geodesy

LECTURE NOTE on G E O D E S Y

prepared by Joenil Kahar

for

Research Center for Seismology,

Volcanology, and Disaster Mitigation

at Graduate School of Environmental Studies

NAGOYA UNIVERSITY April, 2004

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Preface

This lecture note is prepared for Research Center for Seismology, Volcanology, and Disaster Mitigation, Graduate School of Environmental Studies, Nagoya University. At present, combination of geodetic data and seismological data are often to use on research on Seismology, Volcanology that is related to Disaster Mitigation. Geodetic data, which are usually used, obtained from Global Positioning System observations, spirit leveling which is combined with gravity measurements data. Therefore general knowledge about geodesy for geophysicists is necessary.

This note is started by introducing the history of geodesy that is related to the determination of the figure of the earth, scope of geodesy related to structure of the International Association of Geodesy. Coordinate Systems in Geodesy is a basic for positioning systems in geodesy; it is consists of earth coordinate system, geodetic coordinate system, celestial and natural coordinate systems, geodetic datum and its transformation, and computation of geodetic coordinate from geocentric coordinate. Positioning in geodesy is started by introducing concept of astronomical positioning, and System. Data processing is an important knowledge in geodesy. Least squares method as a tool for data processing consists of least squares adjustment, least squares prediction and adjustment in step; introducing of propagation law of variance and co-variance is also take a part in the least square solution. The determination of the geoid as figure of the earth is discussed in physical geodesy. Finally, this note is discussed about the application of geodesy for geophysics.

My gratitude should be expressed to Dr. Fumi Kimata, Associate Professor at the Research Center for Seismology, Volcanology and Disaster Mitigation at Graduate School of Environmental Studies, Nagoya University, who make this lecture note successfully prepared during my stay as Visiting Professor. My thank is also addressed to Dr. H.Z. Abidin, Associate Professor at the Department of Geodetic Engineering, the Institute of Technology in Bandung, Indonesia that supplied many materials on GPS, and also to Mr. M. Irwan, a doctorate student at Nagoya University, who gave valuable suggestions. My thank is also duo to Mr. K. Prijatna who supplied gravity anomaly data derived from GPM98. At last but not the least, I would like to express my ssincere love and gratitude to my wife, Etty Suharti, who gave her moral support during our stay in Nagoya , Japan

Joenil Kahat

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Contents

Preface

1. Introduction

2. Coordinate Systems in Geodesy

3. Positioning

4. Least Squares Method

5. Physical Geodesy

6. Applications of Geodesy for Geophysics

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1. INTRODUCTION

History – Figure of the Earth. Since many centuries, the earth is the only place for everybody lives. The story of

mankind told that during very early times they only concern with the surrounding where they lives. Interaction between a group to another group made the concern expanding and matching, so that the first group require the knowledge about another world that belongs to the other as well as the other group require the knowledge of the world of the first group belongs to. Then, the interest to the knowledge about the earth as the only place for living becomes demand for some scientist and, or philosophers. In very early stage, the Earth was known as a flat surface and known as flat earth model. The theory about the shape of the earth as a sphere came out from Pythagoras (around 500 BC) a famous Greek mathematician, then supported by Aristotle (384-322 BC), a Greek philosopher. Eratosthenes (276-195 BC), an Egyptian astronomer measuring the Earth sphere size using simple technique - as we think at present. Figure(1.1) shows Eratosthenes technique to determine radius of the spherical earth model. He observed that once a year the Sun at noon is directly overhead of a borehole in Aswan (Syena). At the same time he measured shadow of a tower in Alexandria at the north of Aswan, to determine θ. Because he knows the distance dS between these two places, he found the radius R of the Earth sphere.

Figure 1.1: Erathostenes measuring technique

The unit of measure used by Eratosthenes was “stadia” and the distance between Aswan and Alexandria is 5000 stadia. Nobody knows the conversion of stadia to meter

θ

R

Aleksandria

AswandS

t

b

θ

θ

R

Aleksandria

AswandS

t

b

θ

θ

R

Aleksandria

AswandS

t

b

θ

θ

R

Aleksandria

AswandS

t

b

θ

θ

R

Aleksandria

AswandS

t

b

θ

R

Aleksandria

AswandS

R

Aleksandria

AswandS

Aleksandria

AswandS

t

b

θt

b

θt

b

θt

b

θt

b

θ

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correctly, but it is about 185 meter for one stadia. By using this conversion Eratosthenes obtained the radius of the earth 15.5% larger than the radius of the earth that is obtained from satellite technology at present.

Poseidonius (135 – 50 BC), a Greek philosopher also determined Earth’ size using arc measurements between Rhodes and Alexandria by result 11% too large, and then one century later, it was about 827 AD, the Arabian caliph Abdullah al Mamun gave and answer only about 3.6% larger [http://www.zianet.com/globalcogo/ge002.pdf].

According to Newton (1642 – 1727), two bodies of m1 and m2 attract one another with gravitational force by magnitude F which is shown in figure 2, where gravitational force F in dyne (= gram cm sec-2), G is the Newton gravitational constant 6.67 x 10-8 dyne cm 2 gr-2 and r12 is the distance between those two bodies.

212

21N r

mGmF = (1.1)

Apply the equation (1.1) to any mass m in rest at the surface of that the spherical earth of mass M. The spherical earth attracts any the mass m by magnitude FN

2N RGMmF = (1.2)

where R is the radius of the earth. According to the Newton’s Second Law, FN = m aE (1.3) means that the gravitational force of the earth FN produce the gravitational acceleration aN of spherical earth model

2N RGMa =

(.1.4) Let gravitational force of the earth FN attracts a unit mass with the distance from earth center r, the equation (1.1) becomes

2E rGMF = (1.5)

Fm1m2F

r12

Figure 1.2: Illustration of Newton’s Law

Fm1m2F

r12Fm1

m2Fr12

Fm1m2F

r12Fm1

m2Fr12

Fm1m2F

r12Fm1

m2Fr12

Fm1m2F

r12

Figure 1.2: Illustration of Newton’s Law

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This equation (1.3), (1.2) and (1.3) shows that gravitational force field differs from the gravitational acceleration field by a scale m [Vanicek & Krakiwsky, 1986, p. 75].

Daily life shows to us that the earth is a rotating body, which rotates through its rotational axis. Angular velocity ω of the earth rotation is 2π rad. per day. Moritz used ω = 7 292 115 x 10-11 rad s-1 as one of constants parameter to define Geodetic Reference System 1980 [Moritz, 1988] The rotation of the earth through its rotational axis produces centrifugal force Fc that acting to mass m

pmF 2

c ω= (1.6) where p is the distance of mass m to rotational axis. This force produces centrifugal acceleration af with the same direction of the centrifugal force, that is is perpendicular to the rotational axis of the earth, as shown figure (1.3). The magnitude of the centrifugal acceleration ac is easily obtained by utilizing the Newton’s Second Law as shown in eqn. (1.6). Thus any point P will subject to gravity acceleration ag as resultant of gravitational acceleration aF and centrifugal acceleration ac with direction outward of the earth’s center. Distance p is maximum at the equator, so that the values of ac is also maximum. The value centrifugal force Fc is about 0.034% of the value of gravity force Fg at the equator as shown in Heiskanen & Moritz [1967, p. 75], Torge [1989, p. 38] and Vanicek & Kraakiwwsky [1986, p.74]. This ratio is also can be used for the value of centrifugal acceleration and gravity acceleration. The centrifugal acceleration is pa 2

c ω= (1.7) This value is also the same to the value of centrifugal force acting to unit mass.

From the above discussion we know that gravity force vector is sum of gravitational

force and centrifugal force vectors and also gravity acceleration vector is sum of gravitational and centrifugal forces vectors.

Figure 1.3:Gravitational accelaration aN , andcentrifugal acceleration ac producegravity acceleration γ

acp P.

ω

rotational axis

aN γ

acp P.

ω

rotational axis

aN

acp P.

ω

rotational axis

aN

acp P.

ω

rotational axis

aN

p P.pp P. P. P.

ω

rotational axis

ωωω

rotational axis

aNaN γ

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cNg FFF

ρρρ+= (1.8a)

and cNg aaa

ρρρ+= (1.8b)

The equation (1.8) shows that gravity force field differs from the gravity acceleration field by a scale m .

By considering that the earth rotates through its rotational axis, one will improve the idea about the shape of the earth. Figure of the earth becomes ellipsoidal earth that is an ellipsoid of revolution, which is constructed by an ellipse, rotates through its semi minor axis. Geodesists use the ellipsoid of revolution as a model of the earth as reference surface for geodetic position. There are two parameters to define ellipsoidal earth model, that is the equatorial radius (=a) and semi minor axis (=b) or flattening (= f). The relation among a, b and f is shown in the eqn. (1.9). Table 1 shows some reference ellipsoids, which their parameters were obtained the data available at around the year of the reference ellipsoid, was defined.

a

baf −= (1.9)

Using idea of the ellipsoid revolution as figure of the earth is taken by assuming that

the earth is a homogenous density. As a matter of fact, it is not true. Therefore the figure of the earth is called the geoid, a level surface that closed to mean sea level. In the box we can see the development of ideas about of the earth started from the earth sphere, then Earth ellipsoid then geoid.

Table 1: Parameters of some Reference Ellipsoid

Reference Ellipsoid Semi major axis a

(meters) 1/Flattening

(f--1) Bessel (1841) 6377 397 299.15 Clark (1866) 6378 206 294.18 Helmert (1906) 6378 200 298.3 Hayford (1909) 6378 338 297 Krassovski (1942) 6378.245 298.3 Fischer (1960) 6378 166 298.3 Ref. Ellipsoid. 1967 6378 160 298.247 WGS 1972 6378 135 298.26 Ref. Ellipsoid. 1980 6378 137 298.257 WGS 1984 6378 137 298.257

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Scope of Geodesy and IAG From the history of geodesy, which has been described briefly, we may conclude that

geodesy is a science, which has a main objective to determine the shape, and the size of the earth included determination of gravity field. In its development the study of geodynamic phenomena such as earth rotation, crustal movements, sea and earth tide is also included. Studying of the shape and the size of other planets are conducted by geodetic technique. To reach the objective, the knowledge about (1) positioning is a part of geodesy, which should be studying and to be developed. In classical geodesy, two-dimensional (2D) positioning consists of geodetic positioning and astronomic positioning. For three-dimensional (3D) positioning, the 2D positioning should completed by height; they are geodetic height, the height above ellipsoid for the geodetic positioning, and

ω

ω

Spherical EarthHomogeneous and non-rotational body

Earth Ellipsoid of RevolutionHomogeneous and rotational body

GeoidNon-homogeneous and rotational body

ωωωω

ωωωω

Spherical EarthHomogeneous and non-rotational body

Earth Ellipsoid of RevolutionHomogeneous and rotational body

GeoidNon-homogeneous and rotational body

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orthometric height, the height above the geoid, a gravity equipotential surface (is also called a level surface), which globally is closed to mean sea surface. Therefore the knowledge or (2) the earth’s gravity field study becomes important. What kind of data and information required for positioning and the earth’s gravity field study? Positioning means to determine position any point relative to a reference point. It requires distance from the point to the reference point (direct distance with the horizontal component that is horizontal distance and the vertical component that is vertical distance), direction, which has two components: horizontal component and vertical component). In classical geodesy, astronomical positioning is conducted by star observations. Parallel to positioning, to study the earth gravity field requires gravity data at the whole of the earth surface. Why do geodesists study the earth’ gravity field? The knowledge the geometry of gravity field is needed to make possible the transformation of the geodetic observations conducted at the earth surface, which is influenced by gravity field, into the geodetic position.

The earth is not a static body. The earth is composed by some layers, from the core to the outermost layer. The outmost layer, which is called the lithosphere comprising the earth crust and upper mantle. This layer consists of some large plates, which are moving during the time. Tectonic earthquakes is evident of the movements of the earth’s crust. The dynamic of the earth’s interior is as shown by volcano eruptions. The dynamic activities of the earth interior will affect deformation at the earth’s surface. The rotation of the earth is irregular motion. The angular velocity of the earth affects the shape of the earth. All are evidence that the earth is dynamic body. The phenomena of the geodynamics problems can be studied from geodetic observations, therefore (3) geodynamics study using geodetic data is also one of scientific activities of geodesists. The development of science and technology, particularly space science and technology influence geodesist’s activities. The use of geodetic satellite for geodetic works such as satellite positioning, the global gravity field and geodynamic study become more accurate and faster. Therefore (4) the use of space technology and its development become another activity of geodesists. Thus, at present geodesists use many data and information, which are very huge and non-homogenous data. This requires the knowledge development about data processing, and other development of applied mathematics and physics for geodesy. This is the activity of (5) the development of theory and methodology in geodesy.

In relation to the structure of the International Association of Geodesy (IAG), before the General Assembly XXIII of the International Union of Geodesy and Geophysics (IUGG) held at Sapporo, June 3 – July 11, 2003, there were 5 sections: (1) Positioning, (2) Advance Space Technology, (3) Determination of Gravity Field, (4) General Theory and Methodology, and (5) Geodynamics. At the General Assembly XXII in 1999 at Birmingham, IAG Review Committee agreed to evaluate the structure of IAG and propose a new structure at IUGG, Sapporo. The new structure of IUGG since Sapporo 2003 is [Andersen, 2004]

Commission 1: Reference Frame; Commission 2: Gravity Field; Commission 3: Earth Rotation and Geodynamics; Commission 4: Positioning and Application; Inter-Commission Committee on Planetary Geodesy (ICCPG);

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Inter-Commission Committee on Theory (ICCT). The new structure seems more efficient compare to the previous structure. Section 4

at the previous structure has task to handle the development of Theory and Methodology. In the new structure it is handled by Inter-Commission Committee by reasoning that theory is an integrated part of the all commission. The ICC on Theory in the new structure is the solution. As matter of fact that geodetic technique is also used to determine the shape and size of other planets, which is called planetary geodesy. One commission cannot handle this task only. Therefore establishment of ICC on Planetary Geodesy in the new structure is the solution to handle the planetary geodesy problem.

Geodesy and other disciplines of earth sciences have closed relations. Studying the earth planet is an integrated task of the disciplines of earth sciences. Therefore integration of geodesy together with geophysics disciplines under the International Union of Geodesy and Geophysics (IUGG) is the solution.

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2. COORDINATE SYSTEMS IN GEODESY – GEODETIC DATUM

Flat Earth Coordinate System First of all let us consider the idea that the earth is a flat surface and a point P0 is

located at the horizontal plane (H) through at that point. If we state that coordinate of the point P0 consists of abscise x, and ordinate y, it is not clearly define, because there are many coordinate systems show the same abscise and ordinate values as shown in figures 2.1 (a) and (b).

Figure 2.1: Position of P0 on two coordinate systems

Polar coordinates of P0 on (X,Y) system and (X’,Y’) system are equal, x = r sin α and y = r cos α on (X,Y) system x’ = r sin α and y’ = r cos α on (X’,Y’) system Therefore if we want clearly define of coordinate system for P0 we should have

another point P1. The direction angle from P0 to P1 measured clockwise from direction axis Y(+) should be known for orientation of coordinate system, and horizontal distance d01 between P0 and P1 for scaling, see figures 2.1 (c) and (d). Point P0 may be called as initial point of the coordinate system, and the values of abscise x, and ordinate y show the position P0 from the origin O, which is exactly shown by its polar coordinate system P (r,α). Then we may introduce how to define geodetic datum in the plane geodesy,

X

.α

Y

r

P0 (x,y)

OX’

.α

Y’

rO’P0 (x’.y’)

ground surface

horizontal plane (H)P0

(a) (b)

(c)

d01

.

X

.α

Y

r

P0 (x,y)

O

P1α01

//Y ground surface


P1

d01

(d)

(x1,y1)`

h1

X

.α

Y

r

P0 (x,y)

OX’

.α

Y’

rO’P0 (x’.y’)

ground surface


(a) (b)

(c)

d01

.

X

.α

Y

r

P0 (x,y)

O

P1α01

//Y ground surface


P1

d01

(d)

(x1,y1)`

h1

X

.α

Y

r

P0 (x,y)

OX’

.α

Y’

rO’P0 (x’.y’)

ground surface


(a) (b)

(c)

d01

.

X

.α

Y

r

P0 (x,y)

O

P1α01

//Y ground surface


P1

d01

(d)

(x1,y1)`

X

.α

Y

r

P0 (x,y)

OX’

.α

Y’

rO’P0 (x’.y’)

ground surface


(a) (b)

(c)

d01

.

