Introduction

Landslides are natural hazards causing damage to infrastructure, economic losses, and can lead to loss of life (Froude and Petley 2018). Advanced differential interferometric synthetic aperture radar (A-DInSAR) techniques, e.g., PSI (persistent scatterer interferometry), have been proven to be useful for long-term monitoring of slow moving landslides with weekly to monthly sampling and mm-precision (Feretti et al. 2001; Zhao and Lu 2018; Solari et al. 2020). Regional (Raspini et al. 2018) and nationwide A-DInSAR datasets based on Sentinel-1 SAR data are publicly available in various regions and countries (Dehls 2017; Kalia et al. 2017, 2021). Continental-scale A-DInSAR datasets are provided by the European Ground Motion Service (Costantini et al. 2022). A review of these services is given by Crosetto et al. (2020). The frequency of the update of these datasets ranges from every consecutive Sentine-1 acquisition (Raspini et al. 2018) to yearly updates (Kalia et al. 2021; Constantini et al. 2022). Due to the regular updates, state-of-the-art A-DInSAR processing, and open data policy of these services, operational information products for specific applications like landslide hazard assessment become possible.

The use of the large A-DInSAR datasets is often done by visual inspection of the velocity field and the time series. The huge amount of measurement points (e.g., persistent scatterer, PS) makes visual interpretation, subjective, time-consuming, and error prone due to outliers. The question arises how spatial and temporal patterns of the PSI velocity and time series can be detected in a semi-automatic way. Therefore, several InSAR post-processing techniques have been proposed by the scientific community, e.g., based on a sequential series of statistical tests to classify PS time series into pre-defined classes (Berti et al. 2013) or machine learning to estimate the probability of accelerations/decelerations induced by slope instability and subsidence (Confuorto et al. 2022). Chaussard et al. (2014) use temporal mode principal component analysis (PCA) to compute a set of uncorrelated principal components ranked by the percentage of variance explained and constrained by an orthogonal basis, which effectively captures variance but may overlook and mix some data trends. Cohen-Waeber et al. (2018) use independent component analysis (ICA) to maximize the statistical independence of an arbitrary number of independent components. Other techniques include, e.g., the inverse velocity approach to estimate the time of slope failure (Carlà et al. 2019), wavelet analysis to quantify and correlate (intermittent) periodical signals (Haghshenas and Motagh 2016, Tomás et al. 2016; Liu et al. 2022), and spatial clustering to identify clusters of deforming PS and/or distributed scatterers (Lu et al. 2012; Barra et al. 2017; Xi 2017; Kalia 2018). However, the spatial and temporal characteristics of A-DInSAR datasets vary by orders of magnitude (depending on the number of acquisitions, the spatial measurement density), which can have an influence on the ability of these post-processing techniques regarding the extraction of meaningful information. Thus, the scalability of three post-processing techniques (Berti et al. 2013; Barra et al. 2017; Liu et al. 2022) are assessed in this work. Therefore, all three post-processing techniques are applied and parameterized using a Sentinel-1 PSI dataset from descending track 139 which is processed in the framework of the Ground Motion Service Germany. Subsequently, the same techniques and parameters are evaluated on a second independent Sentinel-1 PSI dataset from descending track 37, which is also processed in the framework of the Ground Motion Service Germany.

The rationale for the use of these three post-processing techniques is as follows. First, clusters of deforming PSs are detected to focus the attention of an end-user to specific areas of the PSI dataset characterized by a high reliability. Second, an exemplarily chosen PS time series, within this specific area, is analyzed regarding a time-lag to a potential triggering factor. The climatic water balance (climatic water balance = precipitation − potential evapotranspiration, Thornthwaite and Mather 1957) is used as potential triggering factor, instead of the often used precipitation, because the deep seated sliding surface is approximately 100 m below the Earth’s surface in the investigated landslides. Thus, it is hypothesized that rainfall water can reach the sliding surface (and cause an acceleration of the deformation by decreasing the friction coefficient at the sliding surface) only in times with high precipitation and low evapotranspiration.

