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Integer linear programming for mining systems
Published in Amit Kumar Gorai, Snehamoy Chatterjee, Optimization Techniques and their Applications to Mine Systems, 2023
Amit Kumar Gorai, Snehamoy Chatterjee
The ordinary kriging was used for the grade estimation. The spatial correlation of the data was measured using the variogram analysis. Both the omnidirectional and direction variograms were studied for checking the anisotropy. A spherical variogram model was fitted for modelling the spatial continuity of all lithology. A cross-validation exercise optimally chose variogram’s parameters. After fitting the variograms, the search for anisotropy was carried out. The variograms were calculated at eight different directions with 22.50 tolerance. The directional difference was not obtained for the anisotropic modelling. Therefore, the isotropic model for ordinary kriging was used for modelling purposes. The blocks of the deposit for all seven lithologies were estimated using the ordinary kriging estimation technique. The ore grade map of the deposit was estimated bench-wise. A total number of 14 ore grade maps were generated for attribute Fe. Figure 5.17 shows the grade maps of Fe for the 1172 m bench. The number of blocks within the deposit that could fall within the ultimate pit of the deposit is 49603, and the total volume, tonnage, and average grade are 198,412,000 m3, 567.46 Million Tonne (MT), and 64.86%. The generated orebody model was used for production scheduling.
Estimation of grids, areas and volumes of intersection
Published in Martin Lloyd Smith, Geologic and Mine Modelling using Techbase and Lynx, 2020
The nugget effect is not actually a model, but is included in this discussion since it is almost always included as the first element of a nested model in kriging routines. As explained in Chapter 3, the nugget effect represents that portion of the variance in a variogram which cannot be explained by other models. Usually, the nugget effect accounts for very short-scale variability and sampling and analytic error and must be estimated by interpolating a slope from the first few lags of the experimental variogram to h = 0. Occasionally, the experimental variogram values fluctuate consistently around a constant value implying that there is no spatial correlation. This is referred to as a pure nugget effect and can be modeled as shown in Example 5.9; using kriging with a pure nugget effect is equivalent to a simple average. Inexperienced modelers working with difficult data and limited variogram programs often mistake erratic experimental variograms for pure nugget effects. It should be noted that pure nugget effects are uncommon, and when they are encountered, they are more likely to be the result of bad data or data taken from mixed populations. A true pure nugget effect will display very little fluctuation between lags from the sill/variance of the data.
Points
Published in Christopher M. Gold, Spatial Context: An Introduction to Fundamental Computer Algorithms for Spatial Analysis, 2018
In order to find the values γ1,1, γ1,2, etc. we must be able to have a model variogram to fit the experimental one. Common models are: circular; exponential; Gaussian; and linear – geostatistical packages have functions to fit these models to an experimental variogram. Several terms are worth remembering. The ‘range’ of the model is the distance h at which the variogram does not change significantly any more – points further away have no meaningful influence – a bit like a ‘counting circle’, while the ‘sill’ is the variance value at that point (see Figure 102). The ‘nugget’ is the variance at the origin: if this is not zero then in theory repeated sampling at the same location would give different results – this may be due to measurement errors or surface variation smaller than the sampling interval. For high-precision surveys, for example, a significant nugget effect would be worrying, but for samples of soil chemistry it would be expected.
Determination of representative volume element with consideration of linear anisotropy using geostatistics approach
Published in European Journal of Environmental and Civil Engineering, 2023
Wen Zhang, Shuonan Wang, Shengyuan Song, Chun Tan, Zhifa Ma, Bo Shan, Peihua Xu
A variogram is a description of the spatial continuity of the data. The experimental variogram is a discrete function calculated using a measure of variability between pairs of points at various distances. The exact measure used depends on the variogram type selected. To evaluate the randomness of fracture frequency in different directions, we must use a basic model to fit this series of the γ*(h) values. In Figure 12, the curves go flat and the experimental variogram values measured by changing the value of lag h in each direction are almost equal to the nugget value. According to the characteristics of these curves, we can conclude that the fracture frequency variogram model in the specified direction is the pure nugget effect model, which is the simplest variogram. The general formula for this model is as follows: where C0 (C0 > 0) is the priori variance. This model is equivalent to the regionalized variable being a random distribution. The covariance function between samples is equal to 0 for all lag distance h, that is, the spatial autocorrelation of variables does not exist. The variogram in the direction fluctuates minimally; thus, the experimental variogram of the fracture frequency in each direction can be regarded as a constant.
Evaluation of the potential corrosivity of groundwater using an Analytic Hierarchy Process-based index
Published in Urban Water Journal, 2023
Since they can produce reliable results, the most preferred variogram models in the literature are the exponential model, the spherical model, and the Gaussian model. In this study, different experimental variograms were used to map the spatial distribution of the corrosive activity of groundwaters. To get the most accurate spatial distribution map, different peak value models (the spherical model, the exponential model, the Gaussian model, and the linear model) are tested. The exponential model has been determined as the most appropriate model according to the standardized root mean square residual (SRMR). In the next step, the spatial distribution maps drawn in vector format were converted to raster maps. The raster maps have been divided into four different subclasses by the corrosion class and reclassified by assigning sensitivity values. Spatial distribution maps of the data obtained from methods such as Langelier saturation index (LSI), Aggressive index (AI), and Larson–Skold index (LS), which are widely used in the literature to describe the corrosivity of water, were also produced.
A semi-analytical random shakedown solution for pavements with spatial variability
Published in International Journal of Pavement Engineering, 2022
R. Rahmani, S. M. Binesh, M. Rafiei
As illustrated in Figure 4, when , the variogram exhibits a nugget effect (some value greater than zero), meaning that there is a weak spatial structure even at small separation distances. In addition, the variogram shape (rapid changes) for is linked directly to the roughness of the field. However, as levels up, the intercept value on the vertical axis moves to zero, indicating much more similarity/continuity between adjacent (neighbour) locations. Moreover, as moves from 1a to 16a, the initial slope of the experimental variograms flattens out meaning that a larger area of sample locations are spatially auto-correlated.