Geodesy

Geodesy is the science that deals with the study of the shape and dimensions of the earth’s reference surface, its gravitational field and geodynamic phenomena such as the displacement of the poles, the terrestrial tides and the movements of the crust (theoretical geodesy). Geodesy also deals with the elaboration of theories and operational procedures aimed at the knowledge, description, measurement and representation of more or less extensive areas of the Earth (operational geodesy).

The science that studies the tools and methods of operation, calculation and design necessary for the graphic representation of a more or less extensive part of the earth’s surface is instead topography (from the Greek topos, place and graphein, to write).

One of the most important aspects of cartography is to give a representation of the entire surface of the territory to be represented in a single and shared reference system. The territory is therefore represented both from an altimetric and a planimetric point of view. The altimetric representation is obtained by attributing a height to each generic point of the physical surface of the Earth; this height is given by the distance, along the vertical passing through the point, from a reference surface called geoid. The planimetric representation is made by relating the points of the earth’s surface with the points of the flat Cartesian system, called cartographic projection, through an intermediate mathematical surface of passage represented by the reference ellipsoid.

 

The earth, the geoid and the ellipsoid

One of the first problems that geodesists faced was that of defining a mathematical surface of the Earth that would allow to correspond the points of the physical surface of the Earth with the points of a flat Cartesian system. To do this, an orthogonal Cartesian triplet (X, Y, Z) was chosen that could represent a reference system to which the coordinates of the mathematical surface could be referred (fig. 1).

 

 

This triad can be defined as follows:

• origin of the triad in the center of gravity of the land mass
• Z axis coincident with the earth’s rotation axis
• X axis orthogonal to Z in the center of gravity of the earth’s mass in the plane containing the intersection point between the Greenwich meridian and the equator.

This system is called a geocentric system.

Given a point P on the physical surface of the Earth, the plane of the terrestrial meridian passing through P is defined as the plane containing the Earth’s rotation axis and point P.

Each point P on the surface of the Earth can be identified in two different ways:

• as a function of its X, Y, Z coordinates in the geocentric system
• as a function of a pair of terrestrial geographic coordinates represented by the terrestrial latitude and the terrestrial longitude.

The terrestrial latitude is the angle that the vertical passing through the point P forms with a generic plane orthogonal to the earth’s rotation axis, in particular with the equatorial plane; the terrestrial longitude is, instead, the dihedral angle that the plane, containing the point P and the earth’s rotation axis, forms with a longitude reference plane, which is the one defined by the earth’s rotation axis and the plane containing the X axis, that is the meridian plane passing through Greenwich (Galletto, Spalla, 1999).

To define the mathematical expression of the Earth, the geodesists took its gravitational field as a starting point, starting from the assumption that the force of gravity exists in every point of the Earth, given by the sum of the Newtonian force of attraction and the centrifugal force. Therefore, a surface which is always perpendicular to the lines of force of the gravitational field was assumed as the mathematical surface of the Earth; but since there are an infinite number of these lines, one in particular was chosen, that is, the one passing through the mean sea level at a precise point on the earth’s surface. This surface is called a geoid.

In a reductive way, it can be stated that the geoid is the surface that would be obtained by extending the sea surface below the emerged lands in the absence of accidental or periodic perturbations (tides, winds, currents, etc.) or of hydrostatic equilibrium assumed by the surface of the oceans, also known as the surface of the mean sea level (measured by the tide gauge).

Any point in space before being brought back to the plane of the map is imagined transferred onto the geoid, projecting it vertically, therefore according to the vertical of the place. However, the mathematical formulation of the geoid is very complex in relation to the fact that it includes not only geometric but also mechanical quantities such as the density of the different points within the mass of the Earth. Then, other reference surfaces have been defined that approximate the geoid, for which it is possible to identify simpler mathematical expressions.

The ellipsoid, which represents the most suitable mathematical form to represent the surface of the Earth, is none other than the surface generated by the rotation of an ellipse of semi-axes a and b around the Z axis, coinciding with the Earth’s rotation axis. This ellipsoid is called geocentric (fig. 2).

