#### 2.1. The Vector Inclination Method

The method proposed by [

31]—here named the vector inclination method (VIM)—starts with the assumption that (i) a point on the surface of a sliding soil or rock mass will move in a direction that is parallel to the slope of the sliding surface beneath, provided (ii) the mass moves as a rigid body. A third assumption is that (iii) a single sliding surface exists. This means that theoretically VIM has a limited or null applicability to flow-type landslides where the movement vectors vary with depth and the superficial displacement does not exactly reflect the movements on the basal plane. To use this method a displacement monitoring must be implemented and measurement points (MPs), taken along a cross-section of choice, must be available. Then, the method consists of (see

Figure 1a):

drawing the cross-section of the landslide intersecting the MPs and marking with arrows the vectors of movement (V_{i}) measured at each point; note that the first vector should be representative of the movement close to the main scarp.

drawing the normals to each vector and finding the intersections between two consecutive normals (O_{i});

drawing the bisection lines between two consecutive normals (blue dotted line in

Figure 1a);

drawing a line parallel to the first movement vector (V_{1}), beginning from the back scarp and ending at the intersection with the first bisection line, and finding P_{1}, which represent the first point of the sliding surface.

a second line, parallel to the second vector (V_{2}) is drawn from P_{1} to P_{2}. This procedure is repeated for each movement vector;

the whole procedure is then repeated starting from the toe and working upslope to find a second sliding surface to be interpolated with the first one.

In [

32], an alternative graphical solution, which provides the same results but with circular interpolation, was suggested (

Figure 1b). The first two steps of the procedure are the same as above but, instead of drawing the bisection line, a circle is centered on O1 and drawn passing through P1 (blue dotted circle in

Figure 1b); then, a second circle, centered on O2 and passing through P2 is drawn (magenta dotted circle in

Figure 1b) and so on, for every MP, until the LSS is obtained (red curve in

Figure 1b).

Since it is a graphical method, there is no way to precisely compute the margin of variability due to error, which is mostly due to graphical limitations and, ultimately, to the scale. Therefore, we can estimate a vertical error bar on the location of the LSS as less than 1% of the total length of the landslide section. The trueness of the LSS, that is, the divergence between the real and the calculated LSS, is impossible to calculate because it can vary along the body of the landslide, and depends on the quality and representativeness of the monitoring data (and not only by VIM). Furthermore, eventually, because the real LSS is unknown, it is, at best, the result of a reconstruction, the accuracy of which is often undetermined.

In the following section we applied this approach to a number of landslides. Since the inclination of the vectors is crucial, we have been careful to maintain a 1:1 ratio between vertical and horizontal distance when calculating the LSS on the different cross sections (or else the vectors inclinations should be modified accordingly). As this method is completely graphical, the slope profiles and the movement vectors are the only data required to reproduce our results.

#### 2.2. Satellite Interferometry

The proposed methodology begins with the generation of a deformation map for the selected landslide areas. The ground deformation maps are derived by analyzing large stacks of SAR (Synthetic Aperture Radar) satellites images by means of differential radar interferometry, a well-established remote sensing technique that exploits the phase shift of the back-scattered electromagnetic signal between two or more coregistered images, acquired over the same target area. The returning phase values contain a number of terms, among which are contributions related to displacement, decorrelation, stereoscopic effects, and atmospheric artefacts. The main idea behind any interferometric approach is based on isolating the phase term that is actually related to a variation of the sensor-to-ground path length.

Over the study areas, ESA (European Space Agency) C-band ERS, Envisat and Sentinel-1 (center frequency 5.4 GHz, wavelength 5.6 cm) satellite data in ascending and descending geometries were acquired and processed using different multi-interferometric approaches, in order to obtain comprehensive views of the deformation field of the landslides and to make it possible to infer key features concerning its mechanism and behavior. Historical ERS (1992–2000) and recent Envisat (2003–2010) datasets have been processed in the framework of the Special Plan of Remote Sensing of the Environment (

Piano Straordinario di Telerilevamento Ambientale, PST-A). This project, led by the Italian Ministry of the Environment, was aimed at creating a set of layers, among which is a database of ground deformation measurements over Italy by means of multi-interferometric processing [

37], to assess the stability evolution of the territory. ERS and Envisat images, acquired every 35 days with an incidence angle of 23° and with a ground resolution of 25 × 5 m, were processed using the PSP (Persistent Scatterer Pairs) [

