Even though the reliable integration path guide derived in the previous section is implemented for unwrapping, unwrapping errors can still occur, especially in isolated components. Here, I detect and correct unwrapping errors using residuals from multiple interferograms.
As the three-dimensional (3D) ground deformation included in all interferograms is almost common, the projected LOS displacement in different interferograms with similar LOS directions should be comparable. In this case study, two interferograms observed from the west (i.e., descending left-looking and ascending right-looking) should have comparable LOS displacement, along with the two interferograms from the east (i.e., ascending left-looking and descending right-looking). While the contribution of the difference in the incidence and azimuth angles and atmospheric noise exists, it should be mostly smaller than the unwrapping error (~ 12 cm in this case), or smooth signals must exist that are distinguishable from the unwrapping errors. The difference (residual) of the preliminary unwrapped phases between the two interferograms clearly exhibits unwrapping errors in few isolated components (Fig. 8a, c). SNAPHU can output a connected component file that contains identification numbers for each isolated component (Chen and Zebker 2002; Yunjun et al. 2019). The unwrapping errors can be corrected by manually adding ± 2π × n into the components with suspected unwrapping errors to produce small and smooth residual (Fig. 8b, d). As a result, reliable unwrapped phases are obtained (Fig. 9). Some components in the caldera floor (the lower right part in the image) are not corrected, because they are irrelevant to the displacement lineaments.
Even if left-looking observations are not available, different from this study, this unwrapping error detection approach is feasible if multiple independent interferograms containing comparable LOS displacements are available (e.g., in adjacent paths with different incidence angles, or with the same path and incidence angle but on different acquisition dates). Redundancy increases the reliability of unwrapping results.
Another possible approach to detect the unwrapping error is to implement residuals from 3D decomposition utilizing four different LOS directions (Morishita et al. 2016). An unwrapping error (~ 12 cm in L-band) in an interferogram leads to more than a few centimeters residuals in all interferograms (Fig. 10a–d), while ~ 2 cm smooth residuals remain because of independent atmospheric noise. After correcting for the unwrapping errors, the characteristic residuals disappeared (Fig. 10e–h).
While the residuals highlight the candidates of interferograms and components therein with unwrapping errors, the exact interferogram including the error cannot be uniquely identified from the residuals and hence needs to be empirically determined, for example, from phase continuity from surrounding pixels.
If the same unwrapping errors are accidentally included in the same components of multiple interferograms from similar LOS directions (either just after automatic unwrapping processing or by a false unwrapping error correction), the errors are canceled out and remain undetected in the residual. However, in that case, decomposed (i.e., horizontal and vertical) displacements using different LOS directions directly reflects the errors and could be a clue to detect the errors. An unwrapping error of 12 cm in the LOS direction would result in an error of at least ~ 8 cm in the east–west or up–down directions depending on the incidence angle, although it might be difficult to distinguish this type of error from the true displacements.
Similar approaches to detect and correct unwrapping errors exploiting loop phase closure were developed (Biggs et al. 2007; Yunjun et al. 2019; Morishita et al. 2020; Oliver-Cabrera et al. 2021). The difference of the proposed approach from the above-mentioned approaches is the exploiting interferograms observed from different LOS directions (i.e., containing different LOS displacements projected from the same displacement field). The identical LOS displacements in multiple interferograms lead to similar difficulties in unwrapping and, therefore, may result in the same unwrapping errors which are undetectable. However, different LOS displacements in multiple interferograms bring different difficulties in unwrapping (i.e., success in an interferogram, failure in another one), enhancing the chance to detect and correct the unwrapping errors induced from complicated deformation.