REPORT DOCUMENT PAGE

GEODESY FOR THE LAYMAN

DEFENSE MAPPING AGENCY

BUILDING 56 U S NAVAL OBSERVATORY DMA TR 80-003

WASHINGTON D C 20305

16 March 1984

FOREWORD

1. The basic principles of geodesy are presented in an elementary form. The formation of

geodetic datums is introduced and the necessity of connecting or joining datums is

discussed. Methods used to connect independent geodetic systems to a single world

reference system are discussed, including the role of gravity data. The 1983 edition of

this publication contains an expanded discussion of satellite and related technological

applications to geodesy and an updated description of the World Geodetic System.

2. The Defense Mapping Agency is not responsible for publishing revisions or identifying

the obsolescence of its technical publications.

3. DMA TR 80-003 contains no copyrighted material, nor is a copyright pending. This

publication is approved for public release; distribution unlimited. Reproduction in whole

or in part is authorized for U.S. Government use. Copies may be requested from the

Defense Technical Information Center, Cameron Station Alexandria, VA 22314.

FOR THE DIRECTOR:

 VIRGIL J JOHNSON

 Captain, USN

 Chief of Staff

DEFENSE MAPPING AGENCY

The Defense Mapping Agency provides mapping, charting and geodetic support to the Secretary

of Defense, the Joint Chiefs of Staff, the military departments and other Department of Defense

components. The support includes production and worldwide distribution of maps, charts,

precise positioning data and digital data for strategic and tactical military operations and weapon

systems. The Defense Mapping Agency also provides nautical charts and marine navigational

data for the worldwide merchant marine and private yachtsmen.

The agency is under the direction and control of the Under Secretary of Defense for Research
and Engineering and its director is responsible to the Joint Chiefs of Staff for operational matters.

The Defense Mapping Agency maintains close liaison with civilian agencies of the U.S.

government which are engaged in mapping, charting and geodetic (MC&G) activities. To further

the advances in MC&G, the agency works with various national and international scientific and

operational organizations.

PREFACE

GEODESY FOR THE LAYMAN, first published in 1959, was written by Lt Col Richard K.

Burkard (retired). Since that time and through this, its fourth revision, a host of authors have

revised and supplemented the original work. The publication attempts to develop an appreciation

of the scope of geodesy and to generate some understanding of the problems encountered in the

science. The basic principles are presented through discussions of various geodetic topics such

as: ellipsoids, geoid, horizontal surveying, leveling, geodetic datums and gravity. This edition

provides a revised chapter on Satellite Geodesy to present the results of growth in this field and a

new chapter on Other Developments in Geodesy to present the effect of recent applications of

new technology to Geodesy. Also, the chapter on the World Geodetic System has been

extensively revised and placed last in keeping with the World Geodetic System’s role as the end

product of much of the endeavor described in the preceding chapters.

TABLE OF CONTENTS

FRONT MATTER

PREFACE

INTRODUCTION

CHAPTERS:

I History of Geodesy

II Figure of the Earth

III Geodetic Surveying Techniques

IV Geodetic Systems

V Physical Geodesy

VI Satellite Geodesy

VII Other Developments in Geodesy

VIII The World Geodetic System

BIBLIOGRAPHY

APPENDIX, Spherical Harmonics
INTRODUCTION

What is geodesy? Who needs it and why? These are some of the questions asked by many

people. Actually, geodesy is nothing new having been around for centuries. Webster defines

geodesy as "that branch of applied mathematics which determines by observation and

measurement the exact positions of points and the figures and areas of large portions of the

earth’s surface, the shape and size of the earth, and the variations of terrestrial gravity." It is a

specialized application of several familiar facets of basic mathematical and physical concepts. In

practice, geodesy uses the principles of mathematics, astronomy and physics, and applies them

within the capabilities of modern engineering and technology. A thorough study of the science of

geodesy is not a simple undertaking. However, it is possible to gain an understanding of the

historical development, a general knowledge of the methods and techniques of the science, and

the way geodesy is being used to solve some Department of Defense (DoD) problems.

In the past, military geodesy was largely involved with the practical aspect of the determination

of exact positions of points on the earth’s surface for mapping or artillery control purposes while

the determination of the precise size and shape of the earth was a purely scientific role. However,

modern requirements for distance and direction require both the practical and scientific

applications of the science to provide the answers to problems in such fields as satellite tracking,

global navigation and defensive missile operations.

Chapter I HISTORY OF GEODESY

Man has been concerned about the earth on which he lives for many centuries. During very early

times this concern was limited, naturally, to the immediate vicinity of his home; later it expanded

to the distance of markets or exchange places; and finally, with the development of means of

transportation man became interested in his whole world. Much of this early "world interest" was

evidenced by speculation concerning the size, shape, and composition of the earth.

The early Greeks, in their speculation and theorizing, ranged from the flat disc advocated by

Homer to Pythagoras’ spherical figure-an idea supported one hundred years later by Aristotle.

Pythagoras was a mathematician and to him the most perfect figure was a sphere. He reasoned

that the gods would create a perfect figure and therefore the earth was created to be spherical in

shape. Anaximenes, an early Greek scientist, believed strongly that the earth was rectangular in

shape.

Since the spherical shape was the most widely supported during the Greek Era, efforts to

determine its size followed. Plato determined the circumference of the earth to be 40,000 miles

while Archimedes estimated 30,000 miles. Plato’s figure was a guess and Archimedes’ a more

conservative approximation. Meanwhile, in Egypt, a Greek scholar and philosopher,

Eratosthenes, set out to make more explicit measurements.

He had observed that on the day of the summer solstice, the midday sun shone to the bottom of a

well in the town of Syene (Aswan). Figure 1. At the same time, he observed the sun was not

directly overhead at Alexandria; instead, it cast a shadow with the vertical equal to 1/50th of a

circle (7° 12'). To these observations, Eratosthenes applied certain "known" facts (1) that on the

day of the summer solstice, the midday sun was directly over the line of the summer Tropic Zone

(Tropic of Cancer)-Syene was therefore concluded to be on this line; (2) the linear distance
between Alexandria and Syene was 500 miles; (3) Alexandria and Syene lay on a direct

northsouth line.

FIGURE 1 ERATOSTHENES’ METHOD FOR DETERMINING THE SIZE OF THE EARTH

From these observations and "known" facts, Eratosthenes concluded that, since the angular

deviation of the sun from the vertical at Alexandria was also the angle of the subtended arc, the

linear distance between Alexandria and Syene was 1/50 of the circumference of the earth or 50 x

500 = 25,000 miles. A currently accepted value for the earth’s circumference at the Equator is

24,901 miles, based upon the equatorial radius of the World Geodetic System (Chapter VIII).

The actual unit of measure used by Eratosthenes was called the "stadia." No one knows for sure

what the stadia that he used is in today’s units. The measurements given above in miles were

derived using one stadia equal to one-tenth statute mile.

It is remarkable that such accuracy was obtained in view of the fact that most of the "known"

facts and his observations were incorrect: (1) although it is true that the sun at noon is directly

overhead at the Tropic of Cancer on the day of the summer solstice, it was erroneously

concluded that Syene lay on the line. Actually, Syene is 37 miles to the north; (2) the true

distance between Alexandria and Syene is 453 miles and not 500; (3) Syene lies 3° 30' east of the

meridian of Alexandria; (4) the difference of latitude between Alexandria and Syene is 7° 5'

rather than 7° 12' as Eratosthenes had concluded.

Another ancient measurement of the size of the earth was made by the Greek, Posidonius. He

noted that a certain star was hidden from view in most parts of Greece but that it just grazed the

horizon at Rhodes. Posidonius measured the elevation of the same star at Alexandria and

determined that the angle was 1/48th of circle. Assuming the distance from Alexandria to

Rhodes to be 500 miles, he computed the circumference of the earth as 24,000 miles. While both

his measurements were approximations when combined, one error compensated for another and

he achieved a fairly accurate result.

Revising the figures of Posidonius, another Greek philosopher determined 18,000 miles as the

earth's circumference. This last figure was promulgated by Ptolemy through his world maps. The

maps of Ptolemy strongly influenced the cartographers of the middle ages. It is probable that

Columbus, using such maps, was led to believe that Asia was only 3 or 4 thousand miles west of

Europe. It was not until the 15th century that his concept of the earth's size was revised. During

that period the Flemish cartographer, Mercator, made successive reductions in the size of the

Mediterranean Sea and all of Europe which had the effect of increasing the size of the earth.

The telescope, logarithmic tables, and the method of triangulation were contributed to the

science of geodesy during the 17th century. In the course of the century, the Frenchman, Picard,

performed an arc measurement that is modern in some respects. He measured a base line by the

aid of wooden rods, used a telescope in his angle measurements, and computed with logarithms.

Cassini later continued Picard's arc northward to Dunkirk and southward to the Spanish

boundary. Cassini divided the measured arc into two parts, one northward from Paris, another

southward. When he computed the length of a degree from both chains, he found that the length

of one degree in the northern part of the chain was shorter than that in the southern part. Figure

2. This unexpected result could have been caused only by an egg-shaped earth or by

observational errors.
FIGURE 2 CASSINIS’ ELLIPSOID; HUYGEN’S THEORETICAL ELLIPSOID

The results started an intense controversy between French and English scientists. The English

claimed that the earth must be flattened, as Newton and Huygens had shown theoretically, while

the Frenchmen defended their own measurement and were inclined to keep the earth egg-shaped.

To settle the controversy, once and for all, the French Academy of Sciences sent a geodetic

expedition to Peru in 1735 to measure the length of a meridian degree close to the Equator and

another to Lapland to make a similar measurement near the Arctic Circle. The measurements

conclusively proved the earth to be flattened, as Newton had forecast. Since all the computations

involved in a geodetic survey are accomplished in terms of a mathematical surface (reference

ellipsoid) resembling the shape of the earth, the findings were very important.

CHAPTER II FIGURE OF THE EARTH

The expression "figure of the earth" has various meanings in geodesy according to the way it is

used and the precision with which the earth’s size and shape is to be defined. The actual

topographic surface is most apparent with its variety of land forms and water areas. This is, in

fact, the surface on which actual earth measurements are made. It is not suitable, however, for

exact mathematical computations because the formulas which would be required to take the

irregularities into account would necessitate a prohibitive amount of computations. The

topographic surface is generally the concern of topographers and hydrographers.

The Pythagorean spherical concept offers a simple surface which is mathematically easy to deal

with. Many astronomical and navigational computations use it as a surface representing the

earth. While the sphere is a close approximation of the true figure of the earth and satisfactory

for many purposes, to the geodesists interested in the measurement of long distances-spanning

continents and oceans-a more exact figure is necessary. The idea of flat earth, however, is still

acceptable for surveys of small areas. Plane-table surveys are made for relatively small areas and

no account is taken of the curvature of the earth. A survey of a city would likely be computed as

though the earth were a plane surface the size of the city. For such small areas, exact positions

can be determined relative to each other without considering the size and shape of the total earth.

Ellipsoid of Revolution

Since the earth is in fact flattened slightly at the poles and bulges somewhat at the equator, the

geometrical figure used in geodesy to most nearly approximate the shape of the earth is an

ellipsoid of revolution. The ellipsoid of revolution is the figure which would be obtained by

rotating an ellipse about its shorter axis. Figure 3.

FIGURE 3 ELEMENTS OF AN ELLIPSE

An ellipsoid of revolution is uniquely defined by specifying two dimensions. Geodesists, by

convention, use the semimajor axis and flattening. The size is represented by the radius at the

equator-the semimajor axis-and designated by the letter, a. The shape of the ellipsoid is given by

the flattening, f, which indicates how closely an ellipsoid approaches a spherical shape. The

difference between the ellipsoid of revolution representing the earth and a sphere is very small.

Figure 4.
FIGURE 4 THE EARTH’S FLATTENING

The ellipsoids listed below have had utility in geodetic work and many are still in use. The older

ellipsoids are named for the individual who derived them and the year of development is given.

The international ellipsoid was developed by Hayford in 1910 and adopted by the International

Union of Geodesy and Geophysics (IUGG) which recommended it for international use.

At the 1967 meeting of the IUGG held in Lucerne, Switzerland, the ellipsoid called GRS-67 in

the listing was recommended for adoption. The new ellipsoid was not recommended to replace

the International Ellipsoid (1924), but was advocated for use where a greater degree of accuracy

is required. It became a part of the Geodetic Reference System 1967 which was approved and

adopted at the 1971 meeting of the IUGG held in Moscow. It is used in Australia for the

Australian Geodetic Datum and in South America for the South American Datum 1969.

The ellipsoid called GRS-80 (Geodetic Reference System 1980) was approved and adopted at

the 1979 meeting of the IUGG held in Canberra, Australia. The ellipsoids used to define WGS

66 and WGS 72 are discussed in Chapter VIII.

NAME EQUATORIAL

RADIUS

FLATTENIN

G WHERE USED

Krassowsky (1940) 6,378,245m 1/298.3 Russia

International (1924) 6,378,388 1/297 Europe

Clarke (1880) 6,378,249 1/293.46 France, Africa

Clarke (1866) 6,378,206 1/294.98 North America

Bessel (1841) 6,377,397 1/299.15 Japan

Airy (1830) 6,377,563 1/299.32 Great Britain

Everest (1830) 6,377,276 1/300.80 India

WGS 66 (1966) 6,378,145 1/298.25 USA/DoD

GRS 67 (1967) 6,378,160 1/298.25 AustraliaAustralia

South America

WGS 72 (1972) 6,378,135 1/298.26 USA/DoD

GRS 80 (1979) 6,378,137 1/298.26

The possibility that the earth’s equator is an ellipse rather than a circle and therefore that the

ellipsoid is triaxial has been a matter of scientific controversy for many years. Modern

technological developments have furnished new and rapid methods for data collection and since

the launching of the first Russian sputnik, orbital data has been used to investigate the theory of
ellipticity.

