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  • Articles  (48)
  • geostatistics  (48)
  • 1990-1994  (26)
  • 1985-1989  (22)
  • 1950-1954
  • Geosciences  (48)
  • Electrical Engineering, Measurement and Control Technology
  • 1
    Electronic Resource
    Electronic Resource
    On the geologic structure of the Explora Escarpment (Weddell Sea, Antarctica) revealed by satellite and Shipboard data evaluation (1994)
    Springer
    Marine geophysical researches 16 (1994), S. 325-345 
    Details
    ISSN: 1573-0581
    Keywords: Bathymetry ; gravity ; magnetics ; satellite data ; Gondwana breakup ; geostatistics ; geophysical inverse theory
    Source: Springer Online Journal Archives 1860-2000
    Topics: Geosciences , Physics
    Notes: Abstract Bathymetric, gravity, and magnetic data from Antarctic expeditions with RV POLARSTERN and satellite altimeter data from the Geosat Geodetic Mission are analysed using methods from geostatistics and geophysical inverse theory. The Explora Escarpment represents the edge between the Antarctic Continental Shelf and the Weddell Abyssal Plain. It is an important link in the reconstruction of Gondwana breakup, but a feature as large as the 2000 m deep Wegener Canyon was only discovered in 1984, when extensive bathymetric, gravimetric, and magnetic surveys with RV POLARSTERN began. Geostatistics, the theory of regionalized variables, is applied to integrate dense surveys of Wegener Canyon and sparse observations in adjacent areas into maps with full coverage of the 230 km by 330 km area at 10°–20° W/70°–72° S. The resultant highresolution bathymetric and gravity maps reveal detailed structures of the Explora Escarpment. Using geophysical inversion, the gravity terrain effect is calculated. Satellite data are used for their better coverage, but have much lower resolution. Nevertheless, the structures of Wegener Canyon and other more prominent features appear with surprisingly good correlation also in the Geosat altimeter data. While it was initially supposed that Wegener Canyon is purely an erosional structure, the magnetic map now provides evidence of the ‘canyon's’ tectonic origin.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01203971
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  • 2
    Electronic Resource
    Electronic Resource
    A neural network approach to geostatistical simulation (1994)
    Springer
    Mathematical geology 26 (1994), S. 491-503 
    Details
    ISSN: 1573-8868
    Keywords: conditional simulation ; geostatistics ; neural network
    Source: Springer Online Journal Archives 1860-2000
    Topics: Geosciences , Mathematics
    Notes: Abstract Neural networks offer a non-algorithmic approach to geostatistical simulation with the possibility of automatic recognition of correlation structure. The paper gives a brief overview of neural networks and describes a feedforward, back-propagation network for geostatistical simulation. The operation of the network is illustrated with two simple one-dimensional examples which can be followed through with hand calculations to give an insight into the operation of the network. The convergence of the network is described in terms of the variogram calculated from the values at each of the output nodes at each iteration.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02083491
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  • 3
    Electronic Resource
    Electronic Resource
    Bias in estimating fractal dimension with the rescaled-range (R/S) technique (1994)
    Springer
    Mathematical geology 26 (1994), S. 531-555 
    Details
    ISSN: 1573-8868
    Keywords: geostatistics ; fractals ; spatial analysis ; permeability modeling
    Source: Springer Online Journal Archives 1860-2000
    Topics: Geosciences , Mathematics
    Notes: Abstract Fractal geostatistics are being applied to subsurface geological data as a way of predicting the spatial distribution of hydrocarbon reservoir properties. The fractal dimension is the controlling parameter in stochastic methods to produce random fields of porosity and permeability. Rescaled range (R/S)analysis has become a popular way of estimating the fractal dimension, via determination of the Hurst exponent (H). A systematic investigation has been undertaken of the bias to be expected due to a range of factors commonly inherent in borehole data, particularly downhole wireline logs. The results are integrated with a review of previous work in this area. Small datasets. overlapping samples, drift and nonstationariry of means can produce a very large bias, and convergence of estimates of H around 0.85–0.90 regardless of original fractal dimension. Nonstationarity can also account for H〉1, which has been reported in the literature but which is theoretically impossible for fractal time series. These results call into question the validity of fractal stochastic models built using fractal dimensions estimated with the R/Smethod.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02089240
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  • 4
    Electronic Resource
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    Evaluation and comparison of spatial interpolators II (1994)
    Springer
    Mathematical geology 26 (1994), S. 589-603 
    Details
    ISSN: 1573-8868
    Keywords: kriging ; geostatistics ; spatial estimation ; inverse-distance estimation
    Source: Springer Online Journal Archives 1860-2000
    Topics: Geosciences , Mathematics
    Notes: Abstract The performance of several variations on ordinary kriging and inverse distance estimators is evaluated. Mean squared errors (MSE) were calculated for estimates made on multiple resamplings from five exhaustive data bases representing two distinctly different types of estimation problem. Ordinary kriging, when performed with variograms estimated from the sample data, was more robust than inverse-distance methods to the type of estimation problem, and to the choice of estimation parameters such as number of neighbors.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02089243
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  • 5
    Electronic Resource
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    Multivariable spatial prediction (1993)
    Springer
    Mathematical geology 25 (1993), S. 219-240 
    Details
    ISSN: 1573-8868
    Keywords: geostatistics ; kriging ; cokriging ; cross-variogram ; best linear unbiased prediction ; generalized least squares
    Source: Springer Online Journal Archives 1860-2000
    Topics: Geosciences , Mathematics
