ADJUSTMENT COMPUTATIONS

Transcription

1 ADJUSTMENT COMPUTATIONS Adjustment Computations: Spatial Data Analysis, Fourth Edition. C. D. Ghilani and P. R. Wolf 2006 John Wiley & Sons, Inc. ISBN:

2 ADJUSTMENT COMPUTATIONS Spatial Data Analysis Fourth Edition CHARLES D. GHILANI, Ph.D. Professor of Engineering Surveying Engineering Program Pennsylvania State University PAUL R. WOLF, Ph.D. Professor Emeritus Department of Civil and Environmental Engineering University of Wisconsin Madison JOHN WILEY & SONS, INC.

3 This book is printed on acid-free paper. Copyright 2006 by John Wiley & Sons, Inc. All rights reserved. Published by John Wiley & Sons, Inc., Hoboken, New Jersey Published simultaneously in Canada. No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, scanning, or otherwise, except as permitted under Section 107 or 108 of the 1976 United States Copyright Act, without either the prior written permission of the Publisher, or authorization through payment of the appropriate per-copy fee to the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, (978) , fax (978) , or on the web at Requests to the Publisher for permission should be addressed to the Permissions Department, John Wiley & Sons, Inc., 111 River Street, Hoboken, NJ 07030, (201) , fax (201) , or online at go/ permission. Limit of Liability/ Disclaimer of Warranty: While the publisher and author have used their best efforts in preparing this book, they make no representations or warranties with respect to the accuracy or completeness of the contents of this book and specifically disclaim any implied warranties of merchantability or fitness for a particular purpose. No warranty may be created or extended by sales representatives or written sales materials. The advice and strategies contained herein may not be suitable for your situation. You should consult with a professional where appropriate. Neither the publisher nor author shall be liable for any loss of profit or any other commercial damages, including but not limited to special, incidental, consequential, or other damages. For general information on our other products and services or for technical support, please contact our Customer Care Department within the United States at (800) , outside the United States at (317) or fax (317) Wiley also publishes its books in a variety of electronic formats. Some content that appears in print may not be available in electronic books. For more information about Wiley products, visit our web site at Library of Congress Cataloging-in-Publication Data: Ghilani, Charles D. Adjustment computations : spatial data analysis / Charles D. Ghilani, Paul R. Wolf. 4th ed. p. cm. Prev. ed. entered under Wolf. ISBN (cloth) ISBN (cloth) 1. Surveying Mathematics. 2. Spatial analysis (Statistics) I. Wolf, Paul R. II. Title. TA556.M38W dc22 Printed in the United States of America

4 CONTENTS PREFACE ACKNOWLEDGMENTS xix xxiii 1 INTRODUCTION Introduction / Direct and Indirect Measurements / Measurement Error Sources / Definitions / Precision versus Accuracy / Redundant Measurements in Surveying and Their Adjustment / Advantages of Least Squares Adjustment / Overview of the Book / 10 Problems / 10 2 OBSERVATIONS AND THEIR ANALYSIS Introduction / Sample versus Population / Range and Median / Graphical Representation of Data / Numerical Methods of Describing Data / Measures of Central Tendency / Additional Definitions / 18 v

5 vi CONTENTS 2.8 Alternative Formula for Determining Variance / Numerical Examples / Derivation of the Sample Variance (Bessel s Correction) / Programming / 29 Problems / 30 3 RANDOM ERROR THEORY Introduction / Theory of Probability / Properties of the Normal Distribution Curve / Standard Normal Distribution Function / Probability of the Standard Error / % Probable Error / % Probable Error / Other Percent Probable Errors / Uses for Percent Errors / Practical Examples / 44 Problems / 47 4 CONFIDENCE INTERVALS Introduction / Distributions Used in Sampling Theory / Distribution / t (Student) Distribution / F Distribution / Confidence Interval for the Mean: t Statistic / Testing the Validity of the Confidence Interval / Selecting a Sample Size / Confidence Interval for a Population Variance / Confidence Interval for the Ratio of Two Population Variances / 63 Problems / 65 5 STATISTICAL TESTING Hypothesis Testing / Systematic Development of a Test / Test of Hypothesis for the Population Mean / Test of Hypothesis for the Population Variance / 74

