Astro-geodetic network adjustment

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The astro-geodetic network adjustment is a special adjustment of large-scale surveying networks , which reduces the influence of the gravitational field on the measurements in a mathematically strict manner and uses the stabilizing effect of Laplace azimuths .

Influences of the gravitational field

The “disruptive” influences of the earth's gravity field are in particular the vertical deviations , that is, the deflection of the vertical direction by mountains, valleys and irregularities in the geology . The vertical deviations, which in principle exist everywhere, but have hardly any effect in the lowlands , increase to amounts above 0.01 ° in the high mountains - whereas modern precision measurements in terrestrial geodesy achieve about 100 times the accuracy. If the perpendicular disturbances are not taken into account, ie not attached to the measurements as a small “tilt”, they have a distorting effect on the coordinates of a surveying network.

Leading geodesists and mathematicians ( Gauß , Helmert , Laplace , Liesganig and others) had been thinking of these effects since the beginning of the 19th century, which at that time could deform the network points determined purely geometrically (by angle measurement) by a few mm ... cm per kilometer, but were missing give them the means to determine the vertical deviations in practice. On the other hand, some influences partially offset each other, for example in the case of symmetrical mountain valleys or the staking out of tunnels on both sides . Therefore, it was only with the development of astronomical measuring instruments suitable for use in the field - and with the general increase in measurement accuracy  - that the "pressure of practice" on theory was great enough to gradually take into account the astro-geodetic vertical deviations in the network compensation from the middle of the 20th century to introduce national surveys.

Strict network adjustment according to Helmert

The Berlin geodesist Friedrich Robert Helmert laid the theoretical basis for astro-geodetic network adjustment before the turn of the century (first publication in 1886), but it was mathematically awkward.

In the 19th century, the increasing application of the equalization calculation to the networks of basic surveying (which until then was based on purely geometric fields , simple division of the triangular excesses (angle sum 180 °) or graphically analyzed "displacement" of the measurement errors ) prompted Helmert, in addition to the consideration the geometric equations of condition also to eliminate the physical influences on a network as far as possible.

Thus, from the “activation” and activation of the points (the gradual expansion of the network to include newly measured points), a mathematically strict projection onto a uniform calculation area , because the measuring points are at different sea levels . In addition, Helmert found that a purely geometrically balanced network is only "geodetically error-free" if those small contradictions in the directions (ellipsoidal azimuths ) of the network sides, which can be analyzed using the Laplace equation , are minimized, which result from statistical error propagation . He therefore also recommended the measurement of additional Laplace azimuths at certain intervals in order to be able to "correct" the terrestrial measurements due to the vertical deviations .

Helmert's approach, however, had to use an approximation that is no longer absolutely necessary with today's calculation methods: First, a network should be calculated provisionally (by purely geometric network expansion) on a sphere or the reference ellipsoid in order to obtain good coordinates from all measurement points. Then the perpendicular deviation would have to be measured or computationally interpolated on as many network points as possible so that it could be taken into account over all points in a second large calculation. If available, the Laplace azimuths should finally stabilize the large-scale orientation.

This multi-stage procedure means on the one hand that the network connection given by the geodetic measurements is loosened somewhat in order to eliminate even the smallest, geometrically ascertainable contradictions ; on the other hand, this results in shifts in the network points (back then up to a few decimeters, now mostly only in the centimeter range), which would actually require further network calculations due to their slight non-linearity . The last-mentioned correction can generally be omitted due to its insignificance.

Further development by K. Ledersteger

Helmert's methodology was significantly further developed and critically revised by the Austrian geodesist Karl Ledersteger (1900–1972). Around the same time with several other researchers (including the Russian Magnitzki) and on the occasion of the major project Central European Network , between 1940 and 1960 he put the method on a geometrically perfect basis ("natural network", normal spheroid as theoretical earth figure ). They could later serve for the gradual improvement of the European network , especially in the ED79 . Accurate astronomical Laplace azimuths , the calculation and interpolation of vertical and gravity perturbations using methods of geophysics (especially gravity reductions and isostasis ), through regional geoid determination (gravimetric and astrogeoid ) also contribute to the alignment of the land surveys to the idealized earth figure (middle earth ellipsoid ) and not least because of the satellite geodesy that has been establishing itself since 1960 .

The main difference between a classical network adjustment and its astro-geodetic refinement is:

  • Reduction of all measurements due to deviation from the perpendicular ; where this is not measured, it can be estimated from the shape of the terrain ,
  • a theoretically unambiguous projection of the measuring points on the calculation surface (1-stage "Helmert projection" on the ellipsoid, 2-stage " Pizzetti projection of the surface points" on the geoid and ellipsoid),
  • which is a prerequisite for a mathematically strictly "natural network" (diction K. Ledersteger),
  • the reduction of the scale-forming baselines to the earth ellipsoid - what precise knowledge of the geoid verfordert (see Torges " Zentimetergeoid " realized only in the coming years)
  • the inclusion of as many Laplace points as possible in the surveying network.

Calculation of larger networks

The implementation of the first such network adjustments in the 1970s had to be limited to smaller networks, but this has already proven itself in individual mountain valleys (e.g. test networks of the BEV in the Salzburger Raurisertal or the HsBW in the Allgäu ) (over 50% increase in accuracy to 1 to 2 cm, at point "displacements" from 2 to 10 cm).

Larger networks also mean a very large number of so-called normal equations , which are necessary for minimizing errors ( least squares method ). The inversion of such large matrices of several thousand rows and columns has only been possible with modern computer programs from the last 1–2 decades. Previously, frame networks were used to reduce the number of unknowns .

Bavarian geodesists chose a different approach than Ledersteger (Berlin and TU Wien ), especially at what is now the Bundeswehr University in Munich, which led to the term Integrated Geodesy and the OPERA program system (Günter Hein, approx. 1975). In Italy, A.Marussi (after whom the symposium series is named) and M.Hotine dealt with multidimensional theories, and the Austrian geodesists Helmut Moritz and Hans Sünkel (Berlin and Graz University of Technology) developed corresponding applications of the collocation , with which the classic geometric surveying networks can be expanded to include physical effects ( interference potential , vertical and severe interference ) and covariance analyzes .

One of the most important large-scale applications of the astro-geodetic network adjustment was the recalculation of the European network (see also ED50 and ED77 ) for the network version ED79 . Today, however, the method has taken a back seat because the integration of data from satellite geodesy (GPS, SLR , altimetry) and VLBI promises even greater progress.

See also

Literature and Sources

  • Karl Ledersteger : Astronomical and Physical Geodesy . In: Wilhelm Jordan, Otto Eggert, Max Kneissl (Hrsg.): Handbuch der Vermessungskunde. Volume V, Verlag JB Metzler, Stuttgart 1969, DNB 456892842 , pp. 79-155, 455ff, 705ff.
  • WA Magnitsky , WW Browar, BP Shimbirew: Teorija figury zemli. Moscow 1961. (Russian)
    • German: theory of the figure of the earth. Publishing house for construction, Berlin 1964, DNB 453177972 . (Textbook)
  • Bernhard Heck : Calculation methods and evaluation models of the national survey , 3rd edition. Wichmann-Verlag, Karlsruhe 2003
  • Federal Office for Metrology and Surveying : Austria's astro-geodetic work for the ED79. ÖKIE special issue, Vienna 1982.

Individual evidence

  1. ↑ K. Ledersteger: Handbuch der Vermessungskunde , Volume V, Chapter III The problem of the natural network . Vienna / Stuttgart 1969.