Γεωδαισία

2007

-

i

2 13 2.1 15 2.2 16 2.3 18 2.3.1 18 2.3.2 19 2.4 24 2.4.1 Newton- 24 2.4.2 26 2.4.3 29 2.5 30 2.5.1 31 2.5.2 32 36 3 - 37 3.1 39 3.1.2 40 3.1.3 42 3.1.4 () 44 3.1.5 45 3.2 46 3.3 ..

47

53 4 55 4.1 57 4.2 59 4.2.1 59 4.2.2. 60 4.3 63 4.4 65 4.4.1 67 4.5 69 4.6 70 4.7 : 70 4.8 71 4.9 72 4.9.1. 87' 72 4.9.2 () 73

11 3 1.1 - 5 1.2 5 1.3 - 9

4.9.3 50 (European Datum 1950)` 74 4.10 74 4.11 74 76 5 77 5.1 79 5.2 79 5.2.1 79 5.2.2 83 5.3 () 85 5.3.1 87 5.3.2 - 87 5.4 88 5.5 91 5.6 91 5.7 : 92 5.8 I 93 5.9 94 98 6 996.1 1016.1.1 1016.1.2 - 1026.1.3 1026.2 1046.2.1 1046.2.2 1056.2.3 1076.2.4 1096.2.5 1116.2.6 1126.2.6.1 1126 .2.6.2 1126.3 1146.3.1 1146.3.2 x 1166.3.3 A posteriori o 1176.3.4 1186.3.4.1 1186.3.4.2 119 122 7 1237.1 1257.1.1 1257.1.2 125

ii

iii

7.1.3 1267.2 1297.2.1 1297.2.2 1297.2.3 1307.2.4 1327.3 1367.3.1 1367.3.2 1367.3.3 1377.3.4 1387.3.5 1387.3.6 1407.3.7 1417.3.8 1427.4 1437.4.1 1437.4.2 1447.4.3 148 150 8 1518.1 1538.2 - - 1538.2.1 1558.2.2 1578.2.3 - 1588.2.4 1608.2.5 - 163

, (), () . , , . , , , , - . ( ) / / , , , , . . . , .

2006

1

1.

.

3

1.1

,

, (), ()

.

,

.

(...) .

, ,

.

, ,

.

1.2

,

,

, .

.

(900-800 ..)

.

, 580-500 .. .

( ).

, ,

. (384-322 .)

,

,

5

, , ,

-.

(276-192

..) ,

.

(.1.1)

( )

, ,

7 12 ,

..

5000

,

. -

, .

1.1. , 3 .. .

6

6267 km 7380 km

, = 157.7 m = 185 m,

2% + 15.5%

(6371 km).

, 1670 .., Picard (1620-1683)

6372km +0.01% Rm.

Newton

.

, (1672)

I. Richer,

2.8mm

Gayenne ( , . , 5 )

, 15 Newton

.

, , , Newton

(1643-1727) Huygens (1629-1695) (

) 1/230 1/578,

1/298.

Baeyer Bessel , -

(1831-1833), Bessel (1840)

, a=6378397,155m f=1/299.153.

.

19 20

, ( -

7

, )

.

: Helmert (1907)

=6378200m f=1/298.3 J.F. Hayford (1909)

=6378388m f=1/297.0.

,

. (IUGG)

(1924) Hayford

.

( Datum 1950, E.D.50),

Potsdam.

GRS

1967, GRS 1980 =6378137 m

f=1/298.257.

1950,

,

,

,

,

-

, .

8

1.3

, ,

,

.

( )

.

/ , .

, (

), ,

.

.

. ()

( ) () ()

(). (h)

.

,,h

.

,

.

.

9

Laser GPS (Global

Positioning System)

, /

( )

(X,Y,Z) (,,h)

.

( 4 ).

( ) ,

(

). ,

,

-

.

, ,

,

.

..., ( )

() ,

,

...,

,

,

, .

,

10

,

.

(

) ,

TRANSIT

GPS,

.

.

11

2.

.

13

2.1

( :

, , , ).

4.5 -

. , -

.

2.1. . , -

NASA 70% ,

(...)

28% .

.

. ,

R = 6371km. (, )

.

:

- : 5.110 18 cm2

- : 5.9742 10 27 gr

- : 2.67gr/cm3 (

1.027gr/cm3).

.

- 11180km/sec.

15

2.2

. 4 :

E : Fe-Ni.

1220km. 13 gr/cm3 .

E : Fe-Ni

2265 km. : 9.9 gr/cm3 12.7 gr/cm3 .

: -

~2850km. 5.7 gr/cm3

.

km

2.2

km

:

.

5km 65km .

3.3 gr/ cm3

.

-

Mohorovii ( Moho).

( ~10km ,

km).

16

2.3

400km -

. ( 50-100 km)

.

().

.

.

( ).

50 -100km 300-400km

. , ( 9000F

, ).

.

, .

.

: (compressional) P (shear) S.

. P.

17

. ,

.

2.3

. .

, ,

.

2.3.1

, ,

.

.

29.8km/sec.

,

(, ). ,

, 100 000

0.01 0.05. 0.017,

.

,

. , ,

,

Kepler.

:

18

() .

()

. ,

.

() ()

(a) -

:

2/a3 = .

.

( 365,242199 ).

147 090 000km .

4 .

,

2.4 152 000 000 km

2 Kepler .

2.3.2

H ,

0.997270 (. ) .

().

19

() 23 27

.

-

2.5

2.6

23 27

22.1 24.5

700 000 .

-

,

.

,

.

20

.

. ,

2.7

23.5 ,

465m/sec. , ,

. 2

12 .

.

.

,

)

( 14000 )

3000

(

. .

25.800 -

21

2.8

23. 5

47

& &

.

(

). , ,

( ) 90

.

( ).

5 .

. 18.6

17".

