3. Integrated analysis of spatial and attribute data - RTC, Regional ...

International Training Course on “Climate Analysis and Applications”10-19 October 2011WMO RTC-Turkey FacilitiesAlanya, Antalya, TURKEYFUNDAMENTALS OFGISDr. Arzu ErenerSelcuk University, Department of GeomaticsEngineeringemail: ae76@hotmail.com.tr1

Content1. Fundamentals of GIS2. What is GIS?3. Geo-Spatial Data4. The Nature of Geographic Data5. Organizing Geographic Data For Analysis6. The Components of GIS7. Spatial Data Models8. GIS Analysis Functions9. Principles of GeoStatistical Analysis2

What is GIS?A widely accepted definition is that:“A GIS is a computer based system that provides the following four setsof capabilities to handle georeferenced data” (Aronoff, 1989):inputmanagement (storage and retrieval)manipulation and analysisoutput3

What is GIS?‣One of the best ways to introduce GIS is to consider the generic types ofquestions they have been designed to answer.‣These include questions about location, patterns, trends and conditions:Where are particular features found?What geographical patterns exist?Where have changes occured over a given time period?4

What is GIS?‣Where do certain conditions apply?‣What will the spatial impacts be if an organisation takes certain action?5

‣A GIS professionals were then asked to find appropriate sources of spatialdata that could be used to represent these criteria.‣A range of sources was identified including:7

It is one of the most important elements of a GISIs it ‘spatial’ or ‘geographic’?Spatial Data: Data with an associated spatial location (with respect to agiven reference frame)‣medical images are referenced to the human body‣engineering drawings are referenced to a mechanical object‣architectural drawings are referenced to a buildingGeographic data/geo-spatial data: data whose underlying referenceframe is the earth surface.NOTE: Spatia data and Geo-spatial data often used interchangeably8

The Nature of Geographic DataGeographic data are commonly characterized as having four majorcomponents Geographic Position: geographic data are recorded in a coordinatesystem Attributes: characteristics of geographic features Relationships: the relationships between the geographic features Time: geographic data are referenced to a point in time or a period oftime9

Geographic PositionThe Nature of Geographic Data‣A gis requires that a common coordinate system be used for all the data settherefore they can overlay one another.10

AttributesThe Nature of Geographic Data‣spatial attribute: the physical dimension or the width of a road‣non-spatial attribute: a class, such as a vegetation type and thedescriptive information, such as the name of the owner of a parcel11

The Nature of Geographic DataSpatial RelationshipsThese relationships are generally very numerous, may be complex, and are important.• Spatial relationships:‣• Topological relationships(independent on coordinates)for example: next to, inside‣• Metrical relationships(dependent on coordinates):for example: ten kilometres away• Non-spatial relationships:for example: Owner of, builder of.what it is near ?12

TimeThe Nature of Geographic DataGeographic data are referenced to a point in time or a period of timeMLC map of 2006 MLC map of 200913

Organizing Geographic Data ForAnalysis‣In GIS a data layer consists of logicallyrelated geographic features and theirattributes‣the organizing principal of data layers is beto group similar feature types..‣each layer is registered positionally toother layers through a common coordinatesystem‣in principle, the number of layers isunlimited, restrictions being imposed only bystorage space14

There may be three main reasons for segregating geographic information intoseparate layersOrganizing Geographic Data ForAnalysis‣ to simplify the combination of features‣ to perform manipulation and analysis on multiple data sets‣ to let a smaller scale database lead to a larger scale database showing moredetail15

The Components of GIS‣All of these components need to be in balance for the system to be successful.‣No one part can run without the other.16

The Components of GIS‣People include a plethora of positions including:‣GIS managers,‣database administrators,‣application specialists,‣systems analysts, and‣programmers.17

The Components of GISMethods include‣how the data will be retrieved,‣input into the system,‣stored,‣ managed,‣transformed,‣analyzed, and finally presented in a final output.18

The Components of GISHardware consists of:‣enough power to run the software,‣enough memory to store large amounts of data,‣and input and output devices such as scanners,‣digitizers,‣GPS data loggers,‣media disks, and printers. (Carver, 1998)19

