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1.
Variograms calculated from binary variables, such as from two lithologies, tend to show sinusoidal forms with decreasing amplitudes for increasing lag distances. This cyclicity is observed often when analyzing drill-hole data for rock sequences with alternating lithologies, and the variograms are thus labeled “hole-effect variograms.” Such variograms show a variety of forms: (1) Low to moderate variation in lithologic-body dimensions causes variograms to have strong cyclicity with decaying amplitude. (2) Variograms with one or more peaks and troughs usually result from a binary variable for which lithologies are about equally abundant but possibly large variations exist in the size of lithologic bodies. (3) Variograms show poor cyclicity if one lithology has highly variable body sizes and the other has moderately variable body dimensions. (4) Variograms that attain a plateau at short lag distances represent extremely high or low sandstone fraction, high variability in size of the most abundant lithology, and low variability in the other. Information about the dimensions of lithologic bodies makes it possible to approximate characteristics of the variogram of the lithology variable without numerous wells. Conversely, a hole-effect variogram of lithology may be used to estimate lithologic dimensions. 相似文献
2.
Common variogram models, such as spherical or exponential functions, increase monotonically with increasing lag distance. On the other hand, a hole-effect variogram typically exhibits sinusoidal waves that form peaks and troughs, thereby conveying the cyclicity of the underlying phenomenon. In order to incorporate this cyclicity into a stochastic simulation, hole effects in the experimental variogram must be fitted appropriately. In this paper, we recommend use of several multiplicative-composite variogram models to fit hole-effect experimental variograms. These consist of a cosine function to provide wavelength and phase of cyclicity, multiplied by a monotonic model (e.g., spherical) to attenuate amplitudes of the cyclical peaks and troughs. These composite models can successfully fit experimental lithology-indicator variograms that contain a range of cyclicities, although experimental variograms with poor cyclicity require special considerations. 相似文献
3.
Common variogram models, such as spherical or exponential functions, increase monotonically with increasing lag distance. On the other hand, a hole-effect variogram typically exhibits sinusoidal waves that form peaks and troughs, thereby conveying the cyclicity of the underlying phenomenon. In order to incorporate this cyclicity into a stochastic simulation, hole effects in the experimental variogram must be fitted appropriately. In this paper, we recommend use of several multiplicative-composite variogram models to fit hole-effect experimental variograms. These consist of a cosine function to provide wavelength and phase of cyclicity, multiplied by a monotonic model (e.g., spherical) to attenuate amplitudes of the cyclical peaks and troughs. These composite models can successfully fit experimental lithology-indicator variograms that contain a range of cyclicities, although experimental variograms with poor cyclicity require special considerations. 相似文献
4.
Kriging with imprecise (fuzzy) variograms. I: Theory 总被引：2，自引：0，他引：2
Imprecise variogram parameters are modeled with fuzzy set theory. The fit of a variogram model to experimental variograms is often subjective. The accuracy of the fit is modeled with imprecise variogram parameters. Measurement data often are insufficient to create good experimental variograms. In this case, prior knowledge and experience can contribute to determination of the variogram model parameters. A methodology for kriging with imprecise variogram parameters is developed. Both kriged values and estimation variances are calculated as fuzzy numbers and characterized by their membership functions. Besides estimation variance, the membership functions are used to create another uncertainty measure. This measure depends on both homogeneity and configuration of the data. 相似文献
5.
Anisotropic hole-effect modeling 总被引：1，自引：0，他引：1
The regionalization of tungsten grades at the 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. 相似文献
6.
On the practice of estimating fractal dimension 总被引：11，自引：0，他引：11
Coastlines epitomize deterministic fractals and fractal (Hausdorff-Besicovitch) dimensions; a divider [compass] method can be used to calculate fractal dimensions for these features. Noise models are used to develop another notion of fractals, a stochastic one. Spectral and variogram methods are used to estimate fractal dimensions for stochastic fractals. When estimating fractal dimension, the objective of the analysis must be consistent with the method chosen for fractal dimension calculation. Spectal and variogram methods yield fractal dimensions which indicate the similarity of the feature under study to noise (e.g., Brownian noise). A divider measurement method yields a fractal dimension which is a measure of complexity of shape. 相似文献
7.
