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Biased Kriging: A Theoretical Development

  • Carol Spease
  • James R. Carr
Part of the Computer Applications in the Earth Sciences book series (CAES)

Abstract

Kriging was developed to be a best linear unbiased estimator using a theoretical development to assure a minimum variance of estimation error. The Lagrangian function which assures this minimization constrained such that the weights (λ) sum to one (unbiasedness) is
$$ L({\lambda _i},\mu ) = {\sigma ^2} - 2\sum\limits_i {{\lambda _i}\sigma \left( {{x_O}{x_i}} \right)} + \sum\limits_i {\sum\limits_j {{\lambda _i}{\lambda _j}} \sigma \left( {{x_O}{x_j}} \right) - 2} \mu \left( {\sum {{\lambda _i} - 1} } \right) $$
(A)
In (A), biasedness can be introduced by changing (∑λ-l) to (∑λ-N), where N is the new sum of weights. Yet, differentiating either equation with respect to λ and ∑μ results in formula
$$ \sum\limits_i {\sum\limits_j {{\lambda _i}} \sigma \left( {{x_i}{x_j}} \right) - } \mu = \sum\limits_i {\sigma \left( {{x_O}{x_i}} \right)} $$
(B)
Hence, the same kriging system is used except N is introduced in the right-hand vector instead of 1. This allows each covariance value, σ, in (B) to be computed using a variogram, as with unbiased kriging. Biased kriging is useful for favoring a particular portion of a histogram. By allowing the sum of weights to be greater than one, as an example, the high end of the histogram can be favored.

Keywords

Lagrangian Function Peak Acceleration Foundation Design Earthquake Resistant Design Reserve Estimation 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. Knudsen, H.P., and Kim, Y.C., 1977, A short course on geostatistical ore reserve estimation: Dept. Mining and Geol. Engineering, Univ. Arizona, 224 p.Google Scholar
  2. Matheron, G., 1963, Principles of geostatistics: Econ. Geology, v. 58, no. 8, p. 1246–1266.CrossRefGoogle Scholar
  3. Trifunac, M.D., and others, 1973, Strong motion earthquake accelerograms, digitized and plotted data, v. II. Corrected accelerograms and integrated ground velocity and displacement curves, pts C-S: California Inst. Technology, prepared for the National Science Foundation, February 1973, distributed by the National Technical Information Service, United States Department of Commerce, 2000 p.Google Scholar

Copyright information

© Plenum Press, New York 1988

Authors and Affiliations

  • Carol Spease
    • 1
  • James R. Carr
    • 1
  1. 1.University of MissouriRollaUSA

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