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Global and Local Multiscale Analysis of Magnetic Susceptibility Data

机译：磁化率数据的全局和局部多尺度分析

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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(α), 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 amphibole units and the lowest values for variegated units. Faults are found to correspond to changes for H also when they are of intra-lithological type.
机译：地球物理测井曲线通常显示出复杂的行为，似乎表明它具有多重分形的性质。多重分形是高度间歇性的信号，具有明显的主动脉冲和被动区域，仅通过二阶统计量无法令人满意地表征它们。他们需要高阶统计分析。与具有均匀缩放的单形分形相反，多形分形可以包括许多类型的奇点。在这里，我们描述了如何使用多尺度分析来描述深达9000 m的深井（德国KTB）的磁化率数据缩放特性。多尺度分析以统计方式描述了度量或分布的局部和全局奇异行为。全局分析允许估算各种Holder指数的全局重新分配。因此，它导致了光谱D（α）的定义，称为奇异光谱。局部分析与局部估计Lipschitz规律性的可能性有关，即在多重分形信号的支持的每个点上。两种方法在KTB磁化率数据中的应用表明Holder指数序列与深度和岩性单位之间存在有意义的关联。 Holder指数的片麻岩单位值最高，闪石单位的中间指数最高，杂色单位的最低值。断层为岩性时，断层也对应于H的变化。

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