Gis Concepts 3/5

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Introduction to basic concepts on Geographical Information Systems
Autor: Msc. Alexander Mogollón Diaz
http://www.agronomia.unal.edu.co

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Gis Concepts 3/5

  1. 1. Concepts and Functions of Geographic Information Systems (3/5) MSc GIS - Alexander Mogollon Diaz Department of Agronomy 2009
  2. 2. Concepts and Functions of GIS .PPT Topic #1 Topic #2 Topic #3 1 A GIS is an information system GIS is a technology 2 Spatial Data modelling Sources of data for geodatasets Metadata 3 Geospatial referencing Coordinate transformations 4 Database management 5 Spatial Analysis
  3. 3. Functionalities of GIS INPUT QUERY - DISPLAY - MAP ANALYSE STRUCTURE MANAGE TRANSFORM
  4. 4. Transformations for building a spatial model / gDB <ul><li>Of the geometric data (coordinates, cell definition) </li></ul><ul><li>Of the attribute data </li></ul><ul><ul><li>change of units </li></ul></ul><ul><ul><li>combination of attributes: e.g. time = distance/speed </li></ul></ul><ul><ul><li>... </li></ul></ul>
  5. 5. Transformation of coordinates and Geospatial refence systems <ul><li>Spatial reference system is required to define geometric location and shape; uses ‘coordinates’ </li></ul><ul><li>Geospatial reference system (‘ coordinate reference system – CRS ’) is required for modelling entities and terrain occurring on/below/above the surface of the Earth </li></ul><ul><li>A/D-conversion, remote sensing, ... provide data about location and shape in a technical spatial reference system </li></ul><ul><ul><li>Transformation of coordinates towards a geospatial reference system is imperative </li></ul></ul><ul><li>Two classes of geospatial reference systems </li></ul><ul><ul><ul><li>Geographic </li></ul></ul></ul><ul><ul><ul><li>Projected </li></ul></ul></ul><ul><ul><ul><li>Very many variants exist of both </li></ul></ul></ul><ul><li>Transformations required for vertical integration </li></ul>
  6. 6. Planet Earth
  7. 7. The Earth’s shape is irregular Positioning needs simplification <ul><li>Planet earth = a 3D-body, spherical but (abstraction from relief) </li></ul><ul><li>When abstraction is made from relief, the Earth can be described by: </li></ul><ul><ul><li>The geoid (equipotential surface of gravity force - mean sea level) or by </li></ul></ul><ul><ul><li>A sphere slightly flattened at the poles (spheroid/ellipsoid) </li></ul></ul>
  8. 8. Geoid versus Ellipsoid <ul><li>Geoid </li></ul><ul><ul><li>3D-physical datamodel of the Earth’s surface, based on measurements of the gravity force </li></ul></ul><ul><li>Local and Global Ellipsoids </li></ul><ul><ul><li>Mathematical 3D-models of the Earth’s surface </li></ul></ul><ul><ul><li>Global ellipsoids are defined to represent the full Earth with acceptable accuracy </li></ul></ul><ul><ul><li>Local ellipsoids are defined to represent a part of the Earth’s surface only, with high accuracy </li></ul></ul>
  9. 9.
