SAR imaging through turbulence Synthetic aperture radar (SAR) imaging through a turbulent ionosphere Semyon Tsynkov 1 1 Department of Mathematics North Carolina State University, Raleigh, NC https://stsynkov.math.ncsu.edu tsynkov@math.ncsu.edu +1-919-515-1877 International Conference Advances in Applied Mathematics in memoriam of Prof Saul Abarbanel Tel Aviv University, December 18–20, 2018 S. Tsynkov (NCSU) SAR imaging through turbulence Tel Aviv University, 19/12/2018 1 / 23
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) SAR imaging through turbulence Tel Aviv University, 19/12/2018 2 / 23
In memory of Saul Abarbanel S. Tsynkov (NCSU) SAR imaging through turbulence Tel Aviv University, 19/12/2018 3 / 23
What I plan to accomplish in this talk Spaceborne SAR imaging is a vast area: ◮ A lot of challenging issues lack attention by mathematicians; ◮ Yet any attempt to do a broad overview will be superficial. Instead, I would like to: ◮ Very briefly outline the key aspects of SAR imaging; ◮ Focus on the effect of ionospheric turbulence on spaceborne SAR; ◮ Present some recent findings that are unexpected/intriguing; ◮ Point out the related misconceptions in the SAR literature; ◮ Identify the important questions that require subsequent work. S. Tsynkov (NCSU) SAR imaging through turbulence Tel Aviv University, 19/12/2018 4 / 23
New research monograph (2017) S. Tsynkov (NCSU) SAR imaging through turbulence Tel Aviv University, 19/12/2018 5 / 23
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) SAR imaging through turbulence Tel Aviv University, 19/12/2018 6 / 23
Schematic: 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) SAR imaging through turbulence Tel Aviv University, 19/12/2018 7 / 23
Conventional SAR data inversion Interrogating waveforms (fields assumed scalar) — 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) SAR imaging through turbulence Tel Aviv University, 19/12/2018 8 / 23
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 ) – generalized ambiguity function (GAF) or imaging kernel: � � e − 2 i ω 0 ( R n y − R n e − i α 4 t ( R n y − R n z ) / c z ) / c dt . W = χ n S. Tsynkov (NCSU) SAR imaging through turbulence Tel Aviv University, 19/12/2018 9 / 23
Factorization of the GAF and resolution analysis Convolution form I = W ∗ ν is convenient for analysis: ◮ Yet actual processing is done for the entire dataset – not for each y . 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 ( y 1 − z 1 ) . N sinc 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 ∆ R = B τ π . B τ 2 π is the compression ratio of the chirp (or TBP); must be large. S. Tsynkov (NCSU) SAR imaging through turbulence Tel Aviv University, 19/12/2018 10 / 23
Azimuthal reconstruction Linear variation of the instantaneous frequency along the chirp ω ( t ) = ω 0 + 2 α t yields the range factor of the GAF: � � e − i α 4 t ( R y − R z ) / c dt = e − 2 i ( ω ( t ) − ω 0 )( y 2 − z 2 ) sin θ/ c dt . W R ( y , z ) = χ χ In the azimuthal factor, there is a linear variation of the local L SA n wavenumber along the array, k ( n ) = k 0 RN : � L SA n � RN ( y 1 − z 1 ) = n e 2 ik ( n )( y 1 − z 1 ) . n e 2 ik 0 W A ( y , z ) = Can be attributed to a Doppler effect in slow time n . Can be thought of as a chirp of length L SA in azimuth. Compression ratio of the chirp in slow time: = 2 L 2 λ 0 = 2 π c L SA 1 SA R ≫ 1 , where . ∆ A λ 0 ω 0 2 L 2 SA ≫ 1 is the Fraunhofer distance of the synthetic array. λ 0 S. Tsynkov (NCSU) SAR imaging through turbulence Tel Aviv University, 19/12/2018 11 / 23
Schematic: 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) SAR imaging through turbulence Tel Aviv University, 19/12/2018 12 / 23
Ionospheric distortions of SAR images What the radar actually measures is travel times. Distances are determined from times given that c is fixed. What if the key assumption “DISTANCE=VELOCITY × TIME” fails? EM waves in the ionosphere are subject to temporal dispersion. Dispersion relation: ω 2 = ω 2 pe + c 2 k 2 , pe = 4 π e 2 N e where ω 2 is the Langmuir frequency. m e Group and phase velocities: v gr < c (delay), v ph > c (advance). Mismatch between the received signal and the matched filter. Can be reduced by adjusting the matched filter: ◮ One needs real-time total electron content (TEC) and gradients; ◮ Can be obtained with the help of dual carrier probing. S. Tsynkov (NCSU) SAR imaging through turbulence Tel Aviv University, 19/12/2018 13 / 23
Dual carrier probing Imaging kernel with mismatch: � � e − 2 i ω 0 ( R n y / c − T ph ( x n , z ,ω 0 )) e − i α 4 t ( R n y / c − T gr ( x n , z ,ω 0 )) dt , W ( y , z ) = χ n where � R z � R z 4 π e 2 � � 1 1 1 ∓ 1 R z T ph , gr ( x , z , ω 0 )= v ph , gr ( s ) ds ≈ N e ( s ) ds = . m e ω 2 ¯ c 2 v ph , gr 0 0 0 There are also quadratic phase errors due to change in pulse rate. Lead to deterioration of image resolution and sharpness. The mismatch also yields a shift of the entire image in range: 4 π e 2 TEC 1 ∆ R = R z H . m e ω 2 2 0 Probing on two carrier frequencies, ω 0 and ω 1 , creates two shifts. Registering two shifted images yields their relative displacement and allows to solve for the unknown TEC. S. Tsynkov (NCSU) SAR imaging through turbulence Tel Aviv University, 19/12/2018 14 / 23
Transionospheric SAR after the correction The new imaging kernel: � τ/ 2 � e − 2 i ω 0 T n α n T n e − 4 i ˜ gr t dt , W ( y , z ) = ph − τ/ 2 n T n ph , gr = T ph , gr ( x n , y , ω 0 ) − T ph , gr ( x n , z , ω 0 ) . where Factorization: � B ( y 2 − z 2 ) sin θ � � ω 0 ( y 1 − z 1 ) L SA � W ( y , z ) ≈ τ N sinc sinc . v gr ¯ R ¯ v ph The overall quality of the image is basically restored. 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) SAR imaging through turbulence Tel Aviv University, 19/12/2018 15 / 23
Recommend
More recommend