X

.α

Y

r

P0 (x,y)

O

P1α01

//Y ground surface


P1

d01

(d)

X

.α

Y

r

P0 (x,y)

OX’

.α

Y’

rO’P0 (x’.y’)

ground surface


(a) (b)

X

.α

Y

r

P0 (x,y)

OX’

.α

Y’

rO’P0 (x’.y’)

ground surface


X

.α

Y

r

P0 (x,y)

OX’

.α

Y’

rO’P0 (x’.y’)

X

.α

Y

r

P0 (x,y)

O X

.α

Y

r

P0 (x,y)

O

.α

Y

r

P0 (x,y)

O

α

Y

r

P0 (x,y)

O

Y

r

P0 (x,y)

OX’

.α

Y’

rO’

X’

.α

Y’

rO’

.α

Y’

rO’

α

Y’

rO’

Y’

rO’P0 (x’.y’)

ground surface


ground surface


ground surfaceground surface


horizontal plane (H)horizontal plane (H)P0

(a) (b)

(c)

d01

.

X

.α

Y

r

P0 (x,y)

O

P1α01

//Y ground surface


P1

d01

(d)(c)

d01

.

X

.α

Y

r

P0 (x,y)

O

P1α01

//Yd01

.

X

.α

Y

r

P0 (x,y)

O

P1α01

//Y .

X

.α

Y

r

P0 (x,y)

O

P1α01

//Y

X

.α

Y

r

P0 (x,y)

O

P1

X

.α

Y

r

P0 (x,y)

O X

.α

Y

r

P0 (x,y)

O

.α

Y

r

P0 (x,y)

O

α

Y

r

P0 (x,y)

O

Y

r

P0 (x,y)

O

P1P1α01

//Yα01α01

//Y ground surface


P1

d01

ground surface


ground surface


ground surfaceground surface


horizontal plane (H)horizontal plane (H)P0

P1

d01

P1

d01

P1

d01d01d01

(d)

(x1,y1)`

h1

- - 11 -

1. define a horizontal plane in the field by defining vertical distance of chosen initial point to the plane (may be defined as zero);

2. define direction angle from initial point to a certain point in the field (may be defined as 0o or 90o), if we want direction Y(+) is approximately to the north, then the direction angle may be found by compass.

3. define coordinate value for the initial point; 4. measure the distance between the initial point to above mentioned certain point. Plane geodesy only be used for a small area, because we might assume gravity

direction be parallel. Therefore a horizontal plane which perpendicular to gravity direction in the small area, can be used as reference surface.

The value of P0 (x,y) is the two dimensional (2D) coordinate of point P0 which is located at the horizontal (H) as reference surface on the horizontal plane, or the height above the reference surface, which is denoted by h, is zero. In the three dimensional (3D) coordinate system the position of P is (x,y,0) and position of P1 in 3D coordinate system is P1 (x1,y1,z1), where z1 = h1 represents the height of P above the reference surface, see figure 4d. In the 3D coordinate system

Geodetic Coordinate System

In geodesy, position of any point on the earth’s surface is also defined by its

geographic coordinates, latitude and longitude, as shown in figures (2.2 a and b). Fig.(2.2a)5a shows geographic coordinate for a spherical earth. Transforming this geographic coordinate to 3D Cartesian coordinate system will be

x = R cos φ cos λ y = R cos φ sin λ (2.1) z = R sin φ

X

Figure 2.2: Geographic Coordinate System

(a) (b)

Greenwich meridian

equator

S

λ

P(φ,λ)

φ

NZ

X

Y

O

Z

Y

Greenwich meridian

equator

N

λφ

P(φ,λ)

O

O1X


(a) (b)

Greenwich meridian

equator

S

λ

P(φ,λ)

φ

NZ

X

Y

O

Z

Y

Greenwich meridian

equator

N

λφ

P(φ,λ)

O

O1


(a) (b)

Greenwich meridian

equator

S

λ

P(φ,λ)

φ

NZ

X

Y

O

Z

Y

Greenwich meridian

equator

N

λφ

P(φ,λ)

O

O1

(a) (b)

Greenwich meridian

equator

S

λ

P(φ,λ)

φ

NZ

X

Y

O

Greenwich meridian

equator

S

λ

P(φ,λ)

φ

NZ

X

Y

Greenwich meridianGreenwich meridian

equatorequator

S

λ

P(φ,λ)

φ

NZ

X

YS

λ

P(φ,λ)

φ

NZ

X

YS

λ

P(φ,λ)

φλ

P(φ,λ)

φλ

P(φ,λ)

φ

P(φ,λ)

φ

P(φ,λ)P(φ,λ)

φφ

NZ

NZ

X

Y

O

Z

Y

Greenwich meridian

equator

N

λφ

P(φ,λ)

O

O1

Z

Y

Greenwich meridian

equator

N

λφ

P(φ,λ)

O

O1

Z

Y

Greenwich meridian

equator

N

λφ

P(φ,λ)

O

Z

Y

Greenwich meridian

equator

N

λφ

P(φ,λ)

O

Z

Y

Greenwich meridianGreenwich meridian

equatorequator

N

λφ

P(φ,λ)

Oλλ

φ

P(φ,λ)

φφ

P(φ,λ)

O

O1

- - 12 -

where R is radius of spherical earth. Geometry of latitude at ellipsoidal earth is differing from the geometry of the latitude at spherical earth as shown at fig. (2.2b). Transformation from geographic coordinate system to 3D Cartesian coordinate system will be

sin ab z

sin cos y cos cos x

2

2φν

=

λφν=λφν=

(2.2)

where a, and b are semi-major and semi-minor axes of earth ellipsoid, ν is prime vertical radius of curvature (length of O1P at the fig. (2.2b) which is obtained from

φ+φ

=ν2222

2

sinbcosa

a (2.3)

For spherical earth R = a = b, then ν = R.

Equations (2.2) and (2.3) shows transformation from geodetical geographic coordinate system to 3D Cartesian coordinate system of point P, which is located on the surface of spherical earth and ellipsoidal, earth respectively. The height of P from that reference surface is denoted by h, the transformation equations become

x = {R + h} cos φ cos λ y = {R + h} cos φ sin λ (2.4) z = {R + h} sin φ

for spherical earth system, and

sin hab z

sin cos }h{ y cos cos }h{x

2

2φ

+ν=

λφ+ν=λφ+ν=

(2.5)

for ellipsoidal earth system. The 3D coordinate of any point, which is referred to a reference ellipsoid, denoted by geodetic latitude φ, geodetic longitude λ and height above ellipsoid h is named geodetic coordinate system.

Ellipsoidal earth is a model of the “real earth”. Therefore geodetic coordinate system is an artificial model of natural coordinate system.

Celestial Coordinate System Celestial coordinate system is the most important coordinate system in geodesy. This

system may be used as reference system for two-dimensional case of geodetic coordinate

- - 13 -

system and natural coordinate system. The sky where the stars and others celestial body located may be considered as a surface of a unit sphere which is called celestial sphere. The center of the celestial sphere is the earth. The position of stars on celestial sphere in a system called celestial coordinate system.

Extending the rotational axis of the earth will intersect the celestial sphere at the north celestial sphere and the south celestial sphere. Expanding the equatorial plane of the earth will intersects the celestial sphere the celestial equator. Any small circle at the celestial sphere, which is parallel to the celestial equator, is called a celestial parallel. Any plane through the rotational axis of the earth intersects the celestial sphere is called a celestial meridian . Horizontal plane through an arbitrary point at the earth intersects the celestial sphere a celestial horizon. The line through the observer perpendicular to the horizontal plane is called the vertical of the point. Upward direction the vertical intersects the celestial sphere a point that is called astronomical zenith of that point, and downward direction of the vertical is called direction of the plumb line, which intersects the celestial sphere at the nadir; the direction of the plumb line is also called gravity direction of that point. Figure (2.3) shows geometry of celestial coordinate system, (a) for equatorial system, and (b) for horizon system or called as LA system for local astronomical system [Vanicek & Krakiwsky, 1986, p. 295].

Figure 2.3: Geometry of celestial coordinate system

Thus, astronomical position of the arbitrary point at the earth is represented by

position of the astronomical zenith of that point at the celestial sphere, they are astronomical latitude and astronomical longitude.

Natural Coordinate System Astronomical geographic coordinate of any point at the earth surface, which is

denoted by astronomical latitude (Φ) and astronomical longitude (Λ) is obtained by star observations. Determination of astronomical position of any point at the earth surface will be discussed latter. Reference surface if this 2D coordinate system is the geoid, a

Earthhorizon

NCP

SCPeast

west

celestial meridianof observer

ΦΦ

zenith

nadir

vertical

northsouth

celestial equator

NCP

SCP

Earth

celestial parallel

celestial equator

celestial meridian

rotational axis

Φ

(a) (b)

Earthhorizon

NCP

SCPeast

west


ΦΦ

zenith

nadir

vertical

northsouth

celestial equator

NCP

SCP

Earth

celestial parallel

celestial equator

celestial meridian

rotational axis

Φ

Earthhorizon

NCP

SCPeast

west


ΦΦ

zenith

nadir

vertical

northsouth

celestial equator

Earthhorizon

NCP

SCPeast

west


ΦΦ

zenith

nadir

vertical

northsouth

celestial equator

Earthhorizon

NCP

SCPeast

west


ΦΦ

zenith

nadir

vertical

northsouth

celestial equator

Earthhorizon

NCP

SCPeast

west


ΦΦ

zenith

nadir

vertical

northsouth

celestial equator

horizonhorizon

NCP

SCP

NCP

SCPeast

west

east

west

east

west


ΦΦ

zenith

nadir

vertical

northsouth

celestial equator


ΦΦ

zenith

nadir

vertical

northsouth

celestial equator


ΦΦ

zenith

nadir

vertical

northsouth

celestial equator

celestial meridianof observercelestial meridianof observer

ΦΦ

zenith

nadir

vertical

northsouth

celestial equator

ΦΦ

zenith

nadir

vertical

northsouth

celestial equator

ΦΦΦΦ

zenith

nadir

vertical

northsouth

celestial equator

zenith

nadir

verticalvertical

northsouth

celestial equatorcelestial equator

NCP

SCP

Earth

celestial parallel

celestial equator

celestial meridian

rotational axis

Φ

NCP

SCP

Earth

celestial parallel

celestial equator

celestial meridian

rotational axis

NCP

SCP

Earth

celestial parallel

celestial equator

celestial meridian

rotational axis

NCP

SCP

Earth

celestial parallel

celestial equator

celestial meridian

rotational axis

NCP

SCP

Earth

celestial parallel

celestial equatorEarth

celestial parallelcelestial parallel

celestial equator

celestial meridiancelestial meridian

rotational axisrotational axis

ΦΦΦ

(a) (b)

- - 14 -

gravity equipotential surface closed to global mean sea level. If the point is outside of the geoid, then the height of that point above the geoid is called orthometric H, which is obtained from spirit leveling and gravity measurements. The 3D coordinate system (Φ,Λ, H) is called natural coordinate system.

Figure 2.4: Deviation of natural coordinate from geodetic coordinate system at P

Figure (2.4) shows the deviation of natural coordinate system from geodetic coordinate system, that consists of geoid undulation N and deflection of the vertical ε. The deflection of the vertical ε has two components, they are the meridian component ξ and the parallel component η, see figure (2.5).

Figure 2.5: Deflection of the vertical and its components

ε

NCP

SCP

φ

za

zg

Φ

Earth

astronomicalmeridian

geodeticmeridian

λ Λ

η

ξε

NCP

geodeticmeridian

zg

zg

astronomicalmeridean

celestial equator

ε

NCP

SCP

φ

za

zg

Φ

Earth


geodeticmeridian

ε

NCP

SCP

φ

za

zg

Φ

Earth


geodeticmeridian

ε

NCP

SCP

φ

za

zg

Φ

Earth

ε

NCP

SCP

φ

za

zg

Φ

Earth

NCP

SCP

φ

za

zg

Φ

NCP

SCP

φ

za

zg

Φ

NCP

SCP

φ

za

zg

Φ

NCP

SCP

φ

za

zg

Φ

NCP

SCP

φ

za

zg

NCP

SCP

φ

NCP

SCP

NCP

SCP

φ

za

zg

ΦΦ

Earth


geodeticmeridian

astronomicalmeridianastronomicalmeridian

geodeticmeridiangeodeticmeridian

λ Λ

η

ξε

NCP

geodeticmeridian

zg

zg


celestial equatorλ Λ

η

ξε

NCP

geodeticmeridian

zg

zg


celestial equatorλ Λλ Λ

η

ξε

NCP

geodeticmeridian

zg

zg


η

ξε

NCP

geodeticmeridian

zg

zg


ε

NCP

geodeticmeridian

zg

zg


NCP

geodeticmeridian

zg

zg


NCP

geodeticmeridian

zg

zg

NCP

geodeticmeridian

zg

zg

zg

zg


celestial equator

H

h

ref. ell

geoid

earth’s surface

verticalnormal to ellipsoidzenith

ε

level surfacesP

N

ε is the deflection of the vertical from the normal to ellipsoid

N is the geoid undulation

H

h

ref. ell

geoid

earth’s surface


ε

level surfacesP

N



H

h

H

h

ref. ell

geoid

earth’s surface


ε

level surfacesP

N



ref. ell

geoid

earth’s surface


ε

level surfacesP

N

ref. ell

geoid

earth’s surface


ε

level surfacesP

N

ref. ell

geoid

earth’s surface


ε

level surfacesP

ref. ell

geoid

earth’s surface


ε

level surfacesP

ref. ell

geoid

earth’s surface


ε

level surfacesP

ref. ell

geoid

earth’s surface


ε

level surfacesP

ref. ell

geoid

earth’s surface


ε

level surfacesP

ref. ell

geoid

earth’s surface


ε

level surfaces

ref. ell

geoid

earth’s surface


ε

level surfaces

geoid

earth’s surface


εverticalnormal to ellipsoidzenith

ε

level surfaceslevel surfacesP

N



- - 15 -

Relation of two systems are shown in equation (2.6) ξ = Φ − φ η = (Λ−λ) cos λ (2.6) N = h – H The latest equation is used because ε is very small so that cos ε ≈ 1. Figure (2.5) shows geometry of deflection of the vertical is shown in celestial sphere, ε2 = ξ2 + η2 (2.7)

Geodetic Datum Reminding to the matters have been described in the flat earth coordinate system, that

to establish a coordinate system we should define coordinate value for chosen initial point and the value should be referred to chosen reference surface. Following this idea, to establish geodetic datum for establishing a geodetic coordinate system, we have to do the items below,

1. define an ellipsoid of revolution as reference surface, and semi-minor of the ellipsoid defined parallel to the rotational axis of the earth,

2. define geodetic coordinate value (φ, λ, h) for a chosen initial point, 3. measure the distance from the initial point to another point, 4. measure the azimuth from the initial point to the other point. Defining an ellipsoid of revolution as reference surface means that we should define

the figure of the reference ellipsoid (defining semi-major a, and semi-minor b or flattening f). For orientation of the reference ellipsoid, semi-minor axis is defined parallel to the rotational axis of the earth. The rotational axis of the earth is defined as the z-axis of the conventional terrestrial system (CTS) which the center of the earth is defined as the origin of the (CTS). The z axis of CTS is the straight line through its center to the Conventional International Origin (CIO) and CIO is defined as the mean position of the instantaneous pole measure during 1900 – 1905 [Vanicek & Krakiwsky, 1986, p.44]. The geodetic coordinate (φ, λ, h) of initial point usually is adopted from the astronomical geographic coordinate (Φ, Λ) and orthometric height H of the point is adopted as the geodetic height h. Defining the geodetic coordinate of the initial point means that the center of reference ellipsoid from earth’s center is defined, this will be shown later. Measuring the distance between the initial point and the other point is for scaling the geodetic coordinate system. Measuring the azimuth from the initial point to another point is usually to measure astronomical azimuth between these points. Because the deflection of the vertical shows the difference of the astronomical geographic coordinate to the geodetic latitude and longitude of arbitrary point at the earth surface, then it will affect to the difference between the astronomical azimuth and the geodetic azimuth of the point to another point, ∆Aij as shown in the eqns. (2.7 a and b). The equation is obtained by considering that the semi-minor of reference ellipsoid is parallel to the rotational axis of the earth.

- - 16 -

ijijij AA ∆−=α (2.7a) ijijiijiiiij Z)A cos - Asin (tan A ηξ−Φη−=∆ (2.7b)

The eqns (2.7a and b) tell us that to obtain geodetic azimuth of an arbitrary point to another point, the measured astronomical azimuth between these points should be reduced by ∆A. The reducing value is depending on the deflection of the vertical at the arbitrary point and measured zenith distance of the point to the other point. The difference between measured zenith distance and is

ijiijiij Asin A cos Z η−ξ−=∆ (2.8a) εij = - ∆Zij = Zij - zij (2.8b)

where Zij is measured zenith distance and zij is geodetic zenith distance. The eqn (2.8a) shows that if Aij is zero then ∆Zij becomes - ξi and for Aij is π/2 then ∆Zij becomes - ηi.