  10. 10. Geoid versus Ellipsoid
  11. 11. Geospatial locations are expressed relative to an ellipsoid (1) <ul><li>Geographic coordinates: </li></ul><ul><ul><li>Expressed as angles with respect to 2 of 3 axes through the gravity point of the ellipsoid </li></ul></ul><ul><ul><li>LONGITUDE: 0° (Greenwich) to 180° East and 0° to 180° West measured in the horizontal plane </li></ul></ul><ul><ul><li>LATITUDE: 0° (Equator) to 90 North and 0° to 90° South) measured in the vertical plane </li></ul></ul><ul><ul><li>Degrees-Minutes-Seconds or Decimal Degrees: 20° 15’ 15” = 20,2525 </li></ul></ul>
  12. 12. Geospatial locations are expressed relative to an ellipsoid (2) LON LAT 45° N; 120°E
  13. 13. Geospatial locations are expressed relative to an ellipsoid (3) <ul><li>Several ellipsoids are in use ! </li></ul><ul><ul><li>Major radius or major semi-axis a </li></ul></ul><ul><ul><li>Minor radius or minor semi-axis b </li></ul></ul><ul><ul><li>Flattening f of the ellipsoid: a-b/a = 1/f </li></ul></ul>
  14. 14. Frequently used ellipsoids S
  15. 15. Geospatial locations are expressed relative to an ellipsoid (4) <ul><li>Geodetic datum = further specification of the ellipsoid </li></ul><ul><ul><li>Initial location </li></ul></ul><ul><ul><li>Initial azimuth to define the north direction </li></ul></ul><ul><ul><li>Distance between geoid and ellipsoid at the initial location </li></ul></ul><ul><ul><li>Basis for conversion between LON-LAT and geocentric coordinates (x,y,z) </li></ul></ul><ul><ul><li>A given point has different LON-LAT when expressed against different ellipsoids ! </li></ul></ul><ul><ul><li>A given point has different geocentric coordinates x,y,z when expressed against different datums, even if the ellipsoid is identical </li></ul></ul>
  16. 16. From geographic coordinates to projected coordinates <ul><li>Common GIS-systems model the geographic reality in planimetric 2D </li></ul><ul><ul><li>traditional map view </li></ul></ul><ul><ul><li>Carthesian X-Y coordinates, meters </li></ul></ul><ul><li>LON-LAT (angles, 3D) need to be transformed into X-Y (2D) </li></ul><ul><li>Such a transformation = a projection </li></ul><ul><li>Projected coordinates = map coordinates </li></ul>
  17. 17. From geographic coordinates to projected coordinates <ul><li>gDB may store </li></ul><ul><ul><li>Geographic coordinates or </li></ul></ul><ul><ul><li>Projected coordinates </li></ul></ul><ul><li>Distances, lengths and areas cannot be expressed in geographic coordinates = Essential for most queries and spatial analyses </li></ul><ul><li>If geographic coordinates are stored, most often run time or “on the fly” transformation into projected coordinates is done by the GIS-software when querying, analysing the gDB </li></ul>
  18. 18. Arc distances
  19. 19. Computation of arc-distances
  20. 20. From geographic coordinates to projected coordinates <ul><li>From LON-LAT to X-Y = mathematical, analytical operation </li></ul><ul><li>1. Shape of earth needs to be parameterised by means of a geodetic datum </li></ul><ul><ul><li>global or local approximation of the geoid </li></ul></ul>
  21. 21. From geographic coordinates to projected coordinates <ul><li>From LON-LAT to X-Y = mathematical, analytical operation </li></ul><ul><li>2. One of very many projection functions needs to be choosen </li></ul><ul><ul><li>Cylinder, plane or cone as projection surface </li></ul></ul><ul><ul><li>Tangent or secant at selected locations </li></ul></ul><ul><ul><li>Normal, transversal, arbitrary </li></ul></ul><ul><ul><li>False easting, false northing </li></ul></ul>
  22. 22. Plane – cone - cylinder Tangent - secant Normal – transversal - oblique
