High-precision calculating the normal height as the coordinate line's length

In the literature, there are discrepancies about the direction in which the telluroid points should be plotted from the ellipsoid for the subsequent calculation of the segment of normal height. There are three options: forceline of the normal field back, coordinate line of the spheroidal system, normal to the ellipsoid. Theoretically, the normal gravity field can be successfully used as an orthogonal coordinate system, since its force lines and level surfaces can serve as natural coordinate lines and coordinate surfaces. However, a normal force line does not have two characteristics that would be constant at each of its points with a change in only the third value, as in a conventional orthogonal coordinate system. The normal to the reference ellipsoid plays an important role in solving geometric problems of geodesy, but is of little use in physical matters. It is more convenient to use a curvilinear coordinate system associated with a family of ellipsoids confocal to the reference one, especially since it contains closed expressions for the normal potential of gravity and all derivative elements. The method used so far for calculating the value of the normal height is based on the expansion of the normal gravity in a series using higher derivatives with respect to the geodetic coordinates at the point on the surface of the reference ellipsoid, the expansion error naturally increases with distance from the ellipsoid. This Yeremeyev's formula is often considered as the definition of the normal height while it is only working formula For the first time, the question of the need to study and refine the method for calculating the normal height was raised by M. Pick and M. I. Yurkina in 2004. In their joint publication, the normal height is refined with respect to the gradient solution, taking into account the expression of the normal potential in the spheroidal system u, v, w (Niven), but there is no calculation of the length of the normal force line segment. M. I. Yurkina in 2004 gave a similar expression in the system Heiskanen-Moritz, also indicating an explicit expression for the length of the segment of the coordinate line in the same system, however, the control calculations were not performed, so inaccuracies remained unnoticed in the proposed formulas for the auxiliary quantities, resulting in a low accuracy of the expression for the normal height Hγ. A detailed way contains three steps: 1. Standard calculation by the above Yeremeyev's formula and the corresponding third spheroidal coordinate w′ or b′. This is the first approximation. 2. Refinement of the third spheroidal coordinates of the points on the telluroid from the Molodensky's condition: the reduced latitude u could be precised consequently. 3. Evaluation of the curvilinear integral from w = w0 to w: where [Here was the inaccuracy in the Yurkina M. I. (2004) To refine the height to fractions of a millimeter, this step is absent in the paper Jurkina M. I., Pick M. (2004) Návrh na zpřesnění výpočtu normálních výšek].