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

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## Abstract

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.

### Key words

auto-correlation cross-correlation lithogeochemistry multiple correlation coefficients trend eigenvector amplitude eigenvector### List of symbols

*c*_{i}Scalar factor that minimizes the discrepancy between and

*U*_{i}*D*Radius of circular neighborhood used for estimating auto- and cross-correlation coefficients

*d*Distance for which transition matrix

*U*is estimated*d*_{ij}Distance between observed values

*i*and*j**E*Expected value

*E*_{i}Row vector of residuals in the standardized model

*F*(*d*_{ij})Quadratic function of distance

*d*_{ij}F(*d*_{ij})=*a*+*bd*_{ij}+*cd*_{ ij }^{2}*L*Diagonal matrix of the eigenvalues of

*U**λ*_{i}Eigenvalue of the matrix

*U*;*i*th diagonal element of*L**N*Number of observations

*p*Number of variables

*Q*Total predictive power of

*U**R*Correlation matrix of the variables

*R*_{0j}Variance-covariance signal matrix of the standardized variables at origin;

*j*is the index related to*d*and*D*(e.g.,*j*=1 for*d*=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 the

*m*th variable*S*_{i}Column vector

*i*of the signal values*s*_{k}^{2}Residual variance for variable

*k**T′*_{i}Amplitude vector corresponding to

*V*_{i};*i*th row of*T*=*V*^{−1}*T*Total variation in the system

*U*Nonsymmetric transition matrix formed by post-multiplying

*R*_{01}^{−1}by*R*_{ij}*U*_{i}Component

*i*of the matrix*U*, corresponding to the*i*th eigenvector*V*_{i};*U*_{i}=*λ*_{i}V_{i}T_{i}*U**_{i}Component

*U*_{i}multiplied by*c*_{i}*U*_{ij}Sum of components

*U*_{i}+*U*_{j}*V*_{i}Eigenvector of the matrix

*U*;*i*th column of*V*with*U′V*=*VL**w*Weighting factor; equal to the ratio of two eigenvalues

*X*_{i}Random variable at point

*i**x*_{i}Value of random variable at point

*i**y*_{i}Residual of

*x*_{i}*Z′*_{i}Row vector

*i*for the standardized variables*z*_{i}Standardized value of variable

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