Sentinel-1 PSI dataset and study area

Sentinel-1 PSI dataset

This study is based on wide-area Sentinel-1 PSI datasets from two tracks of the Ground Motion Service Germany (Kalia et al. 2021). The PSI data used in this study is part of the 3rd update of the German Ground Motion Service (released on 12.04.2022). The PSI technique (Feretti et al. 2001) belongs to the group of advanced differential interferometric stacking techniques. The PSI technique uses a stack of wrapped interferograms, all referring to a single reference image, to estimate the residual ellipsoidal height, deformation rate, and time series for coherent pixels (persistent scatterer, PS). The estimates are relative to a reference point and surface deformations are measured in the satellites line-of-sight (LoS) direction. The wide-area Sentinel-1 PSI processing is performed by a modified version of the PSI GENESIS processor (Adam et al. 2013; Goel et al. 2016; Adam 2019). The modifications include geodetic corrections, e.g., plate tectonic motion, solid earth tides (Rodriguez Gonzalez et al. 2018), and tropospheric phase mitigation by a simulation and subtraction of the tropospheric delay using the ECMWF (European Centre for Medium-Range Weather Forecasts) ERA-5 numerical weather forecast model (Adam 2019). The modifications are set-up that it is able to provide a high precision of deformation rate estimates, even over rural areas and large distances, e.g., more than 100 km (Parizzi et al. 2021). An improved version of the 3 arc-second SRTM C-band DEM (Wendleder et al. 2016) was used for topographic phase correction. After PSI processing a Global Navigation Satellite System (GNSS) calibration (Parizzi et al. 2020) was applied to tie the PSI results to the same geodetic reference frame. The GNSS velocities used for calibration are based on post-processed GNSS time series data from 243 continuous GNSS stations from the SAPOS network spread across Germany (Brockmeyer et al. 2021 unpublished). Table 1 shows the characteristics of the Sentinel-1 PSI data used in this study. Figure 1 shows the footprint of the processed Sentinel-1 PSI datasets used in this case study. The Sentinel-1 PSI dataset based on track 139 is used for parametrization of the post-processing techniques. Subsequently, the parameterized post-processing techniques are applied to the PSI dataset based on track 37 to assess the transferability of the post-processing techniques.

Table 1 Characteristics of the Sentinel-1 SAR datasets used for PSI processing
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Fig. 1
figure 1

Location of the processed Sentinel-1 PSI datasets from track 139 and 37 (A), location of study area and the Piesport and Trittenheim landslide (B). The red inset in A shows the extent of B. A digital elevation model serves as background

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Study area

The study area has a size of 30 km2 and is located at the river Moselle in Germany (Fig. 1). River Moselle flows roughly from south-west to north-east. River Moselle is meandering strongly through the Hunsrück Slate (Devonian strongly tectonically modified clay-slate, Rogall 2014) and fluvial erosion is producing undercut slopes. The undercut slopes are not in gravitational equilibrium. Very slow (1.6 m/a > landslide velocity > 16 mm/a) and extremely slow (< 16 mm/a) landslides with deep seated sliding surfaces and extremely rapid (5 m/s) rock falls are present in the study area (Cruden and Varnes 1996; Rogall 2014). The elevation range is from 78 m to more than 400 m above sea level. Two landslides are investigated in this case study, the “Piesport landslide” and the “Trittenheim landslide” (Fig. 1B). The exposition of these landslides varies from south, south-west (Piesport landslide) to north, north-west (Trittenheim landslide). Due to the heading angle of the satellite orbit, in some parts of the landslides, with an exposition approximately north–south the LoS sensitivity to measure a downslope motion is zero. The Sentinel-1 PSI dataset from track 139 shows three clusters of moving PSs (indicated by red arrows in Fig. 2A). These are also visible in the Sentinel-1 PSI dataset from track 37 (Fig. 2B). In the study area, 3993 PS are detected using track 139 and 4632 PS using track 37. PSI velocities range from −18.01 to +2.46 mm/a (track 139), resp. from −14.41 to + 3.08 mm/a (track 37).

Fig. 2
figure 2

Sentinel-1 PSI dataset from track 139 superimposed on the temporal mean amplitude image (A), Sentinel-1 PSI dataset from track 37 superimposed on an optical image (B), ESRI hybrid reference layer superimposed on Corine landcover 5 ha (C). A landslide map (Rogall 2014) is superimposed in A, B, and C

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Highest PS densities are in built-up areas, e.g., the villages Trittenheim, Neumagen-Dhron, Piesport, and Minheim (Fig. 2). Most of the landslide areas are covered by vineyards, intersected with small roads, traffic signs, and railings. The upper parts of the landslides are mostly covered by forest. At the lower part of the Piesport and at the top of the Trittenheim landslide, a village is located (Fig. 2C). This built-up infrastructure causes a coherent point scattering in the SAR time series. In consequence, these pixels are detected as PS during the PSI processing. In general, different objects and processes (e.g., railing, stonewall, soil creeping, sliding mass) can contribute to the reconstructed deformation time series of the PS phase center.

