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  • 1
    Electronic Resource
    Electronic Resource
    Spatial rainfall estimation by linear and non-linear co-kriging of radar-rainfall and raingage data (1989)
    Springer
    Stochastic environmental research and risk assessment 3 (1989), S. 51-67 
    Details
    ISSN: 1436-3259
    Keywords: Spatial rainfall estimation ; kriging ; ordinary co-kriging ; disjunctive co-kriging
    Source: Springer Online Journal Archives 1860-2000
    Topics: Architecture, Civil Engineering, Surveying , Energy, Environment Protection, Nuclear Power Engineering , Geography , Geosciences
    Notes: Abstract The feasibility of linear and nonlinear geostatistical estimation techniques for optimal merging of rainfall data from raingage and radar observations is investigated in this study by use of controlled numerical experiments. Synthetic radar and raingage data are generated with their hypothetical error structures that explicitly account for sampling characteristics of the two sensors. Numerically simulated rainfall fields considered to be ground-truth fields on 4×4 km grids are used in the generation of radar and raingage observations. Ground-truth rainfall fields consist of generated rainfall fields with various climatic characteristics that preserve the space-time covariance function of rainfall events in extratropical cyclonic storms. Optimal mean areal precipitation estimates are obtained based on the minimum variance, unbiased property of kriging techniques under the second order homogeneity assumption of rainfall fields. The evaluation of estimated rainfall fields is done based on the refinement of spatial predictability over what would be provided from each sensor individually. Attention is mainly given to removal of measurement error and bias that are synthetically introduced to radar measurements. The influence of raingage network density on estimated rainfall fields is also examined.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01543427
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  • 2
    Electronic Resource
    Electronic Resource
    Co-kriging for stochastic flow models (1989)
    Springer
    Transport in porous media 4 (1989), S. 585-598 
    Details
    ISSN: 1573-1634
    Keywords: Geostatistics ; kriging ; groundwater ; inverse problem
    Source: Springer Online Journal Archives 1860-2000
    Topics: Geosciences , Technology
    Notes: Abstract Co-kriging equations for log-transmissivity and heads are derived for a two-dimensional stochastic model. The behavior of the weights as a function of the unknown value of mean hydraulic gradient J are discussed and the procedure is illustrated by studying the ‘screening’ effects of adjacent measurements and added head measurements. In addition, the bias of the estimator for head values is studied when J is also estimated.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00223629
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  • 3
    Electronic Resource
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    Interval-valued random functions and the kriging of intervals (1988)
    Springer
    Mathematical geology 20 (1988), S. 145-165 
    Details
    ISSN: 1573-8868
    Keywords: interval-valued data ; random function ; kriging
    Source: Springer Online Journal Archives 1860-2000
    Topics: Geosciences , Mathematics
    Notes: Abstract Estimation procedures using data that include some “values” known to lie within certain intervals are usually regarded as problems of constrained optimization. A different approach is used here. Intervals are treated as elements of a positive cone, obeying the arithmetic of interval analysis, and positive interval-valued random functions are discussed. A kriging formalism for interval-valued data is developed. It provides estimates that are themselves intervals. In this context, the condition that kriging weights be positive is seen to arise in a natural way. A numerical example is given, and the extension to universal kriging is sketched.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00890251
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  • 4
    Electronic Resource
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    Spatial prediction and ordinary kriging (1988)
    Springer
    Mathematical geology 20 (1988), S. 405-421 
    Details
    ISSN: 1573-8868
    Keywords: kriging ; nugget effect ; range ; sill ; variogram
    Source: Springer Online Journal Archives 1860-2000
    Topics: Geosciences , Mathematics
    Notes: Abstract Suppose data {Z(s i ):i=1, ..., n} are observed at spatial locations {s i :i=1, ..., n}. From these data, an unknownZ(s 0) is to be predicted at a known locations 0c, or, ifZ(s0) has a component of measurement error, then a smooth versionS(s 0) should be predicted. This article considers the assumptions needed to carry out the spatial prediction using ordinary kriging, and looks at how nugget effect, range, and sill of the variogram affect the predictor. It is concluded that certain commonly held interpretations of these variogram parameters should be modified.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00892986
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  • 5
    Electronic Resource
    Electronic Resource
    New distance measures: The route toward truly non-Gaussian geostatistics (1988)
    Springer
    Mathematical geology 20 (1988), S. 459-475 
    Details
    ISSN: 1573-8868
    Keywords: Projection theorem ; kriging ; scalar product ; heteroscedasticity ; outlier resistance ; variogram ; distance measure
    Source: Springer Online Journal Archives 1860-2000
    Topics: Geosciences , Mathematics
