Surrogates and Kriging Part I: Kriging Ralf Lindau

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Surrogates and KrigingPart I Kriging Ralf
Lindau
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Contents

• Stepwise Kriging (the future)
• Simple Kriging (the past) avoiding negative
  weights
• Victorian LWP
• Stefanian Permeability

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Summary of 18.5.2009

• Distingish between two types
• 1. downscaling of averages (true downscaling)
• 2. downscaling of point measurements
  (interpolation)
• Example for average downscaling
• Precipitation from 60 to 30 min
• Three methods for point downscaling
• 1. Linear interpolation plus noise
• 2. Stepwise kriging
• 3. Stepwise spatio-temporal data construction

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Stepwise Kriging

• The covariances of a new kriging point to all old
  observation points are correct by definition.
• However the explained variance is smaller than 1
  (normalized case).
• This leads to an underestimation of the
  correlation.
• Thus
• Do not use the kriging technique several times in
  series for all intermediate points.
• But
• Predict only a single point
• Correct its variance by adding noise
• Consider in the next step the predicted value as
  an old one.

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Sequential vs Stepwise

• J.A. Vargas-Guzman, T.-C. Jim Yeh, 1999
• Sequential kriging and cokriging Two powerful
  geostatistical approaches
• Stochastic Environmental Research and Risk
  Assessment, 13, 416-435.
• The sequential estimator is shown to produce the
  best, unbiased linear estimate, and to provide
  the same estimates and variances as classic
  simple kriging or cokriging with the simultaneous
  use of the entire data set. However, by using
  data sets sequentially, this new algorithm
  alleviates numerical difficulties associated with
  the classical kriging or cokriging techniques
  when a large amount of data are used.
• The expression Sequential Kriging means here
• Decomposition of a huge matrix into several
  smaller matrices, which can be solved easier.
• The presented method of Stepwise Kriging means
  something else
• Add the information and correct the variance
  stepwise, data point by data point.

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Stepwise Kriging

• We proudly present
• The algorithm is readily programmed and awaits
  eagerly to work.

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• So far the future.
• Now, the past two examples for simple kriging.
• 1. Clouds (Victor)
• 2. Soil (Stefan)
• Most important difference to OK
• No trend no gradient, only (normalized)
  anomalies.
• No negative weights dont panic, no more
  discussion about that.

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Negative weights in literature

• Deutsch, C.V., 1996 Correcting for Negative
  Weights in Ordinary Kriging, Computers
  Geociences, 22, (7), 765-773.
• Negative weights in ordinary kriging (OK) arise
  when data close to the location being estimated
  screen outlying data. Depending on the variogram
  and the amount of screening, the negative weights
  can be significant there is nothing in the OK
  algorithm that alerts the kriging system about
  the zero threshold for weights. Negative weights,
  when interpreted as probabilities for
  constructing a local conditional distribution,
  are nonphysical. Also, negative weights when
  applied to high data values may lead to negative
  and nonphysical estimates. In these situations
  the negative weights in ordinary kriging must be
  corrected.
• An algorithm is presented to reset negative
  kriging weights, and compensating positive
  weights to zero. The sum of the remaining nonzero
  weights is restandardized to 1.0 to ensure
  unbiasedness. The situations when this correction
  is appropriate are described and a number of
  examples are given.
• Deutsch presented a steamroller method to avoid
  negative kriging weights.
• My method is slightly -) more elegant.

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Autocorrelation

• Difficulties arise for cumulus clouds, where only
  a small number of non-zero observations may
  occur.
• In this case the autocorrelation function is not
  derivable.
• Solution Kriging is based on the mean
  autocorrelation for all cumulus cases.

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Victorian LWP
measurement
measurement
krig
error
error
krig
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Concluding the total variance

• Lines have a smaller variance than the
• entire field.
• Why?
• Divide a field into several lines.
• Then the total variance is equal to
• internal
• the mean variance inside the lines
• plus external
• the variance of the lines means
• Technically, there are more short distances
• in lines than in the entire area. Multiply the
• relative frequencies of distances with their
• corresponding variance. Then the expected
• total variances (lines/area) can be concluded.
  

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Stefanian Permeability
Original
Kriged (the truth)
Semivariogram
Org
0.697
0.378
0.319
Krig
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Nice effects of kriging

• Half of the variance is attributed to errors.
• This error variance can be interpreted in two
  ways
• Pure inaccuracy of measurements. (Observation
  error)
• Point measuremnts are less representative for the
  entire grid box. (Small scale variability)
• For both cases the reduction of variance ís
  necessary.
• The kriged field is superior compared to the the
  original.
• It can be called Truth for this spatial
  resolution

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Only the edge
Original
Kriged
0.839
0.553
0.286
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Random 31
Original
Kriged
0.670
0.425
0.245
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True, Edge, 31
Total Error Remain
True
0.697 0.319 0.378
0.839 0.553 0.286
Edge
0.670 0.425 0.245
31
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Summary
Stepwise kriging is ready to go.

• It is the difficulty to calculate total and error
  variance from a reduced data amount, which
  defines mainly how much variability is remaining
  in the obtained kriging result.
• Strong smoothing may occur. But the main reasons
  are underestimation of the total variance or
  overestimation of the error variance. Not the
  kriging method itself.