Prof. Dr.-Ing. Ralf Bill GI Basics
Spatial Analysis 1
Rostock University, Chair for Geodesy and Geoinformatics X 2007
Introduction
Geom. Meth.
Topol. Meth.
Set Methods
Statistic Meth.
Models
Summary
Spatial Analysis
Prof. Dr.-Ing. Ralf Bill and Dr. Edward NashRostock University
Rostock University, Chair for Geodesy and Geoinformatics SpatialAnalysis - 2 -
Content
- Basic terms- Geometrical methods- Topological methods- Set methods- Statistical methods- Models
- Basic terms- Geometrical methods- Topological methods- Set methods- Statistical methods- Models
Ha
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Introduction
Prof. Dr.-Ing. Ralf Bill GI Basics
Spatial Analysis 2
Rostock University, Chair for Geodesy and Geoinformatics SpatialAnalysis - 3 -
Some definitions of the term 'Analysis'
• Analysis = scientific study of problems or correlations• Analysis = division, decomposition of compounds into their
components (opposite of synthesis!)• Analysis = systematic study of an object• Analysis = scientifically dissolving and studying
=> qualitative analysis = according to properties etc. => quantitative analysis = according to amount, number, order etc.
Introduction
Rostock University, Chair for Geodesy and Geoinformatics SpatialAnalysis - 4 -
Basic problem of spatial analysis
• Given:• User-defined task and an information system with observations A, B, C, ...
• Search:• establish function(s) through which the available data may be involved
and manipulated to provide the required output (e.g. presentations such as maps, graphs, reports, …) related to the problem
Link: U = f (A, B, C ...)
• Functions f• Selection• Boolean operations• Algebraic terms• Reclassification• Polygon overlay with functions between data
Introduction
Prof. Dr.-Ing. Ralf Bill GI Basics
Spatial Analysis 3
Rostock University, Chair for Geodesy and Geoinformatics SpatialAnalysis - 5 -
5 questions a GIS can answer!
• 1. Location: What is at a given location?• The first of these questions seeks to find out what exists at a particular location.
A location can be described as a place name, zip code or address.• 2. Condition: Where does something occur?
• Using spatial analysis the second question seeks to find a location where certain conditions are satisfied (e.g., an unforested section of land at least 2,000 square meters in size, within 100 meters of a road, and with soils suitable for supporting buildings).
• 3. Trends: What has changed since ...?• The third question might involve a combination of the first two and seeks to find
the differences within an area over time.• 4. Patterns: What spatial patterns exist?
• You might ask this question to determine whether cancer is a major cause of death among residents near a nuclear power station. Just as important, you might want to know how many anomalies there are that don't fit the pattern and where they are located.
• 5. Modeling: What if ...?• "What if ..." questions are posed to determine what happens, for example, if a
new road is added to a network. Answering this type of question requires geographic as well as other information.
Source: http://volusia.org/gis/whatsgis.htm
Summ
ary
Rostock University, Chair for Geodesy and Geoinformatics SpatialAnalysis - 6 -
Analysis-Synthesis-Simulation-Prognosis
• Analysis = dissecting, decomposing compounds into its components
• Synthesis = merging single components to a higher order
• Simulation = realistic imitation of technical processes
• Prognosis = assessment in advance (forecast)
AnalysisSynthesis
SimulationPrognosis
Introduction
Prof. Dr.-Ing. Ralf Bill GI Basics
Spatial Analysis 4
Rostock University, Chair for Geodesy and Geoinformatics SpatialAnalysis - 7 -
Spatial analysis
• has mathematical foundations• Coordinate geometry• Numerical methods• Topology and graph theory• Set theory• Relational algebra • Statistics• …
• offers in contrast to CAD/DB/IS ..• Polygon overlay• Geo-statistical analysis• Spatial aggregation• Selective spatial search• Topological analysis• …
Introduction
Rostock University, Chair for Geodesy and Geoinformatics SpatialAnalysis - 8 -
Spatial analysis methods
• Geometrical Methods• Computed geometry• Polygon overlay• Generation of zones• Triangulation/neighbour-
hood graphs
• Topological Methods• Network analysis• Neighbourhood analysis• Site planning
• Statistical Methods• Descriptive statistics• Analytical statistics• Geostatistics
• Set Methods• Boolean, relational and
fuzzy-algebra• Sort and search• Aggregation
• Models and Simulations• Cartographic modeling• System analytical
approaches• Dispersion- and
simulation models
Introduction
Prof. Dr.-Ing. Ralf Bill GI Basics
Spatial Analysis 5
Rostock University, Chair for Geodesy and Geoinformatics SpatialAnalysis - 9 -
Geometric methods
• Mathematical background• Metric and coordinate systems • Computed geometry
• Cross-sections in 2 and 3 dimensions• Spatial search and clipping-algorithms
• Specific analysis functions• Point-in-polygon problem• Polygon overlay• Triangulation and Thiessen-polygons• …
Geom
etricm
ethods
Rostock University, Chair for Geodesy and Geoinformatics SpatialAnalysis - 10 -
Geometry and coordinate systems
• Vector data (points, lines and polygons in x,y,[z])
• Raster data (pixels)
x |H|Northing
y |R|Easting
Geodeticalcoordinate system
100 gon
y
x
90
Mathematicalcoordinate system
Coordinate systems
[0,0]1,1
[0,n]1,n
n,1[n,0]
Column
Row
Geom
etricm
ethods
Prof. Dr.-Ing. Ralf Bill GI Basics
Spatial Analysis 6
Rostock University, Chair for Geodesy and Geoinformatics SpatialAnalysis - 11 -
Metrics and distance definitions
• Different possibilities to describe the distance d between points P,Q• Commonly used distance functions:
• Vector data: Euclidean Distance: dE = sqrt((xi-xj)*(xi-xj)+ (yi-yj)*(yi-yj))
• Raster data: with d1=|i-k|, d2=|j-l| for P(i,,j), Q(k,l) in pixel coordinates• City-Block-Distance: d4 = d1+d2
• Chessboard distance: d8 = max(d1,d2)
• Euclidean Distance: dE = sqrt (d1*d1+d2*d2)
A metric on a set X is a projection d : X*X on R0 with the following properties for any P, Q, T from X:
d(P,Q) = 0 if P=Q Idempotenced(P,Q) = d(Q,P) Symmetryd(P,Q) <= d(P,T) + d(T,Q) Triangle inequality
A pair (X,d) is called metric space.
Geom
etricm
ethods
N.4
N.8
Rostock University, Chair for Geodesy and Geoinformatics SpatialAnalysis - 12 -
Problem of definition? Which distance to take?