    Notes: Abstract The projection or minimum error norm algorithm does not require that the distance measure be a variogram. In non-Gaussian cases, the traditional variogram distance measure leading to minimization of an error variance offers no definite advantage. Other distance measures, more outlierresistant than the variogram, are proposed which fulfill the condition of the projection theorem. The resulting minimum error norms provide the same data configurations ranking as traditionally obtained from kriging variances. A case study based on actual digital terrain data is presented.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00892989
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  • 6
    Electronic Resource
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    Bootstrapped models for intrinsic random functions (1988)
    Springer
    Mathematical geology 20 (1988), S. 699-715 
    Details
    ISSN: 1573-8868
    Keywords: Regionalized variables ; kriging ; interpolation ; sample reuse ; nonparametric error estimates ; confidence intervals
    Source: Springer Online Journal Archives 1860-2000
    Topics: Geosciences , Mathematics
    Notes: Abstract Use of intrinsic random function stochastic models as a basis for estimation in geostatistical work requires the identification of the generalized covariance function of the underlying process. The fact that this function has to be estimated from data introduces an additional source of error into predictions based on the model. This paper develops the sample reuse procedure called the “bootstrap” in the context of intrinsic random functions to obtain realistic estimates of these errors. Simulation results support the conclusion that bootstrap distributions of functionals of the process, as well as their “kriging variance,” provide a reasonable picture of variability introduced by imperfect estimation of the generalized covariance function.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00890586
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  • 7
    Electronic Resource
    Electronic Resource
    Notes on the robustness of the kriging system (1988)
    Springer
    Mathematical geology 20 (1988), S. 189-203 
    Details
    ISSN: 1573-8868
    Keywords: kriging ; variogram ; robustness ; estimation variance
    Source: Springer Online Journal Archives 1860-2000
    Topics: Geosciences , Mathematics
    Notes: Abstract The robustness of the kriging system with respect to uncertainty of the theoretical variogram is investigated. Inequalities for possible changes of the kriging estimator and the estimation variance are derived. Results of a numerical study show that changes of kriging weights can be predicted partly with the help of the maximal kriging weight.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00890253
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  • 8
    Electronic Resource
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    Bayesian kriging—Merging observations and qualified guesses in kriging (1987)
    Springer
    Mathematical geology 19 (1987), S. 25-39 
    Details
    ISSN: 1573-8868
    Keywords: geostatistics ; kriging ; Bayesian statistics
    Source: Springer Online Journal Archives 1860-2000
    Topics: Geosciences , Mathematics
    Notes: Abstract Frequently a user wants to merge general knowledge of the regionalized variable under study with available observations. Introduction of fake observations is the usual way of doing this. Bayesian kriging allows the user to specify a qualified guess, associated with uncertainty, for the expected surface. The method will provide predictions which are based on both observations and this qualified guess.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01275432
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  • 9
    Electronic Resource
    Electronic Resource
    Production of conditional simulations via the LU triangular decomposition of the covariance matrix (1987)
    Springer
    Mathematical geology 19 (1987), S. 91-98 
    Details
    ISSN: 1573-8868
    Keywords: geostatistics ; kriging ; conditional simulation ; fast Fourier transform
    Source: Springer Online Journal Archives 1860-2000
    Topics: Geosciences , Mathematics
    Notes: Abstract This paper reviews the turning band method and fast Fourier transform method of producing a nonconditional simulation of a multinormal random function with a given covariance structure. A review of the two common methods of conditioning the simulation to honor the data shows that they are formally equivalent. Another method for directly pondering a conditional simulation based on the LU triangular decomposition of the covariance matrix is presented. Computational and implementation difficulties are discussed.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00898189
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  • 10
    Electronic Resource
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    Random vectors and spatial analysis by geostatistics for geotechnical applications (1987)
    Springer
    Mathematical geology 19 (1987), S. 467-479 
    Details
    ISSN: 1573-8868
    Keywords: random vectors ; spatial analysis ; vector variogram ; kriging ; rock fracture orientation ; structural modeling
    Source: Springer Online Journal Archives 1860-2000
    Topics: Geosciences , Mathematics
    Notes: Abstract Geostatistics is extended to the spatial analysis of vector variables by defining the estimation variance and vector variogram in terms of the magnitude of difference vectors. Many random variables in geotechnology are in vectorial terms rather than scalars, and its structural analysis requires those sample variable interpolations to construct and characterize structural models. A better local estimator will result in greater quality of input models; geostatistics can provide such estimators: kriging estimators. The efficiency of geostatistics for vector variables is demonstrated in a case study of rock joint orientations in geological formations. The positive cross-validation encourages application of geostatistics to spatial analysis of random vectors in geoscience as well as various geotechnical fields including optimum site characterization, rock mechanics for mining and civil structures, cavability analysis of block cavings, petroleum engineering, and hydrologic and hydraulic modelings.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00896915
