Gis Concepts 3/5

Concepts and Functions of

Geographic Information Systems(3/5)

MSc GIS - Alexander Mogollon Diaz

Department of Agronomy

2009

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Concepts and Functions of GIS

.PPT Topic #1 Topic #2 Topic #31 A GIS is an information

systemGIS is a technology

2 Spatial Data modelling Sources of data for geodatasets

Metadata

3 Geospatial referencing Coordinate transformations

4 Database management

5 Spatial Analysis

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Functionalities of GIS

INPUT

QUERY - DISPLAY - MAP

ANALYSE

STRUCTURE

MANAGE

TRANSFORM

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Transformations for building a spatial model / gDB

• Of the geometric data (coordinates, cell definition)

• Of the attribute data– change of units– combination of attributes:

e.g. time = distance/speed– ...

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Transformation of coordinates and Geospatial refence systems

• Spatial reference system is required to define geometric location and shape; uses ‘coordinates’

• Geospatial reference system (‘coordinate reference system – CRS’) is required for modelling entities and terrain occurring on/below/above the surface of the Earth

• A/D-conversion, remote sensing, ... provide data about location and shape in a technical spatial reference system

– Transformation of coordinates towards a geospatial reference system is imperative

• Two classes of geospatial reference systems1. Geographic2. Projected• Very many variants exist of both

• Transformations required for vertical integration

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Planet Earth

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The Earth’s shape is irregular

Positioning needs simplification

• Planet earth = a 3D-body, spherical but (abstraction from relief)• When abstraction is made from relief, the Earth can be described by:

– The geoid (equipotential surface of gravity force - mean sea level) or by – A sphere slightly flattened at the poles (spheroid/ellipsoid)

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Geoid versus Ellipsoid

• Geoid– 3D-physical datamodel of the Earth’s surface, based

on measurements of the gravity force

• Local and Global Ellipsoids – Mathematical 3D-models of the Earth’s surface– Global ellipsoids are defined to represent the full

Earth with acceptable accuracy– Local ellipsoids are defined to represent a part of the

Earth’s surface only, with high accuracy

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Geoid versus Ellipsoid

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Geospatial locations are expressed relative to an ellipsoid (1)

• Geographic coordinates:– Expressed as angles with respect to 2 of 3 axes

through the gravity point of the ellipsoid – LONGITUDE: 0° (Greenwich) to 180° East and 0° to

180° West measured in the horizontal plane – LATITUDE: 0° (Equator) to 90 North and 0° to 90°

South) measured in the vertical plane– Degrees-Minutes-Seconds or Decimal Degrees:

20° 15’ 15” = 20,2525

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LON

LAT

45° N; 120°E

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• Several ellipsoids are in use !– Major radius or major semi-axis a– Minor radius or minor semi-axis b– Flattening f of the ellipsoid: a-b/a = 1/f

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Frequently used ellipsoids

S

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• Geodetic datum = further specification of the ellipsoid– Initial location– Initial azimuth to define the north direction– Distance between geoid and ellipsoid at the initial location– Basis for conversion between LON-LAT and geocentric coordinates (x,y,z)

– A given point has different LON-LAT when expressed against different ellipsoids !– A given point has different geocentric coordinates x,y,z when expressed against different datums, even if the ellipsoid is identical

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From geographic coordinates to projected coordinates

• Common GIS-systems model the geographic reality in planimetric 2D – traditional map view– Carthesian X-Y coordinates, meters

• LON-LAT (angles, 3D) need to be transformed into X-Y (2D)

• Such a transformation = a projection

• Projected coordinates = map coordinates

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• gDB may store– Geographic coordinates or– Projected coordinates

• Distances, lengths and areas cannot be expressed in geographic coordinates = Essential for most queries and spatial analyses

• 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

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Arc distances

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Computation of arc-distances

Degrees Radians Degrees RadiansA LAT1 51,25 0,894481 LON2 2,97 0,894481B LAT2 49,58 0,865334 LON2 5,85 0,865334

cos(d) = sin(LAT1)*sin(LAT2) + cos(LAT1)*cos(LAT2)*cos(LON1-LON2)cos(d) 0,999403d 0,03456 Radians Degrees 1,980116D = d*111.32 km (lenght of 1 degree at the equator

