handbook for transformation of datums, projections, grids and common coordinate systems
TRANSCRIPT
TEC-SR-7
Handbook for Transformation of Datums, Projections, Grids and Common Coordinate Systems
January 1996
19970417 058
Approved for public release; distribution is unlimited.
U.S. Army Corps of Engineers Topographic Engineering Center 7701 Telegraph Road Alexandria, VA 22315-3864
US Army Corps of Engineers Topographic Engineering Center
.Ota
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1. AGENCY USE ONLY (Leave blank) 2. REPORT DATE January 1996
3. REPORT TYPE AND DATES COVERED Handbook May 1991 - July 1995
4. TITLE AND SUBTITLE Handbook for Transformation of Datums, Projections, Grids and Common Coordinate Systems
6. AUTHOR(S) See Preface
7. PERFORMING ORGANIZATION NAME(S) AND ADDRESS(ES) U.S. Army Topographic Engineer Center 7701 Telegraph Road Alexandria, VA 22315-3864
9. SPONSORING /MONITORING AGENCY NAME(S) AND ADDRESS(ES)
5. FUNDING NUMBERS QE5113UD01
8. PERFORMING ORGANIZATION REPORT NUMBER
TEC-SR-7
19. SPONSORING / MONITORING AGENCY REPORT NUMBER
11. SUPPLEMENTARY NOTES
Beneficial comments or information that may be of use in improving this document should be addressed to the Standards Division of the Digital Concepts & Analysis Center at the above address.
12a. DISTRIBUTION/AVAILABILITY STATEMENT Approved for public release; distribution is unlimited.
12b. DISTRIBUTION CODE UNLIMITED
13. ABSTRACT (Maximum 200 words) This document provides Army orginizations and agencies with general guidance on selecting the appropriate methods for shifting between local geodetic datums and the World Geodetic System (WGS), and for converting Cartesian and map projection coordinates to and from geodetic coordinates. This guidance is provided to aid the Army community in selecting datum shift algorithms, in developing, selecting, and maintaining software using these algorithms, and in implementing this software to support operational units. Equations are furnished for map projections and datums commonly used within the Army, and references are provided for other, less commonly encountered, map projections and datums.
14. SUBJECT TERMS Cartography Geodesy
Datums Coordinates
Projections
17. SECURITY CLASSIFICATION OF REPORT
Unclassified
NSN 7540-01-280-5500
18. SECURITY CLASSIFICATION OFTHISPAGE
Unclassified
19. SECURITY CLASSIFICATION OFABSTRACT
Unclassified
15. NUMBER OF PAGES 170
16. PRICE CODE
20. LIMITATION OF ABSTRACT
Unlimited
Standard Form 298 (Rev. 2-89) Prescribed by ANSI Std. Z39-18 298-102
TABLE OF CONTENTS
LIST OF FIGURES LIST OF TABLES EXECUTIVE SUMMARY PREFACE
1. SCOPE 1.1 Scope 1.2 Applicability 1.3 Application guidance
2. REFERENCED DOCUMENTS 2.1 Government documents 2.1.1 Specifications, standards and handbooks 2.1.2 Other government documents, drawings and
publications 2.2 Non-government publications 2.3 Order of precedence
3. DEFINITIONS AND UNITS 3.1 Acronyms 3.2 Terms 3.2.1 Convergence of the meridian (y) 3.2.2 Coordinate 3.2.3 Datum 3.2.3.1 Horizontal datum 3.2.3.2 Vertical datum 3.2.4 Earth-fixed 3.2.5 Elevation (orthometric height, H) 3.2.6 Ellipsoid 3.2.7 Equator 3.2.8 Eguipotential surface 3.2.9 Geocentric Cartesian coordinates 3.2.10 Geodetic coordinates (geodetic position) 3.2.11 Geodetic height (ellipsoidal height, h) 3.2.12 Geodetic latitude (<j>) 3.2.13 Geodetic longitude (A,) 3.2.14 Geographic coordinates 3.2.15 Geoid 3.2.16 Geoid separation (N) 3.2.17 Grid reference system 3.2.18 Map projection 3.2.19 Map scale 3.2.20 Meridian 3.2.21 Military Grid Reference System (MGRS) 3.2.22 Orthometric height 3.2.23 Parallel 3.2.24 Prime (initial) meridian 3.2.25 Reference ellipsoid 3.2.26 Scale factor (projection) 3.3 Units
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PAGE
ix xi
xiii xv
1 1 1 1
2 2 2 2
4 4
5 5 5 5 5 5 6 6 6 6 6 6 6 6 6 6 7 7 7 7 7 7 7 7 7 7 8 8 8 8 8 8
TABLE OF CONTENTS
PAGE 3.4 Sign conventions 8 3.5 Unit conversion factors 8 3.5.1 Degrees and radians 9 3.5.2 Specifying the unit of angular measure 9
4. FUNDAMENTAL CONCEPTS 10 4.1 Introduction 10 4.2 Reference surfaces 11 4.2.1 Reference ellipsoid 11 4.2.2 Geoid 11 4.2.3 Relationships among topography, the geoid, 11
and the reference ellipsoid 4.3 Earth-fixed coordinate systems 13 4.3.1 Cartesian coordinates (X, Y, Z) 13 4.3.2 Cartesian coordinate system/reference ellipsoid
relationship 13 4.3.2.1 First eccentricity and flattening 15 4.3.2.2 Ellipsoid parameters 15 4.3.3 Geodetic coordinates (<j), X, h) 15 4.3.3.1 Latitude and longitude limits 15 4.3.4 Coordinate conversion 15 4.3.4.1 Geodetic to Cartesian coordinate conversion 16 4.3.4.2 Cartesian to geodetic coordinate conversion 16 4.3.4.2.1 Finding X 16 4.3.4.2.2 Finding <j> 16 4.3.4.2.3 Calculating h 17 4.4 Representation of Geodetic Coordinates 18 4.5 Height Relationships 18 4.5.1 Elevations 18 4.5.2 Geoid separation 18 4.5.3 The relationships among H, h, and N 18 4.5.3.1 Notational use of H and h 18
5. GEODETIC SYSTEMS AND DATUMS 19 5.1 Introduction 19 5.2 Geodetic (horizontal) datums 19 5.2.1 Background 19 5.2.2 Geodetic datums and WGS 84 19 5.3 Vertical datums and elevations 20 5.3.1 Background 20 5.3.2 Vertical datums and mean sea level (MSL) 20 5.3.3 Relationship between local, vertical and horizontal
datums 20 5.4 World Geodetic System 1984 (WGS 84) 20 5.5 WGS 84 referenced elevations 21
6. DATUM SHIFTS 22 6.1 Introduction 22 6.1.1 Shifting between two local datums 22 6.1.1.1 Shifting from NAD 27 to NAD 83 22 6-2 Seven-parameter geometric transformation 23 6.2.1 Transformation to WGS 84 Cartesian coordinates 23
iv
TABLE OF CONTENTS
6.2.2 Transformation to LGS Cartesian coordinates 6.2.3 Parameter Values 6.2.4 Accuracy of seven-parameter transformation
PAGE 24 24 24
24 6.3 Three-parameter (AX, AY, AZ) geometric transformation
6.3.1 Transformation to WGS 84 Cartesian coordinates 25 6.3.1.1 Three-step method: Transformation to 25
WGS 84 geodetic coordinates 6.3.2 Transformation to local geodetic coordinates 6.3.2.1 Three-step method: Transformation to
local geodetic coordinates 6.3.3 Molodensky shifts 6.3.4 AX, AY, AZ shift values
6.3.5 Accuracy of AX, AY, AZ shift parameters 6.3.6 Local datum distortion 6.4 WGS 72 to WGS 84 transformation 6.4.1 Selecting a conversion method 6.4.2 Direct WGS 72 to WGS 84 transformation 6.4.2.1 Conversion equations 6.4.2.2 Estimated errors (0^,0^,0^) 32
6.4.3 Two-step WGS 72 to WGS 84 transformation 32 6.5 Approximating geodetic heights for datum
transformation 32 6.6 Multiple Regression Equations 6.7 Vertical datum shifts 6.7.1 Satellite-derived WGS 72 elevations based
on measured WGS 72 geodetic heights 6.7.2 Survey-derived elevations 6.8 Unlisted datums
26 26
26 28
28 29 29 30 30 30
33 33 33
33 33
7. MAP PROJECTIONS 7.1 Introduction 7.1.1 The map projection process 7.1.2 Properties of projections 7.1.3 Mapping equations 7.1.4 Conformal projection 7.1.5 Scale factor 7.1.6 Map scale 7.1.7 Convergence of the meridian 35 7.1.8 Information sources 35 7.2 Mercator projection 7.2.1 Meridians and parallels 7.2.2 Mercator mapping equations 7.2.2.1 Finding (x,y) 7.2.2.2 Point scale factor and convergence 37
of the meridian 7.2.2.3 Finding (<|),A,) 7.2.3 Accuracy 7.2.4 Area of coverage 7.3 Transverse Mercator (TM) projection
34 34 34 34 34 34 34 35
35 35 36 36
37 38 38 38
TABLE OF CONTENTS
7.3 7.3. 7.3,
7.3,
7.3.
7.3. 7.3.
1 2 2.1
2.2
2.2.
2.2, 2.3
7.3.2.4
7.3.2.5
7.3.2.6
7 .3 .3 7 .3 .4 7 .4 7 .4 .1 7 .4 .1 .1 7 .4 .1 .2 7 .4 .2 7 .4 .3 7 .4 .3, .1 7 .4 .3. .2 7 .4 .3. .3 7 .4 .4 7 .4 .5 7. .5 7 .5, .1 7, .5, .1. .1 7, .5, .1. ,2 7. .5. .2 7. .5. .2. 1 7. .5. .2. 2 7. .5. .3 7. .5. ,4 7. .5. ,5 7. ,6 7. ,6. 1 7. 6. 1. 1 7. 6. 1. 2 7. 6. 1. 3 7. 6. 1. 4 7. 6. 1. 5 7. 6. 2 7. 6. 3 7. 7 7. 7. 1 7. 7. 2
Meridians and parallels Transverse Mercator mapping equations Finding (x,y) Finding ity,X)
1 Finding the footpoint latitude (§x) 2 Finding (ty,X) from tyx
Point scale factor as a function of x and tyx
Point scale factor as a function of «|>, X)
Convergence of the meridian in terms of x and §x
Convergence of the meridian in terms of «t>, X)
Accuracy- Area of coverage Universal Transverse Mercator (UTM) projection UTM zones Finding the UTM zone and central meridian Non-standard width UTM zones Reference source UTM equations Finding UTM coordinates (xUTM, y^) Computing geodetic coordinates Discontinuity Accuracy Area of coverage Lambert Conformal Conic projection Lambert Conformal Conic with two standard parallels Finding (x,y)
Finding (§,X) with two standard parallels Lambert Conformal Conic with one standard parallel Finding (x,y) Finding (<}>,A,) with one standard parallel Scale factor and convergence Accuracy Area of coverage Polar Stereographic projection Polar Stereographic mapping equations Finding (x,y) Finding ((|),A,) Alternate method for finding <|) Finding the point scale factor Finding the convergence of the meridian Accuracy Area of coverage Universal Polar Stereographic (UPS) projection Universal Polar Stereographic (UPS) mapping equations Finding ((j),A,)
PAGE 38 39 39
40
40
41 42
42
42
42
43 43 43 43 44 45 45 45 45 45 46 46 46 46 47 47
48 49 49
50 50 50 50 . 51 51 51
51 53 54 54 54 54 54 55 55
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TABLE OF CONTENTS
n n ^ . . . PAGE 7.7.3 Finding point scale factor and convergence 57
of the meridian 7.7.4 Accuracy 57 7.7.5 Area of coverage 57 7.8 The U.S. Military Grid Reference System (MGRS) 57 7.8.1 MGRS coordinates in the UTM area 57 7.8.2 MGRS coordinates in the UPS area 58 7-9 The World Geographic Reference System (GEOREF) 63 7.10 Non-standard grids 63 7.11 Modification to map projections 63
8. SELECTIONS OF TECHNIQUES, SOFTWARE DEVELOPMENT, 66 AND TESTING
8-1 The general conversion process 66 8.1.1 Application of the general conversion process 70 8.1.2 Procedural examples 70 8.1.2.1 Example 8.1 Horizontal geodetic coordinate-based
datum shifts 70
8.1.2.2 Example 8.2 WGS 84 Cartesian coordinate to local latitude, longitude, and elevation 71
8.1.2.3 Example 8.3 Local latitude, longitude, and elevation to WGS 84 latitude, longitude, and elevation 72
8.2 Choosing a datum shift method 73 8.2.1 Parameter availability 74 8.2.2 Accuracy 74 8.2.3 Implementation issues 75 8.2.3.1 Preferred shift methods 76 8.2.3.2 Software testing 76 8.3 Error Analysis 77 8.3.1 System model 77 8.3.2 Error estimates 78 8.3.2.1 Error types 79 8.3.3 Reformulation 79 8.3.4 Error propagation 80 8.3.5 Example 81 8.4 Numerical examples 85 8.4.1 Example 8.4 Convert WGS 84 geodetic coordinates
to Universal Transverse Mercator coordinates in the NAD 27 reference system using a three-step datum shift 85
8.4.2 Example 8.5 Convert WGS 84 geodetic coordinates to geodetic coordinates in the NAD 27 reference system using the standard Molodensky method 88
8.4.3 Example 8.6 Convert WGS 84 geodetic coordinates to Mercator projection coordinates 90
8.4.4 Example 8.7 Convert Mercator projection
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TABLE OF CONTENTS
PAGE coordinates in the WGS 84 reference system to WGS 84 geodetic latitude and longitude 90
8.4.5 Example 8.8 Convert Universal Transverse Mercator coordinates in the NAD 27 reference system to NAD 27 geodetic latitude and longitude 92
8.4.6 Example 8.9 Convert WGS 84 latitude and longitude to Lambert Conformal coordinates using two standard parallels 94
8.4.7 Example 8.10 Convert Lambert Conformal projection (with two standard parallels) coordinates in the WGS 84 reference system to WGS 84 geodetic latitude and longitude 96
8.4.8 Example 8.11 Convert geodetic latitude and longitude on the International Ellipsoid to Universal Polar Stereographic coordinates 98
8.4.9 Example 8.12 Convert Universal Polar Stereographic coordinates to geodetic latitude and longitude 99
8.5 List of test points
9. NOTES 9.1 Intended use 9.2 Subject term (key word) listing
101
103 103 103
APPENDTPRq PAGE
A. Reference Ellipsoid Parameters 105 B. Ax, AY, AZ Datum Shift Parameters 107 C. Datum List ^.29 D. WGS 84 Geoid Separation Computation 145 E. Seven-parameter Geometric Datum Shifts 149 F. old Hawaiian Datum with International Ellipsoid 151
vixi
LIST OF FIGURES
PAGE
4.1 The relationship between the reference 12 ellipsoid, the geoid, and the physical surface of the earth.
4.2 The geometric relationship between 14 Cartesian and geodetic coordinates.
6.1 Relationship between coordinate axis in a 23 seven-parameter geometric transformation.
6.2 Relationship between coordinate axis in a 25 three-parameter geometric transformation.
7.1 Meridians and parallels in the Mercator projection. 36
7.2 Meridians and parallels in the Transverse Mercator Projection. X0 is the central meridian. 39
7.3 Meridians and parallels (dashed) and 44 a Universal Transverse Mercator Grid.
7.4 Meridians and parallels in the Lambert 47 Conformal Conic Projection.
7.5 Meridians and parallels in the Universal 51 Polar Stereographic Projection.
7.6 Meridians and parallels imposed on a 55 Universal Polar Stereographic grid.
7.7 UTM area grid for MGRS 59
7.8 UTM area grid for MGRS 60
7.9 North UPS area grid for MGRS 61
7.10 South UPS area grid for MGRS 62
8.1 A map projection to geodetic datum 66 conversion. Height information for H and h is not included in the conversion.
8.2 Coordinate conversions within a datum. 67
8.3 Datum shifts. 68
8.4 The general conversion process. 69
8.5 System Model 84
ix
LIST OF FIGURES
PAGE
D.l Coordinate System Associated with Geoid Separation Bi- Linear Interpolation Method 146
x
LIST OF TABLES
Page
3 .1 Conversion Factors 9
6.1 Selection of WGS 72 to WGS 84 Transformation Method 30
7.1 Non-standard Zone Limits
8.1 Accuracy Guidance for Equations
8.2 Probability Level Conversion Factors
A.l Reference Ellipsoids
B.l Ellipsoid Center Shift Transformation Parameters
C.l Countries and their Associated Datums
45
79
81
106
108
130
D.l WGS 84 Geoid Separations 147
E.l Seven Parameter Local Geodetic System to WGS 84 150 Datum Transformation
XI
EXEC! JTTVF, ST JMMAR Y
This document provides Army organizations and agencies with general guidance on selecting the appropriate methods for shifting between local geodetic datums and the World Geodetic System (WGS), and for converting Cartesian and map projection coordinates to and from geodetic coordinates. This guidance is provided to aid the Army community in selecting datum shift algorithms, in developing, selecting, and maintaining software using these algorithms, and in implementing this software to support operational units. Equations are furnished for map projections and datums commonly used within the Army, and references are provided for other, less commonly encountered, map projections and datums.
Beneficial comments (corrections, recommendations, additions, deletions) or information that may be of use in improving this document should be addressed to the Standards Division of the Digital Concepts & Analysis Center (DCAC) at the Topographic Engineering Center (TEC).
Mail: U.S. Army Topographic Engineering Center, ATTN: CETEC-PD-DS, 7701 Telegraph Road, Alexandria, VA 22315-3864.
Fax: ATTN: Standards Division, DCAC (703) 355-2991
Phone: (703) 355-2761
NEW PHONE NUMBERS! voice (703)428-6761 fax (703)428-6991
Xlll
PREFACE
This report was funded under the U.S. Army Topographic Engineering Center's (TEC) Digital Topographic Data Standards Program.
This report was co-authored at TEC by the following individuals:
Frederick Gloeckler Richard Joy Justin Simpson Daniel Specht
The report is based on a draft of the Military Handbook, Datums, Projections, Grids and Common Coordinate Systems, Transformation of (MIL-HDBK-600008), and was revised during the period July 1994 through July 1995. This Technical Report uses material from MIL-HDBK- 600008 co-authored by:
James Ackeret Fred Esch Chris Gard Frederick Gloeckler Daniel Oimoen Juan Perez MAJ Harry Rossander Justin Simpson Joe Watts Tom Witte
This work was performed under the previous supervision of Juan Perez, and Richard A Herrmann and the current supervision of John W. Hale, Chief, Standards Division and Regis J. Orsinger, Director, Digital Concepts and Analysis Center.
Mr. Walter E. Böge was Director and COL Richard G. Johnson was Commander and Deputy Director of TEC at the time of publication of this report.
xv
1. SCOPE
1.1 Scope. This Technical Report provides methods and parameter values to shift positions between approximately 113 geodetic datums and provides methods for converting between geodetic coordinates, Cartesian coordinates and map projection coordinates. Guidance is provided on selecting methods appropriate to the application, and on developing and testing software to implement these conversions.
1.2 Applicability. The methods provided in this Technical Report are used for a wide range of Army mapping, charting, and positioning applications. Equations are furnished for the map projections commonly used within the Army. References are provided for other map projections that occasionally may be encountered. Special high accuracy applications such as engineering, construction, and real estate boundary surveys are outside the scope of this Technical Report. Appendix B provides datum shift parameters for approximately 113 datums that are currently available from the Defense Mapping Agency (DMA). However, many more local datums exist. For guidance on datum shift applications or for datum shift parameters not covered by this Technical Report, contact TEC [address given in the preface].
1.3 Application guidance. Transformation methods and parameters should be appropriate to the application. A system analyst should do a thorough evaluation of the system requirements before developing detailed specifications. This evaluation should consider the intended use, data types, formats, and required accuracy of the output mapping, charting, and geodesy (MC&G) data, as well as the source and accuracy of the input data. Also, any hardware constraints must be considered. The methods presented in this Technical Report can then be examined for how well they can meet or be adapted to meet the requirements. Where more than one method will satisfy a basic requirement, give priority to Army- preferred methods for standardization purposes. Additional application guidance is furnished in Section 8.
2. REFERENCED DOCUMENTS
2.1 Government documents.
2.1.1 Specifications, standards, and handbooks. The following specifications, standards, and handbooks form a part of this document to the extent specified herein.
SPECIFICATIONS. None
STANDARDS.
MIL-STD-2401 Department of Defense World Geodetic System(WGS) (1994)
STANAG 2211 Geodetic Datums, ellipsoids, grids and (Fifth Edition) grid references (1991)
HANDBOOKS. None
2.1.2 Other Government documents, drawings, and publications. The following other Government documents, drawings, and publications form a part of this document to the extent specified herein. Unless otherwise specified, the issues are those cited in the solicitation. The date in parentheses indicates for each document the edition that was used in preparation of this Technical Report.
Document TD/Agenny Title
DMA Department of Defense Glossary of Mapping, Charting, and Geodetic Terms, Fourth Edition. (1981)
DMA TM 8358.1 Datums, Ellipsoids, Grids, and Grid Reference Systems, (1990)
DMA TM 8358.2 The Universal Grids: Universal Transverse Mercator (UTM) and Universal Polar Stereographic (UPS) (1989)
DMA Instruction 8000.1 with change 1, 4 Aug 92
Geodetic and Geophysical Sign Conventions and Fundamental Constants. (1988)
Geological Survey- Professional Paper 1395
NOAA Manual NOS NGS 5
NSWC/DL TR-3 624
Map Projections - A Working Manual. (Snyder, 1987)
State Plane Coordinate System of 1983 (Stem, 1989)
Map Projection Equations. (Pearson, 1977)
U.S. Department of Commerce, Coast and Geodetic Survey, Special Publication No. 251
Conformal Projections in Geodesy and Cartography.' (Thomas, 1979)
Engineer Technical Letter No. 1110-1-147
Federal Register v. 55, no. 155, Friday, August 10, 1990 Docket No. 900655-0165
Engineering and Design Conversion to North American Datum of 1983 (1990)
Notice To Adopt Standard Method for Mathematical Horizontal Datum Transformation (NOAA, 1990)
Copies of; DMA Instruction 8000.1 and GS Professional Paper 1395 are available from USGS, Branch of Distribution, Box 25286, Denver CO 80225. Copies of DoD Glossary of Mapping, Charting, and Geodetic Terms; DMA TM 8358.1; and DMA TM 8358.2 are available from the Director, DMA Combat Support Center, ATTN: CCOR, 6001 MacArthur Boulevard, Bethesda MD 20816-5001. Copies of the NOAA Manual NOS NGS 5 and the Coast and Geodetic Survey Special Publication No. 251 are available from National Geodetic Information Branch, N/CG174, National Geodetic Survey, 1315 East-West Highway, Silver Spring MD 20910. Copies of NSWC/DL TR 3624 (Accession Document Number = AD A037 381 on paper, or GIDEP #E151-2353 on cartridge) are available on from Defense Technical Information Center, Bldg. 5, Cameron Station, Alexandria, VA 22304-6145. Copies of ETL 1110-1-147 are available through the Department of the Army, U.S. Army Corps of Engineers, Washington," D.C. 203140-1000. Copies of STANAG 2211 and MIL-STD 2401 are available from the Defense Printing Service, ATTN: DODSSP, 700 Robbins Ave, Bid 4D Philadelphia PA 19111.
