Discussion on the orthometric height realization

In the theory of the orthometric height, the mean value of gravity along the plumbline between the geoid and the earth's surface is defined as the integral mean. To determine the mean gravity from the gravity observations realized at the physical surface of the earth, the actual topographical density distribution and vertical change of gravity with depth have to be known. In Helmert's (1890) definition of the orthometric height, the assumption of the linear change of normal gravity is used adopting the constant topographical density distribution. The mean value of gravity is then approximately evaluated so that the observed gravity of a point at the earth's surface is reduced to the mid-point of the plumbline by Poincaré-Prey's gravity gradient. To avoid the problems related to the determination of mean gravity, Molodensky (1945) formulated the different concept. In his theory of the normal height, the mean value of the normal gravity along the ellipsoidal normal between the ellipsoid surface and telluroid is considered. The mean normal gravity is then evaluated explicitly without any hypothesis about the topographical density distribution and vertical gradient of actual gravity. In this paper, the corrections to Helmert's orthometric height are formulated based on the comparison of the integral mean of gravity and Poincaré-Prey's gravity reduction. As follows from the results of the numerical investigation, the orthometric heights can also be determined with a reasonable accuracy if the sufficient information about topographical density and gravity are available.