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.