 

 

Given a point P on the surface of the ellipsoid, the meridian plane of P is defined as the plane containing the axis of rotation Z and Point P; P meridian is defined as the intersection of said plane with the surface of the ellipsoid. The rather simple mathematical expression of the ellipsoid allows to easily match its points with those of a system of plane Cartesian coordinates. The expression

in which a = 6378388.00 m represents the equatorial half-axis of the ellipsoid and b = 6356912.00 m the polar half-axis. The values of a and b reported above are those determined by the geodesist Hayford in 1909 and still used to define the international reference ellipsoid. The geodesists of the various nations decided to adopt not the geocentric ellipsoid, but an ellipsoid of the same size and shape as the geocentric one, but slightly rotated and translated with respect to it, in such a way as to achieve the condition of tangency to the geoid in a barycentric point of the territory to be represented (Galletto, Spalla, 1999).

For Italy, the Italian geodesists chose at the end of the 19th century to create this coincidence between the national ellipsoid and the geoid at the Monte Mario Observatory in Rome (Roma 40 system) and subsequently at Potsdam in Germany (ED 50 system ).

 

Cartographic projection systems

The geographical maps are representations on the plane (in the form of a graphic or computerized elaborate) of the earth’s physical surface at an assigned scale 1 / n, according to pre-established rules and conventional signs. The projections are the methods adopted to report and transform the geographical grid into a flat grid, in order to obtain the representation of a part or the entire earth’s surface. Therefore, the projections underlie the use of geometric means and can be divided into:

• pure perspective projections: when the projection occurs on a plane tangent to the ellipsoid at a given point. Depending on the position of the projection center, there will be centrographic, stereographic, scenographic, orthographic projections, etc. (fig. 3)

 

• pure cylindrical projections: when the projection of the points of the ellipsoid takes place on an enveloping surface (cylindrical or conical), placed tangent to the ellipsoid and the projection center at the center of the ellipsoid or along the direction normal to the line of tangency (fig. 4)

 

 

Among the latter, the best known is Mercator’s isogonic cylindrical projection (fig. 5). In this projection, meridians and parallels are straight and perpendicular to each other; the parallels instead of approaching in the polar regions, move away, resulting more dense at the equator than towards the poles. Therefore, the meridians remain equidistant, while in reality they gradually approach each other towards increasing latitudes. Consequently, the parallels move away from each other in the proportion of how much the distance of the meridians is greater on the map than in reality (Pigato, 2000).

 

Gauss representation

By Gaussian map we mean a system of Cartesian plane coordinates N and E and two functions f and g which relate a generic point P of the ellipsoid, given through the geographic coordinates (latitude and longitude) φ and λ, and the corresponding point P ‘ of the Cartesian system. The representation of Gauss was chosen for the official Italian cartography.

To pass from the ellipsoidal system to a plane system N, E, Gauss obtained transformation formulas f (φ, λ) and g (φ, λ) by imposing the following conditions:

• the ellipsoidal equator must turn into the abscissa axis E
• the ellipsoidal meridian assumed as the origin of the longitudes must be transformed into the axis of the ordinates N
• an arc of length m on the origin meridian must transform into a segment of equal length on the ordinate axis N
• the angle α formed by two directions coming out from a point on the ellipsoid must remain the same as that of the corresponding directions shown in the map
• the deformation coefficient, although varying from point to point, must be the same in all directions coming out of a point.

 

From these analytical conditions the two functions si f (φ, λ) and g (φ, λ) have been obtained which, applied to the coordinates, generate a projection similar to that which would be obtained by projecting the points of the ellipsoid, from the center of the ellipsoid, on a cylinder tangent to the ellipsoid along the meridian origin of the longitudes (fig. 6) (Galletto, Spalla, 1999).

 

 

The Gauss cartography is consistent and therefore the angles measured on the map correspond perfectly with the respective angles measured on the ground; the lengths measured on the paper are instead slightly deformed compared to those measured on the reference surface.