111,32D = 220,4265 km

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From LON-LAT to X-Y = mathematical, analytical operation

1.Shape of earth needs to be parameterised by means of a geodetic datum

– global or local approximation of the geoid

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From LON-LAT to X-Y = mathematical, analytical operation

2.One of very many projection functions needs to be choosen– Cylinder, plane or cone as projection surface– Tangent or secant at selected locations– Normal, transversal, arbitrary– False easting, false northing

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Plane – cone - cylinder

Tangent - secant

Normal – transversal - oblique

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Choice of the projection function• From 3D to 2D => deformation cannot be

avoided– shape– direction– area– distance

• Local datum and projection function are choosen in order to minimise deformation for the study area– position and shape of study area– conditioned by objective of cartography (density

mapping requires ‘true’ areas) and type of analysis

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Projection creates geometric distortion

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Conformal projections

Shape and/or direction is preserved

Distances and areas are distorted

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Mercator-projection = conformal

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Transverse Mercator-projection = conformal

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Universal Transverse Mercator projection

•Secant cylinder at 80° North and South•60 strips of 6° East-West•Central meridian: X = 500.000 m•Equator: Y = 0 m for N.Hemisphere•Equator: Y = 10.000.000 m for S.Hemisphere•Applied to various ellipsoids

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Standard projected coordinate system for the Philippines

• Ellipsoid: Clark’s spheroid of 1866– Semi-major axis = 6.378.206,4 m– Semi-minor axis = 6.356.583,8 m

• Projection: Philippines Transverse Mercator– UTM

• Zone 50 (114 – 120 °East)• Zone 51 (120 – 126 °East)• Further subdivided in 6 subzones with central meridian

– 117 °East– 119 °East– 121 °East– 123 °East– 125 °East

• False northing = 0; False easting = 500.000 meters

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Lambert conformal conical projection

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Standard projected coordinate system for Belgium

Belgian Datum = local orientation of Hayford’s ellipsoid of 1909, recommended as International ellipsoid in 1924

Projection function: Lambert 72/50• Conformal conical projection with 2 secant parallels

– 49°50’0.0204” and 51°10’0.0204”– Longitude of central meridian: 4°22’2.952”– Latitude of origin: 90°– Fase easting: 150.000,013 meter– False northing: 5.400.088,4398 meter

Vertical reference system = TAW (average low tide level in Oostende (North Sea Channel)

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Other coordinate systems: examples

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A gDB

• Can have one coordinate reference system only (effective or virtual)

• The coordinates in all geodatasets must be expressed according to that system– Vertical integration

– Horizontal integration

• Most commonly, the choosen coordinate system is– Geographic coordinates (LON-LAT) or

– National coordinate system (from the National Mapping Agency, used for printing topographic / military maps)

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Vertical integration

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Horizontal integration

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Transformation of coordinates for vertical/horizontal integration

1. Analytical conversion between geographic coordinates expressed according to different geodetic datums = datum conversion

2. Analytical conversion of geographic coordinates (e.g. from GPS) in projected coordinates and vice versa = (inverse) projection

3. Analytical conversion between different types of projected coordinates (e.g. between Philippine and Belgian system)

4. Numerical coordinate transformation (e.g. geo-referencing, using control points)

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Numeric coordinate transformation

• Numeric coordinate shifts, based on control points, for vertical and horizontal integration of geodatasets in a gDB– systematic shifts (e.g. conversion of digitiser/scan

coordinates in projected coordinates)– non-systematic shifts: rubber sheeting, edge

matching

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Geo-referencing• When coordinates are expressed according to

an analytical reference system, the term ‘georeferenced data’ is used.

• A/D conversion using tablet digitising or scanning provide digitiser and scan coordinates. Also raw satellite images are not georeferenced.

• Transformation of « technical » coordinates into geographical or projected coordinates = georeferencing.

