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1-1- INTRODUCTIONINTRODUCTION

These notes discuss the precise definitions of, These notes discuss the precise definitions of, and transformations between, the coordinate and transformations between, the coordinate systems to which coordinates of stations on or systems to which coordinates of stations on or above the surface of the earth are referred. To above the surface of the earth are referred. To define a coordinate system we must specify: define a coordinate system we must specify:

• a) the location of the origin, a) the location of the origin,

• b) the orientation of the three axes, b) the orientation of the three axes,

• c) the parameters (Cartesian, curvilinear) which c) the parameters (Cartesian, curvilinear) which define the position of a point referred to the define the position of a point referred to the coordinate system.coordinate system.

• The earth has two The earth has two different periodic different periodic motions in space. It motions in space. It rotates about its axis, rotates about its axis, and it revolves about and it revolves about the sun (see Figure 1-1). the sun (see Figure 1-1). There is also one There is also one natural satellite (the natural satellite (the moon) and many moon) and many artificial satellites, which artificial satellites, which have a third periodic have a third periodic motion in space: orbital motion in space: orbital motion about the earth. motion about the earth. These periodic motions These periodic motions are fundamental to the are fundamental to the definition of systems of definition of systems of coordinates and coordinates and systems of timesystems of time. .

• Terrestrial coordinate systems are earth fixed and Terrestrial coordinate systems are earth fixed and rotate with the earth. They are used to define the rotate with the earth. They are used to define the coordinates of points on the surface of the earth. coordinates of points on the surface of the earth. There are two kinds of terrestrial systems called There are two kinds of terrestrial systems called geocentric systems and topocentric systems (see geocentric systems and topocentric systems (see Figure 1-2). Figure 1-2).

• Celestial coordinate systems do not revolve but may Celestial coordinate systems do not revolve but may rotate with the earth. They are used to define the rotate with the earth. They are used to define the coordinates of celestial bodies such as stars. There coordinates of celestial bodies such as stars. There are four different celestial systems, called the are four different celestial systems, called the ecliptic, right ascension, hour angle, and horizon ecliptic, right ascension, hour angle, and horizon systems. systems.

• The orbital system does not rotate with the earth, but The orbital system does not rotate with the earth, but revolves with it. It is used to define the coordinates revolves with it. It is used to define the coordinates of satellites orbiting around the earth. of satellites orbiting around the earth.

1.1 POLES, PLANES AND AXES 1.1 POLES, PLANES AND AXES • The orientation of axes of coordinate systems can be The orientation of axes of coordinate systems can be

described in terms of primary and secondary poles, described in terms of primary and secondary poles, primary and secondary planes, and primary, primary and secondary planes, and primary, secondary and tertiary axes. secondary and tertiary axes.

• The primary poleThe primary pole is the axis of symmetry of the is the axis of symmetry of the coordinate system, for example the rotation axis of coordinate system, for example the rotation axis of the earth. the earth. The primary planeThe primary plane is the plane is the plane perpendicular to the primary pole, for example the perpendicular to the primary pole, for example the earth's equatorial plane. earth's equatorial plane.

• The secondary planeThe secondary plane is perpendicular to the primary is perpendicular to the primary plane and contains the primary pole. It sometimes plane and contains the primary pole. It sometimes must be chosen arbitrarily, for example the Greenwich must be chosen arbitrarily, for example the Greenwich meridian plane, and sometimes arises naturally, for meridian plane, and sometimes arises naturally, for example the equinoctial plane. example the equinoctial plane. The secondary poleThe secondary pole is the intersection of the primary and secondary is the intersection of the primary and secondary planes. planes. The primary axisThe primary axis is the secondary pole. is the secondary pole.

• The tertiary axisThe tertiary axis is the primary pole. is the primary pole. The The secondary axissecondary axis is perpendicular to the other two is perpendicular to the other two axes, chosen in the direction, which makes the axes, chosen in the direction, which makes the coordinate system either right-handed or left-coordinate system either right-handed or left-handed as specified.handed as specified.

• We will use either the primary plane or the We will use either the primary plane or the primary pole, and the primary axis to specify the primary pole, and the primary axis to specify the orientation of each of the coordinate systems orientation of each of the coordinate systems named above. named above.

Primary plane

Earth’s equatorial Plane

The primary pole

The rotation axis of the earthThe secondary

plane

The Greenwich meridian plane

The secondary pole

The primary axis

X, X1 or 1. axis

The tertiary axis

Z, X3 or 3. axis

The secondary axis

(for right-handed system)Y, X2 or 2. axis

The secondary axis

(for-left-handed system)Y, X2 or 2. axis

Figure 1.3- Poles, planes and axes

1.2 SUMMARY OF REFLECTION 1.2 SUMMARY OF REFLECTION AND AND ROTATIONROTATION MATRICES MATRICES

1.2.1 Orthogonal Transformations1.2.1 Orthogonal TransformationsThe matrix equation The matrix equation

• Y = A X Y = A X

where A is a matrix and X and Y are columnwhere A is a matrix and X and Y are column vectors, can be regarded as a linear vectors, can be regarded as a linear transformation, in which case the matrix A is transformation, in which case the matrix A is called the transformation matrix. If the two called the transformation matrix. If the two vectors X and Y have the same length, then both vectors X and Y have the same length, then both the transformation and the matrix are said to be the transformation and the matrix are said to be orthogonal. Orthogonal matrices have the property orthogonal. Orthogonal matrices have the property that the product of the matrix and its transpose (or that the product of the matrix and its transpose (or vice versa) is the identity matrix, that is vice versa) is the identity matrix, that is

• AATT A = A A A = A ATT = I. = I.

• From this property it follows that the determinant From this property it follows that the determinant of an orthogonal matrix is either +1 or -l. There of an orthogonal matrix is either +1 or -l. There are two kinds of orthogonal transformations called are two kinds of orthogonal transformations called reflections and rotations. The determinant of reflections and rotations. The determinant of reflection matrices is -1, and the determinant of reflection matrices is -1, and the determinant of rotation matrices is +1. rotation matrices is +1.

• There are two interpretations of the linear There are two interpretations of the linear transformation above. The first is that the transformation above. The first is that the transformation describes the relationship between transformation describes the relationship between two coordinate systems, in which case X and Y are two coordinate systems, in which case X and Y are the same vector, but their elements refer to the the same vector, but their elements refer to the two different systems. The second is that the two different systems. The second is that the transformation describes the relationship between transformation describes the relationship between different vectors X and Y in the same coordinate different vectors X and Y in the same coordinate system. In these notes, we are interested only in system. In these notes, we are interested only in the first interpretation. the first interpretation.

1.2.2 Right and Left Handed Cartesian 1.2.2 Right and Left Handed Cartesian Coordinate Systems Coordinate Systems

• A three dimensional Cartesian coordinate system A three dimensional Cartesian coordinate system can be orthogonally transformed in only six can be orthogonally transformed in only six different ways. It can be rotated about each of its different ways. It can be rotated about each of its axes. Each of its axes can be reflected. In such a axes. Each of its axes can be reflected. In such a coordinate system, the vectors X and Y will have coordinate system, the vectors X and Y will have only three elements. Let us define the axis to only three elements. Let us define the axis to which the first, second, and third elements of X which the first, second, and third elements of X and Y are referred as the 1-axis, 2-axis, and 3-axis and Y are referred as the 1-axis, 2-axis, and 3-axis respectively (we could equally well label them the respectively (we could equally well label them the xxll, x, x22, x, x33 axes or x, y, z axes). axes or x, y, z axes).

• These three axes may define either a right-handed or a These three axes may define either a right-handed or a left-handed coordinate system. Right handed systems left-handed coordinate system. Right handed systems follow the right hand rule: if the fingers of the right hand follow the right hand rule: if the fingers of the right hand are curled around any axis so that the thumb points in are curled around any axis so that the thumb points in the positive direction, then the fingers will point from a the positive direction, then the fingers will point from a second axis to the third axis, numbered in cyclic fashion. second axis to the third axis, numbered in cyclic fashion. Grasping the 1-axis, the fingers point from the 2-axis to Grasping the 1-axis, the fingers point from the 2-axis to the 3-axis. Grasping the 2-axis, the fingers point from the 3-axis. Grasping the 2-axis, the fingers point from the 3-axis to the 1-axis. Grasping the 3-axis, the fingers the 3-axis to the 1-axis. Grasping the 3-axis, the fingers point from the 1-axis to the 2-axis. Left-handed point from the 1-axis to the 2-axis. Left-handed coordinate systems follow the left hand rule, which coordinate systems follow the left hand rule, which differs from the above only in that the left hand is used. differs from the above only in that the left hand is used.

Y Y

Z

O

X

Z

O

X

Figure 1.4a- Rigth-handed system Figure 1.4b- Left-handed system

• 1.2.3 Reflections 1.2.3 Reflections • If we denote a reflection of the kIf we denote a reflection of the kthth axis by P axis by Pkk , then , then

the following expressions define the three reflection the following expressions define the three reflection matrices: matrices:

;; ; ; (2.3)(2.3)

• Note that reflection matrices commute (e.g. PNote that reflection matrices commute (e.g. P22PP33 = = PP33PP22), so that it makes no difference in what order a ), so that it makes no difference in what order a sequence of reflections are performed. Note also sequence of reflections are performed. Note also that an odd number of reflections changes the that an odd number of reflections changes the handedness of the coordinate system.handedness of the coordinate system.

P1

1 0 0

0 1 0

0 0 1

P2

1 0 0

0 1 0

0 0 1

100

010

001

3P

• Rotations Rotations • In a Cartesian coordinate system with the axes x, In a Cartesian coordinate system with the axes x,

y, z the position of a point P is determined by its y, z the position of a point P is determined by its position vector position vector

(2.5)(2.5)

• xxPP, y, yPP, z, zPP are real numbers (Fig. 1.5). are real numbers (Fig. 1.5).

X

x

y

zP

P

P

P

Figure 1.5- Cartesian coordinate system

• If we denote a rotation of angle If we denote a rotation of angle about the 1 about the 1stst axis by Raxis by R11((), a rotation of angle ), a rotation of angle about the 2 about the 2ndnd axis by Raxis by R22((), and a rotation of angle ), and a rotation of angle about the about the 33rdrd axis by R axis by R33((), then the following expressions ), then the following expressions define the three rotation matrices: define the three rotation matrices:

R1

1 0 0

0

0

( ) cos sin

sin cos

cos0sin

010

sin0cos

)(2R

R3

0

0

0 0 1

( )

cos sin

sin cos

• Note that rotation matrices do not commute. The Note that rotation matrices do not commute. The product of several rotations is performed from product of several rotations is performed from right to left, for example in right to left, for example in

RR11(() R) R22(() R) R33((), ),

the rotations are performed about the 3-axis of the rotations are performed about the 3-axis of the original system, the 2-axis of the transformed the original system, the 2-axis of the transformed system, and the 1-axis of the doubly transformed system, and the 1-axis of the doubly transformed system, to yield the final triply transformed system, to yield the final triply transformed system. system.

