# Height (geodesy)

In geodesy, the height is the vertical distance between a certain point and a reference surface . If this point is on the surface of the earth or terrain , it is also referred to as the terrain height or the geographic height . With the height as the third coordinate - in addition to the geographical latitude and longitude or the right and high value of a Cartesian coordinate system - the position of each point on, above or below the surface of the earth can be clearly described. The most important height definitions: ellipsoidal height h, normal height H N and orthometric height H

Various geometric figures can serve as the height reference surface in the sense of higher geodesy , for example the geoid , a quasigeoid or a nationally valid reference ellipsoid adapted to the respective country . The mean sea ​​level resulting from long-term level measurements by a suitable coastal station was usually set as the zero level of such a reference area . Depending on the country or application, different height definitions and different zero levels are used (see height above sea level ).

## Height definitions

Generally it is expected that

1. a height is a geometric quantity and is measured in units of length, and
2. no water flows between points of the same height.

Due to the different gravity at the equator and at the poles, heights cannot be geometrically correct (1.) and physically correct (2.) at the same time.

In order to fulfill point 2, points must have the same gravity potential and therefore lie on an equipotential surface of gravity . Only the force of gravity is stronger at the poles 1/189 than at the equator, so that they are closer together at the poles by 1/189.

Therefore, some purely geometrically or physically defined heights are used:

1. Ellipsoidal heights (GPS heights) as purely geometrically defined heights, expressed in a unit of length ,
2. Geopotential levels as purely physical heights, the difference between two gravitational potentials .

When leveling , you get different height differences when you level along different paths. The reason for this so-called theoretical loop closure error is that the height transfer occurs along the non-parallel equipotential surfaces, but the differences are measured in meters. In order to eliminate the contradictions, it is necessary to take the gravitational field into account for large areas with greater height differences . Various metric height systems that take the gravity into account have been developed for practical use:

• Normal orthometric or normal spheroid heights
• Normal heights
• Orthometric heights.

Between height systems there are notable differences in the high mountains to reach orders of magnitude of centimeters to tens of centimeters per kilometer. The irregularities in the earth's gravity field have been researched since the end of the 19th century under the terms vertical deviation or gravity anomaly and geoid , and today they are measured with sufficient accuracy.

### Ellipsoidal heights

Geometrically defined heights are now known as the ellipsoidal height h . They indicate the distance of a point from a geodynamically defined reference ellipsoid along the normal ellipsoid . However, two points of the same ellipsoidal height do not lie on the same equipotential surface , so that water can flow between them.

Ellipsoidal heights can be determined directly using GPS . A simple conversion from leveled to ellipsoidal heights without knowledge of the serious disturbances is not possible. Alternatively, ellipsoidal heights can be determined by creating a spatial polygon .

### Geopotential Levels

A geopotential level C is the negative gravity potential difference between a surface point on the earth and the geoid . Points with the same geopotential level form an equipotential surface .

${\ displaystyle C = W_ {0} -W_ {P} = - \ int _ {P_ {0}} ^ {P} {\ vec {g}} \ \ mathrm {d} {\ vec {s}}}$ Since it is a gravitational potential difference, the SI unit is joules per kilogram (J / kg) or (m² / s²). In some cases geopotential units  (gpu) are also used as a unit (1 gpu = 10 J / kg). In the past, geopotential levels were also given in the unit geopotential meter  (gpm) and derived geopotential decameter  (gpdm). 1 gpm = 10 gpdm corresponds to 9.80665 J / kg. The amount corresponds to the dynamic amount. Geopotential levels can be determined from leveled height differences and gravity measurements . ${\ displaystyle \ Delta n}$ ${\ displaystyle g}$ ${\ displaystyle \ Delta C = \ int _ {1} ^ {2} g \ \ mathrm {d} n}$ or.

