Qiuming Cheng

The separation of geochemical anomalies from background by fractal methods

Lithogeochemical data (major oxides and trace elements) from 1233 surface samples in the Mitchell-Sulphurets precious-metal district (≈120 km2 in area), British Columbia, were analyzed using fractal and multifractal models. Log-log plots for element concentration-area and perimeter-area relations were employed to separate geochemically anomalous areas from background. The values used for perimeters and areas are the lengths and enclosed areas of geochemical isopleths obtained by interpolation. Elements and oxides, including Au, Cu, As, Ag, K2O and SiO2 within alteration zones associated with copper and gold porphyry system(s) in the district, show power-law type element concentration-area and perimeter-area relations, which can be fitted as straight lines on the log-log graphs. Separate relations inside and outside the potassic, sulphidic and silicic alteration areas can be used to delineate the anomalies. The slopes on the graphs show that Au is much more irregularly distributed than Cu and As in the porphyry systems.

Spatial and scaling modelling for geochemical anomaly separation

A spatial and scaling approach with a user-friendly windows program is introduced which can be used to assist exploration geologists and geochemists in geochemical data analysis and anomaly separation. It can also be used for image enhancement and classification. Statistics are calculated and optimized within a variable-sized moving window centred at an arbitrary sample location. The moving window has both variable size and shape determined by three parameters: r (size), β (ratio of long and short axes), and θ (orientation). It calculates five optimal indexes for each sample location: the optimal statistic U(r0,β0,θ0), optimal size r0, shape indexes β0 and θ0, and scaling index α (singularity exponent). These indexes characterize the entities present in an image from different angles and, therefore, can be analyzed by means of multivariate techniques to assist in image enhancement and classification. The user-friendly program prepared can be used in conjunction with GIS (Geographic Information System) software such as ArcView to implement the spatial and scaling method. It has been applied to the stream sediment geochemical data set (923 grid samples) for gold mineral exploration in the Habahe map sheet, Altay Shan, Xinjiang, northwestern China. The spatial and scaling method provides better results than the ordinary moving average method.

Integrated Spatial and Spectrum Method for Geochemical Anomaly Separation

A new approach for separating geochemical anomalies from background has been developedon the basis of integration of spatial and spectrum analysis. A map generated from geochemicaldata can be transformed into a frequency domain in which a spatial concentration-area fractalmethod can be applied to distinguish the patterns on the basis of the power-spectrum distribution.Distinct classes can be generated, such as lower, intermediate, and high power-spectrum valuesapproximately corresponding to background, anomalies, and noises of geochemical values ina spatial domain. An irregular filter then can be constructed on these distinct patterns withthe background and noises related to low- and high-power-spectrum values being removed.The image converted back to a spatial domain with the filter applied can show patterns which,after the removal of background and noise, mainly reflect a residual area that representsanomalous or atypical geochemical patterns. This method is demonstrated using a case studyof soil geochemical data from the Mudik area, on the island of Sumatra, Indonesia. The resultsobtained from this method in comparison with those obtained from other methods have shownthat the newly developed method can separate overlapping populations without using a singlecutoff value.

Multifractality and spatial statistics

The concepts of fractals and multifractals have been increasingly applied in various fields of science for describing complexity and self-similarity in nature. Fractals and multifractals are a natural consequence of self-similarity resulting from scale-independent processes. In the present paper, a theoretical investigation is developed to illustrate: (1) the characteristics of multifractality as measured by the parameter τ″(q); (2) relationships between multifractality and spatial statistics including semivariogram and autocorrelation in geostatistics, indexes used in lacunarity analysis and correlation coefficients. It can be shown that these statistics primarily are related to multifractality as determined by τ″(1). This is an important result because not only does it provide the link between multifractals and spatial statistics but it also shows that statistics based on second-order moments are restrictive in that they only characterize a multifractal measure around the mean value. In applications where extreme values need to be taken into account, the entire multifractal spectrum should be used rather than local properties of the spectrum around the mean only; alternatively, statistics defined on the basis of higher-order moments can be employed for analysis of extreme values. These theoretical results are illustrated by means of application to Landsat TM imagery (bands 1 to 7) from the Mitchell–Sulphurets mineral district, northwestern British Columbia, Canada. It is shown that variations of the TM data bands 1 to 3 in this area can be approximated by fractals but for those of bands 4, 5 and 7, multifractal models with different fractal spectra must be used. The multifractality of these images is evaluated and several methods of spatial analysis are applied to the dataset including a new version of principal component analysis in conjunction with the newly defined statistical parameters based on multifractality.

