UNCLASSIFIED SECURITY CLASSIFICATION OF THlS PAGE

I REPORT DOCUMENTATION PAGE ~ o n i Approved OM8 No. 0704-0188

I

la . REPORT SECURITY CLASSIFICATION UNCLASSIFIED

23. SECURITY CLASSIFICATION AUTHORITY Not Appl icable (N/A)

2b. DECLASSIFICATION I DOWNGRADING SCHEDULE N / A .., . .

4. PERFORMING ORGANIZATION REPORT NUMBER(S)

1 b. RESTRICTIVE MARKINGS

None q

3 . DISTRIBUTION /AVAILABILITY OF REPORT

Approved f o r p u b l i c release; d i s t r i b u t i o n un l im i ted .

5. MONITORING ORGANIZATION REPORT NUMBER(S) I DMA TR 8350.2-A

6a. NAME OF PERFORMING ORGANIZATION 6b. OFFICE SYMBOL (If applicable)

~ u i l d i n g 56 I Uni ted States Naval Observatory

Same 7r. N A M E OF MONITORING ORGANIZATION

Defense Mapping Agency I N/A 6c ADDRESS (City, State, a d Z IPCo&J

Same 7b. ADDRESS (City, State, a d ZIP Cod.)

- Washington, DC 20305-3000

8a. NAME OF FUNDING / SPONSORING 8b. OFFICE SYMBOL ORGANIZATION (If applicable)

13a. T Y P E OF REPORT 13b. TIME COVERED 14. DATE OF REPORT (Year, Month, Day) Techni ca l FROM N/A TO N/A 1 December 1987

16. SUPPLEMENTARY NOTATION

Same 9. PROCUREMENT INSTRUMENT IDENTIFICATION NUMBER

N/A I N/A 8c ADDRESS (City, State, a d ZIP Code)

N/A

This techn ica l repor t , P a r t I o f a th ree-par t supplement t o the Department o f Defense (DoD) World Geodetic System 1984 (WGS 84) Technical Report, D M TR 8350.2, presents the methods, techniques, and data used i n WGS 84 development. The development o f WGS 84 was i n i t i a t e d f o r the purpose o f p rov id ing the more accurate geodet ic and g r a v i t a t i o n a l data requ i red by DoD nav iga t i on and weapon systems. The new system represents the Defense Mapping Agency's modeling o f the ea r th from a geometric, geodet ic, and g r a v i t a t i o n a l s tandpoint us ing data, techniques, and technology a v a i l a b l e through e a r l y 1984.

1 1. TITLE (Include Securrty Claru'fication) Supplement t o Department of Defense World Geodetic System 1984 Technical Report: Pa r t I - Methods, Techniques, and Data Used i n WGS 84 Development

12. PERSONAL AUTHOR(S) N /A

N/A 10. SOURCE OF FUNDING NUMBERS

Report prepared by the DMA WGS 84 Development Committee.

I

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See Reverse Page.

17. COSATI CODES

UNCLASSIFIED

19. ABSTRACT (Continue on reverse if necewary and identify by block number)

FIELD

0 8 2 2

a u~clnss lF lEDNN~lMlTEo tij: SAME AS RPT. DTIC USERS

2Za. NAME OF RESPONSIBLE INDIVIDUAL D r . Randall W. Smith

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DMSC/PIO

18. SUBJECT TERMS

Angular Velocity of the Ear th , Astronomic Nutation, Average Value of Theoretical Gravity, Bessel ian Epoch, Ce les t i a1 Ephemeris Pole, Conventional I n e r t i a l System, Conventional T e r r e s t r i a l Pole, Conventional T e r r e s t r i a l System, Coordinates, Coordinate Accuracy, Coordinate Systems, Datums, Datum S h i f t s , Datum Transfor- mations, Datum Transformation Mu1 t i p l e Regression Equations, Degree Variances, Dynamical El 1 i p t i c i t y , Earth Gravi ta t ional Constant, Earth Gravi ta t ional Model , Ear th l s .Mass , Ea r th ' s Moments of I n e r t i a , Ear th ' s Rotation, E l l ipso id Constants, El 1 ipsoid F la t t en ing , E l 1 ipsoidal Gravity Formula, El 1 ipsoid Parameters, El 1 ipsoid Semimajor Axis, Equatorial Gravity, F la t t en ing , General Precession, Geodesy, Geodetic, Geodetic Heights, Geoids, Geoid Heights, Geoid Undulations, Gravi ta t ion, Gravitat ional Coef f ic ien t s , Gravi ta t ional Model, Gravi tat ion'al Potent ia l , Gravity, Gravity Anomaly Degree Variances, Gravity Formula, Gravity Po t en t i a l , Instantaneous Te r r e s t r i a l System, Jul ian Epoch, Local Datums, Local Geodetic Datums, Local Geodetic Systems, Local Geodetic System-to-WGS 84 Datum S h i f t s , Local Geodetic System-to-WGS 84 Datum Transformations, Mass of t he Ea r th ' s Atmosphere, Mean Datum S h i f t s , Molodensky Datum Transformation Formulas, Mu1 t i p l e Regression Equa- t i ons , Normal Gravity, Normal Gravity Formula, Normal Gravity Po t en t i a l , Nutation, Polar Motion, Precession, Reference Frames, Reference Systems, Theoretical Gravity, Theoretical Gravity Formula, Theoretical Gravity Potent ia l , World Geodetic System, World Geodetic System 1972, WGS 72, WGS 72-to-WGS 84 Transformation, World Geodetic System 1984, WGS 84, WGS 84 Coordinates, WGS 84 Coordinate Accuracy, WGS 84 Coordi- nate System, WGS 84 E l l i p so id , WGS 84 El l ipso ida l Gravity Formula, WGS 84 Earth Gravi ta t ional Model, WGS 84 Geoid, WGS 84-to-Local Geodetic System Datum Transfor- mations, WGS 84 Reference Frame, WGS 84 Reference System.