Centroid distanceAny distance
Minimum distance
Geom
etricm
ethods
Prof. Dr.-Ing. Ralf Bill GI Basics
Spatial Analysis 7
Rostock University, Chair for Geodesy and Geoinformatics SpatialAnalysis - 13 -
Approximation of spatial objects
• Origin
• Simplification ofgeometry:
• to search and to index storage for the computer
• to approximate geometric algorithms
x
2D-approximation
3D-approximation
4D-approximation
y
x
y
x
y
x
yHigher approximation
x
Geometryy
x
Point Line Polygon
Geom
etricm
ethods
Rostock University, Chair for Geodesy and Geoinformatics SpatialAnalysis - 14 -
Rough tests with MER
• MER = minimum enclosing axis-parallel rectangle(also called BoundingBox)
• Concept for quick data access and preprocessing• e.g. used for point in polygon test and intersection
Point in Polygon IntersectionMER
Point1
2
3
4
g1
g2MER
MER
Polygon
Geom
etricm
ethods
Prof. Dr.-Ing. Ralf Bill GI Basics
Spatial Analysis 8
Rostock University, Chair for Geodesy and Geoinformatics SpatialAnalysis - 15 -
Computational geometry
• Point in rectangle
• Geometric intersection
• Minimum enclosing rectangle
• Spatial operators• Area size and perimeter
• Intersection and shortest distance
• Union and difference
• Inclusion
XMIN,YMIN
RP(x,y)
XMAX,YMAX
2-dimensional
3 2
1 4
h
g
s
x
y
3 2
1 4h
g
s
y
H
Gz
x
3-dimensional
Geom
etricm
ethods
Rostock University, Chair for Geodesy and Geoinformatics SpatialAnalysis - 16 -
Point-in-polygon (vector)
• Jordan Theorem:• Every polygon R separates a plane into 2 disjunct regions (inner and
outer). If the number of real intersections of a ray through X with the edges of the polygon odd-numbered, then X is inside R, else outside.
• Algorithm • Choose test ray through X parallel to coordinate axis.• Choose point t on test ray outside R.• Check if TX touches vertex.
Yes: shift T in y-direction (only real intersections are to be taken).• Count number of real intersections of TX with (n-1) polygon edges.
• If mod(number,2)=0, X is outside.If mod(number,2)=1, X is inside
• with [mod (N,2) = N-(int)(N/2)*2
y
x
MER
PT
i X
R
Geom
etricm
ethods
Prof. Dr.-Ing. Ralf Bill GI Basics
Spatial Analysis 9
Rostock University, Chair for Geodesy and Geoinformatics SpatialAnalysis - 17 -
Point-in-polygon (raster)
• Assumptions:All cells inside polygon have the attribute 1All cells outside polygon have the attribute 0
• Algorithm:Check for row l (l=1,m) if row index l = point
row index i.Yes: check for col index k (k=1,n) if col index
k = point col index j.
Yes: check if cell attribute = 1.Yes: Point is inside, else outside.
If point (i,j) is not in the same row / col of polygon increase row index and restart.
123456789
101112
= 0= 1
Row
sl =
1,m
1 72 3 4 5 6 8 9 10 11 12Columns k = 1,n
= point to be checked
R
Geom
etricm
ethods
Rostock University, Chair for Geodesy and Geoinformatics SpatialAnalysis - 18 -
Polygon overlay: Two polygons
• 1. Edge intersections:• Divide all intersecting edges of the
starting polygons at their intersections.
=> List of all nodes and edges: no further intersection of polygons.
• 2. Polygon formation:• Link individual edges to form new,
closed polygons.=> List of all polygons.
• 3. Overlay identification:• Check polygons: which were the
original polygons?=> transfer attributes (copy or link).
12
3 4
1 34
56
81011
11 Objects
POLYGON OVERLAY
RESULT-Object class 3
Object class 1 Object class 2AREA- AREA-
4 Objects 4 Objects
7
2
4
9
32
1
Geom
etricm
ethods
Prof. Dr.-Ing. Ralf Bill GI Basics
Spatial Analysis 10
Rostock University, Chair for Geodesy and Geoinformatics SpatialAnalysis - 19 -
Variations for polygon overlay
Source: Alan Murta (http://www.cs.man.ac.uk/aig/staff/alan/software/gpc.htm)l
Difference Intersection
Exclusive-or Union
Geom
etricm
ethods
Rostock University, Chair for Geodesy and Geoinformatics SpatialAnalysis - 20 -
Polygon overlay: Line with polygon
Class 1: - All power lines of an energy provider
Class 2: - All parcels that are
owned by the municipality
Class 3: - All lines on parcels owned by the municipality
LINE-Object class 1
AREA-Object class 2
3 Objects 4 Objects
6 Objects
23
4
1
2
3
1 2 34
56
RESULT-Object class 3
INTERSECTION
1
Geom
etricm
ethods
Prof. Dr.-Ing. Ralf Bill GI Basics
Spatial Analysis 11
Rostock University, Chair for Geodesy and Geoinformatics SpatialAnalysis - 21 -
Polygon overlay: Raster with raster
y
x
z Feature class 1
+
=
Feature class 2
Feature class 3
Geom
etricm
ethods
Rostock University, Chair for Geodesy and Geoinformatics SpatialAnalysis - 22 -
Buffer- or zone generation (vector)
• Example:• Travel time problem
(geometric)
• All in the yellow area need 20 min to the next station
• All in the read area only 10 min
Point
Line
Polygon Inner buffer
Circular buffer
Narrow buffer
Outer buffer
Square buffer
Wide buffer
20
20
20
10
10
10
Geom
etricm
ethods
Prof. Dr.-Ing. Ralf Bill GI Basics
Spatial Analysis 12
Rostock University, Chair for Geodesy and Geoinformatics SpatialAnalysis - 23 -
Buffer- or zone generation (Raster)
a) Original matrix b) Buffer zone 20mraster size 10m city block metric