The Components of GISSoftwareAll packages must be capable of:‣ data input,‣ storage,‣ management,‣ transformation,‣ analysis, and output.Some examples for the software are:‣ ArcInfo‣ Intergraph‣ MapInfo‣ Smallworld‣ SICAD20

Data InputThe Components of GISThe most time consuming and costly aspectProcedure of encoding dataThe creation of an accurate and well documented data-base is critical to theoperation of GISḊata quality information includes the:‣date of collection‣the positional accuracy‣classification accuracy‣complateness‣and the method used to collect andencode the data21

Data InputData QualityErrors in the data set can addunpleasant andcostly hoursto implementing a GIS and the results and conclusions of the GIS analysismost likely will be wrong.Several guidelines to look at include:LineagePositional Accuracy Attribute Accuracy Logical ConsistencyCompletenes22

Data QualityLineageThe source material from which the data were derivedhistory, the source of data andprocessing stepsInclude all dates of the source material and updates and changes made to it.(Guptill, 1995)23

Positional AccuracyData Qualityexpected deviance in the geographic location of an object from its true ground positionincludes measures of thehorizontal andvertical accuracyThere are two components of positional accuracy:the bias andthe precision24

Attribute AccuracyData QualityAn attribute is a fact about some location, set of locations, or features on the surfaceof the earth.The source of error usually lies within the collection of these facts.CompletenesA check to see if relevant data is missing with regards to the features and the attributes.Grouped into two categories:Complateness of coverageClassification25

SPATIALDATAMODELS26

GIS Spatial Data ModelsThere are two fundamental approaches of representing geographic features:‣Raster Data Model and‣Vector Data ModelRaster DataThe real worldVector Data27

GIS Spatial Data ModelsRepresentation of Raster and Vector Data Model28

Comparison of Vector and Raster Models29

Raster Data ModelIn the raster data model, land cover is represented by a regular grid of aquareEach cell will have a value corresponding to its land cover type.Raster data are good at:‣representing continuous data (e.g., slope, elevation, chemicalconcentrations)‣representing multiple feature types (e.g., points, lines, and polygons) assingle feature types (cells)‣rapid computations ("map algebra") in which raster layers are treated aselements in mathematical expressions‣analysis of multi-layer or multivariate data (e.g., satellite image processingand analysis)‣hogging disk space30

Raster Data Model‣The size of the file increases rapidly with resolution.‣Since the file size is related to the area of coverage, it increases by thesquare of the increase in resolution.31

Raster Data ModelRaster file can be achieved by using various methods of data compression such as:‣ 1. Traditional raster encoding‣ 2. Run_Length raster encoding‣3. Quadtree raster encoding32

1. Traditional Raster EncodingIt repeats storing the same value for each cell33

1. Traditional Raster Encoding34

2. Run_Length raster encoding‣In runlength encoding the adjacent cells along a row that have the same valueare treated as a group termed a run.35

3. Quadtree raster encoding‣provides a more compact raster representation.‣Here consider that you dont use one size cells instead you use a variable sized gridcell dividing the area.‣A coarse resolution (large cells) is used to encode large homogenious areas.‣A finer resolution is used for areas of high spatial variability.36

3. Quadtree raster encodingRoot : the point from which all other branches expended.Leaf : the point from which there is no further branching.Nodes: All other points in the tree.İllustration of the components of quadtree37

Vector Data ModelIn the vector data model, features on the earthare represented aspointslines / routespolygons / regionsTINs (triangulated irregular networks)Every position in the map space has a uniquecoordinate valueVector data are good ataccurately representing true shape andsizerepresenting non-continuous data (e.g.,rivers, political boundaries, road lines,mountain peaks)creating aesthetically pleasing mapsconserving disk space38

Vector Data ModelSpaghetti Model‣It is the simpliest vector data model‣ In this model the paper map is translated line-for-line into a list of x,ycoordinates‣ Although all the spatial features are recorded, the spatial relationshipsbetween these features are not encoded39

