A Mathematical Tutorial on. Margaret Cheney. Rensselaer Polytechnic Institute and MSRI. November 2001

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1 A Mathematical Tutorial on SYNTHETIC APERTURE RADAR (SAR) Margaret Cheney Rensselaer Polytechnic Institute and MSRI November 2001 developed by the engineering community key technology is mathematics unknown in mathematical community close connections with tomography, integral geometry, microlocal analysis, seismic inversion 1

2 History 1951 Carl Wiley, Goodyear Aircraft Corp. mid- 50s late 60s first operational systems DoD sponsorship: U. of Illinois, U. of Michigan, Goodyear Aircraft, General Electric, Philco, Varian NASA sponsorship (unclassified!) first digital SAR processors 1978 SEASAT-A 1981 beginning of SIR (Shuttle Imaging Radar) series 1990s satellites sent up by many countries SAR systems sent to Venus, Mars, Titan 2

3 Outline The wave equation The incident wave: beamforming A linearized scattering model The received signal The reconstruction method: matched filters = backprojection = migration Resolution The state of the art Current research and open problems 3

4 Mathematical Model We should use Maxwell s equations. A simpler model: the scalar wave equation ( 2 1 ) c 2 (x) 2 t U(t, x) = 0 Earth is plane x 3 = 0 All constants = 1 4

5 The Incident Wave Field at (t, x) due to a source at the origin at time 0: δ(t x ) G 0 (t, x) = x satisfies ( 2 2 t ) G 0 (t, x) = δ(t)δ(x) But the signal from the antenna is more complicated... δ(t) P(t) = e iωt p(ω)dω δ(x) J(x) 5

6 Field emanating from antenna satisfies ( 2 t 2 )Uin (t, x) = P(t)J(x) Solve: U in (t, x) = G 0 (PJ) = = P(t x y ) J(y)dy x y e iω(t x y ) p(ω)j(y)dωdy. x y In this formula: Write point on antenna as y = γ(s) + q For x far from the antenna, q << x γ(s) : x y = x γ(s) ( x γ(s)) q + O(q/ x γ(s) ) 6

7 Uγ(s) in (t, x) e iω(t x γ(s) ) x γ(s) ) p(ω) e iω( x γ(s)) q J(γ(s) + q)dq dω. Antenna beam pattern at each frequency is Fourier transform of current density on antenna! U in e iω(t x γ(s) ) γ(s) (t, x) p(ω) j(ω( x γ(s)), s) dω x γ(s) 7

8 Key example: Take J = characteristic function of rectangle j = L/2 L/2 D/2 D/2 e iω[ x γ(s)] (q1 ê 1 (s)+q 2 ê 2 (s)) dq1 dq 2 sinc[ω(d/2)cos θ 1 ] sinc[ω(l/2)cos θ 2 ] where cos θ j = [ x γ(s)] êj and sinc x = sin x x. Main lobe of beam is perpendicular to antenna To get angular width, set argument of sinc = π, solve for angle. Antenna has length L Angular width of main lobe ωl 4π = 2λ L width of antenna footprint = ωl 4π R = 2λ L R. Get a more focused beam with: bigger antenna higher frequency 8

9 Linearized scattering theory ( 2 c 2 (x) 2 t ) U(t, x) = P(t)J(x) ( 2 2 t )Uin (t, x) = P(t)J(x) write U = U in + U sc, c 2 (x) = 1 + V (x), subtract: ( ) 2 t 2 U sc (t, x) = V (x) t 2 U(t, x), write as Lippman-Schwinger integral equation U sc (t, x) = G 0 (t τ, x z)v (z) τ 2 U(τ, z)dτdz. 9

10 single-scattering or Born approximation U sc (t, x) G 0 (t τ, x z)v (z) τ 2 U in (τ, z)dτdz useful: makes inverse problem linear not necessarily a good approximation δ(t τ x z ) γ(s) (t, x) V (z) x z U sc e iω(τ z γ(s) ) j(ω( z γ(s)), s)ω 2 p(ω)dωdτdz z γ(s) e iω(t ( x z + z γ(s) ))j(ω( z γ(s)), s) p(ω) dω z γ(s) x z V (z)dz At center of antenna, Uγ(s) sc (t, γ(s)) e iω(t 2 z γ(s) )j(ω (z γ(s)), s) p(ω) z γ(s) 2 dω V (z)dz 10

11 This data is of the form D(t, s) = e iω(t 2 z γ(s) ) a(z, s, ω)dωv (z)dz From D, want to reconstruct V. D is an oscillatory integral, to which techniques of microlocal analysis apply ( C. Nolan & M.C.) seismic inversion problem with constant background but limited data D(t, s) depends on two variables. Assume V (z) = Ṽ (z 1, z 2 )δ(z 3 h(z 1, z 2 )). if a(z, y, ω) = 1, want to reconstruct V from its integrals over spheres or circles (J. Fawcett, L.-E. Andersson, A. Louis & E.T. Quinto, M. Agranovsky & E.T. Quinto) character of problem depends on frequency content of a 11

12 High frequency case microwave frequencies, ω 10 9, λ cm can form narrow beam high resolution, space-borne doesn t penetrate foliage Low frequency case VHF: ω 10 7, λ 10m cannot form beam penetrates foliage but resolution is low airborne 12

