A TUTORIAL ON (CONVENTIONAL) SYNTHETIC APERTURE …

1 A TUTORIAL ON (CONVENTIONAL). SYNTHETIC APERTURE RADAR. (SAR). Margaret Cheney June 2000. 1. Outline The wave equation The incident wave: beamforming A linearized scattering model The received signal The reconstruction method: matched fil- ters Resolution: azimuthal and range Other SAR systems Open problems 2. Mathematical Model We should use Maxwell's equations. A simpler model: the scalar wave equation ! 2 1. 2 t2 U (t, x) = 0. c (x). Earth is plane x3 = 0. 3. The Incident Wave Field due to a point source: (t |x y|/c0). G0(t , x y) =. 4 |x y|. satisfies ! 2 1 2. 2 t G0(t , x y) = (t ) (x y). c0. But the signal from the antenna is more com- plicated.

2 4. Assume that the signal sent to the antena is P (t) = A(t)ei 0t The field emanating from a point with such a time history satisfies the equation ! 1. 2 2 t2 Uy (t, z) = P (t) (z y). c0. and is thus given by Uy (t, z) = (G0 P )(t, z y). Z. (t |z y|/c0). = P ( )d . 4 |z y|. P (t |z y|/c0). =. 4 |z y|. A(t |z y|/c0) i 0 (t |z y|/c0). = e . 4 |z y|. 5. But the antenna is not a point! x = center point on antenna is y = x + q For z far from the antenna, |q| << |z x|: x) q + O(|z x| 1), |z y| = |z x| (z d Then A(t |z x|/c0 + z d x q/c0 + ). Uy (t, z) . 4 |z x|. d ei 0(t |z x|/c0)eikz x q Since A is slowly varying, P (t |z x|/c0) ikz x q d Uy (t, z) e 4 |z x|.

3 6. Consider the antenna to be a distribution of point sources: Write a point on the antenna as q = s1e 1 +s2e 2. Z L/2 Z D/2. Uxin(t, z) = Ux+s1e 1+s2e 2 (t, z)ds1ds2. L/2 D/2. Z L/2 Z D/2. P (t |z x|/c0) ikz x (s d 1e 1+s2e 2). e ds1ds2. L/2 D/2 4 |z x|. P (t |z x|/c0). w(z d x). 4 |z x|. where . d kD d kL d w(z x) = 2 Dsinc z x e1 2 Lsinc z x e2. 2 2. is the antenna beam pattern 7. Summary of incident wave The incident waveform is P (t |z x|/c0). w(z d x), 4 |z x|. where P (t) = A(t)ei 0t and w is a product of sinc functions Main lobe of beam is perpendicular to antenna Width of main lobe 2 /L. Get a more focused beam with: bigger antenna higher frequency 8.

4 Linearized scattering theory ! 1. 2 2 t2 (t, x) = 0. c (x). (t, x) = in(t, x) + sc(t, x). Turn the wave equation into an integral equa- tion: Z Z. sc(t, x) = G0(t , x z)V (z) 2 ( , z)d dz where V (z) = c21(z) 12 . c0. Use Born approximation: sc(t, x) B (t, x), Z Z. B (t, x) = G0(t , x z)V (z) 2 in(t, x)d dz Z. V (z). = t2 in(t |x z|/c0, z)dz. 4 |x z|. 9. SAR uses a sequence of pulses Antenna moves along flight track: xn = nvT. X. in ( , z) = in n ( , z), n where in in n ( , z) = Uxn ( nT, z). Z Z. = Uxn+q ( nT, z)d2q antenna P ( nT |z xn|/c0) dxn). w(z . 4 |z xn|. 10. The received signal The Born approximation: Z. V (z). B. (t, x) = t2 in(t |x z|/c0, z)dz.

5 4 |x z|. The nth received signal is Sn(t) B n n (t nT, x ): Z. 02P (t nT 2|z xn|/c0). B n n (t nT, x ) . 4 |z xn|. V (z) dxn)dz. w(z . 4 |z xn|. where we have made the approximation t2P (t, x) 02P (t, x). 11. Inversion scheme: matched filter processing An image from the nth look is formed by Z. In(y) = P (t nT 2|y xn|/c0)Sn(t)dt Why is this called a matched filter? The Born approximation: Z. 02P (t nT 2|z xn|/c0). B n n (t nT, x ) . 4 |z xn|. V (z) dxn)dz. w(z . 4 |z xn|. If V (z) = (z y), then B n n n (t nT, x ) P (t nT 2|y x |/c0) . dn ). 02w(y x (4 |y xn|)2. Use |z xn| R0. 12. Why does matched filter processing give us an image?