THE DEFENSE MAPPING AGENCY

With some 9,000 employees in more than 50 locations around the world,

the Defense Mapping Agency (DMA) provides Mapping, Charting, and Geodetic

(MUG) support to the Secretary of Defense, the Joint Chiefs of Staff, the

Military Departments, and other DoD Components. This includes production

and worldwide distribution of maps, charts, precise positioning data, and

digital data for strategic and tactical military operations and weapon

systems. DMA also provides nautical charts and marine navigational data

for the worldwide Merchant Marine and private yachtsmen.

The Defense Mapping Agency was established in 1972, when MC&G

functions of the Defense community were combined into this Joint DoD

Agency. DMA is under the direction and control of the Assistant Secretary

of Defense (Command, Contro 1, Communications, and Intel 1 igence) . The

Director of DMA, a flag officer, is responsible to the Joint Chiefs of

Staff for operational matters.

DMA maintains close liaison with civilian agencies of the United

States Government engaged in MC&G activities and works closely with

various national and international scientific and operational

organizations in the MC&G field.

i i i

T h i s page is intentionally blank.

ACKNOWLEDGMENTS

R e s p o n s i b i l i t y f o r t h e development o f WGS 84 was ves ted i n a DMA

WGS 84 Development Committee o p e r a t i n g i n i t i a l l y under t h e gu idance o f

Dr. Mark M. Macomber/Hq DMA/DH, and l a t e r , Dr. Kenneth I. DaughertylDMA

Systems Center . The Committee was composed o f Dr. R i c h a r d J. Ander le ,

M r . Ra lph L. Kulp, and M r . Mark G. Tannenbaum, Naval Su r face War fare

Center ( NSWC) ; Dr. Thomas M. Davis, Naval Oceanographic O f f i c e

(NAVOCEANO); D r . P a t r i c k J. F e l l , Dr. Benny L. K lock , Dr. Muneendra Kumar,

and M r . F ran B. Varnum, Defense Mapping Agency Hydrographic/Topographic

Cente r (DMAHTC); and Mr. C lyde R. Greenwal t and M r . Haschal L. White,

Defense Mapping Agency Aerospace Center (DMAAC). Mr. B. L o u i s Decker

(DMAAC) was Chairman of t h e Committee.

The Committee expresses i t s a p p r e c i a t i o n t o Mrs. Caro l A. Malyevac and

M r . C. H a r r i s Seay, NSWC, and M r . L o u i s Abramov i tz , M r . L. Thomas

Appelbaum, M r . John A. Banger t , M r . James M. Bar th , Mr. A r c h i e E. Car l son ,

M r . R o b e r t E. C a t u l l e , Mrs. I n e z J. D i m i t r i j e v i c h , Mr. C a r l E. Draper,

Mrs. Mary S. Ealum, Mrs. Joyce E. Fox, Mr. Dav id M. Gleason, M r . John L.

Goodwin, M r . John Hopkins, Mr. Donovan N. Huber, M i s s B e a t r i c e Je rn igan ,

M iss Mar tha R. Lahr, Mrs. Mary M. M a r t i e , M r . Kenneth Nelson, M r . Rober t

M. Perlman, Mr. Dona ld A. Richardson, Mr. M e l v i n E. S h u l t z , M r . N e i l J.

Simmons, Dr. R a n d a l l W. Smith, Dr. W i l l i a m L. S t e i n , M r . George T. S t e n t z ,

M r . B a l f o u r R. Su t ton , Mr. Rober t W. Valska, M r . Denn is H. Van Hee, and

M r . James F. Vines o f DMA f o r t h e i r a s s i s t a n c e . The many c o n t r i b u t i o n s o f

o t h e r s u p p o r t i n g p e r s o n n e l o f DMA and NSWC a r e a l s o g r a t e f u l l y acknow-

ledged. I n a d d i t i o n , a s p e c i a l t hanks i s ex tended t o t h e many o r g a n i z a -

t i o n s and i n d i v i d u a l s i n t h e U n i t e d S t a t e s and abroad who p r o v i d e d d a t a

and t e c h n i c a l e x p e r t i s e i n s u p p o r t o f t h e WGS 84 development e f f o r t . I n

p a r t i c u l a r , Mr. John A. Gergen, Mrs. E l i z a b e t h G . Wade, and M r . L a r r y D.

Hothem, o f t h e N a t i o n a l Geodet ic Survey (NGS), and Dr. Dennis D. McCarthy,

o f t h e U n i t e d S t a t e s Naval Observa to ry (USNO), a r e c i t e d f o r t h e i r

suppor t .

This page is intentionally blank.

DEFENSE MAPPING AGENCY BUILDING 56. U.S. NAVAL OBSERVATORV

WASHINGTON. DC 20305.3000

DMA TR 8350.2-A 1 December 1987

DEFENSE MAPPING AGENCY TECHNICAL REPORT 8350.2-4

Supplement t o Department of Defense World Geodetic System 1984 ~ e c h n i c a l Report:

Par t I - Methods, Techniques, and Data Used i n WGS 84 Development

FOREWORD

1. This t e c h n i c a l r e p o r t , Pa r t I of a th ree -pa r t supplement t o t h e Department of Defense (DoD) World Geodetic System 1984 ( W G S 84) Technical Report , DMA TR 8350.2, p r e s e n t s t h e methods, techniques , and d a t a used i n WGS 84 development. WGS 84 was i n i t i a t e d t o provide more accura te geodet ic and g r a v i t a t i o n a l da ta r equ i red by DoD naviga t ion and weapon systems. The new system represen t s t h e Defense Mapping Agency's (DMA) geometric, geodet ic , and g r a v i t a t i o n a l modeling of t h e Earth using d a t a , techniques, and technology ava i l ab le through e a r l y 1984.

2 . Readers shou ld be aware t h a t DMA r e v i s e s t e c h n i c a l pub l i ca t ions p e r i o d i c a l l y only t o s a t i s f y i n t e r n a l requirements, and it does not n o t i f y ex te rna l r ec ip ien t s of such changes.