13 Inversion scheme: matched filter = backprojection = migration I(x) = P(t 2 x γ(s) )D(t, s)dtds Why is this called a matched filter? Take V (z) = δ(z x) in D(t, s) = e iω(t 2 z γ(s) ) p(ω)j(ω(z γ(s)), s)dω V (z) z γ(s) 2dz j is nearly independent of ω D(t, s) P(t 2 x γ(s) ) = data due to a point scatterer at position x. In case P = δ: The matched filter backprojects the data over the sphere with radius t/2 and center γ(s). Image at x is formed by adding data corresponding to all spheres passing through x. 13

14 Why does matched filter processing give us an image? D(t, s) = e iω(t 2 z γ(s) ) a(z, s, ω)dωv (z)dz = FV Apply I(x) = e i ω(t 2 x γ(s) ) p( ω)d ωd(t, s)dtds F D Note that phase is negative of phase of D. Compare with Fourier inversion, Radon transform inversion,... Use e it( ω ω) dt = δ( ω ω) to get I(x) = W(x, z)v (z)dz where W(x, z) = e 2iω( x γ(s) z γ(s) ) p(ω)a(z, s, ω)dωds is the point spread function or generalized ambiguity function of the imaging system. 14

15 I(x) = W(x, z)v (z)dz W(x, z) = e 2iω( x γ(s) z γ(s) ) p(ω)a(z, s, ω)dωds Main contribution comes from: (high frequency case) x = z only (low frequency case) x = z and x = reflection of z wavefront relation of F is a folding canonical relation artifacts 15

16 Resolution in high frequency case How close is W to a delta function? W(x, z) = e 2iω( x γ(s) z γ(s) ) p(ω)a(z, s, ω)dωds Compare with δ(x z) = e i(x z) ξ dξ. G. Beylkin (JMP 85) approach: 1. Do Taylor expansion of exponent: x γ(s) z γ(s) = (x z) Ξ(x, s, z) near x = z, Ξ(x, s, z) z γ(s) 2. Make change of variables (ω, s) ξ = 2ωΞ(x, s, z): W(x, z) = e i(x z) ξ (s, ω) p(ω(ξ))a(z, s(ξ), ω(ξ)) dξ ξ This is kernel of a pseudodifferential operator puts singularities in the correct locations. 16

17 W(x, z) = e i(x z) ξ (s, ω) p(ω(ξ))a(z, s(ξ), ω(ξ)) dξ ξ Resolution is determined by support of p(ω(ξ)) and a(z, s(ξ), ω(ξ)), i.e., by bandwidth and angular extent of survey. Resolution via Fourier transforms b b e iρr dρ = 2sin br r corresponds to resolution 2π/b. 17

18 Along-track (azimuthal) resolution W(x, z) = with ξ 2ω e i(x z) ξ (s, ω) p(ω(ξ))a(z, s(ξ), ω(ξ)) dξ ξ (z γ(s)). If flight track is γ(s) = (0, s, H) and x z = (0, x 2 z 2,0), need only ξ 2 2ω z 2 γ(s) 2 R = ω R 2(z 2 s) But 2max z 2 s = width of antenna footprint = ωl 4π R = 2λ L R = effective length of the synthetic aperture So max ξ 2 ω R 4πR ωl = 4π L So resolution in along-track direction is 2π 4π/L = L 2 18

19 Along-track resolution is L/2. This is... independent of range! independent of λ! better for small antennas! These are all explained by noting that when a point z stays in the beam longer, the effective aperture for that point is larger. In range direction, want broad frequency band get largest coverage in ξ. 19

20 The state of the art (high frequency) Can correct for antenna motion interferometric SAR (two flight passes or two antennas) topographic information changes (glacier flow, volcano movement) polarimetric SAR (uses full Maxwell s equations) MTI = Moving Target Indicator SAR in wave mode: Use subaperture information to get ocean motion 20

21 Current research and open problems Foliage-penetrating (FOPEN) SAR low frequencies poor directivity tomographic image formation from integrals over circles or spheres want bare earth topography want to find tanks under trees want to estimate forest biomass, trunk volume, tree health Ground-penetrating radar (GPR) land mines unexploded ordinance (UXO) Ultra-wideband (UWB) SAR 21

22 Improved modeling dispersion (FOPEN) multiple scattering rough terrain, wave-guiding structures wave propagation in random media, clutter Image interpretation thickness of ice, species of trees, surface roughness,... Automatic Target Recognition (ATR) (school bus or tank?) sensor fusion, use of multiple frequency bands 22

23 Theoretical issues Uniqueness theorem for backscattering Reconstruction for full nonlinear inverse problem limited aperture time domain (including dispersion!) What are theoretical limits? 23

24 References M. Cheney, A mathematical tutorial on SAR, SIAM Review, SIAM Review 43 (2001) ; cheney/sar.ps G.W. Stimson, Introduction to Airborne Radar, Scitech Publishing, New Jersey, C. Elachi, Spaceborne Radar Remote Sensing: Applications and Techniques, IEEE Press, New York, G. Franceschetti and R. Lanari, Synthetic Aperture Radar Processing, CRC Press, New York, L.J. Cutrona, Synthetic Aperture Radar, in Radar Handbook, second edition, ed. M. Skolnik, McGraw-Hill, New York,

Books. B. Borden, Radar Imaging of Airborne Targets, Institute of Physics, 1999.

Books. B. Borden, Radar Imaging of Airborne Targets, Institute of Physics, 1999. Books B. Borden, Radar Imaging of Airborne Targets, Institute of Physics, 1999. C. Elachi, Spaceborne Radar Remote Sensing: Applications and Techniques, IEEE Press, New York, 1987. W. C. Carrara, R. G.