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  • 1
    Digitale Medien
    Digitale Medien
    Springer
    Natural resources research 9 (2000), S. 53-64 
    ISSN: 1573-8981
    Schlagwort(e): Geophysical data integration ; geostatistics ; mineral exploration ; interpolation
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Geologie und Paläontologie
    Notizen: Abstract Geophysical data are used routinely in mineral exploration to delineate the geology of an area.Because geophysical attributes are sparsely sampled, interpolation methods are used to gridthe individual sets of data. Whereas traditional techniques interpolate constituent attributes ofinterest independently, frequently resulting in simplistic geological models, alternativetechniques interpolate by integrating secondary data sets collected in the explored area. Traditionaltechniques include minimum curvature and ordinary kriging. Alternative integrated interpolationtechniques include standardized ordinary cokriging, collocated cokriging, and kriging withexternal drift. Application of these techniques to a specific exploration area in interpolatinggravity measurements (primary data) using weakly correlated aeromagnetics (secondary data)suggests that when gravity is well sampled, all techniques generate similar results. However,when gravity is undersampled, the secondary data can contribute substantially to identifyinggeophysical anomalies. Considering the high cost of geophysical data collection, this observationcan make geophysical data collection more efficient and cost effective. The case studyshows how, in the absence of a strong correlation, cokriging varieties tend to produce mapssimilar to those generated by traditional techniques. Kriging with external drift is shown togenerate a gravity map reproducing all gravity anomalies. The technique produces excellentresults when features identified in the secondary data need to be reflected in the map of theprimary data.
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  • 2
    Digitale Medien
    Digitale Medien
    Springer
    Biogeochemistry 50 (2000), S. 111-136 
    ISSN: 1573-515X
    Schlagwort(e): carbon sequestration ; geostatistics ; mountain forest soils ; national assessment ; soil organic carbon
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Chemie und Pharmazie , Geologie und Paläontologie
    Notizen: Abstract Soil organic carbon (SOC) has been identified as the main globalterrestrial carbon reservoir, but considerable uncertainty remains as toregional SOC variability and the distribution of C between vegetationand soil. We used gridded forest soil data (8–km × 8–km)representative of Swiss forests in terms of climate and forest typedistribution to analyse spatial patterns of mineral SOC stocks alonggradients in the European Alps for the year 1993. At stand level, meanSOC stocks of 98 t C ha−1 (N = 168,coefficient of variation: 70%) were obtained for the entiremineral soil profile, 76 t C ha−1 (N =137, CV: 50%) in 0–30 cm topsoil, and 62 t Cha−1 (N = 156, CV: 46%) in0–20 cm topsoil. Extrapolating to national scale, we calculatedcontemporary SOC stocks of 110 Tg C (entire mineral soil, standarderror: 6 Tg C), 87 Tg C (0–30 cm topsoil, standarderror: 3.5 Tg C) and 70 Tg C (0–20 cm topsoil, standarderror: 2.5 Tg C) for mineral soils of accessible Swiss forests(1.1399 Mha). According to our estimate, the 0–20 cm layers ofmineral forest soils in Switzerland store about half of the Csequestered by forest trees (136 Tg C) and more than five times morethan organic horizons (13.2 Tg C). At stand level, regression analyses on the entire data set yielded nostrong climatic or topographic signature for forest SOC stocks in top(0–20 cm) and entire mineral soils across the Alps, despite thewide range of values of site parameters. Similarly, geostatisticalanalyses revealed no clear spatial trends for SOC in Switzerland at thescale of sampling. Using subsets, biotic, abiotic controls andcategorial variables (forest type, region) explained nearly 60%of the SOC variability in topsoil mineral layers (0–20 cm) forbroadleaf stands (N = 56), but only little of thevariability in needleleaf stands (N = 91,R 2 = 0.23 for topsoil layers). Considerable uncertainties remain in assessments of SOC stocks, due tounquantified errors in soil density and rock fraction, lack of data onwithin-site SOC variability and missing or poorly quantifiedenvironmental control parameters. Considering further spatial SOCvariability, replicate pointwise soil sampling at 8–km × 8–kmresolution without organic horizons will thus hardly allow to detectchanges in SOC stocks in strongly heterogeneous mountain landscapes.
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  • 3
    Digitale Medien
    Digitale Medien
    Springer
    Transport in porous media 41 (2000), S. 173-195 
    ISSN: 1573-1634
    Schlagwort(e): carbonate reservoir characterization ; geostatistics ; fractals ; artificial neural networks
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Geologie und Paläontologie , Technik allgemein
    Notizen: Abstract In this study, a carbonate oil reservoir located in the southeast part of Turkey was characterized by the use of kriging and the fractal geometry. The three-dimensional porosity and permeability distributions were generated by both aforementioned methods by using the wireline porosity logs and core plug permeability measurements taken from six wells of the field. Since classical regression (lognormal or polynomial) and geostatistical techniques (cross variograms) fail to estimate permeability from wireline log-porosity data, the use of artificial neural networks (ANNs) is proposed in this study to generate permeability data at uncored intervals of porosity logs. For both of the methods, kriging and fractal techniques, the validation of the estimated/simulated data with known wellbore data resulted with acceptable agreements, especially for porosity. Also the comparison of both methods at unsampled locations show better agreements for porosity than permeability.
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  • 4
    Digitale Medien
    Digitale Medien
    Springer
    Mathematical geology 32 (2000), S. 87-108 
    ISSN: 1573-8868
    Schlagwort(e): heterogeneity ; geostatistics ; optimization ; inversion ; nonlinearity
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Geologie und Paläontologie , Mathematik
    Notizen: Abstract This paper describes a new method for gradually deforming realizations of Gaussian-related stochastic models while preserving their spatial variability. This method consists in building a stochastic process whose state space is the ensemble of the realizations of a spatial stochastic model. In particular, a stochastic process, built by combining independent Gaussian random functions, is proposed to perform the gradual deformation of realizations. Then, the gradual deformation algorithm is coupled with an optimization algorithm to calibrate realizations of stochastic models to nonlinear data. The method is applied to calibrate a continuous and a discrete synthetic permeability fields to well-test pressure data. The examples illustrate the efficiency of the proposed method. Furthermore, we present some extensions of this method (multidimensional gradual deformation, gradual deformation with respect to structural parameters, and local gradual deformation) that are useful in practice. Although the method described in this paper is operational only in the Gaussian framework (e.g., lognormal model, truncated Gaussian model, etc.), the idea of gradually deforming realizations through a stochastic process remains general and therefore promising even for calibrating non-Gaussian models.
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  • 5
    Digitale Medien
    Digitale Medien
    Springer
    Mathematical geology 32 (2000), S. 367-379 
    ISSN: 1573-8868
    Schlagwort(e): random functions ; simulation ; geostatistics
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Geologie und Paläontologie , Mathematik
    Notizen: Abstract This paper presents a new method of constructing random functions whose realizations can be evaluated efficiently. The basic idea is to “blend,” both stochastically and linearly, a limited set of independent initial realizations previously generated by any chosen simulation method. The blending stochastic coefficients are determined in such a way that the new random function so generated has the same mean and covariance functions as the random function used for generating the initial realizations.
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  • 6
    Digitale Medien
    Digitale Medien
    Springer
    Mathematical geology 32 (2000), S. 389-419 
    ISSN: 1573-8868
    Schlagwort(e): shale ; geostatistics ; Bayes' theorem ; Frontier Formation ; Erlangian ; tide-influenced delta
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Geologie und Paläontologie , Mathematik
    Notizen: Abstract Dimensions of shales and other geologic bodies that affect fluid flow through reservoirs and aquifers are often estimated from analogous deposits exposed in outcrops. Shale lengths observed in outcrops are biased because the finite length of outcrops truncates longer shales and long shales tend to be overrepresented in the sample. Shale length distributions can be modeled using an Erlangian probability density function based on termination frequency. Termination frequency can be calculated from outcrop observations even if the shales are incompletely exposed. Termination frequency is unbiased regardless of underlying shale length distribution and outcrop size and shape. Complex length distributions can be represented by a weighted sum of Erlangian models, each with a distinct termination frequency. The proportions and termination frequencies of the component models can be estimated using Bayes' theorem. Subsamples of the outcrop area can be analyzed to quantify spatial trends in termination frequency and thus shale length. These methods can be applied to estimate length distributions of any geologic object exposed in outcrops or other spatially limited samples. In many cases estimated termination frequency is normally distributed, and its variance can be estimated using closed-form expressions. Shale length distributions for the Frewens Sandstone were modeled using one- and two-component Erlangian probability density functions. The Frewens Sandstone is a tide-influenced deltaic sandstone body within the Cretaceous-age Frontier Formation, exposed in central Wyoming, USA. Length observations and Erlangian models agree reasonably well. Because the models correct for the effects of shale truncation in limited outcrop exposures, predicted means and proportions of long shales are substantially greater than observed values. There is a vertical trend and pronounced vertical cyclicity in termination frequency, indicating that shales become shorter upward and intervals of longer and shorter shales alternate. Along strike, termination frequency decreases away from the sandstone-body axis, reflecting greater preserved shale lengths toward sandstone-body margins. No correlation was observed between distance along dip and termination frequency. Termination frequencies and histograms of shale length for adjacent, perpendicular walls are similar, indicating that there is no anisotropy in shale length.