Vector Data Model‣The data model is really a mapexpressed in cartesiancoordinates.‣The spagethi model is efficientfor digitally reproducing maps.‣However, inefficient for types ofspatial analysis since any spatialrelationship must be derived bycomputation.40

Vector Data ModelTopological Model‣Topology is geometry without coordinates‣Topology is the mathematical method used to define spatial relationships.‣Mental maps are mostly topologicalNotes: The instruction “Go ahead until you hit the next intersection and turn left”is topological41

Topological ModelTopological relationships will not be affected by continuous distortions42

Topological ModelNodes, Arcs, and PolygonsTopological data structures are often based on nodes, arcs, and polygons‣Arc: a series of points that starts and end at a node.‣Node: intersection point where two or more arcs meet.‣Polygon: comprised of a closed chain of arcs that represent the boundries ofthe area.43

Topological ModelOther names for Nodes, Arcs, and Polygons Nodes: vertices, 0-cells Arcs: chains, links, edges, 1-cells Polygons: faces, areas, 2-cells44

Topological ModelArc-Node data model45

Some Basic Topological RelationshipsSpatial queries can be processed much more quickly using topology tablesthan they can be done by calculation from the coordinate data.‣Connectivity describes, if two points are connected by a line‣Enclosure describes, if an entity is enclosed by another‣Adjacency describes, if two polygons share a common border‣Contiguity is the spatial relationship of adjacencyForexample: biologists might be interested in the habitats that occur next to each other (Contiguity).46

Triangulated Irregular Network (TIN)‣A vector-based topological data model‣A TIN represents the terrain surface as a set of interconnected triangles‣each triangle's surface are defined by the X,Y,Z coordinates of thethree corner points and represented by a plane‣ each piece of the triangle will fit with its neighboring pieces andtherefore, the surface will be continuous47

Triangulated Irregular Network (TIN)Delanuay Triangulation:‣ Of the algorithms available, the Delanuay Triangulation is widely used‣ Artificial boundary points are defined to form a perimeter around theedges of the data set area‣ The triangulation starts with any pair of the artificial boundary pointsknown as inital known neighbours (A and B in the Figure)48

Triangulated Irregular Network (TIN)Delanuay Triangulation:‣The search for the next neighbour is made by constructing a circle with the baseAB diameter and searching to the clockwise to find if any point falls within this circle‣ If no data point lies within the circle, the circle is increased in size to perhaps twicethe area of the original circle49

GIS ANALYSIS FUNCTIONS50

A Classification of GIS Analysis Functions:1. Maintenance and analysis of the spatial data‣ Format transformations‣ Geometric transformations‣ Transformations between map projections‣ Conflation‣ Edge matching‣ Editing of graphic elements‣ Line coordinate thinning2. Maintenance and analysis of attribute data‣ Attribute editing functions‣ Attribute query functions51

A Classification of GIS Analysis Functions:3. Integrated analysis of spatial and attribute data Retrieval/classification‣ retrieval‣ classification‣ measurement Overlay operations Neighbourhood operations‣ search‣ line in polygon and point in polygon‣ topographic functions‣ thiessen polygons‣ interpolation‣ contour generation Connectivity functions‣ contiguity measures‣ proximity‣ network‣ spread‣ seek‣ intervisibility‣ illumination‣ perspective view52

A Classification of GIS Analysis Functions:4. Output formatting‣ Map annotation‣ Text labels‣ Texture patterns and53

1. Maintenance and analysis of the spatial dataFormat Transformation: the transformation of files into the data structure andfile formats used internally by the GISGeometric Transformation: Used to assign ground coordinates to a map ordata layer whitin the GIS or to adjust one data layer so that it can becorrrectly overlayed on another of the same area. (the procedure is calledregistration)1. Registration by Relative Position2. Registration by Absolute Position54

1. Maintenance and analysis of the spatial dataGeometric Transformation:1. Registration by Relative PositionOne data layer, termed as the slave, is registred to a second data layer, termedthe master.2. Registration by Absolute PositionEach data layer is seperately registered to the same geographic coorinate system(such as UTM coordinates)55