• The transformation to a second Cartesian The transformation to a second Cartesian coordinate system with identical origin and the coordinate system with identical origin and the axes x’, y’, z’, which is generated from the first axes x’, y’, z’, which is generated from the first one by a rotation around the z-axis by the angle one by a rotation around the z-axis by the angle , , can be realized through the matrix operation can be realized through the matrix operation

XXPP = R = R33(() X) XPP

(2.7)(2.7)

• The representation is valid for a right-handed The representation is valid for a right-handed coordinate system. When viewed towards the coordinate system. When viewed towards the origin, a counter-clockwise rotation is positive. origin, a counter-clockwise rotation is positive. Any coordinate transformation can be realized Any coordinate transformation can be realized through a combination of rotations. The complete through a combination of rotations. The complete transformation is transformation is

XXPP = R = R11(() R) R22(() R) R33(() X) XPP

(2.8)(2.8)

• Finally, the matrix for a general rotation by the Finally, the matrix for a general rotation by the angles angles , , , , is is

R = RR = R11(() R) R22(() R) R33(()) (2.9)(2.9)

• The relation between the position vectors in two The relation between the position vectors in two arbitrarily rotated coordinate systems is thenarbitrarily rotated coordinate systems is then

• XXPP = R X = R XPP ; X ; XPP = R = RTT X XPP (2.10)(2.10)

R

cos cos cos sin sin

sin sin cos cos sin sin sin sin cos cos sin cos

cos sin cos sin sin cos sin sin sin cos cos cos

R( , , )

1

1

1

(2.11)

In geodesy the rotation angles are often very small, thus allowing the use of the linearized form for R. With cos 1 and sin (in radians), neglecting higher order terms, it follows that

If the rotation angles are all so small that their cosines can be assumed to be unity, then the rotation matrices become commutative. This is the case for differential rotations, for example.

• The above expressions define positive rotations, The above expressions define positive rotations, which are right-hand rotations for right-handed which are right-hand rotations for right-handed coordinate systems and left-hand rotations for coordinate systems and left-hand rotations for left-handed coordinate systems. A right-hand left-handed coordinate systems. A right-hand rotation is related to the right hand rule given rotation is related to the right hand rule given above: if the fingers of the right hand are curled above: if the fingers of the right hand are curled around the rotation axis so that the thumb points around the rotation axis so that the thumb points in the positive direction, then the fingers curl in in the positive direction, then the fingers curl in the direction of a right hand rotation. A similar the direction of a right hand rotation. A similar statement for left hand rotations is obvious. statement for left hand rotations is obvious.

1.2.5 Inverse Transformations 1.2.5 Inverse Transformations • The inverse of a transformation A (denoted AThe inverse of a transformation A (denoted A-1-1) is ) is

the transformation which returns conditions to their the transformation which returns conditions to their original state, that is original state, that is

AA-1-1 A = A A A = A A-1-1 = I. = I. • Reflections are self-inverse, that is Reflections are self-inverse, that is

PPkk P Pkk = I = I

• Common sense tells us that the inverse of a positive Common sense tells us that the inverse of a positive rotation is a negative rotation, that is rotation is a negative rotation, that is

and this conclusion is verified by taking the and this conclusion is verified by taking the orthogonal property orthogonal property

AATT A = I A = I from which it is evident that for orthogonal matricesfrom which it is evident that for orthogonal matrices

AA-1-1 = A = ATT

kk PP 1

)()(1 kk PP

and for each of the above expressions for rotation and for each of the above expressions for rotation matrices it can be shown thatmatrices it can be shown that

• Applying the rule for the inverse of productsApplying the rule for the inverse of products[A B][A B]-1-1 = B = B-1-1 A A-1-1

we have we have

• A product transformation consisting of one rotation A product transformation consisting of one rotation and one reflection commutes only if the rotation and one reflection commutes only if the rotation and reflection refer to the same axis, that is and reflection refer to the same axis, that is

PPjj R Rkk = R = Rkk P Pjj if j = k if j = k otherwise otherwise

PPjj R Rkk = R = Rkk-1-1 P Pjj if j if j k. k.

)()( kTk RR

)()()()()()( 1 jk

Tj

Tkkj RRRRRR

2. 2. TERRESTRIAL COORDINATE SYSTEMSTERRESTRIAL COORDINATE SYSTEMS

• In this chapter we will discuss terrestrial In this chapter we will discuss terrestrial geocentric and terrestrial topocentric coordinate geocentric and terrestrial topocentric coordinate systems. systems.

• We first discuss terrestrial geocentric systems We first discuss terrestrial geocentric systems using only Cartesian coordinates, and considering using only Cartesian coordinates, and considering in detail what is meant by "in detail what is meant by "the earth's axis of the earth's axis of rotationrotation" and "" and "the Greenwich meridianthe Greenwich meridian". Then ". Then the relationship between Cartesian and curvilinear the relationship between Cartesian and curvilinear coordinates is described. Geodetic datums are coordinates is described. Geodetic datums are discussed. Finally terrestrial topocentric systems discussed. Finally terrestrial topocentric systems are considered, with attention paid to what is are considered, with attention paid to what is meant by "meant by "the surface of the earththe surface of the earth". ".

2.1 TERRESTRIAL GEOCENTRIC SYSTEMS 2.1 TERRESTRIAL GEOCENTRIC SYSTEMS

In the introduction it was stated that for terrestrial In the introduction it was stated that for terrestrial geocentric systems: geocentric systems:

• a) the origin is near the centre of the earth, a) the origin is near the centre of the earth, • b) the primary pole is aligned to the earth's axis of b) the primary pole is aligned to the earth's axis of

rotation, rotation, • c) the primary axis is the intersection between the c) the primary axis is the intersection between the

primary plane and the plane containing the primary plane and the plane containing the Greenwich meridian, Greenwich meridian,

• d) the systems are right-handed. d) the systems are right-handed.

• The last specification is unambiguous. As we shall The last specification is unambiguous. As we shall see the other three are not. We will first discuss see the other three are not. We will first discuss problems in defining the earth's axis of rotation and problems in defining the earth's axis of rotation and the Greenwich meridian. Then we will discuss the Greenwich meridian. Then we will discuss translations of the origin from the centre of the translations of the origin from the centre of the earth. earth.

2.1.1 Polar Motion and Irregular 2.1.1 Polar Motion and Irregular Rotation of the Earth Rotation of the Earth

• We think of the earth as rotating about a fixed axis at We think of the earth as rotating about a fixed axis at a uniform rate. In fact, the axis is not fixed and the a uniform rate. In fact, the axis is not fixed and the rate is not uniform. rate is not uniform.

• Over 80 years ago, it was discovered that the direction Over 80 years ago, it was discovered that the direction of the earth's rotation axis moves with respect to the of the earth's rotation axis moves with respect to the earth's surface. This polar motion is principally due to earth's surface. This polar motion is principally due to the fact that the earth's axes of rotation and maximum the fact that the earth's axes of rotation and maximum inertia do not coincide. The resultant motion is inertia do not coincide. The resultant motion is irregular but more or less circular and irregular but more or less circular and counterclockwise (when viewed from North), with an counterclockwise (when viewed from North), with an amplitude of about 5 meters and a main period of 430 amplitude of about 5 meters and a main period of 430 days (called the Chandler period). days (called the Chandler period).

• The earth rotation parameters cannot be described The earth rotation parameters cannot be described through theory but must be determined through actual through theory but must be determined through actual observations by the international time and latitude observations by the international time and latitude services. For the last 80 years, or so, these services services. For the last 80 years, or so, these services were based mainly on astronomical observations.were based mainly on astronomical observations.

• Two international organizations, the International Two international organizations, the International Polar Motion Service (IPMS) and the Bureau Polar Motion Service (IPMS) and the Bureau International de 1'Heure (BIH) routinely measured International de 1'Heure (BIH) routinely measured this motion through astronomic observations; the this motion through astronomic observations; the IPMS from five stations at the same latitude, and IPMS from five stations at the same latitude, and the BIH from about 40 stations scattered the BIH from about 40 stations scattered worldwide. The results were published as the worldwide. The results were published as the coordinates of the true rotation axis with respect coordinates of the true rotation axis with respect to a reference point called the Conventional to a reference point called the Conventional International Origin (CIO) which is the average International Origin (CIO) which is the average position of the rotation axis during the years position of the rotation axis during the years 1900-1905, Figure 2-1 shows the polar motion 1900-1905, Figure 2-1 shows the polar motion during 1969 as determined by IPMS and BIH. during 1969 as determined by IPMS and BIH.

• On January 1, 1988 the International Earth On January 1, 1988 the International Earth Rotation Service (IERS) took over this task. The Rotation Service (IERS) took over this task. The principle observation techniques now used are principle observation techniques now used are Laser ranging to satellites and to the moon and Laser ranging to satellites and to the moon and Very Long Baseline Interferometry (VLBI). Very Long Baseline Interferometry (VLBI).

İstasyonİstasyon BoylamBoylam EnlemEnlem

Carloforte, İtalyaCarloforte, İtalya 88 18 18 44 44 3939 08 08 0808.941.941

Gaitehersburg, Gaitehersburg, Maryland-ABDMaryland-ABD

-77-77 11 11 57 57 3939 08 08 1313.202.202

Kitab, Özbekistan, (eski Kitab, Özbekistan, (eski USSR)USSR)

6666 52 52 51 51 3939 08 08 0101.850.850

Mizusawa, JaponyaMizusawa, Japonya 141141 07 07 51 51 3939 08 08 0303.602.602

Ukiah, California- ABDUkiah, California- ABD -123-123 12 12 3535

3939 08 08 1212.096.096

Şekil 2.6- Kutup hareketi

Tablo 2.3 Kasım-Aralık 1990 için kutup hareketi parametreleriTablo 2.3 Kasım-Aralık 1990 için kutup hareketi parametreleri

POLE COORDINATES, UT1-UTC, AND GPS-UTC FROM BIH, CIRCULAR BPOLE COORDINATES, UT1-UTC, AND GPS-UTC FROM BIH, CIRCULAR B

------------------------------------------------------------------------------------------------------------------------------------------------

MJDMJD X-POLE Y-POLE UT1-UTC GPS-UTC DATE REMARKSX-POLE Y-POLE UT1-UTC GPS-UTC DATE REMARKS

(") (") (") (") (S) (S) (S)(S)

48199. 48199. 0.2260 0.2260 0.1371 0.1371 -0.25656-0.25656 6. 90 6. 90 11 411 4 DEFDEF

48204. 48204. 0.2073 0.2073 0.1253 0.1253 -0.26741 -0.26741 6. 90 11 96. 90 11 9 DEFDEF

48209. 48209. 0.1928 0.1928 0.1138 0.1138 -0.27872 -0.27872 6. 90 6. 90 11 1411 14 DEFDEF

48214. 48214. 0.1777 0.1777 0.1044 0.1044 -0.28971 -0.28971 6. 90 11 196. 90 11 19 DEFDEF

48219. 48219. 0.1623 0.1623 0.0963 0.0963 -0.30094 -0.30094 6. 90 11 246. 90 11 24 DEFDEF

48224. 48224. 0.1436 0.1436 0.0900 0.0900 -0.31241 -0.31241 6. 90 11 296. 90 11 29DEFDEF

48229. 48229. 0.1251 0.1251 0.0845 0.0845 -0.32413 -0.32413 6. 90 12 46. 90 12 4DEFDEF

48234. 48234. 0.1073 0.1073 0.0799 0.0799 -0.33518 -0.33518 6. 90 12 96. 90 12 9 DEFDEF

48239. 48239. 0.0904 0.0904 0.0747 0.0747 -0.34529 -0.34529 6. 90 12 146. 90 12 14 DEFDEF

48244. 48244. 0.0737 0.0737 0.06980.0698 -0.35502 -0.35502 6. 90 12 196. 90 12 19 DEFDEF

48249. 48249. 0.0550 0.0550 0.06780.0678 -0.36508 -0.36508 6. 90 12 246. 90 12 24 DEFDEF

48254. 48254. 0.0346 0.0346 0.0681 0.0681 -0.37551 -0.37551 6. 90 12 296. 90 12 29 DEFDEF

• Over 40 years ago irregularities in the rotation of the Over 40 years ago irregularities in the rotation of the earth were discovered (other than polar motion). There earth were discovered (other than polar motion). There are three types of irregularities: seasonal variations are three types of irregularities: seasonal variations probably due to meteorological changes or earth tides; probably due to meteorological changes or earth tides; secular decrease due to tidal friction; and irregular secular decrease due to tidal friction; and irregular fluctuations. The seasonal variation is the only one of fluctuations. The seasonal variation is the only one of these presently being taken into account, and it is more these presently being taken into account, and it is more or less reproducible from year to year, and produces a or less reproducible from year to year, and produces a displacement along the equator of up to 15 meters with displacement along the equator of up to 15 meters with respect to a point rotating uniformly . throughout the respect to a point rotating uniformly . throughout the year (see Figure 2-2). year (see Figure 2-2).

