The 3-D Global SpatialData Model
Principles and Applications
Second Edition
Earl F. Burkholder
/0\ CRC Press\Cf*' J Taylor &. Francis Group
Boca Raton London NewYork
CRC Press is an imprint of the
Taylor & Francis Group, an informa business
Contents
Preface to the Second Edition xix
Preface to the First Edition xxi
Acknowledgments xxiii
Author xxv
List ofAbbreviations xxvii
Introduction xxxi
Chapter 1 The Global Spatial Data Model (GSDM) Defined 1
Introduction 1
The GSDM 2
Functional Model Component 3
Computational Designations 5
Algorithm for Functional Model 10
Stochastic Model Component 14
The GSDM Covariance Matrices 14
The GSDM 3-D Inverse 16
BURKORD: Software and Database 18
Summary 18
References 19
Chapter 2 Featuring the 3-D Global Spatial Data Model 21
Introduction 21
The GSDM Facilitates Existing Initiatives 22
U.S. National Academy of Public Administration Reports 22
National Oceanic and Atmospheric Administration 23
Coalition of Geospatial Organizations 24
Other Applications 25
Dynamic Environments 25
Static Environments 25
Information Provided by the GSDM 26
Summary 26
References 27
Chapter 3 Spatial Data and the Science of Measurement 29
Introduction 29
Spatial Data Defined 29
Coordinate Systems Give Meaning to Spatial Data 30
Spatial Data Types 32
Contents
Spatial Data Visualization Is Well Defined 34
Direct and Indirect Measurements Contain Uncertainty 34
Fundamental Physical Constants Are Held Exact 34
Measurements/Observations Contain Errors 35
Measurements Used to Create Spatial Data 35
Taping 35
Leveling 35
EDMI 35
Angles 36
GPS and GNSS 36
Remote Sensing 38
Photogrammetry 38
LiDAR 38
Logistics 38
Errorless Spatial Data 39
Sources of Primary Spatial Data 41
Observations and Measurements 41
Errorless Quantities 42
Derived Spatial Data Are Computed from Primary Spatial Data 42
Establishing and Preserving the Value of Spatial Data 43
Summary 44
References 44
Chapter 4 Summary of Mathematical Concepts 47
Introduction 47
Conventions 48
Numbers 48
Fractions 48
Decimal 48
Radian 49
Sexagesimal 50
Binary 50
Unit Conversions 51
Coordinate Systems 51
Significant Digits 52
Addition and Subtraction 52
Multiplication and Division 53
Avoid Mistakes by Working with Coordinate Differences 54
Logic 54
Arithmetic 55
Algebra 55
Axioms of Equality (for Real Numbers A, B, and C) 56
Axioms ofAddition (for Real Numbers A, B, and Q 56
Axioms of Multiplication (for Real Numbers A, B, and C) 56
Boolean Algebra 56
Contents 'x
Geometry 56
Point 57
Distance 57
Dimension 57
Line 57
Plane 57
Angle 58
Circle 58
Ellipse 58
Triangle 58
Quadrilateral 58
Rectangle 59
Square 59
Trapezoid 59
Parallelogram 59
Polygon 59
Pythagorean Theorem 59
Solid Geometry 60
Sphere 60
Ellipsoid 60
Cube 60
Polyhedron 60
Tetrahedron 60
Pyramid 60
Equation of a Plane in Space 60
Equation of a Sphere in Space 61
Equation of an Ellipsoid Centered on the Origin 61
Conic Sections 61
Vectors 62
Trigonometry 62
Trigonometric Identities 63
Law of Sines 64
Law of Cosines 65
Spherical Trigonometry 65
Calculus 68
Example 68
Differential Calculus Equations 70
Integral Calculus Equations 70
Probability and Statistics 71
Introduction 71
Standard Deviation 72
Measurement 73
Errors 73
Blunders 74
Systematic Errors 74
Random Errors 74
X Contents
Error Sources 74
Personal 74
Environmental 75
Instrumental 75
Accuracy and Precision 75
Computing Standard Deviations 76
Standard Deviation of the Mean 76
Confidence Intervals 77
Hypothesis Testing 78
Matrix Algebra 79
Models 80
Functional 80
Stochastic 80
Error Propagation 81
Error Ellipses 87
Least Squares 88
Linearization 89
Procedure for Nonlinear Solution 90
Applications to the GSDM 90
References 91
Chapter 5 Geometrical Models for Spatial Data Computations 93
Introduction 93
Conventions 94
Two-Dimensional Cartesian Models 97
Math/Science Reference System 98
Engineering/Surveying Reference System 98
Coordinate Geometry 99
Forward 99
Inverse 100
Intersections 100
Line-Line (One Solution or No Solution If Lines Are Parallel).... 102
Line-Circle (May Have Two Solutions, One Solution, or
No Solution) 102
Circle-Circle (May Have Two Solutions, One Solution, or