3 . This p u b l i c a t i o n con ta ins no copyrighted m a t e r i a l and i s approved f o r pub l i c r e l e a s e . Dis t r ibu t ion i s unl imited. Copies may be requested from DMA a s indica ted i n t h e PREFACE.

Colonel, ,&AF Chief of S ta f f

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PREFACE

This t e c h n i c a l r e p o r t i s Pa r t I o f a t h r e e - p a r t supplement t o t h e

Department o f Defense (DoD) World Geodet ic System 1984 (WGS 84) Technica 1

Report, DMA TR 8350.2. The l a t t e r p rov ides a b r i e f d i s cuss ion o f WGS 84

and i t s r e l a t i o n s h i p s w i t h l o c a l geode t i c systems. Th is r e p o r t ( P a r t I o f

t h e supplement) d iscusses WGS 84 and t h e methods, techniques, and da ta

used i n deve lop ing t h e parameters and p roduc ts t h a t d e f i n e it. Consider-

a b l e space i s devoted t o t h e d i scuss ion o f t h e WGS 84 Reference Frame,

E l l i p s o i d , E l l i p s o i d a l G r a v i t y Formula, Ea r t h G r a v i t a t i o n a l Model, Geoid,

and methods and procedures f o r o b t a i n i n g WGS 84 coord ina tes . P a r t I 1 of

t h e supplement con ta i ns f i n a l va lues f o r t h e WGS 84 d e f i n i n g parameters

and products , t h e i r accurac ies, and formulas r e l e v a n t t o t h e implementa-

t i o n . Most o f P a r t I 1 c o n s i s t s o f g r a p h i c a l and t a b u l a r m a t e r i a l s f o r

c o n v e r t i n g 83 l o c a l geode t i c systems t o WGS 84. P a r t I 1 i s l a r g e l y s e l f -

con ta i ned i n t h a t t h e contents , w i t h a few except ions, compl

WGS 84. A l though WGS 84 i s e s s e n t i a l l y an u n c l a s s i f i e d

assoc ia ted Ea r t h G r a v i t a t i o n a l Model (and i t s cor responding

degree ( n ) and o rde r (m) 18 a re CLASSIFIED. These m a t e r i

P a r t I 1 1 o f t h e supplement, t h e CLASSIFIED p o r t i o n o f WGS 84.

e t e l y d e f i n e

system, t h e

geo id ) above

a l s comprise

Requesters r e q u i r i n g a d d i t i o n a l i n f o rma t i on , data, o r P a r t I 1 1 o f t h e

supplement shou ld f o rwa rd t h e i r reques t t o :

D i r e c t o r Defense Mapping Agency ATTN: PR B u i l d i n g 56 Un i t ed S ta tes Naval Observatory Washington, DC 20305-3000

S i m i l a r l y , r eques te r s r e q u i r i n g t h e p o s i t i o n i n g o f s i t e s o f i n t e r e s t

d i r e c t l y i n WGS 84 v i a S a t e l l i t e P o i n t P o s i t i o n i n g u t i l i z i n g ground-based

PREFACE (Con t ' d )

r e c e i vers shoul d t r a n s m i t t h e i r r e q u i rements t o t he above address. Other

WGS 8 4 - r e l a t e d reques ts o r ques t ions may a l s o be r e f e r r e d there .

Considerable exp lana to ry m a t e r i a l has been i n c l u d e d i n t h i s p o r t i o n o f

t h e supplement ( P a r t I) i n an e f f o r t t o make WGS 84 more understandable

and usable by t he numerous DoD personnel who a re t h r u s t each y e a r i n t o a

mapping, c h a r t i n g , and geodet i c (MC&G) environment and must accompl ish

MC&G t asks w i t h l i t t l e o r no geodet i c background. Al though t h i s

exp lana to ry m a t e r i a l makes t h i s p u b l i c a t i o n l eng thy and somewhat

cumbersome, and i s n o t needed by t h e p r a c t i c i n g geodesis t , the requi rement

o f MC&G newcomers f o r such data i s deemed paramount.

S ince WGS 84 i s comprised o f a c o n s i s t e n t s e t o f parameters,

o r g a n i z a t i o n s i n v o l v e d i n a DoD a p p l i c a t i o n should n o t make a s u b s t i t u t i o n

f o r any o f t h e WGS 8 4 - r e l a t e d parameters i n an a t tempt t o improve t h e

accuracy. Such a s u b s t i t u t i o n may l e a d t o l e s s accura te WGS 84 products .

TABLE OF CONTENTS

PAGE

ACKNOWLEDGMENTS ................................................. PREFACE ......................................................... TABLE OF CONTENTS ............................................... LIST OF TABLES .................................................. LIST OF FIGURES .................................................

1 . INTRODUCTION ............................................. References ...............................................

2 . WGS 84 AND ASSOCIATED REFERENCE FRAMES ................... 2.1 Genera l ............................................

.................... 2.2 Conven t iona l T e r r e s t r i a l System

2.2.1 D e f i n i t i o n ................................. 2.2.2 M o d i f i c a t i o n o f t h e NSWC 92-2 Coord ina te

System ..................................... 2.2.2.1 General ..........................

.................... 2.2.2.2 O r i g i n Change

2.2.2.3 R e - O r i e n t i n g i n Long i tude ........ ............... 2.2.2.4 Sca le M o d i f i c a t i o n

2.2.2.5 Summary/Comments ................. 2.3 The WGS 84 C o o r d i n a t e System ....................... 2.4 Conven t iona l I n e r t i a l System ....................... 2.5 Ins tan taneous T e r r e s t r i a l System ...................

v

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x i x

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1-1

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2-1

2-1

2-1

2-1

TABLE OF CONTENTS ( C o n t ' d )

PAGE

2.6 Mathemat i ca l R e l a t i o n s h i p Between t h e C I S . ITS. and t h e WGS 84 Coord ina te System

(EBIH-Def ined CTS) .................................

