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70-19 , 34-8

OBENSON, G abriel Francis Tambe, 1934— DIRECT EVALUATION OF THE 'S ANOMALY FIELD FROM ORBITAL ANALYSIS OF ARTIFICIAL EARTH SATELLITES.

The Ohio S ta te U n iv e rsity , Ph.D. , 1970 Geophysics

University Microfilms, A XEROX Company, Ann Arbor, Michigan i

THIS DISSERTATION HAS BEEN MICROFILMED EXACTLY AS RECEIVED DIRECT EVALUATION OF THE EARTH'S GRAVITY ANOMALY FIELD FROM ORBITAL ANALYSIS OF ARTIFICIAL EARTH SATELLITES

DISSERTATION

Presented in Partial Fulfillment of the Requirements, for the Degree Doctor of Philosophy in the Graduate School of The Ohio State University

By

Gabriel Francis Tambe' Obenson, B.Sc., M. Sc., A.R.I.C.S.

The Ohio State University 1970

A pproved b

Department of Geodetic Science Obenenkongho, Obenafa, Nzealu,

Orang, Obenafa & Besem. AC KNOW LEDGEM EN TS

I wish to thank Dr. R. H. Rapp for providing valuable guidance during the preparation of this dissertation and Dr. U. A. Uotila for

the many pertinent discussions 1 had with him.

My thanks also go to the Instruction and Research Computer

Center for providing free computer time for the extensive program­ ming that was involved in this study.

This work was supported, in part, through an Ohio State Uni­

versity Research Foundation project sponsored by the Air Force

Cambridge Research Laboratories, Bedford, Massachusetts. VITA

October 28, 1934 ...... Born —Mamfe', Cameroon, West Africa

1 9 6 1 ...... B.Sc., Photogrammetric Engineering, ITC Delft, Holland

1 9 6 3 ...... Diploma, Comprehensive Planning, Institute of Social Studies, The Hague, Holland

196 5 ...... A. R. I.C. S., Professional Associate of the Royal Institution of Chartered Surveyors, B ritain

1968 ...... M. Sc., Geodetic Science, The Ohio State University, Columbus, Ohio

1961-1965 ...... Senior Surveyor, Survey Department, Kaduna — Nigeria, West Africa

196 6 ...... Senior Surveyor, Lands & Surveys Depart­ ment, Buea, West Cameroon, West Africa

196 8 ...... Teaching and Research Associate, The Ohio State University, Columbus, Ohio

196 9 ...... Research Associate, The Ohio State Uni­ versity, Columbus, Ohio

PUBLICATIONS

"Production of 1/50,000 Scale Maps of Northern Nigeria. 11 C hartered Surveyor (A. R. I.C. S.) Thesis, 1965.

"Photogrammetric Mapping — Practical Hints. " Northern Nigeria Survey, 1965. "Prediction of Mean Gravity Anomalies of Large Blocks from Sub-Bloc Means." M.Sc. Thesis, 1968.

"Prediction Accuracies of 5°x5° Mean Anomalies from 1° xl° Means at Different Latitudes. " Reports of the Department of Geodetic Science, No. 117, The Ohio State University, Nov., 1968.

FIELDS OF STUDY

Major Field: Geodesy Geodetic Dr. I. I. M ueller Physical Geodesy Dr. I. I. M ueller Satellite Geodesy Dr. I. I. M ueller Geometric Geodesy Dr. R. H. Rapp Mathematical Projections in Geodesy Dr. R. H. Rapp Adjustment Computations Dr. U. A . Uotila

Minor Field: Photogrammetry Dr. S. K. Ghosh Mathematical Statistics Dr. T. A. Willke TABLE OF CONTENTS

P age

ACKNOWLEDGEMENTS...... m

VITA ...... iv

LIST OF TABLES ...... v iii

LIST OF FIG URES ...... x

INTRODUCTION ...... 1

Chapter

1. REPRESENTATION OF THE EARTH'S GRAVITATIONAL POTENTIAL F IE L D ...... 6

1. 1 General 1. 2 P o ten tia l of a L ev e l E llip so id 1. 3 Total Disturbing Potential from Gravity Anomalies 1.4 Total Disturbing Potential from Potential Coefficients 1. 5 Numerical Check 1. 6 Su m m ary

2. GENERATION OF SATELLITE ...... 13

2. 1 General 2. 2 Cowell's Method of Generation 2. 3 Method of Variation of Parameters for Orbit G en eration 2.31 Evaluation of S, T, W 2. 4 Components of the Disturbing Force from Potential Coefficients 2. 5 Check Computations

v i TABLE OF CONTENTS (Continued) P age 3. PARTIAL DIFFERENTIAL EQUATIONS AND ADJUSTMENT M O DELS...... 37

3. 1 General 3. 2 Evaluation of the Partial Derivatives 3. 2. 1 Numerical or Variant Method 3.2.2 Variational Partial Derivatives 3. 3 Adjustment Models 3.3.1 General 3.3.2 Minimization of Quadratic Sums 3.3.3 Variance-Covariance Matrix of the Parameters and Observables 3.3.4 Special Cases 3. 4 Relevant Matrices for Satellite Analysis 3. 5 Some Additional Notes 3. 6 Su m m ary 4. NUMERICAL RESULTS AND ANALYSIS ...... 81 4. 1 General 4. 1. 1 Definition of Parameters and Weighting 4. 2 Wrong and Station Coordinates 4. 3 W rong A n om alies 4.4 Wrong Orbital Elements, Gravity Anomalies and Station Coordinates 4. 5 Wrong Orbital Elements and Gravity Anomalies 4. 6 Analysis of the Results 5. CONCLUSION ...... 116

APPENDIX A. Potential C oefficients ...... < . 118

B. 15°xl5° Equal Area Mean Gravity Anomalies ...... 121

C. 15°xl5° Equal Area Terrestrial Mean Gravity Anomalies . 124

D. Set Up for Integration of Cowell's Equations of Motion. 129

REFERENCES 131 LIST OF TABLES

Table Page

1 Comparison of Satellite Positions from Potential Coefficients and Gravity Anomalies ...... 32

2 Comparison of Disturbing Forces, S, T, W ...... 33

3 Comparison of the Generalized Stokes' Function and Its Derivatives for Various Evaluation Points .... 35

4 Satellites First U se d ...... 82

5 Comparison of True and Predicted Differences in Right Ascension, a*, Declination, 6 *, and Range, r*, for Wrong Orbital Elements and One Station ...... 8 6

6 Recovered Errors of Orbital Elements and Station Coordinates from One Revolution of One Satellite . . . 90

7 Check of Variational Equations for Wrong Gravity Anomalies Alone for One Revolution of One Satellite . 91

8 Comparison of Predicted and True Differences in Right Ascension, a*, Declination, 6 *, and Range, r*, Using Wrong Orbital Elements, Station Coordinates, and Gravity Anomalies of Three S a tellites ...... 94

9 Recovered Errors in Orbital Elements from One Revolution Each of Three Satellites ...... 95

10 Recovered Errors in Station Coordinates from One Revolution Each of Three Satellites ...... 95

11 Recovered 15°xl5° Equal Area Mean Gravity Anomaly Errors from One Revolution Each of Three S a t e l l i t e s ...... 96

v iii LIST OF TABLES (Continued)

Table - Page

12 Second Set of Satellites Used . 101

13 Comparison of True and Predicted Differences for Right Ascension, av, and Declination, 6 V, of Vanguard 2 ...... 103'

14 Recovered Input Errors of Orbital Elements from About 15 Revolutions Each of Five Satellites .... 104

15 Recovered Error 3 in Mean Gravity Anomalies from About 15 Revolutions Each of Five Satellites .... 105

16 Conversion Terms for Anomalies from International Gravity Formula System to GRS 1967 ...... 125

17 15°xl5° Equal Area Terrestrial Mean Anomalies . . . 127

i * LIST OF FIGURES

T able Page

1 Components of the Disturbing Force at a Satellite Position, P ...... 14

4 2 Differences Between True and Predicted Differences in Right Ascension, a , Declination, 6 , and R ange, r * ...... 88

3 Schematic Form of the Observation Equations and Normal E quations ...... 93

4 Recovered Mean Gravity Anomaly Errors in Relation to Satellite O rb its ...... 100

5 Differences of (Predicted-True) Differences in Right & a , Ascention, a , and Declination, o , for Vanguard 2 . . 104

6 Recovered Mean Gravity Anomaly Errors from Five Satellites ...... 110

x INTRODUCTION

Gravity anomalies have significant uses in geodesy, for investi­ gating the earth's gravitational potential field and for absolute position determination of points on the earth's surface. They also have signifi­ cant uses in non-geodetic fields such as geology and space mechanics.

However, accurately observed gravity anomalies are not currently available on a world-wide basis because of observational and other physical problems. To obtain anomalies in those areas where there are no observed values, various techniques have been used and are being used, both geophysical and statistical. The geophysical methods include model type anomalies, Uotila [1962], and the statistical methods include least squares techniques of either (a) fitting surfaces through known values, Uotila [1967], or (b) mathematical interpolation and ex­ trapolation, Moritz [1964], Obenson [1968], Rapp [1964].

Currently, gravitational potential coefficients of the earth are being increasingly used to compute gravity anomalies and other geo­ detic quantities. The advent of satellites designed for geodetic purposes has greatly facilitated the evaluation of these potential coefficients. It has therefore been a natural development to combine the results of potential coefficients obtained from satellite analysis with the available

1 anomalies to obtain some sort of unique values for the coefficients and

the anomalies. The procedures and problems involved are dealt with in

W. Kaula [1966a]; R. Rapp [1967, 1968, 1968a]; and W. Kohnlein [1967].

The idea of obtaining gravity anomalies directly from satellite

orbit analysis is largely due to the work of K. Arnold [1967] . He has

given detailed theoretical formulations based on mean orbital elements

of geodetic satellites for filling the unknown parts of the earth's gravity

anomaly field. Others who have proposed alternate methods of studying this problem are K. Koch, who has proposed first computing density anomalies, from which gravity anomalies are later obtained, Koch[1968],

Koch and Morrison [1969], and R. Rapp [ 1967a], who has indicated, in

general terms, how anomalies can be evaluated from topocentric obser­

vations of satellite positions.

The purpose of this paper is to develop practical formulations, in

terms of topocentric right ascension, declination, and range, to obtain directly from analysis of satellite orbits perturbed by the earth's gravi­

tational potential alone the following quantities:

a) corrections to assumed osculating orbital elements at

a reference ;

b) corrections to a priori estimates of gravity anomalies; and

c) geocentric rectangular coordinates of some ground stations

from which observations are made. Since no actual observations of right ajscension, declination and

range are used in this study, all data will be generated. Thus rectang­ ular coordinates of some stations and a set of potential coefficients up

to (14, 14), Rapp [1968], from which gravity anomalies are derived, as

well as another set of actual gravity anomalies are adopted. True orbits

of certain satellites whose osculating elements at some initial time are

assumed correctly known are generated using the potential coefficients

and gravity anomalies. Errors are then introduced into the elements,

gravity anomalies, station coordinates, and approximate orbits gen­

erated. The results from these approximate orbits are compared with

appropriate values from the assumed true orbits and an adjustment done

in an attempt to recover the errors introduced.

One of the advantages of this direct approach is that the variance-

covariance matrix of the adjusted anomalies is readily available for

further analysis. Also, since the earth's gravity anomaly field can be

represented by a finite number of mean gravity anomaly blocks, it is

theoretically possible to avoid truncation errors in computing certain

geodetic quantities, which currently afflict potential coefficients. Be­

fore the actual adjustment equations are derived in Chapter 3, a general

treatment of some relevant material is given: representation of the

earth's gravitational potential field and generation of satellite orbits

affected by the earth's gravitational potential alone. It should be noted, however, that more detailed expositions of some of these subjects can

be found in many other references including those quoted here.

Chapter 1 deals with the representation of the earth's gravitational

potential field at a satellite point. This potential field is divided into

two parts: a much larger component, V, due to a sphere approximating

the earth; and a smaller component referred to as the total disturbing potential, R^.. The total disturbing potential is also made up of two com­

ponents: the first, due to the oblateness of the earth, expresses the de­ viation of the potential of an ellipsoid from that of the sphere defined

above. The second component, providing a much smaller effect than the

first, expresses the deviation of the potential of the earth from that of

an ellipsoid. The equation for Rj. is derived from both potential co­

efficients and gravity anomalies.

In Chapter 2 the generation of a satellite orbit by the method of

Variation of Parameters from gravity anomalies and potential coeffi­

cients is discussed in detail. Cowell's method of orbit generation is

given only in general terms.

Chapter 3, which is the main contribution of this study, gives

detailed derivations of variational equations relating the satellite posi­

tion at any time with gravity anomalies and its position at a given initial

tim e.

All the equations derived in Chapters 1 to 3 are directly appli­

cable only under the assumptions made in deriving them. In order to use them for practical evaluations certain necessary additions must be made to the equations. The main additions are described at the end of

C hapter 3.

In Chapter 4 numerical checks of the equations derived in Chap­ ter 3 are presented. Results of the recovery of the input errors are discussed and analyzed.

Formulas are numbered consecutively in each chapter, the first number indicating the chapter. An attempt has been made to limit

rederivation of equations found in other publications; but where they are given in a form not satisfactory for this exposition, a detailed derivation of them is given here.

All computations in this study were done on the IBM 360/75. CH APTER 1

REPRESENTATION OF THE EARTH'S GRAVITATIONAL POTENTIAL FIELD

1. 1 G eneral

The earth's gravitational potential field has been variously repre­ sented in different forms. In this study it will be written as a sum of three components, the last two of which are associated with a level ellipsoid and free air gravity anomalies. In this way it will be possible to evaluate the geopotential at a satellite position using equations which will later be suitable for computation of anomalies directly from an­ alyzing the satellite orbits.

Before these equations are derived, we will first look at the form of the potential of a level ellipsoid.

1. 2 Potential of a Level Ellipsoid

For an ellipsoid with flattening f, which has the same character­ istics as the earth:

( 1 ) its center of gravity coincides with that of the earth;

(2 ) it rotates at the same angular , «g, around the same minor axis as the earth; (3) its mass, M, is equal to that of the earth,

the potential, U, at a point above its surface is given by Heiskanen and Moritz [1967, p. 73] as:

(1.1) U = — r [‘ "X ( t ) ” J2nPzn<-to

r and are the geocentric distance and latitude respectively of the p o in t;

k is the gravitational constant;

(1.2) J2n = (-l)n+1 (2n+1)(2n+3) ( X - n + J z ) *

n = 2 , . . . where it can be shown that:

(1 .3 ) J2 = |- f - j m - j f 2 + j y fm + . . .

4 7 4 (1 .4 ) J4 = - - f 2 + - fm + . . .

m = uEZae^ (1 -f)/kM;

^2n are ordinarY associated Legendre functions of degree 2n;

ae is the equatorial radius;

e is the eccentricity. .

For the computations done in this study, the ellipsoid used had the following parameters: 8

a e = 6378160 m

f = 1/298.247

kM = 3. 98603 x 1 0 ^ m ^ s e c

1. 3 Total Disturbing Potential from Gravity Anomalies

The gravitational potential, V, due to the earth at a satellite posi­ tion can be written as:

(1. 5) Y = Vj + Rt w here:

( 1 . 6 ) Vi = is the potential due to a sphere.

(1 .7 ) R(. = V2 + V 3 is the total disturbing potential.

V 2 indicates the deviation of the potential of an ellipsoid from the

spherical approximation given by V^.

V 3 , the remaining disturbing potential, indicates the deviation of the potential of the actual earth from that of the ellipsoid and can be expressed in terms of the generalized Stokes' Function relating the dis­

turbing potential, Tg, used in geodesy, to some free air gravity anom­ a lie s , A g.

The division of into two components is completely arbitrary.

However, these components together with Vj must rigorously add up to

the total gravitational potential. 9

Associating only the zonal part of the potential, U, of the level ellipsoid of Section 1.2 with V 2 of Eqn. (1.7), we get:

V 3 , the disturbing potential at any point P, above the surface of this level ellipsoid, is then given by:

t1-9* V3 = Jp J J AgS(r,^)dq q

R = mean earth radius; dq is an element of area on the integration surface; Ag are free air gravity anomalies with respect to the level ellipsoid whose surface is q; S(r,^) is the generalized Stokes' Function given by:

(1. 10) S(r, ^) = t " 3D - tcos $ ^5 + 3l5n tcos^ +

(1. 11) cos ip = sincp sincpq + cos cpcoscpq cos(Xq - X)

( 1 . 1 2 ) t = — r -2 ^ (1. 13) D = (1 - 2tcos^ + t c )

For problems dealing with the external gravity field of the earth free air gravity anomalies are the most appropriate ones to use since their evaluation does not involve transportation of m asses inside the geoid. 10

In this study r, cp, and X are geocentric coordinates of the satel­ lite position and cp and X are the geocentric coordinates of the differ- 4 H ential area, dq, hence of the center of a gravity anomaly block.

Having defined all the necessary quantities, the total disturbing potential, Rt, from J2 , J4 and gravity anomalies is given by:

(1.14) Rt = ^ [ j 2(^ ) P 2 + J4(^r) P4

+ s H AsS(r>Wq q

1.4 Total Disturbing Potential from. Potential Coefficients

If the earth is represented by a set of potential coefficients of maximum order N, the potential due to the earth at a satellite point is g iv en by: N n kM (1. 15) V = r [ " l m l (Cnmcosm ^ + ^nms inmX)- n - 2 m = 0

Pnm n™(8 in

C, S are fully normalized potential coefficients;

Pnm are fully normalized Associated Legendre Functions.

The total disturbing potential in this case is given by: N n lrlV/r n ( 1 . 16) Rt = ^r ^ ( " ! ? ) X ^ nmCOSmX + ^ im ®inmX)Pn m (sincp) n=2 m=0 11

1. 5 N u m erical C heck

For a point in space with coordinates r = 8000km, cp =70°, X =40°,

Rt was computed from Eqns. (1. 14) and (1. 16). The anomalies used in

Eqn. (1. 14) were 5°x5° mean anomalies computed from the potential coefficients of Appendix A.

Rj. =71.5 m^sec"^ from 14, 14 potential coefficients.

= 71.4 mcsec from 5°x5° gravity anomalies.

The difference of O .lm ^sec^is largely due to the fact that Stokes'

Function was assumed constant for each gravity anomaly block and hence evaluated only at its center. Both equations are therefore accepted as giving reasonably close values.

1. 6 Sum m ary

By splitting the earth's gravitational potential into two main com­ ponents — a much larger part due to a sphere, which approximates the earth; and a much smaller part, the total disturbing potential, which is due to the difference between the potential of the actual earth and that of the sphere — it is now possible to discuss the methods of generation of satellite orbits using these principles. In order to simplify computa­ tions and introduce equations which would later be used to investigate evaluation of gravity anomalies from the generated satellite orbits, the total disturbing potential is further divided into two components: the first associated with the potential of a level ellipsoid, less the'spherical effect, and the second with the disturbing potential at any point from free air gravity anomalies. •

Section 1.4, like other sections where potential coefficients are mentioned, was introduced merely to provide a numerical check on the equations derived in earlier sections. For this chapter the check is given in Section 1.5. C H APTER 2

GENERATION OF SATELLITE ORBITS

2. 1 General

Having defined the disturbing potential, we now continue with the formulation of the mechanics of the satellite's motion under the influ­ ence of the earth's gravitational potential alone. Because the purpose of this study is to obtain gravity anomalies from satellite orbits, we will develop in detail (Section 2. 3) suitable equations to achieve this p u rp o se.