• Because of this seasonal variation, the Greenwich Because of this seasonal variation, the Greenwich meridian (the plane containig the earth's rotation axis meridian (the plane containig the earth's rotation axis and the center of the transit instrument at Greenwich and the center of the transit instrument at Greenwich Observatory) does not rotate uniformly. A fictitious zero Observatory) does not rotate uniformly. A fictitious zero meridian which does rotate uniformly (so far as the meridian which does rotate uniformly (so far as the effects of polar motion and seasonal variations are effects of polar motion and seasonal variations are concerned) is called the Mean Observatory or Greenwich concerned) is called the Mean Observatory or Greenwich mean astronomic meridian. Its location was defined by mean astronomic meridian. Its location was defined by the BIH. the BIH.

• The mean Greenwich meridian; also named the The mean Greenwich meridian; also named the Greenwich Mean Observatory (GMO) is defined Greenwich Mean Observatory (GMO) is defined through the nominal longitudes of all observatories through the nominal longitudes of all observatories which contributed to the former international time which contributed to the former international time service (Bureau International de l`Heure (BIH)). service (Bureau International de l`Heure (BIH)). Because of the inclusion of new satellite observation Because of the inclusion of new satellite observation techniques and additional stations for the techniques and additional stations for the determination of time and pole coordinates during the determination of time and pole coordinates during the last decade, this definition was no longer valid in a last decade, this definition was no longer valid in a rigorous sense. It was put on a broader foundation with rigorous sense. It was put on a broader foundation with the introduction of the new International Earth the introduction of the new International Earth Rotation Service (IERS) . Rotation Service (IERS) .

• In future, the CTS will be defined by a global network In future, the CTS will be defined by a global network of fundamental stations whose coordinates are of fundamental stations whose coordinates are determined by the most precise observation determined by the most precise observation techniques. Coordinate changes, caused by recent techniques. Coordinate changes, caused by recent crustal movement, are also considered. In 1984 the crustal movement, are also considered. In 1984 the Bureau International de 1`Heure (BIH) introduced such Bureau International de 1`Heure (BIH) introduced such a terrestrial reference system, named BTS (BIH a terrestrial reference system, named BTS (BIH Terrestrial System), and this system is now maintained Terrestrial System), and this system is now maintained as the IERS Terrestrial Reference Frame (ITRF) . as the IERS Terrestrial Reference Frame (ITRF) .

2.1.2 Average and Instantaneous 2.1.2 Average and Instantaneous Terrestrial Systems Terrestrial Systems

The average terrestrial (A.T.) system is the ideal The average terrestrial (A.T.) system is the ideal world geodetic coordinate system (see Figure 2-3): world geodetic coordinate system (see Figure 2-3):

• a) a) Its origin is at the centre of gravity of the earth.Its origin is at the centre of gravity of the earth.• b) b) Its primary pole is directed towards the CIO (the Its primary pole is directed towards the CIO (the

average north pole of 1900-1905), and its primary average north pole of 1900-1905), and its primary plane is the plane perpendicular to the primary pole plane is the plane perpendicular to the primary pole and containing the earth's center of gravity (the and containing the earth's center of gravity (the average equatorial plane). average equatorial plane).

• c) Its secondary plane is the plane containing the c) Its secondary plane is the plane containing the primary pole and the Mean Observatory. The primary pole and the Mean Observatory. The intersection of these two planes is the secondary intersection of these two planes is the secondary pole, or primary axis. pole, or primary axis.

• d) It is a right-handed system. d) It is a right-handed system.

• We can We can then define then define the position the position vector Rvector Rii of of a terrain a terrain point i in point i in terms of its terms of its Cartesian Cartesian coordinates coordinates

x, y, z x, y, z as as

2-12-1

.

i

A T

x

R y

z

Figure 2.3a – Average Terrestrial

Coordinate System

The instantaneous terrestrial (I.T.) system is The instantaneous terrestrial (I.T.) system is specified as follows: specified as follows:

• a) Its origin is at the centre of gravity of the a) Its origin is at the centre of gravity of the earth. earth.

• b) Its primary pole is directed towards the true b) Its primary pole is directed towards the true (instantaneous) rotation axis of the earth. (instantaneous) rotation axis of the earth.

• c) Its primary axis is the intersection of the c) Its primary axis is the intersection of the primary plane and the plane containing the true primary plane and the plane containing the true rotation axis and the Mean Observatory. rotation axis and the Mean Observatory.

• d) It is a right-handed system.d) It is a right-handed system.

• The main characteristic of these two systems is The main characteristic of these two systems is that they are geocentric systems having their that they are geocentric systems having their origins at the centre of gravity of the earth and origins at the centre of gravity of the earth and the rotation axis of the earth as the primary pole. the rotation axis of the earth as the primary pole.

• By means of rotation matrices the coordinates of By means of rotation matrices the coordinates of a point referred to the instantaneous terrestrial a point referred to the instantaneous terrestrial system are transformed into the average system system are transformed into the average system by the following equation (see Figure 2-4): by the following equation (see Figure 2-4):

2-22-2

TI

PP

TAz

y

x

yRxR

z

y

x

.

12

.

)()(

• where (xwhere (xPP , y , yPP ) are expressed in arcseconds, and ) are expressed in arcseconds, and the rotation matrices are the rotation matrices are

a clockwise (negative) rotation about the x axis, a clockwise (negative) rotation about the x axis, and and

a clockwise (negative) rotation about the y-axis, a clockwise (negative) rotation about the y-axis,

)cos()sin(0

)sin()cos(0

001

)(1

PP

PPP

yy

yyyR

)cos(0)sin(

010

)sin(0)cos(

)(2

PP

PP

P

xx

xx

xR

1

10

01

10

10

001

10

010

01

)()( 12

PP

P

P

P

P

P

P

PP

yx

y

x

y

y

x

x

yRxR

TI

PP

TAz

y

x

yRxR

z

y

x

.

12

.

)()(

The inverse is The inverse is

and because of the orthogonal characteristic of and because of the orthogonal characteristic of rotation matrices, that is rotation matrices, that is

RR-1-1(() = R) = RTT(() = R(-) = R(-))

2-32-3

TA

PP

TIz

y

x

xRyR

z

y

x

.

21

.

)()(

1

2 1

. .

( ) ( )P P

I T A T

x x

y R x R y y

z z

• Astronomik gözlemlerle bulunan kutupsal anlık Astronomik gözlemlerle bulunan kutupsal anlık koordinatlar astronomik enlem, astronomik boylam ve koordinatlar astronomik enlem, astronomik boylam ve astronomik azimut yine kutup hareketi parametrelerine astronomik azimut yine kutup hareketi parametrelerine göre düzeltilerek ortalama kutuba indiregenmiş göre düzeltilerek ortalama kutuba indiregenmiş koordinatlar elde edilir. Bu indirgemelerkoordinatlar elde edilir. Bu indirgemeler

eşitlikleri ile hesaplanır. Eşitliklerde eşitlikleri ile hesaplanır. Eşitliklerde TT (T ölçme (T ölçme anındaki) anlık kutba göre yapılan astronomik anındaki) anlık kutba göre yapılan astronomik gözlemelerle belirlenmiş anlık astronomik azimut, gözlemelerle belirlenmiş anlık astronomik azimut, TT anlık astronomik enlem, anlık astronomik enlem, TT anlık astronomik boylam, anlık astronomik boylam, ortalama kutba (CIO) indirgenmiş astronomik azimut, ortalama kutba (CIO) indirgenmiş astronomik azimut, indirgenmiş astronomik enlem ve indirgenmiş astronomik enlem ve indirgenmiş indirgenmiş astronomik boylamdır. astronomik boylamdır.

sec)cossin( PPTP yx

cossin PPTP xy

tan)cossin( PPTP yx

• Yukarıdaki eşitliklerle hesaplanmış indirgeme Yukarıdaki eşitliklerle hesaplanmış indirgeme değerleri ile bu indirgenmiş büyüklüklerdeğerleri ile bu indirgenmiş büyüklükler

eşitlikleri ile hesaplanır.eşitlikleri ile hesaplanır.

• 1990 yılının kasım ve aralık ayları için beşer 1990 yılının kasım ve aralık ayları için beşer günlük aralıklarla BIH tarafından belirlenen kutup günlük aralıklarla BIH tarafından belirlenen kutup hareketi parametreleri aşağıda Tablo 2.3 de hareketi parametreleri aşağıda Tablo 2.3 de verilmektedir.verilmektedir.

PT

PT

PT

2.2.2 2 Geodetic Systems Geodetic Systems

• In terms of Cartesian coordinates, the geodetic (G) In terms of Cartesian coordinates, the geodetic (G) coordinate system is that system which is coordinate system is that system which is introduced into the earth such that its three axes introduced into the earth such that its three axes are coincident with or parallel to the corresponding are coincident with or parallel to the corresponding three axes of the average terrestrial system (see three axes of the average terrestrial system (see Figure 2-3). The first situation defines a geocentric Figure 2-3). The first situation defines a geocentric geodetic system while the latter non-geocentric geodetic system while the latter non-geocentric system is commonly referred to as a relative system is commonly referred to as a relative geodetic system, whose relationship to the average geodetic system, whose relationship to the average terrestrial system is given by the three datum terrestrial system is given by the three datum translation components translation components

0

0

0

0

z

y

x

r

and in vector equation form, the relationship is and in vector equation form, the relationship is

where the position vector ri is referred to the where the position vector ri is referred to the geodetic system, that is geodetic system, that is

and and

2-42-4

ii rrR 0

G

i

z

y

x

r

GTAz

y

x

z

y

x

z

y

x

0

0

0

.

• In this section we first describe the Cartesian (x, In this section we first describe the Cartesian (x, y, z) and curvilinear (latitude, longitude, height) y, z) and curvilinear (latitude, longitude, height) coordinates for a point on the reference ellipsoid. coordinates for a point on the reference ellipsoid.

• We then develop expressions for its position We then develop expressions for its position vector in terms of various latitudes. vector in terms of various latitudes.

• Finally the transformation from geodetic Finally the transformation from geodetic coordinates coordinates ((, , , h) to (x, y, z) and its inverse , h) to (x, y, z) and its inverse are discussed. are discussed.

• The specific ellipsoid used in geodesy as a reference The specific ellipsoid used in geodesy as a reference surface is a rotational ellipsoid formed from the surface is a rotational ellipsoid formed from the rotation of an ellipse about its semi-minor axis b rotation of an ellipse about its semi-minor axis b (Figure 2-5). The semi-major axis a and the flattening (Figure 2-5). The semi-major axis a and the flattening

2-52-5

are the defining parameters of the reference ellipsoid. are the defining parameters of the reference ellipsoid.

• Other useful parameters associated with this Other useful parameters associated with this particular ellipsoid are the first eccentricity particular ellipsoid are the first eccentricity

2-62-6

and the second eccentricity and the second eccentricity

2-72-7

a

baf

2

222

a

bae

2

222

b

bae

A Cartesian coordinate system is superimposed on A Cartesian coordinate system is superimposed on the reference ellipsoid (see Figure 2-5) so that: the reference ellipsoid (see Figure 2-5) so that:

• a) The origin of the Cartesian system is the centre of a) The origin of the Cartesian system is the centre of the ellipsoid. the ellipsoid.

