# A Scalable and Consistent Method for Multi-Component Gravity-Gradient Data Processing
## 2 Methodology
### Convolutional Equivalent Layer in Rotated Coordinates  
Consider that the observed gravity-gradient data are distributed over a horizontal plane with constant vertical coordinate \(z_{c}=z_{i},i=1,\ldots,N\), at a regularly spaced grid of points along the \(x\) and \(y\) axes of our Cartesian coordinate system, which we refer to as the _north-east system_ in which the \(x\)-axis points north and the \(y\)-axis to the east. In practice, however, gravity-gradient data are often referenced to a different coordinate system whose horizontal axes are not aligned with geographic directions. We refer to this as the _data system_, which is aligned with the acquisition geometry, with local coordinates \(\tilde{x}\)-axis and \(\tilde{y}\)-axis (Figure 1). The data system is rotated relative to the north-east system by an angle \(\psi\) from north. To relate these two systems, we apply a coordinate rotation. The gravity-gradient tensor \(\tilde{\mathbf{G}}\) defined in the data system can be obtained from the tensor \(\mathbf{G}\), which in turn is defined in to north-east system by Equation (1) through the transformation:  
\[\tilde{\mathbf{G}}=\mathbf{R}\mathbf{G}\mathbf{R}^{\top}, \tag{12}\]  
where \(\mathbf{R}\) is a rotation matrix given by  
\[\mathbf{R}= \begin{bmatrix}\cos\psi&\sin\psi&0\\ -\sin\psi&\cos\psi&0\\ 0&0&1\end{bmatrix} . \tag{13}\]  
Let us consider an equivalent layer located on a horizontal plane below the grid of observations, with constant vertical coordinate \(z_{c}>z_{i}\), having one point mass directly below each observation point, so that the number of point masses \(M\) is equal to that of observation points \(N\). In this case, ref. [34] described the particular structure of the sensitivity matrix \(\mathbf{H}\) (Equations (7) and (9)); however, they did not mention that this specific structure, namely, the block-Toeplitz Toeplitz-block (BTTB) structure, emerges only when \(\mathbf{H}\) is defined in the coordinates of the data system (Figure 1). For this reason, all computational operations in this work were performed in the data system, which is aligned with the \(\tilde{x}\)  
Figure 1: Schematic representation of a regular grid of gravity-gradient data (black dots) measured on a horizontal plane, with constant grid spacings \(\Delta x\) and \(\Delta y\). These data are aligned with a local Cartesian system having horizontal axes \(\tilde{x}\) and \(\tilde{y}\) (data system). The data system, in turn, is rotated by an angle \(\psi\) with respect to the north-east system, with \(x\)- and \(y\)-axes pointing north and east, respectively. Both systems share the same \(z\)-axis pointing vertically downward. and \(\hat{y}\)-axes (the acquisition geometry). These operations include the calculation of the BTTB matrix eigenvalues of \(\mathbf{H}\), parameter estimation, prediction of gravity-gradient tensor components, and gravity-gradient data transformations. After completing all computational operations in the data system, the results should be transformed back to the north-east system, which is aligned with geographic directions. This is conducted using the following transformation:  
\[\mathbf{G}=\mathbf{R}^{\top}\mathbf{\bar{G}}\ \mathbf{R}. \tag{14}\]  
We stress that the north-east system was used solely for visualizing the results, including the predicted gravity-gradient data, estimated parameters, and transformed data, since it corresponds to the spatial arrangement of the original observations. To estimate the mass distribution \(\mathbf{\hat{m}}\), we solve the normal equations (Equation (11)) iteratively by using the _standard conjugate gradient least squares (CGLSs) method_ ([37], p. 166) that requires three matrix-vector products: one out of the loop and two per iteration.