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Height System

1. Geometric height system2. Physic height system


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Geometrics system

1. Reference field ; ellipsoid2. Used for theoretical purposes.

terrain

h

ellipsoid


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Physic height system

Reference field : geoid Based on geopotential numbers,

C= Wo - W = gdn

Used for practice and scientific purposes,

due to realistic connection withmeasurement in the Earth's surface.


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Bidang-bidang equipotensial gayaberat


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Spirit leveling

Measure the difference in height, HAB, betweentwo points A and B. If The rod reading l1 =AAand l2=BB, the height difference:

HAB= l1 - l2. ..(5)

A

A

B

B

HAB


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Aand Bfar apart, the measurement must be appliedrepeatedly.

AB HA HB .(6)

reason: leveling increment is different from correspondingincrement HB of HB, due to non-parallelism of level surface.

n

A

B

HA

HB

Geoid

Leveling and orthometric height

HB


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if AB far apart :

HAB = H1 +H2 +H3 + ..

. HAB = H HA HB If the measurement in circuit ,

- expected : H = 0

- reality : H 0, called : misclosure


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Increment of potential WbyW:

- W= g n = g HB ..(7)

Where, g : gravity at leveling stationg : gravity on plumb line of B at HB

Hence,HB = g/g n n (8)

Since (8) expresses a physical relation, there isno direct geometric relation between levelingresult and orthometric height.


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If gravityg is also measured, thenW = -g n (7)

ObtainedWB WA = -g n (8)

Leveling + gravity measurement : potentialdifferent (physical quantities)


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This integral is independent of the pathintegration. So that,different leveling linesconnecting the point A and B will give the sameresult.

If leveling return to A, then

A

B

B

A

g n ........................(9)B AW W

0 ...............(10)B

B A

A

g n W W


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On the other hand, the measured height difference, that is,the sum of leveling increments

nAB = n = ..(11)

depends on the path of integration, thus not in general

zero for a circuit.

0 = misclosure (12)

Potential differences = leveling + gravity measurement

Basic of theory of height : orthometric height must beconsidered as quantities derived from potential differences.

B

A

dn

dn


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Height systemO

O B

B

' '

0 ' 0 '

'

o B o B

B B B

B Bo

oB

B B

gdH g dH gdH g dH

gdHCH

g g


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The height =C/G.

The height system differ according to

the G- Orthometric height system, G=gm

- Dynamic height system, G=0

- Normal height system, G=n


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Geopotential number

Ois a point at sea level (on the geoid) and Abeanother point, connected to O by leveling line.Potential difference between Aand O:

g dn = Wo WA= C .(1)

which is difference between potential at thegeoid and at the point A, called potentialnumber of A (adopted 1955).

Potential number is independentof the levelingline. The same for all points of level surface,thus considered as natural measure of height.


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the geopotential number C is measured ingeopotential unit (g.p.u), where

1 g.p.u = 1 kgal meter = 1000 galmeter

Since g0.98 kgal,C gH 0.98 H .(2)

the geopotential numbers in g.p.u are almostequal to height above sea level in meter.


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Othometric height

Vertical distance along the plumb line frompoint on the earths surface to the geoid,

height above sea level

B

B

O

O


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Hort=C/gmWhere gm is mean gravity along the plumblinebetween geoid and the ground.

gm=g+0.0424H

Helmert height (1980),

..........................(3)

0.0424CH

g H


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Orthometric correction

Added to convert the measured heightdifference into difference in orthometricheight,

(4)

( )

AAB B

AB AB AB ort

Booom mA mB

BAAB o o oA

H H HH n OC

ggg

OC n H H


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The gravity reduction of Poincare and Prey

To convert the leveling result intoorthometric height, needed the gravity g

inside the earth (eq. 4). Since g cannot bemeasured, it must be computed from thesurface gravity, done by reducing themeasured by Poincare and Prey reduction.