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Numeric transformation of coordinates after A/D-conversion via digitization

• Digitisation provides (Xi,Yi) of point objects, nodes, vertices• Xi,Yi are digitizer-coordinates, expressed according to a technical,

flat reference system• Xi,Yi must be transformed into a projected reference system

X

Y

(0,0)

•Xo = f(Xi,Yi); Yo = f(Xi,Yi)

(Xi,Yi)

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Numeric transformation of digitizer- to gDB-coordinates

• AFFINE polynomial transformation function f = popular– Xo = A + BXi + CYi – Yo = D + EXi + FYi– 2 * 3 unknowns: A, B, C and D, E, F– 2 * 3 equations required to compute the unknonws – Equations are derived from 3 control points (3X and 3Y) (GCP)– 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

– System of equations has one single EXACT solution for A ... F– Transformation error is apparently 0

• If more than 3 GCP are available, more equations than unknowns– System of equations has more than one solution for A ... F– Best solution for A ... F can be found by the Least-Squares method– Transformation error can be computed (RMSe - ROOT MEAN SQUARE ERROR)

• If RMSe is sufficiently low– Parameterised AFFINE equations can be applied to all input-points (point objects, nodes,

vertices). Result = transformed output-geodataset

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AFFINE transformation of digitizer- to gDB-coordinates

GCP # Xg (m) Yg (m) Xd (inch) Yd (inch)1 100000 120000 3,02 20,712 120000 120000 18,79 20,653 120000 100000 18,75 4,994 100000 100000 2,99 4,9

X1 100000 = A + B*3,02 + C*20,71 A ?2 120000 = A + B*18,79 + C*20,65 B ?3 120000 = A + B*18,75 + C*4,99 C ?4 100000 = A + B*2,99 + C*4,9

Y1 120000 = D + E*3,02 + F*20,71 D ?2 120000 = D + E*18,79 + F*20,65 E ?3 100000 = D + E*18,75 + F*4,99 F ?4 100000 = D + E*2,99 + F*4,9

RMS_X ?RMS_Y ?RMS_XY ?

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AFFINE-transformation & RMSe: X

Coefficients

A Intercept 96223,86B Variable 1 1268,636C Variable 2 -2,819892

RESIDUAL OUTPUT

Observation Predicted Y Residuals1 99996,74 3,2606392 120003,3 -3,2918833 119996,7 3,2940314 100003,3 -3,262787

Squared Residuals10,6317610,8364910,8506410,64578

Means Squared residuals10,74117

RMS_X Root Means Square Residuals3,277372

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AFFINE-transformation & RMSe: Y

CoefficientsD Intercept 93728,13E Variable 1 -1,207367F Variable 2 1271,026

RESIDUAL OUTPUT

Observation Predicted Y Residuals1 120047,4 -47,423642 119952,1 47,878073 100047,9 -47,909324 99952,55 47,45489

Squared Residuals2249,002

2292,312295,3022251,966

Means Squared residuals2272,145

RMS_Y Root Means Square Residuals47,66702

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AFFINE-transformation & RMSe: X & Y

RESIDUAL OUTPUT X

ObservationPredicted Y Residuals1 99996,74 3,2606392 120003,3 -3,2918833 119996,7 3,2940314 100003,3 -3,262787

RESIDUAL OUTPUT Y

ObservationPredicted Y Residuals1 120047,4 -47,423642 119952,1 47,878073 100047,9 -47,909324 99952,55 47,45489

RESIDUALS XY1 47,53562 47,99113 48,022424 47,56692

SQUARED RESIDUALS1 2259,6342 2303,1463 2306,1534 2262,612

MEAN SQUARED RESIDUALS2282,886

RMS_XY 47,77956

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Judgement of the RMSe

• To be based on the spatial detail (scale for A/D-converted analog documents) of the source document– 1 mm distortion and/or digitizing error on a 1:50.000

analog map = 50 meter RMSe

• To be based on the intended use of the output-geodataset– Requirements for vertical and horizontal integration

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• Translation: – Xo = A + Xi

– Yo = D + Yi

• Change of scale: – Xo = BXi

– Yo = EYi

• Rotation: – Xo = BXi + CYi

– Yo = EXi +FYi

• AFFINE = All combined– Xo = A + BXi + CYi

– Yo = D + EXi + FYi

AFFINE = polynomial transformation of the 1st order

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• Xo = a0+(a1Xi+a2Yi)+(a3Xi2+a4.XiYi+a5Yi

2)+…

• Yo = b0+(b1Xi+b2Yi)+(b3Xi2+b4.XiYi+b5Yi

2)+…

• Order of the polynomial p determines the minimum number of required GCP to find the polynomial coefficients: N = (p+1)*(p+2)/2

Polynomial transformations of higher orders (rubber sheeting, warping)

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Numeric transformation of coordinates after A/D-conversion via Scanning

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Numeric transformation of scan- to gDB-coordinates

• A scanned document is not georeferenced• Scan-coordinates are relative to the reference system

of the scan-device• Transformation of the scan-coordinates is necessary,

using GCP– Regular cell-raster is distorted– A new ‘empty’ cell-raster is created according to the output-

reference system– 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

• Neirest neighbour• Other algorithms

• Also valid for remotely sensed images !

