## 2 Study Area & Data Sets
### InSAR & MAI Data Processing  
In this study, the measurement of KAlauea's summit inflation was performed by co-eruptive interferometric pairs of COSMO-SkyMed acquired from ascending and descending orbits. Although the X-band SAR is limited by high temporal decorrelation due to its short wavelength, the COSMO-SkyMed SAR system is able to maximize the temporal resolution with its four-satellite constellation. We used strip-map mode images, which have a resolution of approximately 3 m in range and azimuth directions and a single frame is 40 \(\times\) 40 km. Since KAlauea's summit and rift zones lie in a single image, the measurement of 3D displacements was conducted for the area of interest. In Figure 1, the region outlined in white delineates the area investigated using 3D surface deformation and the focus of the magma source modeling. Table 1 lists the interferometric pairs used in this study, with perpendicular (B\(\perp\)) and temporal (B\({}_{\rm t}\)) baselines. All the reference and repeat pass images were collected before and after the period of unrest (21 April to 17 May 2015) to measure the recent magma intrusion at KAlauea. From the data sets, eight and seven co-eruptive pairs were selected from ascending and descending orbits, respectively, to generate the conventional InSAR and MAI interferograms. Although most of the image pairs from the ascending path have perpendicular baselines shorter than 300 m, certain pairs from the descending path have baselines longer than 400 m because they preserved the interferometric coherence over 0.7 at the non-vegetated area. In temporal baseline criteria, image pairs with the time durations of less than four months were used for this study. As a first step in mapping 3D displacements of surface deformation, two forward-looking and two backward-looking single-look complex (SLC) images for the MAI processing and two full-aperture SLC images for the conventional InSAR processing were generated for each image pair. The Doppler center of each full-aperture SLC image was determined by averaging Doppler centroids of the reference and repeat path SAR images. The center frequency of the sub-aperture SLC images were determined by the fraction factor, which is the normalized squint [11,14]. The fraction 0.5 was applied in our MAI processing. During the generation of InSAR and MAI interferograms, sixteen looks in the range and azimuth dimensions were applied for noise reduction and enhancement of computational efficiency, respectively. Using the multi-looking process, the final MAI and InSAR interferograms have approximately 32 m pixel spacing, whereas the pixel spacing of the SLC image is approximately 2 m in the range and azimuth dimensions. As already demonstrated by previous studies, the accuracy of the MAI phase depends primarily on the interferometric coherence of the forward- and backward-looking interferograms. Interferometric coherence, in turn, can be improved through the multi-looking and phase filtering process of an interferogram. Thus, accurate along-track displacements were obtained by applying multi-looking and phase filtering to the sub-aperture interferograms before creating the final MAI interferogram. In addition, we used multi-temporal SAR acquisitions for stacking co-eruptive interferograms. Similar to the noise reduction by multi-looking process, stacking of multiple observations enabled us to measure the precise and accurate surface displacements in both line-of-sight (LOS) and along-track directions. The InSAR stacking was performed by averaging all the InSAR interferograms, whereas the MAI stacking was carried out according to the recently proposed optimal MAI stacking approach [16]. The topographic phase contributions were removed using the simulated interferogram from the SRTM digital elevation model (DEM). The DEM, with a resolution of 30 m, was interpolated to approximately 15 m to match the pixel spacing of the interferograms after the first multi-looking process.