multiscale filter methods applied to grace and hydrological data

Kerstin WolfAG GeomathematikTU Kaiserslautern

GRACE Science Team Meeting 2007:Joined International Symposium:GFZ-Potsdam, 15/-17/10/2007

Multiscale Filter Methods Applied to GRACE and Hydrological Data

Willi Freeden, Helga Nutz, Kerstin Wolf



Overview

1. Motivation

2. Time-Space Analysis Using Tensor Product Wavelets

3. Comparison of GRACE and Hydrological Models (WGHM, H96, LaD)

4. Outlook



Comparison

1. Motivation

Time Series of Hydrological Models(WGHM, H96, LaD)

Time Series of Satellite Data(GRACE)

Comparative Analysis in Time and Space Domain

All results are computed with data provided from GFZ-Potsdam (1.3)



Realization of a Time-Space Multiscale Analysisby use of Tensor Product Wavelets

Raw Data:

• Time Series of Spherical Harmonic Coefficients

• Time Series of Water Columns

Determination of Temporally and Spatially Local Changes

Pure and Hybrid Parts

Tensor Wavelet Analysis

Based on Legendre Wavelets in the Time Domain and Spherical Wavelets in the Space Domain

1. Motivation



Realization of a Time-Space Multiscale Analysisby use of Tensor Product Wavelets

Raw Data:

• Time Series of Spherical Harmonic Coefficients

• Time Series of Water Columns

Determination of Temporally and Spatially Local Changes

Pure and Hybrid Parts

Tensor Wavelet Analysis

Based on Legendre Wavelets in the Time Domain and Spherical Wavelets in the Space Domain

1. Motivation

Why do we apply wavelets?



SphericalHarmonics

Dirac-Function

Ideal localization in the frequency domain

But: Not any localization in the

space domain

Ideal localization in the space domain

But: Not any localization in the frequency domain

1. Motivation

Uncertainty Principle (in space domain):



SphericalHarmonics

Dirac-Function

Ideal localization in the space domain

But: Not any localization in the frequency domain

Uncertainty Principle (in space domain):

Solution:

Wavelets

1. Motivation

(Locally) spatial changes only have local influence

Regional changes have an effect on all coefficients

variations of these coefficients cannot be assigned to single regional effects

+

-

Ideal localization in the frequency domain

But: Not any localization in the

space domain



Overview

1. Motivation



4. Outlook



2. Time-Space Analysis

Signal F

Smoothed Part

DetailsFilter Method:

Multiscale Analysis

Filter:Scaling Function

Filter:Wavelets



Why do we distinguish four parts?connection of temporal and spatial filters via a tensor product:


Multiscale Analysis in Time Multiscale Analysis in Space

smoothing detail smoothing detail

pure hybridsmoothingin time and space

detailin time and space

smoothing in timedetail in space

detail in time smoothing in space




the higher the scale the finer the details

Graphical Representation of a Multiscale Analysis

Original Signal F

Det

aile

d P

arts

Smoothed Parts

+

+

+

+

+

+

+

+

+

… …

1st Hybrid

2nd Hybrid

Pure



Waveletsymbol f. scales 2-5 Wavelet f. scales 2-5


Example of a Wavelet Filter: CuP-Wavelet (cubic polynomial)(filter for the detailed information)



Maximum of the absolute values of the 1st hybrid wavelet coefficients ( ) based on a time series of 47 GRACE-data sets (Feb. 03 – Dec. 06) computed with CuP-wavelet in time and space

Scale 3

Scale 6Scale 5

Scale 4




Time dependent courses of the 2nd hybrid wavelet coefficients ( ) based on GRACE data (Feb. 03 – Dec. 06) with CuP-wavelet in time and space

Lilongwe(13°S 33°O)

Kaiserslautern(49°N 7°O)


Manaus(3°S 60°W)

Dacca(23°N 90°O)



Overview

1. Motivation



4. Outlook



3. ComparisonGRACE - Hydrological Models

Maximum of the absolute values of the pure wavelet coefficients ( ) computed out of a time series (Feb. 03 – Dec. 06) with CuP-wavelet in time and space at scale 4

GRACE

H96

WGHM

LaD




GRACE-WGHM (corr = 0.79) GRACE-H96 (corr = 0.74)

GRACE-LaD (corr = 0.75)

Local correlation of the pure detail parts calculated with CuP-wavelet ( ) at scale 4 in time and space.In brackets: global correlation coefficient computed on the continents.




Time dependent courses of the pure detail parts calculated with CuP-wavelet ( ) at scale 4 in time and space.

oo

bad correlation

good correlation




Global correlation coefficients calculated using the pure wavelet coefficients ( ) on the continents



1) Further analysis with different (band- / non-bandlimited) wavelets in the time and space domain:

2) Further analysis for the comparison of the hydrological models and the GRACE data:

• shannon wavelet, Abel-Poisson wavelet, Gauß-Weierstraß wavelet,…

• local calculations for regions of great accuracy (e.g. the Mississippi delta)

Aim:

to state an ‘ideal’ reconstruction of the signal in view of extraction of the hydrological model from the GRACE data

4. Outlook



Thank you

for your attention!

multiscale filter methods applied to grace and hydrological data

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