Gravity field recovery in high latitude areas

(1)

Gravity field recovery Gravity field recovery

in in

high latitude areas high latitude areas

M. Weigelt, W. Keller, N. Sneeuw M. Weigelt, W. Keller, N. Sneeuw

[email protected]

Geodätische Woche 2007 - Geodetic Week 2007

Session 1

(2)

Outline

Motivation for local gravity field recovery Slepian functions - an empirical base

• ^Approach

• ^Properties

Test Scenario: CHAMP in high latitude areas

(3)

Motivation – Influence of the ground track

June 2003 January 2003

Error spectrum:

Ground track:

(4)

Motivation – Data density

January 2003 June 2003

(5)

Slepian functions

Spatial concentration of bandlimited functions (Simons et al. 2005) :

• ^given:

• ^approach:

• yields an algebraic eigenvalue problem:

λ g = D g with D =

R

Y _lm Y _nk d Ω

S =

L

l =0

l

m = − l

g _lm Y ^lm

λ =

R S ² d Ω

Ω S ² d Ω = maximum

− 1

(6)

Potential and Orthogonality Slepian representation of the potential :

double orthogonality (Gilbert & Slepian 1977)

T = GM R

(L+1)

²

i=1

β _i S _i = GM R

(L+1)

²

i=1

β _i

L

l=0

l

m= − l

R

r

^l+1

g _lm ⁱ Y ^lm

g _lm ⁱ g _lm ^j d Ω = δ _ij

g _lm ⁱ g _lm ^j d Ω = λ _ij δ _ij

(7)

Localization property

indication of spatial concentration by eigenvalues

(8)

Spherical harmonics ↔ Slepian coefficients transformation:

influence of one Slepian coefficient, e.g., :

β _i =

L

l=0

l

m= − l

K _lm g _i ^lm K _lm =

( L +1)

²

i =1

β _i g _lm ⁱ

g _lm ³

(9)

Edge effect

(10)

Test scenario

monthly CHAMP solutions in high-latitude areas residual signal:

GGM02s as reference

T ( λ, φ, r ) − T _Lmax ^SH ( λ, φ, r )

(11)

Processing scheme

Energy balance approach

Slepian analysis Spherical harmonic

analysis

GGM02s

+ _

_

+ +

+ _ +

long wavelength part

transformation

(12)

Spherical harmonic vs. Slepian functions w.r.t. GGM02s

(13)

Spherical harmonic vs. Slepian functions w.r.t. GGM02s

(14)

Spherical harmonic vs. Slepian functions w.r.t. GGM02s

(15)

Affected coefficients

January 2003

June 2003

(16)

Conclusions

Slepian functions as a empiricial base function

• maximum spatial concentration of a bandlimited function

• double orthogonality

• spatial concentration indicated by eigenvalue

improvement in high latitude areas

dependency on a priori information (edge effect)

(17)

Gravity field recovery Gravity field recovery

in in

high latitude areas high latitude areas

M. Weigelt, W. Keller, N. Sneeuw M. Weigelt, W. Keller, N. Sneeuw

[email protected]

Geodätische Woche 2007 - Geodetic Week 2007 Session 1

THANK YOU

Gravity field recovery in high latitude areas

Gravity field recovery Gravity field recovery

in in

high latitude areas high latitude areas

M. Weigelt, W. Keller, N. Sneeuw M. Weigelt, W. Keller, N. Sneeuw

[email protected]

Geodätische Woche 2007 - Geodetic Week 2007

Session 1

Outline

Motivation for local gravity field recovery Slepian functions - an empirical base

• Approach

• Properties

Test Scenario: CHAMP in high latitude areas

Motivation – Influence of the ground track

Error spectrum:

Ground track:

Motivation – Data density

January 2003 June 2003

Slepian functions

Spatial concentration of bandlimited functions (Simons et al. 2005) :

• given:

• approach:

• yields an algebraic eigenvalue problem:

λ g = D g with D =

R

Y lm Y nk d Ω

S =

L

l =0

l

m = − l

g lm Y lm

λ =

R S 2 d Ω

Ω S 2 d Ω = maximum

− 1

Potential and Orthogonality Slepian representation of the potential :

double orthogonality (Gilbert & Slepian 1977)

T = GM R

(L+1)

i=1

β i S i = GM R

(L+1)

i=1

β i

L

l=0

l

m= − l

R

r

l+1

g lm i Y lm

g lm i g lm j d Ω = δ ij

g lm i g lm j d Ω = λ ij δ ij

Localization property

indication of spatial concentration by eigenvalues

Spherical harmonics ↔ Slepian coefficients transformation:

influence of one Slepian coefficient, e.g., :

β i =

L

l=0

l

m= − l

K lm g i lm K lm =

( L +1)

i =1

β i g lm i

g lm 3

Edge effect

Test scenario

monthly CHAMP solutions in high-latitude areas residual signal:

GGM02s as reference

T ( λ, φ, r ) − T Lmax SH ( λ, φ, r )

Processing scheme

Energy balance approach

Slepian analysis Spherical harmonic

analysis

GGM02s

+ _

• ^Approach

• ^Properties

• ^given:

• ^approach:

Y _lm Y _nk d Ω

g _lm Y ^lm

R S ² d Ω

Ω S ² d Ω = maximum

β _i S _i = GM R

β _i

^l+1

g _lm ⁱ Y ^lm

g _lm ⁱ g _lm ^j d Ω = δ _ij

g _lm ⁱ g _lm ^j d Ω = λ _ij δ _ij

β _i =

K _lm g _i ^lm K _lm =

β _i g _lm ⁱ

g _lm ³

T ( λ, φ, r ) − T _Lmax ^SH ( λ, φ, r )