urp 3182 l-16 terrain analysis

7/24/2019 URP 3182 L-16 Terrain Analysis

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Lecture 16: Terrain Analysis

URP 3182 GIS and Remote Sensing Studio

1

October 05, 2015

Course Teacher:Md. Esraz-Ul-Zannat

Assistant ProfessorMd. Mokhlesur Rahman,Lecturer

Department of Urban and Regional PlanningKhulna University of Engineering & Technology



Acknowledgement

These slides are aggregations for betterunderstanding of GIS. I acknowledge the

contribution of all the authors and photographers, power point slides from where I tried to accumulatethe info and used for better presentation.

2



Outline

ArcGIS and 3D AnalysisConcept of 3D GIS and 3D Data ModelBasic Methods for Representing a SurfaceSpatial Interpolation

Terrain/Surface Analysis

3



There are 5 basic ArcGIS desktopmodules. Each module contains adifferent methods of dealing withyour GIS data. Those modules are:

ArcView

ArcEditor

ArcInfo

ArcMap ArcCatalog ArcToolbox ArcScene ArcGlobe

1 2 3 4 5

1

3

2

4

5

ARC GIS M ODULES

4



Basic, Standard, and Advanced (formerly ArcView, ArcEditor, andArcInfo) are licensing levels for ArcGIS for Desktop applications.

Basic provides data visualization, query, analysis, and integrationcapabilities along with the ability to create and edit simplegeographic features.

Standard includes all the functionality of Basic and adds a

comprehensive set of tools to create, edit, and ensure the qualityof your data.

Advanced includes all the functionality of Standard and addsadvanced spatial analysis, data manipulation, and high-end

cartography tools.ArcMap and ArcCatalog are the core applications delivered with alllicensing levels of ArcGIS for Desktop; ArcScene and ArcGlobe arepart of the ArcGIS 3D Analyst extension.

ARC GIS FOR DESKTOP B ASIC , S TANDARD , AND A DVANCED

5



a

b

c

Question:What is ArcView?

An ArcMap document

An application to view data on the internet

A licence level within ArcGIS desktop

Before going to the next slide

Skip

PLEASE SOLVE THE PROBLEM

Q-1

6

d A document of ArcMap



SOLUTION

Q-1

7

Question:What is ArcView?

c A licence level within ArcGIS desktop



MilitaryAnalyst

ImageAnalysis

Schematics

ArcScan

Many Specialist Tools

Integrated intoCommon Framework

StreetMap

TrackingAnalyst

ArcGISDesktop

GeostatisticalAnalyst

BusinessAnalyst

ArcPress

SpatialAnalystPublisherSurveyAnalyst

Maplex

3D Analyst

3 rd PartyExtensions

ARC GIS EXTENSIONS

8



Extensions add more capabilities to ArcGIS for Desktop withextensions.

Analysis Key BenefitsNote: Unless noted, extensions can be used with ArcGIS forDesktop Basic, Standard, and Advanced.*Requires ArcGIS for Desktop Advanced**Requires ArcGIS for Desktop Standard or Advanced

ArcGIS 3D Analyst Analyze your data in a realisticperspective.

ArcGIS Geostatistical Analyst Use advanced statistical toolsto investigate your data.

ArcGIS Network Analyst Perform sophisticated routing,closest facility, and servicearea analysis.

ArcGIS Schematics Represent and understand yournetworks to shorten decisioncycles.

ArcGIS Spatial Analyst Derive answers from your datausing advanced spatialanalysis.

ArcGIS Tracking Analyst Reveal and analyze time-basedpatterns and trends in yourdata.

Business Analy st OnlineReports Add-In

Directly access demographicreports and data from Business

Analyst Online (BAO) for tradeareas and sites created in thedesktop.

Productivity Key Benefits

ArcGIS Data Interoperability Eliminate barriers to data useand distribution.

ArcGIS Data Reviewer Automate, simplify, andimprove data quality controlmanagement.

ArcGIS Publisher Freely share your maps anddata with a wide range ofusers.

ArcGIS Workflow Manager ** Better manage GIS tasks andresources.

ArcScan for ArcGIS (included with ArcGIS forDesktop Standard and

Advanced)

Increase efficiency and speedup raster-to-vector dataconversion time.

Maplex for ArcGIS (included with ArcGIS forDesktop Advanced)

Create maps that communicatemore clearly with automaticallypositioned text and labels.

Solution Based Key Benefits

ArcGIS Defense Solutions (includes ArcGIS Military

Analyst, Grid Manager, andMOLE)

Create workflows, processes,and symbology to supportdefense and intelligenceplanning.

Esri Aeronautical Solution * Use the full power of GIS to

efficiently manageaeronautical information.

Esri Defense Mapping * Efficiently manage defensespecification-compliantproducts.

Esri Nautical Solution * A GIS-based platform fornautical data and chartproduction.

Esri Production Mapping ** Standardize and optimize yourGIS production.

Esri Roads and Highways ** Easily manage, visualize, andanalyze transportationnetworks.

ARC GIS EXTENSIONS

9

http://www.esri.com/software/arcgis/extensions/3danalyst/index.html

http://www.esri.com/software/bao-reports/index.html

http://www.esri.com/software/arcgis/extensions/aero-solution/index.html

http://www.esri.com/software/arcgis/extensions/roads-and-highways/index.html



http://www.esri.com/software/arcgis/extensions/production-mapping/index.html

http://www.esri.com/software/arcgis/extensions/nautical-solution/index.html

http://www.esri.com/software/arcgis/extensions/defense-mapping/index.html

http://www.esri.com/software/arcgis/extensions/aero-solution/index.html

http://www.esri.com/software/arcgis/defense-solutions/index.html

http://www.esri.com/software/arcgis/extensions/maplex/index.html



http://www.esri.com/software/arcgis/extensions/arcscan/index.html

http://www.esri.com/software/arcgis/extensions/arcgis-workflow-manager/index.html

http://www.esri.com/software/arcgis/extensions/publisher/index.html

http://www.esri.com/software/arcgis/extensions/arcgis-data-reviewer/index.html

http://www.esri.com/software/arcgis/extensions/datainteroperability/index.html





http://www.esri.com/software/arcgis/extensions/trackinganalyst/index.html

http://www.esri.com/software/arcgis/extensions/spatialanalyst/index.html

http://www.esri.com/software/arcgis/extensions/schematics/index.html

http://www.esri.com/software/arcgis/extensions/networkanalyst/index.html

http://www.esri.com/software/arcgis/extensions/geostatistical/index.html

http://www.esri.com/software/arcgis/extensions/3danalyst/index.html



3D Analyst (ArcMap)

- ArcScene

- ArcGlobe

Spatial Analyst (ArcMap) as well

EXTENSIONS FOR 3D D ATA

10



Interactive 3D and Global viewing

Construction and analysis of 2.5D TIN and raster surfaces

Creation of 3D vector feature

3D animation

Support for textured 3D symbols

3D ANALYST EXTENSION

11



Interactive “Fish tank” view

Good for a small scale range

Best at rendering geometry

ARC S CENE

12

A G



Interactive Global and 3D viewingOptimized to multi-scale viewingGreat for very large raster dataAlso support vector and 3D symbols

ARC GLOBE

13



Integrated raster and vector spatial analysis tools.Extension product that adds functionality to ArcMap,ArcToolbox, and ArcObjectsOver 300 functions and operatorsAnalysis on all raster formatsAnalysis on all vector formatsFull support of selectionsOn the fly projectionsGreat developer tools

S PATIAL ANALYST EXTENSION

14



a

b

c

Question:What are the main two types of data structure used in ArcGIS?