    Notes: Abstract For spatial prediction, it has been usual to predict one variable at a time, with the predictor using data from the same type of variable (kriging) or using additional data from auxiliary variables (cokriging). Optimal predictors can be expressed in terms of covariance functions or variograms. In earth science applications, it is often desirable to predict the joint spatial abundance of variables. A review of cokriging shows that a new cross-variogram allows optimal prediction without any symmetry condition on the covariance function. A bivariate model shows that cokriging with previously used cross-variograms can result in inferior prediction. The simultaneous spatial prediction of several variables, based on the new cross-variogram, is then developed. Multivariable spatial prediction yields the mean-squared prediction error matrix, and so allows the construction of multivariate prediction regions. Relationships between cross-variograms, between single-variable and multivariable spatial prediction, and between generalized least squares estimation and spatial prediction are also given.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00893273
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  • 6
    Electronic Resource
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    Artificially intelligent geostatistics: A framework accommodating qualitative knowledge-information (1993)
    Springer
    Mathematical geology 25 (1993), S. 261-279 
    Details
    ISSN: 1573-8868
    Keywords: geostatistics ; qualitative knowledge-information ; artificial intelligence
    Source: Springer Online Journal Archives 1860-2000
    Topics: Geosciences , Mathematics
    Notes: Abstract The purpose of this paper is to stress the need to examine Al-based models and techniques in dealing with qualitative knowledge, information, and expertise in geostatistics. A model of artificially intelligent geostatistics is proposed as a general framework. The model focuses on the “Geostatistician,” an abstraction of the “collective” knowledge and intelligence of the geostatisticians, be they theoreticians and/or practitioners. Dynamic aspects of the model are examined in the context of an explicit knowledge formalism, integrating geostatistical knowledge, symbolic non-algorithmic techniques for knowledge-information representation and inference, and standard numerical data processing. Two implementations of related computer systems are given together with case studies.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00901419
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  • 7
    Electronic Resource
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    Computationally efficient generation of gaussian conditional simulations over regular sample grids (1993)
    Springer
    Mathematical geology 25 (1993), S. 439-451 
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    ISSN: 1573-8868
    Keywords: block Toeplitz structure ; Cholesky factorization ; geostatistics ; Monte Carlo simulations ; spatial random field
    Source: Springer Online Journal Archives 1860-2000
    Topics: Geosciences , Mathematics
    Notes: Abstract The generation over two-dimensional grids of normally distributed random fields conditioned on available data is often required in reservoir modeling and mining investigations. Such fields can be obtained from application of turning band or spectral methods. However, both methods have limitations. First, they are only asymptotically exact in that the ensemble of realizations has the correlation structure required only if enough harmonics are used in the spectral method, or enough lines are generated in the turning bands approach. Moreover, the spectral method requires fine tuning of process parameters. As for the turning bands method, it is essentially restricted to processes with stationary and radially symmetric correlation functions. Another approach, which has the advantage of being general and exact, is to use a Cholesky factorization of the covariance matrix representing grid points correlation. For fields of large size, however, the Cholesky factorization can be computationally prohibitive. In this paper, we show that if the data are stationary and generated over a grid with regular mesh, the structure of the data covariance matrix can be exploited to significantly reduce the overall computational burden of conditional simulations based on matrix factorization techniques. A feature of this approach is its computational simplicity and suitability to parallel implementation.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00894778
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  • 8
    Electronic Resource
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    Generalized covariance functions in estimation (1993)
    Springer
    Mathematical geology 25 (1993), S. 525-540 
    Details
    ISSN: 1573-8868
    Keywords: geostatistics ; linear model ; best linear unbiased estimation ; experimental variogram ; restricted maximum likelihood
    Source: Springer Online Journal Archives 1860-2000
    Topics: Geosciences , Mathematics
    Notes: Abstract I discuss the role of generalized covariance functions in best linear unbiased estimation and methods for their selection. It is shown that the experimental variogram (or covariance function) of the detrended data can be used to obtain a preliminary estimate of the generalized covariance function without iterations and I discuss the advantages of other parameter estimation methods.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00890244
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  • 9
    Electronic Resource
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    A method for seafloor classification using directional variograms, demonstrated for data from the western flank of the Mid-Atlantic Ridge (1993)
    Springer
    Mathematical geology 25 (1993), S. 901-924 
    Details
    ISSN: 1573-8868
    Keywords: classification ; spatial structure analysis ; geostatistics ; bathymetry ; Atlantic ; automatization
    Source: Springer Online Journal Archives 1860-2000
    Topics: Geosciences , Mathematics
    Notes: Abstract Seafloor classification is aimed at quantitatively characterizing seafloor properties such as roughness and anisotropy, and at using such spatial characteristics to distinguish geological provinces automatically. From geostatistical principals, a variogram method is developed for seafloor classification and it is demonstrated for data from the western flank of the Mid-Atlantic Ridge at 25°45′N to 26°40′N. This study uses HYDROSWEEP bathymetric data which have been ping-edited to flag erroneous data records, and navigation corrected. The classification method can handle the resultant data gaps inside the survey swaths as well as interpret data from several swaths. For a suite of test areas representative of different geological provinces, directional variograms are calculated, and characteristic parameters are extracted for the classification. Examples include a sediment pond, abyssal hill terrain in several segments and of variable spacing, inside and outside corners of ridge discontinuities, and mixed morphological forms. The dependency of the results on random or regular subsampling and on the size of the test area is investigated.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00891050
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  • 10
