Copyright "by

Urho Antti Kalevl Uotila

i960 INVESTIGATIONS ON THE GRAVITY FIELD

AND SHAPE OF THE EARTH

DISSERTATION

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

By

URHO ANTTI KALEVI UOTILA, M.Sc.

******

The Ohio State University 1959

Approved hy:

W , R Adviser Department of Geology (Geodetic Sciences) ii

PREFACE

Geodetic science, as Dr. Welkko A. Heiskanen has long maintained,

has had a great need for a world-wide geodetic system which would

connect the different regional geodetic systems of the world. In

recent years, the need of such a globe-spanning system has also been

increasingly recognized by the Armed Services, particularly the

United States Air Force, as a basis for obtaining the accurate

distances and directions so important to our national defense In this

Jet flight era. These needs, the academic and the military, were

admirably combined with the establishment In 1950 of the World-Wide

Gravity Project by Dr. Heiskanen, under the sponsorship of the United

States Air Force, at the Mapping and Charting Research Laboratory of

The Ohio State University Research Foundation.

Under this significant program, observational gravity data from

various agencies all over the world were, and continue to be, channeled into Columbus, adjusted to one reference system (the

Potsdam system), translated Into mean free air gravity anomalies, and finally used for computations of the undulations of the geold and the deflection of the vertical.

Because of the vast amounts of observational data handled by the

Laboratory and the complexity of the computations Involved, the writer Ill had the opportunity to develop a variety of new methods and techniques particularly as pertains to high-speed computational operations. The more important of these, which are covered In some detail in this dissertation, are;

1. A new method for high-speed computations of the undulations of the geold.

The writer, insofar as he knows, was the first to develop a practical method, based on mean anomalies, for the high-speed computer determination of the undulation of the geold. This method was used in computing the Internationally known "Columbus Geoid". It was also used in detailed investigation of the undulations of the geoid for a number of regional areas in the northern hemisphere. How­ ever, the results of these investigations have not, as yet been released for publication.

2. A method for high-speed computations of the deflection of the vertical.

Again, the writer believes himself to have been the first to develop a high-speed computer method for the numerical Integration of the effect of distant areas on the deflection of the vertical.

3. A template method for manual computations of the deflection of the vertical.

With this practical method, the numerical Integration of the effect of distant areas on the computation point is manually obtained by the use of templates of original design in conjunction with mean anomaly stereographic charts. iv

4. System for uniform high-speed handling of gravity data.

The writer presents, in detail, a system he has conceived and organized for the high-speed handling of vast amounts of gravity data with IBM machines.

5. A high-speed integration method for the determination of the shape of geopotential surfaces at high elevations.

The writer has for the first time determined the shape of geopotential surfaces at elevations greater than 100 nautical miles, using this new, high-speed integration method.

6. Correlations of free air anomalies and elevations on land.

The writer shows, by means of sample graphical analyses of free air anomalies from various parts of the world, the existence, within limited areas,of a general relationship between free air anomalies and elevation. Such correlations have been useful in estimating mean free air anomalies for regions in and near areas where only a few gravity observations are available.

7. General technique for determination and estimation of world free air gravity anomalies.

The writer describes in this dissertation a general technique he has devised and used, which takes into consideration free air anomaly elevation correlations and various methods of extrapolation and inter- o o polatlon, for the determination and estimation of 1 x 1 mean free air gravity anomalies wherever possible throughout the world. With this technique he has determined and estimated approximately 1 0 ,0 0 0 l°x 1° mean free air anomalies. These anomalies, however, have not V

as yet been released for publication. It should be noted that these

seme 1°X 1° mean free air anomalies have also served as the base

material for recent investigations of the Army Map Service and other

agencies in the United States.

8. Relating national reference stations to the same vorld

gravity system.

As a prerequisite to the many types of geodetic computations

carried out under the World-Wide Gravity Project, it was necessary

for the writer to relate the various national reference stations of

the world to a single world gravity system.

There are many individuals and organizations to whom the writer

is indebted in the preparation of this dissertation. Foremost among

these is Dr. Weikko A. Heiskanen, Director of the Finnish Geodetic

Institute (in absentia), Director of the Institute of Geodesy,

Fhotogrammetry and Cartography of the Ohio State University, and

Supervisor of the World-Wide Gravity Project, who, as adviser to the writer, has not only provided constructive criticism, giving liberally of his time, but has been a source of understanding inspiration as well. The other members of the reading committee, Dr. Paul M. Pepper,

Ohio State University, and Dr. C. Tsuboi, Tokyo University, furnished constructive advice for which the writer is grateful.

High-speed computational facilities have been made available in the numerical Computation Laboratory of the Ohio State University.

The programming of the IBM 650 has been done by Mr. Marvin Hardenbrook whom the writer wishes to thank for his excellent cooperation. Hie Ti writer it very grateful to Mr. Lasti A. Kivioja and to Mr. Clarence R.

Johnson for helping the writer in many cosqputatlons, and to the latter again for checking the dissertation in regard to correct English expression.

The writer wishes to express his gratitude to Dr. Paul M. Pepper end Arthur S. Cosier, former directors of the Mapping and Charting

Research laboratory and to the Geophysics Research Directorate, Air

Force Cambridge Research Center, Air Researoh and Development Comand,

Bedford, Massachusetts, which is administering Air Force Contract Kb.

AF19(6o 4)-1963, under which a part of this dissertation has been executed.

The writer is also very grateful to Mr* Lynn Freisner for drawing most of the illustrations, to Mrs. Mary Ann Humphrey for her excellent typing, and to all others who have assisted with this dissertation. TABLE OP CONTENTS

Section Page

PREFACE

INTRODUCTION 1

1 GRAVITY POTENTIAL AND GRAVITY FIELD OF THE EARTH...... 1+

1.1 Potential...... 4

1.11 Potential of A t t r action...... 5

1.12 Potential of Centrifugal Force ...... 13

1.13 Geopotential...... 13

1.14 Spheropotential...... 13

1.2 Gravity...... 14

1.3 Gravity Formulas ...... 16

1.31 Theoretical Gravity Formula...... 16

1.32 Computations of Gravity Formula...... 19

2 GRAVITY MEASUREMENTS...... 24

2.1 types of Gravity Measurements...... 24

2.2 Absolute Gravity Measurements...... 24

2.3 Relative Gravity Measurements...... 26

3 GRAVITY REFERENCE SYSTEM ...... 27

3.1 Potsdam System ...... 27

3*2 Standardization Lines...... 26

3*3 Gravity Reference Stations ...... 29

3*4 Relating and Recording ofGravity D a t a ...... 3?

vii viii

TABU OF CONTENTS (cont'd)

Section Page

4 GRAVITY ANOMALIES ...... 41

4.1 T^pe of Anomalies ...... 41

4.2 Mean Free Air Anomalies ...... 44

5 GRAVITY MATERIAL...... 65

6 COMPUTATIONS OF UNDULATIONS OF THE CBSOID...... 66

6.1 Basic Formulas...... 66

6.2 Numerical Computations...... 69

6.21 Oeneral Technique...... 69

6.22 Selection of the Size of Surface Elements...... TO

6.23 High Speed Computation Method ...... 74

6.231 Computational Technique ...... 74

6.232 Computations of Stokes' Coefficients...... 7?

6.2321 Coefficients for l°x 1° Squares...... 75

6.2322 Coefficients for 5 X 5 Squares...... 8l

6.233 Summation of the Products: Stokes' Coefficients X Gravity Anomalies ...... 83

6 .2 3 3 1 l°x 1° Squares...... 83

6.2332 5°x 5° Squares...... 85

6.234 Total Undulations of the G e o l d ...... 88

6.24 Accuracy of Undulation Values ...... 88 ix

TABLE OP CONTENTS (cont'd)

Section Page

7 COMPUTATIONS OP DEFECTIONS OF THE VERTICAL ...... 92

7.1 Basic Formulas ...... 92

7.2 Numerical Computations...... 94

7 • 21 General Technique...... 94

7.22 Selection of the Size of Surface Elements...... 95

7.23 Manual Computation Method...... 97

7*231 Base Map and Templates...... 9?

7.232 Estimation of Effect of the Nearest Neighborhood .... 100

7-233 Computation of Effect of the Area 3 -20 ...... 103

7-234 Computation of Effect of the Area 20°< \|r < l8o ° ...... 111

7-235 Total Deflection of the Vertical...... 121

7-214- High Speed Computation Method...... 121

7.241 Computation of Vening Meinesz' Coefficients ...... 121

7.242 Summations of the Products: Vening Meinesz* Coefficients X Gravity Anomalies ...... 123

7-243 Total Deflection of the Vertical...... 127

8 ANOMALIES IN UNSURVEYED AREAS...... 130

8 .1 Estimation of Anomalies...... 130

6.2 Balancing the Gravity Field...... 134 TABLE OF corns NTS (cont’d)

Section Page

9 COMPUTATIONS AT HIOH ELEVATIONS...... 143

9*1 Shape of Equlpotentlal Surfaces ...... 143

9*2 Gravity Anomalies ...... 146

10 RECENT DEVELOPMENTS...... 152

REFERENCES...... 153

AUTOBIOGRAPHY...... l6l LIST OF TABLES

Table Title Page

1 Corrections A 7 and AS to the International gravity formula ...... 22

2 Recent absolute measurements of gravity ...... 25

3 Gravity values of the reference stations...... 31-34

^ Curvature correction to Bouguer anomaly ...... 44

5 Elevation correlation analyses ...... 53-57

6 Mean anomalies of squares of various sizes extra­ polated from a single observed anomaly ...... 60

T Extrapolated effect of a l°x 1° mean anaom&ly ...... 64

8 Differences between to£al gndulatlon values obtained by using 1 X 1 mean anomaly areas of various sizes ...... 73

9 Radii of circular rings used in computations of the effect of nearest neighborhood of the computation point on the undulation of the g e o i d ...... 79

10 Effect of Rice's circular rings on the undulation of the geoid ...... 80

11 Rice's circular rings ...... 102

12 Selected values of function- 104

13 Selected circular rings with corresponding - v a l u e s ...... 116

14 Function used In connection with computation at high elevation ...... 147

xi LIST OP FIGURES

Title

Schematic concept of gravimetric geodesy. . .

2 Spherical coordinates ......

3 Spherical triangle ......

k Gravity vector......

5 Attraction of a plate ......

6 Relationship hetveen free air anomalies and , station elevations in the area 36 N-3 6 30 N and 110 W-ll8°30'W ......

7 Relationship hetveen free air anomalies and ( station elevations in the area 36 N-36 30 N and 118 30 W - 1 1 9 ° W ......

8 Relationship hetveen free air anomalies (and station elevations in the area 36 30 N-37 N and 118°W-H8°30 W ......

9 Relationship hetveen free air anomalies and station elevations in the area 36 30 *N-37 N and H8°30'w-119oVf......

10 Relationship hetveen free air anomalies and station elevations in the area 36 N-37 N and 118°W-1 1 9 W ......

11 Relationships hetveen free air anomalies and station elevations in various parts of the vorld......

12 Coefficient of extrapolation for determining a 1 X 1 mean anomaly from a point value . . .

13 Coefficient of extrapolation for determining a 1 X 1 mean anomaly from a computed l x l mean anomaly ......

Geoid and spheroid ......

xii xiii

LIST OF FIGURES (cont'd)

Figure Title Pege

15 Standard mean error of gravimetrically computed undulation of geold. (usual t signs are omitted, and unit : 1 meter)...... 90

16 Deflection of the vertical ...... 92

17 Location of points used In determining effect of inner circle ...... 101

18 Template shoving equal azimuthal lines of 10 degree Intervals and circular distance zones with the value of - (qp a39 0 ) ...... 105

19 Stereographlc base map for 1°X 1° mean anomalies .... 106

20 Zigzag borderline...... 107

21 Template with deflection coefficients for l°x 1° squares...... 109

22 Stereographlc base map for l°x 1° mean anomalies .... 110

23 Templates of curves and a-curves for 1+0 latitude, for the hemisphere where the computation point Is l o c a t e d ...... 112

21+ Stereographlc base map of northern hemisphere...... 113

25 Template of y-curves and a-curves for 1+0 latitude for the hemisphere where the antlpode of the computation point Is located...... 111+

26 Stereographlc base map of southern hemisphere...... Tig

27 Templateqvlth deflection coefficients for 5 x 5 squares (corresponding to figure 2 3) ...... 117

2d Stereographlc base map of northern hemisphere...... lid

29 Tenp£ateovith deflection coefficients for 5 x 5 squares (corresponding to figure 2 5) ...... 119

30 Stereographlc base map of southern hemisphere...... 120 xiv

LIST OP FIGURES (cont'd)

Figure Title Page

31 Borderlines of 6°x 10° areas ...... 128

32 Borderlines of 40°x 70° areas...... 128

33 Inadmissible spherical harmonics ...... 135

34 Corrections to mean anomalies resulting from balancing the gravity field unit 0 .1 mgal., northern hemisphere...... 140

35 Corrections to mean anomalies resulting from balancing the gravity, field unit 0 .1 mgal., southern h emisp h e r e...... 141

36 Corrections to N-valueB along 40° parallel resulting from balancing the gravity field...... 142

37 Elements in Poisson's integral formula ...... 144

38 Function used In connection with computation at high elevation...... 148

39 Separation of geopotential and spheropotentlal surfaces along a selected parallel at various elevations between 0 - 1 8 0 E ...... 150

40 Separation of geopotential and spheropotentlal surfaces along a selected parallel at various elevations between 0 -l8o w ...... 151 LIST CF DIAGRAMS

.•grain Title Page

I Flow chart for high-speed computation of Stokes' coefficient* for 1 X 1 square* Inside a 20 x 30 a r e a ...... 77

II Flow chart for high-speed computation of Stokes' and Vening Meinesz’ coefficients for 5 x 5 squares around the world ...... 82

III Flow chart for high-speed summation of products: Stokes' coefficients X gravity anomalies, using l x l squaresinside 20 X 30 area ...... 84

IV Flow chart for high-speed summation of products: Stokes' coefficients and gravity anomalies or Vening Meinesz' cogffigients and gravity anomalies, using 5 x 5 squares around the w o r l d ...... 86

V Flow chart for high-speed computation of Vening Meinesz V coefficients for l x l squares Inside a 40 X 70 a r e a ...... 124

VI Flow chart for high-speed sumatlen of Vening Meinesz' coefficients x mean anomalies inside M0°X 70° area ...... 125

xv IHTRODUCTIOW

When establishing the world-wide gravity project in Columbus

eight years ago, Dr. Heiskanen stated his conviction that the project,

either alone or together with astronomic determinations, could produce

the following results:

1. A general World Geodetic System, and convert the existing geodetic systems (North American, European, Indian, and so on) to this system.

2. The geodetic coordinates, on the World Geodetic System, of any needed point in the world, where astronomical observa­ tions exist or which is plotted on a local map with a reliable grid.

3. The distances and directions along the reference ellipsoid, between any required points in the world.

4. The conversion of these distances and directions from the ellipsoid to the geold.

5. The control of maps on 1:100 000 and smaller scale.

6. The real shape of the geold.

7. The reduction of triangulation base lines from the geoid to the reference ellipsoid.

8. The corrections of the closure errors of the triangles caused by the deflections of the vertical.

9 . The corrections to the used reference ellipsoid.

10. The correction to the used gravity formula. (12)

Dr. Heiskanen further stated that new knowledge about the

structure of the earth's interior close to the earth's surface could also be obtained.

1 Since 19^1 the determination of the earth'* gravity field t a d the determination of the geometry of epheropotential and the oospvta- tione of the shape of geopotential surfaces at higher elevations have also heen added to this list.

The hasie idea of the gravimetric method is that mass anomalies # visible or invisible, cause the gravity anomalies,Ag, geold undulations,

1, and deflections of the vertical as shovn in Figure 1. Gravity anomalies, Ag, can be observed and from these, undulations of the geold end deflections of the vertical can be computed. The basic

Mountain Geoid

Mass Normal to. Plumb Line Normal to Surplus Ellipsoid Ellipsoid lumb Line

Fig. 1. Schematic Concept of Gravimetric Geodesy mathematical formulas for computations of undulation of the geold using gravity anomalies mere derived by Stokes in 1 & 9 (h6 ). Corre­ sponding formulas for co^utatlons of the deflections of the vertical were derived by Toning Neiness in 1 9 2 6 (5*) •

There are several prerequisites of the gravimetric method (1 8 )

1. The gravity mfield around the whole earth should be known*. 2. All gravity observations should be in the seme system. 3. Either all gravity measurements should be reduced to sea level using a suitable reduction method or theoretical gravity should be computed at observation points.

4. A gravity anomaly is the difference between the measured gravity value as reduced to sea level and theoretical gravity obtained from a standard gravity formula. This means that the theoretical gravity formula should fit actual gravity values as well as possible.

5* The gravity anomaly field should be extended into unsurveyed areas as far as possible by interpolation and extrapolation. There­ fore, it is most important for the gravimetric method to ascertain mathematically as well as physically, the most probable mean gravity anomalies for unsurveyed areas.

In the following chapters the writer will cover these prerequi­ sites of the gravimetric method in greater detail. He will, in addition, present a short resume of accepted potential theory and a detailed discussion of computational methods, including a manual method as well as high-speed computation methods. In the last chapter he will discuss briefly the computations of the shape of geopotential and the geometry of spheropotentlal surfaces at high elevations. k

1. GRAVITY POTENTIAL AND (HIAVITY FIELD OF THE EARTH

1.1 Potential

Gravity of a rotating body Is divided Into two parts:

(l) attraction of mass and (2) centrifugal force. The attraction Is

expressed by Newton's formula

k - ^ 4 ^ . (1) D

where m and u are masses, f Is constant, and D Is the distance between

masses. The constant f has the value 6.6 7*10" 3 In the C.G.S. system.

In connection to gravity the mass p. Is assumed to be unit mass (= l).

The potential of attraction of a sphere at an external point Is

* - H * . <8 >

2 The centrifugal force Is e r where a> is the angular velocity and r Is

the perpendicular distance from point P to the axis of rotation. The

potential of the centrifugal force on the sphere is

2 2

Z m S -k JLm. (3)

The potential of a rotating sphere at point F outside of Its

surface is W * V + Z.

In the c u e of the earth, everything is not so simple, because the shape of the earth is not spherical and because there are Irregular density variations Inside the earth. The potential of the real earth Is called geopotentlal. A close

approximation for the shape of the earth Is a spheroid which Is a mathematically defined solid. The potential of the spheroid Is called

a spheropotentlal.

Gravity of the earth can be measured and Its direction, which Is

always perpendicular to the geopotentlal surface, can be determined.

It Is also possible to compute mathematically the theoretical gravity or normal gravity on the mathematical surface if necessary factors are known.

The potential of gravity, W, Is

W = V + Z , (h ) where V Is potentled. of the attraction of the earth's mass and Z Is the potential of the centrifugal force. Gravity g Itself Is

S = grad W . (5)

1.11 Potential of Attraction

The potential of attraction of the earth Is

+ o where f » gravitational constant, dm ■ a mass element, e = the distance from the mass element to the computation point, and o means integration over the earth. This formula can also be written into the form

(7) where f* the gravitational constant,

p and p'- geocentric distances, f 6 and 0 ■ complements of reduced

latitude, X. and \ ■ longitudes,

7 « angular distance between the

point and mass element, o' ■ density fVtr&eiy of the mass element.

The distance e Is expressed as Fig. 2. Spherical Coordinates e.V9*+ g*2-- 299' cosT ' which Is equal to

or e» §*\// cosv* , 1 2 -Jhas a form of (1 + a)

which may be expressed in series form vhen a < 1

(/+d) = / - i oC-Z-f- oC2-^oC3 od— • • * *

If £ « q or - q so that q < l,then e 0 :§•« Z'/v- (-2y.a»r+

+ 4 a3c&Y • »• • £ + f p,(t*st)+ fz%(cK>f)+2*%(ttsfj+ffifar)-- 7 where

£« / ff(co»Y)=- a » r

%(oo*,t)= \ ( c o s * / - -5 )

;§ fcos/> V -7 z d s T )

( c - V - 4 c ~ V + & ) . m The values are called Legendre polynomials (30). The general formula for P is n % 6i ~ r j - f & h ) ^ r ~ 2+

't-‘t(Sn^0(2n^) °°*r ~ 3%t%»tj3/$Z%fa-SJ + ■■■ or (1 5) ■p, r i / d n(cos'f~/)r>£ % ( c e s r j = 2 nn ! j

In case the polar coordinates cure

used, the angular distance takes the

form

d m COS Y~ COS$C0i

so that P^ (cos 7) is

7}Coodt)= o o s & c d s $ + Fig. 3» Spherical Triangle or as customarily written « T) C o o s Y ) = Tf>Q CcosYj-^o(cos&J+ Tj>f (a>$ify7>j(cos>#'JC0sfy— %)p 8 where 7ft0 (oost?J=* oosf and su?, it

Similarly It can be shown that

fi (oosi')= 7%i0 (ooi &J%'0 {<** £ j + (<*S 1>J$/(c£st!jG»i(){-Xl+

? , 2 (u>i^H.,z(c,>s'^'-( C Q S 2 f > ~ V , where

( c o s = % ( c o s 3 t t — )

T i ' , (co&itj=\/3 CCS if si* tf

These harmonics are called Laplace or associated spherical harmonics.

If cos f » t and - 1; *1 “ *2 “ *3 = ’ * * “ 2 tlie 6*neral formula for coefficients of associated spherical harmonics is (3 0)

where

The terms, where n - 1, 2, 3, 4 ... and v * 0, are called zonal harmonics, the terms, where n - 1 , 2 , 3, **• and n » v, are called sectorial spherical harmonics, and all remaining terms are called tesserai spherical harmonics. The n-value determines the degree of spherical harmonics and v the order.

When a function is developed in spherical harmonics up to nth degree, the number of terms is as follows n + 1 zonal spherical harmonics

2n sectorial spherical harmonics

n(.v-I) tesserai spherical harmonics o the total mother of terms Is (n + 1) . For example, when A. Frey (A3)

dereloped the topography of the earth In terms of spherical harmonies

up to l6th degree, the number of coefficients vas 2 8 9 .

If p ■ p', the reciprocal of distance expressed In terms of

spherical harmonics Is

"C = ~ p X i - (%") ft, (oo*$}1na{cM$JCO**(>!-)■) 9 * 0 ^*0 J and If p - p', then

~ k - i f ' Z - L ( p T Q .

