Geomatics for Urban and Regional Analysis 2024-2025 PDF

GEOMATICS FOR URBAN AND REGIONAL ANALYSIS (2024/2025) G. Bitelli Geomatics for Urban and Regional Analysis How to express the coordinates of a point on the Earth? (...

GEOMATICS FOR URBAN AND REGIONAL ANALYSIS (2024/2025) G. Bitelli Geomatics for Urban and Regional Analysis How to express the coordinates of a point on the Earth? (Prof. Gabriele Bitelli) Geographical coordinates (lat, lon) Short review of Geodesy and Cartography Points on the equatorial plane have latitude zero. If you go North, latitude values increase (positive values); everything South Longitude values (X-values) range between -180 of the equator has negative latitude values. and +180 degrees. The Greenwich Meridian (or Latitude values range between -90 and +90 prime meridian) is a zero line of longitude; it passes degrees. through the Royal Observatory in Greenwich, UK Geographic coordinate systems use an ellipsoid to approximate all locations on the surface of the earth. How to express the coordinates of a point on the Earth? Cartographical (projected) coordinates The framework increasing diffusion of geographical information, but not always the people who use the coordinates have expertise in reference systems to which they relate it is crucial to have standards and shared conventions to avoid errors and very serious misunderstandings the advent of digital mapping and satellite positioning systems permitted the revision of the reference systems in use in the different countries/regions, where a large number of coordinate systems coexisted, and the adoption of a global reference system for a point on a boundary, the surveyor must use a unique set of coordinates Coordinates are pairs (X, Y -> East, North, in meters) in a two-dimensional space. the determination of the position of a point has traditionally been split Whereas triplets (X, Y, Z) of points not only has a position but also has height referenced to a into planimetric coordinates and elevation vertical datum. Earth’s shape and ellipsoid The shape of the Earth cannot be properly represented by a sphere but rather Ellipsoids by an ellipsoid, because the centrifugal force of the Earth’s rotation “flattens it out”. flattening   a b b a e2  a b It is really close to a a 2 2 sphere: considering first eccentricity a sphere of 6 a2 meters in diameter, the ellipsoid would be achieved by squeezing it for 1 a (meters)  cm at each pole Bessel (1841) 6377397 1:299.2 Clarke (1880) 6378243 1:293.5 Helmert (1906) 6378140 1:298.3 Hayford (1909) = Internazionale 1924 6378388 1:297.0 Ellipsoid = closed mathematical surface of which all plane cross GRS80 6378137 1:298.257222101 sections are either ellipses. An ellipsoid is symmetrical about three From the WGS84 6378137 1:298.257223563 mutually perpendicular axes that intersect at the center. advent of GPS The International 1924 and the Bessel 1841 ellipsoids are used in Europe, while in the ellipsoid is a simple artificial surface on which it is easy to make calculations North America the GRS80, and, decreasingly, the Clarke 1866 ellipsoid, are used. on the reciprocal position of the points or to have their coordinates, expressed as In Russia and China the Krasovsky ellipsoid is mostly used, and in India the Everest ellipsoid. (latitude, longitude) GEOMATICS FOR URBAN AND REGIONAL ANALYSIS (2024/2025) 1 G. Bitelli  GEOMATICS FOR URBAN AND REGIONAL ANALYSIS (2024/2025) G. Bitelli What about the heights? (a) definition of the direction perpendicular to geoid (positioning a plumbline) (b) equipotential surfaces: the geoid is the level surface Many corresponding to the mean sea level ellipsoids have been The knowledge of the Earth’s gravitational field (treated by geodesy and in proposed particular by gravimetry) is fundamental for taking height measurements with a over the physical meaning (see water flow…) centuries… 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: heavy rock, such as an iron ore deposit, will have a stronger attraction than lighter materials. A force line of the gravitational field (“vertical”) can be expressed by the plumbline. Among all the possible equipotential surfaces of the gravitational field, the one that has been chosen as reference surface is defined by the mean sea level: the geoid. On the geoidal surface the height