## 4 Magma Source Modeling
When compared to the results of [20], our modeled source is at a reasonable depth but slightly deeper than that of the previous study. The volume change in the magma source differed significantly from the estimate of the previous study. Assuming a uniform dislocation from the overall rectangular magma source, the estimated volume increase is approximately 6.03 \(\times\) 10\({}^{6}\) m\({}^{3}\) using the width, length, and opening model parameters. This value is almost half the size of the volume change in the previous study [20]. Based on the difference in volume changes, we confirmed that the scale of the previous intrusion was much larger than that of the recent event. This finding is further supported by the different durations of the intrusions. The previous intrusion lasted for approximately six months (28 March to 6 October 2006) [20], whereas this minor event lasted only approximately two weeks. \begin{table}
\begin{tabular}{c c c c c c c c c c c} \hline \hline  & & & & & & & & & & & \\ \hline
**Width** & **Length** & **Depth** & **Dip** & **Strike** & **Pos-X*** & **Pos-Y*** & **Strike-Slip** & **Dip-Slip** & **Opening** \\
**(km)** & **(km)** & **(km)** & **(deg.)** & **(deg.)** & **(log.)** & **(lon.)** & **(lat.)** & **(x10\({}^{\text{-}\text{43}}\) km)** & **(x10\({}^{\text{-}\text{43}}\) km)** & **(x10\({}^{\text{-}\text{43}}\) km)** \\ \hline
4.019 \(\pm\) & 1.612 \(\pm\) & 4.247 \(\pm\) & 4.757 \(\pm\) & 67.475 \(-\)155.287 \(\pm\) & 19.382 \(\pm\) & 0.0165 \(\pm\) & 0.0055 \(\pm\) & 0.9314 \(\pm\) \\
0.08 & 0.16 & 0.05 & 0.41 \(\pm\) & 1.02 & 7.31e\({}^{\text{-}\text{44}}\) & 0.13 & 9.24e\({}^{\text{-}\text{46}}\) & 1.24e\({}^{\text{-}\text{48}}\) & 9.56e\({}^{\text{-}\text{48}}\) \\ \hline \hline \end{tabular}
* The ground coordinates at the origin (top left-hand corner) of the X and Y axes are \(-\)155.35208 \({}^{\text{*}}\)and 19.44708 \({}^{\text{*}}\), respectively. ** The angle between the model and a horizontal plane (0 to 90).  
\end{table} 
Table 2: The best-fit model parameters of Kálaea’s magma source using the Okada dislocation model. Figure 5: Inversion results of the rectangular dislocation (Okada) model for Kálaea’s summit area using 3D measurements. The first row shows the (**a**) observed, (**b**) synthetic, and (**c**) residual deformation of the east component. The second row shows the (**d**) observed (**e**) synthetic, and (**f**) residual deformation of the north component. The third row shows the (**g**) observed, (**h**) synthetic, and (**i**) residual deformation of the up component. Based on the dislocation model, the best-fit X and Y positions of the magma source were approximately \(-155.287\) and \(19.382\) degrees in the ground coordinates, which differ slightly from the results in the previous study (\(-155.29\) degrees longitude and \(19.38\) degrees latitude) [20]. As observed in the histograms of the model parameters in the Okada model (Figure 6), the X and Y positions (longitude and latitude coordinates) did not simply converge on a unique value. In particular, the parameter of the Y-position featured a multimodal histogram with relatively large uncertainties. The length and opening parameters also showed multiple peaks with large standard deviations. Although several parameters did not converge on a unique solution due to relatively large uncertainties, the 3D measurements provided an improved magma source model relative to the InSAR measurements by reducing biases. The enhancement of the parameter estimation for the magma source model using 3D measurements was demonstrated by [17]. Figure 7 shows the inversion results of the prolate spheroid model from the 3D displacement maps. In Figure 7, the left column is the inputs of east, north, and up components, which are the same as the dislocation model. The center and right columns are synthetic and residual displacements based on the spheroid model, respectively.