Multifractal modeling and spatial statistics

In general, the multifractal model provides more information about measurements on spatial objects than a fractal model. It also results in mathematical equations for the covariance function and semivariogram in spatial statistics which are determined primarily by the second-order mass exponent. However, these equations can be approximated by power-law relations which are comparable directly to equations based on fractal modeling. The multifractal approach is used to describe the underlying spatial structure of De Wijs 's example of zinc values from a sphalerite-bearing quartz vein near Pulacayo, Bolivia. It is shown that these data are multifractal instead of fractal, and that the second-order mass exponent (=0.979±0.011 for the example) can be used in spatial statistical analysis.     

Spatial and multivariate analysis of geochemical data from metavolcanic rocks in the Ben Nevis area, Ontario

A study of the lithogeochemistry of metavolcanics in the Ben Nevis area of Ontario, Canada has shown that factor analysis methods can distinguish lithogeochemical trends related to different geological processes, most notably, the principal compositional variation related to the volcanic stratigraphy and zones of carbonate alteration associated with the presence of sulphides and gold. Auto- and cross-correlation functions have been estimated for the two-dimensional distribution of various elements in the area. These functions allow computation of spatial factors in which patterns of multivariate relationships are dependent upon the spatial auto- and cross-correlation of the components. Because of the anisotropy of primary compositions of the volcanics, some spatial factor patterns are difficult to interpret. Isotropically distributed variables such as CO ₂ are delineated clearly in spatial factor maps. For anisotropically distributed variables (SiO ₂ ), as the neighborhood becomes smaller, the spacial factor maps becomes better. Interpretation of spatial factors requires computation of the corresponding amplitude vectors from the eigenvalue solution. This vector reflects relative amplitudes by which the variables follow the spatial factors. Instability of some eigenvalue solutions requires that caution be used in interpreting the resulting factor patterns. A measure of the predictive power of the spatial factors can be determined from autocorrelation coefficients and squared multiple correlation coefficients that indicate which variables are significant in any given factor. The spatial factor approach utilizes spatial relationships of variables in conjunction with systematic variation of variables representing geological processes. This approach can yield potential exploration targets based on the spatial continuity of alteration haloes that reflect mineralization.List of symbols c _i Scalar factor that minimizes the discrepancy between andU _i - D Radius of circular neighborhood used for estimating auto- and cross-correlation coefficients - d Distance for which transition matrixU is estimated - d _ij Distance between observed valuesi andj - E Expected value - E _i Row vector of residuals in the standardized model - F(d _ij) Quadratic function of distanced _ij F(d _ij)=a+bd _ij+cd _ij ² - L Diagonal matrix of the eigenvalues ofU - _i Eigenvalue of the matrixU;ith diagonal element ofL - N Number of observations - p Number of variables - Q Total predictive power ofU - R Correlation matrix of the variables - R _0j Variance-covariance signal matrix of the standardized variables at origin;j is the index related tod andD (e.g.,j=1 ford=500 m,D=1000 m) - R _1j Matrix of auto- and cross-correlation coefficients evaluated at a given distance within the neighborhood - R _m ² Multiple correlation coefficient squared for themth variable - S _i Column vectori of the signal values - s _k ² Residual variance for variablek - T _i Amplitude vector corresponding toV _i;ith row ofT=V ^–1 - T Total variation in the system - U Nonsymmetric transition matrix formed by post-multiplyingR ₀₁ ^–1 byR _ij - U _i Componenti of the matrixU, corresponding to theith eigenvectorV _i;U _i= _iV_iT_i - U* _i ComponentU _i multiplied byc _i - U _ij Sum of componentsU _i+U _j - V _i Eigenvector of the matrixU;ith column ofV withUV=VL - w Weighting factor; equal to the ratio of two eigenvalues - X _i Random variable at pointi - x _i Value of random variable at pointi - y _i Residual ofx _i - Z _i Row vectori for the standardized variables - z _i Standardized value of variable     

Appreciation of Contributions by Georges Matheron and John Tukey to a Mineral-Resources Research Project

Georges Matheron (1930–2000) and John Tukey (1915–2000) were among the most prominent mathematical statisticians of the 20th century. Both men produced numerous important new theoretical and practical results. This personal appreciation of their work concentrates on contributions to mineral-resources research and describes their influence on my work in mineral-resource evaluation studies at the Geological Survey of Canada (1966–1983).     

Multifractal modeling and spatial point processes

The multifractal model can be applied to spatial point processes. It provides new, approximately power-law type, expressions for their second-order intensity and K (r) functions. The box-counting and cluster dimensions are different but mutually interrelated according to multifractal theory. This approach is used to describe the underlying spatial structure of gold mineral occurrences in the Iskut River area, northwestern British Columbia. The box-counting and cluster dimensions for the example are estimated to be 1.335±0.077 and 1.219±0.037, respectively. The relatively strong clustering of the gold deposits is reflected by the fact that both values are considerably less than the corresponding Euclidean dimension (=2).     

A conceptual model for kimberlite emplacement by solitary interfacial mega-waves on the core mantle boundary

If convection in the Earth's liquid outer core is disrupted, degrades to turbulence and begins to behave in a chaotic manner, it will destabilize the Earth's magnetic field and provide the seeds for kimberlite melts via turbulent jets of silicate rich core material which invade the lower mantle. These (proto-) melts may then be captured by extreme amplitude solitary nonlinear waves generated through interaction of the outer core surface with the base of the mantle. A pressure differential behind the wave front then provides a mechanism for the captured melt to ascend to the upper mantle and crust so quickly that emplacement may indirectly promote a type of impact fracture cone within the relatively brittle crust. These waves are very rare but of finite probability. The assumption of turbulence transmission between layers is justified using a simple three-layer liquid model. The core derived melts eventually become frozen in place as localised topographic highs in the Mohorovicic discontinuity (Moho), or as deep rooted intrusive events. The intrusion's final composition is a function of melt contamination by two separate sources: the core contaminated mantle base and subducted Archean crust. The mega-wave hypothesis offers a plausible vehicle for early stage emplacement of kimberlite pipes and explains the age association of diamondiferous kimberlites with magnetic reversals and tectonic plate rearrangements.     

A Comparison of Modified Fuzzy Weights of Evidence,Fuzzy Weights of Evidence,and Logistic Regression for Mapping Mineral Prospectivity

Weights of evidence and logistic regression are two of the most popular methods for mapping mineral prospectivity. The logistic regression model always produces unbiased estimates, whether or not the evidence variables are conditionally independent with respect to the target variable, while the weights of evidence model features an easy to explain and implement modeling process. It has been shown that there exists a model combining weights of evidence and logistic regression that has both of these advantages. In this study, three models consisting of modified fuzzy weights of evidence, fuzzy weights of evidence, and logistic regression are compared with each other for mapping mineral prospectivity. The modified fuzzy weights of the evidence model retains the advantages of both the fuzzy weights of the evidence model and the logistic regression model; the advantages being (1) the predicted number of deposits estimated by the modified fuzzy weights of evidence model is nearly equal to that of the logistic regression model, and (2) it can deal with missing data. This method is shown to be an effective tool for mapping iron prospectivity in Fujian Province, China.