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1.
Clifford R. Stanley 《Mathematical Geosciences》1990,22(8):933-956
Recent publications have demonstrated that the composition of a system can be described by the angles which define the vector resulting from the projection of the composition onto the unit hypersphere. Although published trigonometric relationships allow determination of the angles in any number of dimensions, no general hyperdimensional recursive formulae exist for the calculation of these quantities and their variances and covariances. A general methodology for calculating the angles which describe the compositions of systems in any number of dimensions is presented. These angles can be used to calculate statistics (central moments and resultant vectors) describing the central tendency and dispersion among rock compositions, as well as to quantify the angular differences between compositions. Equations that relate the variances and covariances of these angular variables to the variances and covariances of the actual component proportion variables are presented, allowing projection of these measurements from one reference frame to another (cartesian to spherical and vice versa). 相似文献
2.
An iterative least-squares technique to fit circular and elliptical conical surfaces to orientation data from folds is presented. A statistical model is used which assumes that each data point is an observation from a Fisher distribution. The mean of this distribution is assumed to lie on the curve to be fitted. Estimates of variances and covariances for the fitting parameters are calculated, and confidence intervals for the cone axis and half apical angle are estimated from variances and covariances. A normal test with null hypothesis that the cone angle is 90° determines if a conical model fits the data better than a cylindrical model. AnF test is used to determine whether an elliptical cone is a better model than a circular cone. In this fashion, macroscopic folds are classified into cylindrical, circular conical, or elliptical conical folds. Examples of these three types of fold are given. The Wynd Syncline near Jasper, Alberta is the first natural elliptical conical fold described as such. 相似文献
3.
Observed S—O bond lengths for sulfate tetrahedral oxyanions isolated from crystalline solids correlate with the tetrahedral angles impressed upon these ions by their environments. An algorithm is presented which can account for more than 90 percent of the S—O bond length variations from a consideration only of the average of the three angles common to each bond. Extended Hückel molecular orbital theory serves to rationalize these correlations by relating changes in bond overlap population (i.e., bond strength) to angular distortions of sulfate ions observed in a large number of natural and synthetic compounds. 相似文献
4.
Ross M. Renner 《Mathematical Geology》1991,23(4):549-563
Given a compositional dataset in the absence of any prior information on any mixing process which may have formed it, a complete analysis of mixtures determines three distinct types of estimates in order. These are: (i) the estimate of the number of endmembers or fixed source compositions, of which all the sample compositions of the dataset must be approximate mixtures; (ii) the estimated compositions for each of these chosen number of endmembers; and (iii) the estimated contributions of each of these endmember estimates to each sample. Traditionally, the estimate for the number of endmembers has been assessed either by mapping or by inspection of the coefficients of determination between the observed and estimated variables. Mapping entails the plotting on a map of the region from which the samples were taken, either the contours of the contributions of each endmember to each sample, or some other portrayal of the distribution of endmember abundances. Because it requires the complete analysis, assessment by this method is too elaborate except for final confirmation and display. Alternatively, choosing a number of endmembers, which result in suitability high coefficients of determination for all or most variables, may account for elements which are not part of the conjectured mixing process or, worse, may result in the identification of endmembers which may never in fact have existed. Such an error is similar to overspecifying a multiple regression model. So, the obvious starting point from which to assess the validity, or otherwise choice of endmember numbers, is to examine the matrix of residuals. The differences between the logratio-transformed observed and estimated data form an array of residual logratios. A linear combination of these may be formed for each sample, which, under a random perturbation assumption, should follow a univariate normal distribution. Whether or not this scalar is normal can be readily tested. It can also be examined graphically for such desirable qualities as symmetry when the test for normality may be too severe. This procedure is employed to assess the decompositions of the U.S.G.S. Mid-Pacific data and the Nazca Plate Surface sediments.This paper was presented at the 18th Geochautauqua, Newark, Delaware, 13–14 October 1989. 相似文献
5.
This paper extends the concept of dispersion variance to the multivariate case where the change of support affects dispersion covariances and the matrix of correlation between attributes. This leads to a concept of correlation between attributes as a function of sample supports and size of the physical domain. Decomposition of dispersion covariances into the spatial scales of variability provides a tool for computing the contribution to variability from different spatial components. Coregionalized dispersion covariances and elementary dispersion variances are defined for each multivariate spatial scale of variability. This allows the computation of dispersion covariances and correlation between attributes without integrating the cross-variograms. A correlation matrix, for a second-order stationary field with point support and infinite domain, converges toward constant correlation coefficients. The regionalized correlation coefficients for each spatial scale of variability, and the cases where the intrinsic correlation hypothesis holds are found independent of support and size of domain. This approach opens possibilities for multivariate geostatistics with data taken at different support. Two numerical examples from soil textural data demonstrate the change of correlation matrix with the size of the domain. In general, correlation between attributes is extended from the classic Pearson correlation coefficient based on independent samples to a most general approach for dependent samples taken with different support in a limited domain. 相似文献
6.