- - - - - - - - - - - - - - - - - - - - - - - -- - - - - - - - - - - - - - - 0 0 0 0 0 - - - -- - - - - - - - - - - - - - - 0 0 0 0 0 - - - -- - - - - 0 - - - - - - - - - 0 0 0 0 0 - - - -- - - - - - - - - - - - - - - 0 0 0 0 0 - - - -- - - - - - - - - - - - - - - 0 0 0 0 0 - - - -
Geom
etricm
ethods
Rostock University, Chair for Geodesy and Geoinformatics SpatialAnalysis - 24 -
Interpolation, Neighbourhood graphs, Centroids
70
80
90
90
100
110
70
8090
100
110
Point values Points Polygon values
Centroid-generation
Neighbourhood-graph
Isoline-Inter-polation
Geom
etricm
ethods
Prof. Dr.-Ing. Ralf Bill GI Basics
Spatial Analysis 13
Rostock University, Chair for Geodesy and Geoinformatics SpatialAnalysis - 25 -
Delaunay-Triangulation/Thiessen-Diagrams
3
2
1
4
5
0
Concept: Circumference of 3 pointsdoes not contain any further points
(or: Voronoi-Diagram and Dirichlet-Tesselation)
Enlargement
2
34
5 0
Input data
Thiessen-Polygon
Geom
etricm
ethods
Rostock University, Chair for Geodesy and Geoinformatics SpatialAnalysis - 26 -
Neighbourhood graphs (Raster)
a) Original matrix b) Distance transformation c) Main axiswith special given metric transformation
- - - - - - - - - - - - 18 15 12 11 10 9 10 11 12 15 18 21 - - - - - - - - - - - -- - - - - - - - - - - - 17 14 11 8 7 6 7 8 11 14 17 20 - - - - - - - - - - - -- - - - - - - - - - - - 16 13 10 7 4 3 4 7 10 13 16 19 - - - - - - - - - - - -- - - - - 0 - - - - - - 14 12 9 6 3 0 3 6 9 12 15 18 - * - - - 0 - - - - - -- - - - - - - - - - - - 11 10 9 7 4 3 4 7 10 13 16 17 - - * * - - - - - * * -- - - - - - - - - - - - 8 7 6 7 7 6 7 8 11 12 13 14 - - - * * - - - * * * -- - - - - - - - - - - - 7 4 3 4 7 9 10 11 10 9 10 11 - - - - * * * * * - - -- - 0 - - - - - - - - - 6 3 0 3 6 9 11 8 7 6 7 8 - - 0 - - - * - - - - -- - - - - - - - - - - - 7 4 3 4 7 10 10 7 4 3 4 7 - - - - - * * - - - - -- - - - - - - - - 0 - - 8 7 6 7 8 11 9 6 3 0 3 6 - - - - - * - - - 0 - -- - - - - - - - - - - - 11 10 9 10 11 12 10 7 4 3 4 7 - - - - - * - - - - - -- - - - - - - - - - - - 14 13 12 13 14 14 11 8 7 6 7 8 - - - - * * - - - - - -
Geom
etricm
ethods
Prof. Dr.-Ing. Ralf Bill GI Basics
Spatial Analysis 14
Rostock University, Chair for Geodesy and Geoinformatics SpatialAnalysis - 27 -
Neighbourhood graphs (Raster)
6
45
2
3
1
* * * * 6 6 6 6 6 6 6 6 6 * * * * * * * * 6 6 6 6 6 6 6 6 6 * * * * * * * * 6 6 6 6 6 6 6 6 6 * * * ** * * * 6 6 6 6 6 6 6 6 4 4 4 4 45 5 5 5 5 6 6 6 6 6 6 4 4 4 4 4 45 5 5 5 5 5 6 6 6 6 4 4 4 4 4 4 45 5 5 5 5 5 5 6 6 4 4 4 4 4 4 4 45 5 5 5 5 5 5 2 2 4 4 4 4 4 4 4 45 5 5 5 5 5 5 2 2 2 4 4 4 4 4 4 45 5 5 5 5 5 2 2 2 2 2 4 4 4 4 4 45 5 5 5 5 5 2 2 2 2 2 2 4 4 4 4 45 5 5 5 5 2 2 2 2 2 2 2 2 4 4 4 45 5 5 5 2 2 2 2 2 2 2 3 3 3 3 3 *1 1 1 1 1 2 2 2 2 2 3 3 3 3 3 3 *1 1 1 1 1 1 2 2 2 3 3 3 3 3 3 3 *1 1 1 1 1 1 1 3 3 3 3 3 3 3 3 3 *1 1 1 1 1 1 1 1 3 3 3 3 3 3 3 3 *1 1 1 1 1 1 1 1 3 3 3 3 3 3 3 3 *1 1 1 1 1 1 1 1 3 3 3 3 3 3 3 3 *1 1 1 1 1 1 1 1 3 3 3 3 3 3 3 3 *1 1 1 1 1 1 1 1 * * * * * * * * *1 1 1 1 1 1 1 1 * * * * * * * * *
Point distribution Neighbourhood graph* * * * 6 6 6 6 6 6 6 6 6 * * * * * * * * 6 6 6 6 6 6 6 6 6 * * * * * * * * 6 6 6 6 6 6 6 6 6 * * * ** * * * 6 6 6 6 6 6 6 6 4 4 4 4 45 5 5 5 5 6 6 6 6 6 6 4 4 4 4 4 45 5 5 5 5 5 6 6 6 6 4 4 4 4 4 4 45 5 5 5 5 5 5 6 6 4 4 4 4 4 4 4 45 5 5 5 5 5 5 2 2 4 4 4 4 4 4 4 45 5 5 5 5 5 5 2 2 2 4 4 4 4 4 4 45 5 5 5 5 5 2 2 2 2 2 4 4 4 4 4 45 5 5 5 5 5 2 2 2 2 2 2 4 4 4 4 45 5 5 5 5 2 2 2 2 2 2 2 2 4 4 4 45 5 5 5 2 2 2 2 2 2 2 3 3 3 3 3 *1 1 1 1 1 2 2 2 2 2 3 3 3 3 3 3 *1 1 1 1 1 1 2 2 2 3 3 3 3 3 3 3 *1 1 1 1 1 1 1 3 3 3 3 3 3 3 3 3 *1 1 1 1 1 1 1 1 3 3 3 3 3 3 3 3 *1 1 1 1 1 1 1 1 3 3 3 3 3 3 3 3 *1 1 1 1 1 1 1 1 3 3 3 3 3 3 3 3 *1 1 1 1 1 1 1 1 3 3 3 3 3 3 3 3 *1 1 1 1 1 1 1 1 * * * * * * * * *1 1 1 1 1 1 1 1 * * * * * * * * *
Starting pointDistance 1 unitDistance 2 unitsDistance 3 unitsDistance 4 units
Quasi-Thiessen-Diagram resp. Delaunay-triangulation
Geom
etricm
ethods
Rostock University, Chair for Geodesy and Geoinformatics SpatialAnalysis - 28 -
2.5D- and 3D-analysis functions
• 3D-Interpolation (poss. with time 4D)• Isolines• Slope and gradient• Exposition, aspect, hill-shading• Analysis inside objects• Buffer in 3D• Surface, network flows and path analysis• Volumes and deposit calculation• Cross-sections in2D and 3D, profiles• Line-of-sight• ..
Profile from View1
Elev
atio
n
Distance
Vertical exaggeration 7.3 X
770.20
800.
171
3.8
150.0 300.0 450.0 600.0
738.
876
3.8
#
Geom
etricm
ethods
Prof. Dr.-Ing. Ralf Bill GI Basics
Spatial Analysis 15
Rostock University, Chair for Geodesy and Geoinformatics SpatialAnalysis - 29 -
Topological methods
• Mathematical background• Adjacency and incidence
• Algorithms and applications• best path, best site, Travelling Salesman Problem• Shortest path• Floyd-Warshall-Algorithm• Dijkstra-Algorithm
Topologicalmethods
Rostock University, Chair for Geodesy and Geoinformatics SpatialAnalysis - 30 -
Basics of topology (Adjaceny/Incidency)
Edge1234567
fromAABEEED
toBDEACDC
weight7613125
AB
E
CD
1(p=7)
3(p=
1)
4(p=3)
5(p=1)7(p=5)
6(p=2)
2(p=
6)
Assessment matrix :
A
00300
B
70000
E
01000
C
00105
D
60200
ABECD
Adjacence matrix BTB :
A
3....