Edge-matching :Maintenance and analysis of the spatial data‣An editing procedure to adjust the position of features extending across adjacentmap sheet boundaries.‣This function ensures that all features that cross adjacent map sheets have thesame edge locations.‣Links are used when matching features in adjacent coverages.56

Maintenance and analysis of the spatial dataEditing Functions :Editing functions are used to add, delete, or manipulate the geographic position offeatures.Sliver or splinter polygons are thin polygons that occur along the borders of polygonsfollowing digitizing and the topological overlay of two or more coverages.57

Maintenance and analysis of the spatial dataLine Coordinate ThinningUsed to reduce the quantity of coordinate data that must be stored in the GIS whenmore coordinates are entered than needed (Aranoff, 1989)58

2. Maintenance and analysis of attribute dataThis group of functions is used to edit, check, and analyze the non-spatialattribute dataAttribute editing functions - allow attributes to be retrieved, examined andchanged.Can add new attributes and delete old ones.Attribute query functions - retrieve records in the attribute databaseaccording to a set of conditions specified by the operator (Aranoff, 1989).Forexample: attributes of forest coverdata stored in two tables can queried together togenerate a report of the total area of forest more than 30 years old.59

Attribute query functions2. Maintenance and analysis of attribute data60

3. Integrated analysis of spatial and attribute dataThe power of GIS lies in its ability to analyse spatial and attribute datatogether.The group of functions have been subdivided into four categories:‣Retrieval/classification‣Overlay operations‣Neighbourhood operations‣Connectivity functions61

3. Integrated analysis of spatial and attribute dataRetrieval/classification‣Retrieval operations involved selective search, manipulation and output of data‣Information from database tables can be accessed directly through the map, ornew maps can be created using information in the tabular database.Commonly used retrievalfunctions are:‣ browsing‣query windowgenerationFor example:“select all polygons of a certain type of soil within 20km diameter radius of aspecified location”62

3. Integrated analysis of spatial and attribute dataClassification is the procedure of identifying a set of features belonging to agroup and assigning a name to that group.63

3. Integrated analysis of spatial and attribute data‣Generalization, called map dissolve, is the process of making a classification lessdetailed by combining classes.‣Generalization is often used to reduce the level of classification detail to make anunderlying pattern more apparent64

3. Integrated analysis of spatial and attribute data•Measurement functions is the calculation of distances between points, length oflines, perimeters and areas of polygons and the size of a group of cells with thesame class.Spatial measurementsinclude:‣distances betweenpoints‣• length of lines‣• perimeters andareas of polygons‣• the size of a groupof cells with the sameclass‣• 3-D measurements65

3. Integrated analysis of spatial and attribute dataOverlay Operation‣Overlay analysis is a technique of deriving new information from two or morelayers of data covering the same area‣Overlay operations are usually performed more efficiently in raster basedsystems.‣Overlay operations include:‣ Arithmetic overlay‣ Logical overlay66

3. Integrated analysis of spatial and attribute dataOverlay OperationArithmetic operations include:‣• addition‣• subtraction‣• multiplication‣• divisionA logical overlay involves finding those area where a specified set ofconditions occur (or not ocur ) together67

3. Integrated analysis of spatial and attribute dataAddition68

Subtraction3. Integrated analysis of spatial and attribute data‣Image subtraction is often used to identify changes that have occurredbetween images collected on different dates69

3. Integrated analysis of spatial and attribute dataMultiplicationThe precedure is being used to convert the data from units of inches to units inmilimiters by multiplying each rainguige value by 25.4 mm/inc70

3. Integrated analysis of spatial and attribute dataLogical Overlay OperationsExample: Desirable areas for groving a certain type of crop might bedefined as:“ those areas that have an agricultural land and have fertile soil “If the land use and soil data are represented as separate data layers then, alogical overlay operation can identify the locations where these conditions occurtogether71

