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• 1
Electronic Resource
Springer
Mathematical geology 24 (1992), S. 455-461
ISSN: 1573-8868
Keywords: statistics ; image analysis ; strain ; estimation ; structural geology ; tensors
Source: Springer Online Journal Archives 1860-2000
Topics: Geosciences , Mathematics
Notes: Abstract Procedures are outlined for the selection of a least squares best-fit circle to data points defined by rectangular Cartesian coordinates. Equations are derived to allow fitting of circles centered on the x-axis as well as off-axis Mohr circles. These procedures are applicable to the estimation of second-order tensors such as stress and strain by means of Mohr's diagram.
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• 2
Electronic Resource
Springer
Mathematical geology 24 (1992), S. 463-478
ISSN: 1573-8868
Keywords: multivariate (co)variogram ; (dis)similarity ; metric ; cluster analysis
Source: Springer Online Journal Archives 1860-2000
Topics: Geosciences , Mathematics
Notes: Abstract The multivariate variogram and the multivariate covariogram are used as spatial weighting functions for forming spatially homogeneous groups automatically. The groups are created after either deflating similarities between distant samples with the multivariate covariogram or by inflating dissimilarities between distant samples with the multivariate variogram. These approaches can be seen as generalization of the Oliver and Webster proposal. Two data sets show the efficiency of the two weighting functions when compared to the classical approach which does not take spatial information into account. In one case study, the weighting of similarities by the multivariate covariogram showed more interpretable results than the weighting of dissimilarities by the multivariate variogram.
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• 3
Electronic Resource
Springer
Mathematical geology 24 (1992), S. 539-553
ISSN: 1573-8868
Keywords: entropy ; spatial disorder ; bivariate Gaussian distribution ; maximum entropy ; non-parametric relative entropy estimator
Source: Springer Online Journal Archives 1860-2000
Topics: Geosciences , Mathematics
Notes: Abstract In this paper, entropy is presented as an alternative measure to characterize the bivariate distribution of a stationary spatial process. This non-parametric estimator attempts to quantify the concept of spatial ordering, and it provides a measure of how Gaussian the experimental bivariate distribution is. The concept of entropy is explained and the classical definition presented, along with some important results. In particular, the reader is reminded that, for a known mean and covariance, the bivariate Gaussian distribution maximizes entropy. A “relative entropy” estimator is introduced in order to measure departure of an experimental bivariate distribution from the bivariate Gaussian. Two case studies are presented as examples.
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• 4
Electronic Resource
Springer
Mathematical geology 24 (1992), S. 499-537
ISSN: 1573-8868
Keywords: earth's heat flow ; geological modeling ; finite difference schemes ; model validation
Source: Springer Online Journal Archives 1860-2000
Topics: Geosciences , Mathematics
Notes: Abstract A preliminary problem to solve in the set-up of a mathematical model simulating a geophysical process is the choice of a suitable discrete scheme to approximate the governing differential equations. This paper presents a simple technique to test finite difference schemes used in the modeling of geophysical processes occurring in a geological structure. This technique consists in generating analytical solutions similar to the ones characterizing a geophysical process, given general information on some relevant parameters. Useful information for the choice of the discrete scheme to employ in the mathematical model simulating the original geophysical process can be obtained from the comparison between the analytical solution and the approximated numerical solutions generated by means of different discrete schemes. Two classes of numerical examples approximating the differential equation that governs the steady state earth's heat flow have been treated using three different finite differences schemes. The first class of examples deals with media whose phenomenological parameters vary as continuous space functions; the second class, instead, deals with media whose phenomenological parameters vary as discontinuous space functions. The finite difference schemes that have been utilized are: Centered Finite Difference Scheme (CDS), Arithmetic Mean Scheme (AMS), and Harmonic Mean Scheme (HMS). The numerical simulations showed that: the CDS may yield physically inconsistent solutions if the lattice internodal distance is too large, but in case of phenomenological parameters varying as a continuous function, this pitfall can be avoided increasing the lattice node refinement. In case of phenomenological parameters varying as a discontinuous function, instead, the CDS may yield physically inconsistent solutions for any lattice-node refinement. The HMS produced good results for both classes of examples showing to be a scheme suitable to model situations like these.
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• 5
Electronic Resource
Springer
Mathematical geology 24 (1992), S. 631-644
ISSN: 1573-8868
Keywords: blind mineralization ; superimposed information ; gold lodes ; multiple dimension ; quantitative prediction ; weighted matrix analysis
Source: Springer Online Journal Archives 1860-2000
Topics: Geosciences , Mathematics
Notes: Abstract The ability to quantitatively predict Au lodes in Au mineralization series with data from large-scale mineralization information is greatly needed. This paper discusses how to (1) classify the oreforming information, (2) set up the mineralization information model, (3) divide the statistical units within the minimum area of the mineralization anomalies, (4) select the comprehensive ore-forming (controlling) information variables, and (5) carry out the quantitative prediction using some newly proposed statistical models. Finally, the quantitative prediction results for Au lodes in a Au deposit, the Aohan Banner in Mongolia, are provided. Among the three first-grade prediction targets, two were tested and have been found to have industrial significance.