B
-12...
E
-1-14..
C
00-12.
D
-10-1-13
ABECD
Incidence matrix B :
A
110-1000
1234567
B
-1010000
E
00-11110
C
0000-10-1
D
0-1000-11
A
114300
ABECD
Shortest path sums according to Floyd-Warshall:
B E C D
7111000
811100
92105
63200
Shortest path (Floyd-Warshall-Algorithm) Topologicalmethods
Prof. Dr.-Ing. Ralf Bill GI Basics
Spatial Analysis 16
Rostock University, Chair for Geodesy and Geoinformatics SpatialAnalysis - 31 -
Network Analysis: 3 Categories of tasks
Best Path
Start-pointt
End point
Best Location
Location
Start-point
Traveller-Problem
Travelling Salesman Problem:
- Operations research- Linear Optimisation- Graph Theory
Best site in terms of reachability and commuter-belt:- topologic algorithms- polygon overlay- 2D- Median
Best path- geometrically shortest path- topologically best path- cheapest path
Topologicm
ethods
Rostock University, Chair for Geodesy and Geoinformatics SpatialAnalysis - 32 -
Goals for shortest distance
• Minimise distance (ABC=57)• Minimise travel time (ABC=57)• Minimise junctions (AC=61)• Minimise turns (especially left turns) (AC=61)• Minimise cost including fixed points (via D => ADC=59)
=> Optimal path is dependent on the factors considered=> Application: Routing, car navigation systems
D C
A B
14
2121
45
3661 32
Example: Path from A to C
Topologicalmethods
Prof. Dr.-Ing. Ralf Bill GI Basics
Spatial Analysis 17
Rostock University, Chair for Geodesy and Geoinformatics SpatialAnalysis - 33 -
Path problems in networks and graphsTopologic
methods
Round-trips
Steiner network
Min.spanning tree
Distance tree
Shortest path Centre problems
Path problems
Rostock University, Chair for Geodesy and Geoinformatics SpatialAnalysis - 34 -
Shortest path between 2 nodes
Example: Hike starting in A to mountain cabin H. Estimation of walking time [h]. From D to E you can take the chair-lift. Which is the shortest walking time from A to H?
Example: Hike starting in A to mountain cabin H. Estimation of walking time [h]. From D to E you can take the chair-lift. Which is the shortest walking time from A to H?
Solution: Stepwise approachAccumulation of sequential shortest ways
C
F
H
E
G
A
BD
2,0
3,0
1,5
2,5
1,0 3,5
0,5
2,0
2,5
1,53,0
1,5
1,0
0,5
0,5
Topologicm
ethods
Prof. Dr.-Ing. Ralf Bill GI Basics
Spatial Analysis 18
Rostock University, Chair for Geodesy and Geoinformatics SpatialAnalysis - 35 -
Shortest path between two nodes
The shortest path from A to H is ABDEGH = 5,5 hrs
Topologicm
ethods
--H (ABDEGH, 5.5)8
L(ABDEFH)=7.0HF (ABDEF, 4.5)7
L(ABDEGH)=5.5HG (ABDEG, 4.0)6
L(ABDEF)=4.5; L(ABDEG)=4.0F, GE (ABDE, 3.0)5
L(ACE)=3.5; L(ACF)=5.0; L(ACG)=5.0E, F, GC (AC, 3.0)4
L(ABDE)=3.0; L(ABDG)=5.5; L(ABDH)=6.0E, G, HD (ABD, 2.5)3
L(ABC)=4.0; L(ABD)=2.5; L(ACD)=3.5C, DB (AB, 1.5)2
L(AB)=1.5; L(AC)=3.0B, CA (A, 0)1
Outgoing Paths and Total LengthUnvisitedNeighbours
Source Node(Path, Length)
Step
Rostock University, Chair for Geodesy and Geoinformatics SpatialAnalysis - 36 -
Minimum spanning tree
• Example: 6 cities should be linked via glass-fibre network.
• Goal: All cities have to be connected to the network with minimum cost.
• Solution: Order the edges according to their evaluation
• PU(22), RT(24), PT(30), TU(36), QR(38), PQ(39), RS(42), QU(43), ST(57), RU(60), QT(61), PR(62), SU(65), PS(78), QS(84)
U T
P
Q R
S
36
38
2230
39
43 6178
60
60
42
5765
24
U
P
Q R
S
T
62
Topologicalmethods
Prof. Dr.-Ing. Ralf Bill GI Basics
Spatial Analysis 19
Rostock University, Chair for Geodesy and Geoinformatics SpatialAnalysis - 37 -
Minimum spanning tree
• If a circle is created, eliminate the relevant edges
• PU, RT, PT, (TU eliminated), QR, (PQ eliminated), RS
• Remaining sequence is the minimum spanning tree
Algorithm definition :
Given a connected network (E,K) resp. a planar graphwhere each edge ki K is evaluated with di >= 0.
A graph with the edges k1, k2, … kr is calledminimum spanning tree, if
Σ di = minimum.r
i=1
Topologicalmethods
Rostock University, Chair for Geodesy and Geoinformatics SpatialAnalysis - 38 -
Round trip problem (Traveling salesman problem)
• Example: Business man from Frankfurt (F) has to go by car to Kassel (K), Nürnberg (N), Stuttgart (S) and Würzburg (W) and return to F.
K
F
S
NW
190
190
310
220
120
230
105
165
R = min ?3
R : F - K - N - W - S - F
190+310+105+165+220=990
R : F - K - N - S - W - F
190+310+190+165+120=975
R : F - K - W - N - S - F
190+230+105+190+220=935
R : F - W - K - N - S - F
120+230+310+190+220=1070
R : F - K - W - S - N - F
190+230+165+190+225=1000
Topologicalmethods
Prof. Dr.-Ing. Ralf Bill GI Basics
Spatial Analysis 20
Rostock University, Chair for Geodesy and Geoinformatics SpatialAnalysis - 39 -
Traveling salesman problem TSP
• In general: complete net (graph) with n nodes
• n --> 1/2*(n-1)*(n-2)*...*2*1 = 1/2(n-1)! , ne N
• Procedure becomes very time-consuming with an increasingnumber of nodes
Example: Calculation time for round trip problem, dependent on n
nodes n 6 10 11 12 13 14
time t 0,001 [s] 4 [s] 40 [s] 8 [min] 2 [h] 1 [day]
+
Topologicm
ethods
Rostock University, Chair for Geodesy and Geoinformatics SpatialAnalysis - 40 -
Search for optimal site
• Along a line (polygon)• Inside polygon• In space
=> Application in infrastructure planning
• Sites for companies, schools, hospitals• Sites for airports, sewage plants,...