${\ displaystyle \ Delta C = \ sum _ {i} g_ {i} \ cdot \ Delta n_ {i}}$ ### Dynamic highs

Dynamic heights H Dyn are usually converted from the geopotential heights with the normal gravity at sea level at 45 ° latitude into the dimension meters . They express the distance that the equipotential surfaces would have. However, the actual (metric) distance varies by about due to the lower gravitational acceleration at the equator compared to the poles . ${\ displaystyle \ gamma _ {0} ^ {45}}$ ${\ displaystyle \ gamma _ {0} ^ {45}}$ ${\ displaystyle 5/1000}$ ${\ displaystyle H_ {Dyn} = {\ frac {C} {\ gamma _ {0} ^ {45}}}}$ With ${\ displaystyle \ gamma _ {0} ^ {45} = 9 {,} 80665 \, \ mathrm {\ frac {m} {s ^ {2}}}}$ Dynamic heights are unusable for geodetic practice because of the large dynamic corrections. However, they result directly from rescaling the geopotential elevation. They are important in synoptic meteorology and atmospheric research ( main pressure surfaces ).

### Orthometric heights

The orthometric height H results from the distance along the curved plumb line between a point on the earth's surface and the geoid . The geopotential levels are converted using the mean acceleration due to gravity along the plumb line. The gravity cannot be measured in the interior of the earth, so it can only be calculated by setting up a hypothesis about the mass distribution. Orthometric heights are therefore subject to hypotheses. Points of the same orthometric height generally do not lie on the same level surface. ${\ displaystyle {\ bar {g}}}$ ${\ displaystyle H = {\ frac {C} {\ bar {g}}}}$ With ${\ displaystyle {\ bar {g}} = {\ frac {1} {H}} \ int _ {0} ^ {H} g \ \ mathrm {d} H}$ The difference between the ellipsoidal and the orthometric height is called geoid undulation . It is up to 100 m globally, within Switzerland z. B. maximum 5 m. ${\ displaystyle N}$ ${\ displaystyle N = hH}$ ### Normal heights

Normal heights describe the distance between a point and the quasigeoid along the slightly curved normal plumb line (see top figure) . They were developed by the Soviet geophysicist Mikhail Sergejewitsch Molodensky and - unlike orthometric heights - can be determined without hypotheses: ${\ displaystyle H_ {N}}$ ${\ displaystyle H_ {N} = {\ frac {C} {\ bar {\ gamma}}}}$ The mean normal gravity is used for the conversion of the geopotential values : ${\ displaystyle {\ bar {\ gamma}}}$ ${\ displaystyle {\ bar {\ gamma}} = {\ frac {1} {H_ {N}}} \ int _ {0} ^ {H_ {N}} \ gamma \, \ mathrm {d} {H_ { N}}}$ The deviation between the ellipsoidal height and the normal height is called the height anomaly or quasigeoid height and is between 36 and 50 m in Germany: ${\ displaystyle \ zeta}$ ${\ displaystyle \ zeta = h_ {e} -H_ {N}}$ Orthometric and normal heights differ because of the deviation of the actual severity from the normal severity . The differences can be up to a meter or more in the high mountains; in the lowlands they are often only in the millimeter range; in the old federal states they are −5 to +4 cm. ${\ displaystyle {\ bar {g}}}$ ${\ displaystyle {\ bar {\ gamma}}}$ ### Normal orthometric heights

If no gravity measurements are available, the gravity correction of the observed height differences can only be carried out with the normal gravity. The derived heights are then called normal-orthometric heights or spheroid-orthometric heights H Sph . The deviations from normal heights are small, as the corrections only differ due to the small portion of the surface open air gradient.

${\ displaystyle H_ {Sph} = {\ frac {C ^ {*}} {\ bar {\ gamma}}}}$ With ${\ displaystyle C ^ {*} = \ int _ {0} ^ {1} \ gamma \ \ mathrm {d} n}$ ## Corrections

The actual measurand of the height measurement is not heights above sea level , but height differences . In national surveys, these are usually determined by leveling . In order to convert the measured height differences into one of the height definitions, corrections must be made. ${\ displaystyle \ Delta H}$ ${\ displaystyle dn}$ ${\ displaystyle E}$ ${\ displaystyle \ Delta H_ {12} = H_ {2} -H_ {1} = \ int _ {1} ^ {2} \ \ mathrm {d} n + E_ {12}}$ ### Dynamic correction

The leveled height differences can be converted into dynamic height differences through dynamic correction.