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  • 7
    ISSN: 1435-0157
    Schlagwort(e): Key words hydraulic properties ; heterogeneity ; geophysical methods ; geostatistics ; Italy
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Geologie und Paläontologie
    Beschreibung / Inhaltsverzeichnis: Résumé Dans le cadre de plusieurs études réalisées pour caractériser l'aquifère sur le site expérimental près de Montalto Uffugo (Italie), ce travail analyse la distribution spatiale de la conductivité hydraulique de formations non consolidées présentes dans cette région. La technique géostatistique du krigeage a été appliquée, avec une dérive externe, aux données de résistivitéélectrique et de conductivité hydraulique. La fiabilité de la méthode d'estimation a été testée en établissant un modèle, basé sur la méthode des cellules, qui simule l'écoulement souterrain, à partir des valeurs estimées de la conductivité hydraulique. Les résultats obtenus indiquent que cette méthode d'estimation présente un degré de fiabilité acceptable.
    Kurzfassung: Resumen Dentro del marco de los estudios destinados a la caracterización del acuífero del campo experimental de aguas subterráneas cercano a Montalto Uffugo (Italia), el presente trabajo estima la distribución espacial de la conductividad hidráulica de los depósitos no consolidados subyacentes. El estudio se ha llevado a cabo mediante la técnica geoestadística del krigeado con deriva externa de los datos de resistividad eléctrica y de conductividad hidráulica. Se ha verificado la fiabilidad de la estimación mediante un modelo de simulación del flujo de aguas subterráneas, basado en el método de las celdas, que utiliza los valores estimados de la conductividad hidráulica. Los resultados obtenidos indican que el método de estimación empleado posee un grado de fiabilidad aceptable.
    Notizen: Abstract  In the framework of various studies to characterize the aquifer at the groundwater experimental field near Montalto Uffugo, Italy, the present work estimates the spatial distribution of hydraulic conductivity of the unconsolidated deposits that underlie the area, by applying the geostatistical technique of kriging with external drift to electrical-resistivity and hydraulic-conductivity data. The reliability of the estimation method was tested by implementing a model, based on the method of cells, that simulates groundwater flow, with the estimated values of hydraulic conductivity. The results obtained indicate that the estimation method used has an acceptable degree of reliability.
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  • 8
    Digitale Medien
    Digitale Medien
    Springer
    Mathematical geology 31 (1999), S. 651-684 
    ISSN: 1573-8868
    Schlagwort(e): space–time models ; geostatistics ; time series ; trend models ; stochastic simulation
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Geologie und Paläontologie , Mathematik
    Notizen: Abstract Geostatistical space–time models are used increasingly for addressing environmental problems, such as monitoring acid deposition or global warming, and forecasting precipitation or stream flow. Each discipline approaches the problem of joint space–time modeling from its own perspective, a fact leading to a significant amount of overlapping models and, possibly, confusion. This paper attempts an annotated survey of models proposed in the literature, stating contributions and pinpointing shortcomings. Stochastic models that extend spatial statistics (geostatistics) to include the additional time dimension are presented with a common notation to facilitate comparison. Two conceptual viewpoints are distinguished: (1) approaches involving a single spatiotemporal random function model, and (2) approaches involving vectors of space random functions or vectors of time series. Links between these two viewpoints are then revealed; advantages and shortcomings are highlighted. Inference from space–time data is revisited, and assessment of joint space–time uncertainty via stochastic imaging is suggested.
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  • 9
    ISSN: 1573-8868
    Schlagwort(e): conditional probabilities method ; self-calibrated method ; stochastic inversion ; probabilistic assessment ; geostatistics ; stochastic hydrology ; probability fields
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Geologie und Paläontologie , Mathematik
    Notizen: Abstract The conditional probabilities (CP) method implements a new procedure for the generation of transmissivity fields conditional to piezometric head data capable to sample nonmulti-Gaussian random functions and to integrate soft and secondary information. The CP method combines the advantages of the self-calibrated (SC) method with probability fields to circumvent some of the drawbacks of the SC method—namely, its difficulty to integrate soft and secondary information or to generate non-Gaussian fields. The SC method is based on the perturbation of a seed transmissivity field already conditional to transmissivity and secondary data, with the perturbation being function of the transmissivity variogram. The CP method is also based on the perturbation of a seed field; however, the perturbation is made function of the full transmissivity bivariate distribution and of the correlation to the secondary data. The two methods are applied to a sample of an exhaustive non-Gaussian data set of natural origin to demonstrate the interest of using a simulation method that is capable to model the spatial patterns of transmissivity variability beyond the variogram. A comparison of the probabilistic predictions of convective transport derived from a Monte Carlo exercise using both methods demonstrates the superiority of the CP method when the underlying spatial variability is non-Gaussian.
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  • 10
    Digitale Medien
    Digitale Medien
    Springer
    Mathematical geology 31 (1999), S. 47-65 
    ISSN: 1573-8868
    Schlagwort(e): geostatistics ; prior distribution ; likelihood function ; posterior distribution ; variogram ; covariance ; range
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Geologie und Paläontologie , Mathematik
    Notizen: Abstract This paper shows the application of the Bayesian inference approach in estimating spatial covariance parameters. This methodology is particularly valuable where the number of experimental data is small, as occurs frequently in modeling reservoirs in petroleum engineering or when dealing with hydrodynamic variables in groundwater hydrology. There are two main advantages of Bayesian estimation: firstly that the complete distribution of the parameters is estimated and, from this distribution, it is a straightforward procedure to obtain point estimates, confidence regions, and interval estimates; secondly, all the prior information about the parameters (information available before the data are collected) is included in the inference procedure through their prior distribution. The results obtained from simulation studies are discussed.
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  • 11
    ISSN: 1573-8868
    Schlagwort(e): geostatistics ; spatial interpolation ; spatial pattern ; surface-fitting algorithms
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Geologie und Paläontologie , Mathematik
    Notizen: Abstract A factorial, computational experiment was conducted to compare the spatial interpolation accuracy of ordinary and universal kriging and two types of inverse squared-distance weighting. The experiment considered, in addition to these four interpolation methods, the effects of four data and sampling characteristics: surface type, sampling pattern, noise level, and strength of small-scale spatial correlation. Interpolation accuracy was measured by the natural logarithm of the mean squared interpolation error. Main effects of all five factors, all two-factor interactions, and several three-factor interactions were highly statistically significant. Among numerous findings, the most striking was that the two kriging methods were substantially superior to the inverse distance weighting methods over all levels of surface type, sampling pattern, noise, and correlation.
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  • 12
    Digitale Medien
    Digitale Medien
    Springer
    Biogeochemistry 42 (1998), S. 169-187 
    ISSN: 1573-515X
    Schlagwort(e): aridisols ; desertification ; erosion ; geostatistics ; Larrea tridentata ; nitrogen ; phosphorus ; Prosopis glandulosa ; soil heterogeneity
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Chemie und Pharmazie , Geologie und Paläontologie
    Notizen: Abstract Geostatistical analyses show that the distribution of soil N, P and K is strongly associated with the presence of shrubs in desert habitats. Shrubs concentrate the biogeochemical cycle of these elements in ‘islands of fertility’ that are localized beneath their canopies, while adjacent barren, intershrub spaces are comparatively devoid of biotic activity. Both physical and biological processes are involved in the formation of shrub islands. Losses of semiarid grassland in favor of invading shrubs initiate these changes in the distribution of soil nutrients, which may promote the further invasion and persistence of shrubs and cause potential feedbacks between desertification and the Earth's climate system.
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  • 13
    ISSN: 1573-8868
    Schlagwort(e): variograms ; geostatistics ; geological controls ; reservoir characterization
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Geologie und Paläontologie , Mathematik
    Notizen: Abstract To derive meaningful results from a geostatistical study, it is extremely important to establish the relationship between geology and variograms. However, it is not always easy to establish such a relationship, mainly because of the inadequacy of quantitative or qualitative information. Large amounts of reliable well-log porosity data and the detailed geological information from a complex carbonate reservoir located in the Eastern Province of Saudi Arabia provided an ideal case study of the relationships between geology and variograms. The reservoir under consideration is in the form of a gently dipping, elongated anticline and consists of three productive zones which have been subdivided into several lithologically distinct layers. Results, which indicate an excellent match between geology and porosity variograms, are summarized as follows: (1) use of stratigraphic distances resulted in considerable improvement in the behavior of variograms, (2) structurally controlled geometrical anisotropy is the most obvious feature of variograms, and (3) layers consisting of more complex lithologic framework provided variograms with relatively greater nugget variances and shorter ranges. These results provide further insight into geology and form a meaningful basis for estimation and simulation of the reservoir. They also could be used as a general guide in the geostatistical study of similar carbonate reservoirs.
