Geostatistics

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Geostatistics
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Agricultural fields often have spatial
relationships due to soils, topology or
management practices. Samples taken near each
other are more likely to be similar than samples
far apart.
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Spatial relationships violate the underlying
assumptions of classical statistics samples can
be taken randomly from a population described by
a normal distribution.
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Geostatistics are mathematical procedures to
recognize and describe spatial relationships that
might exist in a field. To recognize spatial
trends, the sample position is just as important
as the measured value.
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Spatial trend example
Field transect
Value
Distance
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In order to describe a spatial trend within a
field, multiple samples must be taken within the
range of that trend.
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Apparent spatial trend
Field transect
Value
Distance
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Soil Property Variability
Source Wollenhaupt, et al., 1997
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Grid sampling ranges

• 1 ac/sample - 64 m
• 2.5 ac/sample - 100 m
• 4 ac/sample - 127 m

Common commercial sampling resolutions may not
adequately describe spatial relationships.
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Spatial trends

• Isotropic - trend is a function of distance from
  a known (sampled) point only
• Anisotropic - trend is a function of both
  distance and direction from a known point

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Isotropic variability

• Variation in soil properties as function of
  distance can be estimated by comparing samples at
  a range of separation distances.

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Sample pair distances
Each distance is called a lag (h).
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Lag determination
The distance is calculated between all possible
sample pairs. For non-uniform sampling patterns,
the distances are assigned to a lag class for
calculation of the semivariance. eg 10-25 m,
25-50 m, etc.
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Variability at each lag
Semivariance (?) is calculated to describe the
expected deviation from sample values (z) as a
function of distance (h) between sample pairs.
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Semivariance vs. lag
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Semivariogram
A mathematical model of the semivariance as a
function of lag is called a semivariogram. The
model is normally determined by least squares
regression.
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Semivariogram models

• Linear - ?(h) C0bh
• Spherical -
• ?(h) C0C11.5(h/a)-0.5(h/a)3
• Exponential -
• ?(h) C0C11-e(-h/a)
• C - coefficients, a - range

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Semivariogram
Range
Sill
Nugget
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Semivariogram descriptors

• Nugget - variability at zero distance, represents
  sampling and analytical errors
• Range - the extent of spatial trends, distance
  beyond which sampling is random
• Sill - variability of spatially independent
  samples

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Semivariogram
Spatially dependent
Spatially independent
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Semivariogram uses

• Use range to determine maximum sampling distances
• The sill indicates intra-field variability
• The model can be used for interpolation of values
  in unsampled areas