TO HELMERT ’ S ORTHOMETRIC HEIGHT DUE TO ACTUAL LATERAL VARIATION OF TOPOGRAPHICAL DENSITY

Helmert (1890) used Poincaré-Prey’s gravity gradient for the definition of the orthometric height. According to this approach the gravity value needed for the evaluation of the height is obtained from the observed gravity at the earth surface reduced to the mid-point between the earth surface and the geoid, considering that the gravity gradient is constant along the plumbline. Moreover, the mean topographical density 3 o g.cm 67 . 2 − = is assumed to approximate the actual distribution of topographical density. The correction to Helmert’s orthometric height due to the lateral variation of topographical density has been introduced by Vaní ek et al. (1995). In this paper, some numerical aspects of this correction are investigated.


HELMERT'S ORTHOMETRIC HEIGHT
The fundamental formula for a definition of the orthometric height ( ) reads (e.g., Heiskanen and Moritz, 1967, eqn. 4-21) : where is the geopotential number, and ( ) Ω g is the mean value of the actual gravity along the plumbline between the physical surface of the earth and the geoid surface.The geocentric position is given by the geocentric spherical coordinates φ and λ ; , and the geocentric radius r ; ) ( ) . In eqn.
(1), ( ) further denotes the geocentric radius of the earth surface, and O Ω stands for the total solid angle is given by the difference of the actual gravity potential o W of the geoid and the actual gravity potential referred to the physical surface of the earth, so that : (2) Helmert ( 1890) defined the approximate value of the mean gravity ( ) Ω g along the plumbline by using Poincaré-Prey's gravity gradient.It reads : is the normal gravity gradient.According to this theory the gravity gradient is considered to be constant along the plumbline within the topography.Thus, the mean value of the gravity ( ) Ω g is evaluated directly for the mid-point of the plumbline (Heiskanen and Moritz, 1967, eqn. 1-14) ) and from the expression for the mean curvature of the equipotential surface ( ) the Bruns formula for the actual gravity gradient can be found (Heiskanen and Moritz, 1967) : In the above equations, denotes the mean value of the angular velocity of the earth spin, G is Newton's gravitational constant, and are the second partial derivatives of the gravity potential in the local astronomical coordinate system x , y , z , where the zaxis coincides with the outer normal of the local equipotential surface.The normal gravity gradient Furthermore, the mean curvature of the ellipsoid surface ( ) is given by (e.g., Bomford, 1971) ≅ Ω Ω , (9) Poincaré-Prey's gravity gradient can finally be found in the form (e.g., Vaní ek and Krakiwsky, 1986) : . ( 10)

EFFECT OF LATERAL VARIATION OF TO-POGRAPHICAL DENSITY ON HELMERT'S OR-THOMETRIC HEIGHT
The correction to Helmert's orthometric height due to the laterally varying topographical density ( ) Ω ρ is given by the following approximate expression (Vaní ek et al., 1995) : is the normal gravity referred to the ellipsoid surface.According to Martinec (1993), the laterally varying topographical density ( ) Ω ρ is given by where ( ) Ω g r denotes the geocentric radius of the geoid surface.

NUMERICAL INVESTIGATION
In most of the topography the actual lateral density varies from

H
ranges between -1.9 cm and +3.4 cm.Since the sufficient information about the accuracy of the density model is not available, the error estimation of the numerical result is not employed.

CONCLUSIONS
Based on the theoretical analysis in the previous paragraph (Fig. 1) it has been estimated that the effect of the lateral topographical density variation on the orthometric height can reach up to a several decimeters.As further follows from the result of the numerical investigation on GPS/leveling points this effect represents a change in orthometric height of only a few centimeters (see Fig. 3).It is because the leveling benchmarks are situated in regions of which the elevations are usually less than 2000 meters.
Helmert's orthometric heights are usually used for a definition of the geodetic vertical datum.When the proper density model is available, the accuracy of Helmert's orthometric heights can be improved.The correction to Helmert's orthometric height due to the lateral variation of topographical density can then be applied, especially when the high accuracy is required such as the determination of the orthometric heights on the leveling points.
radii of curvature of the ellipsoid in North-South and East-West directions.Under the following assumption : , value o .However, larger topographical density variations up to 20-30% are encountered in some local and regional geological structures.Regarding eqn.(11), it can be estimated that the variation of topographical density can cause centimeters and decimeters errors in orthometric height.The relation between the change of orthometric height Fig.1.Since the correction to the orthometric height due to the lateral variation of topographical density increasing exponentially with the height, this correction can approximately reach

Figure
Figure.1-Change of orthometric height