6 CONTENTS vii 5.5 Test of Hypothesis for the Ratio of Two Population Variances / 77 Problems / 81 6 PROPAGATION OF RANDOM ERRORS IN INDIRECTLY MEASURED QUANTITIES Basic Error Propagation Equation / Generic Example / Frequently Encountered Specific Functions / Standard Deviation of a Sum / Standard Deviation in a Series / Standard Deviation of the Mean / Numerical Examples / Conclusions / 94 Problems / 95 7 ERROR PROPAGATION IN ANGLE AND DISTANCE OBSERVATIONS Introduction / Error Sources in Horizontal Angles / Reading Errors / Angles Observed by the Repetition Method / Angles Observed by the Directional Method / Pointing Errors / Estimated Pointing and Reading Errors with Total Stations / Target Centering Errors / Instrument Centering Errors / Effects of Leveling Errors in Angle Observations / Numerical Example of Combined Error Propagation in a Single Horizontal Angle / Use of Estimated Errors to Check Angular Misclosure in a Traverse / Errors in Astronomical Observations for an Azimuth / Errors in Electronic Distance Observations / Use of Computational Software / 123 Problems / 123

7 viii CONTENTS 8 ERROR PROPAGATION IN TRAVERSE SURVEYS Introduction / Derivation of Estimated Error in Latitude and Departure / Derivation of Estimated Standard Errors in Course Azimuths / Computing and Analyzing Polygon Traverse Misclosure Errors / Computing and Analyzing Link Traverse Misclosure Errors / Conclusions / 140 Problems / ERROR PROPAGATION IN ELEVATION DETERMINATION Introduction / Systematic Errors in Differential Leveling / Collimation Error / Earth Curvature and Refraction / Combined Effects of Systematic Errors on Elevation Differences / Random Errors in Differential Leveling / Reading Errors / Instrument Leveling Errors / Rod Plumbing Error / Estimated Errors in Differential Leveling / Error Propagation in Trigonometric Leveling / 152 Problems / WEIGHTS OF OBSERVATIONS Introduction / Weighted Mean / Relation between Weights and Standard Errors / Statistics of Weighted Observations / Standard Deviation / Standard Error of Weight w and Standard Error of the Weighted Mean / Weights in Angle Observations / Weights in Differential Leveling / 166

8 CONTENTS ix 10.7 Practical Examples / 167 Problems / PRINCIPLES OF LEAST SQUARES Introduction / Fundamental Principle of Least Squares / Fundamental Principle of Weighted Least Squares / Stochastic Model / Functional Model / Observation Equations / Elementary Example of Observation Equation Adjustment / Systematic Formulation of the Normal Equations / Equal-Weight Case / Weighted Case / Advantages of the Systematic Approach / Tabular Formation of the Normal Equations / Using Matrices to Form the Normal Equations / Equal-Weight Case / Weighted Case / Least Squares Solution of Nonlinear Systems / Least Squares Fit of Points to a Line or Curve / Fitting Data to a Straight Line / Fitting Data to a Parabola / Calibration of an EDM Instrument / Least Squares Adjustment Using Conditional Equations / Example 11.5 Using Observation Equations / 198 Problems / ADJUSTMENT OF LEVEL NETS Introduction / Observation Equation / Unweighted Example / Weighted Example / Reference Standard Deviation / Unweighted Example / Weighted Example / 213