Figure 7 below shows a representation of the geographic grid, that is the complex of lines that represent the transforms of the meridians and parallels: note that the transform of the central meridian is a straight line. It can easily be seen from figure 7 how the central meridian is represented without undergoing any deformation and how instead the deformation grows rapidly moving away from the central meridian.

 

To limit the deformations, the cartographic representations usually used limit the extension of the spindle (portion of an ellipsoid between two meridians) which is represented in a single system. The Gaussian representation is called an inverse cylindrical conformal, also known as a transverse Mercator projection.

 

The UTM (Universal Transverse Mercator) system

In the UTM system, the Earth is divided into 60 time zones of width equal to 6 ° of longitude, numbered from 1 to 60 proceeding from West to East and giving the number 01 to the time zone corresponding to the Greenwich antimeridian (fig. 8). Each time zone has been divided into 20 bands of width equal to 8 ° of latitude each, identified by a capital letter; each zone, identified by the intersection of a spindle with a band, is further subdivided into squares of 100 km on each side, with lines parallel to the N and E axes identified by two capital letters. A point is identified by alphanumeric coordinates (number of the spindle, by the letter of the band, by the pair of letters of the square and finally by its plane coordinates referred to the SW vertex of the square of 100 km per side).

Unlike the Gaussian representation, in UTM cartography the North coordinate originates on the equator, while, in order to eliminate the use of negative numbers for the abscissas of numbers located to the West of the respective central meridians, a fictitious shift was used of the origin of the abscissa, establishing a false origin and attributing to the points on the central meridian of each zone a conventional value from the east coordinate equal to 500 km. The coordinates E (East) and N (North) are then determined, defined by: N = y; E = 500 ± x (De Toma, 1999).

 

 

Within a spindle, the linear deformation reaches the maximum value on the external meridians of the spindle: the linear deformation modulus, defined as the ratio between an infinitesimal linear element on the paper and the corresponding element measured on the ellipsoid, reaches the value of 1.0008, which means that considering two points at a distance of 1000 m on the ellipsoid there is on the map, between the correspondents of these points according to f and g, a distance equal to 1000.80 m.

To limit this deformation, a contraction factor equal to 0.9996 is introduced, i.e. the whole representation is reduced by 4 / 10,000. There is therefore a linear deformation module of 0.9996 on the central meridian, of 1.0004 on the extreme meridians and a unit module on two lines close to the meridians that have about 2 ° of difference in longitude with respect to the central meridian: in this way the relative deformation does not exceeds the value of 4 / 10,000, which is lower than the graphical error committed in the drafting of the map. From a geometric point of view, the application of the contraction factor corresponds to using a cylinder that is no longer tangent, but slightly smaller, and therefore secant, with respect to the ellipsoid (Fig. 9).

 

 

ROME 40: The Italian GAUSS-BOAGA system

The official Italian cartography, proposed in 1940 by prof. Boaga, like the UTM system, also uses the Gaussian representation, but only involves the use of two time zones, called the West and East time zones, which coincide approximately respectively with the 32 and 33 time zones of the UTM system and have respectively meridians placed at 9 ° and 15 ° east of Greenwich as central meridians. The point of emanation (geometric place in which the normal to the ellipsoid and the vertical, understood as the line of force of the earth’s gravitational field, are coincident) for the calculation of the geographical coordinates of all the vertices of the Italian geodetic network was assumed the vertex of Roma Monte Mario (Roma 40 system), to which, following accurate astronomical observations, the following geographic coordinates were attributed:

φ = 41 ° 55’25 “.51 λ = 12 ° 27’08” .40

The International Ellipsoid proposed by Hayford oriented to Monte Mario was chosen as ellipsoid. A double false origin was established, one for each time zone, attributing to the points on the central meridian of the west zone a conventional value of x equal to 1500 km and those on the central meridian of the east zone a value of 2520 km. The coordinates E and N were then determined, defined by:

