Geodesy and Cartography

• Geomatics for Urban and Regional Analysis is taught by Prof. Gabriele Bitelli
• Short review of Geodesy and Cartography

Geographical coordinates

• These are used to express the coordinates of a point on Earth.
• The longitude values range between -180 and +180 degrees.
• The Greenwich Meridian is a zero line of longitude.
• Points on the equatorial plane have latitude zero.
• Latitude values range between -90 and +90 degrees.
• Geographic coordinate systems use an ellipsoid to approximate all locations on the surface of the earth.

Cartographical coordinates

• Coordinates are pairs (X, Y -> East, North, in meters) in a two-dimensional space.
• Triplets (X, Y, Z) of points defines position and height referenced to a vertical datum.
• The x and y are the coordinates of the point, and the Z is the height

Framework of Geographic Information Systems

• The increasing diffusion of geographical information could cause errors.
• Standards conventions are crucial to avoid errors and misunderstandings.
• Digital mapping and satellite positioning systems permitted the revision of reference systems.
• A global reference system was adopted where coordinate systems coexisted.
• For a point on a boundary, the surveyor must use a unique set of coordinates.
• The determination of the position of a point has traditionally been split into planimetric coordinates and elevation.

Earth's, shape and ellipsoid

• The centrifugal force of the Earth's rotation "flattens it out".
• The shape of the Earth cannot be properly represented by a sphere but by an ellipsoid.
• An ellipsoid squeezed for 1 cm at each pole approximates Earth's shape.
• An ellipsoid is a closed mathematical surface where all plane cross sections are ellipses.
• Ellipsoids are symmetrical about three mutually perpendicular axes that intersect at the center.
• An ellipsoid is a simple artificial surface to make calculations on reciprocal position or coordinates.

Ellipsoids metrics

• Bessel 1841 has a of 6377397 meters, and α of 1:299.2
• Clarke 1880 has a of 6378243meters, and α of 1:293.5
• Helmert 1906 has a of 6378140 meters, and α of 1:298.3
• Hayford 1909 = Internazionale 1924 has a of 6378388 meters, and α of 1:297.0
• GRS80 has a of 6378137 meters, and α of 1:298.257222101
• WGS84 has a of 6378137 meters, and α of 1:298.257223563
• International 1924 and the Bessel 1841 ellipsoids are in Europe.
• In North America the GRS80, and Clarke 1866 ellipsoid are used.
• In Russia and China the Krasovsky ellipsoid is mostly used.
• The Everest ellipsoid is used in India.

Spheroids and where they are implemented

• Many ellipsoids have been proposed over the centuries
• Airy 1940 is implemented in England with 6377563 meters 6356256.91
• Bessell 1841 is implemented Central Europe, Chile, and Indonesia with 6377397.155 meters 6356078.96284
• Clarke 1866 is implemented North America and the Philippines with 6378206.4 meters 6356583.8
• Everest 1830 is implemented in India, Burma, and Pakistan with 6377276.3452 meters 6356075.4133
• GRS 1980 is a global geodetic Reference System with 6378137.0 meters 6356752.31414
• Krasovsky 1940 is implemented in Former Soviet Union and some East European countrie 6378245.0 meters 6356863.0188
• WGS 84 is a global World Geodetic System with 6378137.0 meters 6356752.31424517929

Heights

• Gravimetry is fundamental for taking height measurements with a physical meaning
• The gravitational pull varies from place to place because of differences in density.
• The density of the earth's crust is not uniformly the same.
• A force line of the gravitational field ("vertical") can be expressed by the plumbline.
• The equipotential surface chosen as reference is the mean sea level called the geoid.
• The height is assumed to be always 0.00 m on the geoidal surface.
• The geoid is a continuous, irregular, smooth surface, found by oceans surface approximation.
• It responds to physical, not mathematical, considerations.
• The geoid is not a simple mathematical surface and bulge or dip below or above the ellipsoid.
• Overall these differences are small (~100 meters)
• The altimetric position of points on the physical surface is referred to the geoid (height a.m.s.l.).
• Earth needs a shape that allows the planimetric positioning of each point /relative calculations.
• As reference surface that defining the geographical coordinates of a point, is used the ellipsoid of rotation as it approximates the geoid.
• The height is measured with reference to mean sea level (geoid).
• The geoid bulge or dip below or above the ellipsoid has overall small differences (max ~100 meters).
• The force of gravity is perpendicular to the geoid and a reference surface.
• Geoid is quite irregular at a fine scale of detail. Geometric relations and calculations over it are impossible.