2.2 Non-Government publications.
Heiskanen, Weikko A., and Helmut Moritz; Physical Geodesy; Institute of Physical Geodesy, Technical University, Graz, Austria; 1967.
Krakiwsky, Edward J.; Conformal Map Projections in Geodesy; Department of Surveying Engineering, University of New Brunswick, Fredrickton, N.B., Canada; September, 1973.
Rapp, Richard H.; Geometric Geodesy - Part I; Department of Geodetic Science and Surveying, The Ohio State University, Columbus, Ohio; 1984.
Rapp, Richard H.; Geometric Geodesy - Part II; Department of Geodetic Science and Surveying, The Ohio State University, Columbus, Ohio; 1987.
2.3 Order of precedence. MIL-STD-2401 shall take precedence for all matters concerning WGS 84 and datum shift parameters. Nothing in this document, however, supersedes applicable laws and regulations unless a specific exemption has been obtained.
DEFINITIONS AND UNITS
3.1 Acronyms. The acronyms used in this Technical Report
Bureau International de 1'Heure continental United States Conventional Terrestrial Pole Defense Mapping Agency Department of Defense Digital Terrain Elevation Data earth-centered, earth-fixed Global Positioning System local geodetic system Military Grid Reference System mapping, charting and geodesy multiple regression equation mean sea level North American Datum 1927 North American Datum 1983 National Geodetic Survey National Geodetic Vertical Datum of 1929 National Oceanic and Atmospheric Administration
National Ocean Survey Naval Surface Weapons Center/Dahlgren
Laboratory- Transverse Mercator Universal Polar Stereographic Universal Transverse Mercator World Geodetic System 1972 World Geodetic System 1984
äefi ned as foil
a. BIH b. CONUS c. CTP d. DMA e. DoD f. DTED g- ECEF h. GPS l. LGS 3 • MGRS k. MC&G 1. MRE m. MSL n. NAD 27 o. NAD 83 P. NGS q. NGVD 29 r. NOAA
s. NOS t. NSWC/DL
u. TM v. UPS w. UTM X. WGS 72 y. WGS 84
3.2 Terms.
3.2.1 Convergence of the meridian (y) . The angle between the projection of the meridian at a point on a map and the projection North axis (typically represented by the y axis of the projection). The convergence of the meridian is positive in the clockwise direction.
3-2-2 Coordinate. Linear or angular quantities that designate the position that a point occupies in a given reference frame or system. Also used as a general term to designate the particular kind of reference frame or system, such as Cartesian coordinates or spherical coordinates.
3.2.3 Datum. Any numerical or geometrical quantity or set of such quantities specifying the reference coordinate system used for geodetic control in the calculation of coordinates of points on the earth. Datums may be either global or local in extent. A local datum defines a coordinate system that is used only over a
region of limited extent. . A global datum specifies the center of the reference ellipsoid to be located at the earth's center of mass and defines a coordinate system used for the entire earth.
3.2.3.1 Horizontal datum. A horizontal datum specifies the coordinate system in which latitude and longitude of points are located. The latitude and longitude of an initial point, the azimuth of a line from that point, and the semi-major axis and flattening of the ellipsoid that approximates the surface of the earth in the region of interest define a horizontal datum.
3.2.3.2 Vertical datum. A vertical datum is the surface to which elevations are referred. A local vertical datum is a continuous surface, usually mean sea level, at which elevations are assumed to be zero throughout the area of interest.
3.2.4 Earth-fixed. Stationary with respect to the earth.
3.2.5 Elevation. Vertical distance measured along the local plumb line from a vertical datum, usually mean sea level or the geoid, to a point on the earth. Often used synonymously with orthometric height.
3.2.6 Ellipsoid. The surface generated by an ellipse rotating about one of its axes. Also called ellipsoid, of revolution.
3.2.7 Equator. The line of zero geodetic latitude; the great circle described by the semi-major axis of the reference ellipsoid as it is rotated about the semi-minor axis.
3.2.8 Equipotential surface. A surface with the same potential, usually gravitational potential, at every point; a level surface.
3.2.9 Geocentric Cartesian coordinates. Cartesian coordinates (X, Y, Z) that define the position of a point with respect to the center of mass of the earth.
3.2.10 Geodetic coordinates (geodetic position). The quantities of latitude, longitude, and geodetic height (<|), A, ,h) that define the position of a point on the surface of the earth with respect to the reference ellipsoid. Also, imprecisely, called geographic coordinates (See also geographic coordinates). ,
3.2.11 Geodetic height (ellipsoidal height, h) . The height above the reference ellipsoid, measured along the ellipsoidal normal through the point in question. The geodetic height is positive if the point is outside the ellipsoid.
3-2-12 Geodetic latitude l§) The angle between the plane of the Equator and the normal to the ellipsoid through the computation point. Geodetic latitude is positive north of the equator and negative south of the Equator (See figure 4.2).
3-2;13 Geodetic loncrit-.nriP (1.) The angle between the plane of a meridian and the plane of the prime meridian. A longitude can be measured from the angle formed between the local and prime meridians at the pole of rotation of the reference ellipsoid, or by the arc along the Equator intercepted by these meridians. See figure 4.2.
3-2-14 Geographic coordinates. The quantities of latitude and longitude that define the position of a point on the surface of the earth. See also geodetic coordinates.
3.2.15 Geoid- The equipotential surface of the earth's gravity field approximated by undisturbed mean sea level of the oceans. The direction of gravity passing through a given point on the geoid is perpendicular -to this equipotential surface.
3-2-16 Geoid separation (N) The distance between the geoid and the mathematical reference ellipsoid as measured along the ellipsoidal normal. This distance is positive outside, or negative inside, the reference ellipsoid. Also called geoidal height; undulation of the geoid. See figure 4.1.
3-2-17 Grid reference svsfpm A plane-rectangular coordinate system usually based on, and mathematically adjusted to, a map projection in order that geodetic positions (latitudes and longitudes) may be readily transformed into plane coordinates and the computations relating to them may be made by the ordinary methods of plane surveying (see also 3.2.21).
3-2-18 Nap projection. A function relating coordinates of points on a curved surface (usually an ellipsoid or sphere) to coordinates of points on a plane. A map projection may be established by analytical computation or, less commonly, may be constructed geometrically.
3.2.19 Map scale. The ratio between a distance on a map and the corresponding actual distance on the earth's surface.
3.2.20 Meridian. A north-south reference line, particularly a great circle through the geographical poles of the earth, from which longitudes and azimuths are determined; or the intersection o± a plane forming a great circle that contains both geographic poles of the earth, and the ellipsoid.
3-2-21 Military Grid Rcfprence System (MfiRfn. MGRS is a system developed by DMA for usage with UTM and UPS projections.
3.2.22 Orthometric height. See elevation.
3.2.23 Parallel. A line on the earth, or a representation thereof, that represents the same latitude at every point.
3.2.24 Prime (initial) meridian. A meridian from which the longitudes of all other meridians are reckoned. This meridian, of longitude 0°, was traditionally chosen to pass through the Greenwich Observatory in Greenwich, England. For new refined coordinate systems, the location of the prime meridian is defined by the International earth Rotation Service, Paris, France.
3.2.25 Reference ellipsoid. An ellipsoid whose dimensions closely approach the dimensions of the geoid; the exact dimensions are determined by various considerations of the section of the earth's surface concerned. Usually a bi-axial ellipsoid of revolution.
3.2.26 Scale factor (projection). A multiplier for reducing a distance in a map projection to the actual distance on the chosen reference ellipsoid.
3.3 Units. In this Technical Report all distances are expressed in meters. Angles are expressed in degrees, radians or the combination of degrees, minutes, and seconds.
3.4 Sign conventions. The sign conventions used in this Technical Report are defined as follows:
a. Elevation. Positive outside or above the vertical datum, negative inside or below the datum.
b. Geodetic height. Positive outside the reference ellipsoid, negative inside the reference ellipsoid.
c. Geodetic latitude. Positive in the northern hemisphere, negative in the southern hemisphere.
d. Geodetic longitude. Zero at the prime meridian, positive 0° to 180°, eastward from the prime meridian, and negative 0° to 180°, westward from the prime meridian.
e. Geoid separation. Positive outside the reference ellipsoid, negative inside the reference ellipsoid.
f. Cartesian z. Positive in the northern hemisphere.
3.5 Unit conversion factors. The standard unit of distance measurement is the meter. However, MC&G products in many parts of the world were produced using other units of measurement. The user must ascertain the units used in non-standard products and appropriately scale them for use in the equations given in this
technical report. Some commonly used conversion factors are listed in Table 3.1.
TABLE 3.1 Conversion Factors.
1 meter
1 International foot 1 U.S. Survey foot
1 Int'1 Nautical mile
1 Int'l Statute mile
1 degree 1 degree 1 minute 6400 U.S. mils
3.28083333333 U.S. Survey feet 3.28083989501 International feet 0.3048 meter (Exact) 1200/3937 meter (Exact) 0.30480060960 meter 1852 meters (Exact) 6076.10333333 U.S. Survey feet 6076.11548556 International feet 1609.344 meters (Exact) 5280 International feet (Exact)
60 minutes 3600 seconds 60 seconds 360 degrees
3-5.1 Degrees and radians, A circle contains 360 degrees ox 27C radians. If an angle contains d degrees or, equivalently, r radians, then d and r are related by
r = 7Cd 180 (3.1)
3.5.2 Specifying the unit of angular msaanrp. in this Technical Report the unit of angular measure will be specified if it matters. A general rule of thumb is that angles must be in radians in an equation that was derived using calculus and trigonometry. Radians are the usual units in computer evaluation of trigonometric functions.
4. FUNDAMENTAL CONCEPTS
4.1 Introduction. An understanding of geodesy helps one understand how to represent the locations of points on or about the surface of the earth. Terms printed in boldface are defined in Section 3.2.
a. The location of a point is described by a set of coordinates. Some types of coordinates, such as Cartesian (see Section 4.3.1), uniquely identify three-dimensional locations of points in space. Their use is computationally efficient in many applications. However, users often have difficulty visualizing the relationship of Cartesian coordinates to the surface of the earth. Geodetic coordinates (see Section 4.3.3) make visualization easier by providing horizontal coordinates on the surface of a figure approximating the shape of the earth and a vertical (height) coordinate above or below the reference surface.
b. Geodetic coordinates are defined by a geodetic reference system. A geodetic reference system includes, but is not limited to, a reference surface (usually an ellipsoid) and a set of parameters positioning and orienting the reference surface with respect to the earth. Over the years, many geodetic reference systems have been developed for various localities. These systems are often called local datums and are not rigorously tied to each other.
c. When applications require that two or more points be located in a common reference system, transformation between two coordinate systems may be needed. For example: both weapon and target must be located in the same reference system if the indirect fire computations are to be valid. As the ranges of weapons and target acquisition systems increased and positional data were derived from multiple sources, commanders often were confronted with the problem of using positions on more than one reference system. With the advent of satellite geodesy one can establish global geodetic reference systems and take measurements that allow local datums to be related to the global system. The standard global reference system for DoD is the World Geodetic System 1984 (WGS 84). The process of relating one reference system to another is called a datum shift, or datum transformation.
10
d. Coordinates generally are represented alpha- numerically (e.g. latitude 38° 51' 22.45"N and longitude 77° 26' 34.56"W). Position data for an area often are represented graphically, as on a map. The planar representation on a map provides a distorted view of features on the curved earth's surface. Geodesists have developed various mathematical methods to minimize certain types of map distortions, depending on the intended use of the map. The process of relating coordinates of points on a curved surface (usually an ellipsoid or sphere) to coordinates of points on a plane is called map projection. The preferred map projections for DoD are the Universal Transverse Mercator (UTM), used over most of the landmass, and Universal Polar Stereographic (UPS), used in the polar regions.
4-2 Reference snrfanss. Reference surfaces are used as the bases for geodetic coordinate systems and elevations.
4.2.1 Reference ellipsoid, A reference surface called the reference ellipsoid (mathematically described in Section 4.3.2) is used for defining geodetic coordinates (Section 4.3.3) because it is regularly shaped, is mathematically tractable, and can approximate the surface of the earth. Since the earth is flattened at the poles, an oblate ellipsoid fits the shape of the earth better than does a sphere. An ellipsoid of revolution is the reference surface most commonly used in geodesy and is used throughout this Technical Report. A geodetic coordinate has a simply defined relationship with the reference ellipsoid and any other geodetic coordinate because of the regularity and relative simplicity of the reference ellipsoid's shape. The parameters for the commonly used reference ellipsoids are listed in Appendix A. A reference ellipsoid is often chosen to fit the geoid.
4.2.2 Geoid. The geoid is a closed surface of constant gravity, potential approximated by mean sea level (MSL) and the theoretical extension of MSL through land areas. For purposes of this technical report, the geoid is considered a reference surface from which elevations are measured. A reference ellipsoid may fit the geoid; for example, the maximum difference between the WGS 84 geoid model and the WGS 84 ellipsoid found in Appendix D, Table D.l is 102 meters.
4.2.3 Relationships among tonoaranhv. the aeoiri. and the reference ellipsoid. The relationships among topography, the geoid, and the reference ellipsoid are illustrated in figure 4.1.
11
HÄ (0 *— ü
-H AJ CO Xi >i O) Ä -H n. 0)
X! (1) Ä O 4-) •H .
XJ T> CD c-o (0 O
CD *. fil •d •H *. o .*-* 0) K 0) *-* (1) Ö Ä o 4J -H
.D «. m •o > -H a) O <H CQ CD D.
-H M-l rH o rH d) a
•H (ü Xi U CQ Ö a <ü o k ■H (1) 4->
IW <Ö CD iH r( CD
u 0) Xi CD 4J 4J
o • es- Q) 25 CD S • 4JÄ Ö CD ■D o XI rt-H
m .u a CD m
-H M Ä CD (0 toi a Ö 4-> CD O CQ
-H IW 4J OT5 <Ö •H
i-H CD O CD U CD M <Ö o>
U-) (U fc "O Xi 3 Ö E-< CQ (0
0) u Öl
-H tu
12
4.3 Earth-fixed coordinate Systems. There are many coordinate systems used to represent positions on or near the earth. The coordinate systems used in this Technical Report are all earth-fixed. A coordinate system is earth-fixed if the axes are stationary with respect to the rotating earth. Two primary, earth-fixed coordinate systems, Cartesian and geodetic, are discussed in Sections 4.3.1 and 4.3.3, respectively. Conversions from one to the other are presented in Section 4.3.4.
4.3.1 Cartesian coordinates (X. Y. 7,). The Cartesian- coordinate system used in this Technical Report is a right-hand, rectangular, three-dimensional (X, Y, Z) coordinate system with an origin at (0,0,0). When the origin is also at the mass center of the earth, the coordinate system is said to be geocentric.
4.3.2 Cartesian coordinate system/reference ellinso-id relationship. The relationship of the Cartesian coordinate system to the reference ellipsoid is shown in figure 4.2. The Z-axis, the axis of rotation (semi-minor axis) of the ellipsoid, is nearly parallel to the axis of rotation of the earth. The Z-coordinate is positive toward the North pole. The X-Y plane lies in the equatorial plane, the plane swept out by the semi-major axis as the ellipse is rotated. The X-axis lies along the intersection of the plane containing the prime (initial) meridian and the equatorial plane. The X-coordinate is positive toward the intersection of the prime meridian and equator. The reference ellipsoid satisfies the equation
X2 Y2 Z2
Here, a and b,. respectively, denote the semi-major and semi-minor axes of the ellipsoid. This surface can be generated by rotating an ellipse defined by
X2 Z2
Iä + bäsl (4-2)
about the Z-axis. Ideally, for global applications, the Z-axis coincides with the mean axis of rotation of the earth and the origin is the center of mass of the earth. Often, in practice, an ellipsoidal surface is fitted to a local area so that the origin and axes vary slightly from ideal coincidence.
13
Reference Ellipsoid \
P (X,Y,Z) ((|>Xh)
Figure 4.2 The geometric relationship between Cartesian and geodetic coordinates.
14
4.3.2.1 First eccentricity and flattening. Two important constants are the (first) eccentricity £ and flattening f that are related to a and b by
a2 - b2
e2 = —T"^- (4.3) a2
a - b a
Frequently a and f are given, rather than a and b, as the defining parameters of a reference ellipsoid
4.3.2.2 Ellinsoid parameters. Appendix A contains a list of reference ellipsoids and their parameters. These can be divided into two categories: those, such as the Bessel Ellipsoid, that are used to approximate the shape of the earth for a local area and those, such as the WGS 84 Ellipsoid, that are used to yield a global approximation to the shape of the earth.
4.3.3 Geodetic coordinates (()), X, h) . Geodetic coordinates consist of geodetic latitude (<()), geodetic longitude (A,) , and geodetic height (h). Their relationship to the reference ellipsoid is shown in figure 4.2. For an ellipsoid that satisfies equation 4.1, the normal SP intersects the ellipsoid at Q. The angle between the normal SP and the equatorial (X-Y) plane is called the geodetic latitude ((|)) of point P. The meridian plane containing point P is defined as the half-plane containing the Z- axis and point P. The angle between the prime meridian (X-Z) plane and the meridian containing point P is the geodetic longitude {X) of point P. Geodetic longitude is not defined when P lies on the Z-axis. The distance from Q to P is called the geodetic height (h) .
4.3.3.1 Latitude and longitude limits.
Latitude limits: -90° to +90° -90° at the South Pole
0° at the Equator +90° at the North Pole
Longitude limits: 0° at the Prime meridian 180°W (-180°) to 180°E (+180°)
4.3.4 Coordinate conversion. Two basic earth-fixed coordinate systems, geocentric Cartesian coordinates and geodetic coordinates, have been introduced in Sections 4.3.1 and 4.3.3. The following section explains how coordinates in one system can
15
be transformed to the other. In Section 6, transformations between coordinates on two different reference ellipsoids are discussed. Map projections are treated in Section 7 and the general conversion process is summarized in Section 8.1.
4.3.4.1 fteottet-.ic: to Cartesian coordinate conversion. If geodetic coordinates (§, X, h) are known, then the Cartesian coordinates are given by
X = (RN + h) cos § cos X (4.4)
Y = (RN + h) cos <|) sin X
Z = (i2 RN + hJ Sln ^
where RN, the radius of curvature in the prime vertical, is given
by
RN = a2 — (4.5)
ya2 cos2 0 + b2 sin2 <})
Vl - £2 sin2 §
See Figure 4.2. Equation sets 4.4 and 4.5 are exact. Calculating h from elevation (H) is discussed on section 6.5.
4.3.4.2 Cartesian to geodetic coordinate conversion.
4.3.4.2.1 Finding X. In the reverse case, when (X, Y, Z) are known but (<j) ,X, h) are unknown,
for the longitude (X) is given bv Y
X * 0,all Y, X = arctan — (4.6)
X = 0, Y > 0, X = 90° E (+90°) X = 0, Y < 0, X = 90° W (-90° or 270° E) X = 0, Y=0,X,is undefined
Equation set 4.6 is exact. Take care to ensure that the evaluation of the arctan function yields a value of X lying in the desired quadrant.
4.3.4.2.2 Finding 0. When (X, Y) is not (0, 0), the general equations for <j> and h are more involved. The method presented for finding 0 is an iterative procedure of Bowring. Following Rapp (1984, pp. 123-124), one finds an initial approximation to a variable ß given by
16
3. Z
bVx2 + Y2
Once ß0 is obtained, substitute it for ß in the equation
, Z + £12 b sin3 ß tan (p = —, : c
Vx2 + Y2 - a e2 cos3 ß (4.8)
Here £2 and 6'2 are given by
£'2 =
a2_j b2
a? a2 - b2 £2
- a." - w £2 = — = 2f - f2 (4.9)
b2 "1-62
This approximation to (J) is substituted into
tan ß = (1 - f) tan (]) (4.10)
to give an updated approximation to ß. This procedure, of using the latest approximation of ß (equation 4.10) to produce an updated approximation of 0 (equation 4.8), and then using this new value of 0 to update ß, can be continued until the updated value of (J)'is sufficiently close to the previous value. For many purposes, one iteration is sufficient when using the Bowring method, so that § can be found directly from equation 4.8, using the original value of ß0. Rapp (1984, p. 124) states that for terrestrial applications, one such iteration yields values accurate to within 0.1 millimeter. In the polar case (X = 0, Y = 0, Z), the latitude (<j)) is defined to be +90° (90°N) for positive Z values and -90° (90°S) for negative Z values.
4.3.4.2.3 Calculating h. Once 0 has been calculated, in non-polar areas h can be found from
Vx2 + Y2 h = 1~ " RN (4.11) cos (p N .
In polar regions it is preferable to use
h = Titty" RN + £2RN (4.12)
where RN is defined in equation 4.5.
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Note: For many applications h is not required and need not be calculated. An exception to this statement is the computation of height or elevation for the Global Positioning System (GPS).
4.4 Representation of Geodetjr Coordinates Latitude is given before longitude. Latitude and longitude usually are numerically represented as: decimal degrees (e.g. 44.44° N); integer degrees and decimal minutes (e.g.44° 44.44'S); or integer degrees, integer minutes and decimal seconds (e.g. 44° 44' 44.44" E). For clarity, numeric fields representing degrees, minutes and seconds should be ended by the appropriate symbol, respectively " °."' " ' ", and " " ". Numeric fields with a value of zero should be indicated by a "00", and not left blank. The hemisphere of both latitude and longitude must be shown, preferably by placing an "N" or "S" at the end of numeric latitude and an "E" or "W" after numeric longitude.
4.5 Height Relationships.
4.5.1 Elevations. Throughout this Technical Report, the term elevation (H) is used to denote the distance of a point above the geoid or vertical reference surface as measured along the plumb line. A plumb line follows the direction of gravity and is perpendicular to all eguipotential surfaces of the Earth's gravity field that intersect it. Points lying outside the geoid are defined as having positive elevation. For the purposes of this Techncial Report, elevation and orthometric height are considered equivalent. In practice, the reference surface for measuring elevations may not exactly coincide with the geoid.
4.5.2 Geoid separation. Geoid separation (N) is the distance from a reference ellipsoid to the geoid, measured along the ellipsoidal normal. This is equivalent to the geodetic height (see Section 4.3.3) of a point on the geoid. Geoid separation is positive when the geoid lies outside the ellipsoid.
4.5.3 The relationships among H. h. and N. Geodetic height (h), geoid separation (N), and elevation (H) are depicted in figure 4.1 and are related by equation 4.13.
h « H + N (4.13)
4.5.3.1 Notational use of H and h. This Techncial Report uses the classical notation of H for elevation and h for geodetic height.
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5. GEODETIC SYSTEMS AND DATUMS
5.1 Introduction. A geodetic system serves as the framework for determining coordinates of points with respect to the earth. Modern global geodetic systems, such as the World Geodetic System 1984 (WGS 84)., (see Section 5.4), have been established using techniques of satellite geodesy. Typically, they are defined by a geocentric Cartesian coordinate system with the Z-axis along the mean rotation axis of the earth and the X- axis through a longitude reference.
a. DoD World Geodetic Systems have an associated reference ellipsoid and a geoid model. The geoid model is sometimes used as a reference surface for elevations. World Geodetic Systems provide a consistent framework for determining geodetic positions.
b. Local geodetic systems are established by a variety of techniques. Typically, they consist of a geodetic datum (see Section 5.2) for horizontal control and a vertical datum (see Section 5.3) for elevations. Any given local geodetic system covers only a small fraction of the earth's surface. Local geodetic datums and vertical datums generally are independent of each other.