is assumed to be always 0.00 m. The geoid is a continuous, irregular. smooth surface, found with good approximation based on the surface of the oceans. It responds to physical, not mathematical or geometrical physical surface considerations. The geoid (or any equipotential surface) 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.”). geoid The Earth needs to be represented by a shape that allows the planimetric positioning of each point and the relative calculations through simple mathematical processes. For this reason, the ellipsoid of rotation, which is the artificial geometric surface that better approximates the geoid, has been used as the reference surface defining the geographical coordinates of a point, whilst for the height we ellipsoid refer to mean sea level (geoid). The geoid bulge or dip below or above the ellipsoid. Overall these differences are small (max ~100 meters). Summing up… The force of gravity is at any point perpendicular to the geoid, which would then be a reference surface ideal from the physical point of view. Unfortunately, the geoid is actually quite irregular at a fine scale of detail and it is impossible to define simple geometric relations and calculations over it: the ellipsoid is definitely more simple for this purpose (relief very exaggerated) GEOMATICS FOR URBAN AND REGIONAL ANALYSIS (2024/2025) 2 G. Bitelli  GEOMATICS FOR URBAN AND REGIONAL ANALYSIS (2024/2025) G. Bitelli N geoidal undulation = h - H Models of geoidal undulation (A.K.A. “geoid models”) Ellipsoidal (h) vs From red (85.4 m) to Orthometric (H, a.m.s.l.) NB GPS provides ellipsoidal heights magenta (-107.0 m) height The geoidal undulation is actually measured and interpolated using gravitational and other measurements. Examples: in Bologna N is about 40 m: H  hGPS - 40 Global models EGM96 EGM2008 National/regional models ITALGEO (Italy) If the accuracy of these models, with which N is known, were very high, it would be simple and very fast to take altimetric measurements with the GPS, finding H from h. Online geoid calculators Online geoid calculators https://geographiclib.sourceforge.io/cgi-bin/GeoidEval N values at the point provided by different models A local estimate of the N model Ellipsoids and Geoid (Italy) In Italy, Politecnico di We have several different estimates of ellipsoids because Milano, mainly of irregularities and slight deviations that are quite through gravimetric variable across the Earth’s surface measurements, produced a «local Before remote satellite observation (GPS introduction), estimate of the we had to use a different ellipsoid for different regions to geoid» (ITALGEOxxx), account for irregularities (see Geoid) and reduce afterwards further positional errors refined (090, 095, 099, 005…), today maintained by IGM This apply at Institute continental or country level Contour lines of geoidal undulation ITALGEO95 (interval: 0.5 m) GEOMATICS FOR URBAN AND REGIONAL ANALYSIS (2024/2025) 3 G. Bitelli  GEOMATICS FOR URBAN AND REGIONAL ANALYSIS (2024/2025) G. Bitelli How to express the position of a point How Lat / Long are expressed? on the ellipsoid? Geographical Coordinates Latitude (φ) Latitude and longitude can be measured in degrees, Longitude (λ) minutes, seconds (e.g. 56° 34’ 30”); minutes and seconds are base-60, like on a clock We can also use decimal degrees (more common in GIS) where minutes and seconds are converted to a decimal Example: 45° 52’ 30” = 45.875° The Geographic Graticule/Grid Each degree of latitude represents about 110 km, although that varies slightly because the earth is not a perfect sphere. The length corresponding to a degree in longitude instead differs greatly depending on the position (latitude) Datums The concept of Datum Horizontal Datums Definition: a three dimensional surface from which latitude, A Geodetic datum or geodetic system is a coordinate system, longitude and elevation are calculated and a set of reference points, used to locate places on the A datum provides a frame of reference for placing specific Earth. locations at specific points on the ellipsoid Datums are used in geodesy, navigation, and surveying by An ellipsoid only gives you a shape, a datum gives you cartographers and satellite navigation systems to translate locations of specific places on that shape. positions indicated on maps to