Department of Geography and Urban Studies, Temple University GUS 0265/0465 Applications in GIS/Geographic Data Analysis

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Department of Geography and Urban Studies, Temple
UniversityGUS 0265/0465Applications in
GIS/Geographic Data Analysis
Lecture 6 Point Interpolation
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Interpolating a Surface From Sampled Point Data

• Assumes a continuous surface that is sampled
• Interpolation
• estimating the attribute values of locations that
  are within the range of available data using
  known data values
• Extrapolation
• estimating the attribute values of locations
  outside the range of available data using known
  data values

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Interpolation
Estimating a point here interpolation
Sample data
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Extrapolation
Sample data
Estimating a point here extrapolation
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Sampling Strategies for Interpolation
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Linear Interpolation
If A 8 feet and B 4 feet then C (8
4) / 2 6 feet
Sample elevation data
A
C
B
Elevation profile
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Non-Linear Interpolation
Sample elevation data
Often results in a more realistic interpolation
but estimating missing data values is more complex
A
C
B
Elevation profile
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Exact Interpolation
Sample elevation data
Interpolated surface passes through the original
data points
Elevation profile
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Inexact Interpolation
Sample elevation data
Interpolated surface does not pass through all of
the original data points
Elevation profile
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Global Interpolation
Uses all known sample points to estimate a value
at an unsampled location
Sample data
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Local Interpolation
Uses a neighborhood of sample points to estimate
a value at an unsampled location
Sample data
Uses a local neighborhood to estimate value, i.e.
closest n number of points, or within a given
search radius
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Trend Surface
Global method Inexact Can be linear or
non-linear Multiple regression (e.g. predicting a
z elevation value dependent variable with x and
y location values independent variables)
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1st Order Trend Surface
In one dimension z varies as a linear function
of x
z
z b0 b1x e
x
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1st Order Trend Surface
In two dimensions z varies as a linear function
of x and y
z
y
z b0 b1x b2y e
x
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2nd Order Trend Surface
In one dimension z varies as a non-linear
function of x
z b0 b1x b2x2 e
z
x
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Higher Order Trend Surfaces in Two Dimensions
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Splines
Local method Can be exact or inexact Non-linear De
rived from the use of flexible rulers for
drafting Fits a piece-wise polynomial function
to a neighborhood of sample points Typically
produces a smooth surface
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Thin Plate Spline
Minimizes curvature while ensuring interpolated
surface passes through the sampled points Can
adjust the tension of the spline to address
overshoot problem where estimates are too high
due to the modeled curvature of the surface
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Splines
In one dimension
In two dimensions
z
y
y
x
x
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Inverse Distance Weighted (IDW)
Local method Exact Can be linear or
non-linear The weight (influence) of a sampled
data value is inversely proportional to its
distance from the estimated value
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Inverse Distance Weighted (IDW)

• In English
• Find the neighboring sample points of the target
  location (i.e. through n nearest neighbors or a
  search radius)
• Find the distance from each sample point to the
  target location
• Weight each sample point according to the inverse
  of its distance from the target location taken to
  the r exponent
• Average the weighted attribute values of the
  sample points and assign the resulting value to
  the target location

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Inverse Distance Weighted (IDW) Example
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IDW Closest 3 neighbors, r 2
4
3
160
2
200
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Inverse Distance Weighted (IDW) Example
Weights
1 / (42) .0625 1 / (32) .1111 1 / (22) .2500
A BC
A 100
4
B 160
3
2
C 200
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Inverse Distance Weighted (IDW) Example
Weights
1 / (42) .0625 1 / (32) .1111 1 / (22) .2500
A BC
The weight inverse of the distance squared
A 100
4
B 160
3
2
C 200
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Inverse Distance Weighted (IDW) Example
Weights
Weights Value
1 / (42) .0625 1 / (32) .1111 1 / (22) .2500
.0625 100 6.25 .1111 160 17.76 .2500
200 50.00
A BC
Total .4236
A 100
4
6.25 17.76 50.00 74.01
B 160
3
74.01 / .4236 175
2
C 200
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Kriging
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Kriging

• A geostatistical method of interpolation that
  incorporates three components of variation
• A spatially correlated component
• A drift or trend structure
• Random error

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The Nature of Spatial Variation
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Calculating Semivariance
Kriging uses the semivariogram Semivariance is a
measure of the degree of spatial dependence
between samples The sampled semivariance is the
average difference in value between observations
a certain distance apart.
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Semivariogram
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Semivariogram Models
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Ordinary Kriging
estimates the value at an unsampled location by a
weighted average of the values of nearby sampled
points. weights the sampled points by the
semivariance of the distance between each of the
sampled points and the unsampled location, AND
the semivariances of the distances among all
paired combinations of sampled points. results
in the estimation of kriging variance, typically
mapped as the kriging standard deviation, which
may be interpreted as the reliability of the
estimate at each location.
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Ordinary Kriging
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Ordinary Kriging
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Other Types of Kriging
Block Kriging Non-Linear Kriging Ordinary Kriging
with Anisotropy Co-Kriging Universal
Kriging Indicator Kriging