38], PSInSAR [

39], and SqueeSAR [

40] approaches, some of the most advanced techniques specifically implemented for the processing of multitemporal radar imagery. The current Sentinel-1 constellation (2014) operates in the TOPS (Terrain Observation with Progressive Scans; [

41]) imaging mode with an incidence angle ranging from 36° to 40°. Developed within the Copernicus initiative, the Sentinel-1 mission [

42] is a constellation of two twin satellites, Sentinel-1A and Sentinel-1B. Launched in April 2014 and in April 2016, respectively, they share the same orbital plane and offer an effective revisiting time of six days (12 days for each single sensor), which is extremely suitable for interferometric applications. With respect to previous SAR satellites: Sentinel-1 data exhibit some favorable characteristics: regional-scale mapping capability, systematic and regular SAR observations and rapid product delivery. Interferometric wide-swath Sentinel-1 images, featuring a spatial resolution of 4 × 14 m, were processed by means of the SqueeSAR algorithm.

The PSInSAR approach [

39] is the first method implemented for the processing of temporal series of coregistered SAR images acquired over the same target area. The main idea behind PSInSAR is to identify point-like targets, PS (Permanent Scatterers). PS are radar-bright, point-like targets with a dominant signal-to-noise ratio within the radar image, such as pixels corresponding to man-made objects, outcropping rocks, debris areas or buildings, which register a stable radar signal across the whole interferometric stack. The PSP algorithm [

38] is based on the identification and analysis of PS working only with pairs of points (“arcs”). By comparing the SAR signal at a given point only with the signal in other points close to it, the PSP approach makes the method insensitive to any spatially correlated signals, such as atmospheric or orbit phase contributions. PSP is able to identify PS in natural terrain and PS characterized by non-linear movements. The SqueeSAR [

40] approach, a second-generation PSInSAR algorithm, allows for the identification of both point-wise coherent scatterers (i.e., PS) and partially coherent Distributed Scatterers (DS). DS are statistically homogeneous groups of pixels sharing similar reflectivity. DS correspond to homogeneous ground surfaces, such as bare soil, deserts, debris-covered slopes and scattered outcrops. Through multi-interferometric approaches (Multi-temporal Interferometric Synthetic Aperture Radar, MT-InSAR), the LOS (Line Of Sight) deformation rate can be estimated with an accuracy theoretically lower than 0.1 mm/y, at least for very stable PS over a long-time span [

43]. PSI analysis is designed to generate time-series of ground deformations for individual reflectors. The accuracy of a single measurement in correspondence to each SAR acquisition ranges from 1 to 3 mm [

43]. Displacements are calculated relative to a stable PS within the frame and to a unique reference image.

InSAR-based displacements are one-dimensional measurements and, so, the resulting datasets can estimate only a percentage of the three-dimensional real motion of a landslide (i.e., the component corresponding to the projection along the satellite LOS). Under the assumption of absence of N-S deformation components, the availability of two different orbits of acquisition gives the opportunity to calculate the E-W and vertical components of the true movement vector.

A procedure to decompose satellite InSAR data has been described in [

44,

45]. The entire dataset is resampled into grids (with cell size generally ranging from 75 × 75 m to 50 × 50 m). Within each cell, the averaged ascending (V

_{A}) and averaged descending (V

_{D}) velocities are combined to derive the vertical (V

_{V}) and East-West (V

_{E}) ground velocity components, by solving the following formulas (Equation (3)):

where θ

_{A} is the LOS incidence angle in ascending orbit and θ

_{D} the LOS incidence angle in descending orbit.

In this paper, we use the vertical and horizontal displacement vectors measured from the satellite to apply VIM. The cross-sections were selected to be possibly aligned with the direction of the expected movement (i.e., the steepest slope) and, at the same time, to intercept the highest number of MPs. This means that, in general, the cross-section is not aligned E-W (as with the movement vectors); therefore, the E-W movement vectors were reprojected along the cross-section, considering that the maximum movement is expected along the steepest slope direction. It is useful to remember the first assumption made in

Section 2.1, i.e., the mass will move in a direction that is parallel to the slope of the sliding surface beneath; in fact, since the horizontal movements are reprojected along the cross-section where the maximum movement is expected, if a landslide experiences radial movements (i.e., the sides of a landslide diverge from the coaxial displacement either because they spread laterally or are dragged towards the central section, such as in [

46,

47], respectively), when these will be reprojected along the coaxial cross-section, their horizontal component will be underestimated with respect to the vertical one.