22

2.9 1996-2000. 1890-2000.[RS]

( ).

: 14 ( Chandler),

. -

.

)

. ,

IERS (International

,

.

, , 9

, 1899 (International Latitude Service, ILS

Earth Rotation Service).

23

.4

Isaac Newton 17 .

Newton :

,

,

= m =F/m

Newton 1Newton : 1kg

,

.

22.4.1 Newton

, , Galileo Galilei

Johanes Kepler .

, .

1

.

Galileo.

2

.

.

F

3

.

1m/sec2.

24

. Newton

m.

, ,

g =9.81m/sec2 981gal ( Galileo 1gal = 1cm/sec2 1mgal

= 10-3 gal), .

Newton,

,

.

.

F

F

m1

m2r

:

FG m m

r= 1 22

m1 m2 r.

F.

G

. .

Newton ,

,

. ,

, , ,

.

,

. ,

.

25

m , :

G m / R2

R R~6371km,

, .

.

, , .

.

, .

, , ,

. ,

g

.

2.4.2

.

. ,

,

.

.

.

, ,

. .

R = 6371.009km GM = 398600441.5106 m3/sec2 ( International Association of Geodesy, IAG, 1995),

:

26

F = 982.022 [cm/sec2] m,

m .

, . ,

. F .

,

.

.

.

.

, , .

, ,

.

:

f = p 2 m,

: p

.

: ]

0.35%

m

f = 3.392[cm /sec2 m

. ,

.

.

pr

f

&

2.10

27

( = 0)

. 0.35% .

.

(,

)

. , ,

.

rg

,

gals. 980.3gals.

rg

, ,

1gal (1gal = 10-6 gal).

rg

,

.

.

.

, Bureau Gravimetrique

International (BGI), International Union of Geodesy and

Geodynamics (IUGG), . rg , ,

.

rg

:

, ,

5200mgal.

28

,

( ),

1% g 30km.

:

rg

[ ]d 0.308rg = mGal / m dH

,

,

.

2.4.3

(), , ,

(

) .

. ,

.

, ,

.

. r

,

,

.

:

( ).

29

2.5

. , , ,

.

:

( ),

( sec )

( ).

, ,

.

, ,

,

.

: ( )

( ).

(), .

, , , ,

( ).

,

..

30

2.5.1

,

A AB

2.11

. ( )

( ).

( ),

.

. .

, , ,

(

) , . ,

.

, ,

( -

)

- ( 2.11).

.

,

, .

31

, , -

. , ,

,

- .

.

0.5

22 397 .

1.5millisec

.

, 4-40cm,

.

2.5.2

4.5 -

.

.

2.13

,

-

.

( ).

.

32

.

.

250 . .

()

.

2.13

.

, , ,

, .

.

( , ,

).

, ,

.

280-240.

200.

180-140 . 65 .

:

, ,

, (

)

,

,

33

2.14 ., , 1999]

,

... (

...).

.

.

[

,

, , ,

( )

( ) (. 2.15) (

)

34

0cm

25cm

50cm

75cm

100c

m

92-70

2330

'

0'

20Km

30'

00'

2200

'

223 0

'

230 0

'

2.15 1970-1995

35

. ., ., . , , , 1991.

2. ., ., ., ,

3. , ,

1. earch/EarthRot.html

1.

, , , 2006. B.. ., 1997. pp. 304, , .

http://www.geo.tudelft.nl/fmr/res . http://imagine.gsfc.nasa.gov/docs/science/science.html2

3. http://stardate.org/resources/ssguide/earth.html 4. http://www-istp.gsfc.nasa.gov/stargaze/Sintro.htm 5. http://www.windows.ucar.edu/ 6. http://observe.arc.nasa.gov/nasa/entries/entry_5.html

ntro.html

l rgaze/Sintro.htm

tml

7. http://cse.ssl.berkeley.edu/lessons/indiv/beth/beth_i8. http://www.iers.org/iers/earth/ 9. http://www.deos.tudelft.nl/general/toc.shtml 10. http://www.colorado.edu/geography/gcraft/notes/datum/datum_f.htm11. http://www-istp.gsfc.nasa.gov/sta12. http://164.214.2.59/GandG/pubs.html 13. http://www.auslig.gov.au/geodesy/datums/aboutdatums.htm 14. http://www.earth.nasa.gov/ 15. http://www.csr.utexas.edu/gmsl/main.h16. http://einstein.gge.unb.ca/tutorial/tutorial.htm

36

3. .

.

37

3.1

,

.

. (..)

.

72% 28%

. ,

. ,

,

.

,

, ,

...,

. ( 6371 km)

() 9 km,

1/1400.

1400 mm,

1mm, .

(, , , -

...) ,

, (.. ),

o

. , ,

,

, ,

.

,

39

.

(

) 6371 km .

,

.

.

,

.

()

,

() .

(...),

. ,

... ,

, :

( ), (

) ( ).

3.1.2

,

, -

.

40

( ). ,

, ( )

.

()

.

..., ()

.

(..) , ,

,

( 20cm

), ,

, ,

, .

20.000 100m,

, ,

, 1-2 mm

.

,

,

,

.

...,

cm.

41

3.1. .

.

()

1m

.

..., 1m..

3.1.3

,

.

, ,

.

.

, (

) 100 , 15

30m .

42

(

) .

.

() ().

, ...

.

.

(, , ,

..). ,

. .

(),

,

. , , ..

, (

)

,

, .

.

(h).

3.2.

.

43

3.1.4 ()

, a,

, f,1 b, -

e. 3.1, -

.

Bessel (),

Hayford ( ),

(ED50) GRS 80 (Geodetic Reference System 1980)

( 1987).

, ,

1980,

.

,, -

:

(3.1) + + =

2 2 2x y z

12 2 2a a b

().

,

, ( )

/ -

.

100m, 30m.

