ADJUSTMENT COMPUTATIONS

Spatial Data Analysis

Sixth Edition

 

 

CHARLES D. GHILANI, PhD

Professor Emeritus of Engineering

The Pennsylvania State University

 

 

 

 

 

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Cover Page

PREFACE

No observation is ever exact. As a corollary, every observation contains error. These statements are fundamental and accepted universally. 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 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 only 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 this 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 navigation satellite systems (GNSSs) such as GPS and geographic information systems (GISs). These systems rely heavily 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 observations for compliance with these standards unless they adjust their data using least squares. This edition discusses these newer methods of classifying maps and control no matter the source. Clearly modern surveyors must be able to apply the method of least squares to adjust their observed data, and they must also be able to perform a statistical evaluation of the results after making the adjustments.

In the sixth edition, the author has included instructional videos to accompany Chapters 1 through 25 and Appendixes A through C in the book. These videos provide instructional lessons on the subject matter covered in the book. They also demonstrate the solution of example problems using spreadsheets and the software that accompanies the book. Additionally, this book discusses proper procedures to compute accuracy estimates following both the Federal Geographic Data Committee and ASPRS Digital Geospatial Data and control surveys. For instructors who adopt this text in their classes, an Instructor's Manual to Accompany Adjustment Computations is available from the publisher's website. This manual includes detailed solutions to all the problems in the book along with suggested course outlines and exams, which can be used in their courses. It is available to all instructors who adopt this book. To obtain access the manual contact your regional Wiley representative.

The software STATS, ADJUST, and MATRIX will run on any PC-compatible computer in the Windows environment. The first package, called STATS, performs basic statistical analyses. For any given set of observed data, it will compute the mean, median, mode, and standard deviation, and develop and plot the histogram and normal distribution curve. It will also compute critical values for the t, χ2, and F distributions. New features include its ability to compute critical values for the τ distribution, confidence intervals for the population mean variance, and ratio of two variances, and perform statistical test for the population mean, variance, and ratio of two variances.

The second package, called ADJUST, contains programs for performing specific least squares adjustments covered in the book. When performing least squares adjustments, ADJUST allows the user to select either data snooping or the tau criterion for post-adjustment blunder detection. The program contains a variety of coordinate transformations and allows user to fit points to a line, parabola, or circle. A new feature in MATRIX is its ability to perform unweighted and weighted least squares adjustments with a single command. 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 on the companion website demonstrate the use of functions in developing modular programs.

The chapters of this book are arranged in the order found most convenient in teaching college courses on adjustment computations. The content in this book can be covered in two or three typical undergraduate, college-level courses. 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. Applications 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, GNSS networks, and conventional three-dimensional surveys are included. The subject of error ellipses and error ellipsoids are 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 adjustments, 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 χ2 distribution, t 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 surveying, 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 follow the discussions on data analysis.

Besides being appropriate for use as a textbook in college classes, this book will be of value to practicing surveyors and geospatial information managers. With the inclusion of video lessons, it is possible for practitioners to learn the subject matter at their leisure. The author hopes that through the publication of this book, least squares adjustment and rigorous statistical analyses of surveying data will become more commonplace, as it should.

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 Pennsylvania State University–Wilkes-Barre. The students in these classes have provided data for some 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 Dr. Paul R. Wolf, who wrote the first two editions of this book, 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 New Mexico State University, Dr. Herbert Stoughton of Metropolitan State College, Dr. Joshua Greenfeld, Dr. Steve Johnson of Purdue University, Mr. Brian Naberezny, Mr. Preston Hartzell of the University of Houston, and Mr. Edward Connolly of TBE Group, Inc. The suggestions and contributions of these people were extremely valuable and are very much appreciated.