Lecture 5Lecture 5Lecture 5Lecture 5

GeostatisticsGeostatistics

Lecture OutlineLecture OutlineLecture OutlineLecture Outline

Spatial EstimationSpatial EstimationSpatial Interpolation Spatial Interpolation Spatial PredictionSpatial PredictionSamplingSamplingSpatial Interpolation MethodsSpatial Interpolation MethodsSpatial Prediction MethodsSpatial Prediction MethodsI l i R S f i h A GISI l i R S f i h A GISInterpolating Raster Surfaces with ArcGISInterpolating Raster Surfaces with ArcGIS

Spatial Estimation/PredictionSpatial Estimation/PredictionSpatial Estimation/PredictionSpatial Estimation/Prediction

Spatial Prediction: Estimate Spatial Prediction: Estimate values at values at unsampledunsampled locations.locations.Why do we need to do this?Why do we need to do this?–– Resource limitations (time and Resource limitations (time and

money).money).You can’t measure every singleYou can’t measure every single–– You can t measure every single You can t measure every single location.location.

–– Access or safety constraints.Access or safety constraints.yy–– Missing or unsuitable samples.Missing or unsuitable samples.

Spatial InterpolationSpatial InterpolationSpatial InterpolationSpatial Interpolation

The prediction of variables at unmeasured The prediction of variables at unmeasured pplocations based on the sampling of the locations based on the sampling of the samesamevariables at known locations.variables at known locations.Usually used to estimate air and waterUsually used to estimate air and waterUsually used to estimate air and water Usually used to estimate air and water temperature, soil moisture, elevation, population temperature, soil moisture, elevation, population density, etc.density, etc.

Spatial PredictionSpatial PredictionSpatial PredictionSpatial Prediction

The estimation of variables at unsampledThe estimation of variables at unsampledThe estimation of variables at unsampled The estimation of variables at unsampled locations, based partially on locations, based partially on otherothervariablesvariablesvariables.variables.Ex. Use elevation to measure temperature.Ex. Use elevation to measure temperature.

C bi l ti d t t l tC bi l ti d t t l t–– Combine elevation and temperature layers to Combine elevation and temperature layers to better predict temperature at unknown better predict temperature at unknown locationslocationslocations.locations.

The MAUPThe MAUPThe MAUPThe MAUP

MAUP: The Modifiable ArealMAUP: The Modifiable ArealMAUP: The Modifiable Areal MAUP: The Modifiable Areal Unit ProblemUnit Problem–– A potential source of error that A potential source of error that pp

can affect spatial studies can affect spatial studies which utilize aggregate data which utilize aggregate data sourcessourcessources.sources.

–– Commonly applied to Commonly applied to demographic analysis.demographic analysis.

–– Can also be applied to physical Can also be applied to physical geography and GIS.geography and GIS.

SamplingSamplingSamplingSampling

Two characteristics of sampling:Two characteristics of sampling:Two characteristics of sampling:Two characteristics of sampling:–– Location of samples (How they are spread Location of samples (How they are spread

around)around)around)around)–– Number of samples (How many can you Number of samples (How many can you

afford?)afford?)afford?)afford?)

Sometimes we can’t control sampling. i.e. Sometimes we can’t control sampling. i.e. You may be limited to occurrences of anYou may be limited to occurrences of anYou may be limited to occurrences of an You may be limited to occurrences of an event.event.

Common Sampling PatternsCommon Sampling PatternsCommon Sampling PatternsCommon Sampling Patterns

a)a) Systematic Sampling:Systematic Sampling:a)a) Syste at c Sa p gSyste at c Sa p gSimple, uniform intervalsSimple, uniform intervalsRandomly pick first Randomly pick first observation then selectobservation then selectobservation, then select observation, then select observations at intervals observations at intervals from sampling frame.from sampling frame.

b)b) Random Sampling:Random Sampling:Randomly placed samplesRandomly placed samplesRandomly placed samples.Randomly placed samples.Each observation has the Each observation has the same opportunity to be same opportunity to be

l dl dselected.selected.

Common Spatial Sampling PatternsCommon Spatial Sampling PatternsCommon Spatial Sampling PatternsCommon Spatial Sampling Patterns

c)c) Cluster Sampling:Cluster Sampling:c)c) Cluster Sampling:Cluster Sampling:Groups samplesGroups samples

d)d) Adaptive Sampling:Adaptive Sampling:h lh lHigher sampling Higher sampling

densities where densities where feature of interest isfeature of interest isfeature of interest is feature of interest is more variable.more variable.