  23. 23. Choice of the projection function <ul><li>From 3D to 2D => deformation cannot be avoided </li></ul><ul><ul><li>shape </li></ul></ul><ul><ul><li>direction </li></ul></ul><ul><ul><li>area </li></ul></ul><ul><ul><li>distance </li></ul></ul><ul><li>Local datum and projection function are choosen in order to minimise deformation for the study area </li></ul><ul><ul><li>position and shape of study area </li></ul></ul><ul><ul><li>conditioned by objective of cartography (density mapping requires ‘true’ areas) and type of analysis </li></ul></ul>
  24. 24. Projection creates geometric distortion
  25. 25. Conformal projections Shape and/or direction is preserved Distances and areas are distorted
  26. 26. Mercator-projection = conformal
  27. 27. Transverse Mercator-projection = conformal
  28. 28. Universal Transverse Mercator projection <ul><li>Secant cylinder at 80° North and South </li></ul><ul><li>60 strips of 6° East-West </li></ul><ul><li>Central meridian: X = 500.000 m </li></ul><ul><li>Equator: Y = 0 m for N.Hemisphere </li></ul><ul><li>Equator: Y = 10.000.000 m for S.Hemisphere </li></ul><ul><li>Applied to various ellipsoids </li></ul>
  29. 29. Standard projected coordinate system for the Philippines <ul><li>Ellipsoid: Clark’s spheroid of 1866 </li></ul><ul><ul><li>Semi-major axis = 6.378.206,4 m </li></ul></ul><ul><ul><li>Semi-minor axis = 6.356.583,8 m </li></ul></ul><ul><li>Projection: Philippines Transverse Mercator </li></ul><ul><ul><li>UTM </li></ul></ul><ul><ul><ul><li>Zone 50 (114 – 120 °East) </li></ul></ul></ul><ul><ul><ul><li>Zone 51 (120 – 126 °East) </li></ul></ul></ul><ul><ul><ul><li>Further subdivided in 6 subzones with central meridian </li></ul></ul></ul><ul><ul><ul><ul><li>117 °East </li></ul></ul></ul></ul><ul><ul><ul><ul><li>119 °East </li></ul></ul></ul></ul><ul><ul><ul><ul><li>121 °East </li></ul></ul></ul></ul><ul><ul><ul><ul><li>123 °East </li></ul></ul></ul></ul><ul><ul><ul><ul><li>125 °East </li></ul></ul></ul></ul><ul><ul><ul><li>False northing = 0; False easting = 500.000 meters </li></ul></ul></ul>
  30. 30. Lambert conformal conical projection
  31. 31. Standard projected coordinate system for Belgium <ul><li>Belgian Datum = local orientation of Hayford’s ellipsoid of 1909, recommended as International ellipsoid in 1924 </li></ul><ul><li>Projection function : Lambert 72/50 </li></ul><ul><li>Conformal conical projection with 2 secant parallels </li></ul><ul><ul><li>49°50’0.0204” and 51°10’0.0204” </li></ul></ul><ul><ul><li>Longitude of central meridian: 4°22’2.952” </li></ul></ul><ul><ul><li>Latitude of origin: 90° </li></ul></ul><ul><ul><li>Fase easting: 150.000,013 meter </li></ul></ul><ul><ul><li>False northing: 5.400.088,4398 meter </li></ul></ul><ul><li>Vertical reference system = TAW (average low tide level in Oostende (North Sea Channel ) </li></ul>
  32. 32. Other coordinate systems: examples
  33. 33. A gDB <ul><li>Can have one coordinate reference system only (effective or virtual) </li></ul><ul><li>The coordinates in all geodatasets must be expressed according to that system </li></ul><ul><ul><li>Vertical integration </li></ul></ul><ul><ul><li>Horizontal integration </li></ul></ul><ul><li>Most commonly, the choosen coordinate system is </li></ul><ul><ul><li>Geographic coordinates (LON-LAT) or </li></ul></ul><ul><ul><li>National coordinate system (from the National Mapping Agency, used for printing topographic / military maps) </li></ul></ul>
  34. 34. Vertical integration
  35. 35. Horizontal integration