References ............................................... ......................................... . 3 WGS 84 ELLIPSOID

3.1 General ............................................ ........................ 3.2 The E q u i p o t e n t i a l E l l i p s o i d

3.3 D e f i n i n g Parameters ................................ 3.3.1 Semimajor A x i s ( a ) .........................

........ 3.3.2 E a r t h ' s G r a v i t a t i o n a l Constant (GM)

3.3.2.1 GM (Wi th E a r t h ' s Atmosphere I n c l u d e d ) ........................

3.3.2.2 GMA o f t h e ~ a r t h ' s Atmosphere ....................... 3.3.2.2.1 C a l c u l a t e d as a

P roduc t o f G and MA ...............

3.3.2.2.2 I m p l i e d b y Atmospher ic Correc- t i o n t o G r a v i t y Values ...............

3.3.2.3 GM Wi th E a r t h ' s Atmosphere Exc luded (GM') ...................

3.3.3 Normal ized Second Degree Zonal G r a v i t a -

t i o n a l C o e f f i c i e n t (T= ) ................. 290

3.3.4 Angular V e l o c i t y o f t h e E a r t h .............. ........... 3.4 D e r i v e d Geometr ic and P h y s i c a l Constants

x i i

TABLE OF CONTENTS (Cont'd)

PAGE

3.4.1 General .................................... 3-14

3.4.2 Fundamental Derived Constant ............... 3-14

................ 3.4.3 Derived Geometric Constants 3-15

................... 3.4.3.1 Semiminor Axis 3-15

....................... 3.4.3.2 Flattening 3-16

.............. 3.4.3.3 Linear Eccentricity 3-16

........ 3.4.3.4 Polar Radius of Curvature 3-16

........... 3.4.3.5 Meridian Arc Distances 3-17

3.4.3.6 Circumference of the Equator ..... 3-17

...................... 3.4.3.7 Mean Radius 3-17

3.4.3.8 Surface Area and Volume of the WGS 84 Ellipsoid ................. 3-18

3.4.3.9 Other Derived Geometric Constants ........................ 3-19

3.4.3.10 Numerical Results ................ 3-20

................. 3.4.4 Derived Physical Constants 3-20

3.4.4.1 Theoretical (Normal) Potential of ............. the WGS 84 Ellipsoid 3-20

3.4.4.2 Zonal Gravitational (Spherical Harmonic) Coefficients ........... 3-21

3.4.4.3 Theoretical Gravity at the ............ Equator and the Poles 3-23

3.4.4.4 Gravity Flattening ............... 3-23

3.4.4.5 Mean Value of Theoretical Gravity .......................... 3-23

3.4.4.6 Mass of the Earth ................ 3-24

xiii

TABLE OF CONTENTS (Cont'd)

PAGE

3.4.4.7 Numerical Results ................ 3.4.5 Relevant Miscel laneous Constants/Conversion

Factors .................................... ................ 3.4.5.1 Velocity of Light

3.4.5.2 Dynamical Ellipticity ............ 3.4.5.3 Numerical Values .................

3.4.6 Moments of Inertia ......................... ............... 3.4.6.1 Geometric Solution

3.4.6.2 Dynamic Solution ................. ................ 3.4.6.3 Numerical Results

3.4.7 Geocentric Radius and Radii of Curvature ... 3.4.8 Ellipsoidal Arc Distances ..................

............ 3.4.8.1 Meridian Arc Distance

3.4.8.2 Arc Distance Along a Parallel of Latitude .........................

................. 3.4.8.3 Numerical Values

References ............................................... ....................... 4 . WGS 84 ELLIPSOIDAL GRAVITY FORMULA

............................................ 4.1 General

4.2 Analytical and Numerical Forms ..................... 4.2.1 Formulas ...................................

................................. 4.2.2 Derivation

4.3 Average Value of Theoretical Gravity ( ? ) ......... 4.3.1 Average Value (WGS 84 Ellipsoid) ...........

TABLE OF CONTENTS (Cont 'd )

PAGE

4.3.2 D e r i v a t i o n o f 7 (WGS 84 E l l i p s o i d ) ........ 4.4 Atmospheric E f f e c t s ................................

4.4.1 T h e o r e t i c a l Cons idera t ions ................. 4.4.2 Atmospheric Co r rec t i on t o Measured

G r a v i t y .................................... ......................... 4.5 G r a v i t y Anomaly Conversion

4.5.1 Recommended Approach ....................... 4.5.2 A l t e r n a t i v e Approach .......................

4.6 Comments ........................................... ............................................... References

................... 5 . WGS 84 EARTH GRAVITATIONAL MODEL (EGM)

5.1 General (Ph i losophy Under ly ing t h e S o l u t i o n ) ....... .................. 5.2 Data Sets Used i n EGM Development

5.2.1 General .................................... 5.2.2 Doppler ....................................

...................................... 5.2.3 Laser

5.2.4 Surface G r a v i t y ............................ 5.2.5 A l t i m e t r i c Geoid He igh ts ...................

.......... 5.2.6 NAVSTAR Globa l P o s i t i o n i n g System

.......... 5.2.7 Lumped G r a v i t a t i o n a l C o e f f i c i e n t s

5.3 Dete rmina t ion o f G r a v i t a t i o n a l C o e f f i c i e n t s K 3 ..................................... n. m' n. m

5.3.1 Parameters o f t he Normal Equat ions ......... ........... 5.3.2 Sur face G r a v i t y Normal Equat ions

TABLE OF CONTENTS (Cont'd)

PAGE

5.3.3 Altimetric Geoid Height Normal .................................. Equations

5.3.4 Results .................................... ...... 5.4 Evaluation of the WGS 84 EGM (Through n=m=41)

5.4.1 General .................................... 5.4.2 Degree Variances ...........................

........... 5.4.3 Mean Gravity Anomaly Comparisons

.............. 5.4.4 Mean Geoid Height Comparisons

........................ 5.4.5 Coefficients Errors

5.5 Independent Spherical Harmonic EGM Determination .......................................... (n=m=180)

5.6 Summary/Availability of WGS 84 EGM Coefficients .... References ...............................................

6 . WGS 84 GEOID ............................................. 6.1 General ............................................ 6.2 Formulas and Representations .......................