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Q the point wheregis to be computed, sog = gQ. Pis the corresponding surface point. So Pand Qsituated on the plumb line. Gravity at P ,gp, ismeasured.

provided the gravity gradient g/h inside the earthwere known.

P

Q

P0

Earths surface

GeoidW=Wo

H=Hp

,P

Q P Q

gg g dH

h


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The gravity gradient can be obtained by Bruns formula,

2

2

0

0

3 9

2 4 2 ,

normal free-air gradient, 2 2

if `, then 4with 2.67 / , 66.7 10 c.g.s unit and

variation of / ,with latitude is neglecting,

0.3086 0.2238

g

gJ kh

Jh

g

gJ J kh hg cm k x

h

gh

Q P

0.0848 gal/km.

g g 0.0848( )P QH H


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Dynamic heights

It has height dimension but no geometrical meaning. All points in the same level surface have the same

dynamic height.

The dynamic correction is very large, so these systemnot suitable for practical purpose


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The dynamic height is written

Hdyn

= C /0.(13)

0 is normal gravity for standard latitude,

usually 4545= 980.629 galsFor Indonesia E= 978.049 gals

Obviously, dynamic height differ fromgeopotential number only in the scale of unit.The dynamic height has no geometricalmeaning. Hence the geopotential numbers arein general preferable to the dynamic heights.


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Dynamic correction

Dynamic correction : to convert the measured heightdifference, nAB , into a difference of dynamic height.

1 1 1( ) ( )

, : dynamic correction

= ..........

B B

dyn dyn dyn o oA AAB B B o o oA A

B Bo

ooA A

dynAB AB AB AB

B Boo

ABoAoA

H H H C C gdn g dn

gdn dn

H n DC DC

g gDC dn n ...............(14)


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The dynamic correction also used for computing

difference of geopotential numbers.

CB CA = 0 nAB + 0 DCAB(15)


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Normal Height

1. Depend on reference ellipsoid used.2. Assume Gravity field of the Earth to be normal:

W=U,g=:T=0:

*

0

0 0

*

0

*

0

2 2

0 2

*, * ,

1*, *

*

1 ( )*

2 3( ) 1 (1 2 sin )

H C

H

H

dCW W C dH H

C H dH H

z dzH

z f m f z z a a


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*

2 2

0 2

32

0* 2

0

2 3* 2 * *

0* 2

* *22

0 2

* 2

0 0 0

2 3

( ) 1 (1 2 sin )

1 2 3(1 2 sin )

3

1 2 1 (1 2 sin )

1 (1 2 sin )

1 (1 2 sin )

H

z f m f z z a a

zz f m f z

H a a

H f m f H H H a a

H Hf m f

a a

C C CH f m f a a

2


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Normal height differences :

Normal correction :

* * *

* *0 0 0

0 0 0

-

- - --

AAB B AB AB

BA A

A AABA

H H H n NC

gNC n H H


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Comparison of Different Height System

Common form of height system:

Orthometric height system, G=

Dynamic height system, G=Normal height system, G=

Height =

int0

po

geoid

C

G

C W W g dn

g

0, E


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The geopotential number: the most direct result ofleveling.

The Orthometric height height above sea level.Orthometric correction 15 cm per 1 km ofmeasured height difference.

The dynamic height: the same level surface havethe same dynamic height. Because gravity variationfrom the equator to pole by about 5000 mgals, thedynamic height correction can be very large, thus

not suitable for practical purposes.


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From practical point of view, the desiredrequirement height system:

1. Misclosures be eliminated2. Correction to the measured height be as

small as possible and simple.3. Compared to height difference, thecorrection 1:10.000


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Accuracy

Leveling: the most accurate geodeticmeasurement (height), STD error 1mm perkm distance, increase with square root ofdistance.

distance gravity station of 2-3 km in levelcountry, 1-2 km in moderate hill and .3-0.5 kmin the mountainous.

Dynamic and normal height as accurate as C,cause normal gravity is errorless.Orthometric height affected by imperfectknowledge of gravity.

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