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%U
%U
%U
%U%U%U%U%U%U%U%U%U%U%U
%U %U%U
%U%U
%U
%U
%U %U
%U
%U
%U %U%U
%U%U
%U%U%U
%U
%U
%U %U %U%U
%U%U %U
%U
%U%U%U
%U
%U%U%U%U%U
%U
%U
%U
#Y
#Y
Topologicalmethods
Prof. Dr.-Ing. Ralf Bill GI Basics
Spatial Analysis 21
Rostock University, Chair for Geodesy and Geoinformatics SpatialAnalysis - 41 -
Centre problems
Solution: Mean:X = 1/7Σ pos(i) = 7Minimum of quadratic distance = 34Median: M = 50%-Quantil = 5Minimum of absolute values = 32
A B C D E F G0 1 2 3 5 7 10 15 16
• given: road in residential area with residents A,B,C,D,E,F,G• find site with minimal distance to all residents
Topologicalmethods
Rostock University, Chair for Geodesy and Geoinformatics SpatialAnalysis - 42 -
Set methods
• Set theory• Boolean logic• Kleenean logic• Fuzzy-Set theory• Relational algebra• Sort and search• Mathematical functions• Aggregation• others
Set m
ethods
Prof. Dr.-Ing. Ralf Bill GI Basics
Spatial Analysis 22
Rostock University, Chair for Geodesy and Geoinformatics SpatialAnalysis - 43 -
Basics of set theory
Complementary law:A 0 = AA S = AA Ā = SA Ā = 0
U
U
U
U
A U B = B U A
Absorption law:
Distributive law:
A (B C) = (A B) C
A (B C ) = (A B) C
U U U U
U U U U
A (A B) = AU U
A ( A B ) = AU
U
A ( B C ) = (A B) (A C)
A ( B C ) = (A B) (A C)
U
U U
U U UU U
UU
A B = B A
U U
Commutative law:
Associative law:
A and B are sets.
A B is the intersection of the two sets. The intersection contains all elements that are in both sets, A and B.
If A B have no common elements then this is the empty set 0.
A U B is the union of the two sets, containing all elements that appear at least in one of A and B.
The complimentary set Ā contains all elements from the universal set R that are not included in A.
U
U
Set m
ethods
Rostock University, Chair for Geodesy and Geoinformatics SpatialAnalysis - 44 -
Boolean logic – binary decisions
Truth tables for Boolean operators in programmingA B NOT A A AND B A OR B A XOR B1 1 0 1 1 01 0 0 0 1 10 1 1 0 1 10 0 1 0 0 0
1 = ”true"; 0 = "false".
Venn Diagrams
A AND B
A OR B
(A AND B) OR C
A AND (B OR C)
A NOT B
A XOR BBinary logic
(true=1/false=0)
False
True
Set m
ethods
Prof. Dr.-Ing. Ralf Bill GI Basics
Spatial Analysis 23
Rostock University, Chair for Geodesy and Geoinformatics SpatialAnalysis - 45 -
Example: Set operations in raster data
Identify areas where the following three criteria are fulfilledt:- Feasible soil conditions- Water depth smaller 3 m- more than 200 m away from mangroves
Mathematical assumptions related to metrics- Raster cell size is 200 m- Chess board distance resp. N.8-Neighbourhood
Water depthin m
0 1 2 3 4 40 1 2 3 4 40 1 2 2 3 30 1 1 2 2 20 0 1 1 1 10 0 0 0 0 00 0 0 0 0 0
Mangroves(1=Mangroves, 0=no,2= 200m-Zone)
1 1 0 0 0 01 1 0 0 0 00 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 00 1 1 0 0 01 1 1 0 0 0
AND =
Result(1=true, 0=false)
0 0 0 0 0 00 0 0 0 0 00 0 0 0 0 00 0 1 0 0 10 0 0 0 0 10 0 0 0 0 00 0 0 0 0 0
Soil conditions(1=good, 0=bad or
no data)
0 0 1 1 1 10 1 1 0 1 10 0 1 0 1 10 0 1 0 0 1 0 0 0 0 0 10 0 0 0 0 00 0 0 0 0 0
AND
Set m
ethods
Rostock University, Chair for Geodesy and Geoinformatics SpatialAnalysis - 46 -
Kleenean logic
• Sometimes true/false is not enough:• Is the point inside the polygon? (statement “the point is inside the
polygon”)
true
false
maybe
Set m
ethods
Prof. Dr.-Ing. Ralf Bill GI Basics
Spatial Analysis 24
Rostock University, Chair for Geodesy and Geoinformatics SpatialAnalysis - 47 -
Fuzzy logic
• The world is not always clear cut• This uncertainty is modelled using multi-valued logic (“Fuzzy Sets”)
• Even true/false/maybe is not always enough
False
True
Probability-value s
0
1
True False
If ( A with s=x) AND ( B with s=y) THEN ( C with s=z)
Set m
ethods
Rostock University, Chair for Geodesy and Geoinformatics SpatialAnalysis - 48 -
Selective queries in GIS-data base
DESCRIPTIVE
- all parcels with value > 100.000.-
TOPOLOGIC- all parcels bordering x-road on left-hand side
ExamplesGEOMETRIC- all landmarks in
COMBINATIONall parcels along x-road (left side) with value > 100.000 €inside
125 126 127100.000.-
80.000.-
150.000.-
x - Straße
70.000.- 100.000.- 90.000.-
145 146 147
125 126 127100.000.-
80.000.-
150.000.-
x - Straße
70.000.-100.000.-
90.000.-
145 146 147
125 126 127100.000.-
80.000.-
150.000.-
x - Straße
70.000.-100.000.- 90.000.-
145 146 147
DATA
125 126 127100.000.-
80.000.-150.000.-
x - Straße
70.000.- 100.000.- 90.000.-
145 146 147
Set m
ethods
Prof. Dr.-Ing. Ralf Bill GI Basics
Spatial Analysis 25
Rostock University, Chair for Geodesy and Geoinformatics SpatialAnalysis - 49 -
Relational operators
Selection Projection Product Union
abc
xy
aabbcc
xyxyxy
Intersection Difference (natural) join Division
a1a2a3
b1b1b2
a1a2a3
c1c2c3
a1a2a3
b1b1b2
c1c1c2
aaabc
xyzxy
xz
a
Set m
ethods
Rostock University, Chair for Geodesy and Geoinformatics SpatialAnalysis - 50 -
