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

By Kai Borre

This quantity comprises chosen papers via Torben Krarup, some of the most vital geodesists of the 20^{th} century. His writings are mathematically good based and scientifically appropriate. during this extraordinary number of papers he demonstrates his infrequent cutting edge skill to give major subject matters and ideas. smooth scholars of geodesy can study much from his collection of mathematical instruments for fixing genuine problems.

The assortment includes the recognized book "A Contribution to the Mathematical starting place of actual Geodesy" from 1969, the unpublished "Molodenskij letters" from 1973, the ultimate model of "Integrated Geodesy" from 1978, "Foundation of a concept of Elasticity for Geodetic Networks" from 1974, in addition to a number of pattern atmosphere papers at the conception of adjustment.

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The very process of iteration can be described as follows: 1. Select a preliminary point P1 on C 2. Compute i1 = xi1 − xiσ = ai y1α − xiσ 3. Compute ∆y α = y2α − y1α by solving the symmetric linear equation system −gij xj2 − xjσ + gij aj1,α ∆y α = gij ai1,α gij xj2 − xjσ = 0 i 1 (11) 4. Compute y2α = y1α + ∆y α 5. If P2 is not close enough to Pµ , then put P1 = P2 and start again at 2. 2 The Adjustment Procedure in Tensor Form 21 Until now we have only considered adjustment by elements. Perhaps it would be appropriate to tell something about adjustment by correlates.

37) n=0 Since K(P, Q) is a potential as a function of P and as a function of Q, it can be expanded as follows: ∞ (2n + 1) K(P, Q) = n=0 R2n+2 An n+1 Pn (cos w) rPn+1 rQ for all P, Q ∈ Ω. (38) We see that K(P, Q) is a function of rP , rQ and w only, and that it is symmetric and harmonic in Ω and regular at inﬁnity. We shall just ﬁnd the conditions for its being of positive type, and therefore we must investigate the expression N N K(Pi , Qk )xi xk i=1 k=1 = K(i, k)xi xk i ∞ k (2n + 1)An R2n+2 = n=0 ∞ i n An R2n+2 = n=0 m=0 i k n k Pn (cos wik ) xi xk rin+1 rkn+1 Rnm (θi , λi )xi Rnm (θk , λk )xk rin+1 rkn+1 + m=1 ∞ n An R2n+2 = n=0 m=0 i i k S nm (θi , λi )xi S nm (θk , λk )xk rin+1 rkn+1 Rnm (θi , λi )xi rin+1 n 2 + m=1 i S nm (θi , λi )xi rin+1 2 .

If R is a rotation (we regard only rotations about O) then φ(RP ) is also a potential regular in Ω. We shall then deﬁne Σφ for a given potential φ as the set of potentials φ(RP ) for all rotations R and interpret the mean at a point as the mean over Σφ using as a measure µ the invariant measure for the group of rotations G normalized so that µ(G) = 1. When we use this interpretation of M {·}, the reasoning leading to formula (30) runs without diﬃculty. Now one could ask whether it is reasonable to represent the disturbing potential for a non-spherical planet as a rotation invariant stochastic process.