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Forward GCP-based transformation distorts the raster

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3

Xo = f(Xi,Yi); Yo = f(Xi,Yi): NOT valid

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Backward/Inverse polynomial transformation of scan- to gDB-coordinates

1. Creation of a new ‘empty’ rasterstructure in the output-coordinate system

2. Calibration of the inverse polynomial transformation– Xi = f(Xo,Yo)– Yi = f(Xo,Yo

3. 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

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Resampling1. GCP are used to calibrate an inverse polynomial

transformation function, e.g. AFFINE– Xi = G + HXo + IYo

– Yi = K + LXo + MYo

2. 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

3. Xi,Yi is the ‘nearest neighbour’– 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

– Also bi-linear and curbic re-sampling are possible

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Resampling is necessary after transformation of scan- into gDB-coordinates

R = input raster; R’ = output rasterAntrop & 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)

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Numeric coordinate transformation

• Similar systematic numeric transformation is applicable to coordinates coming from other data sources – Remotely sensed images– Theodolites, tachymeters with digital reading

– Global Positioning Systems (GPS)

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(Non-)systematic numeric coordinate-transformations

• Previous numeric polynomial coordinate transformations are based on GCP

• One set of coefficients A, B, C, … is computed and applied to all input-points to obtain the output-coordinates

• Such transformations are systematic

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Non-systematic transformations for further improvement of the positional quality of the georeferenced

geodatasets

• First step in georeferencing is most often a systematic transformation of coordinates – Polynomial function of low order

• The result is often not of sufficient quality or not sufficiently fit for use (vertical/horizontal integration in the gDB)

• In a next step, non-systematic transformation can be performed to make the geodataset geometrically more conformal to the reference geodataset

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Non-systematic coordinate transformations

• Edge-matching• Rubber-sheeting

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Rubber-sheeting

12

3

GCP1: Xi1,Yi1 -> Xo1,Yo1



GCPA...GCPF: Xi = Xo; Yi = Yo

A B

DE

F

G

C

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Rubber-sheeting• Point-by-point correction of the location and shape of objects or of

resampling of cell attributes

• Based on 2 linear “piece wise” TIN-interpolations (7.PPT), 1 for X and 1 for Y

• Z-value to interpolate = Xo resp. Yo

• Result = not-constant translation/rotation/change of scale

• Shifts decrease with increasing distance

• Both forward (for vectorial geodatasets) and backward (for raster datasets)

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Edge-matching• Special case of rubber sheeting• Applied for horizontal integration of

adjacent (A/D converted) map sheets or images (mosaicking)

• Definition of links between coinciding points on two map sheets

• Differential displacement of points based on (mostly inverse distance; TIN) interpolation

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Edge-matching

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Summary of important items• Geospatial reference systems

– Based on a geodetic datum (LON-LAT) and (possibly) a projection function to convert LON-LAT (angles - 3D) into planimetric coordinates (X,Y – 2D)

– Projection leads to distortion of one or more of shape, direction, area, distance

– If the national standards are not used, a rational, functional choice of datum and projection function is required

– The datum for elevation is most often the geoid (approximated by mean sea level)

• Transformation of coordinates– Between parameterised geographic and/or projected coordinate systems

is an analytical operation which does not need external ground truth– Between technical coordinates and projected coordinates is a numeric

operation based on ground truth (GCP)• There are systematic and non-systematic numeric transformation functions• Systematic transformation is most often based on a polynomial function• Non-systematic transformation (rubber sheeting and edge-matching) is based

on TIN-interpolation– The latter is also valid for projected coordinates which need correction

Questions or remarks ?

Thank you …

Gis Concepts 3/5

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