Image and raster

Raster and vector

Image and Shape file


Skip


Q-2

15



SOLUTION

Q-2

16

b Raster and vector

Question:What are the main two types of data structure used in ArcGIS?



Outline



17

W 3D GIS



WHAT IS 3D GIS

The earth is not flat. In the real world, surfaces with the verticaldimension do exist.

The complexity of analyzing three-dimensional data increasesexponentially relative to two-dimensional data. Consequently, this analysis is better performed by morespecialized software. 3D GIS has been created to address and viewsuch data.Increasing speed and computational efficiency have enhancedopportunities for developing the 3D GISExtend capabilities of GIS to build, visualise, and analyse data in 3DPerform interactive perspective viewing and navigation, includingpan and zoom, rotate, tilt, fly-through simulations, and exportutilities for display on the Web

18

W 3D GIS



WHAT IS 3D GISConstruct surface models such as TINs and Raster from any dataExtrude buildings and vector features from a surfaceAerial photographs can be draped onto a 3D model to project amore realistic lookSupply analytical functions to calculate slope, aspect and hillshading to enable the following:

– evaluate the steepest path

– perform visibility analysis – conduct volumetric and cut-fill computations – construct interpolation of surface z-values – create vertical profiles along linear features

19

W 3D GIS



Simulation of complex systems provide understanding on how thesystem operates different perspectives, aided by high qualityvisualization and interactionObservation of system features that would be too small or toolarge to be seen on a normal scale systemAccess to situations that would otherwise be dangerous or tooremote or inaccessible

Enable high degree of interaction which is important to aidunderstandingProvide a sense of immersion of the environment –where the usercan appreciate the scale of change and visualize the impact of abuilding design on the external environment and the inhabitants.

WHY DO WE NEED 3D GIS

20




Allow to export to popular multimedia format such as video (.avi or.mpeg) or VRML (.vrl or .vrml) that provide the following benefits:

- Do not need to know 3D GIS, simply use intuitive and easy to useinterface to operate the 3D model.

– Inherent flexibility/adaptability – these multimedia are 3D cross-platform display and non-browser specific which enable expensivedata to be used more widely

– Fast and slow time simulation – Ability to control timescale byincorporating a sequence of captured events into the keyframes(or snapshots) of the motion video


21

UNDERSTANDING 2D 2 5 D 3D & 4D



Two-dimensional (2D) is based on the Cartesian coordinate (x,y)

system. 2D mapping is limited to representation of data on planarsurfaces.Two and one half-dimensional (2.5D) is a common format used bymost GISs. They are generally for modelling of surfaces or terrainthrough (x,y) and attribute values.Three-dimensional (3D) - The 2D point, line, and polygon vectorrepresentation of objects extends to include a volume element in3D space. More efficient to handle holes or voids withinvolumetric bodies (e.g. caves).

Four-dimensional GIS (volumes over time) is important forshowing processes that occur in nature and through time.

UNDERSTANDING 2D, 2 .5 D, 3D & 4DGEOGRAPHIC FEATURES IN GIS

22

S PATIAL DATA



Spatial Features are two types:a) Discrete Features

These features don’t exist between observationsForm separate entitiesIndividually distinguishableExample: Wells, roads, land use types etc.

b) Continuous FeaturesExists spatially between observationsExample: Precipitation, elevation etc.

S PATIAL DATA

Well Location

Rainfall Map

Elevation 23

3D DATA AND Z VALUE



3D DATA AND Z-VALUE

3D data has a specified z-value,while 2D data does notZ-value can be: elevation, rainfall,temperature, population, ……

S URFACES

Surfaces involve a third 'z' dimension (height/elevation/magnitude,quantity) in addition to x,y planimetric location. Any type ofcontinuous data can be represented as a surface, whether it beground elevation, barometric pressure, rainfall, crop yield, noiselevels, population density, sales intensity, land value, income, crime

rates, etc.24

3D S URFACES



3D S URFACES A 3D surface model is a digital representation of features, eitherreal or hypothetical, in three-dimensional space.Examples of 3D surfaces are a landscape, an urban corridor, gasdeposits under the earth, or a network of well depths to determinewater table depth. A 3D surface is usually derived, or calculated, using speciallydesigned algorithms that sample point, line or polygon data and

convert them into a digital 3D surface. ArcGIS can create and storethree types of surface model: Raster, TIN, and Terrain.

Raster TIN Terrain25

3D S URFACES



3D S URFACES The two main methods of creating surface models areinterpolation and triangulation. There are several interpolation

methods, such as Inverse Distance Weighted , Spline , Kriging,and Natural Neighbors for creating raster and TriangulationMethods for TIN and Terrain.Conversion between Terrain, TIN, and raster surface models arealways possibleRaster, TIN and Terrain surfaces are all types of a functional surface which are actually 2.5D. A functional surface is continuous, and alllocations on the surface may have only one elevation, or z, valueper x, y coordinate. True 3D surfaces are sometimes known as solid

model surfaces, and ArcGIS handles these through multipatchfeatures.In contrast to a functional surface, which has surface continuity,are solid model surfaces, than can model and store true 3D, ormultiple z-values per x, y coordinate. 26

S URFACE CONTINUITY (2 5D VS 3D)

http://webhelp.esri.com/ARCGISDESKTOP/9.2/index.cfm?TopicName=How_IDW_works

http://webhelp.esri.com/ARCGISDESKTOP/9.2/index.cfm?TopicName=How_Spline_works

http://webhelp.esri.com/ARCGISDESKTOP/9.2/index.cfm?TopicName=How_Kriging_works

http://webhelp.esri.com/ARCGISDESKTOP/9.2/index.cfm?TopicName=Natural_Neighbor_%283D_Analyst%29

http://webhelp.esri.com/ARCGISDESKTOP/9.2/index.cfm?TopicName=What_is_a_functional_surface%3F







http://webhelp.esri.com/ARCGISDESKTOP/9.2/index.cfm?TopicName=How_Kriging_works

http://webhelp.esri.com/ARCGISDESKTOP/9.2/index.cfm?TopicName=How_Spline_works








S URFACE CONTINUITY (2.5D VS . 3D)Functional surfaces are considered continuous. This can becontrasted with a discontinuous surface, where different z-values

could be obtained depending on the approach direction. Anexample of a discontinuous surface is a vertical fault displacing thesurface of the earth.

Depending on whether you approach this vertical fault from theright or left along this discontinuous surface, it's possible toobserve different z-values at the same x,y location.

27



a

b

c

Question:Why BTM Projection System is Used in Bangladesh?

