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On the numerical stability in the derivation of Slepian base functions

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On the numerical stability in the derivation of Slepian base functions

Matthias Weigelt Tonie van Dam Lin Wang

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Outline

• Slepian representation

• Meissl scheme

• Dependency on the data distribution

• Combination GRACE + GPS

(3)

Spatial and spectral concentration problem

Objective: base function suitable for local modelling in area R

(4)

Spatial and spectral concentration problem

Basic problem: no concurrent spatial and spectral limitation Objective: base function suitable for local modelling in area R

(5)

Spatial and spectral concentration problem

Basic problem: no concurrent spatial and spectral limitation Approach by Simons et al. 2005:

• given:

Objective: base function suitable for local modelling in area R

(6)

Spatial and spectral concentration problem

Basic problem: no concurrent spatial and spectral limitation Approach by Simons et al. 2005:

• given:

Objective: base function suitable for local modelling in area R

• maximize:

(7)

Spatial and spectral concentration problem

Basic problem: no concurrent spatial and spectral limitation Approach by Simons et al. 2005:

• given:

Objective: base function suitable for local modelling in area R

• maximize:

• Solution is an eigenvalue problem:

(8)

Slepian representation

• Spherical harmonic representation:

(9)

Slepian representation

• Spherical harmonic representation:

• Slepian representation:

(10)

Slepian representation

• Spherical harmonic representation:

• Slepian representation:

• Transformation:

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Localizing property

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MEISSL SCHEME FOR SLEPIANS

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Meissl scheme for spherical harmonics

From Rummel and van Gelderen (1995):

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Meissl scheme for spherical harmonics

From Rummel and van Gelderen (1995):

• surface mass density

• equivalent water height

• displacements

…

(15)

Meissl scheme for spherical harmonics

From Rummel and van Gelderen (1995):

• surface mass density

• equivalent water height

• displacements

…

Transformation:

(16)

Transformation algorithm for Slepians

How to transform from to ?

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Transformation algorithm for Slepians

How to transform from to ? Trivial approach:

(18)

Transformation algorithm for Slepians

How to transform from to ? Trivial approach:

Direct transformation:

(19)

Transformation algorithm for Slepians

How to transform from to ? Trivial approach:

Direct transformation:

(20)

Transformation algorithm for Slepians

How to transform from to ? Trivial approach:

Direct transformation:

(21)

Transformation algorithm for Slepians

How to transform from to ? Trivial approach:

Direct transformation:

(22)

DEPENDENCY ON DATA

DISTRIBUTION

(23)

Motivation

Slepian base function depend implicitly on the data distribution:

Examples:

(24)

Motivation

Slepian base function depend implicitly on the data distribution:

Examples:

(25)

Motivation

Slepian base function depend implicitly on the data distribution:

Examples:

[m] [m]

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Why could the data distribution change?

(27)

Why could the data distribution change?

(28)

Why could the data distribution change?

(29)

Why could the data distribution change?

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How does the estimate change?

Given: Slepian base functions based on a grid Reference:

Case I: observations on the same grid

Case II: observations on a (dense) satellite track

(31)

How does the estimate change?

Given: Slepian base functions based on a grid Reference:

Case I: observations on the same grid

Case II: observations on a (dense) satellite track

(32)

How does the estimate change?

Given: Slepian base functions based on a grid Reference:

Case I: observations on the same grid

Case III: observations on a (sparse) satellite track

(33)

How does the estimate change?

Given: Slepian base functions based on a grid Reference:

Case I: observations on the same grid

Case III: observations on a (sparse) satellite track

(34)

OUTLOOK:

COMBINATION OF GRACE AND GPS

(35)

Approach

Given: SH from GRACE and terrestrial GPS in local area Methodology:

• Convert SH to Slepian coefficients

• Update Slepian coefficients by the terrestrial GPS observations with the help of a Kalman filter

• Kalman filter is implemented as continuous Wiener process acceleration (CWPA) model

Data:

• GRACE GFZ Rel.05 with GAC restored

• Height observations of GPS time series of 12 stations

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GPS site distribution

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Preliminary results

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Conclusion

• The direct conversion between specific and non- specific Slepian coefficients is possible.

 Meissl scheme for Slepians (analogue to SH)

 trivial approach numerically more efficient

• Data distribution is “not” critical for the estimation of

the Slepian coefficients but for the derivation of the

Slepian base function

Références

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