Structure and Applications of Covariance Functions in Geodesy

Abstract

Geodetic data processing relies on proper functional models relating observations to unknowns. Such models are usually adequately formulated. In addition, stochastic modelling of observations is of utmost importance. In both models covariance functions play a key role in the computation of geodetic quantities and their relevant precision.

This research is directed towards: (1) the development of empirical covariance functions from real and simulated data using different models.

(2) investigation of various mathematical models for the prediction of heights as an example of deterministic quantities.

(3) The development of criterion matrices using suitable covariance functions and their use in analytical design of levelling networks.

Empirical covariance functions for real and simulated levelling networks are developed and their use for prediction and analytical design is tested.

The main conclusions are: i) Covariance functions for deterministic quantities take the form of an straight line function. However, for small areas (i.e. less than 9 km2) it was found that a negative gradient straight line is adequate for levelling networks.

ii) The method of least squares prediction is found to be the best model for data densification in levelling networks.

iii) Covariance functions describing the behavior of errors in levelling networks can be fully described by straight line functions. However, the exponential models are suitable for use with two dimensional networks.

iv) The method of least squares used for the design of levelling networks gives different solutions when using the criterion matrix or its inverse. However, both solutions are equivalent as far as the decision of rejecting the observation(s) with the least contribution to the precision of the network.