2.2 Cowell's Method of Orbit Generation

In this method the acceleration of a satellite, of unit m ass, mov­ ing under gravitational forces alone is given by:

(2.1) Xi = VV n + VRti

VVj and VRj- are the gradients of the potentials discussed in the previous chapter;

i = x,y, z expresses a rectangular coordinate system in whose directions the components of the acceleration, Eqn. (2. 1), are given.

The resultant set of equations are second order differential equations and are solved by numerical integration.

13 / N (Ascending Node)

F ig u r e 1

Components of the Disturbing Force at a Satellite Point, P

In Figure 1, O is the center of gravity of the earth, x, y, z is a

right-handed orthogonal rectangular coordinate system. The x-axis is in the direction of the vernal equinox, the z-axis is the rotation axis of the earth, and the x-y plane is formed by the celestial equator.

Adopting the coordinate system shown in Figure 1, the accelera­ tion of the satellite in Cartesian coordinates becomes:

X X 'l o o" ’ b " -kM y y + R3(-a.p) 0-1 0 R2 (6 p-90°)R3(180°) L " r 3 _z_ _ z .0 0 1 ..S. whe r e :

Op and 6 p are the right ascension and declination respectively of

the satellite. 15

The rotation matrices are defined in Mueller [1969, p. 43, Eqns.

3.20 and 3.21].

B, Li, S are the components of VRj. and they are defined as follows,

F ig u re 1:

B is positive northwards along the meridian;

L is positive eastwards along the prime vertical;

S is directed along the radius vector and positive away from

the center of the earth.

H ence:

B i . ' r Sep

1 T dRt ± j r coscp dX

s n i

Noting that:

9 = 6p

coscXp = x / (x 2 + y 2 ) ^

sinOp = y/(x 2 + y 2 ) ^

c o s 6p = (x^ + y 2 ) ^ / r

s in 6p = z / r

r = (x^ + y 2 + z 2 ) ^ and multiplying out the m atrices, Eqn. (2.2) simplifies to: The derivatives of the total disturbing potential, Rj., are devel­ oped later in the next section. Integration of these equations gives the elements required for evaluating the variational equations of Chapter 3

(See Appendix D. )

2. 3 Method of Variation of Parameters for Orbit Generation

In this method, the disturbing force, VR^., is applied to a set of independent orbital parameters instead of to the Cartesian coordinates as in Cowell's method. The method of transforming the second order differential equations of Cowell's method to first order time derivative of certain elements can be found in, for example, Moulton [1914] or

Dobson, et al. [1962].

Resolving the perturbing force, VRj, into another set of three orthogonal components:

S defined e a r lie r on page 15;

T perpendicular to the radius vector, and positive in the the direction of flight;

W normal to the orbital plane and positive northwards; 17 the Gaussian Equations of motion relating the time derivatives of the

Keplerian orbital elements with these components are given as follows,

Moulton [1914, p. 404]:

^ = 2 a / e S sin v + dt v lkM (l -e ) r T)

de _ /_p_ So sin v + /I ------r + — P cos v + , — er . T_ dt V kM

— = — I- —— W cos(u + v) dt a v kM(l-e )

du> U-5) - — S cos v + £jLE sin v T - — W sin(w + v) cot i dt L e ep p

dft sin(w + v) W dt = Ia vJ— kMkM(l (l -e^) sin i

dM /~a~ f\ 2 r 1 - e^ c o s v ) S dt ~ n V kM V

_ (i - e 2 ) n r / + r T sin v kM w here:

kM x/2 (2 . 6 ) n = is the ; l 1 ^

(2.7) p = a(l - e )

The component, S, of the perturbing force, VRt, is defined in Eqn. (2.3). 18

The components T and W are obtained from B and L by a plane rotation through the angle /c, Figure 1, the spherical angle at P between the direction of the meridian and that of the satellite orbit. Hence:

(2. 8 ) T = B cos/c + L sin/c

(2.9) W = B sin/c - L cos/c

In matrix form:

“ - r cos/c sin/c d R t T r coscp dcp

sin/c -cos/c dR t (2 . 10) W = r coscp dX

dR t S 0 _

From the rectangular spherical triangle NMP, Figure 1:

cos(co+v)sin i (2 . 1 1 ) cos/c = coscp

co s 1 (2 . 1 2 ) sin/c = coscp

(2. 13) sincp = sin(oo+v)sin i

Substituting for K into Eqn. (2. 10), the equations for the components of

VRfc becom e:

dRt (2. 14) S = - g j .

1 dRt cos(a>+v)sin i dRt (2. 15) T = - COS 1 r dcp coscp r cos^cp dX

1 dRfc co s i ' 1 dRt (2. 16) W = — • ~v 1 • —- - 5— cos(u>+v)sin i r dcp coscp r c o s 2(p SX 19

Substituting for Rj. from Eqn. (1.7) into Eqns. (2. 14) to (2. 16), and not-

dy 2 ing that - 0 :

dv2 dV3 (2. 17) S = + -=r-— dr dr

cos(w+v)sin i dV 2 cos(co+v)sin i &V 3 (2 . 18) T = • — + • —— r coscp o

cos i dV 3 + r co s2cp dX

cos i dV? cos i dV 3 cos(w+v)sin i dV 3 (2. 19) W ------— - + ------• —------=------• — r coscp dcp l' coscp ocp r cos^cp oX

2.3.1 Evaluation of S, T, W

Differentiating Eqn. (1.8):

&V o kM (2 .20) ------ocp r G ) 1 ' . £ * ( ? ) * ' .

Now:

(2 . 2 1 ) P 2 = | sin2cp - I

_ 35 . 4 15 . o * 3 (2 . 2 2 ) P4 = — sin4cp sin^cp + g

Hence:

dP? (2.23) ^ = 3 sincpcoscp

dP4 35 o 15 (2.24) = — sinJcpcoscp - — sincpcoscp

_ coscp ain^cp - 15 sincp) Ct 20

Therefore:

a v 2 kM coscp (2.25) 3 ( — ) J2 sincp dcp r

35 . •, 15 . + ( — 1 J4 I y sinJcp sinJt - - y sincp

Substituting for sincp from Eqn. (2. 13):

, ^v 2 kM coscp ( ae \ 3 I — J J2 sin(w+v)sin i (2' 26) 4 + ^4 ®i-n^(co+v)sin^i

—15 sin(w+v)s • - t ^ s in i

S V 2 kM (2' 27) 'ST'T 3 ( t ) J2P2 + 5 ( ~ JJ4p4

Though some of the following equations are found in Heiskanen and Moritz [1967] , they are also treated here because of their basic importance to later derivations.

Differentiating Eqn. (1.9):

dV3 r (2.28) Ag S(r, Tj/) dq d

Similarly:

dV3 _R (2.29) S(r,V) dq “55T 477 21

)V3 R f f a ,1 5 7 = 5 J J A g^ _S(r,«) d, q

dS(r,^) _ dS(r,M ' f t p (2.31) dcp * dtp

Differentiating Equation (1. 11):

h ip (2.33) -sin ijJ ^ = coscpsincpq - sin

coscpcoscpQsin(X

In this study the azimuth of any element dq from the sub-satellite point is defined as a. Introducing this azimuth into the spherical triangle of

* Figure 1 and using well-known trignometric relationships, we obtain:

(2.35) sin^cosa = coscpsincpq - sincpcos 9 qCos(Xq - X)

(2.36) sin^sina = coscpqSin(Xq - X)

Inserting these into the preceding equations, we obtain:

-co sa

coscpqsina

are given in Heiskanen and Moritz

[1967] as: ,2.39) ^ 2 * 1 * -t2 a i a f ( Z + & . 8 + 3 -1 +tco«»t D Srfr^ \ D D D t d) s;n2^a i n ^ i p

1 - t cobt)/ + D \ ■3^n 2 )

72 1 - t2 4 (2.40) = . 1 + Z- + 1 - 6D or •H- D D

-T cos*/ ( 13 + 6«» i_I±££2LL2)

From Eqns. (2.37), (2.38), (2.39), and with Eqn. (2.40):

^V3 R C C -2 / 2 6 <2-41) 7 7 “ 7 ; J J Agt sin*‘:09a C p + 5 - 8

+3 D + t COS*/ - 1 _ 3 i n 1_- .tco»»+_p j D sin ^ 2 '

(2 42) ^ 3 = R_c^scp j- y Ag t2 8in<,sina ^ +i . g q

+3 D +tco.*-.! . U n 1 - t c o s * - - f D \ dq D s i n ^ 2 /

+ 1 - 6D (2- 43> 7 7 = I I

- 7 cos*/ (13 + 6«n 1 - tc°.» tD )] dq 23

Having now obtained the derivatives of V 2 and V 3 with respect to the di­ rections, r, 9 , and X from Eqns. (2.41), (2.42), and (2.43), the com­ ponents of the disturbing force, VR^, are:

2 (2.44) S = ^ ^2 35sin 2(oo+v)sin2i - lj-

+ ^ ( ^ — ^ ) J4 ^ 35sin^(cj+v)sin^i - 30sin2(o>+v)sin2i + 3^-

- 5 f S I A^ 2 [ i ^ + i + i- 6 d

7 , f , o . / n 1 “ tcosV' + D^ t c o B T p ( 1 3 + 6 >Kn ------^------J dq

, - -r-v ™ kM . , . . (2.45) T = - —5- cos(u+v)sin 1 3Jo ( — ) sin(oH-v)sin i x L * 1

Z'& f 35 15 1 + J4 ( j -j sin (w+v)sin i - sin(w+v)sin ir

cos(t*)+v)sin iff. . 1 T3 / ^ 2 6 + 47TCOStp J J ( ,5 3 + 5 q

- 8 + 3 PtTcos/ ~ 1 - 3in dq D sin ^ 2 J

cos + Ijj y y &g sinV'sina t3 (4 j + £ - 47TcOS(p q

3 D +, * c° 8/ - 1 - 3in 1 - t c o . » t D \ D sin V' 2 / 24

, _ x 2 (2. 46) W = - ^ cos i 3J2 TT sin(w+v)sin i r (?)

J4 sin 3(u>+v)sin3i - sin(aH-v)sin ij"

cos — \ \ Ag sin^cosa t 3 ( + — - 4 7 T c o scpstp J J \ D3 D q

, _ D + t c o s t / / - 1 _ n 1 - t cos^ + D \ , + 3 ------5------3i/n ) dq D sin ip 2 /

cos(co+v)sin 47TCOS9— j* § Ag si“i|'sina ‘ 3 ( ^ + 5 - 8 q

, _ D + t cos|/ - 1 _ n 1 - t C O ST p + D \ , + 3 ------5------3i)n-— ------1 ) dq D sin Tp 2 /

The terms sin^cosa and sin^sina are defined in Eqns. (2. 35) and (2. 36).

To obtain the Greenwich longitude, X, of the satellite,required in eval­ uating the generalized Stokes 1 function and its derivatives, the rotation of the earth has to be considered. If the longitude, Xj^, of the ascending node at time tQ of the satellite is known, then if ct^ is its right ascension at time, t:

(2.47) X = Xj^j + (otp -£2) - ug(t - tQ) where:

cog is the earth's angular velocity, and

(2.48) tan(Op - J2) = tan(w+v)cos i 25

If the longitude of the ascending node is not known, then the longitude of the satellite can be computed from the following equation:

(2. 49) X = (Op - 8 ) + S2 - GAST where (otp - £2) is given by Eqn. (2. 48);

GAST, the Greenwich Apparent Sidereal Time, is given by S. A.O.

Reports [1966, Vol. 1, p. 19] .

GAST = 100?075542 + 360?985647348 r + 0?29 x 10" 12 T2

-4.°392 x 10-3 sin (12? 1128 - 0?052954 r)

(2.50) +0?053 x - 3 sin 2(12?1128 - 0?052954 T)

-0?325 x 10-3 sin 2(280?0812 + 0?985647 t )

-0?05 x 10 ' 3 sin 2(64? 3824 + 13? 176398 t) where:

T = MJD - 33282.0 days

r = JD - 243328. 5 days

JD is the Julian Day of the time for which the longitude is required;

MJD, Modified Julian Day, is an interval of time measured from Julian 1 Day 2400000. 5.

The above equation gives GAST to 0.2 arsecs and implies a UTI time system. 26

2.4 Components of the Disturbing Force from Potential Coefficients

Equation (1. 16) gives the total disturbing potential in terms of potential coefficients. At the satellite point, P, the components, S,

T, W, of the disturbing force are given by Eqns. (2. 14), (2. 15), and

(2 . 16).

Differentiating Eqn. (1. 16):

dftt kM ^ f &e \ n — (2'5I) = " 7T 2 (n+1) \ t ) 2 (Cnmcos m n m

+ Snmsin m X) Pnm (sincp)

b R t kM V / ae \n V -

i 2 - 5 2 ) dcp — Z (^ c°9m X n m

dlPnm (sin(P)l + Snmsin m X) dtp

dRt kM V f a e \ n V — (2- 53) aT = ' T 2 V T j 2 m X n m

4

- Snm cos m X) Pnmf sincp)

The limits of thesummations of n and m in the above and any subse­ quent equations are as indicated in Eqns. (1. 15) and ( 1. 16), that is m=0 to m=n and n= 2 to n=N.

The fully normalized associated Legendre function, Pnm( sincp), is given as: 27

(2.54) Pnm ( sincp) = + 1) i(n M + ”m)! !l 2 -“co sm

, ni (2n-2i)l_____ ro4«min_m“2i ' ' • I / • \ 1 / \ t I SinCp J I xl (n-i)! (n-m-2i)l i =0 where:

50m> Kronecker delta, = 1, if m = 0

= 0 , if m 4 0

k is the greatest integer (n-m )/2

Differentiating Eqn. (2. 54) with respect to cp we obtain:

(coscp)m +* ^ • <2- 55> £ ■ y<* - 6)<2n + » | r ? = } t 2 ' n i =0

( - iji ~ ~ ~211 ~^~ |H r [sincp] (n_m "2i- 1) i \ (n-i)! (n-m - 2i)| TJ

k ‘

m{cos(p)m_* • sincp £ (- 1)1

i=0

[8 intp](n'm - 2i)

In actual computations recursive equations are much faster than the summation equations given above for the associated Legendre functions and their derivatives. Writing Eqn. (2.54) as:

(2.56) Pnm = Qnm^nm 28 we have:

(n - m)! % (2.57) Q nm (2 - *—om )< 2n + 1) (n + m)!

Pnm is the conventional associated Legendre function.

Recursively:

<2- 58> P„0 ■ sincpP„_ o - ^ Pn. 2>0

(2.59) P nn = (2n - 1) coscpPn _1>n_1

(2.6°) Pnm = Pn.2,m + <2 n ~ c° 8tPP m -l

The recursive forms of the derivatives of these equations are:

, 0 , 1V dPn, 0 (2n - 1) f . dpn - l ,0 , (2.61) 2— = ------sincp------2— + coscp P i q dcp n L dcp ’

n - 1 dpn - 2 , Q n dcp

dPn n T ‘dpn-l n -1 (2.62) 2— = (2n - 1) coscp —------2------sincpP , , dcp L dcp t n i,n i

dpn ,m dpn - 2 ,m , ' f dpn- 1, m - 1 i— = ------1— + (2n - 1) coscp ------2------dcp dcp L dcp

- sincp P. 1 n - 1, m - 1 J 29 where:

0 , 0

d p 0 , 0 0 dcp

P i } o = sincp

d p l ,0 — = coscp dcp

P i 1 = coscp

d p l , l — = -sin cp dcp and

n - 2 , m = 0 if m > n -2

d P _n ■■ ?^ • m « .p v ------2— = 0 if m > n - 2 . dcp

In order to check Eqns. (2. 54) to (2. 63) computations were-done to evaluate Pnm(sincp) and dPnm(sincp)/dcp using the summation and re­ cursive forms for cp = 40°. The results agreed exactly to ten decimal places for values of n and m up to 14, although the summation equations took twice as long to evaluate.

Substituting Eqns. (2. 51) through (2.63) into Eqns. (2. 14) through

(2. 16), the components of the perturbing force are obtained. Thus: 30

(2.64) S= - ™ £ (n+ 1) ( ^ ) £

+ Snmsin m X) P nm ( sincp)

kM cos(u>+v)sin (2.65) T = cos m X coscp n m

+ Snm sin m X) [Pnm (sincp)] - Q

m(Cnmsin m X - Snmcos m X) Pnm( sincp )

(2.66) W = ^ . ^ 2 ( ^ 0 I ^ n m c o s m x n m

d r— cos(u+v) sin i + SnmsmmX) — [Pnln(.imp)] + —

( ^ ) 1 / m

With the equations for S, T, and W given above, Eqns. (2.44) to

(2.46) and (2.64) to (2.66), it is now possible to generate a required

satellite orbit using the Gaussian equations of motion, Eqn. (2. 5). This , is done by numerical integration. \ 31

2. 5 Check Computations

If the gravity anomalies fhat appear in Eqns. (2. 44) through (2.46) were derived from the potential coefficients of Eqns. (2.64) through

(2.66), then the values for S, T, W (and consequently the satellite co­

ordinates, x, y, z) would be the same from both groups of equations.

To check that this is so, computations were done with the following

representations of the anomolous gravity field:

H219: 14, 14 potential coefficients;

G5: 1640 5°x5° equal area anomalies, computed at the center of each block from the 219 potential coefficients.

G10: 410 10°xl0° equal area anomalies, computed at the center of each block from the 219 potential coefficients.

G!5: 184 15°xl5° equal area anomalies, computed at the center of each block from the 219 potential coefficients.

Equal area blocks were chosen because, as compared to conven­

tional representation, the number of blocks is smaller and statistical

analysis of the data is relatively simpler.

The potential coefficients and 15°xl5° anomalies are given in

Appendixes A and B, respectively.

Satellite OBI of Table 4, Section 4. 1, was used in the computa­

tions done in this section.

Numerical integration of Eqn. (2. 5), for this section as well as

for Sections 4. 2 through 4.4, was done using Hamming's variable-step

predictor-corrector integration subroutine, IBM [1969]. The accuracy input required for the subroutine was from 5-10 m in the semi-major axis of the orbital ellipse, 10“^ radians in the other parameters, and an initial integration step size of 20 secs.

Results for S, T, W for the same satellite positions are given only for representations H219 and G15. They are shown in Table 2.

Table 1 gives the differences in x, y, z coordinates of the satel­ lite position from H219 of G5, G10, and G15.

Table 1

Comparison of Satellite Positions from Potential Coefficients and Gravity Anomalies. Differences from potential coefficient positions.

Time After tQ Elem ent G5-H219 G10-H219 G15-H219

0 Ax 0 m 0 m 0 m Ay 0 0 0 A z 0 0 0

10m 4 0 s Ax 0 0. 03 0. 07 Ay 0 .0 1 0.03 0.05 i A z 0 -0. 03 o o -o 52m Ax -0 .6 3 -1 .0 4 Ay / - 0 . 86 -0 .9 9 A z 0.49 2. 23 72m 4 0 s Ax 2. 53 Ay // 2 . 18 Az 3.83 150m Ax 14. 14 Ay // -7. 11 Az 14.05 33

Table 2

Comparison of Disturbing Forces, S, T, W. The Radial Distance is in Terms of Earth Radii.