To get more accurate and precise projection System

To get rid of the problem of falling two UTM zone

UTM does not provide accurate data for Bangladesh


Skip


Q-3

28

d Bangladesh is far away from the central meridian



SOLUTION

Q-3

29

Question:Why BTM Projection System is Used in Bangladesh?

b To get rid of the problem of falling two UTM zone



Outline



30

3 BASIC METHODS FOR REPRESENTING A S URFACE



DEM (digital elevation model): set of regularly spaced sampled groundpoints in the x and y dimensions (although spacing not necessarily the same ineach) accompanied by an elevation measure (z dimension). The DEM

terminology was introduced by USGS.Two concepts used for determiningelevation at points within the grid cells:

Lattice: each point represents a value on the surface only at the center ofthe grid cell Surface grid considers each sample as a square/rectangular cell with a

constant surface value.TIN (Triangulated Irregular Network) a set of adjacent, non-overlappingtriangles with x, y coordinates and z vertical elevations for their vertices, alongwith topological relationship between the triangles and their adjacentneighbors.

Contour lines: lines of equal elevation, drawn at a given interval (e.g. every 6or 25 feet)

The general term digi ta l te rra in m od el (DTM) may be used to refer to any of theabove surface representations when in digital form.DEM sometimes used synonymously with DTM and DSM.

3 BASIC METHODS FOR REPRESENTING A S URFACE

31

S TORING & CONVERTING S URFACE DATA



3-D surfaces are normally stored in one of two forms within ArcGISas a GRID, which is ArcInfo's general raster formatas a TIN which is a vector format for surfaces

However, when you download data from the Internet, surfacedata may be in other formats, such as

DEM format, as originally developed by USGSSDTS (Spatial Data Transfer Standard) format, which is an FGDC (FederalGeographic Data Committee) standard

E00 which is ESRI’s text formatted for distributing coverages and GRIDS Points and breaklinesConversion to GRID or TIN is generally required for display oranalysis within the ArcGIS system

Generally, ArcToolbox has capabilities for converting these formats to GRIDsor TINs

Contour lines can be stored as vector lines in a coverage,shapefile, or geodatabase,

can only be used for map display but not analysis, so this is not arecommended format for surface storage.

S TORING & CONVERTING S URFACE DATA

32

DIGITAL E LEVATION MODEL



a sampled array of elevations (z)that are at regularly spacedintervals in the x and y directions.

two approaches for determiningthe surface z value of a locationbetween sample points.

In a lattice , each mesh pointrepresents a value on the

surface only at the center ofthe grid cell. The z-value isapproximated by interpolationbetween adjacent samplepoints; it does not imply anarea of constant value.A surface grid considers eachsample as a square cell with aconstant surface value.

Advantages• Simple conceptual model

• Data cheap to obtain• Easy to relate to other

raster data• Irregularly spaced set of

points can be convertedto regular spacing byinterpolation

Disadvantages• Does not conform to

variability of the terrain• Linear features not well

represented

DIGITAL E LEVATION MODEL

33

GRID AS A S TORAGE METHOD



GRIDs are ESRI’s raster data format Use for storing DEMS or other data in raster format

GRID stores data as either :Integer: in which case there is an associated value attribute table (VAT)which contains one record for each different value in the raster (thus thereare normally substantially fewer records in the VAT table than there arecells in the raster); this record stores the value itself, a count of thenumber of cells with that value, and any additional attributes the user

wishes to to attach. Thus, the values could be codes for soil type and theVAT could contain fertility measures, soil name, construction suitabilitycodes, etc. If you select a record in the VAT, all cells with that value willhighlight in the View or Scene.Floating point : (number with a decimal point) in which case there is noVAT table, and simply one decimal value per cell

Integer GRIDS are generally substantially faster to process.

GRID AS A S TORAGE METHOD

34

TRIANGULATED IRREGULAR NETWORK



AdvantagesCan capture significant slopefeatures (ridges, etc)Efficient since require fewtriangles in flat areasEasy for certain analyses:slope, aspect, volume

DisadvantagesAnalysis involvingcomparison with other layersdifficult

a set of adjacent, non-overlapping trianglescomputed from irregularlyspaced points, with x, yhorizontal coordinates andz vertical elevations.

TRIANGULATED IRREGULAR NETWORK

35

A MESH OF TRIANGLES IN 2-D



A MESH OF TRIANGLES IN 2-D

Triangle is the onlypolygon that is always

planar in 3-D

Points Lines Surfaces

36

TIN T RIANGLES IN 3-D



TIN T RIANGLES IN 3-D

(x3, y 3, z 3)

(x1, y 1, z 1)(x2, y 2, z 2)

x

y

z

Projection in (x,y) plane

37

TIN AS A S TORAGE METHOD



TINs

are the most useful method for representing a continuous

surface in a vector GIS system.data sets comprising any combination of contours, breaklinesand point elevations (either DEM or massed points) can becombined as input to create a TIN

TINS are especially useful for analytical purposesGood model for representing surfaces

slope and aspect easily derivedsimplify the calculation of surface area and volume

TIN AS A S TORAGE METHOD

38

DELAUNEY T RIANGULATION



DELAUNEY T RIANGULATION

Developed around 1930 to design the triangles efficiently

Geometrically related to theissen tesselations

Maximize the minimum interior angle of triangles that can beformed

No point lies within the circumcircle of a triangle that is containedin mesh

YesMore uniform representation of terrain No

39

INPUTS FOR C REATING A TIN



INPUTS FOR C REATING A TIN

Mass Points Soft Breaklines Hard Breaklines

• Mass Points define points anywhere on landscape• Hard breaklines define locations of abrupt surface change (e.g.streams, ridges, road kerbs, building footprints, dams)• Soft breaklines are used to ensure that known z values along alinear feature are maintained in the tin.

40

CREATING A TIN



No Breaklines Soft Breaklines Hard Breaklines

The Data

TheTriangulation

TheSurface

3D View

Break lines Linear features which define and control surface behavior in terms

of smoothness and continuity.

CREATING A TIN

41

CONTOUR (ISOLINES ) LINES



CONTOUR (ISOLINES ) LINES Advantages

Familiar to many peopleEasy to obtain mental picture of

surfaceClose lines = steep slopeUphill V = streamDownhill V or bulge = ridgeCircle = hill top or basin

DisadvantagesPoor for computer representation:no formal digital modelMust convert to raster or TIN foranalysisContour generation from point datarequires sophisticated interpolationroutines, often with specializedsoftware such as Surfer fromGolden Software, Inc., or ArcViewSpatial Analyst extensionridge

valley hilltop

Contour lines, or isolines, of

constant elevation at aspecified interval,

42

TERRAINS



43

A terrain dataset is a multiresolution , TIN-based surface built from measurements storedas features in a geodatabase. They're typically made from lidar, sonar, and photogrammetricsources. Terrains reside in the geodatabase, inside feature datasets with the features usedto construct them

Terrains have participating feature classes and rules, similar to topologies. Commonfeature classes that act as data sources for terrains include the following:• Multipoint feature classes of 3D mass points created from a data source such as lidaror sonar• 3D point and line feature classes created on photogrammetric workstations usingstereo imagery• Study area boundaries used to define the bounds of the terrain dataset