Datum Shift and Transformation By adopting classical approach that the astronomical coordinate at the initial point Φ,

Λ and orthometric height H as the geodetic coordinate φ, λ, h at the point or ξ = 0, η = 0 and N = 0 yields the reference ellipsoid from the center of the earth is known or in other words that the reference ellipsoid is non-geocentric ellipsoid, see figure (2.6).

Figure 2.6: Position and Orientation of Reference Ellipsoid (RE) Let Cartesian coordinate of arbitrary point P on the RE-1 system is (x, y ,z). which

system is a reference ellipsoid with the parameter a and f. And let the Cartesian

Datum point

RE-new

RE-1

geoid

O

O’X’ ,Y’

X,Y

// CTSZ’

CTSZ

Datum point

RE-new

RE-1

geoid

Datum point

RE-new

RE-1

geoid

Datum point

RE-new

RE-1

geoid

Datum point

RE-newRE-new

RE-1RE-1

geoidgeoid

O

O’X’ ,Y’

X,Y

// CTSZ’

// CTSZ’

CTSZ

CTSZ

- - 17 -

coordinate of the point on new RE system which has parameter a’ and f’ is (x’, y’, z’). According to the eqn. (2.5) and because b = a(1 – f)

{ } 'sin 'h')f'-(1 'z

'sin ' cos }'h'{ y'' cos ' cos }'h'{'x

2 φ+ν=

λφ+ν=λφ+ν=

(2.9)

and from eqn (2.3)

'sin)'f2('f1

'a'2 φ−−

=ν (2.10)

The axes of the RE-1 and the new systems are parallel. For simplification we may write the equations (2.4) for RE-1 system, and (2.9) for the new system as

x

=

zyx

and x’

=

z'y'x'

and let the coordinate of the center of the two systems from the earth center respectively as

x0

=

0

0

0

zyx

and '0x

='0

'0

'0

zyx

and coordinate of the arbitrary point in the earth center system is

X =

ZYX

Thus we have x0 = X - x and '

0x = X - x’ or x0 -

'0x = x’ - x

x’ = x + dx (2.11)

- - 18 -

Displacement of the center ellipsoid dx is called datum shift. If x0 = 0 , then '0x is the

position the center of RE-1 from the earth center. Therefore, in general the eqn. (2.3) is closely related to the position the center of ellipsoid.

Figure (2.7) shows geometry of six transformation parameters (without scale change

parameter). Coordinates of an arbitrary point on (X’,Y’,Z’) system will be obtained from (X,Y,Z) system by the equation

+

=

dzdydx

zyx

1R-R

R1R-R-R1

'z'y'x

xy

xz

yz

(2.12)

If all rotation parameters Rx, Ry and Rz are zero, we will have again the eqn (2.11). And then if Sc is a small-scale change from (X,Y,Z) system to (X’,Y’,Z’) system the eqn (2.12) becomes

( )

+

+=

dzdydx

zyx

1R-R

R1R-R-R1

Sc 1 'z'y'x

xy

xz

yz

(2.13a)

No unit for scale change parameter, and rotation parameters are in radian unit. Because Sc, and all rotation parameters Rx, Ry and Rz, are small values, so that multiplication of Sc with the all rotation parameters may be neglected, then the eqn (2.13a) will be

+

++

+=

dzdydx

zyx

Sc1R-R

RSc1R-R-RSc1

'z'y'x

xy

xz

yz

(2.13b)

X’

Z

Z’

X

YY’

dx

dy

dzO

O’

Rx

Ry

Rz

Figure 2.7: Geometry of six transformation parameters

X’

Z

Z’

X

YY’

dx

dy

dzO

O’

Rx

Ry

Rz

X’

Z

Z’

X

YY’

dx

dy

dzO

O’

Rx

Ry

Rz

X’

Z

Z’

X

YY’

dx

dy

dzO

O’

Rx

Ry

Rz

X’

Z

Z’

X

YY’

dx

dy

dzO

O’

Rx

Ry

Rz

X’

Z

Z’

X

YY’

dx

dy

dzO

O’

Rx

Ry

Rz

Figure 2.7: Geometry of six transformation parameters

- - 19 -

The datum transformation equations (2.13b) show that an arbitrary point P that has coordinate values on the two systems (X,Y,Z) and (X’, Y’, Z’) may be written in a set of three linear equations,

x’ = (1 + Sc) x + (Rz) y + (-Ry) z + dx y’ = (-Rz) x + (1 + Sc) y + (Rx) z + dy (2.13c) z’ = (Ry) x + (-Rx) y + (1+ Sc) z + dz

We cannot determine the seven parameters, dx, dy, dz, Rx, Ry, Rz, and Sc from the three linear equations. At least, there are four more equations must be required to determine the seven transformation parameters. It means that there are three points have Cartesian coordinate values in the two systems.

The purpose of determination of the seven transformation parameters is to obtain geodetic coordinate of geodetic points in a new datum from the geodetic coordinates in the old datum. Originally the coordinate values come out from two sets of the geodetic coordinates of the points in the both datum. Thus, data input for datum transformation parameters determination are two ellipsoidal parameters, the radius equator a and the polar flattening f for each reference ellipsoid, two sets of geodetic coordinate values of at least of three geodetic points. Let (φ’, λ’, h’) is geodetic coordinate in new geodetic datum, (φ, λ, h) is geodetic coordinate in old datum, then the parameter of ellipsoid are (a’, f’) and (a, f) in the new and the old datum respectively. Cartesian coordinates of the geodetic points, which have geodetic, coordinate values in the both datum, (x’, y, z’) for the new datum and (x, y, z) for the old datum that is obtained from (2.9). Usually more than three geodetic points that have geodetic coordinate in the both system, so that we have redundant data to compute the transformation parameters. Therefore computation of the transformation parameters should be done by least squares adjustment technique, which will be described later. The Cartesian coordinate of other geodetic points in the new datum is computed by using the eqn. (2.9). Finally the geodetic coordinate of the other geodetic points is computed the Cartesian coordinates which will described below.

Computation of Geodetic Coordinate from Cartesian Coordinate Computation of Cartesian coordinate from geodetic coordinate has been shown in the

eqn. (2.5) and (2.9) which is rewrite here with a new number,

sin hab z


2

2φ

+ν=

λφ+ν=λφ+ν=

(2.14a)

or

- - 20 -

{ } sin hf)-(1 z


2 φ+ν=

λφ+ν=λφ+ν=

(2.14b)

or

{ } sin h)e -(1 z


2 φ+ν=

λφ+ν=λφ+ν=

(2.14c)

where prime vertical radius of curvature ν is rewrite from the eqns (2.3) and (2.10)

φ+φ

=ν2222

2

sinbcosa

a (2.15a)

or

φ−−

=ν2sin)f2(f1

a (2.15b)

or

φ−

=ν22 sine1

a (2.15c)

Eqn (2.14) is the equation for computation Cartesian coordinate from geodetic coordinate. Computation geodetic coordinate from Cartesian is a little bit complicated. Geodetic longitude λ is obtained from

xy tan =λ (2.16)

Let 22 yx + = p, then from (2.14c) we have

h = ν−φ cos

p (2.17)

and

( )( ){ }p

zhe1

h tg 2 +ν−+ν

=φ (2.18a)

because 2e11

− = ! + e2 , ( 2.18a) becomes

pze

h1 tg 2

+νν

−=φ (2.18b)

- - 21 -

The eqns. (2.17) and (2.18) show that computation of φ is depending on h. , so that the two values is determined iteratively, by taking h = 0 to get approximate value of latitude [Heiskanen & Moritz, 1967, p. 183; Heitz, 1988, p.199]. The iteration algorithm is

(1) compute approximate value of latitude φo by taking h =0, using (2.18); (2) compute prime vertical radius of curvature ν, using (2.15); (3) compute the ellipsoidal height h, using (2.17); (4) compute latitude φ, using (2.18) (5) repeat the computation from (2) and until φ and h remain practically

constant. Other methods for computation geodetic coordinates from Cartesian coordinates may be found in some publications such as Borkowski [1987, 1989], Jones [2002] and Vermeille [2002].

- - 22 -

3. POSITIONING Astronomical Positioning In the previous subject we have discussed that position of any point at the surface is

represented by its astronomical zenith on the celestial sphere. The position of the point is certainly astronomical position, they are astronomical latitude and astronomical longitude. The position system of that point on the celestial sphere is equatorial system, which refers to the celestial meridian of Greenwich for the longitude, and the celestial equator for the latitude. Position determination of this point is conducted by doing star observation at that point as observer station. Position of a star at the celestial sphere is represented by its right ascension (α) and declination (δ). The initial point of this coordinate system in celestial coordinate system is the vernal equinox (γ). Declination of star is spherical distance along celestial meridian of the star measured from the celestial equator to the star, in the same way as latitude, having positive value for a star at north of equator, and negative value for southern star. Right ascension (α) is measured anticlockwise from the vernal equinox along the celestial equator to the meridian of the star, in the same way as longitude in time unit, see figure (3.1c). Thus position system of stars in the celestial coordinate system is equatorial system or AP system for apparent place system [Vanicek and Krakiwsky, 1986, p. 298].

A set of figure 3.1 shows astronomical spherical triangle, which is basic triangle for astronomical positioning system, which built by three points, NCP for north of celestial pole, Za for astronomical zenith of observer and S for stars. The important elements of the triangles as shown in figures (3.1a, b, and d), are co-latitude (90 – Φ), hour angle (h), co-declination (90 - δ), zenith distance (z) of a star and astronomical azimuth (AS) of the star. Figure 3.1c shows that hour angle of observer (h) is measured westward from the meridian of observer to the meridian of a star, and other important elements are Local Apparent Sidereal Time (LAST) which is measured eastward from of the observer meridian to the vernal equinox meridian and Greenwich Apparent Sidereal Time (GAST) which is measured eastward from the Greenwich Meridian to the vernal equinox meridian. Basic formula found from the astronomical triangle is, (see fig. 3.1d):

cos z = sin (Φ) sin (δ) + cos (Φ) cos (δ) cos (h) (3.1)

And other basic formula is, see fig (3. 1c): LAST = h + α (3.2)

Let hour angle of an observed star h = 0, it means the star located in the same meridian of observer, or the star in culminating position, then from the eqn (3.1)

cos z = cos (Φ - δ) or cos z = cos (δ - Φ)

Thus we have two equations related to observed zenith distance of a star, declination of the observed star and latitude of observer,

- - 23 -

Figure 3.1: Astronomical Spherical Triangle Φ = z + δ (3.3a) and Φ = δ − z (3.3b)

These equations are used for determination of latitude of the observer. According to Mueller [1969] the latitude of the observer may be found from a pair of stars in upper culmination at the north and the south of observer ( ) ( ){ }NSNS2

1 zz −+∂+∂=φ (3.4a) and for a pair of stars in lower culmination at the north and the south of observer ( ) ( ){ } π+−+∂−∂=φ 2

1NSNS2

1 zz (3.4b)

(c) (d)

(a) (b)

1 = meridian of V. Equinox2 = meridian of Greenwch

3 = meridian of Star

4 = meridian of Observer

ZaS

celestial equator

NCP

SCP

1 2 3

δ

4

Φ

Za

z

horizon

NCP

SCP east

west Φ

Φ

south north

S

AS

h

NCP

Za

[π/2 – Φ]

[π/2 – δ]

z

..

α

hS

. NCP

γ

Za

celestial equator

Gr

GAST.

LAST

Λ

S

(c) (d)

(a) (b)




ZaS

celestial equator

NCP

SCP

1 2 3

δ

4

Φ

Za

z

horizon

NCP

SCP east

west Φ

Φ

south north

S

AS

h

NCP

Za

[π/2 – Φ]

[π/2 – δ]

z

..

α

hS

. NCP

γ

Za

celestial equator

Gr

GAST.

LAST

Λ

(c) (d)

(a) (b)




ZaS

celestial equator

NCP

SCP

1 2 3

δ

4

Φ

Za

z

horizon

NCP

SCP east

west Φ

Φ

south north

S

AS

h

NCP

Za

[π/2 – Φ]

[π/2 – δ]

z

..

α

hS

. NCP

γ

Za

celestial equator

Gr

GAST.

LAST

Λ

(c) (d)

(a) (b)




ZaS

celestial equator

NCP

SCP

1 2 3

δ

4

Φ

Za

z

horizon

NCP

SCP east

west Φ

Φ

south north

S

AS

h

NCP

Za

[π/2 – Φ]

[π/2 – δ]

z

..

α

hS

. NCP

γ

Za

celestial equator

Gr

GAST.

LAST

Λ

(a) (b)




ZaS

celestial equator

NCP

SCP

1 2 3

δ

4

Φ

Za

z

horizon

NCP

SCP east

west Φ

Φ

south north

(a) (b)




ZaS

celestial equator

NCP

SCP

1 2 3

δ

4

Φ

Za

z

horizon

NCP

SCP east

west Φ

Φ

south north





1 = meridian of V. Equinox11 = meridian of V. Equinox2 = meridian of Greenwch22 = meridian of Greenwch



3 = meridian of Star33 = meridian of Star

4 = meridian of Observer44 = meridian of Observer

ZaS

celestial equator

NCP

SCP

1 2 3

δ

4

Φ

Za

z

horizon

NCP

SCP east

west Φ

Φ

south north

ZaS

celestial equator

NCP

SCP

1 2 3

δ

4

Φ

ZaS

celestial equatorcelestial equator

NCP

SCP

1 2 3

δ

4

Φ

NCP

SCP

1 2 3

δ

4

Φ

NCP

SCP

1 2 3

δ

NCP

SCP

1 2 3

δ

NCP

SCP

1 2 3

δ

NCP

SCP

11 22 33

δδ

44

ΦΦ

Za

z

horizon

NCP

SCP east

west Φ

Φ

south north

Za

z

horizon

NCP

SCP east

west Φ

Φ

south north

Za

z

horizon

NCP

SCP east

west Φ

ΦZa

z

horizon

NCP

SCP east

west Φ

ΦZa

zz

horizonhorizon

NCP

SCP east

west

NCP

SCP east

west

NCP

SCP east

west

NCP

SCP east

west

NCP

SCP

NCP

SCP east

west

east

west

east

west

east

west ΦΦ

ΦΦ

south north

S

AS

h

NCP

Za

[π/2 – Φ]

[π/2 – δ]

z

..

α

hS

. NCP

γ

Za

celestial equator

Gr

GAST.

LAST

ΛS

AS

h

NCP

Za

[π/2 – Φ]

[π/2 – δ]

z

S

AS

h

NCP

Za

[π/2 – Φ]

[π/2 – δ]

z

AS

h

NCP

Za

AS

h

NCP

Za

AS

h

NCP

ZaZa

[π/2 – Φ]

[π/2 – δ]

zz

..

α

hS

. NCP

γ

Za

celestial equator

Gr

GAST.

LAST

Λ ..

α

hS

. NCP

γ

Za

celestial equator

Gr

GAST.

LAST

..

α

hS

. NCP

γ

Za

celestial equator

Gr

GAST.

LAST

..

α

hS

. NCP

γ

Za

celestial equator

Gr

GAST.

..

α

hS

. NCP

γ

Za

celestial equator

Gr

GAST.

.α

hS

. NCP

γ

Za

celestial equator

Gr

GAST.

α

hS

. NCP

γ

Za

celestial equator

Gr

GAST

α

hS

. NCP

γ

Za

celestial equator

Gr

GAST

α

hS

. NCP

γ

Za

celestial equator

Gr

GAST

hhhS

. NCP

γ

Za

celestial equator

Gr

GAST

S

. NCP

γ

Za

celestial equator

GrS

. NCP

γ

Za

celestial equator

S

. NCP

γ

Za

celestial equator

. NCP

γ

Za

. NCP

γ

Za

. NCP

γ

Za

. NCP

γ

Za

. NCP

γ

. NCP

γ

. NCP. NCP. NCP. NCP

γ

Za

celestial equator

Gr

GASTGAST.

LAST

Λ

S

- - 24 -

The astronomical longitude of the observer is obtained from, see figure (3.1c) Λ = LAST - GAST (3.5)

By observing the star in culmination position, from the equations (3.2) and (3.5) then we have Λ = α - GAST (3.6) α, the right ascension of the observed star may be found in Apparent Places for Fundamental Stars, and GAST is obtained through the Universal Time by synchronizing a local time piece at the observer with a time standard by means of HF radio time signals.

In the previous subject that is related to geodetic datum we know astronomical azimuth (=A) from the datum point to another point of a horizontal geodetic networks is adopted as the geodetic azimuth. To determine the astronomical azimuth to the other geodetic point, for the first instance we should determine astronomical azimuth from the geodetic datum to a star. From astronomical spherical triangle,

ΦδΦ

=costan -h cossin

h sinA tan S (3.7)

If a star at meridian of observer, the hour angle h becomes zero so that the astronomical azimuth of star AS will be 0o or 180o . Thus, if can we pointing the stars precisely in the observer’s meridian plane, then by measuring horizontal angle from the observer’s meridian plane to the other geodetic point will obtain astronomical azimuth of the observer to the other geodetic point. To obtain a precise result of the astronomical azimuth as well as astronomical positioning we must refer particular textbook that describe this matter more detailed, such as Mueller [1969].

