Authors’ Reply to “Comments on The Estimation of Mineralized Veins: A Comparative Study of Direct and Indirect Approaches,” by M. Dagbert

Abstract

There are two main points stemming from Dagbert’s comments (2001). First, he indicates a bias occurs when estimating average grade for large blocks by combination of direct point grade estimates. Second, he proposes an “improved” indirect point estimate to minimize the problem of inconsistent point estimates (negative or huge point grade estimates) frequently observed with the indirect method. He concludes that the improved indirect method should be used for point estimation rather than the direct method even in the presence of positive grade-thickness correlation. Marcotte and Boucher (2001) results clearly indicate the indirect point grade estimates are more accurate in the case of negative grade-thickness correlation coefficient, whereas, the direct estimates are more accurate in the case of a positive grade-thickness correlation coefficient. From here on, we assume, otherwise indicated, a positive grade thickness correlation. The first point concerning the large block bias constitutes a necessary and useful reminder as indeed the combination of direct point estimates is biased for large block grade. To see this, consider the true block grade: \batchmode \documentclass[fleqn,10pt,legalpaper]{article} \usepackage{amssymb} \usepackage{amsfonts} \usepackage{amsmath} \pagestyle{empty} \begin{document} \[G\_{v}\ =\ \frac{\frac{1}{v}{\int}\_{v}T(x)G(x)dx}{\frac{1}{v}{\int}\_{v}T(x)dx}\ =\ \frac{\frac{1}{v}{\int}\_{v}A(x)dx}{\frac{1}{v}{\int}_{v}T(x)dx}\] \end{document}(1) where T(x) and G(x) are the thickness and the grade at point x; A(x) is the accumulation. The combination of direct point estimates gives: \batchmode \documentclass[fleqn,10pt,legalpaper]{article} \usepackage{amssymb} \usepackage{amsfonts} \usepackage{amsmath} \pagestyle{empty} \begin{document} \[G\_{v,dir}{\ast}\ =\ \frac{\frac{1}{v}{\int}\_{v}T{\ast}(x)G{\ast}(x)dx}{\frac{1}{v}{\int}_{v}T{\ast}(x)dx}\] \end{document}(2) where the * indicates an estimate. Each term in the integral of the numerator of equation (2) is a biased estimate of the corresponding term in equation (1). It can be shown (Journel and Huijbregts, 1978), that the bias term in the numerator of equation (2) is approximately E[eT(x)eG(x)], where eT(x) = T(x)−T*(x) and eG(x) = G(x)−G*(x). Thus, \batchmode \documentclass[fleqn,10pt,legalpaper]{article} \usepackage{amssymb} \usepackage{amsfonts} \usepackage{amsmath} \pagestyle{empty} \begin{document} \(\frac{1}{v}\ {\int}_{v}T{\ast}(x)G{\ast}(x)dx\) \end{document} underestimates \batchmode \documentclass[fleqn,10pt,legalpaper]{article} \usepackage{amssymb} \usepackage{amsfonts} \usepackage{amsmath} \pagestyle{empty} \begin{document} \(\frac{1}{v}{\int}_{v}T(x)G(x)dx\) \end{document} by the quantity \batchmode \documentclass[fleqn,10pt,legalpaper]{article} \usepackage{amssymb} \usepackage{amsfonts} \usepackage{amsmath} \pagestyle{empty} \begin{document} \(\frac{1}{v}{\int}\_{v}[e\_{T(x)}e_{G(x)}]dx\) \end{document} which is normally positive for positive grade-thickness correlation. Note that for point estimation, the integrals disappear and the direct point grade estimate is simply G*(x), therefore unbiased. For small blocks, T(x) as …