  36. 36. Transformation of coordinates for vertical/horizontal integration <ul><li>Analytical conversion between geographic coordinates expressed according to different geodetic datums = datum conversion </li></ul><ul><li>Analytical conversion of geographic coordinates (e.g. from GPS) in projected coordinates and vice versa = (inverse) projection </li></ul><ul><li>Analytical conversion between different types of projected coordinates (e.g. between Philippine and Belgian system) </li></ul><ul><li>Numerical coordinate transformation (e.g. geo-referencing, using control points) </li></ul>
  37. 37. Numeric coordinate transformation <ul><li>Numeric coordinate shifts, based on control points, for vertical and horizontal integration of geodatasets in a gDB </li></ul><ul><ul><li>systematic shifts (e.g. conversion of digitiser/scan coordinates in projected coordinates) </li></ul></ul><ul><ul><li>non-systematic shifts: rubber sheeting, edge matching </li></ul></ul>
  38. 38. Geo-referencing <ul><li>When coordinates are expressed according to an analytical reference system, the term ‘georeferenced data’ is used. </li></ul><ul><li>A/D conversion using tablet digitising or scanning provide digitiser and scan coordinates. Also raw satellite images are not georeferenced. </li></ul><ul><li>Transformation of « technical » coordinates into geographical or projected coordinates = georeferencing . </li></ul>
  39. 39. Numeric transformation of coordinates after A/D-conversion via digitization <ul><li>Digitisation provides (Xi,Yi) of point objects, nodes, vertices </li></ul><ul><li>Xi,Yi are digitizer-coordinates, expressed according to a technical, flat reference system </li></ul><ul><li>Xi,Yi must be transformed into a projected reference system </li></ul>X Y (0,0) <ul><li>Xo = f (Xi,Yi); Yo = f (Xi,Yi) </li></ul>(X i ,Y i )
  40. 40. Numeric transformation of digitizer- to gDB-coordinates <ul><li>AFFINE polynomial transformation function f = popular </li></ul><ul><ul><li>Xo = A + BXi + CYi </li></ul></ul><ul><ul><li>Yo = D + EXi + FYi </li></ul></ul><ul><ul><li>2 * 3 unknowns: A, B, C and D, E, F </li></ul></ul><ul><ul><li>2 * 3 equations required to compute the unknonws </li></ul></ul><ul><ul><li>Equations are derived from 3 control points (3X and 3Y) (GCP) </li></ul></ul><ul><ul><li>GCP = ground control point = point location that can be unambiguously detected and located on both the dataset which must be transformed and on the reference geodataset or reality </li></ul></ul><ul><ul><li>System of equations has one single EXACT solution for A ... F </li></ul></ul><ul><ul><li>Transformation error is apparently 0 </li></ul></ul><ul><li>If more than 3 GCP are available, more equations than unknowns </li></ul><ul><ul><li>System of equations has more than one solution for A ... F </li></ul></ul><ul><ul><li>Best solution for A ... F can be found by the Least-Squares method </li></ul></ul><ul><ul><li>Transformation error can be computed (RMSe - ROOT MEAN SQUARE ERROR) </li></ul></ul><ul><li>If RMSe is sufficiently low </li></ul><ul><ul><li>Parameterised AFFINE equations can be applied to all input-points (point objects, nodes, vertices). Result = transformed output-geodataset </li></ul></ul>
  41. 41. AFFINE transformation of digitizer- to gDB-coordinates
  42. 42. AFFINE-transformation & RMSe: X
  43. 43. AFFINE-transformation & RMSe: Y
  44. 44. AFFINE-transformation & RMSe: X & Y
  45. 45. Judgement of the RMSe <ul><li>To be based on the spatial detail (scale for A/D-converted analog documents) of the source document </li></ul><ul><ul><li>1 mm distortion and/or digitizing error on a 1:50.000 analog map = 50 meter RMSe </li></ul></ul><ul><li>To be based on the intended use of the output-geodataset </li></ul><ul><ul><li>Requirements for vertical and horizontal integration </li></ul></ul>
