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Inhaltsverzeichnis

Frontmatter

1. Linear Equations

Without Abstract

2. The Adjustment Procedure in Tensor Form

3. The Theory of Rounding Errors in the Adjustment by Elements of Geodetic Networks

Without Abstract

4. A Contribution to the Mathematical Foundation of Physical Geodesy

Without Abstract

5. A Remark on Approximation of T by Series in Spherical Harmonics

Without Abstract

6. On the Geometry of Adjustment

Without Abstract

7. Remarks to the Discussion Yesterday

Without Abstract

8. Letters on Molodenskiy’s Problem

Without Abstract

9. On the Spectrum of Geodetic Networks

Abstract
The spectrum of a geodetic network is defined as the eigenvalues of the matrix A T A of normal equations corresponding to the observation equations Ax = b in the coordinates.
We find the meaning of and some general properties for the eigenvectors of the symmetric matrices A T A and AA T and some results concerning the distribution of the eigenvalues when the network is relatively large.
The ultimate goal for these investigations is to get a deeper insight into the relations between netform and netquality. Unfortunately the results are rather fragmentary because we have only recently arrived at what we consider the core of the problem after years of research.

10. Mathematical Geodesy

Without Abstract

11. Foundation of a Theory of Elasticity for Geodetic Networks

Without Abstract

12. Integrated Geodesy

Without Abstract

13. On Potential Theory

Without Abstract

14. La Formule de Stokes Est-Elle Correcte?

15. Some Remarks About Collocation

Abstract
It is here proposed to apply the statistical theory not to the gravity field itself but to its cause, the mass distribution inside the Earth. Through an example, the paper presents part of mathematical tools necessary for the realization of this idea. In connection with the same example, it also is demonstrated how an error estimation not using statistical arguments can be found.

16. Apropos Some Recent Papers by Willi Freeden on a Class of Integral Formulas in the Mathematical Geodesy

17. S-Transformation or How to Live Without the Generalized Inverse—Almost

18. Integrated Geodesy

Summary
The paper ranges over the principles of Integrated Geodesy, characterizing its peculiar approach to geodetic problems like the adjustment and the combination of different measurements of geometric and gravimetric nature. Particular care is paid to the formulation of observation equations (distances, angles, etc.) including the description of the “local frame” by a suitable matrix formalism.

19. A Measure for Local Redundancy—A Contribution to the Reliability Theory for Geodetic Networks

Without Abstract

20. A Convergence Problem in Collocation Theory

Summary
Collocation theory allows the approximation of the anomalous potential T, harmonic in a region Ω, by a smoother function \( \hat T \) harmonic in a larger domain Σ and agreeing with measurements performed on T at discrete points. The smoothing least-squares collocation method is a part of collocation theory in which a hybrid norm is minimized, norm that depends upon a parameter λ that can be interpreted as the relative weight of the norm of \( \hat T \) in Σ and in Ω.
The problem of the behaviour of \( \hat T \) when the number of measurements tends to infinity and contemporarily λ → ∞ is analyzed: the convergence to the correct solution is proved under suitable hypotheses.

21. Non-Linear Adjustment and Curvature

Without Abstract

22. Mechanics of Adjustment

Without Abstract

23. Angelica Returning or The Importance of a Title

Without Abstract

24. Evaluation of Isotropic Covariance Functions of Torsion Balance Observations

Abstract
Torsion balance observations in spherical approximation may be expressed as second-order partial derivatives of the anomalous (gravity) potential T:
$$ T_{13} = \frac{{\partial ^2 T}} {{\partial x_1 \partial x_3 }}, T_{23} = \frac{{\partial ^2 T}} {{\partial x_2 \partial x_3 }}, T_{12} = \frac{{\partial ^2 T}} {{\partial x_1 \partial x_2 }}, T_\Delta = \frac{{\partial ^2 T}} {{\partial x_1^2 }} - \frac{{\partial ^2 T}} {{\partial x_2^2 }} $$
where x 1, x 2, and x 3 are local coordinates with x 1 “east,” x 2 “north,” and x 3 “up.” Auto- and cross-covariances for these quantities derived from an isotropic covariance function for the anomalous potential will depend on the directions between the observation points. However, the expressions for the covariances may be derived in a simple manner from isotropic covariance functions of torsion balance measurements. These functions are obtained by transforming the torsion balance observations in the points to local (orthogonal) horizontal coordinate systems with first axis in the direction to the other observation point. If the azimuth of the direction from one point to the other point is α, then the result of this transformation may be obtained by rotating the vectors
$$ \left[ {\begin{array}{*{20}c} {T_{13} } \\ {T_{23} } \\ \end{array} } \right] and \left[ {\begin{array}{*{20}c} {T_\Delta } \\ {2T_{12} } \\ \end{array} } \right] $$
the angles α − 90° and 2(α − 90°) respectively.
The reverse rotations applied on the 2×2 matrices of covariances of these quantities will produce all the direction dependent covariances of the original quantities.

25. Contribution to the Geometry of the Helmert Transformation

Abstract
Considering the fact that the determination of the Helmert transformation of one point set to another point set is a non-linear problem of adjustment, a geometrical theory for this problem is treated, and as a result of this theory a simple and numerically strong method for the computation of the parameters of the Helmert transformation is presented.

26. Letter on a Problem in Collocation Theory

Without Abstract

27. Approximation to The Earth Potential From Discrete Measurements

28. An Old Procedure for Solving the Relative Orientation in Photogrammetry

Abstract
We describe the problem of relative orientation in terms of homogeneous coordinates concluding in a least squares problem in the observed image coordinates. The solution determines a rotational matrix for each image; these rotational matrices bring the images back to the normal position. The explicit formula for the rotational matrices is derived using properties of ‘nearly’ orthogonal matrices. The procedure is augmented by a special preliminary iteration step in order to cope with large rotations.
The method is described through a complete Pascal program.

Backmatter

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