Searches in large data base
• Common approaches • e.g. name=Bill in 10,000 data sets
• linear search (O|n|, mean of 5,000 operations, worst case 10,000)
• logarithmic search after build-up of ordered lists (O|log(n)|, 14 operations)
• Tree search methods• used when search space can be represented as a tree• start at the root and run along edges to the next node
representing a hit• can be divided into order in which nodes of the tree are
visited
Set m
ethods
Prof. Dr.-Ing. Ralf Bill GI Basics
Spatial Analysis 26
Rostock University, Chair for Geodesy and Geoinformatics SpatialAnalysis - 51 -
Sorting large data bases
• Sorting procedures are some of the most common procedures in IT-applications• by name• by size• by frequency
• Sorting process such as quicksort, heapsort, shellsort• Order O|n²| to O|n|
• Sorting with Divide/Sort and Merge
Set m
ethods
Rostock University, Chair for Geodesy and Geoinformatics SpatialAnalysis - 52 -
Classification (Generation of spatial clusters)
- Given dataProperty AProperty BProperty CProperty D
- Extreme values- Parallel epiped class- Minimum-Box-
Classification
- Distance Classification(Euclidean Distance)
- Shortest Distance-classification
- Maximimum-Likelihood-Classification (Lines ofequal probability)
Set m
ethods
Prof. Dr.-Ing. Ralf Bill GI Basics
Spatial Analysis 27
Rostock University, Chair for Geodesy and Geoinformatics SpatialAnalysis - 53 -
Reclassification
ac
ab bcad
ae
A
A BA
A
AB
1 2 3
Legend : a,c,d,e – Deciduous Forestb – Coniferous Forest
Reclassification :
Set m
ethods
Rostock University, Chair for Geodesy and Geoinformatics SpatialAnalysis - 54 -
Aggregation
Set m
ethods
Region
Governmentdistrict
Federal state
County
Parish
District X
State Y
County V
Commune U
Region W
X
W
V
U
Data collection and management
Commune codesin Germany (8 digits)
0 8 1 1 8 0 0 1= Affalterbach
Fede
ral s
tate
Gov
ernm
.dis
trict
Reg
ion
Cou
nty
Par
ish
Agg
rega
tion
Prof. Dr.-Ing. Ralf Bill GI Basics
Spatial Analysis 28
Rostock University, Chair for Geodesy and Geoinformatics SpatialAnalysis - 55 -
Statistic methods
• Mathematical foundations• Describing Statistics• Analytical Statistics• Univariate -, bivariate and multivariate Statistics
• Advanced geostatistical methods• Interpolation methods• Variogram and Kriging
Statisticalm
ethods
Rostock University, Chair for Geodesy and Geoinformatics SpatialAnalysis - 56 -
Describing statistics/diagrams
Project Part Nodes Lines Curves Arcs Circles Areas
ATKIS 1 3380 4016 795 12 0 1383Hochheim 1 3090 3671 685 0 0 1208
45004000350030002500200015001000500
0
Number of geometric primitives:
Nodes Lines Curves Arcs Circles Areas
Statisticalm
ethods
Prof. Dr.-Ing. Ralf Bill GI Basics
Spatial Analysis 29
Rostock University, Chair for Geodesy and Geoinformatics SpatialAnalysis - 57 -
Analytical statistics and testing
• Univariate statistics• Average, standard deviation …
• Bivariate statistics• Correlation, covariance• Regression• Cross-correlation
• Multivariate statistics• Cluster analysis• Factor analysis• Multivariate regression• …
Statisticalm
ethods
Rostock University, Chair for Geodesy and Geoinformatics SpatialAnalysis - 58 -
Interpolation
• in raster
• in triangle
• in lines
525
3 (x,y,z)
1 (x,y,z)2 (x,y,z)
P (x,y,?)
1 2
3 4
P (x,y,?)
1
2
(x,y,z)(x,y,z)
(x,y,z)
(x,y,z)
(x,y,z)
(x,y,z)
P (x,y,?)
Dig
ital T
erra
in M
odel
s
Statisticalm
ethods
Prof. Dr.-Ing. Ralf Bill GI Basics
Spatial Analysis 30
Rostock University, Chair for Geodesy and Geoinformatics SpatialAnalysis - 59 -
Interpolation approaches
01 5 10 15
10
5
1
Linear interpolation
01 5 10 15
10
5
1
Polynomial interpolation
01 5 10 15
5
1
10
Compound cubic poly.
01 5 10 15
10
5
1
Akima interpolation
Statisticalm
ethods
Rostock University, Chair for Geodesy and Geoinformatics SpatialAnalysis - 60 -
Interpolation/Approximation of surfaces
• TIN-Interpolation• Interpolation with area summation• Interpolation with minimum least squares methods• Piece wise linear polynomes• Polynom interpolation• Kriging
Nearest neighbour
Minimal curvature
Inverse Distance
Spline
Polynomregression
Area-Summation
Kriging
TIN-Interpolation
Statisticalm
ethods
Prof. Dr.-Ing. Ralf Bill GI Basics
Spatial Analysis 31
Rostock University, Chair for Geodesy and Geoinformatics SpatialAnalysis - 61 -
Example: statistical methods in DTM
Dots: digitised elevation samples (topographic map) Statisticalm
ethods
Rostock University, Chair for Geodesy and Geoinformatics SpatialAnalysis - 62 -
Interpolation – an example
X Y Z7 6 85 1 21 1 24 3 20 4 24 5 17 3 42 6 106 3 43 3 11 3 3
Statisticalm
ethods
Prof. Dr.-Ing. Ralf Bill GI Basics
Spatial Analysis 32
Rostock University, Chair for Geodesy and Geoinformatics SpatialAnalysis - 63 -
Triangle interpolation
• natural coordinate system • Interpolation approach
y
x
v
u
12
33 (0,1)
Arbitrarycoordinate system triangular coordinates
Natural
1 (0,0) 2 (1,0)
1 2
3
L = 0 3
L = 1/33
