Page - 18 - in Contributions to GRACE Gravity Field Recovery - Improvements in Dynamic Orbit Integration, Stochastic Modelling of the Antenna Offset Correction, and Co-Estimation of Satellite Orientations

In practice, the expansion is terminated at some finite upper degree N. In eq. (3.4.1), Cnm(λ,θ)and Snm(λ,θ)are functionals of the fully normalized Legendre polynomi- als P¯nm(cosθ) Cnm(λ,θ)= cos(mλ) P¯nm(cosθ) (3.4.2) Snm(λ,θ)= sin(mλ) P¯nm(cosθ) . (3.4.3) The gravitational attractiongdue to Earth’s gravitational potential V is g(r,λ,θ)=∇V(r,λ,θ)= [ ∂V ∂x ∂V ∂y ∂V ∂z ]T . (3.4.4) See e.g. Mayer-Gu¨rr (2006) for a compact description of the derivatives in eq. (3.4.4). The gravitational potential is superimposed by other potential fields, the disturbing potentials T, some of them due to the effects described in sections 3.1 and 3.2. The largest disturbing potential on Earth is the centrifugal potential Z caused by Earth’s rotation. The total potential is then the sum of all contributing potentials, e.g. U=V+∑ i Ti (3.4.5) 3.4.1 Level Surfaces A gravitational field can be described not only by Stokes coefficients, but also by sets of level surfaces, where U(r,θ,λ)= constant . (3.4.6) One such equipotential surface is the geoid W, defined as the equilibrium surface described by the world’s oceans if they were at rest and continued through the continents. Specifically, the geoid is the constant level surface described by the above criteria in the potential due to the superposition of Earth’s static and centrifugal potential W(r,θ,λ)=V(r,θ,λ)+Z(r,θ,λ)= constant. (3.4.7) The geoid can be described in the form of geoid heights, the height of the geoid level surface with regard to some reference surface, such as the GRS80 reference ellipsoid. The time variability in gravity fields can then be described in terms of changes in the geoid height from one point in time to another. A more intuitive representation of the temporal variability of the gravity field can be achieved by employing equivalent water heights (EWHs). As many of the short-period changes of the potential are due to hydrological signals, the idea of EWHs is to represent these changes as variations in the thickness of a thin layer of liquid water at Earth’s surface (Wahr, Molenaar, and Bryan, 1998). Chapter3 Gravity and Other Signals18