## 3 Results and Discussion
### Grimsv\(\ddot{o}\)tn Volcano, Iceland  
Grimsv\(\ddot{o}\)tn is located in Iceland approximately 200 m below the northwestern portion of the Vatnajokull icecap (See Figure 10). This volcano is among the most active in Iceland. It has erupted 29 times over the last 211 years, where five have occurred within the last 100 years (e.g., 1938, 1983, 1998, 2004, and 2011) [62]. Its 2011 eruption began on 21 May 2011, nearly 13 months after the now infamous 2010 Eyjafjallajokull eruption. Like its predecessor, it produced large ash clouds that extended 20 km into the atmosphere and disrupted European air traffic for several days [67].  
Since Grimsv\(\ddot{o}\)tn is a subglacial volcano, GPS data is used to identify deformation consistent with a magmatic intrusion. GPS data acquired by Icelandic Meteorological Office stations posted on the Icelandic Institute of Earth Science (IES) website was used to estimate the source parameters for this episode [68,69]. The GPS stations (GRIM and GFUM) are located about 3 km from the center of the Grimsv\(\ddot{o}\)tn caldera. Both sensors show continuous uplift at the site following the 2004 eruption, which increases at an almost linear rate between 2005 and 2011. This same pattern of activity was also observed for the five years before the 2004 eruption.  
Surface deformation was modeled using a spherical source in a semi-infinite elastic half space. The vertical and radial displacements at the surface are given by  
\[\Delta h(r) = C\frac{d}{(r^{2}+d^{2})^{\frac{3}{2}}} \tag{26}\] \[\Delta r(r) = C\frac{r}{(r^{2}+d^{2})^{\frac{3}{2}}} \tag{27}\]  
Figure 10: Location of the Grimsv\(\ddot{o}\)tn Volcano (red triangle).  
where \(r\) is the radial distance from the source, \(d\) is the source depth, and \(C\) is a scale factor [70]. The intrusion depth was determined by taking the ratio of Equations 26 and 27 and solving for \(d\), which is computed via \((\Delta h(r)/\Delta r(r))r=d\). Vertical and horizontal displacements are measured directly from the GPS records and the location of the intrusion is assumed to reside beneath the caldera center (_i.e._, \(r=3.0\) km). Resulting source parameters derived eight months before and one month after the 2011 eruption are shown in Table 4. The analysis techniques and assumptions used to determined the source parameters for this episode are similar to those used to study the 1998 eruption [25].  
The vent location search area is defined using Equation 26 and the pre-eruptive source parameters shown in Table 4. Resulting deformation profiles are shown in Figure 11. Based on the modeling results, a search radius of 17 km is used to constrain the vent location process, which is the point where the displacement profiles begin to approach zero.  
\begin{table}
\begin{tabular}{l c c c c} \hline \hline
**Sample** & \(\Delta\)**h** & \(\Delta\)**r** & **d** & **C** \\ \hline Pre-eruption & 40 \(mm\) & 36 \(mm\) & 3.33 \(km\) & 0.0011 \(km^{3}\) \\ Post-eruption & 250 \(mm\) & 468 \(mm\) & 1.60 \(km\) & \(-\)0.0061 \(km^{3}\) \\ \hline \hline \end{tabular}
\end{table}
Table 4: Estimated Mogi source parameters derived from GFUM 240 days before and 30 days after the 2011 Grimsvötn Eruption, where positive or negative values of C indicate uplift or subsidence.  
Figure 11: Modeled deformation field preceding the 2011 Grimsvötn eruption.