Time H219, G15, After tQ r

0 secs 1. 51372 30.093 17.571 S -76.651 -76.697 T 61. 351 61. 347 W -255. 113 -255.095

1000 1.46856 16.683 49.435 S -254.639 -254.635 T 87.243 87.237 W -165.709 -165.702

2000 1. 32903 -4.661 80.088 S ' -489.560 -489.363 T -42.021 -42.222 W 68.246 68.304 3000 1.15594 -28.327 125.119 S -284.003 -285.253 T -217.976 -217.895 W 722.449 723.657

4000 1. 09011 -25.958 195.508 S -482.655 -482.892 T 313.494 314.360 W 837.135 836.357

5000 1. 20682 4. 516 247. 152 S -733.005 -732.911 T -63.325 -63.332 W- 102.888 -102.869 6000 1. 38060 26.830 287. 392 S -165.892 -165.801 T -119.502 -119.452 W -332.468 -332.520 7000 1.49418 32.765 327.173 S -40.144 -40.164 T 17.410 17.406 W -289.046 -289.046

8000 1. 50550 24.688 2 . 190 s -105.736 -150.694 T 89.018 89.023 W -217.137 -217.129

9000 1.41163 7.089 32. 214 s -382.223 -382.245 T 49.369 49.441 W -84.496 -84.526 34

Both tables give an adequate check on the group of equations for

S, T, W mentioned earlier. However, because the values are not exactly the same in all cases, it can be concluded from Table 1 by comparing the 3rd, 4th, and 5th columns that the orbit generated using larger anomaly blocks is less accurate than that generated from either potential coefficients or smaller blocks.

The difference between the "mean gravity anomaly" * and

"potential coefficient" ^ orbits is probably largely due to the fact that the generalized Stokes' Function and its derivatives and the mean gravity anomalies are computed only at one point inside each gravity anomaly block instead of taking their integral mean for the whole block.

To illustrate the difference in values for Stokes' Function,

Table 3 was obtained by applying Eqns. (1. 10), (2. 39), and (2.40) for a point in space with coordinates r = 7500km, cp = 70°, X = 40°, to a 5°x5° band between longitudes 0° and 5° and from 0° to 90° latitude.

Each 5°x5° block was divided into l°xl° sub-blocks; and computations were done at the center of each sub-block and a weighted mean taken, the weighting factor being the cosine of the mid-latitude of the sub- block. These mean values are given in the 5th column of Table 3.

Orbits generated by using gravity anomalies. Orbits generated from potential coefficients. The 4th column shows the values computed at the center of the 5°x5° block and are the ones used in evaluating the satellite orbit.

Table 3

Comparison of the Generalized Stokes' Function and Its Derivatives for Various Evaluation Points

Elem ent Middle Value Mean Value

5° 0° S(r,W -1.44502 -1.44277 dS(r, ip) /dr 0 . 56490xl0 "6 0.56421x10-6 dS(r,^)/d^ -0.26824 -0.27042

25 0 S(r, Tp) -0.85207 -0.84972 dS(r,VO/dr 0.46183x10“^ 0.46092x10-6 dS(r,^)/d^ -3.46581 -3.46702

45 0 S(r , $ ) 1.08569 1.08646 dS(r,^)/dr -0.12260x10-6 -0. 12395x10-6 dS(r -8.82315 -8.82678

60 0 3.91054 3.90391 dS(r,^)/dr -1.34913xl0 "6 -1. 35180x10-6 dS(r,^)/d^ -16.89702 -16.88952

75 0 S(r,^) 6.96747 6.92695 dS(r,^)/dr -3.58968x10-6 -3.55487x10-6 dS(r,^)/d^ -27.00568 -26.68691

90 0 S ( r , V ) 4.58443 4.78328 dS(r,^)/dr -1.73919x10-6 -1.86871x10-6 dS(r,^)/d^ -19.07257 -19.73548

Note: cpq and Xq represent the top left corner of each 5°x5° block. Although for this example the differences appear small, Koch

[ 1968 a], has shown that even better results for satellite positions are obtained by using values from the 5th column rather than those from the 4th column. This process, however, increases not only computer time but also storage space requirements. On the other hand, to use mean anomalies from the sub-blocks directly would give even more accurate results.

The results given here adequately check the main equations derived in this chapter. We now stop using potential coefficients completely and turn our attention to deriving expressions suitable for obtaining gravity anomalies from satellite analysis. CHAPTER 3

PARTIAL DIFFERENTIAL EQUATIONS AND ADJUSTMENT MODELS

3. I General

1 J O bserved 1 and predicted satellite coordinates will not be identi­ cal, due to errors. These errors arise, for the special case of this study, from the fact that errors are introduced into an assumed cor­ rect set of gravity anomalies, station coordinates, and orbital ele­ ments at a reference epoch. In this chapter, we will examine the effect of small errors in the orbital elements, gravity anomalies, and station coordinates on the predicted topocentric right ascension, decli­ nation, and range. The equations so obtained would only be the first terms of the familiar Taylor's series expansion of a function about some approximate values and, together with the differences of ob­ served and predicted values, would be used to obtain the corrections to the initially approximately known quantities.

^Observed values are those obtained from the orbit generated by using assumed true parameters. 2 Predicted values are those obtained from the orbit generated by using altered parameters; these are obtained by introducing small errors into the true parameters.

37 38

At any time, t, the errors in the observed topocentric quantities, right ascension, a*, declination, 6*, and range, r*, are given generally as:

|— — I i— da* dat det d<5* (3. 1) = C 11 C 12 dit du>j. dr* d n t mmJ dM*. t du i du2 dui where:

U j,U 2 ,U 2 are the Cartesian geocentric coordinates of an obser­ vation station.

d(a*, 6*, r*)t 11 d(a,e,i,w, M)t

d(a*, 6*, r*)t C 12 = ------51 d (u i,u 2,u 3) But:

da da*. de de*.^ c di c (3.2) B “ •t dw dw*.j ° Cc dfl dfl*. Cc dM dM* t-t. dAgj dAg2

dAgN where:

d(a, e, i, w,S2, M)t A = d(a, e,i,w,fi,M )t(

d(a, e, i, w,n, M)fc B = ^(Agj, Ag2, . AgN)

Hence approximating the partials as finite differences:

i — 1 i A a*. A a* l I 1 1 AA e*- ° A ■ ° (3.3) 6 A * C 11A 1 CUB ! c 12 a ‘ 1 ° 1 1 A“

That is:

A a* d(a*, 6*, r*)t (3.4) A 6* d[(a,. . . , M)t0> A gj. . . AgN] Ar* ( } • • • > )( r ) t

A Mt d(a*, 6*, r*)t 1 dA g° 9( u i, u2, u3) J : dAg Au j Au2 A u 3

The quantities on the left hand side of the above equation can be ob­ tained by taking the difference between the observed values at time, t, and those obtained at the same time by using the approximately known values of the required quantities. If the number of such equations is equal to the number of unknowns, an inversion of these equations gives the corrections to the initial quantities: the six Keplerian elements at the reference epoch, the gravity anomalies, and the station coordinates.

The total number of unknowns would depend on the size of the anomaly blocks used and the number of stations from which observations are done. Better results would normally be obtained by having more equations than unknowns so that a least squares adjustment can be perform ed. 41

3. 2 Evaluation of the Partial Derivatives

3. 2. 1 Numerical or Variant Method

The partioned full matrix of Equation (3. 3) can be obtained in several different ways. In this section, the numerical or variant approach, which is a very straightforward procedure, is explained.

For any element, say, e^, of Equation (3.3), one can form the partial derivatives from the basic numerical definition of differentiation, that is:

3a*t a*t(at 0 ,eto+Ae,. . .Mt 0 ,A g x, . . . AgN , u 1,u 2 ,u 3) 3ej. A e co

a>:'t(at 0)et0, • • -Mt0 , Ag . . . . AgN , u 1,u 2 ,u 3) A e

3<5*t 6*t(at 0 >et0+A e> • • - Mt0 >Agi> • • • AgN >u i>u2>u 3)

deto A e

6*t(at 0 >et0, •••Mto,Ag1,.. . AgN, u p u ^ ^ )

A e

3rt r *t(at 0 >et0 +A e>- • *Mt 0 >Agl> • • • AgN>u l>u2>u 3> deto Ae

r * t K 0 > «tQ> • • • Mt0 »Ag p • • • AgN >u i>u2>u 3>

A e 42

A numerical approximation of the desired partial derivatives is thus obtained by:

a) incrementing a typical element at the reference epoch,

tQ, and integrating the equations of motion to the re­

quired tim e, t,

b) subtracting the nominal orbit values at the same time, t,

(i.e. , the orbit with the nominal elements without incre­

mentation), and

c) dividing the result from b) by the increment.

The choice of the size of the increment is completely arbitrary but must be such as to produce meaningful changes in the topocentric values and give the best approximations to the desired partial s.

Although simple to formulate, this method may prove inconven­ ient if there are a large number of elements or unknowns in the func­ tional relations of a*, 6*, and r*. In this case, one numerical inte­ gration of a set of variational equations which contains all the desired variables will be more convenient. This is discussed in detail in the next section and is the method used in this study.

3. 2. 2 Variational Partial Derivatives

In this method the partial derivatives of the satellite position, a*,

6*, and r*, at time, t, with respect to the orbital elements, gravity anomalies, and station coordinates are obtained by numerically integrating a set of variational equations. This approach avoids the non- linearity associated with variant orbits of Section 3. 2. 1, i. e. , perturb­ ing the initial conditions individually and approximating the partials with the difference quotients.

Since all the critical parts of these equations pertinent to this study were not explicitly found in any of the references studied, de­ rivation of all equations is treated in great detail here. Although the treatment is primarily based on orbital elements, gravity anomalies, and station coordinates, it is clear that the method can be used with any other elements appearing in the equations of motion, e. g. , ae, earth's semi-major axis, or kM.

From Eqn. (3. 1):

The topocentric coordinates, a*, 6*, r* are given by the familiar relation:

X* c o s 6*cosa*

(3.7) Y* = r* cos 6*sina*

sin 6* 44

i where:

X* x - 4

(3.8) Y* = y - 7)

z* z - C

x, y, z are the geocentric Cartesian coordinates of the satellite at tim e, t;

£ ,7), £, the geocentric station coordinates, at the same time, t, and in the same coordinate system as x,y,z, are obtained from the equation:

i U 1

(3.9) V = R 3(-0) u 2

Z u3 where 0 is the Greenwich Apparent Sidereal Time at the time, t.

Differentiating Eqn. (3. 7) with respect to X*, Y*, Z* and simpli­ fying, we obtain Veis [I960, Eqn. (83)]:

da* - s e c 6*sina* s e c 6*cosa* 0 dX*

(3. 10) d6* -sin 6*cosa* -sin 6*sina* cosfi* dY* r**

dr* r* co s 6 *cosa* r*cos 6*sina* r*sin 6* dZ* 45 where from Eqn. (3.8):

dX* dx -

(3. 11) dY* = dy - dr)

dZ* dz - ' J X 1

By differentiating Eqn. (3.9), we get:

d i c o s 0 -s in 0 0 du i

(3. 12) d r , = sin0 cos 9 0 du2

d C 0 0 1 du3 t t

The geocentric Cartesian coordinates x,y,z of the satellite are obtained from orbital elements given at any time, t, by, Kaula [1966,

Eqn. (3. 22)]:

' X a(cosE -e)

(3. 13) y = [R3(f2)Ri( -i)R 3( -w) ] ^ a(l-e2)^sinE

z 0

The product of the rotation matrices is:

(3. 14) R3(n)Ri (-i)R3(-«)t =

cosficosu - sinftcos i sinto, -cosftsinw- cos i cosusinft, sin i sinfl

sinftcosu + cosftcos i sinu,- sinftsinio + cos i coswcosfi,-sin i cos £2

sin i sinu> sin i costo cos i Differentiating Eqns. (3. 13) and (3. 14)

dx / c o s E - e \ X a N. S -a sinE \

dy = R 3(-J2)Ri(-i)R3(-u) |( 1-e2)^sinE J da -fae( l-e^)_1/^sinEj de +j a(l-e^) cosE J dE

dz V 0 J V o J V 0 J L— —1t

- -1 - ~| sinDsin i sinco sin i cosusinfZ cos i sinfi a(cosE -e)

-cosftsin i sino -cosflsin i cosgj -cosftcos i a( l - e 2)^ sin E di.

cos i sinoj cos i cosu -sin i

(3. 15)

-cosf 2sinw-sinf2cos i c o s o j -cosf^cosw+cos i sinwsinfl 0 a(cosE -e)

-sinf2sinu+cosS 2cos i costo -sinf2cosu-'cos i sinucosS 2 0 a( 1-e^) ^ sin E doj^

sin i cosu -sin i sinw 0 0

-sinftcosu-cosflcos i sinu> sinftsinu-cos i cosucosfi sinicosfi a(cosE -e)

cosS 2coscj-sinficos i sinu -cosf 2sinu-cos i coswsinft sinfisin i a( l-e^J^sinE

0 ,0 0 0

o 47

In order to express dE in terms of the known elements, we differentiate

Kepler's Equation:

M = E - e sin E

From the equation above:

(3. 16) dM = (1 - e cos E) dE - sin E de

Therefore:

dM sin E de dill — — — —■ - ■ i■ -f* 1 1 - e cos E 1 - e cos E

Substituting for r = a( 1 - e cos E):

(3. 17) dE = - dM + ------de r r

Substituting for dE into Eqn. (3. 15) and then with Eqn. (3. 12) into

Eqn. (3. 11), Eqn. (3. 10) becomes: 00

r 0 0 np 0 - Z np Q SOD - Ixip 0XIXS - tylP [/ [a s o o ^ (z a - i) z«] 0 _0U X S- g s o D -1/ [3 «Tszb-]_

0 ‘o ‘o b P 3 uTS^/(za -i)3 i uxsyuxs ‘yuxsmsoD x soD-muxsysoD- ‘muxs x soDyuxs-msoDysoD (9 -[J S O D )B i uxsysoD ‘ysoDmsoD i soD-muxsyuxs ‘muxs x soDysoD-msoDyuxs-

0 muxs x uxs- ‘msoD x uxs ^mp auiS^(za-Ija 0 ‘ysoonuis x soD-nsooyuis- ‘cosod x soDysoD+muxsyuxs- (a-^soojB 0 ‘yuxsmuxs x soo+nsoousoo- 'nsoo x soDyuxs-muxsysoo-

x uxs- msoD x s o d m uxs x s o d *xp 3 uxsyT(za-i)B x soDysoo- ‘msOD x uxsysoo- ‘muxs x uxsysoo- (81 '£) (a- 3 SOD)-e y u x s x s o d ‘yuxsmsoo x uxs ‘muxs x uxsyuxs

*ap x /[3sod3uxs^/(z3-I)zb] + 3 Uxse/i_(2a-i)aB- [ j /( 3 zuis2b) + *]-_

0 ;;:g u i s :;:j ‘ * 9 SOD;;;X>UXS;;;.X ‘^gSOD^TDSOD^J T;j» «l *JP~ ♦Hp 3uxs^(zs-i) # g s o D ‘ ^ g u x s ^ x iu x s - ‘ ;;sgUXS;;:r S O D - *9P I ( 9 - 3 S O D ) J 0 S ;;9D 9S ;;;X)SOD ‘^gDas^ruxs- _**>P_ 49

Equation (3. 18) is the explicit form of Eqn. (3.6).

Referring to Eqn. (3. 2),

Let:

D.[ a !b ]

(3. 19)

d(a M)t

d [ (a a 8n ]

Matrix D expresses the variation of the osculating elements at any time, t, with respect to their values at a reference epoch and gravity anomalies.

The functional relations expressed by the Gaussian equations of m otion, Eqn. (2. 5), show that:

at = at [(»f e»i»w»°»M)t0,A gi AgNJ

(3.20)

Mt = Mt [(a , e,i,w , J2, M)tQ>Ag l ASn

Hence:

a a Aa e e A e i i Ai (3.21) — + GO 10 A (o n a Af2 _AM_ t - L 50

The second vector on the right of the equals sign is given by integrating

Eqn. (2.5) from tQ to t. Differentiating Eqn. (3. 21), we obtain:

da da dA a de de dA e di di dAi (3. 22) — + dto dw dAcj dfi dfi dAfl _dM _dM _dA M t ^ t_to

Since each element is a function of all other elements and gravity anomalies, the second vector on the right of the equals sign is then given as:

SAa^. SAa^. SAa^. SAa^. SAa^. dA a daf Sat Set SMt SA g 1 c)A gN *■0 lo dAe

dAi SAifc (3.23) dMt Sat o dAg° dA w

dA« SA Mf. SA Mj. dAM Sat Set co co _dAgN_ t-tQ t-t. 51

Hence:

c)A aj. a^. ^A af. i^Aaj c)A at da 1 + dat 3al ’ de*. dMt ’ ^A g l ?)A gN *•0 Lo de ^ e t 1 + ^et di (3.24) du> dMf d Q dA g° r)A Mt tiAMt i^Mt ^Mt dMt da<- 1 + SMt0’ ^Ag l ’ * ' ^ g N t-t.

dA gN

da*

Df

dAgN

Interchanging the processes of integration and differentiation in Eqn.

(3. 24) the D matrix at any time is obtained by differentiating the equa­ tions of orbital motion with respect to each variable, and then numer­ ically integrating the resultant equations. Because of the form of the equations of motion, the derivation is rather lengthy and involved, but clear. 52

From Eqn. (2.5):

ir= (e sin v s + p/rT)

Hence:

d ( ^ ) = 2a / ------— 5- — (e sin v S + p/rT ) da V dt J VkM(l-e2) _ 2a

+ -4 e 2 sin v S + ep/rT + S sin v + {

(3.25) + cos E _ 2e^^J- de + e S cos v dv

- T dE + e sin v dS + p/r dT r

— = / p~ S sin v + ( cos v + — ) T dt v kM

Hence:

de -e S sin v + ( cos v + — ) T )- da + S sin v l k { 1 -e 2

+ [ HE c o s v + - 1 t + 1 - 2 e cos E - (2e + cos E) cos v T (1 - e2) (3.26)

, 2a e /r+p , er + ------( — — cos V + de + ( S cos v - iiE sin v T ) dv P \ P ') T1

a e sin E (cos v + e) _ f r+p e r 1, + ------T dE + sin v dS + ( —— cos v + — ) dT P \ P P di_ / a = r /a / ------*- W cos (u> + v) dt V kM( 1 - e 2) V

53

Hence:

. cos(co+v) , , = - / a ~ W^----- da -( cos(u+v)cos£ St. a- vkM( 1- e 2 ) L L

(3.27) + - - - g c o s ( oj + v ) ^ de - sin(u>+v)(doj+dv) + cos(w+v)dEj-

+ cos(co+v)dW

1 r +p r S cos v + — — T sin v W sin(co+v)cot i St = VvP*- kM e ep p

Hence:

1 ) 1 r “f■p r — } = f ~ P ~ — -s S cos v + —— T sin v W sin(co+v)cot if da St / v kM .2a t e ep p

S cos v r+p _ . er S cos v =5------Tx mu sin v v .+ ------5—W sin(u)+v)cot i + y 1-e 2 p ( l- e 2)2' p(l-e2) e 2

a(r+p)( 1 -3e ) 2a acosE \ 2aer -T sin v + + ~2~ W sin(co+v)cot i e V P ep

+ — cos E sin(co+v)cot i w i de + -I — S sin v + T cos v P J Le ep

( 3 . 28 )

r 1 r W cos(w+v)cot i r d v W cos(

r o r+p 1 + — W sin(oo+v) c o se c 6 i di + —— sin v dT - — cosv dS p ep e

sin(w+v)cot i dW P

Stt sin(ea+v) j * W S t a v kM( 1-e^)1-e 2 am 1 54

Hence:

sin(cj+v) a sin(oj+v) W da + cos E -a Vi/ k M (l-e 2) 2a sin i r sin i

ae sin(w+v)\ , cos(co+v) . . , . sin(w+v)cos i + ------;— :---- I d e + ------r—:— (dco+dv) ------1------d i p sin i J sin l sin^i (3.29) + ae sin E sin(to+v) dE + sin(co+v) dW r sm i sin i

/ a f Z v 1-e 2 \ ( 1-e2) / a /r+p\ . _

~ W = n ' v kM K T ~ — COS7 S T->/kM \—J s i n v T

Hence:

3n ^at J_ ___ 2r ( l - e 2) cos v ) S < ! r > 2a ^aT” + 2a VkM

da

3n ^at 1+e 2 cos E + — — cos v ) S 2a det

T ■in r ['{1+e.Z) ( r+ P ^l I 1 -e 2 ( 2er.. C°S E de e^ \ P / e \p( 1-e2 ) 1 -e 2

(3. 30) 3n ^at , 3n Jw / a J" . - 21 • dW " • 3mT " ^ r e sin E,s co o

+ T sin v sin E| dE - VS + T £i£ccs v} dv

N _ 3n V dat JA 3n dat 3n b i df2 2a L dAg; gJ “2 a * di dl j= 1 J to 55

The additional term

•3n Sa*. 2a d (element or gravity anomaly) arises from the fact that n in the equation for also varies due to perturbations and therefore has to be differentiated separately. It is obtained by differentiating Eqn. (2.6).