Figure 3 shows the PSI LoS velocity and an exemplary PS deformation time series from track 139 and 37 at the Trittenheim landslide. The estimated velocities and the time series pattern from track 139 and 37 are very similar. Due to the shallower incidence angle of track 37, the LoS sensitivity to measure a downslope vector is higher than from track 139. Both time series show an acceleration in the beginning of 2020 (Fig. 3).

Fig. 3
figure 3

Sentinel-1 PSI LoS velocity from track 139 (A) and track 37 (B). Corresponding deformation time series based on track 139 (C) and based on track 37 (D). The location of the deformation time series is indicated by the white arrows in A and B. An acceleration in the beginning of 2020 is visible in both deformation time series (black arrow in C and D)

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The Trittenheim landslide has a width of 500 m and a length of 1.700 m. The upper scarp has a height of up to 25 m with an inclination of 45–60°. The landslide type can be characterized as deep-seated continuous creep. The landslide can be differentiated into three sections: a rupture area, a middle deformation zone, and a landslide toe. The sliding surface is several tenth of meters below the Earth’s surface. At nearby similar landslides, the sliding surface was drilled at 55 and 65 m below the Earth’s surface. The subsurface of the landslide consists of the Hunsrück Slate. At the bottom of the landslide, a main road the so-called K86 is present. Several roads show distinct cracks, dry stone walls are tilted, and river Moselle has been narrowed by 25 m, indicating a landslide deformation over long time spans. In the 1990s, the Trittenheim Bridge has been reconstructed due to damages caused by deformation of the landslide. It has to be noted that no recent terrestrial measurements exist in this area, and land clearance activity alongside newly built-up roads did not show recent cracks and deformations; thus, the current state of activity is unknown to public authorities (Rogall 2014).

The Piesport landslide is one of the largest landslides in the Moselle Valley. It has a width of 2.700 m and a length of 700 m. The upper scarp has a height of 30–40 m and a steep slope with an inclination of 80°. The landslide type can be characterized as deep-seated continuous creep. In the eastern part of the Piesport landslide, distinct cracks and deformations of retaining walls and roads are observed (Rogall 2014). The deformation of the sliding mass in the eastern part has led to a narrowing of 30 m in this part of river Moselle. While active landslide deformations are confirmed in the eastern part, the western part is considered inactive (Rogall 2014).

Methodology

This paper uses three different methods to analyze PS deformation time series. Active deformation area (ADA) mapping described in the “Active deformation area mapping” section is used to identify significantly deforming clusters of PS time series. Then, a series of statistical tests, described in the “Time series classification” section, are applied to the PS time series to (i) identify an acceleration date and (ii) quantify a time-lag w.r.t. a potential landslide triggering factor. Finally, a wavelet analysis is performed to analyze correlation and time-lag between intermittent periodical signals of surface deformation and potential landslide triggering factor. All results are verified by a second independent Sentinel-1 PSI dataset, from another track of the Ground Motion Service Germany.

Active deformation area mapping

The mapping of active deformation areas (ADA) is proposed by Barra et al. (2017), Tomás et al. (2019), and Navarro et al. (2019). The ADA can be obtained by the software ADA Tools developed by Navarro et al. (2019). The rationale behind the ADA detection is that a loss of few information is accepted in order to (i) decrease the general noise level and (ii) increase the usability of InSAR stacking results. The ADA-mapping approach consists of two steps (i) ADA detection and (ii) ADA extraction.