5-2 Geodetic (horizontal) dat.nma. A geodetic datum has a earth-fixed reference ellipsoid that may have been fit, in some manner, to the surface of the earth in the area of interest. Geodetic datums have parameters that define the size and shape of the ellipsoid.
5.2.1 Background. Historically, local geodetic datums have an origin, on the surface of the earth, that relates the geodetic coordinate system to the ellipsoid. For these local datums, the ellipsoid semi-minor axis generally does not coincide with the earth's mean rotation axis. However, the North American Datum 1983 (NAD 83) was developed using the same satellite geodesy techniques used for global geodetic systems. This modern datum has an origin at the center of the reference ellipsoid, and the semi-minor axis coincides with the mean rotation axis of the earth. Without reference to the word vertical, the word datum, as used in this Technical Report, means geodetic (horizontal) datum.
5.2.2 Geodetic datums and wns fi4. Over the years, hundreds of geodetic datums have been developed for various locations. Currently, 113 local geodetic datums have been tied to WGS 84. Coordinate transformations from local datums to WGS 84 are discussed in Section 6.
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5.3 Vertical datums and elevations. By definition, a vertical datum is a surface of zero elevation. Elevations are measured (positive upward) from the vertical datum. Ideally, a vertical datum would closely approximate the geoid.
5.3.1 Background. As a practical matter, it is impossible to access the geoid surface directly for use as a vertical datum. Historically, tide gage measurements were averaged over many years to establish the local mean sea level (MSL) references for vertical datums. The National, Geodetic Vertical Datum of 1929 (NGVD 29), which is based on tide gage measurements and precise leveling surveys, is estimated to be within a few meters of the geoid. There is greater uncertainty in the relationships between the geoid and other local vertical datums.
5.3.2 Vertical datums and mean sea level (MSL). The exact relationships between different vertical datums are unknown. Therefore, when shifting datums, all elevations are considered referenced to MSL regardless of the vertical datum used to establish them. For Army applications, two exceptions to this rule are:
a. Conversion of elevations derived from WGS 72 geodetic height measurements to WGS 84 coordinates (see Section 6.7.1) .
b. Civil works, engineering and construction applications are outside the scope of this Technical Report.
5.3.3 Relationship between local vertical and horizontal datums. Generally, vertical datums and horizontal datums are independently defined. For example, NGVD 29 is the major vertical datum, while the North American Datum 1927 (NAD 27) and NAD 83 are the major horizontal datums used in North America.
5.4 World Geodetic System 1984 (WGS 84). A world geodetic system provides the basic reference frame, geometric figure and gravimetric model for the earth, and provides the means for relating positions on various local geodetic systems to an earth- centered, earth-fixed (ECEF) coordinate system.
a. WGS 84 is the current standard DoD geodetic system. WGS 84 is the latest in a series of DoD-developed world geodetic systems. It is a replacement for WGS 72, the previous DoD standard ECEF world geodetic system.
b. The origin of the WGS 84 coordinate system is the center of mass of the earth. The WGS 84 Z-axis is parallel to the direction of the Conventional Terrestrial Pole (CTP), as defined by the Bureau international de l'Heure (BIH). The WGS 84 prime meridian is parallel to the BIH Zero meridian. The X-axis is the intersection of the plane of the WGS 84 reference meridian and the CTP equatorial
20
plane. The epoch is 1984.0. The Y-axis completes a right-handed, earth-fixed Cartesian coordinate system.
c. The WGS 84 reference ellipsoid is a geocentric ellipsoid of revolution. The WGS 84 coordinate system origin and axes also serve as the geometric center and the X, Y, and Z axes of the WGS 84 ellipsoid. WGS 84 ellipsoid parameters are given in Appendix A.
d. The WGS 84 geoid model can be used to convert between elevations and geodetic heights (see Section 5.5).
e. Datum shift techniques and parameters needed to relate positions on many local datums to WGS 84 were also developed as part of WGS 84 (see Section 6 and Appendices B and E).
f. Only those portions of WGS 84 related to the purpose of this Technical Report were described above. A comprehensive view of WGS 84 can be obtained from DMA TRs 8350.2 (1991), 8350.2-A (1987), and 8350.2-B (1987).
5.5 WGS 84 referenced elevations. Sometimes elevations must be generated from geodetic heights, without reference to a local vertical datum. For example, satellite positioning systems, like the Global Positioning System (GPS), establish coordinates geometrically and cannot directly measure elevations. An elevation can be obtained from a WGS 84 geodetic height using:
"■ ~ ^WGS 84 ~ NWGS 84- (5.1)
where NWGS 84 is the value of WGS 84 geoid separation for the WGS 84 geodetic position of the point. Appendix D contains a method for interpolating NWGS 84 from a table of grided values. Table D.l is a list of NWGS 84 on a 10° x 10° grid. It is useful for moderate accuracy mapping and charting applications. TEC will furnish qualified users with either digital 30' x 30' or 1° x 1° NWGS 84 tables or spherical harmonic expansion tables for survey and high accuracy applications. Requests should be sent to the address in the preface.
21
6. DATUM SHIFTS
6.1 Introduction. Several methods are available to shift coordinates from one geodetic datum to another. To develop these shifts, coordinates of one or more physical locations must be known on both datums. Typically, for the local datum to WGS 84 shifts presented in this Technical Report, the WGS 84 coordinates of local datum survey control points were determined using Doppler satellite observations. Two classes of datum shift methods are geometric transformation and polynomial fitting techniques. Only geometric transformations are discussed here.
a. The generalized geometric transformation model assumes the origins of the two coordinate systems are offset from each other; the axes are not parallel; and there is a scale difference between the two datums. This seven- parameter model is discussed in Section 6.2. Data from at least three well-spaced positions are needed to derive a seven-parameter geometric transformation.
b. The number of parameters in a geometric transformation model can be reduced. The commonly used three- parameter model neglects rotations between coordinate systems axes and scale differences between datums. Only the origins are offset. The three-parameter model is discussed in Section 6.3. Data from one or more positions are required to derive a three-parameter geometric transformation.
c. The WGS 72 to WGS 84 transformation, given in Section 6.4, is a special five-parameter geometric transformation. When WGS 84 superseded WGS 72 as the DoD-preferred reference system, this transformation was created specifically to transform WGS 72 coordinates to WGS 84 coordinates.
d. Heights used for datum shifting are treated in section 6.5. Shifting of vertical datums is discussed in Section 6.7.
6.1.1 Shifting between two local datums. The datum shift methods given in this Technical Report are for shifting from a local datum to WGS 84 and from WGS 84 to a local datum. When shifts are needed between two local datums, shift the coordinates on the first local datum to WGS 84, then shift the WGS 84 coordinates to the second local datum.
6.1.1.1 Shifting from NAD 27 to NAD 83. The software package NADCON has been recommended as the standard method for transformations between the North American Datum of 1927 (NAD27) and the North American Datum of 1983 (NAD 83) by The
22
Federal Geodetic Control Committee (Federal Register.) Corpscon, a package incorporating NADCON, is mandated for the Army Corps of Engineers (ETL 1110-1-147.) Because NADCON is a survey tool it is beyond the scope of this Technical Report. Questions can be addressed to the Survey Division of TEC at the address in the preface, (703)355-2798, or to the National Geodetic Survey at (301)713-3178.
6.2 Seven-parameter geometric transformation. The seven- parameter transformation, the most general transformation between local and global Cartesian coordinates, is considered in DMA TR 8350.2-A (1987). The underlying assumption for this seven- parameter model is that the local and WGS 84 Cartesian axes differ by seven parameters: three rotational parameters (GO, E, \j/), a
scale change AS, and three origin shift parameters (AX, AY, AZ) . See Figure 6.1.
JWGS 84
LGS
WGS 84
X» GS 84
FIGURE 6.1 Relationship between coordinate axes in a seven-parameter geometric transformation.
6-2.1 Transformation to WGS 84 Partesian coordinates the seven-parameter model, the local and WGS 84 Cartesian coordinates are related by
For
^WGS 84 ~ ^LGS
*WGS 84 = YLGS
JWGS 84
+ AX + COY
+ AY LGS
LGS
VZLGS
JLGS cox^o + ezIP;c + ASY = ZLGS + AZ + \|AXLGS - 8Y LGS
+ ASX
+ AS^
+ ASZ
LGS
LGS
LGS
(6.1:
23
The subscript LGS designates Local Geodetic System coordinates and the subscript WGS 84 designates World Geodetic System 1984 coordinates.
6.2.2 Transformation to LGS Cartesian coordinates. The equations for converting WGS 84 geocentric Cartesian coordinates to local Cartesian coordinates are
^LGS = XWGS 84 - AX - COYWGS 84 + V^WGS 84 ~ ASXWGS 84 (6.2)
YLGS = ^WGS 84 ~ AY + COXWGS 84 - £ ZWGS 84 - ASYWGS 84
^LGS = ^"WGS 84 ~ AZ - \|/XWGS 84 + E YWGS 34 _ ASZWGS Bi
6.2.3 Parameter values. The parameters AX, AY, AZ, CO, £,
\|/, and AS, for the European 1950 and Ordinance Survey of Great Britain 1936 datums are tabulated in Appendix E. Use of seven- parameter datum shifts for some applications in Europe and Great Britain is proscribed by STANAG 2211. Since the values of the rotational parameters CO, £, and \J/are given in the table in arc seconds, these values must be converted to radians before using equation sets 6.1 and 6.2.
6.2.4 Accuracy of seven-parameter transformation. The RMS differences between WGS 84 surveyed and transformed coordinates using the seven-parameter datum transformations are given in Appendix E. For some datums, the accuracies for the seven- parameter transformation are not an improvement over those for the three-parameter transformation. Equation sets 6.1 and 6.2 provide good approximations to a rigorous orthogonal transformation when CO, E, \J/, and As are small.
6.3 Three-parameter(AX. AY. AZ) geometric transformation. Frequently, it is not practical to determine the parameters for a seven-parameter geometric datum shift or implement such a shift method. Often, only an origin shift, three-parameter model is used as shown in figure 6.2. The three-parameter datum transformation satisfies the requirements of most mapping and charting applications. The three-parameter model can be applied directly to Cartesian coordinates (see Sections 6.3.1 and 6.3.2) or used to compute shifts in geodetic coordinates via the Molodensky equations (see Section 6.3.3)
a. The datum shift parameters (ÄX, AY, AZ) are the coordinates of the origin of the local reference ellipsoid in the WGS 84 Cartesian coordinate system.
b. Datum shift parameters for 113 local geodetic systems are listed in Appendix B, Table B.l. The area covered by the datum shift parameters, the local datum ellipsoid, and
24
the estimates of the errors in the datum shift parameters also are given. Derivation of ÄX, ÄY, and ÄZ datum shift parameters is discussed in DMA TR 8350.2-A (1987).
LGS
WGS 84
■WGS 84
FIGURE 6.2 Relationship between coordinate axes in a three-parameter geometric transformation.
6.3.1 Transformation to WGS 84 Cartesian coordinates. In
certain cases, AX, AY, and AZ datum shift parameters can be applied directly to convert local geodetic system coordinates to WGS 84 Cartesian coordinates as follows:
^WGS 84 = ^LGS + AX
^WGS 84 = ^LGS + AY
^WGS 84 = ^LGS + AZ
(6.3)
Equation set 6.3 is a subset of equation set 6.1 with (0, \|f, £, and AS set to zero.
6.3.1.1 Three-step method: Transformation to WGS 84 geodetic; coordinates. Local geodetic coordinates can be shifted to WGS 84 coordinates in three steps:
a. Convert local geodetic coordinates to local geocentric Cartesian coordinates using equation set 4.4. Ensure that local ellipsoid parameters are used.
b. Shift the local geocentric Cartesian coordinates to WGS 84 geocentric Cartesian coordinates using equation set 6.3.
25
c. Convert WGS 84 geocentric Cartesian coordinates to WGS 84 geodetic coordinates using the methods discussed in Section 4.3.4.2. Ensure that WGS 1984 ellipsoid parameters are used.
6.3.2 Transformation to local gaodetic coordinates. To convert WGS 84 Cartesian coordinates to the local geodetic system, apply AX, AY, and AZ datum shifts as follows
XLQS = XWGS 84 - AX (6.4)
*LGS = ^WGS 84 ~ Ax
ZLGS = ^WGS 84 " ^Z
Equation set 6.4 is a subset of equation set 6.2 with CO, \|/, £, and
AS set to zero.
6.3.2.1 Three-sten method: Transformation to local geodetic coordinates. To shift WGS 84 geodetic coordinates to local geodetic coordinates
a. Convert WGS 84 geodetic coordinates to WGS 84 geocentric Cartesian coordinates using equation set 4.4. Insure that WGS 84 ellipsoid parameters are used.
b. Shift the WGS 84 geocentric Cartesian coordinates to local geocentric Cartesian coordinates using equation set 6.4.
c. Convert the local geocentric Cartesian coordinates to local geodetic coordinates using a method of Section 4.3.4.2. Insure that local ellipsoid parameters are used.
6.3.3 Molodensky shifts. The standard Molodensky method is an approximation to the three-step transformation methods of Sections 6.3.1.1 and 6.3.2.1. To use the Molodensky methods to transform local geodetic coordinates to WGS 84 geodetic coordinates, calculate A0, AX, and Ah shifts using the standard Molodensky formulas Rapp, 1987.
A<J> = [- sin <|> cos A, AX - sin $ sin A, AY + cos <|) AZ (6.5)
£2 sin 0 cos 0 + W
AX =
+ sin(j)cos(j)(2N + £'2 M sin2 ())) (l-f)Af]/[M + h]
[-sin X AX + cos X AY]
[ (N + h)cos <|>]
26
Ah = cos (() cos A, AX + cos <]) sin A, AY + sin (j) AZ
- W Aa + — sin2 (j) (Af)
in which
w2 = 1 - £2 sin2
M = a(l - ■ e2)
W3
N = a W
(j, =
X = <I>LGS
^LGS
E2 = a2 -
0
b2
2
a2 - b2 e2 £'2 =
b2 " i - e2
The units of A0 and AX are radians. Note that N = RN of equation set 4.5.
Note: Most applications do not require shifting geodetic heights, so Ah does not need to be computed or applied. Geodetic heights, which are used to compute datum shift parameters, are a special construct that may not be well related to the local geodetic system. Do not use Ah to shift local geodetic heights. If this is a requirement, seek competent geodetic council.
a. Apply the A(j), AX, and Ah shifts as follows:
^WGS 84 = <|>LGS + A(t>LGS (6.6)
^WGS 84 = ^LGS + A^LGS
■h«GS 84 = •"■LGS + ^-LGS
where A(j)LGS, AX,LGS, AhLGS are computed from equation set
6.5 using <])LGS, XhGS, hLGS, semi-major axis and flattening
for the local geodetic system. AX, AY, and AZ are found
in Appendix B. Aa and Af are calculated as follows;
27
Aa - aWGS 84 a^s
Af = £wGS 84 ~ -"-LGS
Note: See section 6.5 for use of h LGS.
b. The Molodensky formulas can also be used to convert from WGS 84 to the local geodetic system:
<t>LGS = ^WGS 84 + A(1>WGS 84 < 6 • 7 >
^LGS = ^WGS 84 + ^^WGS 84
•"■LGS = "WGS 84 + ^'■WGS 84
where A(|)WGS 84, A\,GS 84, and AhWGS 84 are computed using 0WGS 84,
^WGS 84' nwGs 84, WGS 84 semi-major axis and flattening, and the signs of AX, ÄY, and ÄZ, as given in Appendix B, are
reversed. Aa and Af are calculated as follows
Aa = aLGS - aWGS 84
Af = £LGS ~~ ^WGS 84
Note that ellipsoid parameters a and f are found in Table A.l, Appendix A.
6.3.4 AX. AY. and AZ shift values. Appendix B (Table B.l)
contains the AX, AY, and AZ values for the three-parameter methods of converting local geodetic systems to WGS 84. The accuracies, in terms of error estimates for AX, AY, and AZ, are given in Appendix B. A further discussion of the accuracy of these parameter shifts will be given in Section 6.3.5.
6.3.5 Accuracy of AX. AY, and AZ shift parameters. Error
estimates for AX, AY, and ÄZ are tabulated in Appendix B. These estimates include the errors associated with the Doppler stations as well as the residual differences between transformed local system coordinates and the reference WGS 84 coordinates used to develop the datum shifts. (A Doppler station refers to a position where the WGS 84 coordinates were obtained by observing and processing of TRANSIT satellite data.)
a. For 38 datums, datum shifts were developed using a single Doppler station. For these datums, no measure of internal consistency of the datum shifts is calculable, and DMA has assigned an accuracy value of 25 meters in each coordinate. These error statistics do not reflect
28
any errors in the coordinates used to compute the datum shift.
b. Datum shifts for eight datums were calculated from sources other than Doppler stations for WGS 84 geodetic coordinates. No error statistics are provided for these datums.
c. Geodetic coordinate difference error estimates can be approximated from rectangular coordinate errors using the following spherical relationships:
0>A<t> = "V (°Ax sin0 cosA,)2 + (CAY sin(|) sinA,)2 + (GAz cos <)))2
G^x = -\j ( O^ sinA,)2 + (CAY COSA,)2 (6.8)
°Ah = v (°Ax cos(|) cosA,)2 + (CAY cos(j) sinA,)2 + (GAz sin(|))2
Note that these relationships neglect any correlations that may exist between the error estimates for AX, AY and AZ.
6.3.6 Local datum distortion. For some local datums, such as the Australian Geodetic datums, one set of Ax, AY, and Az datum shift parameters provides a consistent measure of the datum shifts throughout the datum. For other datums, Ax, AY, and Az shift parameters for the whole local datum, called mean-value datum shift parameters, poorly fit one or more regions within the datum. Regional variation in the datum shift parameters may result from using different, poorly connected surveys to establish the geodetic control on the local datum; from rotation of the local datum; and from scale differences. An extreme example of regional datum shift variation is the AY component for the ARC 1950 datum. The mean-value AY is -90 meters. However, AY ranges from -5 to - 108 meters for different countries covered by the ARC 1950 datum. Appendix B (Table B.l) contains regional AX, AY, and AZ datum shift parameters for 17 local datums. For these datums, one should base the selection of mean or regional AX, AY, and Az shift values on the accuracy required for the application. Only mean value three-parameter shifts are available for the remaining datums listed in Table B.l.
6.4 WGS 72 to WGS 84 transformation. WGS 84 has replaced WGS 72 as the accepted geodetic reference system for most DoD applications. The WGS 84 system, developed through a more extensive set of satellite-derived and surface data than was available at the start of WGS 72, is an improved geometric and gravitational model of the earth. Whenever possible, WGS 72 coordinates should be converted to the WGS 84 reference system.
29
6.4.1 Selecting a conversion method. Before selecting a WGS 72 to WGS 84 conversion method, find the source of the WGS 72 coordinates. Select the conversion method using the guidance in Table 6.1.
TABLE 6.1. Selecting a WGS 72 to WGS 84 Transformation Method.
Source of WGS 72 Coordinates Transformation Method
1. Doppler satellite station. Direct (6.4.2)
2. Local datum to WGS 72 transformation. Local datum to WGS 84 Local datum coordinates known. transformation
(6.2, 6.3, 6.6)
3. Local datum to WGS 72 transformation. Two-Step (6.4.3) Local datum coordinates unknown. Shift method known
4. Local datum to WGS 72 transformation Direct (6.4.2) using localized datum shift parameters derived for nearby WGS 72 Doppler station. Local datum coordinates unknown.
5. Unknown. Direct (6.4.2) Accuracy unknown
6.4.2 Direct WGS 72 to WGS 84 transformation. The direct WGS 72 to WGS 84 transformation reflects fundamental changes between the WGS 72 and WGS 84 systems. Changes include a shift in coordinate system origin, a shift in longitudinal reference (initial meridian), a change in system scale, and changes in ellipsoidal parameters. For WGS 72 coordinates not originating in the WGS 72 system (i.e., non-Doppler-derived coordinates), take care when employing the direct WGS 72 to WGS 84 transformation, as any inaccuracies and uncertainties inherent in the WGS 72 coordinates will be directly transferred into the derived WGS 84 coordinates. These inaccuracies and uncertainties are generally a result of local geodetic system to WGS 72 coordinate transformations using mean rather than localized datum shifts. Thus, every effort should be made to find the source of all WGS 72 coordinates before doing a direct WGS 72 to WGS 84 transformation.
6.4.2.1 Conversion equations. It is important to note that the relationship between the WGS 72 coordinates and the WGS 84 coordinates cannot be described by a three-parameter model (see Section 6.3).
30
a. The equations for the direct WGS 72 to WGS 84 geodetic coordinate transformation are
0WGS 84 = ^WGS 72 + A<l> (6.9)
AMGS 84 = KlGS 72 + ^A
•kwGS 84 = -h-WGS 72 + Ah
where
A = 4.5 cos(|>WBS 7a + Af sin 20Wf;,q 7, aWGS 72 Q Q
(arc seconds]
AA, = 0.554 (arc seconds)
Ah = 4.5 sin(|)WGS 72 + aWGS 72 Af sin2 (|)WGS 72 - Aa + Ar (meters)
Af = 0.3121057 x 10"7
awGs 72 = 6378135 (meters)
Aa = 2.0 (meters)
Ar = 1.4 (meters)
7C Q =
180 • 3600
b. The equations for the direct WGS 84 to WGS 72 geodetic coordinate transformation are:
9wGS 72 = $WGS 84 + A<))
A«GS 72 = AwGS 84 + AA, (6.10)
■h-WGS 72 = h-wGS 84 + Ah
where
Arh _ -4.5 cosmos »4 Af sin 2(j)WBS R4 AY - - n - z (arc seconds)
»WGS 84 U U
AA, = -0.554 (arc seconds)
Ah = -4.5 sin<|> - aWGS 84 Af sin2 (|)WGS 84 + Aa - Ar (meters)
Af = 0.3121057 x 10-7
31
aWGS 84 = 6378137 (meters)
Aa =2.0 (meters)
Ar =1.4 (meters)
% Q = 180 • 3600
Elevation based on the WGS 84 geoid model will be different from elevations based on the WGS 72 geoid model due to differences in the geoid models (see Section 6.7.1 for conversion).
6.4.2.2 Rstimated errors (G^.g^.O^) . The estimated errors in a direct WGS 72 to WGS 84 transformation are
GA$ = G&l = 3 m (6.11)
a^h = 3 m to 4 m
These error estimates do not include errors in the original WGS 72 coordinates. The accuracy estimate for G^ reflects discrepancies between WGS 72 and WGS 84 geoid models.