The derivatives for S, T, and W are obtained by differentiating their respective equations.

From Eqn. (2.44):

2 ✓ v „ a, \ 4 S = j \. 5 ^2 ^3sin^(co+v)sin^i-l^ + ^ J4 ^35sin^(w-fev/ sin^i• 4-

-30sin^(to+v)sin^i + sj- - ^ AgjGj

Hence:

2 dS = ^2 ^3sin^(w+v)sin^i-1^ + J4 ^35sin^(w+v)sin^i

2 -30sin^(u+v)sin^i + 3j- dr + 9 J2 sin(a)+v)cos(w+v)sin^i

+ -|^35sin^(co+v)sin^i cos(u+v)- 15sin(w+v)cos(co+v)

(3.31) n I Jq r X3- g\ 2 ^ s in^ij- (dw+dv) + — 9 J2 sin^(w+v)sin i cos i

+ ^ “^4 ^35sin^(cj+v)sin^i cos i-15sin^(w+v)sini cos ij-

N N

- S i A6jdGi - S I G i ^ h j=l j=l 56

From Eqn. (2.45):

_ -kM . . . . /ae\^ /ae\^f35 T = — y cos(u+v)smi 3 J 2~ i j sin(w+v)sini + J4( — ) 1 — sin 3 (w+v)sin3i r

N Aq cos(eo+v)sin —15 sin(w+v)sxn • / x x ■inij- (cosCLaimj/): c* 47T c o scp — I

N Aq cos AgjF j (sinasin^) j 4tt c o scpj

Hence: .

kM , , » • . dT = — t- cos(ai+v)sini 12J2 ( ¥ ) sin(co+v)sini +^^ 4 ^ T ^ r

— ■ sin^(co+v)sin^i - sin(co+v)sin ij -1 dr {

kM . . . —-sin(u+v)sini 3J2^-—^ sin(cj+v)sini + J4(-^-) sin3(w+v)sin3i -r^

N 15 Aq sin(co+v)sinin i V . _ sin(w+v)sini (sin^sina) 47Tcoscp — Z 6i i J j=l

-kM . . —5-cos(oo+v)sim 3J2 -t cos(<*)+v)sini rL m (3. 32)

+ J4^“ ^ sin^(U)+v)cos(w+v)sin3i - -^■cos(w+v)sini|- (du>+dv)

kM , • . j-cos(w+v)cos 1 b ^ TA' ) sin(u)+v)sini u r

J4 •^70sin3(oj+v)sin3i - 15sin((*)+v)sin ij- N 57 Aq cos(co+v) AgjFj(cosasin^)j 4 7 T C O S j=l N ' to s.p I ^jFjfsinV/sinaJjj di 3=1 N

+ ^rrcoscp [ c o s(“ +v)sin 1 Y A8 jF j(sinV/3ina)j j=l N N

+ cos i Y j AgjFj(sin^/sina)j dcp + — s^n * ^ AgjFjd(sin^cosa)j

j=l j=l N N + ^gjFjdlsmU'smalj + ^ Agjtsinj/cosaljdFj j=l j=l

N N

+ Aq_cos_i \ /\a-tsinTl/cosa.)\dF\ + - 7 - — ^— / F; cos(w+v)sini(sin^cosa)i 4 ttcoscp A J J J 4Fcoscp L i J|_ J j=l j=l

+ cos i(sin^/sina)j dAgj

From Eqn. (2.46):

W = —— cos i jj^ 2 ^ T ^ sin(w+v)sin i + ■j-^T-sin^u+vJsin^i

15 . sin(w+v)sin i]> + Y AgjF j< ^ c o s a ) j 2 j=l

N Aq cos(u)+v)sin i \ A _ . . . . . 47rcOS»------2 / 80 i (S1“* S ’j 3=1 58

Hence:

kM dw = —— cos 1 12J2 sin(co+v)sini + 6 J4 sin^(w+v)sin^i r

-kM — sin(to+v)sin i dr + 2 ~ cos 1 cos(co+v)sin i ‘ [3J2 (t )

+ J4 ( — J "I sin^(w+v)cos(w+v)sin^i - -^cos(w+v)sin *}]i

N Aq sin(w+v)) sin i V . _ . . , . . + (dui+dv) 4ttcos^ A8 jFJ (sm^sina)j j=l O /f 'kM . . r,_ M . / a e\ J35 . 3 . x . . 3. —2" sin 1 |J5 J2 \ ~ ^ ~ J sm(oj+v)sini + J4 J — sin-,(u)+v)sinJi

N 15 1 I Aq sin i V' . _ . . — — sin(w+v) sin i j / - “irrcostp L J sin^cosa)j 3=1

N 2 Aq cos(w+v)cos i V . _ . . . , . kM ?. T, _ /ae\ . , , , 5 5 ^ L ASjF j<81nj - ^ - c o s ^ i |.3J 2 j sm (u+v) 1=J (3.33) + J4 sin^(w+v)sin^i - 4p-sin(w+v)'}] di N N ^ |||[ c o s i £ AgjFj(sin^cosa)j- cos(

3=1 j=l

N AgjFjlsinV-sinatj] df + % AgjFjd(.ln*cosa)j

3=1 59 N N Aq cos(oj+v)sin i V . . Aq cos i V . . . , , 3 5 ^ 5 Z + 4 t t c o s ( ( , Z j=l j=l

N N Aq cos u+v sin 1 V . / ■ . . v . Aq V , * 3— ------/ Ag(sm^sina);dFi + — / Fi cos x(sin^cosa); 47TCOSCP L t & r j j 47TCOS9 A ' Y J j=l j=l

- cos(w+v)sin i(sin^/sina)j dAg;

G and F in the above equations are given respectively by:

1-t cos^+D (3. 34) G = t2 [ i ^ + i + 1 - 6 D - t cos ip 13 + 6 in L D3 D £ !) ]

1-t COS I p - D _ 3 /gn 1-t cos^+D &

Aq = area, in radians^, of the blocks.

Differentiating G we obtain:

. _ f — f l-t^ 4 . _ — , f , r, / /» 1 “t cos^+D dG = -^2t ~ '3" + “ + 1 “ “ * co s^ I 13 + 61!n

+ t 2 I[ - I * - c o s * ( 13 + 6 tn 1-t C°s»+D~) + 6i ,5.°s,2» ...11 dt L d3 2 y (1-t cos^+D)J J (3.36)

■3( 1-t2) 4 ^ 6 t cos ip +T2 dD D4 ( l -t cos^+D).

1-t cos^+D)\ 6t £s i m p c o s i p j^t s i ^ 13 + 6in d ip / ’ TT^t(1-t cosV'+D) . 60

Differentiating F we also obtain:

dF = 3T2 [ A + - - 8 + 3 - 3«n c o s * + D _D^ D D s in ^ 2

— / c o s t y ^ COS Ip \ d t \Dsin^ 1-t cos^+D/. (3.37) 2 2 1 1-t cos^-D 1 - 3t3 + —TT------~------=-----£---- +------~ dD _D4 d 2 Dsin2^ D2sin2^ (l-tcosV'+D)

3^3 ( t . 2cos^(l-t-cosy/-P) + __t_sin £_\ ^ \Dsin^ Dsin3^ 1-tcos^+D)/

From Eqn. (1. 12):

(3. 38) dt = - - dr r

From Eqn. (2. 37) and Eqn. (2. 38):

(3. 39) dip = -cosadcp-coscpsinadX

Differentiating Eqn. (1. 13) and substituting for d i/s and d t we obtain:

^ t(sin^cosa) , t(sin^sina) t(cos^-t) (3.40) dD= ------^------d

Substituting Eqns. (3.38), (3.39), and (3.40) into Eqns. (3.36) and

(3. 37) we obtain:

dG = - + ^ + 1 - 6 D - ~ t c o a i f / ^13 + 6 ^n ^ 7

+ T S J $ . c o s y ( + 6 in 1-T c o s ^+DN) + ~ f j C O B ^ I d 3 \ 2 / (l-tcos^+D) 61

cos^ -t / 3( 1 -t2) _4_ + ^ + 6t cos ip dr D V D4 D 2 (l-T cos^+D)/J .

1 f3 (l-t-t2) 4 6t cos ip (3.41) + T^(sin^cosa) + 6 + — _D 1 D 4 1-t cos^/+D

6 cos$ 1-t cos^+D' 13 + 6 ^n dcp 1-t cos^/+D

, t 3 / • i ■ \ 1 f3(l-t2) 4 . 6t cos^ + tJ(sin^sina) — -{ —1—;—- + —^ + 6 + —=------— D L D4 d 2 1-t cos^/+D

_ j £ 2 £ S------( 1 3 + 6fa 1-t cos»tDVI dXc 1-t cos^+D \ 2 / J

-3 t2 2 , 6 „ . , D+t cos^-1 „ „ 1-t cos^+D dF = .53 + D ' 8 + 3 D.in** ‘ 3Jn 2------

, — f cos^ cos^ , cos^-t / 2 2 1-t cos^/-D IDsin2?// (1-Tcos^+D) D \D 4 D2 D 2 sin2^

‘ —.')].] dr + 3t^(sin^cosa) Dsin2^/ (1-t cos^+D)/J . _Dsin2/^ (3.42)

2cos (1 -t cos^-D) + V ^ + 2 D sin4^ (1-t cosf+D) D \D 4 D 2 D sin2^

1 -t cos^-D __ Y dcp D2 sin2^/ (1 -t cos^+D) J

3t3(si„^sina)coscp ^ ‘ (1-t cos^/+D)

J 1-t cos^-D _____ 1_ + 4 ( \ ^ + 2 d X D \D 4 d 2 D sin2^ D 2 sin2^ (1-tcos^+D) )] 62

.i By differentiating the equation for tan v, i.e.

/o i. (1-e2)/^sinE 3.43 tan v = i ------cosE -e and substituting for dE from Eqn. (3. 17), the following result is found for dv:

(3.44) dv = f±) 2 ( l - e2)1/=dM +2llM 2i^de \ r/ r 2( l - e 2)/2

The geocentric distance, r, of the satellite is given by:

r = a( 1-e cosE)

Differentiating this equation and substituting for dE from

Eqn. (3. 17) we obtain:

r a i a2e sinE (3. 45) dr = - da + - (a e sin 2E-r cosE) de + ------dM

Differentiating Eqn. (2. 12):

/■a a l s j sin(u+v)cos i , cos(w+v)sin i (3. 46) dcp = -----1------di + ------1 :------(dco+dv) T coscp coscp

By differentiating Eqn. (2. 36) we obtain:

(3.47) d(sinV'sina) = -coscpqCos(Xq-X)dX

Differentiating Eqn. (2.49):

(3.48) dX = d(Op-O) + dfi-d(GAST)

From Eqn. (2.48):

,■> se c 2(co+v)cos i . , , , . tan(w+v)sin i (3. 49) d(Op-B) = v (dw+dv) ------■ ■ ■ dx se c 2(Op-£2) s e c 2(Op-J2) 63

Therefore:

(3. 50) dX = dfi-d(GAST) + se— ^tY}?,os 1 (du>+dv) SeC^Op-Q) tan(co+v)sin i ------=------d i sec^(Op-f2)

Hence:

(3. 51) d(sin^sina) = -coscpacos(XQ-X) j d£2 + sec (t*>+v)cos i ^ w+(jvj 4 L sec^(ap-n)

tan(co+v)sin i *-=—* ------di-d(GAST) sec^(

In Eqn. (3. 51) the error d(GAST) is made up principally of the error in timing the satellite observations as well as the accuracy of such formulas as Eqn. (2. 50) used to compute GAST. In this study

GAST is assumed to be correct and so no error to it need be evaluated.

Differentiating Eqn. (2.35):

(3.52) d( sin^/cosa) = -cos^dcp-(sin^sina)sincpdX

dcp and dX are given in Eqns. (3.45) and (3. 50) respectively.

This ends the derivation of the partials for S, T, and W, and it is now possible to express their differentials as functions of the Keplerian elements and gravity anomalies. Thus, for example:

(3.53) 5§=|S .|: + |S .|S + ...... de ov oe ocp oe

Hence, the D-matrix, or Eqn. (3. 19), i.e. , the partial deriva­ tives of the orbital elements at any time with respect to their values at a reference epoch, and gravity anomalies, can be computed by numerically integrating Eqns. (3. 25) through (3. 30), e.g. 64

(3.54) dat = 1 + dMf “ I s:;° © ' 'tj > -* -•[“ I a t & )J fj=to> t 0 +At, . . . t0 +At

N tj

At is the time interval between any two consecutive integration steps.

d(a, e , i ?u, £2, M)t Or generally: d(a, e,i,w,Q, M)t

It is clear that at tQ this matrix is equal to the identity matrix.

The partials with respect to the anomalies are similarly com­

puted by numerical integration. d(a, e, i, u, Q, M)t Thus: 3(Agi, Ag2. . . AgN)

t t

tj-tOJ tQ+At, . . . tj

ti

tj

At tQ this is a null matrix.

Equation (3. 55) is thus sub-matrix A and Eqn. (3. 56) sub-matrix

B of Eqn. (3.2). By combining Eqns. (3. 18), (3.55), and (3.56) appro­ priately the rectangular matrix of Eqn. (3. 3) is then obtained. These are the required partial derivatives relating the position vector, a*,

6*, r*, at any time, t, with the orbital elements at an initial epoch tQ, gravity anomalies and station coordinates. It is clear that these equa­ tions can be used with any method of orbit generation as long as the

Keplerian elements at any time are available, and they give the highest possible accuracy since the variation of all equations related with orbit generation and the whole gravity anomaly field are used in their evalua­ tion. A Fortran IV program to obtain Eqn. (3. 3) assuming du^d^jduj, to be zero, that is, Eqns. (3.55) and (3.56) together with the relevant parts of Eqn. (3. 18), can be obtained from the Dept, of Geodetic Science. 66

3. 3 Adjustment Models 1

3.3.1 General

A system of equations relating certain observables and un­ knowns can be solved by using generalized least squares models.

This involves minimization of certain quadratic forms which, if the weighting factor is regarded as inversely proportional to the variance of the observations, is often referred to as the minimum variance method. There are other adjustment techniques, but the minimum variance models are less restrictive than, for example, the maxi­ mum likelihood estimates, where the same result would be obtained by assuming that the corrections are unbiased and normally distrib­ uted. Although it is less restrictive, statistical analysis of least squares adjustment, principally of non-linear forms, is extremely involved. This is the more so if the non-linear terms of the Taylor's series expansion are significant, or if residuals after adjustment are not random but have a definite sinusoidal or other pattern of discern- . able wave-length.

Nevertheless, using least squares techniques, observables are defined as all those variables which either can be directly observed or are certain parameters, quasi observables, which have variances

*Most of the discussion in this section is adapted largely from Uotila [ 1967a]. 67 and hence weights. Although in the actual expression of the solution equation there is a distinction between the weights of the observables and those of the parameters, the definition for observables given above is adopted here to avoid any confusion about the degrees of freedom if such models are used.

For. the general case, the mathematical model has the form:

(3.57) F(La,Xa) = 0 where:

F is a set of functions relating observables and unknowns;

L is a set of observations either adjusted, La, or observed, L^;

X is a set of unknowns either adjusted, Xa, or approximate, X°.

We define further the following matrices:

Vn a set of residuals to be added to iJ 3 to obtain La;

Vx a set of corrections to the approximate values X° to obtain Xa;

W misclosure vector;

P£ weight matrix of L*3;

Px weight matrix of some or all X°.

Linearizing Eqn. (3. 57), we obtain

* (3. 58) BY + AX + W = 0 68 where:

B = 3F dLa

dF A = . a x a

W = F (Lb, X °)

3.3.2 Minimization of Quadratic Sums

Solution by least squares requires the minimization of the quad­

ratic form, V'PV, which in this case is the total weighted sum of

squares of the residuals. That is:

(3. 59) V'PV = V'^P^V^ + V'XPXVX = minimum

The condition given by Eqn. (3. 59) is enforced by minimizing the * function:

(3. 60) cp = V ^ P ^ + V’XPXVX - 2K’(BVg + AVX + W)

where K is a set of Lagrange multipliers.

Differentiating Eqn. (3. 60) with respect to V$ and Vx we obtain respectively:

1 dcp 2 a y r PiV, -B'K (3.61) J ^ - A'K

Letting Eqn. (3.61) be zero, according to least squares pro­ cedures, we obtain with Eqn. (3. 58): 69

BVg + AVX + W = 0

(3.62) - B'K = 0

PXVX - A'K = 0

From the second equation of Eqn. (3.62):

(3.63) V & = P£ “I B'K

Substituting this value into the first equation of Eqn. (3. 55), we obtain:

(3.64) BPj;-1 B'K + AVX + W = 0

Defining:

(3. 65) M = B P ^ 1 B '; and solving for K, we get:

(3. 6 6 ) K = -M - 1(AVX + W)

Substituting for K in the third equation of Eqn. (3.62), we obtain:

(3.67) PXVX = -A 'M -1 AVX - A'M " 1 W

Hence:

(3.68) Vx = -[A'M -1 A + P x ] _1 A 'M -1 W

Substituting for Vx in Eqn. (3.66):

(3.69) K = M "1 A(A'M "1 A + P X) ' 1A'M "1 W - M "1 W

The corrections are then given by substituting for K in Eqn. (3. 63).

The variance of unit weight, m02, is then given by:

7 V'PV (3.70) m 0 2 = — where:

(3. 71) df = degrees of freedom . 3. 3. 3 Variance-Covariance Matrix of the Parameters and Observables

If, for example:

(3.72) Y = f(X) then:

(3. 73) Y = G y G' U y where:

_ dY V _ G = / = variance-covariance matrix of X ax ’ Z,x"'X

Using this principle in Eqn. (3. 58) for W we obtain:

(3.74) ^ = m 0 2(M+APx - 1A') JW

From Eqns. (3.63), (3.69), and (3. 74)

(3.75) y = P^-iB'fM-iAtA'M-iA+P^-iA'M-^M-1] • V£

y [M - 1A(A'M-1A+PX) - 1A ,M - 1-M -1] B P f 1 'W The variance-covariance matrix of the adjusted parameters and ad­ justed observables are given respectively by:

(3.76) > = (A'M^A+PxpiA'M - 1 / M ‘ 1A(A'M- l A + I >x)-1 Xa W 71

3.3.4 Special Cases

1. If the unknowns are regarded strictly as such, then Px = 0.

Hence, some of the equations above simplify to:

(3.78) Vx = -(A'M^A^A'M^W

(3.79) / = mQ M 'W

(3.80) ^ a= m 0 2 (A'M"1A )"1 yXa

(3.81) df = number of observation equations minus number of unknowns.