Singularity analysis of ore-mineral and toxic trace elements in stream sediments

Hydrothermal processes in the Earths crust can result in ore deposits characterized by high concentrations of metals with fractal or multifractal properties. This paper shows that stream sediments in the neighborhoods of ore deposits also can have singular properties for ore-mineral and associated toxic trace elements. We propose a new local singularity mapping method for assembling element concentration values from stream sediment samples to delineate anomalous areas induced by buried mineral deposits, which are often missed in ordinary geochemical surveys and mapping. Applied to the Gejiu area, Yunnan Province, China, which contains world-class size hydrothermal deposits enriched in tin and other elements, non-linear anomalies for tin and arsenic are identified: (1) many relatively small singularity anomalies in about 10% of the study area; and (2) a large high-concentration anomaly in the eastern part of the area where mining occurs. The ore-mineral and toxic elements within these anomalies describe Pareto-type frequency distributions. Spatial proximity of anomalies of the first kind to the ore deposits (mines and prospective mines) indicates that singularity mapping provides a useful new tool for mineral prospecting. The relation of the second kind of anomaly to mining activities indicates that fractal modeling also can provide useful input for decision-making in environmental protection.

A spatial analysis method for geochemical anomaly separation

Abstract One purpose of using statistical methods in exploration geochemistry is to assist exploration geologists in separating anomalies from background. This always involves two types of negatively associated errors of misclassification: type I errors occur when samples with background levels are rejected as background; and type II errors occur when samples with anomalous values are accepted as background. A new spatial statistical approach is proposed to minimize errors of total misclassification using a moving average technique with variable window radius. This method has been applied for geochemical anomaly enhancement and recognition as demonstrated by a case study of Au and Au-associated data for 698 stream sediment samples in the Iskut River area, northwestern British Columbia. Similar results were obtained using the fractal concentration-area method on the same data. By employing spatial information in the analysis, the process of selecting anomalies becomes less subjective than in more traditional approaches.

Conditional Independence Test for Weights-of-Evidence Modeling

Weights-of-evidence modeling is a GIS-based technique for relating a point pattern for locations of discrete events with several map layers. In general, the map layers are binary or ternary. Weights for presence, absence or missing data are added to a prior logit. Updating with two or more map layers is allowed only if the map layers are approximately conditionally independent of the point pattern. The final product is a map of posterior probabilities of occurrence of the discrete event within a small unit cell. This paper contains formal proof that conditional independence of map layers implies that T, the sum of the posterior probabilities weighted according to unit cell area, is equal to n, being the total number of discrete events. This result is used in the overall or “omnibus test” for conditional independence. In practical applications, T generally exceeds n, indicating a possible lack of conditional independence. Estimation of the standard deviation of T allows performance of a one-tailed test to check whether or not T-n is significantly greater than zero. This new test is exact and simpler to use than other tests including the Kolmogorov-Smirnov test and various chi-squared tests adapted from discrete multivariate statistics.

Fuzzy Weights of Evidence Method and Its Application in Mineral Potential Mapping

This paper proposes a new approach of weights of evidence method based on fuzzy sets and fuzzy probabilities for mineral potential mapping. It can be considered as a generalization of the ordinary weights of evidence method, which is based on binary or ternary patterns of evidence and has been used in conjunction with geographic information systems for mineral potential mapping during the past few years. In the newly proposed method, instead of separating evidence into binary or ternary form, fuzzy sets containing more subjective genetic elements are created; fuzzy probabilities are defined to construct a model for calculating the posterior probability of a unit area containing mineral deposits on the basis of the fuzzy evidence for the unit area. The method can be treated as a hybrid method, which allows objective or subjective definition of a fuzzy membership function of evidence augmented by objective definition of fuzzy or conditional probabilities. Posterior probabilities calculated by this method would depend on existing data in a totally data-driven approach method, but depend partly on experts knowledge when the hybrid method is used. A case study for demonstration purposes consists of application of the method to gold deposits in Meguma Terrane, Nova Scotia, Canada.

Multifractal modeling of fractures in the Lac du Bonnet Batholith, Manitoba

Algorithms to analyze fractures by multifractal modeling are discussed. This approach is more general than fractal modeling. For a small (~0.11 km2), triangular test area near the Atomic Energy Canada Limited Underground Research Laboratory in the Lac du Bonnet Batholith, it is shown that the surface fractures have a box-counting dimension of 1.977 ± 0.009, which is slightly greater than the second-order mass exponent (= 1.926 ± 0.016). The measure used is total length of exposed surface fractures per cell of variable size. A relatively stable estimate of the multifractal spectrum of fracture intensity was obtained after correcting for local lack of exposure. These results are in agreement with one-dimensional semivariograms estimated for cells measuring 10 m on a side. Approximate isotropy of the measure is demonstrated.

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.