Calculation of Uncertainty in the Variogram 总被引：6，自引：0，他引：6
There are often limited data available in early stages of geostatistical modeling. This leads to considerable uncertainty in statistical parameters including the variogram. This article presents an approach to calculate the uncertainty in the variogram. A methodology to transfer this uncertainty through geostatistical simulation and decision making is also presented.The experimental variogram value
for a separation lag vector h is a mean of squared differences. The variance of a mean can be calculated with a model of the correlation between the pairs of data used in the calculation. The data here are squared differences; therefore, we need a measure of a 4-point correlation. A theoretical multi-Gaussian approach is presented for this uncertainty assessment together with a number of examples. The theoretical results are validated by numerical simulation. The simulation approach permits generalization to non-Gaussian situations.Multiple plausible variograms may be fit knowing the uncertainty at each variogram point,
. Multiple geostatistical realizations may then be constructed and subjected to process assessment to measure the impact of this uncertainty. 相似文献
8.
Principal component analysis (PCA) is commonly applied without looking at the spatial support (size and shape, of the samples and the field), and the cross-covariance structure of the explored attributes. This paper shows that PCA can depend on such spatial features. If the spatial random functions for attributes correspond to largely dissimilar variograms and cross-variograms, the scale effect will increase as well. On the other hand, under conditions of proportional shape of the variograms and cross-variograms (i.e., intrinsic coregionalization), no scale effect may occur. The theoretical analysis leads to eigenvalue and eigenvector functions of the size of the domain and sample supports. We termed this analysis growing scale PCA, where spatial (or time) scale refers to the size and shape of the domain and samples. An example of silt, sand, and clay attributes for a second-order stationary vector random function shows the correlation matrix asymptotically approaches constants at two or three times the largest range of the spherical variogram used in the nested model. This is contrary to the common belief that the correlation structure between attributes become constant at the range value. Results of growing scale PCA illustrate the rotation of the orthogonal space of the eigenvectors as the size of the domain grows. PCA results are strongly controlled by the multivariate matrix variogram model. This approach is useful for exploratory data analysis of spatially autocorrelated vector random functions. 相似文献
9.
Xavier Emery 《Mathematical Geology》2002,34(1):79-100
This paper presents a conditional simulation procedure that overcomes the limits of gaussian models and enables one to simulate regionalized variables with highly asymmetrical histograms or with partial or total connectivity of extreme values. The philosophy of the method is similar to that of sequential indicator technique, but it is more accurate because it is based on a complete bivariate model by means of an isofactorial law. The resulting simulations, which can be continuous or categorical, not only honor measured values at data points, but also reproduce the mono and bivariate laws of the random function associated to the regionalized variable, that is, every one or two-point statistic: histogram, variogram, indicator variograms. The sequential isofactorial method can also be adapted to conditional simulation of block values, without resorting to point–support simulations. 相似文献
10.
Covariance and variogram functions have been extensively studied in Euclidean space. In this article, we investigate the validity
of commonly used covariance and variogram functions on the sphere. In particular, we show that the spherical and exponential
models, as well as power variograms with 0<α≤1, are valid on the sphere. However, two Radon transforms of the exponential model, Cauchy model, the hole-effect model and
power variograms with 1<α≤2 are not valid on the sphere. A table that summarizes the validity of commonly used covariance and variogram functions on
the sphere is provided. 相似文献
11.
Subhash Lele 《Mathematical Geology》1995,27(5):673-692
Two important problems in the practical implementation of kriging are: (1) estimation of the variogram, and (2) estimation of the prediction error. In this paper, a nonparametric estimator of the variogram to circumvent the problem of the precise choice of a variogram model is proposed. Using orthogonal decomposition of the kriging predictor and the prediction error, a method for selecting, what may be considered, a statistical neighborhood is suggested. The prediction error estimates based on this scheme, in fact, reflects the true prediction error, thus leading to proper coverage for the corresponding prediction interval. By simulations and a reanalysis of published data, it is shown that the proposals made in this paper are useful in practice. 相似文献
12.