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  • 14
    Digitale Medien
    Digitale Medien
    Springer
    Mathematical geology 30 (1998), S. 801-816 
    ISSN: 1573-8868
    Schlagwort(e): geostatistics ; stochastic simulation ; Monte Carlo simulation
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Geologie und Paläontologie , Mathematik
    Notizen: Abstract Simulated annealing (SA) is being increasingly used for the generation of stochastic models of spatial phenomena because of its flexibility to integrate data of diverse types and scales. The major shortcoming of SA is the extensive CPU requirements. We present a perturbation mechanism that significantly improves the CPU speed. Two conventional perturbation mechanisms are to (1) randomly select two locations and swap their attribute values, or (2) visit a randomly selected location and draw a new value from the global histogram. The proposed perturbation mechanism is a modification of option 2: each candidate value is drawn from a local conditional distribution built with a template of kriging weights rather than from the global distribution. This results in accepting more perturbations and in perturbations that improve the variogram reproduction for short scale lags. We document the new method, the increased convergence speed, and the improved variogram reproduction. Implementation details of the method such as the size of the local neighborhood are considered.
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  • 15
    Digitale Medien
    Digitale Medien
    Springer
    Mathematical geology 30 (1998), S. 95-108 
    ISSN: 1573-8868
    Schlagwort(e): geostatistics ; maximum likelihood estimation ; spatial covariance ; sampling distribution ; mean square error
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Geologie und Paläontologie , Mathematik
    Notizen: Abstract In this paper, the maximum likelihood method for inferring the parameters of spatial covariances is examined. The advantages of the maximum likelihood estimation are discussed and it is shown that this method, derived assuming a multivariate Gaussian distribution for the data, gives a sound criterion of fitting covariance models irrespective of the multivariate distribution of the data. However, this distribution is impossible to verify in practice when only one realization of the random function is available. Then, the maximum entropy method is the only sound criterion of assigning probabilities in absence of information. Because the multivariate Gaussian distribution has the maximum entropy property for a fixed vector of means and covariance matrix, the multinormal distribution is the most logical choice as a default distribution for the experimental data. Nevertheless, it should be clear that the assumption of a multivariate Gaussian distribution is maintained only for the inference of spatial covariance parameters and not necessarily for other operations such as spatial interpolation, simulation or estimation of spatial distributions. Various results from simulations are presented to support the claim that the simultaneous use of maximum likelihood method and the classical nonparametric method of moments can considerably improve results in the estimation of geostatistical parameters.
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  • 16
    Digitale Medien
    Digitale Medien
    Springer
    Mathematical geology 30 (1998), S. 259-274 
    ISSN: 1573-8868
    Schlagwort(e): cokriging ; small sample size ; pedotransfer functions ; geostatistics ; parameter uncertainty ; measurement error
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Geologie und Paläontologie , Mathematik
    Notizen: Abstract Lack of large datasets in soil protection studies and environmental engineering applications may deprive these fields of achieving accurate spatial estimates as derived with geostatistical techniques. A new estimation procedure, with the acronym Co_Est, is presented for situations involving primary and secondary datasets of sizes generally considered too small for geostatistical applications. For these situations, we suggest the transformation of the secondary dataset into the primary one using pedotransfer functions. The transformation will generate a larger set of the primary data which subsequently can be used in geostatistical analyses. The Co_Est procedure has provisions for handling measurement errors in the primary data, estimation errors in the converted secondary data, and uncertainty in the geostatistical parameters. Two different examples were used to demonstrate the applicability of Co_Est. The first example involves estimation of hydraulic conductivity random fields using 42 measured data and 258 values estimated from borehole profile descriptions. The second example consists of estimating chromium concentrations from 50 measured chromium data and 150 values estimated from a relationship between chromium and copper concentrations. The examples indicate that in situations where the size of the primary data is small, Co_Est can produce estimates which are comparable to cokriging estimates.
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  • 17
    ISSN: 1435-0157
    Schlagwort(e): Key words geophysical methods ; groundwater exploration ; geostatistics ; unconsolidated sediments
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Geologie und Paläontologie
    Beschreibung / Inhaltsverzeichnis: Résumé Une campagne géophysique a été entreprise pour préciser la profondeur du mur de l'aquifère dans le secteur sud de Jackson Hole (Wyoming, États-Unis). Des mesures audio-magnétotelluriques (audio MT) sur 77 sites de ce secteur ont fourni des logs de résistivitéélectrique du sous-sol ; les variations de la lithologie en fonction de la profondeur en ont été déduites. Un niveau géoélectrique à 100–600 ohm.m, dénommé "aquifère de Jackson", a servi à définir des dépôts superficiels quaternaires saturés en eau et non consolidés. La profondeur médiane de la base de l'aquifère de Jackson est de l'ordre de 61 m, à partir des 62 sites ayant fourni suffisamment de données de résistivité. Les valeurs audio MT mesurées ont été krigées afin d'estimer la profondeur de la base de la formation aquifère dans la partie sud de Jackson Hole. Les cartes d'isovaleurs d'estimation krigées indiquent que la profondeur de la base de la formation aquifère de Jackson est faible dans la partie centrale de la zone d'étude à l'est et à l'ouest des monts Gros Ventre, plus grande dans le secteur ouest près de la zone faillée de Teton, et faible sur la bordure sud de Jackson Hole. Les profondeurs estimées vont de 30 m au sud, près des confluences des rivières Spring et Flat avec la rivière Snake, à 210 m à l'ouest près de la ville de Wilson (Wyoming).
    Kurzfassung: Resumen Se llevó a cabo una campaña geofísica para determinar la profundidad del basamento de un acuífero libre en la zona sur de Jackson Hole, Wyoming, EEUU. USA. Medidas audio-magnetotelúricas (ATM) en 77 lugares de la zona de estudio dieron lugar a registros de resistividad eléctrica del subsuelo, que se usaron para inferir los cambios litológicos con la profundidad. Los depósitos superficiales, saturados y no consolidados de edad cuaternaria, el acuífero de Jackson, forman una capa de resistencia geoeléctrica entre 100–600 ohm-m. La profundidad media de la base del acuífero de Jackson se estima en unos 61 m (200 pies), a partir de 62 registros con medidas suficientes. Los valores ATM fueron krigeados para obtener una medida de la profundidad del basamento del acuífero en la zona sur de Jackson Hole. Los mapas de isoprofundidades obtenidos por krigeado indican que el acuífero es poco profundo en la parte central de la zona de estudio, cerca de los Gros Ventre Buttes orientales y occidentales, más profundo al oeste, cerca del sistema de fallas de Teton, y menos profundo en el borde sur de Jackson Hole. Las profundidades van desde 30 m (100 pies) en el sur, en la confluencia entre los desfiladeros Spring y Flat con el Río Snake, hasta 210 m (700 pies) al oeste, cerca de la ciudad de Wilson, Wyoming.
    Notizen: Abstract  A geophysical survey was conducted to determine the depth of the base of the water-table aquifer in the southern part of Jackson Hole, Wyoming, USA. Audio-magnetotellurics (AMT) measurements at 77 sites in the study area yielded electrical-resistivity logs of the subsurface, and these were used to infer lithologic changes with depth. A 100–600 ohm-m geoelectric layer, designated the Jackson aquifer, was used to represent surficial saturated, unconsolidated deposits of Quaternary age. The median depth of the base of the Jackson aquifer is estimated to be 200 ft (61 m), based on 62 sites that had sufficient resistivity data. AMT-measured values were kriged to predict the depth to the base of the aquifer throughout the southern part of Jackson Hole. Contour maps of the kriging predictions indicate that the depth of the base of the Jackson aquifer is shallow in the central part of the study area near the East and West Gros Ventre Buttes, deeper in the west near the Teton fault system, and shallow at the southern edge of Jackson Hole. Predicted, contoured depths range from 100 ft (30 m) in the south, near the confluences of Spring Creek and Flat Creek with the Snake River, to 700 ft (210 m) in the west, near the town of Wilson, Wyoming.
