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Multiscale Filter Methods Applied to GRACE and
Hydrological Data Willi Freeden, Helga Nutz,
Kerstin Wolf
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Overview

1. Motivation
2. Time-Space Analysis Using Tensor Product Wavelets
3. Comparison of GRACE and Hydrological Models
   (WGHM, H96, LaD)
4. Outlook

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Overview

1. Motivation
2. Time-Space Analysis Using Tensor Product Wavelets
3. Comparison of GRACE and Hydrological Models
   (WGHM, H96, LaD)
4. Outlook

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1. Motivation
Time Series of Hydrological Models (WGHM, H96,
LaD)
Time Series of Satellite Data (GRACE)
Comparison
Comparative Analysis in Time and Space Domain
All results are computed with data provided from
GFZ-Potsdam (1.3)
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1. Motivation
Realization of a Time-Space Multiscale
Analysis by 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
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1. Motivation
Realization of a Time-Space Multiscale
Analysis by 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
Why do we apply wavelets?
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1. Motivation
Uncertainty Principle (in space domain)
Spherical Harmonics
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
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1. Motivation
Uncertainty Principle (in space domain)
Solution
Spherical Harmonics
Dirac- Function
Wavelets
Ideal localization in the space domain But Not
any localization in the frequency domain
Ideal localization in the frequency domain But
Not any localization in the space domain

(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
-
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Overview

1. Motivation
2. Time-Space Analysis Using Tensor Product Wavelets
3. Comparison of GRACE and Hydrological Models
   (WGHM, H96, LaD)
4. Outlook

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2. Time-Space Analysis
Filter Scaling Function
Signal F
Filter Method Multiscale Analysis
Filter Wavelets
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2. Time-Space Analysis
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
hybrid
detail in time and space
detail in time smoothing in space
smoothing in time and space
smoothing in time detail in space
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2. Time-Space Analysis
Graphical Representation of a Multiscale Analysis
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2. Time-Space Analysis
Example of a Wavelet Filter CuP-Wavelet (cubic
polynomial) (filter for the detailed information)
Waveletsymbol f. scales 2-5
Wavelet f. scales 2-5
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2. Time-Space Analysis
Table of influenced region (size of
details) Computation via number of stripes of
the spherical harmonics which are dependent of
the Legendre polynomials
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2. Time-Space Analysis
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 4
Scale 6
Scale 5
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2. Time-Space Analysis
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 (13S 33O)
Manaus (3S 60W)
Kaiserslautern (49N 7O)
Dacca (23N 90O)
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Overview

1. Motivation
2. Time-Space Analysis Using Tensor Product Wavelets
3. Comparison of GRACE and Hydrological Models
   (WGHM, H96, LaD)
4. Outlook

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3. Comparison GRACE - 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
WGHM
GRACE
H96
LaD
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3. Comparison GRACE - Hydrological Models
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.
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3. Comparison GRACE - Hydrological Models
o
o
bad correlation
good correlation
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3. Comparison GRACE - 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 6
WGHM
GRACE
H96
LaD
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3. Comparison GRACE - Hydrological Models
GRACE-WGHM (corr 0.63)
GRACE-H96 (corr 0.60)
Local correlation of the pure detail parts
calculated with CuP-wavelet ( ) at scale 6
in time and space. In brackets global
correlation coefficient computed on the
continents.
GRACE-LaD (corr 0.61)
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3. Comparison GRACE - Hydrological Models
Global correlation coefficients calculated using
the pure wavelet coefficients ( ) on the
continents
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4. Outlook

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
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Thank you for your attention!