T

TERRAINS



T New format for surface data introduced with ArcGIS 9.2Intended to support massive amounts of data from sources such asLIDAR and SONAR

Includes support for the LAS LIDAR data formatSupported only within ArcMAP and ArcGlobe, not ArcSceneSimilar more to a topology

Stored in a geodatabase (personal, file, or SDE) feature dataset

and contain rules as to how feature classes within the featuredataset are used to construct a surfaceData is retrieved and used to build a TIN on-the-fly

See: ArcToolbox/3D Analyst Tools/Terrains 44

VIEWING & P ROCESSING S URFACE DATA



V & P S D To go beyond simple display of rasters or contour lines, you will need to use the 3-D

Analyst or Spatial Analyst extensions.GRIDS can be displayed in ArcMap (and also ArcScene and ArcGlobe)

But Spatial Analyst extension is required to analyze GRIDS

Tools available in ArcToolbox/Spatial Analyst Tools TINS require 3-D Analyst extension to display and analyze

ArcScene is used to display and analyze TINS in Version 8 ArcGlobe partially replaces ArcScene in Version 9

Faster display of large amounts of data, but

will not support subterranean views —use ArcScene for thisArcToolbox/3-D Analyst Tools has TIN analysis toolsNote: certain raster tools from Spatial Analyst are listed here also

Contour lines can be created from a TIN using ArcScene or from a GRID usingSpatial Analyst, or with tools in ArcToolbox (if you have the extensions), andstored as a shapefile , coverage or gdb feature class

See ArcToolbox/Spatial Analyst Tools/Surface/ContourOr ArcToolbox/3D Analyst Tools/Raster Surface/Contour (same tools)

ArcScene and ArcToolbox will convert between all these formatsSee ArcToolbox/3D Analyst/Conversion :

Import from Raster Export to Raster

Import from TIN Export to TIN45



a

bc

Question:Which of the following might be considered as the fourthdimension in GIS??

Location

Space

Time


Skip


Q-4

46

d Scale



SOLUTION

Q-4

47

c Time

Question:Which of the following might be considered as the fourthdimension in GIS??



Outline



48

INTERPOLATION



Critical component for raster surface creationUsed to create GRIDs (ArcGIS format for rasters) which contain

equally spaced cells from irregularly spaced point data.Five methods (each with additional options) are available to dointerpolation

Inverse Distance WeightingSplineTrendNatural NeighborKriging

49



First Law of Geography

• “Everything is related to everything else, but near thingsare more related than distant things. ”

– Waldo Tobler (1970)

• This is the basic premise behind interpolation, andnear points generally receive higher weightsthan far away points

Waldo Tobler

50

http://en.wikipedia.org/wiki/File:Waldotobler1.jpg





What is a spatial interpolation?

Interpolation predicts values for cells in a raster from a limited number of sample datapoints. It can be used to predict unknown values for any geographic point data: elevation,rainfall, chemical concentrations, noise levels, and so on .

In this example the input points happen to fall on cell centers - this is unlikely in practice. Oneproblem with creating rasters by interpolation is that the original information is degraded tosome extent - even when a data point falls within a cell, it is not guaranteed that the cell will

have exactly the same value.Interpolation is based on the assumption that spatially distributed objects are spatiallycorrelated ; in other words, things that are close together tend to have similar characteristics. For instance, if it is raining on one side of the street, you can predict with a high level ofconfidence that it is also raining on the other side of the street. You would be less sure if itwas raining across town and less confident still about the state of the weather in theneighbouring province.

On the left is a point dataset of known values. On theright is a raster interpolated from these points.

Unknown values are predicted with a mathematicalformula that uses the values of nearby known points.

51



Interpolation vs. Extrapolation

• Interpolation is prediction within the range of our data – E.g., having temperature values for a bunch of locations

all throughout PA, predict the temperature values at allother locations within PA

• Note that the methods we are talking about are strictlythose of interpolation , and not extrapolation

• Extrapolation is prediction outside the range of our data – E.g., having temperature values for a bunch of locations

throughout PA, predict the temperature values inKazakhstan

52



What is a spatial interpolation?

Visiting every location in a study area to measure the height, magnitude, or concentration ofa phenomenon is usually difficult or expensive . Instead, dispersed sample input pointlocation s can be selected and a predicted value can be assigned to all other locations. Inputpoints can be either randomly, strategically, or regularly spaced points containing height,concentration, or magnitude measurements .A typical use for point interpolation is to create an elevation surface from a set of sample

measurements. Each point represents a location where the elevation has been measured.The values between these input points are predicted by interpolation.

There are effectively two types of techniques for generating raster surfacesDeterministic Models use a mathematical function to predict unknown values and result inhard classification of the value of features.Statistical Techniques produce confidence limits to the accuracy of a prediction but are more

difficult to execute since more parameters need to be set.

The resulting grid is a prediction ofwhat the elevation is at any locationon the actual surface.

53



Methods of Interpolation

• Deterministic methods – Use mathematical functions to calculate the values at unknown locations based

either on the degree of similarity (e.g. IDW) or the degree of smoothing (e.g. RBF) inrelation with neighboring data points.

– Examples include:• Inverse Distance Weighted (IDW)• Radial Basis Functions (RBF)

• Geostatistical methods – Use both mathematical and statistical methods to predict values at all locations

within region of interest and to provide probabilistic estimates of the quality of theinterpolation based on the spatial autocorrelation among data points.

• Include a deterministic component and errors (uncertainty of prediction) – Examples include:

• Kriging• Co-Kriging

54



Deterministic Models

Deterministic models include Inverse Distance Weighted (IDW), Natural Neighbours, andSpline. You can also develop a trend surface using polynomial functions to create acustomized and highly accurate surface.

In contrast to Deterministic Models are Statistical methods and are based on statisticalmodels that include autocorrelation (statistical relationships among the measured points).Not only do these techniques have the capability of producing a prediction surface, but theycan also provide some measure of the certainty or accuracy of the predictions. Statisticalmodels include Ordinary Kriging, Simple Kriging, and Universal Kriging.

55



Inverse Distance Weighting (IDW)

The Inverse Distance Weighting interpolator assumes that each input point has a localinfluence that diminishes with distance . It weights the points closer to the processing cellgreater than those further away. A specified number of points, or all points within a specifiedradius can be used to determine the output value of each location. Use of this methodassumes the variable being mapped decreases in influence with distance from its sampledlocation.

The Inverse Distance Weighting (IDW) algorithm effectively is a moving average interpolatorthat is usually applied to highly variable data. For certain data types it is possible to return tothe collection site and record a new value that is statistically different from the originalreading but within the general trend for the area. Examples of this type of data include soilchemistry results, environmental monitoring data, and consumer behaviour observations. Itis not desirable to honour local high/low values but rather to look at a moving average ofnearby data points and estimate the local trends.

