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Correction of paleointensity results for anisotropic TRM is based on the premise that in a weak magnetic field a TRM vector ($$\mathbf{TRM}$$) is related to the applied field vector ($$\mathbf{B}$$) by: \[ \mathbf{TRM}=\mathbf{\chi}_{TRM}\mathbf{B}, \] where $$\mathbf{\chi}_{TRM}$$ is the TRM anisotropy tensor, which is assumed to be temperature invariant (Veitch et al., 1984; Selkin et al., 2000). The anisotropy tensor can be experimentally determined by giving a specimen a full TRM or ARM in 6 different directions ($$\pm x$$, $$\pm y$$, $$\pm z$$).
To correct a paleointensity estimate, a unit vector in the direction of the ancient field ($$\mathbf{\hat{B}_{Anc}}$$; the hat denotes a unit vector) must be determined.
Given a unit vector in the direction of the characteristic NRM direction ($$\mathbf{\hat{M}_{ChRM}}$$), $$\mathbf{\hat{B}_{Anc}}$$ can be calculated from:
\[
\mathbf{\hat{B}_{Anc}}=\frac{ \mathbf{\chi}^{-1}_{TRM}\mathbf{\hat{M}_{ChRM}} } { \left| \mathbf{\chi}^{-1}_{TRM}\mathbf{\hat{M}_{ChRM}} \right| }.
\]
$$\mathbf{\hat{M}_{ChRM}}$$ is determined from the free-floating PCA fit to the NRM steps from the selected Arai plot segment.
The paleointensity correction factor, $$c$$, which is the ratio of a magnetization gained in the direction of $$\mathbf{\hat{B}_{Lab}}$$ to a magnetization gained in the
direction of $$\mathbf{\hat{B}_{Anc}}$$ can then be calculated as:
\[
c=\frac{ \left| \mathbf{\chi}_{TRM}\mathbf{\hat{B}_{Lab}} \right| } { \left| \mathbf{\chi}_{TRM}\mathbf{\hat{B}_{Anc}} \right| }.
\]
The anisotropy corrected paleointensity estimate is simply given by:
\[
B_{Anc} = c{B_{Lab}}\left|b\right|.
\]
Useful Note...
Two methods to correct for anisotropy have been outlined in the literature (Veitch et al., 1984;
Selkin et al., 2000). The method outlined above is that
derived from Veitch et al., (1984), but both methods yield identical paleointensity estimates.
The method of Selkin et al., (2000), however, can have a detrimental effect of some selection
statistics and the method outlined above should be the preferred approach (Paterson, 2013).
The anisotropy tensor, $$\mathbf{\chi}_{TRM}$$, can be mathematically represented by a 3$$\times$$3 matrix, which has 6 independent elements. \[ \mathbf{\chi}_{TRM}= \begin{pmatrix} s_1 & s_4 & s_6 \\ s_4 & s_2 & s_5 \\ s_6 & s_5 & s_3 \end{pmatrix} \] For convenience, we can define a column matrix, $$\mathbf{s}$$, which contains the 6 independent elements, $$s_j, j=1\ldots6$$.
The TRM acquired when a specimen is placed in series of positions with respect to the applied field, can be expresses as \[ \mathbf{TRM}_i=\mathbf{A}_{i,j}\mathbf{s}_j, \] where $$i$$ denotes the $$i^{th}$$ measurement position and $$\mathbf{A}$$ is know as the design matrix and depends on the experimental design (i.e., the sequence of axes along which the field is applied).
