Chapter 16 - Spatial Interpolation

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Chapter 16 - Spatial 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
Local
Stochastic
Deterministic
Stochastic
Deterministic
Regression (inexact)
Kriging (exact)
Trend surface (inexact)
Thiessen (exact) Density estimation(inexact) Inver
se distance weighted (exact) Splines (exact)
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Global Interpolation

• Global 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.
• Two types of global Trend surface and regression
  methods

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Trend Surface Analysis

• Approximate points with known values with a
  polynomial equation.
• See Box 16.1
• 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
Find data point within neighborhood
Choose model
Evaluate point value
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Thiessen Polygon (nearest neighbor)

• Any point within a polygon is closer to the
  polygons 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 g\classes\4650_5650\data\3-22\Soil_poll.dbf
  and import it to the project.
• After importing the table into the project, you
  need to create an event theme based on this table
• Go to Tools gt Add XY Data and make sure the
  Easting is shown in X and Northing is in
  Y. (Dont worry the Unknown coordinate
• Click on OK then the point theme will appear on
  your project.

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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
time2)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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Extent and Cell Size

• Go to Spatial Analyst gt 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 gt Distance gt Allocation.
• In Assign to, select soil_poll Event and
• Change the default cell size to 0.1
• click OK to create cell in temporary folder.

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Join Tables

• Join soil_poll Events to Alloc3 grid file by
  ObjectID in Alloc3 and OID in soil_poll
  Events.
• Click Advanced button. Two options are
  available for joining tables.
• Open Attribute of Alloc3 (name may vary) and
  view the joined fields.

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Zinc Concentration
Symbolize the grid with two-color ramp based on
Zn concentration
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Density Estimation

• Simple method divide total point value by the
  cell size
• Kernel estimation associate each known point
  with a kernel function, a probability density
  function.

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Exercise

• Compute density of the sampling points from
  previous dataset.
• If you use the xy-event points for calculation,
  you might receive error message.
• Convert this layer to shape file before using
  Density function from Spatial Analyst.
• Select your cell size (such as 1,) and search
  radius as 5.

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Density function output
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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

estimated value at an unsampled point n number
of control points used to estimate a grid
point kpower to which distance is
raised ddistances from each control points to an
unsampled point
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Computing IDW
6
Z140
Z260
4
Z440
Z350
2
Use k 1
2 4 6 X
Do you get 49.5 for the red square?
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Exercise - generate a Inversion distance
weighting surface and contour

• Spatial Analyst gt Interpolate to Raster gt Inverse
  Distance Weighted
• Make sure you have set the Output cell size to 0.1

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Contouring

• create a contour based on the surface from IDW

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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) xiltxltxi1
• pj(xi)pj(xi) j0,1,,,,
• i1,2,,,,,,k-1

i1
x1
xk1
x0
xk
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
• select Surface gt Interpolate Grid
• Define the output area and other parameters
• Select Spline in Method field, Zn for Z
  Value Field and regularized as type

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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
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Z1 Z2 Z3 Z4 Z5
10
20 30 35 40 50
10 20 30 40 50
Zi values of the attribute at control
points hmultiple of the distance between control
points nnumber of sample points
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Semivariance
h1, h2 h3 h4
21.88 91.67 156.25 312.50
(Z1-Z1h)2
100 25 25 25 175 8
225 100 100 425 6
400 225 625 4
625 625 2
(Z2-Z2h)2
(Z3-Z3h)2
(Z4-Z4h)2
sum 2(n-h)
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Modifications (in real world)

• Tolerance - direction and distance needed to be
  considered

10m
1m
20o
5m
A
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semivariance

• the semivariance increases as h increases
  distance increases -gt semivariance increases
• nearby points to be more similar than distant
  geographical data

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data no longer similar to nearby values
sill
range
h
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kriging computations

• we use 3 points to estimate a grid point
• again, we use weighted average

w1Z1 w2Z2w3Z3
estimated value at a grid point
Z1,Z2 and Z3 data values at the control
points w1,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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Example
distance
1 2 3 g
Z1(1,4)50
0 3.16 2.24 2.24
1 2 3 g
Z3(3,3)25
0 2.24 1.00
0 1.41
Z(2,2)?
0
Z2(2,1)40
w10.00w231.6w322.422.4 w131.6w20.00w322.410.
0 w122.4w222.4w30.0014.1
?
?10h
h
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• 0.15(50)0.55(40) 0.30(25)
• 37

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Homework 6 due next Thursday midnight.

• Task 1 Chapter 16 tasks
• Task 2
• Calculate volume of contaminated Pb soil in
  Thiessen polygon exercise based on range of every
  50 ppm, assuming soil density of 1.65 g/cm3 and
  only the top 1-foot soil is considered.
• Use IDW to compute the volume of the contaminated
  Pb
• Use Kriging (if its working) to compute
  concentration of Pb
• Compare these three methods and see the
  differences (use same output cell size for all
  three methods)
• In Doc file, describe your selection of cell
  size, search radius and results from different
  choice of cell sizes (if you have time to create
  layers with different cell sizes.