Biased Kriging: A Theoretical Development

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.