• b) The primary pole (z-axis) of the Cartesian system b) The primary pole (z-axis) of the Cartesian system is the semi-minor axis of the ellipsoid. The primary is the semi-minor axis of the ellipsoid. The primary plane is perpendicular to the primary axis and is plane is perpendicular to the primary axis and is called the equatorial plane. called the equatorial plane.

• c) Any plane containing the semi-minor axis and c) Any plane containing the semi-minor axis and cutting the surface of the ellipsoid is called a cutting the surface of the ellipsoid is called a meridian plane. The particular meridian plane meridian plane. The particular meridian plane chosen as the secondary plane is called the chosen as the secondary plane is called the Greenwich meridian plane. The secondary pole (x-Greenwich meridian plane. The secondary pole (x-axis) is the intersection of the equatorial plane and axis) is the intersection of the equatorial plane and the Greenwich meridian plane. the Greenwich meridian plane.

• d) Td) The y-axis is chosen to form a right-handed he y-axis is chosen to form a right-handed system, and lies in the equatorial plane, 90° system, and lies in the equatorial plane, 90° counterclockwise from the x-axis. counterclockwise from the x-axis.

Figure 2.5b : The Ellipsoidal (Geodetic) Cartesian Coordinate System

• The equation of this ellipsoid, in terms of Cartesian The equation of this ellipsoid, in terms of Cartesian coordinates is coordinates is

2-82-8

• where where

2-92-9

or or 2-102-10

2

2

2

100

01

0

001

b

a

a

SE

12

2

2

22

b

z

a

yx

12

2

2

22

b

z

a

yx

zyxX T

• The latitude of a point is the acute angular distance The latitude of a point is the acute angular distance between the equatorial plane and the ellipsoid between the equatorial plane and the ellipsoid normal through the point measured in the meridian normal through the point measured in the meridian plane of the point. The line perpendicular to the plane of the point. The line perpendicular to the ellipsoid at a point is called the ellipsoid normal at ellipsoid at a point is called the ellipsoid normal at the point. Ellipsoid normals only pass through the the point. Ellipsoid normals only pass through the geometric center of the ellipsoid in the equatorial geometric center of the ellipsoid in the equatorial plane or along the semi-minor axis. Therefore plane or along the semi-minor axis. Therefore there are two different kinds of latitude. The angle there are two different kinds of latitude. The angle between the ellipsoid normal at the point and the between the ellipsoid normal at the point and the equatorial plane is called the geodetic latitude equatorial plane is called the geodetic latitude . . The angle between the line joining the point to the The angle between the line joining the point to the centre of the ellipse, and the equatorial plane is centre of the ellipse, and the equatorial plane is called the geocentric latitude called the geocentric latitude . There is also a . There is also a third latitude, used mostly as a mathematical third latitude, used mostly as a mathematical convenience, called the reduced latitude convenience, called the reduced latitude (see (see Figure 2-6). Figure 2-6).

• The longitude The longitude of a meridian plane is the of a meridian plane is the counterclockwise angular distance between the counterclockwise angular distance between the Greenwich meridian plane and the meridian plane Greenwich meridian plane and the meridian plane of the point, measured in the equatorial plane of the point, measured in the equatorial plane (see Figure 2-5).(see Figure 2-5).

• The ellipsoid height h of a point is its linear The ellipsoid height h of a point is its linear distance above the ellipsoid, measured along the distance above the ellipsoid, measured along the ellipsoidal normal at the point (see Figure 2-8).ellipsoidal normal at the point (see Figure 2-8).

Figure 2.6 : Figure 2.6 : Geodetic, Geodetic, Geocentric and Geocentric and Reduced Reduced LatitudeLatitude

2.2.2.2.11 The Position Vector in Terms of The Position Vector in Terms of the the Geodetic Latitude Geodetic Latitude

• Consider a point P on the surface of the ellipsoidConsider a point P on the surface of the ellipsoid

• N is the radius of curvature of the ellipsoid surface N is the radius of curvature of the ellipsoid surface in the plane perpendicular to the meridian plane in the plane perpendicular to the meridian plane (called the prime vertical plane). (called the prime vertical plane).

b

ac

2

22 cos1 e

c

V

cN

212222

2

)sincos( ba

aN

2-22

22 sin1 e

a

W

aN

2 21 cosV e 2 21 sinW e

We now refer the position vector P to a system with We now refer the position vector P to a system with the primary axis in the Greenwich meridian plane, the primary axis in the Greenwich meridian plane,

sin/

sincos

coscos

22 ab

N

z

y

x

r 2-24

2.2.2.2 2.2 The Position Vector in Terms of The Position Vector in Terms of the the Geocentric and Reduced Geocentric and Reduced Latitudes Latitudes

• From Figure 2-6 the position vector of the point P From Figure 2-6 the position vector of the point P in terms of the geocentric latitude in terms of the geocentric latitude is is

where where is the magnitude of is the magnitude of . .

sin

sincos

coscos

0)(3 r

z

x

R

z

y

x

r

r

2-25

r

• From Figure 2-6 the reduced latitude From Figure 2-6 the reduced latitude of the point of the point P is the geocentric latitude of both the points Q P is the geocentric latitude of both the points Q and R, where Q is the projection of P parallel to the and R, where Q is the projection of P parallel to the semi-minor axis to intersect a circle with radius semi-minor axis to intersect a circle with radius equal to the semi-major axis, and R is the equal to the semi-major axis, and R is the projection of the point P parallel to the semi-major projection of the point P parallel to the semi-major axis to intersect a circle with radius equal to the axis to intersect a circle with radius equal to the semi-minor axis. semi-minor axis.

• The position vector of P in terms of the reduced The position vector of P in terms of the reduced latitude latitude is is

sin

sincos

coscos

0)(3

b

a

a

z

x

R

z

y

x

r 2-26

2.2.2.32.3 Relationships between Relationships between Geodetic, Geodetic, Geocentric and Geocentric and Reduced Latitudes Reduced Latitudes

From equations 2-24, 2-25, and 2-26 From equations 2-24, 2-25, and 2-26

Canceling the cos Canceling the cos term, term,

2-272-27

2-282-28

2-292-29

costancostancostan2

2

a

b

a

b

x

z

tantana

b

tantanb

a

tantan2

2

a

b

2.2.2.42.4 The Position Vector of a Point The Position Vector of a Point Above Above the Reference Ellipsoid the Reference Ellipsoid Let us consider a terrain point i, as depicted in Figure Let us consider a terrain point i, as depicted in Figure 2-8, whose coordinates are the geodetic latitude 2-8, whose coordinates are the geodetic latitude and longitude and longitude , and the ellipsoid height h. The , and the ellipsoid height h. The projection of i onto the surface of the ellipsoid is along projection of i onto the surface of the ellipsoid is along the ellipsoidal normal defined by the unit vector the ellipsoidal normal defined by the unit vector

• The position vector of i is then the sum of two The position vector of i is then the sum of two vectors, namely vectors, namely

2-302-30where where is defined by equation 2-24 and is defined by equation 2-24 and is the is the unit vector defined by equation 2-68c , that is unit vector defined by equation 2-68c , that is

zpi uhrr

Zu

pr Zu

sin

sincos

coscos

zu

• Thus Thus

or or

2-312-31

• Now the position vector Now the position vector in equation 2-31 refers in equation 2-31 refers to a coordinate system whose origin is at the to a coordinate system whose origin is at the geometrical centre of the ellipsoid. geometrical centre of the ellipsoid.

sin

sincos

coscos

sin/

sincos

coscos

22

h

ab

N

z

y

x

ri

sin)/(

sincos)(

coscos)(

22 haNb

hN

hN

z

y

x

ri

ir

x N h ( ) cos cos y N h ( ) cos sin

zb

aN h e N h

2

221sin ( ) sin

b

ac

2

2

222

a

bae

2

222

b

bae

22 cos1 e

c

V

cN

22 sin1 e

a

W

aN

• If this ellipsoid defines a relative geodetic system, then its If this ellipsoid defines a relative geodetic system, then its centre will not in general coincide with the centre of centre will not in general coincide with the centre of gravity of the earth. The expression for the position vector gravity of the earth. The expression for the position vector in the average terrestrial system is, from equation 2-4in the average terrestrial system is, from equation 2-4

• or or

2-322-32

This expression gives the general transformation from This expression gives the general transformation from relative geodetic coordinates (relative geodetic coordinates (, , , h) to average terrestrial , h) to average terrestrial coordinates (x, y, z), given the size of the ellipsoid (a, b) coordinates (x, y, z), given the size of the ellipsoid (a, b) and the translation components (xand the translation components (x00, y, y00, z, z00). ).

GiTAi r

z

y

x

r )()(

0

0

0

..

sin)/(

sincos)(

coscos)(

)(22

0

0

0

..

..

haNb

hN

hN

z

y

x

z

y

x

r

TA

TAi

2.2.2.2.55 Transformation from Average Transformation from Average Terrestrial Cartesian Terrestrial Cartesian

to Geodetic to Geodetic Coordinates Coordinates • A very useful transformation is the inverse of A very useful transformation is the inverse of

equation 2-32. Given the average terrestrial equation 2-32. Given the average terrestrial coordinates (x, y, z), the translation components coordinates (x, y, z), the translation components (x(x00, y, y00, z, z00), and the size of the ellipsoid (a, b), ), and the size of the ellipsoid (a, b), compute the relative geodetic coordinates (compute the relative geodetic coordinates (, , , h)., h).

• First we translate the origin from the centre of First we translate the origin from the centre of gravity to the centre of the ellipsoid. From equation gravity to the centre of the ellipsoid. From equation 2-32 2-32

2-332-33

0

0

0

..z

y

x

z

y

x

z

y

x

TAG

• The longitude The longitude is computed directly from is computed directly from

2-342-34

The latitude The latitude and ellipsoid height h are more and ellipsoid height h are more difficult to compute since N is a function of difficult to compute since N is a function of , from , from equation 2-22 equation 2-22

22 yxp

p

z

b

atan

x

y1tan

32

32

cos

sintan

aep

bez

Np

h cos

22 cos1 e

c

V

cN

22 sin1 e

a

W

aN

b

ac

2

2

222

a

bae

2

222

b

bae

2.3 GEODETIC DATUMS 2.3 GEODETIC DATUMS • There are two natural figures of the earth (see Figure There are two natural figures of the earth (see Figure

2-9); the topographic or physical surface of the earth 2-9); the topographic or physical surface of the earth including the surface of the oceans (the terrain), and including the surface of the oceans (the terrain), and the equipotential surface of the earth's gravity field the equipotential surface of the earth's gravity field which coincides with an idealized surface of the which coincides with an idealized surface of the oceans (the geoid). oceans (the geoid).

• Control measurenents (e.g. distances, angles, spirit Control measurenents (e.g. distances, angles, spirit levelling) are made between points on the terrain levelling) are made between points on the terrain which we call control points. These measurements which we call control points. These measurements are used to determine the geometrical relationship are used to determine the geometrical relationship between the control points in a computation called between the control points in a computation called network adjustment. Other points are then related to network adjustment. Other points are then related to the network of control points through further the network of control points through further measurements and computations called densification. measurements and computations called densification. The classical approach is to treat the vertical The classical approach is to treat the vertical measurements, networks and computations measurements, networks and computations separately from the horizontal measurements, separately from the horizontal measurements, networks and computations. However the unified networks and computations. However the unified three dimensional approach is currently gaining three dimensional approach is currently gaining favour. favour.

• Even simpler surfaces than the ellipsoid (such as Even simpler surfaces than the ellipsoid (such as the sphere or the plane) can be sufficient the sphere or the plane) can be sufficient approximations to the geoid if the area under approximations to the geoid if the area under consideration is sufficiently small, and/or the consideration is sufficiently small, and/or the control application permits lower orders of control application permits lower orders of accuracy. accuracy.