Geodetic Positioning

Figure 3.2: A Geodetic Network with Laplace Stations L0 , L1, and L2

L0

L1

P0

P1

L2

P2

L0

L1

P0

P1

L0

L1

P0

P1

L0

L1

P0

P1

L0

L1

P0

P1

L2

P2

- - 25 -

Before satellite technology is applied to geodesy, geodetic positioning was held by geodetic surveys at the earth’s surface. There are three geodetic networks: geodetic traverse, triangulation and trilateration. It is quite often by the reasoning of precision and optimization of the networks, two from the three kinds of networks or the three kinds of networks are combined. Because the measurements are conducted at the earth’s surface, all the measured directions (vertical and horizontal) and distances should be reduced to chosen reference ellipsoid. Figure 3.2 shows a triangulation networks with two Laplace stations Lo and L1. In the stations astronomical azimuth are measured. If Lo is initial point of the network, where geodetic azimuth is defined as geodetic azimuth, but in L1 geodetic azimuth and astronomical azimuth from L1 to P1 should satisfy Laplace condition in the eqn (2.8). Processing of the geodetic networks may be divided becomes two parts; they are processing of horizontal networks and vertical networks. Usually the processing is conducted independently. Horizontal (2D) coordinates of the geodetic points are computed in a chosen projection plane. For the first instance the geodetic latitude and longitude of datum point must be converted become a plane Cartesian coordinate on the projection plane. After the position of other points are computed by using plane coordinate system, then they are converted become 2D geodetic coordinates on the reference ellipsoid. Height differences among the geodetic points usually are computed from trigonometric height difference measurements, and mean sea level at datum point is used as reference surface.

Precise height difference measurements are conducted by spirit leveling, as part of determination of optometric height will be described latter.

Satellite Positioning System Terrestrial positioning that has been described previously was time and cost

consuming work. Development of electronic and satellite technology drastically change geodetic work performance. The first satellite navigation system was transit system established for US Navy. This system was known as US Navy Navigation Satellite System (US-NNSS) and operation for military used in 1964 and released for civil used in 1967. The satellites of this system had polar orbits by altitude about 1000 km, and the period of around 100 minutes. There were six orbits with one satellite operation in each orbit by using Doppler effect. Because this system is only equipped by one satellite for each orbit, so that navigation and positioning require long operational time. This system was terminated in 1996, few years after Navigation Satellite Time And Ranging (NAVSTAR), the name given to Global Positioning System (GPS) satellites, was declared for civil used by the US Government.

GPS consists of three segments: the satellites, the control system, and the users., see fig. (3.3) The Global Positioning System orbital configuration is designed in order the system can give information where we are at any time, see fig. (3.4). GPS originally developed to meet US military requirements. In 1978 four Block I satellites were in orbits and became 11 satellites in 1985. Launching of the first of 28 Block II satellites was started at the beginning of 1989. As matter of fact that civilian demand for the using of GPS which was developed for military purpose came after the downing of Korean Airlines Flight over the territory of the Soviet Union in 1983. So that President Reagan announced that GPS would be made available for international used and confirmation

- - 26 -

was made by President Clinton in 1995 by a letter to the International Civil Aviation Organization (ICAO). The use of GPS for surveying community started around 1984 [http://www.rand.org/publications/MR/MR614/MR614.appb.pdf]

Figure 3.3: GPS Segments

Figure (3.4): GPS orbital constellation

SATELITTES

. 24 satelittes. Orbital period : 12 hr. Altitude : 20200 km

CONTROL SYSTEM. Time synchronization. Orbit prediction. Data injection. Satellite health monitoring

USERS. Observe GPS signals

. Compute position, velocity,time information,

or other parameters

© Hasanuddin Z. Abidin, 1998

SATELITTES

. 24 satelittes. Orbital period : 12 hr. Altitude : 20200 km

CONTROL SYSTEM. Time synchronization. Orbit prediction. Data injection. Satellite health monitoring

USERS. Observe GPS signals

. Compute position, velocity,time information,

or other parameters


- - 27 -

Figure 3.5 : GPS Control System GPS satellites have six orbital planes with four satellites per orbit, inclined 55o with a

nominal altitude of 20 200 km, and orbit period 12 hours. Each GPS satellite transmits two carrier waves L1 on 1575.42 MHz (λ = 19 cm), and L2 on 1227.6 MHz (λ = 24.4 cm). The L1 and L2 are modulated by P (precision) code of 110.23 MHz (λ = 30 m) and navigation message of 50 MHz. The L1 is also modulated by C/A (Clear Access) code of 1.023 MHz (λ = 300 m). The carrier frequencies and modulations are controlled by on-board atomic clock. The A/C and P codes, also called pseudo-random noise (PRN) code have function for ranging information. The navigation message code informs position of satellite (broadcast ephemeris), UTC, ionosphere correction, etc.

The purpose of the control system is to monitor the health of the satellites, determine their orbits and the behavior of their atomic clocks, and inject the broadcast message into the satellites. The control system consists of monitor stations on Diego Garcia, Ascension Island, Kwajalein, and Hawaii, and a master control station operates at the Consolidated Space Operations Center, Colorado Springs in Colorado, see fig. (3.5).

The user segment consists of all users, military and civilian. For precise geodetic work, carrier or code frequency phase is measured and recorded for future processing. More detailed information about the all segments can be found at Wells, et al. [1987]

GPS navigation and position determination is based on measuring the distance from the precise location of satellites in their orbits to the user, in three dimensional Cartesian coordinates as well as geodetic coordinates (latitude, longitude and ellipsoidal height). Geometry of GPS positioning and navigation by measuring distance from satellite to a receiver may be described as follow, see fig (3.6). Distance measurement from satellite S1 to the receiver will form a sphere, and distance measurement from satellite S2 to the


Hawaii

Ascension DiegoGarcia

Kwajalein

CapeCarnaval

Master Control Station and Monitor Station, Colorado Spring, USA

Master Control Station

Ground Antenna Station © Hasanuddin Z. Abidin, 1998

Hawaii


Kwajalein

CapeCarnaval



Ground Antenna Station © Hasanuddin Z. Abidin, 1998

Hawaii


Kwajalein

CapeCarnaval



Ground Antenna Station

- - 28 -

receiver form second sphere, which form a circle as intersection with the first circle. Distance measurement from satellite S3 intersects the circle at two points. Distance measurement from a fourth satellite will locate 3D position of the receiver in coordinate system used by the satellites. The measuring distance from five or more satellites or yields more accurate and reliable position. In order GPS can be used anytime at anywhere above the earth surface, the 24 satellites constellations are designed that there are at least five satellites above the horizon of users at any time and anywhere.

Figure 3.6: Geometry of satellite positioning by measuring distance There are two kinds of GPS measurements, pseudo-range measurements and carrier

phase measurements. These measurements are influenced by some errors: time offsets of satellite clock and receiver clock, ionosphere and troposphere effects, and noise on carrier waves. The distance between satellite and receiver which is obtained from peseudo-range measurements is

p = c dτ (3.10) where velocity of light in vacuum is c = 2.99792458 x 108 m s-1, and dτ is travel time from satellite to receiver that is obtained from satellite clock and receiver clock. The eqn. (3.10) is “true distance between satellite and receiver” if there are not error influenced the measurement. Time scale of satellite clock and receiver clock is different, so that they should be refer to an ideal scale that is GPS time scale τ . Let signal is transmitted from satellite at ts in the time scale of satellite, and received by receiver at Tr in the time scale of receiver, thus (3.1) originally is

S3

S2 S1

S3

S2 S1

- - 29 -

p = c (Tr – ts) (3.11) In the GPS time scale they should be τs and τr If dt and dT are the offset error of satellite clock and receiver clock respectively, so that we have τs = ts – dt and τr = Tr – dT , then the eqn. (3.11) become p = c (τr – τs ) + c (dT – dt) (3.12a) p = ρ + c (dt – dT) (3.12b) The measurement is also influenced by ionosphere error dion , troposphere error dtrop , and ephemeris errors of satellite dρ, then the (3.12) become p = ρ + dρ + c (dt – dT) + dion + dtrop (3.13) The eqn. (3.13) is basic equation for pseudo-range measurement. The distance between satellite and receiver which is obtained from phase measurements is ρ = λ Φmeas (3.14) λ is wavelength, Φmeas is measured phase consists of Fr(Φ) and Int (Φ) the fraction part and integer part of phase measurements respectively, .Φmeas = Fr(Φ) + Int (Φ) (3.15) At lock-on time, at epoch t0 there are the unknown number of cycles, N(t0), therefore at epoch t we have total phase Φtotal = Fr(Φ) + Int(Φ) + N(t0) (3.16) Let Φ in length unit is defined as Φ = λ Φtotal, so that (3.16) becomes Φ = ρ + Nλ (3.17) N is called cycle ambiguity. By considering that the measurement is influenced by the offset error of satellite clock and receiver clock, ionosphere error dion , troposphere error dtrop , and ephemeris errors of satellite dρ, (3.17) Φ = ρ + dρ + c (dt – dT) – dion + dtrop + Nλ (3.18) The eqn. (3.18) is basic equation for carrier phase measurement. The differences of this equation with the pseudo-range equation are the cycle ambiguity N, and minus sign of ionosphere effect. Further explanation about the basic equations may be found at Abidin [2000}, Wells, at al., [1987].

- - 30 -

GPS has many applications, not only for military, geodesy, surveys & mapping purposes, but also for research in geophysics, such as geodynamics and deformation studies, meteorology and atmospheric studies, oceanography. GPS is also used for aeroplane navigation, marine and on land transportation. This because of GPS can be used independent of weather conditions, at static and kinematics modes, and can be used at various platforms, such as car, train, vessel, aeroplane and even satellite. But we should know that GPS has some disadvantages, because this system can not be used in the place where GPS satellite signal can not reach GPS antenna, such as inside the room, in tunnel or underneath the water.

Figure 3.7: The IGS Tracking Network To support geodetic and geophysical research activities the International Global

Positioning System (GPS) Service for Geodynamics (IGS) conducted three-months campaign during June through September 1992, and continued through a Pilot-service until the establishment of the IGS in 1993 by the IAG. The primary objective of the IGS is to provide a service to support, through GPS data and data products, geodetic and geophysical research activities. In January 1999 the name of the service was changed to International GPS Service (IGS). To provide GPS data IIGS established IGS Tracking Network, see fig. (3.7) At present (March 28, 2004) there are 366 stations over the world [http://igscb.jpl.nasa.gov, 2004].

http://igscb.jpl.nasa/goc

- - 31 -

4. LEAST SQUARES METHOD Introduction

In previous subject, such as horizontal networks and vertical networks, we have discussion about field measurements, which carried out for determination of horizontal position of geodetic points or their heights from an initial point. In other words we have a group of measurements for determining a group of unknown parameters.

A simple example of this problem is determination of horizontal distance between two points from a series of repeated measurements. The horizontal distance is only one unknown parameter should be determined from the series of repeated measurements. Before doing field measurement, we should check that we have a calibrated instrument to carry out distance measurements. After we have the series of measurements, then we should check that the measurements are not influenced by mistakes or blunders. If there are a measurement value has a big different from other values, this value may be categorized as a mistake or a blunder, and then rejected from the measurement series. The measurements values may be influenced by systematic errors. Systematic errors may be occurred by observation method, and also by outside factor such as influencing atmosphere to measurements. The systematic errors can be eliminated by method of observation, or can be modeled for the case of atmospheric factor. Let blunders have been rejected and the influence of systematic errors on our measurements have been eliminated. If our measurements are free from errors, the measurement values will be equal, but they are not exactly equal. It means that there is another kind of errors influence the measurements, which is occurred randomly, and called random error. Although our measurement values are not equal, but they show a trend value as the most probable value or the expected value of the unknown parameter that is the distance between the two points. It is make sense if we choose the average value of the repeated measurements as the most probable value of the unknown parameter, and by assuming that the measurements have of equal weight, the unknown parameter will be

X = ∑n

1in

1λ (4.1)

The average value X is also called sample mean or mean value of a set n-repeated measurement. To determine the unknown parameter X , each measurement value iλ should be added by correction value vi , X = iλ + vi (4.2a) or vi = X - iλ (4.2b)

The statement “should be added by correction value” has the same meaning with “should be subtracted by error value”, because error = - correction Substitution of (4.2b) to (4.1) yields

- - 32 -

∑ =n

1i 0v (4.3)

Let the “true value” of X is X then we have “true random error” which is

simplified by “true error” that influence of each measurement is ει . We have “true error variance” , then we called as “error variance” of the measurements is

∑ε=σn

1

2i

2

n1 (4.4)

To obtain X , we should subtract ει from measurement value iλ ,

X = iλ - εi (4.5a)

and because of (4.2), then the eqn (4.5a) becomes εi = X - X - vi (4.5b)

Summation of (4.5b) is

n - n

1i

n

1i ∑∑ =ε λ X (4.6a)

and by considering (4.1), the eqn. (4.6a) becomes

n

1i =ε∑ n ( X - X ) (4.6b)

Squaring (4.5b), =ε2

i X2 - 2 X X - 2 vi X + X 2 + 2 vi X + 2iv

Because of (4.3), then we have

∑εn

1

2i = n X2 – 2 n X X + n X 2 + 2

iv

= n ( X - X )2 + ∑n

1

2iv

Substituting (4.6b) to the above equation,

∑εn

1

2i = ∑

∑+

ε

n

1

2i

2n

1i

vn

(4.7a)

The true errors ε are normally distributed, ∑ εε ji = 0 for i ≠ j , so that

- - 33 -

( )2i∑ε = ∑ε

n

1

2i

thus (4.7a) becomes

∑εn

1

2i = ∑

∑+

ε n

1

2i

n

1i

vn

(4.7b)

or

(n – 1) ∑εn

1

2i = ∑

n

1

2iv (4.7c)

Finally the error variance (4.4)

1n

vn

1

2i

2

−=σ

∑ (4.8)

This is formula for computation of error variance from correction values of n repeated measurements.

Propagation Law of Variance-covariance and Cofactor Let 0λ is approximate value of a measurement value λ, so that we have 0λλλ −=∆ (4.9a)

λ∆ is a random variable as well as λ, but not 0λ because it is a constant. The error

variance of λ∆ should be the same as the error variance of λ, 22

λλ σ=σ∆ (4.9b) We make the formula more general, if X = a λ + a0 (4.10a) a and a0 are constants 222

X a λσ=σ (4.10b) Then let X = [ ]t21 xx and L = [ ]n21 .. λλλ t are matrices of random variables, X = A L + A0 (4.11a) A and A0 are matrices of constants,

- - 34 -

A =

n22221

n11211

a..aaa..aa

and A0 =

02

01

aa

t

XX A AΣ=Σ (4.11b) ΣXX is matrix variance-covariance of X and Σ is matrix variance of L

ΣXX =

σσσσ

22x1x2x

2x11x2

1x and Σ =

σ

σσ

2n

22

21

..00..........0..00..0

ΣXX has co-variance σx1x2 and σx2x1 elements, then it is called variance-covariance matrix of X. These elements occur because the random parameters x1 and x2 are correlated parameters due to each of them is obtained from the same set of L. Variance matrix ΣLL is a diagonal matrix, because L has independent random variables of λ i , therefore Σ does not have covariance elements. If matrix X has u elements x1, x2, . . . . . . xu then variance-covariance matrix of X becomes a square matrix (u x u).

Let matrix X (u x 1) consists of two matrices Y (u1 x 1) and Z (u2 x 1), so u = u1 + u2.

