STRUČNÝ PREHĽAD TEÓRIE VÝŠOK A BRIEF OVERVIEW TO THEORY OF HEIGHTS STRUČNÝ PREHĽAD TEÓRIE VÝŠOK A BRIEF OVERVIEW TO THEORY OF HEIGHTS

V minulosti bolo v rôznych častiach sveta formulovaných a prakticky použitých mnoho typov výškových systémov. V súčasnosti sú prevažne používané ortometrické a normálne výšky. Ortometrické výšky sú prirodzené výšky nad hladinou mora, teda nad geoidom. Definícia ortometrických výšok vyžaduje aspoň teoretickú znalosť rozloženia hustoty topografických más medzi zemským povrchom a geoidom. Z tohto dôvodu Molodensky v roku 1945 sformuloval teóriu normálnych výšok založenú na princípe, že výšky môžu byť počítané bez hypotézy o rozložení hustoty topografickej hmoty. Tento článok sa zaoberá teoretickými aspektmi definície výšok.

In various parts of the world many kinds of height systems have been formulated and practically used in the past. Nowadays, orthometric heights and normal heights are widely used. The orthometric heights are the natural heights above sea level, that is, above the geoid. The definition of the orthometric heights required the knowledge of the topographical masses density distribution between the Earth's surface and the geoid, at least theoretically. For this reason Molodensky in

Introduction
In a classical sense of Gauss and Listing, the geoid is defined as an equipotential (level) surface whose geopotential value is W o .
First, C. F. Gauss (1828) stipulated that the Earth's boundary surface should be a level surface [4]. Later, F. W. Bessel (1837) defined the Earth's boundary surface as the equipotential surface approximating the mean calm ocean levels [2]. Finally, J. B. Listing (1873) named this surface "geoid" [7]. The geoid is then basic equipotential surface for the definition of the natural orthometric heights.
To define an actual value of the orthometric height, the distribution of the density of the topographical masses between the geoid and the Earth's surface has to be known.
F. R. Helmert (1890) defined the orthometric height based on a hypothesis of density distribution [6]. The constant topographical density o ϭ 2.67 g.cm Ϫ3 is assumed in this hypothesis.
M. S. Molodensky (1945) formed theory of the normal heights without using any hypothesis about density distribution of topographical masses [9].  Bruns' formula (2.1) states that the constant difference dW(r, ⍀) of potentials of two closed equipotential surfaces is equal to the negative product of the gravity acceleration g(H, ⍀) and the distance dH(⍀) along the plumb line between these equipotential surfaces.

Ortometrická výška
The geopotential number C(H, ⍀) is defined by applying Bruns' formula to the difference between the gravity potential W o on the geoid and the gravity potential W(H, ⍀) on the Earth's surface [5]

Orthometric height
The actual orthometric height H O (⍀) is defined as a length of the plumb line between the geoid and the Earth's surface (Fig.1). According to Bruns' formula, the difference between two potentials is constant and independent of a path of integration. Consequently, the integration along the plumb line between the geoid and the Earth's surface is equal to the integration over the Earth's surface from the zero-height point O (H ϭ 0, ⍀Ј) on the geoid (gauge station) to the point P(H, ⍀Ј) According to the theorem of a mean integral value, the integral on the right-hand side of the equation (3.1) takes the following form where g ෆ (H, ⍀) stands for the mean value of the gravity along the plumb line between the geoid and the Earth's surface. can not be measured, it has to be computed from the surface gravity. This is done by reducing the observed value g(H, ⍀) of gravity according to the hypothesis of the topographical density distribution.