9 x CONTENTS 12.6 Another Weighted Adjustment / 213 Problems / PRECISION OF INDIRECTLY DETERMINED QUANTITIES Introduction / Development of the Covariance Matrix / Numerical Examples / Standard Deviations of Computed Quantities / 226 Problems / ADJUSTMENT OF HORIZONTAL SURVEYS: TRILATERATION Introduction / Distance Observation Equation / Trilateration Adjustment Example / Formulation of a Generalized Coefficient Matrix for a More Complex Network / Computer Solution of a Trilaterated Quadrilateral / Iteration Termination / Method of Maximum Iterations / Maximum Correction / Monitoring the Adjustment s Reference Variance / 249 Problems / ADJUSTMENT OF HORIZONTAL SURVEYS: TRIANGULATION Introduction / Azimuth Observation Equation / Linearization of the Azimuth Observation Equation / Angle Observation Equation / Adjustment of Intersections / Adjustment of Resections / Computing Initial Approximations in the Resection Problem / Adjustment of Triangulated Quadrilaterals / 271 Problems / 275

10 CONTENTS xi 16 ADJUSTMENT OF HORIZONTAL SURVEYS: TRAVERSES AND NETWORKS Introduction to Traverse Adjustments / Observation Equations / Redundant Equations / Numerical Example / Minimum Amount of Control / Adjustment of Networks / Test: Goodness of Fit / 300 Problems / ADJUSTMENT OF GPS NETWORKS Introduction / GPS Observations / GPS Errors and the Need for Adjustment / Reference Coordinate Systems for GPS Observations / Converting between the Terrestrial and Geodetic Coordinate Systems / Application of Least Squares in Processing GPS Data / Network Preadjustment Data Analysis / Analysis of Fixed Baseline Measurements / Analysis of Repeat Baseline Measurements / Analysis of Loop Closures / Minimally Constrained Adjustment / Least Squares Adjustment of GPS Networks / 327 Problems / COORDINATE TRANSFORMATIONS Introduction / Two-Dimensional Conformal Coordinate Transformation / Equation Development / Application of Least Squares / Two-Dimensional Affine Coordinate Transformation / Two-Dimensional Projective Coordinate Transformation / 353

11 xii CONTENTS 18.7 Three-Dimensional Conformal Coordinate Transformation / Statistically Valid Parameters / 362 Problems / ERROR ELLIPSE Introduction / Computation of Ellipse Orientation and Semiaxes / Example Problem of Standard Error Ellipse Calculations / Error Ellipse for Station Wisconsin / Error Ellipse for Station Campus / Drawing the Standard Error Ellipse / Another Example Problem / Error Ellipse Confidence Level / Error Ellipse Advantages / Survey Network Design / Example Network / Other Measures of Station Uncertainty / 385 Problems / CONSTRAINT EQUATIONS Introduction / Adjustment of Control Station Coordinates / Holding Control Station Coordinates and Directions of Lines Fixed in a Trilateration Adjustment / Holding the Direction of a Line Fixed by Elimination of Constraints / Helmert s Method / Redundancies in a Constrained Adjustment / Enforcing Constraints through Weighting / 403 Problems / BLUNDER DETECTION IN HORIZONTAL NETWORKS Introduction / A Priori Methods for Detecting Blunders in Observations / Use of the K Matrix / Traverse Closure Checks / 411

12 CONTENTS xiii 21.3 A Posteriori Blunder Detection / Development of the Covariance Matrix for the Residuals / Detection of Outliers in Observations / Techniques Used in Adjusting Control / Data Set with Blunders / Some Further Considerations / Internal Reliability / External Reliability / Survey Design / 430 Problems / GENERAL LEAST SQUARES METHOD AND ITS APPLICATION TO CURVE FITTING AND COORDINATE TRANSFORMATIONS Introduction to General Least Squares / General Least Squares Equations for Fitting a Straight Line / General Least Squares Solution / Two-Dimensional Coordinate Transformation by General Least Squares / Two-Dimensional Conformal Coordinate Transformation / Two-Dimensional Affine Coordinate Transformation / Two-Dimensional Projective Transformation / Three-Dimensional Conformal Coordinate Transformation by General Least Squares / 449 Problems / THREE-DIMENSIONAL GEODETIC NETWORK ADJUSTMENT Introduction / Linearization of Equations / Slant Distance Observations / Azimuth Observations / Vertical Angle Observations / Horizontal Angle Observations / Differential Leveling Observations / Horizontal Distance Observations / 460

