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Chapter 16 - Spatial Interpolation

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Title: Chapter 16 - Spatial Interpolation


1
Chapter 16 - Spatial Interpolation
  • Triangulation
  • Inverse-distance
  • Kriging (optimal interpolation)

2
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

3
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.

4
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)
5
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

6
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

7
Local, deterministic methods
Define an area around the point
Find data point within neighborhood
Choose model
Evaluate point value
8
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)

9
Thiessen polygons
Delauney Triangulation
10
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

11
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.

12
This is what you might see on screen
13
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.

14
Creating Polygon Theme
15
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.
16
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.

17
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.

18
(No Transcript)
19
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.

20
Zinc Concentration
Symbolize the grid with two-color ramp based on
Zn concentration
21
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.

22
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.

23
Density function output
24
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
25
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?
26
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

27
Contouring
  • create a contour based on the surface from IDW

28
IDW and Contouring
29
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

30
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

31
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
32
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

33
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

34
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

35
semivariance
20
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
36
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)
37
Modifications (in real world)
  • Tolerance - direction and distance needed to be
    considered

10m
1m
20o
5m
A
38
semivariance
  • the semivariance increases as h increases
    distance increases -gt semivariance increases
  • nearby points to be more similar than distant
    geographical data

39
data no longer similar to nearby values
sill
range
h
40
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
41
  • 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)

42
  • 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.

43
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
44
  • 0.15(50)0.55(40) 0.30(25)
  • 37

45
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.
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