6.2.1 Formulas ................................... 6.2.2 Input Data .................................

............................ 6.2.3 Representations

6.3 Geoid Height Interpolation Method .................. 6.4 Analysis/Accuracy of WGS 84 Geoid .................. 6.5 Availability of WGS 84 Geoid Height Data ........... 6.6 Comments ........................................... References ...............................................

TABLE OF CONTENTS (Cont'd)

PAGE - ........ 7. METHODS/PROCEDURES FOR OBTAINING WGS 84 COORDINATES.

7.1 General................. .............................. 7.2 The Four Basic Methods.......... ......................

7.2.1 Satellite Point Positioning Directly in WGS 84................ .....................

7.2.2 NSWC 92-2 to WGS 84 Transformatio~,..... ...... 7.2.3 WGS 72-to-WGS 84 Transformation...............

7.2.4 Local Geodetic System-to-WGS 84 Datum Transformation Techniques ..................... 7.2.4.1 General.. ........................... 7.2.4.2 Basic Data..........................

7.2.4.2.1 Local Geodetic System and WGS 84 Coordinates at the Same Sites.......

7.2.4.2.2 DMA-Developed Local Geodetic System Geoid ................. Heights

7.2.4.3 Datum Transformation Formulas.......

7.2.4.3.1 The Seven Parameter ... Datum Transformation.

7.2.4.3.2 The Molodensky Datum Transformation . Formulas................

7.2.4.3.3 Datum Transformation Multiple Regression ............... Equations

7.2.4.3.4 Datum Transformations Via Direct Coordinate Conversion ...........:..

xvii

TABLE OF CONTENTS (Cont 'd )

PAGE .

7.2.4.4 Ana l ys i s o f Loca l Geodet ic System-to-WGS 84 Datum ........... Trans fo rmat ion Techniques

7.2.4.5 S e l e c t i o n o f P r e f e r r e d (Recommended) ........................... Technique

7.2.4.6 A v a i l a b i l i t y o f M u l t i p l e Regression ........................... Equat ions

7.3 Spec ia l Loca l Geodet ic System-to-WGS 84 Datum S h i f t s ................................................

References .................................................. 8 . WGS 84-TO-LOCAL GEODETIC SYSTEM DATUM TRANSFORMATIONS .......

References .................................................. 9 . ACCURACY OF WGS 84 COORDINATES ..............................

9.1 General ............................................... 9.2 WGS 84 Coord inates Determined v i a S a t e l l i t e P o i n t ........................................... P o s i t i o n i n g

9.3 WGS 84 Coord inates Determined From t h e WGS 72-to-WGS 84 Trans fo rmat ion .......................

9.4 WGS 84 Coord inates Determined From Loca l Geodetic ................ System-to-WGS 84 Datum Transformat ions

9.5 NAVSTAR GPS-Derived WGS 84 Coord inates ................

.................................................. References

APPENDIX . TRANSFORMATION OF E C I (CIS. EPOCH 52000.0) COORDINATES TO WGS 84 (CTS. ECEF) COORDINATES ...............

.................................................... ACRONYMS

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TITLE PAGE

2.1 Range of Candidate Z-Axis Bias Values for NSWC 92-2 Coordinate System ...........................................

2.2 Range of Candidate Scale Corrections for NSWC 92-2 Coordinate System ...........................................

2.3 Quantities Used in Converting Doppler System Coordinates (NSWC 92-2) to WGS 84.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2.4 Formulas and Parameters to Transform NSWC 92-2 Coordinates to WGS 84 Coordinates .......................................

3.1 Defining Parameters of the WGS 84 Ellipsoid and Their Accuracy Estimates (1 a). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3.2 Reference Ellipsoid Constants. .............................. 3.3 Estimated Values for the Universal Gravitational Constant

(G) and Mass of the Earth's Atmosphere (MA). . . . . . . . . . . . . . . . . 3.4 Effect of GMA on Theoretical Gravity ........................ 3.5 Comparison of Second Degree Zonal Gravitational

Coefficients............. ................................... 3.6 Yearly Angular Velocity Values of the Earth's Rotation.. . . . . 3.7 WGS 84 Ellipsoid -- Defining Parameters and Derived

Geometric Constants ......................................... 3.8 WGS 84 Ellipsoid -- Defining Parameters and Derived

Physical Constants . . . . . . . . . . . . . . . . . . . . . . . . : . . . . . . . . . . . . . . . . .

3.9 Relevant Miscellaneous Constants and Conversion Factors ..... 3.10 Principal Moments of Inertia Equations - Geometric

Solution........................ ............................ 3.11 Principal Moments of Inertia Equations - Dynamic Solution ... 3.12 WGS 84 - Related Moment of Inertia Values.. .................

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TABLE TITLE

WGS 84 Ellipsoidal Gravity Formula............... ...........

PAGE - 4-16

The WGS 84 Ellipsoidal Gravity Formula and the Defining and Derived Parameters Used in Its Derivation...............

Values of Theoretical Gravity - Surface of WGS 84 Ellipsoid ................................................... Atmospheric Corrections for Measured Gravity (When Forming WGS 84 Gravity Anomalies) ................................... Equations for Converting Gravity Anomalies to the WGS 84 Ellipsoidal Gravity Formula ................................. Satellites Providing Doppler Data for WGS 84 EGM ................................................. Development

Gravity Anomaly Degree Variances (cn). ...................... Comparison of 3Ox3" Equal-Area Mean Gravity Anomal ies Computed from Earth Gravitational Models with Those Derived from Terrestrial Data...... ......................... Comparison of 5Ox5" Equal-Area Mean Gravity Anomalies Computed from Earth Gravitational Models with Those Derived from Terrestrial Data..... .......................... Comparison of 10°xlOO Equal-Area Mean Gravity Anomal ies Computed from Earth Gravitational Models with Those Derived from Terrestrial Data............... ................ Comparison of 3Ox3" Equal-Area Mean Geoid Heights Computed from Earth Gravitational Models with Those Derived from SEASAT-1 Geoid Height Data.. ................................ Comparison of 5Ox5O Equal-Area Mean Geoid Heights Computed from Earth Gravitational Models with Those Derived from SEASAT-1 Geoid Height Data .................................. Comparison of 10°xlOO Equal-Area Mean Geoid Heights Computed from Earth Gravitational Models with Those Derived from .......................... SEASAT-1 Geoid Height Data........

Number of Satellites Used in EGM Development That are Sensitive to WGS 84 Gravitational Coefficients (Excluding Lumped Data Sets) ...........................................

LIST OF TABLES ( C o n t ' d )

TABLE TITLE PAGE -

C o r r e l a t i o n Matrix-WGS 84 E a r t h G r a v i t a t i o n a l Model ................................................ C o e f f i c i e n t s

WGS 84 Geoid H e i g h t D i f f e r e n c e s - T r u e ( C a l c u l a t e d ) Values ................. Minus Values I n t e r p o l a t e d From a 1°x10 G r i d

WGS 84 Geoid H e i g h t D i f f e r e n c e s - T r u e ( C a l c u l a t e d ) Values ............. Minus Values I n t e r p o l a t e d From a 3 0 ' x 3 0 1 Gr id . .

D i s t r i b u t i o n o f Geoid H e i g h t D i f f e r e n c e s G r e a t e r Than One Mete r -T rue ( C a l c u l a t e d ) Values Minus v a l u e s I n t e r -

................................ p o l a t e d t r o m a 3 0 ' x 3 0 1 Gr id .

...... Geoid H e i g h t Dif ferences-WGS 84 Minus WGS 72..........

......... Technique Used t o D i r e c t l y P lace t h e NNSS on WGS 84

Procedure f o r S o l v i n g D i r e c t l y f o r WGS 84 C o o r d i n a t e s ....... WGS 84 Coord ina tes Determined V ia Dopp ler S a t e l l i t e P o i n t P o s i t i o n i n g Minus WGS 84 Coord ina tes f r o m NSWC 92-2 t o WGS 84 T r a n s f o r m a t i o n ....................................... Formulas and Parameters t o T rans fo rm WGS 72 Coord ina tes t o WGS 84 Coord ina tes .......................................

... D i f f e r e n c e Between WGS 84 and WGS 72 Geodet ic Coord ina tes

L o c a l Geode t i c Systems R e l a t e d t o Wor ld Geodet ic ................................................. System 1984

Development Procedure - DMA-Developed L o c a l Geodet ic ........................................ System Geoid H e i g h t s

S tandard and Abr idged Molodensky Datum T r a n s f o r m a t i o n Formulas - L o c a l Geode t i c System t o WGS 84 . . . ............... D i f f e r e n c e s Between Use o f S tandard Versus Abr idged Molodensky Datum T r a n s f o r m a t i o n Formulas t o Cornpute Loca l ........... Geode t i c System-to-WGS 84 Datum Shifts.. . . . . . . . . .

L o c a l Geode t i c System-to-WGS 84 Mean Datum S h i f t s ( x, T , X? ) For Ma jo r Datums ............................. Mean Datum S h i f t s ( X , T , X? ) and T h e i r D i f f e r e n c e s - L o c a l Geode t i c System-to-WGS 84 and WGS 72 ................

LIST OF TABLES (Cont'd)

TABLE - 7.12

7.13

7.14

TITLE PAGE - RMS Differences Between "Observed (Doppler)" WGS 84 ......... Coordinates and Computed WGS 84 Coordinates........

Investigative Results - Local Geodetic System-to-WGS 84 ................................ Datum Transformations.......

Difference Between "True" and Computed WGS 84 Coordinates as a Function of Datum Transformation Parameters (or ........................... Technique) Used..................

Techniques Available for Transforming Local Geodetic System Coordinates ( 4, A, H ) to WGS 84 Coordinates

Advantages, Disadvantages, and/or Characteristics of Available Local Geodetic System-to-WGS 84 Datum Transfor- ........................................... mation Techniques

Local Geodetic System-to-WGS 84 Datum Transformation Multiple Regression Equations (A$, AX, N ) - North ...................... American Datum 1927 (NAD 27) to WGS 84

RMS Differences Between "True" WGS 84 Coordinates and WGS 84 Coordinates Computed Using Local Geodetic System-to-WGS 84 Datum Transformation Multiple Regression Equations ........................................ Multiple Regression Equation for DMA-Developed Local Geodetic System Geoid Heights - North American Datum 1927

RMS Difference With Which Geoid Height Multiple Regression Equations Reproduce DMA-Developed Local Geodetic System ............................................... Geoid Heights

Non-Doppler Derived Datum Transformation Parameters - Local Geodetic Systems to WGS 84....... ........................... Methods for Obtaining WGS 84 Coordinates for Sites of Interest... ................................................. WGS 84 "True" Coordinates Minus WGS 84 Coordinates Calcu- lated Using Different Local Geodetic System-to-WGS 84 Datum Transformations - NAD 27 (CONUS). ...........................