    Electronic Resource
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    Spatial variability and interpolation of daily precipitation amount (1992)
    Springer
    Stochastic environmental research and risk assessment 6 (1992), S. 209-221 
    Details
    ISSN: 1436-3259
    Keywords: geostatistics ; precipitation ; water balance models ; semivariogram ; kriging ; spatial variation
    Source: Springer Online Journal Archives 1860-2000
    Topics: Architecture, Civil Engineering, Surveying , Energy, Environment Protection, Nuclear Power Engineering , Geography , Geosciences
    Notes: Abstract Daily precipitation amounts show spatial variation over sub-continential regions. Point measurements, representative for regions of land, have to be interpolated towards unobserved locations. In this study four days in 1984 were selected to investigate the spatial variability of daily precipitation amount in North-western Europe in relation to the meteorological conditions. Data were interpolated using Kriging. Crossvalidation was used to compare interpolated values with measured values. Large differences in the spatial structure of daily precipitation amount are obsered as a result of different meterological conditions. Stratification of the study area into a coastal, a mountainous and an interior stratum proved to be successful, reducing the Mean Squared Error of Prediction with up to 55%.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01581451
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  • 11
    Electronic Resource
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    Spatial variability and interpolation of daily precipitation amount (1992)
    Springer
    Stochastic environmental research and risk assessment 6 (1992), S. 304-320 
    Details
    ISSN: 1436-3259
    Keywords: geostatistics ; precipitation ; water balance models ; semivariogram ; kriging ; spatial variation
    Source: Springer Online Journal Archives 1860-2000
    Topics: Architecture, Civil Engineering, Surveying , Energy, Environment Protection, Nuclear Power Engineering , Geography , Geosciences
    Notes: Abstract Daily precipitation amounts show spatial variation over sub-continential regions. Point measurements, represntative for regions of land, have to be interpolated towards unobserved locations. In this study four days in 1984 were selected to investigate the spatial variability of daily precipitation amount in north-western Europe in relation to the meteorological conditions. Data were interpolated using kriging. Crossvalidation was used to compare interpolated values with measured values. Large differences in the spatial structure of daily precipitation amount are observed as a result of different meteorological conditions. Stratification of the study area into a coast, a mountain and an interior stratum proved to be successful, reducing the Mean Squared Error of Prediction with up to 55%.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01581623
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  • 12
    Electronic Resource
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    Adding bounds to kriging (1992)
    Springer
    Mathematical geology 24 (1992), S. 171-176 
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    ISSN: 1573-8868
    Keywords: geostatistics ; kriging
    Source: Springer Online Journal Archives 1860-2000
    Topics: Geosciences , Mathematics
    Notes: Abstract The ordinary kriging interpolation algorithm is extended by the inclusion of explicit lower and upper bounds on the estimate. The associated estimation variance is written as the ordinary kriging variance plus a non-negative correction term.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00897030
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  • 13
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    Evaluation and comparison of spatial interpolators (1992)
    Springer
    Mathematical geology 24 (1992), S. 381-391 
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    ISSN: 1573-8868
    Keywords: sampling ; geostatistics ; estimators ; interpolation
    Source: Springer Online Journal Archives 1860-2000
    Topics: Geosciences , Mathematics
    Notes: Abstract This study evaluates 15 different estimators to determine their relative merits in estimating block concentrations at contaminant waste sites. The evaluation was based on 54 subsets of data drawn from an exhaustive set of 19,800 data. For each subset, 198 block estimates were made with each estimator. The measurements of estimation quality were a linear loss function and a more standard statistic, the mean square error. The linear loss function showed that seven of the estimators produced scores close enough to be within the same statistical population. Results based on the mean square error were similar. The surprising results of this study were that inverse distance and inverse distance squared both produced better scores than kriging.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00891270
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  • 14
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    The fractal character of photos of slabbed cores (1992)
    Springer
    Mathematical geology 24 (1992), S. 73-97 
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    ISSN: 1573-8868
    Keywords: Fourier transform ; spatial analysis ; geostatistics ; reservoir property distribution
    Source: Springer Online Journal Archives 1860-2000
    Topics: Geosciences , Mathematics
    Notes: Abstract Core photos have been analyzed statistically and found to be fractional Gaussian noise (fGn) in both horizontal and vertical directions. This fractal character, fGn, is the same as that previously found to describe vertical and horizontal well logs. An equation for the two-dimensional spectral density of core photos is presented that generates computer photos very similar to “real” core photos.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00890089
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  • 15
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    Efficient updating of kriging estimates and variances (1992)
    Springer
    Mathematical geology 24 (1992), S. 129-133 
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    ISSN: 1573-8868
    Keywords: geostatistics ; kriging
    Source: Springer Online Journal Archives 1860-2000
    Topics: Geosciences , Mathematics
    Notes: Abstract This short note presents a method for efficiently updating ordinary kriging estimates and variances when one or more additional samples are incorporated into the kriging system. First, the foundation linear algebra result is presented. Then the update equations are derived. Finally, an illustrative application of updating is briefly discussed.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00890091
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  • 16
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    The effects of sampling design parameters on block selection (1992)