A second theory, more complicated than triaxiality, proposed that satellite orbital variations

indicate additional flattening at the south pole accompanied by a bulge of the same degree at the

north pole. It is also contended that the northern middle latitudes were slightly flattened and the

southern middle latitudes bulged in a similar amount. This concept suggested a slight pearshaped

earth and was the subject of much public discussion. Modern geodesy tends to retain the

ellipsoid of revolution and treat triaxiality and pear shape as a part of the geoid separation (to be

discussed later).

Geoid

It was stated earlier that measurements are made on the apparent or topographic surface of the

earth and it has just been explained that computations are performed on an ellipsoid. One other

surface is involved in geodetic measurement-the geoid. In geodetic surveying, the computation

of the geodetic coordinates of points is performed on an ellipsoid which closely approximates the

size and shape of the earth in the area of the survey. The actual measurements made on the

surface of the earth with certain instruments are referred to the geoid, as explained below. The

ellipsoid is a mathematically defined regular surface with specific dimensions. The geoid, on the

other hand, coincides with that surface to which the oceans would conform over the entire earth

if free to adjust to the combined effect of the earth’s mass attraction and the centrifugal force of

the earth’s rotation. As a result of the uneven distribution of the earth’s mass, the geoidal

surface is irregular and, since the ellipsoid is a regular surface, the two will not coincide. The

separations are referred to as geoid undulations, geoid heights, or geoid separations.

The geoid is a surface along which the gravity potential is everywhere equal and to which the

direction of gravity is always perpendicular. The later is particularly significant because optical

instruments containing leveling devices are commonly used to make geodetic measurements.

When properly adjusted, the vertical axis of the instrument coincides with the direction of

gravity and is, therefore, perpendicular to the geoid. The angle between the plumb line which is

perpendicular to the geoid (sometimes called "the vertical") and the perpendicular to the ellipsoid

(sometimes called "the normal") is defined as the deflection of the vertical. Figure 5 shows the

north-south component of the deflection of the vertical.

FIGURE 5 DEFLECTION OF THE VERTICAL

Chapter III GEODETIC SURVEYING TECHNIQUES

The dictionary defines the verb survey as, "To determine and delineate the form, extent, position,

etc., of, as a tract of land, by taking linear and angular measurements, and by applying the

principles of geometry and trigonometry." One of the functions of the science of geodesy was

defined in the Introduction as the determination of the exact positions of points on the earth’s

surface. It was further explained that modern technology has brought with it a positioning

problem insofar as long and intermediate range defensive weapons are involved. The operations

to be discussed in this chapter are closely associated with the entire problem of How far? In what

direction?, etc. Four traditional surveying techniques (1) astronomic positioning, (2)

triangulation, (3) trilateration, and (4) traverse are in general use for determining the exact
positions of points on the earth’s surface. In recent years, modern technological developments

have added several new methods utilizing artificial earth satellites. Other methods relevant to

geodetic surveying are being developed and are discussed in Chapter VII. Another field of

endeavor, Photogrammetry, has contributed to geodetic surveying for many years but is not

discussed in this publication except for the observation of satellites by cameras which is included

in Chapter VI.

Horizontal Positioning

Astronomic Position Determination

The position of a point can be obtained directly by observing the stars. Astronomic positioning is

the oldest positioning method. It has been used for many years by mariners and, more recently,

by airmen for navigational purposes. Explorers have often used the astronomic method to locate

themselves in uncharted areas. Geodesists must use astronomic positions along with other types

of survey data such as triangulation and trilateration to establish precise positions. Single

astronomic positions not interconnected by geodetic surveys cannot be related to each other with

sufficient accuracy for the computation of distance and direction between points.

As the name implies, astronomic positions are obtained by measuring the angles between the

plumb line at the point and a star or series of stars and recording the precise time at which the

measurements are made. After combining the data with information obtained from star

catalogues, the direction of the plumb line (zenith direction) is computed.

While geodesists use elaborate and very precise techniques for determining astronomic latitude,

the simplest method, in the northern hemisphere, is to measure the elevation of Polaris above the

horizon of the observer. For the purposes of this publication, astronomic latitude is defined as the

angle between the perpendicular to the geoid and the plane of the equator. Figure 6.

Astronomic longitude is the angle between the plane of the meridian at Greenwich (Prime

Meridian) and the astronomic meridian of the point. Figure 6.

FIGURE 6 ASTRONOMIC COORDINATES

Actually, astronomic longitude is measured by determining the difference in time-the difference

in hours, minutes, and seconds between the time a specific star is directly over the Greenwich

meridian and the time the same star is directly over the meridian plane of the point. Shortwave

radio equipment is used to obtain time signals which can be referred to Greenwich Mean Time

while chronometers (very accurate clocks) are used to measure the time at the point. By referring

to a star catalogue, the exact Greenwich Mean Time the star was over the Prime Meridian is

obtained. The difference between the time at the point and the time at Greenwich is used to

compute the astronomic longitude of the point. Since a point of the earth rotates through 360° in

24 hours, the difference in local time between two points can be easily converted into difference

in longitude.

Another astronomic observation related to horizontal positioning is the astronomic azimuth. Very

accurate azimuths are used in the controlling of the orientation of first-order triangulation which

is the next topic to be discussed. Referring again to Figure 6 and to point P, the astronomic

azimuth of some other point Q as seen from P is defined as the angle between the meridian plane
of point P and the plane containing both Q and the perpendicular to the geoid at P. This angle is

reckoned from north at P clockwise from O° to 360°.

Astronomic observations are made by optical instruments-theodolite, zenith camera, prismatic

astrolabe-which all contain leveling devices. When properly adjusted, the vertical axis of the

instrument coincides with the direction of gravity and is, therefore, perpendicular to the geoid.

Thus, astronomic positions are referenced to the geoid. Since the geoid is an irregular,

nonmathematical surface, astronomic positions are wholly independent of each other.

Triangulation

The most common type of geodetic survey is known as triangulation. It differs from the plane

survey mentioned earlier in that more accurate instruments are used, instrumental errors are

either removed or predetermined so that they can be compensated for in the computations and

more rigorous procedures are employed to reduce observational errors. Another very important

difference is that all of the positions established by triangulation are mathematically related to

each other.

Basically, triangulation consists of the measurement of the angles of a series of triangles. The

principle of triangulation is based on simple trigonometric procedures. If the distance along one

side of a triangle and the angles at each end of the side are accurately measured, the other two

sides and the remaining angle can be computed. Normally, all of the angles of every triangle are

measured for the minimization of error and to furnish data for use in computing the precision of

the measurements. Figures 7. Also, the latitude and longitude of one end of the measured side

along with the length and direction (azimuth) of the side provide sufficient data to compute the

latitude and longitude of the other end of the side.

The measured side of the base triangle is called a base line. Measurements are made as carefully

and accurately as possible with specially calibrated tapes or wires of invar, an alloy highly

resistant to changes in length resulting from changes in temperature. The tapes or wires are

checked periodically against standard measures of length (at the Bureau of Standards in the

United States and corresponding agencies in other countries). The geodimeter and tellurometer,

operating on electro-optical and electronic principles respectively, have replaced the older

methods of base measurement in the recent surveys. The work can be completed more rapidly

and accurately than with wire or tape. The laser equipped geodimeter has proven to be the most

accurate and it can measure much longer distances without losing accuracy.

To establish an arc of triangulation between two widely separated locations, a base line may be

measured and longitude and latitude determined for the initial point at one end. The locations are

then connected by a series of adjoining triangles forming quadrilaterals extending from each end.

Figure 7. With the longitude, latitude, and azimuth of the initial points, similar data is computed

for each vertex of the triangles thereby establishing triangulation stations or geodetic control

stations. The coordinates of each of the stations are defined as geodetic coordinates. Figure 8.

FIGURE 7 A SIMPLE TRIANGULATION NET

FIGURE 8 GEODETIC COORDINATES

Triangulation is extended over large areas by connecting and extending series of arcs and
forming a network or triangulation system. The network is adjusted in a manner which reduces

the effect of observational errors to a minimum. A denser distribution of geodetic control is

achieved in a system by subdividing or filling in with other surveys. Figure 9 serves to illustrate,

in a general manner, the major triangulation networks which have been established.

FIGURE 9 FUNDAMENTAL GEODETIC NETWORKS (HORIZONTAL CONTROL)

There are four general orders of triangulation. First-Order (Primary Horizontal Control) is the

most accurate triangulation. It is costly and time-consuming using the best instruments and

rigorous computation methods. First-Order triangulation is usually used to provide the basic

framework of horizontal control for a large area such as for a national network. It has also been

used in preparation for metropolitan expansion and for scientific studies requiring exact geodetic

data. Its accuracy should be at least one part in 100,000.

Second-Order, Class I (Secondary Horizontal Control) includes the area networks between the

First-Order arcs and detailed surveys in very high value land areas. Surveys of this class

strengthen the US National Horizontal Control Network and are adjusted as part of the network.

Therefore, this class also includes the basic framework for further densification. The internal

closures of Second-Order, Class I triangulation should indicate an accuracy of at least one part in

50,000. The demands for reliable horizontal control surveys in areas which are not in a high state

of development or where no such development is anticipated in the near future justifies the need

for a triangulation classified as Second-Order, Class II (Supplemental Horizontal Control). This

class is used to establish control along the coastline, inland waterways and interstate highways.

The control data contributes to the National Network and is published as part of the network. The

minimum accuracy allowable in Class II of Second-Order is one part in 20,000.

Third-Order, Class I and Class II (Local Horizontal Control) is used to establish control for local

improvements and developments, topographic and hydrographic surveys, or for such other

projects for which they provide sufficient accuracy. This triangulation is carefully connected to

the National Network. The work should be performed with sufficient accuracy to satisfy the

standards of one part in 10,000 for Class I and one part in 5,000 for Class II. Spires, stacks,

standpipes, flag poles and other identifiable objects located to this accuracy also have significant

value for many surveying and engineering projects.

The sole accuracy requirement for Fourth-Order triangulation is that the positions be located

without any appreciable errors on maps compiled on the basis of the control.

Normally, triangulation is carried out by parties of surveyors occupying preplanned locations

(stations) along the arc and accomplishing all the measurements as they proceed. When distances

between two points were too long for conventional methods, connections were sometimes made

by a method known as flare triangulation. Stations were occupied on either side of the gap and

magnesium flares were parachuted from aircraft or "shot" into the air from ships at suitable

points between them. Intersections of lines were made simultaneously at all of the stations and

reasonably accurate "bridges" established. A connection of this type was established between

Norway and Denmark. However, satellite geodesy (Chapter VI) has solved the problem of

bridging wide gaps.

Trilateration
Another surveying method that has been used involves the use of radar and aircraft. The

SHORAN, HIRAN and SHIRAN electronic distance measuring systems have been applied to

performing geodetic surveys by a technique known as trilateration. Figure 10. Since very long

lines (to 500 miles) could be measured by these systems, geodetic triangulation networks have

been extended over vast areas in comparatively short periods of time. In addition, the surveys of

islands and even continents separated by extensive water barriers have been connected by the

techniques. The Canadian SHORAN network connecting the sparsely populated northern coastal

and island areas with the central part of the country and the North Atlantic HIRAN Network

tying North America to Europe are examples of the application of the trilateration technique.

Figure 11 shows these and several other trilateration networks (SHORAN and HIRAN) which

have been established throughout the world. SHIRAN has been used in the interior of Brazil.

FIGURE 10 A TRILATERATION NET

FIGURE 11 MAJOR TRILATERATION SURVEYS SHORAN AND HIRAN

Only distances are measured in trilateration and each side is measured repeatedly to insure

precision. The entire network is then adjusted to minimize the effects of the observations errors.

The angles of the triangles are computed so the geodetic positions are obtained as in

triangulation.

Traverse

The simplest method of extending control is called traverse. The system is similar to dead

reckoning navigation where distances and directions are measured. In performing a traverse, the

surveyor starts at a known position with a known azimuth (direction) to another point and

measures angles and distances between a series of survey points. With the angular

measurements, the direction of each line of the traverse can be computed; and with the

measurements of the length of the lines, the position of each control point computed. If the

traverse returns to the starting point or some other known position, it is a closed traverse,

otherwise the traverse is said to be open. Figure 12.

FIGURE 12 AN OPEN TRAVERSE; A CLOSED TRAVERSE

Since electronic distance measuring equipment has become available, the accuracy of traverse

surveys has increased significantly. The tellurometer (microwave) has been used in Australia to

complete a network (Australian Geodetic Datum) covering that continent. The average loop

length is about 900 miles; the average loop closure of this work is 2.2 parts per million. The laser

equipped geodimeter has been used to produce internal accuracies better than one part per

million in establishing the transcontinental traverse in the United States. The traverse consists of

a series of high-precision length, angle and astronomic azimuth determinations running

approximately east-west and north-south through the conterminous states, forming somewhat

rectangular loops. Figure 13. This traverse will be the "backbone" of a re-adjustment of the

horizontal control network in this country.

