Geostatistics

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Geostatistics

• GLY 560 GIS for Earth Scientists

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Introduction

• Premise
• One cannot obtain error-free estimates of
  unknowns (or find a deterministic model)
• Approach
• Use statistical methods to reduce and estimate
  the error of estimating unknowns (must use a
  probabilistic model)

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Estimator of Error

• We need to develop a good estimate of an unknown.
  Say we have three estimates of an unknown

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Estimator of Error

• An estimator that minimizes the mean square error
  (variance) is called a best estimator
• When the expected error is zero, then the
  estimator is called unbiased.

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Estimator of Error

• Note that the variance can be written more
  generally as

• Such an estimator is called linear

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BLUE

• An estimator that is
• Best minimizes variance
• Linear can be expressed as the sum of factors
• Unbiased expects a zero error
• is called a BLUE(Best Linear Unbiased Estimator)

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BLUE

• We assume that the sample dataset is a sample
  from a random (but constrained) distribution
• The error is also a random variable
• Measurements, estimates, and error can all be
  described by probability distributions

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Realizations
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Experimental Variogram

• Measures the variability of data with respect to
  spatial distribution
• Specifically, looks at variance between pairs of
  data points over a range of separation scales

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Experimental Variogram
After Kitanidis (Intro. To Geostatistics)
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Experimental Variogram
After Kitanidis (Intro. To Geostatistics)
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Small-Scale Variation Discontinuous Case
Correlation smaller than sampling scale Z2 cos
(2 p x / 0.001)
After Kitanidis (Intro. To Geostatistics)
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Small-Scale VariationParabolic Case
Correlation larger than sampling scale Z2 cos
(2 p x / 2)
After Kitanidis (Intro. To Geostatistics)
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Stationarity

• Stationarity implies that an entire dataset is
  described by the same probabilistic process that
  is we can analyze the dataset with one
  statistical model
• (Note this definition differs from that given by
  Kitanidis)

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Stationarity and the Variogram

• Under the condition of stationarity, the
  variogram will tell us over what scale the data
  are correlated.

Correlated at any distance
Uncorrelated
g(h)
Correlated at a max distance
h
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Variogram for Stationary Dataset

• Range maximum distance at which data are
  correlated
• Nugget distance over which data are absolutely
  correlated or unsampled
• Sill maximum variance (g(h)) of data pairs

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Variogram Models
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Kriging

• Kriging is essentially the process of using the
  variogram as a Best Linear Unbiased Estimator
  (BLUE)
• Conceptually, one is fitting a variogram model to
  the experimental variogram.
• Kriging equations may be used as interpolation
  functions.

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Examples of Kriging
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Final Thoughts

• Kriging produces nice (can be exact)
  interpolation
• Intelligent Kriging requires understanding of the
  spatial statistics of the dataset
• Should display experimental variogram with
  Kriging or similar methods