The expression of the distance can he also written In series form

- 9 U + ■§■ (<**?)+■•■ 7 (») sad the potential can be written

U- Vo + **'"' • (1Q)

In this way It Is possible to define terms of different degree. The zero degree term la

V 0 m (ffcU>'= -jr M } < U > where M la the mass of the earth. This Is also the potential of a nonrotating, homogeneous sphere. i o

The first degree term Is

V - ji ffs'V( < * * * ) 4 t{Fs'a* Ydm'- M

xx* + yy* + zz* In rectangular coordinates: cos y ■ *-*■---- P • P* and relations between polar coordinates and rectangular coordinates are expressed by the following formulas

x - p * sin $ cos \

y * p . sinl7 sin \

z ■ p • cos $

Ue\ing the rectangular coordinates the first degree term of the potential, V, is

V, = {x.^X.'dbn'+ yfjfy'alni'-t- 7. ^ 2.'c L m 'J . (13)

The rectangular coordinates of the center of gravity, designated by x , y t and z , can be expressed; c c c v _ flfjt'J'*' _ fffx’dr * ' W * " ' ' M ------y, Mit ' = Hffu'dm' {Vt) 7 Ilf et**’ M — SIz''elm1 Jffz'e//)* ' C J K J m ' ~ M If the origin of the coordinates coincides with the center of the mass, meaning x « y„ « z ■ 0, the first degree term V, « 0. c c C 1 The second degree term is

V g m J j f y' T z ( < # * T ) c Im = jf“ ^ 11

and using the rectangular coordinates, it takes on the following form

£ 2 K * ) c b > >

+ + Blc.y]]JiC.'y'clrr^, -/■ (16)

6 9zfffyVet™'+ fff-z't'dnt'J .

The moments of inertia about the axes are given by the formulas:

About x-axis A — jJJ" ( + Z * ) q L*Y}

" y-axls £} = jjj(£'*+ z'*) oLfh' U 7 )

z-axia C =jff(x'Z+ y '2)oLn>' .

The products of inertia are given as follows;

1)m fffy 'zdm> E*fff ~2* K* dm 1 :i8)

f~- fff Z y ' c L r h * . The second degree term can now be expressed

i - f i { z t{B+c-24)+y2(c+A-2 B) + 'z2( A + B -2C ) +6yz'D*+6z z E + 6)cy'?:rJ t and using polar coordinates instead of rectangular ones, the formula is

\£« 4* 2^ (l-3cosl?)+(3Eoe$} + 3 (20)

3 D (B-A) c e s 2 ) ( -f

^ F s / n 2 \ “] s / n 2^ J . 12

If the earth were an exact ellipsoid of the revolution end the

mass were symmetrically disposed with respect to the center of mass,

then A would equal B. There is, however, some triaxiallty or some

other irregularities and A and B are not equ.l-

If the product of Inertia F ^ 0, there is present some ellipticity

of the equator. It is very possible that it is so because the earth

is not liquid throughout,. If the products of inertia D and E were not

zero, the axes of rotation and inertia would not coincide. This would

cause the axis of rotation to revolve around, the axis of Inertia. This

type of revolving actually occurs, but its amplitude suggests that the

first order second degree harmonics are absent (2).

The potential of the attraction now takes on the following form (30)

003 &) where 9 (22)

and

where a is the density. 13

1.12 Potential of Centrifugal Force

The potential of the centrifugal force, wh^n rectangular 12-2 2 coordinates are used, is z = — a. ',x + y ) 2 (23) 1 2 2 2 v and in the polar coordinate system z=-ajpsinv

1.13 Geopotential

As has been already mentioned, the geopotential is the potential, W, of the real earth. It is expressed by the following formula IV- i zfp* (I~ 3cjOS*t)')+ (cos*#- y CojV"

+ &) + S/” * 1 2V>

+ L (-%r) C Vm c o s e> >+ 5\/ni; sin t> ^j P (cos£) />■* i •* lllS where pQ = R - \/a2b f

Kg and are here constants which are connected tc m s s dis'tribut'on.

The terms and P ^ are incorporated in. the first par i of the expression^ therefore, should, net be. included in the last summation

The term PQ1 is also inadmissible in the last summation, as previously 4 1 explained in Section 1.11 .

1.14 Spheropotential

Spheropotential, U, is the potential of the spheroid, or the potential of the ideal earth. If the potential of the irregularities is designated by T, the following relation is obtained

W = U + T „ (25) 14

U is the regular part of the geopotentl&l and Is given by the follow­

ing formula

2 9 9 3 (2 6) J.* 6 0 Q O + '2j M

where t is geocentric latitude•for the spheroid p ■ r, where r is a

function of latitude. Comparing formulas 24 and 26 the potential of

the irregularities is quite obviously

T- n*2T L (-§?) * (yw cos v) +SV Ca» it), (27) Occasionally, as in triaxial spheroid, some terms from this summation

are included in regular part of the potential.

1.2 Gravity

Using the same definitions as

before, the following formula

provides an expression for the

gravity vector

o _ - T 2 W _ J (2 8)

and the exact value for gravity is

Fig. 4. Gravity Vector

* m s R s r Ik* M i n term under the square root alga is sad frequently the

following definition is used:

(30)

So* level is an equlpotentlal surface on which the gsopotsntial is constant colled tf. The surface which coincides with sea level and its o continuation under the continents and has a constant potential is called the geold.

In geodesy a spheroid is, by definition, a smooth equlpotentlal surface approximating the geold, usually a surface of revolution and being synmetrlcal in regard to the plane of equator. It coincides closely with an ellipsoid of revolution, is an equlpotentlal surface of the regularised earth, and has the same potential as that of the geold. Therefore the potential of the spheroid is

U « W * constant. o (31)

The measure of gravity associated with the spheroid is called normal or theoretical gravity. , Its vector expression is * «

(3a)

»' ti> h 11 'V*1 x*r0 if o n l y l e w r d . g r « . conl huaonlea hay. been lnoluded in U. In that case, the following formula for normal gravity-is obtained.

(33) 16 which can be derived by aid of formula 2 6 .

1.3 Qravlty Formula

1*31 Theoretical Qravlty Formula

Inserting the values

u o . = < » ) and at the pole, where cp = 90° and p = b

+ & # ) . ( 3 5 ) a ^ Vj Using flattening a » —— and b * a(l-a), one obtains from formulas A 3k and 35

(36) o C f / * o C J = (l+2oc)+T?fa -

The theoretical gravity at the equator is

Y* — —14^*1 (37) e & 5 %lo and at the pole

(30)

% * * > * • From 3fc, 35/ and 36 the following formulas are derived for these quantities

*«-•#/' + IS - # * # ) (39) V f/-*-2cC + 3ota- 17

Ftom these formulas It Is easy to obtaln(l9)

(UO) where

This Is called the extended Clairaut's formula. The original formula,

was derived by Clalraut In 174-9 and it expresses an Important relation between gravity and flattening. The B-value in formula 1*0 Is related to the shape of the spheroid. The following formula gives the distance from the center of the earth to a point on the surface of the spheroid, which has a potential U - WQ (19)

The corresponding distance to a point on the surface of the ellipsoid of revolution is (14)

(43)

The difference between those two distances is

i a [ol (§« -p) - 5 7 sIn 2 P ( w )

If the ellipsoid Is assumed to coincide with the spheroid, the differ ence Is zero and the following expression is obtained for &g

(U5) 18

The formula for theoretical gravity on a spheroid can be expressed by

(19)

(^6) e

which alsc can be written

l>7)

where cp is the geographical latitude and p = pg + p^„ If the geocentric

latitude is used instead of the geographic, the formula, has the form

X - t t {>+(?+y<*p-pH) s b * p - }. < « )

Using the formulas 26 and 33 another expression Is obtained

~ f { t + ^ ' 3c,iV/- (1- $ n Z t)

+ p e ? ( s m & ~

From formula k2 the value for r can be computed„ Using this value and

expressing q. in terms of a and p, the following formula is derived for

theoretical gravity

W ^ i'* V> ~U Zs^/3 ^ 9+zs/3+§§- } 7 (50) • {>+(p^ 7< - 3S)s V/ * (3Z- 7o?)sm v*}

Formulas 50 and 18 give the value of 6

ft - i (7a3- iwe + p ^) (51) 19

In practical compute,!ions the y and f3 values are computed from observed gravity anomalies, and the corresponding6 -jj-—\ term and the flattening value are derived by using formulas and 51* Of course, if the reference surface is given, £ and -jp -terms can be computed by aid of "formulas to, k'J, and 51 but the equator value has always to be computed from actual observations.

1.32. Computations of Gravity Formula

Until 1930 the most frequently used gravity formulas were the following ones

Helmert, 1901 973. 0 3 0 (/■*■ 0.0C5302s!r)S-0.00000'73inS2

Helmert, 1 9 1 5 f - 9 7 6 . M 2 ( l + 0 «>32gSsi»‘t

Bowie, 1917 978,o3<)(t-t-eXOOSZ^sh?^-0.0 0 0 0 0 7»r?2*P)Cm sec?

On the basis of extensive gravity material, Dr. Heiskanen in 1928 derived the following formula lf= 97Z.ovq((+0.oo5289s>'>7,P-OGoooo7'3i‘n'2y)cm (53) sec2

In 192k the International. Union of Geodesy and Geophysics (lUGG) adopted the International Ellipsoid as a reference system for triangu- lation and. astronomic observation?. The par .unetere of this ellipsoid are

a = 6378 388 meters

f = 1/2 9 7 ,0 20

In Stockholm, 1930, the International gravity formula corresponding to

the adopted International ellipsoid vas accepted as follows (5)

/ = ^16.on^O+0.oo52d8^iiK)cmi^5k)

The equator value of gravity is the same as Dr- Heiskanen derived and

the rest of the formula has been changed using the formulas 1+0 , 1+7,

and 51 to correspond to reference surface as requested by Cassinis-

Theoretical gravity is expressed by formula

t = 4 ( > + (* > V - ^ 2 9> ). w The difference between the observed gravity value and the theoretically computed value is called the gravity anomaly

Ag = g - 7,

If the adopted theoretical gravity formula is not correct, it causes systematical error in gravity anomalies at each latitude. A correction can be computed for the theoretical gravity formulas using the least square solution for

2 £ (Ag - A 7) = minimum j (55) where A 7 is the correction for theoretical gravity and can be expressed

aV*- a.4 + 9*

a 4 (56) 21

Making the following notations

A Yi

f t * a Y q A

z- - / a * |* y. A & f c + A t y ]

the error equation is as follows

X + us ‘in’cp+ -zsfo22 < P - A £ = — v . (57)

By checking the different terms in the formulas, it is easy to see

that the main terms are A y and y - A£ - Usually only these terms are 6 6 included in the error equations, which now take the form (11)

V (58)

where

Table 1 gives the results of several computations of this type

performed by the writer in 1956, using free air anomalies.

The fi-terms are given for various longitudinal zones in Table 1,

but these values are merely recordings of the partial results rather

than the corresponding values for the zones. As is well kncwn, In computations of the undulation values even for a single point, the gravity must be known around the whole world. In the same way the gravity field around the whole world should be used in computations of the flattening of single meridian or longitudinal zone (31J- 2?

Table 1. CORRECTIONS A 7e AND A£ TO THE INTERNATIONAL GRAVITY FORMULA Q 4;

Longitudinal A y e , using Zones l0 x l 0 squares AS • 103 e ■ 103 t o +- 1643 +1 1 .6 -0 .0 0 7 1 5.2813 45- 90 1007 -12.3 +0.0173 5.3057 90-135 935 + 5-9 -0.0133 5.2751 135-180 287 +11.3 -0.0 0 3 6 5-2848 0 - 45W 528 + 3-1 +0 .0 2 7 8 5.3162 1 1 -r 0 U VO 1052 - 5-5 +0 .0 1 0 7 5.2991 90-135 833 - 2 .5 +0.0042 5 .2926 135-180 192 - 6 .9 +0.0105 5.2989

0- 90E 2649 + 1 .8 +0.0034 5.2918 90-180 1221 + 7 .2 -0 .0 1 0 6 5.2 7 7 8 0 - 90W 1579 - 3-2 +0 .0 1 8 8 5.3072 90-180 1024 - 3-3 +0.0057 5-2941

0-180E 3870 + 4.5 -0.0015 5 .2 8 6 9 0-180W 2603 - 2.4 +0.0105 5.2 9 8 9 Northern 5369 - 0.7 +O.OO63 5.2947 Southern 1104 + 5 .6 +0.0137 5-3021

Whole Earth 6679 + 0 .6 +0 .0 0 5 0 5.2934

Using 5°x 5° Squares 0- 90E 316 + 1 .8 +0 .0 0 1 1 5 .2895 90-180 247 + 2 .8 -0 .0 0 2 8 5.2856 0- 90W 337 - 2 .2 +0.0075 5.2959 90-180 203 + 0.9 +0 .0 0 0 2 5 .2 8 8 9

Whole Earth .1103 + 0.7 +0 .0 0 1 8 5-2902 23

The new gravity formula from 1°X 1° mean anomalies is

9 7 8. 0W 6(\ + 0.0O5293y3M7'<&-0*00O0O39$in22 cP)Cfi9 (50)

Corresponding flattening value 1/297is very close to the flattening value 1/297.338 obtained by Bullard (I*) by using Bullen's density dis­ tribution and assuming hydrostatic equilibrium. Because the corrections were very small, the international gravity formula is used in Columbus,

Ohio, in the practical computations of the undulations of the geoid and the deflection of the vertical. 2 . GRAVITY MEASUREMENTS

2.1 Types of Gravity Measurements

Gravity at a point can be determined accurately only by measuring

the acceleration of gravity at the place. The measurements can be done

by different methods, for example, by (l) measuring the acceleration

of a falling body, (2 ) measuring the period of an oscillating body,

(3) balancing the mass by using elastic methods, (k) measuring the

pressure of gas. The two first ones belong to the dynamic group, the

two last ones to the static group.

All gravity measurements are either "absolute" or "relative"

determination of acceleration due to gravity. In absolute measurements

gravity is measured independently from other gravity measurements, while

in relative measurements, gravity differences are measured between a new

point and reference point, called a base station. Knowing the gravity

at the base station, the gravity value at the new point can be computed

using the measured gravity difference.

2.2 Absolute Gravity Measurements

Absolute determination of the gravity has been made by many

scientists, but usually only at one or two points in a country. It is

laborious to carry out these measurements, and specially trained people are required to perform them. The best accuracy obtained in absolute gravity measurements is about -3 mgal. The accuracy can be improved by repeating the observations several times.

2b Table 2. RECENT ABSOLUTE MEASUREMENTS OF GRAVITY (1*0)

Value of Gravity Difference from Site Author Method Value on Potsdam System Potsdam System Mgal Mgal Mgal

Geodiltlsches KUhnen and Furt- Reversible Inst., Potsdam wilngler (1906) Pendulum 98 263.3* 98 27^.00 -10.7 National Bureau of Standards, Heyl and Cook Reversible Washington, D.C. (1936) Pendulum 980 081.6 980 099.I -17-5 National Physical Laboratory, Clark Reversible 981 183.2 -0 .6 981 195.7 -12.5 Teddington (1939) Pendulum (2.13) All Union Scien­ tific Research Institute of Agaletzki and Reversible Metrology, Egorov Pendulum 981 918.7 ±0.4 981 930.8 -12.1 Leningrad Body Falling tl It Agaletzki in moving 981 921.5 -1.6 - 9-3 chamber If fl Martsinyak Falling Rod 981 923.3 -2 .2 - 7.5 Instituto de Geodesia, Reversible Buenos Aires Baglietto Pendulum 979 696. (8-3) 979 704.6 - 8 .6

Result of KUhnen and Furtwfingler!s measurements as revised by Berroth, 19^9* ro VJI The results of a few of the modern absolute gravity measurements

are given In Table 2. Most of these values are mean valueB of several repeated measurements at the point.

2.3 Relative Gravity Measurements

It is much easier to measure gravity differences between stations

than to make an absolute determination of gravity at any point. Even when pendulum apparatus is used, as in an absolute determination of the gravity, the accuracy of measuring the differences is much better than that in absolute determinations. Modern gravimeters have revolutionized the procedures of relative gravity measurements. The time needed for a measurement at one site has decreased from several hours to 10-15 minutes and measuring accuracy has Increased from -1 mgal to ^0.1 mgal. The reading accuracy of some of the modern gravimeters is ^0 .0 0 1 mgal, and most of them have the accuracy of -0 .0 1 mgal.

Relative gravity measurements have become very popular around the whole world because of these fast and accurate gravimeters, and because it is possible to use less experienced persons to operate the Instruments 3. GRAVITY REFERENCE SYSTEM

3.1 Potsdam System

For making relative gravity measurements, each nation has a funda­ mental reference station from which all gravity measurements originate.

These national gravity reference stations around the world form a net of stations which are tied to each other by measured gravity differences.

One of these stations is the Pendelsaal of the Geodetic Institute in

Potsdam, and this station has been selected as a world fundamental reference station for gravity measurements. All the gravity values which cure referred to this station are said to be in the Potsdam system.

This system is defined as follows: "The acceleration of free fall has the value, g = 981*27*4-00 gal (exactly) at the point midway between the pillars in the northeast corner of the Pendelsaal on the ground floor of the Geodetic Institute in Potsdam, having the geodetic coordinates

o * qp = 52 2 2.86N (approximately) o ^ X * 13 0^.06E (approximately)

h = 8 6.214- meters {iber NN (exactly)" (*4-0 )

Table 2 gives a representative sampling of modern absolute gravity values as well as corresponding value based on the Potsdam system and the difference between them. The differences are systematical, and between -7-5 to -17*5 mgal. Because in many scientific studies it is necessary to know that absolute gravity values are as nearly correct as possible, the International Union of Geodesy and Geophysics passed

27 28

In the XI General Assembly of the Union in Toronto, 1957/ the following

Resolution

The International Union of Geodesy and Geophysics Considering that for various physical purposes the best available value for the correction to the Potsdam system is frequently required and

Considering that definitive correction have not yet been determined:

Recommends that for the time being the correction to the Potsdam system be assumed to lie between -10 and -12 mgal on the basis of existing experimental data and connexion between absolute stations. ( )

This assumption, however, will not be a handicap in the computations where gravity anomalies are used, because the anomalies are differences between observed and theoretical gravity values and the formulas for theoretical gravity values have been derived from gravity values re­ ferred to Potsdam. If the value at Potsdam were corrected by -10 to

-12 mgal, then all observed values and consequently all theoretical gravity values would change in about the same amount; thus, the anomalies would remain practically unchanged.

3-2 Standardization Lines

Gravity differences are measured between Potsdam and the reference stations throughout the world, sometimes directly and sometimes via other stations. While tying the national reference stations to each other and to Potsdam, scientists noticed that different measuring instruments were giving systematically different answers between the same stations (5k and 5 5)• It was soon realized that the Instruments were calibrated in different places and occasionally by using different methods, i.e., the instrument had a different value for a mi Hi gal.

Therefore, it haB been necessary to establish long standardization

lines in order to secure unified calibration for all instruments

(1*0 and 56). One of these lines has been established for America.

It starts from Point Barrow in Alaska on the Arctic Ocean and goes

through Panama to Punta Arenas in Chile at the southern tip of South

America. One similar standardization line has been established for

Europe and Africa: from Hammerfest, Norway, to Capetown, South Africa.

One line has been proposed for Asia starting from a place located close to Salekhard, USSfy and going to Colombo in Ceylon; and one is measured

for the Pacific Ocean area from the island of Hokkaido, Japan, south­ ward through Australia and New Zealand to McMurdo Sound, Antarctica.

Along these standardization lines the interval between the stations

is selected so that the gravity difference is about 200 mgal. All of these stations have been or will be measured by several pendulum apparatuses, and between them, secondary stations are established which are measured by gravimeters.

After standardization lines are measured, it is relatively easy to calibrate any instrument on that line to give gravity differences in standard milligals. In this way there will be only one milligal instead of U.S. Coast and Geodetic Survey's, Woollard's, Dominion Observatory’s,

Martin's, Morelli's, Kneissl's, etc., milligals.

3 .3 Gravity Reference Stations

When the national reference stations were tied to each other and to

Potsdam several times, quite different gravity values were obtained for 30 the same station. One good example Is the well-known base station at

Cambridge, England, which recently received a +3.8 mgal correction (6).

Therefore it is necessary in world-wide studies that all national reference stations be brought into one uniform system. Dr. Woollard and his group (53) have tied many hundreds of base gravity stations to each other around the whole world. Also it is necessary to keep a card catalogue in which the different values are recorded for the reference station. A typical card has the following information (51): name, coordinates, and observed gravity of the station, year of observation or analysis, name of the observer or analyser, reference stations applied, and remarks, and a code number for each value of the reference station.

The values which the writer determined in Mapping and Charting

Research Laboratory, 1956, for the various national reference stations are given in Table 3 .

These values were selected without any additional adjustment computations but using the existing adjusted values and all observa­ tions. The list of the publications used in this work is too numerous to be presented here, but Morelli's (38), Hirvonen's (21), and

Woollard's (53) outstanding works should be mentioned.