1 aabf =

( ) 1/ 22 2 2a / ae b =

44

, ,

.

3.1.

.. a 1/f

Everest (1830) 6377276 300,8

Airy (1830) 6376542 299,3

Bessel (1841) 6377397 299,15

Clarke (1866) 6378206 294,98

Clarke (1880) 6378245 293,5

Helmert (1907) 6378200 298,3

Hayford International (1924) 6378388 297,0

Krassowsky (1942) 6378245 298,3

GRS (1967) 6378160 298,247

WGS (1972) 6378135 298,26

Gaposhkin (1973) 6378140 298,256

GRS (1980) 6378137 298,25722

3.1.5

... ,

... , ( )

,

...

.

45

R ( R =6371 km),

:

(3.2) 2222 RZYX =++

O

, -

.

3.2

,

.

,

-

, (

10km x 10km),

, -

.

(...).,

.

,

.

... ,

( ) ..

max=104m, ... 1m.

46

Nmax 102m =1,6x10-5R.

Nmax 1,5x104m 2,4x10-3R, .

3.3. ... .

...,

, : ,

, ,

.

.

,

..., (.3.3)

... .

3.3. .

() ...

,

( ...).

10 10.

47

()

( ..). (h)

,

() .

: (3.3)

h = Ho + N

.

(3.3) .

-

,

GPS, -

, -

( ) .

,

. ()

( ) ( )

, , .

( ) .

,

(

).

48

0.

3.4. .

, (. 3.4,)

x y, ,

() , y

x y

.

x,y

Ho, .

x

y

.

,

.

.

, ,

.

( )

.

49

. 3.5,

, ...,

R.

3.5.

(,) .

' ..

():

=

=

() Ho h :

() P

,

0 90

0 90 (

).

()

Greenwich (

)

0 360 . ( 0

180 0 -180 ).

50

...

...,

.

...

... ...,

(.

3.6).

3.6.

(,) ...

, ...

:

:

: :

. 0

() + 90 (. ) 0 () 90 (. ).

Greenwich ,

0 360 . ...,

() ,

h ...

.

.

51

.

, ,

, (. 3.7)

, :

3.7.

(,).

:

:

:

... ()

, () (,

),

(,)

Ho. ( ).

52

1. ., ., ., , 2005.

2. Smith J., Introduction to Geodesy. J Wiley and Sons, Inc. 1997 3. orge W., ..., 2000

4. Torge W., Geodesy. Walter de Gruyter, 3rd Edition, Berlin, New York 2001

53

4.

.

55

4.1

,

.

..., /

.

,

,

( ) .

. ,

,

.

, ,

,

() ,

.

,

.

,

(),

...

..., .

, ()

(

).

, ,

:

57

) ,

() ().

) .. ,

(, ).

)

(, , , ).

.

, (,

, , ),

, ( )

.

, , ,

, ( ()

),

.

, .

(.. )

,

,

.

. ,

87 ( 87).

,

.

58

4.2

.

(

),

()

.

.

. 17 ,

.

4.2.1

(.4.1)

.

y. x y

. P

x y , (

) (x,y) P.

(,)

4.1.

59

(, ) (. 4.1),

P ()

oy.

.

:

= + = =

2 2x yy xarcsin arccos

(4.1) = =

x siny cos

4.2.2

. . ()

.

. 2, P

X (), Y () Z ().

4.2.

60

()

, ,

(. 4.3).

4.3.

:

= =

=

cosY sin

= + = =

=

2 2

Yarccos arcsin (4.2)

(. 4.4) P r,

, .

(4.3).

4.4

61

= = r cos cos

Y r sin cos = rsin

2 2 2r = + +

= arcsinr

= arccos = arcsinrcos rcos

(4.3)

,

( R)

.

(.4.5).

4.5

P,

(a f), () :

( ) ( ).

P(X,Y,Z,t), P

t ,

, , .

(. mm/ ).

62

4.3

,

-

, , -

, .

3

(2 1 ),

6 (3 3

).

:

(,,), (,,h)

(x,y)

0.

.

( )

.

, ,

( ),

.

,

( ) , ...,

(

),

.

63

, , ,

.

(3 , 1

)

DATUM, , .

.

()

() (), ()

(). . ,

,

10 ( ).

.

GPS,

(ITRS) .

,

,

() .

(),

. (

)

.

.

: , .

64

4.4

, ,

Greenwich (.4.6).

, (

) ,

, ,

Greenwich (.4.6).

P ...

X,Y,Z, () -

(,) P ( P ) -

h.

.

P (0-90)

(0- -90).

()

Greenwich P.

Greenwich 0-360 .

() (h)

..., (. 4.6).

4.6 . P

65

:

(3) (3)

, (3), (1)

(2) a f.

() ,

, ( , .)

.

:

a) ITRF (International Terrestrial Reference Frame),

, ,

1900-1906,

Greenwich, .

( 1500) , / (VLBI, SLR GPS).

b) WGS84 (World Geodetic System 1984),

GPS ,

() GPS . 1994

ITRF

ITRF

GRS 80, ()

.

5,

66

4.4.1

, ,

.

(

)

, (

),

, .

( GRS 80) ,

(

).

:

(o,o,ho) ( ) ..

(a,f) .

DATUM .

(,) () , ,

, . =, = = ()

,

,

. ho=Ho,

.

.

67

, () .

( ) -

1ppm.

(, )

, .

.

...

,

.

()

,

.

(

), ...,

(,)

(.. , ) -

()

(,) (

(,)

(x,y).

() (o) ... ( )

(h),

.

(x,y,

68

Ho)

.

.

,

.

,,. () ,,h. ()

( H0 )

.

x,y, ( ), H .

() (h=H+N),

.

4.5

( )

.

( ) ,

.

-

.

WGS 84,

GPS,

69

.

()

.

4.6

C,

.

