# Mathematical Foundation of Geodesy: Selected Papers of by Kai Borre

By Kai Borre

This quantity includes chosen papers by means of Torben Krarup, the most vital geodesists of the 20^{th} century. His writings are mathematically good based and scientifically suitable. during this notable choice of papers he demonstrates his infrequent cutting edge skill to give major subject matters and ideas. sleek scholars of geodesy can study much from his collection of mathematical instruments for fixing real problems.

The assortment includes the recognized e-book "A Contribution to the Mathematical origin of actual Geodesy" from 1969, the unpublished "Molodenskij letters" from 1973, the ultimate model of "Integrated Geodesy" from 1978, "Foundation of a idea of Elasticity for Geodetic Networks" from 1974, in addition to various pattern atmosphere papers at the thought of adjustment.

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**Example text**

Let us jump into the new problem: By ﬁrst using Moritz’ prediction formula with a given covariance function C(P, Q) we can ﬁnd ∆gP for all points of some surface. Solution of the corresponding boundary value problem will then give us the disturbing potential T . But can we, by using another covariance function K(P, Q) deﬁned in a domain including the outer space of the Earth and describing the covariance between values of T at the points P and Q, ﬁnd the potential T directly from the measured values of ∆gi ?

This is what I have tried to do in the third section. Here it appears that again the theory of Hilbert spaces with reproducing kernel is a valuable tool and that the central theorem—the Runge theorem—is a special case of a theorem, known as a theorem of the Runge type, which is well-known in the theory of linear partial diﬀerential equations and which concerns the approximation of solutions to a partial diﬀerential equation in some domain by solutions to the same equation in another domain containing the ﬁrst one.

For n = 100 we shall have the condition number of about 106 and for n = 1000 we get 1010 , numbers that give reason to some precaution. A classical example in the literature of least-squares problems is one where the matrix A in (4) is given by ⎡ ⎤ 1 1 1 1 1 ⎢ 0 0 0 0⎥ ⎢ ⎥ ⎢0 0 0 0⎥ ⎢ ⎥ A=⎢ (13) 0 0⎥ ⎢0 0 ⎥ ⎣0 0 0 0⎦ 0 0 0 0 26 3 The Theory of Rounding Errors in the Adjustment by Elements where = 10−5 and the calculations are performed with nine decimals. The matrix of the normal equations is then ⎤ ⎡ 1+ 2 1 1 1 1 ⎢ 1 1+ 2 1 1 1 ⎥ ⎥ ⎢ T 2 ⎢ 1 1+ 1 1 ⎥ (14) A A=⎢ 1 ⎥ ⎣ 1 1 1 1+ 2 1 ⎦ 1 1 1 1 1+ 2 and the eigenvalues are 2 , 2 , 2 , 2 , and 5 + 2 , but the matrix of the normal equations will be calculated so as to consist of unity coeﬃcients overall and will thus be singular and it will not be possible to ﬁnd the solutions, no matter how accurate a method is used for solving the normal equations.