Coherence Linearity and SKP-Structured Matrices in Multi-Baseline - PowerPoint PPT Presentation

Coherence Linearity and SKP-Structured Matrices in Multi-Baseline PolInSAR Stefano Tebaldini and Fabio Rocca Politecnico di Milano Dipartimento di Elettronica e Informazione IGARSS 2011, Vancouver

Introduction The availability of Multi-baseline PolInSAR data makes it possible to Track N decompose the signal into ground-only and volume-only contributions HH HV Track n VH VV Track n y n  w i   Track 1 HH HV Polarization w i VH VV Re{y n (w 1 )} Re{y n (w 2 )} Re{y n (w 3 )} HH HV VH VV Im{y n (w 1 )} Im{y n (w 2 )} Im{y n (w 3 )} Volume-only contributions 60 50 Properties of the vegetation layer 40 30 • Vertical structure 20 10 • Polarimetry 0 -10 200 600 1000 1400 1800 2200 Decomposition Ground-only contributions 60 50 • Phase calibration 40 30 • Digital Terrain Model 20 10 • Ground properties 0 -10 200 600 1000 1400 1800 2200

Introduction Polarimetric SAR Interferometry (PolInSAR) • The coherence locus is assumed to be a straight line in the complex plane • G/V decomposition is carried out by fitting a straight line in each interferometric pair imaginary part imaginary part imaginary part Volume coherence Ground coherence Measured coherences real part real part real part Algebraic Synthesis • The data covariance matrix is assumed to be structured as a Sum of 2 Kronecker Products • G/V dec is carried out by taking the first 2 terms of the SKP decomposition of the data covariance matrix Scope of this work: Compare the two approaches from the algebraic and statistical points of view

Model of the acquisitions We consider a multi-polarimetric and multi-baseline (MPMB) data • Monostatic acquisitions: up to 3 independent SLC images per track Track N Track n y n  w i   Polarization w i HH HV Re{y n (w 3 )} Track n Re{y n (w 1 )} Re{y n (w 2 )} VH VV Track 1 HH HV Im{y n (w 1 )} Im{y n (w 2 )} Im{y n (w 3 )} VH VV HH HV VH VV

Coherence linearity PolInSAR is based on the variation of the interferometric coherence w.r.t. polarization Σ     H w w  Σ   H E y y i nm j w w , nm n m Σ Σ nm i j H H w w w w    i nn i j mm j   y HH VV n       w i ≠ w j  Multiple Scattering Mechanisms (MSM) y y HH VV   n n       2 y HV w i = w j  Equalized Scattering Mechanisms (ESM) n Coherence linearity (*): RVoG model => ESM coherences describe a straight line in the complex plane                   v g w , w 1 w w imaginary part Volume coherence Ground coherence ESM coherences Polarization Volume Ground depending factor coherence coherence real part (*) Papathanassiou and Cloude, “Single Baseline Polarimetric SAR Interferometry

Coherence linearity PolInSAR is based on the variation of the interferometric coherence w.r.t. polarization Σ     H w w  Σ   H E y y i nm j w w , nm n m Σ Σ nm i j H H w w w w    i nn i j mm j   y HH VV n       w i ≠ w j  Multiple Scattering Mechanisms (MSM) y y HH VV   n n       2 y HV w i = w j  Equalized Scattering Mechanisms (ESM) n Coherence linearity (*): RVoG model => ESM coherences describe a straight line in the complex plane Multiple baselines: one line per interferometric pair n = 1 m = 3 n = 1 m = 4 n = 1 m = 2 Volume coherence Ground coherence imaginary part imaginary part imaginary part ESM coherences real part real part real part (*) Papathanassiou and Cloude, “Single Baseline Polarimetric SAR Interferometry

The SKP structure Without loss of generality, the received signal can be assumed to be contributed by K distinct Scattering Mechanisms (SMs), representing ground, volume, ground-trunk scattering, or other     K   s k ( n , w i ) : contribution of the k-th SM in y w s n ; w n i k i  Track n , Polarization w i k 1 Hp: the data covariance is structured as a Sum of Kronecker Products K W K  E  yy H        K   C k  R k y w s n ; w n i k i  k 1 k  1 k-th Scattering Mechanism Covariance matrix Covariance matrix Each SM is among polarizations: among tracks: represented by a EM properties Vertical structure Kronecker Product R k : interferometric coherences of the k-th SM alone [NxN] C k : polarimetric correlation of the k-th SM alone [3x3] Note that R k , C k are positive definite

