Takafumi Kubota Okayama University

1
Using Geometric Anisotropy in Variogram Modeling

• Takafumi Kubota Okayama University
• Tomoyuki Tarumi Okayama University

2
Outline

• Motivation
• Variogram
• Anisotropy
• 2 types of anisotropy
• Detection anisotropy
• Fit to ellipse by least square method
• Collection anisotropy
• Change coordinate and Kriging
• Applying to practical data
• Methodology of Cross-Validation
• Results of Cross-Validation
• Concluding remarks and future works

3
Motivation

• Gostatistical data analysis
• Spatial prediction
• Thiessen polygons
• Inverse Distance Weighted (IDW)
• Spline interpolation
• Kriging
• variogram
• Variogram (model)
• Predicted values depend on variogram model
• Variogram models depend on directions -gt
  anisotropy
• Detect and correct anisotropy
• Second order stationarity

4
Variogram

• Criteria of spatial autocorelation
• Variance of characteristic value in pairs of
  observations
• Spatial disposition of corresponding points
  (vector)
• distance
• direction
• Assume isotropic variogram
• Spatial disposition(vector -gt scholar)
• distance (only)
• Parameters
• nugget effect
• sill
• range

5
gamma
Variogram cloud
Total.catch (Scallop data)
distance
Empirical variogram
Theoretical variogram
observations
6
anisotropy
7
2 types of anisotropy

• geometric anisotropy
• Sill values are same
• Range values are different
• -gt linear transformation
• zonal anisotropy
• Range values are same
• Sill values are different

8
Detection of anisotropy

• Directions ( )
• 0, 45, 90, 135 degree from North direction
• Tollerance
• 22.5 degree in each direction
• Calculate range values in each direction ( dj )
• 70, 100, 80, 60 (km)
• Least square method

b
a
9
Change coordinate and Kriging

• Change coordinate by psiR, psiA( linear
  transformation )
• Ex.
• psiR 2
• psiA 30 degree from North
• Spatial prediction by Kriging
• Using linear transformed data (as isotropic data)
• Variogram cloud
• ( Empirical variogram )
• Theoretical variogram
• Kriging ( using theoretical variogram as
  regression weight)

10
Applying to practical data

• Scallop data
• 148 scallops
• 1990 survey cruise in the Atlantic continental
  shelf of Long Island, New York
• Change coordinate
• Lat, long -gt Euclidean distance
• Transforming natural logarithm (total catch)
• Collection of Regression model (total catch)

11
Methodology of Cross-Validation (anisotropy)

• Prediction ( anisotropy)
• Remove ith data
• Detection of anisotropy by remaining 147 data
• Calculate ( , d) in each direction
• Fit to ellipse by least square method
• Correction of anisotropy by (psiR, psiA)
• Predict the characteristic value at point of i by
  Kriging
• Comparing between observed value and predicted
  value ( in each parameter)
• Parameters of anisotropy
• Number of direction (ndir) 3, 4, 5, 6
• Tolerance ndir / 90 degree
• Parameter and method in calculating theoretical
  variogram
• Cut off value half of the maximum distance in
  each pair
• Fit theoretical variogram directly to variogram
  cloud
• Model of theoretical variogram spherical model

12
Methodology of Cross-Validation (isotropy)

• Prediction ( anisotropy)
• Remove ith data
• Detection of anisotropy by remaining 147 data
• Calculate ( , d) in each direction
• Fit to ellipse by least square method
• Correction of anisotropy by (psiR, psiA)
• Predict the characteristic value at point of i by
  Kriging
• Comparing between observed value and predicted
  value ( in each parameter)
• Prediction (isotropy)
• Remove ith data
• Predict the characteristic value at point of i by
  Kriging
• Comparing between observed value and predicted
  value

13
Results of Cross-Validation
14
Concluding remarks and future works

• Spatial prediction (Kriging) by using
  (geometric) anisotropy in variogram modeling
• Result (Scallop data)
• ndir 4 and 6 correction of anisotropy gives
  good (small error) prediction
• ndir 3 and 5 correction of anisotropy gives
  bad (large error) prediction
• Cause
• (data dependent problem) directional variograms
  in each ndir (3, 4, 5, 6) contains particular
  directional dependence
• Large range toward to Northern East
• (methodology problem) a part of data is used in
  calculating directional variogram
• Future works
• Relation between correction of anisotropy and
  data size

15
Thank you for your Attention!
Takafumi Kubota kubota_at_law.okayama-u.ac.jp