FIGURE 13 TRANSCONTINENTAL TRAVERSE

Celestial Techniques
Celestial methods in geodesy involves the determination of an observer’s position from

observations of the moon, stars and satellites. Celestial triangulation permits the extension of

long arcs across oceans and inaccessible space terrain. Satellites also permit a determination of

the shape of the earth and provide important knowledge of its gravitational field (discussion of

satellite geodesy is reserved for Chapter VI). All of the celestial methods possess one common

characteristic-observed data is not affected by the direction of the vertical at the observation

point.

Geodetic experiments have been performed in the past with the solar eclipse, star occultation and

moon-position camera methods, but for various reasons the experiments did not produce useful

geodetic results. The three techniques have been replaced by the observation and tracking of

artificial earth satellites.

Vertical Positioning

Vertical surveying is the process of determining heights-elevations above the mean sea level

surface. As noted earlier, the geoid corresponds to the mean level of the open sea. In geodetic

surveys executed primarily for mapping purposes, there is no problem in the fact that geodetic

positions are referred to an ellipsoid and the elevations of the positions are referred to the geoid.

However, geodetic data for missiles requires an adjustment in the elevation information to

compensate for the undulations of the geoid above and below the regular mathematical surface of

the ellipsoid. The adjustment uses complex advanced geodetic techniques. One method based on

Stokes’ Theorem is mentioned in the discussion of physical geodesy (Chapter V).

Precise geodetic leveling is used to establish a basic network of vertical control points. From

these, the height of other positions in the survey can be determined by supplementary methods.

The mean sea level surface used as a reference (vertical datum) is determined by obtaining an

average of the hourly water heights for a period of several years at tidal gauges.

There are three leveling techniques-differential, trigonometric, and barometric-which yield

information of varying accuracy. Figure 14. Differential leveling is the most accurate of the three

methods. With the instrument locked in position, readings are made on two calibrated staffs held

in an upright position ahead of and behind the instrument. The difference between readings is the

difference in elevation between the points.

FIGURE 14 METHODS OF ELEVATION DETERMINATION

The optical instrument used for leveling contains a bubble tube to adjust it in a position parallel

to the geoid. When properly "set up" at a point, the telescope is locked in a perfectly horizontal

(level) position so that it will rotate through a 360 arc. The exact elevation of at least one point in

a leveling line must be known and the rest computed from it.

Trigonometric leveling involves measuring a vertical angle from a known distance with a

theodolite and computing the elevation of the point. With this method, vertical measurements

can be made at the same time horizontal angles are measured for triangulation. It is, therefore, a

somewhat more economical method but less accurate than differential leveling. It is often the

only practical method of establishing accurate elevation control in mountainous areas.

In barometric leveling, differences in height are determined by measuring the difference in
atmospheric pressure at various elevations. Air pressure is measured by mercurial or aneroid

barometers, or a boiling point thermometer. Although the degree of accuracy possible with this

method is not as great as either of the other two, it is a method which obtains relative heights

very rapidly at points which are fairly far apart. It is widely used in the reconnaissance and

exploratory surveys where more exacting measurements will be made later or are not required.
Chapter IV GEODETIC SYSTEMS

A datum is defined as any numerical or geometrical quantity or set of such quantities which

serve as a reference or base for other quantities. In geodesy two types of datums must be

considered: a horizontal datum which forms the basis for the computations of horizontal control

surveys in which the curvature of the earth is considered, and a vertical datum to which

elevations are referred. In other words, the coordinates for points in specific geodetic surveys and

triangulation networks are computed from certain initial quantities (datums).

Horizontal Geodetic Datums

A horizontal geodetic datum may consist of the longitude and latitude of an initial point (origin);

an azimuth of a line (direction) to some other triangulation station; the parameters (radius and

flattening) of the ellipsoid selected for the computations; and the geoid separation at the origin. A

change in any of these quantities affects every point on the datum. For this reason, while

positions within a system are directly and accurately relatable, data such as distance and azimuth

derived from computations involving geodetic positions on different datums will be in error in

proportion to the difference in the initial quantities.

Orientation of Ellipsoid to Geoid

Single Astronomic Position Datum Orientation

Selection of the reference ellipsoid provides the radius and flattening factors. The simplest means

of obtaining the other factors to establish the geodetic datum is to select a station, preferably one

located near the center of a triangulation network, to serve as the datum origin. Then the

astronomical coordinates of the station and the astronomical azimuth of a line from the station to

another control station are observed. The observed astronomical coordinates and azimuth are

adopted without any correction as the geodetic coordinates and azimuth of the datum origin on

the reference ellipsoid. Further, the geoid and ellipsoid are assumed to coincide at that point.

This means that the deflection of the vertical and the separation between the ellipsoid and geoid

are defined as zero at the origin. By using this method of orientation, the normal to the ellipsoid

is arbitrarily made to coincide with the plumb line at the datum origin. Figure 15.

FIGURE 15 SINGLE ASTRONOMICAL STATION DATUM ORIENTATION

Although the computed positions will be correct with respect to each other in this type of

orientation, the entire net will be shifted with respect to the axis of the earth. This is not

significant for local use of the positions but may introduce large systematic errors as the survey

is expanded.

It should be noted that although the deflection and undulation are defined as zero at the origin,

deflections will occur at other positions within the network. When comparing the geodetic

latitude and longitude of any other point in the net with the corresponding astronomical latitude

and longitude of that point, differences will appear between the two sets of values.

A datum oriented by a single astronomical point may produce large systematic geoid separations.
The ellipsoid is not earth-centered and its rotational axis is not coincident with the axis of the

earth. The, inconvenience of such an orientation is that the positions derived from different

astronomically oriented datums are not directly comparable to each other in any geodetic

computation. The Tokyo Datum, discussed later, is an example of this type of datum orientation.

Astronomical-Geodetic Orientation

The deflections of the vertical at a number of Laplace stations can be used for a second type of

datum orientation known as the astro-geodetic orientation. A Laplace station is defined as a

triangulation or traverse station at which a geodetic (Laplace) azimuth is derived from an

astronomic azimuth by use of the Laplace equation. The Laplace equation expresses the

relationship between astronomic azimuth, geodetic azimuth and the astronomic longitude and

geodetic longitude. Although it is not in the definition, the astronomic latitude is normally

observed at each Laplace station. In an orientation of this type, a correction is made at the origin

(initial point) which in effect reduces the sum of the squares of the astro-geodetic deflections at

all the Laplace stations to a minimum. Figure 16. One of the Laplace stations in the adjustment is

arbitrarily selected as the origin.

FIGURE 16 ASTRO-GEODETIC DATUM ORIENTATION

The deflection of the vertical-the angle between the plumb line and the normal to the ellipsoid-is

usually resolved into a north-south component which is equal to the difference between

astronomic and geodetic latitude; and an east-west component proportional to the difference

between astronomical and geodetic longitude. (Figure 16 shows only the north-south

component.) The Laplace equation provides a means of reconciling the azimuth differences

resulting from the use of two separate reference surfaces. Laplace equations are introduced into

triangulation adjustments to control the azimuth and orient the ellipsoid. Therefore instead of a

zero deflection at the origin as with a single astronomic position, there is a deflection of the

vertical. Similarly, the geoid separation can be determined at the origin and the ellipsoid reoriented so that a best average fit is provided for the ellipsoid and the geoid in the area of the

Laplace stations used. Consequently, astro-geodetically oriented datums are applicable over

larger areas than those oriented by a single astronomic position.

The astronomical-geodetic orientation has the disadvantage that the deflections of the vertical

remain relative. If the ellipsoid is changed, the deflections of the vertical will also change.

Secondly, it is necessary to assume a specific orientation of the reference ellipsoid with respect

to the geoid before computing the astro-geodetic deflections. Figure 17. The orientation is fixed

by the initial values of the datum origin from which the geodetic coordinates were computed.

Any change in these initial quantities will change the deflection of the vertical at each point.

Consequently, the astro-geodetic deflection of the vertical depends upon a specific geodetic

datum and the use of geodetic data developed by this method is limited to relatively small areas.

FIGURE 17 ASTRO-GEODETIC DEFLECTIONS ARE RELATIVE

Discrepancies Between Datums

In areas of overlapping geodetic triangulation networks, each computed on a different datum, the

coordinates of the points given with respect to one datum will differ from those given with

respect to the other. The differences occur because of the different ellipsoids used and the
probability that the centers of each datum’s ellipsoid is oriented differently with respect to the

earth’s center. In addition, deflection errors in azimuth cause a relative rotation between the

systems. Finally, a difference in the scale of horizontal control may result in a stretch in the

corresponding lines of the geodetic nets.

In view of the different orientation of ellipsoid centers, the relative rotation between the systems,

and the scale differences; the computation of geodetic information from one datum to another

unconnected datum is quite impossible. Regardless of the accuracy of the individual datums for

computation within themselves, there is no accurate way to perform distance and azimuth

computations between unconnected geodetic systems.

With the development of both intermediate and long range defensive weapon systems, geodetic

problems have become more critical than ever before. To satisfy military requirements, it Is

necessary to provide detailed cartographic coverage of areas of strategic importance and to

accomplish geodetic computations between these areas and launch sites which are often on

unrelated datums. Both of these requirements necessitate unification of major geodetic datums

by one or a combination of existing methods.

Datum Connection

There are three general methods by which horizontal datums can be connected. The first method

is restricted to surveys of a limited scope and consists of systematic elimination of discrepancies

between adjoining or overlapping triangulation networks. Figure 18. This is done by moving the

origin, rotating, and stretching networks to fit each other. The method is usually used to connect

local surveys for mapping purposes. This method of datum transformation or datum reduction

can only be used where control exists for common points in different systems.

FIGURE 18 DATUM REDUCTION CAN JOIN OVERLAPPING TRIANGULATION

SYSTEMS

In addition to the classic method of datum transformation described above, there is the

gravimetric method of Physical Geodesy (Chapter V) and the methods of Satellite Geodesy

(Chapter VI). These methods are used to relate large geodetic systems to each other and/or to a

world system (Chapter VIII). Both the gravimetric and satellite methods produce necessary

"connecting" parameters from reduction of their particular observational data.

Major Datums Before World War II

By 1940, every technically advanced nation had developed its own geodetic system to an extent

governed by its economic and military requirements. Some systems were developed by the

expansion and unification of existing local surveys and others by new nationwide surveys

replacing outdated local ones. Normally, neighboring countries did not use the same geodetic

datum. There was no economic requirement for common geodetic information and the use of

common datums was contrary to the military interests of each country. The only surveys of an

international nature based on one datum were the few measurements of long arcs accomplished

for the purpose of determining the size and shape of the earth. The net result was that there were

many different surveys of varying size which differed from each other remarkably. The national

maps based on the surveys also differed widely. Figure 19 illustrates the confusion of geodetic

information in Southeast Asia.
FIGURE 19 AN EXAMPLE OF THE MANY DATUMS IN THE SOUTHEAST ASIA AREA

As military distance requirements increased, positioning information of local or even national

scope became unsatisfactory. The capabilities of the various weapon systems increased until

datums of at least continental limits were required.

The Major Datums Today

The best solution (to meeting the increased military distance requirements) was the establishment

of a "single" datum for a large area and adjusting all local systems to it. The North American,

European, and Tokyo Datums were (initially) selected for this purpose. (WGS 60, Chapter VIII).

Other major geodetic datums of the world include the Arc and Adindan Datums in Africa, the

Australian Geodetic, South American 1969, Indian and the Russian Pulkovo 1942 Datum. Figure

20.

FIGURE 20 MAJOR GEODETIC DATUM BLOCKS

The North American Datum, 1927

The first official geodetic datum in the United States was the New England Datum, adopted in

1879. It was based on surveys in the eastern and northeastern states and referenced to the Clarke

1866 Ellipsoid. Through the years this datum was extended to the south and west and in 1901,

the extended network was officially designated the United States Standard Datum. The

triangulation station Meades Ranch in Kansas was selected as the origin. In 1913, Canada and

Mexico formally agreed to base their triangulation networks on the United States system. The

datum was then renamed the North American Datum. Adjusting new surveys to fit into the

network created many problems and, therefore, during the five-year period 1927-1932 all

available first-order data were adjusted into a system now known as the North American 1927

Datum. The origin is still Meades Ranch and the datum is computed on the Clarke 1866

Ellipsoid which was oriented by a modified astro-geodetic method. The system not only

incorporated Canada and Mexico but there are connections to the South American Datum 1969

through the West Indies and Central America. A new adjustment of the system is currently

(1982) in work. The finished product will be called the North America Datum 1983 (NAD 83). It

will be a geocentric datum, i.e., the center of the ellipsoid will be referenced to the Earth’s center

of mass. The minor axis of the ellipsoid will be oriented towards a universally accepted mean

pole (a mean value for the Earth’s axis of rotation).

The European Datum

The initial point of this system is located at Potsdam, Germany. Numerous European national

systems have been joined into a large datum based upon the International Ellipsoid which was

oriented by the astro-geodetic method. The US Map Service, now known as the Defense

Mapping Agency Hydrographic/Topographic Center, connected the European and African

triangulation chains and filled the gap of the African arc measurement from Cairo to Cape Town.

This work related the Adindan Datum in North Africa, which roughly follows the twelfth

Parallel, and the Arc Datum, extending from the Equator to the Cape, to the European Datum.