• The introduction of a new surface (the ellipsoid) The introduction of a new surface (the ellipsoid) has a price. The horizontal control network (that has a price. The horizontal control network (that is the coordinates of the points of the network) is is the coordinates of the points of the network) is to be referred to the ellipsoid. Therefore before to be referred to the ellipsoid. Therefore before network computations can begin, the control network computations can begin, the control measurements must first be reduced so that they measurements must first be reduced so that they too "refer" to the ellipsoid. too "refer" to the ellipsoid.

• It is important to distinguish between the datum It is important to distinguish between the datum (the coordinate surface or ellipsoid surface) and (the coordinate surface or ellipsoid surface) and the coordinates of the points of the network the coordinates of the points of the network referred to the datum. It is a common but referred to the datum. It is a common but confusing practice (particularly in North America) confusing practice (particularly in North America) to use the term "datum" for the set of coordinates to use the term "datum" for the set of coordinates

Figure 2-9. Meridian Section of The Earth

• In the classical approach vertical measurements and In the classical approach vertical measurements and networks are referred networks are referred toto a a coordinate surface or coordinate surface or (vertical) datum which is the geoid. Rather (vertical) datum which is the geoid. Rather than using the than using the geoid as the coordinate surface or datum for geoid as the coordinate surface or datum for the the horizontal measurements and networks as well, a third, horizontal measurements and networks as well, a third, unnatural figure of the earth is introduced the ellipsoid of unnatural figure of the earth is introduced the ellipsoid of rotation discussed earlier. The reason a mathematical rotation discussed earlier. The reason a mathematical figure like the ellipsoid is used as the horizontal datum is figure like the ellipsoid is used as the horizontal datum is to simplify the computations required both for network to simplify the computations required both for network adjustment and densification. adjustment and densification.

• Correction terms are necessary in these computation to Correction terms are necessary in these computation to account for the fact that the datum is not the geoid. An account for the fact that the datum is not the geoid. An ellipsoid can be chosen to approximate the geoid closely ellipsoid can be chosen to approximate the geoid closely enough that these correction terms can be assumed enough that these correction terms can be assumed linear and for some applications even ignored. For a well-linear and for some applications even ignored. For a well-chosen ellipsoid (see Figure 2-9), the geoid-ellipsoid chosen ellipsoid (see Figure 2-9), the geoid-ellipsoid separation (geoid height) is always less than 100 metres, separation (geoid height) is always less than 100 metres, and the difference between the geoid and ellipsoid and the difference between the geoid and ellipsoid normals at any point (deflection of the vertical) is usually normals at any point (deflection of the vertical) is usually less than 5 less than 5 arcseconds, very rarely exceeding arcseconds, very rarely exceeding 1 1 arcminute.arcminute.

• Even simpler surfaces than the ellipsoid (such as the Even simpler surfaces than the ellipsoid (such as the sphere or the plane) can be sufficient approximations sphere or the plane) can be sufficient approximations to the geoid if the area under consideration is to the geoid if the area under consideration is sufficiently small, and/or the control application sufficiently small, and/or the control application permits lower orders of accuracy. permits lower orders of accuracy.

• The introduction of a new surface (the ellipsoid) has a The introduction of a new surface (the ellipsoid) has a price. The horizontal control network (that is the price. The horizontal control network (that is the coordinatas of the points of the network) is to be coordinatas of the points of the network) is to be referred to the ellipsoid. Therefore before network referred to the ellipsoid. Therefore before network computations can begin, the control measurements computations can begin, the control measurements must first be reduced so that they too "refer" to the must first be reduced so that they too "refer" to the ellipsoid.ellipsoid.

• It is important to distinguish between the datum (the It is important to distinguish between the datum (the coordinate surface or ellipsoid surface) and the coordinate surface or ellipsoid surface) and the coordinates of the points of the network referred to coordinates of the points of the network referred to the datum. It is a common but confusing practice the datum. It is a common but confusing practice (particularly in North America) to use the term (particularly in North America) to use the term "datum" for the set of coordinates "datum" for the set of coordinates

2.3.1 Datum Position Parameters 2.3.1 Datum Position Parameters • In order to establish an ellipsoid as the reference In order to establish an ellipsoid as the reference

surface for a system of control we must specify its surface for a system of control we must specify its size and shape (usually by assigning values to the size and shape (usually by assigning values to the semi-major axis and flattening) and we must semi-major axis and flattening) and we must specify its position with respect to the earth. A specify its position with respect to the earth. A well-positioned ellipsoid will closely approximate well-positioned ellipsoid will closely approximate the geoid over the area covered by the network for the geoid over the area covered by the network for which it is in datum. The parameters to which we which it is in datum. The parameters to which we assign values in order to specify the ellipsoid assign values in order to specify the ellipsoid position we call the datum position parameters. position we call the datum position parameters.

• In three-dimensional space, any figure (and In three-dimensional space, any figure (and particularly our ellipsoid) has six degrees of particularly our ellipsoid) has six degrees of freedom, that is six ways in which its position with freedom, that is six ways in which its position with respect to a fixed figure (in our case the earth) can respect to a fixed figure (in our case the earth) can be changed. Thus there are six datum position be changed. Thus there are six datum position parameters. parameters.

• AnotherAnother way way of of looking at this is to consider two three-looking at this is to consider two three-dimensional Cartesian coordinate systems, one fixed to dimensional Cartesian coordinate systems, one fixed to the ellipsoid and one fixed to the earth. In general the the ellipsoid and one fixed to the earth. In general the origins of the two systems will not coincide, and the axes origins of the two systems will not coincide, and the axes will not be parallel. Therefore, to define the will not be parallel. Therefore, to define the transformation from one system to the other we must transformation from one system to the other we must specify the location of one origin with respect to the other specify the location of one origin with respect to the other system, and the orientation of one set of axes with system, and the orientation of one set of axes with respect to the other system, that is three coordinates, respect to the other system, that is three coordinates, and three rotation angles. These six parameters provide and three rotation angles. These six parameters provide a description of the six degrees of freedom and assigning a description of the six degrees of freedom and assigning values to them positions the ellipsoid with respect to the values to them positions the ellipsoid with respect to the earth. They· are our six datum position parameters. A earth. They· are our six datum position parameters. A datum then is completely specified by assigning values to datum then is completely specified by assigning values to eight parameters - the ellipsoid size and shape, and the eight parameters - the ellipsoid size and shape, and the six datum position parameters. six datum position parameters.

• There are in fact two kinds of datum position parameters There are in fact two kinds of datum position parameters in use. One kind is obtained by considering the ellipsoid-in use. One kind is obtained by considering the ellipsoid-fixed and earth- fixed coordinate systems to have their fixed and earth- fixed coordinate systems to have their origins in the neighbourhood of the geocentre. The other origins in the neighbourhood of the geocentre. The other kind is obtained by considering the ellipsoid-fixed and kind is obtained by considering the ellipsoid-fixed and earth-fixed coordinate systems to have their origins near earth-fixed coordinate systems to have their origins near the surface of the earth at a point we call the initial point the surface of the earth at a point we call the initial point of the datum. of the datum.

• In the first (geocentric) case the earth-fixed system In the first (geocentric) case the earth-fixed system is the Average Terrestrial system of section 2.1.2, is the Average Terrestrial system of section 2.1.2, and the ellipsoid-fixed system is the geodetic and the ellipsoid-fixed system is the geodetic system of Equation 2-31 (except that here we system of Equation 2-31 (except that here we assume the geodetic and average terrestrial axes assume the geodetic and average terrestrial axes are not in general parallel). In this case the datum are not in general parallel). In this case the datum position pararmeters are the Average Terrestrial position pararmeters are the Average Terrestrial coordinates of the ellipsoid origin (xcoordinates of the ellipsoid origin (x00, y, y00, z, z00 of of Equation 2-32) and three rotation angles (say Equation 2-32) and three rotation angles (say 1, 1, 2, 2, 3) required to define the misalignment 3) required to define the misalignment between the axes. It is of course highly desirable between the axes. It is of course highly desirable that the ellipsoid be positioned so that these angles that the ellipsoid be positioned so that these angles are as small as possible, particularly that the two are as small as possible, particularly that the two axes of symmetry (the ellipsoid minor axis and axes of symmetry (the ellipsoid minor axis and earth's average rotation axis or Average Terrestrial earth's average rotation axis or Average Terrestrial z-axis) be parallel. z-axis) be parallel.

• In the second (topocentric) case the earth-fixed In the second (topocentric) case the earth-fixed system is a local astronomic system at the initial system is a local astronomic system at the initial point, and the ellipsoid-fixed system is a local point, and the ellipsoid-fixed system is a local geodetic system at the same point (local astronomic geodetic system at the same point (local astronomic and geodetic systems are discussed in section 2.4). and geodetic systems are discussed in section 2.4).

• Before proceeding further let us consider the geometry Before proceeding further let us consider the geometry in the neighbourhood of a point on the earth's surface. in the neighbourhood of a point on the earth's surface. Figure 2-10 is an exaggerated view of the geodetic Figure 2-10 is an exaggerated view of the geodetic meridian plane at such a point, showing the sectioned meridian plane at such a point, showing the sectioned ellipsoid, geoid, several equipotential surfaces related ellipsoid, geoid, several equipotential surfaces related to the geoid, and the terrain. A particular ellipsoid to the geoid, and the terrain. A particular ellipsoid normal intersects the ellipsoid, geoid, and terrain at normal intersects the ellipsoid, geoid, and terrain at Q,P and T respectively. There are three "natural" Q,P and T respectively. There are three "natural" normals corresponding to this ellipsoid normal; the normals corresponding to this ellipsoid normal; the surface gravity vertical (perpendicular to the surface gravity vertical (perpendicular to the equipotential surface at T, passing through T), the equipotential surface at T, passing through T), the geoid gravity vertical (perpendicular to the geoid geoid gravity vertical (perpendicular to the geoid passing through P), and the plumbline (perpendicular passing through P), and the plumbline (perpendicular to all equipotential surfaces between terrain and to all equipotential surfaces between terrain and geoid, passing through T). In general, the plumbline is geoid, passing through T). In general, the plumbline is curved while the others are straight lines, and none of curved while the others are straight lines, and none of these three actually lie in the geodetic meridian plane these three actually lie in the geodetic meridian plane - they are shown here as projections onto this plane. If - they are shown here as projections onto this plane. If the curvature of the plumbline is ignored the two the curvature of the plumbline is ignored the two gravity verticals are parallel. gravity verticals are parallel.

Figure 2-10. Oriantation of Ellipsoid To Geoid

• The astronomic meridian plane is the plane containing one The astronomic meridian plane is the plane containing one of the gravity verticals and a parallel to the Average of the gravity verticals and a parallel to the Average Terrestrial z-axis. The angle between the gravity vertical Terrestrial z-axis. The angle between the gravity vertical and the parallel to the A.T. z-axis is the astronomic co-and the parallel to the A.T. z-axis is the astronomic co-latitude (latitude ( ). The angle between the astronomic meridian ). The angle between the astronomic meridian plane and a reference meridian plane (Greenwich) is the plane and a reference meridian plane (Greenwich) is the astronomic longitude astronomic longitude . The angle between the ellipsoid . The angle between the ellipsoid normal and the gravity vertical is the deflection of the normal and the gravity vertical is the deflection of the vertical, which can be resolved into a component vertical, which can be resolved into a component in the in the geodetic meridian plane and a component geodetic meridian plane and a component in the in the geodetic prime vertical plane (the plane perpendicular to geodetic prime vertical plane (the plane perpendicular to the geodetic meridian plane which contains the ellipsoid the geodetic meridian plane which contains the ellipsoid normal). Thus corresponding to the two gravity verticals, normal). Thus corresponding to the two gravity verticals, there are two sets of values for the astronomic latitude and there are two sets of values for the astronomic latitude and longitude and deflection components, and if the curvature longitude and deflection components, and if the curvature of the plumbline is ignored, these two sets are equal. of the plumbline is ignored, these two sets are equal.