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  • 18
    Digitale Medien
    Digitale Medien
    Springer
    Biogeochemistry 36 (1997), S. 261-274 
    ISSN: 1573-515X
    Schlagwort(e): boreal forests ; carbon balance ; geostatistics ; GIS ; soil carbon ; spatial variability
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Chemie und Pharmazie , Geologie und Paläontologie
    Notizen: Abstract For confidently estimating the amount of carbon stored in boreal forestsoil, better knowledge of smaller regions is needed. In order to estimatethe amount of soil C in forests on mineral soil in Finland, i.e. excludingpeatland forests, and illustrate the regional patterns of the storage,statistical models were first made for the C densities of the organic and0–1 m mineral soil layers. A forest type, which indicated siteproductivity, and the effective temperature sum were used asexplanatory variables of the models. In addition, a constant C densitywas applied for the soil layer below the depth of 1 m on sortedsediments. Using these models the C densities were calculated for atotal of 46673 sites of the National Forest Inventory (NFI). The amountof the soil C was then calculated in two ways: 1) weighting the Cdensities of the NFI sites by the land area represented by these sites and2) interpolating the C densities of the NFI sites for 4 ha blocks to coverthe whole land area of Finland and summing up the blocks on forestedmineral soil. The soil C storage totalled 1109 Tg and 1315 Tg, whencalculated by the areal weighting and the interpolated blocks,respectively. Of that storage, 28% was in the organic layer, 68% inthe 0–1 m mineral soil layer and 4% in the layer below 1 m. The totalsoil C equals more than two times the amount of C in tree biomass and20% of the amount of C in peat in Finland. Soil C maps made usingthe interpolated blocks indicated that the largest soil C reserves arelocated in central parts of southern Finland. The C storage of theorganic layer was assessed to be overestimated at largest by 13% andthat of the 0–1 m mineral soil layer by 29%. The largest error in theorganic layer estimate is associated with the effects of forest harvestingand in the mineral soil estimate with the stone content of the soil.
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  • 19
    ISSN: 1573-8868
    Schlagwort(e): geostatistics ; regionalization ; fuzzy ; kriging ; hydrogeology
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Geologie und Paläontologie , Mathematik
    Notizen: Abstract In many instances hydrogeological parameters obtained by conventional methods for selected localities within an aquifer or an aquitard are not sufficient for adequate regionalization at the scale of the entire layer. Here, we demonstrate an application of the fuzzy kriging method in regionalization of hydrogeological data, in which the set of conventional, crisp values is supplemented by imprecise information subjectively estimated by an expert. It is believed that such an approach eventually may reflect the real-world conditions more closely than a traditional crisp-value approach, because the former does not impose exactness artificially on phenomena which are diffuse by their nature. Spatial interpolation was done for the thickness of one of the major aquitards (till and glaciolacustrine clay) in northwestern Germany. The dataset consists of 329 crisp values from boreholes supplemented by 172 imprecise values defined as fuzzy numbers. It is demonstrated that the reliability of regionalization was higher, compared to regionalization performed with the crisp dataset only. Fuzzy kriging was performed with FUZZEKS (Fuzzy Evaluation and Kriging System) developed at the Ecosystem Research Center at the University of Kiel.
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  • 20
    Digitale Medien
    Digitale Medien
    Springer
    Mathematical geology 28 (1996), S. 385-405 
    ISSN: 1573-8868
    Schlagwort(e): multiresolution approximation ; nested analysis of variance ; sampling ; quadtrees ; remote sensing ; geostatistics ; environmental mapping
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Geologie und Paläontologie , Mathematik
    Notizen: Abstract A quadtree-based image segmentation procedure (HQ) is presented to map complex environmental conditions. It applies a hierarchical nested analysis of variance within the framework of multiresolution wavelet approximation. The procedure leads to an optimal solution for determining mapping units based on spatial variability with constraints on the arrangement and shape of the units. Linkages to geostatisiics are pointed out, but the HQ decomposition algorithm does not require any homogeneity criteria. The computer implementation can be parameterized by either the number of required mapping units or the maximum within-unit variance, or it can provide a “spectrum” of significances of nested ANOVA. The detailed mathematical background and methodology is illustrated by a salt-affected grassland mapping study (Hortobágy, Hungary), where heterogeneous environmental characteristics have been sampled and predicted based on remotely sensed images using these principles.
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  • 21
    Digitale Medien
    Digitale Medien
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    Mathematical geology 28 (1996), S. 477-499 
    ISSN: 1573-8868
    Schlagwort(e): Fennoscandia ; kriging ; indicators ; indicator principal components simulation ; geostatistics
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Geologie und Paläontologie , Mathematik
    Notizen: Abstract The Ryytimaa dolomite formation in western Finland is mined for fertilizing purposes in agriculture. Ordinary kriging, indicator kriging, and indicator principal components simulation were used to map the dolomite quality. The fertilizing properties depend on the relative amounts of MgO and neutralizing CaO in dolomite. Based on the chemical composition the dolomite is divided into six quality classes according to the Finnish legislation. This quality classification is only partially in accordance with the three populations inferred from the distributions of the MgO, CaO, and neutralizing CaO contents. These populations probably represent compositionally different dolomite layers, now forming complicated fold interference structures that are visible on the kriged maps. The mapping of the quality, and thus, selective mining of the dolomite deposit is possible using the quality classification based on the three populations. In contrast, selective mining of the dolomite based on the legislation is difficult—at least with techniques used in this study.
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  • 22
    Digitale Medien
    Digitale Medien
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    Mathematical geology 28 (1996), S. 829-842 
    ISSN: 1573-8868
    Schlagwort(e): geostatistics ; flow simulation ; streamtubes ; scale up ; gridding ; accuracy and precision ; modeling uncertainty
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Geologie und Paläontologie , Mathematik
    Notizen: Abstract The combination of geostatistics-based numerical geological models and finite difference flow simulation has improved our ability to predict reservoir performance. The main contribution of geostatistical modeling has been more realistic representations of reservoir heterogeneity. Our understanding of the physics of fluid flow in porous media is reasonably captured by flow simulators in common usage. Notwithstanding the increasing application and success of geostatistics and flow simulation there remain many important challenges in reservoir forecasting. This application has alerted geoscientists and physicists that geostatistical/flow models in many respects, are, engineering approximations to thereal spatial distribution andreal flow processes. This paper reviews current research directions and presents some new ideas of where reserach could be focused to improve our ability to model geological features, model flow processes, and, ultimately, improve reservoir performance predictions.
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  • 23
    ISSN: 1573-8868
    Schlagwort(e): heterogeneity ; Monte-Carlo analysis ; uncertainty ; geostatistics ; conditioning ; self-calibrated method
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Geologie und Paläontologie , Mathematik
    Notizen: Abstract Transmissivity and head data are sampled from an exhaustive synthetic reference field and used to predict the arrival positions and arrival times of a number of particles transported across the field, together with an uncertainty estimate. Different combinations of number of transmissivity data and number of head data used are considered in each one of a series of 64 Monte-Carlo analyses. In each analysis, 250 realizations of transmissivity fields conditioned to both transmissivity and head data are generated using a novel geostatistically based inverse method. Pooling the solutions of the flow and transport equations in all 250 realizations allows building conditional frequency distributions for particle arrival positions and arrival times. By comparing these fresquency distributions, we can assess the incremental gain that additional head data provide. The main conclusion is that the first few head data dramatically improve the quality of transport predictions.
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  • 24
    Digitale Medien
    Digitale Medien
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    Mathematical geology 28 (1996), S. 137-160 
    ISSN: 1573-8868
    Schlagwort(e): structure function ; correlation function ; least-squares prediction and collocation ; geostatistics ; measure-of-interpolation error
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Geologie und Paläontologie , Mathematik
    Notizen: Abstract In 1963, Gandin published a monograph on “optimum interpolation for the objective analysis of meteorological fields, ≓ a method that is similar mathematically to geodetical least-squares prediction and collocation, simple kriging, and spectral interpolation. The common problem is the interpolation or extrapolation or estimation of a continuous spatial property from finitely many observations. Gandin 's method is presented in an inverse-theoretical context with focus on a methodological comparison with related methods. Underlying mathematical assumptions as well as geological implications are discussed. An introductory overview of inverse methods in the earth sciences is given, with emphasis on methods with a structure analysis step.
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  • 25
    Digitale Medien
    Digitale Medien
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    Mathematical geology 28 (1996), S. 295-313 
    ISSN: 1573-8868
    Schlagwort(e): sample design ; geostatistics ; estimation error ; criterion based on variance
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Geologie und Paläontologie , Mathematik
    Notizen: Abstract A number of criteria based on kriging variance calculations may be used for infill sampling design in geologic site characterization. Searching for the best new sample locations from a set of candidate locations can result in excessive computation time if these criteria and the naive rekriging are used. The relative updated kriging estimate and variance for universal kriging estimation are demonstrated as a simple kriging estimate and variance, respectively. The updated kriging variance is demonstrated as the multiplication of two kriging variances. Using these updated kriging variance equations can increase the computational speed for selecting the best new sample locations. The application results for oil rock thickness in an oilfield indicate that minimizing the average relative updated kriging variance is a useful alternative to the other criteria based on kriging variance in optimal infill sampling design for geologic site characterization.
