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7年弱前 (2009/11/26)にアップロードin学び

Introduction to basic concepts on Geographical Information Systems

Autor: Msc. Alexander Mogollón...

Introduction to basic concepts on Geographical Information Systems

Autor: Msc. Alexander Mogollón Diaz

http://www.agronomia.unal.edu.co

- Concepts and Functions of

Geographic Information Systems

(3/5)

MSc GIS - Alexander Mogollon Diaz

Department of Agronomy

2009 - Concepts and Functions of GIS

.PPT

Topic #1

Topic #2

Topic #3

1

A GIS is an information

GIS is a technology

system

2

Spatial Data modelling

Sources of data for

Metadata

geodatasets

3

Geospatial referencing

Coordinate transformations

4

Database management

5

Spatial Analysis

2 - Functionalities of GIS

INPUT

S

T

A

T R M

N

R A A

A

U N N

L

C S A

Y

T

F G

S

U O E

E

R R

E M

QUERY - DISPLAY - MAP

3 - 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

– ...

4 - 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 model ing 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 systems

1.

Geographic

2.

Projected

•

Very many variants exist of both

•

Transformations required for vertical integration

5 - Planet Earth

6 - 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)

7 - Geoid versus Ellipsoid

• Geoid

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

on measurements of the gravity force

• Local and Global El ipsoids

– Mathematical 3D-models of the Earth’s surface

– Global el ipsoids are defined to represent the ful

Earth with acceptable accuracy

– Local el ipsoids are defined to represent a part of the

Earth’s surface only, with high accuracy

8 - 9
- Geoid versus Ellipsoid

10 - 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 el ipsoid

– 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

11 - Geospatial locations are expressed

relative to an ellipsoid (2)

45° N; 120°E

LAT

LON

12 - Geospatial locations are expressed

relative to an ellipsoid (3)

• Several ellipsoids are in use !

– Major radius or major semi-axis a

– Minor radius or minor semi-axis b

– Flattening f of the el ipsoid: a-b/a = 1/f

13 - Frequently used ellipsoids

S

14 - Geospatial locations are expressed

relative to an ellipsoid (4)

• Geodetic datum = further specification of the ellipsoid

– Initial location

– Initial azimuth to define the north direction

– Distance between geoid and el ipsoid 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 el ipsoids !

– A given point has different geocentric coordinates x,y,z when expressed against different datums, even if the el ipsoid is identical

15 - 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

16 - From geographic coordinates to

projected coordinates

• 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

17 - Arc distances

18 - Computation of arc-distances

19 - From geographic coordinates to

projected coordinates

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

20 - From geographic coordinates to

projected coordinates

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

21 - Plane – cone - cylinder

Tangent - secant

Normal – transversal - oblique

22 - 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

23 - Projection creates geometric

distortion

24 - Conformal projections

Shape and/or direction is

preserved

25

Distances and areas are distorted - Mercator-projection = conformal

26 - Transverse Mercator-projection =

conformal

27 - 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

28

•Applied to various ellipsoids - 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

29 - Lambert conformal conical

projection

30 - Standard projected coordinate system

for Belgium

Belgian Datum = local orientation of Hayford’s el ipsoid of

1909, recommended as International el ipsoid in 1924

Projection function: Lambert 72/50

• Conformal conical projection with 2 secant paral els

– 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)

31 - Other coordinate systems: examples

32 - A gDB

• Can have one coordinate reference system only

(effective or virtual)

• The coordinates in al 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)

33 - Vertical integration

34 - Horizontal integration

35 - 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)

36 - 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

37 - 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.

38 - 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

Y

(X ,Y )

i

i

(0,0)

X 39

•Xo = f(Xi,Yi); Yo = f(Xi,Yi) - 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

40 - AFFINE transformation of digitizer- to gDB-

coordinates

41 - AFFINE-transformation & RMSe: X

42 - AFFINE-transformation & RMSe: Y

43 - AFFINE-transformation & RMSe: X & Y

44 - 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

45 - AFFINE = polynomial

transformation of the 1st order

• Translation:

– Xo = A + Xi

– Yo = D + Yi

• Change of scale:

– Xo = BXi

– Yo = EYi

• Rotation:

– Xo = BXi + CYi

– Yo = EXi +FYi

• AFFINE = Al combined

– Xo = A + BXi + CYi

– Yo = D + EXi + FYi

46 - Polynomial transformations of higher

orders (rubber sheeting, warping)

•

Xo = a0+(a1Xi+a2Yi)+(a3Xi2+a4.XiYi+a5Yi2)+…

•

Yo = b0+(b1Xi+b2Yi)+(b3Xi2+b4.XiYi+b5Yi2)+…

•

Order of the polynomial p determines the minimum number of required

GCP to find the polynomial coefficients: N = (p+1)*(p+2)/2

47 - Numeric transformation of coordinates

after A/D-conversion via Scanning

48 - 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 cel -raster is distorted

– A new ‘empty’ cel -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 cel s in the output raster

• Neirest neighbour

• Other algorithms

• Also valid for remotely sensed images !

49 - Forward GCP-based transformation distorts the

raster

1

2

3

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

50 - 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 ‘fil ’

the empty cel s of the output raster with (a

combination of) the value(s) of the corresponding

cel (s) in the input raster

51 - Resampling

1.

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-cel (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 cel value of the cell

in which Xi,Yi is located is attributed to the output cel with

midpoint Xo,Yo

–

Also bi-linear and curbic re-sampling are possible

52 - Resampling is necessary after transformation

of scan- into gDB-coordinates

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)

53

R = input raster; R’ = output raster

Antrop & De Maeyer, 2005 - 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)

54 - (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

55 - 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

56 - Non-systematic coordinate

transformations

• Edge-matching

• Rubber-sheeting

57 - Rubber-sheeting

A

B

C

1

2

G

3

E

F

D

GCP1: X ,Y -> X ,Y

i1

i1

o1

o1

GCP2: X ,Y -> X ,Y

i2

i2

o2

o2

58

GCP3: X ,Y -> X ,Y

i3

i3

o3

o3

GCPA...GCPF: X = X ; Y = Y

i

o

i

o - Rubber-sheeting

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

resampling of cel 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)

59 - 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

60 - Edge-matching

61 - 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

62 - Thank you …

Questions or remarks ?