L = 2/33
L = 13
L=
2/3
L=1/3
L=0
L=1
11 1 1
L= 2/
32
L= 1/
32
11
L= 0
2
L= 12
x y
z
1
2
3
x y
z
1
2
3
P P
a. linear b. cubic
Rostock University, Chair for Geodesy and Geoinformatics SpatialAnalysis - 64 -
Triangle interpolation
• Interpolation approach
• Problem: triangulationz=0 z=10
z=5
z=0
2.5
5.07.5
5.02.5
7.5
CubicInterpolation
z=0 z=10
z=5
z=0
2.5
5.0
7.5
LinearInterpolation
2.5
7.5
5.0
z=0 z=10
z=5
z=0
2.5
LinearInterpolation
z=0 z=10
z=5
z=0
2.5
5.0
7.5
LinearInterpolation
2.5
7.5
5.0
0.02.55.0
7.5
Statisticalm
ethods
Prof. Dr.-Ing. Ralf Bill GI Basics
Spatial Analysis 33
Rostock University, Chair for Geodesy and Geoinformatics SpatialAnalysis - 65 -
Triangle interpolation - linearS
tatisticalmethods
Rostock University, Chair for Geodesy and Geoinformatics SpatialAnalysis - 66 -
Interpolation/approximation in grids/raster
• Interpolation by area summation• Interpolation by least squares method• Piecewise linear polynomes• Polynom interpolation• Kriging
Nearest neighbour
Minimal curvature
Inverse Distance
Spline
Polynom regression
Area-Summation
Kriging
Other methods
Statisticalm
ethods
Prof. Dr.-Ing. Ralf Bill GI Basics
Spatial Analysis 34
Rostock University, Chair for Geodesy and Geoinformatics SpatialAnalysis - 67 -
Interpolation - nearest neighbour
• Transferring the z-component from nearest neighbour• Assumes a sufficiently dense distribution of points
Statisticalm
ethods
Rostock University, Chair for Geodesy and Geoinformatics SpatialAnalysis - 68 -
Interpolation - minimum curvature
• Application especially in geo-sciences• Thin deformable plate through all points• Smooth surface• Iterative solution of an equation system
Statisticalm
ethods
Prof. Dr.-Ing. Ralf Bill GI Basics
Spatial Analysis 35
Rostock University, Chair for Geodesy and Geoinformatics SpatialAnalysis - 69 -
Interpolation - inverse distance weightingS
tatisticalmethods
Weight: w(di) = 1/dip
Height: h = Σi=1,n hi*wi / Σ widi = distance to point ip = power (1, 2, .., inf)hi = height of point in = # of neighboring points
Rostock University, Chair for Geodesy and Geoinformatics SpatialAnalysis - 70 -
Approximation - polynomial regression
Statisticalm
ethods
Prof. Dr.-Ing. Ralf Bill GI Basics
Spatial Analysis 36
Rostock University, Chair for Geodesy and Geoinformatics SpatialAnalysis - 71 -
Multilog
0,8
1
1,2
1,4
1,6
1,8
2
0 0,2 0,4 0,6 0,8 1 1,2
Area summation - multilogarithmic kernelS
tatisticalmethods
Rostock University, Chair for Geodesy and Geoinformatics SpatialAnalysis - 72 -
Thin Plate Spline
0,6
0,8
1
1,2
1,4
1,6
1,8
2
0 0,2 0,4 0,6 0,8 1 1,2
Area summation - thin plate spline as kernel
Statisticalm
ethods
Prof. Dr.-Ing. Ralf Bill GI Basics
Spatial Analysis 37
Rostock University, Chair for Geodesy and Geoinformatics SpatialAnalysis - 73 -
Area summation - cubic splines as kernel
Natural Cubic Spline
0,8
11,2
1,4
1,6
1,82
2,2
2,42,6
2,8
33,2
3,4
0 0,2 0,4 0,6 0,8 1 1,2
Statisticalm
ethods
Rostock University, Chair for Geodesy and Geoinformatics SpatialAnalysis - 74 -
Area summation - multiquadratic kernel
MultiquadricInvers Multiquadric
0,6
0,8
1
1,2
1,4
1,6
0 0,2 0,4 0,6 0,8 1 1,2
Statisticalm
ethods
Prof. Dr.-Ing. Ralf Bill GI Basics
Spatial Analysis 38
Rostock University, Chair for Geodesy and Geoinformatics SpatialAnalysis - 75 -
Splines - cubic polynomesS
tatisticalmethods
Rostock University, Chair for Geodesy and Geoinformatics SpatialAnalysis - 76 -
Geostatistics: Variogram I
Assumption: The spatial variability of a random sample Z can be explained as a sum of 3 components:
Z(x) = m(x) + e'(x) + e''(x)
with: m(x) = trend, e‘(x) = random component, e‘‘(x) = random noise/error
Variogram determines the influenceof each point on a random variable
g(h)=1/(2n) Σ(z(xi)-z(xi+h))²
Analogous: Correlation functions
C
C a
h
g(h)
0
1
a = rangec0= nugget (noise)c1= sill (maximum variance)
Statisticalm
ethods
Prof. Dr.-Ing. Ralf Bill GI Basics
Spatial Analysis 39
Rostock University, Chair for Geodesy and Geoinformatics SpatialAnalysis - 77 -
Geostatistics: Variogram II
• Typical estimation function for Variogram
Linear Regression: γ(h) = c0 + b hSpheric Model:
γ(h) = c0 + c1{3h/2a - 0.5 (h/a)³) für 0 < h < aγ(h) = c0 + c1 für h >= a
Gauss Model: γ(h) = c0 + c1 (1 - exp(-h/a)²)
200
100
g(h)
00 10 20 30 40 50 h
g(h)=13,16+4,15h
Example:Variogram estimationwith linear model
Statisticalm
ethods
Rostock University, Chair for Geodesy and Geoinformatics SpatialAnalysis - 78 -
Geostatistics: Kriging I
• Kriging is an exact interpolator. Individual samples are used with a range-dependent weight derived from the variogram.
• Example: 5 Samples with measured values (3,4,2,4,6) and distance between each others and to target point 0.
1 2 3 4 5 01 0.0 5.0 9.8 5.0 3.2 4.32 5.0 0.0 6.3 3.6 4.4 2.93 9.8 6.3 0.0 5.0 7.2 5.54 5.0 3.6 5.0 0.0 2.3 1.05 3.2 4.4 7.2 2.3 0.0 2.0
• Function shall be a spheric model with c0=2.5, c1=7.5 and a=10.0.
