# Combining InSAR and GNSS to Track Magma Transport at Bastic Volcano
## Inverse Models
### Static Inversion  
Given the location of the eruptive fissure, the geometry of the displacement source can be described by nine parameters, which define a quadrangle-shaped intrusion, linked to the surface by a single fissure. This quadrangle can be curved in the along strike and dip directions [16; 51].  
To determine the static geometry of the magma intrusion which fed the 26 May 2016 eruption, we used ground deformation data from interferograms along four different LOS, daily solutions of the permanent GNSS network and the GNSS campaign network. Of the eight available interferograms, five span the total eruption period (CSKA and CSKD, S1 D, S1 A and S1 A2, on Figure 3). S1 A and S1 A2 are taken along the same LOS but the pre-eruptive period spanned by S1 A is longer. We inverted S1 D, CSK D and CSK A but had two different options for the fourth interferogram in the ascending direction: (1) using the data closest to the eruption time (S1 A2) and (2) using the interferogram (S1 A) most consistent with the acquisition dates of the three other interferograms. We chose to invert the interferograms directly rather than use the 3D components reconstruction of the displacement field in order to reduce the errors induced from using several interferograms spanning different time periods. We decided not to use the whole dataset because the use of different data in the same LOS could have led to over weighting of one LOS.  
In total we run seven inversions (Table 1). With inversions 01, 02a and 02b, we checked whether the inflation during the 10 days of unrest could be neglected by comparing the influence of interferograms S1 A and S1 A2 on the inversion results (Table 1). In inversion 04a, 04b and 05, we tested whether the addition of GNSS data, which adds constraints to the north-south displacement, led to a different, better constrained source. Inversions 02a and 02b (and 04a and 04b) were conducted with the same initial conditions in terms of data used, weighting and parametrization, in order to test the variability of the inversion due to the random search. Lastly, in inversions 03, 04a and 04b, we tested the weighting method to investigate if different inverted sources were obtained and to gain insight into the origin of discrepancies between independent datasets.  
\begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline \multirow{2}{*}{**Model**} & \multirow{2}{*}{**Covariance Weighting**} & \multirow{2}{*}{**Family**} & \multirow{2}{*}{**GNSS**} & \multicolumn{6}{c}{**InSAR**} \\ \cline{5-10}  & & & & & **S1D** & & **S1 A** & & **S1 A2** & **CSKD** & **CSKA** & **Total** \\ \hline
**Inv01** & No & F1 & _78_ & **97** & _81_ & **75** & **97** & **83** & 93.5 \\
**Inv02a*** & No & F2 & _73_ & **95** & **82** & 72 & **95** & **82** & 92.5 \\
**Inv02b** & No & F2 & _75_ & **96** & **83** & 76 & **95** & **83** & 93.0 \\
**Inv03** & \(\chi^{2}_{ref}\) & F2 & _73_ & **96** & **86** & 77 & **96** & **87** & 94.1 \\
**Inv04a** & \(\chi^{2}_{ref}\) & F1 & **77** & **95** & **82** & 74 & **95** & **87** & 93.5 \\
**Inv04b** & \(\chi^{2}_{ref}\) & F2 & **71** & **96** & **83** & 72 & **96** & **85** & 93.5 \\
**Inv05** & No & F1 & **83** & **94** & **67** & _57_ & **95** & **75** & 90.3 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Covariance weighting (_Cd_ is defined based on Equations (4) and (5), ’No’ means that no specific additional weighting is applied whereas \(\chi^{2}_{ref}\) means that the misfit is scaled by the misfit of the null model), geometry of the best model (for families definition see Section 5.1.1) and % of Explained data (see Equation (9)) computed for each interferogram spanning the eruption. Numbers in bold are for data used in the inversion, values in small italics are computed a posteriori for data not used in the inversion. Inv01 was computed with InSAR data S1 A2, which does not cover the pre-eruptive period. Inv 01 to 03 were computed without including the GNSS data. Inv02a* is the preferred model presented in [PERSON] et al. [16].