1 geog4650/5650 – fall 2007 spatial interpolation triangulation inverse-distance kriging (optimal...
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GEOG4650/5650 – Fall 2007Spatial Interpolation
Triangulation Inverse-distance Kriging (optimal interpolation)
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What is “Interpolation”? Predicting the value of attributes at
“unsampled” sites from measurements made at point locations within the same area or region
Predicting the value outside the area - “extrapolation”
Creating continuous surfaces from point data - the main procedures
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Types of Spatial Interpolation
Global or Local Global-use every known points to estimate unknown value. Local – use a sample of known points to estimate unknown
value. Exact or inexact interpolation
Exact – predict a value at the point location that is the same as its known value.
Inexact (approximate) – predicts a value at the point location that differs from its known value.
Deterministic or stochastic interpolation Deterministic – provides no assessment of errors with
predicted values Stochastic interpolation – offers assessment of prediction
errros with estimated variances.
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Classification of Spatial Interpolation Methods
Global
Deterministic Stochastic
Local
Deterministic Stochastic
Thiessen (exact)Density estimation(inexact)Inverse distance weighted (exact)Splines (exact)
Kriging (exact)Regression (inexact)
Trend surface (inexact)
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Global Interpolation use all available data to provide
predictions for the whole area of interest, while local interpolations operate within a small zone around the point being interpolated to ensure that estimates are made only with data from locations in the immediate neighborhood.
Trend surface and regression methods
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Trend Surface Analysis Approximate points with known
values with a polynomial equation. Math equation – you don’t want to
know…. Local polynomial interpolation –
uses a sample of known points, such as convert TIN to DEM
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Local, deterministic methods Define an area around the point to
be predicted finding the data points within this
neighborhood choosing a math model evaluating the point
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Thiessen Polygon (nearest neighbor) Any point within a polygon is closer to
the polygon’s known point than any other known points.
One observation per cell, if the data lie on a regular square grid, then Thiessen polygons are all equal, if irregular then irregular lattice of polygons are formed
Delauney triangulation - lines joining the data points (same as TIN - triangular irregular network)
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Thiessen polygons
Delauney triangulation
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Example data set soil data from Mass near the village of
Stein in the south of the Netherlands all point data refer to a support of
10x10 m, the are within which bulked samples were collected using a stratified random sampling scheme
Heavy metal concentration measured
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Exercise: create Thiessen polygon for zinc concentration
Create a new project Copy “classfiles\GEOG4650-Li\data\10-24\
Soil_poll.dbf” and add it to the project. After add the table into the project, you
need to create an event theme based on this table
Go to Tools > Add XY Data and make sure the “Easting” is shown in “X” and “Northing” is in “Y”. (Don’t worry the “Unknown coordinate”
Click on OK then the point theme will appear on your project.
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Exercise1) Try to plot this point data based on Zinc concentration(Try “Graduate Color” and “Graduate Symbol”)
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This is what you might see on screen
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Create a polygon theme The next thing you need to do is
provide the Thiessen polygon a boundary so that the computing of irregular polygons can be reasonable
Use ArcCatalog to create a new shapefile and name it as “Polygon.shp”
Add this layer to your current project. Use “Editor” to create a polygon.
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Creating Polygon Theme
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Notes: 1)Remember to stop Edits, otherwise your polygon theme will be under editing mode all the time
2)Remember to remove the “selected” points from the “Soil_poll_data.txt”. If you are done so, your Thiessen polygons will be based on the selected points only.
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Thiessen Polygon from ArcToolBox
In Arctoolbox | Analysis Tools | Proximity | Create Thiessen Polygon
Make sure the soil_poll event is the Input Features and output to your own folder. Select “All” for Output Fields.
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Result from Thiessen polygon
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Create Thiessen Polygons from Spatial Analyst: Set Extent and Cell Size
Go to “Spatial Analyst > Options” and click on tab and use “Polygon” as the “Analysis Mask”.
If the Analysis Mask is not set, the output layer will have rectangular shape.
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Thiessen Polygon from Spatial Analyst
Select Spatial Analyst > Distance > Allocation.
In “Assign to”, select “soil_poll Event” and click OK to create cell in temporary folder.
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Vector vs Raster Results Polygons from “Analysis Tools” are
vector polygons with attributes. Polygons from “Spatial Analyst” are
raster polygons with same values inside of each polygon, required to be converted to vector and NO attributes..
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Zinc Concentration
Plot thiessen polygons using zinc concentrations from the attribute table.
Before you plot the map, trim thiessen polygons based on polygon.shp.
