Reliability Analysis of GPS Geodetic Control Networks

By Chris McAlister

Supervised by Dr. Jinling Wang

October 2004


Reliability theory was introduced in 1968 by Dutch geodesist Professor Willem Baarda (Baarda, 1968). Lachapelle and Ryan (2000) state that: "Reliability refers to the controllability of observations, that is, the ability to detect blunders and to estimate the effects that undetected blunders may have on a solution." This can be explored further as reliability theory is comprised of two main components: Internal and External Reliability. Internal reliability relates to the amount of gross error in an observation, not detectable at a certain probability level; External Reliability relates to the effect of non-detectable blunders on the estimated quantities (for example coordinates).


The application of Baarda's work to a GPS situation has initially been hindered by Baardas mathematical generalisations, which make the assumption that the dataset to be checked would have uncorrelated measurements, which is untrue of GPS. Steps have been taken by academics such as Wang & Chen (1994) and Schaffrin (1997), among others, to provide an unabridged and more comprehensive version of Baarda's work, which can be applied to the correlated observables of GPS. With the increasing use of GPS as the major observation tool for large geodetic control networks, the application of these unabridged versions has become necessary to ensure the integrity of results. This is advantageous to the surveying community, as it provides a means to understanding the effects of input data on output data, as well as the accuracy and precision of results in terms of the resolving power of the network.


The main purpose of applying reliability theory to GPS is to gain an understanding of, and to derive statistics about the quality of a GPS network. Pelzer (1979) says that "The quality of a geodetic network is described by its accuracy and reliability", and currently in NSW, and possibly other places, little is known about the reliability component of most large, and particularly geodetic, control networks.


To complete the necessary testing required for this thesis, a Least Squares program incorporating Reliability Theory from Baarda, Wang and Chen and Schaffrin's papers was developed. This was undertaken by the NSW Department of Lands Survey Branch, and is called ORCA. More information pertaining to ORCA can be gained from the Department of Lands, Panorama Ave, Bathurst, NSW.


The statistical tests present in ORCA are a practical and appropriate method for monitoring the presence and effects of gross errors in GPS networks. The use of such tests also allow for network optimisation at a design stage, which is advantageous when dealing with geodetic networks that require large amounts of time, economic and personnel resources to observe.


There were two significant outcomes from this thesis:


  1. The effectiveness of reliability theory is influenced by two major factors. The first is the choice of values for the test statistics. Pertinent values must be selected so that the values are functions of the number of observations, redundancies, intended accuracy and perceived precision of the network. The second is the spread of redundancy across a network. There needs to be appropriately placed redundancies such that the network redundancy numbers are high and uniformly spread.


2.      Using the results from conducting reliability theory analyses on different datasets, a series of network design guidelines have been recognised. Various recommendations relating to network design optimisation can be made based on the research, testing and conclusions arising from this thesis. The main points recognised were that:


          A design stage analysis should be undertaken on a network

          Determination of the approximate error detection levels from Marginally Detectable Error values.

          Determination of the effect of errors on coordinates using the external reliability matrix.

          Use of the correlation matrix to determine where undesirably high correlations exist.

          Perimeter stations should be connected by at least three baselines.

          Introducing eccentric marks will improve the redundancy of a network.


Reliability Theory proved to be an effective method for network monitoring and a useful design tool when applied to GPS geodetic control networks. It should be considered when designing GPS control networks, especially those with specific requirements and error budgets.




Baarda, W. (1968) A Testing Procedure for use in Geodetic Networks, Delft, Computing Centre of the Delft Geodetic Institute, 97 pages.


Lachapelle, G. and S. Ryan (2000) Statistical Reliability Measures for GPS, IMA Workshop on Mathematical challenges in GPS, University of Minnesota, 16 -18 August, 7 pages.


Pelzer, P. H. (1979) Some Criteria for the Accuracy and the Reliability of Networks, XVII General Assembly of the International Union of Geodesy and Geophysics, Canberra, Australia, 2 -15 December: 19 pages.


Schaffrin, B. (1997) Reliability Measures for Correlated Observations, Journal of Surveying Engineering, August: 126-137.


Wang, J. and Y. Chen (1994) On the Reliability Measures of Observations, Acta Geodaetica et Cartographica Sinica, English Edition: 42-51.


Wang, J. and Y. Chen (1999) Outlier Detection and Reliability Measures for Singular Adjustment Models, Geomatics Reasearch Australasia, No. 71: 52-72.


Further Information

For more information, please contact:


Dr. Jinling Wang




School of Surveying and Spatial Information Systems Engineering

University of New South Wales




Phone: +61-2-9385-4188

Fax: +61-2-9313-7493



Ms. C. E. McAlister



For more information on ORCA, please contact:


Mr. Tony Watson

Mr. Case Bosloper



Department of Lands

PO Box 143




Phone: +61-2-6332-8200