (0). u(t, z) = u (0) (t, z) + u (1) (t, z), Linearization: higher order terms ε (1) u (1) are dropped. S. Tsynkov (NCSU) Spaceborne SAR ICERM, October 18, 2017 30 / 37
Schematic for the scattering geometry orbit (flight track) 3 x H θ L SA R R z ground track R R z x 1 L 0 φ z ψ beam footprint 2 S. Tsynkov (NCSU) Spaceborne SAR ICERM, October 18, 2017 31 / 37
Solution for the scattered field The field is governed by the wave (d Alembert) equation: ( n 2 c 2 2 t 2 ) (u (0) + u (1) ) = 0. The field and its normal derivative are continuous at the interface. Interface conditions hold separately for the zeroth and first order. Zeroth order solution by Fresnel formulae. May involve strong refraction. First order solution using the separation of variables rendered by Fourier transform in time t and along the surface (z 1, z 2 ). Uncoupled ODEs in the direction z 3 are solved in closed form. The inverse transform is done by stationary phase; it accounts for any reflection angles φ and ψ if the distance to the surface is large. S. Tsynkov (NCSU) Spaceborne SAR ICERM, October 18, 2017 32 / 37
Surface retarded potential Scattered field in the time domain: u (1) (t, x ) = M(ε(0), θ, φ) 4π 2 RR P ( t R z + R ) z ε (1) (z 1, z 2 )dz 1 dz 2, c c where M = ( cos φ+ ε (0) sin 2 φ )( cos θ cos φ ε (0) sin 2 φ+ ε (0) sin 2 θ )( ). cos θ+ ε (0) sin 2 θ It is a surface retarded potential obtained with no inconsistent assumptions and no use of the first Born approximation: Convolutions are two-dimensional by design; The scattering is linear yet not necessarily weak. For backscattering (x = x ), the previous expression simplifies: u (1) (t, x) = M(ε(0), θ, θ) 4π 2 R 2 c P ( t 2 R ) z ε (1) (z 1, z 2 )dz 1 dz 2. c S. Tsynkov (NCSU) Spaceborne SAR ICERM, October 18, 2017 33 / 37
What is the SAR observable quantity? The reflection coefficient for monostatic imaging (as P iω 0 P): M(ε (0), θ, θ) ν(z 1, z 2 ) = iω 0 4π 2 R 2 ε (1) (z 1, z 2 ). c Yet is it the reflectivity ν(z 1, z 2 ) itself that we actually see? The scattered field is given by almost the Fourier transform at the Bragg frequency k θ = 2k 0 sin θ (R z is linearized): u (1) (t, x) iω 0 M(ε (0), θ, θ) 4π 2 R 2 c iω 0 M(ε (0), θ, θ) 4π 2 R 2 c ( P ( A t 2 R z c t 2 R z c ) ε (1) (z 1, z 2 )dz 1 dz 2 The true mechanism of surface scattering is resonant. ) e 2ik 0z 2 sin θ ε (1) (z 1, z 2 )dz 2 dz 1. If ν(z 1, z 2 ) has no spectral content at or around k θ, then there is no backscattering, and no monostatic SAR image can be obtained. S. Tsynkov (NCSU) Spaceborne SAR ICERM, October 18, 2017 34 / 37
Physical interpretation of SAR observables The image: I(y) = dz 1 dz 2 W A (y 1, z 1 )W R (y 2, z 2 )ν new (z 1, z 2 ). ν new is obtained by shifting and band limiting the spectrum of ν: ˆν new (z 1, k) = ˆν(z 1, k + k θ )χ β (k), 2B sin θ where k θ = 2k 0 sin θ is the Bragg frequency, and β = c. ν new varies slowly in space, on the scale R = πc B, because all its spatial frequencies β 2 k θ B/c k 0 or B ω 0. Alternatively, the observable quantity ν new (z 1, z 2 ) can be derived as a window Fourier transform (WFT) of ν: ν new (z 1, z 2 ) = τ sin θ W R (z 2 z )ν(z 1, z )e ik θz dz. R ν new (z 1, z 2 ) is a slowly varying amplitude of the Bragg harmonic e ik θz 2 in the spectrum of ν computed on a R size sinc ( ) window. S. Tsynkov (NCSU) Spaceborne SAR ICERM, October 18, 2017 35 / 37
What does it mean for imaging? The actual quantity reconstructed by SAR is not the reflectivity per se but rather its particular slowly varying spectral component. Yet the reconstruction itself is enabled by the presence of high frequencies. Implications for angular coherence in the case of wide apertures: The WFT along different directions for different antenna positions. Vector extension is possible that would account for polarization. Anisotropic and/or lossy materials can be considered. Other models include Leontovich and rough surface. S. Tsynkov (NCSU) Spaceborne SAR ICERM, October 18, 2017 36 / 37
Thank you for your attention! S. Tsynkov (NCSU) Spaceborne SAR ICERM, October 18, 2017 37 / 37