• If the ellipsoid is positioned so that its geocentric axes are If the ellipsoid is positioned so that its geocentric axes are paralled to the Average Terrestrial axes (that is paralled to the Average Terrestrial axes (that is 11==22==33= = 0) then 0) then

• = = - - 2-402-40• = (= ( - - ) cos) cos 2- 412- 41• where (where (, , ) are the common geodetic coordinates of Q, P ) are the common geodetic coordinates of Q, P

and T. and T.

2

• The distance between the ellipsoid and geoid, measured The distance between the ellipsoid and geoid, measured along the ellipsoid normal (QP) is the geoid height N*. along the ellipsoid normal (QP) is the geoid height N*. The distance between the ellipsoid and terrain, measured The distance between the ellipsoid and terrain, measured along the ellipsoid normal (QT) is the ellipsoid height h. along the ellipsoid normal (QT) is the ellipsoid height h. The distance between the geoid and terrain, measured The distance between the geoid and terrain, measured along the plumbline (P'T) is the orthometric height H. If along the plumbline (P'T) is the orthometric height H. If the curvature of the plumbline is ignored the curvature of the plumbline is ignored

• h = N* + Hh = N* + H 2-422-42• Given a point some distance from T, the angle between Given a point some distance from T, the angle between

the geodetic meridian plane and the plane containing this the geodetic meridian plane and the plane containing this point and the ellipsoid normal QPT is the geodetic point and the ellipsoid normal QPT is the geodetic azimuth azimuth of that point with respect to Q, P or T. (Actually of that point with respect to Q, P or T. (Actually this is the azimuth of the normal section, and is related to this is the azimuth of the normal section, and is related to the geodetic azimuth by small corrections, The angle the geodetic azimuth by small corrections, The angle between the astronomic meridian plane and the plane between the astronomic meridian plane and the plane containing this point and the corresponding gravity containing this point and the corresponding gravity vertical is the astronomic azimuth A of that point with vertical is the astronomic azimuth A of that point with respect to either P or T depending on which gravity respect to either P or T depending on which gravity vertical is used. Because the deflection of the vertical is vertical is used. Because the deflection of the vertical is small, then for all such points the difference small, then for all such points the difference

• = A - = A - 2-43 2-43 is nearly constant, and is the angle between the geodetic is nearly constant, and is the angle between the geodetic and astronomic meridian planes. and astronomic meridian planes.

• Returning to the topocentric datum position Returning to the topocentric datum position parameters, it is natural to specify that our local parameters, it is natural to specify that our local geodetic system at the initial point have its origin on geodetic system at the initial point have its origin on the datum surface, that is on the ellipsoid. In the the datum surface, that is on the ellipsoid. In the classical (non-three-dimensional) approach the classical (non-three-dimensional) approach the orthometric height H enters into horizontal networks orthometric height H enters into horizontal networks only in the reduction of surface quantities to the only in the reduction of surface quantities to the geoid, therefore it is natural to take our local geoid, therefore it is natural to take our local astronomic system at the initial point to have its astronomic system at the initial point to have its origin on the geoid. Denoting quantities at the initial origin on the geoid. Denoting quantities at the initial point by zero subscript, we then see that the six point by zero subscript, we then see that the six datum position parameters are in this case the datum position parameters are in this case the geodetic coordinates of the local astronomic origin geodetic coordinates of the local astronomic origin ((00, , 00, N, N00*) and the rotation angles required to define *) and the rotation angles required to define the transformation between the local geodetic and the transformation between the local geodetic and local astronomic systems (local astronomic systems (00, , 00 , ,00)·)·

• 2.3.2 Establistment of a Datum 2.3.2 Establistment of a Datum • We have seen that a datum is defined by assigning We have seen that a datum is defined by assigning

values either to the eight parameters (a, b, xvalues either to the eight parameters (a, b, x00, y, y00, z, z00, , ww11, w, w22, w, w33) or to the eight parameters (a, b, ) or to the eight parameters (a, b, 00, , 00, , N*N*00, , 00, , 0,0, 00). However, an arbitrary set of values ). However, an arbitrary set of values will not in general result in a satisfactory datum. We will not in general result in a satisfactory datum. We recall that it is important that a datum closely app recall that it is important that a datum closely app roximate the geoid over the area of the network for roximate the geoid over the area of the network for which it is a datum, and that the geocentric axes of which it is a datum, and that the geocentric axes of the geodetic coordinate system be closely parallel to the geodetic coordinate system be closely parallel to the Average Terrestrial Axes, particularly that the the Average Terrestrial Axes, particularly that the axes of symmetry be parallel. The process of axes of symmetry be parallel. The process of assigning values to the eight datum parameters in assigning values to the eight datum parameters in such a way that these characteristics are obtained is such a way that these characteristics are obtained is called establishment of a datum. called establishment of a datum.

• To begin with, in establishing a datum values are always To begin with, in establishing a datum values are always assigned to the topocentric set (a, b, assigned to the topocentric set (a, b, 00, , 00, N*, N*00, , 00, , 0,0, 00) ) rather than the geocentric set (a, b, xrather than the geocentric set (a, b, x00, y, y00, z, z00, w, w11, w, w22, w, w33) ) because it is the set which is related to the geodetic and because it is the set which is related to the geodetic and astronomic measurements which we must use in astronomic measurements which we must use in establishing the datum. We see that we must somehow establishing the datum. We see that we must somehow choose values for (a, b, choose values for (a, b, 00, , 00, N*, N*00, , 00, , 0,0, 00) so that the ) so that the values of (N*,values of (N*,, , ) elsewhere in the network are not ) elsewhere in the network are not excessive (the datum approximates the geoid), and so that excessive (the datum approximates the geoid), and so that ww11 = w = w22 = w = w33 = 0 (the axes are parallel). Additionally for = 0 (the axes are parallel). Additionally for networks of global extent we require that xnetworks of global extent we require that x00 = y = y00 = z = z00 = 0, = 0, in vhich case the datum is termed a geocentric datum. in vhich case the datum is termed a geocentric datum. Otherwise the datum is a local datum. Otherwise the datum is a local datum.

• The problem of approximating the geoid can be ignored, in The problem of approximating the geoid can be ignored, in which case the values which case the values

N*N*00 = = 00 = = 00 = 0 = 0

are assigned, which forces the ellipsoid to intersect and be are assigned, which forces the ellipsoid to intersect and be tangent to the geoid at the initial point. tangent to the geoid at the initial point.

• The geoid can be approximated in two vays, by The geoid can be approximated in two vays, by choosing values of (a, b, N*choosing values of (a, b, N*00, , 00, , 00) such that either ) such that either values of (values of (, , ) or values of N* throughout the network ) or values of N* throughout the network are minimized (Vanicek, 1972). Note that values of are minimized (Vanicek, 1972). Note that values of (N*,(N*,, , ) are available throughout the netvork only if ) are available throughout the netvork only if some adjusted network already exists, which points some adjusted network already exists, which points up the iterative nature of datum establishment - a up the iterative nature of datum establishment - a "best fitting" datum can be established only as an "best fitting" datum can be established only as an improvement on an already existing datum. improvement on an already existing datum.

• The classical method of "ensuring" that the axes of The classical method of "ensuring" that the axes of symmetry are parallel is to enforce the Laplace symmetry are parallel is to enforce the Laplace azimuth condition at the initial point, that is to assign azimuth condition at the initial point, that is to assign a value to a value to 00 according to according to

00 = A = A00 - - 00 = = 00 tan tan 00 2-44 2-44

where Awhere A00 is an observed astronomic azimuth. is an observed astronomic azimuth.

• This condition forces the geodetic and astronomic This condition forces the geodetic and astronomic meridians to be parallel at the initial point, and thus meridians to be parallel at the initial point, and thus forces both axes of symmetry to lie in this common forces both axes of symmetry to lie in this common plane. However, the axes of symmetry can still be plane. However, the axes of symmetry can still be misaligned within the meridian plane. The solution misaligned within the meridian plane. The solution to this dilemma has been to apply the Laplace to this dilemma has been to apply the Laplace condition at several geodetic meridians parallel to condition at several geodetic meridians parallel to their corresponding astronomic meridians. In their corresponding astronomic meridians. In essence this constrains the adjusted network to essence this constrains the adjusted network to compensate for misalignment of the datum, rather compensate for misalignment of the datum, rather than ensuring that the datum minor axis is parallel than ensuring that the datum minor axis is parallel to the earth's rotation axis. Note that enforcing the to the earth's rotation axis. Note that enforcing the Laplace condition throughout the network presumes Laplace condition throughout the network presumes the existance of an adjusted network, which again the existance of an adjusted network, which again points up the iterative nature of datum points up the iterative nature of datum establishment.establishment.

Godetic DatumGodetic Datum

Geocentric Godetic DatumGeocentric Godetic Datum

Reference Ellipsoid

Geocentric and Non-Geocentric (Relative) Godetic DatumsGeocentric and Non-Geocentric (Relative) Godetic Datums

Non-Geocentric Non-Geocentric (Relative) (Relative) Geodetic Geodetic DatumDatum

• In terms of Cartesian coordinates, the geodetic In terms of Cartesian coordinates, the geodetic (G) coordinate system is that system which is (G) coordinate system is that system which is introduced into the earth such that its three axes introduced into the earth such that its three axes are coincident with or parallel to the are coincident with or parallel to the corresponding three axes of the average corresponding three axes of the average terrestrial system (see Figure 2-3). The first terrestrial system (see Figure 2-3). The first situation defines a geocentric geodetic system situation defines a geocentric geodetic system while the latter non-geocentric system is while the latter non-geocentric system is commonly referred to as a relative geodetic commonly referred to as a relative geodetic system, whose relationship to the average system, whose relationship to the average terrestrial system is given by the three datum terrestrial system is given by the three datum

translation componentstranslation components

0

0

0

0

z

y

x

r

and in vector equation form, the relationship is and in vector equation form, the relationship is

where the position vector rwhere the position vector rii is referred to the is referred to the geodetic system, that is geodetic system, that is

and and

2-42-4

ii rrR 0

G

i

z

y

x

r

GTAz

y

x

z

y

x

z

y

x

0

0

0

.

Eksenlerin paralel olması durumu ise genellikle bir Eksenlerin paralel olması durumu ise genellikle bir relatif jeodezik sistemirelatif jeodezik sistemi tanımlar. Ayrıca eksenlerin tanımlar. Ayrıca eksenlerin paralelliği de tam olarak sağlanamayabilir, bu paralelliği de tam olarak sağlanamayabilir, bu durumda eksen dönüklükleri söz konusudur. Bu en durumda eksen dönüklükleri söz konusudur. Bu en genel durumda elipsoid dik koordinatları x, y, z ile genel durumda elipsoid dik koordinatları x, y, z ile Ortalama Yersel Sistem arasındaki bağıntı vektörel Ortalama Yersel Sistem arasındaki bağıntı vektörel olarak (Şekil 2.10),olarak (Şekil 2.10),

(2.48)(2.48)

,, , , (2.49)(2.49)

şeklindedir.şeklindedir.

