Parametric Estimation and Model Selection based on Amplitude-only Data in PS-Interferometry. (How Many Lines or Points are There?)

Transcription

1 Parametric Estimation and Model Selection based on Amplitude-only Data in PS-Interferometry (How Many Lines or Points are There?), Richard Bamler, Michael Eineder and Bert Kampes

2 DLR s PSIC4 results: Marseille average displacement rate mm/y

3 DLR s PSIC4 results: Gardanne PSI technique is well established mm/y

4 single dominant point scatterer simple estimation optimal estimator: periodogram 8 m LOS

5 single dominant nearly point scatterer simple estimation optimal estimator: periodogram 8 m LOS

6 two dominant point scatterers standard PS estimation fails PS density reduces 100 m 8 m 256 m LOS

7 tomography is a frequency estimation problem phase needs to be corrected for displacement and APS B eff x resolution: ca. 10 m parametrically difficult if number of PS is unknown model selection problem r = 2 m R + x Rm R ( x h) m h x 2 R m R m LOS h

8 PS: two dominant scatterers model x intensity only i = a = a a + a a + 2 a 0 0 a 1 a 1 cos cos ( B ( h0 h1 ) + ( ϕ0 ϕ1) ) ( B h + ϕ) B eff a 1 h a 0 no phase information required only two dominant scatterers other scatterers contribute to clutter resolution: (4m).. 30 m R m a 0 LOS h

9 single plate scatterer model = 2m =1m = 3m i = 2 sin( B + ϕ) 2 a B + ϕ a=const a =1.0 : width of sinc() a: scale φ: shift

10 two dominant scatterers model a0 = 2.0 a1 = 0.5 a = 1.0 a1 0 = 1.5 a0 = 1.0 a1 = 1.0 ( B + ϕ) 2 2 i = a + a + a a cos h h=const=12m h: periodocity and shape a: scale and shape/offset

11 two dominant scatterers model h = 2m h = 0. 5m h = 10m ( B + ϕ) 2 2 i = a + a + a a cos h a=const: a0 = 1.0 a1 = 1.5 h: periodocity and shape a: scale and shape/offset

12 two scatterers better model than single obvious example (1) with real data sinc is not periodic but the data are single scatterer assumption -> wrong sinc-fit two scatterers assumption -> cos-fit better describes the data

13 two scatterers better model than single single scatterer assumption obvious example (2) with real data sinc always goes to zero but the data don t -> wrong sinc-fit two scatterers assumption -> cos-fit better describes the data

14 two scatterers better model than single obvious example (3) with real data better LMS fit single scatterer assumption -> good sinc-fit two scatterers assumption -> cos-fit better describes the data

15 limits in h estimation lower limit: (4m) but often better 4 similar sinc and cos fit sinc(.) a=1 =2.3m cos(.) a 0 =0.7 a 1 =0.3 =3m upper limit: ~30m 4 caused by the limited number of observations 4 the limited sampling frequency (along the baseline) 4 the influence of clutter cos(.) a 0 =0.7 a 1 =0.7 h=2m cos(.) a 0 =0.7 a 1 =0.3 h=1m sinc(.)

16 model selection problem sinc-fit: 3 parameters cos-fit: 4 parameters theory by D.S. Sivia ( Data Analysis, A Bayesian Tutorial ) two hypotheses: 4 A: flat panel (sinc-fit) 4 B: two dominant scatterers (cos-fit)

17 Bayesian theory on model selection posterior ratio = prob prob ( A D) > 1: A preferred ( B D) < 1: B preferred A, B: scatterer configuration model D: data (radar amplitude) prob prob ( A D) ( B D) = = prob prob ( D A) prob( A) prob( D) ( D B) prob( B) prob( D) a priory knowledge (=1) likelihood function prob(d) cancels prob prob ( D A) ( D B) = prob prob ( D A, a, h, ϕ) ( D B, a, a, h, ϕ) 0 0 goodness-of-fit Ockham factor f B min max max f A( a, a, h, σ a, σ h ) min max min max max ( a, a, a, a, h, σ, σ, σ ) 0 0 best predictions parameter range a, h, a, a 0 1,ϕ uncertainty permitted by the data σ,,, a σ h σ a σ 0 a1 1 1 a min max max, a, h a 0 a 1 h

18 simple model selection goodness-of-fit ratio: 2 count: 63 out of per cent of PS candidates

19 simple model selection goodness-of-fit ratio: 1.5 count: 2267 out of per cent of PS candidates

20 simple model selection goodness-of-fit ratio: 1.3 count: out of per cent of PS candidates

21 simple model selection goodness-of-fit ratio: 1.2 x count: out of per cent of PS candidates

22 application: reference network by hypotheses testing: inconsistent points and arcs are rejected quite expensive due to iterative processing now reference net takes advantage from scatterer configuration prediction

23 applications continued complements classical tomography 4 classical ca. 10m resolution 4 ampl. parametric: (4)-30m 4 no APS and displacement influence 4 number of scatterers (1 or 2) SCR estimation and PS detection 4 more accurate: SCR(time) 4 better PS density estimation 4 scatterer configuration 4 model selection for PS estimation high precision calibration based on natural scatterers (no CR) SCR max SCR min clutter

24 summary parametric estimation and detection of scatterer configuration 4 single dominant scatterer: const or sinc-fit 4 two dominant point scatterers: cos-fit complements full tomography 4 uses amplitude data only 4 no correction of APS and displacement necessary 4 therefore usable before estimation 4 complements resolution: (4)-30m (~10m for classical spectral methods) 4 number of scatterers (1 or 2) theoretically optimal, computationally effective and robust algorithm 4 Bayesian theory for model selection 4 robust: Least Median Squares