. ,,C ,, H.

0.1,

.

.

4.7

, ()

, -

,

, .

,

. x y -

( y x

).

, ,

70

.

(X,Y,Z) ,

(,y) ( x,y)

( )

.

4.8

()

(,,) ,

.

,

-

, .

, ,

,

. -

.

.. .

,

/ -

.

.

,

( )

. ,

, ,

,

.

71

()

(,,h)

(,,).

4.9

87 , ,

.

1990,

(),

,

.

.

4.9.1 87 ()

.

:

=38 04 33"8107

=23 55 51"0095

h=481,743m

N=7,000m.

(GRS 80) -

:

87 , .

72

=+199,652m

= - 74,759m

= -246,055m

,

.

.

( )

() =24

0=0.9996,

/.

4.9.2 ()

1889

.

=37 5820",1000

=0 0 00",0000

= 359 46 13",1000 ( )

= 0m (. h=Ho)

=234258"815 Greenwich.

Bessel.

( )

() .

.

,

Hatt,

( 130 ) 3030.

73

3.

4.9.3 ED 50 (European Datum 1950)

ED50 (European Datum 1950) Potsdam

ayford.

,

6. =21

=27 =0.9996.

.

4.10

ITRS ITRF

WGS 84,

GPS .

4.11

()

.

,

()

, 1933-1978.

,

R287, =14,665m. ,

,

.

74

(.. , ),

.

.

.

(h) ,

h=Ho+,

.

75

1. ., ., ., , 2006,

2. Smith J., Introduction to Geodesy. J Wiley and Sons, Inc. 1997 3. orge W., ..., 2000

4. Torge W., Geodesy. Walter de Gruyter, 3rd Edition, Berlin, New York 2001

76

5.

.

77

79

5.1

.

.

,

.

5.2

:

..

(, , ).

5.2.1 EDM (Electronic Distance

Measurements)

, , ,

( ) GPS (Global Positioning

System) ( ).

( ) .. 1cm, (p.p.m., parts per million), ..

1cm 50m :

1/5000=(1cm/50x100cm) = 210-4 = 20010-6 200 p.p.m.

(EDM)

p.p.m. GPS

0,1 p.p.m., 1cm 100km. E.D.M.

(EDM : Electronic Distance Measurements)

,

,

.

:

.

( )

( ) (

) (t)

( 1nsec=10-9 sec).

D = .t

: = c/n

c c=299.792.458m/sec n o

.

m -

,

.

.

, ( )

()

()

() () (. 5.1).

5.1.

80

D = + . /2=+

K .

EDM

Laser

.

G.P.S. To GPS

(Global Positioning System)

.

.

,

(-) - r=ct, c t o

.

,

4

X,Y,Z ,,h 4

, .

5.1. GPS

81

82

24+3 ,

20200km 4

24 (. 5.1).

,

GPS

.

,

, 3m, , ,

1 cm 2 cm . GPS ,

,

,

.

GLONASS (GLObal Navigation System)

15 19099km

.

.

VLBI

VLBI (Very Long Baseline Interferometry)

(quasars).

, ,

(inertial survey systems) -

.

GPS GPS

, .. , .

5.2.2

, ,

(. 5.2).

.

( =200g.0000).

5.2. :

B , ,

.

(. 5.2) ..

83

5.2. =

,

= . . ..

,

.

36 .

1cc 1c .

()

.

() (

) (. 5.3).

5.3. () ()

-

, .

84

cc c

p.p.m. ( ), ..

1cc 1cc/636620cc= ~1,5x10-6 1,5 ppm.

(),

() ..

(1cc-20cc) -

~ 1c

(1g).

5.3 () 1

()

() (5.1)

=+ (5.1)

,

5.3. -

(. 5.3),

( ) .

85

1 () . () . 1m.

, =-, (.

5.4) 1mm/km. ,

-, . (-).

.

5.4. .

(D)

(z) (. 5.5), (5.2) :

=Dcosz+YO-Y (5.2)

, :

YO= , = .

S

.5.5 .

(5.2),

(S2/2R) (-kS2/2R), k= 0.16 -

R . 500 m

[(1-k)S2/2R] 1.6 cm.

:

=Dcos z+(1-k)S2/2R+Y0-Y (5.3)

86

87

5.3.1.

(

)

, .

.

.. , ,

, .

.

5.3.2

, ,

..:

, ,

( g)

.

() , ,

..

. .

()

.

88

() ,

.

( )

.

5.4

(),

.

.

GPS

.

,

,

, .

() ( ) ,

S

S , ,

.

(.5.6).

.

,

( )

.

5.6. . ()

()

. .

.(.5.7).

.

5.7. . ()

( )

S

D

.

(.5.8 ) .

89

5.8. .

(

) (

EDM) (.5.4), -

,

-

() .

5.4. :

90

91

( ),

-

:

( )

..

(1:5000 1:10.000)

(..

).

5.5

.. ,

,

,

.

GPS.

5.6 ..

.

() .. .

.., (-

) ,

.

.. ...

92

5.7 :

..

.

, ( )

.

5.8

,

, .

:

.

( )

. .

: .

: .

: , ,

(.. ).

: ...

... (.. ).

: =

""

(. 5.9). .

5.9

5.10

() :

.

(z). (. 5.10). : +z= 100g (900)

: (A),

( ) .

93

(...)

.

( ): (AB)

B A.

: ... (..

) ,

.

.

: ...

.

5.9

.

"-" ,

.

:

( ) ( ) .

.

, ,

.

(Imperial System). , , -

(Systeme Internationale d

Units) SI, .

94

SI *

, . :

(Mm)

(Km)

(hm)

(dam) (dm)

(cm)

(mm)

(m)

()

(nm)

Angstrom

()

106m 103m 102m 101m 10-1 m 102m 10-3 m 10-6 m 10-9 m 10-10 m

Km, m, cm mm**.