    Springer
    Mathematical geology 24 (1992), S. 329-343 
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    ISSN: 1573-8868
    Keywords: sampling ; geostatistics
    Source: Springer Online Journal Archives 1860-2000
    Topics: Geosciences , Mathematics
    Notes: Abstract Cost-effective spatial sampling strategy requires balancing sampling costs with the expected benefits from improved information. A contaminated site numerical model was used to test various single-phase sampling schemes, which were evaluated based on the quality of block selections from interpolated values. Different sample set sizes, different sampling patterns, and two levels of sampling precision were used. The sample set size was the only one of these factors observed to be significant. Bias was also examined. Modest levels (〈20%) had minimal impact; the effects of higher levels of bias varied with the selection level concentration.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00893753
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  • 17
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    Interrelations between geostatistics and information theory and their practical use (1991)
    Springer
    Mathematical geology 23 (1991), S. 3-7 
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    ISSN: 1573-8868
    Keywords: geostatistics ; information theory
    Source: Springer Online Journal Archives 1860-2000
    Topics: Geosciences , Mathematics
    Notes: Abstract Well-defined analytical interrelations exist between the geostatistical estimation variance performing an expression of degree of geological exploration and the task-oriented entropy as a concept of information theory. The example of normal distribution shows that this relationship can be expressed in mathematical terms. An extensive practical use could consist in the improvement of exploration optimization on the common basis of continuous geological parameters as is demonstrated by means of an example.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02065960
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  • 18
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    Estimation of covariance parameters in kriging via restricted maximum likelihood (1991)
    Springer
    Mathematical geology 23 (1991), S. 119-135 
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    ISSN: 1573-8868
    Keywords: Gaussian random field ; covariance estimation ; geostatistics ; eigenvalue decomposition
    Source: Springer Online Journal Archives 1860-2000
    Topics: Geosciences , Mathematics
    Notes: Abstract In kriging, parametric approaches to covariance (or variogram) estimation require that unknown parameters be inferred from a single realization of the underlying random field. An approach to such an estimation problem is to assume the field to be Gaussian and iteratively minimize a (restricted) negative loglikelihood over the parameter space. In doing so, the associated computational burden can be considerable. Also, it is usually not easy to check whether or not the minimum achieved is global. In this note, we show that in many practical cases, the structure of the covariance (or variogram) function can be exploited so that iterative minimizing algorithms may be advantageously replaced by a procedure that requires the computation of the roots of a simple rational function and the search for the minimum of a function depending on one variable only. As a consequence, our approach allows one to observe in a straightforward fashion the presence of local minima. Furthermore, it is shown that insensitivity of the likelihood function to changes in parameter value can be easily detected. The note concludes with numerical simulations that illustrate some key features of our estimation procedure.
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    URL: http://dx.doi.org/10.1007/BF02065971
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  • 19
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    Monitoring well placement using conditional simulation of hydraulic head (1991)
    Springer
    Mathematical geology 23 (1991), S. 1059-1080 
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    ISSN: 1573-8868
    Keywords: conditional simulation ; contaminant migration ; geostatistics ; plume interception
    Source: Springer Online Journal Archives 1860-2000
    Topics: Geosciences , Mathematics
    Notes: Abstract A theoretical technique for conditioning simulations in frequency domain is developed and then applied to a hydraulic-head data set from Pittman, Nevada. Frequency-domain simulation rapidly generates simulations while requiring minimal computer memory. This makes it possible by using a personal computer to create large numbers of simulations of a physical parameter field for use in studying stochastic processes. In our application, groundwater flowlines are constructed from the simulations of the hydraulic head field. Then, the crossings of the flowlines at a transect down-stream from a contaminant point source generate histograms for predicting the probability of plume interception by groundwater monitoring wells. The simulation process is discussed in detail for the Pittman study site.
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    URL: http://dx.doi.org/10.1007/BF02066111
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  • 20
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    Orthonormal residuals in geostatistics: Model criticism and parameter estimation (1991)
    Springer
    Mathematical geology 23 (1991), S. 741-758 
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    ISSN: 1573-8868
    Keywords: geostatistics ; hypothesis testing ; parameter estimation
    Source: Springer Online Journal Archives 1860-2000
    Topics: Geosciences , Mathematics
    Notes: Abstract In linear geostatistics, models for the mean function (drift) and the variogram or generalized covariance function are selected on the basis of the modeler's understanding of the phenomenon studied as well as data. One can seldom be assured that the most appropriate model has been selected; however, analysis of residuals is helpful in diagnosing whether some important characteristic of the data has been neglected and, ultimately, in providing a reasonable degree of assurance that the selected model is consistent with the available information. The orthonormal residuals presented in this work are kriging errors constructed so that, when the correct model is used, they are uncorrelated and have zero mean and unit variance. It is suggested that testing of orthonormal residuals is a practical way for evaluating the agreement of the model with the data and for diagnosing model deficiencies. Their advantages over the usually employed standardized residuals are discussed. A set of tests are presented. Orthonormal residuals can also be useful in the estimation of the covariance (or variogram) parameters for a model that is considered correct.