The ADA detection starts with a filtering of the PS deformation time series regarding isolated, single PSs, noise PSs, and spatial outliers. Therefore, a priori defined thresholds regarding the maximum distance used to define a PS as isolated and the accepted noise-level are used. The spatial criteria is based on two conditions and uses a moving window around the PSs (e.g., twice the spatial resolution of the input SAR data). The first condition is based on the presence of at least one more PS within the search window. The second condition is that at least two PS are moving within the search window. The threshold used to classify a moving PS is based on the standard deviation (σ) of the deformation rate of all PSs. The initial parameters suggested by Barra et al. (2017) uses a threshold of 1.5σ. If the absolute value of the deformation rate is higher than the threshold, it is classified as “active”; otherwise, it is considered as “stable.” PS classified as “active” are the input for the next step, the ADA extraction.

The ADA extraction begins with the construction of a circular polygon around the “active” PS. Then the intersecting polygons are merged to form a spatial cluster of PS. If the cluster consists of a minimum number of PS, it is considered as a significant deformation area. For each extracted ADA, several attributes are included, e.g., the number of aggregated PS, mean, maximum and minimum deformation rate of the aggregated PS, and a quality index. The purpose of the quality index is to provide information on the reliability of each ADA. It is based on a combination of the spatial and temporal noise of the aggregated PS.

Time series classification

In order to semi-automatically extract information from the PS time series, a series of conditional sequence of statistical tests is performed using the method proposed by Berti et al. (2013). The approach classifies the PS time series into six a priori defined deformation classes: “uncorrelated,” “linear,” “quadratic,” “bilinear,” “discontinuous with constant velocity,” and “discontinuous with variable velocity.” In this case study, the focus is in particular to identify PS with an acceleration and to semi-automatically extract the date when the acceleration began. This time series pattern corresponds to the class “bilinear.” Given that the PS time series has a significant linear trend, a bilinear model is tested. Therefore, a changepoint regression is used. A changepoint regression, also known as piecewise or segmented regression, divides the time series into intervals and a separate linear regression is fitted for every interval (Main et al. 1999; Steven 2001). The segments are used to identify whether a change in the slope exists and when the change occurs. Based on the approach proposed by Main et al. (1999), the PS time series t1, …, tn is split into two intervals, divided by a breakpoint tb. The breakpoint is moved along the time series from b = 5 to b = n −5, with n = number of deformation measurement. Thus, a minimum of 5 deformation measurements are required to build an interval. This threshold has been chosen, to prevent very short intervals at the beginning or at the end of the PS time series. For every breakpoint, a two-line unconstrained model is fitted for the intervals t1, …, tb and tb+1, …, tn and the Bayesian information criterion (BIC, Schwarz 1978) is calculated to assess the goodness of fit:

$$\mathrm{BIC}({t}_{b})=\mathrm{ln}\left(\frac{\mathrm{RSS}}{n}\right)+\frac{\left(k+1\right)}{n}\mathrm{ln}\left(n\right)$$

where RSS is the residual sum of squares and k is the number of model parameters (in the case of a two-line regression k = 3). The BIC is also calculated for a single linear and a quadratic regression (BICL and BICQ, with k = 1 and k = 2). The BIC is used for model selection and uses a penalty term for the number of parameters in the model. By doing so, the BIC approaches overfitting by finding the best model that is fitting the data (low RSS value) using only a few parameters (low k). If the minimum value of BIC(tb) is smaller than BICL and BICQ, the bilinear regression outperforms the quadratic and linear regression (Berti et al. 2013). The date of the minimum value of BIC(tb) is the date of the breakpoint for the bilinear regression.

Wavelet analysis

In order to analyze potential (intermittent) periodical signals of the PS time series, a wavelet analysis is performed (Grinsted et al. 2004). As only the non-linear component of the PS time series is of interest in the wavelet analysis, the original PS time series is decomposed into two components (a linear and a non-linear component). The linear component is estimated by using a least-squares fitting. The residuals of this linear trend are assigned as the non-linear component.

First, a continuous wavelet transform is used because it is capable to detect periodical patterns in low SNR time series. The continuous wavelet coherence is calculated as follows (Grinsted et al. 2004):

$${W}_{t}\left(\tau , s\right)={\int }_{-\infty }^{\infty }x\left(t\right){\Psi }_{\tau ,s}^{*}\left(t\right)dt \tau ,s \in Rs\ne 0$$

where \(\Psi\) is the daughter wavelet, \({\Psi }^{*}\) is the complex conjugate of \(\Psi\), \(\tau\) is the translation parameter, and s is the scaling factor. The result of the continuous wavelet coherence is visualized as a 2D graph with the X-axis representing the date of the time series and the Y-axis representing the frequency of the periodical signals. High power values, visualized in the 2D graph, indicate the existence of significant periodical patterns at corresponding timespans.