Geoidal undulation

• Represented by N = h - H.

Ellipsoidal/Orthometric heights

• Ellipsoidal (h) is obtained by using the GPS
• Orthometric (H, a.m.s.l.)height is obtained by gravitational measurement
• GPS provide ellipsoidal heights

Geoid models (geoidal undulation)

• The range varies in Bologna from red (85.4 m) to magenta (-107.0 m)
• Geoidal undulation are measured and interpolated using gravitational measurements.
• Global models include EGM96 and EGM2008.
• National/regional models include ITALGEO (Italy).
• It would be simple and very fast to take altimetric measurements using the GPS, finding H from h as long as the accuracy of these models is high.

Online geoid calculators

• Can be computed online at https://gnss-reflections.org/geoid?station=smm3
• Can be computed online at https://geographiclib.sourceforge.io/cgi-bin/GeoidEval

Local estimates

• Politecnico di Milano produced a local estimates of the geoid called ITALGEOxxx
• IGM Institute maintains the local estimate of the geoid

Ellipsoids and Geoid irregularities

• Estimates of ellipsoids are different because of irregularities/slight deviations across Earth's surface.
• Before remote satellite observation (GPS introduction), it was necessary to use a different ellipsoid . To account for irregularities and reduce positional errors
• Continental or country level adjustments were necessary
• North America used the red ellipsoid to fit the geoid; Europe used the blue ellipsoid

Latitude and Longitude

• How to express the position of a point on the ellipsoid?
• The latitude angle between the ellipsoid's normal in P and the equatorial plane
• 90° < < +90° or 90°S ≤ ≤ 90°N cost. parallel
• The longitude angle between the meridian plane in P and the meridian plane of a reference point.
• 180° ≤ x ≤ +180° or 180° W ≤ x ≤ 180° E x = cost. → meridian

How Lat / Long are expressed

• Latitude and longitude are measured in degrees, minutes, seconds, where minutes and seconds as base-60. -Example: 56° 34' 30"
• In GIS is more common to use decimal degrees, where minutes and seconds are converted. -Example: 45° 52' 30" = 45.875°

Geographic Graticule or Grid

• Each degree of latitude represents about 110 km, but varies slightly.
• The length corresponding to a degree in longitude differs on the latitude
• At the equator one degree of longitude is 111.321KM while at 60 degrees latitude it is 55.802KM Datums

Concept of Datum

• A Geodetic datum or geodetic system is a coordinate system, and a set of reference points, that locate places on the Earth.
• Used in geodesy, navigation, and surveying.
• Used by cartographers/satellite navigation systems to translate map positions to their real position on Earth.
• A horizontal datum is associated with an ellipsoid defining latitude and longitude coordinates.
• Local datums can give a more accurate representation of the area of coverage.
• The global WGS84 datum is becoming increasingly adopted.

Horizontal Datums

• A three-dimensional surface from which latitude, longitude and elevation are calculated
• A datum provides a frame of reference for placing specific locations at specific points on the ellipsoid
• An ellipsoid only gives you a shape, a datum gives you locations of specific places on that shape
• A datum, origin/orientation must be defined, with a set surveyed points

Datums in Italy (before WGS84)

• Datum ROMA40
  • Ellipsoid: International 1924
  • Orientation in Roma Monte Mario. Datum ED50 (European Datum 1950)
• Ellipsoid: International 1924 with European orientation (defined in Potsdam, near Bonn

Satellite based global Datum

• The old pre-GPS datums were local based for accuracy near the touching point.
• With the advent of GPS, a new global ellipsoid was through satellite measurements, with the center coincident with the mass center of the Earth.
• This gives an ellipsoid mapping the whole Earth, that all Latitude/Longitude measurements agree with the datum .
• Coordinate measurements are linked to reference in outer space

