Adjustment Computations : Statistics and Least Squares in Surveying and GIS

by Paul R. Wolf (Emeritus, Univ. of Wisconsin-Madison); Charles D. Ghilani (Pennsylvania State Univ. )

• ISBN13: 9780471168331
• ISBN10: 0471168335
• Edition: 3rd
• Format: Hardcover
• Copyright: 1997-02-01
• Publisher: Wiley-Interscience

Summary

This book examines every aspect of least square adjustment. It defines terms and introduces readers to the fundamentals of errors and describes methods for analyzing them. It also illustrates the application of least squares in adjusting a wide range of survey types and provides detailed coverage of applications of least squares to GPSs and GISs.

Author Biography

PAUL R. WOLF, PhD, Emeritus Professor, Department of Civil and Environmental Engineering, University of Wisconsin, is the author of Elements of Photogrammetry and the bestselling Elementary Surveying, now in its ninth edition. CHARLES D. GHILANI, PhD, is Associate Professor in the College of Engineering and Chair of the Surveying Program at Pennsylvania State University.

Table of Contents

Preface	xiii	(4)
Acknowledgments	xvii
1 Introduction	1	(12)
1.1 Introduction	1	(1)
1.2 Direct and Indirect Measurements	2	(1)
1.3 Measurement Error Sources	2	(1)
1.4 Definitions	3	(1)
1.5 Precision Versus Accuracy	4	(3)
1.6 Redundant Measurements in Surveying and Their Adjustment	7	(1)
1.7 Advantages of Least-Squares Adjustment	8	(2)
1.8 Overview of the Book	10	(1)
Problems	10	(3)
2 Measurements and Their Analysis	13	(22)
2.1 Introduction	13	(1)
2.2 Sample Versus Population	13	(1)
2.3 Range and Median	14	(1)
2.4 Graphical Representation of Data	15	(3)
2.5 Numerical Methods of Describing Data	18	(1)
2.6 Measures of Central Tendency	19	(1)
2.7 Additional Definitions	19	(3)
2.8 Alternative Formula for Determining Variance	22	(2)
2.9 Numerical Examples	24	(6)
2.10 Derivation of the Sample Variance (Bessel's Correction)	30	(1)
2.11 STATS Program	31	(1)
Problems	31	(4)
3 Random Error Theory	35	(18)
3.1 Introduction	35	(1)
3.2 Theory of Probability	35	(3)
3.3 Properties of the Normal Distribution Curve	38	(2)
3.4 Standard Normal Distribution Function	40	(3)
3.5 Probability of the Standard Error	43	(3)
3.6 Uses for Percent Errors	46	(1)
3.7 Practical Examples	46	(3)
Problems	49	(4)
4 Confidence Intervals and Statistical Testing	53	(28)
4.1 Introduction	53	(2)
4.2 Distributions Used in Sampling Theory	55	(4)
4.3 Confidence Interval for the Mean: t Statistic	59	(3)
4.4 Testing the Validity of the Confidence Interval	62	(1)
4.5 Selecting a Sample Size	62	(1)
4.6 Confidence Interval for a Population Variance	63	(2)
4.7 Confidence Interval for the Ratio of Two Population Variances	65	(2)
4.8 Hypothesis Testing	67	(3)
4.9 Test of Hypothesis for the Population Mean	70	(3)
4.10 Test of Hypothesis for the Population Variance: o2	73	(2)
4.11 Test of Hypothesis for the Ratio of Two Population Variances	75	(4)
Problems	79	(3)
5 Propagation of Random Errors in Indirectly Measured Quantities	81	(16)
5.1 Basic Error Propagation Equation	81	(5)
5.2 Frequently Encountered Specific Functions	86	(1)
5.3 Numerical Examples	87	(4)
5.4 Conclusions	91	(1)
Problems	92	(5)
6 Error Propagation in Angle and Distance Measurements	97	(26)
6.1 Introduction	97	(1)
6.2 Error Sources in Horizontal Angles	97	(1)
6.3 Reading Errors	98	(2)
6.4 Pointing Errors	100	(1)
6.5 Estimated Pointing and Reading Errors with Total Stations	101	(1)
6.6 Target-Centering Errors	101	(3)
6.7 Instrument Centering Errors	104	(4)
6.8 Effects of Leveling Error in Angle Measurement	108	(2)
6.9 Numerical Example of Combined Error Propagation in a Single Horizontal Angle	110	(1)
6.10 Use of Estimated Standard Error to Check Angular Misclosure in a Traverse	111	(1)
6.11 Errors in Astronomical Observations for Azimuth	112	(5)
6.12 Errors in Electronic Distance Measurements	117	(1)