In order to analyze potential periodical similarities between two time series, a cross wavelet transform and a wavelet coherence are calculated. The cross wavelet transform is calculated by multiplication of the first continuous wavelet transform (based on the PS time series) with the complex conjugate of the second continuous wavelet transform (based on the climatic water balance). The resulting amplitude has high values where both continuous wavelet transforms have high values. These high values represent time-spans where both time series have an (intermittent) periodical signal. The resulting phase represents a potential time lag between two (intermittent) periodical signals. The cross wavelet transform is calculated as follows (Liu et al. 2011):

$${W}_{xy}\left(\tau ,s\right)={{W}_{x}\left(\tau ,s\right) W}_{y}^{*}(\tau ,s)$$

where \({W}_{y}^{*}\) is the complex conjugate of \({W}_{y}\), and the power spectrum of the crossed wavelet is \({\left|{W}_{x}\right|}^{2}\).

The wavelet coherence is defined as coherence of the two continuous wavelet transforms. It is calculated by the normalized cross-correlation coefficient and a scaling factor between the two continuous wavelet transforms. The wavelet coherence can show additional similarities between two (intermittent) periodical signals, compared to the cross wavelet transform. The wavelet coherence is calculated as follows (Nourani et al. 2019):

$$R_{xy}\left(\tau,s\right)=\frac{\left|S\left(W_{xy}\left(\tau,s\right)\right)\right|}{\sqrt{\left|(S\left(W_x\left(\tau,s\right)\right)\right|^2\left|S(W_y\left(\tau,s\right))\right|^2)}}$$

where S is a smoothing operator. The difference between cross wavelet transform and wavelet coherence is that the cross wavelet transform performs best when the power level of the signal is similar in both continuous wavelet transforms. On the other hand, the wavelet coherence is able to deal with different signal power levels, because of a normalization and smoothing. As a consequence, the wavelet coherence has a lower resolution in space and time with regard to the cross wavelet transform but can handle different levels of signal power.

In the case study, two continuous wavelet transforms are computed, the first is based on a PS time series, and the second is based on the climatic water balance, which is a potential landslide triggering factor. The climatic water balance data is based from a meteorological station located 8 km north of the PS time series analyzed in the wavelet analysis. Because the sliding surface of the landslide is several tenth of meters below the land surface, the climatic water balance is used instead of precipitation measurements, which is often used as a potential triggering factor in landslide analysis. In order to investigate on a relationship between (intermittent) periodical signals of the climatic water balance and landslide deformation, two continuous wavelet transforms are combined by cross wavelet transform and wavelet coherence.

A requirement for the wavelet analysis is a regular sampling in time across both time series datasets. The PSI time series has a nominal 6-day temporal sampling rate, while the climatic water balance has a daily temporal sampling rate. In the PSI observation time span (1.4.2015–30.12.2020, track 139), 350 acquisitions are theoretically possible with a 6-day sampling. As the 6-day sampling was not possible before the start of the Sentinel-1B satellite (25.04.2016) and acquisition gaps exists especially at the beginning of the SAR missions due to the commissioning phase, the actual dataset consists of 279 out of the theoretical 350 acquisitions (80%). Thus, at first the PSI time series is filled up by a piecewise linear interpolation between acquisition gaps. Second, the climatic water balance was resampled from daily to 6-day average. Therefore, the subsequent 6-day climatic water balance average was calculated for each PSI observation date. After these pre-processing steps, the continuous wavelet transform of the PSI and climatic water balance time series are calculated. Subsequently, the continuous wavelet transform of the PSI and climatic water balance time series are the input for the cross wavelet transform and wavelet coherence.

Results

The results section first reports on the ADA-mapping results used to identify clusters of deforming PS (“ADA-mapping results” section). Subsequently, the time series of these PS is analyzed. Using one exemplary PS deformation time series, the classification results are provided in the “Time series classification results” section and the results of the wavelet analysis are presented in the “Wavelet analysis” section.

