surface models interpolation
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Interpolation, Surface Models, and Geostatistical Methods
Creating continuous surfaces from measurements at discrete points
Surface Models - Topics
Examples of useInput dataInterpolation– Thiessen Polygon example– Inverse Distance Weighted Example– Splines and Kriging
Kriging methods
Elevation &Contours - the most common example of interpolated surface
Digital Elevation Models
Digital representation of the continuous variation of relief over spaceForm the base, or backbone, of landscapeApplied to hydrological models, visibility analysis, erosion studiesInterpolation is fundamental to DEM production
DEM Derivatives
Slope - the steepness of the land– used in many environmental models,
especially those involving water and erosion
Aspect - the direction the slope is facing– provides shaded relief image of surface as
well as being useful for analysis
Hillshade, & Visibility
Interpolation in EpidemiologyExamples
Disease incidence – cancer by county– Lyme disease by state
Risk mapsVector densityOther environmental factors– temperature– concentration of mercury
Choropleth version
Kriged version
Lyme Disease Forecast
Risk Map from Oak Ridge National Labs, Tennessee, U.S. using splineinterpolation
Oak Ridge K-25 Gaseous Diffusion Plant
HighHigh
ModerateModerate
LowLow
Spatial distributionof nymphs in Rhode Island , 1993, based on ordinary kriging estimate of point samples of tick densities in forested habitats(Nicholson & Mather, 1996).
Input PointsAlso called mass points for DEMsInput points vary in terms of– number of points
• high density, low density– location of points
• random, systematic, stratified– accuracy of points and the “support”
They are the basis for the model creation
Input Points - the points from which interpolation is done
Number of Control Points
Location of Control Points
Spatial Sampling Options
systematic - grid systematic - profile
random stratified random
Types of Interpolation
Results are not correct or incorrect so much as they are plausible or absurd.
• Mark Monmonnier
Thiessen PolygonMoving averagesSplinesOptimal interpolation (kriging)
Thiessen Polygons - Overview
Input is irregularly spaced control pointsAssigns a value to a set of polygons, one polygon for each pointThe size and shape of the polygons divides the study area in such a way that the area within each polygon is closer to the polygon’s control point than to any other control point.
Thiessen Polygon example
Thiessen PolygonsPolygons with straight sides that surround one of a set of points insuch a way that all of the area within the polygon is closer to theenclosed point than to any other point in the set.
They supply a way in which you can estimate the amount ofsomething that occured at any point, based on the values providedby a set of input points measured at discrete locations.
Creation of Thiessen Polygons
Extend the midpoint lines until they meet, thus forming the Thiessen polygons. Any place within each polygon can be assumed to have the same value as the point inside the polygon.
Imagine that each of the input points is connected to its nearest neighbors. Then, a perpendicular line is drawn at the midpoint of each of the connecting lines
This set of seven points shows the location of a set of input points used to measure rainfall.
Inverse Distance Weighted Interpolation
An commonly available example of a weighted moving average.Control points may have regular or irregular spacingGoal is to estimate a set of unknown Z values based on the values of the known control points.Next is an example of the required calculations for one Z value.
Inverse Distance Weighted Interpolation Example
The general formula
Z = point to estimate z = known points d = distance from Z to n nearest z’s ∑
∑
=
== n
iai
n
iai
i
d
dz
Z
1
1
1
100z1
Z
z3
z2
z4
110
120
130
Calculate the numeratori 1 2 3 4z 100 120 110 130d2 8 8 2 10z/d2 12.5 15 55 13 ∑ 2d
z = 95.5
Calculate the denominator1/d2 = 1/8 +1/8 + 1/2 + 1/10 = .85
Distance, d, is calculated as straight line distance from each z to Za2 + b2 = c2
E.g. from Z to z1
d1 = , d2 = 844 +
Z = 95.5/.85 = 112
SplinesCharacteristics– Simulate a flexible ruler as used by
draftsmen of old.– Break points allow local changes to be
made without affecting the whole.– Used to smooth digitized contour lines for
better graphic depiction.– Thin plate splines are used to create DEMs
Splines - Good and BadPositives– Handle large amounts of data efficiently– Retain local features– Aesthetically pleasing
Negatives– Too smooth– Artifacts/ very high or low values– No direct estimate of error
Optimal Interpolation - KrigingCharacteristics– recognizes that a single smooth
mathematical equation will not suffice to depict natural variation
– three parts• drift or structure of general trend• small variations, random but related (spatially
autocorrelated)• noise - not autocorrelated and not related to
underlying structure
Kriging - Good & BadPositive– good where relatively few, expensive
control points must be used for interpolation, e.g. ore samples
– accounts for number and location of control points
– provides an estimate of potential error in output
Negative– high computational requirements– too much noise undermines accuracy
Variography
Method used for modeling the spatial structure of the input dataThe variogram is central to this method – it graphs the differences (semi-variance) among input point values across different distances and directionsThe shape of the variogram provides information about the nature of the spatial autocorrelationModeling may be performed with assumption of isotropy (differences same in all directions) or anisotropy (directional trend in differences)
semi-variance –measure of the degree of spatial dependence between samples; squared diff of points at distance h
sill – maximum variance, values not autocorrelatedafter this point
range – distance (lag) at which sill is reached
nugget – noise or intrinsic error
spatial lag examples
SemivariogramSemivariogram of distances between samples of distances between samples (Nicholson & (Nicholson & MatherMather, 1996), 1996)
Sem
ivar
ianc
eSe
miv
aria
nce
11
22
33
44
1000010000 2000020000 3000030000 4000040000 5000050000Distance (meters)Distance (meters)
Types of kriging
Ordinary – estimated mean, AKA punctual kriging. Most commonly used type, overallSimple – known meanUniversal – polynomial regression with x, y as independent variables, used when trend is presentIndicator, probability, disjunctive– non-linear forms, check for values above a particular valueCo-kriging – involves multiple variables
Hypothetical Surface
This is reality.
100 randomly located sample points hold Z values for those places
Data DistributionTrue distribution of Z value
Distribution of 285point sampleDistribution of 100 point sample
Trend analysisN-S trend blue; E-W trend green. 2nd order polynomial
More focusedGlobal
Variography
Before trend removal
Range: 8800 m –distance where autocorrelation ends
Sill: 41.4 m – squared difference at 8800 m
Center is lowest lag distance. Blue/green = similarity; Orange/red = dissimilarity
After trend removal
Surface from 100 random points, default parameters
Goal – Good model
Prediction Error IdealsMean error = 0
RMS and average standard error are small
RMS standardized error = 1
Mean=0.01916; RMS= 11.32; RMS Stnd=0.9216
Surface from 285 random points, default parametersMean=0.010462; RMS= 9.159; RMS Stnd=0.7312
Surface from 285 random points, trend removedMean=-0.00826; RMS= 7.491; RMS Stnd=1.405
Surface from 285 random points, trend removed
Surface from 100 random points, default parameters
Surface from 285 random points, default parameters
Using additional input points and removing trends improved the model, i.e. reduced prediction error
Prediction standard error is lower closer to original input points. Measures how well the surface matches the model.
Predicted – Actual values
Error measures how well the surface matches a set of independent values.
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