6.13 Use of Spreadsheets	118	(1)
Problems	119	(4)
7 Error Propagation in Traverse Surveys	123	(18)
7.1 Introduction	123	(1)
7.2 Derivation of Estimated Error in Latitude and Departure	124	(1)
7.3 Derivation of Estimated Standard Errors in Course Azimuths	125	(1)
7.4 Computing and Analyzing Polygon Traverse Misclosure Errors	126	(6)
7.5 Computing and Analyzing Link Traverse Misclosure Errors	132	(4)
7.6 Conclusions	136	(1)
Problems	136	(5)
8 Error Propagation in Elevation Determination	141	(14)
8.1 Introduction	141	(1)
8.2 Systematic Errors in Differential Leveling	141	(3)
8.3 Random Errors in Differential Leveling	144	(5)
8.4 Error Propagation in Trigonometric Leveling	149	(4)
Problems	153	(2)
9 Weights of Observations	155	(14)
9.1 Introduction	155	(2)
9.2 Weighted Mean	157	(2)
9.3 Relation Between Weights and Standard Errors	159	(1)
9.4 Statistics of Weighted Observations	160	(2)
9.5 Weights in Angle Measurements	162	(1)
9.6 Weights in Differential Leveling	163	(1)
9.7 Practical Examples	164	(3)
Problems	167	(2)
10 Principles of Least Squares	169	(32)
10.1 Introduction	169	(1)
10.2 Fundamentals Principle of Least Squares	170	(2)
10.3 Fundamentals Principle of Weighted Least Squares	172	(1)
10.4 Stochastic Model	173	(1)
10.5 Mathematical Model	173	(1)
10.6 Observation Equations	174	(2)
10.7 Systematic Formulation of the Normal Equations	176	(4)
10.8 Tabular Formation of the Normal Equations	180	(1)
10.9 Using Matrices to Form the Normal Equations	180	(4)
10.10 Least-Squares Solution of Nonlinear Systems	184	(3)
10.11 Least-Squares Fit of Points to a Line or Curve	187	(3)
10.12 Calibration of an EDM Instrument	190	(1)
10.13 Least-Squares Adjustment Using Conditional Equations	191	(3)
10.14 Example Using Observation Equations	194	(1)
Problems	195	(6)
11 Adjustment of Level Nets	201	(16)
11.1 Introduction	201	(1)
11.2 Observation Equation	201	(1)
11.3 Unweighted Example	202	(2)
11.4 Weighted Example	204	(3)
11.5 Reference Standard Deviation	207	(2)
11.6 Another Weighted Adjustment	209	(3)
Problems	212	(5)
12 Precisions of Indirectly Determined Quantities	217	(12)
12.1 Introduction	217	(1)
12.2 Development of the Covariance Matrix	217	(4)
12.3 Numerical Examples	221	(1)
12.4 Standard Deviations of Computed Quantities	222	(3)
Problems	225	(4)
13 Adjustment of Horizontal Surveys: Trilateration	229	(20)
13.1 Introduction	229	(2)
13.2 Distance Observation Equation	231	(2)
13.3 Trilateration Adjustment Example	233	(6)
13.4 Formulation of a Generalized Coefficient Matrix for a More Complex Network	239	(1)
13.5 Computer Solution of a Trilaterated Quadrilateral	239	(4)
13.6 Iteration Termination	243	(2)
Problems	245	(4)
14 Adjustment of Horizontal Surveys: Triangulation	249	(26)
14.1 Introduction	249	(1)
14.2 Azimuth Observation Equation	249	(3)
14.3 Angle Observation Equation	252	(2)
14.4 Adjustment of Intersections	254	(5)
14.5 Adjustment of Resections	259	(6)
14.6 Adjustment of Triangulated Quadrilaterals	265	(4)
Problems	269	(6)
15 Adjustment of Horizontal Surveys: Traverses and Networks	275	(22)
15.1 Introduction to Traverse Adjustments	275	(1)
15.2 The Observation Equations	275	(1)
15.3 Redundant Equations	276	(1)
15.4 Numerical Example	277	(5)
15.5 Minimum Amount of Control	282	(1)
15.6 Adjustment of Networks	283	(7)
15.7 X2 Test: Goodness of Fit	290	(1)
Problems	291	(6)
16 Adjustment of GPS Networks	297	(38)
16.1 Introduction	297	(1)
16.2 GPS Measurements	298	(3)
16.3 GPS Errors and the Need for Adjustment	301	(1)
16.4 Reference Coordinate Systems for GPS Measurements	301	(2)
16.5 Converting Between the Terrestrial and Geodetic Coordinate Systems	303	(5)
16.6 Application of Least Squares in Processing GPS Data	308	(1)
16.7 Network Preadjustment Data Analysis	309	(5)
16.8 Least-Squares Adjustment of GPS Networks	314	(7)
Problems	321	(14)
17 Coordinate Transformations	335	(22)
17.1 Introduction	335	(1)
