# A Scalable and Consistent Method for Multi-Component Gravity-Gradient Data Processing
## 2 Methodology
### Classical Equivalent Layer  
Let \(\mathbf{g}^{\alpha\beta}\) be an \(N\times 1\) vector whose \(i\)th element is a gravity-gradient measurement of the \(\alpha\beta\)-component \(g_{i}^{\alpha\beta}\), \(\alpha,\beta\in\{x,y,z\}\), at the observation point \((x_{i},y_{i},z_{i})\), \(i\in\{1,\ldots,N\}\). According to the equivalent-layer principle, the observed data vector \(\mathbf{g}^{\alpha\beta}\) can be approximated by  
\[\mathbf{d}^{\alpha\beta}(\mathbf{m})=\mathbf{H}^{\alpha\beta}\mathbf{m}\,, \quad\alpha,\beta\in\{x,y,z\}\,, \tag{2}\]  
where \(\mathbf{d}^{\alpha\beta}\) is the predicted gravity-gradient data vector of size \(N\times 1\) due to a discrete set of \(M\) point masses located below the observation points, at the coordinates \((x_{j},y_{j},z_{c})\), \(j\in\{1,\ldots,M\}\), on a horizontal plane with constant vertical coordinate \(z_{c}>z_{i}\), \(\forall\,i\in\{1,\ldots,N\}\). In Equation (2), \(\mathbf{m}\) is the \(M\times 1\) parameter vector whose \(j\)th element is the mass of the \(j\)th point mass and \(\mathbf{H}^{\alpha\beta}\) is an \(N\times M\) sensitivity matrix whose element \(ij\) is defined as  
\[h_{ij}^{\alpha\beta}= \begin{cases}c_{g}\gamma\frac{3(\alpha_{i}-\alpha_{j})^{2 }-r_{ij}^{2}}{r_{ij}^{5}}&\text{if}\quad\alpha=\beta\\ c_{g}\gamma\frac{3(\alpha_{i}-\alpha_{j})(\beta_{i}-\beta_{j})}{r_{ij}^{5}}& \text{if}\quad\alpha\
eq\beta\end{cases} \,, \tag{3}\]  
where \(\alpha,\beta\in\{x,y,z\}\), \(c_{g}=10^{9}\) is a transformation constant from \(1/\text{s}^{2}\) to eotvos, \(\gamma\) is the gravitational constant in \(\text{m}^{3}/(\text{kg}\,\text{s}^{2})\), and \(r_{ij}\) is the Euclidean distance between the \(i\)th observation point and the \(j\)th point mass given by \[r_{ij}=\left[(x_{i}-x_{j})^{2}+(y_{i}-y_{j})^{2}+(z_{i}-z_{c})^{2}\right]^{\frac{1} {2}}. \tag{4}\]  
The classical equivalent-layer technique applied to gravity-gradient data involves estimating a single mass distribution \(\hat{\mathbf{m}}\) that simultaneously fits all the \(\alpha\beta\)-components of the gravity-gradient data. It can be accomplished by minimizing the data-misfit function  
\[\Phi(\mathbf{m})=\frac{1}{N_{t}}\|\mathbf{g}^{o}-\mathbf{H}\,\mathbf{m}\|_{2}^ {2}\,, \tag{5}\]  
which is subjected to some constraint, where \(N_{t}\) is the total number of observations, \(\|\cdot\|_{2}\) denotes the Euclidean norm, \(\mathbf{g}^{o}\) is an observed data vector containing all measured \(\alpha\beta\)-components of the gravity-gradient data, and \(\mathbf{H}\) is a sensitivity matrix depending on the specific airborne system. For example, if we consider gravity-gradient data measured by the FTG system, \(\mathbf{g}^{o}\) is a \(N_{t}\times 1\) vector given by  
\[\mathbf{g}^{o}=\left[\mathbf{g}^{xx\top},\,\mathbf{g}^{xy\top},\,\mathbf{g}^{ xz\top},\,\mathbf{g}^{yy\top},\,\mathbf{g}^{yz\top}\right]^{\top}, \tag{6}\]  
and \(\mathbf{H}\) is the \(N_{t}\times M\) matrix composed by the sub-matrices \(\mathbf{H}^{\alpha\beta}\), \(\alpha,\beta\in\{x,y,z\}\) (Equation (3))  
\[\mathbf{H}=\left[\mathbf{H}^{xx\top},\,\mathbf{H}^{xy\top},\,\mathbf{H}^{xz \top},\,\mathbf{H}^{yy\top},\,\mathbf{H}^{yz\top}\right]^{\top}, \tag{7}\]  
where \(N_{t}=5N\). Otherwise, if we consider the two components measured by the Falcon AGG system, \(\mathbf{g}^{o}\) is a \(N_{t}\times 1\) vector given by  
\[\mathbf{g}^{o}= \begin{bmatrix}\mathbf{g}^{xy\top}\\ \mathbf{g}^{uv}\end{bmatrix} , \tag{8}\]  
and \(\mathbf{H}\) is the \(N_{t}\times M\) matrix  
\[\mathbf{H}= \begin{bmatrix}\mathbf{H}^{xy\top}\\ \mathbf{H}^{uv}\end{bmatrix}