Through common survey stations, a datum transformation was derived between the old Russian

Pulkovo 1932 and European systems. This extended the European Datum eastward to the 84th

meridian. In 1946 the Pulkovo 1932 system was united with a basic Siberian network and the
new datum was designated the "1942 Pulkovo System of Survey Coordinates" (Pulkovo Datum

1942). Additional ties across the Middle East connected the European with the Indian Datum.

The Tokyo Datum

The third of the initially selected datums has its origin in Tokyo. It is defined in terms of the

Bessel Ellipsoid and oriented by means of a single astronomic station. With triangulation ties

through Korea, the Tokyo Datum is connected with the Manchurian Datum. Unfortunately,

Tokyo is situated on a steep geoid slope and the single-station orientation has resulted in large

systematic geoid separations as the system is extended from its initial point.

For military distance and direction problems limited to continental areas or smaller, the large

datums were satisfactory. However, while they are improvements over the limited national

datums, they too have serious deficiencies which prevent them from providing the geodetic

information required for intercontinental ballistic missiles.

It is true that the North American Hiran tie permits connection of the European Datum and the

North American Datum. This, however, does not completely solve the problem, for both the

North American and European Datums are relative. Figure 21. While in each case the ellipsoid

chosen is an adequate fit in the area of the origin, neither provides a good fit for the entire earth.

Also, the process of connecting various datums by means of intervening datums or triangulation

ties allows errors to accumulate which do not always provide agreement with newly observed

data.

FIGURE 21 THE PREFERRED DATUMS ARE RELATIVE DATUMS

The South American Datum 1969

In 1965 the Working Group for the Study of the South American Datum was asked by the

Committee for Geodesy of the Cartographic Commission of the Pan American Institute of

Geography and History (PAIGH) to establish one consistent geodetic datum for the entire

continent of South America. In June 1969, the "South American Datum 1969" was accepted by

the Commission at the IX General Assembly of PAIGH in Washington, D.C. This datum is

computed on the GRS 67 Ellipsoid as listed in Chapter II. CHUA, the National datum point of

Brazil was taken to be the origin. A vast amount of triangulation, Hiran, astronomic, and satellite

data were incorporated in the solution, and the datum now provides the basis for a homogeneous

geodetic control system for the continent.

The Australian Geodetic Datum

The survey net of Australia consists of 161 sections which connect 101 junction points and form

58 loops. Virtually all the surveys are of the traverse type in which distances were determined by

electronic measuring equipment. There are 2506 stations and the total length of the traverse is

33,100 miles. A complete readjustment of the geodetic network was made in 1966 using the

GRS 67 Ellipsoid as listed in Chapter II. Tasmania has been connected and a tie to New Guinea

and the Bismarck Archipelago was effected by a traverse up cape York and the USAF Hiran

network of 1965. This tie placed an additional 135 points on the Australian Geodetic Datum.

The Indian Datum
The Indian Datum has been used for India and several adjacent countries in Southeast Asia. It is

computed on the Everest Ellipsoid with its origin at Kalianpur in Central India. Derived in 1830,

the Everest Ellipsoid is the oldest of the ellipsoids in use and is much too small. As a result, the

datum cannot be extended too far from the origin or very large geoid separations will occur. For

this reason and the fact that the ties between local triangulation in Southeast Asia are typically

weak, the Indian Datum is probably the least satisfactory of the major datums.

Vertical Datums

Just as horizontal surveys are referred to specific original conditions (datums), vertical surveys

are also related to an initial quantity or datum. As already noted, elevations are referred to the

geoid because the instruments used either for differential or trigonometric leveling are adjusted

with the vertical axis coincident to the local vertical. As with horizontal datums, there are many

discrepancies among vertical datums. There is never more than 2 meters variance between

leveling nets based on different mean sea level datums; however, elevations in some areas are

related to surfaces other than the geoid; and barometrically determined heights are usually

relative.

In the European area, there are fewer vertical datum problems than in Asia and Africa. Extensive

leveling work has been done in Europe and practically all of it has been referred to the same

mean sea level surface. However, in Asia and Africa the situation has been different. In places

there is precise leveling information available based on mean sea level. In other areas the zero

elevation is an assumed elevation which sometimes has no connection to any sea level surface.

China has been an extreme example of this situation where nearly all of the provinces have had

an independent zero reference. There is very little reliable, recent, vertical data available for

much of the area of Africa and Asia including China.

The mean sea level surface in the United States was determined using 21 tidal stations in this

country and five in Canada. This vertical datum has been extended over most of the continent by

first-order differential leveling. Concurrent with the new adjustment of the horizontal network,

mentioned previously, is the readjustment of the vertical network. Countries of North and

Central America are involved. In the conterminous United States 110,000 kilometers of the basic

network are being releveled.
CHAPTER V PHYSICAL GEODESY

Most of the discussion to this point has been concerned with geometrical geodesy. This aspect of

the science is involved with strictly geometrical relationships measured in various waystriangulation, trilateration, electronic surveys, etc.-for the purpose of deducing the size and shape

of the earth and the precise location of specific positions on the earth’s surface. Geometric

geodesy considers the geoid through the use of astrogeodetic methods. For example, the astrogeodetic orientation method was discussed in the previous chapter.

This chapter introduces another branch of geodetic science known as physical geodesy. Physical

geodesy utilizes measurements and characteristics of the earth’s gravity field as well as theories

regarding this field to deduce the shape of the geoid and in combination with arc measurements,

the earth’s size. With sufficient information regarding the earth’s gravity field, it is possible to

determine geoid undulations, gravimetric deflections, and the earth’s flattening.

In using the earth’s gravity field to determine the shape of the geoid, the acceleration of gravity is

measured at or near the surface of the earth. It might be interesting to compare the acceleration

measured by the gravimetrist and the acceleration experienced in an airplane. In an airplane, an

acceleration is simply called a G force and is measured by a G meter. A G factor of one is used

to indicate the acceleration due to the attraction of the earth and is considered a neutral condition.

The gravity unit used and measured in geodesy is much smaller. A G factor of one is

approximately equal to one thousand gals, a unit named after Galileo. The still smaller unit used

in geodesy is the milligal (mgal) or one-thousandth part of a gal. Thus, in geodesy we are dealing

with variations in acceleration equal to one millionth of a one G aircraft acceleration. The most

accurate modern instruments permit measurement of acceleration changes of one hundred

millionth part of the well known G factor or better.

Gravity Measurements

An attracting force, called gravitational force, operates between the earth and every object that is

located within, on, or above the earth’s surface. At the same time, any object within or on the

earth’s surface pursues a circular path as the earth rotates on its axis. If the body is on the earth’s

surface, it follows the circular path because it is held onto the earth’s surface by the gravitational

force. However, the body as it pursues a circular path exerts an outward force called the

centrifugal reaction. The centrifugal reaction can be experienced by revolving an object at the

end of a string. The sum of the gravitational force and centrifugal reaction acting on a body is

called gravity. Since the gravitational force is much stronger than the centrifugal reaction,

gravity causes an object to have weight and, if the object is free to move, to fall with increasing

speed (i.e., the body accelerates) toward the center of the earth. The acceleration experienced by

that object as it moves toward the earth’s center is called the acceleration of gravity, which is the

quantity observed when gravity measurements are made.

Two distinctly different types of gravity measurements are made: absolute gravity measurements

and relative gravity measurements. If the value of acceleration of gravity can be determined at

the point of measurement directly from the data observed at that point, the gravity measurement

is absolute. If only the differences in the value of the acceleration of gravity are measured
between two or more points, the measurements are relative.

Absolute measurement of gravity-Until the middle of the 20th Century, virtually all absolute

measurements of gravity were made using some type of pendulum apparatus. The most usual

type of apparatus contained a number of pendulums that were swung in a vacuum. By measuring

the peroid of the pendulums, the acceleration of gravity could be computed. In 1818, Kater

developed the so-called reversible pendulum that had knife edge pivots at both ends. These

pendulums were flipped over (reversed) during the measurements and, using this procedure, a

number of important error sources were eliminated. Still, there were numerous other problems

and error sources associated with pendulum measurements of absolute gravity, and the results

obtained were not sufficiently accurate to meet the needs of geodetic gravimetry. Consequently,

in recent years, the pendulum method has been superseded by the ballistic method which is based

on timing freely falling bodies. The acceleration of gravity can be determined by measuring the

time taken by a body to fall over a known distance.

In the United States, the basic falling body apparatus was developed in the early 1970’s jointly by

J.A. Hammond of the Air Force Geophysics Laboratory and J.E. Faller of the Joint Institute for

Laboratory Astrophysics. In the so-called Hammond-Faller apparatus, a corner cube reflector

falls in a vacuum while distance and time are measured continuously by a laser beam in

conjunction with a photo multiplier tube. This apparatus weighed about 800 kilograms and

considerably more accurate than the best absolute pendulum apparatus. Hammond recently has

completed fabrication of a somewhat smaller and more accurate version of the original

Hammond-Faller apparatus. The new instrument weighs about 700 kilograms when packed for

shipment in nine units. Hammond’s apparatus has been used to establish very accurate values for

absolute gravity at a number of sites within the United States. Faller is also developing a more

refined falling body apparatus. The most elaborate and probably the most accurate ballistic

apparatus has been developed by Sakuma. The equipment occupies two rooms at the

International Bureau of Weights and Measures at Sevres, France, and unlike the other

instruments described here, is not portable. Sakuma’s apparatus is a rising and falling body

apparatus. A body is projected upward and allowed to rise and fall in a partial vacuum.

Measurements of time and distance are made during both the rise and fall. Certain error sources

cancel out when such a procedure is used. An Italian group, Instituto di Metrologia "G.

Colonetti" has worked with Sakuma to develop a miniaturized, portable version of Sakuma’s

apparatus. This portable version, generally known as "the Italian Apparatus," has been used to

make very accurate measurements of absolute gravity at a number of sites in Europe and the

United States since 1976. A French group, in 1977, advertised its plans to manufacture a version

of the Italian apparatus for sale.

Over the years, absolute gravity measurements have been made at only a few key sites, and these

few measurements have served chiefly to establish datum and scale for relative gravity

measurements. The reasons for the sparsity of absolute gravity measurements is that the

necessary measuring equipment is very bulky and costly, and a single measurement requires days

of painstakingly careful work under near laboratory conditions. This, however, may change in

the next decade or so. As absolute equipment continues to be miniaturized and made more

portable, absolute gravity measurements are becoming more commonplace.

Relative measurement of gravity-Solution of some of the problems of gravimetric geodesy

requires a knowledge of the acceleration of gravity at very many points distributed uniformly
over the entire surface of the earth. Since absolute gravity measurements have been too

complicated and time consuming and, until recently, could not be obtained with sufficient

accuracy, relative gravity measurements have been used to establish the dense network of gravity

measurements needed. The earliest relative gravity measurements were made with reversible

pendulums. Since the theory of relative pendulum measurements is somewhat simpler than that

of absolute pendulum measurements, better accuracy was obtained for the former. However, the

equipment was bulky, measurements were time consuming, and better accuracy was desired than

could be obtained with pendulums. The most accurate relative pendulums to be developed were

the Gulf quartz pendulum and the Cambridge invar pendulum. These two instruments were used

as late as 1969.

Modern relative gravity measurements are made with small, very portable, and easily used

instruments known as gravimeters (gravity meters). Using gravimeters, highly accurate relative

measurements can be made at a given site, known as a gravity station, in half-an-hour or less.

Modern gravimeter-type instuments were first developed in the 1930’s. Although at least 28

different kinds of gravimeters have been developed, only two types have been used extensively.

The LaCoste-Romberg gravimeters are used for most geodetic work today, although the Worden

gravimeters have been used extensively for such work in the past.

The heart of all modern gravimeters consists of a weight suspended by a very sensitive spring.

Changes in length of the spring due to changes in the acceleration of gravity as the gravimeter is

moved from place to place are translated by the mechanisms of the gravimeter into "dial reading"

differences which are proportional to gravity differences. The relative measurements at each

gravity station consists of reading the gravimeter dial when the spring has been adjusted to a null

or equilibrium position. The constant of proportionality relating dial reading differences to

gravity difference is known as the calibration constant or calibration factor. The dial reading of

the gravimeter at each site is multiplied by the calibration factor to obtain a gravity value. Each

instrument has a unique calibration factor which must be determined empirically. This is done by

the manufacturer. Many gravimeter users redetermine and periodically check the calibration

factor by taking dial readings over a so-called calibration line. A calibration line is a series of

well described monumented, reoccupied sites where the acceleration of gravity has been

determined very accurately and over which the value of gravity varies significantly. By

comparing the dial readings to the known gravity values at points along the calibration line, the

calibration factor can be computed. In this manner, the scale of relative gravity surveys is

controlled by the calibration factor. For the most precise work, it cannot be assumed that the

calibration factor is constant, and more complicated calibration procedures must be used.

There are two other important considerations when relative gravity measurements are made: drift

and base station connections. Gravimeter drift is a phenomenon related to certain instrumental

instabilities that cause the dial reading to change slowly with time even when the accelration of

gravity remains constant. Although the amount of drift is very small for most modern

gravimeters, it still must be determined and eliminated from the relative gravity measurements.

