On nonnegative weights of linear kriging estimation

The technique of linear kriging as a grade estimation method of mining blocks is reason¬ably established in the mining industry. During the kriging process, negative kriging weights ([]i) are sometimes encountered. Due to negative kriging weights, negative block grades may be obtained in some extreme situations, which makes no practical sense. To guarantee nonnegative weights, an additional constraint is required in the kriging system of equations. Unfortunately, the nonnegative weight constraint results in a set of nonlinear system of equations. In this paper, a derivation of the resulting set of nonlinear equations is presented and a numerical algorithm is developed for the solution of the equations. Because the algorithm developed does not remove negative weights arbitrarily, optimal nonnegative weights are always guaranteed. Results obtained using the developed algorithm were compared with those using the standard linear kriging, in terms of computer, time and the estimation results. Both the computer time and the kriging variance were found to be higher with nonnegative constraint, as expected. However, some systematic pattern was observed in the kriged values (i.e., over-or underestimation) with or without the nonnegative constraint. The study also demonstrated when one should generally expect negative kriging weights in practice, and what practical impact the negative weights can have on the kriging estimates.