6 The received signal was Z. 02P (t nT 2|z xn|/c0). B n n (t nT, x ) . 4 R0. V (z) dxn)dz. w(z . 4 R0. so the matched filter output is Z. 02V (z). In(y) Wn(y, z) 2. dz, (4 R0). where R. dxn) P (t nT 2|y xn|/c ). Wn(y, z) = w(z 0. n P (t nT 2|z x |/c0)dt Wn is the point spread function of this single- look imaging system: if V (z) = (z z0), then In(y) = cWn(y, z0) would be the resulting im- age of V . 13. R. dxn). Wn(y, z) = w(z P (t nT 2|y xn|/c0). P (t nT 2|z xn|/c0)dt Plan: Try to make Wn(y, z) (y z) by good choice of P . Key idea of SAR: Improve the point spread function by improved by summing over n, , by combining information from multiple looks.

7 Z. X 02V (z). I(y) = In(y) W (y, z) 2. dz, n (4 R0). with X. W (y, x) = Wn(y, z). n W is the generalized ambiguity function of the SAR system We want to choose P so as to make W as close to a delta function as possible. 14. How close is W to a delta function? Recall P (t) = A(t)ei 0t so X. W (y, z) = dxn) . w(z . n Z ! 2|y xn|. 2|y xn| i 0 (t nT c0 ). A t nT e . c0. ! 2|z xn|. 2|z xn| i 0 (t nT c0 ) dt A t nT e c0. Change variables t nT t Use the fact that A is slowly varying to pull A out of the sum 15. Then W (y, z) WR(y, z)WA(y, z). where Z ! ! 2|y xn| 2|z xn|. WR(y, z) = A t A t dt c0 c0. and X n n WA(y, z) = ei2k(|y x | |z x |)w(z.)

8 Dxn). n WR controls the range ( cross-track ) res- olution involves the fast time t WA controls the azimuthal ( Doppler or along-track ) resolution. involves the slow time xn 16. Azimuthal resolution X n n WA(y, z) = ei2k(|y x | |z x |)w(z . dxn). n Use q q |xn y| = 2 + (xn )2 = R. R0 1 + (xn /R )2. 2 0 s 0. (xn 2 )2. R0 + + . 2R0. q n 2 n 2 (xn 2 z2)2. |x z| = R0 + (x2 z2) R0 + + . 2R0. Then X n 2. WA(y, z) eik(2x2 z2 z2 )/R0 w(z . dxn). n Use xn 2 = nvT : X . ikz22 /R0 2ikz2 vT /R0 n dxn). WA(y, z) e e w(z . n 17. X . ikz22 /R0 2ikz2 vT /R0 n dxn). WA(y, z) e e w(z . n Limits on n are controlled by w: n contributes if z2 R0/L < xn 2 < z2 + R0 /L.

9 , z remains in the beam while the antenna moves a distance of 2 . Leff = R0. L. This is the effective length of the SYNTHETIC APERTURE Obtain sin(kz2 Leff /R0). WA(y, z) . kz2vT /R0.. ikz22 /R0 kD d e 4 LDsinc z x e1. 2. 18. Get resolution by setting kz2 Leff /R0 = . Solve for 2z2 and use = 2 /k: R0. 2z2 =. Leff But Leff = R0(2 /L), so the resolution is 2z2 = L/2. Note: resolution is .. independent of range! independent of ! 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. 19. Range resolution We still have freedom to choose A.

10 We would like to make A a delta function, or a short pulse. But a short pulse has little energy, so the scat- tered field is weak and is drowned out by noise. Instead, use pulse compression: transmit a complex waveform and compress the received signal (with a matched filter). The most common modulated pulse is a chirp, which uses the notion of instantaneous fre- quency 20. Instantaneous frequency of F (t) = ei (t). Fourier transform: Z Z. f ( ) = F (t)e i t dt = ei( (t) t)dt Apply the method of stationary phase: The leading order contribution comes from t for which d 0 = ( (t) t). dt Call = d /dt the instantaneous frequency of F . 21.