Logical Operations on Raster: LOGICAL AND72

Logical Operations on Raster: LOGICAL OR73

Overlay Analysis‣One of the major factors affecting the performance of the overlayoperations is the data model‣Vector and raster models differ significantly in the way overlayingoperations are implemented74

Overlay AnalysisVector overlay generates new: nodes arcs polygons• The vector overlay generally requires significantly more processingtime than raster overlay• The new polygons inherit the attributes from the original input layers75

Vector overlayThe process of subdividing polygons is termed clippingClipping makes overlay operations more complex in vector domain than in rasterdomain.76

Raster overlay‣Raster overlay processing involves retrieving and comparing thecorresponding pixels from both layers‣The overlay operation is carried out two layers at a time‣There is no need to calculate intersections of boundaries or make anymodifications to take the feature boundaries77

Comparison of Vector and Raster OverlayOperations‣Overlay operations are easier in raster data‣ However, sparse data sets require as much processing as denselypopulated ones‣ In vector data, only the data of interest are processed‣ In many GISs, a hybrid approach, which takes the advantage of thecapabilities of both data models, is used78

Connectivity (Network) Operations:Many types of analysis can be carried out on networks for;‣ transportation‣ planning‣ utility management‣ airline scheduling‣navigation etc.For example;‣ finding the shortest path through a network between selected point‣ finding the parts of the network which can be reached within a giventravel time from a selected point79

Connectivity (Network) Operations:‣Each connectivity function must include the following:‣1. a specification of the ways spatial elements (such as roads) areinterconnected‣2. a set of rules that specify the allowed movement along theseinterconnections‣3. a unit of measurement‣ The widely used connectivity operations are:‣ contiguity measures‣proximity analysis‣ network operations80

Connectivity (Network) Operations:Contiguity measures evaluate the characteristics of spatial units that areconnected A contiguous area consists of a group of spatial units that are connectedForexample seach a land unit to be used as a park might be specified as a contiguous land unit offorest having minimum area of 1000 square km.81

Connectivity (Network) Operations:Proximity‣Proximity is a measure of distance between features‣It is most commonly measured in units of length but can be measuredin other units, such as travel time or noise level‣Four parameters must be specified to measure proximity:‣1. target location (e.g. a road, a school, a park)‣2. unit of measure (e.g. distance in meters)‣3. function to calculate proximity (e.g straight line distance,travel time)‣4. area to be analyzed82

Connectivity (Network) Operations:ProximityForexample: The 300 ft buffer zone drawn around the roads definesthe forest area where logging is not permitted.83

Connectivity (Network) Operations:Proximity‣A buffer zone is an area of a specified width drawn around one or moregeographic features‣Buffering creates a new area, enclosing the buffered object84

Network FunctionsConnectivity (Network) Operations:‣A network is composed of interconnected linear features‣ Networks are commonly used for moving resources from one location toanother‣ Network analysis operations are performed on topologically structuredvector data‣ A GIS is used to perform thee principal types of network analysis:‣ prediction of network loading‣ route optimization‣ resource allocation85

Network FunctionsConnectivity (Network) Operations:86

Connectivity (Network) Operations:Neighbourhood Operations‣Neighbourhood operations evaluate the characteristics of the area surrounding aspecified location‣For example:“Counting the number of residential buildings within a 5 km radius of a fire station”87

Connectivity (Network) Operations:Neighbourhood Operations‣Every neighbourhood function requires the specification of at least three basicparameters:1. target location(s)2. specification of the neighbourhood3. function to be performed88

OUTPUT FUNCTIONSMap AnnotationTitles, Legends, Scale Bars, and North Arrows are the simplest forms ofdepicting information concerning the map.Graphic SymbolsGraphic Symbols areused to portray thevarious entities depictedon the map.89

The principles of geostatistical analysis90

Geostatistical analysis‣Sample points taken at different locations in a landscape and creates(interpolates) a continuous surface.‣The sample points are measurements of some phenomenon such as:‣oil spill,‣ elevation heights‣Soil and rock properties‣Climate measurements‣Etc.‣The Geostatistical Analyst derives a surface using the values from themeasured locations to predict values for each location in the landscape.91