ADA-mapping results

Four clusters of moving PSs are classified as ADA in the study area. Two of them are located in the Piesport landslide, one in the Trittenheim landslide (Fig. 4), and the third is located in a quarry. The ADA in the Trittenheim landslide consists of 20 PS with a mean deformation rate of 5.65 mm/a in LoS. The maximum PS velocity of this ADA is 18.01 mm/a in LoS. The two ADA at the Piesport landslide consists of 4, resp. 3 PS, with a maximum velocity of 7.09 resp. 5.28 mm/a in LoS. All three ADA are characterized by a high quality index, indicating a reliable ADA. It has to be noted that several isolated but deforming PS are not classified as ADA (Fig. 4).

Fig. 4
figure 4

Sentinel-1 PSI mean velocity, detected ADA and landslide boundary of the Trittenheim (A) and Piesport landslide (B). A shaded relief digital elevation model serves as background

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Concerning the LoS deformation rate, the ADA in the Trittenheim landslide is deforming faster than the ADA in the Piesport landslide. Both ADA are in large landslide areas, with respect to the average landslide size in the Moselle Valley. Both ADA are located in landslides where soil creeping is present. The creeping soil is part of the sliding mass. The sliding mass is intersected from top to bottom by a road with adjacent stone-walls and metal railings (Fig. 3).

Focusing on the Trittenheim landslide, a comparison of the PSI velocity with a geological transect shows that at least three PS velocity clusters can be differentiated (Fig. 5). First, a cluster with very low deformation rates slower than −5 mm/a is present at the landslide top. Followed by slightly higher velocities in the rupture area (approximately 250–300 m.a.s.l.) with only few PS points. Then a cluster with the highest velocities of up to −18 mm/a located in the middle deformation zone (approximately 200–250 m.a.s.l.). Finally, a cluster with low velocities at the landslide toe (150–200 m.a.s.l.). In the landslide toe area, a federal street (K86) is present. This is a highly vulnerable element and even low velocities, e.g., −5 mm/a, can be of interest for safety measures.

Fig. 5
figure 5

Transect of the Trittenheim landslide (A, modified after Rogall 2014), PSI LoS deformation rates vs. elevation (B). The PS velocities shown in B correspond to the PS shown in C

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Regarding the northern ADA located at the Piesport landslide, the deformation rates are in general slower with regard to the Trittenheim landslide. At the top of the landslide (approximately 320–400 m.a.s.l.), no PS is present. The highest velocities in the rupture area (> 300 m.a.s.l.) are slower than −4 mm/a. In the middle deformation zone, velocities reach −7 mm/a (Fig. 6). Below < 250 m.a.s.l. velocities start to decrease rapidly.

Fig. 6
figure 6

Transect of the Piesport landslide (A, modified after Rogall 2014), PSI LoS deformation rates vs. elevation (B). The PS velocities shown in B correspond to the PS shown in C

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All PS deformation time series of both ADA in landslide areas show a linear deformation trend spanning the entire observation time span. The exemplary chosen PS shows a long-term linear trend as well as a periodical signal and abrupt changes of the deformation date (Fig. 3). The following sections shows the results based on the exemplary chosen PS time series.

Time series classification results

In order to detect characteristic patterns of the PS time deformation time series, a sequence of statistical tests is performed. Especially PS characterized by an acceleration within the time series are investigated. The motivation is a quantification of a potential time lag between strong rainfall events during times of low potential evapotranspiration and an accelerated deformation. As an example, the deformation time series of a single PS shows an acceleration and is classified as bilinear with a breakpoint on 11.03.2020 (the location of the PS is shown in Fig. 3A by the white arrow). A comparison of the PSI time series with a potential landslide triggering factor, e.g., the climatic water balance, shows high positive values in February 2020, approximately 1 month before the acceleration started. That means, a high precipitation coinciding with a little potential evapotranspiration was present before the detected acceleration. It is hypothesized that the kinematic behavior of the landslide mass is affected by a higher water content and a lower friction coefficient at the deep seated sliding surface. Figure 7 shows the overall deformation trend covering almost 6 years of measurements and the detected acceleration. A time-lag of 42 days can be observed between the start of the previous positive climatic water balance period (29.01.2020, red bar in Fig. 7) and the PSI breakpoint (11.3.2020, red dot in Fig. 7). Until the breakpoint, the slope of the linear regression shows a velocity of v = 10.1 mm/a. After the breakpoint, a velocity of v = 14.1 mm/a is estimated by a linear regression. The corresponding BIC, based on the 5-measurement segments, shows the minimum of 1.3 at the 11.03.2020 and marks the breakpoint of the changepoint regression (Fig. 7B).

