Revista Brasileira de Cartografia N 55/01

GEOID-QUASIGEOID CORRECTION IN FORMULATION OF THE FUNDAMENTAL FORMULA OF PHYSICAL GEODESY

Robert Tenzer1 Petr Vanek2

1Department of Geodesy and Geomatics Engineering, University of New Brunswick

P.O. Box 4400, Fredericton, New Brunswick, Canada, E3B 5A3; Tel.: + 1 506 458 7167 rtenzer@unb.ca

2Department of Geodesy and Geomatics Engineering, University of New Brunswick

P.O. Box 4400, Fredericton, New Brunswick, Canada, E3B 5A3; Tel.: + 1 506 458 7167 vanicek@unb.ca

ABSTRACT To formulate the fundamental formula of physical geodesy at the physical surface of the Earth, the gravity anomalies are used instead of the gravity disturbances, because the geodetic heights above the geocentric reference ellipsoid are not usually available. The relation between the gravity anomaly and the gravity disturbance is defined as a product of the normal gravity gradient referred to the telluroid and the height anomaly according to Molodenskys theory of the normal heights (Molodensky, 1945; Molodensky et al., 1960). Considering the normal gravity gradient referred to the surface of the geocentric reference ellipsoid, this relation is redefined as a function of the normal height (Vanek et al., 1999). When the orthometric heights are practically used for the realization of the vertical datum, the geoid-quasigeoid correction is applied to the fundamental formula of physical geodesy to determine the precise geoid. Theoretical formulation of the geoid-quasigeoid correction to the fundamental formula of physical geodesy can be found in Martinec (1993) and Vanek et al. (1999). In this paper, the numerical investigation of this correction at the territory of Canada is shown and the error analysis is introduced. Keywords: Geoid-Quasigeoid Correction, gravity, height. 1. BASIC THEORY

The gravity disturbance ( )[ ]trg referred to the Earths surface :O ( ) ( ) ( )+ OHrr gt , where :O ( )gr denotes the geocentric radius of the geoid surface, is defined by (Heiskanen and Moritz, 1967, Eq. 2-146)

:O ( )[ ] ( )[ ] ( )[ ]= ttt rrgrg , (1) where ),( rg is the actual gravity, and ),( r is the normal gravity of the geocentric reference ellipsoid. The normal gravity is defined according to Somigliana Pizzettis theory of the normal gravity field generated by the ellipsoid of revolution (Pizzetti, 1894 and 1911; Somigliana, 1929).

A geocentric position is represented by the geocentric radius r ; )( )+ ++ ,0r , and the geocentric spherical coordinates and ; ( ),= ,

( )20;2/2/OO . Since the evaluation of the normal gravity

( )[ ]tr at the Earths surface would require the knowl-

edge of the geodetic height ( )h above the geocentric reference ellipsoid, the gravity disturbance ( )[ ]trg is transformed into the gravity anomaly ( )[ ] trg accord-ing to the following equation (Vanek et al., 1999)

:O ( )[ ] ( )[ ] ( )[ ] ( ) ottt rrgrg += ( )

( )

( )

=

N

n, Hr

orr

. (2)

In Eq. (2), ( )NH denotes the normal height, ( ) o is the normal gravity referred to the surface of the geocen-tric reference ellipsoid ( ) or:2/,2/ , and

( ) n/, r is the normal gravity gradient. 2. GEOID-QUASIGEOID CORRECTION

When the orthometric heights ( )OH are used instead of the normal heights ( )NH , the geoid-quasigeoid correction is applied to Eq. (2). To define this correction, the commonly used Helmerts orthomet-

Revista Brasileira de Cartografia N 55 58

ric heights ( )OH and Molodenskys normal heights ( )NH are briefly introduced.

The fundamental formula for a definition of the orthometric height reads (Heiskanen and Moritz, 1967)

:O ( )( )[ ]( )

=grCH tO , (3)

where ( )[ ]trC is the geopotential number, and ( )g is the mean value of the gravity along the plumbline between the geoid and the Earths surface.

For the Helmert (1890) orthometric height, the mean value of the gravity ( )g is evaluated using Poincar-Preys gravity gradient (Heiskanen and Moritz, 1967, Eq. 4-25)

:O ( ) ( )[ ]( )

( )

( )

=

O,21 H

Hrgrgg

trr

t

( )[ ] ( )( )

( )

=

O

n,

21 H

rrg

trr

t

,

( ) OoG2 H , (4) where ( )[ ]trg is the observed gravity at the Earths surface, G is Newtons gravitational constant, and o is the mean topographical density [ ]3o g.cm67.2 = .