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  • 11
    Electronic Resource
    Electronic Resource
    Estimation of the generalized covariance function: II. A response surface approach (1987)
    Springer
    Mathematical geology 19 (1987), S. 769-783 
    Details
    ISSN: 1573-8868
    Keywords: Cross-validation ; geostatistics ; intrinsic random functions of orderk ; kriging
    Source: Springer Online Journal Archives 1860-2000
    Topics: Geosciences , Mathematics
    Notes: Abstract A procedure is proposed that employs first-moment estimation (kriging), cross-validation, and response surface analysis to estimate parameters of a generalized covariance function. Results from application of this procedure to two data sets are given.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00893014
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  • 12
    Electronic Resource
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    An interpolation method taking into account inequality constraints: II. Practical approach (1986)
    Springer
    Mathematical geology 18 (1986), S. 53-73 
    Details
    ISSN: 1573-8868
    Keywords: kriging ; mapping ; quadratic programming ; isopachs
    Source: Springer Online Journal Archives 1860-2000
    Topics: Geosciences , Mathematics
    Notes: Abstract Common features of models for interpolation, consistent with a finite number of inequality constraints on the range of values of a variablez, are discussed. A method based on constrained quadratic minimization yielding kriging estimates when no constraints exist, is presented. A computationally efficient formulation of quadratic minimization is obtained by using results on duality in quadratic programming. Relevant properties of the optimal interpolator are derived in a simple, self-contained way. The method is applied to mapping of horizon depth and estimation of thickness of an oil-bearing formation.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00897655
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  • 13
    Electronic Resource
    Electronic Resource
    An interpolation method taking into account inequality constraints: I. Methodology (1986)
    Springer
    Mathematical geology 18 (1986), S. 33-51 
    Details
    ISSN: 1573-8868
    Keywords: kriging ; splines ; mapping ; prediction
    Source: Springer Online Journal Archives 1860-2000
    Topics: Geosciences , Mathematics
    Notes: Abstract Several kinds of data can provide information about a variable measured on a one- or two-dimensional space; at some points, the value is known to be equal to a certain number. At other points, the only information may be that the variable is greater or smaller than a given value. The theory of splines provides interpolating functions that can take into account both equality and inequality data. These interpolating functions are presented. The parallel between splines and kriging is reviewed, using the formalism of dual kriging. Coefficients of dual kriging can be obtained directly by minimizing a quadratic form. By adding some inequality constraints to this minimization, an interpolating function may be calculated which takes into account inequality data and is more general than a spline. The method is illustrated by some simple one-dimensional examples.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00897654
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  • 14
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    A sensitivity analysis for universal kriging (1986)
    Springer
    Mathematical geology 18 (1986), S. 653-676 
    Details
    ISSN: 1573-8868
    Keywords: approximate analysis ; covariance model ; kriging ; perturbation surface ; sensitivity analysis
    Source: Springer Online Journal Archives 1860-2000
    Topics: Geosciences , Mathematics
    Notes: Abstract Universal kriging is an interpolation method for producing contour maps from irregularly spaced sample points, taking into account the trend (or polynomial drift), which is of known form. It assumes a known covariance model to express correlation of points a short distance apart. A sensitivity analysis examines how the fitted surface will change for given perturbations in the covariance model. We develop a simple approximate analysis in preference to exact analysis and show that it is adequate for small perturbations. For large data sets, a dramatic reduction in computer time is possible using approximate analysis. Possible extensions of this work are noted.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00897095
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  • 15
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    Determination of support in soil sampling (1986)
    Springer
    Mathematical geology 18 (1986), S. 529-537 
    Details
    ISSN: 1573-8868
    Keywords: geostatistics ; kriging ; nugget effect ; soil sampling ; support
    Source: Springer Online Journal Archives 1860-2000
    Topics: Geosciences , Mathematics
    Notes: Abstract Short-range variation of measurements of a soil characteristic is a function of the support (i.e., size, shape, and orientation of the physical sample taken at a sample point) of the soil samples. This short-range variance often shows up as a substantial component of the “nugget effect” in experimental semivariograms or estimated generalized covariance functions. Proper choice of support may substantially reduce short-range variance (or nugget) of measurements made at sample points and thereby reduce estimation variance in point or block kriging based on these measurements. Appropriate variance formulas and an example are given for determination of support when support is an array of vertical cores (or bucket auger samples) of equal depth and diameter.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00914253