Fig. 7
figure 7

Climatic water balance vs. PSI LoS deformation time series (A). The PSI detrended time series is two-times exaggerated for visualization. The acquisition date where an acceleration is detected is shown as red dot. The earliest date with positive climatic water balance before the acceleration is shown as red bar. Bayesian information criterion (BIC) for successive five-measurement segments (black points) (B), for a quadratic regression (green line), and for a linear regression (red line) of the PS deformation time series from track 37

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Wavelet analysis

The same PS time series used in the “Time series classification results” section is used in the wavelet analysis. In order to calculate the wavelet transform, the PS deformation time series is detrended and gap filled (Fig. 8A, bottom). The continuous wavelet transform shows a periodical signal with a yearly frequency from the beginning of 2018 until the end of 2020 (Fig. 8A, top). The power of this signal reaches a value of 16. A periodical signal with a 2-year frequency starting in 2016 is also visible, with slightly higher power values (Fig. 8A, top). However, the cone of influence indicates that most of the 2-year signal is not reliable, because the time series is too short. The continuous wavelet transform of the climatic water balance time series shows a strong yearly periodical signal throughout the entire time series (Fig. 8B, top). The power values of this signal reach a value of 30. The yearly periodical signal is clearly visible in the climatic water balance time series plot (Fig. 8B, bottom).

Fig. 8
figure 8

Continuous wavelet coherence of detrended and gap-filled PSI time series (A) and continuous wavelet coherence of climatic water balance (B). The black line in the continuous wavelet coherence shows the 5% significance level against red noise. The cone of influence, visualized as transparent gray areas, shows the time–frequency range which can be affected by edge distortions

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The results of the cross wavelet transform show a high correlation between both continuous wavelet transforms. The cross wavelet transform shows a common yearly (and 2 years) periodical signal (Fig. 9A). Both signals are mostly in phase which is visualized by the arrows pointing to the right. If the periodical signals were in anti-phase, the arrows would point to the left. After 2018, the arrows are pointing slightly downwards, meaning the deformation has a lag to the climatic water balance.

Fig. 9
figure 9

Cross wavelet transform of detrended PSI time series and climatic water balance (A) and wavelet coherence of detrended PSI time series and climatic water balance (B)

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The wavelet coherence shows that a significant correlation starts in 2017 until the end of the time series (Fig. 9B). The 2-year periodical signal is not significant based on the wavelet coherence. The slight downward pointing of the arrows indicate that there is a time-lag between the time series of ~ 10 days (time-lag = 10° ∙ π/180° ∙ 365 days/(2π) = 10.14 days). Three other time–frequency areas with lower periods are also detected by the wavelet coherence, which are not detected by the cross wavelet transform.

Discussion

The ADA-mapping approach based on track 139 successfully detects all visible deformation clusters (Piesport, Trittenheim landslide, quarry) after adjusting the thresholds (Table 2). The ADA-mapping result, in the landslide areas, using default and adjusted thresholds is shown in Fig. 10. Based on the default thresholds, only a part of the southern cluster of deforming PS in the Trittenheim landslide is detected (Fig. 10A). The northern cluster of deforming PS, located in the Trittenheim landslide, is not detected by default thresholds using track 139. The ADA-mapping results based on track 37 also show that default thresholds underestimated the cluster of deforming PS (Fig. 10B). Regarding the Piesport landslide, only the ADA-approach using track 139 and adjusted threshold is able to detect the cluster of deforming PS (Fig. 10C). The reason for the failure of the ADA-approach based on track 37 is the low spatial PS density (Fig. 10D).

Table 2 Default and adjusted thresholds for the ADA-mapping
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Fig. 10
figure 10

ADA-mapping result based on default and adjusted thresholds based on track 139 (A, C) and 37 (B, D)

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Due to the low PS density caused by the rural landcover in the landslide areas, relatively large distances between deforming PS are present. This affects the spatial thresholds used in the ADA-mapping approach. The increased distance threshold detected the visually identified clusters but also detected many other implausible ADA. Thus, the factor for the standard deviation filter threshold is also increased. The threshold for velocity class 1 was increased from 10 to 16 mm/a, because this is the threshold used to distinguish very slow from extremely slow landslides (Cruden and Varnes 1996). The minimum ADA size was lowered from 5 to 3 PS due to the low PS density. However, results show that the thresholds needed an adjustment to be successful and the question of transferability is still open, e.g., how a spatially large PSI dataset with various landcover types and as a consequence various PS densities can be exploited by using fixed thresholds.

