Global and Local Multiscale Analysis of Magnetic Susceptibility Data

Geophysical well-logs often show a complex behavior which seems to suggest a multifractal nature. Multifractals are highly intermittent signals, with distinct active bursts and passive regions which cannot be satisfactorily characterized in terms of just second-order statistics. They need a higher-order statistical analysis. In contrast with monofractals which have a homogeneous scaling, multifractals may include singularities of many types. Here we describe how a multiscale analysis can be used to describe the magnetic susceptibility data scaling properties for a deep well (KTB, Germany), down to about 9000 m. A multiscale analysis describes the local and global singular behavior of measures or distributions in a statistical fashion. The global analysis allows the estimation of the global repartition of the various Holder exponents. As such, it leads to the definition of a spectrum, D(a), called the singularity spectrum. The local analysis is related to the possibility of estimating the Lipschitz regularity locally, i.e., at each point of the support of a multifractal signal. The application of both approaches to the KTB magnetic susceptibility data shows a meaningful correlation between the sequence of Holder exponents vs. depth and the lithological units. The Holder exponents reach the highest values for gneiss units, intermediate ones for amphibolite units and the lowest values for variegated units. Faults are found to correspond to changes for H also when they are of intralithological type.