Multifractal modelling and spectrum analysis: Methods and applications to gamma ray spectrometer data from southwestern Nova Scotia, Canada

Multifractal theory was developed for handling scale invariant fields instead of geometry only1―4]. From a multifractal point of view, some fractal models, ordinary physical processes and relevant probability distribution types can be considered as special cases of multifractal models which provides new insight into the interrelationships between systems and subjects. For example, the low order moment exponents τ (0), τ (1), τ (2) or τ ″(1) obtained by means of the moment method determi…

Multifractal modelling and spectrum analysis: Methods and applications to gamma ray spectrometer data from southwestern Nova Scotia, Canada

Multifractal spectrum, autocorrelation/semivariogram and power spectrum are three different functions characterizing a field or measure from different aspects. These functions are interrelated in such that the autocorrelation/semivariogram and power spectrum are related to the low order statistical moments (0 to 2 nd) which may determine the local multifractality (τ″(1)) of a multifractal measure. A better understanding of the interrelationships among these three functions is important because, on one hand, the multifractal modelling characterizes a multifractal measure in a more detailed manner since it involves moments of all orders. On the other hand, the commonly used semivariogram and power spectrum functions can be used as alternatives to study the dominant multifractal properties around the mean measure. Moreover, semivariogram and power-spectrum functions provide spatial and spectral information, which is highly valued in geological applications. A new fractal relation found between area and power-spectrum will be useful for investigation of new methods of spatial-spectral analysis for pattern recognition, anomaly separation, classification, etc. These results have been demonstrated with the case study of modelling gamma ray spectrometer data from the mineral district, southwestern Nova Scotia, Canada. The results have shown that the values of uranium (U), thorium (Th) and potassium (K) may possess monofractal properties whereas the ratios of these values show high multifractality. The values of the ratios U/K and U/Th show relatively large variances and may provide more information for distinguishing the distinct phases of the granites, country rocks as well as possible gold mineralization-associated regional hydrothermal alteration zones. In addition, the power spectra for U, Th, K, U/Th and U/K consistently show two distinct power-law relationships for two ranges of wave number 12⩽ω⩽160 km and 0⩽ω⩽12 km. These properties might provide useful thresholds for separating the power-spectrum values into two types of patterns to reflect different influences of possible geological processes such as hydrothermal alteration in the study area.