  46. 46. <ul><li>Translation: </li></ul><ul><ul><li>X o = A + X i </li></ul></ul><ul><ul><li>Y o = D + Y i </li></ul></ul><ul><li>Change of scale: </li></ul><ul><ul><li>X o = BX i </li></ul></ul><ul><ul><li>Y o = EY i </li></ul></ul><ul><li>Rotation: </li></ul><ul><ul><li>X o = BX i + CY i </li></ul></ul><ul><ul><li>Y o = EX i +FY i </li></ul></ul><ul><li>AFFINE = All combined </li></ul><ul><ul><li>X o = A + BX i + CY i </li></ul></ul><ul><ul><li>Y o = D + EX i + FY i </li></ul></ul>AFFINE = polynomial transformation of the 1st order
  47. 47. <ul><li>X o = a 0 +(a 1 X i +a 2 Y i )+(a 3 X i 2 +a 4 .X i Y i +a 5 Y i 2 )+… </li></ul><ul><li>Y o = b 0 +(b 1 X i +b 2 Y i )+(b 3 X i 2 +b 4 .X i Y i +b 5 Y i 2 )+… </li></ul><ul><li>Order of the polynomial p determines the minimum number of required GCP to find the polynomial coefficients: N = (p+1)*(p+2)/2 </li></ul>Polynomial transformations of higher orders (rubber sheeting, warping)
  48. 48. Numeric transformation of coordinates after A/D-conversion via Scanning
  49. 49. Numeric transformation of scan- to gDB-coordinates <ul><li>A scanned document is not georeferenced </li></ul><ul><li>Scan-coordinates are relative to the reference system of the scan-device </li></ul><ul><li>Transformation of the scan-coordinates is necessary, using GCP </li></ul><ul><ul><li>Regular cell-raster is distorted </li></ul></ul><ul><ul><li>A new ‘empty’ cell-raster is created according to the output-reference system </li></ul></ul><ul><ul><li>Based on the established transformation function, cell values are resampled from the input raster to compute the values for the cells in the output raster </li></ul></ul><ul><ul><ul><li>Neirest neighbour </li></ul></ul></ul><ul><ul><ul><li>Other algorithms </li></ul></ul></ul><ul><li>Also valid for remotely sensed images ! </li></ul>
  50. 50. Forward GCP-based transformation distorts the raster 1 2 3 Xo = f (Xi,Yi); Yo = f (Xi,Yi): NOT valid
  51. 51. Backward/Inverse polynomial transformation of scan- to gDB-coordinates <ul><ul><li>Creation of a new ‘empty’ rasterstructure in the output-coordinate system </li></ul></ul><ul><ul><li>Calibration of the inverse polynomial transformation </li></ul></ul><ul><ul><ul><li>Xi = f(Xo,Yo) </li></ul></ul></ul><ul><ul><ul><li>Yi = f(Xo,Yo </li></ul></ul></ul><ul><ul><li>Use of the calibrated transformation function to ‘fill’ the empty cells of the output raster with (a combination of) the value(s) of the corresponding cell(s) in the input raster </li></ul></ul>
  52. 52. Resampling <ul><li>GCP are used to calibrate an inverse polynomial transformation function, e.g. AFFINE </li></ul><ul><ul><ul><li>X i = G + HX o + IY o </li></ul></ul></ul><ul><ul><ul><li>Y i = K + LX o + MY o </li></ul></ul></ul><ul><li>By means of this function, for the mid point of every output-cell (Xo,Yo) the corresponding point (Xi,Yi) in the input raster is computed </li></ul><ul><li>Xi,Yi is the ‘nearest neighbour’ </li></ul><ul><ul><li>With ‘nearest neighbour resampling’, the cell value of the cell in which Xi,Yi is located is attributed to the output cell with midpoint Xo,Yo </li></ul></ul><ul><ul><li>Also bi-linear and curbic re-sampling are possible </li></ul></ul>
  53. 53. Resampling is necessary after transformation of scan- into gDB-coordinates R = input raster; R’ = output raster Antrop & De Maeyer, 2005 Xi,Yi = Xo,Yo (change of resolution only) Xi,Yi <>Xo,Yo (nearest neighbour) Xi,Yi <> Xo,Yo (bilinear interpolation) Xi,Yi <> Xo,Yo (cubic convolution)
  54. 54. Numeric coordinate transformation <ul><li>Similar systematic numeric transformation is applicable to coordinates coming from other data sources </li></ul><ul><ul><li>Remotely sensed images </li></ul></ul><ul><ul><li>Theodolites, tachymeters with digital reading </li></ul></ul><ul><ul><li>Global Positioning Systems (GPS) </li></ul></ul>
