Geostatistics

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Geostatistics ? Statistics for Geoscientists

• What is it?
• Exploratory (observational) DA, like TSA (as
  hopposed to analytical and numerical methods)
• Dates back to the 60s (Krige (50s), Matheron),
  gproblems in mining
• Overlaps partly with GIS and spatial statistics
  in ggeneral
• While in principal interpolation of irregular
  data, gthere are some points to note
• spatial data location and volume
• incorporates uncertainty (probabilistic method)
• Basis Samples close in space are more similar
  than samples far apart or Variability depends
  only on distance.
• How is it done?
• (look at standard statistics of data)
• determine structure of data (its variability as
  a gfunction of space experimental variogram)
• fit model functions to experimental variogram
• interpolate data - Kriging
• verify / iterate
• (analyse continuous field)

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Step 1 Model Spatial Dependence Characteris
e variability - Variogram Example 1 Rainfall
Data
We have average July rainfall data irregularly
sampled at 262 meteorological stations (1961
1990). We want a continuous rain surface that
could then be used, e.g., for agricultural
suitability assessment (a model or prediction for
precipitation). We note well distributed samples
(support) with a directional trend of the
variable or attribute amount of rain. We now
need to characterise the spatial variability
(continuity) in detail, ending up with a
mathematical description.
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Variogram surface variability(distance,
direction)
Omnidirectional semivariogram variability(distanc
e) Close samples are more similar than distant
ones!!
Lag statistics representativeness of individual
lags
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Another technique h-scatterplot Plots values of
all pairs of samples within a certain distance
range (lag) vs. each other
Perfect correlation 45º line
Again Continuity decreases with distance!
Ideal to detect outliers manipulation of sample
data often required (e.g., non-normal
distributions, sparse sampling)
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Anisotropy requires directional variograms! We
observed highest continuity in 95º
and lowest in 5º. The isotropic
(omnidirectional) variability is somewhere in
between.
These are sample (experimental) variograms. Is
this enough to capture the spatial nature of the
undelying field? Covariogram, other estimators,
lags, angular ranges,
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Example 2 Elevation Data
227 elevation samples with clearly more complex
spatial dependence
as confirmed by the surface variogram.
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Characteristics of the variogram (and therefore
of the data)
Nugget variance at zero distance which should
be zero but isnt Range Distance at which
max. variance is reached (data considered
decorrelated) Sill Level of max.
variability Anisotropy might thus be manifested
by varying range with direction (constant sill
so-called geometric anisotropy). This is observed
with the elevation data
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Surface variogram Main axes of anisotropy
ellipse are at 132º and 42º
However, the variograms are too irregular Change
angular range (tolerance)
Lags could have been changed too For now, lets
use these experimental variograms not only to
describe the data but to model (interpolate) it
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Step 2 Model Fitting Express variability as a
continuous function Elevation data Isotropic
model
Fit visually and mathematically (algorithmically)
a smooth continuous model to surfaces
variability, I.e., to the experimental
variogram(s) (not yet to the surface itself!) Two
steps 1) Visual fit 2) optimise fit using
algorithm
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Omnidirectional semivariogram for isotropic model
Choosing lag width 95, we obtain a first rough
visual model using a single spherical model
without nugget
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Addition of a nugget and subsequent adjustment of
structure 2
looks better. Fit at close distances is
extremely important! Statistical support for all
lags is good. Would this model also fit the
variogram obtained from smaller lags?
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Different structures (functions) for small and
large lags
Gaussian for high lags, spherical for low ones,
e.g. Model fit, using WLS, finally gives
optimised combined function. In reality, we have
to take into account anisotropy
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Step 3 Surface Interpolation (Prediction) Krigin
g and Validation Example Rainfall
Data
We have a model function for the variogram, now
we want to use it to model the continuous data
(field)
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Ordinary kriging does this by minimising the
variance error between the model (analytical
variogram) and the estimate BLUE. Output
Kriged surface plus field of estimated variances.
New attribute is estimated at each grid cell
(pixel/voxel) based on local neighbours.
Cross validation to judge fit Use model to
predict actual samples
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Predictions at sample points
Variances at sample points
are higher where points are more dispersed.
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The real thing kriged rain field (w/o cross
validation)
And its variance field lowest, where points are
dense.
This does NOT tell about the quality of the
prediction!
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All we can do is decide wether were satisfied
with the fit of the model to the sample data
points, taking their spatial distribution into
account. Superimposing the actual samples on the
prediction might be of further help
In reality, one would of course take real-world
knowledge about the processes into account
fitting different variograms to different regions
of the data might be required. Frequently, cross
validation, variance surface and above plot would
lead to new variogram models iterative
exploratory data analysis!
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Other Issues
Geological Constraints (Data Integration)
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Real 3D examples
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Gold ore grade conc. at mine shafts
Visualising uncertainty where are more samples
needed?
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Sample space