# Susceptibility and Remanent Magnetization Estimates from Orientation Tools in Borehole Imaging Logs
## 2 Materials and Methods
Since the majority of sedimentary rocks has very low content of ferromagnetic minerals, which are responsible for retaining magnetization, we assume that remanent magnetization is not preserved in these rocks. This assumption considers that the resolution of magnetometers used in wells are not fine enough to register  
Figure 2: Graphic illustration of some important definitions. (**a**) Spatial relationship between the components, \(F_{x}\), \(F_{y}\) and \(F_{z}\), of the magnetic field described in terms of three parameters: magnitude \(F\), inclination \(I\) and declination \(D\). The sign convention for field and its components is: x positive to north, y positive to east and z positive downward. (**b**) Representation of the meaning of four tool angles, SDEV, HAZI, RB and P1 NO, which describes tool’s orientation in the geographical coordinate system (modified from [20]). the weak magnetic field disturbance caused by such low content of ferromagnetic minerals. In these cases, Equation (1) simplifies to:  
\[F_{\alpha}=H_{\alpha}+\chi_{\alpha}H_{\alpha} \tag{2}\]  
where \(\alpha\) may represent any of the \(x\), \(y\) and \(z\) directions. In this case, the magnetic susceptibility \(\chi\) can be easily calculated by:  
\[\chi_{\alpha}=\frac{F_{\alpha}}{H_{\alpha}}-1. \tag{3}\]  
The second approach assumes the presence of a natural remanent magnetization \(M_{R}\) caused by the existing magnetic field at the time that rocks with high content of ferromagnetic minerals were formed. This case leads to a situation where there is only one equation and two unknowns \(\chi\) and \(M_{R}\) as:  
\[F_{\alpha}=H_{\alpha}+\chi_{\alpha}H_{\alpha}+M_{R_{\alpha}}. \tag{4}\]  
Figure 3: Example of the amplitude fluctuation caused by tool’s rotation in a typical well. The strong amplitude fluctuations occurring in the horizontal components shown in (a) were greatly reduced after correction, as exhibit in (b). However, if we assume \(\chi\) and \(M_{R}\) as constant along some measured interval, we will have \(n\) measurements of \(F_{\alpha}\) and can also get \(n\) values for \(H_{\alpha}\) in the same interval. Although fundamental to get the estimates, this assumption seems geologically reasonable as you can make your measured interval \(n\) as small as two samples (0.3 m). This allows us to construct a system of \(n\) equations and two unknowns \(\chi\) and \(M_{R}\):  
\[ \begin{cases}F_{\alpha_{1}}-H_{\alpha_{1}}=\chi_{\alpha}H_{\alpha_{1}}+M_{R_{ \alpha}}\\ \vdots\\ F_{\alpha_{n}}-H_{\alpha_{n}}=\chi_{\alpha}H_{\alpha_{n}}+M_{R_{\alpha}}\end{cases} , \tag{5}\]  
which in matrix form becomes:  
\[ \begin{bmatrix}H_{\alpha_{1}}&&1\\ &\vdots\\ H_{\alpha_{n}}&&1\end{bmatrix}\begin{bmatrix}\chi_{\alpha}\\ M_{R_{\alpha}}\end{bmatrix} = \begin{bmatrix}F_{\alpha_{1}}-H_{\alpha_{1}}\\ \vdots\\ F_{\alpha_{n}}-H_{\alpha_{n}}\end{bmatrix} . \tag{6}\]  
This system can be solved for \(\chi_{\alpha}\) and \(M_{R_{\alpha}}\) using the least squares method. So, the mean value of \(\chi\) and the magnitude of vector \(M_{R}\) will be given by:  
\[\chi=\frac{\chi_{x}+\chi_{y}+\chi_{z}}{3}\quad\text{and}\quad M_{R}=\sqrt{M_{R _{x}}^{2}+M_{R_{y}}^{2}+M_{R_{z}}^{2}}. \tag{7}\]  
Additionally, the inclination \(I_{R}\) and declination \(D_{R}\) of the remanent magnetization may be estimated by rearranging Equation (4) in terms of the remanent magnetization, and making use of the spatial relations shown in Figure 2a.