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  • 16
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    Matheronian geostatistics—Quo vadis? (1986)
    Springer
    Mathematical geology 18 (1986), S. 93-117 
    Details
    ISSN: 1573-8868
    Keywords: deterministic ; “estimation variance” ; interpolation ; geostatistics ; kriging ; least-squares prediction ; ore deposit assessment ; probabilistic ; semivariogram ; statistical inference
    Source: Springer Online Journal Archives 1860-2000
    Topics: Geosciences , Mathematics
    Notes: Abstract Components of geostatistical estimation, developed as a method for ore deposit assessment, are discussed in detail. The assumption that spatial observations can be treated as a stochastic process is judged to be an inappropriate model for natural data. Problems of semivariogram formulation are reviewed, and this method is considered to be inadequate for estimating the function being sought. Characteristics of bivariate interpolation are summarized, highlighting kriging limitations as an interpolation method. Limitations are similar to those of inverse distance weighted observations interpolation. Attention is drawn to the local bias of kriging and misplaced claims that it is an “optimal” interpolation method. The so-called “estimation variance,” interpreted as providing confidence limits for estimation of mining blocks, is shown to be meaningless as an index of local variation. The claim that geostatistics constitutes a “new science” is examined in detail. Such novelties as exist in the method are shown to transgress accepted principles of scientific inference. Stochastic modeling in general is discussed, and purposes of the approach emphasized. For the purpose of detailed quantitative assessment it can provide only prediction qualified by hypothesis at best. Such an approach should play no part in ore deposit assessment where the need is for local detailed inventories; these can only be achieved properly through local deterministic methods, where prediction is purely deductive.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00897657
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  • 17
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    An application of disjunctive kriging for earthquake ground motion estimation (1986)
    Springer
    Mathematical geology 18 (1986), S. 197-213 
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    ISSN: 1573-8868
    Keywords: disjunctive kriging ; nonlinear geostatistics ; kriging ; earthquake ground motion
    Source: Springer Online Journal Archives 1860-2000
    Topics: Geosciences , Mathematics
    Notes: Abstract For earthquake ground motion studies, the actual ground motion distribution should be reproduced as accurately as possible. For optimal estimation of ground motion, kriging has been shown to provide accurate estimates. Although kriging is accurate for this application, some estimates it provides are underestimates. This has dire consequences for subsequent design for earthquake resistance. Kriging does not provide enough information to allow an analysis of each estimate for underestimation. For such an application, disjunctive kriging is better applied. This advanced technique quantifies the probability that an estimate equals or exceeds particular levels of ground motion. Furthermore, disjunctive kriging can provide improved estimation accuracy when applied for local estimation of ground motion.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00898283
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  • 18
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    A parametric study of robustness of kriging variance as a function of range and relative nugget effect for a spherical semivariogram (1986)
    Springer
    Mathematical geology 18 (1986), S. 477-488 
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    ISSN: 1573-8868
    Keywords: kriging ; nugget effect ; range ; semivariogram ; geostatistics
    Source: Springer Online Journal Archives 1860-2000
    Topics: Geosciences , Mathematics
    Notes: Abstract In geostatistics, an estimation of blocks of a deposit is reported along with the variance of error made in their estimation. This calculation is based on the model chosen for the semivariogram of the deposit so that mistakes in its estimation can manifest themselves in the perception of accuracy with which blocks are known. Changes in kriging variance resulting from various amounts of error in modeling the relative nugget effect and range of the semivariogram are investigated for an extensive set of spherical semivariograms.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00897500
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  • 19
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    Treatment of earthquake ground motion using regionalized variables (1985)
    Springer
    Mathematical geology 17 (1985), S. 221-241 
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    ISSN: 1573-8868
    Keywords: variograms ; covariance ; positive definite conditions ; kriging ; Modified Mercalli Intensity
    Source: Springer Online Journal Archives 1860-2000
    Topics: Geosciences , Mathematics