their real position on Earth. A horizontal datum is associated with an ellipsoid, and defines Where is the origin? latitude and longitude coordinates for each point. Because the How the ellipsoid is oriented? Earth is an imperfect ellipsoid, local datums can give a more accurate representation of the area of coverage than a global For a datum, origin and orientation one, like WGS84. However, as the benefits of a global system must be defined, together with a set outweigh the greater accuracy, the global WGS84 datum is of surveyed measured points becoming increasingly adopted. GEOMATICS FOR URBAN AND REGIONAL ANALYSIS (2024/2025) 4 G. Bitelli  GEOMATICS FOR URBAN AND REGIONAL ANALYSIS (2024/2025) G. Bitelli Datums in Italy (before WGS84) datum ROMA40 Ellipsoid: International 1924 Orientation in Roma Monte Mario, azimuth towards Mount Soratte (astronomical definition 1940) lat 41° 55’ 25.51” lon 0° (12° 27’ 08.4” East from Greenwich) Adjustment of the fundamental Italian geodetic network datum ED50 (European Datum 1950) Ellipsoid: International 1924 European orientation (defined in Potsdam, near Bonn, with a solution that does not cancel the vertical deviation here but minimizes it in respect to other points with known astronomical azimuth and longitude) Local datum Origin of longitudes: Greenwich Global datum Adjustment of the fundamental European geodetic network In this system the Monte Mario point has geographical coordinates: lat 41° 55’ 31.487” Starting from the advent of GPS, a global ellissoid (WGS84) has lon 12° 27’ 10.93” been defined NB at a latitude of 41°: 1” in lat=30.85 m 1” in lon=23.37 m Introduction of a satellite based global Datum WGS84 The old pre-GPS datums were local based: datums were most Z accurate for mapping near the touching point, less accurate as GPS provides the X, Y, Z move away coordinates of a point in a 3D cartesian system centered on P With the advent of GPS, a new global ellipsoid was defined en ta le h the Earth mass centre rid ia n o fo n d a m through satellite measurements, with the center coincident with the mass center of the Earth. or  Y  me latitude, longitude and ellipsoidal This gives an ellipsoid that, when used as a datum, correctly X height (,  and h) referred to the maps the whole Earth such that all Latitude/Longitude geocentric ellipsoid WGS84 measurements from all maps created with that datum agree. Rather than linking points to an initial surface point, the World Geodetic System 1984 (WGS84) is a newer measurements are linked to reference in outer space… ellipsoid/datum, created by the US DOD (Department of Defense, USA) when designing and developing the GPS system WGS84 is a global datum The importance to agree on the datum… Datums and Lat/Long coordinates Different surface datums can result in different lat/long values for the same location on the earth. So, just giving lat and long is not enough! You must add the datum name. Lat/long coordinates calculated with one datum are valid only with reference to that datum. Example for a point in Bologna: in WGS84 datum: lat 44° 29’ 16.77” N lon 11° 19’ 44.07” E in Roma40 datum: lat 44° 29’ 14.42” N lon 11° 19’ 45.01” E NOTE The computation of the coordinates transformation between two Most of the times an hybrid choice is made by using the triplet (φ, λ, H), system is a task usually performed by a GIS software. where H is the orthometric height (referred to geoid → a.m.s.l.). GEOMATICS FOR URBAN AND REGIONAL ANALYSIS (2024/2025) 5 G. Bitelli  GEOMATICS FOR URBAN AND REGIONAL ANALYSIS (2024/2025) G. Bitelli When providing the coordinates of a point, you must add the name of the reference datum. Main concepts Depending on the datum, a couple of coordinates can correspond to a position that may vary of hundreds of meters referring to different reference systems. Reference surfaces for positioning and mapping: geoid, ellipsoid. Datums Coordinate systems: geographic, cartesian 3D, … Map projections Coordinate transformations (see also material provided in Virtuale website) http://www.epsg.org/ A database of definitions of coordinate reference systems and coordinate transformations (global, regional, national or local). The EPSG Geodetic Parameter Dataset is maintained by the Geodesy Subcommittee of the IOGP (International Association of Oil & Gas Producers) Geomatics