Appendix D: Anisotropy in Paleomagnetism

D.1The 15 measurement protocol¶

The Jelı́nek (1978) 15 measurement scheme is illustrated in Figure D.1. This is the procedure recommended in the manual distributed with the popular Kappabridge susceptibility instruments. In the 15 measurement case shown in Figure D.1, the design matrix is:

and

D.2The spinning protocol¶

More recent models of the Kappabridge magnetic susceptibility instruments (e.g., KLY-3S and KLY-4S; see e.g., Pokorný et al. (2004)) measure anisotropy by spinning the specimen around three axes (Figure D.2). For complete measurement and analysis details, see Gee et al. (2008). Here we just give the bare bones explanation.

The specimen is lowered into the measurement region and the susceptibility meter is set to zero. The deviatoric susceptibility is then measured in 64 positions per revolution for multiple (often eight) revolutions (see Figure D.3a). These data must be corrected for instrumental drift (red line in the figure), adjusted to have zero mean and stacked (Figure D.3b). The data can be fit with a best fit theoretical curve (red line in the figure). The theoretical curve can be derived from the complicated design matrix (see Gee et al. (2008) for details). As one example, the measurement recorded at an angle $\theta_i$ (as shown in Figure D.2d) while spinning in Position 1 (Figure D.2a) is given by:

The best fit values for $\chi$ for the entire sequence of data gives the 2D Model for the set of data (see Figure D.3b). There are three such models for the measurement protocol, each yielding estimates of two of the three on-axis $\chi$ values ($\chi_{11}, \chi_{22}, \chi_{33}$). These measurements are all adjusted to zero mean susceptibility so one more measurement is required to determine the bulk susceptibility (the absolute susceptibility measured in the position shown in Figure D.2c or $\chi_{11}$ in Position 3). The three 2D models plus the bulk measurement are combined as shown in Figure D.3d whereby the $\chi_{11}$ position in the 2D model for Position 3 is adjusted to the bulk measurement, and the best-fit 3D model is found that minimizes cross over errors for pairs of $\chi_{11}, \chi_{22}, \chi_{33}$. Once the best fit values for $\chi$ are found and standard deviation, the data can be treated as described in Chapter 13.

D.3Correction of inclination error with AARM¶

The magnitude of ARM is here denoted $M_a$. The particle anisotropy is denoted $a$ and is given by:

where $M_{a_{||}}$ and $M_{a_{\perp}}$ are the magnitudes of the ARM acquired parallel to and perpendicular to the detrital particle long axis respectively. The normalized eigenvalues of the ARM tensor ($q_i$) are defined as:

Stephenson et al. (1986) defined an orientation distribution function for the preferred alignment of particle long axes whose eigenvalues are given by $\kappa_i$ where $\kappa_1 > \kappa_2 > \kappa_3$ as usual. Jackson et al. (1991) collect together the two sources of anisotropy (alignment of particle long axes and individual particle anisotropies) as:

Assuming that the DRM anisotropy is identical to the orientation distribution function of particle long axes we can combine and rearrange Equation 13.23 and Equation D.7 to get the relationship between the flattening factor $f$ and the ARM anisotropy:

From the foregoing, measuring the AARM tensor yields the values for $q$, but determining values for $a$ are more problematic. Vaughn et al. (2005) describe a technique whereby magnetic particles are separated from the matrix, then allowed to dry in an epoxy matrix in the presence of a magnetic field sufficient to fully align the long axes of the magnetic particles (say 50 mT). The AARM parallel to and perpendicular to the axis of alignment therefore gives $a$ by Equation D.5.