In most cases there were no difficulties in selecting suitable base values because where many values existed there were usually the adjusted values or recent gravimeter ties were available. When the existing value was not referred directly to Potsdam, the reference station was checked first and the necessary correction applied to the reference station. After additional gravity ties have been established and new 31 Table 3 . GRAVITY VALUES OF THE REFERENCE STATIONS

Name 2 t K h, m &, Sal 0 1 Aberdeen 5T°o 8!9 - 2 05■7 1 8 .7 981.6993 Akureyri 65 1+0 .3 - 30 1+7-0 1+7 982.31+2 Arkangelsk 61+ 35-0 1+0 3 0 .0 5 982.272 Athens 37 58.3 28 1+3.2 95 980.055 Baku kO 22.0 1+9 50.3 - 1+ 98O.O83 Bangkok 13 1+5*1 100 2 9 .7 3.1+ 978.313 Basel, Binn. k7 32.5 7 35.1 3H+.1+ 980.7637 Basel, Bern. 1+7 33.6 7 3l+.8 277 980.7775 Batavia - 6 0 5 .6 106 5 3 .0 6 978.159 Besanpon hi 15.0 5 59-3 311 980.752 Beyrouth 33 51-0 35 3 0 .0 55 979.690 Bogota, I.G.M. k 38.5 - 71+ 03.9 2592 9 7 7.1+01+ Bologna, I.G.U. 1*1+ 2 9 .8 11 2 1 .3 50 9 8 0.1+511 Bordeaux 1+1+ 5 0 .7 - 0 31.3 71 9 8 0.572 Bouzareah 36 1+7 .8 30 2 .1 31+8 979.920 Breteuil 1+8 1+9 .7 2 13.2 6 5 .9 980.9396 Brisbane -27 2 8 .0 153 0 2 .0 15.2 979.1711+ Brno 1+9 1 2.1+ 16 3 6.O 235 9 8 0.961 Bucarest 1+1+ 2k. 6 26 0 6 .8 83 980.553 Budapest 1+7 2 8 .8 19 03.2 106 980.8533 Buenos Aires -3*+ 3*+.2 - 58 2 6 .2 12 979.705 Cambridge 52 1 2 .9 0 0 5 .8 25 981.2688 Cape Town -33 5 6 .1 18 2 8 .7 38.1+ 979-61+75 Caracas 10 3 0 .6 - 66 5 5 .8 910.7 978.0687 Catania 37 30.1 15 01+.5 1+6 .2 980.01+59 Colombo, Obs. 6 56 79 *9 7 9 7 8.131+5 Copenhagen, Budd. Pil. I 55 W - 3 12 3 0 .1 1+1+.7 981.5575 Cork 51 53.5 - 8 29-5 1 8 .9 9 8 1.21+63 Cracow 50 0 3 .9 19 57-6 205 981.0528 Danzig 51+ 2 2 .1 18 37-0 2 1.1+ 9 8 1.1+1+87 De Blit 52 0 6 .2 5 10.7 2 .1 981.2688 Dehra Dun 30 19.1+ 78 0 3 .2 683 979-063 Dublin, Dunsink 53 2 3 .2 - 6 2 0 .3 8 0 .8 981.3896 32

Table 3 (cont'd)

Name 2 h, m g> 6*1 ■ t Edinburgh ’55.4 - 3(0 7 .6 129.4 981.5838 Florence, Astr. Obs. 43 45.2 11 16.5 184 980.4896 Florence, I.G.M. 43 46.8 11 15.2 48 980.501 Galvay 53 1 6 .6 - 9 03-7 8.3 981.3657 Genoa 44 2 5.I 8 55.3 97-5 980.5572 Gloucester 51 52.0 - 2 14.8 15.0 981.2277 Greenwich 51 2 8 .6 000.3 47 981.1898 Guam 13 2 6 .8 144 3 9 .8 978.538 Hankow 30 33 114 17 19 979-3643 Hanoi 21 01.4 105 51.3 11 978.6856 Helsinki, Obs. 60 09-7 24 57-3 29 981.9144 Helsinki, Phys. New 60 10.6 24 57-5 20.5 981.9158 Helwaa 29 51-5 31 20.4 115 979.2948 Hong Kong 22 1 8 .1 114 10.3 33 978.7721 Honolulu 21 1 8 .4 -157 5 2 .0 978.9410 Irkutsk 52 16.5 104 16.5 462 981.096 Ivigtut 61 1 1 .8 - 48 11-5 32.7 981.96OO Johannesburg -26 11-5 28 0 1 .8 1755-0 978.5514 Karlsruhe 49 0 0 .7 8 24.7 114 980.9558 Kaunas 54 33-7 23 52.4 70.3 981.4911 Kazan, Anc. Obs. 55 47.4 49 07.3 75-6 981.5587 Kazan, Enguh Obs. 55 50.3 48 4 9.I 94 981.563

Kew 51 2 8 .1 - 0 1 8 .8 5 981.2009 Khartoum 15 36.5 32 32.5 381.3 978.3063 Kotelnich 58 18.6 48 2 1 .0 112 981.771 Kyoto 35 0 1 .6 135 47.2 6l .6 979.721 La Plata -34 54.5 - 57 55.9 1 0 .6 979-748 Leningrad, Inst. Met. 59 55-1 30 19-0 4 981.9317 Leningrad, Ast. Obs. 59 56.5 30 17.7 3 981.9324 Leningrad, New Ast. Obs. 59 5 6.2 30 2 0 .9 4 981.9339 Lima -12 01.5 - 77 0 2 .2 144 978.289 Lisbon 38 42.5 - 9 12.2 75 980.0876 33

Table 3 (cont •d)

Name \ h, m g> gal _ 2. * _ a Madrid W 24.5 - 3* 41.2 656 979.9821 Malta, Dock No. 3 35 5 2 .8 14 3 1 .0 0 979.8871 Manila 14 34.7 120 58.9 4 978.3642 Melbourne -37 4 7 .2 144 53.5 49.3 979.981 Meudon 48 48.3 2 13.9 130 980.919 Milan 45 2 8 .7 9 13.7 116 980.5645 Moscow, Obs. 55 45.3 37 34.3 145 981.5598 Moscow, Seism. 55 44.3 37 37.4 124 981.5468 Milne hen 48 0 8 .7 11 3 6 .6 525 980.7327 Newcastle 51+ 48.7 - 1 36.9 57.4 981.5100 Noumea -22 18.2 16627.6 978.8824 Obi-Gann 38 4 2 .7 69 42.3 1333 979.5363 Odessa k6 2 8 .6 30 45-5 51 980.767 Omsk 51* 59-1 73 2 2 .0 79 981.478 Oslo 59 55-2 10 46.6 30.4 981.9284 Ottawa 45 23-6 75 43-0 83 980.6218 Padova 45 24.0 11 52.3 14.3 980.6586 Paris, Anc. Salle 48 5 0 .2 2 20.3 61 980.9430 Pola 44 5 1 .8 13 50.7 28 980.621 Poltava 49 36 34 34 146 981.0067 Potsdam 52 2 2 .8 6 13 04.06 86.24 981.2740 Poznan 52 2 4 .7 1655.7 57 981.2635 Pretoria -25 4 5 .1 28 11.4 407.5 978.630 Pulkowo 59 46.3 30 1 9 .6 75.2 981.8993 Riga 565 7 .1 24 0 7 .0 4.6 981.6589 Rome 41 5 3 .5 12 29.7 49.3 980.3663 Rovaniemi 662 9 .9 25 43-9 88.3 982.3603 Saigon 10 4 6 .9 10641.4 12 978.228 Seonl 22 0 5 .5 77 29 619 978.622 Sevastopol 44 37-3 33 31.4 4 980.667 Shrewsbury 52 41.4 2 43-5 73.5 981.3213 Sidney -33 53-2 151 11.3 39-3 979-6844 Table 3 (cont *d)

Name X h, m Si gal ■ Singapore 1(5l8! 3 103'V 8 17 978.087 Sligo 54 1 6 .2 - 8 2 8 .3 1 2 .1 981.4661 Southampton 50 54.8 - 1 24.2 24 981.1264 Stockholm, R. A. Kartverk 59 19-7 18 02.9 8.6 981.8467 Strasbourg 1+8 35.0 7 46.1 137 980.897 Stuttgart 1+8 1+6 .9 9 10.5 247 980.888 Suez 29 56 32 33.3 2 979.305 Sverdlovsk 56 1+9 .6 60 39-2 252 981.622 Tacubaya 19 21+.3 - 99 11.7 2297.9 977.9415 Tallinn 59 26.3 24 44.4 1+2.5 981.8400 Tananarive -18 55.0 47 33.1 1381.2 978.2265 Tashkent 1+1 19-5 69 17.7 478 980.081 Teddlngton 51 25.2 - 0 20.3 9.3 981.1963 Tiflis 1+1 1+3.1 44 4 7 .8 406 980.1777 Tokyo 35 1+2.6 139 46.0 18 979.801 Toulon 1+3 07.1 6 32.7 0 980.490 Trieste, Oeoph. Obs. 1+5 38.0 13 45.1 4.7 980.6647 Turin 1+5 01+.1 7 41.8 233 980.537 Uccle 50 1+7.9 4 21.5 102 961.1320 Vienna 1+8 12.7 16 21.5 183 980.8595 Vienna, Stemwarte 1+8 13.9 1620.4 236 960.8526 Vladivostok 1+3 0 6 .9 131 53-5 23 9 8 0.486 Warsaw, B.N.M. 52 14.1+ 21 00.2 111 961.2396 Warsaw,Obs. 52 13.1 21 01.8 109 981.239 Washington, D.C. 38 53.6 - 77 02.0 0 980.1190 Zi-Ka-Wei 31 11.5 121 25.7 7 979-440 Zurich 1+7 22.7 8 33.2 463 980.664 35

computations have been carried out using the standard milligal, the values of the reference stations may change slightly, but the changes will seldom, if ever, exceed one milligal.

3.4 Relating and Recording of Gravity Data

It has been already mentioned that gravity measurements made in any

country are referred to a national reference station. If the observed gravity values were given in gravity differences from reference stations,

it would be easy to compute the observed gravity value at the point after the best fitting gravity value in the Potsdam system ha^ been

selected for the reference station. Usually, however, the literature gives not the gravity differences but the computed, relative, observed gravity values. In order to be able to bring these observed values into the Potsdam system, the used reference station and its gravity value to which computations are referred must be known. Therefore, it is necessary that in the lists of the gravity observations--published or unpublished— the following data be given and recorded; latitude, longitude, and elevation of the station; observed gravity value at the station; name, location, and the gravity value of the reference station; and all computed anomalies with indication of assumptions made in these computations.

After the best value has been determined for the national reference station, the gravity observations referring to this station should be adjusted to correspond to the selected value. In the Mapping and

Charting Research Laboratory the reference station and its value have been always recorded along with the observed gravity whenever it has been possible. For convenience in using gravity material, the informa­ tion mentioned above has been recorded on cards for each l°x 1° "square" which corresponds to the area Aqp = 1°, ■ 1°. If there were a single card for each "square^” there would be 64,800 cards in the catalogue.

Many of the "squares" are empty and without a card, but many have more than one card. In this form all gravity information is readily accessible for any square in the world. This card catalogue is easy to keep up-to-date whenever new gravity information Is received, The information has been recorded on the card as it has been given in the source. Before using any observed gravity value it is necessary to check the reference, the station value to which it is referred, and, if there are any differences between the referred and adopted values, to correct the observed gravity to correspond to the adopted value.

Because the number of gravity stations is increasing very rapidly, this card catalogue is becoming more difficult to keep up-to-date, and when the gravity values are used in same computation, they must alwayE first be copied from the cards by hand. Therefore it is suggested that each station should be recorded on an IBM card,- in this way the informa­ tion will be ready to be used in computation or reproduced or printed, whichever Is desired, without new copying work. The writer suggests that the following information be recorded on IBM cards in the Indicated locations.

1. Columns 1-2: collecting agency; for example,

10 * Ohio‘State University

11 » U.S. Navy Hydrographic Office

12 - Aeronautical Chart and Information Center, etc. Columns 3-6: Month and year data were received*.

Columns 7-3: Tear of observation.

Columns 9-10: Classification of data; for example,

01 s unclassified material taken from printed publications

02 - unclassified material taken from unpublished sources

03 = received from a private company and may be released

only with company's permission.

Columns 11-16: Latitude. Columns 11-12 give the degree,

13-11*, the minutes, and 15-16 gives tenths and hundredths of

minutes. Positive means northern latitude and negative means

southern latitude.

Columns 17-21*: Longitude. Column 17 is zero, columns 18-20

give the degrees, columns 21-22 give the minutes, and columns

23-21* give the tenths and hundredths of minutes. Positive sign means that longitude is east from Greenwich and negative sign

means the longitude is read west from Greenwich.

Column 25: Unit of elevation which is given in columns 26-31,

for example,

1 - meters

2 = feet

3 * fathoms

Columns 26-31: Elevation of the station. The decimal point

is between columns 30 and 31-

Column 32: ^Ype of elevation given in columns 26-31.

0 * positive land elevation

9 = depth of the ocean 38

3 = negative land elevation

7 - depth of the lake

10. Columns 33-40: Observed gravity. When observed gravity is

given In gals, the decimal point is between columns 35 and 36.

11. Columns 41-44: Source of information. Each agency would have

a list of publications from which the gravity anomalies are

taken and each publication would be coded by number. For

example,

0256 - Gulatee, B. L.: Survey of India Technical Report

1952, Part III, Geodetic Work, Chapter V, Gravity,

Dehra Dun.

So far there are about four hundred publications listed this

way at OSU.

12. Columns 45-48: Free air anomalies in 0.1 mgal.

13* Column 4 9: Information concerning terrain correction.

0 = no terrain correction added to Bouguer anomaly

1 = terrain correction added ■D 14. Columns 50-52: Density in g/cm used in computation of

Bouguer correction. Decimal point is between columns 50 and 51*

1 5. Columns 53-56: Bouguer anomalies in 0.1 mgal.

16. Columns 57-60: Reference base station. There exists a list of

base stations around the whole world, and a code for each of

the stations will be assigned. A list of the codes shall be

available at each agency.

17• Columns 61-62: Value of the reference station to which

observed gravity is referred. The same base station might have over ten different values depending upon which way the

station has been tied to Potsdam. Or absolute measurements

might have been performed at the station.

18. Columns 63-61*: Apparatus vised in the determination of the

observed gravity. For example,

10 3 pendulum

20 = land gravimeter

21 = underwater gravimeter

22 * sea gravimeter (surface)

23 *= airborne gravimeter

1 9. Columns 65-66: Gravity formula used in computations.

01 = International gravity formula, 1930

02 = Helmert's formula, 1901

03 = Krassowski's formula, 1938

20. Columns 67-6 8: Type of anomaly given in columns 69-7 2. For

example,

10 = Airy-Helskanen lsostatic anomaly, T ■ 30 km

11 = Airy-Helskanen lsostatic anomaly, T ■ 20 km

12 * Airy-Helskanen isostatic anomaly, T = kO km

13 B Airy-Heiskanen isostatic anomaly, T « 60 km

20 = Vening Meinesz regional, T * 30 km and R = 29.05 km

21. Columns 69-7 2: The anomaly Identified in columns 67-68 in

0 .1 mgal.

22. Columns 73-7^: Classification of elevation determination.

01 » bench mark

02 “ altimeter elevation k o

03 = trigonometric elevation

Ok » sounding

05 = radar elevation

23* Columns 75-7 6: Identification of institute, company, or any

larger surveying organization which has made the observationa

Only the organizations which have done quite a bit of gravity

surveys should be Identified. Code numbers have not been set

up yet, but it means a list of organizations with given

numbers.

2k. Columns 77-7 8: Accuracy of the observations in 0.1 mgal.

This will be recorded if it is stated in the publication from

where the observation has been taken.

25* Columns 79-80: Not designed for anything special and can be

used for anything worthwhile recording.

If all collecting agencies used this type of system, the exchange of data would be much easier. k. GRAVITY ANOMALIES

l*.l Type of Anomalies

Observed gravity values themselves are very difficult to compare,

because there are so many reasons differences In rallies may exist.

Usually the values themselves are compared first with the theoretical

gravity value at the same place, and the differences, which are called

gravity anomalies, are then compared.

Theoretical gravity at any place on the mathematical reference

surface is given by the gravity formula. As shown earlier there are

several different gravity formulas but the most widely used 1b tbfc

International Gravity Formula

97S. ovg Q ooooo59&r>QJ>q*)c»t sec* (3*0

The gravity formula gives the theoretical gravity on the reference

surface, but the observation has been made on the physical surface of

the earth. Therefore they must be transferred to the same surface

before computation can be made. There are several types of gravity

anomalies depending on the method which is used for reducing the

observed gravity to sea level or computing the theoretical gravity

value for the observation point. The three principal methods are (3*0:

(l) free air method, (2) Bouguer method, (3) method of generalised

subsurface densities, e.g., the lsostatic method.

In the free air method only the difference in distance from sea

level to the observation point is taken Into account. If the earth is taken as a homogeneous sphere vith mass M, the attraction g at any point at a distance p from the center Is

where f Is the constant of gravitation. By differentiating with respect to p the following rate of change of gravity is obtained

. iiSi.. Sa (61) d P - j - p ^ o When the values g - 980 cm/sec and p ■ 6371 km are used, the rate is

- -0.0003076 Gals per meter d p

Because the radius of the earth is not constant, the rate changes as a function of the latitude. A good average value for practical work is 0.3036 mgal per meter. However, if more accurate values are desired, the latitudinal change and the second order term of height have to be taken into account, and the formula for free air reduction is the following (25) 0.3085507 h+(X0002207h cos2yi

-O.ooooooshcof‘2 - O .o o o o v ttw h * (&) - 0 .0 0 0 0 0 0 // h \ o s 2 91 C m s e c 2 , where elevation, h, is in kilometers.

In the free air reduction only the distance between the station and Bea level has been taken into account. In simple Bouguer reduction the effect of the masses between sea level and the station has been 43 computed as the effect of a horizontal Infinite slab with a thickness equal to the elevation of the station above sea level. The general formula for the attraction of the plat

a = density .

dh = thickness of the plate

a = radius of the plate and t * distance from the point to

the plate. Fig. 5• Attraction of a Plate

If^t -# 0 and) a -» *>, the formula takes the following form

(64) which is the effect of the horizontal infinite slab and is the formula for simple Bouguer reduction. If f = 6 .6 7 3 10 cm g~ sec and a - 2 .6 7 g • cm , the reduction formula gets the following form

-/ = Q . 1119 d h mgal • m (65)

The topography between the station and sea level is not actually in the form of a horizontal infinite slab; therefore! the simple Bouguer reduction needs corrections because of the curvature of the earth and because the actual surface is not as smooth as a slab! but irregular.

Using the given formula the effect of the curvature can be computed for different thicknesses of the layer. Billiard (3) has computed correc­ tions! which are given in Table 4. 1*1*.

Table k, CURVATURE CORRECTION TO BOUGUER ANOMALY

Station Elevation h - km 0 0.5 1 .0 1.5 2.0 2.5 3.0 3-5 k.O l*.5 5 .0

Correction 10”1 mgal 0 -6 -12 -15 -17 -17 -15 -1 1 -6 +2 *-10

These corrections also remove the effect of the part of the Bouguer plate outside Hayford zone 0. This outside effect is very small and is usually included in the isostatic reduction tables(lO).

Topographic correction or terrain correction, which is always positive, can be obtained by using the topographic maps and correction tables which are computed using a modification of formula 63 for attraction of the plate (3).

In isostatic reduction methods, sub-surface densities are generalized. This is like a model earth with the gravity values corrected to correspond to the assumed model. Detailed discussions of the isostatic reductions are not Included here, because free air anomalies are used in all computations included in this paper.

k.2 Mean Free Air Anomalies

A mean anomaly Is defined as an anomaly value which is representa­ tive for the whole area in question. Mean anomalies are generally obtained for l°x 1° "squares" and 5°X 5° "squares." In Chapter 6.22 there will be an explanation of how these mean anomalies are used.

Actually, a l°x 1° "square" is a trapezoid with sides A

^ = 1°. The same is true for 5°X 5° "squares," where the sides are Arp s 5^ and » 5°. The free air anomaly can be considered as an isostatic anomaly, where the thickness of the earth's crust is equal to zero (13)* Because the isostatic compensation does not occur at sea level but at a depth

of 20 to 50 km below it, the free air anomalies are locally correlated to the elevations of the observation points or to the relief of the area (13)* Thus a mountain station has usually a greater (algebraically)

free air anomaly than a neighboring valley station. This correlation is true also at sea, greater depths giving algebraically smaller anoma­ lies than more shallow neighboring waters. This correlation means that when a free air anomaly map is drawn, the topography of the area must also be considered. The free air anomaly can approximately be expressed by the simple formula

Agf * a + b • h (66) where h is the elevation of the station and a as well as b are variables, independent of h. The a-value varies from area to area, but is more regular than Ag^.. Sample graphs from California, given in I 1 figures 6-9,show that the b value is nearly constant in all 10 x 10 squares. The a-value varies from one square to another. In figure 10 all gravity anomalies, used in figures 6-9> are plotted on the same graph. Of course the values no longer fall so well along one straight line. It is still possible to draw a line giving average values for the anomalies at various elevations in the square. In order to draw the best free air anomaly map for this square, one must first construct a map giving a-values and possibly a map giving b-values. By aid of these and a topographic map the final free air anomaly map is constructed. Fig. -6. Relationship "between. Free Air Anomalies and Station Elevations Station "between.and Relationship Anomalies Air Fig. Free -6. METERS 2000 250Qj 300QJ IOOOL I500_ 500_ -200 _ i i i i i i i i i | i i ii i i i i i ] f i i T i i i i i ; i i i i i i r i " in the Area 36°N-36°3 Area the in -100

MILLIGALS o 0 ' n

and ll8°W-ll8°30'w and iif* iintf iif* 100 o A A A A ® e 1 isr*n »•*»<>•

n 200 k6 Fig. 7* Relationship between Free Air Anomalies and Station Elevations Station and Anomalies Air Free between Relationship 7* Fig. 2000 METERS 2500. 3000: 1000 I500_ 500_ -200 _ _ I I ) I 1 I I I I I I —r i— I i I I I I I 1 I I I r 1 I I I ) I I I I I in the Area 36°N-36°3 Area the in a, 9 -100 O ° n 9

a ■"B I MILLIGALS o 9 9 ' 0 n adlQ3 W ° 9 H - W llQ°30 and

II ' T I| I I I 100 it** ii** w ut*« so' 1 I I I — — N * t 9 St* St* so' ST* H ST*

<“ 1 200 Fig. 3. Relationship between Free Air Anomalies and Station Elevations Station and Anomalies Air Free between Relationship 3. Fig.

METERS 2000 3000. 2500_ I00CL 1500, 500_ , 200 I I I | I I Tl i [ i i i i | I I TT l TT I I I I l | i i i i— l i i i I [ I i I I "T~l I I I | I I I I I I I — n in the Area 36°3 Area the in -100 o ' n - MILLIGALS 37°N and 118°W-U8°30 *W 118°W-U8°30 and 37°N 0

100 iis W ' ua*w T“ l 200 Q h Fig. S'• Relationship between Free Air Anomalies and Station Elevations Station and Anomalies Air Free between Relationship S'• Fig. METERS 2000 2500. 3000J K> I500_ 500_ 00 _ , 200 ii i i i i i i i I i i i i i i T i i T i q i r i i'i i | ) i i"T n'l i in the Area 36°3o'n-37°N and ll and 36°3o'n-37°N Area the in -100

MILLIGALS 0

A A 8°30 A & A W- n«*stf iia«w » w i 100 119 °W ‘ 0 » * M S7*N

r 200 -1

9 k Pig. 10. Relationship between Tree Air Anomalies and Station Elevations Station and Anomalies Air Tree between Relationship 10. Pig.