Common Spatial Sampling PatternsCommon Spatial Sampling PatternsCommon Spatial Sampling PatternsCommon Spatial Sampling Patterns

Transect Sampling:Transect Sampling:Transect Sampling:Transect Sampling:Select the transects firstSelect the transects firstSelect sampling pointsSelect sampling pointsSelect sampling points Select sampling points along the transectsalong the transects

Contour Sampling:Contour Sampling:Select sampling sitesSelect sampling sitesSelect sampling sites Select sampling sites along contour linesalong contour lines

Spatial Interpolation MethodsSpatial Interpolation MethodsSpatial Interpolation MethodsSpatial Interpolation MethodsNo one interpolation method is superior for all datasets. No one interpolation method is superior for all datasets. p pp pMethod choice depends on:Method choice depends on:–– Characteristics of the variable to be measuredCharacteristics of the variable to be measured–– Cost of samplingCost of samplingCost of samplingCost of sampling–– Available resourcesAvailable resources–– Accuracy requirements of the usersAccuracy requirements of the users

Methods differ in the mathematical functions used toMethods differ in the mathematical functions used toMethods differ in the mathematical functions used to Methods differ in the mathematical functions used to weight each observation, and the number of weight each observation, and the number of observations used. observations used.

h dh dMethods:Methods:–– Nearest NeighborNearest Neighbor–– IDWIDW–– Fixed DistanceFixed Distance–– SplineSpline

Spatial Interpolation MethodsSpatial Interpolation MethodsSpatial Interpolation MethodsSpatial Interpolation MethodsTypes of Interpolators:Types of Interpolators:

E I lE I lExact InterpolatorsExact Interpolators–– An interpolation method where the estimated value is identical to the An interpolation method where the estimated value is identical to the

observed value at sampling locationsobserved value at sampling locationsInexact InterpolatorsInexact InterpolatorsInexact InterpolatorsInexact Interpolators–– An interpolation method where estimated value is predicted/estimated An interpolation method where estimated value is predicted/estimated

from a measured value.from a measured value.

Methods:Methods:Methods:Methods:Global MethodsGlobal Methods–– Use information at all sampled location to estimate the value for each Use information at all sampled location to estimate the value for each

unknown locationunknown location–– Trend surface interpolationTrend surface interpolation

Local MethodsLocal Methods–– Use information at nearby locations to estimate the value at locations of Use information at nearby locations to estimate the value at locations of

interestinterestinterestinterestGeostatisticalGeostatistical MethodsMethods–– Spatial autocorrelationSpatial autocorrelation

Spatial Interpolation MethodsSpatial Interpolation Methodsi hbi hbNearest NeighborNearest Neighbor

Nearest Neighbor*Nearest Neighbor*Nearest Neighbor Nearest Neighbor ((ThiessenThiessen Polygon)Polygon)Assigns a value to Assigns a value to

l dl dan an unsampledunsampledlocation that is location that is equal to the value equal to the value qqfound at the found at the nearestnearest sample sample location.location.Exact interpolator: Exact interpolator: Value at each Value at each sample point issample point issample point is sample point is preserved.preserved. *Referred to as Natural Neighbor/Voronoi Polygons

in lab exercise.

Spatial Interpolation MethodsSpatial Interpolation Methods( i i h d)( i i h d)IDW (Inverse Distance Weighted)IDW (Inverse Distance Weighted)

Uses distance and values to Uses distance and values to nearby known points.nearby known points.Reduces the contribution of Reduces the contribution of distant points.distant points.Weight of each sample point is Weight of each sample point is g p pg p pan inverse proportion to the an inverse proportion to the distance.distance.Further points = less weightFurther points = less weightFurther points = less weightFurther points = less weightCloser points = more weightCloser points = more weightExactExact InterpolatorInterpolator

l h k ( )Value at each known point (50,52,30) are averaged, with the weights based on the distances (d1, d2, d3) from the interpolated point.

Spatial Interpolation MethodsSpatial Interpolation Methodsi d di d S lii d di d S liFixed Radius and SplineFixed Radius and Spline

Fixed Radius:Fixed Radius: Spline:Spline:Fixed Radius:Fixed Radius:Cell values estimated Cell values estimated based on based on averageaverage of of

b lb l

Spline:Spline:Used to interpolate along a Used to interpolate along a smooth curve.smooth curve.

nearby samples.nearby samples.Depends on search radius.Depends on search radius.

Force a smooth line to pass Force a smooth line to pass through a set of points.through a set of points.