2. If B and are identity matrices and Px = 0, then

M = I

Hence:

(3.82) Vx = ~(A'A)~lA 'W

(3.83) ) =m 0 2 (A'A) - 1 L j x &

(3.84) mo2A(A,A)-lA,

3.4 Relevant Matrices for Satellite Analysis

In this section we will indicate the necessary mathematical models relating topocentric satellite positions to orbital elements at a

reference epoch, gravity anomalies, and station coordinates. The

basic model in this case is:

(3.85) "F lt" "-a*t + ft (a^ijW ^M ^u^u^Ujj^Agj, . = = 0 F t = F 2t -6*t + f't (ato, ...... : ...... • a§n )

_F 3t_ _-r*t + f"t(ato, ...... • A§n )_ 72 where:

k = number of stations

t = time of observation, 1, 2 , . . . ,m

The observables are the topocentric coordinates, a*, 6*, r*. Gravity anomalies are also regarded as observables. The parameters are orbital elements at tQ: eto, ij.^, u>to, £2^ , and station coord i­ nates, U},U 2 ,U 3 .

3 f ^ ^ 3 f ^ j 3 f 2 ^ 3 f j j 3 F n 3 F U S F n 3 a * x 3 6 * 1 3 r * x 3 a * 2 3 r * n 3 A g j d A g N

3 f 21 3 f 21 3 f 2 i 3 a * j 3 6 * j ^ A g N

3 f 3 i 3 f 3 i 3 f 3 i B = 3 a * i a r * n d A g N (3.86) 3 F i 2 S f 12 3 a * 2 ^ A g N 1

5 F 3 m * F 3 m 3 a * m d A g N

B 11 I B 12

n = number of topocentric observations at any time t.

s = number of satellites.

m = total number of observation equations.

I 3 F U 3 F U 3 F U 3 f u dF n dF u dF n dF n da. de* dM. da. dM, ^ l k 3u2k ^ k t o l *01 *0 1 o 2 o3

5 f 21 dp 2 1 ^f 2 1 dM, du3k 3a‘ol o3

3 f 31 dF3 1 s*tol ^u3k A =

(3.87) ^F 12 dF 1 2 dMt 3atol lol

^F3m ^F 3m 3a‘ol ^ 3 k

6*0 j - 6^c l

(3.88) W = - r ^ j - r * ^

a *°2 -<**C2

The superscripts, "o" and "c, " mean "observed" and "predicted"

respectively. A a. *ol Ae+ o 1 Ai t-ol Awt ol Aft ol AM* bol (3.89) A a Vx = *02

A q3 Au lk

Au3k

A a*

A6*] Ar*j

Aa*-

(3.90) V* =

Ar* n dAg,

dAgN The weight m atrix is given by:

r a*if 6*., r*£

(3.91) =

-1

, • • • AgN

where:

is the variance-covariance matrix of the 2 observed topocentric coordinates. a*i ,6 r* i

is the variance-covariance matrix of the . gravity anomalies. Ag i , • • • Agjyj 76

If there are no covariances in each of these sub-m atrices, then

P^ is a diagonal matrix; thus:

a*.

r*.

a 2a*,

n =

(3.92) 'Agi

a 2 AgN 77

Performing the differentials with respect to the topocentric ob­

servables, the B jj square sub-matrix of Eqn. (3.86) is:

-1 0 0 0 0 0 ...... 0

0 -1 0 . , ...... 0

0 0 -1 0 ...... 0

(3.93) Bn = = -I -1

-1

-1

Hence:

pn ! 11 (3.94) M .= B 11 !B 12 o I P 22 B' 12

-1 11 11 11 12 22 12

+ B 12 ^ 2 2 12

If the observations of a,*, 6*, r* are regarded as being of unit weight, then the equation for M becomes:

(3.95) M = I + B 1 2 P - ‘ B'12.

If in Eqn. (3.95) P22 = 0

M = I.

This is equivalent to Special Case 2 of Section 3.3.4 and is the one used in the adjustments in Chapter 4. 3. 5 Some Additional Notes

The equations derived in this report are all directly applicable only under the assumptions made to derive them. For practical computations the following additions must be made to the appropriate equations:

1) A complete force model to include air drag, luni-solar

effects, etc. , should be used in generating the satellite

orbits.

2) The generated a*, <5*, r* should be transformed to the

same coordinate system as the observed values.

3) In the adjustment system constraints should be placed

on the adjusted anomalies so as to eliminate the poten­

tial coefficients of zero and first degrees and the ap­

propriate ones of second degree. The additional mathe­

matical model would be of the form:

(3. 57a) G(La) = 0 or

G(Xa) = 0

depending on whether the gravity anomalies are regarded

as observables or as parameters. 79

Explicitly, Eqn. (3. 57a) is as follows: “

G1 AgiAqi-47TAg0 I i

g2 AgisincpqiAqi ■ I i

g3 AgicoscpqicoskqiAqi I __ 1 l ~47T V g4 ^ Ag.coscp^.sinXq.Aq. i

G5 Ag^sincpqiCoscpqiCosXq^Aq^ £ i

g 6 AgiSincpqiCOScpqiSinXqiAqi £ i _ «■ _ where:

Ag^ are the adjusted anomalies;

Ag0 is the mean anomaly over the reference ellipsoid. If the parameters of a mean earth ellipsoid are used then Ag0 =0;

Aq- is the area of each gravity anomaly block. Aq: 1 If n equal area blocks are used then - r - = - = — . 47T n

An adjustment of the type of the mathematical model formed

by Eqns. (3.57) and (3.57a), that is,

F(La, Xa) = 0

G(La or Xa) = 0

can be found in, for example, Uotila [1967a] . t 80

By the use of appropriate equations (not given here) the adjusted anomalies can be developed into terrestrial potential coefficients. Of particular interest is the coefficient J2 . It is evaluated by subtracting from the J2 of the reference ellipsoid used in the main computations the following term:

(3.96) 47TkM 1

The flattening, f, of a mean earth ellipsoid, the J2 of which is equal to that computed above, is evaluated from Eqn. (1.3) iteratively.

3. 6 Summary

In this chapter a set of variational differential equations has been derived from the Gaussian Equations of Orbital Motion, Eqn. (2. 5).

These differential equations give the position of the satellite at any time as a function of its position at a reference epoch and gravity anomalies. Using these equations and any of the adjustment models described in Section 3.4 with generated satellite data, an adjustment will be made to try and recover certain input errors. This follows in the next chapter.

/ CHAPTER 4

NUMERICAL RESULTS AND ANALYSIS

4. 1 General

This being a theoretical study, least squares techniques apply

4 here principally because there are normally more observation equa­ tions than unknowns. If the mathematical models of Chapter 3 are theoretically correct, there being no serious numerical problems, the input errors should be recovered exactly and the estimated standard errors should be zero.

Some results have already been given in Chapters 1 and 2 to check some equations derived in those chapters. We will now pro­ ceed to check the differential equations and solution methods derived in Chapter 3. In this respect the method of observation equations —

Special Cases: No. 2, Section 3.3.4 of Chapter 3 — will be used.

The satellites from Table 4 were adopted.

The 15°xl5° equal area mean gravity anomalies given in Appen­ dix B were taken as representing the. field. Only one station was used.

u l = 3376887m

u2 = 4403992m

U3 = 3136259m

81 82

Orbits for these three satellites were generated for one revolution each and the topocentric coordinates, right ascension, a*, declination,

6 *, range, r*, were computed. The method of generating the orbits is described in Section 2.5. All the data was stored on tape and was re­ garded as the true or "observed" values of the true orbit. Approximate

orbits were generated by introducing errors to the Keplerian elements

and station coordinates alone, Section 4.2; or to the gravity anomalies

alone, Section 4.3; or to all three sets of parameters, Section 4.4.

These gave approximate or "computed" values. Relevant equations from Chapter 3 were used to form observation equations and then, as in

Sections 4.2 and 4,4, to solve them in order to recover the errors

originally introduced.

Table 4

Satellites First Used

Element OBI OB2 OB3

8299053. 05m 7710341. lm 10082517. 2m atto 0.16363 0. 11915 0.04672 % 32! 887 491954 881 351 ito 29514 461910 521 651 wtco

R 9 0 ! 62 9°. 086 351 332 *-o M*. 0.4892rev 0 . 84009rev 0 .9239rev co MJDt 38486 38520 58574 ‘o 83

4. 1. 1 Definition of Parameters and Weighting

The unknown quantities were the Keplerian elements, gravity

anomalies, and station coordinates. The "observed" values were a*,

6*, r* from the true orbit.

The weight matrix of each "observed" quantity in all cases was

taken as one in whatever units the misclosure vector was, the units be­

ing altered in order to obtain relatively more stable normal equations.

There was, however, no reduction of the whole adjustment system to

any one particular type of unit, a* and 6* being in either radians or

degrees and r* in meters. Thus, for the angles the weights were either

the reciprocal of (1 radian)^ or reciprocal of (1 arcsec)^ and for the

range were always the reciprocal of (1 m eter)^. The weight matrix of

the gravity anomalies, Px, was taken as zero. In a practical applica­

tion, where accuracy estimates of the gravity anomalies are available,

the weights of the gravity anomalies must be incorporated in the adjustment either as Px or as part of P^. This depends on whether the

anomalies are treated as observations or parameters. Eqn. (3.93) then 4 reduces to:

1 0“

P 0 = 1

0 1 84

In the equation above, the off-diagonal elements were assumed zero, indicating independence of the observed quantities.

It is, however, realized that, due to the integration process for generating the true orbit, successive satellite position values may in­ herit slowly changing errors by virtue of their proximity in time and space, thus actually creating a correlation between the "observed" quantities. A method which attempts to estimate the off-diagonai ele­ ments from an analysis of residuals after an initial adjustment from actual data is described in Brown and Trotter [1969, p. 83], but it was not applied here.

4. 2 Wrong Orbital Elements and Station Coordinates

In this case, the anomalies were assumed correct, as errors were introduced only into the Keplerian elements of OBI and station coordinates. An approximate orbit was generated; and the values were used to compute the A-matrix, Eqn. (3.87), from Eqns. (3.25) through

(3. 30) with dAg = 0 in Eqns. (3. 31), (3. 32), and (3. 33). The topocentric coordinates from the observation station were computed all around the orbit without regard to actual visibility. If these partial equations are correct, the following relationship must hold exactly:

(4. 1) AX = L. 85 where:

X = vector of introduced errors;

L = misclosure vector obtained by subtracting the true a*, 6*, r* from the approximate values at the same time.

Table 5 shows the results of this check for a*, 6*, and r*.

Figure 2 gives the differences in graphical form. The last column of the table, "percentage difference", gives the critical statistic, q, de­ fined by:

(4.2) q = 100 x | E(Y)i - Yi |/E(Y)i i=l,...,n where:

E(Y)i = ^

Yi = (AX)i

Although it appears to have no definite distribution, q< 10 was used as a criterion for accepting the partial equations as good enough to produce a reliable estimate of the parameters required. This number was chosen after solving several simulated homogeneous equations using different values of q.

From the table referred to above, where q over one revolution is generally less than 5, the A-matrix is acceptable and adequately checks the appropriate equations of Chapter 3. 86

Table 5

Comparison of True and Predicted Differences in Right Ascension, a*, Declination, 6*, and Range, r*, for Wrong Orbital Elements and One Station

Time After True Diff. , Predicted t0, sec Element L Diff. , AX L-AX q, %

0 Act* -5'.'42 2 -5'.'422 0. 000 0 A 6* OL'901 0 ’.'901 0.000 0 Ar* 89. 019m 89. 019m 0. 000 0

500 Act* -7'.'247 -7'.'274 0. 027 0 A 6* 3'.'257 3 L'265 -0.008 0 A r* 58. 048m 58. 356m -0.308 1

1000 A a* -8'.'091 -8'.'178 0. 087 1 A 6* 5 3 3 3 5'.'365 -0.031 1 A r* -10. 360m -10. 758m 0. 399 4

1500 A a* -71'285 -7'.'3 35 0. 050 1 A 6* 3:>453 31'437 0. 014 0 A r* -75. 778m -77. 536m 1. 758 2

2000 A a* -6:'206 -6i'142 -0.065 1 A 6* 1'.'298 1'.'260 0. 038 3 A r * -103. 101m -104. 334m 1. 232 1

2500 A a* -5:'340 - 51'242 -0.098 2 A 6* OJ'059 0'.'048 0. 011 20 A r* -107.836m -107. 875m 0. 038 0

3000 A a* -4 '.'5 38 -4'.'481 -0.058 1 A 6* -0'.'721 -Ot'717 -0.004 0 Ar* -98. 558m -98.423m -0.135 0

3500 A a* -3'.'841 -3'.'805 -0.036 1 A 6* -1 '.'224 - 1'.'217 -0.007 1 A r* -76. 206m -75. 783m -0.423 1

4000 A a* -3'.'346 -3'.'3 24 -0.023 1 A 6* -1L'501 - 1"500 -0.001 0 A r# -42. 667m -43. 019m 0. 352 1 87

Table 5 (Continued)

Time After True Diff. , Predicted t0, sec Element L Diff. , AX L-AX q, %

4500 A a* -3'.'064 - 3 0 72 -0.008 0 A 6* -1"582 -i:'602 -0.020 1 A r* -4. 335m -5. 317m 0. 982 23

5000 A a* -2'.’971 -21*999 0. 028 1 A 6* -1'.'509 -li'546 0. 037 2 A r* 31. 095m 30. 429m 0. 665 2

5500 A a* -3['040 - 31' 051 0. 011 0 A 6* -1 '.’321 -11*346 0. 025 2 A r* 59. 754m 59. 30lm 0.454 1

6000 A a* -3L’251 - 3' 2 4 7 -0. 004 0 A 6* -1 '.’028 -1 L'039 0. 011 1 Ar* 81. 379m 81. 330m 0. 049 0

6500 A a* - 31’585 - 3 5 86 0. 001 0 A 6* -O'.'6 06 -0'.'610 0. 004 1 A r* 96.878m 97. 250m -0.372 0

7000 A a* -4'.'022 -4C031 0. 009 0 CM o o i- < ^ i A 6* Ol'013 o o o 1— 8 A r* 106. 826m 107. 299m -0.473 0

7500 A a* -4'.'532 -4['547 0. 015 0 A 6* O'.'941 O'.'9 5 3 -0.012 1 Ar* 110. 577m i;i. 106m -0.528 0

8000 A a* -5'.'109 -5'.' 141 0.032 1 A 6* 2”381 2L'413 -0.032 1 Ar* 104.’882m 105. 637m -0.756 1

8500 A a* -5i'886 -5:'960 0. 074 1 A<5* 4'.'550 4'.'6 20 -0.070 2 A r* 80.947m 81.429m -0.482 1 88

Graph of (True-Predicted) Differences Time After Reference Epoch

0.05

8000 secs5000

Time After Reference Epoch -0. 05

Meters 2

1

0 5000 8000 secs

1 Time After Reference Epoch

2

Figure 2

Differences Between True and Predicted Differences

A = Right Ascension, a* B = Declination, 6* C = Range, r* Normal equations were then formed from 26 observation equations and solved to obtain estimated input errors:

(4. 3) X = (A1 A)"1 A'L

A As mentioned earlier, X should equal X. Table 6 gives the result of solving for X, from a* and 6* alone and from r*, 6*, a* together. Al-

A though the recovered or estimated errors, X, are not all exactly equal to the input errors, X, the recovery is regarded as satisfactory, par­ ticularly for the orbital elements. Thus the use of q < 10 is confirmed for checking the observation equations.

Although the difference is not very marked, it would appear, by comparing the third and fourth columns of Table 6, that errors in the station coordinates are recovered more satisfactorily from a*, 6*, and r* than from a* and 6* alone. 90

Table 6 Recovered Errors of Orbital Elements and Station Coordinates from One Revolution of One Satellite

A A Input Error, Recovered Error, X, Recovered Error, X, Element X From a*, 6* Only From a*, 6*, andr*

A at -6m -6m -6m Lo A e^ 0.0000005 0.0000003 0.0000003 uo -Of 00005 -Of 00005 -Of 00005 Aito Acjf Of 0005 Of 0005 Of 0005 Lo A fit Of00004 Of 00004 Of 00004 o AMt Of 0006 Of 0006 Of 0006 ‘'O Auj -6m -9m -8m

Au 2 7m 9m 7m

Au 3 10m 8m 9m

4.3 Wrong Anomalies

An approximate orbit was generated, in this case, with errors introduced to the gravity anomalies alone; the orbital elements and station coordinates for the true orbit were used. The A-matrix was obtained by the same methods as in Section 4. 2 in order to check only the relation:

AX = Li.

There was no attempt to try to recover the introduced errors as

in the previous case, but merely a check of the terms containing dAg in

Eqns. (3.31), (3.32), and (3.33). The results of the check of the 91

equation above are given below in terms of the changes in theKeplerian

elements between the true and approximate orbits for two times. Here

again, although the difference

(AX)i - Li is not exactly zero all around the orbit generated, because the percent­ age difference is less than 10, the results are quite satisfactory and the

equations containing dAg accepted as correct.

Table 7

Check of Variational Equations for Wrong Gravity Anomalies Alone for One Revolution of One Satellite

4 Time After True Predicted t0, rev Element Error, L Error, AX q, %

0 Aa 0 0 A e 0 0 Ai 0 0 A w . 0 0 Afi 0 0 AM 0 0

1 Aa 0. 59m 0. 6 lm 3 A e -0.382xl0"6 -0.369xl0-6 3 Ai OL'OIO 0'.'009 10 Aw -1. 658 -1 "627 2 AO 01'239 OL'243 2 AM 0'.'873 O'.'8 37 4 92

4. 4 Wrong Orbital Elements, Gravity Anomalies, and Station Coordinates

For these computations, all three satellites of Table 4, Sec­ tion 4. 1, were used. Approximate orbits of one revolution each were generated and partial equations computed. Figure 3A is a schematic

diagram of these partials and Figure 3B of the normal equations.

As in Section 4. 2, Table 8 gives the check on Eqn. (4. 1) for only a few positions of each satellite, and the explanation therein applies here.

In this case too, the check can be regarded as satisfactory.

The results of the solution of the 205 normal equations formed from 840 observation equations are given in Tables 9, 10, and 11, and also displayed in Figure 4 for the anomalies. For the anomalies, the

0 results indicate generally that, using only one revolution each of three different satellites, the input errors were "fairly well" ^ recovered

only along the orbits for geocentric satellite heights less than 8000km.

This is what would normally be expected.

