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Wavelets as a framework for describing inhomogenity and anisotropy

Wavelets as a framework for describing inhomogenity and anisotropy. Alex Deckmyn Tomas Landelius Anders Höglund. Variational data assimilation. Minimization of a cost function. The importance of the B matrix. B contains the background error covariances:

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Wavelets as a framework for describing inhomogenity and anisotropy

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  1. Wavelets as a framework for describing inhomogenity and anisotropy Alex Deckmyn Tomas Landelius Anders Höglund SMHI-OH

  2. Variational data assimilation Minimization of a cost function SMHI-OH

  3. The importance of the B matrix • B contains the background error covariances: • Weights background state against observations. • Helps to impose balance to the assimilated fields. • Smoothes out the observational information. SMHI-OH

  4. Problems with the B matrix • It is HUGE: • Impossible to store explicitly. • Impossible to fully determine. SMHI-OH

  5. Optimal interpolation Direct local solution to 3D-VAR Local models of B that differ for different locations. SMHI-OH

  6. Need a global model for B Variable substitution Fourier/Wavelet transform SMHI-OH

  7. The Fourier transform + Fast implementation O(n log(n)). - Homogenous; same for all locations. - Boundary problems; periodic. SMHI-OH

  8. Tiling the time-frequency plane Gridpoint Fourier Wavelet STFT SMHI-OH

  9. Discrete wavelet transform Some problems with the DWT - Oscillations near singularities. - Shift variance. - Sensitive to processing. - Lack of directionality. SMHI-OH

  10. Dual tree complex wavelet transform SMHI-OH

  11. The wavelet transform + Very fast implementation O(n).+ Some anisotropy and inhomogenity.+ Boundary problem can be avoided. SMHI-OH

  12. Estimating D Orthogonal T Non-orthogonal T SMHI-OH

  13. Work status and issues ALADIN implementation- Wavelet transform v0.1 (cy30t1_wavbec)Software for estimation of D- FESTATWAV assumes orthogonal T- Treatment of vertical correlationsDesign of boundary wavelets SMHI-OH

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