ulti-baseline SAR interferometry A. onti Guarnieri, S. Tebaldini Dipartimento di Elettronica e Informazione - Politecnico di ilano G. Fornaro, A. Pauciullo IREA-CNR ESA cat-1 project, 3173. Page 1
Classical Interferometry (single baseline) SLAVE B n Slant range ASTER r s ϕ( P) 4π λ ( ) r m r s r m P Azimuth The interferogram phase depends upon the topography Page
SAR acquistion model distributed scatterer We assume the 1D case, at constant azimuth (most demanding) Acquistion geometry odel heigth sensor Bandpass filter (end-to-end SAR IRF) w(r) rumore s(r) f(r) y(r) r range Large-bandwidth reflectivity exp j r SLC s(r ) s(r) Topography affects phase modulation ground range Page 3
Interferometric Phase Estimate s N We need the most accurate and local phase estimate we assume, for example, a linear model (1 parameter) s ϕ n (r;r ) = π b n ν r (r ) r +ψ n (r ) ψ n : phase offset (Atmospheric Phase Screen, baseline error etc) => NUISANCE. s 1 B 1 B 1N ν r : frequency/baseline, related to topography. To be estimated on a local scale. v r = λr f = B n v r tan ( ϑ α ) α Page 4
Single Baseline: LSI frequency estimate* L estimate with Gaussian noise turns to be LS. We estimate the local shift f in 3 steps: yˆ S (1) Synthesis of the slave from the master: ( r) () LS error minimization between the synthesized and the given slave (3) Repeat, starting from the slave. It It is is the the L L for for coherence coherence γ<<1. γ<<1. odello w SLC -aster Slave synthesis f (r) y G S + - s(r) Source reflectivity w S f S (r) y S aster synthesis G S + - min( ) exp(jπ f r r) SLC -Slave Topography f r (r) f * Tebaldini 5 Page 5
Step #1: Slave Synthesis The Slave is synthesized by frequency shifting and filtering in the common band Yˆ S H ( f ) = H ( f ) Y = arg min E Yˆ H S ( f ( f ˆ f ) ) Y ( f ) Y (f) Y (f- f) f f The LS reconstruction filter is LTI in this case: Y (f- f)h (f) f H = F F ( f ( f ) f ) F S ( f ( 1+ σ ) σ w ws ) Y S (f) Page 6
Step #: Error Whitening The error (innovation) between the slave and its synthesis, is frequency-by-frequency independent, but with non-uniform variance. The L estimate minimizes the integral of the error weighted by the inverse variance: ˆ f = arg min f, H Yˆ S ( f ) Y S ( f ) W S ( f ) df 1 1 γ ˆ f Y S 1 f The witening kernel, W S weigths the spectral contribute in the common band according to the coherence, γ. ˆ f Y S (f) Page 7
L frequency estimate (single-baseline) The estimate exploits LTI filters for constant sloped terrain and ignoring edges effects. For fast varying slopes, or small windows, space-adaptive kernels can be preferred. odello w SLC -aster Slave synthesis Error whitening f (r) y H S + - W S s(r) Source reflectivity w S exp(jπ f r r) f S (r) y S SLC -Slave exp ( jπ ˆ f r) aster synthesis r H S + - W S min ( ) Topography f r (r) f Time-varying kernel Page 8
s(r) Interferometry: classical vs PS s(r n ) S(f k ) SLC -aster y + - y S s(r n ) S(f k ) SLC -Slave exp ( jπ ˆ f r) r min( ) f If the sole reliable source is a single scatterer (PS), we get spectral correlation. The L estimate reduces to the search of the maximum of the interfergram s periodogram. [Spagnolini95][Ferretti 96][Lombardini98][Pascazio1][Heineder5] We could even estimate fase shifts φ>π φ Page 9
Slope (frequency) estimators: a comparison 1.5 -.5-1 1 1.5.5 -.5 -.5 periodogram -1 L -1 LSI Estimated frequency -...4.6.8 1 1. -...4.6.8 1 1. -...4.6.8 1 1. 1.75.5.5 RSE True frequency Simulation of a frequency sweep. Notice the biasing of the periodogram. -.5.5.5.75 1 1.5 1.5 f r / B r Page 1
B Interferometry: the role of the baseline s N In multi-baseline interferometry, Atmospheric Phase Screen contributes locally as a phase offset, and it does not affect the slope estimate ( nuisance parameter). s The phase ambiguity can be removed if low-baseline pairs are exploited. s 1 B 1 B 1N There is a optimal baseline for each slope. The larger the number of baselines, the better the estimate. α m 4 m log(rse) 1 m 5 m 5 m Lower bound.5 1 1.5.5 ν r / B r 1 Page 11
The role of the baseline t. Vesuvius DE Bn=1 m Estimated shift.8.6.4. -. -...4.6.8 True shit, f r /B r Bn=3 m Bn=33 m Page 1
ulti-channel Interferometry: joint estimate H 1 y 1 We maximize a log-likelihood that is the superpostion of the SB log-likelihood. s H i. H j y i y j G j i + G i j - W ij N N n= 1 m= n+ 1 ( ν ; y, y γ ) ˆ ν = arg max J, J 1-1.5-1 -.5.5 1 1.5 nm n m nm H N y N J 13-1.5-1 -.5.5 1 1.5 υ max(- ) Σ J 3-1.5-1 -.5.5 1 1.5 ΣJ nm -1.5-1 -.5.5 1 1.5 ν / B r 1 We can assume that kernels G i j, Wi j are dependent only on channels i,j. Page 13
ulti Baseline: joint estimate vs a-posteriori combination Simulations of four datasets: baselines { 1 3 33}..8 A-posteriori combination of the estimates..6.4. -. -...4.6.8.8.6.4. -. -...4.6.8 Proposed method: Joint estimate. The joint estimate is effective in removing slope biasing. The gain increases with the number of baselines Page 14
Experimental Data (LAS VEGAS): Range gradient 6 ENVISAT SLC over Las Vegas. Comparison with SRT (1 arc sec) SRT ENVISAT B.1 1 1 Dato B N [m] T Y1 Y 37. 35 Y3 3. 7 Y4 55. 15 Y5 399. 45 Y6 544. 8 x 1 3 1.5.5 1.5 3 3 4 -.5 4 -.5 5 -.1 5-1 -1.5 6 1 3 4 5 6 7 8 6-1 3 4 5 6 7 8 Page 15
Experimental Data (LAS VEGAS): Range gradient 19 ENVISAT SLC over Las Vegas. Comparison with SRT (1 arc sec) SRT ENVISAT B immagine Delta giorni baseline normale [m] Y1-15 Y 34-359 Y3 68-74 Y4 13-171 Y5 138-14 Y6 9-138 Y7 46 Y8 8 95 Y9-63 114 Y1-598 183 Y11-564 43 Y1-53 61 Y13-46 45 Y14-39 454 Y15-356 461 Y16-183 641 Y17-149 734 Y18-79 888 Y19-45 17 Page 16
Azimuth Gradient SRT ENVISAT B Page 17
DE generation Absolute phase field, unwrapping by IRLS L norm. Page 18
Conclusions The L estimate has been derived for the case of constantsloped terrain, and assuming distributed target. An efficient LTI implementation has been proposed. The estimate is computationally intensive, but much more accurate than the conventional interferogram periodogram and more efficient than the L estimate based on the inversion of the data covariance matrix. In the multi-baseline case, a simple and efficient extension has been proposed by computing the likelihood of i,j indepently upon the others. The joint estimate is much more effective than the a-posteriori combination in removing bias. Page 19