6.4.3 Two-step WGS 72 to WGS 84 transformation. If WGS 72 coordinates are known to have been derived from local geodetic system coordinates using WGS 72 Ax, AY, and ÄZ datum shift parameters, the preferred method for transforming them to the WGS 84 system is to do a two-step conversion. First, do a WGS 72 to local transformation using the inverse of the transformation originally used to shift local datum coordinates to WGS 72. This step will eliminate the inaccuracies introduced using local to WGS 72 datum shifts. Next, do a local geodetic system to WGS 84 conversion to produce coordinates within the WGS 84 system (see Sections 6.2, 6.3, or 6.6). Seppelin (1974) provides mean value local datum to WGS 72 datum shift parameters.
6.5 Annroximat-.incr geodetic heights for datum Transformation. When doing the Molodensky approximation (Section 6.3.3), ellipsoidal height (h) is often unavailable. MSL height (H) can be substituted for h without introducing errors in horizontal position significant to mapping and charting applications. The inpact of this substitution is so small that zero can be used in place of h, if H is unavailable.
NOTE: The generation and usage of local datum ellipsoidal heights are outside the scope of this Technical Report. For such applications, consult TEC at the address in the foreward.
32
6.6 Multiple regression equations (MRE). The use of MREs for datum shifting is not recommended.
6.7 Vertical datum shifts. Because exact relationships between vertical datums are unknown, shifting elevations between different vertical datums should not be attempted, except for WGS 72 elevations computed from measured WGS 72 geodetic heights (see Section 6.7.1.) Otherwise, the following are assumed to be the same:
a. Elevations on different local vertical datums. b. Elevations computed from measured WGS 84 geodetic heights
and WGS 84 geoid separations.
It is estimated that errors resulting from this assumption will be one to two meters. Should the accuracy be insufficient, contact TEC at the address listed in the foreword for assistance.
6.7.1 Satellite-derived WGS 72 elevations based on measurer) WGS 72 geodetic: heights. The WGS 84 geoid model is a closer approximation to the geoid than the older WGS 72 geoid model. Elevations obtained from WGS 72 geodetic height measurements and geoid separations may be improved by converting to WGS 84 using
H = HWGS 72 + NWGS 72 + Ah - NWGS 84 (6.12)
where HWGS 72 is the elevation computed using the WGS 72 geoid, NWGS 72 is tne WGS 72 geoid separation, and Ah is computed using
equation set (6.9).
6.7.2 Survev-derived elevations. Elevations obtained by conventional leveling surveys, for points with WGS 72 geodetic horizontal coordinates, should not be converted to WGS 84 elevations. Simplified procedures to convert WGS 72-derived elevations to WGS 84 are not recommended.
6.8 Unlisted datums. This Technical Report provides datum shift parameters between 113 local datums and WGS 84. Regional shift parameters are also given for portions of many of these datums. Contact TEC at the address given in the foreword for any datums and their respective shift parameters that are not listed in this Technical Report.
33
7. MAP PROJECTIONS
7.1 Introduction. Map projections are used to represent the earth's features or points on a map or in a plane coordinate system. A map projection is a systematic representation of a part of the earth's surface on a plane.
7.1.1 The map projection process. The map projection process can be done in three steps:
(1) The positions of points on the surface of the earth ((j), X, h) are reduced to positions on the reference ellipsoid ((|),
X, h = 0). (2) Next, a map projection is chosen and used to transform
the feature's geodetic coordinates (<j), X, h = 0) into planar map projection coordinates (x, y) .
(3) Finally, the map projection coordinates of the features may be plotted, on a map, or stored for future use.
7.1.2 Properties of projections. Once the earth's surface is projected on a plane, one can observe such properties as distance, angle, direction, shape, and size. None of the projections described in this Technical Report faithfully preserves all these characteristics. Some type of distortion is always introduced by a map projection. However, a given map projection can preserve some of these properties, and if a sufficiently small part of the earth's surface is displayed, then other properties can be represented approximately. Map projections presented in this Technical Report have areas of coverage beyond which the projection should not be used due to increased distortion.
7.1.3 Mapping equations. The map projections discussed in this Technical Report have mapping equations that are defined on a reference ellipsoid and are functions of geodetic coordinates ((j), X) . Geodetic height, h, does not enter into the projection equations since h = 0 on a reference ellipsoid. The equations
that express projection coordinates, x and y, in terms of (<t>, X) are sometimes called mapping equations.
7.1.4 Conformal projection. Each projection presented in this Technical Report has the important property of being conformal. A conformal projection is angle preserving. If two lines on the ellipsoid meet at an angle of 0, then in a conformal projection the image of these lines on the map meet at the same angle, 0.
7.1.5 Scale factor. Two quantities associated with map projections, which are often confused, are the scale factor of the projection and the map scale. The scale factor of the projection
34
is used in precise direction and distance calculations using a map projection's coordinates. The scale factor of a projection is the ratio of arc length along a differentially snail line in the plane of the projection to arc length on the ellipsoid. This number depends on both the location of the point and on the direction of the line along which arc length is being measured. However, for conformal projections, the scale factor of the projection is independent of the direction of the line and depends only upon the location of the point. Since all projections treated in this Technical Report are conformal, the scale of the projection will be referred to as the point scale factor (k).
7.1.6 Map scale- The map scale is the approximate number for converting map distances to terrestrial distances and is usually published on the map. Technically, the map scale is the ratio of a distance on the map to the corresponding distance on the ellipsoid. Although the map scale changes over the area of a map sheet, it is customary to assign a single value to denote the map's scale. This value is the true map scale along at least one line on the map. Usually, the map scale is given by a ratio l:n, so that the larger the map scale, the smaller the value of n. For military maps of scale greater than 1:100,000 (n < 100,000), the map scale changes only slightly over a map sheet, and the map scale approximates the ratio of distance on the map to distance on the ellipsoid.
7.1.7 Convergence of the merirHan. The convergence of the meridian (y), often called grid convergence, is used to convert between direction in the map projection (grid azimuth) and geodetic azimuth. The convergence of the meridian at a point on a map is the angle between the projection of the meridian at that point and the projection North axis (typically represented by the y axis of the projection). The convergence of the meridian is positive in the clockwise direction. Convergence of the meridian should not be confused with declination, which is the difference between geodetic and magnetic north at the point.
7.1.8 Information soiirnfts. Information on map projections can be found in Krakiwsky (1973); NSWC/DL TR-3624 (1977); U.S. Department of Commerce, Coast and Geodetic Survey, Special Publication No. 251 (1979); and Geological Survey Professional Paper 1395 (1987).
7-2 Mercator pro-ient-.ion. The Mercator projection is a. conformal projection for which the point scale factor is one along the equator. The equator lies on the line y = 0. This projection is not defined at the poles.
7-2.1 Meridians and parallels. Meridians and parallels provide the framework for the Mercator projection. Meridians are projected as parallel straight lines that satisfy the equation x = constant. Evenly spaced meridians on the ellipsoid project to
35
evenly spaced straight lines on the projection. Parallels are projected as parallel straight lines perpendicular to meridians and satisfy the equation y = constant. Evenly spaced parallels on the ellipsoid project to unevenly spaced parallels on the projection. The spacing between projected parallels increases with distance from the equator. See figure 7.1.
DU
^0°
o 40 ■■
30°
20°
10°
0°
20 40c 60c 80c 100 120
FIGURE 7.1. Meridians (longitude) and parallels (latitude) in the Mercator projection.
7.2.2 Mercator mapping equations.
7.2.2.1 Finding (x,v). The following discussion of the Mercator mapping equations can be found in Geological Survey Professional Paper 1395 (1987). First, choose a central meridian iXQ) that represents the zero point on the x axis. Given a point (<|), X), in radians, on the ellipsoid, the corresponding point (x, y) in the plane is
x = a (A, - K) (7.1)
y = a In tar\4 fn (fry 1 - £ sin (j)
£ sin <|) •A 1 +
or
-Ü) ^ In 1 + sin 0
1 - sin (j) fc
£ sin 0
E sin <j) J J
(7.2)
36
where a = semi-major axis of the ellipsoid £ = eccentricity X0 = longitude of the central meridian
<)), X and X0 are expressed in radians.
7-2-2-2 Point scale factor and nnnverosrirp of f.hp meridian. The equations for the point scale factor (k) and the convergence of the meridian (y) are
, _ a " N cos <j) (7>3)
where N =
Y = 0
a. Vl - £2 sin2 (|)
7-2.2.3 Finding ((|),A,) . Next, consider the inverse probl of finding (((), X) when (x, y) is known. The longitude can be found from
em
^ = a + K (7.4)
where both X and X0 are expressed in radians. The value of <|> can be found iteratively. Let an intermediate parameter t be defined
t = e a (7.5)
After an initial value of
2
iterate the following equation for (()
, K <Po = T " 2 arctan t • (7.6)
. K Yn+i = 2~ " 2 arctan
v
(1 - £ sin <!>„)' / r . . . -, t\
2
J '.(1 + £ sin <|)n) (7.7)
37
Substitute the initial value (<j>0) into equation 7.7 to find the next candidate ^1. Similarly, substitute §x into equation 7.7 to obtain the updated candidate, (])2. Continue this process until the
difference between successive values of 0n is sufficiently small.
Then set 0 = <J)n. All angles are measured in radians.
7.2.3 Accuracy. The Mercator equations for x, y, k, y, and X are exact. The iterative equation for 0 can be updated until any desired accuracy is obtained.
7.2.4 Area of coverage. In the Mercator projection, as the latitude (<()) approaches the poles, the y coordinate approaches infinity. Area and length distortion increases with distance from the equator. For example, the point scale factor is approximately 2 at 60° latitude and 5.7 at 80° latitude.
7.3 Transverse Mercator (TM) proiection. The Transverse Mercator projection is a conformal projection for which the point scale factor equals one along the central meridian. The line y = 0 is the projection of the equator, and the line x = 0 is the projection of the central meridian. The Universal Transverse Mercator (UTM) projection (see Section 7.4) is a modification of the Transverse Mercator projection.
7.3.1 Meridians and parallels. Both the central meridian and the equator are represented as straight lines. No other meridian or parallel is projected onto a straight line. Since the point scale factor is one along the central meridian, this projection is most useful near the central meridian. Scale distortion increases away from this meridian. (See Figure 7.2.)
38
FIGURE 7.2. Meridians and parallels in the Transverse Mercator Projection, meridian.
A,n is the central
7.3.2 Transverse Mercator manning equations.
7.3.2.1 Finding (x.v). The presentation here follows U.S. Department of Commerce, Coast and Geodetic Survey, Special Publication No. 251 (1979). A point with geodetic coordinates (<|>, X) in radians has Transverse Mercator coordinates (x,y) that are given by
*T A A N A3 cos3 0 „ x = N A cos(}> + g * (1 - t2 + T\2)
N A5cos5(})
(7.8)
120
N A2
(5 - 18t2 + t4 + 14T12 - 58t2T|2)
y = S^ + " 2~ sin 0 cos 0
N A4
24 N A6
sin (j) cos3 0(5 - t2 + 9T|2 + 4T|4)
+ 720 sin <)> cos5 0(61 - 58t2 + t4 + 270T|2 - 330t2Tl2-)
39
where
N = ■ =- (7.9! Vl - £2 sin2 $
t = tan (|) T| = e'cos <J)
e2
£'2 =
e2 =
(i - e2) a2 - b2
a2
S<j, = a[A0(j) - A2 sin 20 + A4 sin 4<j> - A6 sin 6<j) + A8 sin 80] 13 5 175 A - 1 - ^=p2 _ -=-c4 _ —2—p6 _ J-/J p8 A° - J- 4& g^ 256^ 16384c
A2 = g(e2 + -^e4 + 3^g£6 - 4096e8)
A* = 256(£4 + 4^ 128£8)
35 41 Ag = 3072 (£6 " 32£8)
A - - 315 £8 As " 131072 fc
A = A, - A.0 in radians X0 = central meridian in radians
At the poles, where 0 = + or - 7C/2, the Transverse Mercator coordinates are x = 0 and y = SA, a quadrant arc on the ellipsoid.
7.3.2.2 Finding (0. X) .
7.3.2.2.1 Finding the footpoint latitude (fa) . To find (()),
A,), given x and y, first compute the footpoint latitude (fa) using the following iterative method (See Krakiwsky 1973). The equation to be iterated to yield the footpoint latitude (fa) is
SA - y 0' = 0 - 9
G, (7.10) 0 <j)
where
S'A = a[A0 - 2A2cos20 + 4A4cos40 - 6A6cos6(j) + 8A8cos8(j)] (7.11)
and SA AQ, A2, A4, A5, and Ag are defined in equation set 7.9.
Let the first candidate for fa be 0 = y/a. When this value of 0 is substituted into equation 7.10, the updated candidate for 0X is 0'. Similarly, the substitution of 0' for 0 in equation 7.10
40
gives the next updated value of 0'. When successive values of (j)' are sufficiently close, then the footpoint latitude is found by setting (J^ = (J)'. A non-iterative method for finding §± can be found in NOAA Report NOS NGS 5 (1989).
7.3.2.2.2 Finding (ty.X) from (|)1. Once (j^ is determined, (()), X) can be found from the approximations
where
$ = 4>i - Mi Ri 2 l^fj 24 l^Nj + 720 [uj (7.12)
A,= XQ + sec ♦i [ N, " 6 UJ + 120 fej
B3 = 1 + 2tJ + Tli (7.13)
B4 = 5 + 3t? + T|J - 4TJi -9tJnJ
B5 = 5 + 28ti + 24tJ + 6Tli + 8tll\l
B6 = 61 + 90tJ + 46T|i + 45tJ - 252t^J
- 3TlJ - 66tJnJ 90t{t\l + 225tJ TlJ
i2 _ e2
(i - e2) TJ! = £'2 cos2 tyx
tx = tan 0X
N,
R,
Vl - £2 sin2 0! a(l - £2)
(Vl - £2 sin2 (^ )"
7Ü If <!>! = ± y then (|) = ± 90° (a pole) and X is undefined.
41
7.3.2.3 Point scale factor as a function of x and ty±. As a function of x and §lt the point scale factor is given by
where
1 + — i7 + 24
C2 = 1 + Ttf
C4 = 1 + 6T|i + 9T|i - 24tJ Tli
INJ + 24 ^NJ + 720 [NJ
k = 1 +
in which
A2 cos2 0 2 D2 +
A4 cos4 (j) 24 D4 +
A6 cos6 0 720 6
(7.14)
(7.15)
and tx , t\x, and Nj are given by equation set 7.13.
7.3.2.4 Point scale factor as a function of (ty.X) . As a function of (<j), A,) , the point scale factor is given by
(7.16)
D2 = 1 + Tl2
D4 = 5 - 4t2 + 14Tp + 13T|4 - 28t2T|2 - 48t2Tl4
D6 = 61 - 148t2 + 16t4
(7.17)
and A, t, and TJ are given in equation set 7.9.
7.3.2.5 Convergence of the meridian in terms of x and (j^.
The convergence of the meridian, in terms of x and (j^, is
Y = txx N l L
1 - El f2L^2 (2L\2 EL (2L\4 _ JÜs_ (2Lf [NJ
+ 15 {NJ " 315 \jsix) _ (7.18)
where
E2 = 1 + t? - Tl? - 2T\t
E4 = 2 + 5tJ + 2T|2 + 3tJ + t2Tli + 9T|i - 7tJnJ
E6 = 17 + 77tj + 105tJ + 45tJ
(7.19)
and t1# V[1, and Nx are given by equation set 7-13.
7.3.2.6 Convergence of the meridian in terms of ((|), X). The convergence of the meridian, in terms of ((|), X). is
42
A • A f-, r. A2 COS2 0 A4 cos4 (j) A6 cos6 (b ~\ Y = Asm.^1 + F2 * + F4 ^j-^ + F6 315 y J (7.20)
in which
F2 = 1 + 3T|2 + 2T|4
F4 = 2 - t2 + 15T|2 + F6 = 17 - - 26t2 + 2t4
(7.21) 35T|4 - 15t2T]2 - 50t2T|^
and A, t, and T) are given in equation set 7.9.
7.3.3 Accuracy. The Transverse Mercator equations for x, y, <]), X, Sq. tyx, k, and y are approximations. Within 4° of the central meridian, the equations for x, y, (j), and X have an error of less than 1 centimeter. (See Geological Survey Professional Paper 1395, 1987.)
7-3.4 Area of coveraap. The equations given here for the Transverse Mercator projection can be used within 4° of the central meridian. If these equations are used with UTM coordinates, then consult Section 7.4.1 for guidance on the area of coverage for UTM coordinates.
7.4 Universal Transverse Mercator (UTM) projectinn. The UTM projection is a family of projections that differ from the Transverse Mercator projection in these ways (see Figure 7.3):
a. The longitudes of the central meridians (in degrees) are 3 + 6n, n = 0,...59.
b. The point scale factor along the central meridian is 0.9996.
c. The y value, called the northing, has an origin of 0 meters in the northern hemisphere at the equator. The y value, called the southing, has an origin of 10,000,000 meters at the equator and decreases toward the pole in the southern hemisphere.
d. The x value, called the easting, has an origin of 500,000 meters at the central meridian.
7.4.1 UTM zones. For the UTM grid system, the ellipsoid is divided into 60 longitudinal zones of 6 degrees each. Zone .number one lies between 180° E and 186° E. The zone numbers increase consecutively in the eastward direction. It is intended that a UTM projection should be used only in one of the 6-degree zones, plus the overlap area. In each zone, the UTM projection extends for a 40 km overlap into the two adjacent zones. Moreover, UTM coordinates are used only between 80° S and 84° N plus the overlap region. The UTM projection extends to 84° 30' N and 80° 30' S to overlap with the Universal Polar Stereographic (UPS) projections.
43
FIGURE 7.3. Meridians and parallels (dashed) and a Universal Transverse Mercator Grid.
7.4.1.1 Finding the UTM zone and central meridian. To find UTM coordinates from geodetic coordinates, first compute the UTM zone. A standard zone can be computed directly from geodetic longitude. If the geodetic longitude (X) is expressed in positive radians, then the zone (z) is
z = Greatest Integer < (31 + j for 0 ^ X < It (7.22)
z = Greatest Integer < f - 29 J for K < X < 27C
Once the zone (z) is known, the central meridian is given by equation set 7.23. If a longitude lies on a UTM zone boundary, then the user must choose the desired UTM zone. If a map is being used, then the UTM zone appears on the map. The central meridian of zone (z) is
X0 = (6z - 183)
Xn = (6z + 177)
71 180
K 180
with X0 expressed in radians
for (z > 31)
for (z < 30)
(7.23)
44
7.4.1.2 Non-standard width UTM znnps. Non-standard width UTM zones are used in North Atlantic regions and Norway as delineated in Table 7.1 (see DMA TM 8358.1 (1990). These non- standard zones may also be applied in the 30' UPS overlap region.
TABLE 7.1. Non-standard zone limits.
Zone Latitude
Lower | Upper" Longitude
West I East Central Meridian
31 56° N 64° N 0° E 3° E 3° E 32 56 64 3 12 9. 31 72 84 0 9 3 32 72 84 not used 33 72 84 9 21 15 34 72 84 not used 35 72 84 21 33 27 36 72 84 not used 37 72 84 33 42 39
7.4.2 Reference source. The standard references for the UTM projection are DMA TM 8358.1 (1990) and DMA TM 8358.2 (1989).
7.4.3 UTM equations.
7-4.3.1 Finding UTM coordinates (xDTt]. yDTI1) . Having found the central meridian, next compute Transverse Mercator coordinates (x, y), using equation sets 7.8 and 7.9. Then, the UTM coordinates can be found from:
XUTM = 0.9996 XrM + 500,000 (7.24) YUTM = 0.9996 yra (Northern hemisphere) YUTM = 0.9996 yM + 10,000,000 (Southern hemisphere) ^UTM
= 0.9996 kTM «UTM = Y TM
7.4.3.2 Computing geodetic coordinates. To compute geodetic coordinates from UTM coordinates, a zone, and a hemisphere, first solve equation set 7.25 for (x^, yTM), equation set 7.23 for X0i
and then compute (<j), X) from equation set 7.12.
x, •TM (xUTM - 500,000) 0.9996 - 1
YTM - 0.9996 ^YUTM'
(YUTM " 10,000,000) YTM =
k, 0.9996
1 TM _ 0.9996 UTM
YTM = YUTM
(7.25)
(Northern hemisphere)
(Southern hemisphere)
45
7.4.3.3 Discontinuity. Notice that UTM coordinates are discontinuous in y at the equator.
7.4.4 Accuracy. The accuracy of the UTM equations is based on the accuracy of the Transverse Mercator equations. Within the area of coverage, the equations for x, y, (j), and X have an error of less than one centimeter. (See Geological Survey Professional Paper 1395, 1987.)
7.4.5 Area of coverage. The area of coverage for UTM coordinates is defined by zone limits, latitude limits, and overlap.
Zone limits:
Latitude limits:
Zone overlap:
Polar overlap:
6° zones, extending 3° to each side of the central meridian. See Table 7 for exceptions to this general rule.
North: 84° South: 80°
40 km on either side of the zone limits.
30' toward the poles North: 84° 30' South: 80° 30'
7.5 Lambert Conformal Conic projection. The Lambert conformal conic projection is a conformal projection in which the projected parallels are arcs of concentric circles centered at the pole. The projected meridians are radii of concentric circles that meet at the pole. Moreover, there are one or two parallels, called standard parallels, along which the point scale factor is one. (See Figure 7.4.)
46
90°W
45°W
Pole 90°E
45°E
FIGURE 7.4. Meridians and parallels in the Lambert Conformal Conic Projection. (Not drawn to scale.)
7.5.1 Lambert. Conformal Conic with two standard parallels. This presentation of the mapping equations follows Krakiwsky (1973). The first case to be considered is the case of two standard parallels, §± and (t>2. The point scale factor is one along both standard parallels. Choose a fixed central meridian,
7.5.1.1 Finding (x.y) . Begin by fixing a latitude ((|)0) that is below the area of interest. For a given (()), A,) the mapping equations are
where
x = r sin LÄ y = r0 - r cos LA
A = X - Xa
(7.26)
r = K e -Lq
(7.27)
r0 = K e
K =
_Lq0
Nj COS 0X N2 COS §2 -Lq, -Lq.,
Le A Le e = base of natural logarithms ( In e ()>! = first standard parallel (j)2 = second standard parallel a
N =
1 )
Vl - e2 sin2 <})
47
q =
£2 =
In tan n (j)Yl - £ sin 0 Y
■2 IH £ sin (j)
a2 - b2
L = ln(Nx cos <!)]_) - ln(N2 cos (j)2)
q.j = q evaluated at <J)j , j = 0,1,2 Nj = N evaluated at <J>j , j = 1,2
7.5.1.2 Finding (§.X) with two standard parallels. For the reverse case, when (x, y) is given along with (j)0, ^x, (j)2, and X0i
first compute L, r0, K, Nl7 N2, q0, C^, and q2 from equation set
7.27. Then compute X and q from
X =
g =
L + *° (7.28)
In K
in which
r = cos e (7.29)
0 = arctan x
rG- y
Note: When evaluating the arc tangent, the user must take care to insure that the resulting angle lies in the desired quadrant.