Geostatistical Analyst‣A major challenge facing most GIS modelers is to generate the most accuratepossible surface.‣The deterministic methods (eg. IDW) weights the surrounding measured valuesto derive a prediction for each location.‣the weights are based only on the distance between the measured points‣However, the geostatistical methods that are based on statistical models theweights are based not only on the distance between the measured points andthe prediction location but also on the overall spatial arrangement among themeasured points.‣To use the spatial arrangement in the weights, the spatial autocorrelationmust be quantified.92

Calculate the empirical semivariogram‣Kriging, like most interpolation techniques,is built on the assumption that” things thatare close to one another are more alikethan those farther away (quantified here asspatial autocorrelation).”‣The empirical semivariogram is a meansto explore this relationship.93

Calculate the empirical semivariogram‣The Semivariogram/Covariance Cloud allows you to examine the spatialautocorrelation between the measured sample points.‣Since closer locations should be more alike, in the semivariogram the closelocations (far left on the x-axis) should have small semivariogram values (low onthe y-axis)‣As pairs of locations become farther apart (moving to the right on the x-axis ofthe semivariogram cloud), they should become more dissimilar94

Understanding a semivariogram—the range, sill, and nugget‣ Sample locations separated by distances closer than the range are spatiallyautocorrelated, whereas locations farther apart than the range are not.‣range= The distance where the model first flattens out‣sill = The value that the semivariogram model attains at the range (the valueon the y-axis).Theoretically, at zero separation distance (i.e., lag = 0), the semivariogram valueshould be zero.However, at an infinitesimally small separation distance, the difference betweenmeasurements often does not tend to zero.This is called the nugget effect.For example, if the semivariogram model intercepts the yaxis at 1.34, then 95thenugget is 1.34.

Modeling a semivariogram‣spatial modeling of the semivariogram, begin with a graph of the empiricalsemivariogram, computed for all pairs of locations separated by distance h as:Semivariogram = 0.5 * average [ (value at location i - value at location j) 2 ]The empirical semivariogram is a graph of the averaged semivariogram values onthe y-axis and distance (or lag) on the x-axis96

Calculate the empirical semivariogramThe configuration of the points is displayed in orange on the map theThe spatial coordinates each point are given as (X,Y).You will use ordinary kriging to predict a value for location (1, 4) (yellow)which is called the prediction location.97

Calculate the empirical semivariogramThe ordinary kriging model is:‣where s =(X,Y) location; one of the sample locations is s = (1,5),and‣Z(s) is the value at that location; for example, Z(1,5) = 100.‣The model is based on a constant mean m for the data (no trend) and randomerrors e(s) with spatial dependenceThe predictor is formed as a weighted sum of the data:where‣Z(si) is the measured value at the ith location, for example, Z(1,5) = 100;‣ג i is an unknown weight for the measured value at the ith location; ‣s 0is the prediction location, for example, (1,4); and N = 5 for the fivemeasured values98

Calculating the empirical semivariogram‣In a semivariogram, half the differencesquared between the pairs of locations (the y-axis) is plotted relative to the distance thatseparates them (the x-axis).‣The distance between two locations is calculated by using the Euclidean distance:100-105=5 2 =25Theempiricalsemivariance=0.5 timesthe difference squared:0.5 * average[(value atlocation i - value atlocation j) 2 ].99

Binning the empirical semivariogram‣With larger datasets (more measured samples) the number of pairs oflocations will increase rapidly and will quickly become unmanageable.‣Therefore, you can group the pairs of locations, which is referred to asbinning.‣1. form pairs of points‣2. group the pairs with common distance and direction.Shows all possible pairwise links among all12 locations.pairs are groupedbased on common distances anddirections100

Calculating the empirical semivariogram‣In this example, a bin is a specified range of distances.‣That is, all points that are within 1 to 2 meter apart are grouped into the first bin,within 2 to 3 meters apart are grouped into the second bin..so on101