Helmert's orthometric height H O (⍀) is defined as [10]
Helmert's gravity [5] is defined by using Poincaré-Prey's gravity gradient, which is given by the expression in the square brackets.
According to this theory the vertical gradient of gravity is considered to be constant along the plumb line between the geoid and the Earth's surface. Since the gravity gradient is considered to be constant, g ෆ (H/2, ⍀) is evaluated directly for the mid-point of the plumb line H O (⍀).
Integrál v rovnici (2.2) môže byť nahradený konečným počtom výškových rozdielov ⌬H(⍀Ј i ) z nivelácie a konečným počtom tiažových údajov g(H, ⍀Ј i ) z tiažových meraní realizovaných v nivelačnom ťahu, we can obtain Bruns' formula for the gravity gradient Here is the mean value of the angular velocity of the Earth's rotation, G is Newton's gravitational constant, o is the mean density of the topographical masses between the geoid and the Earth's surface and ∂ 2 W(x, y, z)/∂ x 2 , ∂ 2 W(x, y, z)/∂ y 2 , ∂ 2 W(x, y, z)/∂ z 2 are second partial derivatives of the gravity potential in the local astronomical co-ordinate system x, y, z, where the z-axis coincides with the outer normal of the local equipotential surface.
The vertical gradient ∂ ␥ (H, )/∂ H(⍀) of normal gravity generated by the mean ellipsoid of the Earth is with the mean curvature J o () of the ellipsoid given by ᎏ . (3.11) Here M(), N() are the principal radii of curvature of the mean ellipsoid of the Earth [3] where is the geodetic latitude, a, b are the semiaxes of the ellipsoid, and e 2 ϭ (a 2 Ϫ b 2 )/a 2 is the square of the first numerical eccentricity of the ellipsoid.
The Poincaré-Prey's theory of the vertical gradient of the gravity assumes, with sufficient accuracy, the following [10] g(h, ⍀) J(H, ) Х ␥(H, ) J o (). (3.13) The Poincaré-Prey's vertical gradient of gravity is then and the Helmert orthometric height takes the following form

Normal orthometric height
The numerical value of the geopotential number C(H, ⍀Ј) is a result of levelling combined with gravity measurements.
The integral in the equation (2.2) can be replaced by finite elements of the height differences ⌬H(⍀Ј i ) from levelling and by finite discrete gravity values g(H, ⍀Ј i ) from gravity measurements realised in a levelling line, )
The actual gravity values g(H, ⍀Ј) on the Earth's surface are replaced by theoretic values ␥(H, Ј) of the normal gravity in the definition of the normal orthometric height H NO (⍀), which is then given by the following formula

Dynamic height
The dynamic height H D (⍀) is defined as where ␥ o () is the normal gravity on the level rotation ellipsoid for an arbitrary standard latitude, usually ϭ /4.
Obviously, the dynamic height differs from the geopotential number only in a scale and in a unit. The division of the geopotential number by the constant value ␥ o () just converts the geopotential number into a length.

Normal height
The determination of the heights from levelling and gravity measurements without any hypothesis about a density distribution of topographical masses is fundamental principle of Molodensky's theory of the normal height H N (⍀), [9].
Replacing the mean value g ෆ (H, ⍀) of the gravity by the mean value of the normal gravity ␥ ෆ (H, ) along the normal between the reference ellipsoid and the telluriod (Fig.2), we obtain the formula for the definition of the normal height H N (⍀) as The mean value of the normal gravity ␥ ෆ (H, ) is referred on the mid-point of the normal between the telluriod and the reference ellipsoid, so it can be expressed by Taylor series in the following form The first derivative ∂ ␥/∂ H given by equation (3.10) can be rewritten as [5] C(H, ⍀Ј) ϭ ͵ (4.1) Poznámka: J. Vignal [12] navrhol podobný výškový systém, kde stredná hodnota normálneho tiažového zrýchlenia je počítaná z normálneho tiažového zrýchlenia ␥ o () na hladinovom rotačnom elipsoide v zmysle aproximácie vertikálneho gradientu normálneho tiažového zrýchlenia, pričom v tejto teórii je použitá iba prvá parciálna derivácia Taylorovho rozvoja (6.2) The second derivative ∂ ␥ 2 /∂ H 2 can be described in the spherical approximation ᎏ ∂ The surface, whose normal gravity potential U(H N , ) is equal to the gravity potential on the Earth's surface is the telluroid. The quasigeoid (which is not equipotential surface) is given by heights anomalies (⍀) referred on the reference ellipsoid.
Note: Vignal [12] proposed a similar system of heights, whereby the mean value of the normal gravity is evaluated from the normal gravity ␥ o () on the level rotation ellipsoid by means of approximate vertical gradient of normal gravity. In this theory only the first partial derivative of the Taylor series (6.2) is used in the form  According to Heiskanen and Moritz, (1967), the difference between the normal and orthometric height ␦H(⍀) is, with sufficient accuracy, equal to [5] The Z tejto schémy vyplýva, že uvedené výškové systémy môžu byť získané delením geopotenciálnej kóty príslušnou hodnotou tiaže.

Summary
By means of the geopotential number (H, ⍀Ј), we can describe different kinds of heights in a instructive form: From this scheme it is clear that the above mentioned height systems can be obtained by dividing the geopotential number by the relevant value of the gravity.