A C R f G ( )1 0

A

X

Y

Z

C

X

Y

Z

0

0

0

G

x

y

z

Eşitlikteki Eşitlikteki ffoo ölçek faktörüdür. R dönüşüm matrisi olup x, ölçek faktörüdür. R dönüşüm matrisi olup x, y, z eksenleri etrafındaki eksen dönüklükleri sırasıyla y, z eksenleri etrafındaki eksen dönüklükleri sırasıyla xx, , yy, , zz olmak üzere, yukarıda (2.6) da verilen rotasyon (Dönme) olmak üzere, yukarıda (2.6) da verilen rotasyon (Dönme) matrislerinde matrislerinde , , , , yerine sırasıyla yerine sırasıyla xx, , yy, , zz alınarak alınarak

R = RR = R11((xx) R) R22((yy) R) R33((zz)) (2.50)(2.50)

(2.51)(2.51)

şeklindedir. Eksen dönüklüklerinin küçük oldukları kabul edilir şeklindedir. Eksen dönüklüklerinin küçük oldukları kabul edilir ve bazı küçük terimler ihmal edilir ve ölçek faktörü de ve bazı küçük terimler ihmal edilir ve ölçek faktörü de dikkate alınırsa dönüşüm matrisidikkate alınırsa dönüşüm matrisi

(2.52)(2.52)

olur.olur.

100

0cossin

0sincos

cos0sin

010

sin0cos

cossin0

sincos0

001

zz

zz

yy

yy

xx

xxR

0

0

0

0

1

1

1

)1(

f

f

f

fR

xy

xz

yz

• Ortalama yersel sistemden jeodezik sisteme Ortalama yersel sistemden jeodezik sisteme dönüşüm (2-48) eşitliğindendönüşüm (2-48) eşitliğinden

eşitliği ile gerçekleştirilir.eşitliği ile gerçekleştirilir.

))(1( 01 CAfRG

2.3.3 Deflection of The 2.3.3 Deflection of The VerticalVertical

• The angle between the ellipsoid normal and the The angle between the ellipsoid normal and the gravity vertical is the deflection of the vertical, gravity vertical is the deflection of the vertical, which can be resolved into a component which can be resolved into a component in the in the geodetic meridian plane and a component geodetic meridian plane and a component in in the geodetic prime vertical plane (the plane the geodetic prime vertical plane (the plane perpendicular to the geodetic meridian plane perpendicular to the geodetic meridian plane which contains the ellipsoid normal).which contains the ellipsoid normal).

DDeflection of the verticaleflection of the vertical

Şekil 2.15- Helmert ve Pizetti İzdüşümleri ve çekül sapmaları

Çekül sapması bileşenleri Çekül sapması bileşenleri

(2.72)(2.72)

eşitlikleri ile verilmektedir. Şekil 2.16‘da lokal eşitlikleri ile verilmektedir. Şekil 2.16‘da lokal astronomik ve lokal jeodezik sistemlerin z eksenleri astronomik ve lokal jeodezik sistemlerin z eksenleri ile Ortalama Yersel Sistem ve Elipsoidal Sistemin ile Ortalama Yersel Sistem ve Elipsoidal Sistemin eksenleri arasındaki ilişkiler ve birim küre üzerinde eksenleri arasındaki ilişkiler ve birim küre üzerinde çekül sapması bileşenleri gösterilmektedir.çekül sapması bileşenleri gösterilmektedir.

Çekül sapması bileşenleriÇekül sapması bileşenleri

• Astronomik-jeodezik yöntemlerleAstronomik-jeodezik yöntemlerle

• Gravimetrik yöntemlerleGravimetrik yöntemlerle

• Topoğrafik-izostatik yöntemlerleTopoğrafik-izostatik yöntemlerle

belirlenir.belirlenir.

cot)(

cos)(

A

Şekil 2.16- Ortalama Yersel Sistem, Elipsoidal Sistem, Şekil 2.16- Ortalama Yersel Sistem, Elipsoidal Sistem, Lokal Astronomik Sistem veLokal Astronomik Sistem ve Lokal Jeodezik Sistemin z Lokal Jeodezik Sistemin z eksenleri ve Çekül sapması bileşenleri. eksenleri ve Çekül sapması bileşenleri.

• Astronomik-jeodezik yöntemlerle belirlenen çekül Astronomik-jeodezik yöntemlerle belirlenen çekül sapmaları Astro-jeodezik çekül sapması olarak sapmaları Astro-jeodezik çekül sapması olarak adlandırılır. Bir P noktasının astronomik enlem, adlandırılır. Bir P noktasının astronomik enlem, boylam ve azimutu astronomik gözlemelerle boylam ve azimutu astronomik gözlemelerle belirlenir. Bu noktanın elipsoidal enlem ve belirlenir. Bu noktanın elipsoidal enlem ve boylamı yeryüzünde yapılan açı ve kenar ölçmeleri boylamı yeryüzünde yapılan açı ve kenar ölçmeleri ile nirengi ağları kurularak hesaplanır. Astronomik ile nirengi ağları kurularak hesaplanır. Astronomik yöntemlerle bulunan enlem, boylam ve azimut ile yöntemlerle bulunan enlem, boylam ve azimut ile jeodezik ölçülerle elipsoid üzerinde yapılan jeodezik ölçülerle elipsoid üzerinde yapılan hesaplarla bulunan elipsoidal enlem, boylam ve hesaplarla bulunan elipsoidal enlem, boylam ve azimutun farkları ile elde edilen çekül sapması azimutun farkları ile elde edilen çekül sapması bileşenleri “bileşenleri “Astro-jeodezik çekül sapması Astro-jeodezik çekül sapması bileşenleribileşenleri” olarak adlandırılır. ” olarak adlandırılır.

• Hesapların yapıldığı referans elipsoidinin Ortalama Hesapların yapıldığı referans elipsoidinin Ortalama Dünya Dik Koordinat Sistemine göre konumuna göre Dünya Dik Koordinat Sistemine göre konumuna göre astro-jeodezik çekül sapması bileşenleri rölatif çekül astro-jeodezik çekül sapması bileşenleri rölatif çekül sapmaları veya mutlak çekül sapmaları olarak elde sapmaları veya mutlak çekül sapmaları olarak elde edilir. Eğer referans elipsoidinin merkezi yeryuvarının edilir. Eğer referans elipsoidinin merkezi yeryuvarının ağırlık merkezinde ve elipsoidal dik koordinat ağırlık merkezinde ve elipsoidal dik koordinat sisteminin x. y, z eksenleri Ortalama Dünya Dik sisteminin x. y, z eksenleri Ortalama Dünya Dik Koordinat Sisteminin X, Y, Z eksenleri ile çakışık ise Koordinat Sisteminin X, Y, Z eksenleri ile çakışık ise bu elipsoide göre hesaplanan çekül sapmaları bu elipsoide göre hesaplanan çekül sapmaları ““Mutlak Çekül SapmalarıMutlak Çekül Sapmaları” adını alır. Eğer referans ” adını alır. Eğer referans elipsoidinin merkezi yeryuvarının ağırlık merkezinin elipsoidinin merkezi yeryuvarının ağırlık merkezinin dışında ise bu elipsoide göre hesaplanan çekül dışında ise bu elipsoide göre hesaplanan çekül sapmaları “sapmaları “Rölatif Çekül SapmalarıRölatif Çekül Sapmaları” adını alır.” adını alır.

2.2.44 TERRESTRIAL TOPOCENTRIC TERRESTRIAL TOPOCENTRIC SYSTEMS SYSTEMS

In the introduction it was stated that terrestrial In the introduction it was stated that terrestrial topocentric systems are defined as follows: topocentric systems are defined as follows:

• a) the origin is at a point near the surface of the a) the origin is at a point near the surface of the earth, earth,

• b) the primary plane is the plane tangential to the b) the primary plane is the plane tangential to the earth's surface at the point, earth's surface at the point,

• c)c) the primary axis is the north point, the primary axis is the north point, • d)d) the systems are left-handed. the systems are left-handed.

The last two specifications present no problems. However, "the surface of the earth" can be interpreted in three ways to mean the earth's physical surface, the earth's equipotential surface, or the surface of a reference ellipsoid. It is not practical to define a coordinate system in terms of a plane tangential to the earth's physical surface. 'Two kinds of terrestrial topocentric coordinate systems can be defined, however: The system in which the primary pole is the normal to the equipotential surface at the observation station is called a local astronomic system; The system in which the primary pole is the ellipsoid normal passing through the observation station is called a local geodetic system [Krakiwsky 1968].

2.2.44.1 Local Astronomic System .1 Local Astronomic System

A local astronomic (L.A.) system is specified: A local astronomic (L.A.) system is specified:

• a) a) The origin is at the observation station. The origin is at the observation station.

• b) b) The primary pole (z-axis) is the normal to the The primary pole (z-axis) is the normal to the equipotential surface (the gravity vertical) at the equipotential surface (the gravity vertical) at the observation station. The primary plane is the plane observation station. The primary plane is the plane containing the origin and perpendicular to the containing the origin and perpendicular to the gravity vertical. gravity vertical.

• c) c) The primary axis (x-axis) is the intersection of the The primary axis (x-axis) is the intersection of the primary plane and the plane containing the average primary plane and the plane containing the average terrestrial pole and the observation station, and is terrestrial pole and the observation station, and is called the astronomic north. called the astronomic north.

• d) d) The y-axis is directed east to form a left-handed The y-axis is directed east to form a left-handed system.system.

• The position vector of an observed station l, The position vector of an observed station l, expressed in the local astronomic system of the expressed in the local astronomic system of the observation station k, is given byobservation station k, is given by

2-542-54

where where rrklkl is the terrestrial spatial distance, v is the terrestrial spatial distance, vklkl the the vertical angle, and vertical angle, and AAklkl the astronomic azimuththe astronomic azimuth

• If the zenith angle If the zenith angle is measured, the position is measured, the position vector is given byvector is given by

2-552-55

kl

klkl

klkl

kl

AL

ALkl

v

Av

Av

r

z

y

x

r

sin

sincos

coscos

)(

..

..

kl

klkl

klkl

kl

AL

ALkl A

A

r

z

y

x

r

cos

sinsin

cossin

)(

..

..

oror

2-56a2-56a

2-56b2-56b

2-56c2-56c

• where where klkl is the zenith angle which is corrected is the zenith angle which is corrected for refraction (Fig. 2.9).for refraction (Fig. 2.9).

klklkll ArX cossin

klklkll ArY sinsin

klkll rZ cos

Figure 2.9 : Local Astronomic and Average Terrestrial Figure 2.9 : Local Astronomic and Average Terrestrial Coordinate Systems.Coordinate Systems.

• Refraction correction Refraction correction klkl is given by is given by

2-572-57wherewhere

2-582-58

andand ;; ; ; ;;

• RRklkl is the radius of curvature of the geodesic passing is the radius of curvature of the geodesic passing through k and l , Athrough k and l , Aklkl is the azimuth of the geodesic at is the azimuth of the geodesic at k and S is the length of the geodesic. k and S is the length of the geodesic. kkkk is the is the refraction coefficient at K. Corrected zenith distance is refraction coefficient at K. Corrected zenith distance is given bygiven by

2-592-59

where where ** is measured zenith distance (Fig. 2.10). is measured zenith distance (Fig. 2.10).

kklkl k

kl

klkl R

S2

klkl

klAMAN

MNR

22 sincos

3V

cM

V

cN

klkl

Figure 2.10: Measured zenith distance, refraction Figure 2.10: Measured zenith distance, refraction correction and corrected zenith distancecorrection and corrected zenith distance

hP

hK

R

S

K

P

2

*

• Note that the relationship of the local astronomic Note that the relationship of the local astronomic system to the average terrestrial system is given by system to the average terrestrial system is given by the astronomic latitude the astronomic latitude kk and longitude and longitude kk only only after the observed quantities after the observed quantities kk, , kk, A, Aklkl have been have been corrected for polar motion. Thus the position vector corrected for polar motion. Thus the position vector of 2-54 expressed in the average terrestrial system of 2-54 expressed in the average terrestrial system is: is:

2-602-60

where the reflection matrix where the reflection matrix

2-612-61

accomplishes the transformation from a left-handed accomplishes the transformation from a left-handed system into a right-handed system, while the rotation system into a right-handed system, while the rotation matrices matrices

..

223

..