    Notes: Abstract Earthquake ground motion is a spatial phenomenon associated with local random behavior and global attenuation away from a source. By this description, ground motion is a regionalized variable. Analyses of directional variograms for earthquake ground motion reveal ansiotropic attenuation. For some earthquakes, a spatial structure exists in all directions and an average variogram can be developed. For other earthquakes, either a spatial structure can be identified only for restricted directions, or a spatial structure cannot be found to exist in any direction. A valid application of regionalized variables theory to the spatial analysis of earthquake ground motion must therefore be predicated on an initial, exhaustive analysis of estimated variograms. To estimate these data, an application of ordinary kriging is preferable to other more complex regionalized variables techniques to simplify the spatial analysis of earthquake ground motion.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01034751
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  • 20
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    Global estimation variance: Formulas and calculation (1985)
    Springer
    Mathematical geology 17 (1985), S. 785-796 
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    ISSN: 1573-8868
    Keywords: geostatistics ; kriging ; estimation variance
    Source: Springer Online Journal Archives 1860-2000
    Topics: Geosciences , Mathematics
    Notes: Abstract This paper discusses the combination of kriging variances, which have been considered heretofor unfeasible since linearity of the problem and considerable simplifications which follow were overlooked. A simplified expression for global estimation variance is presented and an algorithm discussed with respect to precision and computer cost. A case study is presented, and, finally, an optimum calculation method is recommended.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01034061
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  • 21
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    Single/multiple input-single output transfer function with noise models for constituents of iron ores of Northern Goa, India (1985)
    Springer
    Mathematical geology 17 (1985), S. 755-767 
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    ISSN: 1573-8868
    Keywords: time (spatial) series ; prewhitening ; feedback ; transfer function ; residual variance ; forecasting ; kriging
    Source: Springer Online Journal Archives 1860-2000
    Topics: Geosciences , Mathematics
    Notes: Abstract Parametric time domain time series models for different chemical constituents such as SiO 2,Fe 2 O 3,TiO 2,MnO, and P 2 O 5 of iron ores belonging to banded hematite quartzites in Bicholim Mine (Goa) have been utilized for forecasting Fe 2 O 3 assay values through single/multiple input—single output (Fe 2 O 3 ) transfer function with noise methodology. Statistically significant correlated chemical constituent inputs (say, SiO 2,TiO 2,MnO, P 2 O 5 ) along lateral (horizontal strike direction) and vertical (perpendicular to bedding planes) sections in this mine are useful in modeling as well as forecasting assay values of blocks of ore to be exploited. Parameters of transfer function—noise models are estimated by utilizing time domain methodology, and the usefulness of these models for optimization of mining operations is discussed.
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    URL: http://dx.doi.org/10.1007/BF01031615
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  • 22
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    Fitting variogram models by weighted least squares (1985)
    Springer
    Mathematical geology 17 (1985), S. 563-586 
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    ISSN: 1573-8868
    Keywords: generalized least squares ; kriging ; median polish ; robustness ; stationarity
    Source: Springer Online Journal Archives 1860-2000
    Topics: Geosciences , Mathematics
    Notes: Abstract The method of weighted least squares is shown to be an appropriate way of fitting variogram models. The weighting scheme automatically gives most weight to early lags and down-weights those lags with a small number of pairs. Although weights are derived assuming the data are Gaussian (normal), they are shown to be still appropriate in the setting where data are a (smooth) transform of the Gaussian case. The method of (iterated) generalized least squares, which takes into account correlation between variogram estimators at different lags, offer more statistical efficiency at the price of more complexity. Weighted least squares for the robust estimator, based on square root differences, is less of a compromise.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01032109
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  • 23
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    Some limitations in the geostatistical evaluation of ore deposits (1985)
    Springer
    Geotechnical and geological engineering 3 (1985), S. 155-159 
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    ISSN: 1573-1529
    Keywords: Estimation variance ; geostatistics ; interpolation ; kriging
    Source: Springer Online Journal Archives 1860-2000
    Topics: Geosciences
    Notes: Conclusions The foregoing discussion indicates that geostatistical estimation of ore deposits is not local; it is not objective; it is not sensitive to local data trends; and it is not unrestrained by the range of data values. Kriging, as an interpolation method, is a variant of IDW least squares linear fit. As such, it suffers from the limitations of all IDW linear interpolation methods that employ only data values. The estimation variance, currently used to calculate the confidence limits of values for individual mining blocks, is hypothetical and globally derived. It is more closely related to sampling density than to local variation in the data set. Geostatistical methods, of course, have a real place in ore deposit assessment, e.g. global, comparative evaluation to assist decisions on development and investment. What is questioned here is the validity of employing a global method to assess detail (mining blocks) within an ore deposit.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00881627
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