Question 15 of 23: Simple Kriging – Matrix, Input Vectors, and Weight Sum

Question 4.2: Simple Kriging – Matrix, Input Vectors, and Weight Sum (5 marks)

Question text not reproduced: the examination questions are © Engineers and Geoscientists BC. Open the official past paper (linked at the top of this page) to read the question, then follow the worked solution below.

In the source sketch, sample iii sits closest to the estimation point, with sample i at intermediate distance and sample ii clearly the most distant of the three — that relative geometry is used qualitatively below, since the sketch gives relative positions only, not a coordinate grid or the exact medium-variogram parameters.

4.2.1 The simple-kriging matrix. The square, symmetric n×n matrix of the FITTED variogram (or equivalently covariance) model evaluated between every pair of the n data samples used in the estimate, $$[\Gamma]_{jk} = \gamma(h_{jk}) \quad \text{or} \quad [K]_{jk}=C(h_{jk})$$ — it encodes how correlated the samples are with EACH OTHER (not just with the point being estimated), which is what lets kriging avoid double-counting information from clustered/redundant samples.

4.2.2 Matrix and input vectors. Solving the SK system $$[\Gamma]\,\mathbf{w} = \mathbf{b}, \qquad b_j = \gamma(h_{j0})$$ requires two inputs derived purely from the fitted variogram model and the sample/point geometry: the left-hand matrix [Γ] (sample-to-sample distances h_jk run through the model), and the right-hand vector b (sample-to-ESTIMATION-POINT distances h_j0 run through the same model). Both are obtained by measuring the geometric separations from the sketch/drillhole coordinates and evaluating the already-fitted “medium average representative” variogram at each of those separations — no new field data is needed once the model is fitted. Solving for w gives the weight on each sample; the simple-kriging estimate is then $$Z^*_{SK}=m+\sum_j w_j\,[z(x_j)-m]$$ using the assumed stationary mean m.

4.2.3 Why weights need not sum to unity. Simple kriging assumes a KNOWN, constant mean m and estimates the deviation from that mean as a weighted combination of the samples' own deviations; unbiasedness is guaranteed through the (1−Σw_j) weight implicitly applied to the known mean itself, not by forcing Σw_j=1. Consequently the weights only need to minimize estimation variance, and in general (especially when samples are far from the point, weakly correlated, or redundant with each other) the minimizing weights sum to something LESS than 1, with the shortfall effectively assigned to the known mean — unlike ORDINARY kriging (Question 4.3), which does not assume a known mean and therefore must constrain Σw_j=1 explicitly to remain unbiased.