METERS 2000 3000_ 2500_ I00CL I500_ 500_ -200 _ r I I I I I I I I I' I I I I [ I I I I I I I I I I I I I r r -too in the Area 36°N-37°N and H8°W-119°W and 36°N-37°N Area the in O O O o 9 4 A <9 O MILLIGALS 6 > o O 0

1 I ‘T~l O A a A O T -TTTT|I r i l ' l ' I T T 0 A A A A O I*W l*0 IIB*W ll**30' II** W 100 © 37*N 'N I S ’ 0 3 * « 3 i“ l 200 50 j*.44*N,A«I07#W

tooo- -

iooo-■

too- -

so -SO ♦SO *100 *180 *800 K h / h M

1000-

J- 1---- 1---- 1---- 1---- I04»g*l ---- 1— 1 * OfNtMj •50 o *5b *>00 *iqo*180 +tOO+too tSO -WO ♦So *ioo *80 two *180 *SOO ♦£ Fig. IX. Relationships between Free Air Anomalies and Station Elevations in Various Parts of the World ^ 52

In thle l°x 1° square in California there were many anomaly values available, but this is not always the case around the whole world.

Therefore, it is seldom possible to construct the a-value map for all squares with observations, but nevertheless the free air anomalies should still be reduced to mean elevation level in the square. The best way to proceed in this case is to assume the a-value to be constant for the l°x 1° square. If there are two gravity anomalies with an appreci­ able difference in elevation inside the l°x 1° square, it is possible by graphical or numerical method to determine the value of b by keeping the a-value a constant. Figure 11 shows samples of graphical solution o o of b, elevation correlation, In 1 x 1 squares from various parts of the world. In these samples the b-value is nearly constant and almost the same as the constant in Bouguer reduction, which means that the a-value closely approximates the Bouguer anomaly. There are additional results of elevation correlation analyses around the world in table 5* o o The coordinates refer to northwest corner of the l x l square in question. The squares were selected so that the difference between the highest and lowest station elevation in the square would be at least

250 meters. If an accurate mean elevation of the square was available, it was also recorded in the table.

According to this table the b-values vary from place to place.

There seem to be some areas where the b-value is systematically above or below the mean b-value. The average of the 131 b-values given in the table is bffi = 0.110 mgal/meter. This value is close to the constant used in Bouguer reduction as expected because of the lsostatic equi-

■3 librium. If the density is taken 2 .6 7 g/cm , the Bouguer constant 53

Table 5- ELEVATION CORRELATION ANALYSES Highest Station Station Mean ® X b Elevation Elevation Elevation Remarks mgal/meter m m m

46°N 11^ 0.148 130 810 ^7 12 0 .0 8 0 80 2110 47 11 0.119 180 2960

14 78 0 .1 3 8 590 870 Poor Corr. (4 points) 23 81 0 .1 2 0 370 610 Poor Corr. 23 80 0 .1 1 8 340 610 22 80 0.123 260 625 23 79 0 .1 6 0 360 850 22 79 0 .1 7 0 255 550 23 78 0 .1 1 0 335 1065 22 78 0.147 315 780 22 77 0 .1 1 0 305 760 23 76 0 .1 2 7 170 545 22 76 0 .1 0 0 240 635 23 75 0 .1 0 8 150 590 Poor Corr. (many points) 22 75 0 .1 1 8 185 425 21 75 0 .0 6 2 190 660 Poor Corr. (many points) 21 73 0.055 25 615 21 74 0.093 255 640 20 73 0.105 5 715 19 73 0 .1 0 0 6 620 26 73 0 .1 1 8 215 740 32 76 O.O67 250 1225 Popr Corr. (many points) 31 76 0.105 235 670 35 75 0 .1 1 4 1965 4055 Very Good Corr. (many points) 34 75 0 .1 1 3 685 2805 26°N 91°E 0.124 50 1525 Excellent Corr. Table 5 (cont'd)

Lowest Highest Station Station Mean

26°N 93°E O .136 135 685 Poor Corr. 26 92 0 .1 3 6 85 415 Poor Corr. 28 86 0 .0 8 8 415 4510 Very Good Corr. (many points) 27 86 0.105 50 2120 Very Good Corr. (many points) 28 87 0.109 275 2560 Very Good Corr. (many points) 27 87 0 .1 1 0 60 1775 Very Good Corr. (many points) 27 88 0 .1 1 0 60 1500 33 75 0 .0 9 8 270 2010 34 74 0 .1 3 8 1590 3340 33 74 0 .0 9 2 295 785 33 76 0.097 525 1495 Very Good Corr. 35 74 0.095 1560 2485 Poor Corr. (many points) 31 77 0 .1 1 1 130 2115 Excellent Corr. 31°N 78°e 0.113 360 2165 Excellent Corr.

42°S 171°B 0 .0 6 0 10 735 Poor Corr. (many points) 43 170 0.040 25 745 Poor Corr. 43 171 0.084 100 840 43 172 0 .0 8 2 10 250 42 172 O.O76 130 865 44 169 O.O97 270 550 Poor Corr. (4 points) 44 170 0 .0 7 0 70 720 Very Poor Corr. 45 169 0 .0 8 2 50 430 Very Poor Corr. 45 170 0.046 5 535 Very Poor Corr. 41 171 0 .1 0 8 5 585 Poor Corr. 4l°S 172°E 0 .0 5 6 5 390 Very Poor Corr. 55 Table 5 (cont'd)

Lowest Highest Station Station Mean 9 b Elevation Elevation Elevation Remarks mgal/meter m m m

32°S 21°E 0 .100 535 1455 Excellent Corr. 33 21 0.093 320 535 33 22 0 .100 3 855 Poor Corr. 32 20 0.105 1005 1560 30 18 0.103 225 1385 Excellent Corr. 30 19 0.103 480 970 31 18 0 .110 100 395 30 IT 0.100 5 755 2l<. 26 0 .118 880 1120 Very Poor Corr. (many points) 26 31 0.049 200 1555 22°S 30°E 0.117 340 740 Poor Corr. (many points)

50°N I15°w 0 .150 1055 1360 1660 50 116 0.180 735 1200 1425 50 117 0.160 535 1005 1600 50 120 0.140 285 1260 1195 50 121 0 .125 525 1340 1465 50 122 0.080 6 770 1015 51 116 0 .0 9 0 805 1475 1925 51 118 0 .1 0 0 435 1020 1675 51 119 0.115 355 1190 1345 51 120 0.245 345 695 955 51 121 0.115 350 925 1110 51 122 0 .162 175 525 1250 52 118 0.165 690 955 1925 53 H 9 0 .160 580 1220 1845 50 118 0.155 425 1400 1500 50 119 0 .150 430 1380 1310 51°N 117°W 0.120 540 1350 1825 56

Table 5 (cont'd.)

Lowest Highest Station Station Mean 9 \ b Elevation Elevation Elevation Remarks mgal/meter m m m

52°N 11T°W O.lUO 786 2025 2075 52 119 0.1 1 2 0 1900 1630 53 118 0.117 1225 2010 2150 ^7 88 0 .2 0 0 190 1*50 270 Poor Corr. (many points) kj 89 O .287 190 515 1*05 1*7 90 0.293 190 535 370 Very Poor Corr. (many points) kk 7** 0.180 1*5 505 265 kk 72 0 .1 3 8 130 1*05 275 Poor Corr. (many points) kk 73 O.I7O 135 1*90 1*30

62 11*9 0 .165 80 61*0 1150 Fair Corr. (1* points) 61 150 0 .11*0 10 325 600 Poor Corr. (many points) 62 11*3 0 .01*5 1*70 620 1900 2 Points 62 11*3 0.055 15 71*0 1600 Poor Corr. (3 points) 62 11*8 0 .107 1*20 1005 1500 63 150 0 .0 8 8 175 505 870 61* H*3 0 .100 1*95 955 715 63 ll*5 0 .0 9 0 565 805 1150 61* 11*6 0 .080 385 995 1200 63 ll*6 0 .112 31*5 825 715 65 11*2 0 .0 9 2 225 1205 750 66 11*6 0.100 1*1*0 1110 570 66 11*7 0.090 385 675 700 61* 11*9 0 .0 8 0 1*10 670 1035 Poor Corr. (8 points) 61* 150 0.105 315 710 1020 Poor Corr. (7 points) 62°N li*6°w 0.135 80 71*0 1000 Poor Corr. (10 points) 57

Table 5 (cont'd)

Lowest Highest Station Station Mean

70°N 150UW 0 .21*1* 200 580 Excellent Corr. 70 159 0 .21*0 200 61*0 Excellent Corr. 70 160 0 .21*0 155 510 Excellent Corr. 70 161 0 .2 5 2 200 635 Excellent Corr. 70 162 0 .2 5 2 21*0 1*75 Excellent Corr.

11 67 0.10k 85 1270 11 68 0 .1 1 6 1*50 121*5

9 7* 0 .1 1 7 33 308 660 Poor Corr. 8 71* 0 .0 5 1 1001* 1597 560 Poor Corr. 0 73 0 .1 1 1 826 3277 1815 7 7** 0 .1 1 9 158 21*67 1220 7 73 0 .1 0 2 1100 3820 2615 6 77 0 .2 0 7 32 1*92 71*5 6 75 0 .0 7 6 220 2611 920 6 71* 0 .0 8 2 1576 2931* 2575 5 77 0.157 7 1631* 867 5 76 0 .1 1 2 1*87 3261* 2051* 5 75 0.093 21*1 2600 1670 l* 77 0 .0 9 2 151* 1178 1530 3 ' 78 0.075 0 983 620 Poor Corr. 3 77 0 .0 7 0 967 3367 2385 3 76 0.193 1*28 1028 1815 2 79 0.159 0 1055 270 2 78 0.057 71 3086 2015 2 77 0.095 336 211*1 1785 1°W 78°W 0.097 2668 3153 2512 58

Is 0.1119 mgal/meter. In ocean areas there are very few l°x 1° squares

where many gravity anomalies are available. All current material at

sea indicates that the b-value is close to -O.069 mgal/meter.

The writer has used the following procedures to determine the o o mean free air anomalies of 1 x 1 squares in the World-Wide Gravity

Project:

1. If an area has a good coverage of gravity observations, maps

of a- and b-values are prepared. Mean values, and bm , are estimated I I for the 10 x 10 squares. The mean elevation, hm > is also estimated

for the same squares. The mean free air anomaly of 10 x 10 square

is now

Ag = a + b h nm m m m

0 0 1 * The mean anomaly of 1 X 1 square is a mean of 10 x 10 mean anomalies

inside the square. Sometimes a Bouguer anomaly map is used instead of

the a-value map and bffl is taken as the constant (0 .1119) used in

Bouguer reduction.

2. If an area has many stations at different elevations, but not

enough information for making a map showing a-values in the square, the

b-value is determined graphically and the a-value is kept constant for the square. Using the obtained b-value the anomalies are reduced to mean elevation of the square and the mean of reduced values is the mean anomaly value for the square. 59

3. If there are only a few observations in the square and only

small elevation differences which do not allow the determination of the

b-value, the anomalies are reduced to mean elevation using the b-values

of neighboring squares. If there are no computed b-values in neighbor­

ing squares the b-value 0.1 1 1 9 mgal/meter is used on land and

-O.O687 mgal/meter at sea for reducing the observed anomalies to mean

elevation of the square.

Using the methods described above, the writer estimated mean free

air anomalies for 6679 l°x 1° squares, for which some observational

gravity data were available in 1956 (16).

Mean elevations used by the writer in these estimations were partly read by personnel of the gravity project, partly taken from mean

elevation maps published by various agencies, and partly taken from a

compilation made by the Cincinnati Field Office of the Army Map Service

in 1952.

The computed mean anomalies were plotted by the writer on the

General Bathymetric Charts of the Ocean published by the International

Hydrographic Bureau in Monaco. In some cases it was possible to see a correlation in the mean anomalies of neighboring squares on the basis of mean elevation. Thus the square, which had a higher mean elevation, had an algebraically greater mean free air anomaly than the neighboring square which had a lower mean elevation. This correlation can be applied when estimating the mean free anomaly of an empty square by extrapolation or interpolation methods, but it should always be used with caution. 60

The question now arises of how large an area a Bingle gravity

anomaly can represent. One of the most recent papers concerning this

subject is Hirvonen's (23), where he states that the representation

becomes critical when the length of the side of the square approaches

2?6; "that means that if there is only one observed station inside a

square with s > 2?6, it is better to put mean anomaly equal to zero

than to use the observed anomaly as the representative of the square."

He continues, "However, the best value obviously is the weighted mean

of them both (observed and zero)."

He suggests the following formula for the mean anomaly of the square

(67)

where Ag = point anomaly

Gq = root mean square anomaly of point value

Gg = root mean square anomaly in a square with side S

Agg = mean anomaly of a square with side S

According to this formula and Hirvonen's Q0 and G values, one o o observed gravity anomaly, e.g., +40 mgal, would yield the mean anoma­

lies given in table 6.

Table 6. MEAN ANOMALIES OF SQUARES OF VARIOUS SIZES EXTRAPOLATED FROM A SINGLE OBSERVED ANOMALY

Area

Mean Anomaly in mgal +32 +24 +l8 +13 +8 0.4.

0.2.

' O I 10. 15 d

Fig. 12. Coefficient of Extrapolation for Determining a l°x 1° Mean Anomaly from a Point Value

o\ H 0.4.

0.2 _

Fig. 13- Coefficient of Extrapolation for Determining a l°x 1° Mean

Anomaly from a Computed l°x 1° Mean Anomaly 63

If there is one observed gravity anomaly, Ag, in the middle of the

square, the mean anomaly of that l°x 1° square will be

- O .79 * Ag

For a 3°x 3° square the equivalent formula is

Ag k 0.1*5 * Ag

0 0 0 0 In a 3 x 3 square there are nine l x l squares. The first

central one has the mean anomaly O .79 • Ag and eight others will have the following mean anomaly

Ag « 0.1*0 • Ag I X 1

If d represents the distance in degrees between the center of the 0 0 0 0 first l x l square and the center of the l x l square where the mean anomaly is wanted, the general formula for Ag^, as derived by the writer, is the following

ftd+tJ

Coefficients can be computed beforehand and they are given in figure 12. When the mean anomaly for the first l°x 1° square has been determined, the mean anomalies for the surrounding empty square can be computed by using the coefficients given in figure 13* 6b o o If the point value were +30 mgal, the mean anomalies of 1 X 1 o o squares would be +23 mgal. l x l mean anomalies extrapolated outwards o o from this l x l square are shown In table 7.

Table 7* EXTRAPOLATED EFFECT OF A 1°X 1° MEAN ANOMALY

+23 +12 +8 +6 +5 +^ +b +1* +3 +3 +3 +2 +2 +2 +2 +1

o o Using this method the mean anomalies for each l x l square could be computed. As Hlrvonen has already pointed out, his values of 0 are

hardly representative for the entire earth, because the samples are

from limited areas. These values can be used until new values are

computed by using all available material.

As the graphs presented earlier (figures 6-11) have shown, free air anomalies are correlated to local elevation changes or to relief.

Therefore, Hirvonen's mathematical procedure cannot be used alone.

The observed anomaly should be first reduced to correspond to the mean elevation of the square and then the mean anomaly of the squares should be computed. New determinations of the G-values should be made and then, using these values, the corresponding mean anomalies should be computed for the surrounding empty squares. Possible local elevation correlation between mean anomalies and mean heights of the squares should be determined.

While awaiting the new correlation values, the mean free air anoma­ lies were plotted on topographic maps and by aid of these maps and Hirvonen's correlation numbers the anomaly field was extended by the writer as far into unsurveyed areas as seemed reasonable by observing the correlations in local areas. 5* GRAVITY MATERIAL

The base of the material In the MCRL was the card catalogue which

Tanni (1*7) had started in Finland. During the years many publications

have been received containing new gravity information. Many libraries

have been also checked for material. Gravity Information has been

collected from over U50 publications, and very valuable additional material has been received in manuscript or in tabulated form from various oil companies, the U.S. Coast and Geodetic Survey, from

Dr. Woollard and other scientists around the world.

All this material has been analyzed, related, and the anomalies

computed. The mean free air anomalies of l°x 1° squares have been determined by the writer as described in a previous chapter. A list of these anomalies has been published in a classified paper (16).

It should be noted that these same l°x 1° mean free air anomalies

have also served as the base material for recent investigations of the

Army Map Service and other agencies in the United States.

6 5 6. COMPUTATIONS OF UNDULATIONS OF THE GEOID

6.1 Basic Formulas

As has been mentioned, the potential of the earth is

W = U + T

Where U is the potential of the spheroid, or that of an ideal earth, and

T is the potential of the disturbing masses.

According to formulas 30 and 33

N

If the spheroid is assumed to be a

sphere, then

Fig. 1^. Geold and Spheroid

and the formula takes the form

Potential on the geold is

66 Nov the formula for the anomaly is

/ _ V (69)

This means that gravity anomaly Is equal to the sum of attraction of

mass Irregularities and of so-called Bruhn*s term. T is a harmonic

function, therefore it Is possible to express gravity anomalies In

spherical harmonics.

Gravity anomalies expressed In spherical harmonics are (30)

where

% (*< VAJ ~" i f ; c ( 1 c c +s%osk^ 1>^ (c**® ( n > and &T

q = / f T2)

In practical application these last values can be computed from (30)

; (73) c fnii~ A]fa, ty&Zx/favMJ&t

This formula Is suitable for use with high-speed computing machines.

Development of the gravity field In terms of spherical harmonics Is

in progress in the World-Wide Gravity Project. 6 8

The separation between spheropotential and geopotential surfaces at any elevation can be computed from

(74)

When R = p, the formula gives undulations of the geoid at sea level.

Spherical harmonics have been used for geoidal computations (33, 58)

but it has been considered better (15) to use the formula which Stokes

derived in 1849 (46) because of the existence of large unsurveyed areas

in the world. Stokes' formula is 7T ITT

where

Close to the computation point the function S(f) approaches infinity, and therefore the following formula should be used for com­ puting the effect of the immediate area around the computation point

i r

(77)

where

(78) 6 9

where g: (♦) 1b the mean anomaly of the circular ring. If small surface

elements are used, the function S(i|r) and F(t) can be assumed to be

constant over small area. The integral may be replaced by the sum

(79)

where q is a surface element, or

where Ag {^aO is the mean anomaly of a compartment, m i j

6.2 Numerical Computations

6.21 General Technique

In numerical computations the sum 79 has been divided Into two

S(t) ' 0. > contains constants, m the value of Stokes 1 function, and area factor; the second part

contains the summation of the products of multiplication between

c-values and corresponding mean anomalies. A set of c-values can be

computed beforehand for each latitude where computations will be desired. In fact, it is necessary to compute c-valuea for only half of the sphere at each latitude, because coefficients are symmetrical with respect to the meridian through the computation point. The same set of c-values can be used for the same latitudes in northern and TO

southern hemispheres. After these c-values have been computed for the whole world, the undulation value is simply

(81)

6.22 Selection of the Size of Surface Elements

The use of the mean anomalies and the mean values of the

function In this numerical integration causes certain errors In final results. Of course, the errors depend on the size of the area for which the mean values are taken and the distance from the computation point.

According to Hlrvonen (23) the best accuracy which can be obtained in the gravimetric determination of N is -10 meters. The error Is caused by the lack of gravity information. Therefore, the writer considered the method to be sufficiently accurate if the error caused by the numerical integration was not greater than the order of -1 meter.

Because there are thousands of squares on the surface of the earth, each gives about the same contribution to the results, the accuracy with which each contribution should be computed is of the order of -1 cm. * Formula 79 gives the undulation values of the geold.

Often the function of t(^) Is used instead of S(t). The relation between these is

The values of the function f(t) have been published by Lambert and

Darling (35)* R If the factor ia designated d, the effect of the surface

el«B»nt on tho undulation !■

(02)

where f^ Is the value of f(t) for the area q, Ag^ mean anomaly for the

seas area q. it la practical to use l°x 1° and 5°x 5° "squares,"

because that way it is not necessary to repeat the estimations of the

aean anomalies for the areas In each of the computations. When the

square areas and mean anomalies are used, a better approxlmation of the

effect of the area can be obtained by dividing area into two halves

along the line which gives the m a n distance of the square from the

computation point. The new values of f^ are approximately f + Bf and

f^ - Bf, and the new values of Ag are Ag + Bg and Ag^ - Bg. The new value for the effect Is

(83) where the second term gives an Indication of the error committed In the computation when the whole square Is used Instead of two halves.

The 5f-factor Is approximately^* ^ and according to Hirvonen's statistical Investigation (23), the Bg Is -6 mgal for any site of the square having Vq between 1° and 10°. The error of one square is then

1.? * Vq3* * ot. If, for example, It Is desired to keep the maxlmw error coming from site of the square within -0.01 meters, then for

>/q ■ 1° : ~ * 1250 or t - 1?6 and for '/q*5°»j^“ 10ort» 25°.

The ideal case would be to compute the effect of the olosest tones 72 anomalies of 1°X 1° squares, and beyond > « 25°, mean anomalies of

5°X 5° squares. These confutations are based on the size of "squares" o , o at equator. Most of the computations are made at 30 -60 latitudes, and the corresponding zone for l°x 1° squares is y ■ 1?2 - 13°3 k5° latitude. As a conclusion, it can be stated that if l°x 1° squares are used inside 20°x 30° area around the confutation point and 5°x 5° o squares beyond that in numerical integration at 45 latitude, the computation method contributes hardly more than -1 meter to the final standard error. Of course, in unfavorable cases the error might be greater.

As a check, the undulation values were computed using l°x 1° squares

(a) for 20°x 30° area, (b) for 30°x kO° area, and (c) for U0°x 50° area around the confutation points and 5°x 5° squares were used beyond these areas. The differences in the undulation values are given in table 8.