X =

ZY

; A =

CB

and A0 =

0

0

CB

Thus we have two correlated unknown parameters Y and Z that are obtained from L, Y = B L + B0 (4.14a) and

Z = C L + C0 (4.14b) Variance-covariance matrix of X becomes

ΣXX =

ZZZY

YZYY

ΣΣΣΣ

=

CB

Σ [ ]tt CB =

ΣΣΣΣ

tt

tt

C CB CC BB B

so that we have variance-covariance matrix ΣYZ = B Σ Ct (4.14c) and

ΣΖY = C Σ Bt (4.14d)

- - 35 -

The unknown parameter X is obtained from L, it means that X and L are correlated or dependent random matrices; therefore we may obtain their covariance matrix as shown below

X = A L + A0 (4.15a) and L = I L (4.15b)

I is identity matrix, and according to (4.14) we will have ΣXL = A Σ (4.15c) and ΣLX = Σ At (4.15d)

In the case of L has equal variance elements 2

iσ , we call that all λ i (for i from 1 to n) have equal weight, and Σ an identity matrix. If the variance elements are un-equal, then the measurements have un-equal weight as illustrated below,

measurements λ1 λ2 λ3 . . . . . . . . . λn variances 2

1σ 22σ 2

3σ . . . . . . . . . 2nσ

weights p1 p2 p3 pn

Variance 2iσ shows the accuracy of λ i, if 2

jσ < 2kσ then λ j more accurate than λk, but

pj > pk. Therefore we may have relation between variance and weight as follow k

2kj

2j p p σ=σ = constant (4.16)

This constant value is called variance factor or reference variance, which is denoted by

2oσ . The variance factor may be defined as variance of a measurement that has weight is

equal 1. Inverse of weight is called cofactor which is denoted by q , so that 2

iσ = 2oσ qi (4.17)

In matrix form, weight and cofactor of a set of measurement is a diagonal matrix, denoted by P and Q respectively, Q = P-1 (4.18)

- - 36 -

P =

n

2

1

p..00..........0..p00..0p

and Q =

n

2

1

q..00..........0..q00..0q

From eqn (4.17) we have Σ = 2

oσ Q (4.20)

and we will have similar form for propagation of cofactor as propagation of variance-covariance which is shown in the table below

Table1: Propagation Law

X = AL + A0 Y = BL + B0

Variance-covariance

Cofactor

ΣXX = A Σ At QXX = A Q At ΣYY = B Σ Bt QYY = B Q Bt ΣXY = A Σ Bt QXY = A Q Bt ΣYX = B Σ At QYX = B Q At

ΣXL = A Σ QXL = A Q ΣLX = Σ At QLX = Q At ΣYL = B Σ QYL = B Q ΣLY = Σ Bt QLY = Q Bt

The average value as unknown parameter used at the eqn (4.1) is computed from a set

of repeated equal weight measurements, so that the sample variance at eqn (4.8) is also the sample variance factor 2

oσ ,

X = n21 n1 . . . . .

n1

n1

λλλ +++

The error variance of X that is computed from the propagation law of variance is

2Xσ = 2

on1

σ

(4.21) The number of sample is the weight of X as found from (4.16), pX = n (4.22)

- - 37 -

This equation tell us that the weight of the average value of a set of equal weight measurements is the same to the number of repeated measurements. It means that we will get more accurate result if we measure with a large number of repeated measurements. Theoretically it is true, but practically it is not efficient and we should we investigate this matter statistically.

Least Squares Principle The problem in geodesy or in some other disciplines is how to get accurate unknown

parameters from many set measurements. In the previous discussion we have variance factor as accuracy measures. We will have accurate parameters when we have smallest variance factor that is obtained from the measurements. The smallest variance factor is obtained from the smallest summation result of the square of corrections, or we should have

∑n

1

2iv is minimum for equal weight measurements, and

∑n

1

2iivp is minimum for unequal weight measurements.

This is called least squares principle. We may also say that the least squares principle is how to obtained minimum value of variance.

Is the average value or the mean value of repeated measurements agree with least squares principle? To answer this question let n is the number of unequal weight repeated measurements, and we have correction vi for each iλ ,

vi = X - iλ

∑n

1

2iivp = X2 ∑

n

1ip - 2 X ∑

n

1ii p λ + ∑

n

1ii p λ

Let Φ = ∑n

1

2iivp is a function of X , and if its first derivation Φ’ = 0, then Φ is

minimum.

Φ’ = 2X ∑n

1ip - 2 ∑

n

1ii p λ = 0

X = ∑

∑

n

1i

n

1ii

p

p λ (4.23a)

The eqn (4.23) is mean value of unequal weight of repeated measurements which is determined from least squares principle. If the repeated measurements are equal weight, we may put pi = 1, and then insert this value to (4.23), then we will have mean value for repeated measurements of equal weight,

- - 38 -

X = n

n

1i∑λ

(4.23b)

The eqn (4.23b) is exactly the same with the eqn (4.1), therefore the average value of equal weight of repeated measurements agree with the least squares principle. Variance factor of the unequal repeated measurements is

1n

vpn

1

2ii

2o −

=σ∑

(4.24)

and variance of X is

2Xσ = 2

on

1ip

1σ

∑ (4.25)

In matrix form ∑n

1

2iivp = Vt P V, where V = [ ]n21 v..vv t

Least Squares Adjustment Let unknown parameters X (u x 1) will be determined from L (n x 1) measurements,

V is matrix correction should be added to L to obtain corrected value L~ )X,L~(F = 0 (4.26)

And let Xo is approximate value of X, so that we have

L~ = L + V and X = Xo + x

Then we have

( ) ( )( ) ( )oL,X)L,X( X,LFxX,L~FX

VX,L~FL~ oo

+∂∂

+∂∂ = 0

or AV + Bx + C = 0 (4.27) We follow Mikhail [1976, p. 112]: A is matrix (c x n) ; B is matrix (c x u) ; C is matrix (c x 1) V is matrix (n x 1) ; x is matrix (u x 1) ; 0 is matrix (c x 1) Xo is a non-random matrix and x is a random unknown parameter as well is X. The random unknown parameter x and correction V will be determined least squares principle,

- - 39 -

thus Vt P V should be minimum. Let transpose of Lagrange multiplicator K , multiply to (4.27) and we have Vt P V + 2 Kt (AV + Bx + C ) = Φ (4.28) Φ is function of V and x; the minimum of Vt P V is also the minimum of Φ, and it will be minimum if its derivation to V and to x are zero. 2 Vt P + 2 Kt A = 0t (4.29a) Kt B = 0t (4.29b) From (4.29a) we have V = - P-1 At K (4.30) K is matrix (c x 1) Subtitution of (4.30) into (4.27) (A P-1 At) K = B x + C (4.31) This a set of linear equations with unknown of K (c x 1) K = (A P-1 At)-1 (B x + C) (4.32) Substitution of (4.32) into (4.29b) {Bt (A P-1 At)—1 B} x + {Bt (A P-1 At)—1 C} = 0 (4.33a) This a set of linear equations with unknown x (u x 1). According to Kahar [2001], this equation becomes (Bt PC B)-1 x + Bt PC C = 0 (4.33b) because weight of C is PC = 1−

CCQ = (A P-1 At)—1 x = - (Bt PC B)-1 (Bt PC C) (4.34) From (4.30), (4.31) and (4.34) we obtain Vt P V = Ct PC B x + Ct PC C (4.35) And variance factor

u-n 2

0VPVt

=σ (4.36)

- - 40 -

Derivation of (4.36) is found in Kahar [2001]. Cofactor QXX is obtained by using propagation of cofactor to (4.34), and because of cofactor QCC = 1−

CP then we have QXX = (Bt PC B)-1 (4.37) so that variance of x ΣXX = 2

oσ QXX (4.38) A special case of this problem is A is identity matrix I, it means PC = P , the weight of L, so that notation C at (4.27) may be written as V + Bx + L = 0 (4..39)

Vt P V = (xt Bt + Lt ) P (B x + L) = xt Bt P B x + 2 Lt P B x + Lt P L Let Vt P V = Φ is a function of x, will be minimum if Φξ , its derivative to x is zero, thus (Bt P B ) x + Bt P L = 0 (4.40a) and x = - (Bt P B)-1 Bt P L (4.40b) This matrix represents u linear equations with u unknown parameters of x . We may notice that F in (4.27) is L so that the equation becomes L + V = - B x and P is weight matrix of L. Now let

AV = VC (4.41)

so that the original form in (4.27) becomes VC + B x + C = 0 (4.42) The unknown parameter x is obtained by minimizing CC

tC V P V where PC is weight

matrix of C . In original form C is C = F (L , Xo) (4.43) Let Lo is approximate value of L, so that L = ∆L + Lo

- - 41 -

Linearization of (4.31)

C = ( ) ),(, ooo XLFLXLFL

+∆∂∂

= A ∆L + F(Lo, Xo) From propagation of cofactor we obtain QCC = A Q At because cofactor of ∆L is also cofactor of L , that is Q . Then weight of C becomes PC = (A Q At )-1 (4.44)

Cofactor of adjusted measurements L~ and parameter x may be obtained by implementing the propagation law of cofactor. From (4.40b) cofactor of x, and see also (4.37)

Qxx = (Bt P B)-1 (4.45a) or Qxx = (Bt Q-1 B)-1 (4.45b)

Before determination of L~L~Q we should determine QVV, because

L~ = L + V and from (4.39) V = - (L + Bx) or V = [ B ( Bt Q-1 B)-1 Bt Q-1 - I ] L (4.46) so that QVV = Q - B ( Bt Q-1 B)-1 Bt (4.47a) or QVV = Q – B QXX Bt (4.47b) Then we have L~ = B ( Bt Q-1 B)-1 Bt Q-1 L (4.48) so that L~L~Q = B ( Bt Q-1 B)-1 Bt (4.49a) or L~L~Q = Q – QVV (4.49b)

- - 42 -

Least Squares Prediction A measured distance between two points does not have correlation with other

measured distance between others two points. A measured angle at certain point does not have correlation with other measured angle at other point. Therefore we only have variance matrix for a set measured values in our pervious discussion, it is a diagonal matrix variance. Determination of gravimetric geoid undulation requires gravity anomaly data over the whole earth. Gravity anomaly is the difference between actual gravity value and theoretical gravity value of the earth ellipsoid. If the earth ellipsoid is the good representative of the geoid, the mean of gravity value will be zero, and then certainly the mean of the geoid undulation over the whole earth is also zero. Let us use terminology “signal” for gravity anomaly, geoid undulation because this value will give information about mass density of the earth crust. Each individual gravity anomaly in a set of gravity anomalies at certain area that are obtained from gravity measurements has correlation to other gravity anomaly data of that area This correlation is shown by its correlation function or by its covariance function. The function depends on distance between two values, thus it does not depend on direction between the two values and also it is not depend on position of each gravity value. In this case we call that gravity anomaly is a homogenous signal.

Two closest signals have strong correlation than another two distant signals, or covariance between two closest signals will be larger than the other two distant signals. If the distance of two signals is very distant, we may assume that covariance between the two signals is closed to zero. Gaussian function is a simple model of covariance function. Let a measured signal iλ is disturbed by noise ni , so that we find relation among signal si , noise ni , and measured signal iλ , as shown in figure (4.1)

Figure 4.1: (a) relation among iλ , si and ni (b) a predicted signal P among measured signals iλ = si + ni (4.50)

Signal si and noise ni are independent random variables, so that we do not have covariance between signal and noise, then variance of iλ

21 . . . .

. .

.

.

..

. .

. .

.

.

. .

. .

.

. .. .

k-1 k

. P

1 2

.

..

.

.

. λ

1

n1

s1

λ2 n2

s2 . .

. 3

n3s3

λ3

(a) (b)

- - 43 -

d2

o2i +σ=σ (4.51)

2oσ is variance of signal si and d is variance of noise ni

Let L (k x 1) is a set of measured signals and s (k x 1) is a set of signals that will be filtered from a set of noise n (k x 1), so that (4.50) in matrix form. L = s + n (4.52) Variance-covariance matriks of measured signal will be Σ = C + D (4.53) Variance-covariance matrix of signal s is

C =

o2k1k

k2o21

k112o

C..CC..........

C..CCC..CC

(4.54)

Because we have a set of homogenous signals, the variance value of the all signals are the same, Co . Noise n has variance matrix, because ni and nj are uncorrelated noise, and if variance of individual noise is the same as d, so that variance matrix of noise is D = d I (4.55) Covariance elements of (4.54) is a function distance between two signals i and j (dij) for i is not equal j, so Cij = C(dij). Therefore variance of each signal becomes, see also (4.51)

2σ = C(0) = C0 Then variance-covariance matrix of measured signals L becomes

C =

+

++

dC..CC..........

C..dCCC..CdC

o2k1k

k2o21

k112o

(4.56)

A predicted signal P, see fig (4.1) is computed as follow

- - 44 -

sP = h1 λ 1 + h2 λ 2 + . . . . . . . + hk λ k = ht L The true signal at P is Ps , so that sP has a true error εP and variance of the error is

2Pσ = 2

Pε . We should minimize this variance for obtaining h.

εP = sP - Ps = ht L - Ps = [ ]

−

P

t

sL

Ih

By using propagation law of variance-covariance

2

Pσ = ht C h - 2 otLP C h C + (4.57)

Its derivative to h is zero, so that

-1t

LPt C C h = (4.58)

And predicted signal at P

sP = L C C -1t

LP (4.59a)

If we have m predicted signals so that sP is (m x 1) matrix sP = L C C -1t

LP (4.59b) CLP is covariance matrix (k x m) between measured signal L and predicted signal sP.

The accuracy of each predicted signal is obtained from (4.57); it is a diagonal element of matrix LP

-1tLPoP C C C - C =Σ (4.60)

The predicted signals sP at (4.59) are predicted by filtering measured signals L from noise n . C is used for filtering noise n from L , and together with CLP predicting the predicted signals. CLP is covariance matrix between measured signal L and predicted signals sP . Because we minimizing the variance of the predicted signal error, therefore we call this solution as least square filtering and least squares prediction techniques.

In the previous problem the measured signal L only be influenced by random noise n; now let the measurements are influenced by systematic part Bx, where unknown parameter of the systematic part and B is coefficient matrix, so that to get signal s, the systematic part Bx should be subtracted from measured signal (4.52)

- - 45 -

s = L – Bx - n (4.61a)

Thus, to get signal s the measured signal L should be subtracted by systematic part Bx and random noise n as shown in (4.61a). This equation is identical with (4.39) as shown at (4.61b) s + n + Bx – L = 0 (4.61b) by changing s + n by v, and according to (4.40), the parameter x will be x = (Bt C-1 B)-1 Bt C-1 L (4.62) because P is replaced by C-1 ; and covariance of x is , see (4.37) Cxx = (Bt C-1 B)-1 (4.63) According to (4.56) signal s is obtained after filtering of noise n and L is subtracted by Bx, so that predicted signals sP at (4.59) becomes sP = Bx) - (L C C -1t

LP (4.64) Substitute (4.62) into (4.64),

sP = ( ) L CBBCBB - I C C 1t11t1-tLP

−−−

= Ht L where

Ht = ( ) L CBBCBB - I C C 1t11t1-tLP

−−−

Similar to (4.56) then we obtain the accuracy matrix of predicted signal sP LP

-1tLPoP C C C - C =Σ + LP

tttLP CCB)BC(B B CC 1111 −−−− (4.65)

If B = 0 we will obtain (4.57).

The eqn. (4.62) shows that parameter x is determined from measured signal L, and the eqn. (4.64) is the equation for predict signal outside of observation point. The determination of parameter x and predicted signal sP is conducted by filtering the measured signal by utilizing covariance model of signals and estimated variance of noise by using least squares principle. Least squares solution which is used for determination of parameter x and predicted signal sP from filtered measured signal L is called least squares collocation. More detailed explanation of this technique is described in Moritz [1973].

- - 46 -

Adjustment in Steps Recall (4.39):

V + Bx + L = 0 (4..39) Parameter x from (4,40) x = - (Bt P B)-1 Bt P L (4.40) and Qxx from (4.45) Qxx = (Bt P B)-1 (4.45a) or Qxx = (Bt Q-1 B)-1 (4.45b) and finally L~L~Q from (4.49a)

L~L~Q = B ( Bt Q-1 B)-1 Bt (4.49a) Let (4.39) devided becomes two parts

0 L

L x

B

B V

2

1

2

1

=

−−+

−−+ (4.66)

where B1 = matrix (n1 x u) and B2 = matrix n2 x u L1 = matrix n1 x 1 and L2 = matrix n2 x 1 n = n1 + n2 From (4.66)

=L

−−

2

1

L

L

we have cofactor

=

2

1

Q00Q

Q

If use only the set of n1 data of L2 we will get x1 from V1 + B1 x1 + L1 = 0 (4.67a) or we use only the set of n2 data of L2 we will get x2 from V2 + B2 x2 + L2 = 0 (4.67b) The parameters are

- - 47 -

x1 = - Qx1x1 11

1t1 L Q B − (4.68a)

and x2 = - Qx2x2 2

11

t2 L Q B − (4.68b)

( ) 11

1-1

t11x1x B Q BQ

−= and ( ) 1

21-

2t22x2x B Q BQ

−=

Let the firdt step we use the eqn (4.67a) which give result (4.68a). To obtain parameter of x, we should add ∆x to x1, x = x1 + ∆x (4.69a) or ∆x = x – x1 (4.69b) In the second step we use the n2 data of L2 and coefficient B2 from (4.66), and then combine with (4.69b), to obtain x, V2 = B2 x + L2 x1 = x – x1 We grouping that equation in matrix form,

+

=

∆ 1

222

x-L

x I

B

xV

(4.70a)

or L x B V

)))+= (4.70b)

so that ) L Q B ( Q x -1t

xx)))

−= (4.71)

=

1x1x

2

Q00Q

Q)

(4.72)

-11t

xx ) B QB ( Q))) −= (4.73)

[ ]

==−

IB

Q0

0Q IB ) B Q B ( Q 21-

x1x1

-12t

21-t1

xx)))

= -11x1x2

12

t2 Q B Q B +−

= -11x1x

12x2x Q Q +− (4.74a)

or -1-1

x2x21

1x1xxx ) Q Q (Q += − (4.74b)

- - 48 -

) L Q B ( -1t ))) in (4.71) is

[ ]

=

1

21-

XX

-12t

21-t

x-L

Q0

0Q IB ) L Q B (

11

)))

= 1

-1XX2

-12

t 2 x Q - L Q B

11 (4.75)

Multiplication of Qxx to 1

xxQ− at (4.74a) yields -1

2x2xxx-1

1x1xxx Q Q Q Q I += (4.76a) or -1

2x2xxx-1

1x1xxx Q Q I Q Q −= (4.76b) Insert (4.75) into (4.71), )x Q - L Q B ( Q x 1

-1x1x12

-12

t2xx−=

And by considering (4.76b), 1

-12x2xxx2

-12

t2xx x ) Q Q - I ( ) L Q B ( Q x +−=

= x1 – { }122

-12

t2xx x B L Q B Q +

or x = x1 – { }122 x B L K +

) (4.77)

and correction (4.69) will be { }122 x B L K - x +=∆

) (4.78)

Q B Q K -1

2t2xx=

) (4.79)

K)

is called Kalman gain We can do least squares adjustment in k steps, so that at each steps we have ni data set

of Li for i = 1 to k, so that n1 + n2 + . . . nk = n and the measurements L consists of k groups, where

=

k

2

1

L..

