# Combining InSAR and GNSS to Track Magma Transport at Bastic Volcano
## 5 Results
### Temporal Inversion  
#### 5.2.1 A Need for Geometrical a Priori to Invert for the GNSS Time Series  
We developed three methods of source parametrization to invert the whole GNSS time series. Taking advantage of the SAR acquisition during the eruptive crisis, we compared inversion results using either continuous GNSS data, InSAR data (S1 D1) or both datasets covering the first part of the eruption (until 1:45) with each method (Table 4). \begin{table}
\begin{tabular}{c c c c c c c} \hline  & **GNSS** & **S1 D** & **S1 A** & **CSKD** & **CSKA** & **Total** \\ \hline \(\chi^{2}_{ref}\) & X & 1100 & 500 & 1000 & 470 & 3070 \\ \(\chi^{2}_{ref}/\chi^{2}_{ref}\) \% & X & 36\% & 16\% & 33\% & 15\% & 100\% \\ after weighting & X & 25\% & 25\% & 25\% & 25\% & 100\% \\ \(\chi^{2}_{ref}\) & 4800 & 1100 & 500 & 1000 & 470 & 7870 \\ \(\chi^{2}_{ref}/\chi^{2}_{ref}\) \% & 61\% & 14\% & 6\% & 13\% & 6\% & 100\% \\ after weighting & 20\% & 20\% & 20\% & 20\% & 20\% & 100\% \\ \hline \end{tabular}
\end{table} 
Table 3: Values of \(\chi^{2}_{ref}\) Equation (6) and weight of each dataset in the computation of the misfit. The top three lines correspond to inversions conducted without the campaign and continous GNSS data, while the bottom three lines include the GNSS data. The Ellipse method does not have any a priori on the source location. The inversion of cumulated displacement from GNSS time series data leads to a very good fit (96%) of these data (Table 4). However, the fit on the S1 D1 interferogram falls to 22%. The horizontal location of the deformation source (Figure 9) is comparable to the source determined by the static inversion for the whole eruption. However, the source depth (Figure 9) and pressure (Table 4) are not accurately estimated because of a lack of GNSS stations close to the most deformed area. This leads to an overestimate of the pressure (Table 4) as well as the modeled displacement (Figure 10). The Projected Disk and Subgraph methods, which rely on the a priori geometry determined in the static inversion step, give similar pressurized areas (Figure 9) and fit of InSAR data (Table 4 and Figure 10) when using the F2 model family. The fit for the GNSS data (Table 4) amounts to a high percentage of explained data of \(\sim\)84% and \(\sim\)83% for the Projected Disk and the Subgraph methods, respectively. With both methods, the modeled displacement fields are compatible with the S1 D1 interferogram but the percentage of explained data is slightly higher with the Subgraph (83%) than with the Projected Disk method (77%). With respect to the InSAR data, GNSS data overestimate the overpressure and the intruded volumes, which is related to the GNSS network configuration and coverage as indicated by the inversion of GNSS data always overestimating displacements (Figure 10) in the area which is devoid of GNSS stations. However, these overestimations are not as high with the Subgraph method (36% and 50% on the overpressure and volume, respectively) as with the Projected Disk method ((84% and 45% on the overpressure and volume, respectively), probably because the pressurized area is determined in a more precise way, free of any projection, with the Subgraph method. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Method Data**} & \multicolumn{2}{c}{**Ellipse**} & \multicolumn{2}{c}{**Projected Disk**} & \multicolumn{2}{c}{**Subgraph**} \\ \cline{2-7}  & **GNSS** & **InSAR** & **GNSS** & **InSAR** & **GNSS** & **InSAR** \\ \hline
**Overpressure (MPa)** & 3.2 & 7.0 & 3.5 & 1.9 & 3.4 & 2.5 \\
**Average opening (m)** & 1.3 & 1.3 & 1.0 & 0.6 & 1.0 & 0.7 \\
**Area (10\({}^{6}\) m\({}^{2}\))** & 2.8 & 1.6 & 3.4 & 3.9 & 3.3 & 3.1 \\
**Volume (10\({}^{6}\) m\({}^{3}\))** & 3.6 & 2.1 & 3.5 & 2.4 & 3.3 & 2.2 \\
**GNSS** & **96** & _85_ & **84** & _83_ & **83** & _84_ \\
**InSAR S1 D1** & _22_ & **95** & _77_ & **96** & _83_ & **94** \\ \hline \hline \end{tabular}
\end{table}