Calculating the empirical semivariogram(1.414+2)/2 (12.5+0+0)/3=4.167102

Calculating the empirical semivariogramthe goal is to solve the equations for all ofthe ג is (the weights), so the predictor canbe formed by usingThe matrix formula for ordinary kriging is:‣The gamma matrix ק contains the modeled semivariogram valuesbetween all pairs of sample locations, where ɣ ijdenotes the modeledsemivariogram values based on the distance between the two samplesidentified as the ith and jth locations.‣The vector g contains the modeled semivariogram values between eachmeasured location and the prediction location, where ɣ i0denotes themodeled semivariogram values based on the distance between the ithsample location and the prediction location.103

Fitting a model‣Now you can plot the averagesemivariance versus average distance ofthe bins onto a graph.‣the empirical semivariogram.‣But the empirical semivariogram values cannot be used directly inthe ק matrix because you might get negative standard errors for thepredictions;‣instead, you must fit a model to the empirical semivariogram.‣Once the model is fit, you will use the fitted model whendetermining semivariogram values for various distances.104

Fitting a model to the empirical semivariogram‣Geostatistical Analyst provides the following functions to choose from tomodel the empirical semivariogram:‣Circular,‣Spherical,‣Tetraspherical,‣ Pentaspherical,‣Exponential,‣Gaussian,‣Rational Quadratic,‣Hole Effect,‣K-Bessel,‣J-Bessel, and‣Stable.‣The selected model influences the prediction of the unknown values,particularly when the shape of the curve near the origin differs significantly.‣The steeper the curve near the origin, the more influence the closestneighbors will have on the prediction.‣As a result, the output surface will be less smooth.‣Each model is designed to fit different types of phenomena more accurately.105

Different types of semivariogram modelsThe Spherical modelThe Exponential modelForexample:The Spherical model shows a progressive decrease of spatial autocorrelation(equivalently, an increase of semivariance) until some distance, beyond whichautocorrelation is zero.This model is applied when spatial autocorrelation decreases exponentially withincreasing distance, disappearing completely only at an infinite distance106

Fitting a model‣For simplicity, the model that you will fit is a least-squares regression line, andyou will force it to have a positive slope and pass through zero.‣The formula to determine the semivariance at any given distance in thisexample is: Semivariance = Slope * DistanceSlope (13.5) = slope of the fitted model.Distance (h) = distance between pairs of locations107arise due to unbiasednessconstraints

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‏;ג In order to solve for109

Creation of g vector for the unmeasured locationThe g vector for (1,4) is13.5x2=27110

Make a predictionThe kriging weights for the measured Values used, to calculate a prediction forthe location with the unknown value.kriging weights vector is solved by using ק matrix and g vector by:46.757+10.3257+……-1.679=102.6218111

Make a prediction‣the weights decrease with distance but are more refined than a straight distanceweighting‣since they account for the spatial arrangement of the data.112

Kriging variance‣One of the strengths of using a statistical approach is that it is possible to alsocalculate a statistical measure of uncertainty for the prediction.‣To do so, multiply each entry in the ג vector times each entry in the g vector andadd them together to obtain what is known as the predicted kriging variance.‣The square root of the kriging variance is called the kriging standard error.113

‣The kriging standard error value is 3.6386.‣If it is assumed that the errors are normally distributed, 95 percent predictionintervals can be obtained in the following way:‣Kriging Predictor + 1.96*sqrt(kriging variance)Which means:‣If predictions are made again and again from the same model, in thelong run 95 percent of the time the prediction interval will contain the valueat the prediction location.‣In the example, the prediction interval ranges from:‣ 95.49 to 109.75 (102.62 + 1.96 * 3.64 )114

LABRATORY WORK‣ You will use the mean annualrainfall dataset.‣ The stations are in the easternand north eastern part of Turkey‣ interpolate the rain values at thelocations where values are notknown1. using the default settings of theGeostatistical Analyst.2. By analysing spatialautocorrelations3. Compare the result models‣Exploring your data‣Examining the distribution of your data‣Histogram‣QQ Plots‣Identifying global trends in your data‣Understanding spatial autocorrelation and directional influences115

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