## 2 LiCSAR and LiCSBAS
### Time Series Analysis
#### 2.4.6 Step 1-6: Spatiotemporal Filtering of Time Series  
The derived time series still includes some noise terms, including residual tropospheric noise, ionospheric noise, and orbital errors, which are, in general, spatially correlated and temporally uncorrelated. A spatiotemporal filter (i.e., high-pass in time and low-pass in space) can be applied to attempt to separate these noise components from the displacement time series [3,6]. To achieve  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Noise index** & **Meaning** \\ \hline coh\_avg & Average value of the interferometric coherence across the stack (0–1). \\ n\_unw & Number of unwrapped data used in the time series calculation. \\ vstd & Standard deviation of the velocity (mm/yr) estimated in Step 1-4. \\ maxTlen & Maximum time length of the connected network (years).The larger the value, the better the precision in the velocity estimate (see Section 3.5). \\ n\_gap & Number of gaps in the interferogram network and time series breaks. \\  & Spatiotemporal consistency (mm) [55], which is the minimum root mean \\ stc & square (RMS) of the double differences of the time series in space and \\  & time between the pixel of interest and an adjacent pixel among all \\  & adjacent pixels. \\ n\_ifg\_noloop & Number of interferograms with no loops that cannot be checked by the \\ n\_loop\_err & loop closure and possibly have unidentified unwrapping errors. \\ n\_loop\_err & Number of the unclosed loops after the network refinement in Step 1-2. \\ resid\_rms & RMS of residuals in the small baseline (SB) inversion (mm). \\ \hline \hline \end{tabular}
\end{table}
Table 1: List of the noise indices used for masking.  
Figure 3: Example of the masking of velocities in noisy pixels derived from the time series analysis based upon a suite of indices for the region of the Echigo Plain (Section 4). Masked and unmasked velocities and the mask itself are shown along the top row (along with the average coherence from the input interferometric data), and the other images denote further noise indices used to create the mask (explanations and units given in Table 1). Numbers shown in the parentheses next to the titles of each noise index are the applied threshold value.  
this filtering, we apply a one- and two-dimensional Gaussian kernel in time and space, respectively. Moreover, if the spatial scale of the target deformation signal is sufficiently small compared to the processing area (e.g., subsidence, landslides, or volcanic deformation), removing a ramp (linear, bilinear, or quadratic) from the entire area may help mitigate the noise.  
It is noteworthy that too-strong a spatiotemporal filter (i.e., with long and short filter widths in time and space, respectively) can remove not only the noise components but also the actual displacement signals of interest, especially for cases where the displacement time series is nonlinear and the temporal width of the filter is too long compared to the time scale of the nonlinear displacement [6]. On the other hand, the temporal filter width needs to be sufficiently long, compared to the sampling interval, or no effect is produced. These issues have made separating noise from nonlinear displacements using infrequent data provided by earlier InSAR satellites (e.g., Envisat and ERS) challenging. However, with Sentinel-1, we can apply a relatively short temporal filter width and potentially more easily detect real nonlinear displacement. In LiCSBAS Step 1-6, we use \(3\overline{dt}\) as the default temporal filter width (\(1\sigma\) of Gaussian), where \(\overline{dt}\) is the average temporal sampling interval. For example, if the average temporal baseline of interferograms is 12 days, a temporal filter width of 36 days is applied to the displacement time series, which is sufficiently short to highlight seasonal displacements associated with the hydrological cycle [56; 57].