Compositions, proportions, and equilibrium temperature of coexisting two-feldspar in crystalline rocks are of great importance to classification in petrography and interpretation of petrogenesis. Crystalline rocks are usually composed of 4-6 minerals (phases), depending on their independent chemical components and the equilibrium temperature of crystallizations. In general, number of mineral phases can be determined by the “Phase Rule”. According to the mass balance principle, bulk composition of coexisting two-feldspar could be evaluated from the bulk chemistry of a rock, provided that the compositions of the coexisting mafic mineral phases containing calcium, sodium, and potassium oxides are determined, e.g., by microprobe analysis. The compositions, proportions, and temperature of two-feldspar in equilibrium can thus be simultaneously resolved numerically from bulk composition of the rock, by incorporating the activity/composition relations of the ternary feldspars with the mass balance constraints. Upon the numerical approximation method presented in this paper, better-quality, internally consistent data on feldspar group could usually be obtained, which would be expected more realistic and accurate in consideration of thermodynamic equilibria in the system of crystalline rocks, as well as bulk chemistry of a rock and the composing minerals. 相似文献
7.
A. G. Journel 《Mathematical Geology》1988,20(4):459-475
The projection or minimum error norm algorithm does not require that the distance measure be a variogram. In non-Gaussian cases, the traditional variogram distance measure leading to minimization of an error variance offers no definite advantage. Other distance measures, more outlierresistant than the variogram, are proposed which fulfill the condition of the projection theorem. The resulting minimum error norms provide the same data configurations ranking as traditionally obtained from kriging variances. A case study based on actual digital terrain data is presented.This paper was presented by title at MGUS 87 Conference, Redwood City, California, 14 April 1987. 相似文献
8.
The structures of jadeite and albite composition glasses quenched from 10 and 15 kbar, respectively, have been studied using radial distribution analysis. It is found that aluminum does not undergo a coordination increase at these pressures, in agreement with recent Raman and NMR studies of these glass compositions. Radial distribution functions (RDF's) indicate that the structural response of albite and Jadeite composition melts to pressure up to 10–15 kbar has to do primarily with a distortion of the tetrahedral sites, a slight collapse of the tetrahedral rings forming the framework, and a small loss of short range order resulting in a more densely packed framework structure. The RDF's do not provide any clues concerning T-O-T angular changes at these pressures. It is suggested that the observed structural changes should be closely related to those that take place in melts of rhyolitic composition. Such changes in rhyolitic melts would result in a density increase and viscosity decrease with increasing pressure similar to that observed for melts of albite and jadeite composition. 相似文献
9.
D. B. Clarke 《Contributions to Mineralogy and Petrology》1970,25(3):203-224
New analyses show that Tertiary tholeiitic basalts from the Baffin Bay area have generally primitive characteristics. Comparison of the bulk chemical compositions of the basalts with material of known phase relations suggests that these basalts have properties intermediate between investigated natural and synthetic compositions. A discrete group of these basalts coincides in projection with the composition of the liquid produced by the partial melting of garnet periodotite at 30 kb. The compositions in this group are believed to be parental to the other rocks in the province principally through a mechanism of olivine fractionation. Eclogite fractionation is invoked to account for the chemical differences between the two suites of basalts in Baffin Bay. 相似文献
10.
A spectral approach to simulating intrinsic random fields with power and spline generalized covariances 总被引：4，自引：0，他引：4
This article presents a variant of the spectral turning bands method that allows fast and accurate simulation of intrinsic
random fields with power, spline, or logarithmic generalized covariances. The method is applicable in any workspace dimension
and is not restricted in the number and configuration of the locations where the random field is simulated; in particular,
it does not require these locations to be regularly spaced. On the basis of the central limit and Berry–Esséen theorems, an
upper bound is derived for the Kolmogorov distance between the distributions of generalized increments of the simulated random
fields and the normal distribution. 相似文献
11.