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TABLE TITLE PAGE

9.1 Accuracy ( l a ) of WGS 84 Coord inates a t Doppler Sites-WGS 84 Coord inates Determined Using Molodensky Datum Trans fo rmat ion Formulas and Loca l Geodet ic System-to-WGS 84 Mean Datum S h i f t s ( a, n, a7 ) ....................................... 9-20

9.2 Accuracy ( l a ) o f WGS 84 Coord inates a t Doppler Sites-WGS 84 Coord inates Determined Using Loca l Geodet ic System-to-WGS 84 Datum Trans fo rmat ion M u l t i p l e Regression Equat ions .......... 9-23

9.3 Accuracy ( l a ) o f WGS 84 Coord inates as a Func t i on o f Coord ina te De te rmina t ion Technique .......................... 9-26

A . l Correspondence Between Besse l ian and J u l i a n Epochs Plus M a t r i x f o r Precess ing From B1950.0 t o J2000.0..... .......... A-14

A.2 1980 I A U Theory o f N u t a t i o n - Ser ies f o r N u t a t i o n i n Long i tude (A$) and O b l i q u i t y ( AE) (Mean Equator and

............................................ Equinox o f Date) A-15

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FIGURE TITLE PAGE

2.1 D i f f e r e n c e Between NSWC 92-2 and WGS 84 Reference Frame O r i g i n s ....................................................

2.2 D i f f e r e n c e Between NSWC 92-2 and WGS 84 L o n g i t u d e References (X-Axes) ........................................

2.3 D i f f e r e n c e s Between WGS 84 and NSWC 92-2 D i s t a n c e s and ................. Geode t i c H e i g h t s ( E f f e c t o f Sca le Change).

...... 2.4 P i c t o r i a l R e p r e s e n t a t i o n o f Some As t ronomica l Terms..

2.5 P i c t o r i a l D e f i n i t i o n o f t h e C o n v e n t i o n a l I n e r t i a l System [ S t e l l a r (FK5), Epoch 52000.01.. ...........................

2.6 Downwardly P r o j e c t e d C o n v e n t i o n a l T e r r e s t r i a l Po le (CTP) , C e l e s t i a l Ephemeris Po le (CEP ), and Conven t iona l ~ n e r t i a i System (CIS) Po le ..........................................

2.7 CIS-to-WGS 84 (CIS-to-CTS) T r a n s f o r m a t i o n .................. 3.1 High, Low, and Y e a r l y Average Angu lar V e l o c i t y ( w )

Values. .................................................... 3.2 G e o c e n t r i c Radius, R a d i i o f Curva tu re , and E l l i p s o i d a l

Arc D is tances. . ............................................ 4.1 Atmospher ic C o r r e c t i o n f o r Measured G r a v i t y (When Forming

WGS 84 G r a v i t y Anomal ies) . ................................. 5.1 lo x lo Mean T e r r a i n C o r r e c t i o n s Used i n Development o f

WGS 84 EGM ( U n i t s = M i l l i g a l s ) ............................. 6.1 R e l a t i o n s h i p Between t h e E a r t h ' s P h y s i c a l Sur face, t h e

Mean Sea L e v e l S u r f a c e o f t h e E a r t h ( o r Geo id) , and t h e E a r t h ' s Ma themat i ca l F i g u r e (an E l 1 i p s o i d ) . ................

6.2 S a t e l l i t e - D e r i v e d Height-Above-Mean Sea L e v e l Values ( T e r r e s t r i a l P o i n t s ) . ......................................

6.3 S a t e l l i t e - D e r i v e d Height-Above-Mean Sea L e v e l Values ( S p a t i a l O b j e c t s ) ..........................................

6.4 I l l u s t r a t i o n o f a Nega t i ve WGS 84 Geodet ic H e i g h t (H) ...... 6.5 Two. Techniques f o r O b t a i n i n g H e i g h t Above Mean Sea Leve l

Values.......... ...........................................

x x i v

LIST OF FIGURES ( C o n t ' d )

FIGURE TITLE

Coord ina te System Assoc ia ted w i t h Geoid H e i g h t B i - L i n e a r I n t e r p o l a t i o n Scheme.................. ..................... L o c a t i o n o f Geoid H e i g h t D i f f e r e n c e s Grea te r Than 1 Meter (T rue Minus I n t e r p o l a t e d From a 3 0 ' x 3 0 1 G r i d ) .............. Doppler S t a t i o n s Used i n WGS 84 Development ................ D i f f e r e n c e Between WGS 72 and WGS 84 Reference Frame O r i g i n s .................................................... D i f f e r e n c e Between WGS 72 and WGS 84 Long i tude References ( X-Axes) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . D i f f e r e n c e s Between WGS 84 and WGS 72 D is tances and Geodet ic H e i g h t s ( E f f e c t o f Sca le Change). . . . . . . . . . . . . . . . . . WGS 84 and WGS 72 E l l i p s o i d D i f f e r e n c e s (S ize, Shape). . . . .. H y p o t h e t i c a l R e l a t i o n s h i p Between Two Coord ina te Systems ... Doppler S t a t i o n s (405) Used i n Deve lop ing NAD 27 (CONUS) t o WGS 84 Datum S h i f t s . . ................................... Doppler S t a t i o n s (216) Used i n Deve lop ing NAD 83 (CONUS) t o WGS 84 Datum Sh i f t s . . . ..................................... Doppler S t a t i o n s ( 8 4 ) Used i n Deve lop ing SAD 69 t o WGS 84 Datum Shifts................................ ............... Doppler S t a t i o n s ( 8 5 ) Used i n Deve lop ing ED 50 (Western Europe) t o WGS 84 Datum S h i f t s . . . . . . . . . . . . .*. . . . . . . . . . . . . . . Doppler S t a t i o n s ( 4 7 ) Used i n Deve lop ing ED 50 ( U K / I r e l a n d ) t o WGS 84 Datum S h i f t s . .................................... Doppler S t a t i o n s (40) Used i n Deve lop ing ED 50 (UK On ly ) t o WGS 84 Datum Shifts.................. ................... Doppler S t a t i o n s ( 1 3 ) Used i n Deve lop ing TD t o WGS 84 Datum Shifts...............................................

Doppler S t a t i o n s (105) Used i n Deve lop ing AGD 66 t o WGS 84 Datum Shifts.......... .....................................