As is to be noted, the sign is consistently negative. This is caused by a slope in the gravity anomaly field. Of course, these differences do not give the accuracy of the method of integration but an idea of the effect of variation of the size of the squares in certain areas. 73

Table 8. DIFFERENCES BETWEEN TOTAL UNDULATION VALUES OBTAINED BY USING 1°X 1° MEAN ANOMALY AREAS OF VARIOUS SIZES AROUND THE COMPUTATION POINTS

Location Using l*0°x 50° Using l*0°x 50° Using 30°X 1*0 Minus Minus Minus

25°N 280° -0-11 m -0.12 m -0.01 m

25° 285° -0.12 -0 .2 9 -0 .1 7 °ir\ CVJ 290° -0.20 -0.55 -0.35

30° 280° -0.03 -0.1*2 -O.38 o fO o 285° -o.oi* -0.57 -0.52

30° 290° -0 .0 9 -0.71 -0 .6 2

35° 280° -0 .3 6 -0 .5 0 -0 .11*

35° 285° -0 .3 6 -0.52 -0 .1 6

35° 290° -0.39 -O.Jk -0.15 o o 280° -0.19 -0.25 -0.07 o

-p- o 285° -0 .1 6 -0.25 -0 .0 9

1*0 ° 290° -0.13 -0 .2 7 -0 .11* 6.23 High Speed Computation

6.231 Computational Technique

The manual method of computation of the undulations of

the geold has been explained thoroughly in many publications and can

be found, for example, in Hirvonen's (20), Tanni's (V7), and

Tengstrfim's (h8) publications. Development of high-speed computing machines has made the use of the manual method less frequent. The principles of the manual method can be used in planning the program

for high-speed computing machines. The details of the method used in

high-speed computing machines depend on the type of machine available

and Its capability. Such factors as memory capacity, input rate,

output speed, and the number of the digits are very important in planning the program for these computations. The speed and the cost of the computations on the one hand and the accuracy and quantity on

the other hand are the factors which must be in balance.

At the Ohio State University there is available a type 650 Mag­ netic Drum Data Processing Machine. This machine has all the advantages of stored programming, high component reliability, self-checking, automatic operation, compact design, and ease of operation. It is a

stored-program, modified single-address, numerical, decimal machine.

Xts memory capacity is 2060 words and its maximum input rate is

200 cards or 1 6 ,0 0 0 digits per minute and the maximum output speed,

100 cards (2 6). 75

In the 650 Magnetic Drum Data Processing Machine the computations of the undulations of the geold are divided by the writer into two parts (formula 6l)

f t (1) computations of c(

certain qpQ value. These c(cp', k') will be referred to as

Stokes' coefficients.

(2 ) performing of the summation c(tp', \ ) Ag^ (cp*, \).

These computations are divided into two areas

(1) for the area 20°X 30° around the computation point by using

1°X 1° "squares"

(2 ) for the area outside 20°x 30° area around the computation

point by using 5°X 5° "squares."

6.232 Computations of Stokes1 Coefficients

6.2321 Coefficients for l°x 1° Squares

Stokes' coefficient, as mentioned, includes

where R = 6371 km, 7 * 979*8 Gal* f(t) = \ S ( ), and da is area of the square in radians. The constant part is

For the computations of f(t) the distance has to be determined between the computation point (cp, k) and the surface element in question (qp , k')* If the real shape of the earth la taken into account, the formulas are relatively complicated, hut the error com­ mitted, by using a sphere as an approximation for the earth, is very small (57)• Therefore, in practice the desired distance is computed by aid of the following fundamental formula of spherical trigonometry

COS P « S M < p s ! n Cp* -f- C O S C O S C p 'COSO?*. .

Nov, knowing the distance i(r, f('l') is computed directly from formula 7 2 rather than being taken from tables. It should be remembered that

2 f(i|r) = S( i|r)

The first program is to compute Stokes' coefficients for each

1°X 1° square inside a 20°x 30° area. Actually only one half of the values--because eastern and western halves are symmetrical— has to be computed. The flow chart of these computations is given in diagram I, which is self-explanatory.

This program has been separated into two parts, A and B, only because it is desirable to have five coefficient values on a card rather than one. The latitude of the computation point is read in with the program. As can be easily seen from the flow chart, this program is made for exact latitude degrees and a modification is needed if the computation point is not at an exact degree of latitude. o o o o Close to the computation point 1 < f < 2 the 1 X 1 squares have been divided into four parts and the Stokes' coefficient for the whole square has been taken as an average of the Stokes' coefficient for these four l/2°x l/2° squares. Computation time for a set of Stokes' coefficients (20°x 15°) Is four minutes. This computed set is half of 77.

77

*n ha

ht^ltottol IN A taoNrlNm totoNr m tofaf'to dr> dr 1 topltolA* N M f

Jdl M m A, ^ Ototo to Nat MaUbto <•**(« I'*!* nH hr MM'm MiM IMUMIa S T i l T !lit via. a wd to^ii

Hit u tofli> Unt. III I to atotor O^toi •tota'tototltoal •tart* ktonN. to (■■ totoi to da* A +(*-* A ■*7WNto to tom toMto ■to. Into totol) lllto ptoto toto Tv ipN MM (■total I1M “ |#^rat in torn i ^DS_ fan.

*hn

^BL tomMpato laato Mto to wto kato toktola to Bill Nat Itortolw to ftoHtoi INO / KcrSJL, ItoNMNtoto r- h i a t a l toto (ton pato *1 l-Ntoto toto \ m j /

•m V m &♦ *)-» a to (M tart to •tonaatototoM Hto)•tot to ftoM aril to01 l \ > ■ lA y

b T * •mi Hn 1 ^Dn Him f IMItotok v A NfcattoftoN MUp ton ««toph* / j * n (to itto* 4 toi toto ta*toM*iri \ m MN*)tap Ultoptol 1 Ito 3E.-1___ •m_____ teil telnlte r \ r \ MU^Nto* (A»4) -*A w c s s totontotoa *1 k j -

Dlafra I. now Chart ftr Hj)»4feMd Capitation of Stoke*1

Coofflelwta fir l*^ 1° S g w i Inalda a 20°x 30° Atm a whole set and, therefore, It must be duplicated for other side of the meridian. In the area where 0° < + < 1°, the Stokes' coefficient has not been computed In a similar way because S(t) approaches Infinity.

The effect of the closer squares can be computed from

F f o c l t . (77)

If Aqp Is denoted by a and by b, the formula for the Stokes' coefficient Is ______

3. (. .-JZhF+o. ^ _ / - n /

The effect of the closer areas can be also computed graphically using formula 77, where r* f F(i)dP = 064* - {l+hsln^-oast-bsh^-li*>•?'*' 0 (85)

- § s i n 7P l a ^ f which Is tabulated by Lambert and Darling (35)*

When R » 6371 km, - 979*8 Gal, and Ag^ is In mi Hi gals, then the effect of the one circular ring Is

a N = 6.3023/0* _f r m wwm (*)w m m . When the circle Is divided into 18 sectors and the effect of one com­ partment Is 1 m/mgal, the outer radii of the circular rings are those given in Table 9* The first circle is divided into k quadrants Instead of 18 compartments. 79

Table 9 . RADII OIF CIRCULAR RINGS USED IN COMPUTATIONS OF THE EFFECT OF NEAREST NEIGHBORHOOD OF THE COMPUTATION POINT ON THE UNDULATION OF THE OEOID

Outer Radius Number of Effect of One Degrees Compartments Compartment

0?175 * 5 nm/mgal 0.325 18 1 0.480 18 1 0.6 3 0 18 1 O .775 18 1 0 .9 2 0 18 1 1 .070 18 1 1.225 18 1 1.360 18 1 1.505 18 1 1.645 18 1 1.785 18 1 1.925 18 1 2.065 18 1 2.203 18 1 2.341 18 1 2.480 18 1 2.620 18 1 2 .7 6 0 18 1 2 .900 18 1 3.040 18 1 3.170 18 1 3.300 18 1 3 .430 18 1 3.450 18 1 3.695 18 1 3.830 18 1 80

Ibis graphical method can be used for computing the effect of the immediate area around the computation point and it can be also used for computing the Stokes' coefficient for squares within the distance of

0 ° < i|r < 1°.

When Rice's (1*1*) circular rings are used, the effects of the circular rings will be as shown in table 1 0.

Table 10. EFFECT OF RICE'S CIRCULAR RINGS ON THE UNDULATION OF THE <3E0H>

Effect of Effect of n Inner Radius Circ.. Ring n Inner Radius Circ. Ring

- r xmn/mgal r mm/mgal

0-lk 0 1.36 35 0?357 7 .8 7 15 o?oii7 0 .2 7 36 0.1*23 9 .6 2 16 0.0139 0.32 37 0.501 10.73 IT 0 .0 1 6 5 0.33 38 0.593 13.1*6 l8 0 .0 1 9 6 0.1*5 39 0 .7 0 2 1 5 .6 0 19 0.0 2 3 3 0.52 1*0 0 .8 3 0 1 8 .2 1 20 0.0 2 7 6 0.59 1*1 O .981 21.85 21 0 .0 3 2 8 O .7 1 , 1*2 1 .1 5 8 26.65 22 0 .0 3 8 9 0 .8 5 1*3 1*37 30.51 23 0 .0l*6l 0-97 1*1* 1 .6l 37*12 2k 0 .051*7 1 .21* 1*5 I .90 1*2 .9 2 25 0 .061*9 1.1*3 1*6 2.23 51.17 26 0 ,0 7 7 0 1 .6 2 1*7 2 .6 2 59*63 27 0 .0 9 1 3 1.95 1*8 3.07 69.77 28 0 .1081* 2 .2 8 1*9 3.59 8 1.1*1 29 0 .1 2 9 2 .5I* 50 1*.19 93.32 30 0 .1 5 2 3.25 51 k .87 10 8 .3 8 31 0 .1 8 1 3.70 52 5 .6 5 32 0 .211* 1**69 33 O.25U 5.52 3k 0 .3 0 1 6.57 8l

The final values of Stokes' coefficients are punched on the IBM

cards as follows

Column Explanation

1 - 3 zeros

k - 6 location of first word when reloaded

7 - 9 zeros

10 number of words on card to be loaded

1 1 - 6 0 five words, all same longitude, decreasing latitude

by 1° per value

6l - 65 zeros

6 6- 68 longitude of values

69 - 70 zeros

7 1- 72 latitude of computation point (

73 - 75 zeros

7 6- 78 latitude of the first coefficient on the card (cp )

7 9 - 8 0 zeros

6.2323 Coefficients for 5°X 5° Squares

The method for 5°x 5° squares is similar to that

for l°x 1° squares except in this program the coefficients are not com­

puted for the squares that fall within the rectangle 20° latitude by

30° longitude around the computation point. The flow chart for these

computations is shown in diagram II which is self-explanatory. It

should be noted that in this same diagram there is shown the computation

of two sets of Venlng Melnesz* coefficients for the 5°x 5° squares.

Details of these will be explained later on. The items punched on the IBM cn/rds are xxxn m

> 1 vp 9* initial Dk t u m i d d n w on i. j-nd Into PCCi vith -roproa) output card by 6 t

Gal. sic9 «ad oos^* Corract ldsotlflcat Sat up liertlficatloi Sat >■ - 2.5° la output to D m ■at up eddraaaes on «'( proper part of the output card* uorld ( l/J )

an XIX lee Correct ldent.

Do m m u

Calculate htoies Jtore c - value Keeet store ccafficlertt Increase address on Instructions to etor i first auebar In output cards t;- 6 punch hards

VI XII XVI im Calculate sin OC Store c| -value 0 - w o r d oouot lalrulete eln S® Tee pond: answers, sat and and c -value la punch lend (3) ensterf on card op Identification M i V \ 777 / It. output cards

VIII XIII fcbdify to store Jtore e_ -values Calculate cos CL values la r.ext location of patch ZXXII la punch band (2) and el -value beads ( 9^— upper)

Diagram II. Flow Chart for High-Speed. Computation of Stokes' and

Veiling Keineoz' Coefficient-: for 5° Squares Around the 1 oriel Column Explanation

1 Identification of coefficients

2 Zero

3- 6 Location of first vord when reloaded

7- 9 Zero

10 Number of words on card (to be loaded)

11-70 Six wordB

71 Section of world (1 - 90°N-30°N, 2 - 30°N-30°S,

3 =« 30°S-90°S)

72-73 Latitude of computation point,

7k- 76 Latitude of surface element, cp*

77-80 Longitude of surface element, x/

The time required for computation of one set of Stokes’ coefficients

and Vening Melnesz1 coefficients for 5°X 5° squares is about kO minutes.

6.233 Summation of the Products;

Stokes1 Coefficients x Gravity Anomalies

6.2331 l°x 1° Squares

When computing the effect of a 20°x 30° area on undulation values, all 600 Stokes’ coefficients for the desired latitude are read in the machine. Because the anomalies are punched on the cards so that there are five anomalies on the same card, it is necessary to read 750 mean anomalies of 1°X 1° squares everywhere except at those latitudes which are multiplied by five, where only 600 mean anomalies will be enough. Example: If the undulation values are desired at a 8V

Read in Read blank oard oosffiolenta * h control card * into 1750 band with L]_

No platitude of coefficients lot.of oon

Yes

III i

Road In Read in

750 ■blft anomalies

IV

Shift amsselle* Increase shift Set up multipl.

* instruction again instruction for 25 locations sunaation

Su&mation of ano- nallaa x Stokes' ooofflclanta

Punch answer i Increase mult. *■* 1 § reset • instructions

to aero by one

IK No Store suraatioa Inoreaoe mult. Xes instructions to and identify to am---- <" of sunmation m take next 20 nunbors N,. punch.

Diagram III. Flow Chart for High-Speed Sunsnation of Products:

Stokes' Coefficients X Gravity Anomalies,

Using l°x 1° Squares Inside 2 0 ° x 30° Area 85

55° latitude, the anomalies vill be read in between the 45° and 65°

latitudes, but if the undulation values are desired at 56°, 57°> 58°>

or 59°/ the mean anomalies will be read in between the k'y0 and 70°

latitudes.

When the effect of the 20°x 30° area on the undulation value has

been computed for one point, the computations can be automatically

performed for the next points at a one degree interval along the same

parallel by adding 25 new mean anomalies into the machine. This way

the computations can be performed and answers can be given at a one degree interval between any two points along the same parallel.

The flow chart is given in diagram III. The time required for one

summation is approximately 36 seconds sifter the program, coefficients, and anomalies are read in.

6.2332 5°x 5° Squares

The program, diagram IV, will give the sum of the products of anomalies times corresponding Stokes' coefficients or

Vening Heines z' coefficients for any given distance between 0° and 355° longitude along any parallel divisible by five. o o There are 2592 5 x 5 squares on the surface of the earth; therefore a special arrangement must be made in order to carry out this many multiplications in the 650 machine. Because of the limits of storage locations, it was decided to divide the Vhole world into three sections

Section 1 - Area between 90°N and 30°N latitudes

Section 2 - Area between 30°H and 30°S latitudes

Section 3 “ Area between 30°S and 90°S latitudes. 86

atart point

reset H I

No

Road la ooeffioleet Tm and

XII, las III VII

Bo laa ooafflc. la control m _____ oard raaat instruction Xaa VII

Chack ^ , section Add 1 to both coefficients, ate.

III

Sat laa e_. -ooafflelan " ITT / 1 - -Ol

IV las

Multiply ooaff. Sat and anomaly and add to sow. 863

Diagram IV. Flow Chart far High-Speed Smmatlan of Products:

Stakes* Coefficienta and Gravity Anomalies or Venlng Meinesz.'

Coefficients and Gravity Anomalies,

Using 5°x J° Squares Around the World In each section there are 86k 5°x 5° squares which now can he handled

at the same time In the 650 machine.

The mean anomalies of 5°x 5° squares are punched on the cards, which are seven

Column Explanation

1 Identification of section of the world 1 (90°N-30°E)

2 (30°N-30°3)

3 (30°S-90°S)

2 Zero

3- 6 Location of the first word of card

7- 9 Zeros

10 6 = number of anomalies on card

11-70 6, 10-dlgit words

71-75 Latitude of northwest corner of thesquare for first

anomaly on card

76-80 Longitude of left side of all squares represented

on card

When computations are performed along any parallel at 5° Intervals, the effect of each section is computed separately and then partial results added later on to form the final results. Because the effect of the closer zones is computed using l°x l°anomalles, the Stokes' coefficients for 20°x 30° area around the computation point are replaced by zeros in this 5°x 5° program. 6.231* Total Undulation of the Qeoid

When the effect of the 20°x 30° area and the effect of

area outside 20°x 30° are combined, the result naturally gives the

final value of the undulation of the geold. Because the effect of the

outer area changes slowly from point to point, it is not necessary to perform these computations at each corner point of a l°x 1° grid, but at the corner points of a 5°x 5° grid. A map can be easily drawn

showing the effect of the outer area at any point. Instead of drawing a map, the interpolations can be done in the high-speed computing machine. The parabolic interpolation should give as good or even better values than can be read from the map. Even a straight line

interpolation introduces only small errors. The high-speed methods described under this chapter were developed by the writer and were used in computing the internationally known "Columbus Qeoid" (17) at the Mapping and Charting Research Laboratory of the Ohio State Research

Foundation. They were also used in detailed Investigations of the undulation of the geoid for many regional areas of the world, but the results of these investigations are, at the present time, considered classified information by the United States Air Force (l6).

6.21* Accuracy of Undulation Values

In the numerical integration, the size of the surface elements was so selected that the error caused by the method would not exceed

-1 meter, which was considered to be sufficient accuracy, Inasmuch as the actual accuracy of the results was assumed at best to be +10 meters

(23). Hirvonen estimated the accuracy of the undulations at two points, one in the UFA, one in Europe in 195^- It is, of course, important to know also the accuracy of the new values of the undulations of the geold which have been computed for the larger part of the northern hemisphere.

However, because the results of statistical investigation of the gravity anomaly field used in these computations are not yet in hand, the writer's accuracy estimations are based on Hlrvonen's results of error of representation and related functions.

When the gravity anomalies Ag are known, the geodlal heights can be computed at any point using formula 79

If 2 f (♦) » S (t) is used, the formula will be

M m = 5 W m HV? .

Hlrvonen's formula for the standard mean error is

m = I. 035\/2^ • 0 a , ' (86) (H) V 3 T? where f is the mean value of f(ifr) for the area in question. Q bee two different values: (1) for an observed area Q ■ F^a^q, (2) for an 2 2 unobserved area Q ■> 0 s q, where q is the area of the. square used and 2 s has two values: (1) in an observed area, s - q/n, where n ■ number o of observations per square and (2) in an unobserved area, A * a , whexe

A is the area of a large unobserved gap. F and G are statistical functions given by Hlrvonen. The writer estimated Q-values for all r>°x 5° squares and the mean standard errors were computed along each 0%

*0

Fig. 1?. Standard Mean Error of Oravimetrlcally Computed Undulation

of Qeoid. (Usual - signs are omitted, ,and unit : 1 meter) 91

10 degree parallel at 5 degree Intervals, betveen 0° and 70° northern

latitudes. The results of the computations are given in figure 15.

Oiven mean standard errors are in meters, plus and minus signs are

omitted from the map. The values represent the error caused by the

lack of gravity observations. The agreement is good betveen Hlrvonen's values (23) and these. Additional material received since 195^ seemed to improve the results and decrease the error about two meters. The high accuracy in the East Indies was expected, because Stokes' coef­

ficients close to the zero fall into the largest empty areas. A very detailed accuracy determination might change somewhat the values, o o because only 5 X 5 squares were used in this investigation. 7- COMPUTATIONS OF DEFLECTIONS OF THE VERTICAL

7*1 Basic Formulas

As It has been seen (figure l), the geoid undulates from place to

place. Since the plumbllne Is always perpendicular to the geold, the

angle e, between the normal of the spheroid and the plumbllne, also

changes from place to place. The angle, c, Is defined as a deflection

of the vertical.

From figure 16 one obtains

> -

Because ft and r\ are small angles,

c can be expressed

The components are

(88) c ^ N

Fig. 16. Deflection of the Vertical where £jt e- ' R oI P C C & O L

Venlng Heinesz derived (52) In 1928 the following formulas for deflection of the vertical T ZW ^

§ * m r Y m / f 4 d

Sin ?= - - 3 - 8i)n^+32iir?^H2s'J^

— 32s'm'£ +3$tn'/' Jtdp^ (&*?"% + s/'9'zJ

The values of the functions

' -5/W^ and ^ / d^ L S i r > P d P ?WD.» are tabulated by Sollins (45) •

The formulas 09 are not practical in the neighborhood of the

computation point because with a small value, the expression

For the effect of the closest circular area with radius sin |

s the following formulas have been derived (2 9) V * W d S N 4 . < * h t If V ( ( d? *•»+<*** r Hir *Lo 0(jro

= - t V m ( s ° '5 ° ) (9 1) d n . ' r fK f smiths/nd dPdd

— i n z ( S o + vT'5*2) ^ - ? where is north-south gravity gradient and east-west one. The

Vening Meinesz. formulas can be expressed in the following usable form TT I *

£ J a rr* *F ^ (92) The first parte of the formulas are used for computations of the

effect of closest areas and the rest of the formulas are used for

computing the effect of outer areas.

7-2 Numerical Computations

7-21 General Technique

There have been several attempts to apply the Vening Meinesz'

formulas to the computations of the deflection of the vertical. The

Russian Kasansky (32) computed some deflection of the vertical values

near Moscov and J. E. de Vos van Steenwijk (8) computed In the East

Indian area. Hopmann (26) computed for Potsdam; Rice (W-) for several

astrogeodetlc points in the U.S.A.; and, later, several other scientists for additional points around the world. In none of these computations has the Integration been carried over the whole earth. No practical method of computation of the effect of distant areas has been developed earlier, because there has not been sufficient gravity material avail* able to Justify carrying out numerical integration over the whole earth.

The principles used in computations of geoidal heights are adapted by the writer to the computations of the deflections of the vertical.

There must, of course, be quite a few modifications because the local gravity field has proportionally a much greater effect on the total deflections of the vertical than on the undulation of the geold. It is clear that the square method should also be employed here. The last part of formula 92 is modified by making the substitution

d 4 m e/e* et* S/n t , 95 where do Is a surface element of a solid angle on a unit sphere. Using

the following notation

(93)

the last part of formula 92 takes the form

> C o S o l J a n f (93')

' 0* In numerical integration ? is considered constant over the unit area, end in this way, integration is replaced by summation:

Ar/} = ~ X I - b ' l I^^j) A 2"> to, ) , where

% f a . V -=- [ ■ & * { s x t } (9*0 where in turn q is the area of the square. c'(q>, \) is called the

Vening Meinesz1 coefficient. As In computations of geoldal heights, these coefficients can be computed beforehand for each latitude.