Guocheng Pan 《Mathematical Geology》1993,25(5):603-631
This paper presents a regionalized method for the estimation of a favorability function through generalization of all relevant variables (explanatory and target) into random functions. The new method allows the use of cross-covariance functions in addition to ordinary covariances for extracting spatial joint information, which is virtually overlooked in the conventional analyses. The optimal weights for a favorability equation are derived from solving a generalized eigen-system established by the maximization of covariances between a favorability function and the principal components of a set of pre-selected target variables. Various correlation coefficients may be computed to assist in interpretation of the favorability estimates. Both favorability functions and correlation coefficients may be estimated for a point or a block. The regionalized favorability theory can be compared to cokriging in that both use the sample-sample covariances to account for the sample-sample relations and the point-sample covariances to account for the point-sample configurations. The new technique is demonstrated on a test case study, which involves the integration of geochemical, airborne-geophysical, and structural data sets for the target selection of hydrothermal gold-silver deposits. 相似文献
12.
13.
The formation of gaseous diffusional accretion-decretion disks is an important stage in the evolution of numerous astronomical objects. Matter is accreted onto the object in the accretion part of these disks, while the angular momentum of the accreted matter is transported from the central region to the periphery in the decretion part. Here, we consider general questions connected with the formation and evolution of diffusive accretion-decretion disks in various astrophysical objects. Such disks can be described using nonstationary diffusion models. The phenomenological parameters of these models are the coefficients in the relations for the characteristic turbulent velocity and mean free path of diffusion elements in the disk. We have developed a numerical technique to compute the disk evolution for a number of models (a massive disk, a disk with continuous accretion, a purely decretion disk). Analytical expressions estimating the basic parameters of accretion-decretion disks are presented. We discuss the relationship between the models considered and the classical α model of an accretion disk. 相似文献
14.
O. Brotzen 《Mathematical Geology》1975,7(3):191-214
Large sets of rock compositions can be grouped by the analysis of corresponding point distributions in multidimensional space,
where the separate dimensions represent the chemical variables. The point density around each composition is estimated in
a small, multivariate, rectangular interval. Establishment of the identity of specimens contained in the interval, and systematic
comparison of the point densities surrounding each of them, leads to the recognition of density maxima, which represent the
statistical modes of the rock types present in the set. The remaining specimens by the same operations can be assigned to
the groups formed. Experimental results are given for two sets, one comprising a wide variety of metasedimentary and metavolcanic
rocks from the ore-bearing Skellefte district and adjacent parts of V?sterbotten, Sweden, and the other representing a less
variable population of basic rocks from Norrbotten, Sweden. Linear discriminant analysis and existing geological information
indicate that the groups obtained are statistically and geologically valid. 相似文献
15.
We discuss the concept of components in the Earth's mantle startingfrom a petrological and geochemical approach, but adopting anew method of projection of geochemical and isotopic data. Thisallows the compositional variability of magmatic associationsto be evaluated in multi-dimensional space, thus simultaneouslyaccounting for a large number of compositional variables. Wedemonstrate that ocean island basalts (OIB) and mid-ocean ridgebasalts (MORB) are derived from a marble-cake mantle, in whichdifferent degrees of partial melting of recycled lithosphere,which are heterogeneous in age and composition, contribute tothe magma genesis. This view is supported by the variabilityin the geochemical and isotopic signatures of OIB that are observedon the scale of a single ocean island as well as on that ofan ocean, mostly varying between two extreme compositions, thatare not strictly related to the commonly accepted mantle components(DMM, EMI, EMII, HIMU). Rather they are a distinctive featureof the mantle source sampled at each ocean island and are stronglydependent on the Pb isotope system. We recommend a change inperspective in studies of MORB–OIB geochemistry from onebased on physically distinct mantle components to a model basedon the existence of a marble-cake-like upper mantle. Althoughresembling the statistical upper mantle, this model impliesthat geochemical homogenization can be attained only withinthe limits of local mantle composition, so that a world-wideuniform depleted reservoir cannot be sampled by simply extendingthe volume of the region undergoing partial melting. KEY WORDS: geochemistry; isotope; mantle; OIB 相似文献
16.
A. T. Miesch 《Mathematical Geology》1979,11(4):345-371
Compositional variations in the lavas of Parícutin volcano, Mexico, have been examined by an extended method of Q-mode factor analysis. Each sample composition is treated as a vector projected from an original eight-dimensional space into a vector system of three dimensions. The compositions represented by the vectors after projection are closely similar to the original compositions except for Na2Oand Fe2O3.The vectors in the three-dimensional system cluster about three different planes that represent three stages of compositional change in the Parícutin lavas. Because chemical data on the compositions of the minerals in the lavas are presently lacking, interpretations of the mineral phases that may have been involved in fractional crystallization are based on CIPW norm calculations. Changes during the first stage are attributed largely to the fractional crystallization of plagioclase and olivine. Changes during the second stage can be explained by the separation of plagioclase and pyroxene. Changes during the final stage may have resulted mostly from the assimilation of a granitic material, as previously proposed by R. E. Wilcox. 相似文献
17.