13 xiv CONTENTS 23.3 Minimum Number of Constraints / Example Adjustment / Addition of Slant Distances / Addition of Horizontal Angles / Addition of Zenith Angles / Addition of Observed Azimuths / Addition of Elevation Differences / Adjustment of Control Stations / Results of Adjustment / Updating Geodetic Coordinates / Building an Adjustment / Comments on Systematic Errors / 471 Problems / COMBINING GPS AND TERRESTRIAL OBSERVATIONS Introduction / Helmert Transformation / Rotations between Coordinate Systems / Combining GPS Baseline Vectors with Traditional Observations / Other Considerations / 489 Problems / ANALYSIS OF ADJUSTMENTS Introduction / Basic Concepts, Residuals, and the Normal Distribution / Goodness-of-Fit Test / Comparison of Residual Plots / Use of Statistical Blunder Detection / 501 Problems / COMPUTER OPTIMIZATION Introduction / Storage Optimization / Direct Formation of the Normal Equations / Cholesky Decomposition / Forward and Back Solutions / 511

14 CONTENTS xv 26.6 Using the Cholesky Factor to Find the Inverse of the Normal Matrix / Spareness and Optimization of the Normal Matrix / 513 Problems / 518 APPENDIX A INTRODUCTION TO MATRICES 520 A.1 Introduction / 520 A.2 Definition of a Matrix / 520 A.3 Size or Dimensions of a Matrix / 521 A.4 Types of Matrices / 522 A.5 Matrix Equality / 523 A.6 Addition or Subtraction of Matrices / 524 A.7 Scalar Multiplication of a Matrix / 524 A.8 Matrix Multiplication / 525 A.9 Computer Algorithms for Matrix Operations / 528 A.9.1 Addition or Subtraction of Two Matrices / 528 A.9.2 Matrix Multiplication / 529 A.10 Use of the MATRIX Software / 531 Problems / 531 APPENDIX B SOLUTION OF EQUATIONS BY MATRIX METHODS 534 B.1 Introduction / 534 B.2 Inverse Matrix / 534 B.3 Inverse of a 2 2 Matrix / 535 B.4 Inverses by Adjoints / 537 B.5 Inverses by Row Transformations / 538 B.6 Example Problem / 542 Problems / 543 APPENDIX C NONLINEAR EQUATIONS AND TAYLOR S THEOREM 546 C.1 Introduction / 546 C.2 Taylor Series Linearization of Nonlinear Equations / 546 C.3 Numerical Example / 547 C.4 Using Matrices to Solve Nonlinear Equations / 549

15 xvi CONTENTS C.5 Simple Matrix Example / 550 C.6 Practical Example / 551 Problems / 554 APPENDIX D NORMAL ERROR DISTRIBUTION CURVE AND OTHER STATISTICAL TABLES 556 D.1 Development of the Normal Distribution Curve Equation / 556 D.2 Other Statistical Tables / 564 D Distribution / 564 D.2.2 t Distribution / 566 D.2.3 F Distribution / 568 APPENDIX E CONFIDENCE INTERVALS FOR THE MEAN 576 APPENDIX F MAP PROJECTION COORDINATE SYSTEMS 582 F.1 Introduction / 582 F.2 Mathematics of the Lambert Conformal Conic Map Projection / 583 F.2.1 Zone Constants / 584 F.2.2 Direct Problem / 585 F.2.3 Inverse Problem / 585 F.3 Mathematics of the Transverse Mercator / 586 F.3.1 Zone Constants / 587 F.3.2 Direct Problem / 588 F.3.3 Inverse Problem / 588 F.4 Reduction of Observations / 590 F.4.1 Reduction of Distances / 590 F.4.2 Reduction of Geodetic Azimuths / 593 APPENDIX G COMPANION CD 595 G.1 Introduction / 595 G.2 File Formats and Memory Matters / 596 G.3 Software / 596 G.3.1 ADJUST / 596 G.3.2 STATS / 597 G.3.3 MATRIX / 598 G.3.4 Mathcad Worksheets / 598