There are several observational techniques commonly used, but all involve some kind of looping

back to previously measured points such that the relative gravity survey is continuously closed

back on itself. The difference in dial readings (converted to gravity) between two successive

occupations of the same site is the drift of the gravimeter over the interval of time which has

elapsed between the two occupations. Drift is often considered to occur at a constant rate.
Consequently, the time at which each measurement is made is recorded along with the dial

reading. After each closure in the survey, the total drift is determined and allocated back through

the loop proportionally to the time elapsed between each measurement. For the most accurate

work, drift cannot be assumed to be constant, and more complicated techniques should be

employed. For the most accurate gravity surveys, several gravimeters are used simultaneously,

and a least squares adjustment is applied to compensate for imprecise drift and calibration

determinations.

Since relative gravity surveys can determine only differences in gravity from point to point,

every relative gravity survey must include measurements at one or more reoccupiable points

where acceleration of gravity is known. Such points are called base stations. Then all gravity

difference measurements are computed with respect to the known gravity value at the base

station. Hence, tying a relative gravity survey to a base station establishes the "gravity datum" of

that survey. The earliest "gravity datum" was the so-called Potsdam System. In 1906, reversible

pendulums were used to measure absolute gravity at Potsdam, Germany. The value measured at

this time was later adopted as the initial point for the Potsdam gravity reference system. Using

relative measurements, several points on each continent were connected to Potsdam, and these

served as the fundamental base stations for many relative gravity surveys.

The Potsdam system, however, was found to be in error and, in 1971, was replaced by the

International Gravity Standardization Net 1971 (IGSN71). The IGSN contains 1854 reoccupiable

stations distributed worldwide. The acceleration of gravity at each point was determined by a

least squares adjustment that included a number of absolute gravity measurements and a

multitude of relative gravity measurements that interconnected all stations. The IGSN71

established the basic "gravity datum" for today’s relative gravity surveys. National base networks

exist in many countries. These national base networks usually are connected to the IGSN71 and

contain reoccupiable stations located at convenient sites throughout the country concerned. For

example, the US gravity base network contains approximately 50 stations. The national base

networks usually are established by precise gravimeter measurements. In the future, some

national nets will be tied directly to high accuracy absolute gravity measurements instead of the

IGSN71. Actually any reoccupiable gravity station which has been tied to the IGSN71 or a

national base network can be used as a base station for a new relative gravity survey.

Gravity measurement at sea-So far, only procedures used to measure gravity on land areas have

been discussed. Gravity measurements, however, also are made at sea. The earliest

measurements at sea were made by F.A. Vening Meinesz who, in 1927, installed a pendulum

apparatus in a submarine. The submarine pendulum gravity measurements of Vening Meinesz

are mainly of historical interest today. The first gravimeters installed in surface ships appeared

during the 1950’s. These early ocean surface gravity measurements were only of modest

accuracy and, again, now are mainly of historical value. Reasonably accurate measurements

from gravimeters on surface ships date only from the late-1960’s. Instruments used include

LaCoste-Romberg S Meter, Askania Meter, Bell Meter, and the Vibrating String Gravimeter. All

of these meters are compensated to minimize the effects of oscillatory motion of the ship due to

ocean surface waves. The effects are also eliminated or averaged out by computational

techniques. A big problem with ocean surface measurements is that the forward motion of the

ship adds a centrifugal reaction component to measured gravity which must be eliminated by the

so-called Eotvos correction. Therefore, the ship’s velocity and heading, as well as the ship’s
position, must be known accurately. Near shore, shore based electronic positioning/navigation

systems (such as LORAN) are used. In the deep ocean, satellite navigation and inertial systems

must be used.

Gravity measurement in the air-Problems in airborne gravity measurements are similar to those

encountered for surface ships. The position, velocity, and heading of the aircraft must be known

accurately. Because of the higher aircraft speeds, the Eotvos correction is much larger for

airborne measurements than for surface ship measurements. It also is very difficult to

compensate for spurious aircraft accelerations. In addition, reduction of the gravity value from

aircraft altitude to an equivalent surface value is a problem that has not yet been solved

satisfactorily. In 1959, the US Air Force was instrumental in developing a gravimeter suitable for

use aboard a fixed wing KC-135 aircraft. This gravimeter measured average gravity values along

the path of the aircraft reasonably well, but could not produce point gravity values. The program

was cancelled in the late-1960’s. Recent suggestions have been made that fixed wing airborne

measurements should be restudied. Today, there is an operational airborne gravity measurement

system aboard a S-61 helicopter. The gravity measurement system aboard the helicopter uses a

LaCoste-Romberg S Meter to sense gravity and a Del Norte transponder electronic navigation

system to establish aircraft position, velocity, and heading. The operator of the system, Carson

Helicopters, advertises that it is possible to do 3,000 to 10,000 miles of gravity survey per month

with reasonably good accuracy.

Gravity Anomalies

Gravity measurements provide values for the acceleration of gravity at points located on the

physical surface of the earth. Before these measurements can be used for most geodetic purposes,

they must be converted into gravity anomalies.

A gravity anomaly is the difference between a gravity measurement that has been reduced to sea

level and normal gravity. Normal gravity, used to compute gravity anomalies, is a theoretical

value representing the acceleration of gravity that would be generated by a uniform ellipsoidal

earth. By assuming the earth to be a regular surface without mountains or oceans, having no

variations in rock densities or in the thickness of the crust, a theoretical value of gravity can be

computed for any point by a simple mathematical formula. The theoretical value of gravity at a

point on the ellipsoid’s surface depends both on the size and shape of the ellipsoid and on a

value, computed from observational data, which is accepted as the theoretical value of gravity at

the equator. It varies only with the latitude of the observation point if the figure of the earth is

taken as an ellipsoid of revolution. While there are several formulas for computing Theoretical

Gravity, the one most commonly used is the Gravity Formula 1967 which is based on the

Reference Ellipsoid 1967, the parameters of which correspond to the Geodetic Reference System

1967. A simple formula will convert 1967 theoretical (or normal) gravity to the newer more

accurate 1980 system, GRS 80 (Chapter II). A number of different procedures can be used to

reduce a gravity measurement from the physical surface to sea level and, depending upon which

procedure is used, different types of gravity anomalies result. The most common type of gravity

anomaly used for geodetic applications is the so-called free-air gravity anomaly.

Since there are variations in the densities of the earth’s crustal materials as well as terrain

variations, the observed gravity of the earth varies irregularly from point to point. As a result the

surface known as the geoid is an irregular figure. As explained earlier, the distance between the
mathematical ellipsoid and the actual geoid is called the undulation of the geoid. Since the geoid

is so irregular, geoid. undulations cannot be computed directly but must be determined point by

point. Gravity measurements provide the observational data for these computations.

The word, anomaly, as used in geodesy refers to a deviation from the normal and can be used

either for a single point or to describe a regional or area effect. To represent an area of the earth’s

surface, the gravity measurements (anomalies) within the area are averaged by some specialized

technique to produce a mean value. This may be done for areas of different size such as the one

degree by one degree square. To make use of the anomalies, the observed gravity must be

reduced to a common frame of reference, the geoid-mean sea level. The reductions may take into

account the elevation above (or below) sea level and can account for the mass between the point

and sea level, the surrounding terrain, and the assumed structure of the earth’s crust. Figure 22.

FIGURE 22 FACTORS CONSIDERED WHEN REDUCING OBSERVED GRAVITY

Undulation and Deflections by the Gravimetric Method

The method providing the basis from which the undulations of the geoid may be determined

from gravity data was published in 1849 by a British scientist, Sir George Gabriel Stokes.

However, the lack of observed gravity data prevented its application until recent years. In 1928,

the Dutch scientist, Vening Meinesz, developed the formulas by which the gravimetric deflection

of the vertical (Figure 23) can be computed.

FIGURE 23 PRODUCTS OF THE GRAVIMETRIC METHOD

The computation of the undulations of the geoid and the deflections of the vertical require

extensive gravity observations. The areas immediately surrounding the computation point require

a dense coverage of gravity observations and detailed data must be obtained out to distances of

about 500 miles. A less dense network is required for the remaining portion of the earth. While

the observational requirements for these computations appear enormous, the results well justify

the necessary survey work.

Figure 24 shows how the mass surplus of the mountains and the mass deficiency of the oceans

cause the deflections of the vertical and the undulations of the geoid. A mountain mass attracts

the plumb line from the normal of the ellipsoid. Likewise the mass deficiency of the ocean does

not attract the plumb line. These effects of the mass anomalies contribute to the deflection of the

vertical. However, both deflections of the vertical and undulation values result from density

variations throughout the earth. In the area of mass surplus, the observed gravity (reduced to sea

level considering only the elevation) is generally greater than the theoretical value, and the

anomalies are positive. In the areas of mass deficiency, the observed (reduced in the same

manner) is generally smaller than the theoretical value and the anomalies are negative. Figure 25.

FIGURE 24 GEOID-ELLIPSOID RELATIONSHIPS

FIGURE 25 THE EFFECT OF MASS ANOMALIES ON THE GEOID

The deflections and undulations computed with sufficient gravity information are considered

absolue values referred to an earth-centered reference ellipsoid. In other words, the axis of

rotation for the ellipsoid passes through the earth’s center of gravity.
Effective use of the gravimetric method is dependent only on the availability of anomalies in

sufficient quantity to achieve the accuracy desired. Successful use of Stoke’s integral and

Vening-Meinesz formulas depends on a good knowledge of gravity anomalies in the immediate

vicinity of the point under consideration and a general knowledge of anomalies for the entire

earth.

Geophysical Gravity Predictions

Measured gravity coverage is by no means complete. There are many large regions on the

continents where gravity measurements are lacking or available only in small quantities. Gravity

data for ocean areas has always been sparse, however, Satellite Altimetry (Chapter VII) has

overcome this deficiency. In regions where an insufficient number of gravity measurements

exists, some other approach must be used to obtain or predict the mean gravity anomalies for the

areas.

Correlations exist between variations in the gravity anomaly field and corresponding variations

in geological, crustal, and upper mantle structure, regional and local topography and various

other types of related geophysical data. In many areas where gravity information is sparse or

missing, geological and geophysical data is available. Therefore, the various prediction methods

take into account the actual geological and geophysical cause of gravity anomalies to predict the

magnitude of the anomalies.
Chapter VI

SATELLITE GEODESY

Scientific papers advocating the use of satellites for geodetic purposes were published as early as

1956. Geodetic applications were outlined by the Smithsonian Astrophysical Observatory for

data obtained from Project Vanguard during the 1958-59 International Geophysical Year. Many

techniques and a great deal of knowledge were ultimately derived from this project. With this

information, the constant growth of space technology, the development of electronic distance

measuring devices, and the perfection of electronic data processing equipment, satellites

specifically equipped for geodetic purposes have been developed, launched, observed and the

data utlizied.

The first real geodetic satellite was ANNA-1B launched in 1962. Project ANNA was a truly

cooperative effort involving the Department of Defense (DoD), the National Aeronautics and

Space Administration (NASA), and other civil agencies. Several observational systems were

developed and improved during ANNA. These systems included geodetic cameras, electronic

ranging and Doppler. Knowledge gained from Project ANNA was also useful in the development

of Laser ranging systems.

Observational Systems

Two basic systems have been used for obtaining geodetic information from artificial earth

satellites-optical and electronic. These systems have made it possible to perform various geodetic

measurements to relate known or unknown positions to the earth’s center, to relate unknown

positions to existing triangulation networks, and to relate the triangulation networks to each

other. Important parameters of the earth’s gravitational field and values for the earth’s flattening

have also been obtained.

Historical Systems

Optical tracking with the BC-4, PC-1000, MOTS, or Baker Nunn cameras consisted of

photographic observations of a satellite, or flashing light on the satellite, against a background of

stars. The stars, whose positions were accurately determined, provided a framework on the

photographic plate or film for a determination of precise directions from camera station to

satellite. Geodetic positioning work with cameras was usually performed with one camera

observing simultaneously with one or more other cameras. Figure 26. Camera systems are

weather dependent and that is one major reason why they are little used today. Laser systems

discussed later, are also weather dependent but their extreme accuracy justifies their use and

development.

FIGURE 26 SIMULTANEOUS METHOD

The U.S. Army developed the SECOR (Sequential Collation of Range) system and the first

SECOR transponder was orbited on ANNA-1B in 1962. The SECOR system continued in use

through 1970. The system operated on the principle that an electromagnetic wave propagated

through space undergoes a phase shift proportional to the distance traveled. A ground station
transmitted a phase modulated signal which was received by the satellite-borne transponder and

returned to the ground. The phase shift experienced by the signal during the round trip from

ground to satellite and back to ground was measured electronically at the ground station which

provided as its output a digitized representation of range.

Doppler

A geodetic satellite can also carry electronic signaling equipment to produce the Doppler effect

which can be used for geodetic purposes. The Doppler observational system was derived from

the fact that while a satellite transmitter sends a continuous unmodulated wave at a fixed

frequency, the received signal at the tracking stations exhibits a shift in frequency due to the

relative velocity of the satellite and observing station. A similar phenomenon may be observed

with sound waves, as the source of the sound approaches and recedes from the observer. For

example, the pitch of a train whistle apparently changes as the train approaches and recedes from

the observer. Although the sound waves travel at a constant rate-approximately 1080 feet per

second at sea level-they become crowded together as the source approaches the observer, the

wave lengths become shorter, and the pitch increases. The opposite effect takes place as the

source moves away.

