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Singular value decomposition of tidal harmonics on the rigid Earth

Singular value decomposition of tidal harmonics on the rigid Earth



Adam Ciesielski


In Tidal Analysis the tidal gravity potential is expressed in terms of tidal harmonics, which result from celestial mechanics and astronomical ephemerides (trajectories of objects) in the Solar System. The outcoming gravity time series should be the superposition of all contributions. However, Earth properties such as non-rigidity and anelasticity cause different Earth's response to tidal forcing and result in differences between recordings and theory. Other phenomena, like NDFW/FCN, oceans and atmosphere also contribute to the observed gravity. Currently, tidal harmonics are arranged in wave groups (based on their frequencies) and parameters are kept constant within each group. That may be at variance with the actual recorded signal, therefore we develop a new approach to tidal analysis which abandons wave grouping.

To study the problem I performed a singular value decomposition (SVD). SVD is a factorization of a matrix, a generalisation of diagonalization (eigendecomposition). Its application allows us to perform a pseudoinverse and to study the nature of significant and insignificant linear combinations and their dependency on data time window. In my presentation, I will show how this tool is applied within Tidal Analysis.

I will present and explain SVD in the forward problem of synthetic Earth tide gravity time series computed for rigid, elastic Earth. The results indicate that harmonics from different wave groups may couple. Moreover, they support the abandonment of the standard wave-grouping concept, with SVD as an alternative. I will present the first preliminary results of the inverse problem, using pseudoinverse of computed rigid earth tide signals with synthetic gravity time signal vector. The outcome of the first tests of solving the inverse problem is promising.