Spatial Interpolation, Overview

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Spatial Interpolation, Overview
Krag Caverly, Junior Student
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(No Transcript)
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Spatial Interpolation
Interpolation is the procedure of predicting the
value of an attribute at unsampled site from the
measurements made at point locations within the
same area or region.
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Principle of Spatial Interpolation
Data close together in space (e.g. elevations)
or time (e.g. temperatures) are likely to be
correlated (related). Many interpolation
procedures and methods are being used in
different fields of science. These methods can
be classified into a few categories.
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Global/Local Interpolations

• Global
• global interpolators determine a single function
  which is mapped across the whole region
• a change in one input value affects the entire
  map
• Local
• local interpolators apply an algorithm repeatedly
  to a small portion of the total set of points
• a change in an input value only affects the
  result within the window

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Exact/Approximate Interpolations

• Exact
• exact interpolators honor the data points upon
  which the interpolation is based. the surface
  passes through all points whose values are known
• Approximate
• used when there is some uncertainty about the
  given surface values
• this utilizes the belief that in many data sets
  there are global trends, which vary slowly,
  overlain by local fluctuations, which vary
  rapidly and produce uncertainty (error) in the
  recorded values
• the effect of smoothing will reduce the effects
  of errors in the surface

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Stochastic/Deterministic Interpolations

• Stochastic / statistical
• stochastic methods incorporate the concept of
  randomness
• the interpolated surface is conceptualized as one
  of many that might have been observed, all of
  which could have produced the known data points
• Deterministic
• deterministic methods do not use probability
  theory

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Statistical prediction

• The sample mean (expectation) is
• The Standard Deviation is (How far away are the
  values from the mean value?

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Gradual/Abrupt Interpolations

• Gradual
• a typical example of a gradual interpolator is
  the distance weighted moving average
• Abrupt
• it may be necessary to include barriers in the
  interpolation process

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Spatial InterpolationLocal Interpolation
Voronoi diagram (nearest neighbor)
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Spatial InterpolationLocal Interpolation
TIN linear interpolation
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Spatial InterpolationDeterministic interpolation
Inverse Distance Weighted
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Spatial InterpolationGeostatistical Interpolation
Kriging
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Spatial InterpolationCombined methods
Topogrid IDW with smoothing
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Inverse Distance Interpolation

• The Inverse Distance Weighted (IDW) interpolation
  assumes that each point has a local influence
  that diminishes with distance.
• Procedure
• Compute distances to all the points in the
  dataset
• Compute the weight of each point.
• Weighting function is the inverse power of the
  distance.
• 3. Compute the weighted average

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Inverse Distance Interpolationexample
Use inverse distance interpolation to calculate
point 0
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Inverse Distance Interpolationexample

• Sum of ? 1
• ?4 is the highest
• ?6 is the lowest

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Inverse Distance Interpolationexample
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Linear interpolation using TIN

• We want to compute the value Zp at point P inside
  a triangle, using the three surrounding points
  P1,P2,P3 with the values Z1 ,Z2, Z3 respectively.
  
• Zp ?1?Z1 ? 2?Z2 ? 3?Z3
• The unique weights ? 1, ? 2, ? 3 are called the
  convex coordinates (or barycentric coordinates)
  of P and can be computed using various methods

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TIN for linear interpolation

• Zp ? 1?Z1 ? 2?Z2 ? 3?Z3
• ? 1Area(?PP2P3)/Area (?P 1P 2P 3),
• ? 2Area(?P1PP3)/Area (?P 1P 2P 3),
• ? 3Area(?P1P2P)/Area (?P 1P 2P 3).
• The area of the triangle can be computed from the
  length of the edges using Heron's formula
• when s (abc)/2 or

p2
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TIN for linear interpolation

• Area (? P 1P 2P 4) 1175.00
• Area (? P2P0P1) 725.00
• Area (? P 1P 0P 4) 400.00
• Area (? P 2P 0P 4) 50.00

? 1 Area(? P2P0P1)/Area (?P 1P 2P 4)
0.617 ? 2 0.34 ? 3 0.042 ZP
0.617110.5 0.34123.1 0.042115.4 114.94 m
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Interpolation methods summary

• There is no 'best' interpolation algorithm that
  is clearly superior to all others and appropriate
  for all applications.
• The quality of the resulting surface is
  determined by the distribution and accuracy of
  the original data points, and the adequacy of the
  underlying interpolation model
• The most important criterion for selecting an
  interpolation method are the degree to which
• structural features can be taken into account,
  and
• the interpolation function can be adapted to the
  varying terrain character.