Molodenskys normal height ( )NH reads (Molodensky, 1945)

:O ( )( )[ ]( )

=

trCH N . (5)

The mean value of the normal gravity ( ) along the ellipsoidal normal between the surface of the geocentric reference ellipsoid and the telluroid :O ( ) ( )+ NHro is given by

:O

( ) ( ) ( )[ ] ( )( ) ( )

( )

+

+=

N

N

N

n,

21 H

rHr

Horr

o

( )

( ) ( )

( )[ ] 2N2

2

Nn,

41

+=

Hr

Hrr o

. (6)

The difference between the normal height and orthometric height, i.e., the geoid-quasigeoid correc-tion, can be found beginning with the following relation

:O ( ) ( ) ( )( ) ( )

( )

=

gHHH OON

( ) ( ) ( )( )

o

gH O . (7)

Assuming :O

( )( ) ( )

( )[ ] 0n

,41 2N

2

2

N

+=

Hr

Hrr o

,

( )( )

( ) ( )( ) ( )

( ) 0n,

21

n,

21 NO

N

+==

HrHr

Hrrrr ot

,

(8)

the difference between the mean gravity ( )g and the mean normal gravity ( ) in Eq. (7) becomes

:O ( ) ( ) ( )[ ] ( ) ( )[ ]+ NHrrgg ot ( ) OoG2 H . (9) The expression on the right-hand side of Eq. (9) is ap-proximately equal to the simple Bouguer gravity anom-aly ( )[ ] trg SB (Martinec, 1993)

:O

( )[ ] ( )[ ] ( ) ( )( )

( )

=

=

OSB

n, Hrrgrg

orr

ott

( ) OoG2 H . (10) Regarding Eq. (10), the geoid-quasigeoid correction from Eq. (7) finally takes the following form

:O ( ) ( ) ( )( )[ ]

( ) otrgHHH

SBOON .

(11) Substituting Eq. (11) to Eq. (2), the gravity

anomaly ( )[ ] trg becomes :O

( )[ ] ( )[ ] ( )[ ] ( ) ottt rrgrg +=

( )

( )

( ) ( )[ ]( )

+

=

o

t

rr

rgH

r

o

SBO 1

n,

( )[ ] ( )( )

( )

orrot

rrg

=

=

n,1FA

( ) ( )[ ] trgH SBO , (12) where ( )[ ] trg FA stands for the free-air gravity anom-aly

:O

( )[ ] ( )[ ] ( ) ( )( )

( )

=

=

OFA

n, Hrrgrg

orr

ott

.

(13) Applying the spherical approximation (Heiskanen and Moritz, 1967, Eq. 2-150)

:O ( )( )

( ) R2

n,1

=

orro

r , (14)

Eq. (12) is subsequently rewritten as :O

( )[ ] ( )[ ] ( ) ( )[ ]+= ttt rgHrgrg SBOFA R2 . (15)

The second term on the right-hand side of Eq. (15) defines the geoid-quasigeoid correction to the fundamental formula of physical geodesy (Vanek et al., 1999)

:O ( )[ ] ( ) ( )[ ] tt rgHr SBOR2

. (16)

Revista Brasileira de Cartografia N 55 59

3. ERROR ANALYSIS

The accuracy estimation of the geoid-quasigeoid correction ( )[ ]tr to the fundamental for-mula of physical geodesy depends on the accuracy of the orthometric height, gravity and topographical den-sity.

From Eq. (16) follows the relation between the actual error ( )[ ]tr of the geoid-quasigeoid correc-tion to the fundamental formula of physical geodesy and the error of the orthometric height )(O H and the error

( )[ ] tg rSB of the simple Bouguer gravity anomaly

:O

( )[ ] ( )[ ] ( )[ ]( )[ ] ( )[ ]

+

= tg

t

tH

tt rrg

rH

rr SBO SBO )()(

( )[ ] ( )[ ]+= tgHt rHrg SBO )(R

2)(R2 OSB

( )[ ] ( )[ ]( )[ ]

+

=

t

tgHt rg

rH

rSBO

SBO

)()(

. (17)

Furthermore, the error ( )[ ] tg rSB of the sim-

ple Bouguer gravity anomaly is expressed by :O

( )[ ] ( )[ ]( )[ ] ( )[ ]( )[ ]

( )

+

= O

SBSB

SB

Hrgr

rgrgr ttg

t

ttg

( )[ ]( ) ( )[ ]

( ) ( )[ ]( ) )(OO

O

O

SB

++

+ H

o

o

t

HHr

Hrrg

( )[ ]( ) ( )

+

trg

SB

( )[ ]

( )[ ] ( )[ ]( )[ ]

( ) )(OOSBSB

+

Ht

tgt

t

Hrg

rrg

rg

( )[ ]( ) ( )

+

trg

SB

, (18)

where ( )[ ]tg r