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  • 26
    ISSN: 1573-8868
    Schlagwort(e): electrofacies ; kriging ; classification ; regionalization ; geostatistics
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Geologie und Paläontologie , Mathematik
    Notizen: Abstract Regionalized classification of electrofacies utilizes the statistical relationships between laboratory determined hydrologic properties and field-measured geophysical properties to estimate spatial distributions of porosity, permeability, and diagenetic characteristics. The method, illustrated with an application to the St. Peter Sandstone in the Michigan basin, combines techniques for multivariate analysis and spatial estimation. Core plug and borehole geophysical data are clustered into electrofacies that reflect the hydrologic properties and diagenetic characteristics of the formation. Electrofacies characteristics then are used to assign a class membership probability at locations where only geophysical data are available. Three-dimensional estimation of electrofacies occurrence is done by kriging datasets containing the probability of electrofacies membership at borehole locations. The discretization and kriging geometry allow three-dimensional estimation of hydrologic parameters for a large region that incorporates meter-scale heterogeneity. Finally, permeability and porosity are estimated at each grid location by probability-weighting. Because the electrofacies carry information about both the hydrologic and lithologic properties, the resulting spatial distributions provide an understanding of both the present-day flow characteristics and the extent of processes that control them.
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  • 27
    ISSN: 1573-8868
    Schlagwort(e): Gaussian random fields ; geostatistics ; Monte Carlo simulations ; orthogonal polynomials
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Geologie und Paläontologie , Mathematik
    Notizen: Abstract Consider the problem of generating a realization y1 of a Gaussian random field on a dense grid of points Ω1 conditioned on field observations y2 collected on a sparse grid of points Ω2. An approach to this is to generate first an unconditional realization y over the grid Ω=Ω1 ∪ Ω2, and then to produce y1 by conditioning y on the data y2. As standard methods for generating y, such as the turning bands, spectral or Cholesky approaches can have various limitations, it has been proposed by M. W. Davis to generate realizations from a matrix polynomial approximations to the square root of the covariance matrix. In this paper we describe how to generate a direct approximation to the conditional realization y1, on Ω1 using a variant of Davis' approach based on approximation by Chebyshev polynomials. The resulting algorithm is simple to implement, numerically stable, and bounds on the approximation error are readily available. Furthermore we show that the conditional realization y1 can be generated directly with a lower order polynomial than the unconditional realization y, and that further reductions can be achieved by exploiting a nugget effect if one is present. A pseudocode version of the algorithm is provided that can be implemented using the fast Fourier transform if the field is stationary and the grid Ω1 is rectangular. Finally, numerical illustrations are given of the algorithm's performance in generating various 2-D realizations of conditional processes on large sampling grids.
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  • 28
    Digitale Medien
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    Mathematical geology 27 (1995), S. 421-462 
    ISSN: 1573-8868
    Schlagwort(e): self-similarity ; self-affinity ; variogram-criterion ; inheritance of scaling properties ; geostatistics ; bathymetric data
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Geologie und Paläontologie , Mathematik
    Notizen: Abstract The topographic structure of the ocean bottom is investigated at different scales of resolution to answer the question: Can the seafloor be described as a fractal process? Methods from geostatistics, the theory of regionalized variables, are used to analyze the spatial structure of the ocean floor at different scales of resolution. The key to the analysis is the variogram criterion: Self-similarity of a stochastic process implies self-similarity of its variogram. The criterion is derived and proved here: it also is valid for special cases of self-affinity (in a sense adequate for topography). It has been proposed that seafloor topography can be simulated as a fractal (an object of Hausdorff dimension strictly larger than its topological dimension), having scaling properties (self-similarity or self-affinity). The objective of this study is to compare the implications of these concepts with observations of the seafloor. The analyses are based on SEABEAM bathymetric data from the East Pacific Rise at 13°N/104°W and at 9°N/104°W and use tracks that run both across the ridge crest and along the ridge flank. In the geostatistical evaluation, the data are considered as a stochastic process. The spatial continuity of this process is described by variograms that are calculated for different scales and directions. Applications of the variogram criterion to scale-dependent variogram models yields the following results: Although the seafloor may be a fractal in the sense of the definition involving the Hausdorff dimension, it is not self-similar, nor self-affine (in the given sense). Mathematical models of scale-dependent spatial structures are presented, and their relationship to geologic processes such as ridge evolution, crust formation, and sedimentation is discussed.
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  • 29
    Digitale Medien
    Digitale Medien
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    Mathematical geology 27 (1995), S. 589-608 
    ISSN: 1573-8868
    Schlagwort(e): geostatistics ; sample design
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Geologie und Paläontologie , Mathematik
    Notizen: Abstract Three problem-dependent meanings for engineering “extremes” are motivated, established, and translated into formal geostatistical (model-based) criteria for designing infill sample networks. (1) Locate an area within the domain of interest where a specified threshold is exceeded, if such areas exist. (2) Locate the maximum value in the domain of interest. (3) Minimize the chance of areas where values are significantly different from predicted values. An example application on a simulated dataset demonstrates how such purposive design criteria might affect practice.
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  • 30
    Digitale Medien
    Digitale Medien
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    Mathematical geology 27 (1995), S. 813-830 
    ISSN: 1573-8868
    Schlagwort(e): Lévy-stable distributions ; geostatistics ; well logs ; fractals
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Geologie und Paläontologie , Mathematik
    Notizen: Abstract Borehole measurements of petrophysical properties of sedimentary rock from contrasting geological settings are shown to be consistent with fractional Lévy motion. Specifically, successive increments in the measurements sequences are modeled accurately as having symmetric Lévy-stable distributions. The measurement sequences are statistically selfsimilar for a wide range of spatial scales, limited by the finite length of the measurement sequences at large scales and by the resolution limits of the logging devices at small scales. The measurements sequences display clear evidence of antipersistence (negative dependence in the increments). These results suggest that the fractional-Gaussian-noise model used in reservoir description is an inappropriate model for sedimentary rock, because the rock properties are not stationary and do not have a Gaussian distribution as required for fractional Gaussian noise. In contrast to those of Gaussian-based fractals, the sample paths of the new model increase and decrease in jumps of all magnitudes and mimic geological stratification. These results have fundamental implications for petroleum geostatistics, as many of the statistical techniques developed for Gaussian variables are not valid for Lévy-distributed variables.
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  • 31
    ISSN: 1573-0581
    Schlagwort(e): Bathymetry ; gravity ; magnetics ; satellite data ; Gondwana breakup ; geostatistics ; geophysical inverse theory
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Geologie und Paläontologie , Physik
    Notizen: 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.
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  • 32
    Digitale Medien
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    Mathematical geology 26 (1994), S. 491-503 
    ISSN: 1573-8868
    Schlagwort(e): conditional simulation ; geostatistics ; neural network
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Geologie und Paläontologie , Mathematik
    Notizen: 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.
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  • 33
    Digitale Medien
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    Mathematical geology 26 (1994), S. 531-555 
    ISSN: 1573-8868
    Schlagwort(e): geostatistics ; fractals ; spatial analysis ; permeability modeling
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Geologie und Paläontologie , Mathematik
    Notizen: 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.
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  • 34
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    Mathematical geology 26 (1994), S. 589-603 
    ISSN: 1573-8868
    Schlagwort(e): kriging ; geostatistics ; spatial estimation ; inverse-distance estimation
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Geologie und Paläontologie , Mathematik
    Notizen: 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.
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  • 35
    Digitale Medien
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    Mathematical geology 25 (1993), S. 219-240 
    ISSN: 1573-8868
    Schlagwort(e): geostatistics ; kriging ; cokriging ; cross-variogram ; best linear unbiased prediction ; generalized least squares
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Geologie und Paläontologie , Mathematik
    Notizen: 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.
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  • 36
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    Mathematical geology 25 (1993), S. 261-279 
    ISSN: 1573-8868
    Schlagwort(e): geostatistics ; qualitative knowledge-information ; artificial intelligence
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Geologie und Paläontologie , Mathematik
    Notizen: 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.
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  • 37
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    Mathematical geology 25 (1993), S. 439-451 
    ISSN: 1573-8868
    Schlagwort(e): block Toeplitz structure ; Cholesky factorization ; geostatistics ; Monte Carlo simulations ; spatial random field
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Geologie und Paläontologie , Mathematik
    Notizen: 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.
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  • 38
    Digitale Medien
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    Mathematical geology 25 (1993), S. 525-540 
    ISSN: 1573-8868
    Schlagwort(e): geostatistics ; linear model ; best linear unbiased estimation ; experimental variogram ; restricted maximum likelihood
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Geologie und Paläontologie , Mathematik
    Notizen: 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.
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  • 39
    ISSN: 1573-8868
    Schlagwort(e): classification ; spatial structure analysis ; geostatistics ; bathymetry ; Atlantic ; automatization
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Geologie und Paläontologie , Mathematik
    Notizen: 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.
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  • 40
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    Stochastic environmental research and risk assessment 6 (1992), S. 304-320 
    ISSN: 1436-3259
    Schlagwort(e): geostatistics ; precipitation ; water balance models ; semivariogram ; kriging ; spatial variation
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Architektur, Bauingenieurwesen, Vermessung , Energietechnik , Geographie , Geologie und Paläontologie
    Notizen: 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%.