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    URL: http://dx.doi.org/10.1007/BF02082534
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  • 21
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    Variance as a function of sample support size (1990)
    Springer
    Mathematical geology 22 (1990), S. 107-121 
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    ISSN: 1573-8868
    Keywords: Variogram ; geostatistics ; sample support ; variance
    Source: Springer Online Journal Archives 1860-2000
    Topics: Geosciences , Mathematics
    Notes: Abstract The effect of sample support size on variance is examined and evaluated. Results based on variograms and geostatistics are compared to the classical relationship developed by H. F. Smith in 1938; that is, that the variance is reduced fromV 1 toV 1 /n b as the support area increases from I ton plots for uniformity trials. The exponentb is between zero and one. Theoretical results are based on use of auxiliary functions and account for the size and shape of the sample support and the overall field geometry. Results are given in terms of approximations by rational functions for ease of calculation. Experimental results for uniformity trials, infiltration measurements, and spectral data from satellites are compared to theoretical and empirical results. Applications include not only uniformity trials, but also measurement theory.
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    URL: http://dx.doi.org/10.1007/BF00890300
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  • 22
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    Comparing some approximate methods for building local confidence intervals for predicting regionalized variables (1990)
    Springer
    Mathematical geology 22 (1990), S. 123-144 
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    ISSN: 1573-8868
    Keywords: nonlinear estimators ; geostatistics ; confidence intervals ; indicator kriging ; probability kriging ; disjunctive kriging
    Source: Springer Online Journal Archives 1860-2000
    Topics: Geosciences , Mathematics
    Notes: Abstract Approximate local confidence intervals can be produced by nonlinear methods designed to estimate indicator variables. The most precise of these methods, the conditional expectation, can only be used in practice in the multi-Gaussian context. Theoretically, less efficient methods have to be used in more general cases. The methods considered here are indicator kriging, probability kriging (indicator-rank co-kriging), and disjunctive kriging (indicator co-kriging). The properties of these estimators are studied in this paper in the multi-Gaussian context, for this allows a more detailed study than under more general models. Conditional distribution approximation is first studied. Exact results are given for mean squared errors and conditional bias. Then conditional quantile estimators are compared empirically. Finally, confidence intervals are compared from the points of view of bias and precision.
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    URL: http://dx.doi.org/10.1007/BF00890301
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  • 23
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    Regionalization in geology by multivariate classification (1990)
    Springer
    Mathematical geology 22 (1990), S. 573-588 
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    ISSN: 1573-8868
    Keywords: classification ; regionalization ; Bayes' theorem ; geostatistics
    Source: Springer Online Journal Archives 1860-2000
    Topics: Geosciences , Mathematics
    Notes: Abstract The concept of multivariate classification of “geological objects” can be combined with the concept of regionalized variables to yield a procedure for typification of geological objects, such as rock units, well records, or samples. Numerical classification is followed by subdivision of the area of investigation, and culminates in a regionalization or mapping of the classification onto the plane. Regions are subdivisions of the map area which are spatially contiguous and relatively homogeneous in their geological properties. The probability of correct classification of each point within a region as being part of that region can be assessed in terms of Bayesian probability as a space-dependent function. The procedure is applied to subsurface data from western Kansas. The geologic properties used are quantitative variables, and relationships are expressed by Mahalanobis' distances. These functions could be replaced by other metrics if qualitative or binary data derived from geological descriptions or appraisals were included in the analysis.
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    URL: http://dx.doi.org/10.1007/BF00890505
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  • 24
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    Problems in space-time kriging of geohydrological data (1990)
    Springer
    Mathematical geology 22 (1990), S. 611-623 
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    ISSN: 1573-8868
    Keywords: geostatistics ; geohydrology ; kriging ; spatiotemporal variables
    Source: Springer Online Journal Archives 1860-2000
    Topics: Geosciences , Mathematics
    Notes: Abstract Spatiotemporal variables constitute a large class of geohydrological phenomena. Estimation of these variables requires the extension of geostatistical tools into the space-time domain. Before applying these techniques to space-time data, a number of important problems must be addressed. These problems can be grouped into four general categories: (1) fundamental differences with respect to spatial problems, (2) data characteristics, (3) structural analysis including valid models, and (4) space-time kriging. Adequate consideration of these problems leads to more appropriate estimation techniques for spatiotemporal data.
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    A Bayesian/maximum-entropy view to the spatial estimation problem (1990)
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    Mathematical geology 22 (1990), S. 763-777 
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    ISSN: 1573-8868
    Keywords: spatial estimation ; entropy ; Bayes law ; information ; geostatistics
    Source: Springer Online Journal Archives 1860-2000
    Topics: Geosciences , Mathematics
    Notes: Abstract The purpose of this paper is to stress the importance of a Bayesian/maximum-entropy view toward the spatial estimation problem. According to this view, the estimation equations emerge through a process that balances two requirements: High prior information about the spatial variability and high posterior probability about the estimated map. The first requirement uses a variety of sources of prior information and involves the maximization of an entropy function. The second requirement leads to the maximization of a so-called Bayes function. Certain fundamental results and attractive features of the proposed approach in the context of the random field theory are discussed, and a systematic spatial estimation scheme is presented. The latter satisfies a variety of useful properties beyond those implied by the traditional stochastic estimation methods.