, , .

(m2). , :

1 =102 m2

1 = 103m2

1 (ha) = 104m2

1Km2 = 106m2

, ,

9/16m2, 1 = 0.5625m2.

1791 10-7 . 1799, () , ( 90% 10% ) . 1960 1650763.73 2p10 5d5 Kr86 ( 86). 1983 1/ 299 792 458 sec. sec (). 299 792 458 m/sec. ** SI . : 1 ft (foot, ) = 12in (inches, ) 1 yd (yard, ) = 3ft 1 mile () = 5280ft : 1 inch = 2.54 cm 1ft = 0.3048 m 39.37in = 1 m

95

, , (m3).

(lt), 1lt = 0.001 m3.

SI (rad),

.

2 rad = 1 = 4

, /,

.

, (degrees,

) (grades, ). :

60" = 1 100cc = 1c

60 = 1 100c = 1g

360 = 4 400g = 4

: 123 25 40

: 319g 89c 45cc 319.g 8945

.

(h )

1h = 1/24 = 60min, 1min = 60sec

(, , , )

rad 200g

= = =180 24o

h :

.

/ .

, .

96

,

( ). , ,

: ( )rad = arc sin tan

,

:

arc arc 1sin sin 1tan tan 1

= ( )= ( )= ( )

cc cc

cc cc

cc cc

: arc 1 sin 1 tan 1

arc 1 sin 1 tan 1

cc cc cc

cc

ccrad

=

= = =

1636620

1

1206265

1

cc

:

SI Kelvin () ,

. Celsius (oC)

1C =1oK.

Celsius ( ) :

0C = 273.15oK :

() = 273o.15 + t (oC)

Fahrenheit ( oF).

Celsius Fahrenheit :

t (oC) =[ t (oF) 32o]5/ 9

SI Pascal (Pa). : 1Pa=1N/m2

97

:

101,325 / m2 =101325 Pa. (N=Neuton)

1bar = 105 N/m2

: 1mbar=100 N/m2 1mm Hg 1,33322mbar,

: 760mmHg=1013,25mbar.

(sec). :

60sec = 1min, 60min = 1h = 3600sec

: 1sec=103msec=106sec=109nsec=1012pcosec.

(m=mili, =micro, n=nano p=pico)

1. -., , , 1993

2. -., , ., , --

. ,2000

3. .,., .., .., .., -

, ,, 2 , ..

98

1. -., , , 1993

2. -., , ., , --

. ,2000

3. .,., .., .., .., -

, ,, 2 , ..

99

6. .. -

.. - &.. 1988.

. .

.

.. -

99

101

6.1

, , ,

.

,

, .

, .

.

-

.

6.1.1

,

,

.

,

..., :

...

( ).

, -

, .

.

, .

.

,

.

.

6.1.2

. ,

.

. ..

mm,

.

.

(precision)

.

-

.

(Accuracy)

.

, " - -

".

6.1.3

X xi

( ) ""

i :

i= xi - X (6.1)

X

, , ""

.

1 2x = x =...=xn -

n o , :

102

[ ]1n n

x x = lim lim

n n = 1 (6.2)

ui=xi- (6.3) .

:

.

.

hP(x)=

2 2-h xhy= e

Gauss. .6.1

1 [x] 2

,

.

2 (x)

x x +dx :

2 2-h xe dx (6.4)

y h3 h2 h1.

6.1

Gauss .

x x+dx,

.

103

Gauss , h ,

h .

h .

- + 1 ().

2 2-h xh e dx= 1

+

(6.5) 6.1

.

,

.

, (

) ,

.

6.2

" - -

"

. .

6.2.1

, n ,

1 2 nx , x ...,x

x x :

[ ]n1 2 ni

1

xx x ...+x 1x=x= xn n

+ = = n (6.6) l (6.6) :

104

[ ]ni

1

1x=n n

= = ll l = l (6.7) . i x= l i x= l :

(6.8) [ ]n i1 0= =

(6.9) [ ]n 2i1 min= =

(6.9)

,

.

u .

x

6.2.2

h, , :

1h= 2

(6.10)

.

()

:

[ ]uu=n

(6.11)

, (6.11) . n -

:

105

[ ]=n-1

3 (6.12)

n-1 = r

.( ,

,

s

2

2u-

21y = f(u) =f(x) e 2

=

,

3

[ ]n

, n=

.)

. 2

ui = xi -

, : 2

2(-21 e

2=

x-) (6.13)

,

. (6.2)

1 2 3 . f(u) f(x) + + .

6.2

1, 0

106

f(u) - +

0.6826 :

( )P - x- 68.3% + = ( )P - x 68.3% + = - +

68.3% - +

68.3%.

f(u) (2,+2) (3,+3) :

( )P -2 x- 2 95.4%+ =p p . ( )P -3 x- 3 99.7%+ =p p

, 2, 3

68.3%, 95.4% 99.7% .(.6.3)

6.3

6.2.3

-

, :

sS

1K

s

SK=

ppm

,

. 20cc

107

60g 160g.

ij rad (),

ij.

ppm (parts per million = )

rad1K rad

1K=

0.05m

10

ABS

ABcc. ABS

5000m

1K

ppm.

:

0.05 15000 100000

=

ppm :

61 10 10ppm100000

=

106m 10m. rad

. : ABS

rad cc

AB

10 10 1 BB 636620 63662 S

= = = =

10cc 5000m =0.08m

63662m 1m.

ppm :

6 rad 6 6 cc

10 1010 10 10 15.7 16ppm 636620

= = =

16m 106m.

108

6.2.4

y x1 x2 n

y x1 x2 ..

109

2

2i

1y=x x+ (6.14) y,

y .

