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# Tidal Deformation of the Solid Earth

**Master Thesis defense: Stine Kildegaard Poulsen**

*Vejleder*: C.C. Tscherning, NBI og Klaus Mosegaard, NBI

Tidal forces are a source of noise in many geophysical field observations, but they are useful in revealing information of the interior of the Earth. Tidal forces arise because of the gravitational attraction from external bodies varying over the volume of a body. The Earth's response to the tide generating potential can be derived analytically by the driving force, i.e. the knowledge of the orbital motion of the Earth and the Moon, the Sun and other external objects, combined with the Love numbers, which are related to the rheological properties of the Earth.

By using a two dimensional steady state version of Navier's equation of motion, the tidal deformation of the Earth is solved numerically. The Navier equation of mo- tions is discretized in polar coordinates using a finite difference approximation. The linear system of equations proved to be ill-conditioned, and to subdue the obstacle a Tikhonov regularization is applied.

Assuming the Moon being the only attracting body, and assuming a stationary Earth-Moon relationship, two Earth models are developed. The first model, the ho- mogeneous Earth model, is assumed to be a pure Poisson solid, where the seismic velocities and the elastic parameters are constant throughout the Earth. The second model, the layered model, includes four layers representing an inner core, an outer core, a mantle and a crust. The elastic properties are taken from the isotropic Prelimi- nary Reference Earth Model, [Dziewonski and Anderson, 1981], and averaged in each layer. This gives a realistic Earth model with a fluid outer core.

The Earth models response to the tides are given by the radial and tangential dis- placement fields. Though the compression of the radial displacement field not fully fulfill the picture of the inner structures of the Earth, the results clearly reproduce the physical picture of the solid Earth tides, by means of an expansive and compressive Earth for the radial field and a tractive displacement for the tangential field. The radial displacement field has a maximum expansion of 131 mm and 74 mm for the homo- geneous and layered Earth models, respectively, and the maximum compression are -5 mm and -26 mm for each model. For the tangential displacement field the maxi- mum tractions are found to be 41 mm and 45 mm for the homogeneous model and the layered Earth models, respectively. The layerd Earth model is similar to the analytical derived, with a factor 1.4 to 4 too small, weathers the homogeneous Earth model shows larger displacements, but as expected, it deviates more from the physical picture of the analytic solution.