See Section 4.6. 93

a ! • • • Agi. Ui

OBI

Ln

OB2

‘•n

OB3

- a n ... Agn. u

Figure 3

Schematic Form of Observation Equations, A, and Normal Equations, B, for Wrong Orbital Elements of Three Satellites, Gravity Anomalies, and One Station 94

Table 8

Comparison of Predicted and True Differences in Right Ascension, oc*, Declination, 6*, and Range, r*, Using Wrong Elements, Station Coordinates, and Gravity Anomalies of Three Satellites

Time q, P er­ Satel­ After Ele­ True Diff., Predicted centage lite tQ, sec ment L i Diff. , AX L-AX Diff., %

OBI 0 A a* -51*422 -51*422 0 0 A 6* 01'9 01 01*901 0 0 A r* 89. 019m 89. 018m 0. 001 0 3500 A a* 0!'704 01'6 75 0. 029 0 A 6* 0'.'980 0:'976 0. 004 0 Ar* -62. 953m -63. 114m 0. 161 0 8930 A a* 13!'937 131*726 0.211 2 A 6* -181*363 181*167 0. 096 1 Ar* 157.400m .158. 873m 1.473 1

OB 2 40 A a* 01.136 2 O'.'36 3 -0.001 0 A 6* O'.'94 3 Oi'943 0 0 Ar* -1. 289m -1. 311m 0. 022 0 4055 A a* 2'.'064 21*103 -0.049 2 A 6* -101*877 -101*918 0.041 0 Ar* -139. 039m -140.673m 1.634 1 7115 A a* 11 L'634 111*689 -0.055 0 A 6* 271*933 271*983 -0.050 0 Ar* -185. 655m -184. 694m -0.961 1

OB 3 3650 A a* -01*358 -01*360 0. 002 0 - A 6* 31*175 31*212 -0.037 1 Ar* -16. 015m -16. 129 m 0. 114 1 5840 A a* 11*154 11*177 -0.023 2 A 6* 31'6 24 31*683 -0.059 2 Ar* 33. 314m 34. 004m -0.726 2 7690 A a* 21*162 21*167 -0.005 0 O r-H A 6* -0:'204 I -0.007 3 Ar* 151. 652m 153.641m -1.989 1 95

Table 9 Recovered Errors in Orbital Elements from One Revolution Each of Three Satellites

OBI OB 2 OB 3 Input Recov­ Input Recov­ Input Recov­ ered ered Errors, ered Ele­ Errors, A Errors, A ments X Values, X X Values, X X Values, X

A a* -6.0m -3.0m 8.0m 8.5m -10.0m -7. lm co A e*. 0.5xl0-6 0.4xl0-6 0.3xl0-6 O.lxlO-6 O.lxlO-6 0.0 Lo -0?00005 -0°00003 -0°00004 -0°00005 -0? 00003 -0°00004 Aito * <1 3 -U 0?0005 0^0006 -0?00003 -0?00001 0?00003 0?00005 o

A o !00004 0?00005 0?000001 -0?000004 0°00002 -OtOOOOl Lo AMf 0t0006 0 :0 0 0 5 - 0 : 0 0 0 2 - 0 :0 0 0 1 0 :00 01 -0:0001

These values, as well as those in Tables 10 and 11, were obtained after the adjustment of the system described in Section 4.4.

Table 10 Recovered Errors in Station Coordinates from One Revolution Each of Three Satellites

Input Recovered A Coordinate Errors, X Values, X

Auj -6. 0m -4. 2m

Au2 7. 0 11. 1

Au3 10. 0 15. 0 96 Table 11 Recovered 15°xl5° Equal Area Mean Gravity Anomaly Errors from One Revolution Each of Three Satellites

Input Error Recovered Error , o x,

82.5 60.0 -1 0 .0 0 - 483.43 82.5 180.0 - 10.00 +178.99 82.5 300.0 -10.00 +25.10 67.5 20.0 “10.00 +293.21 67.5 60.0 -1 0 .0 0 +940.54 67.5 100.0 -1 0 .0 0 +266.62 67.5 140.0 -1 0 .0 0 -351.13 67.5 180.0 -1 0 .0 0 +122.43 67.5 220.0 -1 0 .0 0 - 87.15 67.5 260.0 -1 0 .0 0 - 26. 15 67.5 300.0 -10.0.0 +30.12 67.5 340.0 -1 0 .0 0 -256.35 52.5 12.0 -10.00 -527.72 52.5 36.0 -10.00 +1362.94 52.5 60.0 -1 0 .0 0 -3154.59 52.5 84.0 -10 .0 0 +14.80 52.5 108.0 -1 0 .0 0 +148.76 52.5 132.0 -10.00 -18 5 .7 9 52.5 156.0 -1 0 .0 0 +233.13 52.5 180.0 -1 0 .0 0 -64.91 52.5 204.0 -1 0 .0 0 -1 4 .5 3 52.5 228.0 -10.00 - 8.98 52.5 252.0 -10 .0 0 -14.98 52.5 276.0 -1 0 .0 0 -4 7 .6 6 52.5 300.0 -10.00 +34.72 52.5 324.0 -1 0 .0 0 -2 9 .0 6 52.5 348.0 -1 0 .0 0 +176.18 37.5 9.5 -1 0 .0 0 +110.62 37.5 28.4 -1 0 .0 0 -329.87 37.5 47.4 -1 0 .0 0 +430.27 37.5 66.3 -1 0 .0 0 +914.14 37.5 85.3 -1 0 .0 0 +1307.99 37.5 104.2 -1 0 .0 0 -81 1 .4 2 37.5 123.2 -10.00 +405.31 37.5 142.1 -1 0 .0 0 - 35.23 37.5 161.1 -1 0 .0 0 -3 0 .7 6 37.5 180.0 -1 0 .0 0 -1 2 .2 0 37.5 198.9 -10.00 + 1.86 37.3__ 217.9 -1 0 .0 0 __ -7 6 .0 5 37.5 236.8 • -1 0 .0 0 - 51.77 37.5 255.8 -1 0 .0 0 -3 1 .8 9 37.5 274.7 -1 0 .0 0 -1 8 .8 0

. . . _ 3 7 -3 293.7 -10.00 -47.12 Table 11 (Continued)

{J1o A 9- X X

37.5 312.6 -1 0 .0 0 '29.92 37.5 331.6 -10.00 -80.58 37.5 350.5 -1 0 .0 0 -95.75 22.5 8.2 -1 0 .0 0 ” 151.70 22.5 24.5 -10.00 4158.14 22.5 40.9 - 10.00 -255.52 22.5 57.3 -1 0 .0 0 -222.96 22.5 73.6 *10.00 “464.98 22.5 90.0 -10.00 "319.86 22.5 106.4 -10.00 4404.82 22.5 122.7 -10.00 -376.20 22.5 139.1 - 10.00 +49.01 22.5 155.5 - 10.00 -10.82 22.5 171.8 - 10.00 -5 .0 0 22.5 188.2 - 10.00 +119.74 22.5 204.5 - 10.00 '165.46 22.5 220.9 -1 0 .0 0 4426.81 22.5 237.3 - 10.00 -8 2 .0 0 22.5 253.6 -10 .0 0 - 62.06 22.5 270.0 - 10.00 - 35.93 22.5 286.4 -10.00 -21.31 22.5 302.7 -10.00 -31 .1 9 22.5 319.1 -10 .0 0 -27.43 22.5 335.5 _10.00 +51.93 2 2.5 351.8 -10.00 +45.89 7.5 7.5 -10.00 +227.45 7.5 22.5 -10.00 -361.38 7.5 37.5 -1 0 .0 0 +387.71 7.5 52.5 -10.00 +121.11 7.5 67.5 -10.00 +89.00 7.5 82.5 -1 0 .0 0 +56.67 7.5 97.5 -1 0 .0 0 -40.08 7.5 112.5 -10.00 - 8.26 7.5 127.5 -15.00 443.07 7.5 142.5 -15.00 -11.06 7.5 157.5 -15.00 -16.69 7.5 172.5 -15.00 +32.67 7.5 187.5 -1 5 .0 0 -14 2 .4 0 7.5 202.5 -15.00 -495.18 7.5 217.5 -15.00 + 5.25 7.5 232.5 -15 .0 0 - 8 7 .0 5 7.5 247.5 -15.00 -20.64 7.5 262.5 -15.00 - 1.56 7.5 277.5 -1 5 .0 0 -2 2 .2 6 7.5 292.5 —15.00 -86.21 7.5 307.5 -15.00 -26.30 7.5 322.5 -1 5 .0 0 - 98.80 7.5 337.5 - 15.00 -5 1 .6 3 Table 11 (Continued)

o A

- 6 X X

7.5 352.5 -1 5 .0 0 -L15.99 -7 .5 7.5 -1 5 .0 0 -392.95 -7 .5 22.5 - 15.00 +734.21 -7 .5 37.5 -1 5 .0 0 -833.32 -7 .5 52.5 - 15.00 -35 .2 6 -7 .5 67.5 -1 5 .0 0 “62.14 -7 .5 82.5 - 15.00 -8 .4 2 -7 .5 97.5 -1 5 .0 0 “9.97 -7 .5 112.5 -1 5 .0 0 “12.22 -7 .5 127.5 - 15.00 —15.25 -7 .5 142.5 -15.00 - 14.70 -7 .5 157.5 -1 5 .0 0 -1.85 -7.5 172.5 -15.00 + 4. 66 -7 .5 187.5 - 15.00 +39.66 -7.5 202.5 - 15.00 +51.61 -7 .5 217.5 -15.00 -2.28 -7 .5 232.5 -1 5 .0 0 -13.12 -7 .5 247.5 - 15.00 -33.79 -7 .5 262.5 -1 5 .0 0 -282.08 -7 .5 277.5 -15.00 +99.88 -7 .5 292.5 -15.00 -146.36 -7 .5 307.5 _15.00 426.59 -7.5 322.5 -15.00 -5 6 .2 6 -7 .5 337.5 -1 5 .0 0 +139.84 -7 .5 352.5 -15.00 -2^8.95 -2 2 .5 8.2 -15.00 +449.03 -2 2 .5 24.5 -15.00 -500.61 -2 2 .5 40.9 -1 5 .0 0 +360.17 -2 2 .5 57.3 -15.00 495.36 -22.5 73.5 - 15.00 -4 7 .6 7 -2 2 .5 90.0 - 15.00 -1 2 .1 9 -2 2 .5 106.4 - 15.00 -15.81 -22.5 122.7 - 15.00 - 15.45 -22.5 139.1 - 15.00 -1 5 .7 0 -22.5 155.5 - 15.00 -31.32 -2 2 .5 171.8 - 15.00 -29.85 -2 2 .5 188.2 -15.00 - 24.80 -2 2 .5 204.5 -15.00 - 19.18 -2 2 .5 220.9 -1 5 .0 0 -5 0 .7 9 -2 2 .5 237.3 -15.00 - 528.13 -2 2 .5 253.5 -15.00 +682.05 -2 2 .5 270.0 -1 5 .0 0 -381.48 -2 2 .5 286.4 -15 .0 0 +146.21 -22.5 302.7 '15.00 -7 .9 9 -22.5 319.1 -15.00 -190.16 -2 2 .5 335.5 -15.00 -50a 85 -2 2 .5 351.8 -1 5 .0 0 -6 9 .6 4 -3 7 .5 9.5 -1 5 .0 0 '1 5 7 .4 3 99

Table 11 (Continued)

o A X X

-37.5 28.4 ~ -15.00 -17.88 -37.5 47.4 _15.00 _111.32 -37.5 66.3 —15.00 - 44.51 -37.5 85.3 -15.00 -13.93 -37.5 104.2 -15.00 -14.42 -37.5 123.2 -15.00 -25.32 -37.5 142.1 -15.00 -24.01 -37.5 161.1 - 15.00 - 25.12 -37.5 180.0 - 15.00 - 33.49 -37.5 198.9 -15.00 -103.40 -37.5 217.9 -15.00 -622.38 -37.5 236.8 +10.00 +1559.90 -37.5 255.8 +10.00 -1721.69 -37.5 274.7 +10.00 +1004.72 -37.5 293.7 +10.00 -856. 18 -37.5 312.6 +10.00 +464.55 -37.5 331.6 +10.00 +32.57 -37.5 350.5 +10.00 '77.71 -52.5 12.0 rio.oo +106.49 -52.5 36.0 +10.00 -12.78 -52.5 60.0 +10.00 +39.36 -52.5 84.0 +10.00 -37.44 -52.5 108.0 +10.00 +27.58 _ -52.5 132.0 •rio.oo . -3 8 .8 2 ...... -52.5 156.0 +10.00 +66.31 -52.5 180.0 +10.00 -49.85 -52.5 204.0 +10.00 +328.45 -52.5 228.0 +10.00 +989.18 -52.5 252.0 +10.00 -1120.70 -52.5 276.0 +10.00 •r740. 23 -52.5 300.0 +10.00 -282.53 -52.5 324.0 +10.00 -358.68 -52.5 348.0 +10.00 +68.53 -67.5 20.0 +10.00 -374.54 -67.5 60.0 +10.00 +174.13 -67.5 100.0 +10.00 -193.43 -67.5 140.0 +10.00 +307.67 -67.5 180.0 +10.00 -794.36 -67.5 220.0 +10.00 -182.59 -67.5 260.0 +10.00 -142.03 -67.5 300.0 +10.00 +197.22 -67.5 340.0 T10.00 +224.73 -82.5 60.0 +10.00 - 1 6 .5 2 -82.5 180.0 +10.00 +319.61 -82.5 300.0 +10.00 -127.56 -483 PER I GF!

123 =

-3151 1363 ■> RiGFE 1308 *-3^4-30 30K.M 30 40 - 10( -10 -320 405-465

-361

15 -15 -15 -15 -15 -15 159 [-249-393 734-146 -833 -35

-381 -190 -501 PERlGEE^il 151 - 1 5 331-103 1560 1722 5 . 70KM

-1 120

0B3_

Figure 4

Recovered Mean Gravity Anomaly Errors in Relation to Satellite Orbits These values are obtained from Table 11. The shaded areas indicate where the maximum deviation between the input and recovered errors does not exceed 10 mgals. The upper number in each block is the input error and the lower number the recovered error. 100 101

4.5 Wrong Orbital Elements and Gravity Anomalies

For the solution of this section, approximately 30-hour orbits of four different satellites and a 15-hour orbit of a fifth satellite,

Table 12, were used. The orbits were generated using a slightly modified version of the Cowell orbit generator program given in

Cigarski, et al [1967] . A fixed step mode of integration of 100 secs, was used, giving an average computer time of about 5 minutes per

30-hour orbit. The 15°xl5° equal area mean gravity anomalies of

Appendix C, based on actual gravity data, were taken to represent the true gravity anomaly field. The method of their estimation from

2592 5°x5° non-equal area mean gravity,anomalies is described in

Appendix C. The mean anomaly errors were obtained from a pseudo­ normal random sample with variance 400.

Table 12 Second Set of Satellites Used

Element Vanguard 2 Geos 1 Explorer 8 Explorer 19 Midas 4

8298835m 8073835m 7711120m 7861916m 10004915m aco t 0.1645 0.07142 0.119333 0. 1133 0.011262 eto 32? 8722 59?3853 49?945 78?606 95?8566 Xt0 CO. 41?256 188? 269 80? 54 141 ?3 265?698 to 140?815 325?9684 328?355 251 ? 215 40? 2206 ntt o 0 .75334rev 0.680485 0.68208 0.96253 0.38098 MJD. 38574 39130 38532 38576 38088 to 102

Twelve stations around the world., whose coordinates were assumed exactly known, were adopted as observation stations. To approximately simulate practical conditions, limit the number of ob­ servation equations, and save computer time, only topocentric right ascensions and declinations of a satellite position above the horizon of one station at 10-minute intervals were used in the adjustment. The total execution time on the IBM 360/75 for evaluating 214 coefficients for each of 928 observation equations used was about 46 minutes.

Table 13 gives a check of the evaluated observation equations for Vanguard 2. The statistic q, 5th column, is generally less than

3%. The graph of Figure 5 shows the differences (Predicted-True) in arc-secs for a* and 6*, respectively, as functions of time after the reference epoch of Vanguard 2. A comparison of the 5th column of

Table 13 with the 6th column of Table 8 or with the 5th column of

Table 5 shows that better results are obtained with the variational equations if a smaller integration step size is used in generating the satellite orbits.

Results of the solution of the 214x214 normal equations are shown in Table 14 for the 30 orbital elements and in Table 15 and Figure 6 for the 184 15°xl5° equal area mean gravity anomalies. 103

Table 13

Comparison of True and Predicted Differences for Right Ascension, a*, and Declination, 6*, for Vanguard 2

Time After True Diff. , Predicted tQ, min Element L Diff. , AX AX-L q, %

0 A a* -01*770 -01*770 01*000 0. 0 A 6* -0.266 -0.266 0.000 0. 0

206.7 • A a* 47.048 46.779 -0.269 0. 6 A 6* -41.887 -41.236 0. 651 1. 6

401. 7 A a* 318.035 316.472 -1.563 0. 5 A 6* 63.013 62.594 -0.419 0. 7

601. 7 Aa* 240.091 237.931 -2.160 0.9 A 6* -22.883 -21.999 0.884 8.9

806.7 A a* 208.264 205.784 -2.480 1.2 A 6* -157.500 -154.588 2.962 1.9

1001.7 Aa* 280.042 277.311 -2.731 1.0 A 6* -19.276 -19.609 -0.333 1.7

1201.7 A a* 110.396 109.305 -1.391 1.0 A 6* -128.355 -125.905 2.450 1.9

1403.3 A a* 414.230 412.792 -1.438 0. 3 • A 6* 75.551 73.673 -1.878 2.5

1605.0 Aa* 340.100 335.828 -4.272 1. 3 A 6* -96.101 -93.707 2.494 2. 5

1788.3 A a* 21.113 22.627 1. 514 7. 2 A 6* 696.840 691.246 -4.594 0.8 104

Graph of (Predicted-True) Differences A rcsecs v Time After Reference Epoch

150 ^300 m in

Time After Reference Epoch

F igure 5 Differences of (Predicted-True) Differences in Right Ascension, A, and Declination, B

Table 14 Recovered Input Errors in Orbital Elements from About 15 Revolutions Each of Five Satellites

Aa A et A if Aojf t> AMt 4?

co co Lo o Lo m Vang. 2 -5. -0.0000005 0° 00002 0°00003 -0°. 0002 -0“000072 ’ -3. -0.0000007 0.00000 0.00005 -0.0003 -0.00008 Geos 1 -6. -0.0000003 -0.00006 0.0001 -0.0056 -0.000108 -8. -0.0000004 -0.00007 0.0002 -0.0056 -0.000127 Expl. 8 6. -0.0000006 0.00005 0.00004 -0.0003 0.000036 6. -0.0000003 0.00008 0.00005 -0.0003 0.000044

Expl. 19 8. 0.0000005 0.00003 0.0002 -0.0002 0.000108 6. 0.0000007 0.00004 0.0003 -0.0004 0.000211 Midas 4 7. 0.0000004 0.00003 -0.0003 0.00002 0.000072 4. 0.0000001 0.00004 -0.0005 0.00000 0.000057

The top numbers for each satellite are the Input Errors and the bottom numbers the Recovered Errors. 105

Table 15 Recovered Mean Gravity Anomaly Errors and Their Standard Deviations from About 15 Revolutions Each of Five Satellites