16 CONTENTS xvii G.4 Using the Software as an Instructional Aid / 599 BIBLIOGRAPHY 600 INDEX 603

17 PREFACE No measurement is ever exact. As a corollary, every measurement contains error. These statements are fundamental and universally accepted. It follows logically, therefore, that surveyors, who are measurement specialists, should have a thorough understanding of errors. They must be familiar with the different types of errors, their sources, and their expected magnitudes. Armed with this knowledge they will be able to (1) adopt procedures for reducing error sizes when making their measurements and (2) account rigorously for the presence of errors as they analyze and adjust their measured data. This book is devoted to creating a better understanding of these topics. In recent years, the least squares method of adjusting spatial data has been rapidly gaining popularity as the method used for analyzing and adjusting surveying data. This should not be surprising, because the method is the most rigorous adjustment procedure available. It is soundly based on the mathematical theory of probability; it allows for appropriate weighting of all observations in accordance with their expected precisions; and it enables complete statistical analyses to be made following adjustments so that the expected precisions of adjusted quantities can be determined. Procedures for employing the method of least squares and then statistically analyzing the results are major topics covered in this book. In years past, least squares was seldom used for adjusting surveying data because the time required to set up and solve the necessary equations was too great for hand methods. Now computers have eliminated that disadvantage. Besides advances in computer technology, some other recent developments have also led to increased use of least squares. Prominent among these are the global positioning system (GPS) and geographic information systems and land information systems (GISs and LISs). These systems rely heavily xix

18 xx PREFACE on rigorous adjustment of data and statistical analysis of the results. But perhaps the most compelling of all reasons for the recent increased interest in least squares adjustment is that new accuracy standards for surveys are being developed that are based on quantities obtained from least squares adjustments. Thus, surveyors of the future will not be able to test their measurements for compliance with these standards unless they adjust their data using least squares. Clearly, modern surveyors must be able to apply the method of least squares to adjust their measured data, and they must also be able to perform a statistical evaluation of the results after making the adjustments. This book originated in 1968 as a set of lecture notes for a course taught to a group of practicing surveyors in the San Francisco Bay area by Professor Paul R. Wolf. The notes were subsequently bound and used as the text for formal courses in adjustment computations taught at both the University of California Berkeley and the University of Wisconsin Madison. In 1980, a second edition was produced that incorporated many changes and suggestions from students and others who had used the notes. The second edition, published by Landmark Enterprises, has been distributed widely to practicing surveyors and has also been used as a textbook for adjustment computations courses in several colleges and universities. For the fourth edition, new chapters on the three-dimensional geodetic network adjustments, combining GPS baseline vectors and terrestrial observations in an adjustment, the Helmert transformation, analysis of adjustments, and state plane coordinate computations are added. These are in keeping with the modern survey firm that collects data in three dimensions and needs to analyze large data sets. Additionally, Chapter 4 of the third edition has been divided into two new chapters on confidence intervals and statistical testing. This edition has greatly expanded and modified the number of problems for each chapter to provide readers with ample practice problems. For instructors who adopt this book in their classes, a Solutions Manual to Accompany Adjustment Computations is also available from the publisher. Two new appendixes have been added, including one on map projection coordinate systems and another on the companion CD. The software included on the CD for this book has also been greatly expanded and updated. A Mathcad electronic book added to the companion CD demonstrates the computations for many of the example problems in the text. To obtain a greater understanding of the material contained in this text, these electronic worksheets allow the reader to explore the intermediate computations in more detail. For readers not having the Mathcad software package, hypertext markup language (html) files are included on the CD for browsing. The software STATS, ADJUST, and MATRIX are now Windows-based and will run on a PC-compatible computer. The first package, called STATS, performs basic statistical analyses. For any given set of measured data, it will compute the mean, median, mode, and standard deviation, and develop and plot the histogram and normal distribution curve. The second package, called