17.2 Two-Dimensional Conformal Coordinate Transformation	335	(1)
17.3 Equation Development	336	(2)
17.4 Application of Least Squares	338	(3)
17.5 Two-Dimensional Affine Coordinate Transformation	341	(2)
17.6 Two-Dimensional Projective Coordinate Transformation	343	(4)
17.7 Three-Dimensional Conformal coordinate Transformation	347	(6)
17.8 Statistically Valid Parameters	353	(1)
Problems	354	(2)
18 Error Ellipse	357	(18)
18.1 Introduction	357	(2)
18.2 Computation of Ellipse Orientation and Semiaxes	359	(4)
18.3 Example of Standard Error Ellipse Calculations	363	(2)
18.4 Another Example	365	(2)
18.5 Error Ellipse Confidence Level	367	(2)
18.6 Error Ellipse Advantages	369	(3)
Problems	372	(3)
19 Constraint Equations	375	(22)
19.1 Introduction	375	(1)
19.2 Adjustment of Control Station Coordinates	375	(6)
19.3 Holding Control Station Coordinates and Directions of Lines Fixed in a Trilateration Adjustment	381	(4)
19.4 Helmert's Method	385	(5)
19.5 Redundancies in a Constrained Adjustment	390	(1)
19.6 Enforcing Constraints Through Weighting	390	(3)
Problems	393	(4)
20 Blunder Detection in Horizontal Survey Networks	397	(26)
20.1 Introduction	397	(1)
20.2 A Priori Methods for Detecting Blunders in Measurements	398	(2)
20.3 A Posteriori Blunder Direction	400	(2)
20.4 Development of the Covariance Matrix for the Residuals	402	(2)
20.5 Detection of Outliers in Observations	404	(2)
20.6 Techniques Used in Adjusting Control	406	(1)
20.7 Data Set with Blunders	407	(9)
20.8 Some Further Considerations	416	(2)
20.9 Survey Design	418	(2)
Problems	420	(3)
21 General Least-Squares Method and Its Application to Curve Fitting and Coordinate Transformations	423	(18)
21.1 Introduction to General Least Squares	423	(1)
21.2 General Least-Squares Equations for Fitting a Straight Line	423	(2)
21.3 General Least-Squares Solution	425	(5)
21.4 Two-Dimensional Coordinate Transformation by General Least Squares	430	(5)
21.5 Three-Dimensional Conformal Coordinate Transformation by General Least Squares	435	(3)
Problems 453	438	(3)
22 Computer Optimization	441	(18)
22.1 Introduction	441	(1)
22.2 Storage Optimization	441	(3)
22.3 Direct Formation of the Normal Equations	444	(1)
22.4 Cholesky Decomposition	445	(2)
22.5 Forward and Back Solutions	447	(3)
22.6 Using the Cholesky Factor to Find the Inverse of the Normal Matrix	450	(1)
22.7 Sparseness and Optimization of the Normal Matrix	451	(5)
Problems	456	(3)
A Introduction to Matrices	459	(14)
A.1 Introduction	459	(1)
A.2 Definition of a Matrix	459	(1)
A.3 Size or Dimensions of a Matrix	460	(1)
A.4 Types of Matrices	461	(1)
A.5 Matrix Equality	462	(1)
A.6 Addition or Subtraction of Matrices	463	(1)
A.7 Scalar Multiplication of a Matrix	463	(1)
A.8 Matrix Multiplication	464	(3)
A.9 Computer Algorithms for Matrix Operations	467	(3)
A.10 Use of the Program MATRIX	470	(1)
Problems	470	(3)
B Solution of Equations by Matrix Methods	473	(12)
B.1 Introduction	473	(1)
B.2 Inverse Matrix	473	(1)
B.3 Inverse of a 2 * 2 Matrix	474	(2)
B.4 Inverse by Adjoints	476	(2)
B.5 Inverses by Row Transformations	478	(2)
B.6 Numerical Example	480	(3)
Problems	483	(2)
C Nonlinear Equations and Taylor's Theorem	485	(10)
C.1 Introduction	485	(1)
C.2 Taylor Series Linearization of Nonlinear Equations	485	(1)
C.3 Numerical Example	486	(2)
C.4 Using Matrices to Solve Nonlinear Equations	488	(1)
C.5 Simple Matrix Example	489	(1)
C.6 Practical Example	490	(3)
Problems	493	(2)
D Normal Error Distribution Curve and Other Statistical Tables	495	(22)
D.1 Development for the Normal Distribution Curve Equation	495	(8)
D.2 Other Statistical Tables	503	(14)
E Confidence Intervals for the Mean	517	(6)
F Documentation for Software	523	(30)
F.1 Introduction	523	(1)
F.2 Similarities in the Programs STATS and ADJUST	524	(7)
F.3 STATS	531	(1)
F.4 ADJUST	532	(16)
F.5 MATRIX	548	(5)
Bibliography	553	(4)
Index	557