A. T. Miesch 《Mathematical Geosciences》1979,11(4):345-371
Compositional variations in the lavas of Parícutin volcano, Mexico, have been examined by an extended method of Q-mode factor analysis. Each sample composition is treated as a vector projected from an original eight-dimensional space into a vector system of three dimensions. The compositions represented by the vectors after projection are closely similar to the original compositions except for Na2Oand Fe2O3.The vectors in the three-dimensional system cluster about three different planes that represent three stages of compositional change in the Parícutin lavas. Because chemical data on the compositions of the minerals in the lavas are presently lacking, interpretations of the mineral phases that may have been involved in fractional crystallization are based on CIPW norm calculations. Changes during the first stage are attributed largely to the fractional crystallization of plagioclase and olivine. Changes during the second stage can be explained by the separation of plagioclase and pyroxene. Changes during the final stage may have resulted mostly from the assimilation of a granitic material, as previously proposed by R. E. Wilcox. 相似文献
18.
Antonio Possolo 《Mathematical Geosciences》1983,15(1):207-210
The second-degree Taylor polynomial for a ratio is used to derive approximations to the means, variances, and covariances of proportions. 相似文献
19.
It is well known among geologists that closure of an open-number system, as when stratigraphic rock thicknesses are converted to percentages, introduces correlations among the components even in the absence of correlations in the open system. In closed three-component systems the covariances are single-valued functions of the closed variances and are exactly predictable. If the open system has inherent correlation (point correlations) among its components the corresponding closed covariances reflect their presence in a predictable manner. If areal trends are present in the open system, the open covariances are themselves affected, but this trend effect can be completely removed to recover the initial point correlations among the components. Areal trends in open systems strongly influence the structure of the closed variance-covariance matrices, and the situation becomes increasingly complicated if the open system has both point correlations and areal trends. The paper considers the problems involved, and includes Monte Carlo runs to compare computed and predicted variances and covariances as data sets are followed from open systems with correlation but no trend to the closed equivalent of open systems with point correlations and trends. 相似文献
20.
Laboratory measurements of pivoting angles for applications to selective entrainment of gravel in a current 总被引：2，自引：0，他引：2
Important to grain entrainment by a flowing fluid is the pivoting angle of the grain about its contact point with an underlying grain. A series of experiments has been undertaken to determine how this angle depends on grain shape (rollability and angularity), on the ratio of the size of the pivoting grain to those beneath, and on factors such as imbrication. The experiments involved gravel-sized spheres (ball-bearings and marbles), natural pebbles selected for their approximately triaxial ellipsoid shapes, and angular crushed basalt pebbles. The pivoting angles for these grains were measured on an apparatus consisting of a board which can be progressively inclined, the angle of the board being equal to the pivoting angle at the instant of grain movement. The pivoting angles of spheres showed reasonable agreement with a theoretically derived equation, showing much better agreement than in previous studies which utilized sand-sized spheres. A series of measurements with spheres ranging from sand to gravel sizes reveals that the pivoting angles decrease with increasing particle size. Our results are therefore consistent with the earlier studies limited to sand-size spheres. The cause of this size dependence is unknown since moisture and electrostatic binding can be ruled out. Similar size dependencies are also found for the ellipsoidal pebbles and angular gravel. The experiments with ellipsoidal pebbles demonstrated a strong shape dependence for the pivoting angle, being a function of the ratio of the pebble's smallest to intermediate axial diameters. This ratio controls the grain's ability to roll and pivot; with small ratios of these diameters the pebbles tended to slide out of position, whereas with ratios closer to unity (circular cross-section) true pivoting took place and the angles were smaller. Experiments with flat pebbles placed in an imbricated arrangement yielded much larger angles than when the pebbles lay in a horizontal position, the pivoting angle being increased approximately by the imbrication angle. The angular crushed gravel also required high pivoting angles, apparently due to interlocking of the grains resulting from their angularity. Other factors being equal, the measurements of pivoting angles demonstrate that the order of increasing difficulty of entrainment is spheres, ellipsoidal grains, angular grains, and imbricated grains. The results obtained here make possible the quantitative evaluation of these shape effects on grain threshold, as well as evaluation of the selective entrainment of grains from a bed of mixed sizes. 相似文献