The interpolated surface, estimatedusing a moving average technique, isless than the local maximum valueand greater than the local minimumvalue. 56



More on the Inverse Distance Weighting (IDW)

he IDW technique calculates a value for each grid node by examining surrounding data pointsthat lie within a user-defined search radius. Some or all of the data points can be used in theinterpolation process. The node value is calculated by averaging the weighted sum of all thepoints. Data points that lie progressively farther from the node influence the computed valuefar less than those lying closer to the node

A radius is generated around each grid nodefrom which data points are selected to beused in the calculation. Options to control theuse of IDW includePowerSearch RadiusFixed search radius

Variable Search RadiusBarrier

The exponent of distance: Controls the significance of surrounding points on theinterpolated value. A higher power results in less influence from distant points. It can be anyreal number greater than 0, but the most reasonable results will be obtained using valuesfrom 0.5 to 3. The default is 2.

57



Examples of IDW with Different q’s

• Larger q ’s (i.e., power to which distance is raised) yield smoother surfaces • Food for thought: What happens when q is set to 0?

Gold concentrations at locations inwestern PA

q = 1

q=2

q=3

q=10

The Geostatistical Analyst of ArcGIS is ableto tell you the optimal value of q by seeingwhich one yields the minimum RMSE. (Here,it is q=1).

58



Inverse distance weighting (IDW) is a method for multivariate interpolation, a process ofassigning values to unknown points by using values from usually scattered set of knownpoints. Here, the value at the unknown point is a weighted sum of the values of N knownpoints.A general form of finding an interpolated value u at a given point x based on samples u i =u(x i) for i = 0,1,...,N using IDW is an interpolating function:


is a simple IDW weighting function, as defined by Shepard, [1] x denotes an interpolated(arbitrary) point, xi is an interpolating (known) point, d is a given distance (metric operator)from the known point xi to the unknown point x, N is the total number of known pointsused in interpolation and p is a positive real number, called the power parameter.

59



Here weight decreases as distance increases from the interpolated points. Greater valuesof p assign greater influence to values closest to the interpolated point. For 0 < p < 1 u(x)has smooth peaks over the interpolated points xi , while as p > 1 the peaks become sharp.The choice of value for p is therefore a function of the degree of smoothing desired in theinterpolation, the density and distribution of samples being interpolated, and themaximum distance over which an individual sample is allowed to influence the surrounding

ones. For two dimensions, power parameters, cause the interpolated values to bedominated by points far away, since with a density ρ of data points and neighboring pointsbetween distances r0 to R, the summed weight is approximately


60



Natural Neighbourhood Interpolation

The Natural Neighbour method is a geometric estimation technique that uses naturalneighbourhood regions generated around each point in the data set.Like IDW, this interpolation method is a weighted-average interpolation method . However,instead of finding an interpolated point's value using all of the input points weighted bytheir distance , Natural Neighbors interpolation creates a Delauney Triangulation of theinput points and selects the closest nodes that form a convex hull around the interpolationpoint , then weights their values by proportionate area. This method is most appropriatewhere sample data points are distributed with uneven density . It is a goodgeneral-purpose interpolation technique and has the advantage that you do not have tospecify parameters such as radius, number of neighbours or weights.This technique is designed to honour local minimum and maximum values in the point fileand can be set to limit overshoots of local high values and undershoots of local low values .The method thereby allows the creation of accurate surface models from data sets that are

very sparsely distributed or very linear in spatial distribution .

In the natural neighbourhood theinterpolated surface is tightlycontrolled by the original data points

61



How Natural Neighbourhood Interpolation Works?

Very simply, the Natural Neighbour method makes use of an area-stealing, or area-weighting, technique to determine a new value for every grid node . As shown belownatural neighbourhood region is first generated for each data point. Then, at every node inthe new grid, a new natural neighbourhood region is generated that effectively overliesvarious portions of the surrounding natural neighbour regions defining each point. The newgrid value is calculated as the average of the surrounding point values proportionallyweighted according the intersecting area of each point .

A display of the natural neighbourhoodregions around the point file as well as

the created around a grid node.

62



Variations of Natural Neighbourhood Interpolation

Three variations to this basic technique are incorporated into the Natural Neighbourinterpolator are usually available.

3) Light-grey line represents a Slope-based Solution where the grid value is determined byaveraging the extrapolated slope of each surrounding natural neighbour region and areaweighted as in the Linear Solution. By examining the adjacent points, a determination ismade as to whether that point represents a local maximum or minimum value. If such is thecase, a slope value of zero is assigned to that value and the surface will therefore honourthat point by neither overshooting nor undershooting it.

A graph showing the three variations of the NaturalNeighbour Interpolator.1) Black line represents a Constant Value interpolatorin which each grid node takes on the value of theunderlying natural neighbourhood region.2) Mid-grey line represents a Linear Solution, wherethe grid value is determined by averaging the pointvalues associated with surrounding naturalneighbour regions and weighted according to thearea that is encompassed by a temporary naturalregion generated around the grid cell

63



Natural Neighborhood InterpolationThe method is based on Voronoi tessellation of a discrete set of spatial points. This hasadvantages over simpler methods of interpolation, such as nearest neighbor, in that it

provides a more smooth approximation to the underlying "true" function.The basic equation in 2D is:

Natural neighbor interpolation.The colored circles. whichrepresent the interpolatingweights, wi, are generated usingthe ratio of the shaded area tothat of the cell area of thesurrounding points. The shadedarea is due to the insertion of thepoint to be interpolated into the

Voronoi tessellation

where G(x,y) is the estimate at (x,y), wi are the weights andf(xi,yi) are the known data at (xi,yi). The natural neighbourmethod proposes a measure for the computation of theweights, and the selection of the interpolating neighbors

The natural neighbor method utilizes the change tothe Voronoi tessellation to compute weights.

The weights, wi, are by utilization of the area"stolen" from the surrounding points when insertinga new point into the tessellation. Each weight maybe computed by dividing the section of the newtessellated region that lies within the tessellatedregion of each original neighboring tessellatedpolygon.

64



Spline Interpolation

Spline estimates values using a mathematical function that minimizes overall surfacecurvature , resulting in a smooth surface that passes exactly through the input points.Conceptually, it is analogous to bending a sheet of rubber to pass through known points whileminimizing the total curvature of the surface . It fits a mathematical function to a specifiednumber of nearest input points while passing through the sample points. This method is bestfor gently varying surfaces, such as elevation, water table heights, or pollutionconcentrations . There are two spline methods

Spline the Regularized Method

Spline the Tension Method

The Spline tool uses an interpolation method that estimates values using a mathematicalfunction that minimizes overall surface curvature, resulting in a smooth surface thatpasses exactly through the input points.

65



The basic form of the minimum curvature Spline interpolation imposes the following twoconditions on the interpolant:

The surface must pass exactly through the data points.The surface must have minimum curvature —the cumulative sum of the squares of the

second derivative terms of the surface taken over each point on the surface must be aminimum.The basic minimum curvature technique is also referred to as thin plate interpolation. Itensures a smooth (continuous and differentiable) surface, together with continuous first-derivative surfaces. Rapid changes in gradient or slope (the first derivative) can occur inthe vicinity of the data points; hence, this model is not suitable for estimating secondderivative (curvature).The basic interpolation technique can be applied by using a value of zero for the Weight

argument to the Spline tool.


66



The most commonly used splines are cubic spline, i.e., of order 3 —in particular, cubic B-spline and cubic Bézier spline. They are common, in particular, in spline interpolationsimulating the function of flat splines.

