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

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 Molodensky’s 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 (Vaníček 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 Vaníček 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.

The gravity disturbance O ) , ( Ω r g is the actual gravity, and ) , ( φ γ r is the normal gravity of the geocentric reference ellipsoid.The normal gravity is defined according to Somigliana -Pizzetti's theory of the normal gravity field generated by the ellipsoid of revolution (Pizzetti, 1894 and1911;Somigliana, 1929).
A geocentric position is represented by the geocentric radius r ; ) ( ) , and the geocentric spherical coordinates φ and λ ; ( ) λ φ, = Ω , ( ) Since the evaluation of the normal gravity according to the following equation (Vaníček et al., 1999) : In Eq. ( 2), ( ) is the normal gravity referred to the surface of the geocentric reference ellipsoid is the normal gravity gradient.

GEOID-QUASIGEOID CORRECTION
When the orthometric heights The fundamental formula for a definition of the orthometric height reads (Heiskanen and Moritz, 1967) : where is the geopotential number, and ( ) Ω g is the mean value of the gravity along the plumbline between the geoid and the Earth's surface.
For the Helmert (1890) orthometric height, the mean value of the gravity ( ) Ω g is evaluated using Poincaré-Prey's gravity gradient (Heiskanen and Moritz, 1967, Eq. 4-25) : is the observed gravity at the Earth's surface, G is Newton's gravitational constant, and o ρ is the mean topographical density [ ] Molodensky's normal height The mean value of the normal gravity ( ) along the ellipsoidal normal between the surface of the geocentric reference ellipsoid and the telluroid : The difference between the normal height and orthometric height, i.e., the geoid-quasigeoid correction, can be found beginning with the following relation the difference between the mean gravity ( ) Ω g and the mean normal gravity ( ) Ω γ in Eq. ( 7) becomes The expression on the right-hand side of Eq. ( 9) is approximately equal to the simple Bouguer gravity anomaly ) Regarding Eq. ( 10), the geoid-quasigeoid correction from Eq. ( 7) finally takes the following form (11) Substituting Eq. ( 11) to Eq. ( 2), the gravity anomaly where stands for the free-air gravity anomaly : (13) Applying the spherical approximation (Heiskanen and Moritz, 1967, Eq. 2-150) Eq. ( 12) is subsequently rewritten as : The second term on the right-hand side of Eq. ( 15) defines the geoid-quasigeoid correction to the fundamental formula of physical geodesy (Vaníček et al., 1999) :

ERROR ANALYSIS
The accuracy estimation of the geoidquasigeoid correction to the fundamental formula of physical geodesy depends on the accuracy of the orthometric height, gravity and topographical density.
From Eq. ( 16) follows the relation between the actual error of the simple Bouguer gravity anomaly Furthermore, the error of the simple Bouguer gravity anomaly is expressed by : ) ) ) (20) Therefore, the error of the observed gravity is practically equivalent to the error of the free-air gravity anomaly , so that In Eq (20), a , b are the semi-axes and f is the first numerical flattening of the geocentric reference ellipsoid, ω is the mean angular velocity of the Earth's spin, and GM stands for the geocentric gravitational constant.

NUMERICAL INVESTIGATION
The geoid-quasigeoid correction  of the free-air gravity anomalies is only a few microgals (Fig. 3).

CONCLUSION
To increase the accuracy of the geoidquasigeoid correction, the laterally varying topographical density can be used for the computation of the simple Bouguer gravity anomaly.According to the error propagation in Chapter 3, the change of the topographical density causes the largest error of the geoidquasigeoid correction to the fundamental formula of physical geodesy.From the numerical result over the territory of Canada follows that the variation of this correction due to the anomalous topographical density is up to 40 ± microgals.
's surface would require the knowl-edge of the geodetic height ( ) error of the observed gravity at the Earth's surface.The laterally varying anoma- air gravity anomaly is shown in Fig.1.

Figure
Figure.1-Relation between the geoid-quasigeoid correction ( ) [ ] Ω t r χ to the fundamental formula of physical geodesy and the free-air gravity anomalies

Figure
Figure.2-Relation between the error of the geoidquasigeoid correction ( ) [ ] Ω t r χ ε to the fundamental formula of physical geodesy and the error ) ( O Ω H ε of

Figure
Figure.3-Relation between the error of the geoidquasigeoid correction ( ) [ ] Ω t r χ ε to the fundamental formula of physical geodesy and the error

Figure
Figure.4-Error ( ) [ ] Ω t r χ ε of the geoid-quasigeoid correction to the fundamental formula of physical geodesy due to the variation of the topographical density ( ) ( ) Ω ≡ Ω ρ ε δρ .The geoid-quasigeoid correction ( ) [ ] Ω t r χ to the fundamental formula of physical geodesy has been computed at the territory of Canada (Fig. 5).At this territory it varies from -0.425 to + 0.019 [mGal], with the mean value equal to -0.013 [mGal].The magnitude of the correlation between this correction and the variation of the lateral topographical density ( ) Ω δρ is between -0.036 and + 0.032 [mGal], see Fig. 6.