Huaan Fan

ISBN 91-7170-200-8 Royal Institute of Technology (KTH) Division of Geodesy and Geoinformatics 100 44 Stockholm Sweden August 2010

Theory of Errors and Least Squares Adjustment

Huaan Fan, PhD Royal Institute of Technology (KTH) Division of Geodesy and Geoinformatics Teknikringen 72, 100 44 Stockholm, Sweden Phone: +46 8 7907340 Fax: +46 8 7907343 E-mail: hfan@kth.se URL: http://www.infra.kth.se/impgg

With 22 illustrations and 49 numerical examples

ISBN 91-7170-200-8 Copyright c 1997 by Huaan Fan All Rights Reserved Geodesy Report No. 2015 TRITA-GEOFOTO 1997:21 Printed by Universitetsservice AB, Drottning Kristinas väg 53 B, Stockholm October 1997, August 1998, August 1999, May 2000, August 2001, August 2003, October 2005, August 2006, August 2008, August 2009 Printed by AJ E-print AB, Oxtorgsgatan 8, 111 57 Stockholm. August 2010.

To the memory of my father

Fuguan Fan (1928-1996)

who loved and inspired a geodesist

Preface

Theory of errors has been a well de…ned subject within geodesy, surveying and photogrammetry. Presently there are several masterpieces on this subject, such as the famous works by Bjerhammar (1973), Mikhail (1976) and Koch (1980), just to mention a few of them. However, the author still feels the need for a middle-level textbook in this …eld. A textbook that balances between practical applications and pure mathematical treatments; A textbook that integrates classical adjustment methods with modern developments in geodesy, surveying and photogrammetry. To meet the above need, an attempt was started in 1995 that has resulted in this compendium. Naturally, it is neither the author’ ambition nor within his capability to challenge those great works mentioned s above. The compendium will primarily be used for courses given at the Royal Institute of Technology (KTH).

The compendium consists of seven chapters. Chapter 1 deals with basic concepts in theory of errors, such as standard error and error propagation, error ellipse and error ellipsoid, linear equation systems as well as some elementary concepts of statistical analysis. Chapter 2 is devoted to the classical condition adjustment method, including condition adjustment in groups and condition adjustment with unknowns. The method of adjustment by elements is treated in Chapter 3, where adjustment by elements with constraints and sequential adjustment have also been described. Chapters 4-7 deal with diverse topics based on recent developments in theory of errors. These topics include generalized matrix inverses and their applications in least squares adjustment; a posteriori estimation of variance-covariance components; detection of gross and systematic errors; and …nally prediction and …ltering in linear dynamic systems. The essential prerequisite for the compendium is a familiarity with linear algebra, mathematical statistics and basic concepts in surveying. In other words, it is assumed that students have already acquired background knowledge in mathematics and surveying. Therefore, e¤orts have been made to limit discussions on pure mathematical or surveying subjects. In order to keep mathematical derivations brief and elegant, matrix notations have been used exclusively throughout the compendium. Several old concepts in theory of errors (e.g. the classical Gauss-Doolittle’ s table for solving normal equations ) have, in the author’ opinion, become out of date and thus been s excluded. To help readers better understand the theoretical concepts, a number of numerical examples (mostly originating from geodesy and surveying) are provided. For the sake of simplicity, most numerical examples are constructed so that only ideally simple numbers are involved and that they can be solved without needing to use calculators and computers. For those examples with more realistic data, the presented numerical results are obtained based on the author’ own programming on an Intel Pentium...