19 PREFACE xxi ADJUST, contains programs for performing specific least-squares adjustments. Level nets, horizontal surveys (trilateration, triangulation, traverses, and horizontal network surveys), GPS networks, and traditional threedimensional surveys can be adjusted using software in this package. It also contains programs to compute the least-squares solution for a variety of coordinate transformations, and to determine the least squares fit of a line, parabola, or circle to a set of data points. Each of these programs computes residuals and standard deviations following the adjustment. The third program package, called MATRIX, performs a collection of basic matrix operations, such as addition, subtraction, transpose, multiplication, inverse, and more. Using this program, systems of simultaneous linear equations can be solved quickly and conveniently, and the basic algorithm for doing least squares adjustments can be solved in a stepwise fashion. For those who wish to develop their own software, the book provides several helpful computer algorithms in the languages of BASIC, C, FORTRAN, and PASCAL. Additionally, the Mathcad worksheets demonstrate the use of functions in developing modular programs. This current edition now consists of 26 chapters and several appendixes. The chapters are arranged in the order found most convenient in teaching college courses on adjustment computations. It is believed that this order also best facilitates practicing surveyors who use the book for self-study. In earlier chapters we define terms and introduce students to the fundamentals of errors and methods for analyzing them. The next several chapters are devoted to the subject of error propagation in the various types of traditional surveying measurements. Then chapters follow that describe observation weighting and introduce the least-squares method for adjusting observations. Application of least squares in adjusting basic types of surveys are then presented in separate chapters. Adjustment of level nets, trilateration, triangulation, traverses, and horizontal networks, GPS networks, and traditional three-dimensional surveys are included. The subject of error ellipses is covered in a separate chapter. Procedures for applying least squares in curve fitting and in computing coordinate transformations are also presented. The more advanced topics of blunder detection, the method of general least squares, and computer optimization are covered in the last chapters. As with previous editions, matrix methods, which are so well adapted to adjustment computations, continue to be used in this edition. For those students who have never studied matrices, or those who wish to review this topic, an introduction to matrix methods is given in Appendixes A and B. Those students who have already studied matrices can conveniently skip this subject. Least-squares adjustments often require the formation and solution of nonlinear equations. Procedures for linearizing nonlinear equations by Taylor s theorem are therefore important in adjustment computations, and this topic is presented in Appendix C. Appendix D contains several statistical tables including the standard normal error distribution, the 2 distribution, Student s t

20 xxii PREFACE distribution, and a set of F-distribution tables. These tables are described at appropriate locations in the text, and their use is demonstrated with example problems. Basic courses in statistics and calculus are necessary prerequisites to understanding some of the theoretical coverage and equation derivations given herein. Nevertheless, those who do not have these courses as background but who wish to learn how to apply least squares in adjusting surveying observations can study Chapters 1 through 3, skip Chapters 4 through 8, and then proceed with the remaining chapters. Besides being appropriate for use as a textbook in college classes, this book will be of value to practicing surveyors and geospatial information managers. The authors hope that through the publication of this book, least squares adjustment and rigorous statistical analyses of surveying data will become more commonplace, as they should.

21 ACKNOWLEDGMENTS Through the years many people have contributed to the development of this book. As noted in the preface, the book has been used in continuing education classes taught to practicing surveyors as well as in classes taken by students at the University of California Berkeley, the University of Wisconsin Madison, and the Pennsylvania State University Wilkes-Barre. The students in these classes have provided data for many of the example problems and have supplied numerous helpful suggestions for improvements throughout the book. The authors gratefully acknowledge their contributions. Earlier editions of the book benefited specifically from the contributions of Mr. Joseph Dracup of the National Geodetic Survey, Professor Harold Welch of the University of Michigan, Professor Sandor Veress of the University of Washington, Mr. Charles Schwarz of the National Geodetic Survey, Mr. Earl Burkholder of the New Mexico State University, Dr. Herbert Stoughton of Metropolitan State College, Dr. Joshua Greenfeld of New Jersey Institute of Technology, Dr. Steve Johnson of Purdue University, Mr. Brian Naberezny of Pennsylvania State University, and Professor David Mezera of the University of Wisconsin Madison. The suggestions and contributions of these people were extremely valuable and are very much appreciated. To improve future editions, the author will gratefully accept any constructive criticisms of this edition and suggestions for its improvement. xxiii