Once q is found, then 0 can be determined using the Newton - Raphson method, which is explained next. Define functions f and f' , and an initial approximation (j)' by
48
f «I» = -q + r- In 1 + sin (}A fl - £ sin (j>Y 1 - sin §J \l + £ sin <))
(7.30)
, _ 1 - £2
(1- £2 sin2 (|))cos (j)
<J)' = 2 arctan (e<*) - 7C/2
Then successive iterations for (}> are found from the equation
<j)' • =0' - f' ((j)')
(7.31)
In particular, substitute (j)' into this equation to obtain (j)' ', the first approximation to <(). Then set (J)' = <j)'', substitute 0" into the equation to obtain §' ', the second approximation to (j). Continue this procedure until the differences between successive candidates for (j) are sufficiently small.
7.5.2 Lambert Conformal Conic with one standard parallel. In the second case, the point scale factor is one along one standard parallel, <J)0. Fix a central meridian, X0.
7.5.2.1 Finding (x.y) . The mapping equations are
where
x = r sin LA y = r0 - r cos LA
A = X - X0
L = sin (|)0
K = N0 cot (|)0 e -Lq
r = K e
r0 = K e e = base of natural logarithms
M. S
(7.32)
(7.33)
%
In e = 1 ),
Vl - E2 sin2 <])
q = In tan (* ♦ f 1 ' £ Sln * * 1 + £ sin (j)
J -i
qo = q evaluated at <]),
N0 = N evaluated at <j)(
49
7.5.2.2 Findina (§.X) with one standard parallel. To transform in the other direction, suppose that x, y, <j)0, and A,0 are given. First, compute L, K, N0, c^, r, and r0 from equation
set 7.33. Then, as in the previous case, (()), X) can be found from
X = L + *■
g = la* r
T.
(7.34)
in which
r - y r = ——ö (7.35)
cos b o x D = arctan
r0- y
Again, § can be obtained from q by using the Newton-Raphson method (see equation sets 7.30 and 7.31).
7.5.3 Scale factor and convergence. The scale factor and convergence of the meridian are given by
rL k = z 1 (7-36> N cos 9
Y = LX
In the case of two standard parallels, the values of r, L, and N are given in equation set 7.27. For one standard parallel, the values of r, L, and N are found in equation set 7.33.
7.5.4 Accuracy. For the Lambert conformal conic projection with either one or two standard parallels, the equations for x, y, X, q, k, and y are exact. The equation for 0 can be iterated until the desired accuracy is obtained.
7.5.5 Area of coverage. The family of Lambert conformal conic projections and their limiting cases is used worldwide. For a given member of this family, the point scale factor increases as a point moves away from the standard parallel(s).
50
7.6 Polar Stereocrranhic nrrriecfcion. The Polar Stereographic projection is a limiting case of the Lambert conformal conic projection when the one standard parallel approaches a pole. In this conformal projection meridians are straight lines, parallels are concentric circles, and the point scale factor is one at the pole (see Figure 7.5).
135°E 180c
90°E
45°E
135°W 180c
135°W
90°W 90°W
45°W 45°W
35°E
90°E
45°E Figure 7.5. Meridians and parallels in the Universal Polar
Stereographic Projection. Left: South Zone, Right: North Zone (Not drawn to scale.)
7.6.1 Polar Stereocrranhic manning equations. There is considerable diversity in the treatment that various authors give to the polar Stereographic projection. Some authors interchange the x and y axis; others change the sign convention. The discussion here is largely taken from NSWC/DL TR-3624 (1977) since it most closely conforms to DMA TM 8358.2 (1989).
7.6.1.1 Finding (x.v). Given (<]), X), the Polar Stereographic coordinates are
x = r sin X y = -r cos X y = r cos X
(Northern hemisphere) (Southern hemisphere)
(7.37)
in which
r = K tan n _ Jj|>h ( 1 + £ sin 101 >|
1-8 sin |0| (7.38)
2a2 (i - e\
51
|(j)| denotes the absolute value of <j). ((]), X) is expressed in radians.
At the poles, X is not defined. Nevertheless, these equations can be used at the poles since r is zero there. At <J> = ± ft/2, pick an arbitrary value for X. Then, equation sets 7.37 and 7.38 yield Polar Stereographic coordinates of (0, 0).
7.6.1.2 Finding ((j).X,) . The discussion of the conversion . from (x, y) to (<j>, X) follows Krakiwsky (1973). If (x, y) = (0, 0), then 0 = ± 90° and X is undefined. Otherwise, the longitude is given by
X = - arctan — (Northern hemisphere) (7.39)
"V*
X = arctan — (Southern hemisphere)
If (x, y) 5* (0, 0), then x, y or both are non-zero. Use one of these non-zero values to find q from the equations
-q e
-q e
JY_L K I cos X\
X K sin X
when IxI > IyI
when lyl > Ixl
(7.40)
where
K = 2a2 (\ - e\ b l + e
(7.41)
As in the Lambert conformal conic projection, <j) can be found from q using the Newton-Raphson method. Define functions f and f', and an initial approximation <j)0 by
f (0) = -q + 2 ln rl + sin (|lWl - 8 sin <|A e 1 - sin <j)J [l + £ sin <|)J
(7.42)
_ 1 - £2 ' ^ " (1- £2 sin2 (j))cos ())
, q 7C <])0 = 2 arctan e - ~z
52
Then, successive iterations for § are found from the equation
*» = 0»-i " F^7 (7-43)
In particular, substitute <|)0 into this equation to obtain (j^, the first approximation to (|). Then substitute ty± into the equation to obtain (|)2, the second approximation to (j). Continue this procedure until the differences between successive candidates for (|) are sufficiently small. Even though q approaches infinity at the poles, this method is satisfactory for many computers. This procedure yields a positive value of (j). In the southern hemisphere, replace this positive value of (|> by -(|).
7.6.1.3 Alternate method for finding ([). Alternatively, an approximate equation for (j) is given in Geological Survey Professional Paper 1395 (1987).
<|) = % + A sin2% + B sin4% + C sin6% + D sin8% (7.44)
where
% X = 2 ~ z (in radians) (7.45) z x
tan ;r = ——;—TT X * 0 ^ K sin A,
tan f = | y , | y * 0 ^ K cos K
K=^i A - €\ v1 + e,
A . §e> + £* + £* + 3^e»
B = -^-€4 + -^-€6 + 811 £8 D 48° 240° + 11520°
^ 12CT 1120°
D _^279_^8 161280°
a2 - b2
£2 = . a2
53
Since this procedure yields a positive value of (J), replace (|) by -§ in the southern hemisphere.
7.6.1.4 Finding the point, scale fanf.nr. The point scale factor is given by
k = at the poles (7.46)
where
k =
K =
N cos <|)
r = K tan
elsewhere
1L _ M\( 1 + e sin Id)i >p (!-f) 2a2 fi - e\ 7
l + e
v 1 - £ sin |(()| j (7.47)
N = Vl - e2 sin2 0
7.6.1.5 Finding the convergence of the meridian, convergence of .the meridian is
7=A, (Northern hemisphere)
7=—A, (Southern hemisphere)
The
(7.48)
7.6.2 Accuracy. For the Polar Stereographic projection, the equations for x, y, X, and q are exact. The equation for <|> can be iterated until the desired accuracy is obtained.
7-6.3 Area of coverage. There is no general agreement on the area of coverage for the Polar Stereographic projection. Section 7.7.5 gives the area of coverage for UPS coordinates.
7-7 Universal Polar Stereogranhic (UPS) proiection. The Universal Polar Stereographic (UPS) projection is the standard military grid used in polar regions (see Figure 7.6). The main reference for the UPS grid is DMA TM 8358.2 (1989). The UPS grid is a family of two projections that differ from the Polar Stereographic projection in these ways:
a. Both the x and y values (see Figure 7.6), called easting and northing, respectively, have an origin of 2,000,000 meters.
b. The scale factor at the origin is 0.994.
54
180c
90°W 90°E
Figure 7.6
0°
Meridians and parallels imposed on a Universal Polar Stereographic grid (north zone).
The limits of the system are north of 84°N and south of 80° S. In order to provide a 30-minute overlap with the UTM grid, the UPS grid contains an overlap that extends to 83° 30'N and 79° 30'S. Although the UPS grid originally was defined to use the International Ellipsoid, in practice several reference ellipsoids have been used. The ellipsoid appropriate to the selected datum should be used.
7.7.1 Universal Polar Stereoaranhic (UPS) manning equations. For known geodetic coordinates, the UPS easting is given by
xups = 2,000,000 + 0.994r sin X
and the UPS northing is given by
(7.49)
YUPS = 2,000,000 - 0.994r cos X (Northern hemisphere) (7.50) yUPS = 2,000,000 + 0.994r cos X (Southern hemisphere)
The parameter r is computed using equation set 7.38.
7.7.2 Finding (§.X) . if a hemisphere and UPS coordinates (x, y) are known, convert the UPS values to Polar Stereographic coordinates using equation set 7.51, and then transform to geodetic coordinates as described in Sections 7.6.1.2 and 7.6.1.3.
55
XpS = Q^g^ (xups - 2,000,000) (7.51)
YPS = "Ö~994 (
^"PS " 2,000,000)
In particular, at the poles where x = 2,000,000 and y = 2,000,000 then, (|) = + 90° and X is undefined. Otherwise longitude and latitude can be found from
„ (xups - 2,000,000) k = - arctan (Northern hemisphere) (7.52)
VjfUPS ' 'J'-'U / UUU /
r, (xups - 2,000,000) A = + arctan — (Southern hemisphere)
\j npc " / UUU / UUU /
(j) = % + A sin2% + B sin4% + C sin6% + D sin8%
in which
% = f - Z (7.53)
z (xups - 2,000,000) tan r = "P° „ : 1 XUPS * 2,000,000 2 0.994K sin A ups
Z (YUPS - 2,000,000) tan
2 0.994K cos A yups zfr 2,000,000 (Northern hemisphere)
tan 7T = + (YUPS " 2,000,000)
2 0.994K cos A YUPS * 2,000,000 (Southern hemisphere)
2 a2 K =
f i - e \*
v1 + £y
e2 =
b
a2 - b2
a2
The values of A, B, C, and D are found using equation set 7.45.
In the southern hemisphere, replace the computed value of <j) with
56
7.7.3 Findina point scale factor and convergence of the meridian. The point scale factor and convergence of the meridian are
k = 0.994 at the poles (7.54)
r k = 0.994 — r elsewhere
N cos (p
Y = X (Northern hemisphere) y = -X (Southern hemisphere)
where r and N are given in equation set 7.47.
7.7.4 Accuracy. The accuracy of the UPS equations is based upon the accuracy of their Polar Stereographic counterparts. For the Polar Stereographic projection, the equations for x, y, X, and q are exact. The equation for 0 can be iterated until the desired accuracy is obtained.
7.7.5 Area of coverage. The area of coverage for UPS coordinates is defined by zone limits and overlap.
North zone: The north polar area 84° - 90°
South zone: The south polar area 80° - 90 o
UTM overlap: 30' overlap, North: 83° 30' South: 79° 30'
7.8 The U.S. Military Grid Reference System (MGRS). The MGRS is an alpha-numeric system, based upon the UTM and UPS map projections, for identifying positions. An MGRS coordinate consists of a zone designation, alphabetic 100,000-meter grid square designator, and numeric coordinates within the 100,000 meter grid square. MGRS coordinates are defined for the primary UTM and UPS areas, but not for the overlap areas.
7.8.1 MGRS coordinates in the UTM area. In the UTM area, MGRS coordinates are based on the ellipsoid, geodetic latitude, UTM zone, easting and northing as follows:
a. The first two characters of the MGRS coordinate are the two digits of the numeric UTM zone (see sections 7.4.1.1 and 7.4.1.2.) Leading zeros must be included.
b. The third character of the MGRS coordinate is a letter representing a band of geodetic latitude. Beginning at 80° South and proceeding northward, 20 bands are lettered C through X,
57
omitting I and 0. The bands are all 8° high except band X (72°N to 84°N) , which is 12° high.
c. The fourth and fifth characters of the MGRS coordinate are a pair of letters representing the 100,000-meter grid square. The letter pair can be selected from figure 7.7 and 7.8 (from DMA TM 8358.1.) First, reduce the UTM northing by multiples of 2,000,000 until the resulting value is in the range of 0 to 1,999,999 meters. Second, reduce the UTM northing and easting values to the nearest 100,000 meters. Third, locate the corresponding 100,000 meter northing and easting grid lines for the UTM zone number on the figure. The grid square identifier will be immediately above and to the right of the intersection of the easting and northing grid lines on the figure.
Note: The alphabetic method of designating 100,000- meter grid squares has changed over time. Older products may have different grid square designations than those described above. Also, some software was programmed to compute MGRS coordinates based on geographic position and a map of preferred ellipsoids. The boundaries between preferred ellipsoids have changed with time. Therefore, MGRS coordinates computed with such software may not agree with current map products. MGRS computation software should always require input of the ellipsoid. If MGRS coordinates are to be used in conjuction with a map product they should be checked against the map product to verify compatibility.
d. The remainder of the MGRS coordinate consists of the numeric easting and northing values within the 100,000-meter grid square. The left half of the digits is the easting grid value, which is read to the right of the 100,000-meter easting grid line established in step c. above. The right half of the digits is the northing grid value, which is read northward of the 100,000-meter northing grid line established in step c. above. Both the easting and northing grid values are within the range of 0 to 100,000 meters (i.e., zone and grid square designators, only) to 1 meter (i.e., zone and grid designators followed by 10 digits, five for easting grid value and five for northing grid value). Both easting and northing value must have the same resolution and must include leading zeros.
7.8.2 MGRS coordinates in the UPS area. In the UPS area, MGRS coordinates are based on the ellipsoid, geodetic latitude and longitude, easting, and northing as follows:
a. The first character of the MGRS coordinate is A in the Southern and Western Hemispheres, B in the Southern and Eastern Hemispheres, Y in the Northern and Western Hemispheres, and Z in the Northern and Eastern Hemispheres as shown in figures 7.9 and 7.10.
58
SET
6 6
12 1
8 24 3
0 36
42 4
8 54 6
0
» a. a e 5 C (3 e » a. X z ► * = X « >- a-
o >- u oa >- £ 3 >- t CO >- OC >- o- a. * z s £ = X u
K Ä o CJ CO
X 3 X ^ CO oc
o a. z * 5 s £ X £ % CD s £ 3 £ % £ » > z z > s X
■^ s X > 5» *. > a a- c_> > oo g 5 3 > & co > ac •>•
3 a. 3
z =3
z 3
ae =3 5 X
3 43 3 3 3
o 3
CJ 3
OB 3 3
> 3
3 3 3
CO 3
ae 3
o Cw Z z
o a. Z z io 5 £ X «J c^> ^ CO
0D a ft 3 K K 5
SET
5 5
11 17 2
3 29 3
5 41 4
7 53 5
9
ac s X OE
u oe ae cc
o a CO QC s £ 3
ac ac £ ac O ae OC
Z OC
z oc ae
ac o- s X O* o & o
o ©-
oa O ©■
3 & C3> ca- s o- O
Z
Z Z X Z z z z z z oa
z z z 3 z z z se z o_ z z
z z ac 5 Z
X z o z z z
o z z 4» z s z z z
to z. z o z a.
Z z S
Z
ac ac if
X aC
u s ac o
ac 413 ac
« ac 3 ac ac ac
oe ac ae ac
z z ac ac — = X s2 Ä Ä e U <B a z. 3 fis *« oe o> & z Z mi
SET
4 4
10 1
6 22 2
8 34
40 4
6 52 5
8
©- X
a. X
z X z
X X ac X X
X X
O X X X
o X X X X
> X
3 X X
to X
oe
C9 Ck. u
z CJ z
C3 u ac 5 X
o C9 u u o CJ
fc9 CD 19 C9 > a
C9 5 3 oe C3
Cr a. z z £ ac = X «9 £ s o €L> 0D < £ 3 tz CO OC
» a. z - z = ac a X « & £ o c-> oa s £ 3 & CO oe UJ
O- o
a. o z o z a o o Q
&9 C3 o. o o
C3 CO o s C3
3 s CO o oe Q
O- a. z u z
CJ «_> u X u <-> o CO
CJ CJ & 3 CJ CJ Ö oe
CJ
©■ 00
a. as z
00 z 00 CD
ac CD
X so
C9 CD CO so
o CD
OB CD CD
3 CD
to CD oa
a. z z « ac = X tP < « o « do S s 3 & •A oe
SET
3 3
9 15 2
1 27 33
39 4
5 51 57
ac = X O *1 « o Sä CO a s 3 K (2 es ej
ft] r»i s ^ ac >- = X >- >- £ a-
o >■
t-> >- s «c £ 3 £ SC oc >- o- cv. >■
z >- tf ae = X C9 St » a (_) 490 s > 3 5 X
oc o* o. z s ^ i s
X * s- * £ s CD £ i 3 * K £ I 1- o. * z ? *
ac = X > s *£ * (3 CD > * 5 3 3> ' ^ e g C7 >• S: s s *
a: 3 3
X 3
13 3 3 s =3 =) CD
3 ^ 3 3 3 3 3
oe 3
at 3
o_ 3 z
3 z 3 3
JC ,~ X s ts SB O (_) CO R tt 3 K ts oe e> o. Z z CO
SET
2 2
8 14 2
0 26 3
2 38 4
4 50 5
6
OS (K z z
oe ÖC oe DC X oe
(3 DC QC oc
a ^ oa s s g ce to QC
oe
a. o z z o s o- 5- X U o o O" o i-3 CD-
CD & C31 &. co O C7
o* z
a. Z
z z z 4TC- ■ .z z z
X z z z z o z
<-> z CO z >■ z Z £ to z ae
©• z a. Z
z z z z z S z X z z z z z z
CD z Z Z
3 z Z
CO Z
oe Z
o * z z aC X
ac ac ac X
ac ts s ac
a ac
u> CD ac 5 ac
3 ac ae
CO ac
oc ac
o* *= z z * ac a X sa Ä Si o tJ m 5 % E TX sa OC
ac X X
X X X X X X X
CD X X
> X
3 X X X
ac X X
a. X
z X
z X X
o X C3
C9 u u «J £3 CO 3
C9 •s ae C3
C3- C3
a. z Z C9 u
1*1 x X C9 Q o OS «: > CL.
«SI
CO Cl «• o Q X
o o O O oo
o 3 3 o o
ae a
C3* 3
Q. O Q
z o o
r^ m ac 3 X u CJ
O <-> 0D s > o u to ac
u o> Cv 5 Z
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62
b. The second and third characters of the MGRS coordinate are letters representing a 100,000 meter grid square. The letter pair can be selected from figure 7.9 or 7.10 (from DMA TM 8358.1.) Reduce the UPS easting and northing values to the nearest 100,000 meters. Locate the corresponding 100,000 meter northing and easting grid lines for the UTM zone number on the figure. The grid square identifier will be immediatly above and to the right of the intersection of the easting and northing grid lines on the figure.
c. The method for calculating the remainder öf the MGRS coordinate is the same as in paragraph 7.8.1 d.
7•? The World Geographic Reference System (GEORF.F). The GEOREF is an alpha-numeric system for reporting positions for air defense and strategic air operations. It is based upon geodetic coordinates. GEOREF is described in DMA TM 8358.1 (1990).
7-10 Non-standard arirte. The specifications and diagrams of non-standard grids, used as the primary or secondary grid on DMA produced maps, are contained in DMA TM 8358.1 (1990).
7-1l Modifications to man nrrriections. Providing equations for transforming between geodetic coordinates and every local map projection is beyond the scope of this Technical Report. However, many local map projections are minor modifications of the map projections given in this Technical Report. Typically, these modifications include use of false Eastings (E) and Northings (N) at the projection origin, a non-unity scale factor (k0) along the principal line 6f the projection, and units other than meters. Sometimes the local projection origin is offset from the origin used in this Technical Report. Chapter four of DMA TM 8352.1 (1991) gives parameters for several local projections.
If the local projection is based on one of the projections given in this Technical Report convert geodetic latitude (<j)) and longitude (A,) to coordinates in the local projection by using the following projection;
1. Make sure (0, X) are on the ellipsoid for the local projection and determine the ellipsoid's parameters, a and either f or b.
2. Convert (<]), A) to map projection coordinates (Xp, Yp) using the appropriate map projection equations given in this Technical Report. Note that the units for Xp and Yp are in meters.
3. If the local projection has false coordinates, E, N, at the origin, convert them to units of meters if necessary.
63
The map projection coordinates (Xp, Yp) and scale factor (kp) can be converted to local projection coordinates using equations 7.55 if the origin of the local projection is the same as the origin of that projection as used in this technical report. Those origins are central meridian (A,0) and equator for Transverse Mercator and Mercator projections, and standard parallel(s) and equator for the Lambert Conformal projection.
X = K0Xp + E
Y = k0Yp + N ' (7.55)
JC = JC0Kp
7 = YP
where
E = false X-value (in meters) at the origin
N = false Y-value (in meters) at the origin
ko = scale factor along the appropriate line. ( Along the equator for the Mercator projection, along the standard parallel(s) for the Lambert Conformal Conic projection, and along the central meridian for the Transverse Mercator projection.)
5. Convert (X, Y) to the units that are locally used. It is the user's responsibility to determine if this procedure is valid. This can be done by comparing computed mapping coordinates to test points where the relationship between ((j), X) and (X, Y) is known, independently.
The above procedure is intended to be used only when the origin of the local map projection coincides with the origin used in this Technical Report. Some Transverse Mercator projections have origins of ((j)o, Xo) instead of (0, Xo) on the equator. For some of these projections, agreement with test points can be obtained by changing the computation of Y in equation set 7.55 to
64
Y = koYTM + N - koS<|>0 (7 5g\
where
ko = scale factor on the central meridian, A-o
YTM = Transverse Mercator Y value computed from equation set 7.8
N = false Northing in meters of the origin
S<|)o = value computed using equation set 7.9 (arc length in meters on the ellipsoid from the equator to the latitude <))o.)
65
8. SELECTION OF TECHNIQUES. SOFTWARE DEVELOPMENT. AND TESTTNH
8.1 The general conversion process. The general coordinate conversion process can be divided into three basic parts:
1. Conversion between map projection coordinates and geodetic coordinates (see Figure 8.1).
2. Coordinate conversion between geodetic coordinates and Cartesian coordinates within a datum (see Figure 8.2).
3. Coordinate transformation between datums (datum shifts) (see Figure 8.3).
The general conversion process which includes all methods presented in this Technical Report, is described graphically in Figure 8.4.
a. Figure 8.1 represents the map projection coordinate (x,y) to the geodetic coordinate ((j),A,) conversion described in Section 7. As noted in Section 7.1, map projection coordinates do not include the height information necessary for finding the geodetic height (H) or elevation (h). For conversions from map projection to geodetic coordinates, the user may have to provide this
. information. Elevations can often be inferred from contour lines on the map or they can be obtained from DTED or direct measurement. Geodetic height may be measured by using GPS or by calculating from an elevation and geoid separation, or for some applications, may be assumed to be 0 (zero).
A x,y -^ ^ (|>, A,
Figure 8.1. Map projection to geodetic coordinate conversion. Height information for H and h is not included in the conversion.
where
x,y = map projection coordinates 0 = geodetic latitude X = geodetic longitude A = map projection coordinates to/from latitude,
longitude conversion
66
b. Figure 8.2 represents coordinate conversions within a datum as described in Section 4.3.4 and 6.5.