Spatial Prediction MethodsSpatial Prediction MethodsSpatial Prediction MethodsSpatial Prediction Methods

Often generated via a statistical processOften generated via a statistical processOften generated via a statistical process.Often generated via a statistical process.Type of Type of predictive modelingpredictive modeling..P i M th dP i M th dPrimary Methods:Primary Methods:–– Spatial AutocorrelationSpatial Autocorrelation–– Spatial RegressionSpatial Regression–– KrigingKriging

Spatial Prediction MethodsSpatial Prediction Methodsl il iAutocorrelationAutocorrelation

Tendency of nearby objects to vary together.Tendency of nearby objects to vary together.Tendency of nearby objects to vary together. Tendency of nearby objects to vary together. “Everything in the universe is related to “Everything in the universe is related to everything else, but closer things are moreeverything else, but closer things are moreeverything else, but closer things are more everything else, but closer things are more related.” related.” –– Tobler’s First Law of GeographyTobler’s First Law of Geography

Spatial Prediction MethodsSpatial Prediction MethodsS i l iS i l iSpatial RegressionSpatial Regression

Establishes relationships between numerous Establishes relationships between numerous stab s es e at o s ps bet ee u e ousstab s es e at o s ps bet ee u e ousinput variables and presents the relationships in input variables and presents the relationships in a succinct manner.a succinct manner.A i l i hA i l i hA regression analysis has two parts:A regression analysis has two parts:–– The dependent variable, which is the phenomenon The dependent variable, which is the phenomenon

whose level or presence you are trying to predict orwhose level or presence you are trying to predict orwhose level or presence you are trying to predict or whose level or presence you are trying to predict or explain for each location in a study site. explain for each location in a study site.

–– The independent variables which are the knownThe independent variables which are the known–– The independent variables, which are the known The independent variables, which are the known attributes of the locations that influence the level or attributes of the locations that influence the level or presence of the dependent variable.presence of the dependent variable.

Spatial Prediction MethodsSpatial Prediction Methodsi ii iKrigingKriging

Weights the surrounding measured values to derive a prediction for Weights the surrounding measured values to derive a prediction for g g pg g peach location. However, the weights are based not only on the each location. However, the weights are based not only on the distance between the measured points and the prediction location but distance between the measured points and the prediction location but also on the overall spatial arrangement among the measured points.also on the overall spatial arrangement among the measured points.

3 Components:1. Spatial Trend: increase or

d i i bl d didecrease in variable depending on direction. Ex. Temperature to NW

2. Spatial Autocorrelation: Tendency for points near each other to havefor points near each other to have similar values.

3. Random variation of measured points.

Spatial Prediction MethodsSpatial Prediction Methodsi ii iKrigingKriging

Lag Distance:Lag Distance:Lag Distance:Lag Distance:–– For paired points, the distance For paired points, the distance

between two points.between two points.–– Symbolized by Symbolized by hh–– Defines the neighbors.Defines the neighbors.

SemivarianceSemivariance::–– Based on values at nearby sample points.Based on values at nearby sample points.

For a given h there is a value forFor a given h there is a value for semivariancesemivariance–– For a given h, there is a value for For a given h, there is a value for semivariancesemivariance..–– If values are similar to each other for locations at n lags If values are similar to each other for locations at n lags

apart, you will see a smaller value of apart, you will see a smaller value of semivariancesemivariance..S llS ll i ii i i di t b l tii di t b l ti–– So, smaller So, smaller semivariancesemivariance indicates nearby locations are indicates nearby locations are similar to each other or stronger spatial autocorrelation.similar to each other or stronger spatial autocorrelation.

Spatial Prediction MethodsSpatial Prediction Methodsi ii iKrigingKriging

Variogram/SemivariogramVariogram/SemivariogramVariogram/SemivariogramVariogram/Semivariogram–– A graph showing the relationship A graph showing the relationship

between lag distance and between lag distance and semivariance.semivariance.

NuggetNugget–– Initial semivariance.Initial semivariance.–– Where autocorrelation is typicallyWhere autocorrelation is typicallyWhere autocorrelation is typically Where autocorrelation is typically

highest.highest.SillSill–– Points where the variogram levels offPoints where the variogram levels off–– Points where the variogram levels offPoints where the variogram levels off–– Where little spatial autocorrelation Where little spatial autocorrelation

occurs.occurs.RangeRangeRangeRange–– Lag at which Lag at which sillsill is reachedis reached

Spatial Estimation inSpatial Estimation in ArcGISArcGISSpatial Estimation in Spatial Estimation in ArcGISArcGIS

Analysis performed usingAnalysis performed usingAnalysis performed using Analysis performed using Spatial Analyst Tools Spatial Analyst Tools

Interpolation ToolboxInterpolation ToolboxInterpolation Toolbox.Interpolation Toolbox.OROR

Geostatistical Analyst Toolbar:Geostatistical Analyst Toolbar:Geostatistical Analyst Toolbar:Geostatistical Analyst Toolbar: