Multifractal model of magnetic susceptibility distributions in some igneous rocks

Abstract

Abstract. Measurements of in-situ magnetic susceptibility were compiled from mainly Precambrian crystalline basement rocks beneath the Colorado Plateau and ranges in Arizona, Colorado, and New Mexico. The susceptibility meter used measures about 30 cm 3 of rock and measures variations in the modal distribution of magnetic minerals that form a minor component volumetrically in these coarsely crystalline granitic to granodioritic rocks. Recent measurements include 50-150 measurements on each outcrop, and show that the distribution of magnetic susceptibilities is highly variable, multimodal and strongly non-Gaussian. Although the distribution of magnetic susceptibility is well known to be multifractal, the small number of data points at an outcrop precludes calculation of the multifractal spectrum by conventional methods. Instead, a brute force approach was adopted using multiplicative cascade models to fit the outcrop scale variability of magnetic minerals. Model segment proportion and length parameters resulted in 26 676 models to span parameter space. Distributions at each outcrop were normalized to unity magnetic susceptibility and added to compare all data for a rock body accounting for variations in petrology and alteration. Once the best-fitting model was found, the equation relating the segment proportion and length parameters was solved numerically to yield the multifractal spectrum estimate. For the best fits, the relative density (the proportion divided by the segment length) of one segment tends to be dominant and the other two densities are smaller and nearly equal. No other consistent relationships between the best fit parameters were identified. The multifractal spectrum estimates appear to distinguish between metamorphic gneiss sites and sites on plutons, even if the plutons have been metamorphosed. In particular, rocks that have undergone multiple tectonic events tend to have a larger range of scaling exponents.

Modelling of magnetic data for scaling geology

“…The multifractal distribution of sources is another unexplored area, however some work has been carried out for the multifractal distribution of susceptibility (Fedi ; Fedi et al . ; Gettings , ) although its use in the estimation of depth of anomalous sources has not yet been explored. Lovejoy et al .…”

“…Therefore a multi scaling or multifractal distribution of sources (Fedi ; Singh et al . ; Fedi et al ., ; Gettings ; , Bansal et al . ) will be a better choice for estimating the depth using this approach.…”

“…This may not be true since the low wavenumbers should give information about greater depths. Therefore a multi scaling or multifractal distribution of sources (Fedi 2003;Singh et al 2004;Fedi et al, 2005b;Gettings 2005;2012, Bansal et al 2010b) will be a better choice for estimating the depth using this approach. Li and Eaton (2005) found that the combined effect of topography and magnetization also generates the fractal behaviour of the magnetic field.…”

“…There is no doubt that the scaling spectral method provides more reliable depth estimates than the conventional spectral method. The multifractal distribution of sources is another unexplored area, however some work has been carried out for the multifractal distribution of susceptibility (Fedi 2003;Fedi et al 2005b;Gettings 2005Gettings , 2012 although its use in the estimation of depth of anomalous sources has not yet been explored. Lovejoy et al (2001) suggested that anisotropic scaling is suitable for modelling the stratified earth and its magnetic field at higher and intermediate wavenumber ranges.…”

Mapping Curie Depth Across Western Canada From a Wavelet Analysis of Magnetic Anomaly Data

“…However, magnetization typically follows a fractal distribution, meaning that crustal magnetization has some degree of self‐similarity (Maus & Dimri, , ; Pilkington et al, ; Pilkington & Todoeschuck, ). The degree of self‐similarity depends on a number of factors including the lithology and geological processes and is likely to vary within a study area (Bansal et al, ; Gettings, , ; Leonardi & Kumpel, ; Maus & Dimri, , ; Pilkington, ; Pilkington & Todoeschuck, ). Since z b is estimated by analyzing the shape of the PSD of magnetic anomalies (e.g., Spector & Grant, ), self‐similarity is an important characteristic to consider: a fractal distribution is expressed by a negative slope in the PSD of magnetic anomalies on a log‐log plot affecting all wavenumbers, whereas a random distribution has a flat power spectrum.…”