1i 2ix , x i 1iy x x= + (6.15) : 1y x x= + 2 (6.16)

(6.15) ii y

y y+= 1i1i 1 x

x x += ,2i2i 2 x

x x += n :

1 11

2 12

n 1n

y 1 x 2 x

y 1 x 2 x

y 1 x 2 x

y+ (x + ) (x + )

y+ (x + ) (x + )

y+ (x + ) (x + )

= += +

= +M M M

21

22

2n

21

22

2n

x

x

x

(6.17)

(6.16) (6.17) :

1 11

2 12

n 1n

y x x

y x

y x

= += +

= +M M M

(6.18)

(6.18) :

1 11 21 11 21

2 12 22 12 22

n 1n 2n 1n 2n

2 2 2 2 2y x x x x

2 2 2 2 2y x x x x

2 2 2 2 2y x x x

2

+ 2

2

= + += +

= + +M M M M

n :

1 1 2 2 1 2

2 2y y x x x x x x 2 = + + (6.19)

n-1:

1 1 2 2 1 2x x x x x xy y 2 2

2n-1 n-1 n-1 n-1

= + + (6.20)

1 1 2 21 2

x x x x y y2 2x x

, ,

n-1 n-1 n-1

= = 2y=

: 1 2x , x

1 2 1 2x x x x 0 = (6.21)

y,

1 2x , x , :

1

2 2 2 2 2y x + = 2x (6.22)

y :

1

2 2 2 2y x + = 2x (6.23)

1 1 2 2 y x x ... x= + + + (6.24) , : 1 2 x , x ,..., x

1 2

2 2 2 2 2 2y 1 x 2 x x + ... = + + (6.25)

y

, y :

1 2 x , x ,..., x 1 2 y = f(x , x ,..., x )

1 2 x , x ,..., x

1 2 y = f(x , x ,..., x ) (6.26)

y :

1 2 dx ,dx ,...,dx 1 2 x , x ,..., x

1 2 1 2

y y ydy dx dx ... dxx x x

= + + + (6.27)

: 1 2 x , x ,..., x

1 1i 1i 21 2i i iy 1 x x x dy , dx , dx ,..., dx= = = = n : 1 2 x , x ,..., x

1 11 21 1

2 12 22 2

n 1n 2n n

y x x1 2

y x x1 2

y x x1 2

y y y ... x x xy y y ... x x x

y y y ... x x x

= + + + = + + +

= + + +

M M M M

x

x

x

(6.28)

(6.28) (6.18). :

110

1 2

22 22 2

y x x 1 2

y y y + ... x x x

= + + 2x (6.29)

(6.29) 4

1 2 x , x ,..., x .

,

,

1 2 x , x ,..., x

1 2 f(x , x ,..., x )

1 2 x , x ,..., x

1 2

y y y, ,...,x x x

.

6.2.5

(

).

, .

n ,

:

1 2 x , x ,..., x

1 2x x xx= ...n n n

+ + + n (6.30)

:

1 2

2 2 2x x x2 2 2

1 1 1 = ... n n n

+ + +n

2x

(6.31)

,

, 1 2 nx x x

= =...= = , : 2 2 2 2x 2 2 2 2

1 1 1 n = ... n n n n

+ + + = 2 (6.32)

[ ]=n-1

(6.32) :

4

111

[ ]2x

=n n(n-

= 1)

(6.33)

(6.29) (6.33)

y, , : 1 2 y = f(x , x ,..., x )

1 2

22 22 2

y x x 1 2

y y y + ... x x x

= + + 2x (6.34)

6.2.6.

6.2.6.1. :

= 15.20m 0.02m

= 30.15m 0.04m

, - .

:

= 2 +2 = 90.70m

2 2 2 2 = 2 2 0.09m + = :

= =458.3m2

:

22

2 2 2 2 2 2 2E

E E = 0.9m

+ = + =

- = 14.95 m :

2 2(-) = 0.04m + = 6.2.6.2. ,

. S

= 0.03m

= 0.04m AB ABS = 0.03m . ,

AB

cc = 20 B B

, SAB 1000m

30g.

112

B A AB B A AB

X X S sin S cos

= + = +

:

B B B

B B B

2 22 2

X X S A AB AB

2 22 2

S A AB AB

X X X + X S

+ X S

= + = +

22

22

A A

B B

AB AB

AB B AB B

AB AB

X 1 1X XX sin cosS SX S cos = -S sin

= = = = =

B B B

B B B

2 2 2 2X X B S AB B

2 2 2B S AB B

+(sin ) (S cos )

+(cos ) ( S sin )

= += +

2

2 2

. ,

m

B

2 2AB B (S cos ) B

2 2AB B ( S sin )

2cc2. , .

rad

p

B

B

AB

cc ( AB BccS cos

p, AB Bcc

S sinp

).

:

B B B

B B B

22 2 2 2AB B

X X B S cc

22 2 2 2AB B

B S cc

S cos +(sin ) 0.04mp

S sin +(cos ) 0.05mp

= + = = + =

113

6.3.

" - -

", ,

.

.

6.3.1 -

.

.

" - - " n

. : 1 2 x , x ,..., x

1 2x x ...+xx=x=n

+ n (6.35)

, nA nB (nA+ nB=n),

:

A A+1 A+21 2 n n n nA B

A B

x x ...+x x x ...+xx = x =

n n+ +

B (6.36)

x :

A A B B

A B

n x n xx=n n

++ (6.37)

(6.35) (6.37) x .

(6.37) nA nB

:

A B

A B

P P .n n

= = (6.38)

,

Ax

Bx .

114

,

Ax Bx :

A

x

=n

x

=n

:

A

x B

x A

n=

n (6.39) A

2x 2x

=

(6.40)

(6.40)

Ax Bx .