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    Infill-sampling design and the Cost of classification errors (1990)
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    Mathematical geology 22 (1990), S. 915-932 
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    Keywords: sample design ; geostatistics ; spatial classification
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    Notes: Abstract The criterion used to select infill sample locations should depend on the sampling objective. Minimizing the global estimation variance is the most widely used criterion and is suitable for many problems. However, when the objective of the sampling program is to partition an area of interest into zones of high values and zones of low values, minimizing the expected cost of classification errors is a more appropriate criterion. Unlike the global estimation variance, the cost of classification errors incorporates both the sample locations and the sample values into an objective infill-sampling design criterion.
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    Geostatistical analysis of longwall face roof convergence data (1989)
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    Geotechnical and geological engineering 7 (1989), S. 227-237 
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    ISSN: 1573-1529
    Keywords: Roof convergence ; longwall face ; geostatistics ; coal mining ; support advance
    Source: Springer Online Journal Archives 1860-2000
    Topics: Geosciences
    Notes: Summary Roof convergence and the rate of convergence are regionalized variables; the geostatistics can thus be applied to a set of underground observations. This investigation shows that the removal of a slice of coal induces roof movement. The rate of convergence is highest just after the coal removal and decreases with time. The immediate advance of a ‘rigid’ powered roof support decreases the total convergence for a complete production cycle. On the other hand, the timing of the advance of a relatively ‘soft’ support element has no influence on the total convergence for a complete production cycle.
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    Bayesian updating revisited (1989)
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    Mathematical geology 21 (1989), S. 285-308 
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    Keywords: Bayesian updating ; geostatistics ; Kriging ; linear inversion
    Source: Springer Online Journal Archives 1860-2000
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    Notes: Abstract Bayesian updating methods provide an alternate philosophy to the characterization of the input variables of a stochastic mathematical model. Here, a priori values of statistical parameters are assumed on subjective grounds or by analysis of a data base from a geologically similar area. As measurements become available during site investigations, updated estimates of parameters characterizing spatial variability are generated. However, in solving the traditional updating equations, an updated covariance matrix may be generated that is not positive-definite, particularly when observed data errors are small. In addition, measurements may indicate that initial estimates of the statistical parameters are poor. The traditional procedure does not have a facility to revise the parameter estimates before the update is carried out. alternatively, Bayesian updating can be viewed as a linear inverse problem that minimizes a weighted combination of solution simplicity and data misfit. Depending on the weight given to the a priori information, a different solution is generated. A Bayesian updating procedure for log-conductivity interpolation that uses a singular value decomposition (SVD) is presented. An efficient and stable algorithm is outlined that computes the updated log-conductivity field and the a posteriori covariance of the estimated values (estimation errors). In addition, an information density matrix is constructed that indicates how well predicted data match observations. Analysis of this matrix indicates the relative importance of the observed data. The SVD updating procedure is used to interpolate the log-conductivity fields of a series of hypothetical aquifers to demonstrate pitfalls and possibilities of the method.
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    Factorial kriging analysis applied to geological data from petroleum exploration (1989)
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    Mathematical geology 21 (1989), S. 683-691 
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    Keywords: geostatistics ; factorial kriging analysis ; spectral analysis ; petroleum exploration
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    Topics: Geosciences , Mathematics
    Notes: Abstract A regionalized variable, thickness of the reservoir layer, from a gas field is decomposed by factorial kriging analysis. Maps of the obtained components may be associated with depositional environments that are favorable for petroleum exploration.
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    Variography of submarine morphology: Problems of deregularization, and cartographical implications (1989)
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    Mathematical geology 21 (1989), S. 693-713 
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    ISSN: 1573-8868
    Keywords: geostatistics ; variography ; deregularization ; universal kriging ; bathymetry ; SEABEAM survey ; cartography
    Source: Springer Online Journal Archives 1860-2000
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    Notes: Abstract During the German Antarctic Expedition VI (leg 3, December 1987 to March 1988), bathymetric surveys were made in the Weddell Sea by the SEABEAM sonar system. For the first time geostatistical methods were applied in the SEABEAM-postprocessing. The investigations of variography that were necessary prior to the cartographical-geomorphological evaluation shed new light on classical geostatistical concerns. SEABEAM data provide a good example of a mean square, differentiable regionalized variable, where data are sampled over a two-dimensional support due to the technique of the sonar device. By deregularizations of the sample variograms, spatial continuity can be shown to be a property of seafloor depth as well as a point variable. The results are discussed in a sedimentological context. As an application of the regional variogram analyses, large-scale kriged bathymetric maps are presented.