X,Y,Z
c
<D, X, h
B
O, X, H
X,
FIGURE 8.2. Coordinate conversions within a datum,
where
X, Y, Z = Cartesian coordinates <|), X, h = geodetic coordinates (latitude, longitude,
geodetic height) H = elevation B = geodetic height to/from elevation conversion C = geodetic coordinate to/from Cartesian coordinate
conversion
Note that a conversion between Cartesian coordinates (X,Y,Z) and local datum coordinates(§,X,H) is not a direct conversion and requires going through the geodetic coordinates ((|),A,,h) .
c. Figure 8.3 represents the datum shift portion of the general conversion process as described in Section 6. Two attributes of figure 8.3 are important to note.
1. Direct local datum to local datum shifts are not performed.
2. Elevations do not change when changing datums. Exceptions to the second statement are explained in Section 6.7.
d. In general, a direct local datum to local datum shift should not be done. Instead, the local datum to local datum shift should be done in two steps.
1. An (old) local datum to WGS 84 shift.
2. A WGS 84 to (new) local datum shift.
67
Local Datum
*;**,*<*£, »
|Ah «*■ II 3
1: H
W6S 84
X, Y, Z
i c
(j), A,, h
H
Figure 8.3. Datum shifts. Note elevation values (H) do not generally change when
changing datums.
where
X, Y, Z <j), X, h
H C
AA BB CC
Cartesian coordinates geodetic coordinates (latitude, longitude, geodetic height) elevation geodetic coordinate to/from Cartesian coordinate conversion Cartesian coordinate-based datum shifts geodetic coordinate-based datum shifts elevations are unchanged
The general conversion process, which includes all methods presented in this Technical Report, is described graphically in figure 8.4. Figure 8.4 is the combination of figures 8.1 through 8.3,
68
l»oeal Datum
1^
I 3
::■:$£ —
WGS 84
X, Y# Z
t° O r A,, h
1- O r A., H
Figure 8.4. The general conversion process,
A
A
where
x, Y X, Y, Z
<l>, A,, h
H A
B C
AA BB CC
map projection coordinates Cartesian coordinates geodetic coordinates (latitude, longitude, geodetic height) elevation map projection coordinates to/from latitude, longitude conversion geodetic height to/from elevation conversion geodetic coordinate to/from Cartesian coordinate conversion Cartesian coordinate-based datum shifts geodetic coordinate-based datum shifts elevations are unchanged
In figures 8.1 through 8.4, coordinate conversions are denoted as a single letter (A, B, C). They are described in Section 7, Section 6.5 and Section 4.3.4, respectively. Datum shifts are denoted as double letters (AA, BB, CO and are described in Section 6.
69
g. The Cartesian coordinate-based datum shifts (AA) include:
Seven-parameter geometric transformation (6.2) AX,AY,AZ direct application (6.3.1 & 6.3.2)
Both of these datum shift methods are geometric transformations.
h. The geodetic coordinate based datum shifts (BB) include:
Molodensky shifts (6.3.3) Ax, AY, AZ three-step method (6.3.1.1 & 6.3.2.1) Direct WGS 72 to WGS 84 transformation (6.4.2)
i. For the purposes of this Technical Report, the elevation to elevation shift (CO is the equality function ( = ); that is, the elevation for the same location on different datums is considered to be the same (H = H) . An exception to this rule is the WGS 72-derived elevations to WGS 84 elevations shift. Section 6.7 describes the elevation to elevation shift process.
8.1.1 Application of the general conversion process. The general conversion process described above can be used to solve coordinate conversion/datum shift problems in five steps:
a. Locate beginning point and ending point of the problem on figure 8.4.
b. Determine valid paths between beginning and end points.
c. Evaluate alternatives and select a path.
d. Locate the needed equations and parameters in this Technical Report.
e. Implement the selected path.
8.1.2 Procedural examples.
8.1.2.1 Example 8.1. Horizontal geodetic coordinate-based datum shifts ( ((|), X,h) LGS to (<t>> A,)WGS 84) (Local datum geodetic heights (hLGS) typically are not available. It is not recommended they be transformed to WGS 84).
This Technical Report has discussed two methods of doing datum shifts. These methods are the three-step method, and Molodensky shifts. Either the three-parameter Ax, AY, AZ shift or the seven-parameter shift may be used in the three-step method. This example corresponds to a "BB" shift of figures 8.3 and 8.4.
70
a. Approach 1 - Three-step method using a seven-paramet-pr Shift T
1. Compute (X,Y,Z)LGs from (<|), X, h)LGS using equation sets 4.4 and 4.5, and the reference ellipsoid parameters from Table A.l.
2. Compute (X,Y,Z)WGS84 from (X, Y, Z) LGS using the seven- parameter shift equation set 6.1 and shift parameters from Table E.l.
3. Compute ((j), X,)WGS 84 from (X,Y,Z)WGS 84 using equation sets 4.6 through 4.10.
b. Approach 2 - Three-step method using AX, ÄY. AZ shifts
1. Compute (X^ZJLQS from (<J), X,h)LGS using equation sets 4.4 and 4.5. Reference ellipsoid parameters are found in Table A.l.
2. Compute (X,Y/Z)WGS84 from (X, Y, Z) LGS using equation set 6.3, and shift parameters from Table B.l
3. Compute (§, X)WGS 84 from (X,Y,Z)WGS 84 and equation sets 4.6 through 4.10.
c. Approach 3 - Using Molodenskv shift
1. Compute A(() and AX from ((j), X, hLGS) using the Molodensky equation set 6.5,shift parameters from Table B.l, and the local datum's reference ellipsoid parameters given in Table A.l.
2. Compute (<j), A,)WGS 84 from (<J), X,)LGS using equation set 6.6.
8.1.2.2 Example 8.2. WGS 84 Cartesian coordinate to local latitude, longitude, elevation.
Several paths for converting (X/Y,Z)WGS84 to «t>LGS ,^LGS,H) can be traced in figure 8.4. The steps of three approaches are provided below.
a. Approach 1 - Using a seven-parameter datum shift
1. Compute (X,Y,Z)LGS from (X, Y, Z)WGS 84 using the seven- parameter shift equation set 6.2 and parameters from Table E.l.
71
2. Compute (()), A,)LGS from (X,Y,Z)LGS using equation sets 4.6 through 4.10 with the parameters of the local ellipsoid.
3. Compute hWGS 84 from (X,Y/Z)WGS 84 using equations 4.7 through 4.11 (Use equation 4.12 in polar regions).
4. Compute H from hWGS 84 and NWGS 84 using equation 5.1,
resulting in § LGS> A,LGS, H.
b. Approach 2 - Using a direct Ax, ÄY, ÄZ datum shift
1. Compute (X,Y,Z)LGS from (X,Y,Z)WGS84 using the AX,
AY, AZ shift equation set 6.4 and shift parameters from Table B.l.
2. Compute (0, A,)LGS from (X,Y ,Z) LGS using equation sets 4.6 through 4.11 with the parameters of the local ellipsoid.
3. Compute h^g 84 from (X,Y,Z)WGS 84 using equations 4.7 through 4.11. (Use equation 4.12 in polar regions).
4. Compute H from hWGS 84 and NWGS 84 using equation 5.1,
resulting in <J> LGS / Ä-LGS / H.
c. Approach 3 - Using a Molodenskv datum shift
1. Compute (()), X,,h)WGS 84 from (X,Y,Z)WGS 84 using equation sets 4.6 through 4.11. (In polar regions, equation set 4.11 should be replaced by equation set 4.12).
2. Compute (([), A,)LGS from (((), A,)WGS 84 using the Molodensky equation set 6.5, equation set 6.7 and shift parameters from Table B.l.
3. Compute H from hWGS 84 and NWGS 84 using equation 5.1,
resulting in (|>LGS, A,LGS, H.
8.1.2.3 Example 8.3. Local latitude, longitude, elevation ((j)LGS, A,LGS, H) to WGS 84 latitude, longitude, elevation
(YWGS 84' ^WGS 84'^) •
Example 8.3 is a extension of example 8.1. Example 8.3 shows the recommended methods for determining geodetic height used in each of the geometric datum shift methods. The steps of three alternate approaches are provided as follows.
72
a. Approach 1 - Using a seven-parameter datum shift.
1. Compute (X,Y,Z)LGS from (<j), X, H)LGS using equation sets 4.4 and 4.5.
2. Compute (X,Y/Z)WGS84 from (X, Y, Z) LGS using the seven parameter shift equation set 6.1 and shift parameters from Table E.l.
3. Compute ((|), X)WGs 84 from (X, Y, Z)WGS 84 using equation sets 4.6 through 4.10.
4. Take H from the input coordinates resulting in <|)WGS 84, ™WGS 84/ H..
b. Approach 2 - Using a AX. ÄY, ÄZ datum shift
1. Compute (X,Y,Z)LGS from (<J), X, H) LGS using equation sets 4.4 and 4.5.
2. Compute (X,Y,Z)WGS84 from (X, Y, Z) LGS using AX, AY,
AZ shift equation set 6.3, and shift parameters from Table B.l.
3. Compute (<]), X,)WGS 84 from (X,Y,Z)WGS 84 using equation sets 4.6 through 4.10.
4. Take H from the input coordinate resulting in §yGS 84, ^WGS 84/ H.
c. Approach 3 - Using a Molodenskv datum shift
1. Compute ((|>, A,)WGS 84 from (()), X, H)LGS using Molodensky equation set 6.5, equation set 6.6, and shift parameters from Table B.l.
2. Take H from the input coordinate resulting in (j)WGS 84,
AWGS 84 < H.
8.2 Choosing a datum shift method. Several factors should be considered in selecting a datum shift method. These factors include: shift parameter availability, timing requirements, accuracy requirements, software availability, hardware limitations, standardization considerations and potential future requirements. If future requirements are considered as part of the system development, future product improvements may be minimized. These requirements may include the need to provide data to other systems and to meet higher accuracy requirements. The discussion below applies to a wide range of Army applications.
73
Special, high accuracy applications are outside the scope of this Technical Report (see Section 1.2).
The three-parameter shift is recommended for mapping and charting and most other military application. For Western Europe (EUR-M), and Ordnance Survey of Great Britain 1936 (OGB-M), STANAG 2211 specifies the seven-parameter shifts. The use of MRE's is not recommended, although it should be noted that STANAG 2211 specifies the use of MRE's for geodetic and geophysical applications where "data is available in geodetic coordinates."
Parameter sets for Ax, AY, Az shifts are found in Appendix B, and seven-parameter geometric shifts are contained in Appendix E. Be sure to use the same method throughout a project, identifying the method chosen in the project documentation so that future users of the data can replicate or backtrace your results.
8.2.1 Parameter availability. For many datums, only mean-
value three-parameter AX, AY, and AZ shifts are given in this Technical Report. Based on system implementation tradeoffs, the system developer can: apply the shifts directly to Cartesian coordinates (see Sections 6.3.1 and 6.3.2); use the three-step method (see Section 6.3.1.1 and 6.3.2.1); or apply shifts to geodetic coordinates using the Molodensky approximations (see Section 6.3.3). For other datums, some combination of seven- parameter geometric shifts, and mean and regional AX, AY, and AZ shifts are available. Select a parameter set from those that meet or come closest to meeting the accuracy requirements of the output product. Base the final selection on standardization requirements and system implementation tradeoffs.
8.2.2 Accuracy. Errors in the shift parameters are the dominant error source in datum shifting. Errors introduced by any of the datum shift algorithms given in this Technical Report are small compared to the errors in the shift parameters and can generally be neglected. However, take care that accuracy is not degraded by the software and hardware implementation of these algorithms.
a. For the large datums,the regional AX, AY, AZ shifts and seven-parameter shifts are generally considered the most accurate, and mean value AX, AY, AZ shifts have the poorest accuracy. The estimated accuracies of the shift parameters are given in Appendices B and E for each listed datum. The errors in geodetic coordinates may be approximated from the AX, AY, AZ errors listed in Appendix B, using the equation set 6.8.
b. The error estimates and general guidance above must be used judiciously. For a given datum, all types of shift
74
parameters were derived from essentially the same set of control data, and errors were estimated by using a comparison to the same data. For small datums, with poorly distributed control, the better fitting shift type may not be more accurate. The seven-parameter model cannot accurately represent datum rotations and scale changes unless the control is geometrically well distributed across the datum.
c. Regional AX, AY, AZ shifts were developed when there were sufficient data to identify regional variations over the datum as a whole. Regional AX, AY, AZ shifts are more
■ accurate than mean-value AX, AY, AZ shifts.
8-2.3 Implementation issues. System implementation issues may influence the selection of transformation method(s).
a. Computational speed and efficiency are important for some applications, such as real-time navigation. The shift type and method might be selected from those that meet accuracy requirements to minimize computational time. Alternately, shifts might be computed in a background process and applied as fixed values for a period of time or over a given area. The system designer must analyze the impact of such design strategies on system accuracy and performance.
b. For some applications, system computational and data storage capability are considerations in selecting and implementing transformations. The system designer must determine that word precision, range, and accuracy of mathematical functions preserve the required accuracy.
c. Code complexity, modularity, and software maintenance are important system design concerns. While the datum shift algorithms themselves are relatively simple, selection logic often is complex.
d. Both future requirements and present needs should be considered. Factors for consideration include: timing requirements, current accuracy requirements, potential future accuracy requirements (such as requirements to provide data for future systems), effects of improved databases, and the effects of improvements in other ■ inputs. In other words, the savings in future planned product improvements should be evaluated against today's costs.
e. It is possible that a software developer may choose to use a transformation technique that is more accurate than is required. In such a case, the results are not necessarily superior to those arising from a less
75
accurate, but acceptable, technique. The data are simply- degraded less by the more accurate method; the transformed data can be no better than the input data. For example, techniques that achieve geodetic accuracy must be used when geodetic accuracy is needed. Such techniques can be used with a 1:250,000-scale product, but the transformed data still have the accuracy of a 1:250,000 product.
8.2.3.1 Preferred shift methods. Preferred shift methods and shift parameters may change as additional information becomes available. Shift parameters and identification information should be treated as data apart from the algorithm code so that they may be updated without modifying and recompiling the code. Some examples of implementation tradeoffs are:
a. If Cartesian coordinate are already available, AX, AY, AZ shifts can be applied directly (see Sections 6.3.1 and 6.3.2) rather than through the Molodensky approximations (see Section 6.3.3).
b. If the program contains algorithms for converting between Cartesian and geographic coordinates (see Section 4.3.4), the three-step method (see Sections 6.3.1.1 and 6.3.2.1) can be used instead of Molodensky approximations (see Section 6.3.3) to apply AX, AY, AZ shifts.
c. Both the Molodensky approximations (Section 6.3.3) and Cartesian-to-geographic conversion equations (Section 4.3.4.2) become singular at the poles. Selection of one technique or the other will reduce the logic needed to guard against singularity. An alternate equation to compute geodetic height from Cartesian coordinates in polar regions is given as equation 4.12.
8.2.3.2 Software testing. When developing software based on methods discussed in this Technical Report, it is advisable to check the software against test points. Test points can be found in Section 8.5.
a. Issues relating to software development and system performance might force ä software developer to use a special-purpose transformation technique that is not mentioned in this Technical Report. The results of the special-purpose software should be compared to those of the techniques in this Technical Report to insure that the differences are acceptable.
b. Another useful debugging aid is circular testing. This is done by using the output from a transformation as the input for the inverse function; in other words trying to verify the original numbers are returned within
76
acceptable accuracy. Since the code for a transformation and its inverse might contain mistakes that cancel each other, circular testing can not prove that code is correct. At best, it can demonstrate that code is faulty.
c. The following is a list of points that might prove troublesome to coordinate transformation software. Test the code at these points, if it is appropriate:
Poles Polar regions Discontinuity in X (transition between 0,360 degrees or -180, 180 degrees) Origin expressed in ((|), X) $ = 0 X = 0 Both hemispheres in <|) (North and South) Both hemispheres in X (East and West) x = 0 y = 0 All four x-y quadrants Discontinuities in x and y at zone boundaries, equator, etc.
8.3 Error analysis. Error analysis is an important part of the system design process. Error analysis will estimate the accuracy of the system's output products. This is a prediction of how well the system will meet its accuracy requirements. The system analyst can use error analysis to perform system implementation studies. The major steps in error analysis are:
a. Develop the system model.
b. Determine the estimated errors for each source in the model.
c. Reformulate the estimated errors to a common description.
d. Propagate the errors through the model.
Each of these topics is discussed below. An example is given in 8.3.5.
8.3.1 .System model. The system model simply is a description of the data used in the process and all the steps (processes) used to process the data. The data include the raw input data and parameters, like datum shift parameters, used in the processing of the data. Both hardware-driven and software- driven processes should be included in the model. For example, errors in digitizing and plotting are primarily hardware related;
77
coordinate conversion and datum shifting are primarily software processes. For the datum shifting and coordinate conversion, the system model will probably be a simple, serial process. That is, the output of one process will be the input for the next, with no feedback to prior processes. For trade-off studies, alternate system models can be developed to evaluate accuracy versus implementation.
8.3.2 Error estimates. Once the system error model has been described, error estimates must be assigned to each process and data source in the model. When this is done, the analyst will quickly determine which error sources can be neglected and which must be propagated through the model.
a. Estimates of the expected errors in the input data may be provided with the data or in product specifications. Also, the original source or intended use of the data may provide an indication of its accuracy (e.g. digital data derived from a cartographic source will have an accuracy no better than the cartographic source).
b. Without proper guidance, it is often difficult to estimate the error in the data. This is especially true for cartographically derived elevations, because map accuracy standards for elevation allow for error in both the horizontal location and the vertical direction. Therefore, the allowable elevation error is dependent on the topography and may vary within a map sheet.
c. In addition to providing datum shift parameters, Appendices B and E give the estimated errors in the parameters for AX, AY, AZ; and seven-parameter shifts, respectively. (Note: Correlations between parameters, which may be significant, were not available. Therefore, they are not given in the appendices.)
d. The accuracy and behavior of the equations presented in this Technical Report are well defined. Table 8.1 identifies the location of accuracy guidance for the coordinate conversion, datum shift, and map projection equations. Additional references describing equation accuracy and behavior are noted in the text and listed in section 2.
e. The errors introduced by system implementation factors • such as computer word size, accuracy and precision of mathematical functions, truncation of terms, etc. must be quantified.
f. The errors introduced by hardware processes must be determined from equipment specifications, analysis, and/or testing.
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Table 8.1. Accuracy guidance for equations
Subject Accuracy guidance section(si
Coordinate Conversion Equations Cartesian/geodetic conversion Height conversion
4.3.4 4.4.3
Datum shift Equations Seven-parameter geometric transformation AX,AY,AZ shifts Direct WGS 72 to WGS 84 transformation Vertical datum shifts
Map projection Equations Mercator projection Transverse Mercator projection Universal Transverse Mercator (UTM) Lambert Conformal Conic projection Polar Stereographic Universal Polar Stereographic
grid
6.2.4 6.3.1, 6 .3.2 6.4.2.2 6.7, 6.7 .1, 6 7. 2
7.2.3 7.3.3 7.4.4 7.5.4 7.6.2 7.7.4
Appendix B Appendix E Section 6.4.2 .2
Shift Parameter Accuracy values
AX,AY,AZ shifts Seven-parameter geometric transformation Direct WGS 72 to WGS 84 transformation
8.3.2.1 Error types. Depending on the source, error estimates may be described in several ways. Error descriptions may have a bias component and a random (noise) component. A set of error estimates describing the accuracy in multiple dimensions, for example, G^, (JAY, (J^, may have correlations between the parameters. Generally, with MC&G products, the error descriptions do not differentiate between biased, correlated, and independent random errors; the assumption is made that the error estimates are unbiased, independent and random. The resultant error analysis will be less accurate than if the true statistical natures of the error sources were known.
8.3.3 Reformulation. Once the error sources have been quantified, they must be converted to a common frame of reference. The form used for the system output error estimate is often a convenient frame of reference.
79
a. The reference for the error estimates must be established (i.e. the ellipsoid or the map sheet). It may be necessary to scale from one reference to another.
b. The error estimates should be converted to the same units.
c. The datum shift parameter error estimates, GM, GAY, GAZ,
given in Appendix B, can be reformulated to GA(j,, G^, G^
using equation set 6.8.
d. Error estimates typically are linear, circular, or spherical statistical values with associated probability levels. Error estimates should be converted to the same type of statistic at the same probability level. Two independent linear standard deviations (68.27% probability), Gx and Gy, can be converted to a circular standard deviation (39.35% probability), using
if the values of Gx and Gy are reasonably close. Factors to convert between different levels of probability are given in Table 8.2.
8.3.4 Error propagation. Assuming the error estimates are statistically independent and have been converted to the same units and statistical type, they may be propagated through a simple, serial system model using the Root-Sum-Square (RSS) technique.
l
RSS = [{NJ2 + (N2)2 + + (Nn)
2]2 (8.2)
If error estimates have known biases or correlations, then a more sophisticated error propagation technique should be used.
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Table 8.2. Probability Level Conversion Factors
Notation
Error Form
Standard Error (One Sigma)
Probable Error
Map Accuracy Standard
Near Certainty Error
Linear O (68.27%) PE (50%) MAS (90%) 3 a (99.73%)
Circular ac (39.35%) CEP (50%) CMAS (90%) 3.5GC(99.78%)
Spherical as (19.9%) SEP (50%) **SAS (90%) 4(JS (99.89%)
**SAS = Spherical Accuracy Standard
Linear Error (One dimensional)
To From
50% 68.27% 90% 99.73%
50% 1.0000 1.4826 2.4387 4.4475 68.27 0.6745 1.0000 1.6449 3.0000 90 0.4101 0.6080 1.0000 1.8239 99.73 0.2248 0.3333 0.5483 1.0000
Circular Error (Two dimensional)
To From
39.35% 50% 90% 99.78%
39.35% 1.0000 1.1774 2.1460 3.5000 50 0.8493 1.0000 1.8227 2.9726 90 0.4660 0.5486 1.0000 1.6309 99.78 0.2857 0.3364 0.6131 1.0000
Spherical Error (Three Dimensional)
To From
19.9% 50% 90% 99.89%
19.9% 1.000 1.538 2.500 4.000 50 0.650 1.000 1.625 2.600 90 0.400 0.615 1.000 1.600 99.89 0.250 0.385 0.625 1.000
(Note: Errors are assumed to be random and uncorrelated.)
8.3.5 Example. The commander wants a printed product showing the transportation routes near Seoul, Korea with WGS 84, UTM coordinates, at a scale of 1:50,000. The data source is a Class A, l:50,000-scale topographic map on the Tokyo datum. The product will be created by digitizing the transportation routes on the map sheet, shifting the coordinates from the Tokyo datum to WGS 84 using the three-step method, and plotting the shifted data.
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a. The system model is shown in Figure 8.5. In this model, it is assumed that computer-related issues will not degrade coordinate conversion and datum shift calculations. Also, it is assumed that the representation of the digitized data will not change through the rest of the process. Often this assumption is not valid because digital data representation must be changed to a predetermined format. Such data modifications can introduce error.
b. The error estimates for each of the elements in the model are
piemen t Error
Map accuracy (horizontal) 0.02" CMAS(90%) Digitizer ±0.005" UTM to Geodetic conversion <1 cm Geodetic to XYZ conversion 0.0 Datum shift equations 0.0 Datum shift parameters: G^ 5 m
(7AY 3 m
GAz 3 m XYZ to Geodetic conversion 0.1m Geodetic to UTM conversion <1 cm Plotter ±0.01"
Only the map, digitizer, datum shift parameter, and plotter errors are significant and will be considered in the rest of the analysis.
c. The error estimates must be reformulated to the same units and statistical basis. Since the commander wants to know the estimated error on the ground, we will convert to circular error, in meters, on the ellipsoid, at 50 percent probability (CEP).