  55. 55. (Non-)systematic numeric coordinate-transformations <ul><li>Previous numeric polynomial coordinate transformations are based on GCP </li></ul><ul><li>One set of coefficients A, B, C, … is computed and applied to all input-points to obtain the output-coordinates </li></ul><ul><li>Such transformations are systematic </li></ul>
  56. 56. Non-systematic transformations for further improvement of the positional quality of the georeferenced geodatasets <ul><li>First step in georeferencing is most often a systematic transformation of coordinates </li></ul><ul><ul><li>Polynomial function of low order </li></ul></ul><ul><li>The result is often not of sufficient quality or not sufficiently fit for use (vertical/horizontal integration in the gDB) </li></ul><ul><li>In a next step, non-systematic transformation can be performed to make the geodataset geometrically more conformal to the reference geodataset </li></ul>
  57. 57. Non-systematic coordinate transformations <ul><li>Edge-matching </li></ul><ul><li>Rubber-sheeting </li></ul>
  58. 58. Rubber-sheeting 1 2 3 GCP1: X i1 ,Y i1 -> X o1 ,Y o1 GCP2: X i2 ,Y i2 -> X o2 ,Y o2 GCP3: X i3 ,Y i3 -> X o3 ,Y o3 GCPA...GCPF: X i = X o ; Y i = Y o A B D E F G C
  59. 59. Rubber-sheeting <ul><li>Point-by-point correction of the location and shape of objects or of resampling of cell attributes </li></ul><ul><li>Based on 2 linear “piece wise” TIN-interpolations (7.PPT), 1 for X and 1 for Y </li></ul><ul><li>Z-value to interpolate = X o resp. Y o </li></ul><ul><li>Result = not-constant translation/rotation/change of scale </li></ul><ul><li>Shifts decrease with increasing distance </li></ul><ul><li>Both forward (for vectorial geodatasets) and backward (for raster datasets) </li></ul>
  60. 60. Edge-matching <ul><li>Special case of rubber sheeting </li></ul><ul><li>Applied for horizontal integration of adjacent (A/D converted) map sheets or images (mosaicking) </li></ul><ul><li>Definition of links between coinciding points on two map sheets </li></ul><ul><li>Differential displacement of points based on (mostly inverse distance; TIN) interpolation </li></ul>
  61. 61. Edge-matching
  62. 62. Summary of important items <ul><li>Geospatial reference systems </li></ul><ul><ul><li>Based on a geodetic datum (LON-LAT) and (possibly) a projection function to convert LON-LAT (angles - 3D) into planimetric coordinates (X,Y – 2D) </li></ul></ul><ul><ul><li>Projection leads to distortion of one or more of shape, direction, area, distance </li></ul></ul><ul><ul><li>If the national standards are not used, a rational, functional choice of datum and projection function is required </li></ul></ul><ul><ul><li>The datum for elevation is most often the geoid (approximated by mean sea level) </li></ul></ul><ul><li>Transformation of coordinates </li></ul><ul><ul><li>Between parameterised geographic and/or projected coordinate systems is an analytical operation which does not need external ground truth </li></ul></ul><ul><ul><li>Between technical coordinates and projected coordinates is a numeric operation based on ground truth (GCP) </li></ul></ul><ul><ul><ul><li>There are systematic and non-systematic numeric transformation functions </li></ul></ul></ul><ul><ul><ul><li>Systematic transformation is most often based on a polynomial function </li></ul></ul></ul><ul><ul><ul><li>Non-systematic transformation (rubber sheeting and edge-matching) is based on TIN-interpolation </li></ul></ul></ul><ul><ul><li>The latter is also valid for projected coordinates which need correction </li></ul></ul>
  63. 63. Questions or remarks ? Thank you …

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