In Figure 27, the radio source is assumed to be moving in a straight line at a specified speed. The

angle Ø is used to indicate the distance between the position of the satellite at P (time t) and the

point Pm (time tm) where it will be closest to the observer. If the Doppler shift were expressed in

terms of Ø, as Ø decreases, the Doppler shift decreases to zero. As the satellite recedes, the

received frequency decreases and the shift increases until the transmitter is out of range. This

received frequency is actually a function of the transmitted frequency, velocity of propagation,

and the rate of change of the slant range between the satellite and station. From observations at

one station, the satellite period, time and distance of its closest approach, and its relative velocity

can be determined. If observations are made from three or more stations, the orbital parameters

may be derived.

FIGURE 27 DOPPLER SHIFT

Of all the satellite observation systems used so far, Doppler tracking has been the most fruitful.

Reasons for this are: it is passive, not requiring any interrogation or directionally sensitive

antennae at the receiver; the data obtained (Doppler counts) are in digital form; the radio

frequencies used permit all weather day and night tracking; and accuracies achieved have

steadily improved. Much effort has gone into identifying and eliminating the sources of error in

Doppler work. Automatic portable receiving equipment is available from several suppliers. The

GEOCEIVER (geodetic receiver) has played an important role in DoD observational programs.

Since 1967 there have been at least four operational satellites continuously available for Doppler

positioning. The National Geodetic Survey (NGS) will be using results from many Doppler

stations within the United States and its possessions in support of the readjustment of the North

American horizontal datum.

Laser

The laser has been adapted to measuring distances over the earth's surface and for computing

ranges from earth stations to satellites and the lunar surface. The laser instrument is pointed to a
target and then activated by a clock at the appropriate time. The laser beam is reflected at the

target by special reflectors and the returning light is detected photoelectrically, and its time of

flight measured to yield range data. The laser transmitter is mounted adjacent to some type of

telescope or optical device used for receiving the reflected laser beam.

In satellite laser ranging, the interval between the outgoing and returning pulse from the satellite

is measured very accurately and then transformed into a range measurement which is corrected

for atmospheric refraction. Laser ranging is possible even when the satellite is in the earth’s

shadow and during daylight hours.

Simultaneous laser ranging to a near-earth satellite from two sites is used to determine the

coordinates of one laser site relative to the fixed position of the other site and simultaneously the

inter-site distance. NASA has used laser tracking since 1972 to measure the distance between

points in North America. They have been testing the accuracy of laser tracking in measuring the

crustal motion between points on opposite sides of the San Andreas fault and plan to make

repeated measurements of baselines across the fault over a number of years. Simultaneous laser

tracking has also been achieved between an east coast site and Bermuda enabling a determination

of the Bermuda site’s relative location (North American Datum) and the baseline between the

two sites.

Laser ranging data has been incorporated into the development of world geodetic systems by the

Smithsonian Astrophysical Observatory (SAO) and the Department of Defense (DoD). NASA

has also included laser data in their development of gravitational models. Laser data is also being

used for polar motion and earth rotation studies.

Doppler Satellite Surveys

Doppler satellite surveying is a method of determining positions of points on the earth’s surface

by observing the Doppler shift of radio transmissions from satellites of the U.S. Navy Navigation

Satellite System (NNSS). NNSS was developed for the Navy as a worldwide all-weather

navigation system, and provides position fixes at time intervals of 2 hours or less. Observations

of these satellites began in earnest in 1971, when portable tracking receivers became available to

establish precise positions on the earth’s surface. Since then, the application of Doppler satellite

surveying techniques expanded, until today they are used worldwide.

System Description

The NNSS satellites currently operational are in circular polar orbits approximately 1000km

above the earth’s surface. The time to complete an orbit (orbital period) is approximately 105

minutes. The earth’s rotation causes a satellite to cross the Equator on each revolution

approximately 26° in longitude west of the previous crossing. Each satellite transmits what is

referred to as the "broadcast ephemeris, " which describes the satellite's position in space. The

ephemeris is a predicted orbit based on Doppler observations previously acquired by four

tracking stations located in the United States.

Doppler satellite observations, reduced using the broadcast ephemeris, yield point positions with

sufficient accuracy to satisfy the requirements of many NNSS satellite users. There exists a more

accurate "precise" ephemeris which is generated for selected NNSS satellites from Doppler

observations acquired by a tracking network (called TRANET) composed of 15 to 20 stations.
The positioning accuracies for single station observations are greatly improved when using the

precise ephemeris. These ephemerides are computed by the Defense Mapping Agency (DMA) to

support geodetic applications requiring point positions of one-meter accuracy.

Modes of Observation

Points on the earth’s surface can be positioned by various modes (point positioning, simultaneous

point positioning, translocation, semishort-arc or short-arc) using either the broadcast ephemeris

or the precise ephemeris.

In the point-positioning mode, multiple passes collected with a single Doppler receiver are used

with an ephemeris to determine an independent station position in geocentric coordinates

(X,Y,Z) referenced to the Earth-centered satellite coordinate system. Figure 28. The geocentric

coordinates can also be expressed in geodetic coordinates (latitude, longitude, and height above

ellipsoid). Doppler positions determined with the precise ephemeris can be directly transformed

to the World Geodetic System 1972 (WGS-72) discussed in Chapter VIII.

FIGURE 28 POINT POSITIONING

When employing the point-positioning mode as the Doppler surveying technique, the user may

wish to transform the Doppler satellite-derived geocentric coordinates to the local geodetic

system. The process of deriving the coordinate shifts and transforming the Doppler position

requires a thorough understanding of datum transformation concepts and procedures. Occupation

of a station with known local geodetic coordinates allows the transformation parameters to be

derived for subsequent use with Doppler positions that were not established on a station tied to

the local geodetic system.

In the simultaneous point-positioning mode, stations are simultaneously occupied in figures of

two or more. The data are independently reduced as in the point-positioning mode and

differenced to form relative positions. The simultaneous observations are performed during a

common time period, but do not necessarily include common satellite passes.

In the translocation mode, observations are simultaneously collected, usually at two stations.

This mode is employed to obtain very accurate relative positions, even if the precise ephemerides

are not available. When the broadcast ephemeris is used, statistical correlation performed during

data reduction improves the accuracy of the positioning. The principal error sources affecting an

individual satellite position fix are the ephemeris errors and refraction errors. Improved

compensation for these errors is possible when the same signal is received at separate sites. The

maximum spacing between sites is generally limited to approximately 500km (or less if

comparability with existing control is to be maintained) so that desirable portions of satellite

passes can be tracked simultaneously. During processing, enforcement of simultaneity of data

points is optional. When simultaneity is enforced, it is generally referred to as rigorous

translocation. Figure 29.

FIGURE 29 TRANSLOCATION

The short-arc and semishort-arc modes allow for small adjustments in the orbit instead of

holding the satellite ephemeris fixed, as is done for the other methods. The translocation method,

for example, assumes that orbit errors affect positioning of all sites in the same way, whereas the
short-arc technique adjusts the reference orbit while simultaneously solving for positions. In

short-arc processing, six orbital parameters are allowed to adjust. The method becomes

semishort-arc processing when one to five parameters are adjustable.

Short-arc, semishort-arc, translocation and simultaneous point positioning are all used for

relative positioning. In a survey scheme, one or more base station positions are generally held

fixed. These fixed positions usually have a known position on a given geodetic datum. While

data are being collected at the fixed station(s), one or more additional receivers are circulated

among the various unknown stations comprising the net. The data collected simultaneously are

subjected to postprocessing to determine a position relative to the base stations. Figure 30.

FIGURE 30 RELATIVE POSITIONING

Although point positioning is the least accurate mode to use when only the broadcast ephemeris

is available, this technique can be performed with a single receiver and fairly simple

computations. Translocation and short-arc techniques are the most accurate modes when the

precise ephemeris is not available, but for these modes, field and computational procedures are

more complex.

Because NNSS satellites are capable of providing the broadcast ephemeris, it may be more

economical to use the broadcast ephemeris rather than the precise ephemeris to meet the survey

requirements. This can reduce significantly the period of occupation for a station. Data

processing can also be performed more quickly because no time is lost waiting for posttracking

orbital data in order to generate the precise ephemerides.

Harmonic Analysis of Orbital Data

A great deal of study has been done regarding the effect of the earth’s gravitational attraction on

satellite motion. The fact that there are a number of perturbing factors has already been

mentioned-the uneven distribution of the earth’s mass, the oblateness of the earth, atmospheric

drag, the effects of the planets, sun and moon, and electromagnetic effects. The perturbations are

measured by observing the position of the satellite in orbit around the earth. As observational

data accumulates, orbital parameters become more precisely defined and reliable earth-centered

positioning becomes available. An analysis of orbital data can also be used to develop an expression of the earth’s external gravity field for a better interpretation of the shape of the geoidal

surface through spherical harmonics. (The Appendix contains a discussion of spherical

harmonics.) Although a complete analysis of orbital data requires consideration of all perturbing

effects, the earth itself is the only perturbing body of major consequence in the study of nearearth satellite motion. The effects of the sun, moon, and atmosphere are removed so that only the

effects of earth’s shape and uneven mass distribution remain.

The uneven distribution of the earth’s mass causes the force of gravity to vary from point to point

on the surface and in external space. While force of gravity is measured at points on the surface

with highly sensitive instruments, mathematical procedures are required to analyze orbital

perturbations and to express the gravitational potential. The gravitational potential may be

explained in terms of potential surfaces-surface to which the force of gravity is always perpendicular. If the earth were a perfect non-rotating sphere with homogeneous mass distribution, the

potential surface would be spherical in shape. The fact that the earth is shaped more like an
ellipsoid than a sphere causes the potential surface to be shaped more like an ellipsoid. Actually,

the earth is neither spherical nor ellipsoidal. The potential surface bulges where there is

excessive mass and it is depressed in areas of mass deficiency. The undulating surface described

earlier as the geoid is a potential surface of the real earth. The diagram in Figure 31 illustrates the

three surfaces just discussed.

FIGURE 31 POTENTIAL SURFACES

The most convenient way to express the gravitational potential is in terms of a series of spherical

harmonics mentioned above. The coefficients of the various harmonic terms are functions of the

various orbital perturbations. A few are directly related to the shape of the earth and the remainder to the uneven distribution.

While it is possible to derive harmonic coefficients from observed gravity, the method is limited

due to the lack of high quality worldwide gravity coverage. The computation of coefficients from

satellite data also has its limitations. There are many coefficients that are not well defined from

tracking data due to the small magnitude of the orbital perturbations at geodetic satellite

altitudes. In addition, satellites orbiting at different inclinations are needed to reduce the

correlation between the computed coefficients. For best results, the current practice is to combine

tracking data with available surface-gravity data when solving for the spherical harmonic

coefficients of the earth’s gravitational field.

Chapter VII OTHER DEVELOPMENTS IN GEODESY

Many pertinent developments have occurred since the last revision of this publication in

February 1968. This chapter provides a place for subjects of interest that are not discussed in the

other chapters. Some of the areas of new geodetic developments are: satellite laser ranging

(Chapter VI), lunar laser ranging, very long baseline interferometry, satellite radar altimetry, the

NAVSTAR Global Positioning System, satellite-to-satellite tracking, and inertial surveying.

Lunar Laser Ranging

In July 1969, the first men to set foot on the moon performed a number of tasks of scientific

importance. Among these tasks was the deployment of a rack structure carrying an array of 100

fused silica retroreflectors designed to return some of the light of a pulsed laser beam to the

telescope to which the laser equipment is coupled. These retroreflectors are a part of an Apollo

experiment called LURE (Lunar Laser Ranging Experiment). LURE was designed to obtain

extremely accurate measurements of the range from known locations on the surface of the earth

to the lunar based retroreflectors and enable the improvement of our knowledge of the rotation of

the earth and the moon about their center of mass and the moon’s libration and motion about the

earth.

Very Long Baseline Interferometry

Observations of extragalactic radio sources such as quasars, can provide the geodetic information to determine the vector separations between the antennas of two widely separated radio

telescopes. The components of the vector are its length and direction. To accomplish this, it is
necessary to measure very accurately the difference in the time of arrival, recorded at the two

antennas, of a particular wavefront from a given (point) source of radio radiation. The

phenomena called interference, in Very Long Baseline lnterferometry (VLBI), is produced by

electronically superimposing the recorded signals to produce a resultant disturbance or

"interference" pattern. The theoretical expression for the relative phase delay shows it to be a

function of the source direction, the antenna locations, the relative clock error between the two

sites, the time of day, the model atmosphere employed, the earth’s tidal parameters, the radio

frequency at which the observation is made, etc. Proper account must also be taken of the earth’s

rotation. Two of the main limiting factors in the VLBI technique are clock stability and

atmospheric variations. A major goal of VLBI is to reduce the uncertainty in intercontinental

baselines to the centimeter level.

VLBI derived baselines have already contributed scale information to the development of the

DoD World Geodetic System in 1972. Baselines accurate to the centimeter level would function

as standards of comparison for future world systems. Other applications of VLBI include the

determination of polar motion, variations in the earth’s rotation, and the monitoring of motions of

the major plates that compose the earth’s crust.

Satellite Radar Altimetry

The development of orbiting space satellites from which microwave remote sensing of the earth

can be achieved has provided a new instrument to the geodesist which measures directly the

shape of the geoid in the ocean areas. The satellite altimeter consists of a downward ranging

radar which measures the time delay from the transmission to the reception of a pulse of energy.