7«22 Selection of the Size of Surface Elements

In the computation of undulation values it was possible to use mean anomalies of the squares all the way to the computation point without losing very much in accuracy. But the computations of the deflections of the vertical have to be divided into four parts:

1. Estimation of the effect of an inner circle 2. Estimation of the effect of an area where a detailed gravity map and the circular ring method should he used

3- Computations of the effect of an area where 1°X 1° mean anomalies can he used

Computations of the effect of an area where 5°x 5° mean anomalies can be used

The borderlines for the last two was determined by the writer using a formula similar to formula 8 3. By assuming cos a and sin a equal to one, the formula becomes

where df^ is the value of for the area q; Ag^, the mean anomaly for the same area; and &df^, the change of when two halves of squares are used. If 6g is taken as -6 zogal as previously and the

+ " error contribution of one square is kept at the order of -0 .001, then the effect of the closest areas, t up to k®'?, ^*3 to be computed from a detailed anomaly map; the effect of the area i|r - 4?7 - 27°, using mean anomalies of l°x 1° squares; and the effect of the area beyond

+ => 27°, using mean anomalies of 5°x 5° squares. Corresponding border­ lines are if - 4?3 and y * 2^° at 40° latitude. In the practical computations borderlines were fixed at t « 3° and if » 20°. The error which is caused by use of these borderlines in the numerical integra- + " tlon is estimated not to be greater than of the order of -0 .1 . 97

7.23 Manual Computation Method

7.231 Base Map and Templates

The following information is needed for computation of

the deflection of the vertical (see formulas 92 and 9 3)

7m = mean gravity value of the earth

R = mean radius of the earth

sq = radius of the first circle ring used in computations

5x* 5y = gradient inside the first circle Ag - mean anomaly of surface element

= value of the derivative of Stokes1 function over the

surface element

a = azimuth of surface element from the computation point

i|f = angular distance from computation point to surface

element

Mean gravity and radius are given in 6.2321, and the radius of the first circle and gravity gradient are determined at each computation point. The mean anomaly is read for each area unit in question and

f) is computable when the distance, \|r, is known. There now remain o y two values, \Jr = angular distance and a = azimuth, which must be determined from the computation point to area unit in order to perform the numerical integration over the whole earth. If the earth is taken as a sphere, the desired values can he easily computed from fundamental

formulas of spherical trigonometry c c s y — sin*pstn OosPPooscp'oos 4 A

s/n & case* — ccs cpsincp'— sin ^Pc^os *P cosA A (95)

Sinft's/r7c( = O o S ^ s in A A >

where

* Is the latitude of

a surface element, AA Is the longitude difference between computation

point P and the surface element, * is the angular distance between

point P and the surface element.

During Integration these computations have to be performed

thousands of times, and since less accurate values are sufficient In

4 the manual method, it 1b therefore advisable to use the more rapid graphical method than the numerical method for determination of distances and azimuths.

In order to use the graphical method a map projection should be

selected from which the distances and azimuths can be read easily.

Requirements for this type of base map would be

(1) easy to draw equi-distance lines

(2 ) easy to draw equi -azimuthal lines

(3) a suitable scale for showing the necessary gravity

Information.

A natural answer for this problem is a stenographic projection:

It is easy to construct and all equi-distance lines as well as all 99 equi-azimuthal lines are circles on this ‘projection- The fundamental formulas for the polar stereographic projection of uhe sphere follow

y, = r c o sa \ y- r s /o A > ^ r = k. = h. S'',> where $ = co-latitude, r » radius of the parallel of co-latltude , k = radius of the equator in drawing scale, x and y = rectangular coordinates, origin at the pole and x-axis coincides with central meridian, = longitude from the central meridian.

Using the formulas given above, base maps can be drawn in any selected scale. On the templates, ^-curves, or the equi-distance curves, a a-curves, or the equi-azimuthal curves, should be constructed.

The following relations are derived from equations 95

c o s 9 3 = % i r 1 ; t i 5 a (97) ic rL * kr+r*’

When these values are substituted into the formula

t T cos \|f = sin

cos the following expression is obtained

cos? = 2k.i~ costPco a* X {96) k 2 -t r 2

The equation of circles representing the ^-curves is now

(y*+jc?m)(a& P+Sfn¥)-2kjLCos

2 - 2 Ofhvoi = (k-r )c03? A A

The equation of the circle representing the a curves is

(&2+ y t~ r z)ctisy+2kKs/r}

£^ — /c h x n <^p

(10 2 ) _ >c c o i c u r j o L c o s ^ and radius

Kliot = cos cps/'n

When the coordinates of the center and the radius of the circle are known, it is always possible to construct the circular ring by graphi­ cal method, or, if the radius is very big, coordinates can be computed for points along the circular ring which then is constructed by drawing the curve through these points.

7.232 Estimation of Effect of the Nearest Neighborhood

In formula 92, effect of the innermost circle was given in following form 101 dAg and for A t;, was replaced by dy Using the mean values 7 - 979-8 Gal and R = 6371 km, the formulas m yield the values

A # = - 0 * / O 5 3 • So O-So (103) A") = - O'!1 0 5 3 ' S a . ‘d-S,

Horizontal gravity gradients can be computed by using point

values at ^5° azimuth intervals along the circular ring as Rice (MO

did. Unit weight is given to the central gradient and half-unit weight

to each of the outer ones. In this method there is always a certain

ratio between Ax, or Ay, and s^.

Taking this ratio and the given

weights into account, the effect of

gravity field inside the circular

ring can be computed directly from

the following formulas (figure 1 7)

Fig. 17- Location of Points Used in Determining Effect of Inner Circle

= 0.02625^ -Ajtiyo.o/es6(4.9S£- A % E + ) O—S (10U) A r f = 0.02625/Afr-AJE ) ^ 0 . 0 / B S < ( A ^ - A ^ ) 0 "Sj

where Ag is gravity anomaly in mgal. Subindices refer to the azimuths

of the points where the anomalies are read on the circular ring. 102

Under 7 .2 2 It was established that the detailed gravity anomaly map and circular ring Integration method should he used up to 3°dlstance

from the computation point. The last part of formula 92 can be used to

compute the effect of this area 3 ° 2 T

d t d c t A ih f

for A t): replace cos a by sin a.

In his computations Rice (Mi) used circular templates having uni­ form, eitgniar sectors of 10 degrees. He computed the values of zone n radii, so that each compartment had a radial deflection effect of 0 .0 0 1 for a mean anomaly of 1 mgal. The total effect for each 10 degree sector vas found by sumnatlon and then resolved In meridian, £, and prime vertical, r\, components. The same method Is used here.

For* convenience of the reader the radii of Rice's circular rings are given In table 1 1.

Table 11. RICE'S CIRCULAR RINGS

n r km n r km n r km n r km n r km

1 0 .1 1 9 11 0.657 21 3.61*1 31 2 0 .0 9 1*1 109.0 2 0 .11*1 12 0.7 8 0 22 *.320 32 23.83 1*2 12 8.7 3 0 ,1 6 7 13 O .926 23 5.125 33 28.25 *3 151.9 * 0 .1 9 6 11* 1.099 21* 6 .0 8 1 3* 33-*8 1*1* 179.1 5 0 .2 3 5 15 1.301* 25 7 .216 35 39.67 *5 2 1 0 .9 6 0 .2 7 9 16 1.5*7 26 8.560 36 1*7 .0 0 1*6 21*8 .0 7 0 .3 3 1 17 1.8 3 6 27 10.15 37 55.66 *7 291.2 8 0 .3 9 3 18 2.199 28 12.05 38 65.90 1*8 3*1.2 9 0 .1*67 19 2 .5 6 6 29 11*. 29 39 77-97 *9 399-0 10 0 .55* 20 3.0 6 8 30 1 6.91* 1*0 92.22 50 *65.0 r * Inner radius of the zone.

This method has Droved its usefulness In manv nractlcal conmutatlons. 103

7*233 Computation of effect of the area 3°< ^ < 20°

The circular ring method described above could be con­ tinued farther than a 3° distance from the computation point, but the compartments start to become very large and computing the mean anomaly for each of them is quite a task, especially if this estimation of mean anomalies should be carried out for several computation points.

In order to use the mean anomalies of l°x 1° squares which had already been obtained, a method of numerical integration for the area between ijr - 3° and « 20° was developed by the writer utilizing formula

93

The integration procedure is divided into three steps

(l) The constant factor ttt— and the area factor 2tj> o A \ (sin

1.02 * 10~^ cos but if a different size of square is used, the multiplier must be recomputed to correspond to actual area.

The mean anomalies of 1°X 1° squares are multiplied by this factor and the products recorded on the base map which is drawn in the form of a polar stereographic projection. This map is in suitable scale when k, given in formula 96, is selected as 800 mm. A sample of the base map is given in figure 19 but in smaller scale. 101+

(2) The next ?tep concerns the multiplication of the values on the base map by the factor ^ • Because this value of function of distance from computation point to the surface element, the value of the function is known only if this angular distance is known.

But instead of drawing equi-distance curves at 1° interval or so, it is better to draw a template with equi-distance curves for suitable distances where ^ changes from one round value to another.

The writer decided that the lines at 10 and 50 unit intervals would be constructed and used in this numerical integration. A sample of the template is given in figure 18. This template is prepared for

39° latitude. The values of - ^ 8111(1 corresponding to distance, i|r, are given in table 1 2.

Table 12. SELECTED VALUES OF THE FUNCTION -

Outer Outer Outer 8f(i) Radius Radius Radius 8* di|r 8^ * * *

2?82 3 ?56 290 5°3i 11+0 2 .8 5 1+1+0 3.63 230 5.53 130 2 .8 9 1+30 3.70 270 5-79 120 2.92 1+20 3 .7 8 260 6 .0 9 110 2 .9 6 1+10 3 .8 6 250 6.1+3 100 3-00 1+00 3-91+ 21+0 6.81+ 90 3.01+ 390 1+.03 230 7.33 80 3 .0 8 380 1+.13 220 7-95 70 3-13 370 1+.21+ 210 8.71+ 60 3.17 360 1+-35 200 9 .8 1 50 3-22 350 1+.1+8 190 11.3 8 1+0 3.27 31*0 1+.62 l80 13.95 30 3-32 330 1+.76 170 1 9 .2 20 3.38 320 1+-93 160 38 10 3.1+1* 310 5.11 11+0 3.50 300 m

Fig. 18. Te^lmte Shoving Equal Axieuthal Linos of 10° Degree Intervals

and Circular Distance Zones with the Value of - ^ - 39°) 8

Fig. 19. Stereographlc Base lut

m

M

f.*,

1 _ Ir.

rif. 18. Teeplete Shoving Xqual Azlauthal Linns of X0° Degree Intervals

end Circular Dlstaaee Zones with the Value of - " 39°)

F'i ~ 107

(3) Using the base map and constructed templates, it is now possible to compute the values

2.7r r m ' ^ for each square in question.

As with the circular ring method, where the template is divided into uniform angular sectors of 10 degrees, similarly the templates for this computation are drafted with equi-azimuthal lines at 10 degree intervals starting from a 5°ME azimuth. The total effect for each

10 degree sector is found as before by summation of products and then resolved in meridian, |, and prime vertical, q, components.

Because the mean anomalies of 1°X 1° squares are used outside i|r > 3° in this method, the borderline at about ijr = 3° is not circular but a zigzag line as shown In figure 20. The effect of the shaded area

falling between the last full

circle and used squares is com­

puted as follows. Additional

circular rings are drafted so that

they easily overlap the first

squares used in the square method.

All full compartments which do not

Fig. 20. Zigzag Borderline overlap squares are recorded in the same manner as compartments in full circular rings. There are several compartments which partly overlap the squares. The effect of tf a full compartment is 0.001 per milligal, and of course the effect of part of the conqjartment is proportioned, to effect of the full 108

compartment in the ratio of areas In question. For example, if the area which does not overlap is a 1 /3 of the area of a full compartment, tl the effect of this area on deflection Is l/3 X 0 . 0 0 1 per mgal. The mean anomaly Is then read for the non-overlapping area, and is multi­ plied by the ratio between that area and a full compartment. Hie results are added in regular suxmnation In that 10 degree sector.

The described method of t-curves and a-curves is very useful if the deflection of the vertical Is desired at only one point, but if results are desired at several points, for exaaple, Inside a 1° X 1° square, then it Is advisable to compute the effect of the common area between 3° - 20° at the comer points of the square and then inter­ polate the effect of the common area for any point inside this square.

It is easy to understand that in the described method some azimuthal lines inadvertantly divide a square into two halves and some­ times it is difficult to decide quickly which value of - df(t) belongs to certain square; therefore, if these computations are repeated several times at the same latitude, it is advisable to definitely decide in which sector each square belongs and what is the - df(t) value for each square.

Figure 21 shows the templates after marking each square into certain azimuth sectors and determining the - dfQfr) value for each

1° X 1° square. This template is, of course, symmetrical to the central meridian. Summations of products are performed for each 10 degree sector and then resolved in the usual way for 6- and q-com­ ponents . iT7 * • i V

ft#. 21. Ti— pliti with Dofloction Co«fficl«nt» for l°x l°Squftr«t Fig. 22. Stereographic Base Map for l°x 1° Mean Anomalies r&#. 21. T ^ p T t i with Deflection Coefficient* for 1°X l°8auer«f

f V r l ' 1 ° Meevr. Anoina'i i e s Ill

Since the values in the given template are - * 10 1 **** anomalies have been multiplied by 1.02 • cos q>, the answers therefore , " _i^ * are in 1 X 10 units.

7.234 Computation of Effect of the Area 20°< i|r < 1 8 0 °

The outer borderline of the area 3° < ’lf < 20° the effect of which on the deflection of the vertical is computed.with the method described in the previous chapter and using l°x 1° mean anomalies, is not exactly \|r = 20° but the closest zigzag line outside this distance going along parallels and meridians with a multiple of five (as shown in figure 23)* This arrangement is necessary in order to include all areas into computations, because the effect of the outside area, ijr > 2 cP, on the deflection of the vertical is computed using 5°X 5° mean anomalies.

The method itself is very similar to that used in computations of the effect of the area between 3° and 20° distances. There are a few alterations however. The scale factor of the stereographic projection, k, (formula 9 6) is selected as 240 ram. The mean anomalies of 5°X 5° 2 squares are multiplied by the area factor times 10 . This means the multiplier between equator and 5° parallel is 0 .7606. The products are plotted on two base maps, one for the northern hemisphere, and one for the southern hemisphere. A series of templates is drafted as in the previous case, ^-curves are now drafted corresponding to one unit intervals of - . Distances and corresponding - values are given in table 13* K g . 23. Tfceqplates of t-Curves and a-Curves for W)° Latitude,

for the Hemisphere where the Computation Point is Located 113

Fig. 2k- Stereograph!c Base Map of Northern Hemisphere Pig. 23- Teeplates of t-Curves and a-Curvea for ¥)° Latitude,

for the Hemisphere where the Confutation Point la Located

Fi.s- 2**. Stereographic- Base Map * r: \< 1 sph© r*c Fig. 25. Template of ^-Curves and a-Curvea for 40° Latitude for the

W— 1 sphere where the Antlpode of the Computation Point la Located Fig. 26 . Stereographic Base Map of Southern Hemisphere Fig. 25* TeqplA.te of y-Curves and a-Curves for U0° Latitude for the

Hemisphere where the Antipode of the Computation Point Is Located

" " T ' : p. ir=e M a p •%; ^.rr. 116

Table 13- SELECTED CIRCULAR RINGS WITH CORRESPONDING - VALUES

Outer df(ijr) Outer - - * ( * ) . Outer Outer Radius di|f Radius di|r Radius di|r Radius d\|r V3 ♦°

19.5 2 6 .0 10 hO.O 5 77-0 0 20.5 Ik 2 8 .0 9 h5 .0 k 92.5 -1 21.5 13 30.0 8 5 1 .0 3 150.5 -2 2 3 .0 12 32.5 7 5 8 .0 2 171.5 -1 24.5 11 36.0 6 66.5 1 1 80 .0 0

A sample base map Is shown In figures 2k and 26 and a template of

\|f-curves and a-curves in figures 23 and 25* The effect of these outer

zones changes slowly from one computation point to another. Therefore,

it is sufficient to compute the effect of this distant area at corner points of 5°X 5° squares and interpolate values for points inside those o o areas. As in the case of 1 x 1 squares, it Is advisable to draw templates with - ? values and azimuths for each 5 degree parallel needed in the computations. In this way, computations are speeded up.

Samples are given in figures 27 and 2 9. The method is, of course, similar to the method used for the area between 3 and 20 degrees. In actual computations, the template value occupies the upper half of the square and the mean anomaly the lower half.

Summation of the products of map values X template values, is performed for each 10 degree sector and then resolved in the usual way for f- and tj-components. It should be noted that these results must be multiplied by a constant factor 0 .0 0 0 335 in order to have final answers in angular seconds when anomalies are in mllligals. X ^ / •> ss+ ' * \/ A r y\ * V # \ // y/ * S ^ ^ /”nV V o v ' . \ ^^vsrN&^sr* v v * x •*/ /ov\ ' . * x '' *'\* F^v y ^ x i V"X< °'/fc "x^ ° j '';'' * ** #*'■»»*W ' s ? ' * ^* w 'o V rV ^ n^Xv^/' \''\c * v\ v, v . . . >/V*/ \\V \ ^ , 1 , >A.%\*y\y^X* j- ^ % V a ',i)'^V '"\ I I *aT7—r I HnA_1 V — i\> t \ *> _ * '■"*' *■'+ v * y"' / v>/ a /''S L * ' '0 I? . I I t « \ »J k ' ' ' ^ * \X \'b /V ^ * /*s'-/ * A - I X # "J /" * "*1" t ' ^ a • v ' \ fc V ^ X * , / * • r • /. • 1. • i • 1 \ * ' * # // ~ /•- • /' fl /--Ll 1 --* L—1 * -r—“r1 - A » 5 \ 4 . \ \*

Fig. 27* Tenplste with Deflection Coefficients for 5°x 5° Squares

(Corresponding to Figure 23) Fig. 28. Stereographic Base Map of Northern Hemisphere 3 f t &

1 &

•/A , * « I A \ r '

ft tt IT It

Fig. 2 7 . Template with Deflection Coefficients far 5°X 5° Squeree

(Corresponding to Figure 23)

I.’crtneri •lemi sphere ** ** 0 ‘

Fig. 2 9 . Trap late with Deflection Coefficients for 59x 5° Squares

(Corresponding to Figure 25 ) Pig. 30. Stereographic Base Map of Southern Hemisphere ' v

; * « ‘ Jtf '00 * ^ , * *° ■* 3,

Fig. 2 9 . Template with Deflection Coefficients far 5<’X 5° Squares

(Corresponding to Figure 25) 121

7.235 Total Deflection of the Vertical

Deflection of the vertical is by definition (11)

f = <=p'—

/ ^ / \ syj (lO^) ^7 - ( A - A ) co s , 1 1 where (p , \ are astronomical coordinates and cp, \ geodetic coordinates

The signs of the components of the deflection of the vertical depend on

the signs of the coordinates. In computations of the deflections of

the vertical tfLj was taken with a minus sign. When the latitude I f grows northward and the longitude eastward, the signs of the sines and

cosines should be selected so that in computation of £-component the

northside products are substracted from southside products, and in

computations of q-component east side products are subtracted from west-

side products. Total deflection of the vertical is the sum of the partial results.

7* 2k High Speed Computation Method

7*2kl Computation of Venlng Meinesz Coefficients

As was pointed out in the section 7-22 entitled,

Selection of the Size of Surface Elements, the effect of closer areas has to be computed by the circular ring method even when 1°X 1° and

5°X 5° mean anomalies are used in conjunction with the high-speed computer. Of course, computations using the square method can be carried closer than a 3 degree distance from the computation point if squares smaller than l°x 1° are used. It is worthwhile using squares 122 smaller than l°x 1° and reading corresponding mean anomalies if the deflection of the vertical is needed at very many points in a small area. In most cases it is preferable to use the circular ring method for the area inside 6°x 10°, and the high-speed computing machine with

1°X 1° mean anomalies for the area between 6°x 10° and I+0°X 70° and with 5°x 5° mean anomalies for the area outside 1*0 °x 70° •

The Vening Meinesz*coefficients used in the high-speed computing machine are computed according to formula 9^

cf (-CP‘ ^ ~ ~ 2irXm c o s cJy.

i t i and for c (cp , \ ) cos a Is replaced by sin a. This means that there are two sets of coefficients for each latitude. Both sets of the coefficients are symmetrical in respect to meridian, but in the case

1 1 V of c^(tp , \ ) the signs of the coefficients are different east and west of the meridian.

In computations of the Vening Meinesz1 coefficients the main problem is to compute distance and azimuth from the computation point to the surface element in question. In these computations the earth is assumed to be a sphere and the fundamental formulas 95 of spherical trigonometry provide the required distances and azimuths. Knowing the distance, the value is computed from formulas 93 and 90* Sine and cosine values corresponding to computed azimuths are determined.

After computing the area values and the value of the constant, all necessary information is available for computation of Vening Meinesz' coefficients according to formula 9^* 123

Diagram V shows a flow chart for the high-speed computing of

Vening Meinesz'coefficients for 1° X 1° squares for the area 40° x 70° around the computation point. This chart is very similar to the one used in computing Stokes' coefficients. Only one more step is added— azimuth. The final values are punched on the IBM cards in the same order as Stokes' coefficients.

Diagram II shows a flow chart for computing Vening Meinesz' coefficients for 5° X 5° squares. The output on the card is similar to that for the computations of Stokes' functions explained under

6.2323.

7-2U2 Summation of the products; Vening Meinesz'

Coefficients X Mean Anomalies

The flow chart of the summation of Vening Meinesz' coefficients times gravity anomalies inside a 1*0° x 70° area is shown in Diagram VI. Only one set of coefficients is used at a time.