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  • 41
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    Stochastic environmental research and risk assessment 6 (1992), S. 209-221 
    ISSN: 1436-3259
    Schlagwort(e): geostatistics ; precipitation ; water balance models ; semivariogram ; kriging ; spatial variation
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Architektur, Bauingenieurwesen, Vermessung , Energietechnik , Geographie , Geologie und Paläontologie
    Notizen: 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%.
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  • 42
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    Mathematical geology 24 (1992), S. 73-97 
    ISSN: 1573-8868
    Schlagwort(e): Fourier transform ; spatial analysis ; geostatistics ; reservoir property distribution
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Geologie und Paläontologie , Mathematik
    Notizen: 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.
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  • 43
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    Mathematical geology 24 (1992), S. 129-133 
    ISSN: 1573-8868
    Schlagwort(e): geostatistics ; kriging
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Geologie und Paläontologie , Mathematik
    Notizen: 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.
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  • 44
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    Mathematical geology 24 (1992), S. 329-343 
    ISSN: 1573-8868
    Schlagwort(e): sampling ; geostatistics
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Geologie und Paläontologie , Mathematik
    Notizen: 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.
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  • 45
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    Mathematical geology 24 (1992), S. 381-391 
    ISSN: 1573-8868
    Schlagwort(e): sampling ; geostatistics ; estimators ; interpolation
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Geologie und Paläontologie , Mathematik
    Notizen: 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.
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  • 46
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    Mathematical geology 24 (1992), S. 171-176 
    ISSN: 1573-8868
    Schlagwort(e): geostatistics ; kriging
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Geologie und Paläontologie , Mathematik
    Notizen: 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.
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  • 47
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    Mathematical geology 23 (1991), S. 3-7 
    ISSN: 1573-8868
    Schlagwort(e): geostatistics ; information theory
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Geologie und Paläontologie , Mathematik
    Notizen: 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.
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  • 48
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    Mathematical geology 23 (1991), S. 741-758 
    ISSN: 1573-8868
    Schlagwort(e): geostatistics ; hypothesis testing ; parameter estimation
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Geologie und Paläontologie , Mathematik
    Notizen: 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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  • 49
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    Mathematical geology 23 (1991), S. 1059-1080 
    ISSN: 1573-8868
    Schlagwort(e): conditional simulation ; contaminant migration ; geostatistics ; plume interception
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Geologie und Paläontologie , Mathematik
    Notizen: 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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  • 50
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    Mathematical geology 23 (1991), S. 119-135 
    ISSN: 1573-8868
    Schlagwort(e): Gaussian random field ; covariance estimation ; geostatistics ; eigenvalue decomposition
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Geologie und Paläontologie , Mathematik
    Notizen: 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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  • 51
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    Mathematical geology 22 (1990), S. 107-121 
    ISSN: 1573-8868
    Schlagwort(e): Variogram ; geostatistics ; sample support ; variance
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Geologie und Paläontologie , Mathematik
    Notizen: 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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  • 52
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    Mathematical geology 22 (1990), S. 123-144 
    ISSN: 1573-8868
    Schlagwort(e): nonlinear estimators ; geostatistics ; confidence intervals ; indicator kriging ; probability kriging ; disjunctive kriging
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Geologie und Paläontologie , Mathematik
    Notizen: 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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  • 53
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    Mathematical geology 22 (1990), S. 573-588 
    ISSN: 1573-8868
    Schlagwort(e): classification ; regionalization ; Bayes' theorem ; geostatistics
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Geologie und Paläontologie , Mathematik
    Notizen: 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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  • 54
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    Mathematical geology 22 (1990), S. 611-623 
    ISSN: 1573-8868
    Schlagwort(e): geostatistics ; geohydrology ; kriging ; spatiotemporal variables
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Geologie und Paläontologie , Mathematik
    Notizen: 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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  • 55
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    Mathematical geology 22 (1990), S. 763-777 
    ISSN: 1573-8868
    Schlagwort(e): spatial estimation ; entropy ; Bayes law ; information ; geostatistics
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Geologie und Paläontologie , Mathematik
    Notizen: 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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  • 56
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    Mathematical geology 22 (1990), S. 915-932 
    ISSN: 1573-8868
    Schlagwort(e): sample design ; geostatistics ; spatial classification
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Geologie und Paläontologie , Mathematik
    Notizen: 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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  • 57
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    Geotechnical and geological engineering 7 (1989), S. 227-237 
    ISSN: 1573-1529
    Schlagwort(e): Roof convergence ; longwall face ; geostatistics ; coal mining ; support advance
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Geologie und Paläontologie
    Notizen: 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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  • 58
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    Mathematical geology 21 (1989), S. 285-308 
    ISSN: 1573-8868
    Schlagwort(e): Bayesian updating ; geostatistics ; Kriging ; linear inversion
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Geologie und Paläontologie , Mathematik
    Notizen: 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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  • 59
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    Mathematical geology 21 (1989), S. 683-691 
    ISSN: 1573-8868
    Schlagwort(e): geostatistics ; factorial kriging analysis ; spectral analysis ; petroleum exploration
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Geologie und Paläontologie , Mathematik
    Notizen: 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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  • 60
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    Mathematical geology 21 (1989), S. 693-713 
    ISSN: 1573-8868
    Schlagwort(e): geostatistics ; variography ; deregularization ; universal kriging ; bathymetry ; SEABEAM survey ; cartography
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Geologie und Paläontologie , Mathematik
    Notizen: 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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  • 61
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    Mathematical geology 20 (1988), S. 287-311 
    ISSN: 1573-8868
    Schlagwort(e): fuzzy sets ; geostatistics ; methodology ; application ; permeability prediction ; soil liner
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Geologie und Paläontologie , Mathematik
    Notizen: 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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  • 62
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    Mathematical geology 20 (1988), S. 631-654 
    ISSN: 1573-8868
    Schlagwort(e): Fractals ; geostatistics ; fracture network ; granite
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Geologie und Paläontologie , Mathematik
    Notizen: 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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  • 63
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    Mathematical geology 20 (1988), S. 1001-1019 
    ISSN: 1573-8868
    Schlagwort(e): geostatistics ; change of support ; recoverable reserves ; selectivity ; normality index ; Cartier's formula ; affine correction ; discrete Gaussian model ; mosaic model
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Geologie und Paläontologie , Mathematik
    Notizen: 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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  • 64
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    Mathematical geology 19 (1987), S. 25-39 
    ISSN: 1573-8868
    Schlagwort(e): geostatistics ; kriging ; Bayesian statistics
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Geologie und Paläontologie , Mathematik
    Notizen: 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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  • 65
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    Mathematical geology 19 (1987), S. 91-98 
    ISSN: 1573-8868
    Schlagwort(e): geostatistics ; kriging ; conditional simulation ; fast Fourier transform
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Geologie und Paläontologie , Mathematik
    Notizen: 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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  • 66
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    Mathematical geology 19 (1987), S. 99-107 
    ISSN: 1573-8868
    Schlagwort(e): geostatistics ; simulation ; Toeplitz matrices ; block Toeplitz matrices ; matrix polynomial approximation
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Geologie und Paläontologie , Mathematik
    Notizen: 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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    Mathematical geology 19 (1987), S. 183-205 
    ISSN: 1573-8868
    Schlagwort(e): geostatistics ; intrinsic hypothesis ; regionalized variables ; sampling ; spherical semivariogram
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Geologie und Paläontologie , Mathematik
    Notizen: 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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  • 68
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    Mathematical geology 19 (1987), S. 769-783 
    ISSN: 1573-8868
    Schlagwort(e): Cross-validation ; geostatistics ; intrinsic random functions of orderk ; kriging
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Geologie und Paläontologie , Mathematik
    Notizen: 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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  • 69
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    Mathematical geology 18 (1986), S. 529-537 
    ISSN: 1573-8868
    Schlagwort(e): geostatistics ; kriging ; nugget effect ; soil sampling ; support
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Geologie und Paläontologie , Mathematik
    Notizen: 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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  • 70
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    Mathematical geology 18 (1986), S. 635-652 
    ISSN: 1573-8868
    Schlagwort(e): distribution of kriging error ; variogram stationarity ; coal ; geostatistics ; contract risk
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Geologie und Paläontologie , Mathematik
    Notizen: 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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  • 71
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    Mathematical geology 18 (1986), S. 93-117 
    ISSN: 1573-8868
    Schlagwort(e): deterministic ; “estimation variance” ; interpolation ; geostatistics ; kriging ; least-squares prediction ; ore deposit assessment ; probabilistic ; semivariogram ; statistical inference
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Geologie und Paläontologie , Mathematik
    Notizen: 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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  • 72
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    Mathematical geology 18 (1986), S. 287-305 
    ISSN: 1573-8868
    Schlagwort(e): interpolation ; mapping ; geostatistics
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Geologie und Paläontologie , Mathematik
    Notizen: 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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  • 73
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    Mathematical geology 18 (1986), S. 477-488 
    ISSN: 1573-8868
    Schlagwort(e): kriging ; nugget effect ; range ; semivariogram ; geostatistics
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Geologie und Paläontologie , Mathematik
    Notizen: 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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  • 74
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    Geotechnical and geological engineering 3 (1985), S. 155-159 
    ISSN: 1573-1529
    Schlagwort(e): Estimation variance ; geostatistics ; interpolation ; kriging
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Geologie und Paläontologie
    Notizen: 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.