Figure 32. The observed one-way distance from the transmitting antenna to the surface is equal

to one-half the product of the time delay and the speed of light. From this distance or height, the

local surface effects such as tides, winds and currents are removed to obtain the satellite height

(h) above the geoid. Figure 33. With a precise Doppler ephemeris available for the satellite, the

radius (Rsat) to the satellite, determined for the time of each observation, along with the radius

(RØ) to the ellipsoid are readily at hand. It is then possible to compute the geoid height (N) by

subtracting the radius RØ and the satellite height h from Rsat.

FIGURE 32 THE MEASUREMENT OF THE GEOID BY THE SATELLITE ALTIMETER

FIGURE 33 SATELLITE HEIGHT ABOVE THE GEOID

The Skylab spacecraft, launched in 1973, provided the first opportunity for satellite based radar

altimetry. It was basically a research mission for which data was obtained for the designing of

future altimeters. The GEOS-3 altimeter which incorporates many of the design features that

were tested in the Skylab altimeter was launched in 1975 and provided geoid measurements over

the water areas of the earth from 65°N to 65°S. The SEASAT altimeter which was a more

sophisticated instrument with greater measurement capabilities was launched in June 1978 and

added data from 72°N to 72°S.

The NAVSTAR Global Positioning System

Scientists, engineers, and planners have been tasked with making comprehensive studies of

currently available navigation systems as part of an effort to devise a system capable of meeting

the requirements of the United States after 1980. Since the late-1950's both military and civilian
agencies have actively and independently pursued the idea of position determination and

navigation using satellites. This resulted in the development of several systems with a multitude

of specialized equipment responsive to particular mission requirements with varying degrees of

accuracy and capabilities. In order to integrate the independent efforts of the military services,

the Department of Defense issued a memorandum in 1973 naming the Air Force as the Executive

Service for the initial development of a future Defense Navigation Satellite System (DNSS),

designated the NAVSTAR Global Positioning System (GPS).

The GPS concept calls for a precise navigation system divided into three segments: space

segment, control segment and user equipment segment. The space segment will consist of six

orbital planes of satellites at inclinations of 55° in circular orbits at an altitude of 20,200 km.

Figure 34. Each plane is to eventually contain three satellites. Each satellite will broadcast

signals containing information as to its position. This broadcast will include an orbital ephemeris

referenced to the DoD World Geodetic System. The control segment will be the ground stations

necessary to track the satellites, monitor the system operation and periodically provide

corrections to the navigation and time signals. The user segment will consist of the equipment

necessary to convert the satellite signals into useful navigation information. By receiving signals

from four satellites, the user, whether stationary or moving, can calculate his precise time, threedimensional position and, if moving, his three-dimensional velocity. Position determination

alone requires analysis of range information from three of the satellites in view. However, since

the user's receiver clock will not be synchronized to the satellite clock, time of arrival

measurements from four satellites are needed to update the user's clock.

FIGURE 34 GPS SATELLITE CONSTELLATION

When operational, GPS should satisfy the navigational accuracy requirements of many militarytype missions on land, sea or in the air. Agencies also have many requirements for accurate

geodetic positioning for which GPS will satisfy for years to come. These include establishing

and densifying geodetic control, offshore positioning and the geodetic needs of national defense

which brings in global requirements. GPS will also provide an excellent facility for

determination of the position of other satellites and space vehicles while they are in lower earth

orbits. This satellite- to-satellite tracking is discussed next.

Satellite-to-Satellite Tracking

A new technique for using artificial satellites for geodetic purposes is being studied and tested.

This technique uses satellites to track other satellites. There are a number of variations which

may be used for specific purposes such as gravity field investigations and orbit improvement. A

high altitude satellite may act as a relay from ground tracking stations to a low altitude satellite.

In this way, low altitude satellites may be observed when they are not accessible to ground

stations. Figure 35. In this type of tracking, a signal generated by a tracking station is received by

the relay satellite and then retransmitted to a lower altitude satellite. This signal is then returned

to the ground station by the same path. Two low altitude satellites can track one another obsering

mutual orbital variations caused by gravity field irregularities. Several high altitude satellites

with accurately known orbits may be used to fix the position of a low altitude satellite. Figure 36.

These examples present a few of the possibilities for the application of satellite-to-satellite

tracking.
FIGURE 35 SATELLITE-TO-SATELLITE TRACKING (VIA RELAY SATELLITE)

FIGURE 36 SATELLITE-TO-SATELLITE TRACKING (VIA SATELLITE

CONSTELLATION)

Satellite-to-satellite tracking data was first collected and analyzed in a high-low configuration

between ATS-6 and GEOS-3. The data was studied to evaluate its potential for both orbit and

gravitational model refinement. This experiment and others that followed proved this new

technique to be an important tool for space geodesy.

Inertial Surveying

Inertial Navigation is the art and science of determining the position and velocity of a vehicle

solely by means of sensing that vehicle’s accelerations and performing the necessary integrations

to determine the position and velocity on a real-time basis. The inertial system is composed of

precise accelerometers to sense specific force acting on the vehicle and precise gyros to maintain

orientation of the accelerometers in a chosen coordinate frame or to determine the orientation of

the accelerometers with respect to that frame. Computation is performed by a small on-board

computer and the position and velocity of the vehicle are displayed on a real-time basis. In the

two decades that inertial navigation has been used, continued hardware developments have

brought a state-of-the-art in which the inertially determined position of the vehicle is sufficiently

accurate that inertial techniques can be applied to surveying.

At the heart of the inertial surveyor is the inertial measuring unit which contains three sensitive

accelerometers and three precise gyros. The accelerometers are mounted as a mutually

orthogonal triad on a platform which is torqued by the gyros to maintain orientation with the

local vertical and local north, that is, the three axes are oriented north-east-down. The

accelerometers measure the specific force on the vehicle which is the sum of the vehicle’s own

accelerations and the local gravity vector. The digitized output of the accelerometers are

processed in real-time by a digital computer. They are integrated once to give velocity, and

integrated again to give distance travelled along each sensitive axis. The system does not yield

the latitude, longitude and elevation directly. To the computed distances, which are referenced to

inertial space, there must be added the initial position and a conversion to latitude, longitude, and

elevation accomplished. Although high quality accelerometers and gyros are used in the system,

they are still subject to drift and bias. This will cause a misalignment of the platform and errors

in the sensed accelerations, which results in small errors in computed velocities and positions.

The currently available inertial surveying systems must stop or hover at frequent intervals. At

these times, a Kalman filter process corrects for the difference between the indicated velocity

and zero, and calculates normal gravity, elevation, and anomalous gravity, but only at these

points where remaining errors in platform alignment are also corrected by the Kalman filter.

It was stated above that the accelerometers sense the sum of the vehicle’s acceleration and the

local gravity vector and that the vehicle’s accelerations are needed for integration into velocity

and distance travelled. However, a model of the earth’s gravity field is required to remove the

accelerations due to gravity. In current systems, a very simplistic model is used in which only the

downward gravity component resulting from an ellipsoidal earth is computed. Thus the system

cannot correct for deflection of the vertical. Further, these deflections of the vertical result in

erroneous platform alignments which may introduce errors as large as 40 cm in the computed
positions.

Gravity Gradiometry

Gravity gradiometers have been suggested as a means of independently determining the

components of the gravity vector on a real-time basis. A gravity gradient is simply the spatial

derivative of the gravity vector. The gradient can be thought of as the rate of change of a

component of the gravity vector as measured over a small distance. Hence, the gradient can be

measured by determining the difference in gravity at two close but distinct points. This principle

is embodied in several recent moving-base instruments. The gravity gradient at a point is a

tensor, since it is the derivative of each component of the gravity vector taken in each sensitive

axis. Thus, the value of any component of the gravity vector can be known all along the path of

the vehicle if gravity gradiometers are included in the system and their outputs are integrated by

the system computer. In theory, an accurate gravity model will be computed in real-time and a

continuous map of normal gravity, elevation, and anomalous gravity will be available.
Chapter VIII THE WORLD GEODETIC SYSTEM

Because of the inability of the large geodetic systems such as the North American Datum

(NAD), European Datum (ED), and Tokyo Datum (TD), to provide a basis for expression of

inter- continental geodetic information, a unified world system became essential. The

Department of Defense, in the late 1950’s began to develop the needed world system to which

geodetic datums could be referred and compatibility established between the coordinates of

widely separated sites of interest. Efforts of the Army, Navy and Air Force were combined

leading to the DoD World Geodetic System 1960 (WGS 60). In accomplishing WGS 60, a

combination of available surface gravity data, astrogeodetic data and results from HIRAN and

Canadian SHORAN surveys were used to define a best-fitting ellipsoid and an earth-centered

orientation for each of the initially selected datums (Chapter IV). (The datums are relatively

oriented with respect to different portions of the geoid by the astro-geodetic methods already

described.) Figure 21. The sole contribution of satellite data to the development of WGS 60 was

a value for the ellipsoid flattening which was obtained from the nodal motion of a satellite.

Prior to WGS 60, the Army and Air Force had each developed a world system by using different

approaches to the gravimetric datum orientation method. Figure 37. To determine their

gravimetric orientation parameters, the Air Force used the mean of the differences between the

gravimetric and astro-geodetic deflections and geoid heights (undulations) at specifically

selected stations in the areas of the major datums. The Army performed an adjustment to

minimize the difference between astro-geodetic and gravimetric geoids. By matching the relative

astro-geodetic geoids of the selected datums with an earth-centered gravimetric geoid, the

selected datums were reduced to an earth-centered orientation. Since the Army and Air Force

systems agreed remarkably well for the NAD, ED and TD areas, they were consolidated and

became WGS 60.

FIGURE 37 GRAVIMETRIC DATUM ORIENTATION

The Department of Defense World Geodetic System 1966

In January 1966, a World Geodetic System Committee composed of representatives from the

Army, Navy and Air Force, was charged with the responsibility of developing an improved WGS

needed to satisfy mapping, charting and geodetic requirements. Additional surface gravity observations, results from the extension of triangulation and trilateration networks, and large

amounts of Doppler and optical satellite data had become available since the development of

WGS 60. Using the additional data and improved techniques, WGS 66 was produced which

served DoD needs for about five years after its implementation in 1967. The defining parameters

of the WGS 66 Ellipsoid were the flattening (1/298.25), determined from satellite data and the

semimajor axis (6,378,145 meters), determined from a combination of Doppler satellite and

astro- geodetic data. A worldwide 5° x 5° mean free air gravity anomaly field provided the basic

data for producing the WGS 66 gravimetric geoid. Also, a geoid referenced to the WGS 66

Ellipsoid was derived from available astrogeodetic data to provide a detailed representation of

limited land areas.

The Department of Defense World Geodetic System 1972
After an extensive effort extending over a period of approximately three years, the Department

of Defense World Geodetic System 1972 was completed. Selected satellite, surface gravity and

astrogeodetic data available through 1972 from both DoD and non-DoD sources were used in a

Unified WGS Solution (a large scale least squares adjustment). The results of the adjustment

consisted of corrections to initial station coordinates and coefficients of the gravitational field.

The largest collection of data ever used for WGS purposes was assembled, processed and applied

in the development of WGS 72. Both optical and electronic satellite data were used. The

electronic satellite data consisted, in part, of Doppler data provided by the U.S. Navy and

cooperating non-DoD satellite tracking stations established in support of the Navy’s Navigational

Satellite System (NNSS). Doppler data was also available from the numerous sites established by

GEOCEIVERS during 1971 and 1972. Doppler data was the primary data source for WGS 72.

Figure 38. Additional electronic satellite data was provided by the SECOR (Sequential Collation

of Range) Equatorial Network completed by the U.S. Army in 1970. Optical satellite data from

the Worldwide Geometric Satellite Triangulation Program was provided by the BC-4 camera

system. Figure 39. Data from the Smithsonian Astrophysical Observatory was also used which

included camera (Baker Nunn) and some laser ranging.

FIGURE 38 DOPPLER SATELLITE GROUND STATIONS PROVIDING DATA FOR WGS

72 DEVELOPMENT

FIGURE 39 WORLDWIDE GEOMETRIC SATELLITE TRIANGULATION NETWORK, BC4 CAMERAS

The surface gravity field used in the Unified WGS Solution consisted of a set of 410 10° x 10°

equal area mean free air gravity anomalies determined solely from terrestrial data. This gravity

field includes mean anomaly values compiled directly from observed gravity data wherever the

latter was available in sufficient quantity. The value for areas of sparse or no observational data

were developed from geophysically compatible gravity approximations using gravitygeophysical correlation techniques. Approximately 45 percent of the 410 mean free air gravity

anomaly values were determined directly from observed gravity data.

The astrogeodetic data in its basic form consists of deflection of the vertical components referred

to the various national geodetic datums. These deflection values were integrated into

astrogeodetic geoid charts referred to these national datums. The geoid heights contributed to the

Unified WGS Solution by providing additional and more detailed data for land areas.

Conventional ground survey data was included in the solution to enforce a consistent adjustment

of the coordinates of neighboring observation sites of the BC-4, SECOR, Doppler and BakerNunn systems. Also, eight geodimeter long line precise traverses were included for the purpose

of controlling the scale of the solution.

