# Network (geodesy)

Survey network

In geodesy , a surveying network (usually just called a network ) is an arrangement of surveying points that are connected to one another "like a network" through observations (measurements). Networks can for example

As a fixed point field, the points form the basis for further measurements, which are connected to the network and thus determined in a uniform coordinate system. Depending on whether the position or height of the fixed points has been determined, one speaks of a fixed point field or fixed point field.

## General

A geodetic network is used to determine the coordinates of survey points in a chosen reference system . For the determination, observations are made between the individual points of the network. These observations can be:

## Types of geodetic networks

• Triangulation networks : directions or angles to neighboring points are measured from each point. The measurements are planned in such a way that triangles are created between the points . To determine the size of the network, the length must be measured along at least one side of the triangle, which was previously done using basic measurements .
• Trilateration networks: Of the triangles that are formed between the points of the network, only the distances are measured.
• Elevation networks : If only the height of the survey points can be determined, it is sufficient to measure the height differences.
• GPS networks: The spatial vectors (i.e. the coordinate differences in the direction of the x, y and z axes) are determined between the points using GPS measurements.
• Combined networks: two or more of the measurement methods listed above are used.

Until the 1970s -year triangulation were preferably used, since the measurement of angle by means of the former instrument theodolite was much easier than a complicated distance-measuring means Invar baseline measurement or measuring rods, etc. Due to the advancements in the field of electronic distance measurement since 1980 is now the Distance measurement is usually easier than angle measurement . GPS measurements are also being used more and more often. Therefore, today, geodetic networks are usually created as combined networks.

## calculation

In order, on the one hand, to have a control over the measurements carried out, on the other hand, to be able to make qualitative statements about the network and the measurements carried out, the measurements are carried out overdetermined . This means that more measurements ( surplus measurements ) are carried out than are necessary to determine the geometry of the network, followed by a network adjustment . The mathematical optimization of the network structure is called network design (see below).

### Simplified example

If you measure all three angles α, β, γ in a triangle, there is an overdetermination. This can be expressed in the following condition:

α + β + γ = 180 °

If you get a value of 180.1 ° for the angle sum from the measurement , you can estimate that the measurement is worse than if you get a sum of 180.001 °.

### Adjustment calculation and network design

In order to arrive at a plausible result, improvements are made to the three measured angles so that the result then gives the expected value of 180. For example, if the measured angles add up to 180.1 °, the final error is divided into three equal parts:

α improved = α measured - 0.033 °
β improved = β measured - 0.033 °
γ improved = γ measured - 0.033 °

The choice of improvements is normally made using the least squares method , according to which the sum of squares of all improvements is as small as possible. There are two calculation models for this:

• Conditional adjustment : According to the above example, a conditional equation is set up for each excess measurement:
(α + v 1 ) + (β + v 2 ) + (γ + v 3 ) = 180 °
other conditions in triangles can, for example, satisfy the sine law or the law of cosines . The resulting system of equations is solved for all v in such a way that the minimum condition is also fulfilled.
• Mediating adjustment : every measurement that has been made is expressed as a function of the searched coordinates of the survey points.
If you measure the (horizontal) distance between two points, you get:
s² + v 1 = (x 2 –x 1 ) ² + (y 2 –y 1 ) ²
(whereby it is already taken into account here that the observation is again an improvement receives).

Again, one obtains an (overdetermined) system of equations, whereby the coordinates of the measurement points are already the unknowns. A clear solution can be found for this (after linearization ) under the minimum condition.

Example: see adjustment based on mediating observations

In modern, especially large-scale networks, an extensive optimization of the network structure is desirable. In addition to the greatest possible accuracy , this so-called network design also aims to ensure full controllability ( reliability ) of the measurements and is a demanding task in mathematical geodesy. Methods related to this were developed by geodesists Erik Grafarend and Fernando Sansò , among others , and are partly based on multi-dimensional concepts of geometric stochastics .

## Storage and reference systems

In geodetic networks, usually only relative quantities are measured, i.e. quantities that refer to the difference between two or three points (angles, distances, space vectors). This defines the internal geometry of the network. But this does not yet define where the network will be stored in the coordinate system. Example: The distance s  = 35 m between two measurement points is measured . This specifies that the two points must be 35 m apart.
That can mean:

Point 1 has the coordinates x = 100 and y = 100
Point 2 has the coordinates x = 100 and y = 135

or:

Point 1 has the coordinates x = 317 and y = 412
Point 2 has the coordinates x = 282 and y = 412

or ...

