04 Exercises Variogram and Kriging

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Sustainable Use of Underground Energy Resources 2023-2024 Variogram and Kriging Exercise 1 The figure below shows 15 sample locations spaced 2 m apart. The measurements are available in 11 out of 15 points, the others are missing. Calculate the experimental variogram for 2m, 4m, and 10m. Exercise 2 The figure shows 12 sample locations organized on a regular grid with step 1 m. Measurements of a given property ξ are available at each of the sampling points, with values listed in the figure (units omitted). Calculate the directional experimental variogram of ξ for the two directions x and y for the following classes of separation distances: h1 = 1 m ± 0.1 m; h2 = 1.5 m ± 0.1 m; h3 = 2 m ± 0.1 m. Exercise 3 Given the experimental variogram provided in the following table, estimate the parameters of an exponential variogram representing such data. lag - h 4.98 14.94 24.90 34.85 44.81 54.77 64.73 74.69 84.64 gamma(h) 2.43 6.31 8.32 10.68 12.14 12.92 12.65 12.90 13.53 lag - h 94.60 104.56 114.52 124.48 134.44 144.39 154.35 164.31 174.27 gamma(h) 13.97 15.59 16.66 16.81 16.56 16.49 15.56 13.67 15.24 Exercise 4 Given the following dataset of the spatial random function 𝑌 = K(Ω, x), measured in 2 locations x1 = 5 [m] andx2=20[m], and and knowing that the correlation structure of 𝑌 (which is second order statistically stationary) is described by an exponential semi-variogram with unit sill and integral correlation range of 5[𝑚]. Evaluate the 95% confidence interval of your kriging estimate. i) Estimate the value of 𝑌 in x0 = 10 [m] using ordinary kriging (OK). Exercise 5 The quality of groundwater is measured by an indicator (I) at two wells, A and B. Coordinates of these wells and the corresponding values of I are reported in the table below. Compute the ordinary kriging…