0 in Column 80 indicates the g-aet and 1 in Column 80 Indicates the

T)-set. The anomalies used with this program are sorted first in decreasing order on the basis of latitude (Col. 6 9, 7 0) and then In

Increasing order on longitude (Col. 73-80). The anomalies are arranged so that the longitude of the first group is equal to the longitude of first desired summation minus 3? degrees and the longitude of the last group of anomalies is equal to the last desired summation plus 3^°* 12*

II III

S*t up r and Calculate sin Set address on output card to •tor* In punch eea and ator* eoci. ( in punch band) band

XX IV m

Set A to .5° and Stop store In punch band

I Yea XVI XVII XVIII

Punch results, (9>-5)-* **>' Calculate cos ^ , Increase address on ^ No •In > , cos^xcosV0 output cord by 5 V 19.5 / (Identification on and store N . m output card)

XIII XIV VI

Increase to store words Ko next value In Set 9°- 9^19.5° next location of punch band and store

VII

Compute sin cC o*<* Set up storing inatr c *j to put and

Store c ' in punch In proper location band ^ on output cards

XI IX VIII

Compute cos o( and Compute Compute cos & Calculate sin

' Store c* In punch and _r - cos 5®and store band *

Diagram V. Flow Chart for High-Speed Computation of Vening Meinesz*

Coefficients for l°x 1° Squares Inside a **0°x 7 0 ° Area 125

125

Bap alt. tort. lr to p to adUlatl* iatr. mt, _ to ail a iatract, nit, tortnatia back to alt, fra to* tf nutr<.L„ (tJ.a fer eoaffldato to tottatotop m HKtaoal^k^aff^ •tart kitb 0001 iw

■It. lat \ tortnrttoa to Ittf aaftU nbtnct (toltljflj tortnrttoa tor alt. atff.ad aoal.) tortr.ftr

m n. mx. X«

itnl uco alt. E®. aai

8ton la t acetnl laitttoa ad Stop laat lonfltadi itlOB to to rad

nm, m n.

lit,) aito

m n ,

nn, UIIIF. flw fra lalttol alt. fna tad art

Bit, latltoto Hat i t a to atora latrvtla to iton to ftort loe. flat

D lig m T I. flan Chnrt for High-Speed SuBstlon of Vening Meiness'

Coefficient* X Mean Anoeillea Inside 40°x 70° Area 126

In these computations a no-vord control card Is used with the

following Information

Column

1- 6 Zeros

9-10 Latitude

11-17 Zeros

18-20 Longitude of the first summation

21-27 Zeros

26-30 Longitude of the last summation

31-Uo Zeros If one of the lower five summations

Is desired and 0000000020 if the first

summation is desired.

1*1-80 Zeros with signs.

Usually the anomalies are read In using a 1+0°-vlde latitudinal hand. If the anomalies between 70°H and 25°N are used, the sumnations

for 50°, k9, 1*8, 1*7, k6, 1*5° can be calculated using the same anomalies. In the control card mentioned the first sunuation would be for 50°H.

The answers are punched on IBM cards as follows

Column

1-10 1 or 2 (1 for t), 2 for ()

11-20 latitude (OOOOOOOOxx)

21-30 longitude (OOOOOOQxxx)

31 -1*0 result of summation of products

1*1-80 zeros with signs. 127

The summation of products of anomalies X Vening Meinesz* coefficients for 5° X 5° squares is similar to the procedure used in computation of undulation of the geoid as explained under 6.2332. A flow chart of the program is provided under the same section.

7.2^3 Total Deflection of the Vertical

In the section entitled, Selection of the Size of Sur­ face Elements, borderlines were established between the areas where different computation methods should be used In order to obtain certain accuracies. Later it was decided that the circular ring method should be used inside a 6° x 10° area around the computation point; 1° x 1° squares, used In a U0 ° x 70° area around computation point excluding the 6° x 10° area mentioned before; 5° X 5° squares, used outside the

UO° x 70° area. When computing the effects of the different areas, care should be taken to carry out the integration over the whole sur­ face of the earth and to include no area more than once. In practical computations therefore, the 6° X 10° area is a 7° x 11° area, and the

U0° x 70° area is a 45° X 75° area. If A, B, C, D, Figure 31# are the corner points of the 1° X 1° square in which the point is located for which the deflection of the vertical is desired, and the corresponding corner points of the 6° x 10° area are A^^, B1 and then the 7° x 11° area is A^, Bg, C y D^. The effect of this area is computed by the circular ring method. When computing the effect of the k0° x 70° area at point A, the area A^ Ag A^ A^ is excluded. The effect of the area Ag Bg C ^ A^ Ag should be still sub tracted from the effect of the k0° x 70° area. At point B, the corresponding area is A^ B1 B^ B^; at point C, the area Is Bg Cg 128

C^; and at point D, the area is A1 Bg Dg D^.

This procedure is necessary in

order to eliminate the inappli­

cable common area. The effects

of these areas can now be

easily computed with desk calcu­

lators providing the corre­

sponding Vening Meinesz*

coefficients have already been

calculated with a high-speed Fig. 31. Borderline of 6° X 10° Area. computing machine. Of course the outer border of the UO° X 70° area falls in each case into a different location and hardly ever coincides with the borderline of the 40° x 70° area which is left out in the computations of the effect of outer areas, the effect of outer areas is usually computed at each corner point of 5° x 5° square.

E F Ei pi 2 2 1*6 H, " a

E F

Jl If

h JA-:

Fig. 32 Borderlines of U0° x 70° Areas. In Figure 32 E, F, G, and H are comer points of a 5° X 5° square and

E1 4' F1 k* G1 4 ' 311(1 H1 4 corner Points o:f a corresponding 40 x 70 area. Point A is the corner point of the 1° x 1° square; and A^_g, the corner point for the corresponding 40° x 70° area around it.

In this case a similar method to that used for the 7° x 11° area can be employed. However, the effect of outer area changes very slowly from point to point. It, therefore, seems to be possible to interpolate the effect of the area outside 40° X 70° on a component of the deflection of the vertical at any point inside a 5° x 5° square

+ " with an accuracy of at least -0.05. Of course, if the differences between the points are large, a more accurate method Bhould be used. 8 . ANOMALIES IN UNSURVEIED AREAS

8.1 Estimation of Anomalies

In the previous chapters the formulas have been given and practical

computation methods have been explained for computations of undulations of the geoid and of deflections of the vertical. Using these tools of modern geodesy the shape of the earth would be well known if gravity were known everywhere around the earth. As was mentioned in section

6.1 , gravity measurements unfortunately do not cover the whole world but only a part of the surface.

Anomalies in unsurveyed areas will certainly affect undulation and deflection values computed using the gravimetric method. Therefore it is important to know the size of the error caused by the lack of gravity information as well as all possible methods of imporving the accuracy by filling the empty areas with values which are more representative for the area than no information at all.

There are several methods which give some information in unsurveyed areas:

1. Existing gravity Information is reduced isostatically and zero isostatlc anomalies are used for unsurveyed areas.

2. Zero isostatic anomalies are assumed for empty areas and free air anomalies corresponding to zero isostatic anomalies are computed.

3* Free air anomalies are correlated with topography and geology in surveyed areas and the correlation assumed to prevail in unsurveyed areas.

130 131

4. Existing gravity anomalies are developed in terms of spherical harmonics which in turn will give anomalies in unsurveyed areas.

5. Behavior of artificial satellites is observed and lower degree zonal, harmonics are computed from the information so obtained.

6. Deflections of the vertical are computed gravimetrically and the results are compared to astrogeodetic deflections of the vertical.

Existing gravity material shows that there are several areas where a mean Isostatic anomaly is not zero because of upward or downward bend­ ing of the crust or of unknown mass anomalies in the crust, but it has many times been shown (1 5) that mountains and oceans are very close to being in complete lsostatlc equilibrium. Therefore, if gravity observa­ tions are isostatically reduced, the zero anomaly for unsurveyed areas

Is a good estimation.

If the free air anomalies are used Instead of lsostatlc ones, this same above mentioned phenomenon can be used to advantage by computing free air anomalies corresponding to zero lsostatlc anomalies. This means that free air anomalies would be equal to iBostatlc reduction values with reversed signs. These reduction values can be computed point by point using the usual reduction methods.

lsostatlc reduction values are of course correlated to topography, and, therefore, it is also possible to compute average lsostatlc reduction values from the development of earth's topography Into spherical, harmonics. Hirvonen (2 0) employed this method by using the results of A. Prey (1*3) who expressed the altitudes of the earth's topography in spherical harmonics up to the 16th order. Since Hirvonen's application, many empty areas have been covered by gravity observations. 132

The writer has compared 8l6 squares of 5°X 5° observed mean anomalies to the values computed from Prey's results and to the zero anomaly (51)*

The standard deviations were computed between Frey's and observed anomalies and In turn between the zero anomalies and observed anomalies.

In the former c u e It was -l8.lt mgal and In latter one -I7.lt mgal.

This suggests that It would have been as valid to use zero anomalies as the computed values.

It has been seen In section It.2 that free air anomalies are correlated with local topography or relief, but the correlation becomes much smaller when large areas and mean anomalies are In question. This

Is In agreement with Jeffreys' results (28). High-speed computing machines have provided good opportunities for making statistical analyses with vast gravity material. According to Hauls's analysis (33)

In the Army Map Service, there exists a slight correlation between free air anomalies and topography around the whole world.

Gravity Information has been so scarce that it has not been admissible to try to develop gravity anomalies Into spherical harmonics up to a very high order. The necessary formulas have been developed however and are given in chapter 6.1. The Russian Zhongolovlch (5 8) developed the gravity anomalies as spherical harmonics up to the 8th order of the 8th degree. Since Zhongolovlch's development many new areas have been covered by gravity observations. The writer has com­ pared the actual values with anomalies computed by Zhongolovlch and

* noticed systematic differences (5 1)* It* therefore, was decided not to use Zhongolovlch's values for unsurveyed areas. It should be mentioned here that Kaula (3 3) has Just made a new development of 133

gravity anomalies into spherical harmonics using Zhongolovlch's method

and all Ohio State University material available in 1956.

Jeffreys has computed corrections to the international gravity

formula (28) which include lower degree harmonics. The international

gravity formula, expressed in spherical harmonic, is, in milligals

Y = q i 9 ‘770 + 34^6.o‘R0 (&*cR)+5.3'Po ('s'>’c:f5/) (107)

With Jeffreys1 corrections it is y=^79772.5±l-9)-h(3N39-9±So)H0 (sir>?)+53

M k o ± i ^ 2 (sin tpj&s2X+0.3± 068)^2 fewc^5 2 X (lo8)

-f 2 j (s/rt2s)Ht3 fenCP ) 5 U ) 3 2 .

He based his computations on available values around the whole world but used only a part of the gravity material which is available now.

Because of this, his developments have not been used in the Mapping and Charting Research Laboratory.

These lower degree harmonics are being computed from gravity information by many agencies today, but -they can also be determined from the motions of artificial satellites (*vl). Sometimes the arti­ ficial satellites are overemphasized in this connection. They can possibly give information about lower degree zonal harmonics but hardly any information about local disturbances. Results are not yet available, even for lower degree zonal harmonics.

If there is a local unsurveyed area, such as a lake or small sea, it is possible to estimate a gravity anomaly in the area if gravity 13^

field Is known around the area and there are available several astro- geodetic points in same area. Deflections of the vertical are computed for those points by the gravimetric method and the results are compared to astrogeodetlc values. Gravity anomalies In empty areas are adjusted

so that the differences between astrogeodetlc and gravimetric deflec­ tions of the vertical approach a systematical variation. This method is based on the assumption that the effect of the outer areas is constant on deflection of the vertical at each point, which of course is true only in limited areas.

Whatever method is used, the gravity field around the whole earth should be in balance; therefore, anomalies which are added to unsurveyed areas should be adjusted to eliminate inadmlssable spherical harmonics.

8.2 Balancing the Gravity Field

It was pointed out in section 1.11 that if the origin of the coordinate system were to coincide with the gravity center of the earth, the first degree term of the potential would have to be zero and the first order term of the second degree (PQ1) would have to be zero in order not to have any precession of the earth. These harmonics are illustrated in figure 33* This same rule also applies to gravity anomaly field (2h).

The gravity anomaly is expressed in spherical harmonics by the formula 70

(70) v p t 0 (costy = COS

I«r

+90° 180° -90° 0 +90 Tj, (cos i?) = sin'll cos (A- X) ** * i

i

A - X 2 +90* 180° -90 o 2>J s/nitoos -Pcosfa

Fig.. 33- Inadmissible Spherical Harmonics 136 where

% (^ )=io (c<}nu“3S0^ % » sl”^ ) Pn» (“* % •

According to the statement made above concerning the balance of the gravity field, the following terms should disappear by correcting anomalies.

Jj>0 (cos &) f (cos1$) cos 1 j t (cos fijsi'ffk

7£t (cos$)oos'\ and. ^ (cos&J&in'X..

The corresponding coefficients for these terms are given by formula 73 and they are following

AcasJA'&A^S/h#

^ swficos A A&A A 6/K> 2^

(109)

c ^ 2 /" f ftco&A A A &'*> ^

$%, ~ ^ tT ^ t J'Ctsfts/nAa '&AS/n ft 137

In order to set up the equations of conditions, the following simplifi­ cations of notation are made

Ci — cos ft sin b = sin'llc,os X (110) C = siiftft sin X cf — s//?V* cos iftO 0 S X £ « sin2’ft cos ft <5in X

“>b*Zr\ WUAfV,*) (111)

c o j = 21 IE cL(&,)) )) % - Z Z XJ

If the corrections to the anomalies are denoted by v^, the equations of conditions are

i

a , v, ■+ a z vz * ...... ■+ an vn + = o

b>V, + t>2 Vz + ...... ■* Yn + = ° C, V, + Cz vt-t- + Cn Vr, + M c = o (112)

d , V , ■+ dz v ^ + dr) Vn + °^cj — o

« , v , + e * V i * • ...... *°e. * ° 138

Some mean anomalies Ag (2^', \) are based on observed gravity anomalies, some mean anomalies have been extrapolated, and many squares are represented by zero anomalies. It seems natural that if gravity field must be changed in order to bring it into the proper balance, then most changes should occur in gravimetrically unsurveyed areas. There­ fore, for each mean gravity anomaly, a weight was determined which was relative to the accuracy of the mean anomaly. The Q-factors, mentioned in section 6.24, were taken as an inverse weight l/p for each 5°X 5° square in question.

The normal equations for least square solution are

(113)

+ ^ o _

The correlation values, k, are solved in the usual way, and the corrections to mean anomalies are computed from the following formula

As mentioned, the writer has computed Q values for each 5°X 5° square around the whole world. Using MCRL free air anomaly field as It existed In August, 1958, the writer

then formed normal equations and solved for the k-values of the field

ka * +1 1 .9 1 = -13.30 kc = -43.71

k. = -7 6 .8 9 k • -64.54 CL 6

Corresponding corrections v(^\) were then computed. They are

given in figures 34-35* The unit is 1 x 10-1 mgal. It should he noted

that there is a large systematical correction in the Pacific Ocean.

When any undulation values and, in particular, any values of

deflection of the vertical are computed by using an unbalanced gravity

field, the result must be adjusted by adding the effect of these

corrections. For example, the corrections to the components of the

deflection of the vertical at Meades Ranch are II Ag = +0.14 II A t] = +0.24

Corresponding corrections at Potsdam are M Ag = +0.03 I t At) = -0.10

Corrections have also been computed for N values along the 40°N parallel. Results are given in figure 36. They show that the effect of balancing is not very great, but nevertheless, it should be taken

into account in the final results. Whenever values are computed gravl- metrically, the corresponding anomaly field should be always balanced and corrections computed. Pig. 31*. Corrections to Mean Anomalies Resulting from Balancing the

Gravity Field,Unit 0.1 Mgal., Northern Hemisphere SL-AA

w

s r

Fig. 35* Corrections to Mean Anomalies Resulting from Balancing the

Gravity Field,Unit 0*1 Mgal., Southern Hemisphere m e t e r s i. 6 CorrectionsFig. 36. to N-Valuesalong 40°N Parallel Resulting from Balancing theGravity Field + -lOJ + 5. -5. 10 ^ * W 0 6 60* E 120*1 L _-5 +10 10 -1

METERS 9- COMPUTATIONS AT HIGH EIEVATX039S

9*1 Shape of Equlpotentlal Surfaces

When the constants of the regularized earth are given, it Is

possible to compute theoretical gravity value with and without

centrifugal force at any point outside the earth. It is also possible

to compute the geometry of the spheropotential surface if the mentioned

constants are given. The necessary formulas can he found, for example,

in Hlrvonen's technical report (2 2).

When the gravity anomaly field Is known at sea level, It is possible to compute the separation of eq,uipotential surfaces of geo- potentlal and spheropotential In any one of several different methods:

(1) If the gravity anomalies at sea level are developed In terms of

spherical harmonics, the values can he computed from formula 7k given In section 6.1. This formula is

y - 1 7 & ( § ) q (A*)

(2) Instead of using spherical harmonics a formula can he used which corresponds to the Stokes1 formula and can he derived from Polsson's

Integral formula for a sphere (22). (3) When the undulation values are known at sea level, a so-called coating can he computed. Using this method, anomalies can he computed at any elevation, and from these anomalies, corresponding H(p,i^,X.) values (36) (k2) . (k) Curvature and second derivatives are used In the method developed hy Arnold (l) and

TengstrSm (t9) • In this method the curvature of an anomaly field and I k k of the undulations of the geoid are used. The computation formulas are expressed in very simple forms and computations can he performed rapidly, which in some cases, such as in inertial guidance systems, is very important.

In this connection, only the second method Is discussed in any detail here, because the computational problems are very similar to those used in computations of the geoid using Stokes' formula. Poissorfs

D integral formula for a sphere is

d a (115) o'

where H is a harmonic function,

o a sphere with radius R, P a

point at distance p > R from center

of the sphere and

Z>-vka + f z- 2 Q $ c o s ' P Fig. 37* Elements in Poisson's Integral Formula

Hirvonen (22) derived the following formula for separation of equi- potential surfaces of geopotential and spheropotential

// 2//

lf'rrK , ' V A~0 where

S ( $ , £ J = - 3 D- 3? a a s t t ' f a J 6 ? ) < U 7 ) 1*5

By compari'ig this formula with formula 75# it Is easy to see the simi­

larity. The numerical integration can be performed through summations

and the formula takes the form

V 7T Y m ^ / ( l l 8 )

where q is a surface element.

If the elevation above the sea level is denoted by h, the following relations are derived

p = R + h

£ = 1 + £ R R

If £ = t and R is taken outside of summation signs, the formula A is expressed

" M u - * # » r £ £ A j j t ' y s (*,*>■? ( i i 9 )

where

* < * - *+i?3tgL. s c — _

/ ? - o r * z - 2 + o o s ^ 1 2 0

Table 1* shows values of function S(t,i|r) at different h values. These same values are drawn in figure 3@ where it is possible to observe how the effect of areas at different distances changes when the elevation changes. N(p,*f,\) values can be computed In a high-speed computing

machine using the same method and program vhlch were used In computa­

tions of undulations of the geoid. Formula 8l is modified to

(121)

where

(122)

Figures 39* **0 shcw N(p,i^,\) values at selected elevations along a parallel around the world. The profiles are marked with corresponding elevation values. In these computations only 5°X 5° mean anomalies were used. The accuracy of the values can be improved by using l°x 1° mean anomalies in computations of the effect of closer areas to the computation point.

