## 3 Material and Methods
### The Art of Gradient Tensor Analysis
For magnetic data these are \(B_{xx}\), \(B_{xy}\), \(B_{xz}\), \(B_{yy}\), \(B_{yz}\). Refs. [42; 43] explore a different type of tensor representation that is based on amplitudes and phases. A tensor measurement can be transformed using eigenvalues and eigenvectors. Each reading is divided into its orthogonal rotation matrix, invariant eigenvalue amplitudes, and local to the survey reference frame eigenvectors. The amplitude and phase of the tensor are represented by the eigenvalue amplitudes and eigenvector rotations. where:  
\[I_{1}=B_{yy}B_{zz}+B_{xx}B_{yy}+B_{zz}B_{xx}-B_{xy}^{2}B_{yy}^{2}=\lambda_{1} \lambda_{2}+\lambda_{1}\lambda_{3}+\lambda_{2}\lambda_{3}=\frac{-\left(\lambda_{1 }^{2}+\lambda_{2}^{2}+\lambda_{3}^{2}\right)}{2} \tag{11}\]  
\[I_{2}=\det(B)=B_{xx}B_{yy}B_{zz}+B_{xx}B_{yz}^{2}+B_{zz}B_{xy}^{2}-2B_{xy}B_{xz }B_{yz}^{2}=\lambda_{1}\lambda_{2}\lambda_{3} \tag{12}\]  
where \(\lambda_{1}\geq\lambda_{2}\geq\lambda_{3}\). Applying eigenvector analysis to our tensor equation, as shown in the previous equation, we obtain 3 eigenvalues and 3 eigenvectors. The rotation matrix \(R\) which has as its columns the eigenvectors \([\psi_{1},\psi_{2},\ \psi_{3}]\), diagonalises \(\mathbf{B}\) when applied to it. It is straightforward to verify that the following holds:  
\[R^{T}BR=R^{T} \begin{bmatrix}B_{xx}&B_{xy}&B_{zz}\\ B_{yx}&B_{yy}&B_{yz}\\ B_{xx}&B_{xy}&B_{zz}\end{bmatrix} R= \begin{bmatrix}\lambda_{1}&&\\ &\lambda_{2}&\\ &&\lambda_{3}\end{bmatrix}  \tag{13}\]  
The eigenvalues, and any combination thereof, are the tensor rotational invariants. [36] makes extensive use of one such rotational invariant, specifically, the standardized source magnitude. It is defined as:  
\[\mu_{nss}=\sqrt{-\lambda_{2}^{2}-\lambda_{1}\lambda_{3}} \tag{14}\]  
In contrast to magnitude of the tensor (Frobenius norm) 1B1, ref. [36] notes that it is entirely isotropic in the vicinity of a dipole source. It is perfect for inverse operation (total magnetic field calculation). As a result, the previous formulation may be used to determine the horizontal invariants of the FTG map the edge of units (Figure 4). They are characterized as:  
\[I_{H1}=\sqrt{B_{xz}^{2}+B_{yz}^{2}} \tag{15}\]  
\[I_{H2}=\sqrt{B_{xy}^{2}+\left(\frac{B_{yy}-B_{xx}}{2}\right)^{2}} \tag{16}\]  
The gradient tensor determinant of 2-D structures, according to [10], is equal to zero. The first system for which the x-axis corresponds with the strike direction will be obtained if the coordinate system is then rotated such that it lines up with the striking direction., the element \(\frac{\partial^{2}S}{\partial x^{2}}\) becomes zero. Although this does not really occur in practice, the optimal striking direction can be identified if the matrix is rotated around the z-axis to decrease this component. You may compute this rotation angle [44; 45; 46] (Figure 5) using:  
\[\theta=\frac{1}{2}\text{arctan}\left(2\frac{B_{xy}\left(B_{xx}+B_{yy}\right)+ B_{xz}B_{yz}}{B_{xx}^{2}-B_{yy}^{2}+B_{xz}^{2}-B_{yz}^{2}}\right) \tag{17}\]Figure 4: Processed Horizontal invariants from calculated Gradient Tensor of Bou Azzer Inlier. (**A**) Lh1. (**B**) Lh2.