LL

L and

=

k

2

1

Q..00..........0..Q00..0Q

Q

- - 49 -

For k steps adjustment, we modify notation which shown in the above equations, xk = xk - 1 – { }1-kkk x B L K +

) eqn (4.77)

Q B Q K -1

ktkxkxk

))=

( ) Q Q Q xkxk1)-1)x(k-x(kxkxk +=

))

( ) 1k

1-k

tkxkxk B Q BQ

−=

If we only doing the adjustment for one step, than B2 = 0, so that we do not have Kalman gain. For the first step, k = 1, Q Q x1x11x1x =

).

- - 50 -

5. PHYSICAL GEODESY Earth’s Gravity Field The previous subjects we have already discussed briefly about gravity, gravitational

and centrifugal acceleration of the earth as well their forces. These forces are gradient of their potential. Let Vg, a scalar is gravity potential energy of the earth, relation with gravitational potential energy and centrifugal potential energy is

Vg = VN + Vc (5.1)

Gravity force vector is gradient of gravity potential energy gF

ρ = grad Vg

m gρ = grad Vg

Let W is gravity acceleration potential and for simplicity we omit “acceleration”. g

ρ = grad W (5.2a)

or

g = -HW

∂∂ (5.2b)

The positive direction of gravity is opposite direction of positive direction of H, therefore (5.2) has minus sign. Here we change notation for gravity acceleration from ag becomes g. Equivalent to (5.1) and (5.2), then we have Na

ρ = grad V (5.3) and ca

ρ = grad Φ (5.3) where V and Φ are called gravitational and centrifugal potentials after we omit “acceleration” as well for gravity. From (5.2), (5.3) and (5.4) we have a scalar summation W (x, y, z) = V(x, y, z) + Φ(x, y, z) (5.4) Let a Cartesian coordinate has origin at center of the earth, and Z-axis is rotational axis of the earth. Centrifugal acceleration of a point P (x,y,z) according to (1.6) is ac = ω2 p p = 22 yx + (5.5) From (5.3) and (5.5) we have

- - 51 -

Φ = )yx(21 222 +ω (5.6)

Gravitational potential for solid body, which can be applied, to the earth [Heiskanen & Moritz,1967, p. 3]

V = G ∫∫∫ρ

vdv

r (5.7)

then gravity potential (5.4) becomes

W (x, y, z) = G ∫∫∫ρ

vdv

r+ )yx(

21 222 +ω (5.8)

By introducing Laplacian operator

2

2

2

2

2

2

zyx ∂∂

+∂∂

+∂∂

=∆

then according to (5.4) we have ∆W = ∆V + ∆Φ (5.9) Apply to the earth as a rotational solid body we have the generalized Poisson equation [Heiskanen & Moritz, 1967, p. 47] ∆W = - 4πGρ + 2ω2 (5.10) Outside the earth which is assumed as vacuum space ∆W = 2ω2 (5.11) because ∆V = 0 (5.12) The eqn (5.12) is called Laplace equation. Thus, outside of the earth gravitational potential satisfies Laplace equation, so that V can be expressed into a spherical harmonic series. We refer to Heiskanen & Moritz [1967, p. 21 to 32] the gravitational potential of the earth as unit sphere (radius is equal R = 1) is for a point in spherical coordinate (r, θ, λ), see the eqn (2.14), in this case R is replaced by r,

[ ] ) (cosP m sinbm cosar

1),,r(V0n

n

0mnmnmnm1n∑ ∑

∞

= =+ θλ+λ=λθ (5.13)

nma and nmb are spherical harmonic coefficients and nmP is a Legendre functions; n is degree and m is order of Legendre functions. Now let an ellipsoid revolution is a

- - 52 -

normal figure of the earth. Talking about a normal figure of the earth means we talk about Geodetic Reference System which is defined by a = equatorial radius of the earth GM = geocentric gravitational constant J2 = dynamical form factor ω = angular velocity of the earth By inserting of equatorial radius of the earth and the geocentric gravitational constant, (5.13) becomes

( ) ) (cosP m sinKm cosJ ra -1

rGM),,r(V

1n

n

0mnmnmnm

n

θλ+λ

=λθ ∑ ∑

∞

= = (5.14)

For an ellipsoid of revolution as geodetic reference system, expansion of the normal gravitational potential V’ into a series of spherical harmonics have only even degree, in the other words the series does not odd degree harmonics and order m , so that (5.14) becomes,

)( cosPraJ - . . - ) (cosP

raJ - ) (cosP

raJ - 1

rGM),,r('V 2n

2n'2n4

4'42

2'2

θ

θ

θ

=λθ

or

)( cosPraJ - 1

rGM),,r('V 2n

2n

1-n

'2n

θ

=λθ ∑

∞ (5.15)

Coefficients '

n2J can be computed from '2J as explained in Heiskanen & Moritz [1967.

p.73]. Thus by taking '2J = J2 the gravitational normal potential (5.15) can be computed

from the parameters of geodetic reference system a, GM, J2 . Normal gravity potential U that refers to the normal earth, see also (5.4) U (x, y, z) = V’(x, y, z) + Φ’(x, y, z) (5.16) As shown in (5.2), the actual gravity is the actual gravity potential gradient, and similar with this, the normal gravity γ is the normal gravity potential gradient, γ = grad U (5.17a) or

γ = -hU

∂∂ (5.17b)

Again, the positive direction of normal gravity is opposite direction of positive direction of h, therefore (5.2) has minus sign. The normal centrifugal potential Φ’ has the same

- - 53 -

value with the actual centrifugal Φ, because the angular velocity of the normal earth ω’ is also the angular velocity of the actual earth, ω’ = ω and Φ’ = Φ .

Let T, the anomalous potential T (x, y, z) = W (x, y, z) - U (x, y, z) (5.18) and T (x, y, z) = V (x, y, z) – V’ (x, y, z) (5.19) Because V and V’ satisfy Laplace equation, therefore the anomalous potential T also satisfies Laplace equation for the case of outside of the earth, and from (5.14), (5.15) and (5.19),

( ) ) (cosP m sinKm cosJ ra

rGM),,r( T

0n

n

0mnmnmnm

n

θλδ+λδ

−=λθ ∑ ∑

∞

= = (5.20)

Geoid Undulation: Stokes Integral We know that the geoid is a level surface or a gravity equipotential surface, which is

closed to global mean sea level. Thus gravity potential of the geoid is constant, W = Wo. The normal figure of the geoid is represented by an ellipsoid revolution, so that the ellipsoid revolution is also a level surface, which has normal gravity potential as the same as the gravity potential of the geoid,

U = Uo = Wo

Figure 5.1: Geoid-ellipsoid separation N Actual gravity potential of P at the geoid, see fig. (5.1), WP = Wo , and normal gravity potential of Q at the surface of ellipsoid, UQ = Uo , so that UQ = WP . Normal gravity potential at P is obtained by a simple Taylor series

UP = UQ + PQ

NhU

∂∂

P

Q

ellipsoid, U = Uo = Wo

geoid, W = Wo

NPgP

γQ

P

Q


geoid, W = Wo

NPgP

γQ

P

Q


geoid, W = Wo

NPgP

γQ

P

Q


geoid, W = Wo

NPgP

γQ

- - 54 -

Ellipsoidal height h is positive outward, but normal gravity positive inward of ellipsoid so that UP = UQ - γQ NP Because UQ = Uo = Wo = WP so that UP = WP - γQ NP WP – UP = TP = γQ NP and finally

T NQ

PP γ

= (5.21)

This is Bruns formula, basic equation for determination of geoid undulation N from anomalous potential T. The eqn (5.21) shows that geoid undulation N satisfies Laplace equation as well as anomalous potential T.

Gravity anomaly ∆g and gravity disturbance δg are defined as, see fig (5.1) ∆gP = gP - γQ and δgP = gP - γP

Normal gravity at point P may be obtained from normal gravity at point Q by using a simple Taylor series

γP = γQ +Qh

∂γ∂ NP

γP = γQ +Q

P

Q

Th γ

∂γ∂

From the above equation we obtain relation between gravity anomaly and gravity disturbance

δgP = gP - γP = gP - γQ -Q

P

Q

Th γ

∂γ∂

δgP = ∆gP - Q

P

Q

Th γ

∂γ∂ (5.22)

By using spherical approximation, the normal gravity at the surface of the spherical earth

of radius R is 2o RGM

=γ we have

h∂γ∂ =

r∂γ∂ = - 2 3R

GM = - R

2 oγ

- - 55 -

γo is gravity value for the spherical earth, and it is gravity value without centrifugal acceleration, therefore γo has magnitude as the same as gravitational acceleration magnitude. According to (5.2) and (5.17) that gravity disturbance may be concluded as anomalous potential gradient, δg = grad T (5.23a)

δg = -hT

∂∂ (5.23b)

The direction of H and h has a very small direction, which is called the deflection of the vertical. Therefore in this case we may assume that H and h is has the same direction. By using spherical approximation in (5.22), γQ may be replaced by γo , so that at the surface of spherical earth the equation becomes

δg = ∆g + RT2 (5.24a)

or

0gRT2

hT

=∆++∂∂ (5.24b)

This is the fundamental boundary condition in the form of the spherical approximation.

The anomalous potential T in the form of spherical harmonics expansions as shown in (5.20) can be written for a point outside of the earth as [Heiskanen & Moritz, 1967, p.88]

T(r,θ,λ) = ∑∞

=

+

λθ

0nn

1n

),(T rR (5.25)

Tn (θ,λ) is a surface spherical harmonics. For a point outside the spherical earth of radius

R, so that r is > R. Because hT

∂∂ =

rT

∂∂ , then we have

hT

∂∂ = - ∑

∞

=

+

λθ

+

0nn

1n

),(TrR)1n(

r1 (5.26)

Gravity anomaly for a point outside of the earth is obtained from (5.24)

∆g = -hT

∂∂ -

rT2

∑∞

=

+

λθ

−=∆

0nn

1n

),(TrR)1n(

r1g (5.27)

This is expression of gravity anomaly in the form of spherical harmonics series in term of Tn (θ,λ). The gravity anomaly in the form of spherical harmonics series of ∆gn (θ,λ) is

- - 56 -

∆g (r,θ,λ) = ∑∞

=

+

λθ∆

0nn

1n

),(g rR (5.28)

so that we have

),(g1n

R),(T nn λθ∆−

=λθ (5.29)

For n =1, the eqn. (5.27) becomes zero, and (5.29) becomes infinity. Therefore gravity anomaly can never have a first-degree spherical harmonics. Substitution of (5.29) into (5.25) for any point at the geoid (r = R)

T(θ,λ) = ∑∞

=λθ∆

2nn ),(g

1-n1R (5.30)

The surface harmonics of ∆gn(θ,λ) is obtained from the integration of ∆g over spherical earth σ by using formula

∆gn(θ,λ) = ∫ ∫π

=λ′

π

=θ′λ′θ′θ′ψλ′θ′∆

π+ 2

0 0n d d sin ) (cosP ),(g

41n2 (5.31)

Insert (5.31) into (5.30) then we obtain T(θ,λ) at the geoid, which is approximated by the spherical earth of radius R

T(θ,λ) = ∫ ∫ ∑π

=λ′

π

=θ′

∞

=λ′θ′θ′ψ

+λ′θ′∆

π

2

0 0n

2nd d sin ) (cosP

1-n12n ),(g

4R (5.32)

∑∞

=ψ

−+

2nn ) (cos P

1n1n2 is Stokes function which is denoted by S(ψ) so that

T(θ,λ) = ∫ ∫π

=λ′

π


π

2

0 0d d sin )S( ),(g

4R (5.33)

Then geoid undulation N is obtained by inserting (5.33) into Bruns formula (5.20)

∫ ∫π

=λ′

π


πγ=λθ

2

0 0od d sin )S( ),(g

4R ),(N (5.34)

γP is replaced by γo because we use spherical approximation approach in obtaining (5.31). If we use polar coordinate for the position of dσ (ψ,α), see fig. (5.2) the eqn (5.34) becomes

∫ ∫π

=α

π

=ψαψψψαψ∆

πγ=λθ

2

0 0od d sin )S( ),(g

4R ),(N (5.34b)

- - 57 -

The eqns (5.31) and (5.32) is Stokes integral or Stokes formula for the determination of anomalous potential and geoid undulation respectively. Stokes function S(ψ) is

S(ψ) =

ψ

+ψ

ψψ+ψψ

2 sin

2sin ln cos 3 - cos 5 - 1

2sin 6 -

2 cosec 2 (5.35)

and ψ is computed from cos ψ = sin φ sin φ’ + cos φ cos φ’ cos (λ’- λ) (5.36)

Because dσ = sin θ’ dθ’ dλ’ = cos φ’ dφ’ dλ’ , see fig. (5.2) then the equations (5.33) and (5.34) can be written more simple

T(φ,λ) = ∫∫σ

σψλ′φ′∆π

d )( S ),( g4R (5.37)

N(φ,λ) = ∫∫σ

σψλ′φ′∆πγ

d )( S ),( g4

R

o (5.38a)

or

N(φ,λ) = ∫∫σ

σψαψ∆πγ

d )( S ),( g4

R

o (5.38b)

The equation (5.38) tells us that to determine geoid undulation at a certain point P at the geoid requires gravity anomaly at dσ, which are well distributed over the whole of the geoid σ. The Stokes integral (5.33) is the solution of the geodetic boundary problem by considering boundary condition at the geoid is shown by (5.24).

Determination of the Geoid from Surface Data and Geopotential Coefficients Anomalous potential at the surface of the geoid (r = a = R) in spherical harmonic

series, see (5.20) is

θ’=[π/2-φ’]

ψ

λ’-λ

P

NP

θ =[π/2-φ]

α

dσ

Figure 5.2: Geographical coordinatesin spherical triangle

θ’=[π/2-φ’]

ψ

λ’-λ

P

NP

θ =[π/2-φ]

α

dσ

θ’=[π/2-φ’]

ψ

λ’-λ

P

NP

θ =[π/2-φ]

α

θ’=[π/2-φ’]

ψ

λ’-λ

P

NP

θ =[π/2-φ] θ’=[π/2-φ’]

ψ

λ’-λ

P

NP

θ =[π/2-φ]

ψ

λ’-λ

P

NP

θ =[π/2-φ]

λ’-λ

P

NP

θ =[π/2-φ]

P

NP

θ =[π/2-φ]

α

dσdσ

Figure 5.2: Geographical coordinatesin spherical triangle

- - 58 -

( ) ) (sinP m sinKm cosJ R

GM),( T2n

n

0mnmnmnm

φλδ+λδ−=λθ ∑ ∑

∞

= = (5.39)

and geoid undulation is

( ) ) (sinP m sinKm cosJ R),( N2n

n

0mnmnmnm

φλδ+λδ−=λθ ∑ ∑

∞

= = (5.40)

and gravity anomaly at the surface of the geoid, see the eqn (5.27) for r = R

( ) ) (sinP m sinKm cosJ 1 -n

1 ),( g2n

n

0mnmnmnmo

φλδ+λδ

γ−=λφ∆ ∑ ∑

∞

= = (5.41)

To eliminate minus sign in the right side of the eqns (5.39), (5.40) and (5.41) and for the most convenient and the most widely used these, equations are expressed in the form of fully normalized harmonics , so that

) (sinP )msin S m cos C( R

GM),( T2n

n

0mnmnmnm

φλδ+λδ=λφ ∑ ∑

∞

= = (5.42)

) (sinP )msin S m cos C( R),( N2n

n

0mnmnmnm

φλδ+λδ=λφ ∑ ∑

∞

= = (5.43)

) (sinP )msin S m cos C(1-n

1 ),( g2n

n

0mnmnmnmo

φλδ+λδ

γ=λφ∆ ∑ ∑

∞

= = (5.44)

'nmnmnm C - C C =δ and '

nmnmnm S - S S =δ

For parameter geodetic reference system J2 is changed becomes 0,2C , J2 = J2,0 = - 0,2C 5

In general nmnm C - J = and nmnm S - K =

A detailed explanation about conventional harmonic functions and coefficients related to the fully harmonic functions and coefficients can be found in Heiskanen & Moritz, [1967,p. 23 – 60].