On the condition number of covariance matrices in kriging, estimation, and simulation of random fields 总被引：1，自引：0，他引：1
The numerical stability of linear systems arising in kriging, estimation, and simulation of random fields, is studied analytically and numerically. In the state-space formulation of kriging, as developed here, the stability of the kriging system depends on the condition number of the prior, stationary covariance matrix. The same is true for conditional random field generation by the superposition method, which is based on kriging, and the multivariate Gaussian method, which requires factoring a covariance matrix. A large condition number corresponds to an ill-conditioned, numerically unstable system. In the case of stationary covariance matrices and uniform grids, as occurs in kriging of uniformly sampled data, the degree of ill-conditioning generally increases indefinitely with sampling density and, to a limit, with domain size. The precise behavior is, however, highly sensitive to the underlying covariance model. Detailed analytical and numerical results are given for five one-dimensional covariance models: (1) hole-exponential, (2) exponential, (3) linear-exponential, (4) hole-Gaussian, and (5) Gaussian. This list reflects an approximate ranking of the models, from best to worst conditioned. The methods developed in this work can be used to analyze other covariance models. Examples of such representative analyses, conducted in this work, include the spherical and periodic hole-effect (hole-sinusoidal) covariance models. The effect of small-scale variability (nugget) is addressed and extensions to irregular sampling schemes and higher dimensional spaces are discussed. 相似文献
13.
Teacher''s Aide Variogram Interpretation and Modeling 总被引：13，自引：0，他引：13
The variogram is a critical input to geostatistical studies: (1) it is a tool to investigate and quantify the spatial variability of the phenomenon under study, and (2) most geostatistical estimation or simulation algorithms require an analytical variogram model, which they will reproduce with statistical fluctuations. In the construction of numerical models, the variogram reflects some of our understanding of the geometry and continuity of the variable, and can have a very important impact on predictions from such numerical models. The principles of variogram modeling are developed and illustrated with a number of practical examples. A three-dimensional interpretation of the variogram is necessary to fully describe geologic continuity. Directional continuity must be described simultaneously to be consistent with principles of geological deposition and for a legitimate measure of spatial variability for geostatistical modeling algorithms. Interpretation principles are discussed in detail. Variograms are modeled with particular functions for reasons of mathematical consistency. Used correctly, such variogram models account for the experimental data, geological interpretation, and analogue information. The steps in this essential data integration exercise are described in detail through the introduction of a rigorous methodology. 相似文献
14.
Noel Cressie 《Mathematical Geology》1985,17(7):693-702
The relative variogram has been employed as a tool for correcting a simple kind of nonstationarity, namely that in which local variance is proportional to local mean squared. In the past, this has been linked in a vague way to the lognormal distribution, although if {Zt; t D}is strongly stationary and normal over a domain D,then clearly {exp (Zt); t D}will stillbe stationary, but lognormal. The appropriate link is made in this article through a universal transformation principle. More general situations are considered, leading to the use of a scaled variogram. 相似文献
15.
Ute Christina Herzfeld 《Mathematical Geology》1993,25(7):901-924
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°45N to 26°40N. 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. 相似文献
16.
Jane A. Gilotti 《Contributions to Mineralogy and Petrology》1989,101(1):30-45
The mylonite zone at the base of the Särv thrust sheet, Swedish Caledonides, contains diabase dikes which record intense deformation and syntectonic greenschist facies metamorphism. An angular shear strain of 100 is calculated for a single dike which can be followed for 50 m in the mylonite zone and abundant centimeter thick greenschist layers imply shear strains in excess of 1000. This extraordinary amount of deformation is comparable to the largest strains attained during experimental superplastic deformation of metals and alloys and, by analogy, suggests that dike deformation was macroscopically superplastic. The progress of five syntectonic reactions was measured as a function of increasing strain for the continuously exposed dike in order to assess the contribution of reactionenhanced ductility and fluid-rock interactions to strain localization along the thrust. Reaction progress calculations suggest that the breakdown of amphibole to form weaker phyllosilicates (which are added to the incompetent matrix fraction) is the important strain softening mechanism below 100. The ultimate tectonite is a stable biotite-epidote schist comprised of a uniformly fine grain size (< 200 m), constant grain shapes and strain free grains. Below 40, metamorphism was isochemical and shear strain was independent of H2O in the reactions. Petrologic fluid:rock ratios are low and suggest that deformation could have occurred under relatively dry conditions.Deformation micromechanisms were probably dominated by diffusive mass transfer processes throughout the life of the shear zone. The absence of cataclasis and the fine grain size of the protolith basalt suggest that fluids were introduced via grain boundary diffusion. Incongruent pressure solution at low strains and K-metasomatism above 40 also support diffusional flow. Diffusion-accomodated grain boundary sliding is thought to be the dominant micromechanism once the stable biotite-epidote tectonite forms. 相似文献
17.