Statisticalm
ethods
Prof. Dr.-Ing. Ralf Bill GI Basics
Spatial Analysis 40
Rostock University, Chair for Geodesy and Geoinformatics SpatialAnalysis - 79 -
• To be solved: A-1b =
1 2 3 4 5 1 2.500 7.656 9.996 7.656 5.977 1.000 7.039 0.0189 2 ... 2.500 8.650 6.375 7.131 1.000 5.671 0.17623 ... ... 2.500 7.656 9.200 1.000 b = 8.064 = -0.01094 ... ... ... 2.500 5.401 1.000 3.621 0.62125 ... A ... ... 2.500 1.000 4.720 0.1945
... ... ... ... ... 0.000 1.000 -0.1676
• Interpolation of the target point 0 and variance according to
z(x0) = Σ λi z(xi) = 0.0189*3+0.1762*4-0.0109*2+0.6212*4+0.1945*6 = 4.392
σ² = Σ λi bi + h = 0. 0189*7.039+0.1762*5.671-0.0109*8.064+ 0.6212*3.621+0.1945*4.720-0.1676 = 4.044
Geostatistics: Kriging II
λh
λh
Statisticalm
ethods
Rostock University, Chair for Geodesy and Geoinformatics SpatialAnalysis - 80 -
Kriging with linear variogram asumption
Statisticalm
ethods
Prof. Dr.-Ing. Ralf Bill GI Basics
Spatial Analysis 41
Rostock University, Chair for Geodesy and Geoinformatics SpatialAnalysis - 81 -
Comparing quality of interpolation methods
Area: ca. 63haHeight difference: 60mCaptured with: DGPS – 850 pointsMeasuring time: ca. 14 hours
Statisticalm
ethods
Rostock University, Chair for Geodesy and Geoinformatics SpatialAnalysis - 82 -
Quality comparison: Computing time
1
:
5
:
20
Statisticalm
ethods
Prof. Dr.-Ing. Ralf Bill GI Basics
Spatial Analysis 42
Rostock University, Chair for Geodesy and Geoinformatics SpatialAnalysis - 83 -
Quality comparison: Contour line digitising vs. DGPS
Mean terrainslope: 7.2°
Standard deviationmeasured:sG = 1.88mallowed ZIR10: sG = 2.10m
Statisticalm
ethods
Rostock University, Chair for Geodesy and Geoinformatics SpatialAnalysis - 84 -
Quality comparison: Standard deviation (m)based on 80% of points, 20% true error points
0.33 1.49 3.17
0.22 0.69 1.81
0.29 0.77 1.87
1.49 3.17
0.69 1.81
0.77 1.87
Statisticalm
ethods
Prof. Dr.-Ing. Ralf Bill GI Basics
Spatial Analysis 43
Rostock University, Chair for Geodesy and Geoinformatics SpatialAnalysis - 85 -
Models
• Point models (interpolation..)• Line models (net flow calculations..)• Area models (dispersion..)• Simulation• others
Modellling
Rostock University, Chair for Geodesy and Geoinformatics SpatialAnalysis - 86 -
Classification of models
Models
Physicalmodels
Mathematicalmodels
Electricalmodels
Analyticalmodels (flows)
Numericalmodels(finite elements)
Deterministicmodels
Stochasticmodels
Source: G. Teutsch, 1992
Modellling
Prof. Dr.-Ing. Ralf Bill GI Basics
Spatial Analysis 44
Rostock University, Chair for Geodesy and Geoinformatics SpatialAnalysis - 87 -
Geographic models I: Stochastic approaches
• Behaviour of geographic systems is determined to a considerable extent by random processes. For such systems initial hypothesis are defined by probability theory.
Spatial probability models
Geographic decision support
models
- spatial pattern of factory sites- correlation of sales and number of employees,
literacy and social status- autocorrelation between voters of special parties
- Decision about areas for cultivating grain
Modellling
Rostock University, Chair for Geodesy and Geoinformatics SpatialAnalysis - 88 -
Geographic models II: Deterministic approaches
• Behaviour of geographic systems is determined by (pseudo-)physical laws and thus can be predicted exactly.
Cascading models
Space-time-models
- Population migration- Ecosystem stability
- Temperature distribution in soil profiles- Water flow in soil- Heating-up of cities
Models for spatial interaction
Models forspatial assignment
(also called gravitational models)- Movement of consumer capital between regions- Migration of employees (residence to work)(also called transport models)
- Consumers to vendors- Pupils to schools
Modellling
Prof. Dr.-Ing. Ralf Bill GI Basics
Spatial Analysis 45
Rostock University, Chair for Geodesy and Geoinformatics SpatialAnalysis - 89 -
Cartographic modeling
• C.D. Tomlin (1983), (1990), MAP (Map Analysis Package)• Goal: Division of workflow into parts that can be combined and a
definition of a map algebra to process that workflow.• Terms:
Carto-graph.Model
Map-sheet
Map-sheet
Map-sheet
Title
Orientation
Zone
Zone
Zone
Mark
Value
Position
Position
Position
Column-coor-dinate
Row-coor-dinate
Scale
Modellling
Rostock University, Chair for Geodesy and Geoinformatics SpatialAnalysis - 90 -
Workflow in cartographic modeling terms
- Data inter-pretation :
Sheet
Operation
Input Processing Output
SheetSheet
Sheet
- Procedure : I
H
F
C
G
E
A
AlgebraicExpression :
(F+C+(E/A)) 2
- Data interpretationoperations :
1 = f (Position)2 = f (Neighbourhood)3 = f (Zone)
1 2 3
Modellling
Prof. Dr.-Ing. Ralf Bill GI Basics
Spatial Analysis 46
Rostock University, Chair for Geodesy and Geoinformatics SpatialAnalysis - 91 -
Example: Best location for a sports field
• Conditions for a site suitable for a sports field:
• A: slope < 7 %.• B: area > 40.000 sqm.• C: outside residential area (> 50m).• D: traffic infrastructure (< 50m away from existing road
network/public transport).
Modellling
Rostock University, Chair for Geodesy and Geoinformatics SpatialAnalysis - 92 -
Solution: Sports field
A B
C D
E = A AND B AND C AND D
E
Set theory andcartographic model
Modellling
SettlementareasLand use Selection
occupiedBuffer50m outer
Bufferedsettlementareas
DTM Slope Slopemap
Selection< 7 % A
NOT
C
Road net Buffer bothsides 50 m D
Polygon overlay
Recommareas
Area size> 40000 qm
E=Potent.candidates
B
Prof. Dr.-Ing. Ralf Bill GI Basics
Spatial Analysis 47
Rostock University, Chair for Geodesy and Geoinformatics SpatialAnalysis - 93 -
Soil erosion model
R = rain factor = f(precipitation)
K = Erodability of soil = f(soilparticle distribution); soil map
L = slope length factor = f(plot size)
S = slope factor = f(slope) from DTM
C = land use factor = f(crop rotation)
P = Erosion protection
A = mean annual erosion [t/ha]
Universal Soil Loss Equation A = R*K*L*S*C*Pfrom K. Kraus (1991) after Wischmeier/Smith (1978)
Modellling
R
K
A
*
=
L*
S*
C*
P*
Rostock University, Chair for Geodesy and Geoinformatics SpatialAnalysis - 94 -
Erosion model: cascading
Gradient = Δz/Δd mit Δd = Δx+Δy (City-Block-Distanz)
78 72 6974 67 5669 63 44
5.5 5 17 X -111 4 -11.5
78 72 69 71 58 49
74 67 56 49 46 50
69 63 44 37 38 48
64 58 56 29 31 34
68 61 47 21 18 19
74 60 34 12 10 12
z.B.
Modellling
Prof. Dr.-Ing. Ralf Bill GI Basics
Spatial Analysis 48
Rostock University, Chair for Geodesy and Geoinformatics SpatialAnalysis - 95 -
Cost functions for utility planning
Approach:Raster data
Problem: Cost function
Cost efficient path Cost surface
Modellling
New fabrique
Existing cables
Agriculture 1Street 1Uncultivated land 1Coniferous forest 4Decidous forest 5Water 1000Settlement 1000
Weight
Rostock University, Chair for Geodesy and Geoinformatics SpatialAnalysis - 96 -
Status of GIS data analysis
• Geometric operations usually realised• Polygon overlay is a basic function• Topologic operations rather restricted• Set methods such as sort, search, query etc. realised• Simple descriptive statistics realised, interpolations for DTM,
geostatistics rare• Models usually externally realised for special applications
Summ
ary
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