The typical 6 positions of measurement of $$\mathbf{\chi}_{TRM}$$ or $$\mathbf{\chi}_{ARM}$$ ($$\pm x$$, $$\pm y$$, $$\pm z$$) can be represented by the matrix $$\mathbf{P}$$, which contains the unit vectors of the axes along which $$B_{Lab}$$ is applied. \[ \mathbf{P}= \begin{pmatrix} P_{1,1} & P_{1,2} & P_{1,3}\\ P_{2,1} & P_{2,2} & P_{2,3}\\ \vdots & \vdots & \vdots \\ P_{6,1} & P_{6,2} & P_{6,3}\\ \end{pmatrix} \] For each element $$P_{i,j}$$ of $$\mathbf{P}$$, $$i$$ denotes the $$i^{th}$$ measurement position and $$j=1\dots3$$, denotes the Cartesian coordinates of the unit vector along which the remanence is acquired (i.e., $$j = 1 = x$$, $$j = 2 = y$$, $$j = 3 = z$$). The design matrix of such a routine is given by: \[ \mathbf{A}= \begin{pmatrix} P_{1,1} & 0 & 0 & P_{1,2} & 0 & P_{1,3}\\ %1 0 & P_{1,2} & 0 & P_{1,1} & P_{1,3} & 0\\ %2 0 & 0 & P_{1,3} & 0 & P_{1,2} & P_{1,1}\\ %3 P_{2,1} & 0 & 0 & P_{2,2} & 0 & P_{2,3}\\ %4 0 & P_{2,2} & 0 & P_{2,1} & P_{2,3} & 0\\ %5 0 & 0 & P_{2,3} & 0 & P_{3,3} & P_{3,1}\\ %6 P_{3,1} & 0 & 0 & P_{3,2} & 0 & P_{3,3}\\ %7 0 & P_{3,2} & 0 & P_{3,1} & P_{3,3} & 0\\ %8 0 & 0 & P_{3,3} & 0 & P_{3,2} & P_{3,1}\\ %9 P_{4,1} & 0 & 0 & P_{4,2} & 0 & P_{4,3}\\ %10 0 & P_{4,2} & 0 & P_{4,1} & P_{4,3} & 0\\ %11 0 & 0 & P_{4,3} & 0 & P_{4,2} & P_{4,1}\\ %12 P_{5,1} & 0 & 0 & P_{5,2} & 0 & P_{5,3}\\ %13 0 & P_{5,2} & 0 & P_{5,1} & P_{5,3} & 0\\ %14 0 & 0 & P_{5,3} & 0 & P_{5,2} & P_{5,1}\\ %15 P_{6,1} & 0 & 0 & P_{6,2} & 0 & P_{6,3}\\ %16 0 & P_{6,2} & 0 & P_{6,1} & P_{6,3} & 0\\ %17 0 & 0 & P_{6,3} & 0 & P_{6,2} & P_{6,1}\\ %18 \end{pmatrix}. \]
Numerical Tip...
To allow flexibility, the below pseudo-code can be used to easily generate $$\mathbf{A}$$, when $$\mathbf{P}$$ is variable.
for $$i = 1 \rightarrow 6$$ do
index=(3$$\times$$(i-1))+1
A(index,1)=P(i,1)
A(index,2)=P(i,2)
A(index,3)=P(i,3)
A(index+1,1)=P(i,1)
A(index+1,2)=P(i,2)
A(index+1,3)=P(i,3)
A(index+2,1)=P(i,1)
A(index+2,2)=P(i,2)
A(index+2,3)=P(i,3)
end for
The best-fit values for $$\mathbf{s}$$ for the measured data can be obtained through the linear relationship: \[ \mathbf{s}=\left(\mathbf{A}^{T}\mathbf{A}\right)^{-1}\mathbf{A}^{T}\mathbf{TRM}, \] where $$^{T}$$ and $$^{-1}$$ denote the matrix transpose and inverse, respectively. These best-fit value then be used to construct $$\mathbf{\chi}_{TRM}$$ and hence determine $$c$$.
Numerical Tip...
Calculating the inverse of the anisotropy tensor or $$\left(\mathbf{A}^{T}\mathbf{A}\right)$$ is not strictly necessary and can be inefficient
and inaccurate. An alternative approach can be used if it is recognized that $$\mathbf{\chi^{-1}}_{TRM}\mathbf{\hat{M}}_{ChRM}$$ and
$$\left(\mathbf{A}^{T}\mathbf{A}\right)^{-1}\mathbf{A}^{T}$$ are linear problems of the form:
\[
\mathbf{x}=\mathbf{A}^{-1}\mathbf{b}.
\]
Many programming languages support tools that allow the solving of such linear systems without having to calculate the matrix inverse.
For example, the MATLAB command x=A\b or the Python
command x=linalg.solve(A,b) are solutions that do not need to
calculate the inverse of $$\mathbf{A}$$. Such approaches are numerically efficient and more stable and should be used where available.
Statistic: $$\delta{TRM_{Anis}}$$
Report to 1 d.p.
To test for alteration during the measurement of $$\mathbf{\chi}_{TRM}$$ a repeat remagnetization step is performed to the first treatment position. $$\delta{TRM_{Anis}}$$ is the difference between the intensities of the TRM acquired during first heating in position 1 ($$TRM_{P1}$$) and the second heating in position 1 ($$TRM'_{P1}$$) normalized by $$TRM_{P1}$$. \[ \delta{TRM_{Anis}}=\frac{\left|TRM_{P1} - TRM'_{P1}\right|}{TRM_{P1}}\times{100} \]
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