Synthetic Aperture Radar (SAR)

asic Concept of SAR: Consider fied array Synthetic Aperture Radar (SAR) L Angular resolution θ λ /L v(t i ) Σ Claim Equivalent to Moving Antenna: θ Assumes phase coherence in oscillator during each image v Lec.4-1 R1

Synthetic Aperture Radar (SAR) Maintain coherence Ideal point reflector Typical CW pulse signal and echo, jφ Ee Received phase: v = t φ (t) π If scat f( angle): Phase φ (t) = f( λ, point source range) Lec.4- Process a "window" of data, e.g. Received magnitude: E θ R We seek -D source characterization: o, t σ = f(azimuth, range) s, t Magnitude E = f( point source σ s, range) R

Resolution of Unfocused SAR Reconstruction of image: first select R, o of interest W() L θ SAR λ L W() E() observed W E( ) = E( ˆ ) SAR angular resolution: θ λ L λ R θ = D R SAR SAR λ D Lec.4-3 R3

Resolution of Unfocused SAR SAR angular resol ution: θ λ / L λ / R θ = D / R SAR SAR Lateral spatia l resolution = θ SAR R D λ / D ( want smal l D, l arge θ, large L) Range resol ution ct / Note: If antenna is steered toward the target, increasing L, the lateral resolution can be < D. Then repeat for al l R, Lec.4-4 Yi elds image: R R ma min R R > D R ct R4

Phase-Focused SAR θ λ /D () R D L L R + = (R +δ ) R + λ / D λ L λ δ< δ 16 8R 16 L therefore R "farfield" λ δ R and focusing is unnecessary Phase focusing is required if δ >λ 16, so the target is in the S AR near fi eld, i.e., if R < L / λ = λ R / D where L =λ / = D. ( θ λ R/ ) SAR Lec.4-5 S1

Phase-Focused SAR Solution: W() W '() with phase correction φ (t) Pulse π o PRF must be higher; to reconstruct φ (t) we need at least samples per phase wrap. Note: With phase focusing, L increases and spatial resolution in ca be less than D, using beamsteering (e.g. a phased array), t Lec.4-6 Also: L.O. phase drift can be corrected if bright point sources eist in scene. S

Required Pulse-Repetition Frequency (PRF) L Ι () W() Ι ( ) Ι ( ) L λ / L W() Ι() E() ˆ Observed W( ) Ι ( ) E( ) = E ˆ PRF = v / L ( e.g., v = aircraft velocity) Want = λ / L >> λ / D, or D >> L Therefore want PRF >> v/d Lec.4-7 S3

PRF Impacts SAR Swath Width v R min R ma θ y Swath being imaged y Don't want echoes to overlap for successive pulses. Therefore let R 1 D R min / c < = L / v << PRF v ma Implies that large swath widths require large and yield poorer spatia l resolution. D v Lec.4-8 S4

Envelope Delay Focusing Range o Envelope-delay focusing required when δ R > R3 R R θ R θ δ R v Envelope delay R o c ( Target location) Compensation for both envelope delay and phase delay is needed for very high spatia l resolution.e., for smal arge ) and large R. [ i l R (l ] θ Lec.4-9 S5

SAR Intensity Estimates: Speckle For each SAR pie l, the estimated cross-section is: σ= ˆ K w ( i ) E i, j Many subscatters j per piel ij where K = range-dependent constant i = pulse number j = subscattering inde Simplifying: N N j ω+φ t k k k k= 1 k= 1 σ= ˆ ε = ae = r random variable uniformly over π typically P piel j j + 1 Scattering centers Ι m{ ε } k k Re { ε } Re { } ε = {} g.r.v.z.m. Ι m ε = g.r.v.z.m. Lec.4-1 v U1

p r = σ { } r / σ ε e "r " k SAR Intensity Estimates: Speckle Ι m{ ε } r Re { ε } p{r} σ r E[r] =σ π / =σ E r ( π / ) ε ecause we are adding (averaging) phasors, not scalars { } E r r σ / 3 f(n)! pr > r = e ( σ) Thus raw SAR images, ful l spatia l resolution, STD deviation σ / 3 have Q =.53 very grainy images mean intensi ty σ π / o o r / / Lec.4-11 U

Reduction of SAR Speckle Alternate ways to reduce "speckle" by averaging piel intensities: 1) lu r imag e by averaging M piels Q =.53 / M ( using large τ signals yields high resolution, c an be smoo the d) ) Average using spatia l diversity Q =.53 N L N = 3 or more looks possible. Trade against potential resolution in. Lec.4-1 U3

