Geo479/579: Geostatistics Ch13. Block Kriging

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Geo479/579 Geostatistics Ch13. Block Kriging
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Block Estimate

• Requirements
• An estimate of the average value of a variable
  within a prescribed local area
• One method is to discritize the local area into
  many points and then average the individual point
  estimates to get the average over the area
• This method is computationally expensive

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Objective

• See how the number of computations can be
  significantly reduced by constructing and solving
  only one kriging system for each block estimate
• Block Kriging

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Block Kriging

• Block Kriging is similar to the point kriging
• The mean value of a random function over a local
  area is simply the average (a linear combination)
  of all the point random variables contained in
  the local area
• Where VA is a random variable corresponding to
  the mean value over an area A, and Vj are random
  variables corresponding to point values within A

Equation 13.1
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Point Kriging

• In point kriging, the covariance matrix D
  consists of random variables at the sample
  locations and the location of interest

(12.13)
(12.14)
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Point Kriging

• In point kriging, these are point-to-point
  covariances. For block kriging, these are
  point-to-block covariances (the block of
  interest)

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Block Kriging

• Point-to-block covariances required for Block
  Kriging

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Block Kriging

• The covariance between the random variable at the
  ith sample location and the random variable VA
  representing the average value over the area A is
  the same as the average of the point-to-point
  covariances between Vi and the random variables
  at all the points within A

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Block Kriging

• The Block Kriging System
• The average covariance between a particular
  sample location and all of the points within A

Equation 13.3
Equation 13.4
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Block Kriging

• The Block Kriging Variance
• The value C is the average covariance between
  pairs of locations within A

Equation 13.5
Equation 13.6
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Ordinary Kriging Variance

• Calculate the minimized error variance by using
  the resulting to plug into equation (12.8)

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Block Estimates vs. the Averaging of Point
Estimates

• The average of the four point estimates is the
  same as the direct block estimate
• The average of the point kriging weights for a
  sample is the same as the block kriging weight
  for the sample

Figure 13.1
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Varying the Grid of Point Locations within a Block

• When using the Block Kriging approach - How to
  discretize the local area for block being
  estimated?
• The grid of discretizing points should be always
  regular
• The spacing between points may be larger in one
  direction than the other if the spatial
  continuity is anisotropic (Figure 13.2)

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Discretizing Points

• The shaded block is approximated by six points
  located on a 2X3 grid. The closer spacing of the
  points in a north-south direction reflects a
  belief that there is less continuity in this
  direction than in the east-west direction.
  Despite the differences in the east-west and
  north-south spacing, the regularity of the grid
  ensures that each discretizing point accounts for
  the same area, as shown by the dashed line

Figure 13.2
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Discretizing Points

• Discretizing points lt 16, Significant differences
  Discretizing points gt 16, Estimates are similar
  
• Sufficient discretizing points number
• 2D block 4x4 16, 3D block 4x4x4 64

Table 13.2
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Block Kriging vs. Inverse Distance Squared Block
Estimates

• A plus symbol denotes a positive estimation error
  while a minus symbol denotes negative estimation
  error
• The relative magnitude of the error corresponds
  to the degree of shading indicated by the grey
  scale at the top of the figure

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Block Kriging vs. Inverse Distance Squared Block
Estimates
Figures 13.3, 13.4
Figure 13.4
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Case Study

• Comparison of summary statistics for Block
  Kriging and Inverse Distance Weighted
• Inverse Distance Weighted has larger errors
• For Inverse Distance Weighted, there are several
  large overestimation where relatively sparse
  sampling meets much denser sampling
• Inverse Distance Weighted did not correctly
  handle the clustered samples, giving too much
  weight to the additional samples in the
  high-valued areas
• Block Kriging showed some underestimation due to
  its smoothing effect and the positive skewness of
  the distribution of the true block values

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Block Kriging vs. Point Kriging
Figures 13.3, 13.5
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Block Kriging vs. Inverse Distance Squared Block
Estimates
Table 13.3 Estimates
Table 13.4 Errors
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Block Kriging versus Point Kriging
Table 13.5 Errors