No Solution) 103
Perpendicular Offset 104
Area by Coordinates 104
Circular Curves 106
Definitions 106
Degree of Curve 106
Elements and Equations 107
Stationing 109
Metric Considerations 110
Contents xi
Area Formed by Curves 111
Area of Unit Circle 112
Spiral Curves 113
Spiral Geometry 113
Intersecting a Line with a Spiral 116
Computing Area Adjacent to a Spiral 117
Radial Surveying 118
Vertical Curves 121
Three-Dimensional Models for Spatial Data 124
Volume of a Rectangular Solid 124
Volume of a Sphere 124
Volume of Cone 125
Prismoidal Formula 126
Traditional 3-D Spatial Data Models 128
The 3-D GSDM 128
References 129
Chapter 6 Overview of Geodesy 131
Introduction: Science and Art 131
Fields of Geodesy 131
Goals of Geodesy 132
Historical Perspective 137
Religion, Science, and Geodesy 138
Degree Measurement 139
Eratosthenes 139
Poseidonius 140
Caliph Abdullah al Mamun 140
Gerardus Mercator 140
Willebrord Snellius 141
Jean Picard 141
Isaac Newton 141
Jean-Dominique and Jacques Cassini 142
French Academy of Science 142
Meter 143
Developments during the Nineteenth and Twentieth Centuries 143
Forecast for the Twenty-First Century 145
References 146
Chapter 7 Geometrical Geodesy 147
Introduction 147
The Two-Dimensional Ellipse 149
The Three-Dimensional Ellipsoid 154
Ellipsoid Radii of Curvature 154
Normal Section Radius of Curvature 155
Geometrical Mean Radius 155
xii Contents
Rotational Ellipsoid 155
Equation of Ellipsoid 155
Geocentric and Geodetic Coordinates 156
BK1 Transformation 157
BK2 Transformation 158
Iteration 158
Noniterative (Vincenty) Method 159
Example of BK1 Transformation 160
Example of BK2 Transformation—Iteration 161
Example of BK2 Transformation—Vincenty's Method
(Same Point) 162
Meridian Arc Length 163
Length of a Parallel 166
Surface Area of Sphere 166
Ellipsoid Surface Area 167
The Geodetic Line 169
Description 169
Clairaut's Constant 170
Geodetic Azimuths 172
Target Height Correction 174
Geodesic Correction 175
Geodetic Position Computation—Forward and Inverse 175
Puissant Forward (BK18) 176
Puissant Inverse (BK19) 177
Numerical Integration 178
BK18: Forward 178
BK19: Inverse 181
Geodetic Position Computations Using State Plane Coordinates.... 185
GSDM 3-D Geodetic Position Computations 186
Forward—BK3 186
Inverse—BK4 187
GSDM Inverse Example: New Orleans to Chicago 188
References 193
Chapter 8 Geodetic Datums 195
Introduction 195
Horizontal Datums 196
Brief History 196
North American Datum of 1927 (NAD 27) 198
North American Datum of 1983 (NAD 83) 198
World Geodetic System 1984 199
International Terrestrial Reference Frame 200
High Accuracy Reference Network—HARN 202
Contents xin
Continuously Operating Reference Station—CORS 204
NA2011 205
Vertical Datums 205
Sea Level Datum of 1929 (now NGVD 29) 205
International Great Lakes Datum 206
North American Vertical Datum of 1988—NAVD 88 206
Datum Transformations 207
NAD 27 to NAD 83 (1986) 208
NAD 83 (1986) to HPGN 208
NAD 83 (xxxx) to NAD 83 (yyyy) 208
NGVD 29 to NAVD 88 208
HTDP 208
Software Sources 208
7-( 14-) Parameter Transformation 209
3-D Datums 209
References 210
Chapter 9 Physical Geodesy 213
Introduction 213
Gravity 214
Definitions 215
Elevation (Generic) 216
Equipotential Surface 216
Level Surface 216
Geoid 216
Geopotential Number 217
Dynamic Height 217
Orthometric Height 217
Ellipsoid Height 217
Geoid Height 217
Gravity and the Shape of the Geoid 218
Laplace Correction 219
Measurements and Computations 221
Interpolation and Extrapolation 221
Gravity 222
Tide Readings 223
Differential Levels 223
Ellipsoid Heights 224
Time 225
Use of Ellipsoid Heights in Place of Orthometric Heights 225
The Need for Geoid Modeling 227
Geoid Modeling and the GSDM 231
Using a Geoid Model 232
References 234
xiv Contents
Chapter 10 Satellite Geodesy and Global Navigation Satellite Systems 237
Introduction 237
Brief History of Satellite Positioning 240
Modes of Positioning 244
Elapsed Time 244
Doppler Shift 244
Interferometry 246
Satellite Signals 247
C/ACode 249
Carrier Phase 250
Differencing 251
Single Differencing 252
Double Differencing 252
Triple Differencing 252
RINEX 252
Processing GNSS Data 253
