Invariant deformation parameters from GPS permanent networks using stochastic interpolation

Transcription

1 Invarant deformaton parameters from GPS permanent networks usng stochastc nterpolaton Ludovco Bag, Poltecnco d Mlano, DIIAR Athanasos Dermans, Arstotle Unversty of Thessalonk

2 Outlne Startng hypotheses Methodology: >>wth partcular attenton to the new concepts A case study: a dense network n Japan Prelmnary results on EPN analyss

3 The temporal scale Permanent networks provde contnuous tme seres; deformatons are computed at dscrete epochs: a tme model s needed.

4 The spatal scale Permanent networks provde dscrete n space observatons; deformatons are represented by a feld: a spatal model s needed. To study large areas deformatons should be analyzed on the ellpsod surface (curvlnear coordnates) and not on some planar projecton: >>a metrc deformaton functon s requred to restore orthogonal deformaton parameters from curvlnear analyses.

5 The estmaton process Preprocessng Estmaton of the lnear trend for each staton P. Clusterng of one regon n homogeneous networks. Estmaton of the Tsserand parameters for homogeneous networks. Processng Wthn each homogeneous network, modelng of the deformaton at any pont, by: fnte elements (nterpolaton), and/or collocaton (predcton).

6 The temporal scale: a lnear nterpolaton Coordnates on a short tme span are lnear functons of tme. Observaton model for each staton: T T T η( Pk, t) η0k vkt εk; η E, N /,, v ve, vn / v, v Stochastc model, for the network: At frst teraton T C( t ) I 3N3N

7 Estmaton of η 0, v, ε by Least Squares; k k emprcal estmaton of C. dentcal for each epoch, no correlatons between dfferent epochs. k C( t ) Cηη ( t 1 1 ) Cηη ( t )... ( ) 1 2 Cηη t 1 p T Cηη ( t) Cηη ( t)... Cηη ( t) T T ( t ) ( )... ( ) p 1 t p 2 t p p Cη η Cη η Cη η p C >>smplfed but stochastcally rgorous covarance estmaton >> no teratve approach requred.

8 Horzontal deformaton on the surface of the reference ellpsod Horzontal deformaton on ellpsodal surface t t Actual deformaton s 3-dmensonal

9 t t But we can observe only on 2-dmensonal earth surface! Interpolaton t t Extrapolaton Why not 3D deformaton? 3D deformaton: nterpolaton and extrapolaton. Extrapolaton from surface geodetc data s not relable f no addtonal geophyscal data are avalable.

10 x Horzontal deformaton analyss f F x x f( x) xu( x) x x( Pt, ) Spatal nterpolaton x x x( Pt, ) Dscrete geodetc nformaton at GPS permanent statons x x Interpolaton to obtan contnuous dsplacement nformaton Dfferentaton to obtan the deformaton gradent F dsplacement gradent

11 Interpolaton n presence of dscontnutes Separaton of rgd moton from deformatons: pecewse analyss s needed before nterpolaton. x x x( P, t) x x x( P, t) Bad spatal nterpolaton x x( P, t) Good spatal nterpolaton x f( x) x f( x) x x x x( P, t)

12 Separaton of rgd moton from deformaton Apart from nternal deformaton regons are n relatve moton

13 Separaton of rgd moton from deformaton Orgnal RS Optmal RS How to represent the moton of a deformng regon as a whole? By the moton of a regonal optmal reference system. Optmal: such that the correspondng dsplacements are as small as possble.

14 Separaton of rgd moton from deformaton Horzontal moton on earth ellpsod ( sphere): Rotaton around an axs wth angular velocty v ap x vap [ ω] x x v Defnton of optmal reference frame: Mnmzaton of relatve knetc energy of regonal network T v T ap,ap v,ap mn Dscrete Tsserand reference system Soluton: C h [ x] ω C h 2 dx [ x] dt 1 () t () t () t = nerta matrx = relatve angular momentum

15 Deformaton analyss wthn one network Estmaton of the nner deformatons between two network states at two epochs t and t. Deformaton functon: x(') t f((',), t t x ()) t Deformaton gradent n space doman: F f x ( x) Interpolaton of F by fnte elements: Bag and Dermans, IAG Symposa Volumes 131

16 Fnte elements Determnstc nterpolaton Pecewse dsplacements and deformatons. Collocaton Stochastc predcton Contnuous dsplacements and deformaton feld.

17 completely absorbed n estmated deformatons predcton. only functons of observatons uncertantes. Observatons errors Accuraces estmates can be fltered by the predcton. reflect both observatons and sgnals uncertantes.

18 Predcton of F by collocaton Estmated dsplacements ux () t v x t, u () y t v yt at the network ponts P are used to predct dsplacements and/or dsplacement gradent elements at any pont P: ux u uy u x J FI, J11, J12, J21, J22 x y x y y 2. A rotaton s appled to the network to obtan C ( P, Q) 0 uu x y >>Remove: >>emprcal decorrelaton of dsplacements on the sphere.

19 Choce of the model covarance functon for the dsplacements: Exp: 2 2 PQ PQ 0 u 0 0 C ( P, Q) C e, C ( P, Q) C e, C 0 u x x y y >>Legendre Polnomals: N C ( P, Q) c P(cos ), C ( P, Q) c P(cos ), c 0 ux nx n PQ uy ny n PQ j n0 n0 3. Emprcal estmaton of the covarance functon parameters. >>In case of Legendre polynomals: >>applcaton of Not Negatve Least Squares to >>fnd and estmate not zero coeffcents. N

20 Constructon of observaton vector/covarance matrx s ux Cu 0 x, ux ux( P), uy uy( P), C ss u y 0 Cu y 5. Predcton of a sgnal t at any pont P: two approaches A. Exact nterpolaton: B. Smoothng: ˆ 1 t CtsCsss ˆ 1 t C [ C C ] s ts ss where C s the estmated covarance matrx of the dsplacements errors.

