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Multiscale Filter Methods Applied to GRACE and Hydrological Data

Multiscale Filter Methods Applied to GRACE and Hydrological Data Willi Freeden, Helga Nutz, Kerstin Wolf. Overview. Motivation Time-Space Analysis Using Tensor Product Wavelets Comparison of GRACE and Hydrological Models (WGHM, H96, LaD) Outlook. Overview. Motivation

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Multiscale Filter Methods Applied to GRACE and Hydrological Data

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

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

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

  4. 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)

  5. 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

  6. 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?

  7. 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

  8. 1. Motivation Uncertainty Principle (in space domain): Solution: Spherical Harmonics Dirac- Function Wavelets 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 + (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

  9. Overview Motivation Time-Space Analysis Using Tensor Product Wavelets Comparison of GRACE and Hydrological Models (WGHM, H96, LaD) Outlook

  10. Smoothed Part Details 2. Time-Space Analysis Filter: Scaling Function Signal F Filter Method: Multiscale Analysis Filter: Wavelets

  11. 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 detail in space smoothing in time and space

  12. … … Original Signal F Smoothed Parts + + + 1st Hybrid + + + 2nd Hybrid Detailed Parts + + + Pure the higher the scale the finer the details 2. Time-Space Analysis Graphical Representation of a Multiscale Analysis

  13. 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

  14. 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 5 Scale 6

  15. 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 Manaus (3°S 60°W) Lilongwe (13°S 33°O) Kaiserslautern (49°N 7°O) Dacca (23°N 90°O)

  16. Overview Motivation Time-Space Analysis Using Tensor Product Wavelets Comparison of GRACE and Hydrological Models (WGHM, H96, LaD) Outlook

  17. 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

  18. GRACE-WGHM (corr = 0.79) GRACE-H96 (corr = 0.74) GRACE-LaD (corr = 0.75) 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.

  19. Time dependent courses of the pure detail parts calculated with CuP-wavelet ( ) at scale 4 in time and space. 3. Comparison GRACE - Hydrological Models o o bad correlation good correlation

  20. 3. Comparison GRACE - Hydrological Models Global correlation coefficients calculated using the pure wavelet coefficients ( ) on the continents

  21. 4. Outlook Further analysis with different (band- / non-bandlimited) wavelets in the time and space domain: 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

  22. Thank you for your attention!

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