The geoid undulation can be determined by using Stokes integral and spherical harmonic series as shown in (5.39) and (5.40). When we use Stokes integral, we should have one gravity anomaly value at dσ, which represents gravity value of dσ. Stokes function in the Stokes integral is closed to infinity if ψ closed to zero, or dσ very near to computation point P. Therefore for the nearest area to the computation we should have

- - 59 -

small size of dσ with one gravity anomaly value at the dσ. Heiskanen & Moritz [1967. p. 120] shows a practical approach to determine effect of the nearest zone on the geoid computation using Stokes integral It means, when we use Stokes integral, we should have gravity anomaly data in different size of dσ over the whole earth. Geoid undulation, which is determined by using spherical harmonic series, is depending on the degree of harmonic n. At present geopotential coefficients C nm and nmS such as for Earth Gravitational Model 1996 (EGM96) completed to degree n = 360. It means, that geoid undulation at any point P will represent the geoid undulation in the area by the size of 0.5o x 0.5o (0.5o = 180o/degreen n with n = 360), with the P is the center of the angular grid area. The geoid, which is obtained from geopotential data, is usually called global geoid. A global geoid which is obtained from n = m = 360 is more detailed than the geoid undulation that is obtained from n = m = 180. There is another model which is called GPM98 model completed to degree n = 1800. The global geoid which is computed by using this model may be called 0.1o global geoid.

To determine a detailed geoid in a certain area requires surface gravity anomaly data, which can be organized in a small grid, for example 0.1o x 0.1o angular grid. Then Stokes integral is devided into

∫∫σ

σψλθ∆πγ

=λθ d )( S )','( g4

R),(No

= ∫∫σ

σψλθ∆πγ 1

So

d )( S )',' ( g4

R

+ ∫∫σ

σψλθ∆πγ 2

Ho

d )( S )',' ( g4

R = N1 + N2

N1 is computed from surface gravity anomaly data, ∆gs and N2 computed from geopotential coefficients data. The geoid undulation that is obtained from geopotential coefficients data NH , see (5.43) written in integral form

∫∫σ

σψλθ∆πγ

=λθ d )( S )','( g4

R),(N Ho

H = NH1 + NH2

∆gH is gravity anomaly that is computed from geopotential coefficients data, see (5.44).

∫∫σ

σψλθ∆πγ

=1

Ho

H1 d )( S )',' ( g4

R N and ∫∫σ

σψλθ∆πγ

=12

Ho

H2 d )( S )',' ( g4

R N

Because N2 = NH2 then we have N = N1 + NH2 = N1 – NH1 + NH1 + NH2 = N1 – NH1 + NH

N1 - NH1 = ∫∫σ

σψλθ∆πγ 1

So

d )( S )',' ( g4

R - ∫∫σ

σψλθ∆πγ 1

Ho

d )( S )',' ( g4

R

N1 - NH1 = ∫∫σ

σψλθ∆∆πγ 1

HSo

d )( S )',' ( )g-g(4

R (5.45)

- - 60 -

N1 - NH1 = NStokes Finally we have geoid undulation obtained from combination of surface data and geopotential coefficients data as N = NStokes + NH (5.46)

To apply (5.45) we should have gravity anomaly data, which is obtained geopotential coefficients in the same grid with the surface gravity anomaly data. In the grid where the surface gravity data is not available, ∆gS is taken equal to ∆gH. For integral Stokes (5..38) we may use R = 6371 km and γo = 9.797 6 m s-2

Geodetic Reference System 1980 To obtain surface gravity anomaly value at any point P at the geoid, see fig. (5.1) the

gravity value at that point should be subtracted by the normal gravity value of point Q at reference ellipsoid. The normal gravity value is computed by using the formula adopted IAG for Geodetic Reference System 1980 (GRS 1980). The GRS has been adopted at the XVII General Assembly of the IUGG in Canberra 1979 replacing GRS 1967 adopted at the General Assembly of the IUGG, Lucerne, 1967 [Moritz, 2000] . The parameter constants that is defined the GRS 1908 is,

• equatorial radius or the earth: a = 6 378 137 m, • geocentric gravitational constant of the earth (including atmosphere): GM = 3 986 005 x 108 m3 s-2, • dynamical form factor of the earth, excluding the permanent tidal

deformation: J2 = 108 263 x 10-8, • angular velocity of the earth: ω = 7 292 115 x 10-11 rad s-1.

Based on the parameters, some derived geometric constants is semi minor axis: b = 6 356 752.3141 m, first eccentricity (e): e2 = 0.006 694 380 022 90, second extensity (e’): e’2 = 0.006 739 496 775 48, flattening: f = 0.003 352 810 681 18 reciprocal flattening: f-1 = 298.257 222 101 mean radius: R1 = 6 371 008.7714 m radius of sphere of same surface: R2 = 6 371 007.1810 m radius of sphere of same volume: R3 = 6 3712 007.7900 m.

Derived physical constants is normal potential at ellipsoid: Uo = 6 263 686.0850 x 10 m2 s-2 normal gravity at equator: γe = 9.780 326 7715 m s-2 normal gravity at pole: γp = 9.832 186 3785 m s-2

mean value of normal gravity γo = 9.797 644 656 m s-2

Normal gravity at latitude φ γ = 9.780 326 77 (1 + 0.005 3045 sin2 φ - 0.000 0058 sin2 2φ ) (5.47)

- - 61 -

Molodensky’s Approach

Gravity g is measured at the physical surface of the earth. The use of Stokes integral for the determination of the geoid undulation requires gravity anomaly at the geoid and there must be no masses outside of the geoid. Therefore, to obtain gravity anomaly at the geoid, the measured gravity must be reduced to mean sea level as approximate surface of the geoid and masses outside of the geoid must be completely shifted below the mean sea level. The gravity reduction and masses shifting into the geoid will affect or change the geoid itself, which is called indirect effect. Thus the geoid that is computed by Stokes integral is slightly difference than the geoid itself, but we will obtain a surface, which called cogeoid.

To solve the problem of gravity reduction, Molodensky proposed an approach that the geoid undulation is determined using gravity anomalies data at the earth’s surface. If normal figure of the geoid is the level ellipsoid, a normal gravity equipotential surface, then Molodensky introduced a normal figure if the earth’s surface which is called by Hirvonen as telluroid [Heiskanen & Moritz, 1967, p. 292], as shown in figure 5.3.

Po

geoid, WP =Wo

gP

P

γo

Q

gPo

Qo

ζ

Hn

N

Hh

earth’s surface

telluroid

ellipsoid, UQ = Uo= Wo

sea surface

spherop of Q, UQ = WP

geop of P, WP

spherrop of P, UP

spherop of Po UPo

Figure 5.3: Geoid, earth’s surface,ellipsoid and telluroid

N = geoid undulaation

ζ = height anomaly

H = orthometric height

Hn = normal height

h = ellipsoidal heght

h = N + H = ζ + Hn

γQo

Po

geoid, WP =Wo

gP

P

γo

Q

gPo

Qo

ζ

Hn

N

Hh

earth’s surface

telluroid


sea surface


geop of P, WP

spherrop of P, UP

spherop of Po UPo



ζ = height anomaly


Hn = normal height


h = N + H = ζ + Hn

Po

geoid, WP =Wo

gP

P

γo

Q

gPo

Qo

ζ

Hn

N

Hh

earth’s surface

telluroid


sea surface


geop of P, WP

spherrop of P, UP

spherop of Po UPoPo

geoid, WP =Wo

gP

P

γo

Q

gPo

Qo

ζ

Hn

N

Hh

earth’s surface

telluroid


sea surface


geop of P, WP

spherrop of P, UP

Po

geoid, WP =Wo

gP

P

γo

Q

gPo

Qo

ζ

Hn

N

Hh

earth’s surface

telluroid


sea surface

Po

geoid, WP =Wo

gP

P

γo

Q

gPo

Qo

ζ

Hn

N

Hh

earth’s surface

telluroid


sea surface

geoid, WP =Wo

gP

P

γo

Q

gPo

Qo

ζ

Hn

N

Hh

earth’s surface

telluroid


geoid, WP =Wo

gP

P

γo

Q

gPo

Qo

ζ

Hn

N

Hh

gP

P

γo

Q

gPo

Qo

gP

P

γo

Q

gPo

Qo

gP

P

γo

Q

gPo

gP

P

γo

Q

gP

P

γo

Q

gP

P

γo

Q

gP

P

γo

Q

gPo

Qo

ζ

Hn

ζζ

Hn

N

H

NN

Hhh

earth’s surface

telluroid


sea surface


geop of P, WP

spherrop of P, UP

spherop of Po UPo



ζ = height anomaly


Hn = normal height


h = N + H = ζ + Hn


ζ = height anomaly


Hn = normal height



ζ = height anomaly


Hn = normal height


h = N + H = ζ + Hn

γQo

- - 62 -

Because the earth’s surface is not level surface, therefore the telluroid obviously is not a level surface. Height of the earth’ surface above the geoid is orthometric height, which is denoted by H, see also fig. (2.4), and the height of the telluroid above the level ellipsoid is normal height, denoted by Hn. If the geoid is separated from ellipsoid by the geoid undulation, which usually denoted by N, the telluroid is separated from the earth’s surface by height anomaly, denoted by ζ . Because the height of the earth’s surface above the ellipsoid is ellipsoidal height, denoted by h, then we have

h = H + N and h = Hn + ζ

then we have N = ζ + Hn – H (5.48) It is shown in Heiskanen & Moritz [1967, p. 327], that

Hn – H = Hg

o

B

γ∆ (5.49)

Therefore geoid undulation can be obtained from height anomaly

N = ζ + Hg

o

B

γ∆ (5.50)

Where ∆gB is Bouguer gravity anomaly. Height anomaly is determined by using free air gravity anomalies data at the earth

surface. Gravity anomaly of any point P at the earth’s surface is ∆gP = gP - γQ (5.51)

where Q is a point at the telluroid and a normal gravity equipotensial, which is also called spheropotential surface (spherop) of Q. This spheropotensial (spherop) has potential value UQ that is equal to potential value of P, which is also located at an actual gravity equipotential, which is also called geopotential surface (geop) of P. Gravity value at Q is obtained from [Heiskanen & Moritz, 1967, p. 123],

2n

nQ

2Q a

H3H) sin f2mf1 (a2 - 1 [

+φ−++γ=γ (5.52)

γ is normal gravity value at Qo which has latitude φ, and computed from normal gravity formula (5.47). Normal height Hn is obtained from spirit leveling which will be discussed later. Other values a and f are the semi-major axis and flattening of reference ellipsoid respectively; the value of m is [Heiskanen & Moritz, 1967, p. 69]

GM

)f1(a m32 −ω

= (5.53)

- - 63 -

GRS value for m is m = 0.003 449 786 003 08 m s-2 [Moritz, 2000]. Let ζo is approximate value of height anomaly, which is computed from

∫∫σ

σψλ′φ′∆πγ

=ζ d )( S ),( g4

R Q

o (5.54)

Gravity anomaly in (5.54) is the gravity anomalies data at the earth surface, which is not a level surface. We will obtain height anomaly after we use gravity anomalies value at the geopotential surface or level surface belongs to the computation point. Therefore the any gravity anomaly at the earth surface must be added by the correction G1 [Heiskanen & Moritz, 1967, pp. 300 - 307],

∫∫σ

σ

ς

γ+∆

−π

= d R2

3gHH 2R G o

o3o

P2

1 λ (5.55)

2

sin R2oψ

=λ

HP = height of computation P above sea level H = height of dσ above sea level If we put G1 into Stokes integral we will obtain ζ1 that is correction should be added to the approximate height anomaly ζο ,

∫∫σ

σψλ′φ′πγ

=ζ d )( S ),( G4

R 1Q

1 (5.56)

so that ζ = ζο + ζ1 (5.57)

Height Systems and Spirit Leveling The eqn. (5.2) shows that from gradient potential as shown at (5.2), we will obtain

potential difference between Po at the geoid and P at the earth surface, see fig. (5.3)

∫−=−P

PPP

oo

dH gWW (5.58a)

Because orthometric of Po is zero and of P is H, and oPW = Wo then (5.58a) may be

written

∫−=−H

0oP dH gWW (5.58b)

- - 64 -

Figure 5.4: Spirit Leveling Measurement

In the case of spirit leveling measurement as shown in fig. (5.4), the distance between points Po and P1 are between 50 to 150 meters, so that gravity direction at these points may be assumed parallel short. Thus the height difference between these points dn1 yielded from the spirit leveling measurement may be assumed as orthometric height difference between these points, and potential difference between P1 and Po is 11PP dn g WW

o1−=−

where g 1 is average value of gravity values of Po and P1 . Let point Pk and Po at fig. (3.2) is the point P and Po at fig. (5.3), so that we have

∑−=−k

1i1oP dn gWW (5.59)

The eqn (4.59) shows that potential difference between two points can be obtained by combination of leveling and gravity measurements. Let CP is geopotential number of point P defined as CP = Wo - WP (5.60a) or, because (5.58)

CP = ∫H

0dH g (5.60b)

Measuring unit for geopotential number C is geopotential unit (gpu) , where 1 gpu is equal to 1 kgal meter. The eqns. (5.59) and (5.60) shows that potential of point at the earth surface points, which is related to the potential of the geoid, and geopotential

f1

Po

P1

Pk

b1

fk

f2b2

bk

Pk-1

P3

dn1

dn2

dnk

f1

Po

P1

Pk

b1

fk

f2b2

bk

Pk-1

P3

dn1

dn2

dnk

f1

Po

P1

Pk

b1

fk

f2b2

bk

Pk-1

P3

dn1

dn2

dnk

Po

P1

Pk

b1

fk

f2b2

bk

Pk-1

P3

Po

P1

Pk

b1

fk

f2b2

bk

b1

fk

f2b2

bk

Pk-1

P3

dn1

dn2

dnk

- - 65 -

number of that point can be obtained by combination of leveling and gravity measurements. The eqn (5.60b) can be modified as CP =

PHg HP (5.61) because the average gravity value between Po and P, see fig. (5.3)

PHg = ∫

H

0dH g

H1 (5.62)

or

PHg = g + 0.0424 H, where g in gals and H in km [Heiskanen & Moritz, 1967, p. 167]. Because UQ = WP and UQo = Uo = Wo then equivalent with (5.61) and (5.63) we have CP = nHγ Hn (5.63) and the average normal value between Qo and Q is

nHγ = ∫ γnH

0

nn dH

H1 (5.64)

or nHγ is computed from the formula [Heiskanen & Moritz, 1967, p. 170],

nHγ = ( )22nn

2

aH

aH sin 2f - m f 1 - 1

+φ++γφ

In the fig. (5.3) we have two height systems, they are orthometric height and normal

height. There is another height system that is dynamic height. This system is defined as

r

dyn CHγ

= (5.65)

γr is normal gravity at reference parallel; internationally the parallel at the latitude of 45o is chosen as reference parallel. Two points, which are located at the same level surface, have the same dynamic height, so that dynamic height system has no geometrical meaning. Height dynamic difference between two arbitrary points P and Q is

) CC (1HHH PQr

dynP

dynQ

dynPQ −

γ=−=∆

According to the eqns. (5.59) and (5.60a) the height dynamic different become

dn g 1 HQ

Pr

dynPQ ∑

γ=∆ (5.66)

The eqn. (5.66) shows determination of dynamic height difference between two points

- - 66 -

from combination of spirit leveling and gravity measurements, where g is the average gravity value between two leveling stations, which the distance between the two stations (Pi and P2 ) is not more than 150 meters, see fig. (5.4).