Stable isotopic evidence for large-scale seawater infiltration in a regional metamorphic terrane; the Trois Seigneurs Massif,Pyrenees, France 总被引：1，自引：0，他引：1
Oxygen isotopic analyses of 95 metamorphic and igneous rocks and minerals from a Hercynian metamorphic sequence in the Trois Seigneurs Massif, Pyrenees, France, indicate that all lithologies at higher metamorphic grades than the andalusite in isograd have relatively homogeneous
18O values. The extent of homogenization is shown by the similarity of
18O values in metacarbonates, metapelites and granitic rocks (+11 to +13), and by the narrow range of oxygen isotopic composition shown by quartz from these lithologies. These values contrast with the
18O values of metapelites of lower metamorphic grade (
18O about +15). Homogenization was caused by a pervasive influx of hydrous fluid. Mass-balance calculations imply that the fluid influx was so large that its source was probably high-level groundwaters or connate formation water. Hydrogen isotopic analyses of muscovite from various lithologies are uniform and exceptionally heavy at D=–25 to –30, suggesting a seawater origin. Many lines of petrological evidence from the area independently suggest that metamorphism and anatexis of pelitic metasediment occurred at depths of 6–12 km in the presence of this water-rich fluid, the composition of which was externally buffered. Deep penetration of surface waters in such environments has been hitherto unrecognized, and may be a key factor in promoting major anatexis of the continental crust at shallow depth. Three types of granitoid are exposed in the area. The leucogranites and the biotite granite-quartz diorite are both mainly derived from fusion of local Paleozoic pelitic metasediment, because all these rocks have similar whole-rock
18O values (+11 to +13). The post-metamorphic biotite granodiorite has a distinctly different
18O (+9.5 to +10.0) and was probably derived from a deeper level in the crust. Rare mafic xenoliths within the deeper parts of the biotite granite-quartz diorite also have different
18O (+8.0 to +8.5) and possibly represent input of mantle derived magma, which may have provided a heat source for the metamorphism.Contribution No. 4192, Publications of the Division of Geological and Planetary Sciences, California Institute of Technology 相似文献
18.
The variogram is a critical input to geostatistical studies: (1) it is a tool to investigate and quantify the spatial variability of the phenomenon under study, and (2) most geostatistical estimation or simulation algorithms require an analytical variogram model, which they will reproduce with statistical fluctuations. In the construction of numerical models, the variogram reflects some of our understanding of the geometry and continuity of the variable, and can have a very important impact on predictions from such numerical models. The principles of variogram modeling are developed and illustrated with a number of practical examples. A three-dimensional interpretation of the variogram is necessary to fully describe geologic continuity. Directional continuity must be described simultaneously to be consistent with principles of geological deposition and for a legitimate measure of spatial variability for geostatistical modeling algorithms. Interpretation principles are discussed in detail. Variograms are modeled with particular functions for reasons of mathematical consistency. Used correctly, such variogram models account for the experimental data, geological interpretation, and analogue information. The steps in this essential data integration exercise are described in detail through the introduction of a rigorous methodology. 相似文献
19.
R. M. Spang 《Rock Mechanics and Rock Engineering》1976,8(3):185-198
ZusammenfassungMöglichkeiten und Grenzen des Base Friction Konzepts Der vorliegende Beitrag befaßt sich mit einer von Goodman eingeführten neuen Technologie für zweidimensionale Modellversuche in der Felsmechanik. Bei diesen Modellen sollen die im Prototyp durch die Gravitation erzeugten Körperkräfte durch Reibungskräfte simuliert werden, die dieselbe Richtung wie die Erdbeschleunigung besitzen müssen und die entstehen, wenn das zweidimensionale Modell in einer horizontalen Ebene aufgebaut und zwischen dem Modell und der horizontalen Basis eine entsprechende Relativbewegung erzwungen wird.Nach einem kurzen Überblick über die einschlägige Literatur wird anhand eines Stabilitätsdiagramms für einen Einzelkörper an der Schiefen Ebene untersucht, ob die im geklüfteten Fels auftretenden Bruchmechanismen Gleiten, Kippen und Kippen bei gleichzeitigem Gleiten im Base Friction Modell exakt modellierbar sind.Aus dieser Untersuchung ergibt sich, daß in den Bereichen Kippen und Gleiten bezüglich der Kinetik mit Schwerkraftmodellen übereinstimmende Ergebnisse zu erzielen sind. Im Bereich Kippen bei gleichzeitigem Gleiten treten dagegen wesentliche Abweichungen auf, weil die Simulation der Schwerkraft durch Reibungskräfte nicht vollständig gelingt. Fälle, bei denen Aussagen über Geschwindigkeits-und Beschleunigungsverhältnisse im Prototyp gemacht werden sollen, können aus demselben Grund mit einem Base Friction Modell nicht untersucht werden.Nach einer ausführlichen Kritik wird dann der Schluß gezogen, daß das Base Friction Modell zwar als Entwurfshilfsmittel ungeeignet ist, daß es aber als Methode für die Grundlagenforschung in der Felsmechanik mit Aussicht auf Erfolg eingesetzt werden kann.