Datums and Lat/Long coordinates

• Different surface datums results in different lat/long values for the same location.
• Coordinates calculated with a datum are valid only for that datum.
• Example of WGS84 datum in Bologna: lat 44° 29' 16.77" N lon 11° 19' 44.07" E
• Example of Roma40 datum in Bologna: lat 44° 29' 14.42" N lon 11° 19' 45.01" E
• It is suggested, most of the ties to use the triplet with the orthometric height referred to geoid: (φ, λ, Η)

WGS84

• GPS provides the X, Y, Z coordinates in a 3D cartesian system. Or provides latitude, longitude and ellipsoidal height referred to the geocentric ellipsoid WGS84
• World Geodetic System 1984 a newer ellipsoid/datum, created by the US DOD.
• WGS84 is a global datum

Datum agreement

• Agreement on the coordinates transformation is fundamental -This transformation is performed by a GIS software

Main concepts of Geomatics

• Reference surfaces: geoid, ellipsoid, datums

• Coordinate systems: geographic, cartesian 3D

• Map projections

• Coordinate transformations All are available in Virtuale website http://www.epsg.org/ is a database with definitions of coordinate reference systems and coordinate transformations.

• The EPSG codes for Geodetic Parameter Dataset registry of geodetic datums, spatial reference systems/Earth ellipsoids.

• The EPSG code for WGS84 is 4326

Web based interactive conversion

• Example available at : http://twcc.fr/en/
• There is also a UTM Coordinate converter available

Summarizing points

• A geographic point position is always relative to other reference points expressed by means of coordinates
• To determine the point's coordinates, a reference must be defined
• A coordinate system must be related to Earth shape and gravitational field
• The geoid , with constant gravitational field potential, could be the best choice for its physical meaning, because the mathematical formulation is complex
• An ellipsoid approximate the Geoid's shape but it is smooth, regular and expressed by a simple analytical form

Ellipsoidal reference surface

• Defined by shape/orientation.
• Latitude and Longitude are the most used to express a point

Introduction of GPS system, a global datum WGS84 was then defined

• The ellipsoid barycenter coincides with the Earth centre of mass
• axis Z is aligned to the Earth's rotation axis while axis X lies on Greenwich meridian
• It is a Cartesian system with an associated reference ellipsoid.
• Transmitted GPS orbits are given and hence it is the system used by any GPS device.

Geographic coordinates

• These coordinates are used on ellipsoids for a flat representation
• Latitude, longitude and map making used in cartography

Numerical maps

• Nominal scale of a numerical/digital map has errors related to instruments, techniques and generation procedures
• The type of the map must be related to data characteristics

Cartographic representation:

• Classified by mapping properties are based on the characteristics preserved like shape, angles etc
• Compromise projections don't preserve any quantities exactly but they present several reasonably well, and it involves distortion

Projections:

• Many projections of the past were based on geometrical relations like plane/cone/cylinder, so a projection is mathematically done Gauss formulas do
• x = f (λ,φ)
• y = g(λ, φ)

UTM (Universal Transverse Mercator)

• It's the most used map projection system starting from the Gauss formulas with a contraction factor
• Long East = 0.9996xx + FE
• North = 0.9996xy + FN
• each 6 degree longitude wide zone is treated the same-The coordinates are referred to the equator and to the central meridian of the zone, with the FE offset and FN defined at the zone
• the zone can be found for a given location through these tools https://en.wikipedia.org/wiki/Universal_Transverse_Mercator_coordinate_system

Gauss Boaga

• Is a conformal adaptation from Prof. Boaga to better suit Italian mapping need
• Must be applied only to geographical coordinates (φ, λ) in Rome40 datum.
• Related closely to UTM zone that divides area by West (E=1500 Km) and East sector (E= 2520 Km)
• Reminder: points have geographical and cartographical coordinates for different datums with geoidal undulation

Explore the Greenwich Meridian, latitude, longitude, and ellipsoids. Learn about the geometric properties of ellipsoids and their use in geomatics. Discover why local geoid estimates are crucial for accurate surveying.