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 2 D 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 t b i r o ) k c a r t t h g i f l ( n x 3 antenna D 0 x γ n L SA θ d n u o r g k c a r t R H R n z n R y θ L y z beam 0 footprint 2 1 S. Tsynkov (NCSU) Spaceborne SAR ICERM, October 18, 2017 6 / 37
Conventional SAR data inversion Interrogating waveforms — linear chirps: A ( t ) = χ τ ( t ) e − i α t 2 . P ( t ) = A ( t ) e − i ω 0 t , where ω 0 — central carrier frequency, τ — duration, α = B 2 τ — chirp rate. Incident field — retarded potential from the antenna at x ∈ R 3 : u ( 0 ) ( t , z ) = 1 P ( t − | z − x | / c ) . 4 π | 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 ) d z . 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 | ): � P ( t − 2 R y / c ) u ( 1 ) ( t , x ) dt I x ( y ) = χ � � = d z ν ( z ) dt P ( t − 2 R y / c ) P ( t − 2 R 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 ) d z n n � � � � � = W x n ( y , z ) ν ( z ) d z = W ( y , z ) ν ( z ) d z = 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 � � k 0 L SA � W ( y , z ) = W ( y − z ) ≈ τ sinc c ( y 2 − z 2 ) sin θ N sinc ( y 1 − z 1 ) . R 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 ( · ) : π R π Rc Range: ∆ R = π c ◮ Azimuthal: ∆ A = = ; B . ω 0 L SA k 0 L SA What would it be with no phase modulation? The range resolution would be � the length of the pulse τ c . The actual range resolution is better by a factor of τ c = B τ π . ∆ R B τ 2 π is the compression ratio of the chirp (or TBP); must be large. 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 orbit (flight track) x (t) 3 v x (0) γ z x’ γ y ground L SA track H r R y R L R z θ y 0 beam z footprint 2 1 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 | ): �� � � �� t − r 1 + v c cos γ z P u ( 0 ) ( t , z ) ≈ 1 c . 4 π r Linear Doppler frequency shift can be obtained by taking ∂ ∂ t . The scattered field in the original coordinates: � � �� � � � − 2 r 1 + 2 v 1 + v u ( 1 ) ( t , x ′ ) ≈ ν ( z ) P c cos γ z c cos γ z d z . t c The receiving location x ′ is not the same as the emitting location x . Two different factors multiplying the time t and the retarded time 2 r 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 ( v 2 c 2 ) are neglected. Matched filter that accounts for the antenna motion: � � � �� � � � − 2 R y 1 + 2 v 1 + v I x ( y ) = d z ν ( z ) dtP t c cos γ y c cos γ y c χ � �� � � � − 2 R z 1 + 2 v 1 + v · P c cos γ z c cos γ z . t c A simple correction; only geometric info is required. Summation along the synthetic array: � � � I ( y ) = I x n ( y ) = W x n ( y , z ) ν ( z ) d z n n � � � � � = W x n ( y , z ) ν ( z ) d z = W ( y , z ) ν ( z ) d z = 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: � k 0 L SA �� � + 2 L ( y 2 − z 2 ) v W A ( y , z ) ≈ N sinc ( y 1 − z 1 ) . R R c � �� � � �� � original due to Doppler The range factor: typically O ( 1 ) � B � �� � �� � � 1 + ω 0 � c − ( y 1 − z 1 ) v W R ( y , z ) ≈ τ sinc ( y 2 − z 2 ) sin θ . c c α R 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 | π � ( y 1 − z 1 ) + L ( y 2 − z 2 ) 4 + ω 0 c B � v � � � . � � max | W R W A | 8 ω 0 ∆ A 2 α R R c 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: � � � � t − 2 R y I x ( y ) = d z ν ( z ) dtP c χ � �� � � � − 2 R z 1 + 2 v 1 + v · P c cos γ z c cos γ z . t c 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 | π 1 + ω 0 B c � ( y 1 − z 1 ) + v � � � . � cR � � max | W R W A | 8 ω 0 ∆ A α R 2 The term ω 0 c 2 may become large if, e.g., α is small. α R 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 235 GHz ). FMCW interrogating waveforms due to hardware limitations: ◮ FMCW = Frequency Modulated Continuous Wave. S. Tsynkov (NCSU) Spaceborne SAR ICERM, October 18, 2017 16 / 37
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