In order to eliminate this date defect, which means a system of equations with a rank defect in a mediating calculation , additional conditions for storage in space or on the earth's surface must be specified:

• Defining a zero point and a preferred direction: coordinates and the azimuth to a neighboring point are given arbitrarily for a point in the network that is as centrally located as possible .
• Astro-geodetic network: At the zero point ( fundamental point ), the exact direction of the perpendicular in the star coordinate system is determined (from star measurements) and transferred to the earth ellipsoid or a regional reference ellipsoid as geographical latitude and longitude . This means that the perpendicular deviation existing at other points in the network center is set to zero. The altitude of the network results from its sea level (i.e. the geoid height is assumed to be zero), the network scale from one or more measured point distances.
• Astrogeodetic network adjustment: The measurements between the network points are reduced by the effect of the vertical deviation (which is up to 30 centimeters per km in the mountains) and in a later step an area-wide geoid determination is carried out. An average geoid height of e.g. B. 30 m changes the network scale by about 5 millionths (5 mm per km), but allows later comparisons with a world network or with the national survey of neighboring countries.
• Inclusion of existing survey points in the network ( point activation ) and use of the coordinates of these known points.
• Free adjustment : First the network - as it was measured - is adjusted (whereby the rank defect is ignored) and then it is transformedonto existing measurement points with the help of control points .
• Satellite World Network : With methods of satellite geodesy (combination of SLR , GPS and possibly VLBI ) will set up a network with long triangle sides, can be integrated into the later regional networks ( Anfelderung or power-up ). Instead of a regional reference ellipsoid, however, the mean earth ellipsoid must bethe calculation basis - which requires intercontinental cooperation.
• Such measurements and network adjustments have been carried out regularly since the 1990s , with IUGG international commissions responsible for data organization (e.g. the IGS service for GPS and the IVS for radio VLBI ). The coordinates of a few hundred network points determined annually define the ITRF's global network , which also serves geodynamic purposes.

## Significance and history of surveying networks

Extensive surveying networks must have existed in ancient Egypt in order to restore the property boundaries after the annual Nile flood and to build the pyramids. In Greek antiquity, on the other hand, location determination was largely limited to astronomical latitude determination and isolated data from ship navigation .

In the Middle Ages , the measurement and calculation methods were further developed in Arabia , but large-meshed surveying networks only emerged in the course of the nautical portolane (coast and sea ​​maps ). Around 1610 , the Dutch astronomer Willebrord van Roijen Snell (Snellius) developed the mathematical basis for triangulation .

Jean Picard is considered one of the founders of geodesy . He carried out the first triangulations over great distances with telescopes. In the 18th century, the Cassini family triangulated a network encompassing all of France and created the Carte de Cassini on this basis .

In the course of the 18th century, surveying management was transferred to Germany and Austria-Hungary . At the beginning of the 19th century , Gauß , Liesganig and others developed the theory of land surveying and triangulation networks were spanned over large stretches of land. Angle measurements were carried out between prominent points up to 50 km away (on hilltops, mountain peaks, etc.), while the distance measurements had to be limited to a few “baselines”. The nets were initially stored in regional fundamentals, later in cross-state projects ( Potsdam in Germany , the Hermannskogel near Vienna in Austria ). To determine the network scale, distance measurements were carried out approximately every 200–300 km (for example the “Wienerneustädter Basis” near Wiener Neustadt in the level of the southern Vienna Basin or near Josefstadt in Bohemia.)

Based on the triangulation points of the first order network (TP) created in this way, local “network densifications” were carried out later. For this purpose, the “geometer” or “engineer topographer” established a geodetic network in the local area (e.g. a municipality), which was measured and calculated using the first-order TP that had already been calculated. This resulted in further triangulation points of subordinate hierarchy (network of 2nd to 4th or 5th order), which already formed a narrow grid of fixed points every 1 to 3 km. From the 1950s , the increasing demand was taken into account by so-called switch-on points, which were closer than 1 km in the country, in cities even down to 200-300 meters.

At the beginning of the 1970s, the world network of satellite triangulation achieved a previously unattainable resolution (± 4 m at the ground stations). In the 1990s , many of these fixed points were re-measured using a GPS network, which allowed the large-scale network accuracy to be increased to a few centimeters .

Today the measurement of a geodetic network to determine the points in all areas is a standard procedure. It is used in engineering geodesy as well as in cadastral surveying for the division of parcels. This is due on the one hand to the extensive use of electronic measuring devices ( theodolite , electronic distance measurement ) and to the spread of software for calculating a mediating compensation.