Reduction of SAR Speckle Alternate ways to reduce speckle by average piel intensities: 3) Average using frequency diversity, where each band yields an independent image. Note that the same total bandwidth could alternatively yield more range resolution and piels for averaging. Image 1 Image 3 Image Q =.53 N f N ands Lec.4-13 U4

Reduction of SAR Speckle 4) Time averaging works only if source varies. For eample a narrow swath permits high PRF and an increase in N, but unless the antenna translates more than ~λr/d between pulses [R = range, D = piel width (m)], then adjacent pulse returns are correlated. j D R θ λ In general, reasonably smooth images need > 8 levels, so Q = S.D./M (1/3)/4 levels = 1/1, versus Q 1/. To reduce standard deviation by 6, need N ~ > 6 = 36 looks. p D Lec.4-14 U5

Alternative SAR Geometries and Applications Case A. Source Stationary Case. Source Moves Case C. Source rotates ( e.g., geology) v (e.g. satellite) (e.g. moon or spacecraft) v Antenna Moves Antenna Stationary Antenna Stationary Velocity dispersion within inverse SAR (ISAR) source (e.g., a moving car) can displace its apparent position laterally. Lec.4-15 U6

ESTIMATION Eamples taken from Remote Sensing Principles are Nonetheless Widely Applicable Linear and Non-Linear Problems Gaussian and Non-Gaussian Statistics Professor David H. Staelin Massachusetts Institute of Technology Lec.4-16 A1

Linear Estimation: Smoothing and Sharpening Smoothing reduces image speckle and other noise; sharpening or deconvolution increases it. Sharpening can de-blur images, compensating for diffraction or motion induced blurring. Audio smoothing and sharpening are similar. Consider antenna response eample; we observe: 1 T A = G A s T s dω π A s 4 4 π Lec.4-17 A

Linear Estimation: Smoothing and Sharpening Consider antenna response eample; we observe: 1 T A A = G A s T s dωs 4 π 4 π 1 T A A G T ( sma ll soli d angles) 4π 1 T A f = G f T f 4π Cycles/Radian Antenna Spectral Response ( + y y ) i G (, ) e j π f f.e., G f d d, f = y y y Lec.4-18 A3

Eample: Square Uniformly-Illuminated Aperture Lec.4-19 D Aperture Try: D ˆT f 1 T A A G T ( sma ll soli d angles) 4 π 1 T A f = G f T f 4π Cycles/Radian Antenna Spectral Response = E, G ( f ) R G ( ) τ y λ τ λ,fy A 4π T f G f e.g. τ,f D λ λ y λ / D lurring Function This restores high-frequency components, but goes to zero for f > D / λ! A4

Eample: Square Uniformly-Illuminated Aperture ˆT f = A 4π T f G f This restores high-frequency components, but goes to zero for f > D / λ! A 4π T f Try: T ˆ f = W f G f W( f ) f y D λ f, τ λ cycles/radian The window function W f avoids the singularity. The "principal solutio n" uses a bocar W (s). Lec.4- A5

Eample: Square Uniformly-Illuminated Aperture =δ ( f ) = ˆ f = π f W f = W f, since T A f = G f T f 4 π ( ) = -D s nc funct on for a uniform nated rectangular Consi der the restored (sharpened) i mage of a point source, T : Then T 1, so T 4 T G f A ˆT i i ly illumi antenna aperture. y Can clip, transform, iterate ˆT = ˆT ( ) Zero at λ / D Predicts T <! ˆT f Lec.4-1 f Can set to zero (apriori information) A6

e.g. = A + N f ˆT f T f A Truth Sharpening Noisy Images T rms T A ( f ) N f Amplifies white noise ll N f for sma G f ˆT f for point source D/λ f ˆ = 4 πt A G f W f Tˆ f f /! Amplifier Noise D/ λ D/ λ f Lec.4- Therefore optimize W f A7

Sharpening Noisy Images 4π W f Minimize E T f T f + = Q Therefore optimize W f : ( N ( f )) A G f Q/ W = yields: W f optimum = 1 1 * E T f T f T f A o A o A o + N f + N f E T A f o + N f Lec.4-3 [ ] If E T A N =, then W optimum ( f ) A o 1 1 = N E N f 1 + 1 + S E T f A8

Sharpening Noisy Images [ ] If E T N A = The weighting is widely used W opt for N = : opt ( f ), then (1 + N / S) 1 W optimum ( f ) N A o 1 1 = N E N f 1 + 1 + S E T f W opt f D/ λ ( f ) W f wi der beam (lower spatia l resolution), lower sidelobes. f Lec.4-4 Can be used for restoration of blurred images of al l types: A ( ) A photographs TV, T maps, SAR images, filtered speech, etc. A9