Mit 9 Abbildungen 相似文献
SummaryPossibilities and Limitations of the Base Friction Model The paper deals with a new technology for two-dimensional model tests in rock mechanics introduced by Goodman. In this model gravity is simulated by base friction forces. Their direction is to be that of gravity. The two-dimensional model is placed on a horizontal base. A movement of this base relative to the model produce frictional forces which model gravity.First a single model element placed on an inclined plane is analysed resulting in a stability diagram. The purpose of the subsequent analysis is to assess whether the mechanisms of failure being investigated, i. e. sliding, toppling, and toppling combined with sliding, are adequately reproduced by the base friction model.The specific mechanisms toppling and sliding produce kineticly valid results. Considerable errors are to be expected for toppling combined with sliding, because of the specific base friction feature. Mechanisms including velocity and acceleration cannot be modeled by base friction.Finally the limitations of the base friction model as a design tool and instrument for rock mechanics research are discussed.
RésuméPossibilités et limites du base friction model Le présent article traite d'une nouvelle technologie des essais à deux dimensions, introduite par Goodman, sur la mécanique des roches. Par cette technologie les forces en prototype, produites par la gravitation, doivent être remplacées par des forces de frottement. Leur direction doit être la même que celle de la gravitation. Le modèle à deux dimensions est posé sur un plan horizontal. Le mouvement relatif du modèle contre la base horizontale produit des forces de frottement, qui modèlent la gravitation.A l'aide d'un diagramme de stabilité pour un élément isolé sur un plan incliné, on fait des analyses pour savoir si les états de mouvement aparaissant à la rupture de la roche, c'est-à-dire le fait de glisser, basculer, et basculer en même temps que glisser, peuvent être modelés exactement avec le base friction model.Les mécanismes glisser et basculer produisent des résultats qui sont cinétiquement réels. Des fautes remarquables doivent être attendues pour le fait de basculer en même temps que glisser à cause de la mécanique spécifique du modèle. A modeler des vitesses et des accélérations le modèle n'est pas propre.En conclusion on expose une critique détaillée pour savoir jusqu'à quel point cette méthode peut servir comme moyen de dimensionnement dans la pratique et comme instrument pour la recherche en mécanique des roches.
Mit 9 Abbildungen 相似文献
20.
The enstatite-achondrite Khor Temiki has been studied by high-voltage electron microscopy. The normal Khor Temiki lithology has a fine-grained matrix in which individual grains show the well-known effects of unshielded solar irradiation. Intensity of deformation varies greatly from grain to grain; this material has a varied history of impact deformation, and must have formed in an environment similar to that of the lunar regolith. The meteorite is traversed by veins of extra-dark material. This was produced in situ from the normal lithology by intense shock, sufficient to erase its irradiation record almost completely. Instead of the enstatite that dominates the bulk meteorite, optically twinned clinoenstatite is found to be the major mineral in such a shock-vein. It is highly defective, and its electron diffraction patterns contain diffuse orthopyroxene maxima. It is interpreted in terms of inversion of protoenstatite produced by the thermal effect of the shock. Recrystallization phenomena, and the occurrence of enstatite in close association with the clinoenstatite, are described. This occurrence of twinned clinoenstatite is contrasted with those due to quenching of primary protopyroxene in small bodies such as chondrules. The effects of shock have eliminated porosity from the vein material, and indurated it. Less severe such effects, outside the veins themselves, must have contributed to the lithification of the meteorite. 相似文献