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    On the importance of choosing a change of support model for global reserves estimation (1988)
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    Mathematical geology 20 (1988), S. 1001-1019 
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    Keywords: geostatistics ; change of support ; recoverable reserves ; selectivity ; normality index ; Cartier's formula ; affine correction ; discrete Gaussian model ; mosaic model
    Source: Springer Online Journal Archives 1860-2000
    Topics: Geosciences , Mathematics
    Notes: Abstract The practical problem considered here is: how can block distribution in an orebody be forecast from sample data? The task is arduous because information yielded by samples is too often insufficient to allow an accurate evaluation of blocks. In practice, necessary additional information is obtained via a model. Choosing that model is crucial; the value of results reflects the model, i.e., its adequacy to represent reality. In this paper, the importance of choosing the change of support model is illustrated with simulations and practical examples (especially deposits with a skewed sample distribution and a large spike at the origin). An attempt to quantify this importance is made also.
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    Imprecise (fuzzy) information in geostatistics (1988)
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    Mathematical geology 20 (1988), S. 287-311 
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    Keywords: fuzzy sets ; geostatistics ; methodology ; application ; permeability prediction ; soil liner
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    Notes: Abstract A methodology based on fuzzy set theory for the utilization of imprecise data in geostatistics is presented. A common problem preventing a broader use of geostatistics has been the insufficient amount of accurate measurement data. In certain cases, additional but uncertain (soft) information is available and can be encoded as subjective probabilities, and then the soft kriging method can be applied (Journel, 1986). In other cases, a fuzzy encoding of soft information may be more realistic and simplify the numerical calculations. Imprecise (fuzzy) spatial information on the possible variogram is integrated into a single variogram which is used in a fuzzy kriging procedure. The overall uncertainty of prediction is represented by the estimation variance and the calculated membership function for each kriged point. The methodology is applied to the permeability prediction of a soil liner for hazardous waste containment. The available number of hard measurement data (20) was not enough for a classical geostatistical analysis. An additional 20 soft data made it possible to prepare kriged contour maps using the fuzzy geostatistical procedure.
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    Fractal and geostatistical methods for modeling of a fracture network (1988)
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    Mathematical geology 20 (1988), S. 631-654 
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    Keywords: Fractals ; geostatistics ; fracture network ; granite
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    Notes: Abstract The modeling of fracture networks is useful for fluid flow and rock mechanics studies. About 6600 fracture traces were recorded on drifts of a uranium mine in a granite massif. The traces have an extension of 0.20–20 m. The network was studied by fractal and by geostatistical methods but can be considered neither as a fractal with a constant dimension nor a set of purely randomly located fractures. Two kinds of generalization of conventional models can still provide more flexibility for the characterization of the network: (a) a nonscaling fractal model with variable similarity dimension (for a 2-D network of traces, the dimension varying from 2 for the 10-m scale to 1 for the centimeter scale, (b) a parent-daughter model with a regionalized density; the geostatistical study allows a 3-D model to be established where: fractures are assumed to be discs; fractures are grouped in clusters or swarms; and fracturation density is regionalized (with two ranges at about 30 and 300 m). The fractal model is easy to fit and to simulate along a line, but 2-D and 3-D simulations are more difficult. The geostatistical model is more complex, but easy to simulate, even in 3-D.
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    Bayesian kriging—Merging observations and qualified guesses in kriging (1987)
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    Mathematical geology 19 (1987), S. 25-39 
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    Keywords: geostatistics ; kriging ; Bayesian statistics
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    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.
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    Production of conditional simulations via the LU triangular decomposition of the covariance matrix (1987)
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    Mathematical geology 19 (1987), S. 91-98 
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    Keywords: geostatistics ; kriging ; conditional simulation ; fast Fourier transform
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    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.
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    Efficiency of kriging estimation for square, triangular, and hexagonal grids (1987)
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    Mathematical geology 19 (1987), S. 183-205 
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    Keywords: geostatistics ; intrinsic hypothesis ; regionalized variables ; sampling ; spherical semivariogram
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    Topics: Geosciences , Mathematics
    Notes: Abstract Although several researchers have pointed out some advantages and disadvantages of various soil sampling designs in the presence of spatial autocorrelation, a more detailed study is presented herein which examines the geometrical relationship of three sampling designs, namely the square, the equilateral triangle, and the regular hexagon. Both advantages and disadvantages exist in the use of these designs with respect to estimation of the semivariogram and their effect on the mean square error or variance of error. This research could be used to design optimal sampling strategies; it is based on the theory of regionalized variables, in which the intrinsic hypothesis is satisfied. Among alternative designs, an equilateral triangle design gives the most reliable estimate of the semivariogram. It also gives the minimum maximum mean square error of point estimation of the concentration over the other two designs for the same number of measurements when the nugget effect is small relative to the variance. If the nugget effect is large (.90 σ 2 or more), and the linear sampling density is 〉0.85r where r is the range, the hexagonal design is best. This study computes and compares the maximum mean square error for each of these designs.
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    Generating large stochastic simulations—The matrix polynomial approximation method (1987)
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    Mathematical geology 19 (1987), S. 99-107 
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    Keywords: geostatistics ; simulation ; Toeplitz matrices ; block Toeplitz matrices ; matrix polynomial approximation
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    Notes: Abstract An algorithm for producing a nonconditional simulation by multiplying the square root of the covariance matrix by a random vector is described. First, the square root of a matrix (or a function of a matrix in general) is defined. The square root of the matrix can be approximated by a minimax matrix polynomial. The block Toeplitz structure of the covariance matrix is used to minimize storage. Finally, multiplication of the block Toeplitz matrix by the random vector can be evaluated as a convolution using the fast Fourier transform. This results in an algorithm which is not only efficient in terms of storage and computation but also easy to implement.