The SKP decomposition The key to the exploitation of the SKP structure is the existence of a decomposition of any matrix into a SKP Two sets of matrices U p , V p such that: SKP P W    W U V Dec p p  p 1 Theorem: K    W C R Let W be contributed by K SMs according to H1,H2,H3, i.e.: k k  k 1 then , the matrices U k , V k are related to the matrices C k , R k via a linear, invertible transformation defined by exactly K(K−1) real numbers Corollary :     W C R C R If only ground and volume scattering occurs, i.e: g g v v                 1 C 1 U U R a V 1 a V a b b b then , there exist two real 1 2 g g 1 2                  1 numbers ( a,b ) such that: C a b 1 a U a U R b V 1 b V v 1 2 v 1 2

Forested areas: how many KPs ? BIOSAR 2007 – Southern Sweden – P-Band BIOSAR 2008 – Northern Sweden – P – Band and L- Band HH P-Band - HV 60 height [m] 50 30 Height [m] 40 20 30 20 10 10 0 0 -10 -10 200 600 1000 1400 1800 2200 2000 2500 3000 3500 4000 4500 5000 HV L-Band - HV 60 LIDAR Terrain Height 30 height [m] 30 50 Height [m] Height [m] LIDAR Forest Height 40 20 20 30 10 10 20 10 0 0 0 -10 -10 -10 2000 2000 2500 2500 3000 3000 3500 3500 4000 4000 4500 4500 5000 5000 200 600 1000 1400 1800 2200 Ground range [m] slant range [m] TROPISAR – French Guyana – P-Band Courtesy of ONERA HH 60 Height [m] 40 20 0 height 400 600 800 1000 1200 1400 HV 60 Height [m] 40 20 0 400 600 800 1000 1200 1400 Slant range [m]

Forested areas: how many KPs ? BIOSAR 2007 – Southern Sweden – P-Band BIOSAR 2008 – Northern Sweden – P – Band and L- Band HH P-Band - HV 60 height [m] 50 30 Height [m] 40 20 30 20 10 10 0 0 -10 -10 200 600 1000 1400 1800 2200 2000 2500 3000 3500 4000 4500 5000 HV L-Band - HV 60 LIDAR Terrain Height 30 height [m] 30 50 Height [m] Height [m] LIDAR Forest Height 40 20 20 2 KPs account for about 90% of the information 30 10 10 20 carried by the data in all investigated cases 10 0 0 0 -10 -10 -10 2000 2000 2500 2500 3000 3000 3500 3500 4000 4000 4500 4500 5000 5000 200 600 1000 1400 1800 2200 Ground range [m] slant range [m]  2 Layered-models (Ground + Volume) are well TROPISAR – French Guyana – P-Band Courtesy of ONERA suited for forestry investigations HH 60 Height [m] 40 20 0 height 400 600 800 1000 1200 1400 HV 60 Height [m] 40 20 0 400 600 800 1000 1200 1400 Slant range [m]

Forested areas: how many KPs ? BIOSAR 2007 – Southern Sweden – P-Band BIOSAR 2008 – Northern Sweden – P – Band and L- Band HH P-Band - HV 60 height [m] 50 30 Height [m] 40 20 30 20 10 10 0 0 -10 -10 200 600 1000 1400 1800 2200 2000 2500 3000 3500 4000 4500 5000 HV L-Band - HV 60 LIDAR Terrain Height 30 height [m] 30 50 Height [m] Height [m] LIDAR Forest Height 40 20 20 Overview talk: 30 10 10 20 10 0 0 0 P-Band penetration in tropical and boreal -10 -10 -10 2000 2000 2500 2500 3000 3000 3500 3500 4000 4000 4500 4500 5000 5000 200 600 1000 1400 1800 2200 Ground range [m] slant range [m] forests: Tomographical results TROPISAR – French Guyana – P-Band Courtesy of ONERA Friday – 14:40 HH 60 Height [m] Room 1 40 20 0 height 400 600 800 1000 1200 1400 HV 60 Height [m] 40 20 0 400 600 800 1000 1200 1400 Slant range [m]

Coherence linearity and 2KPs: Algebraic connections • Polarimetric Stationarity (PS): • Introduced by Ferro-Famil et al. to formalize the widely considered – RVoG consistent – case where the scene polarimetric properties are invariant to the choice of the passage        Σ Σ H H E y y E y y nn n n mm m m • Always valid after whitening the polarimetric information of each image in such a way as:   Σ I  n nn 3 3  Always retained in the remainder • Under the PS condition the ESM coherence can be decomposed into a weighted sum:   K          K Σ        H k y y C E W C R k R k k nm n m k nm k nm nm   k 1 k 1       K H   w C w ( PS )         w   k w , w w w k K k  nm k nm H w C  1 k k  k 1