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6-26

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7-67

7-68

7-69

7-70

7-71

7-72

7-73

7-74

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7-76

7-77

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LIST OF FIGURES (Con t ' d )

FIGURE TITLE PAGE

Doppler S t a t i o n s (90) Used i n Developing AGD 84 t o WGS 84 ............................................... Datum S h i f t s

............. NAD 27 (CONUS) - Geodetic L a t i t u d e D i f f e r e n c e s

.......... NAD 27 (CONUS) - Geodet ic Long i tude Di f ferences. .

............... NAD 27 (CONUS) - Geodet ic He igh t Di f ferences

Contour Char t - NAD 27 t o WGS 84 Datum S h i f t s (CONUS), AX ..................... Component (Contour I n t e r v a l = 1 Meter )

Contour Char t - NAD 27 t o WGS 84 Datum S h i f t s (CONUS), AY ..................... Component (Contour I n t e r v a l = 1 Mete r )

Contour Char t - NAD 27 t o WGS 84 Datum S h i f t s (CONUS), AZ ..................... Component (Contour I n t e r v a l = 1 Meter )

Contour Char t - L a t i t u d e D i f f e rences , WGS 84 Minus NAD 27 (CONUS; Contour I n t e r v a l = 0.1 Arc Second) ................. Contour Char t - Longi tude D i f f e rences , WGS 84 Minus NAD 27 ................. (CONUS; Contour I n t e r v a l = 0.2 Arc Second)

Contour Char t - Geodet ic He igh t D i f f e rences , WGS 84 Minus NAD 27 (CONUS; Contour I n t e r v a l = 2 Meters) ................ DMA-Developed NAD 27 Geoid (CONUS; Referenced t o ..................... C la r ke 1866 E l l i p s o i d ; U n i t s = Meters)

........... NAD 27 (CONUS) - Geodet ic L a t i t u d e D i f fe rences . .

.......... NAD 27 (CONUS) - Geodet ic Long i tude D i f fe rences . .

............. NAD 27 (CONUS) - Geodet ic He igh t D i f fe rences . .

Maximum L a t i t u d e D i f f e r e n c e s i n Conver t ing WGS 72 Coord inates t o WGS 84 (NAD 27 Area, CONUS) - ( U n i t s = 0.1 Arc Second). ............................................... Maximum Long i tude D i f f e r e n c e s i n Conver t ing WGS 72 Coord inates t o WGS 84 (NAD 27 Area, CONUS) - ( U n i t s = 0.1 Arc Second) ................................................

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LIST OF FIGURES ( C o n t ' d )

FIGURE TITLE

9.3 Maximum Geodet ic H e i g h t D i f f e r e n c e s i n C o n v e r t i n g WGS 72 Coord ina tes t o WGS 84 (NAD 27 Area, CONUS) - ( U n i t s = 2 M e t e r s ) .........................................

9.4 Accuracy ( l a ) o f WGS 84 Geodet ic L a t i t u d e s a t Dopp ler S i t e s (NAD 27 Area, CONUS), L a t i t u d e Determined Us ing Mean Datum S h i f t s (Contour I n t e r v a l = 1 M e t e r ) . .......................

9.5 Accuracy ( l a ) o f WGS 84 Geodet ic Long i tudes a t Dopp ler S i t e s (NAD 27 Area, CONUS), L o n g i t u d e Determined Us ing

............ Mean Datum S h i f t s (Con tou r I n t e r v a l = 1 M e t e r ) .

9.6 Accuracy ( l a ) o f WGS 84 Geodet ic H e i g h t s a t Dopp ler S i t e s (NAD 27 Area, CONUS), Geode t i c H e i g h t Determined Us ing Mean Datum S h i f t s (Contour I n t e r v a l = 1 M e t e r ) ..................

9.7 Accuracy ( l a ) o f WGS 84 Geodet ic L a t i t u d e s a t Dopp ler S i t e s (NAD 27 Area, CONUS), L a t i t u d e Determined Us ing .. Datum T r a n s f o r m a t i o n MREs (Contour I n t e r v a l = 0.5 M e t e r ) .

9.8 Accuracy ( l a ) o f WGS 84 Geodet ic Long i tudes a t Dopp le r S i t e s (NAD 27 Area, CONUS), L o n g i t u d e Determined Us ing

.. Datum T r a n s f o r m a t i o n MREs (Contour I n t e r v a l = 0.5 M e t e r ) .

9.9 Accuracy ( l a ) o f WGS 84 Geodet ic H e i g h t s a t Dopp ler S i t e s (NAD 27 Area, CONUS), Geodet ic H e i g h t Determined Us ing

. Datum T r a n s f o r m a t i o n MREs (Contour I n t e r v a l = 0.5 Meter 1..

A . l ECI (CIS, Epoch J2000.0)-to-WGS 84 (CTS, ECEF) T r a n s f o r m a t i o n .............................................

A.2 C e l e s t i a l Sphere D e p i c t i n g P recess ion o f t h e Equinox ....... ........................... A.3 P recess ion T r a n s f o r m a t i o n M a t r i x

A.4 P recess ion Equa t ions ....................................... ................................ A.5 Ast ronomic N u t a t i o n Diagram

.................. A.6 Ast ronomic N u t a t i o n T rans fo rma t ion M a t r i x

A.7 Ast ronomic N u t a t i o n Equa t ions (1980 I A U Theory) ............ .............................. A.8 Ast ronomic N u t a t i o n Arguments

A.9 S i d e r e a l Time Diagram ......................................

PAGE

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FIGURE

LIST OF FIGURES (Cont'd)

TITLE

Sidereal Time Transformation Matrices..... ................. Sidereal Time Equations .................................... Polar Motion Diagram ....................................... Polar Motion Transformation Matrix.........................

Summary - ECI (CIS, Epoch J2000.0)-to-WGS 84 (CTS, ECEF) Transformation Matrices... ................................. Summary - ECI (CIS, Epoch J2000.0)-to-WGS 84 (CTS, ECEF) Equations .................................................. Summary - Steps in ECI (CIS, Epoch J2000.0)-to-WGS 84 (CTS, ECEF) Transformat ion.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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