Input Recovered E rro r E rro r A cp ° 2 A. ° 2 X, m gals X, m gals a q q -82. 5 6 0 .C000C -70.0 -31.6 20.3 -82. 5 180.CC00C -68.8 -83. 1 19.5 -82.5 300.00000 -60.2 -28.9 20. 1 -6 7 .5 2 0. C000C -38.8 -33.9 27.6 -6 7 .5 60.0C00C -22.4 -84. 1 29.0 -67.5 100.00000 4.7 9.0 26.6 -67.5 140•00000 -0 .4 31.6 21.6 -6 7 .5 180.0C00C 37.3 -0 .7 25.8 -6 7.5"“'“220.00000 -2.5 38.5 23.1 -67.5 260.00000 -17.8 -57.0 26.0 -6 7 .5 300.CC00C 9.0 22.4 25.3 -67. 5 3 4 0 .C000C -16.9 -6.0 28.5 -52.5 12.00000 6.8 -36.7 26.3 -52.5 3 6 .CC00C -0 .3 77.8 25.4 -52.5 60.00000 13.2 92. 1 24.2 -5 2 .5 84.00000 -2 .0 -0 .6 24.7 -5 2 .5 108.CC00C -7 .9 -16.4 20.8 -52. 5 1 32 .C000C -20.5 -23.6 20.6 -52.5 156.00000 9.9 74.7 23.6 -52.5 180.0C00C 8.9 14.9 22.3 1* - 5 2 .5 “ 204.CC00C -10.0 -0 .6 24.9 2* -52.5 228.00000 -1 2 .8 -21.5 21.6 3* -5 2 .5 252-CC000 -23.4 -17.3 23.4 4* -52.5 276.CC00C -5 .9 44.3 23.5 -5 2 .5 300.00000 -13.1 " -15.4 28.5 -52.5 324.C000C -17.8 -17.0 30.0 -52.5 348.C000C -21.0 -15.1 27.3 -3 7.5 9.47368 -17.0 48.1 19.0 -3 7 .5 28.42104“ 18.7 15.4 18.5 -37.5 47.36838 -29.3 -114.7 17.1 -37.5 66.31573' -10.5 -9 .2 16.9 -37.5 85.26311 7.4 12.9 18.9 -3 7 .5 104.21045 19.8 84.3 16.2 -37.5 123. 15781 -31.8 -0 .8 17.3 -3 7 .5 142.10518 -15.5 -95.2 14.1 -37.5 161.05254 -22.7 -28.0 16.0 -37.5 179.99988 13.4 12.1 17.0 -37.5 198.94724 -17.8 -62.0 16.1 -37.5 217.89461 32.1 59.3 15.4 -37.5 236.84195 -8.1 -12.4 17.0 1 See Section 4.6. 2 9q and X a refer to the center of each anomaly block. H H. Table 15 (Continued)

o A A X X a

- 3 7 .5 255.78931 -3.3 22. 1 17.8 -3 7 .5 274.73633 11.2 11.3 17.1 - 3 7 .5 293.68384 - 4 .5 - 1 4 .0 20.4 - 3 7 .5 312.63135 - 1 6 .9 - 1 1 .3 19.4 -3 7 .5 331.57837 29.9 61.5 21.4 - 3 7 .5 350.52588 -25.4 - 7 0 .8 17.6 -22.5 8. 18182 - 2 1 .5 - 1 5 .0 16.1 -2 2 .5 24.54543 8 .7 8.8 17.2 - 2 2 .5 40.90906 42.2 100.2 17.3 -2 2 .5 57.27271 5.2 68.6 17.8 -2 2 .5 73.63634 2 9.0 62.7 17.9 - 2 2 .5 89.99997 5 .6 - 4 0 .9 17.1 -2 2 .5 106.36359 - 1 2 .9 - 1 1 .1 15.2 -2 2 .5 122.72722 - 1 5 .6 - 4 1 .2 13.8 - 2 2 .5 139.09087 - 4 .4 38. 1 14.4 -2 2.5 155.45450 19.9 47.4 15.1 - 2 2 .5 171.81812 - 1 0 .3 - 4 5 .7 14.8 -2 2 .5 1 8 8 .18L75 - 1 8 .3 - 1 3 .7 13.0 - 2 2 .5 204.54538 -24.0 -2 4 .3 15.4 - 2 2 .5 220.90903 - 6 .4 -1 0 .1 13.4 - 2 2 .5 237.27266 -23.6 - 3 7 .6 14.8 - 2 2 .5 253.63628 -33.0 -1 9 .9 14.9 -2 2 .5 269.99976 - 7 .1 - 3 1 .5 16.1 - 2 2 .5 286.36353 - 2 3 .0 - 1 5 .6 14.8 - 2 2 .5 302.72681 11.0 17.8 19.2 -2 2 .5 319.09058 13.8 7 .0 15.1 - 2 2 .5 335.45435 - 4 0 .4 - 3 2 .6 16.3 - 2 2 .5 351.81763 37.1 75.4 16.1 -7.5 7.5C00C 0 .2 - 1 8 . 1 9.4 - 7 .5 22.5C000 37.2 35.5 11.0 - 7 . 5 37.50000 -11.9 - 1 3 .9 9 .4 - 7 .5 52.5C000 27.8 -1 3 .4 11.6 - 7 .5 67.50000 36.8 36. 1 10.5 - 7 . 5 82.50000 -6.4 - 5 .6 13.2 - 7 .5 97.5C00C - 3 9 .6 -3 3 .6 10.7 - 7 .5 112.50000 -14.3 27 .8 11.8 - 7 .5 12 7 . 5C00C 4.1 - 0 .5 10.2 - 7 .5 1 4 2 -5COOO - 1 .4 - 2 5 .8 9 .9 - 7 . 5 157.50000 32.2 2 7 .0 9 .2 - 7 . 5 172.50000 - 0 .5 - 1 .4 10.4 - 7 .5 187.50000 - 5 .1 - 1 0 .2 10.1 - 7 .5 202.50000 12.3 25 .3 10.2 - 7 .5 217.50000 3.3 19.3 10.9 -7.5 232.50000 5.3 14.9 12.4 - 7 .5 2 4 7 .5000C 24 .9 3 1 .4 12.8 - 7 .5 262.50000 42.1 82.6 13.6 Table 15 (Continued)'

A A 9 q xq X X a - 7 .5 27 7 . 5000C -2 9 .7 1.6 12-8 - 7 . 5 292.5CC0C -16.3 - 9 .0 12.0 -7.5 307.50000 2. 1 40.1 10.6 -7.5 322.50000 -2 4 .6 -1 0 .3 12.7 - 7 . 5 33 7 . 5C00C 45.2 84.2 10.2 -7.5 352.50000 2.6 20.4 12-6 7.5 7.50000 -1 2 .2 - 6 .7 9 .4 7.5 22.50000 -1 8 .0 1.5 1C.3 7.5 37.50000 -r 13. 8 - 1 .1 9 .4 7.5 52.50000 1.4 14.4 9.3 7.5 67.50000 -24.3 - 6 .9 8 .7 7 .5 82.50000 - 7 .4 - 1 0 .3 9 .9 7.5 97.50000 -3 .1 -5 8 .3 9 .6 7.5 112.50000 - 6 .4 - 5 .9 10.7 7.5 127.50000 -22.6 - 2 4 .6 9 .2 7 .5 142.50000 23.9 - 2 .2 9 .8 7.5 157.5CC0C 3.7 1.3 9 .3 7.5 172.50000 15.1 20.7 9.1 7 .5 18 7.5 0 0 0 C - 6 .7 - 4 3 .0 8 .6 7 .5 2 0 2 .5C00C 23.9 4 .8 8 .7 7.5 217.50000 26.0 14.5 8.5 - 4 8 .7 9 .4 7 .5 232.50000 -1... 4 .3 7.5 247.5C00C ~ - 2 2 .7 8 .9 7.5 262.50000 -4.5 -2.4 9 .6 7.5 277.50000 ' -1 9 .7 - 1 1 .4 10.2 7.5 292.50000 -10.5 -10.7 1C.3 7.5 307.50000 15.4 4 .6 10.0 7 .5 322.50000 11.3 1.7 9 .6 7 .5 337.50000 -1 0 .4 - 1 2 .5 10.9 7.5 352.50000 - 0 .5 1.6 10.8 2 2 .5 8. 18182 0.5 16.2 4 .9 22.5 24.54543 1.9 5.3 4 .6 22.5 40.90906 -11.3 -10.9“ “4.7 22.5 57.27271 18.9 17.4 4 .7 22 .5 73.63634 -14.1 - 1 4 .8 4 .7 22.5 89.99997 1 . 8 7 .6 5 .2 2 2 .5 106.36359 -1 0 .2 - 4 .8 5 .3 22 .5 122.72722 -11.8 -15.1 4.8 22.5 139.09087 -1 7 .9 - 3 8 .7 5 .7 2 2 .5 155.45450 31.8 9.1 5.3 22.5 171.81812 -56.6 - 4 1 .1 7 .1 22.5 188.18175 3.5 - 6 . 0 4 .6 2 2 .5 204.54538 - 2 . 6 - 1 5 .7 5 .6 Table 15 (Continued)

A A 9 ° a q X X 22.5 220.90903 17.7 26.9 4.4 22.5 237.27266 4.0 9.1 5.7 22.5 253.63628 - 3 .5 -12.9 8. 1 22.5 269.99976 -22.3 - 1 6 . 1 6.1 22.5 286.36353 - 1 6 .7 - 0 . 1 7.9 22.5 302.72681 24.6 36.8 6 .0 22.5 319.09058 -2 2 .1 - 4 .6 6 .8 22.5 335.45435 5.6 1 7 .C 4 .8 22.5 351.81763 25.7 30.6 6. 1 37.5 9.47368 -11.9 -9.2 4-9 37.5 28.42104 6 .5 5.0 4 .8 37.5 47.36838 1.8 2 .0 4 .7 37.5 66.31573 -1 0 .6 - 8 .8 4.1 37.5 85.26311 -1 6 .7 -1 3 .2 4 .3 37.5 104.21045 -1 6 .2 - 3 0 .1 4 .2 37.5 123.15781 -0.4 -0.1 5.6 37.5 142.10518 -1 4 .9 - 2 7 .6 7.4 37.5 161.05254 - 4 .3 - 2 2 .2 6 .6 37.5 179.99988 20.9 26.8 6.2 37.5 198.94724 5.5 - 1 .2 7 .2 37.5 217.89461 - 9 .7 - 1 3 .3 7 .2 37.5 236.84195 11.8 4 .4 8.6 37.5 255.78931 - 3 .9 - 1 0 .9 9.1 37.5 274.73633 18.2 14.3 6 .9 37.5 293.68384 -1.7 - 7 .6 6 .4 37.5 312.63135 C • 7 15.3 5.5 37.5 331.57837 -23.9 -33.7 4.8 37.5 350.52588 16.4 24.0 7.2 52.5 12.CC00C 22.3 28.8 7 .0 52.5 36.CC00C 0 .6 - 3 .4 7.2 52.5 60.00000 -5 .1 - 5 .0 6 .1 5 2.5 84.CCOOO -4 1 .1 - 4 7 . C 5 .9 52.5 1 0 8 .CC00C 14.9 34.5 6 .6 52.5 132.CCOOO 7.4 - 2 6 .9 8 .0 52.5 1 5 6 .0C00C - 0 .9 7 .3 8.1 52.5 180.CG000 16.2 6 .7 7 .8 52.5 204.00000 0 .4 3 .2 8.5 52.5 2 2 8 .CC000 -1 4 .0 - 5 .1 10.8 52.5 252.CC00C -1 5 .9 - 2 1 .0 11.6 52.5 276.00000 -1 0 .0 -7 .3 10.0 52.5 300.00000 -37.5 -46.2 7 .4 52.5 3 2 4 .CC000 - 1 6 .5 - 1 .3 6 .0 52.5 348.00000 - 7 . 7 - 1 6 .9 5 .5 109

Table 15 (Continued)

A A X X a

6 7 .5 20.0000C - 2 .6 - 5 .1 6.1 67.5 6 0 . CC00C - 2 7 .7 - 2 1 .9 5.9 67.5 100.00000 -2 2 .1 - 3 5 .0 6 .3 67.5 140.00000 - 3 2 .0 - 4 6 .5 7.8 67 .5 180.00000 -2 4 .4 - 3 3 .3 9.5 6 7 .5 220.00000 -21.2 - 2 4 .0 8 .0 67 .5 2 6 0 .CC000 14.7 35.4 8.8 67.5 300.00000 -3 5 .4 - 4 8 .5 8 .0 6 7 .5 340.00000 —0.9 8 .6 6.0 82.5 60.00000 - 3 2 .8 - 2 2 .6 6.1 82.5 180.00000 - 0 .2 - 1 1 .6 6 .2 82.5 300.00000 - 2 5 .9 - 4 9 .9 5 .9

The last column, cx, the marginal standard deviation of each estimated anomaly error, is based on a standard error of 1 arcsec for the "observed" right ascensions and declinations. Relatively better values for the Northern Hemisphere underline the fact that more "observations" were done there than in the Southern Hemisphere.

This is so principally because there were more observing stations in the north than in the south. r. ' ’ ’ >. / 1K 7 7 T r l ~\— .-1 0 ///-3 S // -16 / Tl /fte.6// ~lr r H r r t f *. /-24//16,* tw ~// ■ '*y//"I 3./l/i/' -30 ' / 0 -a . -22/r -17 2 S I S ® V1//-J2'; 1 -24 \j—7/ -3-J/-6/ " *—2, —11/ _i i , ri ' ' __ —. -7 .-10, -58. -6V-25/^ -2 J T y r ,

29 6 / —13.-—16 I —A

95 /_28 ijZ-i 67V,-13// : T W S ^ -

-39 //

Figure 6

Recovered Mean Gravity Anomaly Errors From Five Satellites. The shaded blocks indicate where the maximum deviation between the input and recovered errors does not exceed 10 mgals. The top number in each block is the input error and the lower number the recovered error. Both sets of values are taken from Table 15. The asterisks mark the observation stations. £ o Ill

4.6 Analysis of Results

Because it is immediately apparent, in such simulation studies, whether the estimated values are acceptable or not, statistical tests may not be directly applicable. Thus, the setting of confidence limits for

estimated parameters from the true values and such tests as the F-test

for determining the theoretical compatibility between the estimated and

true parameters cannot be usefully applied here. However, in order to

define accuracies, the following two questions will be considered:

1) How precise was the adjustment made?

2) How accurate was the recovery of the input errors?

The first question relates to the internal precision of the adjust­

ment system. It is solved by giving the variance-covariance matrix of

the adjusted parameters, preferably with an a priori variance of unit

weight. Associated with this matrix are the correlation matrices, both

marginal and conditional, which indicate the degree and type of correla­

tion between the estimated parameters. Another useful statistic would

be to compute the rms value of the residuals before and after adjustment.

For this simulation study the deviation of the rm s value of the residuals

after adjustment from zero would give an indication of the accuracy of

the adjustment.

For the purpose of this study the second question is the more per­

tinent one: How accurately were the input gravity anomaly errors re­

covered? Or, what could be regarded as the same thing: Taking into 112 account all of the limiting factors connected with this problem, what, besides an exact recovery, can be accepted as satisfactory? In this respect it would appear that a meaningful evaluation can come only from sources external to the adjustment. The rms value of the differences between the true and estimated errors can be used. It is possible to compute this statistic directly for the anomaly blocks used here since they are all of equal area, which is one of the main reasons equal area anomaly blocks were chosen. This quantity may, however, be deceptive and might blur meaningful conclusions that may be drawn from the actual adjustment. Thus, for example, there may be, physically possible reasons, a few very large differences which would increase this rms value and therefore give a distorted picture of the adjustment.

A seemingly arbitrary approach, but one that is actually based on external knowledge of the problem, would be to set a critical value, q, as being the maximum allowable difference between the estimated and true errors. An analysis based on this approach and possibly combined with some or all of the previously mentioned statistics would give a much better idea on how accurately the input errors were recovered.

This is the method adopted here.

Some of the statistics discussed above are given below:

rms value of the residuals before adjustment = ±8173

rms value of the residuals after adjustment = ±1102 113

The decrease in value shows that the adjustment absorbed about

88 % of the original errors. This isnot verygood for simulation studies but is probably feasible here because of the numerical problems involved.

The rms value of the 184 differences, (True-Recovered) anomaly errors, is computed as ±24. 3 mgals. The rms value of the input errors was ±24. 2 mgals. Taking only these two values, it could be concluded that the adjustment did not improve the anomaly errors. However, as pointed out earlier, this rms value could be deceptive. The fact is that due to the distribution of the observations, most errors were properly recovered (see below) while only a few were not.

Defining the critical value to be 10 mgals, the shaded areas of

Figure 10 indicate those blocks with a difference of 10 mgals or less between the true and estimated errors. The rms value of the differ­ ences of these 102 blocks alone is equal to ±5.7 mgals.

The standard deviations of Table 15 indicate that the input anomaly errors in the northern hemisphere were more precisely recovered than those in the south. This is confirmed by the fact that of the 102 values with q < 10 mgals, 59 of them were in the north. The rms value of the differences of these blocks is ± 6 . 0 mgals, while for the south the rms value of the differences of the 43 blocks is ±5.4 mgals.

The marginal and conditional correlation matrices, respectively, between the values marked with numbered asterisks in Table 15 are given as follows: ]

Marginal Correlation Matrix 1* " 1 -0.667 0. 380 0 . 159

2* 1 -0.416 0. 184

3* 1 -0.485

4* 1 -

Conditional Correlation Matrix 1* "-1 - 0.6 1 0 -0.003 0.471

2* -1 -0.780 -0.312

3* -1 -0.827

4* -1

Only a general conclusion can be made from the above matrices taken together; there is a very high correlation between adjoining ad­ justed anomaly blocks. This correlation falls off as the distance be­ tween the'blocks increases. This is what would normally be expected.

It would appear that an analysis of these matrices from actual adjust­ ments in relation to co-variance functions of the same size anomaly blocks would probably provide some interesting results about the be­ havioral pattern of the adjusted gravity anomalies.

For the sake of completeness in the analysis of the recovered

A gravity anomaly errors, X, assuming that their differences from the corresponding input errors, X, satisfy the basic conditions for an F- test, Hamilton [1964] , the following hypothesis was tested:

Hq : X - X = 0 115

(X-X) > a (X-X) ^ X The value, F , computed from should be distributed m as F.m , n -m , a* where:

is the variance-covariance matrix of the recovered £ parameters;

m = number of recovered parameters being tested;

n-m = degrees of freedom in the adjustment;

a = significance level.

F c was computed as 0. 13.

184, 744 0 . 05 from statistical tables is approximately 1. 2 .

Since Fc < F 744 at the 5% significance level, it cannot be rejected that the recovered errors are theoretically compatible with the input errors. Although these results suggest a theoretical satis­ factory recovery of all the input errors, the validity of the basic as­

sumptions made in carrying out the test is dubious. The rather large discrepancies between X and X in a few blocks makes a further critical examination of the spread of the observations as well as the normal equations necessary. However, taking the results of the adjustments together with the statistics given above and also noting the numerical problems involved, the recovery of the input errors from the data used can be regarded as generally satisfactory and the method feasible, thus achieving the purpose of this study. CHAPTER 5

CONCLUSION

We set out at the beginning of this study to investigate one method of obtaining directly from analyzing satellite orbits differential correc­ tions to a priori estimates of gravity anomalies, as well as differenti­ ally correct satellite orbital elements at a reference epoch and station coordinates. In order to do this, appropriate equations were derived

(Chapters 1-3) and assuming a gravity anomaly field represented by

184 15°xl5° equal area mean gravity anomalies, elements of some sat­ ellites and coordinates of some stations, data were generated to check these equations. Taking into account the limiting factors of this prob­ lem and using some relevant statistical inferences, the recovery of the input gravity anomaly errors (of the last adjustment) could be regarded as satisfactory. It is necessary to note that the method adopted here is not unique; neither are the equations. Although its efficiency and accuracy with respect to other methods were not investigated, particu­ larly in evaluating the variation equations of Chapter 3, the results obtained indicate that it is a feasible method. The method of variation of parameters, although perhaps slower to evaluate, was chosen as the basis for this study because it is the most obviously useful, especially

116 117 for the variational equations which are capable of giving the highest possible accuracy since all of the equations involved in orbit genera­ tion are used in their derivation.

Although the derivation of the required equations is relatively

more complicated than some indirect methods, it has the greatest ad­

vantage of giving gravity anomalies directly. Extrapolating from Fig­ ure 4, the indication that anomaly blocks around satellite orbits with perigee heights less than 1650 kms are satisfactorily recovered, and with the results of Figure 6 , it can be concluded that an analysis of

several low altitude satellites at different inclinations for long periods, with observations properly distributed in time and space, would give much better results. It would, however, be interesting to compare practical results, using all available but uniformly estimated standard errors of a priori gravity anomalies, with solutions from some other m ethods.

