Types

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Types

• Indicator Kriging uses thresholds to create
  binary data (0 or 1 values, also called indicator
  values), and then uses ordinary kriging for this
  indicator data.
• Predictions using indicator kriging are
  interpreted as the probability of exceeding (or,
  depending on how the binary variables are
  defined, not exceeding) a threshold.
• Additional Cutoffs can compensate for the loss
  of information caused by coding data with
  indicator functions, but it requires fitting
  cross-covariances which requires more modeling
  decisions and parameter estimation.
• Indicator kriging is not recommended for data
  having a trend.
•  

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Types

• Probability Kriging is considered an improvement
  over indicator kriging by using the original,
  continuous data for ordinary cokriging of the
  indicator data. Probability kriging uses more
  information than indicator kriging so it can be
  more powerful, but it requires fitting
  cross-covariances which involves more modeling
  decisions and parameter estimation. Probability
  kriging is not recommended for data having a
  trend.
•  
• Disjunctive Kriging is a nonlinear method that
  is more general than ordinary kriging and
  indicator kriging. Disjunctive kriging tries to
  do more than ordinary kriging and indicator
  kriging by considering functions of the data,
  rather than using only the data. As usual, to get
  greater rewards requires stronger assumptions.
  Disjunctive kriging assumes all data pairs come
  from a bivariate normal distribution. This
  assumption can be examined in the Geostatistical
  Wizard.

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Types

• Cokriging uses information on several variable
  types. The main variable of interest is Z1, and
  both autocorrelation for Z1 and
  cross-correlations between Z1 and all other
  variable types are used to make better
  predictions.
• It is appealing to use information from other
  variables to help make predictions, but it comes
  at a price.
• Cokriging requires much more estimation, which
  includes estimating the autocorrelation for each
  variable as well as all cross-correlations.
  Theoretically, you can do no worse than ordinary
  kriging because if there is no cross-correlation,
  you can fall back on just autocorrelation for Z1.
  
• But, each time you estimate unknown
  autocorrelation parameters, you introduce more
  variability, so the gains in precision of the
  predictions may not be worth the extra effort.

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Co-Kriging If y and x are highly correlated we
can use the information about x to improve the
prediction of y. If the primary variable of
interest is y, the x values that were sampled can
be used to improve the y predictions at any point
in the region We first need a formal method for
estimating and modeling the correlation Do this
by extension of the covariogram and variogram for
single variables to cross-covariogram and
cross-variogram
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Co-Kriging ( for two variables) A random process
relating to the primary variable A random
process relating to the secondary variable If
both processes are assumed stationary, then the
cross-covariogram is defined as and the
cross-variogram is defined as
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Sample estimator for cross-variogram is given
by This sample cross-variogram is then fit
with a smooth model.
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Prediction of y is performed through a weighted
average of nearby y and x values Where i 1,
, n measurements of Y and j 1,..,m measurements
of X The weights are a function of distance
modified by the variogram and cross-variogram.
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Natural Neighbor

• Natural neighbor interpolation finds the closest
  subset of input samples to a query point and
  applies weights to them based on proportionate
  areas in order to interpolate a value.
• Its basic properties are that it's local, using
  only a subset of samples that surround a query
  point, and that interpolated heights are
  guaranteed to be within the range of the samples
  used. It does not infer trends and will not
  produce peaks, pits, ridges or valleys that are
  not already represented by the input samples.
• It adapts locally to the structure of the input
  data, requiring no input from the user pertaining
  to search radius, sample count, or shape. It
  works equally well with regularly and irregularly
  distributed data.

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Natural Neighbor

• The natural neighbors of any point are those
  associated with neighboring Voronoi (i.e.
  Thiessen) polygons. Initially, a Voronoi diagram
  is constructed of all the given points,
  represented by the olive colored polygons. A new
  Voronoi polygon, beige color, is then created
  around the interpolation point (red star). The
  proportion of overlap between this new polygon
  and the initial polygons are then used as the
  weights.
• By comparison, a distance based interpolator
• such as IDW (Inverse Distance Weighted)
• would assign similar weights to the northern
  most
• point and to the north-eastern point based on
• their similar distance from the interpolation
  point.
• Natural neighbor interpolation, however, assigns
  
• weights of 19.12 and 0.38 respectively which
• is based on the percentage of overlap.

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Interpolation local TIN

• TIN-based linear interpolation
• Value at a given point is a linear combination of
  values at 3 nearby points that form vertices of a
  triangle
• Delaunay Triangulation - proximal method that
  satisfies the requirement that a circle drawn
  through the three nodes of a triangle will
  contain no other node

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Interpolation points to 2m DEM

• Voronoi polygons

Linear on TIN
Properties function not continuous first
derivatives not continuous
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