The Slepian approach revisited:

Transcription

1 he Slepian approach revisited: dealing with the polar gap in satellite based geopotential recovery Oliver Baur, Nico Sneeuw Geodätisches Institut Universität Stuttgart 3 rd GOCE user workshop 6-8 Noveber 26 Frascati, Italy

2 he polar gap proble GOCE sun-sync inclination: area of polar gap: axi-syetric gap I A 2 " 1.7 " 1 k ˆ.66% z " 6.6 y x

3 he polar gap proble Poor estiability of low order coefficients relative error s l c l s l c l Reason: Ø Loss of orthogonality of spherical haronics due to non-global coverage Ø Low-order Legendre functions contain ore signal at high latitudes

4 he polar gap proble Rule of thub " ax $ f # i l f 2 l Ordnung quadrature f 1 LS adjustent f.5 Grad

5 Slepian approach Philosophy Adapt the paraeterization of the geopotential to GOCE observations band-liited base functions that are optially concentrated in B: Slepian Functions sphere belt B cap C " B C

6 Derivation of Slepian functions Maxiize the spatial concentration of a bandliited function g(x): Ø haronic developent L l g (x) l " l g Y l l( #) Ø axiization # g( x) g( x) 2 B 2 " B " g( x) g( x) 2 2 d" d" g Dg g g ax Ø algebraic EVP g Dg ax g g " Dg g D is real, syetric and positive definit Ø concentration factor < < 1

7 Derivation of Slepian functions Ø kernel D ll' ' 1 YlYl ' ' d" 4# B Ø special case: spherical band D ll' 1 & # $ % 2 % P l P l' sin% d% " D g g D diag{ D, D 1, D 1,..., D L, D L } Blockdiagonal structure, separation by order Diensions D : D : ( L + 1) 2 "( L + 1) ( L + 1) "( L + 1) 2 Solve (L+1) separate algebraic EVP to obtain L(L+1)/2 concentration factors (eigenvalues) and eigenvectors

8 Derivation of Slepian functions Slepian functions bandliited eigenfunctions L S j (cos# ) " glpl(cos# ), j l S G P L Properties: Ø orthonoral on sphere Ø orthogonal on spherical band Ø linear cobination of Legendre functions of sae order Ø order is valid for Legendre and Slepian functions Ø even/odd function for j even/odd

9 Solving the algebraic EVP L 5 : spectru of concentration factors (sorted according to ) 1 7 D ll' 1 2 $ " # # P l P l' sin#d# D g g equal eigenvalues abiguous eigenvectors Rando Slepian functions

10 Slepian functions: calculation Original EVP D g g Diagonalization Nuerical Integration >.5 <.5, g Spectru of concentration factors (L 5, ) Rando Estiation S G P

11 Slepian functions: calculation Original EVP D g g Couting Operators / Matrices D D Couting EVP g g Diagonalization Nuerical Integration Analytically, ridiagonal Matrix Diagonalization # ", g, g Rando Estiation S G P Unique Estiation

12 Slepian functions: calculation Couting EVP g g Analytically, ridiagonal Matrix Diagonalization Spectru of Grünbau eigenvalues (L 5, ), g S G P Unique Estiation

13 Slepian functions: exaples Slepian Functions 1-6, 5, 51 (L 5, ) M M co-latitude co-latitude

14 Slepian vs. Legendre L 5,

15 Gravity recovery transforation of base functions P S G LS estiation Py A PA A x 1 Slep ) ( ˆ back transforation of base functions Leg ˆ Slep ˆ x G x coputation of eigenvectors g g g transfer function (radial continuation) 1) 2)( ( ) ( " # $ % & + l l r R R G r H l l, ) ( ) ( ) ( " # $ $ $ % & r H r H r L l l L M O M L H radial continuation l l r r P H G S G H G S ) ( ) ( :

16 Gravity recovery: siulation Closed-loop siulation Ø EGM96 till L3 (SC7) Ø radial SGG coponent Ø noiseless data Ø Recovery up till L1 Ø Legendre paraetrization Ø Slepian paraetrization (with subsequent backtransforation)

17 Gravity recovery: spectral results relative errors Legendre paraeterization Degree s l c l order Slepian paraeterization Slepian Index b j a j

18 Gravity recovery: spatial results Breitengrad RMS () Legendre in 1 Slepian in 5 Slepian j in 5 Slepian in 1 Slepian j in 1

19 Conclusions Slepian functions: optial in ters of siultaneous spectral and spatial liitation Grünbau coutating operator: uniquely defined Slepian functions Great results in spectral doain ( Slepian world ) In spatial doain siilar results as with Legendre paraeterization, in particular when noisy data No direct benefit for global geopotential recovery Isolation, no solution, of polar gap proble Hence, handle on polar gap proble in stochastic odeling and regularization

20 References 1. Albertella A, Sansò F, Sneeuw N (1999). Band-liited functions on a bounded spherical doain: the Slepian proble on the sphere, J. Geod. 73: Grünbau FA, Longhi L, Perlstadt M (1982). Differential operators couting with finite convolution integral operators: soe non-abelian exaples, SIAM J. Appl. Math. 42: Sions FJ, Dahlen FA, Wieczorek MA (26). Spatiospectral localization on a sphere, SIAM Review, 48(3): Slepian D (1978). Prolate spheroidal wave functions, Fourier analysis and uncertainty. V. he discrete case, Bell. Syst. ech. J. 57(5): Slepian D (1983). Soe coents on Fourier analysis, uncertainty and odeling, SIAM Rev., 25(3):