1. Map error must be scaled from the map to the ellipsoid and changed from CMAS to CEP.
Map Error = 0.02" CMAS x 0.0254 m/in x 50,000 = 25.4 m CMAS x 0.5486 = 13.9 m CEP
2. It is assumed that the digitizer error is a near certainty, circular error specification that must be scaled from the map to the ellipsoid and converted to CEP.
Digitizer error = 0.005" max x 0.0254 m/in x 50,000 = 6.4 m max x 0.3364 = 2.1 m CEP
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3. The datum shift parameter error estimates, O^, GAY,
GAz, must be converted to O^ and G^, then to CEP.
Using the above values for G^, 0AY, and GAz, (j) = 37.5°
and A, = 127°, equation set 6.8 yields GA(^ = 3.34 m
and G&X = 4.38 m.
m CEP = 1.1774 x Gc - 1.1774 x (GA<t, + 0^/2 = 4.5
4. It is assumed that the plotter error is a near certainty, circular error (99.78% probability) specification that must be scaled from the map to the ellipsoid and converted to CEP.
Plotter error = 0.01" max x 0.0254 m/in x 50,000 = 12.7 m max x 0.3364 = 4.3 m CEP
5. The RSS error for the complete system model is
l
RSS = [(13.9)2 + (2.1)2 + (4.5)2 + (4.3)2]2
= 15.4 m CEP
d. From the above analysis, it is evident that the error in the source data is the prime contributor to error in the output product for this application. For other types of applications, such as surveying, this may not be the case.
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Figure 8.5. System Model
Digitize
I Convert UTM to
Geodetic
I Convert Geodetic
to XYZ
I Shift Tokyo Datum
to WGS 84
I Convert XYZ to Geodetic
I Convert Geodetic
to UTM
I Plot
I Transportation
Route Map
Datum Shift Parameters AX, AY, AZ
84
8.4 Numerical Examples.
8.4.1 Example 8.4. Convert WGS 84 geodetic coordinates (<|),A,,h)WGS 84 to Universal Transverse Mercator coordinates
^XUTM'YUTM) in ths NAD 27 reference system using a three-step AX,
AY, AZ datum shift.
This example converts WGS 84 geodetic coordinates to Universal Transverse Mercator projection coordinates in two steps. First, the WGS 84 geodetic coordinates are converted to NAD 27 geodetic coordinates using the three-step AX, AY, AZ datum shift. Geodetic height is needed for this conversion. Next, the NAD 27 geodetic latitude and longitude are converted to UTM coordinates. Geodetic height is not needed for this step.
Beginning WGS 84 coordinates:
$ = 42° 56' 52.163"N = 0.749580918883376 radians
X = 288° 22' 24.326"E = 1.891473118 radians
h = 203.380 m
Step 1: Convert WGS 84 geodetic coordinates to NAD 27 geodetic coordinates using the three-step AX,AY,AZ datum shift.
1.1 Convert ((|),A,,h)WGS 84 to (X,Y,Z)WGS 84 using equation sets 4.4 and 4.5.
a = 6378137
f = 0.00335281066474
£2 = 2f - f2 = 0.00669437999013
b = a(l - f) = 6356752.3142
RN = . a 6388070.57383009 "VI - £2 sin2 0
XWGS 84 = (RN + h) cos0cosA,= 1473933.5413
YWGS 84 = (RN + h) cos<|>sinA,= -4437679.0666
85
ZWGS 84 = U2 RN + h) sin(|) = 4323399.2717
1.2 Convert (X/Y,Z)WGS84 to (X/Y,Z)NAD27 using equation set
6.4 and AX,AY,AZ datum shift values from Table B.l NAS-C.
XNAD 27 = XWGS 84 - AX = 1473933.5413 - (-8.0) = 1473941.5413 m
YNAD 27 = Y
WGS 84 - AY = -4437679.0666 - 160.0 = -4437839.0666 m
ZNAD 27 = Z
WGS 84 - AZ = 4323399.2717 - 176 = 4323223.2717 m
1.3 Convert (X,Y,Z)NAD27 to (()), A,)NAD 2i following equations given in Section 4.3.4.2.
^NAD 27 = arctan ( ^D 27"| = 5.033056560575 radians VANAD 27 )
Calculating $ is an iterative process.
1.3.1 Compute an initial approximation to a variable ß , using equation 4.7. Parameters a and b for the Clarke 1866 ellipsoid are found in Table A.
ßo = arctan [ ,a ] = 0.74788820090716142 radians b VX2 + Y2
1.3.2 Substitute ß0 for ß in equation 4.8 to get an
approximation for <j>.
. f Z + £'2 b sin3 ß 1 9 = arctan [-== K—-J = 0.74958142853674489 radians
VX2 + Y2 - a E2 cos3 ß
e2 = 2f - f2 = 0.'0067686579973
and e2
£'2 = -7 = 0.0068147849459 l - e2
For most applications, this first iteration is sufficient.
Step 2: Convert the NAD 27 geodetic latitude and longitude to Universal Transverse Mercator projection coordinates.
2.1 Find the UTM zone z and the central meridian X0 from equations 7.22 and 7.23.
86
flSO X - 29
v 6Jü z = greatest integer <
z = 19
A,0 = (6z + 177) j^ = 5.078908123303 radians
2.2 Calculate Transverse Mercator projection coordinates (XTM'Y
TM) using equations 7.8 and 7.9.
2.2.1 Calculate terms in equation set 7.9 using 82 and £'2 as calculated in Step 1.3.2.
*• - X " «* - hl - ifi66 - 1635388 •- "-998305681856
A2 = s r2 + I8' + 12886 " «H6") = 0.2542555420215 x 10"2
T K /" ^ ' r7p7 \ A4 = 256 (£4 + 4s6 " nä88) = 0.2698010542695 x 10"*
35 / 41 \ Aß = 3072 (£6 " 32s8) = 0.3502448582027 x 10-*
A8 = - ±1^2 £8 = -0.5044416191579 x 10"11
S(|) = a [A0<|) - A2 sin 2(|) + A4 sin 4(j) - A6 sin 6(|) + A8 sin 80]
S(j)= 4756711.680282
N = RN'= . = 6388250.562973 Vl - £2 sin2 (j)
Note that N = RN differs from RN calculated in Step 1.1, as Step 2 is in the NAD 27 reference frame and Step 1.1 is in the WGS 84 reference frame.
1\ = E'cos <}) =0.6042571576496 x 10"1
t = tan$ = tan(0.74958094822852639) = 0.9308149267480
K = X-XQ= -0.04585156272818 radians
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2.2.2 Calculate the Transverse Mercator projection coordinates (x ,y ).
xTM = N A cos(j) + 7 (1 - t2 + T|2)
N A5cos54> 120 (5 - 18t2 + t4 + 14T12 - 58t2T|2;
N A2 . , . yTM = SA + —2— sin Y
cos 9
N A4
+ 2A sin 0 cos3 <])(5 - t2 + 9T|2 + 4T|4)
N A6
+ -^20" sin <|>cos5 (J) (61 - 58t2 + t4 + 270T12 - 330t2Tl2)
XrM = -214408.9715411
yTM = 4760061.989246
2.3 Convert the Transverse Mercator projection coordinates (x-m,y-m) to Universal Transverse Mercator projection coordinates (XUTM/YUTM) using equation set 7.24 (Northern hemisphere).
XUTM = 0.9996 xTO + 500,000 = 0.9996 (-214408.9715411) + 500,000
YUTM = 0.9996 yTM = 0.9996 (4760061.989246)
The resulting Universal. Transverse Mercator coordinates (xUTM,yUTM) in the NAD.27 reference system are:
xUTM = 285676.792
yUTM = 4758157.964
8.4.2 Example 8.5. Convert WGS 84 geodetic coordinates (<|),A,,h) to geodetic coordinates (<J),A,,h) in the NAD 27 reference system using the standard Molodensky method.
This example converts WGS 84 geodetic coordinates to NAD 27 coordinates using the standard Molodensky method.
Beginning WGS 84 coordinates:
Y = 42° 56'52.163" N = 0.749580918883376 radians X, = 108° 22'24.326" W = 1.891473118 radians h = 203.380
88
Step 1: Compute Aa, Af, W2, M, N, £2, and £'2 using equation set 6.5. Obtain aWGS 84, ANAD 27, fWGS 84, and fNAD27 from Table A.l.
Aa = aWGS 84 - aCLARKE 1866 = -69.4
Af = fwGs 84 - fCLARKE 1866 = 3.726463863 x IO-5
£2 = 2f - f2 = 0.006694379990
£2 E'2 = -—~ = 0.006739496742
i-e2
W = Vl " £2 sin2<j) = 0.9984449805
a W
N = RN = - = 6388070.574
a(i - e2) M = —vP = 6365086-681
Step 2: Compute A(|) and AA,, and Ah using equation set 6.5, and
AX, AY, and AZ datum shifts from Appendix B for datum code NAS-A.
In this example, the signs of AX, AY, and AZ are reversed from those of Appendix B due to the direction of the datum shift (WGS 84 to local coordinates).
A()) = [ (-sin(|)cosAAx - sin(|)sinAAy + cos(j)Az + r^ ^Aa +
sin(|)cos(j){2N + £'2Msin2<|)) (1 - f)Af ] / [M + h]
A(j) = 0.131"
. [-sinXAx + cosXAy] A^ = —77T,—r:—rr^ =2.615" {(N + h)cos(j)]
Ah = cos(|)cosAAx + cos^sinXAy + sin<|)Az - WAa + — sin2f Af
Ah = 28.65 m
Step 3: Compute the NAD 27 geodetic coordinates ((|),A,,h) from equation set 6.7 that results in the following valuesl;
89
^NAD 27 = <1>WGS 84 + A(J) = 42° 56'52.294 "N
KhD 27 = ^WGS 84 + A^= 108° 22'21.711 "W
^NAD 27 = HWGS 84 + AH = 232.03 m
8-4.3 Example 8.6. Convert WGS 84 geodetic coordinates (<J), ^)WGS 84 to Mercator projection coordinates (x,y) .
This example converts WGS 84 geodetic coordinates to Mercator projection coordinates within the WGS 84 reference system. Geodetic height (h) is not used in this conversion. The Mercator projection central meridian X0 is 288° = 5.026548246 radians
Beginning WGS 84 coordinates are
<1>WGS 84 = 42° 56' 52.163 "N = 0.7495809189 radians
Awss 84 = 288° 22' 24.326"E = 5.033057346 radians
Step 1: Use equation 7.1 and WGS 84 datum parameters ä and f from Table A.l to calculate x.
x = a (A, - Xo) = 6378137(5.033057346 - 5.026548246) = 41569.357
Step 2: Calculate £ from f using equation 4.9, then use equation 7.2 to calculate y.
£ = V2f - f2 = 0.0818191908426
y = a In tan (5 + f) • f 1 - e Sin,* .) V4 2) \x + e Sin (j)^
E
= 5274911.868
The resulting Mercator projection coordinates (x,y) are
x = 41569.357
y = 5274911.868
8.4.4 Example 8.7. Converst Mercator projection coordinates (x,y) in the WGS 84 reference system to WGS 84 geodetic latitude and longitude i§,X) .
90
This example converts WGS 84 Mercator coordinates to WGS 84 latitude and longitude.
Beginning Mercator coordinates and central meridian (A,0) :
x = 41569.3572 Y = 5274911.8684 XQ = 288°E = 5.0265482457 radians
Step 1: Compute the longitude (X) using equation 7.4 where a is the semi-major axis of the WGS 84 ellipsoid listed in Table A.l.
^ = ~ + ^o =5.0330657221 radians = 288° 22'24.326"E or
71° 37' 35.674"W
Step 2: Compute an intermediate value t from equation 7.5.
v , 5274911.8684 t = e<" t> = el" 6378137 ' = -43734619376064
Step 3: Compute an initial value of the geodetic latitude (<J>0) from equation 7.6.
9o = 2" - 2 arctan(t) = 0.746233651 radians
Step 4: Iterate equation 7.7 until the updated value of (|) is sufficiently close to the previous value of (j).
£2 = 2f - f2 = 0.00669437999013
£ = 0.0818191908426
t TO+1 = T " 2 arctan ( rl - £ sin d)n"
1 - ^
_1 + £ sin ())r 2
)
<|>1 = 0.7495688569418636
The geodetic latitude will be given by the last value of (j) calculated.
Final (j) = 0.7495809188833704 radians = 42° 56- 52.163 "N
The resulting WGS 84 geodetic latitude and longitude ((j),A,) are:
<|>= 42° 56' 52.163 "N
91
X = 288° 22' 24.326"E or 71° 37' 35.674"W
8.4.5 Example 8.8. Convert universal Transverse Mercator coordinates (xUTM,yuTM) in the NAD 27 reference system to NAD 27 geodetic latitude and longitude ((|), X) .
This example converts NAD 27 UTM coordinates to NAD 27 geodetic latitude and longitude. The UTM coordinates are converted to Transverse Mercator projection coordinates. The Transverse Mercator projection coordinates are then converted to NAD 27 latitude and longitude.
Beginning coordinates are XUTM
= 285/677.332 YUTM = 4,758,154.856 Zone = 19
Step 1: Convert Universal Transverse Mercator coordinates (XUTM#YUTM)
to Transverse Mercator coordinates (xTM,yTM) using equation set 7.25.
1 X™ = o 9996 (Xu™ " 500,000) = -214,408.4314
y™ = 0.9996 Yu™ = 4,760,058.8796
Step 2: Convert Transverse Mercator coordinates (XTM,VTM) to geodetic latitude and longitude(0, X).
2.1 Calculate the UTM zone central meridian (X0) using equation 7.23.
X0 = (6z + 177)[^QJ = 5.078908123303 radians
2.2 Compute the eccentricity (£) and second eccentricity (£') of the Clarke 1866 ellipsoid, the reference ellipsoid for NAD 27, using equation set 7.9.
£2 = 2f - £2 = 0.6768657997291 X 10"2
£2 8'2 = ~ = 0.6814784945915 X 10~2
i - e2
2.3 Calculate the subscripted-A terms of equation set 7.9, using e2 calculated in Step 2.1. The subscripted-A terms are used in the calculation of the footpoint latitude.
92
A° = 1 - 4 £2 " h £4 - 256 £6 " 16384 £* = 0.998305681856
A2 = $
A4 =
A6 = J\J i
315
0i = ^~ = 0.7463005398433 radians a.
g f£2 + - £4 + — £6 _ j22-. £8j = 0.2542555420215 X 10"2
2^g [£4 + 4 £6 - ^2g £8) = 0.2698010542695 X IQ"5
3072 f£6 " 32 £81 = 0-3502448582027 X 10"8
A8 = - 1331
10
572 S
8 = " 0.5044416191579 X 10"11
2.4 Compute the footprint latitude, <j)i( using equation 7.10 and a from Table A.l.
Make an initial estimate for 0i:
a
2.5 Calculate S$ using equation set 7.9 and the estimate of 0, 0L
SA = a[A0(|).- A2 sin 20 + A4 sin 40 - A6 sin 60 + A8 sin 80]
SA = 0.4735829138159 X 107
2.6 Calculate S'A using equation set 7.11 and the estimate of 0, 01.
S'A = a[A0 - 2A2cos 20 + 4A4 cos 40 - 6A6 cos 60 + 8A8 cos 80]
S'A = 0.6364798134276 X 107
2.7 Find an updated value of the footpoint latitude, 0r using equation 7.10.
0i (updated) = 0i (current) V: = 0.7501073756426 radians
2.8 Iterate Steps 2.5 through 2.7 until successive values of 0i are as close as is desired.
93
For this example, the final value of fa is:
Final fa = 0.7501073020563 radians
2.9 Calculate values used in equation set 7.12 using equation set 7.13.
ti = tan fa = tan (0.7501073020563)
Ni = ■ 6,388,261.949762 Vl - E2 sin2 fa
a (1 - £2) Ri = V = 6,365,044.225886
Wl - E2 sin2 fa )
T|i2 = £'2 cos2 fa = 0.3647692467262 X 10"2
B3 = 1 + 2ti2 + Tli2 = 2.740138642336
B4 = 5 + 3ti2 + Tli2 - 4TV " 9t!2!!!2 = 7.579827062321
B5 = 5 + 28ti2 + 24tx4 + 6T]i2 + St!2!]!2 = 47.45050110665
B6 = 61 + 90ti2 + 46TU2 + 45tx4 - 252ti2TU2 - 3TJi4 - eoti2!^4 -
90ti4Tli2 + 225t14Tl1
4 = 172.1890097418
2.10 Calculate the geodetic latitude and longitude (<j),A,) using equation set 7.12.
A rh Zl Nl I"! C2Lf *± (2LV B6 sx\e 1 * = ^ " Rx L2 UJ ' 24 UJ + 720 fej J
(|) = 0.74958094314009481 radians = 42° 56' 52.168"
X - Xo + sec fc [| - f gj + ^ (f J ]
A, = 5.03305669671 radians = 288° 22' 22.464"
8.4.6 Example 8.9. Convert WGS 84 latitude and longitude (<j),A,) to Lambert Conformal coordinates (x,y) using two standard parallels.
This example converts WGS 84 latitude and longitude to Lambert conformal projection coordinates in the WGS 84 reference system.
94
Beginning WGS 84 coordinates are
<t>wGs 84 = 42° 56' 52.163"N = 0.749580918883376 radians
'Kics 84 = 288° 22' 24.326"E = 5.033065722110382 radians
Standard parallels:
<h = 42° 30- 00.000» = 0.7417649320975901 radians
(1)2=43° 00- 00.000« = 0.7504915783575617 radians
Origin:
(J)0= 42° 30- 00.000« = 0.7417649320975901 radians
XQ= 288° 00- 00.000« = 5.026548245743669 radians
Step 1: Calculate eccentricity £ from flattening f.
£ = ^2f - f2 = 0.0818191908426
Step 2: Calculate q at 0, <j>!, <j)2 and (])0 using equation set 7.27
■ x . ' , [ ft $i\ A - e sin (M f q 0i = In tan \- + -^H • ;—**■ 2
where i = null, 1, 2, 0
q(<}>) = 0.8270301921136922
q«t>!) = 0.8164293462094778
q((|)2) = 0.8282703592274435
q(<j)0) = 0.8164293462094778
Step 3: Calculate N at <]>! and <j)2 using equation set 7.27 and a from Table A-l.
N«^) = a ^ •VI - £2 sin2 (^
95
N(0!) = 6387903.46785173
N((j)2) = 6388090.05758619
Step 4: Calculate L from equation set 7.27.
ln(Nicos0i) - InfNocosdb) L = x— = 0.6788029306559900
02 - qi
Step 5: Calculate K from equation set 7.27.
N, cos (b-, Nn cos (b-, K = L e-Lq = L e-Lq = 12076169.25203636
Step 6: Calculate r and r0 from equation set 7.27.
r = KE-^W = 6888432.18075871
r0 = K£-L<i<«y = 6938179.32109766
Step 7: Calculate the Lambert coordinates (x,y) using equation set 7.26.
x = r sin L(A,- X0) =30474.8898082
y = r0 - r cos L(X - X0) = 49814.5521555
8.4.7 Example 8.10. Convery Lambert Conformal projection (with two standard parallels) coordinates (x,y) in the WGS 84 reference system to WGS 84 geodetic latitude and longitude ((j),A,) .
This example converts WGS 84 Lambert Conformal coordinates to WGS 84 latitude and longitude.
Beginning Lambert Conformal coordinates, standard parallels (<}>!, <|>2) and origin ((J)oA0) are
x = 30474.890 y = 49814.552 (]>! = 42° 30'00" = 0.7417649320976 radians (j)2 = 43° OO'OO" = 0.7504915783576 radians <j)0 = 42° 30'00" = 0.7417649320976 radians Xn = 288° OO'OO" = 5.0265482457437 radians
96
Step 1: Compute E, L,r0,k,N1#N2,q^qj, and q2 using equation set 7.27. Obtain a from Table A.l.
£ = V2f - f2 = 0.0818191908426
f2£ to fl - £ sin 0^ Qi = In tan (?-*)-(ff £ sin (Pi^
2
Qo = 0.8164293462094778
<2i = 0.8164293462094778
<& = 0.8282703592274435
„ a
yi - £2 sin2 (jJi
Nx = 6387903.467851727
N2 = 6388090.057586191
In(Nicoscbi) -In(N?cosd)7) L = ~ *J :u- = 0.6788029306559713
<22 "Qi
Nicosd)-, N?cos(l>.> K = TQ La = T r. La = 12076169.25203651 Le ^ Le~Ls2
Step 2: Compute r0 using equation set 7.27.
r0 = Ke-^'ty = 6938179.321097849
Step 3: Compute r and 0 from equation set 7.29.
0 = arctan X_ = 4.4240820859501107 x 10"3 radians
r = ——r^ = 6888432.18091526 COSÖ
Step 4: Compute geodetic longitude (A,) using equation set 7.28.
X = - + %0 = 5.0330657221511255 radians = 288° 22' 24.326"
97
Step 5: Compute q using equation set 7.28.
In (£) q = ^— = 0.8270301920802524
Step 6: Find an initial approximation (§') for the geodetic latitude from equation set 7.30.
<J>' = 2 arctan (e*) - j = 0.746233650926621
Step 7: Compute f(()>') and f'(0') from equation set 7.30.
l+sin«!)' > (\ - £ sin 0' ^ JL-sin(J)' J ^1 + e sin (j)1^
E f ((J)') = -q + j In
f((t>') = -4.5493507336696604 x 10"3
l - e2 f'(ty') =— —~ . , . t T- = 1.357001593456580 (1 - £2 sin2 (J)' )cosq)'
Step 8: Find an updated approximation (<j>") for the geodetic latitude using equation 7.31.
4>" = 0' " ~ VA \ = 0.7495861530575 f ■ ((j)' )
Step 9: Iterate steps 7 and 8 until successive values of (j)" are sufficiently close after many iterations.
Final ((|)). = 0.74958091885881052 radians = 42° 56' 52.163".
The resulting WGS 84 coordinates are:
<|) = 42° 56' 52.163"
X = 288° 22' 24.326"
8.4.8 Example 8.11. Convert geodetic latitude and longitude ((j),A,) on the International Ellipsoid to Universal Polar Stereographic coordinates (x,y) .
This example converts geodetic latitude and longitude to UPS projection coordinates. Coordinates are computed on the International Ellipsoid.
98
Beginning coordinates are
$ = 87° 17' 14.400" S = -1.523451361952464 radians X = 132° 14' 52.303" E = 2.308160619653466 radians
Step 1: Compute the eccentricity (£) of the International Ellipsoid using equation set 7.38 and a and b from Table A.l
£2 = 2f - f2 = 0.0067226700623 £ = 0.08199189022
Step 2: Compute k using equation set 7.38.
e 2a2 r(l - £h;
K = -£- | — —]2 = 12713920.17558161
Step 3: Compute r using equation set 7.38.
r = (% _|Ar\ (1 + £ sin I (b I ^ f = Ktan [j - -*- ' k ~ : TT7 2 = 303059.087879879
V4 2 J ^l - e sin l(|>lj
Step 4: Compute UPS coordinates using equations 7.49 and 7.50.
x = 2,000,000 + 0.994 r sin X = 2222991.410
y = 2,000,000 - 0.994 r cos U 1797464.051
8.4.9 Example 8.12. Convert Universal Polar Stereographic coordinates (X,Y) to geodetic latitude and longitude (0,A,).
This example converts UPS coordinates calculated on the International Ellipsoid to geodetic latitude and longitude on the International Ellipsoid.
