By Banerjee S., Wal M.M., Carlin B.P.

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**Example text**

The potential function of the gravity field of the earth. There are situations where a function may be completely, or at least efficiently described by a finite number of parameters. For example, in order to describe the attractive force of the sun on a satellite orbiting the earth, it is sufficient to know the components of the satelliteto-sun vector. However we need theoretically an infinite number of parameters to describe the corresponding attraction of the earth, due to the presence of unknown variations in the density of the earth masses, whose effect cannot be ignored.

Thus p i=llo:, where o: is the instrument error variance and 4, =6,a : , where the zero non-diagonal elements reflect the statistical independence (no correlation) between different observational errors. A formal justification of the above choice is given by the celebrated Gauss-Markov theorem. As already explained in section 2, a probabilistic or "stochastic" model is attached to the deterministic or "functional" model y=Ax , by setting b=Ax+ v , where v - (0, C ) , meaning that v has zero mean E{v) =O and (variance-)covariance matrix E{vv } =C , where E{ ) is the expectation operator (mean over all possible realizations).

In short one could say that in addition to the three spatial dimensions and gravity, geodesy has conquered one more dimension: time. In addition it has extended from its continental limitations to the oceans. Helmert (1880, p. 3) once stated that in the case of the topographic relief there are no physical laws that govern its shape, in a way that would allow one to infer one part from another, which means that they are not susceptible to modeling in the above sense. Therefore topography is determined by taking direct measurements of all its characteristic features.