A quadratic spline composed of six polynomialsegments. Between point 0 and point 1 a straight line.Between point 1 and point 2 a parabola with second

derivative = 4. Between point 2 and point 3 aparabola with second derivative = -2. Between point 3and point 4 a straight line. Between point 4 and point5 a parabola with second derivative = 6. Betweenpoint 5 and point 6 a straight line.

A cubic spline composed of seven polynomialsegments. This shape used as pulse in the articlePulse (physics)


67



Spline equationThe algorithm used for the Spline tool uses the following formula for the surfaceinterpolation:


where: j = 1, 2, ..., NN is the number of points.λj are coefficients found by the solution of a system of linear equations.rj is the distance from the point (x,y) to the jth point.T(x,y) and R(r) are defined differently, depending on the selected option.

For computational purposes, the entire space of the output raster is divided into blocks or

regions equal in size. The number of regions in x and in y directions are the same, and theyare rectangular in shape. The number of regions is determined by dividing the total amountof points in the input point dataset by the value specified for the number of points. For dataless uniformly distributed, the regions may contain a significantly different number of points,with the value for the number of points being only the rough average. If in any region, thenumber of points is smaller than eight, the region is expanded until it contains a minimum ofeight points. 68




The regularized method creates a smooth, gradually changing surface with values that may lieoutside the sample data range.

Applying the regularized Spline methods allows a surface to over- and under-shoot thesample data rangeUsing a regularized spline the higher the weights, the smoother the surface. Weightsbetween 0 to 5 are the most suitable with typical values of 0, 0.001, 0.01, 0.1, and 0.5.

69




T(x,y) = a1 + a2x + a3y

• where:ai are coefficients found by the solution of a system of linear equations.And,

• where:r is the distance between the point and the sample.

is the Weight parameter.Ko is the modified Bessel function.c is a constant equal to 0.577215.

70




The regularized method creates a smooth, gradually changing surface with values that may lieoutside the sample data range.

The Tension method tunes the stiffness of the surface according to the character of themodelled phenomenon.

It creates a less-smooth surface with values more closely constrained by the sample datarange. For Tension, the higher the weight the coarser the generated surface. The valuesentered have to equal or greater than zero. The typical values are 0, 1, 5, and 10.Both the Regularized and Tension spline methods can be further refined by defining thenumber of points used in the calculation of each interpolated cell. The more input points youspecify, the more each cell is influenced by distant points and the smoother the resultingsurface.

71




T(x,y) = a1

• where:a1 is a coefficient found by the solution of a system of linear equations.And,

• where:r is the distance between the point and the sample.φ 2 is the Weight parameter.Ko is the modified Bessel function.c is a constant equal to 0.577215.

72



Before we do any Geostatistics…

• … Let’s review some basic statistical topics: – Normality – Variance and Standard Deviations

– Covariance and Correlation• … and then briefly re -examine the underlying premise of

most spatial statistical analyses: – Autocorrelation

73



Normality

• A lot of statistical tests – including many in geostatistics – rely on theassumption that the data are normally distributed

• When this assumption does not hold, the results are often inaccurate

N=140

74



The Mean and the Variance

• The mean (average) of a variable is also known as the expected value – Usually denoted by the Greek letter μ – As an aside, for a normally distributed variable, the mean is equal

to the median• The variance is a measure of dispersion of a variable

– Calculated as the average squared distance of the possible valuesof the variable from mean.

– Standard deviation is the square root of the variance – Standard deviation is generally denoted by the Greek letter σ, and

variance is therefore denoted by

75



Standard deviation is a widely used measure of variability or diversity usedin statistics and probability theory . It shows how much variation or " dispersion " there isfrom the average ( mean , or expected value). A low standard deviation indicates that thedata points tend to be very close to the mean , whereas high standard deviation indicatesthat the data points are spread out over a large range of values.

76

Standard Deviation

Dark blue is less than one standard deviation from the mean. Forthe normal distribution , this accounts for 68.27 percent of the set; whiletwo standard deviations from the mean (medium and dark blue) accountfor 95.45 percent; three standard deviations (light, medium, and darkblue) account for 99.73 percent; and four standard deviations account for99.994 percent. The two points of the curve that are one standarddeviation from the mean are also the inflection points .

http://en.wikipedia.org/wiki/Statistics

http://en.wikipedia.org/wiki/Probability_theory

http://en.wikipedia.org/wiki/Statistical_dispersion

http://en.wikipedia.org/wiki/Mean


http://en.wikipedia.org/wiki/Normal_distribution

http://en.wikipedia.org/wiki/Inflection_point

http://en.wikipedia.org/wiki/File:Standard_deviation_diagram.svg









http://en.wikipedia.org/wiki/Statistical_dispersion




http://en.wikipedia.org/wiki/Statistics



Example: Calculation of Mean and Variance

Person Test Score Distance from the Mean (Distance from the Mean) Squared

1 90 15 225

2 55 -20 400

3 100 25 625

4 55 -20 400 5 85 10 100

6 70 -5 25

7 80 5 25

8 30 -45 2025

9 95 20 400 10 90 15 225

Mean: 75 Variance: 445 (Average of theentries in this column)

Standard deviation (Square root ofthe variance): 21.1 77



Covariance and Correlation

• Defined as a measure of how much two variables X and Y changetogether – The units of Cov (X, Y) are those of X multiplied by those of Y – The covariance of a variable X with itself is simply the variance of X

• Since these units are fairly obscure, a dimensionless measure of thestrength of the relationship between variables is often used instead.This measure is known as the correlation . – Correlations range from -1 to 1, with positive values close to one

indicating a strong direct relationship and negative values close to -1indicating a strong inverse relationship

78



Spatial Autocorrelation

• Sometimes, rather than examining the association between twovariables, we might look at the relationship of values within a singlevariable at different time points or locations

• There is said to be (positive) autocorrelation in a variable ifobservations that are closer to each other in space have relatedvalues (recall Tobler’s Law)

• As an aside, there could also be temporal autocorrelation – i.e., valuesof a variable at points close in time will be related

79



Examples of Spatial Autocorrelation

80



Examples of Spatial Autocorrelation (Cont’d)

81

Statistical techniques using a semi variogram for



Statistical techniques using a semi-variogram fordeveloping continuous surface models (Kriging)

Kriging is a geostatistical interpolation technique that considers both the distance and thedegree of variation between known data points when estimating values in unknown areas. Akriged estimate is a weighted linear combination of the known sample values around thepoint to be estimated.

Applied properly, Kriging allows the user to derive weights that result in optimal andunbiased estimates . It attempts to minimize the error variance and set the mean of theprediction errors to zero so that there are no over- or under-estimates . Included with theKriging routine is the ability to construct a semivariogram of the data which is used to weightnearby sample points when interpolating . It also provides a means for users to understandand model the directional (e.g., north-south, east-west) trends of their data . A uniquefeature of Kriging is that it provides an estimation of the error at each interpolated point,providing a measure of confidence in the modeled surface and for this reason it is

considered to be a statistical technique rather than a deterministic method.