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• 6
Electronic Resource
Springer
Mathematical geology 24 (1992), S. 675-691
ISSN: 1573-8868
Keywords: compositional data ; co-regionalization ; regionalized composition ; additive-logratio transformation ; cross-covariance
Source: Springer Online Journal Archives 1860-2000
Topics: Geosciences , Mathematics
Notes: Abstract Like compositions in general, regionalized compositions present the problem of spurious spatial correlation. To avoid this problem, this paper uses the additive-logratio transformation of regionalized compositions, following techniques introduced over the last few years for the statistical analysis of compositional data. It leads to an appropriate definition of a spatial covariance structure to describe spatial dependence between regionalized variables subject to constant-sum constraints in the case of weak stationarity. To illustrate stated problems, simulated data are used.
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• 7
Electronic Resource
Springer
Mathematical geology 24 (1992), S. 663-674
ISSN: 1573-8868
Keywords: weights of evidence ; gold exploration ; discrete multivariate analysis ; Geographic Information Systems
Source: Springer Online Journal Archives 1860-2000
Topics: Geosciences , Mathematics
Notes: Abstract Weights of evidence modeling for combining indicator patterns in mineral resource evaluation is based on an application of Bayes' rule. Two weights are defined for each indicator pattern and Bayes' rule is applied repeatedly to combine indicator patterns. If all patterns are conditionally independent with respect to deposits, the logit of the posterior probability can be calculated as the sum of the logit of the prior probability plus the weights of the overlay patterns. The information to be integrated for gold exploration in Xiong-er Mountain Region comes from a geological map, an interpreted map of a Thematic Mapper (TM) image, and the locations of known gold deposits. Favorable stratigraphic units, structural control factors, and alteration factors are considered. The work was conducted on an S600 I2S image-processing system. FORTRAN programs were developed for creating indicator patterns, statistical calculations, and pattern integration. Six indicator patterns were selected to predict mineral potential. They are conditionally independent according to pairwiseG 2 tests, and an overall chi-square test. The potential area predicted using the 32 known deposits generally coincides with the prospect areas determined by geological fieldwork.
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• 8
Electronic Resource
Springer
Mathematical geology 24 (1992), S. 759-773
ISSN: 1573-8868
Keywords: Extrapolation ; interpolation ; ordinary kriging ; spatial correlation ; universal kriging
Source: Springer Online Journal Archives 1860-2000
Topics: Geosciences , Mathematics
Notes: Abstract In this paper, the effect on Kriging weights of non-bias conditions, when the same residual covariance model is used, has been studied by the l2 norm of the weights difference between Ordinary Kriging and Kriging with a trend model. Four covariance models, in 1-D and 2-D, and in interpolation and extrapolation conditions are examined. Situations in which both algorithms yield the same results are pointed out.
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• 9
Electronic Resource
Springer
Mathematical geology 24 (1992), S. 731-758
ISSN: 1573-8868
Keywords: spatial factors ; autocorrelation ; crosscorrelation ; correlogram ; gridded data ; irregularly spaced data
Source: Springer Online Journal Archives 1860-2000
Topics: Geosciences , Mathematics
Notes: Abstract Spatial factor analysis (SFA) is a multivariate method that determines linear combinations of variables with maximum autocorrelation at a given lag. This is achieved by deriving estimates of auto-/cross-correlations of the variables and calculating the corresponding eigenvectors of the covariance quotient matrix. A two-point spatial factor analysis model derives factors by the formation of transition matrixU comparing auto-/cross-correlations at lag “0,”R 0, with those at a specified lag “d,”R d, expressed asU d=R 0 −1 Rd. The matrixU d can be decomposed into its spectral components which represent the spatial factors. The technique has been extended to include three points of reference. Spatial factors can be derived from the relationship: $$\left[ {\begin{array}{*{20}c} {U_{d1} } \\ {} \\ {\begin{array}{*{20}c} {} \\ {} \\ \end{array} } \\ {U_{d2} } \\ \end{array} } \right] = \left[ {\begin{array}{*{20}c} {R_0 R_{d3} } \\ {} \\ {\begin{array}{*{20}c} {} \\ {} \\ \end{array} } \\ {R_{d3} R_0 } \\ \end{array} } \right]^{ - 1} \left[ {\begin{array}{*{20}c} {R_{d1} } \\ {} \\ {\begin{array}{*{20}c} {} \\ {} \\ \end{array} } \\ {R_{d2} } \\ \end{array} } \right]$$ where the factors ofU d1 andU d2 predict the relationships of the variables over three lag distances and orientations from matricesR d1,R d2, andR d3. Estimates of auto-/cross-correlation can be achieved using irregularly spaced or gridded data from experimental correlograms or function estimates determined from the experimental correlograms. The technique has applications in studies where several variables have different spatial ranges or zones of influence.
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• 10
Electronic Resource
Springer
Mathematical geology 24 (1992), S. 789-805
ISSN: 1573-8868
Keywords: exploration potential ; occurrence modeling ; truncated probability model ; Walker Lake quadrangle ; California-Nevada
Source: Springer Online Journal Archives 1860-2000
Topics: Geosciences , Mathematics
Notes: Abstract Analysis of the distribution of ore-bodies in space can be useful generally in the estimation of mineral resources or the management of exploration. This study is directed to analysis of the undiscovered potential of well known areas of mining district size, referred to as metallogenic units (MUs). The analysis employs an effort-adjusted and truncated probability model for number of occurrences within a subdivision (quadrat) of a MU and Monte Carlo sampling to generate an approximation to the probability distribution for number of occurrences and number of mines within an MU when it is totally explored. Exploration potential for Monitor, Bodie, Aurora, and Camp Douglas MUs (Walker Lake quadrangle of Nevada and California) are estimated to be 9, 4, 7, and 4 mines, respectively.
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