## 4 Magma Source Modeling
When the results of the dislocation model are compared to those of the spheroid model, the residuals were small, demonstrating that the magma model is accurately estimated by the spheroid model. The RMS estimates of the residual displacements were 0.81, 0.79, and 0.41 cm  
Figure 6: Histograms of the magma source parameters estimated from the 3D measurement inputs using the Okada dislocation model. for the east, north, and up components, respectively. Among the 3D components, the residuals of the north component notably decreased by approximately 23%. Therefore, the spheroid magma source matches the 3D measurements and provides the most realistic magma model for the recent magma intrusion. Similar to the north component, the residuals of the east and up components were lower in the spheroid model than in the dislocation model. The residual to observation ratios for the east, north, and up components were approximately 0.45, 0.37, and 0.13, respectively. Figure 7: Inversion results of the prolate spheroid model for the magma source below Kflavae’s summit 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. The best-fit parameters of the spheroid model are summarized in Table 3. The magma source's X- and Y-positions, depth, pressure, major and minor axes, strike, and plunge were determined from the 3D-based modeling. The magma pressure estimated from the modeling has been converted to a volumetric change using the conversion equation in [27]. Although the X- and Y-positions of the magma source are similar to the best-fit model parameters of the dislocation model, _i.e._, \(-155.288\) degrees longitude and \(19.388\) degrees latitude, the depth of the source, approximately \(2.81\) km below the surface, is quite different. The strike and plunge of the source were estimated to be approximately N64E and \(8.3\) degrees, respectively. The orientation angles of the two models are similar, differing by only approximately three degrees. The reservoir's volume change due to magma intrusion was calculated to be \(5.20\times 10^{6}\) m\({}^{3}\) based on the estimated pressure change. Figure 8 shows the histograms of the parameter estimations using a spheroid model. Ten thousand iterations were performed to determine the best-fit model parameters. Unlike the simulation results of the dislocation model, all the parameters converge on unique values with very small standard deviations. Therefore, the spheroid model is in good agreement with the magma source. In particular, the source position and orientation parameters are clearly determined by the spheroid model, unlike the parameters in the Okada model that exhibited multimodal histograms. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline  & & & & & & \\ \hline
**Pos-X*** & **Pos-Y*** & **Depth** & **Volume Change** & **Major** & **Minor** & **Strike** & **Plunge** \\ (**lon.)** & **(lat.)** & **(km)** & **(\(\times 10^{6}\) m\({}^{3}\))** & **Axis (km)** & **Axis (km)** & **(deg.)** & **(deg.)** \\ \hline \(-155.288\pm\) & \(19.388\pm\) & \(2.813\pm\) & & \(3.142\pm\) & \(2.139\pm\) & \(64.265\pm\) & \(8.306\pm\) \\ \(1.042\)e\({}^{-04}\) & \(5.75\)e\({}^{-05}\) & \(0.012\) & & \(0.044\) & \(0.048\) & \(0.603\) & \(0.464\) \\ \hline \hline \end{tabular}  
* The ground coordinates at the origin (top left-hand corner) of X and Y axes are \(-155.35208\) and \(19.44708\) degrees, respectively.  
\end{table} 
Table 3: The best-fit prolate spheroid model parameters of Kilauea’s magma source. Figure 8: Histograms of the magma source parameters estimated from the 3D measurement inputs using prolate spheroid model. Based on the previous studies, at least three to four distinct magma sources are responsible for inducing uplift or deflation on Kflavaea [7, 16, 20, 22].