${\ displaystyle E_ {12} = \ int _ {1} ^ {2} {\ frac {g- \ gamma _ {0} ^ {45}} {\ gamma _ {0} ^ {45}}} \ \ mathrm {d} n}$ ### Orthometric correction

With the orthometric correction, there are two hypothesized, location-dependent components in addition to the strictly determinable dynamic component.

${\ displaystyle E_ {12} = \ int _ {1} ^ {2} {\ frac {g- \ gamma _ {0} ^ {45}} {\ gamma _ {0} ^ {45}}} \ \ mathrm {d} n + {\ frac {{\ bar {g}} _ {1} - \ gamma _ {0} ^ {45}} {\ gamma _ {0} ^ {45}}} H_ {1} - {\ frac {{\ bar {g}} _ {2} - \ gamma _ {0} ^ {45}} {\ gamma _ {0} ^ {45}}} H_ {2}}$ Assuming the mean earth crust density of 2.67 g / cm³, the following applies to mean severity : ${\ displaystyle {\ bar {g}}}$ ${\ displaystyle {\ bar {g}} = g + 0 {,} 424 \ cdot 10 ^ {- 6} \, \ mathrm {s} ^ {- 2} \, H}$ ### Normal correction

Similarly, normal height differences can be calculated with the normal correction. Instead of the mean severity, the hypothesis-free mean normal severity is used here. ${\ displaystyle {\ bar {g}}}$ ${\ displaystyle {\ bar {\ gamma}}}$ ${\ displaystyle E_ {12} = \ int _ {1} ^ {2} {\ frac {g- \ gamma _ {0} ^ {45}} {\ gamma _ {0} ^ {45}}} \ \ mathrm {d} n + {\ frac {{\ bar {\ gamma}} _ {1} - \ gamma _ {0} ^ {45}} {\ gamma _ {0} ^ {45}}} H_ {1} - {\ frac {{\ bar {\ gamma}} _ {2} - \ gamma _ {0} ^ {45}} {\ gamma _ {0} ^ {45}}} H_ {2}}$ ### Normal orthometric correction

In normal orthometric correction, the normal severity is used for dynamic correction instead of the measured severity . ${\ displaystyle g}$ ${\ displaystyle \ gamma}$ ${\ displaystyle E_ {12} = \ int _ {1} ^ {2} {\ frac {\ gamma - \ gamma _ {0} ^ {45}} {\ gamma _ {0} ^ {45}}} \ \ mathrm {d} n + {\ frac {{\ bar {\ gamma}} _ {1} - \ gamma _ {0} ^ {45}} {\ gamma _ {0} ^ {45}}} H_ {1 } - {\ frac {{\ bar {\ gamma}} _ {2} - \ gamma _ {0} ^ {45}} {\ gamma _ {0} ^ {45}}} H_ {2}}$ ## Overview

Definition name →
property ↓
Geopotential Kote Dynamic height Orthometric height Normal height Normal orthometric height Leveled height Ellipsoidal height
Abbreviation C. H Dyn H H N H Sph H
unit m² / s² = J / kg = 0.1 gpu m Note 1 m
Reference area Geoid Quasigeoid Reference ellipsoid
determination Leveling GPS / space polygon
Measurement of the local acceleration due to gravity necessary Yes No
Assumptions about the density distribution in the interior of the earth necessary No Yes No
Level loop closure failure No no on the surface Yes (-) Yes (--)
Equipotential surfaces all heights at height zero none (approximated at height zero) no (-) no (--) no (---)
Note 1 The dynamic height does not indicate the distance from the reference surface.

Red text : disadvantageous property of the respective height definition. "(-), (-), (---)": strength of the disadvantages.

Green text : advantageous property of the respective height definition.