, ,

,

,

n

1x x2

2

1

2

1x , x 1 n2

.

n1

2

1

n2 , . :

1

x

1

=n

2

x

2

=n

(6.41)

1

2

1 2x

n =

2

2

2 2x

n =

(6.42)

1x , x2 1 2 :

2

1

21 12

2 2

P nP n

= = (6.43)

x :

1 1 2 2

1 2

x xx=

++ (6.44)

:

115

[ ][ ]x

x=

(6.45)

0

,

[ ] min.= 6.3.2 x

,

. ,

1

,

.

, :

2ji i2

j j

P nP n

= =i

(6.46)

. iP 1= i = a

priori

.

x ,

, ,

a posteriori .

, .

:

1

x

1

= (6.47)

:

[ ] [ ] [ ]1 1 2 2 n n x x xx= ...

+ + + (6.48)

:

116

[ ] [ ] [ ]1 22 2 2

2 2 21 2 nx x x2 2 2

= ...

+ + +n

2x (6.49)

[ ] [ ] [ ]2 22 2 2

2 o o1 2 nx 2 2 2

1 2

= ...

+ + +2o

n

(6.50)

[ ][ ] [ ]2

2 2 ox o2

=

= (6.51)

[ ]o

x

= (6.52)

6.3.3 A posteriori o

xi i,

,

ix - x = i

i , i i P

. o

,

:

o

[ ]o

P =

n-1 (6.53)

(6.53) (6.52) :

[ ][ ]X

P =

(n-1) P (6.54)

(6.54) o X

.

117

6.3.4

6.3.4.1 :

S (m)

1 200.152 8

2 200.090 4

3 200.125 16

.

.

.

: 31 2 PP P K8 4 16

= = =

2 = 1, 1 = 2, 3 = 4.

:

2 200.152+1 200.090+4 200.125x= 200.128m1+2+4

=

1

2

3

200.152 200.128 0.024m 200.090 200.128 0.038m 200.125 200.128 0.003m

= == = = =

118

:

[ ]o

P = 0.036m

n-1 =

:

X0.0036 0.014m

7= =

6.3.4.2

.

,

.

1 2 3

225 200 250

200 200 240

260 225 260

220 225 180

245 230 275

260 200 220

225 225 280

190 220 225

220 230

245 235

200

210

.

1 2 3 , ,

119

.

1 2 3 , ,

1 2

, ,3

2 2 2

:

10g

1 1 ,=1

1 73 822910

= = :

1

102 g

1i ,i=1

1 0 00249

= = :

11

g,

0 0007

10= =

12g

2 2 ,=1

1 73 821712

= = :

2

122 g

2i ,i=1

1 0 001411

= = :

22

g,

0 0004

12= =

8g

3 3 ,=1

1 73 82418

= = :

3

82 g

3i ,i=1

1 0 00337

= = :

120

33

g,

0 0012

8= =

a

priori :

go , 0 0012=

:

1

2 2

1 2 2

0.0012 3 0.0007

= = =

2

2 2

0.0012 90.0004

= =

2

3 2

0.0012 10.0012

= =

:

g1 1 2 2 3 31 2 3

= 73. 8222

+ + =+ +

a posteriori :

2 2 2

1 1 2 2 3 3 go ,

P P P = 0 0019

2

+ + =

:

[ ]g,X

0.0019 = 0 000513P

= =

121

122

1. ..,.-. ,

1988

2. .. , 2000

3. . , 1979

4. . ,

..., 1967

5. Mikhail E.M and Gracie G. Analysis and Adjustment of Survey Measurements, Van

Nostrand Reinhold Company, New York 1981

6. Moffitt F. and Bouchard H. Surveying 6th Edition Indext Educational Publishers New York

1975

7. Parratt L.G. Probability and experimental errors in Science, Dover Publications Inc. New

York 1971

8. Rainsford H.F. Survey Adjustments and Least Squares Constable & Company Ltd,

London 1975

7.

.

123

7.1.

7.1.1

( )

.

,

.

. ,

.

.

7.1. 2

' ( ). ,

.

'

(x, y), 7.1.

(x, y).

y (

ordinate)

x ( abscissa)

.

125

(

,

).

(. )

.

M'

O

+ y

+ x

xM

yM

7.1 (xM, yM)

7.1.3

(x, y)

( ) ,

7.2.

126

+ y

- y+ x

- xO

x

y

7.2

( )

y ()

. 0 400 0

360 . y (

) ,

.

(

) (x, y) (x, y)

SAB, ,

7.3.

+ y

- y+ x

- xO

x

y

x

y

S

7.3

127

,

y (

)

.

( ) 200g ( 180),

0 g 400 g 0 360.

= + 200 g = + 180 (7.1)

400g 360

400g (

) 0g

400g . , = 356.25 g = 356.25 g + 200 g = 556.25 g = 156.25 g .

7.4 .

+ y

- y

+ x- x

A

B

0g 100g

100g 200g

I IV

I I

200g 300g

300g 400g

O

0g

100g

200g

300g

A

B

A

B

A

B

7.4

128

4

7.1

V

0 g -100 g 100 g -200 g 200 g -300 g 300 g -400 g

x + + - -

y + - - +

sin + + - -

cos + - - +

tan + - + -

7.1

7.2. 7.2.1

.

, ,

.

.

.

7.2.2 ,

, S

.

129

+ y

- y+ x

- xO

x

y

x

y

S

y

x

7.5

7.5 :

xxxxxxyyyyyy

ABAB

ABAB

+==+==

(7.2)

y = SAB cosAB x = SAB sinAB (7.3)

(7.3) (7.2)

ABABAB

ABABAB

sinSxxcosSyy

+=+=

(7.4)

7.2.3 S

.