21 Computaton of the crosscovarance matrx C ts If t u ( P)) [ u ( P) u ( P)] T x t follows C [ C C ] ts tu tu x Id est { Ctu } 1 (,,, ) x k Cu x y x x k yk { C } C ( x, y, x, y ) 0 C ( x, y, x, y ) { C } tu 2k u u k k u u k k tu 1k x x y y x y { C } C ( x, y, x, y ) tu 2k u k k y y y y

22 Deformaton Jacoban t Vect( J ( P)) [ J ( P) J ( P) J ( P) J ( P)] T C [ C C ] ts tu tu x y { C } 1 C ( P, P ) C (, ) (,,, ) tu x k J 11 u x k P P C x y x y u x k u x u k k x x x { C tu } 2 (, ) (, ) (,,, ) x k CJ 12u P P x k C u P P x k Cu x y x x k yk ux y y { C } 3 C ( P, P ) C (, ) (,,, ) 0 tu x k J 21 u x k P P C x y x y u y k u x u y u k k x x x { C tu } 4 (, ) (, ) (,,, ) 0 x k CJ22 u P P x k C u P P y k Cuxu x y x y k yk ux y y The same can be appled to { C tu y }

23 Estmaton of the covarance matrx of the predcted sgnal A. Exact nterpolaton 1 1 Covarance matrx of the predctons: Ctt ˆˆ C C CC C Covarance matrx of the errors: C C T ts ss ss ts tt ˆˆ B. Smoothng 1 Covarance matrx of the predctons: Cˆˆ C ( C C ) tt C 1 Covarance matrx of the errors: C C C ( C C ) C T ts ss ts T tt ts ss ts >>Restore of the decorrelaton nto the predcted sgnals From F ˆ ( P) IJ ˆ ( P) the deformaton parameters are computed.

24 Egenvalues and angles: max ( P), mn ( P), ( P ) F R( ) LR ( ) cos sn 0 cos sn sn cos sn cos t R( ) t F L R( ) 2 1 1

25 Shears ( P), ( P) and scale s 1 FF(, s,, ) sr( ) R( ) GR ( ), G 0 1 shear along drecton Γ R( ) ΓR( ) addtonal scalng (scale factor s) s

26 Covarance propagaton from the deformaton gradent to the deformaton parameters. For example, f max λ mn λ( Vec( F )), under a frst order approxmaton then, smply λ Vec( Fˆ ) L, C LC L Vec( F ) T

27 The metrc on the ellpsod In case of planar coordnates, the computaton [ η, u] F F F [,,,,, ] T 0 max mn s drect, ether by fnte elements or by collocaton. In case of curvlnear (geodetc) coordnates, one more step s needed: [ η, u] F CD F F D [,,,,, ], T T 0 max mn where D s the matrx of the metrc deformaton on the ellpsod surface, here not dscussed nto detals.

28 The case study: Japan network One year (2005) of daly solutons of the Japan PN ( 1200 SP). Preprocessng: rejecton of bundlers, clusterng of the network n homogeneous subregons, selecton of a partcular subregon as a case study, Delaunay trangulaton. Predcton of the deformaton parameters, by fnte elements and collocaton. Comparsons.

29 Orgnal dsplacements Tsserand dsplacements

30 Decorrelaton of the sgnals n longtude and lattude Before decorrelaton After decorrelaton

31 Emprcal estmaton of the model covarances Longtude and lattude dsplacements: comparsons between NNLS Legendre fttng and a typcal choce of exponental covarance

32 R1-SouthWest rotaton R3-North translaton R2-Center compresson Tsserand analyses of R1 and R3 wrt R2. Separate deformaton predctons.

33 Egenvalues of deformaton gradent Fnte elements Collocaton

34 Shears and dlatatons Fnte elements Collocaton

35 Sgnal to nose ratos Fnte elements Collocaton

36 Methodologcal results and conclusons New algorthms for curvlnear (geodetc) coordnates have been studed and mplemented for 1. Tsserand analyss on networks. 2. Dsplacements and deformaton feld predcton, both by fnte elements and by collocaton. 3. LP's NNLS fttng of emprcal covarance functon. 4. Covarance propagaton from tme seres to deformaton parameters. WRT fnte elements, collocaton provdes 1. contnuous estmates, 2. smooth results, 3. more realstc varance-covarance assessment.

37 Frst trals on EPN: class A soluton of GPSW 1570 Orgnal veloctes (n ITRF2005)

38 Tsserand veloctes Note: n ths very frst analyss: all the EPN statons used, no subregon clusterng.

39 Comparsons between ITRF-Tsserand and ETRF A small rotaton due to the no subregon clusterng between class A EPN statons (stable part of Europe).

40 Numercal dfferences (ITRF+Tss)-(ETRF) (ITRF+Tss)-(ETRF+Tss) East North East North E StdDev mn max Applcaton of Tsserand prncple to ETRF coordnates leads to almost dentcal results. What about spatal covarances of Tsserand dsplacements?

41 Emprcal covarances (wthout model estmaton) East and North dsplacements n pseudo-mm: no correlated sgnal to predct. A macro clusterng between subregons would be useful. The relatve moton of subregons could be estmated; to predct deformatons n subregons, denser networks are needed.