Beginning UPS coordinates in the Southern hemisphere are
X = 2222991.410
y = 1797464.051
Step 1: Convert Universal Polar Stereographic coordinates (XUPS,YUPS) to Polar Stereographic coordinates (XPS,YPS) .
99
_ XTTPs - VOODOO = 224337.434205231 A
PS - 0.994
_ YITPS ~ 2000000 = _203758,500100604 XPS ~ 0.994
Step 2: Compute the longitude (X) from equation 7.39.
Xpg A, = TC + arctan Y
A, = 2.308160619653466 radians = 132° 14' 52.303" E
Step 3: Compute k using equation 7.41, and a and b from Table A.l.
e v _ 2§i r (1 - £)-|2 = 12713920.17558161 * " b L (l + er
Step 4: Compute q using equation set 7.40.
IYIPS e-« = K I cos X\
q = 3.73652495299514
Step 5: Compute an initial estimate (<J)0) of the geodetic latitude using equation set 7.42.
(J)o = 2 arctan e* - f = 1.52313176628106 radians
Step 6: Compute f«j)0) and f(^0) using equation set 7.42.
1 f «)>) = -q + 2 In
(1 + sin 4A (1 - 8 sin (JA e ll - sin <j)J 11 + 6 sin (|)J
f«j)0) = -0.006730109061328
f,/(h^ _ 1 ~ ^ -=20.9875717033244 V " (1 - e2 sin2(j)) cos9
Step 7: Compute an updated geodetic latitude (^ using equation
set 7.43. . _ x _ f ^n-i> = 1.523452437445710 <Pn - «Pn-l f , «j)^)
100
Step 8: Iterate steps 6 and 7 until successive values of the geodetic latitude are sufficiently close.
Final (<(>) = 1.523451361952464 radians = 87° 17' 14.4"S.
The resulting geodetic latitude and longitude (<j),A,) coordinates are:
0 = 87° 17' 14.4 "S
X = 132° 14'52.303 "E
8.5 List of test points. The test points listed here are taken from DMA TM 8358.2 (1989) and Snyder (1987).
Mercator
Central Meridian = 180° Ellipsoid = Clarke 1866
<)) = 35° N 51 = 75° W (285° E)
x = 11,688,673.7 m y = 4,139,145.6 m
k = 1.2194146
Universal Transverse Mercator
Central Meridian = 285° E (75° W) Ellipsoid = Clarke 1866
0,= 40° 30' N X = 73° 30' W (286° 30' E)
x = 627,106.5 m y = 4,484,124.4 m
k = 0.9997989
Lambert Conformal Conic
Ellipsoid = Clarke 1866 Standard parallels = 33° N
= 45° N
Origin: <J)0 = 23° N
Xn = 96° W (264° E)
101
<}) = 35° N i, = 75° W (285° E)
x = 1,894,410.9 m y = 1,564,649.5 m
k = 0.9970171
Universal Polar Stereographic
Ellipsoid = International
<))'= 87° 17' 14.400-S X = 132° 14' 52.303 "E
y = 1,797,464.051 m x = 2,222,991.410 m
102
9. NOTES
9.1 Intended use. This Technical Report provides guidance on coordinate conversion and datum shifting for a wide range of Army mapping, charting, and positioning applications. Guidance is provided to aid the Army and others in developing software for applications of datum shifts and other transformations and in implementing this software in support of operational units.
9-2 Subject (kev word) listing
Cartography Charts Coordinate Systems Datums Elevation Geodetic Satellites Geodesy Geodetic Coordinates Geographic Coordinates Grids Hydrographie Surveying Latitude
. Longitude Map Projection Mapping Maps MC&G Mean Sea Level Military Standards Photogrammetry Projections Surveying Topographic Maps Topography UTM
103
APPENDIX A
REFERENCE ELLIPSOID PARAMETERS
10. GENERAL
10.1 Scope. This appendix identifies reference ellipsoids with associated geometric parameters.
20. APPLICABLE DOCUMENTS.
The ellipsoids and associated parameters are taken from DMA TR 8350.2 (Edition 2, Insert 1.) Note: Use semi-major axis a and flattening f to calculate E and E'. Do not use semi-minor axis b, which is a derived geometric constant.
30. GENERAL REQUIREMENTS.
30.1 Application. The ellipsoid furnishes a simple, consistent, and uniform reference system for geodetic coordinates.
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APPENDIX B
AX, AY, and AZ DATUM SHIFT PARAMETERS
10. GENERAL
10.1 Scope. This appendix identifies the ellipsoid center shift transformation parameters used to transform a local geodetic system to WGS 84. An identifier, the reference ellipsoid, AX, AY, AZ
shifts, and the estimated errors (G) in the shifts are listed for each local geodetic system.
20. APPLICABLE DOCUMENTS.
The datum shift parameters and associated information for each local geodetic system are taken from DMA TR 8350.2 (Second Edition, Insert 1.)
30. GENERAL REQUIREMENTS
30.1 Application. The AX, AY, and AZ shifts are used for datum transformations using the three-step method discussed in section 6.3, and shown by equation sets 6.3 and 6.4.
The following symbology is used in Table B.l.
*. WGS 84 minus local geodetic system.
** Derived from non-Dobbler sources, accuracy unknown.
*** Derived from a single Doppler station, accuracy not verified.
+ See Appendix F to transform Old Hawaiian Datum coordinates on the International Ellipsoid.
++ Also known as Hito XVIII, 1963.
+++ Use A = 6,377,483.865 meters for the Bessel 1941 Wllipsoid in Namibia.
& Derived from non-satellite sources, accuracy unknown.
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APPENDIX C
DATUM LIST
10. GENERAL
10.1 Scope. This appendix consists of a list of countries and geographic areas and their associated datums.
20. APPLICABLE DOCUMENTS.
This section is not applicable to this appendix.
30. GENERAL REQUIREMENTS
30.1 Application. The intended use of this appendix is to establish a table of countries or geographic areas for quick reference to their respective datum.
129
TABLE C-l
Countries and Their Associated Datums
COUNTRY/GEOGRAPHIC AREA DÄ12M
Afganistan
Alaska
Alberta
Antigua
Argentina
Ascension Island
Australia
Austria
Bahama Islands
Bahrain Island
Baltra, Galapagos
Islands
Bangladesh
Barbados
Barbuda
Herat North
North American 1927, 1983
North American 1927, 1983
Antigua Island Astro 1943
North American 1927
Campo Inchauspe,
South American 1969
Ascension Island 1958
Australian Geodetic 1966, 1984
European 1950, 1979
Cape Canaveral,
North American 1927
Ain el ABD 1970
South American 1969
Indian
North American 1927
North American 1927
TABLE C-l (cont'd)
Countries and Their Associated Datums
COUNTRY/GEOGRAPHIC AREA DATUM
Belgium European 1950
Belize North American 1927
Bolivia Provisional South American
South American 1969
1956,
Botswana ARC 1950
Brazil Corrego Alegre,
South American 1969 British Columbia North American 1927, 1983
Brunei Timbalai 1948
Burkina Faso Adindan
Point 58
Burundi ARC 1950
Caicos Islands North American 1927
Cameroon Adindan
Minna
Canada North American 1927, 1983
Canal Zone North American 1927
Canary Islands Pico de Las Nieves
TABLE C-l (cont'd)
Countries and Their Associated Datums
COUNTRY/GEOGRAPHIC AREA DATUM
Caribbean
Caroline Islands
Cayman Brae Island
Central America
Chatham Island
Chile
Cocos Island
Colombia
Congo
CONUS
Corvo Island (Azores)
Costa Rica
Cuba
Cyprus
North American 1927
Kusaie Astro 1951
L.C. 5 Astro 1961
North American 1927, 1983
Chatham Island Astro 1971
Provisional South American 1956,
South American 1969, PS Chile 1963
Anna 1 Astro 1965
Bogota Observatory,
South American 1969
Point Noire 1948
North American 1927, 1983
Observatorio Meteorologico 1939
North American 1927
North American 1927
European 1950
TABLE C-l (cont'd)
Countries and Their Associated Datums
COUNTRY/GEOGRAPHIC AREA DATUM
Denmark European 1950
Diego Garcia ISTS 073 Astro 1969
Djibouti Aybella Lighthouse
Dominican Republic North American 1927, 1983
East Canada North American 1927, 1983
East Falkland Island Sapper Hill 1943
Easter Island Easter Island 1967
Eastern United States North American 1927, 1983
Ecuador Provisional South American 1956 South American 1969
Eftate Island Bellevue (IGN)
Egypt European 1950 Old Egyptian
El Salvador North American 1927, 1983
England European 1950, Ordnance Survey of Great Britain 1936
Erromango Island Bellevue (IGN)
TABLE C-l (cont'd)
Countries and Their Associated Datums
COUNTRY/GEOGRAPHIC AREA DATUM
Eritrea
Espirito Santo Island
Ethiopia
Faial Island (Azores)
Federal Republic of
Germany (before 1990)
Federal States of
Micronesia
Finland
Flores Island (Azores)
Florida
France
Gabon
Ghana
Gibraltar
Massawa
Santo (DOS) 1965
Adindan
Graciosa Base SW 1948
European 1950
Kusaie 1951
European 1950, 1979
Observatorio Meteorologico 1939
Cape Canaveral
North American 1927, 1983
European 1950
M'poraloko
Leigon
European 1950
TABLE C-l (cont'd)
Countries and Their Associated Datums
COUNTRY/GEOGRAPHIC AREA DATUM
Gizo Island (New ,
Georia Islands) DOS 1968
Graciosa Island Graciosa Base SW
Grand Canyon North American 1950
Greece European 1950
Greenland (Hayes North American 1927
Peninsula)
Guadacanal Island Gux 1 Astro
Guam Guam 1963
Guatemala North American 1927, 1983
Guinea-Bissau Bissau
Guinea Dabola
Guyana Provisional South American
South American 1969
1956
Hawaii Old Hawaiian
North American 1983
Honduras North American 1927, 1983
TABLE C-l (cont'd)
Countries and Their Associated Datums
COÜNTRY/GEOGRAPHTP ARKA
Hong Kong
Iceland
India
Iran
Iraq
Ireland
Isle of Man
Israel
Italy
Iwo Jima
Jamaica
Japan
Johnston Island
Jordan
Kalimantan Island (Indonesia)
C&TJ2M
Hong Kong 1963
Hjorsey 1955
Indian
European 1950
European 1950
European 1950, Ireland 1965
Ordnance Survey of Great Britain 1936
European 1950
European 1950
Astro Beacon "E"
North American 1927
Tokyo
Johnston Island 1961
European 1950
Gunung Segara
TABLE C-l (cont'd)
Countries and Their Associated Datums
COUNTRY/GEOGRAPHIC AREA DATUM
Kauai Old Hawaiian
North American 1983
Kenya Arc 1960
Kerguelen Island Kerguelen Island 1949
Korea Tokyo
Kuwait European 1950
Lebanon European 1950
Leeward Islands Fort Thomas 1955
Antigua Island Astro 1943
Montserrat Island Astro 1958
Liberia Liberia 1964
Luxembourg European 1950
Madagascar Tananarive Observatory
Madeira Islands Porto Santo 1936
Make Island Mahe 1971
Malawi Arc 1950
TABLE C-l (cont'd)
Countries and Their Associated Datums
COUNTRY/GEOGRAPHTf-. ARFA
Malaysia
Maldives, Republic of
Mali
Malta
Manitoba
Marcus Islands
Marshall Islands
Mascarene Island
Masirah Island
(Oman)
Maui
Mexico
Micronesia
Midway Island
Mindanao Island
DATUM
Timbalai 1948
Gan
Adindan
European 1950
North American 1927, 1983
Astronomic Station 1952
Wake Eniwetok 1960
Reunion
Nahrwan
Old Hawaiian
North American 1983
North American 1927, 1983
Kusaie 1951
Midway Astro 1961
Luzon
TABLE C-l (cont'd)
Countries and Their Associated Datums
COUNTRY/GEOGRAPHIC AREA DATUM
Montserrat Montserrat Island Astro 1958
Morocco Merchich
Namibia Schwarzeck
Nepal Indian
Netherlands European 1950, 1979
Nevis Fort Thomas 1955
New Brunswick North American 1927, 1983
New Foundland North American 1927, 1983
New Zealand Geodetic Datum 1949
Nicaragua North American 1927
Niger Point 58
Nigeria Minna
Northern Ireland Ireland 1965
Northwest Territories North American 1927, 1983
Norway European 1950, 1979
TABLE C-l (cont'd)
Countries and Their Associated Datums
COUNTRY/GEOGRAPHIC AREA DÄIDM
Nova Scotia
Oahu
Okinawa
Oman
Ontario
Paraguay
Peru
Philippines
Phoenix Islands
Pico Island (Azores)
Pitcairn Island
Porto Santo Island
Portugal
Puerto Rico
North American 1927, 1983
Old Hawaiian
North American 1983
Tokyo
Oman
North American 1927, 1983
Chua Astro, South American 1969
Provisional South American 1956,
South American 1969
Luzon
Canton Astro 1966
Graciosa Base SW
Pitcairn Astro 1967
Porto Santo 1936
European 1950
Puerto Rico
TABLE C-l (cont'd)
Countries and Their Associated Datums
COUNTRY/GEOGRAPHIC AREA DATUM
Qatar Qatar National
Quebec North American 1927, 1983
Salvage Islands Selvagem Grade 1938
San Salvador Island North American 1927
Santa Maria Islands Sao Braz (Azores)
Sao Jorge (Azores) Graciosa Base SW
Sao Miguel Island Sao Braz
Sarawak and Sabah Timbalai 1948
Sardinia European 1950, Rome 1940
Saskatchewan North American 1927, 1983
Saudi Arabia Ain El Abd 1970, European 1950,
Nahrwan
Scotland European 1950, Ordnance Survey
Great Britain 1936
of
Senegal Adindan
Shetland Islands European 1950, Ordnance Survey
Great Britain 1936
of
TABLE C-l (cont'd)
Countries and Their Associated Datums
COUNTRY/GEOGRAPHIC AREA
Sicily
Singapore
Somalia
South Africa
South Chile
South Geogia Islands
South Greenland
Spain
Sri Lanka
St. Helena Island
European 1950
Kertau 1948, South Asia
Afgooye
Cape
Provisional South Chilean 1963
ISTS Astro 1968
Qornoq
European 1950, 1979
Kandawala
Astro Dos 71/4
St. Kitts Fort Thomas 1955
Sudan Adindan
Sumatra Island
(Indonesia) Djakarta (Botavia)
Surinam Zanderij
Swaziland Arc 1950
TABLE C-l (cont'd)
Countries and Their Associated Datums
COUNTRY/GEOGRAPHIC AREA DATUM
Sweden European 1950, 1979
Switzerland European 1950, 1979
Syria European 1950, 1979
Taiwan Hu-Tzu-Shan
Tanzania Arc 1960
Tasmania Island Australian Geodetic 1966, 1984
Terceira Islands Graciosa Base SW
(Azores)
Tern Island Astro/Tern Island (Frig) 1961
Thailand Indian
Trinidad and Tobago Naparima, BWI; South American 1969
Tristan da Cunha Tristan Astro
Tunisia Carthage
Turk and Caicos North American 1927 Islands
United Arab Emirates Nahrwan
Uruguay Yacare
TABLE C-l (cont'd)
Countries and Their Associated Datums
COUNTRY/flEOGRAPH-m ARTTA CATJZH
Venezuela
Vietnam
Virgin Islands
Viti Levu Island (Fiji Island)
Wales
Wake Atoll
Western Europe
West Malaysia
Western United States
Yukon
Zaire
Zambia
Zimbabwe
Provisional South American 1956, South American 1969
Indian
Puerto Rico
Indian
Ordnance Survey of Great Britain 1936
Wake Island Astro 1952
European 1950
Kertau 1948
North American 1927, 1983
North American 1927, 1983
Arc 1950
Arc 1950
Arc 1950
APPENDIX D WFS 84 Geoid Separation Computation
10.0 GENERAL.
10.1 Scope. This appendix provides the bilinear method for interpolating geoid separation and WGS 84 Geoid Separations on a 10° x 10° grid. These separations are expressed in meters.
20. APPLICABLE DOCUMENTS.
This section is not applicable to this appendix.
30. GENERAL REQUIREMENTS.
30.1 The Bilinear Interpolation Method of Calculating WGS 84 Geoid Separation (NHGS Bi}_. The bilinear interpolation method uses interpolation in two directions {§,X) to approximate the geoid separation NWGS 84, from four known WGS 84 geoid separations given in a look-up table. The bi-linear interpolation method has the form given in equation set D.I. Figure D.l shows the relationship between the positions of the known NWGS 84 values and the interpolation point.
NP«|>,A,) = a0 + axX + a2Y + a3XY (D.l)
Where
Np((j),A,) = Geoid Separation (N) to be interpolated at Point
P«|>,A.>.
a0 = Nj.
*i = N2 - Nx
a2 = N4 - Nx
a3 = Nx + N3 - N2 - N4
X = a - xr) ( Ar2 - A-! )
Y = ^ - 4M «j)2 - tyj
145
0 = Geodetic latitude of Point P A, = Geodetic longitude of Point P
Nlt N2, N3, N4 = Known geoid heights at grid points used in the interpolation process.
Y
t M^, Xi) Hl{$2, h)
ity.h)
m __ 1 >
N.^Xi)
<> < > ( N2tyl»
» »»
(4>1, X)
Figure F-l Coordinate System Associated with Geoid Seperation Bi-Linear Interpolation Method
30.2 NWGS S4 Tables. NWGS 84 values are tabulated on a 10° x 10° grid in Table D.I. The values in Table D.l are suitable for use in moderate accuracy mapping and charting applications. Qualified users may obtain digitally-stored tables of NWGS 84 values on 30' x 30' and 1° x 1° grids for survey and high accuracy mapping and charting applications.
146
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APPENDIX E
Seven-Parameter Geometrie Datum Shifts
10. GENERAL
10.1 Scope. This appendix provides the data needed to perform seven-parameter geometric datum shifts between two local geodetic systems and WGS 84.
20. APPLICABLE DOCUMENTS.
This section is not applicable to this appendix.
30. GENERAL REQUIREMENTS
30.1 Application. Table E-l lists the coverage area, ellipsoid applicable to the local geodetic system, ellipsoid center shifts (AX, AY, AZ), axes rotations (£, \|/, CO), and scale factor change
(AS) for use in seven-parameter geometric datum shifts between two local geodetic systems and WGS 84. The root-mean square differences between Doppler-derived WGS 84 geodetic coordinates and WGS 84 coordinates, computed from local system coordinates using the seven-parameter model, are provided as an indication of the datum shift consistency.
149
TABLE E.l
Seven-Parameter Local Geodetic System to WGS 84
Datum Transformations
DATUM: EUROPEAN 1950 (EUR-M)
COVERAGE AREA: Western Europe
ELLIPSOID: International
AX = -102 m AY = -102 m AZ = -129 m
£ = 0.413" \|/ = -0.184" CO = 0.385" AS = 2.4664 x 10~6
RMS DIFFERENCES : WGS 84 surveyed, minus computed coordinates
A<j) = 2m AÄ, = 3 m AH = 2m
DATUM: ORDNANCE SURVEY OF GREAT BRITAIN 193 6
COVERAGE AREA: England, Isle of Man, Scotland, Shetland Islands,
and Wales
ELLIPSOID: Airy
AX = 446 m AY = -99 m AZ = 544 m
£ = -0.945" \|/ = -0.261" CO = -0.435" AS = -20.8927 x 10"6
RMS DIFFERENCES : WGS 84 surveyed minus computed coordinates
A(J) = 2m AX = 2 m AH = 1 m
150
APPENDIX F
Old Hawaiian Datum with International Ellipsoid
10. GENERAL
10.1 Scope. This appendix provides the information needed to shift coordinates between the Clarke 1866 and International in the Old Hawaiian datum.
20. APPLICABLE DOCUMENTS
DMA SGG MEMORANDUM FOR RECORD SUBJECT: Old Hawaiian Datum with
International Ellipsoid .13 June 1990
Army Map Service GEODETIC MEMO No. 687, SUBJECT: Hawaiian Islands, UTM Coordinates and International Spheroid Geographic Positions. P.O.# 23832-002, 29 November 1950.
30. GENERAL REQUIREMENTS
30.1 Background. Positions on the Old Hawaiian datum may be on either.the Clarke 1866 ellipsoid or the International ellipsoid. Users of Old Hawaiian datum positions must determine which ellipsoid applies. Datum shift parameters from Old Hawaiian datum to WGS 84 have been determined only for the Clarke 1866 ellipsoid. Old Hawaiian datum coordinates on the international ellipsoid must be converted to the Clarke 1866 ellipsoid before being shifted to WGS 84. The conversion between International and Clarke 1866 ellipsoids has been defined in terms of UTM coordinates. Geodetic latitude and longitude must be converted to UTM before changing ellipsoids. Height conversions should not be attempted as no conversion method is available.
30.2 Application. To convert Old Hawaiian datum UTM coordinates on the International ellipsoid (EINT, NINT) to UTM coordinates on the Clarke 1866 ellipsoid (EC66/ NC66) use:
In UTM Zone 4
Ec66 = EJNT ~ 3 (1.1) NC66 = NINT - 169
In UTM Zone 5
EQ66 = EINT + 13 (1.2)
NC66 = NINT - 169
151
To convert Old Hawaiian datum UTM coordinates on the Clarke 1866 ellipsoid to the International ellipsoid use:
In UTM Zone 4
EINT = Ec66 + 3 (1.3) NINT = Nc66 + 169
In UTM Zone 5
EINT = EC66 - 13 (1.4) NINT = NC66 + 169
30.3 Example. Convert geodetic survey coordinates on the Old Hawaiian datum, International ellipsoid, to WGS 84 geodetic coordinates.
<j)INT = 21° 19' 37.431 "N A,INT = 157° 58' 25.610"W
This position is on the island of Oahu.
a. Convert (J)INT, ^INT to UTM:
Zone = 4
EINT = 606,428 NINT = 2,358,722
b. Convert EINT, NINT to the Clarke 1866 ellipsoid using equation set 1.1, for zone 4.
Ec66 = 606,425 NC66 = 2,358,553
c. Convert Ec66, NC66 to geodetic coordinates on the . Clarke 1866 ellipsoid.
<j)c66 = 21° 19' 37.425 "N ^c66 = 157° 58' 25.631"W
d. AX, AY, AZ shift parameters for datum identifier code OHA-D should be used for shifting geodetic survey- coordinates on Oahu.
e. Shift (|>c66, Ac66 to WGS 84 using, for example, the three-step method.
152
<!>WGS 84 = 21° 19 ' 26.069 "N taras 84 = 157° 58" 15.766"W
153