Spatial Data Types 254
Autonomous Processing 255
Vector Processing 256
Multiple Vectors 257
Traditional Networks 259
Advanced Processing 259
The Future of Survey Control Networks—Has It Arrived? 262
References 265
Chapter 11 Map Projections and State Plane Coordinates 267
Introduction: Round Earth—Flat Map 267
Projection Criteria 268
Projection Figures 270
Permissible Distortion and Area Covered 273
U.S. State Plane Coordinate System (SPCS) 274
History 275
Features 275
NAD 27 and NAD 83 277
Current Status—NAD 83 SPCS 279
Advantages 280
Disadvantages 280
Procedures 281
Grid Azimuth 281
Grid Distance 281
Traverses 284
Loop Traverse 285
Point-to-Point Traverse 285
Contents xv
Algorithms for Traditional Map Projections 285
Lambert Conformal Conic Projection 286
BK10 Transformation for Lambert Conformal Conic
Projection 288
BK11 Transformation for Lambert Conformal Conic
Projection 288
Transverse Mercator Projection 289
BK10 Transformation for Transverse Mercator Projection ....292
BK11 Transformation for Transverse Mercator Projection.... 294
Oblique Mercator Projection 297
BK10 Transformation for Oblique Mercator Projection 299
BK11 Transformation for Oblique Mercator Projection 300
Low-Distortion Projection 302
References 302
Chapter 12 Spatial Data Accuracy 305
Introduction 305
Forces Driving Change 305
Transition 306
Consequences 308
Accuracy 309
Introduction 309
Definitions 311
Absolute and Relative Quantities 311
Spatial Data Types and Their Accuracy 313
Accuracy Statements 313
But Everything Moves 313
Observations, Measurements, and Error Propagation 315
Finding the Uncertainty of Spatial Data Elements 315
Using Points Stored in a XIYIZ Database 317
Example 319
Control Values and Observed Vectors 320
Blunder Checks 321
Least Squares Solution 322
Results 323
Network Accuracy and Local Accuracy 323
References 328
Chapter 13 Using the GSDM to Compute a Linear Least SquaresGNSS Network 329
Introduction 329
Parameters and Linearization 329
Baselines and Vectors 330
xvi Contents
Observations and Measurements 330
Covariance Matrices and Weight Matrices 331
Two Equivalent Adjustment Methods 332
Formulations of Matrices—Indirect Observations 333
Example GNSS Network Project in Wisconsin 336
RINEX Data Used to Build the Wisconsin Network 338
Blunder Checks 338
Building Matrices for a Linear Least Squares Solution 341
/Vector—n, 1 341
B Matrix—n, u 343
Q Matrix—n, n 343
Computer Printouts 345
Notes Pertaining to Adjustment 364
References 364
Chapter 14 Computing Network Accuracy and Local Accuracy
Using the Global Spatial Data Model 365
Introduction 365
Background 366
Summary of Pertinent Concepts 366
Detailed Example Based on Wisconsin Network 368
Conclusion 376
References 376
Chapter 15 Using the GSDM—Projects and Applications 379
Introduction 379
Features 381
The Functional Model 381
The Stochastic Model 381
Database Issues 384
Implementation Issues 385
Examples and Applications 387
Example 1—Supplemental NMSU Campus Control Network...387
Example 2—Hypothesis Testing 400
Example 3—Using Terrestrial Observations in the GSDM 401
Example 4—Using the GSDM to Develop a 2-D Survey Plat....407
Example 5—New Mexico Initial Point and Principal Meridian 410
Example 6—State Boundary between Texas and New
Mexico along the Rio Grande River 417
Example 7 in Wisconsin—Leveling in the Context of the
GSDM (Example in Wisconsin) 427
Example 8—Determining the NAVD 88 Elevation of
HARN Station REILLY 427
Contents xvii
Example 9—Determining the Shadow Height at a Proposed
NEXRAD Installation 432
Example 10—Comparison of 3-D Computational Models 434
Example 11—Underground Mapping 438
Example 12—Laying Out a Parallel of Latitude Using the
GSDM 440
Analogous to Solar Method 441
Analogous to Tangent-Offset Method 442
The Future Will Be What We Make It 443
References 445
Appendix A: Rotation Matrix Derivation 447
Appendix B: 1983 State Plane Coordinate Zone Constants 451
Appendix C: 3-D Inverse with Statistics 459
Appendix D: Development of the Global Spatial Data Model (GSDM) 461
Appendix E: Evolution of Meaning for Terms: Network Accuracyand Local Accuracy 465
Index 473