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    Estimation of the generalized covariance function: II. A response surface approach (1987)
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    Mathematical geology 19 (1987), S. 769-783 
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    Keywords: Cross-validation ; geostatistics ; intrinsic random functions of orderk ; kriging
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    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.
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    Determination of support in soil sampling (1986)
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    Mathematical geology 18 (1986), S. 529-537 
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    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.
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    Matheronian geostatistics—Quo vadis? (1986)
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    Mathematical geology 18 (1986), S. 93-117 
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    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.
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    Disjunctive kriging, universal kriging, or no kriging: Small sample results with simulated fields (1986)
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    Mathematical geology 18 (1986), S. 287-305 
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    Keywords: interpolation ; mapping ; geostatistics
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    Notes: Abstract This paper provides a comparison between linear (universal) and nonlinear (disjunctive) kriging estimators when they are computed from small samples chosen randomly on simulated stationary and nonstationary fields. Point estimation results are reported. In all cases considered, kriging estimators were found better than a local mean estimator, with universal kriging either better than or as good as disjunctive kriging. The latter, which is suited to handle stationary fields, did not provide more accurate estimates because the use of small samples led to inconsistencies in the assumed bivariate model. Universal kriging was particularly better with nonstationary fields.
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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)
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    Mathematical geology 18 (1986), S. 477-488 
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    Keywords: kriging ; nugget effect ; range ; semivariogram ; geostatistics
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    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.
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    Distribution of kriging error and stationarity of the variogram in a coal property (1986)
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    Mathematical geology 18 (1986), S. 635-652 
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    Keywords: distribution of kriging error ; variogram stationarity ; coal ; geostatistics ; contract risk
    Source: Springer Online Journal Archives 1860-2000
    Topics: Geosciences , Mathematics
    Notes: Abstract If a particular distribution for kriging error may be assumed, confidence intervals can be estimated and contract risk can be assessed. Contract risk is defined as the probability that a block grade will exceed some specified limit. In coal mining, this specified limit will be set in a coal sales agreement. A key assumption necessary to implement the geostatistical model is that of local stationarity in the variogram. In a typical project, data limitations prevent a detailed examination of the stationarity assumption. In this paper, the distribution of kriging error and scale of variogram stationarity are examined for a coal property in northern West Virginia.
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    Minimum-variance unbiased quadratic estimation of covariances of regionalized variables (1985)
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    Mathematical geology 17 (1985), S. 195-208 
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    Keywords: geostatistics ; covariance estimation ; optimization
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    Notes: Abstract The parameters of covariance functions (or variograms) of regionalized variables must be determined before linear unbiased estimation can be applied. This work examines the problem of minimum-variance unbiased quadratic estimation of the parameters of ordinary or generalized covariance functions of regionalized variables. Attention is limited to covariance functions that are linear in the parameters and the normality assumption is invoked when fourth moments of the data need to be calculated. The main contributions of this work are (1) it shows when and in what sense minimum-variance unbiased quadratic estimation can be achieved, and (2) it yields a well-founded, practicable, and easy-to-automate methodology for the estimation of parameters of covariance functions. Results of simulation studies are very encouraging.
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    Global estimation variance: Formulas and calculation (1985)
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    Mathematical geology 17 (1985), S. 785-796 
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    Keywords: geostatistics ; kriging ; estimation variance
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    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.
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    The deterministic side of geostatistics (1985)
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    Mathematical geology 17 (1985), S. 1-15 
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    Keywords: geostatistics ; stationarity ; ergodicity ; spatial average ; deterministic kriging
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    Notes: Abstract The probabilistic approach is but one language used by geostatisticians to characterize spatial variability and to express a very simple criterion for goodness of estimation. Notions such as stationarity and ergodicity are important for the consistency of the probabilistic language but are irrelevant to the real problem, that of estimating a well-defined deterministic spatial average. The kriging algorithm is established without any recourse to probabilistic modeling or notation.
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    Two-dimensional simulation by turning bands (1985)
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    Mathematical geology 17 (1985), S. 81-90 
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    Keywords: simulation ; turning bands ; covariance ; semivariogram ; geostatistics
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    Notes: Abstract Journel (1974) developed the turning-bands method which allows a three-dimensional data set with specified covariance to be obtained by the simulation of several one-dimensional realizations which have an intermediate covariance. The relationship between the threedimensional and one-dimensional covariance is straightforward and allows the one-dimensional covariance to be obtained immediately. In theory a dense uniform distribution of lines in three-dimensional space is required along which the one-dimensional realizations are generated; in practice most workers have been content to use the fifteen axes of the regular icosahedron. Many mining problems may be treated in two dimensions, and in this paper a turning-bands approach is developed to generate two-dimensional data sets with a specified covariance. By working in two dimensions, the area on which the data is simulated may be divided as finely as desired by the lines on which the one-dimensional realizations are first generated. The relationship between the two-dimensional and one-dimensional covariance is derived as a nontrivial integral equation. This is solved analytically for the onedimensional covariance. The method is applied to the generation of a two-dimensional data set with spherical covariance.
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    Some limitations in the geostatistical evaluation of ore deposits (1985)
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    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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