82



Kriging is a weighted moving average technique, similar in some ways to Inverse DistanceWeighting (IDW) interpolation . Comparing the two techniques provides insight to thebenefits of Kriging. With IDW each grid node is estimated using sample points which fallwithin a circular radius . The degree of influence each of these points will have on thecalculated value is based upon the weighted distance of each of sample point from the gridnode being estimated . In other words, points that are closer to the node will have a greaterdegree of influence on the calculated value than those that are farther away . The generalrelationship between the amount of influence a sample point has with respect to its distanceis determined by IDW's power (or exponent) setting, graphically represented below.

Effectiveness of Kriging

Decay Curves used by IDWInterpolation (Exponent values isanalogous to Power curves).Most applications use a power(or exponent) of 2.

83



The disadvantage of the IDW interpolation technique is that it treats all sample points thatfall within the search radius the same way.For example, if a power (or exponent ) of 1 is specified, a linear distance decay function isused to determine the weights for all points that lie within the search radius (see abovefigure). This same function is also used for all points regardless of their geographic orientationto the node ( north, south etc.) unless a sectored search is implemented . Kriging on the otherhand, can use different weighting functions depending on, 1 ) the distance and orientation ofsample points with respect to the node , and 2) the manner in which sample points areclustered.

More on Kriging Works?

Unless you developed a sectored search IDWimplements a circular search for averagingvalues.

84



The Semivariogram model to be used. There are two methods for kriging, Ordinary andUniversal.Ordinary kriging can use the following semivariogram models:Spherical — Spherical semivariogram model. This is the default.Circular — Circular semivariogram model.

Exponential — Exponential semivariogram model.Gaussian — Gaussian or normal distribution semivariogram model.Linear — Linear semivariogram model with a sill.Universal kriging can use the following semivariogram models:Linear with Linear drift — Universal Kriging with linear drift.Linear with Quadratic drift — Universal Kriging with quadratic drift.

Methods of Krigging

85



Kriging is similar to IDW in that it weights the surrounding measured values to derive aprediction for an unmeasured location. The general formula for both interpolators is formedas a weighted sum of the data:

Kriging Works Similarly to Inverse Distance Weighting

Where Z (si ) is the measured value at the i th location;? i is an unknown weight for the measured value at the i th location;s0 is the prediction location;N is the number of measured values.

In IDW, the weight, ?i , depends solely on the distance to the prediction location. However, in

Kriging, the weights are based not only on the distance between the measured points andthe prediction location but also on the overall spatial arrangement among the measuredpoints. To use the spatial arrangement in the weights, the spatial autocorrelation must bequantified. Thus, in Ordinary Kriging, the weight, ? i , depends on a fitted model to themeasured points, the distance to the prediction location, and the spatial relationshipsamong the measured values around the prediction location. 86



To make a prediction with Kriging, two tasks are necessary(1) to uncover the dependency rules and(2) to make the predictions.

To realize these two tasks, Kriging goes through a two-step process:(1) the creation of variograms and covariance functions to estimate the statisticaldependence (called spatial autocorrelation) values, which depends on our model ofautocorrelation (fitting a model), and(2) actually predicting the unknown values (making a prediction). It is because of these twodistinct tasks that it has been said that Kriging uses the data twice: the first time to estimatethe spatial autocorrelation of the data and the second time to make the predictions.

Tasks for prediction with Kriging

87



As mentioned above, Kriging uses a different weighting function depending on both thedistance and geographic orientation of the sample point to the node being calculated.The problem is that it is impossible for a user, at a first glance, to know precisely how a dataset varies outward from any one location with respect to distance and direction . There are,however, many techniques available to help determine this, the most popular being avariance analysis.

Generating a Semivariogram

Example of data that has no variance crosswise butvaries greatly along the lengthwise axis of the dataset.

88



Kriging uses a property called the semivariance to express the degree of relationship betweenpoints on a surface. The semivariance is simply half the variance of the differences betweenall possible points spaced a constant distance apart.

The semivariance at a distance d = 0 will be zero, because there are no differences betweenpoints that are compared to themselves. However, as points are compared to increasinglydistant points, the semivariance increases. At some distance, called the Range , the

semivariance will become approximately equal to the variance of the whole surface itself . Thisis the greatest distance over which the value at a point on the surface is related to the valueat another point. The range defines the maximum neighbourhood over which control pointsshould be selected to estimate a grid node, to take advantage of the statistical correlationamong the observations.

Understanding Semivariance

89



The image below shows the pairing of one point (the red point) with all other measuredlocations. This process continues for each measured point.

Semivariance Illustration

Often each pair of locations has a unique distance , andthere are often many pairs of points . To plot all pairsquickly becomes unmanageable. Instead of plotting eachpair, the pairs are grouped into lag bins . For example,compute the average semivariance for all pairs of pointsthat are greater than 40 meters apart but less than 50meters. The empirical semivariogram is a graph of theaveraged semivariogram values on the y-axis and thedistance (or lag) on the x-axis (see diagram below).

Relationship between Variance among measure pointsand distance showing that the more point you use andhence the further away they are the greater thevariance in data that will result. This graph is called asemivariogram.

90



As previously discussed, the semivariogram depicts the spatial autocorrelation of themeasured sample points. Because of a basic principle of geography (things that are closer aremore alike), measured points that are close will generally have a smaller difference squaredthan those farther apart. Once each pair of locations is plotted (after being binned) a model isfit through them. There are certain characteristics that are commonly used to describe thesemodels.

Understanding a semivariogram-the range, sill, and nugget

The range and sillWhen you look at the model of a semivariogram, you will notice that at a certain distancethe model levels out. The distance where the model first flattens out is known as the range.

Sample locations separated by distances closerthan the range are spatially autocorrelated,

whereas locations farther apart than the rangeare not.The value at which the semivariogram modelattains the range (the value on the y-axis) is calledthe sill. The partial sill is the sill minus the nugget(see following section).

91



The nuggetTheoretically, at zero separation distance (i.e., lag = 0), the semivariogram value is zero.However, at an infinitely small separation distance, the semivariogram often exhibits a nuggeteffect, which is some value greater than zero. If the semivariogram model intercepts the y-axis at 2, then the nugget is 2.The nugget effect can be attributed to measurement errors or spatial sources of variation atdistances smaller than the sampling interval (or both). Measurement error occurs because ofthe error inherent in measuring devices. Natural phenonema can vary spatially over a rangeof scales (i.e., micro or macro scales). Variation at micro scales smaller than the samplingdistances will appear as part of the nugget effect. Before collecting data, it is important togain some understanding of the scales of spatial variation that you are interested in.An example of a real semivariogram is shown below.

Understanding a semivariogram-the range, sill, and nugget

92



There are options available via the Advanced Parameters dialog box. Theseparameters are:Lag size — The default is the output raster cell size.Major range — Represents a distance beyond which there is little or nocorrelation.Partial sill — The difference between the nugget and the sill.Nugget —Represents the error and variation at spatial scales too fine todetect. The nugget effect is seen as a discontinuity at the origin.semi-variance values for that location.