Figure (5.5) shows two different lines of spirit leveling. Because the eqn. (5.66)

∑∑ =Q

P221

Q

P1 dn g dn g (5.67)

therefore

∑∑ ≠Q

P2

Q

P1 dn dn (5.68)

This is the evidence that combination of spirit leveling and gravity measurements between two points which is conducted through two or more different lines must give same results, but not for spirit leveling measurements without gravity measurements. Therefore leveling measurements should be added by dynamic height difference correction DcPQ. The eqn (5.66) may be written as

PQQ

P

dynPQ Dcdn H +=∆ ∑

so that difference dynamic height correction between P and Q that should be added to leveling measurements,

DcPQ = ∑ γ−γ

Q

Pr

rdn )g(1 (5.69)

Dynamic height of arbitrary point P may be obtained from orthometric height HP by

giving dynamic correction Dc to the orthometric height, PP

dynP Dc H H +=

P

Q

1

2

Figure 5.5: Two different linesof spirit leveling

P

Q

1

2P

Q

1

2

Figure 5.5: Two different linesof spirit leveling

- - 67 -

From (5.61) we have

HP = PH

P

gC

so that, because (5.61) and (5.65)

DcP = Pr

P HC−

γ

then

DcP = PrHr

H)g(1P

γ−γ

(5.70)

Orthometric height difference between P and Q is

P

dinPQ

dynQPQPQ DcHDcHHHH +−−=−=∆

or ∆HPQ QP

dynPQ DcDcH −+∆= (5.71)

To obtain orthometric height, leveling measurements should be given orthometric height difference correction, thus

PQQ

PPQ OcdnH +=∆ ∑

From (6.69), (5.70) and (5.71)

OcPQ = PQQ

PDcdn +∑ + DcP - DcQ

OcPQ = ∑ γ−γ

Q

Pr

rdn )g(1 + PrH

rH)g(1

Pγ−

γ QrH

rH)g(1

Qγ−

γ (5.72)

Equivalent with the orthometric height difference correction, to obtain normal height difference, we should add normal height difference correction to spirit leveling measurements,

NcPQ = ∑ γ−γ

Q

Pr

rdn )g(1 + n

PrHr

H)(1nP

γ−γγ

nQrH

rH) (1

nQ

γ−γγ

(5.73)

- - 68 -

6. GEODESY APPLICATIONS FOR GEOPHYSICS

There are many countries are located at volcanic and seismic active area, such as

Japan and Indonesia, see fig. (6.1). Natural hazards such as earthquakes, volcanic eruptions and tsunamis are occurred quite often in these countries.

GPS technology to monitor crustal deformation, which is caused by the dynamic activities of the earth interior, GSI, the Geographical Survey Institute of Japan established the nationwide continuous GPS network, called GPS Earth Observation Network (GEONET) in 1993. According to Takemoto [2003], more than one thousands observation sites were established in 2002, and GEONET data became available to researches on line at the GSI web. Hori, et al. [2000] used GEONET data to predict stress field in Japan using GEONET data.

In 2000 there were six active volcanoes eruptions in Japan. One of the eruptions was known 2000 Miyakejima eruption. A lot of buildings were destroyed by the eruption. Many volcanologists and seismologists in Japan investigate the occurrence of the eruptions using GEONET data. Furuya, et al. [2003] studied mass budget of the magma flow at Izu-islands regarding the 2000 volcano-seismic activity in this region, see fig. 6.2. They analyzed gravity and elevation changes which were obtained from leveling measurements and microgravity measurements. By combining the results with ground displacements were obtained from GEONET, magma behavior during 2000 Miyake-Kozu volcano seismic activity was detected. Irwan [2004] detected magma behavior of the Miyakejima volcano on June26 – 27, 2000 by analyzing ground deformation obtained from kinematic GPS. Immediately after the large eruption of the 2000 Miyakejima, Ukawa, et al. [2000] studied subsurface magma movement by analyzing the National Research Institute for Earth Science and Disaster Prevention (NIED) GPS Observation Network constructed at Miyakejima 1995 – 1999 combined with 2-component tiltmeter and 3-component short-period seismometer installed at each GPS station. This investigation succeed to detect magma movement of Miyakejima, which has good agreement with the later investigation done by Furuya, et al. [2003] and Irwan, et al. [2003].

Figure 6.1: Seismicity Map of the World

http://wwwrses.anu.edu.au/seismology/quakes/seimogram.html

- - 69 -

Ground deformation was detected accompanying an earthquake swarm at Hachijo

Island (100 km at the south of Miyakejima) during the period of 13 – 16 August 2002 by analyzing continuous GPS measurements at four sites on the island combined with GEONET data. Eastward ground displacements of 2 – 6 cm were succeeded to detect [Kimata, et al. 2004]. Analysis of 1-Hz GPS data observed at 14 stations of GEONET, associated with the 2003 MJMA 8.1 Tokachi-oki earthquake was conducted by Irwan, et al. [2004]. The analysis yielded that GPS data clearly captured rapid co-seismic ground displacements. The data shown that co-seismic displacements started 15 seconds after the origin time at the station 70 km away from the epicenter, and 40 seconds at the station 240 km away.

The Indonesian Archipelago is located in three major plates systems, Eurasia plate, Australia plate and Pacific-Philippine Sea plates. The region, which is closed to a plate boundary and triple junction of the plates, is a tectonic active region. Subduction of the Australia plate into Eurasian plate, as well as the motion of the Pacific-Philippine Sea plate make this region is surrounded by some ocean trenches such as Sunda trench, Banda trench, New Guinea trench, North Sulawesi trench, Palawan trench and Philippine trench. There are many active fault systems in this region such as Sumatra fault, Palu-Mentano fault, and Sorong fault. Close to plate boundaries there is a volcanic arc along Bukit Barisan in Sumatra island, continue to Sunda Strait (where Krakatau volcano is located), to the islands of Java, Bali, and to Sunda Lesser islands, cross to Banda arc, and continue to the North Sulawesi, Halmahera and Philippine. These plate tectonic, volcanic and trenches systems make the Indonesian Archipelago is a geological and geophysical complex system. Related to this complex system, this region is located in the highest gravity anomaly variations in the world. These are presented by the range of the geoid

Figure 6.2: Miyakejima and Kozushima in Izu Islands

Behavour of magma can be detected between Miyakejima and Kozuma which is located at shaded area

- - 70 -

undulations in this region around – 45 m at Indian Ocean to + 80 m in Papua. Therefore geodynamics of Indonesia is an interesting subject to be investigated by many earth scientists.

In 1989, a Memorandum of Understanding (MOU) between the National Coordination Agency for Surveys and Mapping (abbr.: BAKOSURTANAL in the Indonesian language) and US National Science Foundation (US-NSF) was signed; US side were represented by Scripps Institution of Oceanography (SIO), the California University, San Diego, and the Rensselaer Polytechnic Institute (RPI), New York. In 1992 the MOU was renewed in 1992 where the Institute of Technology in Bandung (ITB) took a part to sign MOU for the Indonesia side. The main objective of the MOU was to investigate crustal movements along the Sumatra fault by using GPS technique. In the first GPS campaign, a GPS team from Japan led by the Earthquake Research Institute, the University of Tokyo was also participated. Then the campaign was expanded to the East part of Indonesia. A report of this study may be found in Bock et al. [2003].

Figure 6.3: A Preliminary Result of Sulawesi GPS Campaign after Sarsito et al. [2002]

Tectonic motions at Sulawesi island which is closed to triple junction of Eurasia,

Australia and Pacific-Philippine Sea plates are being studied since 1997, under cooperation of DEOS (Delft Institute for Earth Oriented Space Research) and the Department of Geodetic Engineering, the Institute of Technology, Bandung. A preliminary result was reported by Sarsito et al, 2002].

- - 71 -

In 1991, under the program of the International Decade for Natural Disaster Reduction (IDNDR), the Disaster Prevention Research Institute (DPRI) of the Kyoto University established research cooperation with some Indonesian Institutions. Implementing Agreements between DPRI and each institution carried implementation of the cooperation. One of the Implementing Agreement was between DPRI and the Research and Development Center for Geotechnology (RDCG), of the Indonesian Institute of Science (LIPI) on “Crustal movements monitoring by GPS and gravity change measurements around volcanoes and active fault system” by the period of 1991 – 1993. In 1994, DPRI and the Directorate General of Geology and Mineral Resources (DGGMR) signed an Implementing Agreement for research on “Volcanoes deformation monitoring by GPS measurements” by the period of 1994 – 1998. Technically, the implementation of the above mentioned researches were carried out by Research Center of Earthquake Prediction (RCEP) of DPRI and the Department of Geodetic Engineering, the Institute of Technology in Bandung [Kahar, 1998].

Figure 6.4: Major Volcanoes of Indonesia Beside GPS survey for crustal motions study, some others natural hazard mitigation

in Indonesia for volcano deformation, land subsidence, and land slide studies using GPS technique are also conducted. Monitoring volcano deformation have been conducted at several volcanoes in Indonesia. Department of Geodetic Engineering, Institute of Technology Bandung in cooperation with Directorate of Vulcanology and Geological Hazard Mitigation has conducted GPS surveys in Guntur, Papandayan, Galunggung, Tangkubanperahu, Kelut, Bromo, Ijen, and Batur volcanoes since 1997. Location of these volcanoes can be seen in the following fig. (6.4). The studies have provided not only the deformation characteristics of the corresponding volcanoes, but also the understanding on strengths and weaknesses of GPS survey technique for volcano deformation study and monitoring [Abidin, 2003]. Combination of GPS and INSAR techniques for studying the deformation of Indonesian volcanoes was also conducted under cooperation of the

- - 72 -

Department of Geodetic Engineering of ITB and Research Center for Seismology, Volcanology and Disaster Mitigation, Nagoya University [Abidin, 2001].

Figure 6.5: PWV in Bandung City on September 02, 2001

It has been discussed briefly that GPS data are influenced by atmospheric conditions, so that GPS observations equations should consider ionosphere and troposphere effects as shown in (3.13) and (3.18). Inversion of this problem is that GPS data at some fixed locations can be used to determine integrated Precipitable Water Vapor (PWV) of the region. This solution is known as GPS Meteorology, that is the application of GPS data to the monitoring and analysis of atmospheric conditions. Figure (6.5) shows PWV in Bandung City, West Java, Indonesia on September 02, 2001 [Kuntjoro, et al., 2002].

Gravity field information which is obtained from geopotential coefficients depend on degree of harmonics n. Figure (6.6) shows the gravity anomaly information in Japan region which is obtained from n = 30 and n = 90 from GPM98 potential coefficients. Information which is given by degree n = 90 more detail than degree n = 30. By analyzing the information combined with tomography, land and sea bottom topography data might be used for study the interior of the earth surface at certain depth.

-11.3 -9.9 -8.6 -7.2 -5.8 -4.5 -3.1 -1.8 -1.1 0.3 1.7 3.0 4.4 5.7 7.0 8.4 9.8

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PETA PWV KOTA BANDUNG13 Juni 2001

Relatif Terhadap Titik Residen (RSDN)

(04) (12.00 - 13.00 WIB)

(03) (11.00 - 12.00 WIB)(02) (10.00 - 11.00 WIB)

(07) (15.00 - 16.00 WIB) (08) (16.00 - 17.00 WIB)

(05) (13.00 - 14.00 WIB) (06) (14.00 - 15.00 WIB)

(01) (09.00 - 10.00 WIB)

X min = 782726.319 ; X max = 800353.876Y min = 9230406.770 ; Y max = 9237836.553

Maksimum = 9.75 mm Minimum = -11.25 mm

SKALA KEABUAN PWV Dalam Satuan Milimeter (mm)

SISTEM KOORDINAT UTM ZONA 48 S

INFORMASI KONTUR

Interval = 0.45 mm

Ujung BerungCicaheumPasteur

Tegal Lega Buah Batu

Antapani

Cibiru



Antapani

Cibiru



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Antapani

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- - 73 -

Figure 6.6: Gravity anomaly in Japan Region derived

from GPM98 [Prijatna, 2004]

nmax = 30 nmax = 30

nmax = 90 nmax = 90

- - 74 -

Above briefly description tells the application of geodesy to geophysics study. But the development of geodesy also requires the knowledge from other disciplines. We have started from Phythagoras 2500 yeas ago that had brilliant opinion that the earth must be a sphere. Aristotle supported this opinion by a reason that God created the earth as an ideal and completed form that is a sphere. Eratosthenes taught us that by a simple technology at his time he succeed measure the size of the earth. By idea of Newton 350 years ago, we learned that the surface of the spherical earth is a gravitational equipotential surface. Then we learned from the earth is a rotational body, therefore we think that the ideal form of the earth is ellipsoid of revolution. The angular velocity of the earth rotation influences the shape of the earth. But the earth is not a homogenous body, therefore the figure of the earth must be deviated from the ellipsoid of revolution that is the geoid, which is known as a gravity equipotential surface. To study the geoid, the knowledge about the interior of the earth is required. By interpreting information given by geodesists, then geophysicists studying the earth interior. We know that this investigation is also required advanced technology. Therefore we may have conclusion that to study about the earth, which was created by God, require thousands of years by collaboration of many disciplines.

- - 75 -

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Abidin, H.Z., M.A. Kusuma, O.K. Suganda, F. Kimata, A. Tanaka, S. Kobayashi, N. Fujii (2001a). Studying the deformation of Indonesian volcanoes by GPS and INSAR techniques. Proceedings of 1st ALOS PI Workshop, Earth Observation Research Center (EORC) NASDA, Tokyo, 28-30 March, pp. 157 - 159.

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Heitz, S. [1988]: Coordinates in Geodesy, Springer, Berlin Heidelberg, New York. Irwan, M., F. Kimata, N. Fujii, S. Nakao, H. Watanabe, S. Sakai, M. Ukawa, E. Fujita,

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Jones, G.C. [2002]: New Solutions for the Geodetic Coordinate Transformation, Jjournal of Geodesy, Vol. 76/08, pp. 437 – 446.

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Kimata, F., M. Irwan, and K. Fukano [2004]: Ground Deformation at Hachijo Island, Japan on 13 – 22 August 2002 Observed by GPS Measurements and Estimated Dike Intrusion Model, Bulletin of the Vulcanological Society of Japan, Vol. 49, No 1, pp. 13 – 22 .

Kuntjoro, W. , Z.L. Dupe, D.S. Permana, B. Setyadji, D. Darmawan, H. Andreas, H.Z. Abidin [2002]: Pemantauan Precipitable Water Vapor Relatif pada Permukaan Troposfir Kota Bandung dengan Teknik Inversion GPS. Disampaikan pada Seminar Sains Antariksa dan Sain Atmosfir dalam Pengembangan Teknologi Kedirgantaraan, 4 - 5 November, Bandung.

Mikhail, E. M. [1976]: Observations and Least Squares. IEP-A dun-Donnelley Publisher.

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Moritz, H. [1973]: Least Squares Collocation, Deutsche Geodatische Kommision, Nr. 75, Muenchen.

Moritz, H.[1988]: Geodetic Reference System 1980, Geodesists Handbook, Bulletin Geodesique, Vol. 62, No. 3.

Mueller, I. I. [1969]: Spherical and Practical Astronomy as Applied to Geodesy. Ungar. Prijatna, K. [2004]: Private Communication Sarsito, D.A., W. J. F. Simons, B.A.C. Ambrosius, H.Z. Abidin, J. Kahar [2002]:

Tectonic Motions of Sulawesi Region: Preliminary Results of the Sulawesi Campaign 2000 GPS Campaign . Presented at the International Workshop on Tsunami Risk and Its Reduction in the Asia Pacific Region, Bandung, Indonesia, 18 – 19 May 2002.

Takemoto, S. [2003]: Activity Report of the National Committee for Geodesy, National Report on Geodesy and Geophysics in Japan from 1999 to 2003 for XXIII IUGG General Assembly, 30 June – 11 July 2003, Sapporo, Japan.

Torge, W. [1989]: Gravimetry. Walter de Gruyter, Berlin and New York 1989. Ukawa, M., E. Fujita, E. Yamamoto, Y. Okada, M. Kikuchi [2000]: The 2000

Miyakejima Eruption: Crustal Deformation and Earthquakes Observed by the NIED Miyakejima Observation Network, Earth Planets Space, 52, XIX – XXVI.

Vermeille. H. [2002]: Direct Traansformation from Geocentric Coordinates to Geodetic Coordinates, Journal of Geodesy, Vol. 76/08, pp. 451 – 454.

Vanicek,P. & E. Krakiwsky [1986]: Geodesy; The Concepts, North-Holland. Amsterdam and New York.

Wells, D.E., N. Beck, D. Delikaraoglou, A. Kleusberg, E.J. Krakiwsky, G. Lachapelle, R.B. Langley, M. Nakiboglu, K.P Schwarz, J.M Tranquilla, P. Vanicek [1987]: Guide to GPS Positioning. Canadian GPS Associates, Fredericton, N.B, Canada.

Websites: http://www.zianet.com/globalcogo/ge002.pdf http://www.rand.org/publications/MR/MR614/MR614.appb.pdf http://igscb.jpl.nasa http://wwwrses.anu.edu.au/seismology/quakes/seismogram.html

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