## 4 Magma Source Modeling  
To define the magma source inducing the surface deformation at KAlauea's summit for the recent intrusion in April-May 2015, we applied the prolate spheroid model ([PERSON] model) [24] and rectangular dislocation model (Okada model) [26], assuming an isotropic, homogeneous, and elastic half-space condition. We assumed a Poisson's ratio of 0.25. To determine the optimal parameters of the magma source, we performed ten thousand simulations via Monte Carlo optimization. All model parameters were not fixed, and each of them was determined between the boundary conditions. Among the total of over 400,000 pixels within the deformation maps, 7% from randomly selected samples  
Figure 4: The horizontal displacement vector field generated from the east and north components of surface deformation. The arrows indicate the magnitude and directions of the horizontal displacements, and the colored map represents the vertical displacements. Figure 3: 3D surface displacement maps retrieved from integrating multi-stacked InSAR and MAI measurements: (**a**) east, (**b**) north, and (**c**) up components of surface deformation. Positive signals represent eastward, northward, and upward movements. were used for the model parameter estimation in each iteration. The upper and lower boundaries of the model parameters were broadly chosen by referencing previous modeling from other studies. The optimum model parameters of the magma source were determined as the model with the minimum misfit. Based on the 3D measurements of surface deformation, we expect that the magma source inducing this minor event should be appropriately modeled by a spheroid model. However, we exploited the dislocation model as well, in order to compare the two modeled results. Additionally, a previous study simulated a magma source using the dislocation model for deformation measured in almost the same position as our observations [20]. Among the four distinct magma sources below the Kilauea summit, the deformation patterns and location in this study are quite similar to the surface deformation observed during the period of March 2006 to October 2006 [20]. Figure 5 shows the inversion results of the dislocation model using 3D displacement maps, reporting the observed (left), modeled (center), and residual (right) displacements for the east, north, and up components. The 3D components were jointly modeled to determine the parameters of the magma source inducing the surface changes. Based on the residual displacement maps in Figure 5, a few centimeters of residuals were observed in the east, north, and up components. The root-mean-square (RMS) values for the residual displacements were 0.88, 1.03, and 0.54 cm for the east, north, and up components, respectively. Based on the RMS estimation, the residuals of the north component are slightly larger than those of the east and up components, but the RMS ratios between the residual and observed displacements of the east and north components were similar, with values of 0.49. The residual of the observation ratio for the up component (0.18) was the smallest among the 3D components. In other words, much of the noise might be introduced by measuring the horizontal displacements. Since the up component is only relevant to the InSAR measurement, we infer that the MAI observation introduced the majority of the errors. The best-fit parameters based on the dislocation model are summarized in Table 2. Based on the results of the 3D-based modeling, the estimated magma source has a dimension of approximately 4.02 km in width and 1.61 km in length. The orientation of this rectangular source is approximately N67E, and the dip angle is small enough to be considered horizontal. The depth of the magma reservoir is approximately 4.25 km below the surface. [20] estimated the depth of this deep source to be 2.9-4.2 km at the 95% confidence level, and their best-fit estimate of depth was approximately 3.6 km.