Kriging Techniques

93



Ordinary KrigingThis method assumes that the data set has a stationary variance but also a non-stationary mean valuewithin the search radius. Ordinary Kriging is highly reliable and is recommended for most data setsSimple KrigingThis method assumes that the data set has a stationary variance and a stationary mean value and requiresthe user to enter the mean value.\Universal KrigingThis method represents a true geostatistical approach to interpolating a trend surface of an area. The

method involves a two-stage process where the surface representing the drift of the data is built in the firststage and the residuals for this surface are calculated in the second stage. With Universal Kriging the usercan set the polynomial expression used to represent the drift surface. The most general form of thisexpression is:F(x, y) = a20 * x2 + a11 * xy + a02 * y2 + a10 * x + a01 * y + a00where a00 is always present but rarely set to zero in advance of the calculation. However, any of the othercoefficients can be set to zero. The recommended setting is a first degree polynomial which will avoidunpredictable behaviour at the outer margins of the data set.Block KrigingAny one of the three Kriging interpolation methods can be applied in one of two forms Punctual or Block.Punctual Kriging (the default) estimates the value at a given point and is most commonly used. BlockKriging uses the estimate of the average expected value in a given location (such as a "block") around apoint. Block Kriging provides better variance estimation and has the effect of smoothing interpolatedresults.

Other Kriging Techniques

94



IDW vs. Kriging

• We get a more “natural” look to the data with Kriging• You see the “bulls eye” effect in IDW but not (as much) in Kriging• Helps to compensate for the effects of data clustering, assigning individual points within a

cluster less weight than isolated data points ( or, treating clusters more like single points) • Kriging also give us a standard error• If the data locations are quite dense and uniformly distributed throughout the area of

interest , we will get decent estimates regardless of which interpolation method we choose.

• On the other hand, if the data locations fall in a few clusters and there are gaps in betweenthese clusters, we will obtain pretty unreliable estimates regardless of whether we use IDW orKriging.

These are interpolation results using the gold data in Western PA (IDW vs. Ordinary Kriging)95

DEM creation by interpolation



Inverse Distance weighted - simple Nearest neighbour – honours raw values

Spline – minimizes curvature -> smooth surface Kriging – uses spatial correlation of points(employing semi-variogram of distance v difference)

96

DEM Interpolation methods



97



•

The choice of methods depends on :

– Speed (IDW – Spline – Krig)– Detail (Krig – Spline – IDW)–

Smoothness (IDW–

Spline–

Krig)– Overall Accuracy (Spline – Krig – IDW)– Insensitivity to Outliers (IDW – Krig – Spline)

*Ranking is subjective*

Interpolation Methods

98

Whi h I l i h d ?



It is not always easy to understand how data behaves beforecommencing with the gridding process and therefore it can be difficult toknow what technique should be used.

o TIN Triangular Irregular Networko NN Natural Neigbouro IDW Inverse Distance Weightingo Kriging

However, there are some questions that can be asked about a data setthat will help determine the most appropriate technique. Thesequestions are listed below.What kind of data is it or what do the data points represent?

Which Interpolation methods to use?

99

Some interpolation techniques can be



Data Type Possible Interpolationo Elevation TIN, NNo Soil Chemistry IDW, Krigingo Demographic NN, IDW, Krigingo Drive Test NNHow accurate is the data?

Some techniques assume that the value at every data point is an exact value andwill honour it when interpolating. Other techniques assume that the value is morerepresentative of an area.Point Value Accuracy Possible InterpolatorVery Accurate NN, TINNot Very Accurate IDW, KrigingWhat does the distribution of the points look like?Some interpolation techniques produce more reasonable surfaces when thedistribution of points is truly random . Other techniques work better with pointdata that is regularly distributed.

p qautomatically applied to certain data types

100

Application of Interpolation Techniques



pp p qIllustrated

101

So lets have a look at some typical point data that you



generate and work out which interpolated works best.

102

Is interpolation processing speed a factor?



Is interpolation processing speed a factor?

All interpolation techniques have certain factors that will influence the speed ofinterpolation. Two factors common to all interpolators is the cell size and the numberof points. The smaller the cell and/or the more points in the data set, the longer ittakes to calculate the surface. However, some interpolators are faster than others.

Interpolator Speed Limiting Factorso TIN Fast Noneo IDW Fast Search and Display Radius sizeo Rectangular Very Fast Search Radius sizeo NN Slow Point distributiono Kriging Slow Number of directions analyzed

103

Is it necessary to over/undershoot the local



yMin. and Max. values?

Some interpolators allow for overshooting and undershooting the local minimum andmaximum values in a data set. This is generally necessary when interpolatingelevation surfaces.Over/Undershoot? InterpolatorsYes TIN, NNNo IDW, Rectangular, Kriging

104

Outline



Outline ArcGIS and 3D AnalysisConcept of 3D GIS and 3D Data ModelBasic Methods for Representing a SurfaceSpatial Interpolation


105

TERRAIN/S URFACE ANALYSIS Slope



p

Aspect

Hillshade

Viewshed

Cut/fill

106

SLOPE • The incline, or steepness, of a surface.

Sl i h f i h i l f h ll



• Slope is the rate of maximum change in z-value from each cell.• Slope can be measured in degrees from horizontal (0 –90), or percent

slope (which is the rise divided by the run, multiplied by 100).• A slope of 45 degrees equals 100 percent slope. As slope angle

approaches vertical (90 degrees), the percent slope approaches infinity.• The slope for a cell in a raster is the steepest slope of a plane defined by

the cell and its eight surrounding neighbors.

107

ASPECT • The compass direction that a topographic slope faces, usually



measured in degrees from north. Aspect can be generated fromcontinuous elevation surfaces.

• The conceptual center of a projection system.

108

HILLSHADE Setting a hypothetical light source and calculating the illumination



g yp g gvalues for each cell in relation to neighboring cells. It can greatlyenhance the visualization of a surface for analysis or graphical display.

Azimuth 315, altitude 45

109

VIEWSHED Viewshed identifies the cells in an input raster that can be seenf b i i li



from one or more observation points or lines.It is useful for finding the visibility. For instance, finding a well-exposed places for communication towers

hillshaded DEM as background

110

CUT/F ILL Understanding cut/fill volumetric analysisC /Fill i h d l f h b



Cut/Fill summarizes the areas and volumes of change between twosurfaces. It identifies the areas and volume of the surface that havebeen modified by the addition or removal of surface material.By taking two surface rasters of a given area from two differenttime periods, the Cut/Fill function will produce a raster displayingregions of surface material addition, surface material removal, andareas where the surface has not changed over the time period.Negative volume values indicate areas that have been filled;positive volume values indicate regions that have been cut.Taking river morphology as an example, to track the amount andlocation of erosion and deposition in a river valley, a series of crosssections need to be taken through the valley and surveyed on aregular basis to identify regions of sediment erosion and

deposition.

111

Before going to the next slide Q-5



a

bc

d

Question:Which of the following problems might 3D data models beapplied to?

Network analysis.

Polygon overlay. Visibility analysis.

Landscape visualization.

Hydrological models. d

Skip


112

Q-5



SOLUTION

c Visibility analysis.

Question:Which of the following problems might 3D data models beapplied to?

113

Arethereanyquestions?



Are there any questions ?

AREA 1

AREA 2

3

12

urp 3182 l-16 terrain analysis

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