A Specific Approach to Least Squares Adjustment of the State Levelling Network

Abstract

In addition to its main purpose: establishing and maintaining the parameters of the height system for the territory of a given state (country), state levelling networks also serve to establish (register) the contemporary (recent) vertical movements of the Earth's crust. The detection of such movements, besides in a purely research sense, is of great practical importance. The displacement of the benchmarks over time plays an essential role in seismic forecasting in the short and long term. Sometimes, not very often, it happens that the duration of the measurements in a single cycle of State levelling network measurement is commensurate or nearly commensurate with the period between the different cycles. Such a fact raises serious issues to be addressed, both in the process of preliminary accuracy estimation of the measurements and in the formation of the adjustment model. A period of ten years or more is long enough for displacements on the order of a few mm (millimeters) to become apparent and to be reliably detected. One possible approach, in such cases, is to apply a modified version of adjustment using the Least Squares Method. It would be appropriate, as additional unknowns, to introduce the velocities of the individual benchmarks of the network into the adjustment model. Thus, taking into account the time of the start of the measurements, preconditions are created for taking into account the dynamic behaviour of the benchmarks during the measurement period. The applied adjustment model is based on the so-called "dynamic" or "kinematic" adjustment model, which also takes into account some technological features in the overall network measurement process.

Geometryczne dane niwelacyjne i niektóre błędy systematyczne w ich przetwarzaniu

“…Its value are -1 for the initial benchmark of a line, 1 for the terminal benchmark of a line and zeros for all other adjusted benchmarks. If we present the matrix Q by equation ( 2), we can rewrite equation (1) as an equation (3).…”

“…The highest order geometric levelling has been used as a main method for establishing of height system for state territories [1][2][3][4] or even for a whole continent [1, [4][5] since the last decades of 19 th century. The method is also widely applied for validation of gravimetric geoid [5] and quasigeoid models [6], verification of chronometric levelling results [7], monitoring of the recent vertical movements of the Earth's crust [3], verification and calibration of GNSS antennas [8], etc. The importance of the precise geometric levelling for civil engineering activities is also well-known [9].…”