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  • 75
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    Mathematical geology 17 (1985), S. 785-796 
    ISSN: 1573-8868
    Schlagwort(e): geostatistics ; kriging ; estimation variance
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Geologie und Paläontologie , Mathematik
    Notizen: 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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  • 76
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    Mathematical geology 17 (1985), S. 195-208 
    ISSN: 1573-8868
    Schlagwort(e): geostatistics ; covariance estimation ; optimization
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Geologie und Paläontologie , Mathematik
    Notizen: 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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  • 77
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    Mathematical geology 17 (1985), S. 1-15 
    ISSN: 1573-8868
    Schlagwort(e): geostatistics ; stationarity ; ergodicity ; spatial average ; deterministic kriging
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Geologie und Paläontologie , Mathematik
    Notizen: 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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  • 78
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    Mathematical geology 17 (1985), S. 81-90 
    ISSN: 1573-8868
    Schlagwort(e): simulation ; turning bands ; covariance ; semivariogram ; geostatistics
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Geologie und Paläontologie , Mathematik
    Notizen: 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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  • 79
    ISSN: 1573-8868
    Schlagwort(e): geostatistics ; reserve evaluation ; bauxite prospecting ; open-pit mining
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Geologie und Paläontologie , Mathematik
    Notizen: Abstract Geostatistical calculations were carried out on two completely exhausted open-pit bauxite mines in the Iharkut bauxite district, Hungary. Fictitious regular drilling grids were laid on the maps, and horizontal variograms were calculated for drilling grids evaluating the bauxite surface, footwall surface, and bauxite thickness. Point kriging was carried out for all three parameters. Bauxite reserves were calculated using block kriging for bauxite thickness. The total estimation variance of the reserves has been established. Results were compared with real reserves obtained from the mine maps.
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  • 80
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    Mathematical geology 16 (1984), S. 19-35 
    ISSN: 1573-8868
    Schlagwort(e): groundwater ; geostatistics ; cokriging ; kriging ; transmissivity ; specific capacity
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Geologie und Paläontologie , Mathematik
    Notizen: Abstract This paper presents a new application of the cokriging technique for constructing maps of aquifer transmissivity from field measurements of transmissivity and specific capacity. The technique is illustrated using data from Yolo Basin, California. Cokriging is well-suited for estimating undersampled variables. To improve the accuracy of the estimation, cokriging considers the spatial auto-correlation of the variable to be estimated and the spatial cross-correlation between the variable to be estimated and other, better-sampled variables. Consequently, in regions that lack data of the variable to be estimated, accurate estimation can still be made on the basis of auto- and cross-correlation. In addition, estimation variances can be obtained with a little additional computation effort.
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  • 81
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    Mathematical geology 16 (1984), S. 407-421 
    ISSN: 1573-8868
    Schlagwort(e): variogram ; geostatistics ; positive-definite ; Bochner's theorem ; Hankel transforms
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Geologie und Paläontologie , Mathematik
    Notizen: Abstract A method based on Bochner's theorem is described for demonstrating the positive-definiteness of variogram models and for generating classes of valid variogram functions.
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  • 82
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    Mathematical geology 16 (1984), S. 249-265 
    ISSN: 1573-8868
    Schlagwort(e): kriging ; moving neighborhood ; global neighborhood ; geostatistics ; automatic contouring
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Geologie und Paläontologie , Mathematik
    Notizen: Abstract The kriging estimator is usually computed in a moving neighborhood; only the data near the point to be estimated are used. This moving neighborhood approach creates discontinuities in mapping applications. An alternative approach is presented here, whereby all points are estimated using all the available data. To solve the resulting large linear system the kriging estimator is expressed in terms of the inverse of the covariance matrix. The covariance matrix has the advantage of being positive definite and the size of system which can be solved without encountering numerical instability is substantially increased. Because the kriging matrix does not change, the estimator can be written in terms of scalar products, thus avoiding the more time-consuming matrix multiplications of the standard approach. In the particular case of a covariance which is zero for distances greater than a fixed value (the range), the resulting banded structure of the covariance matrix is shown to lead to substantial computational savings in both run time and storage space. In this case the calculation time for the kriging variance is also substantially reduced. The present method is extended to the nonstationary case.
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  • 83
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    Mathematical geology 16 (1984), S. 327-350 
    ISSN: 1573-8868
    Schlagwort(e): geostatistics ; estimation variance ; recoverable reserves
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Geologie und Paläontologie , Mathematik
    Notizen: Abstract One of the tasks routinely carried out by geostatisticians is the evaluation of global mining reserves corresponding to a given cutoff grade and size of selective mining units. A long with these recovery figures, the geostatistician generally provides an assessment of the global estimation variance, which represents the precision of the overall average grade estimate, when no cutoff is applied. Such a global estimation variance is of limited interest for evaluating mining projects; what is required is the reliability of the estimate of recovered reserves or, in other words, the conditional estimation variance. Unfortunately, classical linear geostatistical methods fail to provide an easy way to estimate this variance. Through the use of simulated deposits (representing various types of regionalization)the present paper reviews and discusses the effects of changes in cutoff grade and selective mining unit size on the conditional estimation variance. It is shown that, when the cutoff grade is applied to a pointsupport (sample-size)distribution, the conditional estimation variance appears to be readily accessible by classical formulas, once the conditional semivariogram is known. However, the evaluation of the conditional estimation variance seems to be less straightforward for the general case when a cutoff is applied to the average grade distribution of selective mining units. Empirical approximation formulas for the conditional estimation variance are tentatively proposed, and their performance in the case of the simulated deposits is shown. The limitations of these approximations are discussed, and possible ways of formalizing the problem suggested.
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  • 84
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    Geotechnical and geological engineering 1 (1983), S. 331-342 
    ISSN: 1573-1529
    Schlagwort(e): Optimization ; geostatistics ; mining ; exploration
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Geologie und Paläontologie
    Notizen: Summary In this paper mathematical programming models and special solution algorithms are presented for the optimal allocation of data points for regionalized variables. Two models are discussed; the first one minimizes the estimation variance subject to a given number of additional measurement points or additional measurement cost. The second model minimizes the number of additional points or additional cost subject to upper bounds given to the estimation variances. The models are based on the theory of kriging and the properties of the estimation variances, which lead to certain multiobject programming problems. As it can be shown, a special branch and bound algorithm can be used for solving the resulting nonlinear programming problems
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  • 85
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    Mathematical geology 15 (1983), S. 183-195 
    ISSN: 1573-8868
    Schlagwort(e): Computers ; data files ; exploration ; geostatistics ; petroleum ; prospects ; spatial filtering ; trend analysis
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Geologie und Paläontologie , Mathematik
    Notizen: Abstract Computer recognition of prospective areas through the processing of digital exploration data can be effective if the statistical tests for the determination of the prospects are pertinent to the presence of the desired mineral. Where exploration involves the application of polynomial trend analysis to structure contour maps in the search for petroleum and natural gas, standard analysis of variance tests may not indicate the best exploration maps. Variance tests may be completely invalid where isolated dips and clustered samples cause the surfaces generated by some of the most common trend programs to oscillate, creating a false impression of variance. On the other hand, tests that directly compare the position of residual features with areas of known production consistently indicate the best map for the determination of new prospects. They are simple to apply and appear to offer the most opportunity for the automatic recognition of prospective areas.
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  • 86
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    Mathematical geology 15 (1983), S. 517-536 
    ISSN: 1573-8868
    Schlagwort(e): Kriging ; geostatistics ; regression effect
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Geologie und Paläontologie , Mathematik
    Notizen: Abstract One traditional method of estimating ore reserves is to allocate the arithmetic mean of local samples to each block or stope within the deposit. Thirty years ago, Krige showed that this method produced a “bias” in the estimates, in that high values were commonly overestimated and low values underestimated. This bias became known as the “regression effect” because it could usually be corrected by an empirical regression relationship derived from previously mined areas. This paper shows how the regression effect can be reconciled with the geostatistical method of kriging and how a simple regression line may be derived before any areas are mined out. The method is similar to kriging with known mean, but acknowledges that the overall mean of the deposit may not be known precisely.