In order to verify results from the time series classification and the wavelet analysis, a PS deformation time series from another track is used (track 37). The PS is chosen based on the nearest spatial distance to the PS used from track 139 in the results section. Regarding the estimation of a time-lag between the acceleration start and a potential triggering factor (climatic water balance), the changepoint regression from track 139 indicates a 42-day delay. Using the PS from track 37, the changepoint regression failed in finding a reasonable breakpoint date (e.g., ~ begin of 2018 or begin of 2020, Fig. 10A). Thus, no reasonable time-lag is calculated by the time series classification approach using track 37. Reasonable breakpoint dates correspond to low BIC values (Fig. 10B); however, the lowest BIC value lies on October 6th 2020, where no acceleration is visible in the PS time series (Fig. 11A). This raises the question of transferability of this approach regarding breakpoint detection.

Fig. 11
figure 11

Climatic water balance vs. PS deformation time series based on track 37 (A). The PSI detrended time series is two-times exaggerated for visualization. The detected breakpoint date is shown as red dot. Bayesian information criterion (BIC) for successive five-measurement segments (black points) (B), for a quadratic regression (green line), and for a linear regression (red line) of the PS deformation time series from track 37

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The continuous wavelet transform from the PS time series from track 37 shows a similar yearly (and 2-yearly) periodical signal as from track 139 (Fig. 12). The cross wavelet transform and wavelet coherence of track 37 confirm the correlation between PS deformation and climatic water balance regarding a yearly (and two-yearly) periodical signal (Fig. 12C, D). Based on the wavelet coherence, the time-lag is 10 days for 2018–2019. The wavelet verification indicates a good transferability to another PS time series from another track with a similar time series pattern.

Fig. 12
figure 12

Continuous wavelet transform (A) of detrended, gap-filled PSI time series of the PS deformation time series from track 37 (B). The black line in the continuous wavelet transform shows the 5% significance level against red noise. The cone of influence, visualized as transparent gray areas, shows the time–frequency range, which can be affected by edge distortions. (C) Shows the cross wavelet transform of the detrended, gap-filled PSI time series of the PS from track 37 and the climatic water balance shown in Fig. 8B. (D) Shows the corresponding wavelet coherence

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Conclusion

The results of the case study show that information regarding groups of deforming PS, accelerations of deformation, and intermittent periodical signals can be semi-automatically extracted from a wide-area Sentinel-1 PSI dataset from the Ground Motion Service Germany (Kalia et al. 2021). In order to assess the transferability of the three applied post-processing techniques (Berti et al. 2013; Barra et al. 2017; Liu et al. 2022), the approaches are tested on a second independent Sentinel-1 PSI dataset from another track which is also processed in the framework of the Ground Motion Service Germany (track 37). Verification results show that only the cluster of deforming PS in the Trittenheim landslide is detected by using track 37. The active deformation area correctly detected by using track 139 is not detected by using track 37. The reason is the low spatial PS density. Just two deforming PS are present in the Piesport landslide from track 37. This is below the minimum threshold of at least three deforming PS to form a cluster within the ADA-mapping approach.

Regarding the detection of a plausible acceleration date, the changepoint regression is successful by using the PS time series from track 139. It failed by using the PS time series from track 37. The estimated time-lag between acceleration and a potential triggering factor (climatic water balance) is 42 days (based on the acceleration date from track 139).

The wavelet analysis successfully quantified a yearly (and 2 years) periodical signal in both exemplarily chosen PS deformation time series from track 139 and 37. Both wavelet coherence results indicate a time-lag of 11 days between the seasonal acceleration and the seasonal signal of the potential triggering factor (climatic water balance) for 2018 and 2019.

To conclude, the ADA-mapping approach is able to highlight reliable deformation areas and can guide the end-users attention. Once these areas are identified, time series analysis can provide further insights regarding correlation and quantification of time-lags with respect to a potential triggering factor.