N = y for both time zones

E = 1500 ± x for the West time zone

E = 2520 ± x for the East time zone

In this way, the first digit of the East coordinate always corresponds to the number of the zone and is therefore equal to 1 for the West zone and 2 for the East zone. To connect the representations in the two national time zones, an overlapping zone was created by extending the West time zone by 30 ‘in longitude; in this area the trigonometric vertices refer to both the East and the West time zones and the references of the two systems are imprinted on the cartography that represents this zone. To allow the entire representation of the national territory in just two time zones, the East time zone was also extended by 30 ‘in order to include the Salento Peninsula which otherwise would have been represented on a third time zone (fig. 10).

 

 

In fact, in order to keep unchanged the cut of the pre-existing cartography in 1: 25.000 and 1: 100.000 scale, the overlapping area was obtained by extending the West time zone up to the meridian of Rome (Monte Mario) and therefore the meridian of separation between the two time zones is that of longitude 12 ° 27’08 “.40 (De Toma, 1999).

 

ED 50: The European UTM system

In the years following the Second World War, the nations of Western Europe decided to unify their geodetic networks by setting Potsdam, a locality near Berlin (considered to be the center of gravity with respect to Europe), the point of emanation for the calculation of geographical coordinates. . In particular, it was precisely at this point that the coincidence between the normal to the ellipsoid and the vertical (normal to the geoid) was imposed. Since this point is different from that adopted by the Roma 40 system, we are faced with irregular phase displacements between the two systems. Consequently, the geographical coordinates of Rome Monte Mario have undergone small variations, resulting in:

φ = 41 ° 55’31 “.49 λ = 12 ° 27’10” .93

This differentiation in the orientation of the ellipsoid means that the same point of the Italian network has different coordinates in the two systems, although the same representation is taken as a reference. This new system, identified with the name of ED 50 (European Datum, 1950), has adopted the Greenwich meridian as its fundamental meridian (0 ° longitude) (fig. 11).

 

 

Similarly to the Roma 40 system, a false origin was established, attributing to the points on the central meridian of time zones 32 and 33 a conventional value of + 500 km, in such a way as to obtain always positive coordinates also west of the meridian. The ED 50 system also has in common with that of Rome 40 the reference ellipsoid (that is, the Hayford ellipsoid), the Gaussian projection with Cartesian axes represented by the equator and meridians (Pigato, 2000).

 

UTM-WGS 84: The worldwide system

In order to overcome the limitations related to the possibility of only partially representing the physical surface of the Earth of the previous reference systems such as the Rome 40 and the European Datum (ED 50), in 1984 a new geodetic reference system was created capable of cover the entire globe, the World Geodetic System 1984 (WGS84). It is a geocentric global system, defined through spatial observations and consisting of a right-handed Cartesian triad with origin coinciding with the center of mass of the Earth, the Z axis directed towards the conventional North Pole in 1984, the X axis orthogonal to the previous one. and intersecting the Greenwich meridian at 1984 and the Y axis directed so as to complete a right-handed triad (fig. 12).

 

 

Unlike other reference systems, which rely on the Hayford ellipsoid, this new system is associated with the WGS84 ellipsoid, with center and axes coinciding with those of the Cartesian triad.

The WGS84 ellipsoid is defined by the following parameters:

• semi-major axis: a = 6378137.00 m
• crushing: s = 1 / 298.257223563.

The flat representation of the WGS84 system takes place through the UTM cartographic system. The WGS84 system normally represents the reference system for positioning carried out with GPS instruments and its implementation on a global scale has been handled by the United States Department of Defense. In Europe, the implementation of the WGS84 system is constituted by the ETRS89 (EUREF Terrestrial Reference System 1989), while in Italy the WGS84 system was created with the establishment of the high-precision three-dimensional geodetic network, called IGM95, detected with positioning tools Differential GPS.

The coordinates of the Roma Monte Mario point in the WGS84 system are:

• latitude 41 ° 55’27.851 ”
• longitude 12 ° 27’07.658 ” (from Greenwich).