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  • 87
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    Mathematical geology 15 (1983), S. 445-468 
    ISSN: 1573-8868
    Schlagwort(e): indicator kriging ; geostatistics ; nonparametric
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Geologie und Paläontologie , Mathematik
    Notizen: Abstract The indicator approach, whereby the data are used through their rank order, allows a nonparametric approach to the data bivariate distribution. Such rich structural information allows a nonparametric risk-qualified, estimation of local and global spatial distributions.
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  • 88
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    Mathematical geology 15 (1983), S. 259-286 
    ISSN: 1573-8868
    Schlagwort(e): geostatistics ; local recovery ; conditional expectation ; multigaussian kriging ; disjunctive kriging
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Geologie und Paläontologie , Mathematik
    Notizen: Abstract The performance of the geostatistical techniques of disjunctive kriging and multigaussian kriging for the estimation of local recovered reserves are compared using a simulated deposit.
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  • 89
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    Mathematical geology 14 (1982), S. 57-64 
    ISSN: 1573-8868
    Schlagwort(e): estimation ; generalized covariance function ; generalized increments ; geostatistics ; intrinsic random functions ; multicollinearity
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Geologie und Paläontologie , Mathematik
    Notizen: Abstract The estimation of the generalized covariance function, K, is a major problem in the use of intrinsic random functions of order k to obtain kriging estimates. The precise estimation by least-squares regression of the parameters in polynomial models for K is made difficult by the nature of the distribution of the dependent variable and the multicollinearity of the independent variables.
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  • 90
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    Mathematical geology 14 (1982), S. 309-319 
    ISSN: 1573-8868
    Schlagwort(e): Bias ; drift ; semivariogram ; geostatistics
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Geologie und Paläontologie , Mathematik
    Notizen: Abstract When nonlinear drift is present, the nature of the bias in the experimental semivariogram estimator of the semivariogram function is determined by the extent and density of the sampling as well as by the drift function itself. The bias caused by drift may affect the interpretation of the experimental semivariogram over its entire range.
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  • 91
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    Mathematical geology 14 (1982), S. 557-564 
    ISSN: 1573-8868
    Schlagwort(e): geostatistics ; variogram ; outliers
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Geologie und Paläontologie , Mathematik
    Notizen: Abstract Variograms for gold and lead values from the Loraine and Prieska mines, respectively, indicate that data outliers can seriously distort and/or mask the real variogram patterns. Studies show that this problem is best overcome for these mines by logarithmic transformation of the data, and/or a suitable screening out of such outliers, and/or more robust variogram estimation procedures; the benefits are particularly significant when the basic data is limited.
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  • 92
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    Mathematical geology 14 (1982), S. 645-660 
    ISSN: 1573-8868
    Schlagwort(e): geostatistics ; uranium ; production forecast
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Geologie und Paläontologie , Mathematik
    Notizen: Abstract Geostatistical estimation techniques were customized to allow forecasting of production figures at the Silver Bell uranium mine (Uravan District).Surface drill hole data were used to provide a block model of kriged estimators of average uranium grades. Figures for recoverable ore grade and the ore-waste ratio are then deduced from regressive curves previously obtained from underground information and production data. Cross-validations of the entire model were performed and were found positive.
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  • 93
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    Mathematical geology 14 (1982), S. 217-239 
    ISSN: 1573-8868
    Schlagwort(e): geostatistics ; variograms ; anisotropies ; tungsten
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Geologie und Paläontologie , Mathematik
    Notizen: Abstract The regionalization of tungsten grades at the χ deposit represents an ideal case for anisotropic hole-effect variogram modeling. The modeling technique is presented step by step and the consequences of the model on block kriging are indicated.
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  • 94
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    Mathematical geology 14 (1982), S. 387-404 
    ISSN: 1573-8868
    Schlagwort(e): geostatistics ; resource estimation ; geoexploration ; sampling
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Geologie und Paläontologie , Mathematik
    Notizen: Abstract Sampling exploration of uncertain functions to locate critical contour levels is most effective if sampling decisions are made sequentially. A simple sequential exploration strategy, based on pseudo-Bayesian second-moment analysis, is proposed and compared with non-sequential systematic sampling. Repeated application to functions simulated pseudorandomly from stationary random processes on the line and on the plane indicates uniform superiority of the sequential strategy. The method is particularly advantageous when the function of interest,h(X), has an uncertain trend, and in general when the random process that quantifies prior uncertainty onh(X) is highly correlated.
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  • 95
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    Mathematical geology 14 (1982), S. 547-555 
    ISSN: 1573-8868
    Schlagwort(e): geostatistics ; kriging
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Geologie und Paläontologie , Mathematik
    Notizen: Abstract Parallel variogram analyses, block kriging, and follow-up studies were effected for the lead content of part of the Prieska copper-zinc ore body and for the gold content of the highly variable Breef in a section of the Loraine gold mine, based first on untransformed values and second on logarithmically transformed values using the lognormal-de Wijsian model. For both models the effect was also analyzed of using the population mean or ignoring it. Practical follow-up comparisons confirm theoretical considerations and show that on these mines conditional biases can be eliminated conveniently by kriging with mean; also that the lognormal-de Wijsian model with mean gives the best results.
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  • 96
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    Mathematical geology 14 (1982), S. 87-106 
    ISSN: 1573-8868
    Schlagwort(e): coal ; geostatistics ; kriging ; regional estimation ; systematic sampling
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Geologie und Paläontologie , Mathematik
    Notizen: Abstract Two-dimensional systematic sampling of small plots followed by the kriging of those plots may be employed to obtain regional estimates of coal resources and measures of the accuracy of the estimates. The use of sampling makes large savings in computation possible. Two case studies involving the estimation of coal tonnage are discussed.
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  • 97
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    Mathematical geology 13 (1981), S. 321-329 
    ISSN: 1573-8868
    Schlagwort(e): coal washability ; disjunctive kriging ; geostatistics
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Geologie und Paläontologie , Mathematik
    Notizen: Abstract The washability characteristics of coal are dependent on two basic relations: the ash assay vs. relative density curve, and the distribution by weight of the relative densities of coal particles. Armstrong and Whitmore (1980) demonstrated that the ash content and the yield of coal floating at a particular density can be predicted with reasonable accuracy using a simple inverse proportionality relation for the ash assay vs. density curve and a lognormal model for the distribution. In this paper, geostatistical techniques are used in conjunction with the two models to predict the washability characteristics of coal in situ.
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  • 98
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    Mathematical geology 13 (1981), S. 415-428 
    ISSN: 1573-8868
    Schlagwort(e): geostatistics ; hydrologic transport ; kriging ; radioactive waste disposal
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Geologie und Paläontologie , Mathematik
    Notizen: Abstract A significant part of the work of evaluating a geologic formation as a potential repository for hazardous wastes is the modeling of contaminant transport in the surrounding media in the event the repository is breached. The transport equations that are commonly used are deterministic functions. However, because the data can vary within the area being considered, there is a degree of uncertainty associated with the results obtained from the contaminant transport models. There are several ways to incorporate uncertainties into the transport equations, but they assume that distributions and parameters such as variances and covariances are known. This paper discusses the application of geostatistical spatial estimation techniques to estimate quantities used in transport modeling. The techniques are illustrated on data from an electric analog simulation of a two-dimensional ground water system. Geostatistical methods were used to estimate potential and hydraulic conductivity surfaces from data generated from the simulation of the ground water system. Although the two surfaces were highly dependent through Darcy's Law, they were estimated independently. Independent verification of the two surfaces showed that they approximately satisfied the required conservation of mass condition that: ∇ ⋅ v = 0.
    Materialart: Digitale Medien
    Standort Signatur Erwartet Verfügbarkeit
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  • 99
    Digitale Medien
    Digitale Medien
    Springer
    Mathematical geology 13 (1981), S. 455-459 
    ISSN: 1573-8868
    Schlagwort(e): Variograms ; geostatistics ; positive-definite
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Geologie und Paläontologie , Mathematik
    Notizen: Abstract The aim of this short article is to stress the importance of using only positive-definite functions as models for covariance functions and variograms. The two examples presented show that a negative variance can easily be obtained when a nonadmissible function is chosen for the variogram model.
    Materialart: Digitale Medien
    Standort Signatur Erwartet Verfügbarkeit
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  • 100
    Digitale Medien
    Digitale Medien
    Springer
    Mathematical geology 12 (1980), S. 577-588 
    ISSN: 1573-8868
    Schlagwort(e): variogram ; geostatistics ; estimation
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Geologie und Paläontologie , Mathematik
    Notizen: Abstract The application of regionalized variables requires the estimation of the variogram function and the evaluation of its integral. By representing the variogram by a polygonal function the integral may be easily approximated by closed form representations of polygonal integrals. This approach provides a basis for more extensive statistical evaluation not evident in existing approximation methods. This paper provides the closed form representations for two-dimensional variogram functions whose domain is represented by a finite collection of rectangles.
    Materialart: Digitale Medien
    Standort Signatur Erwartet Verfügbarkeit
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