Committee. Downloadable locally as MS Access file or other formats. EPSG Geodetic Parameter Dataset (also EPSG registry) is a public registry of geodetic datums, spatial reference systems, Earth ellipsoids, coordinate transformations and related units of measurement. Each entity is assigned an EPSG code. The code for WGS84 is 4326 Web based interactive conversion: an example (web address to download the program in the Virtuale course page) http://twcc.fr/en/ https://www.ewert-technologies.ca/home/products/utm-coordinate-converter- home/utm-coordinate-converter-downloads.html 62 GEOMATICS FOR URBAN AND REGIONAL ANALYSIS (2024/2025) 6 G. Bitelli  GEOMATICS FOR URBAN AND REGIONAL ANALYSIS (2024/2025) G. Bitelli Summarising… (I) Summarising… (II) Position is relative: a point position is always relative to other An ellipsoidal reference surface is completely defined by fixing its reference points and is expressed by means of coordinates. shape and its orientation To determine the point’s coordinates a reference system has to be Latitude and Longitude are the most common coordinates to defined. express the position of a point on the ellipsoid A terrestrial reference system and the associated coordinates should be related to: - the Earth’s shape - the Earth’s gravitational field which are themselves related, too. The geoid, as a surface with a constant Earth’s gravitational field potential, could be the best choice for its physical meaning, but its mathematical formulation is very complex and it can’t be used for geometrical calculations. To get coordinates with a geometrical meaning, and to make Most of the times, e.g., in Cartography or in classical surveys, an calculations between them, an ellipsoid is used as reference hybrid choice is made by using the triplet (φ, λ, H), where H is the surface. The ellipsoid’s shape approximates the Geoid’s shape orthometric (geoidal) height. (locally or globally), but it is more smooth and regular, and The difference N=h-H between ellipsoidal and geoidal height is expressed by a simple analytical form. 63 named Geoidal undulation. 64 Summarising… (III) By the introduction of GPS system, a global datum WGS84 was defined: the ellipsoid barycenter coincides with the Earth’s center of mass From geographic coordinates on the axis Z is aligned to the Earth’s rotation axis the ellipsoid to a flat the axis X lies on a specific plane (Greenwich meridian) containing the axis z representation: the axis Y completes the triad of axes map making, cartography It is a Cartesian system with an associated reference ellipsoid. It is the reference system in which the transmitted GPS orbits are given and hence it is the system used by any GPS device. How to express the coordinates of a point on the Earth? Cartographical (projected) coordinates The cartographic problem Coordinates are pairs (X, Y -> East, North, in meters) in a two-dimensional space. GEOMATICS FOR URBAN AND REGIONAL ANALYSIS (2024/2025) 7 G. Bitelli  GEOMATICS FOR URBAN AND REGIONAL ANALYSIS (2024/2025) G. Bitelli The reference surface (spheroid or ellipsoid) is not developable on the plane You cannot project a ellipsoidal surface onto a flat surface without some distortion You can project the earth so that certain properties are projected without distortion – Local shapes and angles – Distance along selected directions – Direction from a central point – Area NB Even in the presence of deformations, a map is associated to a unique scale because the deformation effects remains within the “graphical error” of 0.2 mm (pen sign width). Its correspondence in meters is related to the map scale. Example: for a 1:5000 map, 0.2 mm on the map correspond to 1 m on the ground, then if I derive the position of a point from this map I can expect an error of ±1 meter (normally the tolerance is assumed double or triple). Important notes about numerical maps Nominal scale of a numerical/digital map: a numerical map (e.g. a shapefile in GIS describing the topographic base of a city) has a nominal scale of 1:2000 if the precision/accuracy of the described elements and also the thematic contents are the same of a 1:2000 classical/analogical map (“graphical error”) If a numerical map has, as an example, a nominal scale 1:2000, it is absolutely not correct to print the map at a larger scale (1:1000 or 1:500) The scale