# The Survey of Lava Tube Distribution in Jeju Island by Multi-Source Data Fusion
## 3 Methods  
Figure 2 shows the data processing flow to detect LTDPs combining InSAR time series analysis, spatial analysis, and machine learning methods with subsequent interpretation. Since expected deformation caused by undiscovered lava tubes is supposedly small and highly localized among the large InSAR observational coverage, the strategy in this study was to perform investigations based on multiple clues. InSAR time-series analysis, so-called Persistent Scatterers (PS), was the core component in the strategy. Persistent Scatterers were used to monitor the temporal migration on specific scatterers, such as crossing points of roads and lava tubes [50; 51]. Figure 2: (**a**) Overall processing flow to define LTDP and (**b**) PS processor. Regarding the implementation, PS analysis first extracted all pointwise deformations. Then the PS observation points were pre-filtered by the extracted mask through spatial analysis. Classification by machine learning algorithms was then employed to classify all pre-filtered PS points into several categories. The results were interpreted to identify LTDP candidates over undiscovered lava tubes. The PS method exploits strong scatterers in time series interferograms, which are paired by a common master image and a sufficient number of slave images. The observations and error component terms of InSAR to be handled in algorithms of time series analyses are expressed as follows:  
\[\Delta\varphi_{obs}=\Delta\varphi_{Dint}+\Delta\varphi_{atm}+\Delta\varphi_{orb} +\Delta\varphi_{topo}+\Delta\varphi_{ion}+\varphi_{noise} \tag{1}\]  
\[\Delta\varphi_{Dint}=\Delta\varphi_{int}-\varphi_{topo} \tag{2}\]  
where \(\Delta\varphi_{obs}\) is the phase difference of interferogram, \(\Delta\varphi_{Dint}\) is the phase difference by target topography only, \(\Delta\varphi_{orb}\) is the phase difference by inaccurate orbital information, \(\Delta\varphi_{topo}\) is the phase difference by inaccurate base topography, \(\varphi_{noise}\) is the other phase noise, \(\Delta\varphi_{atm}\) is the phase difference by atmospheric phase components, and \(\Delta\varphi_{ion}\) is the phase difference caused by delay in the ionosphere. In different time series InSAR networking methods, for instance, a master to multi-slaves in PS and multi-masters to multi-slaves in small baseline subsets (SBAS) technique [52], the error components were interpolated and canceled accordingly. In this study, we aim to precisely trace individual scatterer's behavior; thus PS became the primary InSAR time series technique. The core idea of the PS algorithm is to discriminate Persistent scatterers with constant responses for amplitude dispersion and to address error estimations using iterative non-linear equations. PS processing was initiated by a master to multi-slave interferogram network construction. Stable PS observations were then taken for the base of network inversion. The network was inverted together with the filter-out of error components, called atmospheric phase screen (APS) in each slave to master pair, which is mainly induced by \(\Delta\varphi_{atm}\) and \(\Delta\varphi_{topo}\) components as described in Equation (1). The APS canceling procedure was iteratively applied by an increasing number of PS points up to the final estimation of \(\Delta\varphi_{Dint}\). Then, the deformations in topography in each PS can be driven on the condition that \(\varphi_{noise}\) is negligible. Although PS requires a large number of image stacks, the accuracy of InSAR deformation is up to 1 mm/year [53]. On the contrary, the PS algorithm often suffers from a lack of observation density and PS variants were often introduced. In this study, the target area has quite large areas' rocky/low vegetation land cover with stable rock scatterers. Therefore, we were able to employ the conventional PS algorithm to observe the temporal migration with high precision. The density of extracted scatterers was also observed to be sufficient for further interpretation. The PS procedure applied in this study is summarized in Figure 2b.