The Unified WGS Solution, as stated above, was a solution for geodetic positions and associated

parameters of the gravitational field based on an optimum combination of available data. The

WGS 72 ellipsoid parameters, datum shifts and other associated constants were derived

separately. For the unified solution, a normal equation matrix was formed based on each of the

mentioned data sets. Then, the individual normal equation matrices were combined and the

resultant matrix solved to obtain the positions and the parameters.
The value for the semimajor axis (a) of the WGS 72 Ellipsoid is 6378135 meters. The adoption

of an a-value 10 meters smaller than thaf for the WGS 66 Ellipsoid was based on several

calculations and indicators including a combination of satellite and surface gravity data for

position and gravitational field determinations. Sets of satellite derived station coordinates and

gravimetric deflection of the vertical and geoid height data were used to determine local-togeocentric datum shifts, datum rotation parameters, a datum scale parameter and a value for the

semimajor axis of the WGS Ellipsoid. Eight solutions were made with the various sets of input

data, both from an investigative point of view and also because of the limited number of

unknowns which could be solved for in any individual solution due to computer limitations.

Selected Doppler satellite tracking and astro-geodetic datum orientation stations were included in

the various solutions. Based on these results and other related studies accomplished by the

Committee, an a-value of 6378135 meters and a flattening of 1/298.26 were adopted.

In the development of local-to WGS 72 datum shifts, results from different geodetic disciplines

were investigated, analyzed and compared. Those shifts adopted were based primarily on a large

number of Doppler TRANET and GEOCEIVER station coordinates which were available

worldwide. These coordinates had been determined using the Doppler point positioning method.

A New World Geodetic System

The need for a new world geodetic system is generally recongnized by the geodetic community

within and without the Department of Defense. WGS 72 no longer provides sufficient data,

information, geographic coverage, or product accuracy for all present and anticipated

applications. The means for producing a new WGS are available in the form of improved data,

increased data coverage, new data types and improved techniques. GRS 80 parameters (Chapter

II), and presently available Doppler, laser and VLBI observations constitute significant new

information which are being utilized. There are now over 1000 Doppler determined station

positions available as compared to the near 100 station values used in WGS 72. Also, an

outstanding new source of data is now being made available from satellite radar altimetry

(Chapter VII). Among the improved techniques now employed is an advanced least squares

method called collocation which provides for a consistent solution from different types of

measurements all relative to the earth’s gravity field, i.e. geoid, gravity anomalies, deflections,

dynamic Doppler, etc.

WGS (initially WGS 72) is the reference system being used by the Global Positioning System.
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APPENDIX

SPHERICAL HARMONICS

A. HARMONIC EXPRESSIONS

Trigonometric functions, sines and cosines, are harmonic in that they repeat themselves after a

certain interval. Harmonic Expressions, formed by combining sines, cosines and arbitrary

constants, are mathematical devices for curve-fitting and interpolation in either two or three

dimensions. An example of a harmonic Expression for a two-dimensional curve is
where A(sub0), A(sub1), A(sub2),..., are arbitrary constants.

The nature of every harmonic expression is that the constant term (AO in the equation above)

equals the mean or average value of all points on the curve. The succeeding terms represent

harmonic deviations from that mean value. Each harmonic term has two parts: the trigonometric

part which is harmonic and controls the frequency of the oscillations, and a constant multiplier

which controls the amplitude. For the second term in equation (1)

coso is the trigonometric part and A(sub1) is the constant multiplier.

 Example. Consider the harmonic expression

which describes the curve shown in Figure A-1. Its first term, which is its

constant term, is 3.

Thus 3 is the mean value of the curve. The second term, y = 2cos(Theta), is a

harmonic of period 360(Degree),

because after this period it repeats itself. Its amplitude is 2, the largest

value for y this term can

attain. The third term, y = 2sin(Theta), traces the same curve as y =

2cos(Theta) except that it is 90(Degree) out

of phase, or shifted 90(Degree). The term y = 4cos2(Theta) has an amplitude

of 4 and a frequency of 2.

The frequency of 2 means that the curve for this term repeats itself twice

every 360(Degree). The last

term, y = -1sin8(Theta), has an amplitude of -1 and a frequency of 8. The

negative constant indicates

that the curve for this term will start on the negative side of y = 0, rather

than on the positive

side. Graphs of the harmonic terms are shown in Figure A-2.

B. SPHERICAL HARMONICS
 In the geosciences, a commonly used form of harmonics are the surface

spherical harmonics.

These functions are much used in geodesy because they are relatively simple

to use and because

the earth’s shape can be approximated by a sphere. The coordinates normally

associated with

spherical harmonics are the spherical coordinates: radius r, co-latitude or

polar distance (Theta), and

longitude (Lambda).
Spherical Coordinates
 1. Legendre Functions

 The basic spherical harmonic form is the Legendre polynomial,

Pn(cOso), given by the

Formula

where n, a nonnegative integer, is the degree of the polynomial. The Legendre

polynomials for

degrees n = 0 through n = 5 are:

Graphs of these polynomials up to n = 4 are shown in Figure A-3.

 Another form of spherical harmonics are the associated Legendre

functions, P(subnm)(cos(Theta)),

where m, a nonnegative integer smaller than or equal to n, is the order of

the function. When

m = 0, this function becomes P(subn0)(cos(Theta)), or simply the Legendre

polynomial P(subn)(cos(Theta)).

Figure A-3

 The surface spherical harmonics are Legendre functions multiplied by cos
m(Lambda) or sin m(Lambda),

where (Lambda) is the longitude,

and so on.

 These resultants are tesseral harmonics, which divide the sphere into

blocks of alternate

positive and negative values, called tesserals. Figure A-4 shows a typical

tesseral harmonic. The

shaded areas represent tesserae where the harmonic is positive, and the nonshaded areas represent negative tesserae.

 A special case arises when m = 0, in which case the sin m(Lambda)

harmonic terms disappear

because sin O(Lambda) = 0. When m = 0, the tesseral harmonic divides the

sphere into zones of alternate

positive and negative values, and is therefore called a zonal harmonic. An

additional special case

arises when m = n. Then the tesseral harmonic divides the sphere into sectors

of alternate positive

and negative values. Hence they are called sectorial harmonics. Examples of

zonal and sectorial

harmonics are shown in Figure A-4.

 2. Surface Spherical Harmonics

 The general expression for the surface spherical harmonic of an

arbitrary function f((Theta),(Lambda)) is
where A(subnm) and B(subnm) are arbitrary constants. For simplicity and

easier handling, the harmonics

in Equation (3), commonly referred to as conventional harmonics, can be

replaced by other functions called fully normalized harmonics. The surface spherical harmonic then

becomes

Figure A-4

where the bar indicates that the harmonic term is fully normalized. The

conventional and fully

normalized terms are related through the following expressions
 In many applications, geopotential coefficients, symbolized by C and S,

are used in harmonic

expressions rather than the A and B coefficients. Like the A and B

coefficients, they can be used in

either conventional or fully normalized form. The relationship between the

two types of coefficients is.

where a is the equatorial radius and GM is the gravitational constant. Fully

normalized geopotential coefficients, C and S, can be obtained from the conventional

coefficients, C-BAR and S-BAR,

by means of an expression similar to Equation (5).

C. GRAVITATIONAL POTENTIAL OF THE EARTH

 The expression for the earth’s gravitational potential, that is, the

potential of the attraction

of the earth, is
where

 The ability to express the gravitational potential by Equation (8) lies

in determining proper

values for the harmonic coefficients, C-BAR and S-BAR. These coefficients are

a function of the mass

distribution of the earth, which unfortunately is not known. Thus, the

coefficients have to be

computed in some other way, such as, from artificial satellite perturbations

or gravity observations on the surface of the earth.

 In the expression for the gravitational potential, the constant term, U

= GM/r, which is the

n = 0 term, equals the mean value. The mean value of the earth’s

gravitational potential

is the value which would occur if the earth were a sphere. As in the twodimensional case, each

term after the mean value represents a harmonic deviation from that mean

value. As was previously stated, the magnitude of the coefficients, C-BAR and S-BAR, is
related to the mass distribution

of the earth. Thus, where the potential is positive with respect to the mean

value, there is a

mass excess. Similarly, there is a mass deficiency where the potential is

negative.

D. ANALYSIS OF LOWER DEGREE HARMONIC TERMS

 The following discussion will analyze in detail some of the lower degree

harmonic terms.

In order to simplify writing, conventional harmonics will be used in this

portion of the discussion.

Equation (8), expressed in terms of conventional harmonics, is

If this equation is expanded, the harmonic terms for degrees 1 and 2 are:
1. Degree n = 1

 Figure A-5 shows a geometric representation of the harmonic term

P(sub10)(cos(Theta)). The solid

circle represents the mean value and the dashed curve is the value of the

potential when only

the mean value term and the term under consideration are combined. Thus, the

dashed curve

shows the effect of an individual term. The hachured part of the figure

indicates the area of the

sphere where the harmonic is positive; and the non-hachured part indicates

where the harmonic

is negative.

Figure A-5, A-6, A-7

is positive in the northern hemisphere and negative in the southern. As is

shown in Figure A-5,

this corresponds to there being a mass excess in the northern hemisphere and

a mass deficiency

in the southern hemisphere. If this were true of the earth, the center of

mass would be north

of the center of coordinates. Since it is always specified that the center of

coordinates and the

center of mass coincide, this harmonic is not admissible in the potential

expression and therefore

must equal zero. In order to make this term zero, the coefficient C(sub10) is

set equal to zero, so

that the entire term disappears.

 The P(sub11) (cos(Theta)) terms are shown in Figures A-6 and A-7. They

represent hemispheres

(eastern and western) of positive and negative values. Existence of the terms

indicate a shift of the

center of mass away from the center of the coordinates in the positive

hemisphere. Each term represents different hemispheres of positive and negative values, shifted by

90(Degree), since cosx and sinx

are 90(Degree) out of phase. The same logic which applied to the harmonic

P(sub10)(cos(Theta)) applies here
and these harmonics are also inadmissible. Thus, their coefficients, C(sub11)

and S(sub11), are set equal

to zero, so that the terms disappear.

2. Degree n = 2

 The zonal harmonic P(sub20)(cos(Theta))is shown in Figure A-8. In the

figure, the constant C(sub20) is

assumed to be negative, thus representing a mass excess in the equatorial

region and a mass

deficiency in the polar regions. Such a mass distribution indicates that the

earth must be flattened

somewhat. Actually, the magnitude of the C(sub20) coefficient is 10(sup2)

times greater than that of any of

the other coefficients. This indicates the extent to which the earth is

flattened and the importance

of this harmonic.

 The next harmonic term in order is the P(sub21)(cos(Theta))cos(Lambda)

term shown in Figure A-9. As

indicated by the figure, it represents opposite northern and southern

quadrants of mass excesses

and deficiencies. This would make the axis of inertia revolve around the axis

of rotation of the

earth, thus causing the earth to wobble on its axis. Since there is really

some wobbling, which

is commonly referred to as precession, the coefficient C(sub21) cannot equal

zero. However, the precession is so small the C(sub21) is also very small. In many cases it is

approximated by zero and

the term disappears. There is also a sin(Lambda) term which has similar

geometry, except that the quadrants are shifted by 90(Degree) in longitude. The same logic applies.

 The next two terms are the sectorial harmonics

P(sub22)(cos(Theta))cos2(Lambda) and P(sub22)(cos(Theta))sin2(Lambda),

shown in Figure A-10. The figure shows sector quadrants which would indicate

ellipticity of the

equator. Each term represents ellipticity in a different direction. Thus, the

two terms must be

combined to obtain the overall ellipticity of the equator. The direction of

the semimajor axis of the

equatorial ellipse is at the point where their sum is a maximum. The

direction of the semiminor

axis is 90(Degree) from the semimajor axis. As shown in Figure A-10, the sum

is a maximum where the

two curves meet. This is only true if their coefficients, C(sub22) and

S(sub22), are equal. The value for the

ellipticity of the equator can be determined from the values of the

coefficients C(sub22) and S(sub22).
 3. Degrees n = 3 and n = 4

 The presence of the zonal harmonic P(sub30)(cos(Theta)) indicates that

the earth is pear-shaped,

as shown in Figure A-1 1. For this reason, this term has been referred to as

the pear-shaped

harmonic. Since values have been computed for its coefficient C(sub30), the

earth is considered to be

somewhat pear-shaped.

 The last harmonic term that will be mentioned here is the zonal harmonic

P(sub40)(cos(Theta)),

shown in Figure A-12. Although it has no recognizable geometric relationship,

it is important

because it is a term of the normal gravity Formula and permits refinement of

the potential on

an ellipsoid.

Figure A-8, A-9

Figure A-10

Figure A-11, A-12

E. APPLICATIONS OF SPHERICAL HARMONICS

 There are an unlimited number of functions which can be expressed in

terms of spherical

harmonics. One of these functions is the normal gravitational potential of

the reference ellipsoid.

 The normal gravitational potential V, which is the gravity potential of

the reference

ellipsoid minus the potential of centrifugal force, can be developed into a

series of zonal spherical

harmonics
where

For this application, the J coefficients are usually used rather than the C

coefficients. There is a

constant relationship between the different coefficients.

 In some applications involving the reference ellipsoid, equation (9) is

truncated at n 2,

giving

Equation (9)

Mathematical expressions for the J2 and J4 coefficients are
where

 Equation(10) shows the relationship between the flattening and the J2

coefficient. An

example of ageodetic application of artificial earth satellites is to

determine J2 from satellite

perturbations,and to use that value in equation (10) to determine the

flattening.

END OF DOCUMENT