<p>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.</p> <p><img src="" alt="" /></p> <p>There are three options:&#160;forceline of the normal field back, coordinate line of the spheroidal system, normal to the ellipsoid.</p> <p>Theoretically, the normal gravity field can be successfully used as an orthogonal coordinate system, since its force lines and level surfaces can serve as <em>natural </em>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</p> <p><img src="" alt="" /></p> <p>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.</p> <p>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 H<sup>&#947;</sup>.</p> <p>A detailed way contains&#160;three steps:</p> <p>1. Standard calculation by the above Yeremeyev's formula and the corresponding third spheroidal coordinate w&#8242; or b&#8242;. This is <em>the first approximation</em>.</p> <p>2. Refinement of the third spheroidal coordinates of the points on the telluroid from the Molodensky's condition:</p> <p><img src="" alt="" /></p> <p>the reduced latitude u could be precised consequently.</p> <p>3. Evaluation of the curvilinear integral from w = w<sub>0</sub> to w:</p> <p><img src="" alt="" /></p> <p><img src="" alt="" /></p> <p>where</p> <p><img src="" alt="" /></p> <p>[Here was the inaccuracy in the <em>Yurkina M. I.</em> (2004) To refine the height to fractions of a millimeter, this step is absent in the paper <em>Jurkina M. I., Pick M.</em> (2004) N&#225;vrh na zp&#345;esn&#283;n&#237; v&#253;po&#269;tu norm&#225;ln&#237;ch v&#253;&#353;ek].</p>
The authors highlight some special issues of the theory of heights. In establishing a global system of normal heights, one of the key matters is the final choice of a system of altitudes to represent elevation marks. In addition to proving the advantages of the mentioned system, it is necessary to eliminate some “white spots” within itself. In 2004, a more accurate way of calculating normal heights as the length of a coordinate line in a spheroidal system was considered. Simultaneously with this, in the papers by foreign researchers, methods of "practically accurate" calculation of the orthometric height were developed, which is associated with increasing knowledge of the earth`s crust upper layers structure. At studying the normal height, it is required to develop methods of its high-precision calculation and explore the properties of various options for setting the corresponding curvilinear integral. An expression is obtained for the normal height as a segment of the coordinate line of the spheroidal system; the one obtained in 2004, which contained inaccuracies, was corrected. The proposed method can be applied at an arbitrary distance from the reference ellipsoid.
Our review of the Russian literature on geodesy caused a desire to consider the texts related to determining the heights of points on the earth’s surface. This topic, seeming simple, is very complex and is a mandatory part of most textbooks for students of geodetic specialties in universities and colleges. The presentation of the heights theory in the course of topography affects not only the specialized departments of construction, polytechnic universities and specialized colleges, but higher geodetic educational institutions as well. The authors review and evaluate the sections on the theory of heights in the domestic educational geodetic literature. Typical inaccuracies in the presentation of elevation systems are analyzed, criticism of the most common clichés among surveyors is given, recommendations are made on the minimum of presentation of the elevation system for non-specialists, and some useful illustrations are provided to make understanding the essence of the phenomenon easier. The article was written basing on the experience of lecturing height systems by employees of the departments of surveying and higher geodesy. We hope to arouse the interest to this topic.
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