7.5, :

130

22

ABABAB

ABAB

ABAB

yxcosy

sinxS

yxarctan

yyxx

yxtan

+===

===

(7.5)

, (7.5),

0g 400g. ,

:

yx

arctan

yx

tan

=

= (7.6)

, x y,

7.6 7.2.

y

x

A

B

AB

AB

AB

AB

A

B

V

AB

AB

7.6

131

g

g

g

g

g

g

g

0300 x0 y

0100 x

0200 y0 x

00 ys

400yx

200yx

200yx

yx

=

= + > 3. -< < + 4. + > +

6. .

.

.

:

1)

(0 -180).

2) (.. )

.

8.2.2

/2 (90 100 g). (- ).

.

- -

.

4, 5, 5, 6,

6, 7, 7

157

90 1 0 .

=90 :

cos cos cos cot cotsin sin sin tan cotcos cos sin tan cot

= =

= = = =

(8.8)

.

8.2.3

( 100 .

) .

R=6371km.

.

( ) (.8.4).

A

A'

B

B'S

R

=

O

8.4

.

158

. :

RS= (rad) (8.9)

: S R .

.

(.8.5), : .

P

P'

I' I

A

A'

h =AA'=h

8.5 .

.

(.8.5).

.

.

159

(.8.4)

(, ).

.5

.

.

( 0 360).

(-90, +90). (.. 3545'30 ).

8.2.4 .

.

.

.

, ,

.

..

r .8.6.

O

R

r

P

P'

K

8.6 .

160

, ,

:

cos= Rr (8.10)

.

(.4.7)

, , . = . :

( ) === cosRrrS ABAB (8.11)

11 rad.

.

A

rr

K

KA=KB=r=RcosP

P'

II'

8.7 .

1: ,

= 23 15' 24" = 24 12' 35" = 38 25' 30" R=6371km.

.

161

.

A

B

= + + =

= + + =

231560

243600

23 2566667

241260

353600

24 2097222

0

0

.

.

:

= =B A 0 95305550.

:

= =0 9530555180

0 0166339564.

. rad

11 :

( )S R km kAB = = =cos cos . . . 6371 38 425 0 0166339546 83 0230 m

.

. (.4.7) .

R=6371km. =

.

(.4.7) :

= A (8.12)

S , ,

:

S R = (8.13)

2: = 38 25' 30" = 41 30' 45"

. .

162

:

A = + + =

= + + =

382560

303600

38 425

413060

453600

41 5125

.

.

:

= = =3 0875 180 0 05388704. . 13 :

S R km km = = = 6371 0 05388704 343314. . 8.2.5 - .

, ,

.

.4.8 ( ,) ( ,).

P

P'

II'

A

BSpA

90-

90-

AB

B

A

A

BAAB

8.8 .

163

S.

.

, .

( )

. .

( ),

() ( )

() ( )

( ) ( ).

:

1. : =, =, =360-2. : = = 90-, = =90- = p = .

(SAB), R ,

= p

(5).

+= cossinsincoscoscos p (8.14)

:

( ) ( ) ( ) ( ) += cos90sin90sin90cos90coscos p (8.15)

+= coscoscossinsincos p (8.16) : , : =+1800+

164

16 p

.

(8.1).

pRS = (8.17) : p

( 5 ), ,

(=) :

App cossinsincoscoscos +=

( ) ( ) ( ) += cos90sinsin90coscos90cos pp

+= coscossinsincossin pp (8.18)

18 .

, ,

(2 ).

:

, ( , ) , SAB

.

1 .

:

1. 17 (p)

S.

2. 18 .

3. 16 :

+= (8.19)

165

.

(

), 8.17,

.

( ) (..

).

: ( / )

. S (.8.8).

= 38 25' 30" , =23 15' 18" , =41 30' 45" , = 24 12' 36" , R=6371km

:

:

=38.425 , =23.255 , =41.5125 , =24.2100

:

955.0== 16 (=p)

.

9984669575.0cos =p 173001124.3)9984669575.0arccos( ==p

- arccosx () x.

x.

rcsinx (x), arctanx (x) ,arccotx (x) .

/

sin-1 , cos-1, tan-1.

166

S 8.17

p .

radp 0553793168.0180

173001124.3 == mkmkmS 627.352821821627.3520553793168.06371 ===

18 .

9742478867.0cossin

sincossincos ==

A

ABAB p

pA

03108265.13)9742478867.0(arccos == (. ' " )

:

=13

= (13.03108265 13)*60 = 1.864959 = 1'

= (1.864959-1)*60 = 51.89754 = 51".9 ()

= 13 1' 51".9

.

, S .

.

= 38 25' 30" , =23 15' 18" , S=352821.627m, = 13 1' 51".9.

:

.

17 .

radm

mR

Sp 0553793168.06371000

627.352821 ===

173001122.31800553793168.0 == p

18 .

167

6627834292.0sin = 5403415125.41)6627834292.0(arcsin ===

16 .

9998610939.0coscos

sinsincoscos ==

p

81750954999879.0)9998610939.0(arccos ===

19 .

...999.532124...209999.24 ==+=

, .

168

exofylo_geodesia_1_fin.pdf

_off.pdf

prologos.pdf

_1_ekswfullo.pdf.

_3_ekswfullo.pdf.

_3__.pdf3.1.2 3.1.3

3.1.4 () ,, 3.1.5

:

_3_bibliografia.pdf

_4_syst_anaf.pdf4.4.1 87\(4.9.2 ()4.9.3 ED 50 (European Datum 1950)

_4_bibliografia.pdf

_5_f.pdf 5.1. GPS 5.2.2 ()5.3.2 5.4 5.5 :

_5_f_1.pdf

_5_bibliografia.pdf

_6_sfalmata.pdf6.1.3

_7_.pdf = + 200 g = + 180 (7.1) 4 7.4 7.1 7.5 7.6 7.2

7.3.3 7.11 7.17

_8__.pdf8.1 8.2 8.1 8.2.1 .8.2.2 8.2.3 8.2.4 .8.2.5 - .

Γεωδαισία

Documents

Transcript of Γεωδαισία