9.2 Gravity Anomalies

The anomaly at height, h, can be computed (22) on the basis of anomalies Ag_ at sea level by the Integral formula

(123)

where 0 -« — 2 C O S & . Function U M d In Connection with Computation at High Elevation

6 t c o m - t . i . . t - eo«* + A t t2 - 2 t coat' 8(^*) - A ♦ X V t co.« ♦ t - s oort - 3 «.♦ in ------

t - 1 +

▼A h*o b - 1 5 2 4 0 n h - 3 0 4 0 0 i» h - i 3 5 2 0 0 n h - 3 7 0 * 1 0 0 8 h * 9 2 6 0 0 0 n f t 1163.040 6 9 2 ,141 408.317 79.364 4 3 .0 8 6 19*932 .2 3 0 7 .995 487,631 3 5 2 .354 7 3 .9 5 5 4 3 .0 2 8 19.947 .3 3 9 5 .765 3 6 ].2 4 5 295.632 78.288 4 2 .9 3 3 1 9.9 3 8 .4 2 9 9 .4 2 ! 283,961 0 4 9 ,2 7 ’ 7 7 .3 8 3 42.801 1 9.9 2 7 .5 241.448 233,203 2 1 3 .229 7 6 .2 6 6 4 2 .6 3 3 19,912 .6 2 0 2 .696 11 ^.9(V > 1 8 5 .436 74.971 4 2 .4 3 0 19.893 .7 174,942 171 .9 ,1 1 1 6 3 .686 7 3.5 2 5 4 2 .1 9 5 19.872 .K 154.070 132.023 146.353 71.961 4 1 ,9 2 8 19.8 4 7 .9 1 3 7 .794 136.35? 132.287 7 0 .3 1 0 4 1 .6 3 2 19.8 1 9 l.o 124.737 123,653 1 2 0 ,679 6 8 .5 9 9 4 1 ,3 0 9 19.7 8 8 I.I ------1U.054 113.276 110.953 6 6 .8 5 4 4 0 .9 6 0 19.7 5 4 1.2 105.076 1 04 ,469 1 0 2 .647 6 5 .0 9 5 4 0 .3 8 9 19.7 1 7 1.3 97.482 97.006 93.604 63.340 4 0 .1 9 7 19.6 7 7 1 .* 9 0 .9 5 6 90.5 7 6 99.4 4 4 6 1 .6 0 3 3 9 .7 8 7 19.634 1.5 83.284 84.977 84,039 39.896 3 9.3 6 1 19.588 ' 1.4 4J.15* 8 0.0 3 4 T9.2M 58.227 3 8.9 2 1 19.339 1.7 7 5.9 0 5 75.697- 7 5 .0 6 4 3 6 .6 0 3 38.469 19.487 1.3 71.981 7 1,4 0 7 7 1 .2 7 4 5 5.0 2 3 3 8 .0 0 7 19.432 1.9 64.460 68.313 67.861 5 3 .5 0 6 3 7 .5 3 6 19.375 ?.C 6 5 .2 3 7 65.159 64.7 7 1 3 2 .0 3 7 3 7 .0 6 0 ?.o 44.888 44.8 5 7 4 4 .7 4 8 4 0 .1 8 0 3 2 .2 4 7 fill#! 6.~ 34.396 34.393 34.347 32.302 2 7 .8 9 4 17 .6 9 6 5.0 27.916 2 7.9 1 ? 27,901 26,853 24.255 16 .6 9 8 4.'.' 23.4?i 2 3.4 7 5 2 3 .4 6 9 2 2 .8 8 7 21.274 15.661 7 . 20.201 00,207 20.2 0 7 19.8 6 9 1 8.8 2 6 14.6 2 9 P." 17.676 17.683 17.486 17.488 1 6 .7 9 3 1 3.6 2 9 9 .0 15.654 15.662 15.667 15.333 13.0 8 2 12.6 7 8 i o.o 13,989 13.997._ 14.003 13 .9 4 6 1 3 .6 2 2 11.7 8 4 1 1.0 12.306 12.595 12.602 12.382 12.361 10.9 4 8 1 2 .0 11.384 11 • 1"3 1 1 .4 0 0 11 .4 0 6 1 1 .2 5 7 10.170 1 ».o 10.737 10.346 1 0.3 3 4 10.378 10.282 9.445 i 4 . r " .4 1 3 9.424 9 .4 3 2 9 .4 7 0 9 .4 1 3 8 .7 7 2 1 5 .0 P., 393 P . 607 8 .6 )1 8.653 8.6 3 2 8,1 4 5 1 6 .* ' 7.8 3 3 ■’.964 ’.873 7.928 7 .9 2 4 7.561 1 7 ,0 7.1 3 6 7.193 7.20-4 7.263 7.2 7 9 7.0 1 6 ; 6. ■; 6.5 7 7 6 .5 9 6 6 .5 9 5 6 • 660 6 .6 8 3 6.505 1 9 .0 6.C18 6 .007 6 .0 1 3 6 .1 0 5 6 .1 4 4 6 .0 2 7 ? ? • 5 .50? 5.511 5 .5 2 0 5.5 9 2 5.641 5 .3 7 8 21.'-' 5 .0 7 J 5.074 5 .0 4 3 5.1 1 7 5 . i 7 l 5 .1 5 6 ? 2 • 4,532 4.591 6 .6 0 0 4 .6 7 6 4 .7 3 7 4 .7 5 8 2 1 . ' 4.163 4.1 77 4 .1 8 5 4 .2 6 3 4 .3 3 0 4 .3 8 2 ? 4 . ‘ 7.731 7.790 3 ,7 9 9 3.8 7 7 3,9 4 8 4.0 2 7 2 1 .0 3.413 3.477 3.4 1 6 3.5 1 5 3 .5 8 9 3.6 9 0 2 6.0 >.M' 2.096 3.095 3.174 3.251 3.3 7 1 2’.0 2.756 2 .7 6 5 2 .7 7 3 2 .8 3 3 2.9 3 2 3.0 6 9 2 P .'' 7 .4 6 ? 2.461 2 ,4 70 2 .5 5 0 2 .6 3 1 2.7 8 1 2 9 .0 0 .1 4 6 0.174 2 .1 8 3 2 .2 6 3 2 .3 4 3 2.5 0 8 3Q.,.o 1.494 1.993 1 . 9 U 1.9 9 2 2 .0 7 5 2.2 4 7 --Ti.C .732 .741 . 749 .829 .9 1 4 1.118 /.o,r .144- .1 6 1 - . 1 5?- .0 7 7 - .0 0 9 .224 4 5 .0 .8 6 3 - .8 3 1 - .9 5 3 — .7 8 0 - .6 9 5 - .4 8 1 - 5(1.0 1.492- ,7 4 5 - ! . 183- 1.319- 1.241- 1 .0 3 0 - 5 5 .o 1.796- 1 .7 9 ''- 1 .7 8 3 - 1 .7 1 9 - 1 .6 4 5 - 1 « 444 — f o . ’} 0.06 3- 2.057- 1 .9 9 7 - 1.929 ■ 1.740- r,5,c 0.213- '.22°- 2 . 2 ’ 1- 2 .1 6 9 - 2 .1 0 6 - 1 .9 3 3 - 0 0 . 3 0 .1 3 3 - 0 .0 9 3 - 2 .2 9 3 - 2 .2 4 5 - ? .1 8 9 - 2 .0 3 2 - 7 5 .0 0 .2 8 4 - 0 .2 6 4 - 2 .2 7 9 - 2 ,2 3 7 - 2 .1 8 3 - 2 ,0 4 9 - 9 0 ,0 2 .1 9 3 - 2 .1 9 4 - ?.190- 2.154- 2*11?— 1.991- H S.’l 2 .C 4 2 - 2 .0 J 9 - ? • C15 ■ • 2 .0 0 3 - 1 .9 7 0 - 1 .8 6 9 - 9C,-. 1 .8 2 9 - 1 .3 2 6 - !.024- l.800- 1.771- 1 .6 9 1 - 9 5 .0 1.567- 1.565- 1 .5 4 3 - 1 .5 4 6 - 1 .5 2 5 - 1 .4 6 4 - i CO.-: 1 .2 6 5 - 1 .2 6 4 - 1 .7 6 3 - 1 .2 5 1 - 1.2 37- 1.196- ’ 0 5 ,0 .9 3 2 - .9 3 1 - .931 - .9 2 5 - .9 1 3 - .8 9 7 - ■llc.o ,5 7 5 - .5 7 5 - .6 74- .5 7 3 - ,5 7 5 - .6 7 3 - 11 5 .0 .2 0 2 - .201- .7.1 )■ .2 1 0 - .216- .2 3 3 - 1 2 0 ,' .179 .177 .17- .1 6 4 .151 .116 125. '■ • 56rt .638 . 156 .5 3 9 .620 .468 ! 30 .0 ,934 .932 .979 .908 .3 8 ? .8 1 4 ll*. 5 1.J4S 1.292 1.239 1.341 i . i i 2 1.149 1 3 0 .0 1 .634 1 .633 1.9 30 1 .599 l .564 1.466 145.0 1.951 1 .948 1 • 344 1 . V 1 0 1 .5 7 0 1.760 150,0 2.236 2.232 '.7 7'! 2 .1 9 0 2 .1 4 7 2.0 2 6 155.0 2.4 8 5 2.4 81 ’.',77 2 .4 3 6 2.7-8 9 2 .2 5 9 If---',0 2 .6 9 4 2 .6 9 0 2 . 6 36 2.441 i.6 9 1 2.4 3 4 16? .0 2.861 2 .8 5 6 2 .8 3 ? 2.8(J7 2 .7 5 5 2.6 1 0 1 7 0 .0 2.9 8 2 2.9 7 7 2 .9 7 ? ? . 926 2 .8 7 2 2.7 2 3 175,0 3.055 3 .030 7.046 2.9 9 8 2 .9 4 4 2 .7 9 ? 1PC.0 3.079 3.075 3.0 70 1.052 2 .9 6 8 2.8 1 3 90 so1 90' 120 ISO ISO

- 0 NAUTICAL MILES - 200 NAUTICAL MILES " 9 0 0 NAUTICAL MILES

Fig. 33. Function Used in Connection with Computation at High Elevation I k 9

Using the same notation as In the previous formulas, the formula takes the following form

A Q = — 7T f / -3 ' d d (12U) % ‘fTT'S** J V /+ * Z - 2 4 Cost* x 6* and for high-speed computing machines the formula is

(125)

vhere

d & (126) t ' J 4//p-R s/ /s-i'1--2-t cost

If more accurate values are desired, it is necessary to consider two additional small terms, as derived by V«ning Meinesz in 1928, to formula 126 in order to eliminate spherical harmonics of the degree

0 and 1. The coefficient is then

Cf^ l> V/+t?-2-tcosp ,3 s>1 S’3

A set of c (q>,\) are computed for certain selected elevations and latitudes. These formulas are suitable to be used in high-speed computing machines and the programs used for computations of undula­ tions of the geoid can be adopted for these computations. METERS METER® -20 0 4 45- 30. IO. O. IO IO.

. Fig. J .

39' Separation NUIA MILES M 0 0 5 NAUTICAL 0 0 2 000 FEET 0 0 ,0 0 0 1 Selected Parallel at Various Elevations between l a c i t u a n 0 2 tto of Geopotential and Spheropotential SurfacesAlong a

s e l i m 30 40 130 140 50 iN X 150 0°-l80°E 0 7

80* 80* E r- L-20 .-1010 - . . .-10 30 45 40 20 10

150 METERS METERS -IO_ ‘ 40 ‘40 30_ 20 30. - J Fig. Separation of Geopotential and Spheropotential Surfaces Along a Along Surfaces Spheropotential and Geopotential of Separation Fig. 9 S 190 ,\ 500 500 NUIA MILfcS NAUTICAL 0 0 2 100,000 Selected Parallel at Various Elevations between 0°-l80°W between Elevations Various at Parallel Selected 200 N A U T I C A LM l L f r S PtET 300 210 220 320 330 240 ^ 340 250 t ~

_ -40 _-40 30 40 20 O S IO to

T£T- 10. RECENT DEVELOPMENTS

In previous chapters the theory and computational methods used in the world-wide gravity project are discussed and explained. During these eight years that the project work has teen in progress, many new ideas concerning physical geodesy have been brought up In scientific meetings and publications; for example, de Graaff-Hunter's model earth and reduction of gravity anomalies presented in the Toronto meeting of

International Association of Geodesy (7). There have also been new theories advanced in which computations'of separations of equipotential surfaces of geopotential and spheropotential are performed on the surface of the earth rather than at sea level (3T)• In practical computations Molodensky's and"Stokes' methods give about the same results, which means that computed space coordinates of a point will also be about the same for the two methods.

Deflections of the vertical can be also computed several ways; for example, Professor Tsuboi has used the Bessel-Fourler series instead of the circular ring method described here (50). Both methods give the same results, but the development of the series can be conveniently used also for computing the deflections of the vertical at high altitudes.

Theories and methods of physical geodesy require still further investigations even though present methods give satisfactory accuracy for today's practical applications. In order to improve our knowledge of the shape of Mother Earth much more observational, data are required around the whole earth.

152 REFERENCES

Arnold, K. Beitrfige zur gravtmetrlachen GeodtLsle, Verttff. d.

Geod. Inst. Potsdam, Nr. 11, Berlin, 1956

Bomford, G., Geodesy. Oxford, 1952.

Bullard, E. C., Gravity Measurements In East Africa. Trans­

actions of the Royal Society of London, Ser. A. V. 235#

1936.

Bullard, E. C., The Figure of the Earth. Monthly Notices of the

Royal Astronomical Society, Geophys. Suppl. 5# No. 6,

pp. 186-192, 19^8.

Cassinis, G., Sur 1'Adaption d'une Formule Internationale pour

la Pesanteur Normale. Bulletin Geodeslque, Nr. 2 6, 1930,

Avril-Mai-Juin.

Cook, A. H., Comparison of the Acceleration due to Gravity at

the National Physical Laboratory, Teddington, the Bureau

International des Polds et Mesures, Sevres, the Physikalisch-

Technlsche Bundesanstalt, Brunswick, and Geodetic Institute, ^ ; Potsdam. Proc. of Royal Society A, Vol. 312, p. M)8,

London, 1952. de Graaff Hunter, J.,Reduction of Gravity Observations for

Stokes' Formula. Advance Report of Special Study Group No.

8, International Association of Geodesy, June 1957* de Vos van Steenvijk, J. E., Plumb Line Deflection and Geoid In

Eastern Indonesia. Publ. of the Netherlands Geodetic Comm.,

19^7. 154 9* Heiskanen, W. A., 1st die Erde eln Drelachsiges Ellipsoid?

Gerlanda Beltr. z. Geophys. Vol. 19# 1928.

10. Heiskanen, W. A., New Isostatic Tables for Reduction of the

Gravity Values Calculated on Basis of Airy's Hypothesis.

Fubl. Isost. Inst, of IAG No. 2, Helsinki, 1938.

11. tHelskanen, W. A., Investigations on the Gravity Formula, Publ.

Isost. Inst, of IAG, No. 1, Helsinki, 1938.

12. Heiskanen, V. A., On the World Geodetic System, Publ.Isost.

Inst, of IAG, No. 2 6, 1951* Publ. Inst, of Geodesy, Photo-

grammetry and Cartography of OSU, No. 1, 1952.

13. Heiskanen, W. A., Research Directed Toward the Application of

Gravity Data to the Determination of the Geoid in Eurasia. **> Final. Report under Contract No. AF19(604)-287, P w t II-

Geodetic Manual, Mapping and Charting Research Laboratory,

Columbus, 1956.

14. Heiskanen, W. A. and U. A. Uotlla, Some Recent Studies on

Gravity Formulas. Contributions in Geophysics: In Honor

of Beno Gutenberg, Pergamon Press, Ltd., London, pp. 200-208,

1958. * 15. Heiskanen, W. A. and F. A. Venlng Meinesz, The Earth and Its

Gravity Field, Mc-Graw-Hill Book Co., Inc., New York, 1958.

16. Heiskanen, W. A., U. A. Uotlla, and 0. tf. Williams, The Mean Free

Air Gravity Anomalies, Geold Contour Curves, and the Average

Deflections of the Vertical. Air Force Survey in

Geophysics No. 9 8, Air Force Cambridge Research Center,

1958. 155

17* Heiskanen, W. A., The Columbus Geold. Trans. Aner. Geophys.

Union, Vol. 38, Ho. 6, 1957-

18. Heiskanen, V. A., On the Gravimetric Undulations of the Geold and

the Deflections of the Vertical. Geoflslca Pura e Applicata,

Vol. 1*1 (1958/HI) pp. 1-18, Milano, 1958.

19. Helmert, F . R., Die Mathematischen und Fhyslkalischen Theorien

der Hfiheren Geoddsie. IX Tells Die Fhyslkalischen

Theorien, Leipzig, 188*4-.

20. Hlrvonen, R. A., The Continental UndulationB of the Geold.

VeriJff. Finn. Geod. Inst., No. 19, Helsinki, 1934.

21. Hlrvonen, R. A., On the Establishment of the Values of Gravity

for the Rational Reference Stations. Publ. Isost. Inst of

IAG No. 19, Helsinki, 1948.

22. Hlrvonen, R. A., Gravity Anomalies and Deflections of the Verti­

cal Above Sea Level. Trans. Amer. Geophys. Union, Vol. 33#

Ho. 6, 1952.

23. Hlrvonen, R. A., On the Precision of the Gravimetric Determina­

tion of the Geold. Trans. Amer. Geophys. Union, Vol. 37*

Ho. I, 1956.

2*4-. Hlrvonen, R. A., The Balancing of the Gravity Field. Technical

Paper Ho. 716-3, Mapping and Charting Research Laboratory,

(Ohio State University Research Foundation) Columbus, 1957*

25* Hlrvonen, R. A., The Hew Theory of Gravimetric Geodesy (I).

Technical Papers Hob. (716)-4-290 and (833)-4-290, Mapping

and Charting Research Laboratory, Columbus, 1959* 156

26. Hopmann, Joseph, {fber die gravimetrlsche und astronomische

Bestiomung von Lotabvelchungen und Ihre Ausvirkung ouf

trigonometrlsche Netze. Verttffentlichungen dee Institutes

ftlr Erdmessung No* 12, 1950.

27* International Business Machines Corporation, 650 Magnetic Drum

Data-Processlng Machine, Manual of Operation. New York,

1955.

28. Jeffreys, H., The Determination of the Earth's Gravitational

Field (Second Paper). Mon. Not. Royal Astr. Soc. Vol. 5#

PP. 55-56, 19^3.

29* Jung, K., Flgur der Erde. Handbuch der Fhyslk. Edited by

S. Flugge, Vol. ^7, Geophysics I, pp. 563-639# Berlin,

Gdttingen, Heidelberg, 1956.

30. Jung, K., Physical Geodesy II. Unpublished Lecture Notes at

Ohio State University, 1959*

31. Jung, K., Clairaut's Theorem and the Flattening of Single

Meridians or Longitude Zones. Forthcoming Technical Paper,

Research Foundation, Ohio State University, 1959*

32. Kasansky, I., Ein Praktischer Versuch der"Gravimetrischen

Bestimmung der Lotabvelchungen. Verh. d. 7, Tag. Balt.

Geod. Konmu, Helsinki, 1935*

33* Kaula, W. M., Statistical and Harmonic Analysis of Gravity.

Army Map Service Technical Report No. 2k, Washington, D.C.,

1959. 157

3^. Lanibert, W. D., The Reduction of Observed Values of Gravity to

Sea Level. Bull. Geodesique Ho. 26, 1930* PP* 128-130.

3?. Lambert, W. D. and F. V. Darling, Tables for Determining the Form

of the Geold and Its Indirect Effect on Gravity. U.S. Coast

and Geodetic Survey Spec. Ho. 199* Washington, D.C., 1936.

36. Lambert, W. D., The Problem of Gravity at Considerable Elevations

Above the Surface of the Earth. Technical Paper 1^5, Mapping

and Charting Research Laboratory, Columbus, 1951*

37* Molodensky, M. S., Hie External Gravitational Field of the Figure

of the Earth (in Russian). Publ. Acad. Sci., Geogr. and

Geophys. Series, Vol. 12, Ho. 3* Moscow, 19^8.

38. Morelli, Carlo, Compersazione della rete Internationale della

Stazione di Referimento per le Misure di Gravita Relative.

Facolta di Scienze e d'Ingegneria dell'Universita di Trieste,

19^60

39. Morelli, Carlo, Contributo alia rete Oravimetrlca Europea.

Instltuto Hazionale di Geoflsica, Roma, 195^*

^0. Morelli, Carlo, Gravimetry. Spec. Study Group Ho. 5, IAG XI

General Assembly, Toronto, 1957. “

4l. O'Keefe, J. A., Ann Eckels and R. Kenneth Squires, Zonal

Harmonics of the Earth's Field and the Basic Hypothesis of

Geodesy. Paper presented in UOth Annual Meeting of American

Geophysical Union, 1959* b2, Orlln, H., Hie Three Components of the External Anomalous Gravity

Field. Paper presented in kOth Annual Meeting of American

Geophysical Union, 1959* 158

1+3 • Prey, A., Darstellung der Ho hen-und Tlefenverh<nisse der Erde

durch elne Entvicklung n&ch Kugelfunktlonen bis zur 16.

Ordnung. Abh. Ges. Wlsa., G&ttlngen, Math.-phys. Kl., N.F.

11, 1 , 1922.

1+1+. Rice, D. A., Deflections of the Vertical from Gravity Anomalies.

Bull. Geodeslq.ue, No. 25* 1952.

1+5. Solllns, A. D., Tables for the Computation of Deflections of the

/ / Vertical from Gravity Anomalies. Bull. Geodesique No. 6,

19^7.

1+6. Stokes, G., On the Variation of Gravity and the Surface of the

Earth. Trans., Cambridge Phil. Soc. 8, p. 672, 161+9*

1+7* Tanni, L., On the Continental Undulations of the Geold as

Determined from the'Present Gravity Material. Publ. Isost.

Inst, of IAG, No. 18, Helsinki, 19*+8.

1+8. Tengstrttm, E . , Outlines of a Method for Determining the Geold In

Sweden by Free-Air Anomalies. Rikets Allmftnna K&rtverk,

Meddelande Nr. 22, Stockholm, 195*+ *

1+9* Tengstrtta, E., Determination of the Earth's Gravity Field and Its

Level Surfaces. Meddelande frfin geodetlska lnstitutlonen

vld Upsala universltet Nr. l+, Stockholm, 1958.

50. Tsubol, Chujl, A New and Simple Method for Calculating the

Deflections of the Vertical from Gravity Anomalies with the

Aid of the Bessel-Fourier Series. Proc. of the Japan

Academy, Vol. 30 (195*+) No. 6. 159

51. Uotlla, Urho A., Determination of the Shape of the Geold, Publ,

of the Institute of Geodesy, Photogrananetry and Cartography

Ho. 7* pp. 90-97* Columbus, 1957-

52. Vening Melnesz, F. A., A Formula Expressing the Deflections of

the Plumb Line In the Gravity Anomalies and Some Formulas

for the Gravity Field and the Gravity Potential Outside the

Geold. Proc. Konlnkl. Akad. Van Wetensch. Amsterdam,

Vol. 31* 1952.

53* WooHard, George P., World-Wide Gravity Measurements Conducted

During the Period June, 19^9 - January, 1952. Woods Hole

Oceanographic Inst*, Ref. Ho. 52-59* 1952| 53-36, 1953*

5^-53* 195fcj 5^-5^* 195*.

54. Wool lard, George P. and W. E. Bonlnl, A Review of the Factors

Affecting the Accuracy of Long-Distance and Intercontinental

Gravimeter Measurements. Trans. Amer. Geophys. Union, Vol.

36, pp. 575-583, 1955.

55. Woollard, G. P., N. C. Harding and J. C. Rose, The Problem of

Calibrating High-Range Geodetic-Type Gravimeters. Trans.

Amer. Geophys. Union, Vol. 38, pp. 12-23* 1955*

56. Woollard, G. P.* J. C. Rose and W. E. Bonlnl, Hie Establishment

of the International Gravity Standard. Trans. Amer. Geophys.

Union, Vol. 37* pp. 143-155* 1956.

57* Zagrebln, D.* Stokes' Formula for the Case of an Ellipsoid Level

Surface. Bull, de l 1Inst. Astr. de l'Adad. des Sciences de

l'U.S.S.R., Ho. 52* Leningrad, 1944. i 6 o

58. Zhongolovlch, I., The External Gravity Field of the Earth and the

Fundamental Constants Connected vith It (in Russian).

Academy of Sciences, U.S.S.R., Moscow,, 1952. l6l

AUTOBIOGRAPHY

I, Urho Antti Kalevi Uotlla, was born In PttytyA, Finland, February 22,

1923* I received my grade school education In the public schools of

Ptiytyil, and my secondary school education at the Second Finnish

Lyceum in Turku, Finland.

In 19^6, I was granted a Bachelor of Science degree and, in 19b 9, a Master of Science degree in Surveying and Geodesy by Finland’s

Institute of Technology.

I was a Surveyor and Geodesist with the Finnish Government before coming to the United States in 1951* I have, since 1952, been employed by The Mapping and Charting Research Laboratory, The Ohio

State University as Research Assistant, 1952-53? Research Associate,

1953-58; and Associate Supervisor, 1958— j under the world wide gravity project and several other projects.

During my stay at The Ohio State University I have completed the requirements for the degree Doctor of Philosophy, in addition, to continuing my regular employment.