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Inverse Distance Weighted the value of an attribute z at some
unsampled point is a distance-weighted average of data point occurring within a neighborhood, which compute:
ki
n
ki
n
i
d
dZZ
/1
/
1
1
Z =estimated value at an unsampled point
n= number of control points used to estimate a grid point
k=power to which distance is raised
d=distances from each control points to an unsampled point
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Computing IDW
2 4 6 X
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4
2
Z1=40
Z2=60
Z4=40Z3=50
24.25 41.12
Do you get 49.5 for the red square?
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Exercise - generate a Inversion distance weighting surface and contour Spatial Analyst >
Interpolate to Raster > Inverse Distance Weighted
Make sure you have set the Output cell size to 50.
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Contouring create a contour
based on the surface from IDW
Spatial Analyst | Surface Analysis | Contour
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IDW and Contouring
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Problem - solution Unsampled point may have a higher data value
than all other controlled points but not attainable due to the nature of weighted average: an average of values cannot be lesser or greater than any input values - solution:
Fit a trend surface to a set of control points surrounding an unsampled point
Insert X and Y coordinates for the unsampled point into the trend surface equation to estimate a value at that point
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Splines draughtsmen used flexible rulers
to trace the curves by eye. The flexible rulers were called “splines” - mathematical equivalents - localized
piece-wise polynomial function p(x) is
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Spline - math functions piece-wise polynomial function
p(x) is p(x)=pi(x) xi<x<xi+1
pj(xi)=pj(xi) j=0,1,,,, i=1,2,,,,,,k-1
i+1
x0
xk
x1 xk+1
break points
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Spline r is used to denote the constraints
on the spline (the functions pi(x) are polynomials of degree m or less
r = 0 - no constraints on function
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Exercise: create surface from spline have point data theme activated Spatial Analyst | Interpolate to
Raster | Spline Define the output area and other
parameters Select “Zn” for Z Value Field and
“regularized” as type and “50” for Output cell size.
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Kriging comes from Daniel Krige, who
developed the method for geological mining applications
Rather than considering distances to control points independently of one another, kriging considers the spatial autocorrelation in the data
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Semivariance ()
10 20 30 40 50
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10Z1 Z2 Z3 Z4
Z520 30 35 40 50)(2
)(1
2
hn
ZZhn
ihii
h
Zi = values of the attribute at control pointsh=multiple of the distance between control pointsn=number of sample points
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Semivariance
hh=1, h=2 h=3 h=4
21.88 91.67 156.25 312.50
(Z1-Z1+h)2
(Z2-Z2+h)2
(Z3-Z3+h)2
(Z4-Z4+h)2
sum2(n-h)
1002525251758
225100100
4256
400225
6254
625
6252
)(2
)(1
2
hn
ZZhn
ihii
h
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semivariance the semivariance increases as h
increases : distance increases -> semivariance increases
nearby points to be more similar than distant geographical data
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data no longer similar to nearby values
h
h
sill
range
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kriging computations we use 3 points to estimate a grid
point again, we use weighted average
Z =w1Z1 + w2Z2+w3Z3
Z= estimated value at a grid point
Z1,Z2 and Z3 = data values at the control pointsw1,w2, and w3 = weighs associated with each control point
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In kriging the weighs (wi) are chosen to minimize the difference between the estimated value at a grid point and the true (or actual) value at that grid point.
The solution is achieved by solving for the wi in the following simultaneous equations
w1(h11) + w2(h12) + w3(h13) = (h1g)
w1(h12) + w2(h22) + w3(h23) = (h2g)
w1(h13) + w2(h32) + w3(h33) = (h3g)
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w1(h11) + w2(h12) + w3(h13) = (h1g)
w1(h12) + w2(h22) + w3(h23) = (h2g)
w1(h13) + w2(h32) + w3(h33) = (h3g)
Where (hij)=semivariance associated with distance bet/w control points i and j.
(hig) =the semivariance associated with the distance bet/w ith control point and a grid point.
Difference to IDW which only consider distance bet/w the grid point and control points, kriging take into account the variance between control points too.
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ExampleZ1(1,4)=50
Z2(2,1)=40
Z3(3,3)=25
Zg(2,2)=?
1 2 3 g
03.162.242.24
02.241.00
01.410
123g
distance
h
=10h
w10.00+w231.6+w322.4=22.4w131.6+w20.00+w322.4=10.0w122.4+w222.4+w30.00=14.1
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=0.15(50)+0.55(40) + 0.30(25) = 37
Z
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Homework 6 – due next Friday midnight (11/2/07)
See website