.. )90()180()(

AL

ko

ko

TA

TAkl

z

y

x

PRR

z

y

x

r

100

010

001

2P

2-622-62

andand

2-632-63

bring the three axes of the local astronomic system bring the three axes of the local astronomic system parallel to the corresponding axes in the average parallel to the corresponding axes in the average terrestrial system. terrestrial system. These equations are given as followsThese equations are given as follows

2-64a2-64a

2-64b2-64b

2-64c2-64c

)90cos(0)90sin(

010

)90sin(0)90cos(

2

ko

ko

ko

ko

R

100

0)180cos()180sin(

0)180sin()180cos(

3 ko

ko

ko

ko

R

kklklkklkl ZYXXX coscossincossin

kklklkklkl ZYXYY sincoscossinsin

kllklkl ZYXZZ sin0*cos

The inverse transformation is The inverse transformation is

2-652-65

2-662-66

• Note that so far no translations have taken place. Note that so far no translations have taken place. We have merely rotated the position vector (rWe have merely rotated the position vector (rklkl) of ) of station 1 with respect to station k into the average station 1 with respect to station k into the average terrestrial system. If the position vector of station k terrestrial system. If the position vector of station k with respect to the centre of gravity in the average with respect to the centre of gravity in the average terrestrial system is (Rterrestrial system is (Rkk))A.TA.T.., then the total position , then the total position vector Rvector Rll of the observed station 1 with respect to of the observed station 1 with respect to the centre of gravity in the average terrestrial the centre of gravity in the average terrestrial system is given by system is given by

2-672-67

..

1

223.. )()90()180()( TAklko

ko

ALkl rPRRr

..322.. ))(180()90()( TAklko

ko

ALkl rRRPr

...... )()()( TAklTAkTAl rRR

The local astronomic coordinate system is unique The local astronomic coordinate system is unique for every observation point. Because of this fact, for every observation point. Because of this fact, this system is the basis for treating terrestrial this system is the basis for treating terrestrial three-dimensional measurements at several three-dimensional measurements at several stations together in one solution. stations together in one solution.

The inverse transformation is given byThe inverse transformation is given by

2-712-71

where where

2-722-72

and the product R is orthogonal.and the product R is orthogonal.

kl

kl

klT

lZZ

YY

XX

R

Z

Y

X

PRRR kk )90()180( 23

This equation can be written as followsThis equation can be written as follows,,

2-73a2-73a

2-73b2-73b

2-73c2-73c

where where X= XX= Xll-X-Xkk , , Y= YY= Yll-Y-Ykk , , Z=ZZ=Zll-Z-Zkk . .

The relations between Cartesian coordinates and The relations between Cartesian coordinates and curvilinear coordinates of the point l with respect to curvilinear coordinates of the point l with respect to the local astronomic system at k are given bythe local astronomic system at k are given by

2-742-74

kkkkkl ZYXX cossinsincossin

kkl YXY cossin

kkkkkl ZYXZ sinsincoscoscos

l

lkl X

YA arctan 222 ZYXrkl

kl

lkl r

Zarccos

• And substituting the right sides of the equations (2-And substituting the right sides of the equations (2-73a,b,c) instead of in these equations (2-74) we get the 73a,b,c) instead of in these equations (2-74) we get the relations between astronomic azimuth, spatial distance relations between astronomic azimuth, spatial distance and zenith angle of the observed station l and cartesian and zenith angle of the observed station l and cartesian coordinates of the points in the Average Terrestrial System coordinates of the points in the Average Terrestrial System and astronomic latitude and longitude of the observing and astronomic latitude and longitude of the observing station k as follows;station k as follows;

2.75a2.75a

2.75b2.75b

2.75c2.75c

These are the basic equations in three dimensional These are the basic equations in three dimensional geodesy, and they represante the relations between the geodesy, and they represante the relations between the cartesian coordinates with respect to Average Terrestrial cartesian coordinates with respect to Average Terrestrial System and measured zeninth distance (corrected for System and measured zeninth distance (corrected for refraction), astronomic azimuth and spatial distance.refraction), astronomic azimuth and spatial distance.

kkkkk

kkkl ZYX

YXA

cossinsincossin

cossinarctan

222 ZYXrkl

kl

kkkkkkl r

ZYX

sinsincoscoscosarccos

2.2.44.2. Local Geodetic System .2. Local Geodetic System A local geodetic (L.G.) system is specified (see A local geodetic (L.G.) system is specified (see Figure 2-11 and 2-12):Figure 2-11 and 2-12):

• a) The origin lies along the ellipsoidal normal a) The origin lies along the ellipsoidal normal passing through the observation station. Note that passing through the observation station. Note that in principle the origin may lie anywhere along the in principle the origin may lie anywhere along the ellipsoidal normal. In practice it is chosen to be at ellipsoidal normal. In practice it is chosen to be at the observation station, at the ellipsoid, or at the the observation station, at the ellipsoid, or at the intersection of the ellipsoidal normal with the geoid. intersection of the ellipsoidal normal with the geoid.

• b) The primary pole (z-axis) is the ellipsoidal normal. b) The primary pole (z-axis) is the ellipsoidal normal. The primary plane is the plane containing the origin The primary plane is the plane containing the origin and perpendicular to the primary pole. and perpendicular to the primary pole.

• c) The primary axis (x-axis) is the intersection of the c) The primary axis (x-axis) is the intersection of the primary plane and the plane containing the semi-primary plane and the plane containing the semi-minor axis of the ellipsoid and the origin, and is minor axis of the ellipsoid and the origin, and is called the geodetic north. called the geodetic north.

• d) The y-axis is directed east to form a left-handed d) The y-axis is directed east to form a left-handed system. system.

Figure 2-11: Geodetic and Local Geodetic Coordinate Systems

• Transformations between local geodetic and local Transformations between local geodetic and local astronomic systems sharing a common origin can be astronomic systems sharing a common origin can be expressed in terms.of the angle between the ellipsoidal expressed in terms.of the angle between the ellipsoidal normal and gravity vertical (the deflection of the vertical) normal and gravity vertical (the deflection of the vertical) and the angle between the geodetic north and astronomic and the angle between the geodetic north and astronomic north.north.

• Given the meridian and prime vertical deflection Given the meridian and prime vertical deflection components components , , respectively, and the geodetic and respectively, and the geodetic and astronomic azimuths astronomic azimuths , A to a particular point, then a , A to a particular point, then a vector in the local astronomic system is transformed into a vector in the local astronomic system is transformed into a vector in the local geodetic system by vector in the local geodetic system by

2-762-76• Note that the order in which the rotations are performed in Note that the order in which the rotations are performed in

this case is not important, since the angles this case is not important, since the angles , , , (A - , (A - ) are ) are small enough that their rotation matrices can be assumed small enough that their rotation matrices can be assumed to commute. Note also that if the Laplace condition is to commute. Note also that if the Laplace condition is enforced at the origin of these local systems, we have.enforced at the origin of these local systems, we have.

2-772-77

..123.. ))(()()()( ALklkGLkl rRRARr

tansin)( A

Similar to equations 2-54 and 2-60 the position Similar to equations 2-54 and 2-60 the position vector from observing station k to observed station vector from observing station k to observed station 1 is given by 1 is given by

2-782-78

and and

2-792-79

where (a, where (a, , r) are the geodetic altitude, azimuth , r) are the geodetic altitude, azimuth and range, and (and range, and (, , ) are the geodetic latitude and ) are the geodetic latitude and longitude.longitude.

kl

klkl

klkl

kl

GL

GLkl

a

a

a

r

z

y

x

r

sin

sincos

coscos

)(

..

..

..

223

.

. )90()180()(

GL

ko

ko

G

Gkl

z

y

x

PRR

z

y

x

r

The inverse transformation is The inverse transformation is

2-802-80

2-812-81

Note that the geodetic system (G) and the average Note that the geodetic system (G) and the average terrestrial system (A.T.) are related by equation 2-4 terrestrial system (A.T.) are related by equation 2-4

2-822-82

where (xwhere (x00, y, y00, z, z00) are the translation components of ) are the translation components of the origin of the geodetic system in the average the origin of the geodetic system in the average terrestrial system. terrestrial system.

Gklko

ko

GLkl rPRRr )()90()180()(1

223..

Gklko

ko

GLkl rRRPr ))(180()90()( 322..

GTAz

y

x

z

y

x

z

y

x

0

0

0

.

Similar to Local Astronomic system the position vector Similar to Local Astronomic system the position vector from observing station k to observed station from observing station k to observed station ll is is given by given by

2-842-84

when the zenith angle is measured as vertical angle.when the zenith angle is measured as vertical angle. is the ellipsoidal azimuth, is the ellipsoidal zenith is the ellipsoidal azimuth, is the ellipsoidal zenith angle, angle, rr is the spatial distance, is the spatial distance, and and are the ellipsoidal are the ellipsoidal latitude and longitude respectively.latitude and longitude respectively.The ellipsoidal zenith angle The ellipsoidal zenith angle is given byis given by

2.852.85where where ββ is the measured zenith angle with respect to is the measured zenith angle with respect to gravity vertical and corrected for refraction, gravity vertical and corrected for refraction, and and are the components of the deflection of the vertical at are the components of the deflection of the vertical at observing station k,observing station k, is the ellipsoidal azimuth. is the ellipsoidal azimuth.

kl

klkl

klkl

kl

GL

GLkl

a

r

z

y

x

r

cos

sinsin

cossin

)(

..

..

)sincos( klkklkklkl

The relations between cartesian coordinates and The relations between cartesian coordinates and curvilinear coordinates in the local geodetic system can curvilinear coordinates in the local geodetic system can be written in similar manner to the equations for the local be written in similar manner to the equations for the local astronomic systemastronomic system

• 2.86a2.86a• 2.86b2.86b• 2-86c2-86c

Note that the relationship of the local geodetic system to Note that the relationship of the local geodetic system to the geodetic (ellipsoidal) system is given by the geodetic the geodetic (ellipsoidal) system is given by the geodetic latitude latitude kk and longitude and longitude kk. Thus the position vector . Thus the position vector of 2-67 expressed in the geodetic system can be written of 2-67 expressed in the geodetic system can be written as follows, as follows,

2-87a2-87a

• 2.87b2.87b

• 2.87c2.87c

klklkll rx cossin

klklkll ry sinsin

klkll rz cos

klr

kklklkklkl zyxxx coscossincossin

kklklkklkl zyxyy sincoscossinsin

kllklkl zyxzz sin0cos

The inverse transformation is given byThe inverse transformation is given by

2.88a2.88a

2.88b2.88b

2.88c2.88c

where where x = xl-xk , x = xl-xk , y= yl-yk , y= yl-yk , z=zl-zk z=zl-zk

The relations between cartesian coordinates and The relations between cartesian coordinates and curvilinear coordinates of the point l with respect curvilinear coordinates of the point l with respect to the local geodetic system at k are given byto the local geodetic system at k are given by

,, , , 2.892.89

kkkkkl zyxx cossinsincossin

kkl yxy cossin

kkkkkl zyxz sinsincoscoscos

l

lkl x

yarctan 222 zyxrkl

kl

lkl r

zarccos

And substituting the right sides of the equations And substituting the right sides of the equations (2-88a,b,c) instead of in these equations we get the (2-88a,b,c) instead of in these equations we get the relations between geodetic azimuth relations between geodetic azimuth , spatial , spatial distance r and zenith angle distance r and zenith angle ’ of the observed ’ of the observed station l and cartesian coordinates of the points in station l and cartesian coordinates of the points in the geodetic system and geodetic latitude and the geodetic system and geodetic latitude and longitude of the observing station k as follows;longitude of the observing station k as follows;

2.90a2.90a

2.90b2.90b

2.90c2.90c

kkkkk

kkkl zyx

yx

cossinsincossin

cossinarctan

222 zyxrkl

kl

kkkkkkl r

zyx

sinsincoscoscosarccos