# Combining InSAR and GNSS to Track Magma Transport at Bastic Volcano
## Inverse Models
### Inversion and Data Weighting Through the Covariance Matrix  
Inversion of ground displacements to determine the best-fitting deformation model requires minimization of the difference between the observations and the model results. This difference is called a misfit and we calculated it in a least square sense (L2-norm) as :  
\[\chi^{2}=(\mathbf{d_{0}}-\mathbf{G}(\mathbf{m}))^{T}\mathbf{C_{d}}^{-1}\left( \mathbf{d_{0}}-\mathbf{G}(\mathbf{m})\right) \tag{1}\]  
where \(\mathbf{d_{0}}\) represents the vector of observed displacements, \(\mathbf{G}(\mathbf{m})\) is the vector of modeled displacements with \(\mathbf{m}\) the set of model parameters and \(\mathbf{G}\) represents the model prediction. \(\mathbf{C_{d}}\) is the covariance matrix for the data, which accounts for correlated noise related to the atmospheric contribution to InSAR data and for the variance of GNSS data. The atmospheric contribution results in correlated random noise, which can be characterized by the autocorrelation function or covariance function [48; 50], expressed as:  
\[C(r)=\sigma_{d}^{2}exp(-r/a) \tag{2}\]  
Figure 4: Use of the different data sets in our modeling strategy. where \(r\) is the distance between two points, \(\sigma_{d}^{2}\) is the noise variance and \(a\) is the correlation length. Following the study of [PERSON] et al. [51], the data covariance matrix \(\mathbf{C_{d}^{SAR}}\) can be computed from this function using a noise variance \(\sigma_{d}^{2}=5\times 10^{-4}\) m\({}^{2}\) and a correlation distance \(a=850\) m. This results in a diagonal dominant matrix where diagonal terms correspond to \(\sigma_{d}^{2}\). In addition to InSAR data, GNSS measurements were used from both permanent and campaign networks. We assumed there was no correlation between the GNSS stations, so that the covariance matrix \(\mathbf{C_{d}^{GNSS}}\) was diagonal and was a function of the variance. \[\mathbf{C_{d}^{GNSS}}=diag\left(\sigma_{d}^{2}\right) \tag{3}\]  
Typical magnitude of the GNSS variance was \(\sigma_{d}^{2}=2\times 10^{-4}\) m\({}^{2}\) for horizontal components and \(\sigma_{d}^{2}=9\times 10^{-4}\) m\({}^{2}\) for vertical components. The data vector \(\mathbf{d_{o}}\) and covariance matrix on data \(\mathbf{C_{d}}\) combining InSAR and GNSS data are given, respectively, as :  
\[\mathbf{d_{o}}=\left( \begin{array}{c}\mathbf{d_{o}}^{GNSS}\\ \mathbf{d_{o}}^{SAR}\end{array} \right) \tag{4}\]  
\[\mathbf{C_{d}}=\left[ \begin{array}{cc}\mathbf{C_{d}^{GNSS}}&\mathbf{0}\\ \mathbf{0}&\mathbf{C_{d}^{SAR}}\end{array} \right] \tag{5}\]  
Inversions consist of two stages : a search and an appraisal stage. In the search stage, the misfit is minimized using a near-neighborhood inversion algorithm [52]. In the appraisal stage, model uncertainties are estimated by resampling the population of models obtained in the search stage following the Bayesian inference [53]. Because the misfit function is a L2-norm, it favors models that explain the largest displacements. The matrix covariance characterizing the data is used to account for errors and intrinsic noise affecting the data. A proper definition of the covariance matrix should weight data according to their own errors in the inversion process. However, InSAR data are projections of a single displacement field along the LOS direction. Due to this projection, a displacement vector will produce LOS displacement of different amplitudes on ascending and descending LOS. Since a null LOS displacement can mean either no displacement or displacement orthogonal to the LOS, there is no reason to give a lower weighting to datum of lower amplitude. To give equal weighting to each dataset and avoid overweighting one LOS over the other or the GNSS data when they were added, we chose to conduct inversions in which misfits of the different datasets were scaled by their reference values.