of a map is related to the actual quality of the data, related to the instruments, techniques and procedures adopted for its generation The use of a numerical map (e.g. in a GIS) must be related to the data characteristics Kind of cartographic representation Projections and preservation of geometrical properties Many projections of the past were based on geometrical relations and approximate the Earth with A classification of the mapping properties is a surface that can be flattened based on the characteristics preserved (from the – Plane real world to the map) – Cone Examples: – Cylinder – Preservation of Local Shape or Angles: Conformal Complex projections are not based on simple surfaces – Preservation of Area: Equal Area A projection is today a mathematical transformation, Compromise projections don’t preserve any an analytical relationship in the form: quantities exactly but they present several x  f ( ,  ) (from geographical coordinates reasonably well y  g ( ,  ) to a 2D x,y system) All kinds of representation involve some distortion GEOMATICS FOR URBAN AND REGIONAL ANALYSIS (2024/2025) 8 G. Bitelli  GEOMATICS FOR URBAN AND REGIONAL ANALYSIS (2024/2025) G. Bitelli Choice of Projections For small areas almost all projections are pretty accurate Principal issues – Optimizing accuracy for legal uses – Fitting sheets for larger coverage Many projections are instead suitable only for global use Italy in historical maps Simple Projection Methods On small scale maps, distortions introduced by different projections can be very substantial Stereographic Lambert Conformal Conic Gauss formulas for mapping  x  a1  a33  a55 x  f ( ,  )   y  g ( ,  )    y   0  d  a2 2  a4 4  a6 6 Equirectangular (Geographic) Mercator Geometrically, it corresponds to a cylindrical projection GEOMATICS FOR URBAN AND REGIONAL ANALYSIS (2024/2025) 9 G. Bitelli  GEOMATICS FOR URBAN AND REGIONAL ANALYSIS (2024/2025) G. Bitelli UTM (Universal Transverse Mercator) UTM is today the most used cartographic representation method, applied to The most used map projection system WGS84 geographical coordinates as well as to other datums (e.g. in Europe European Datum ED50). We can use the notations: UTM(WGS84) UTM(ED50) Starting from the x,y coordinates East = 0.9996×x + FE (East offset) obtained by the Gauss formulas: North = 0.9996×y + FN (North offset) NB: when providing UTM coordinates you must add also the zone number! Longitudine Longitudine meridiano meridiano origine 0.9996 = contraction factor origine UTM zones numbers ( cilinder secant) P (E,N) P (E,N) Note: origine reticolato each 6 degree per Emisfero Nord N E longitude wide zone is N [FN=0] (0,0) Falsa Est origine equatore FE=500000 treated the same origine naturale Falsa Est naturale Falsa FE (FE,FN) Nord Falsa Nord FN emisfero Sud (0,0) equatore FN=10000000 E origine reticolato per Emisfero Sud (0,0) 60 zones of 6°, numbered from the Greenwich antimeridian (Bologna is in the zone 32). The coordinates are referred to the equator and to the central meridian of the zone, with the offsets: FE = 500000 (always) FE and FN were introduced to FN = 0 for the hemisphere North avoid negative values in the E,N =10000000 m for the hemisphere South coordinates Bologna is in zone 32, Ravenna is in zone 33 Gauss Boaga (only for Italian cartography) a point has different geographical and cartographical coordinates for Giovanni different datums. Gauss formulas (conformal projection), Boaga Example (a point in Bologna): adapted by Prof. Boaga to better suit Italian mapping needs. Closely related to UTM, but Italian area is WGS84 lat 44° 30’ 00.0000” lon 11° 20’ 00.0000” divided in two zones, Western and Eastern. UTM(WGS84) N=4930057.324 m E=685495.722 m 32 The values of the False East are 1500 km and 2520 km respectively for the West and ED50 lat 44° 30’ 03.4323” lon 11° 20’ 03.5040” East zones, so that the first digit of the UTM(ED50) N=4930256.512 m E=685578.672 m 32 coordinate E indicates implicitly the zone to which the point belongs. ROMA40 lat 44° 29’ 57.6335” lon 11° 20’ 00.9033” Must be applied only to geographical GAUSS BOAGA N=4930075.941 m E=1685526.349 m coordinates (, ) in Rome40 datum. Regarding the elevation, the geoidal undulation at this point is 39.26 m: 87 Hamsl  hGPS – 39.26 GEOMATICS FOR URBAN AND REGIONAL ANALYSIS (2024/2025) 10 G. Bitelli