An obvious extension would be to perform an analysis to obtain the anomalies of much smaller blocks, say 5°x5°. In this case, al­ though the normal equations would be quite large, making a solution very tricky, the assumption that the kernel of the disturbing potential,

V3, (and its derivatives) is constant over each block would be much more acceptable. APPENDIX A

POTENTIAL COEFFICIENTS TO (14, 14) FROM COMBINED SOLUTION BY RAPP APPROACH, RAPP [1968a]

n m C S 2 0 -434.1750 - 0. 2 2 2.37 0 6 -1 . 342 2 3 0 0.9365 - 0. 1 1 .8 555 0.243 4 3 2 0.7130 -0.5486 . 3 3 0.5 33 1 1.5234 4 0 0.5522 0. 4 1 -0.5513 -0.4470 4 2 0.297 1 0.5844 ... 4 ... 3 0.8729 -0.1975 4 4 0.0 93 6 0.2741 .... 5.. 0 0.0496 0 . 5 1 -0.0816 -0.0635 c ? 0.5 22 3 -0.2134 5 3 — C.3 560 0.0273

_ ..5. 4 -0.0492 0.0744 5 5 0.0859 -0.5689 6 0 -0.1366 0. 6 1 -0.0647 -0.0194 6 2 0.0283 -0.233 5 6 3 -0.0535 0.0602 . . 6 .... 4 -0.0250 -0.4123 6 5 -0.2906 -0.4509 6 6 -0.0087 -0.1843 7 0 0.0702 0. 7 1 0.1289 0.1060 1 7 2 0.3065 0. 1372 7 3 0.179 6 0.0092 7 4 -0.1931 -0.0906 7 5 0.0704 0.0355 7 6 -0.1654 0,0917 7 7 0.0679 -0.0298

8 0 ' 0.0460 0 . ___ 8 1 -0.0433 0.0250 8 2 . 0.0401 0.0981 ... . 8 ...... 3 .... -0.0044 . 0.0380

118 119

Appendix A (Continued)

n m C S p 4 -0 .0922 0.0138 P q -0.0630 0.0818 Ft 6 -0.1070 0.2369 p 7 ______0-0256 _____ ...... 0.0 300____ 8 a -0.1041 -0.0168

q ...... 0 0.0248 0 . 9 i 0.1325 -0.0872 9 -> 0.0125 0.0001 9 3 -0.0 753 -0.0303

9 . 4^. 0.0400 . -0.0154 . . 9 5 -0.0465 0.0253 . - 9 ... .6 _ - 0.0102 0.0432 9 7 0.0421 0.0181 q R 0.20 7 4 -0.0032 9 9 0.0129 -0.0345

. . . 10. . .0 ...... -0.0127 ______0. .. ______10 1 0.0 73 8 -0.064 5 1 0 2 -0.0402 -0.0706 10 3 -0.0499 -0.1314 i n 4 -0.031 0 -0.0366 10 5 -0 .0046 0.0037 10- ___ 6 -0.0622 ___ .-0.0202_____ 10 7 0.0745 -0.0160 1 0 a 0.0439 -0. 1 03 9 10 9 0.0934 -0.0026 10 10 0.0678 -0.0690 11 0 -0.0880 0. 11 I 0.0244 0.0120 11 2 0.0304 -0.0245 11 3 -0.0049 -0.0049 11 4 -0.0196 -0.0597 11 5 0.0087 -0,0615 11 6 0.0339 0.009 8

. . 11.. 7 0.0088 - 0. 080 1 11 6 C.0409 0.0106 11 9 0.0334 0.0060 11 10 -0.0140 -0.0109 11 1 1 0.0752 0.0167 12 0 -0.0068 0.

... .12 _ JL_ -0.0643 -0.0596 12 2 -0.0184 0.0865

12 ...... _3 0.0683 -0.0065 Appendix A (Continued)

n m C s 1 2 4 -0.0176 -0.0169 1? s 0.0318 -0.0677 12 6 0.0033 0 .0 346 12 . 7_ -0.0423 ___ _Q. 0270 ___ 12 s 0 .0 101 0. 0507 . . 12 q -0.01 12 0.0627 12 10 -0.0033 0.0027 1 ? 11 -0.0170 n .0039 12 12 0.0256 -0.0 313 .. n ... 0__ 0.0462 - _...... 0. ____ 13 1 -0.009 8 -0.0172 13 2 -0.000 8 0.006 1 13 3 0.0198 0.0535 1 3 4 0.0092 -0.029 3 13 5 0.0516 -0.04 76 13 6 -0.035.3 0.0308 13 7 0.0078 0.0228 _12‘. 3 . -0.0460 -0.0033 13 9 0.0014 0.0 52 0 1 3 1 0 0.0162 -0.0576 13 11 -0.0434 -0.0011 13 1 2 -0.0137 0.064 4 13 13 -0.0203 0.046 5 14 0 - C . 0 10 3 0 . 14 1 -0.0177 0.0057 1 4 2 -0.0533 0.0006. . 14 3 0.0179 0.0149 14 4 0.0218 -0.0038 14 5 0.0745 -0.0631 ...... 14 _ 6 0.0144 -0.0374 14 7 0.0571 0.0310 14 8 -0.0159 -0.0208 14 9 0.0346 0.0720 14 __ 10 -...... C.0546 ...... -0_. 046 1 ... 14 11 0 .0 1 2 0 -G> 0106 14 12— 0.0254 -0.0073 14 1 3 0.0145 0.0171 . 34...... 1 4_ ... . -a.Q3.Q8_ ...... -0.0075 Note: The value of each coefficient should be multiplied by 10"6 . APPENDIX B 15°xl5° EQUAL AREA MEAN GRAVITY ANOMALIES o o A g mgals Ag mgal

82 5 60 ooooo -1 4 82 5 180 OOOOO -1 3 82 5 300 ooooo 7 7 67 5 20 OOOOO 4 9 67 5 60 ooooo 8 0 67 5 100 OOOOO -18 7 67 5 140 ooooo 9 9 67 5 180 OOOOO 3 4 67 5 220 •ooooo 3 6 67 5 260 OOOOO -7 7 67 5 300 ooooo -9 1 67 5 340 OOOOO 28 0 52 5 12 ooooo 6 7 52 5 36 ooooo 3 5 52 5 60 ooooo -3 3 52 5 84 ooooo -17 9 52 5 108 ooooo -17 0 52 5 132 ooooo 5 6 52 5 156 ooooo 10 7 52 5 180 ooooo 0 9 52 5 204 ooooo 8 9 52 5 228 ooooo -8 8 52 5 252 ooooo 6 2 52 5 276 ooooo -12 4 52 5 300 ooooo -7 7 52 5 324 ooooo 19 5 52 5 348 ooooo 5 4 37 5 9 47368 16 7 37 5 28 42105 14 0 37 5 47 36842 7 3 37 5 66 31579 -1 1 0 37 5 85 26316 -0 4 37 5 104 21 052 -9 4 37 5 123 15789 1 9 37 5 142 10526 13 4 37 5 161 05263 -7 5 37 5 180 ooooo -8 0 37 sJ198 94736 -9 0 37 5 217 89473 -1 1 9 37 5 236 8421 0 -9 3 37 5 255 78947 7 7 37 5 274 73684 -3 4 37 5 293 68421 -12 5 37 5 312 631 58 7 9 37 5 331 57894 16 0 37 5 350 52631 14 9 22 5 8 18182 -2 9 22 5 24 54545 -8 9 22 5 40 90909 7 1 22 5 57 27273 -0 3 22 5 73 63636 -6 5 22 5 90 OOOOO -9 4 22 5 106 36364 -9 2 22 5 122 72727 8 6 22 5 139 09091 2 6 22 5 155 45454 -6 9 22 5 171 81818 -8 2 22 5 188 18182 2 2 22 5 204 54545 7 0 22 5 220 90909 -7 5 22 5 237 27272 -22 3 22 5 253 63636 -2 5 22 5 270 OOOOO 13 8 22 5 286 36363 -27 3 22 5 302 72727 -23 2 22 5 319 09090 -4 7 22 5 335 45454 -6 3 22 5 351 818.18 5 9 7 5 7 50000 11 9 7 5 22 50000 -1 1 7 5 37 50000 8 5 7 5 52 50000 -18 3 7 5 67 50000 -21 0 7 5 82 50000 -30 5 7 5 97 50000 -4 0 7 5 112 50000 22 0 7 5 127 50000 24 0 7 5 142 50000 12 7 7 5 157 50000 0 9 7 5 172 50000 3 2 7 5 187 50000 0 1 7 5 202 50000 4 3 7 5 217 50000 -7 6 7 5 232 50000 -4 2 7 5 247 50000 -3 9 7 5 262 50000 2 0 7 5 277 50000 18 1 7 5 292 50000 6 0 121 122

A p p en d ix B (Continued)

9q \q Ag mgals cp° Ag mgals

7 5 307 50000 -23 4 7 5 322 50000 -11 7 7 5 337 50000 9 9 7 5 352 50000 16 0 -7 5 7 50000 -13 9 -7 5 22 50000 -8 9 - 7 5 37 50000 -5 7 -7 5 52 50000 -14 8 -7 5 67 50000 -15 9 -7 5 82 50000 -35 3 “7 5 97 50000 5 3 -7 5 112 50000 14 2 -7 5 127 50000 6 0 -7 5 142 50000 24 0 -7 5 157 50000 12 5 -7 5 172 50000 1 1 5 “7 5 187 50000 -2 2 -7 5 202 50000 9 3 “7 5 217 50000 -1 5 -7 5 232 50000 -1 2 -7 5 247 50000 -2 5 -7 5 262 50000 4 1 -7 5 277 50000 0 7 -7 5 292 50000 17 6 -7 5 307 50000 -6 2 -7 5 322 50000 -4 2 -7 5 337 50000 -9 6 -7 5 352 50000 0 8 -22 5 8 1 8182 3 8 -22 5 24 54545 9 9 -22 5 40 90909 -1 5 -22 5 57 27273 1 1 3 -22 5 73 63636 12 1 -22 5 90 OOOOO -13 4 -22 5 106 3 6364 -16 6 -22 5 122 72727 1 8 -22 5 139 09091 4 5 -22 5 155 45454 19 2 -22 5 171 81818 19 6 -22 5 188 18182 0 2 -22 5 204 54545 6 6 -22 5 220 90909 -4 2 -22 5 237 27272 1 7 -22 5 253 63636 0 5 -22 5 270 ooooo -1 1 -22 5 286 36363 1 1 2 -22 5 302 72727 ' 9 1 -22 5 319 09090 ' -12 3 -22 5 335 45454 -4 7 -22 5 351 81818 4 8 -37 5 9 47368 8 2 -37 5 28 421 05 15 1 -37 5 47 36842 4 8 -37 5 66 31579 12 6 -37 5 85 26316 1 2 -37 5 1 04 21052 -18 9 -37 5 123 15789 -26 3 -37 5 142 10526 1 7 -37 5 161 05263 6 1 -37 5 180 OOOOO 7 2 -37 5 198 94736 3 2 -37 5 217 89473 -2 9 -37 5 236 84210 -5 5 -37 5 255 78947 4 6 -37 5 274 73684 4 0 -37 5 293 68421 5 4 -37 5 312 63158 -0 8 -37 5 331 57894 -7 4 -37 5 350 52631 1 2 -52 5 12 OOOOO 2 2 -52 5 36 ooooo 14 1 -52 5 60 ooooo 17 2 -52 5 84 ooooo 15 6 -52 5 108 ooooo -0 6 -52 5 132 ooooo -1 9 -52 5 156 ooooo -6 6 -52 5 180 OOOOO • -9 1 -52 5 204 ooooo -7 4 -52 5 228 OOOOO -0 -52 5 252 ooooo 0 0 -52 5 276 ooooo 2 9 -52 5 300 ooooo 3 0 -52 5 324 OOOOO 1 5 -52 5 348 ooooo 1 1 7 -67 5 20 ooooo 5 7 -67 5 60 ooooo 8 3 -67 5 100 ooooo 2 1 -67 5 140 ooooo -12 1 -67 5 180 ooooo -19 6 -67 5 220 ooooo -20 7 -67 5 260 ooooo -1 7 -67 5 300 ooooo 5 9 -67 5 340 ooooo —4 4 -82 5 60 ooooo •4 3 -8 2 5 180 ooooo -26 2 -82 5 300 ooooo -5 5 123

The anomalies were computed from the potential coefficients of

Appendix A at the latitude and longitude indicated for each anomaly.

This location is the center of each 15°xl5° equal area block. The equation used was the following:

Ag(cPqXq) ='Ve^(n - 1)^ (Cnm cos mXq + Snm sin m \ q)Pnm(sincpq) n m where:

"ye is the equatorial gravity

^nm “ ^nmA " SimE

CnmA *s a coefficient from Appendix A

CnmE *s corresponding coefficient of the adopted ellipsoid.

For the ellipsoid used here it follows that:

C20 “ C20A " C20E lu II o C40A - C40E -X-vO lu .. 11 o c 60 lu n iH C ' l 4 , 0 ,o .{

APPENDIX C

15°xl5° EQUAL AREA TERRESTRIAL MEAN GRAVITY ANOMALIES

The anomalies in Table 17 below, given on an absolute system,

GRS 1967, were estimated from 2592 5°x5° anomalies given in Rapp

[1968] and are based on the International Gravity Formula. In first transforming the 5°x5° gravity anomalies to the absolute system, the following relationship was used:

(C. 1) AgAbs = Agl + - 7 Abs m gals where:

(C . 2) 'YI(cpq) = (978049 - 13. 8)( 1 + . 0052884sin2cpq

- 5. 9 x 10_^sin22cp )mgals SI

(C - 3) 7A bs(cPq) = 978031. 8(1 + .0053022sinz cpq

-5.8 x 10”^sin22cpq)mgals

The constant term, 13.8, is approximately the error in the Potsdam system in which anomalies are generally given.

Hence:

(C. 4) AgAbs = A gj + 3. 4 - 13. 5sin2cpq - 0. ls in 22cpq

= A gj + c

124 125 \ The values for c at different latitudes are given in Table 16. In

Table 17, cpq and kq for each mean gravity anomaly refer to the center

of the 15°xl5° equal area block.

Table 16 Conversion Terms for Anomalies from International Gravity Formula System to GRS 1967

cp° c(mgals)

2.5 3. 4 7.5 3. 1 12. 5 2. 7 17. 5 2. 1 22. 5 1.4 27. 5 0. 4 32. 5 -0.6 37. 5 -1 .7 42. 5 -2.9 47. 5 -4. 0 52. 5 -5. 2 57. 5 -6. 3 62. 5 -7. 3 67. 5 -8 .2 72. 5 -8 .9 77. 5 -9 .5 82.5 -9 .9 87.5 -10. 1

The 15°xl5° equal area anomalies, Agjg, were computed from the

5°x5° non-equal area mean values, Agg, from the following equation: n ^ Ag5icoscpmi i=l

/ , coscPmi i 126 where:

9m = mid-latitude of the 5°x5° block;

i = 1, ... ,n is the number of anomalies from which a 15°xl5° equal area anomaly was estimated.

If the number of 5°x5° blocks in the 15°xl5° equal area block was not an integer, the anomaly, Ag^, of the part, a, of the 5°x5° block inside it, was computed from:

— A Area of a g5 = g5 ' Total Area of 5°x5° block

The anomaly of the remaining part of the 5°x5° block in the next

15°xl5° block was then A gg - Agg.

!

I 127

Table 17

159xl5° Equal Area Terrestrial Mean Gravity Anomalies

0 0

9 q A g mgals 9q xq A g mgals -7 5 307 50000 1 2 -7 5 322 50000 -6 0 -7 5 337 50000 -1 8 -7 5 352 50000 2 7 7 5 7 50000 11 8 7 5 22 50000 -1 3 7 5 37 50000 0 7 7 5 52 50000 -13 0 7 5 67 50000 -32 8 7 5 82 50000 -28 0 7 5 97 50000 2 3 7 5 1 12 50000 15 5 7 5 127 50000 18 4 7 5 142 50000 6 0 7 5 157 50000 2 6 - 7 5 172 50000 9 3 7 5 187 50000 4 2 7 5 202 50000 13 1 7 5 217 50000 2 2 7 5 232 50000 4 0 7 5 247 50000 -0 4 7 5 262 50000 6 0 7 5 277 50000 17 6 7 5 292 50000 0 1 7 5 307 50000 -18 1 7 5 322 50000 -7 2 7 5 337 50000 5 2 7 5 352 50000 13 5 2? 5 8 18182 4 1 22 5 24 54544 2 2 22 5 40 90907 2 0 22 5 57 27271 -6 0 22 5 73 63634 -6 8 22 5 89 99997 -1 1 6 22 5 106 36360 -2 7 22 5 122 72723 6 0 22 5 139 09087 1 9 22 5 155 45450 0 6 22 5 171 81813 -0 3 22 5 188 18176 2 4 22 5 204 54539 7 0 22 5 220 90903 -3 8 22 5 237 27266 -7 1 22 5 253 63629 0 5 22 5 269 99976 7 2 22 5 286 36353 -16 6 22 5 302 72705 -10 7 22 5 319 09082 -0 8 22 5 335 45435 2 9 22 5 351 81787 8 i 37 5 9 47368 11 9 37 5 28 421 04 4 9 37 5 47 36839 7 0 37 5 66 31575 -7 5 37 5 85 2631 1 18 9 37 5 104 21 046 5 4 37 5 123 15782 2 8 37 5 142 10518 6 3 37 5 161 05254 -5 7 37 5 179 99989 -3 6 37 5 198 94725 -5 0 37 5 217 89461 -5 1 37 5 236 84196 -5 5 37 5 255 78932 3 6 37 5 274 73657 -2 8 37 5 293 68384 -1 1 9 37 5 312 63135 1 6 37 5 331 57861 14 7 37 5 350 52588 15 9 . 52 5 12 OOOOO 6 3 52 5 36 ooooo 2 4 52 5 60 OOOOO -4 4 52 5 84 ooooo -7 9 52 5 108 OOOOO -0 7 52 5 132 ooooo 3 4 52 5 156 ooooo -0 4 52 5 180 ooooo -5 1 52 5 204 OOOOO 1 2 52 5 228 ooooo -3 5 52 5 252 ooooo 1 5 52 5 276 ooooo -13 0 52 5 300 OOOOO -5 4 52 5 324 ooooo 3 5 52 5 348 ooooo 3 3 67 5 20 ooooo 2 1 67 5 60 ooooo -4 6 67 5 100 ooooo -1 5 67 5 140 ooooo 4 8 67 5 180 ooooo 4 4 67 5 220 ooooo 2 7 67 5 260 ooooo -14 0 67 5 300 ooooo -10 1 67 5 340 ooooo 4 3 82 5 60 ooooo -2 0 82 5 180 ooooo -6 5 82 5 300 ooooo 1 6 APPENDIX D

SET UP FOR INTEGRATION OF COWELL'S EQUATIONS OF MOTION

In order to facilitate the numerical integration of Cowell's Equa­ tions of motion, Eqn. (2.4) is generally rewritten as a system of first- order equations. Thus with x = u, y = v, z = w, Eqn. (2.4) becomes:

Xi ui

Y i vi m z i Wi - *i

vi Y i

wi zi

At i = tQ, that is, at an initial time tQ, *t0>yt0> zt0 >*t0 >Yt0 > z tQ are known either directly or through a set of Keplerian elements. Con­ version from these elements is given by:

129 130

r , _ X — (cosE - e) na

r 2 l/ z . — (1 - e ) sinE y na

z 0 na‘ (D.2) = co ' X -sinE

y (1 - e2)^cosE

z 0

_ _ — —

Rxq is given by Eqn. (3. 14).

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