*3.1. Intercomparison of InSAR and In Situ Displacement Time Series*

The InSAR LoS measurements are independently projected to the vertical as well as combined to calculate the actual vertical motion component, using the ascending track and descending track (VERT-ASC-DES) at the West Point (WP) (see Materials and Methods section). Qualitatively, there is a positive correlation between the vertical displacement VERT-ASC-DES at WP, as calculated by InSAR using the satellite observations and as measured by ground-based extensometers EXT1 and EXT2 (Figure 2). Quantitatively, the least squares correlation coefficient (cc) is much higher for EXT2 (cc = 0.69) than for EXT1 (cc = 0.47). The relatively low cc values are expected given the non-linearity phenomenon and the heterogeneities of the clay layer as described in Section 3.2.

During the three-year period, the ground displacement time-series may be decomposed with the addition of a linear trend component T, a seasonal component S and an irregular residual component. There is a linear trend with a shrinking of about −2.2 mm/yr at EXT1, while the measured motion at EXT2 is merely a cyclic component with a negligible shrinking of about −0.3 mm/yr (Figure 2). During the same period, the linear trend measured by Sentinel-1A/B lies within in situ rates at EXT1 and EXT2, with a global shrinking of about −0.55 mm/yr. Concerning the cyclic part, the expansion is up to three times higher at EXT2 than at EXT1, with average swelling of 9.4 ± 2.0 mm and 3.1 ± 1.0 mm, respectively (Figure 2). Over the same period, the swelling magnitude measured by VERT-ASC-DES is 4.5 ± 0.5 mm.

#### *3.2. Electrical Tomography Survey*

Electrical resistive tomography has been carried out in order to image the lithology stratification and the heterogeneity of the clay layer (i.e., depth, thickness, and fraction of clay) (see Materials and Methods section). Along three 28.5 m long profiles in the WP cell (Figure 1), the resistivity values range from 8 to 100 Ω.m and a three-layer stratified subsurface is highlighted (Figures 3 and 4).

**Figure 2.** Comparison of the in situ vertical displacement of both extensometers (EXT1 and EXT2) to VERT-ASC-DES, the vertical displacement using the ascending and the descending track of Sentinel-1 at West Point (WP). The linear trendlines of the displacements are also shown (dotted lines).

**Figure 3.** (**a**) Vertical cross section of resistivity along P1 profile (EXT1-EXT2). (**b**) Clay depth and thickness at EXT1 and EXT2 deduced from the resistivity profile (see text). E1 sensor (near EXT1, Figure 1) and W5 (near EXT2) are both located at the same 1.2 m depth inside the clay layer.

**Figure 4.** (**a**) Vertical cross section of resistivity along P2 profile (SC1-SC2) showing the SC1 and SC2 geological logs. (**b**) Vertical cross section of resistivity along P3 profile (South-North) showing TR1 drilling.

We used two drill cores along the SC1-SC2 profile to define the clay subsurface material by the resistivity value [30] (Figure 4). A first layer above the depth of 0.59 m ± 0.22 m corresponds to topsoil and backfill with some clay lenses (Figures 3 and 4). A second layer with a mean thickness of 1.5 m ± 0.42 m corresponds to the clay material with a resistivity lower than 17 Ω.m ± 1 Ω.m. Below 2.09 m ± 0.42 m, a third layer corresponds to sandy limestones with a resistivity gradient due to the weathering process.

Along the EXT1-EXT2 resistivity profile (Figure 3), the mean depth and thickness of the clay material layer is 0.46 m ± 0.05 m (min-max, 0.34–0.55) and 1.97 m ± 0.18 m (minmax, 1.70–2.39), respectively. The mean depth of the carbonate layer is 2.43 m ± 0.16 m (min-max, 2.15–2.73). The depth (approximately at 0.5 m) and thickness (about 2.3 m) of the clay layer at EXT1 and EXT2 are very similar (Figure 3). However, the resistivity of the clay layer at EXT2 is much lower compared to EXT1, indicating that the clay fraction is much higher beneath EXT2 [20]. That is also consistent with the expansion difference between the extensometers showing a much higher swelling magnitude at EXT2.

Along the three profiles, the mean depth and thickness of the clay material layer in the West Point (WP) cell is 0.59 m ± 0.22 m and 1.5 m ± 0.42 m.

#### *3.3. Intercomparison of In Situ Soil Moistures and SMOS Satellite Surface Soil Moistures*

We investigated here the soil moisture variations acquired by E1 sensor (near EXT1) and W5 (near EXT2) that are both located at the same 1.2 m depth inside the clay layer (Figure 3). The temporal variations of E1 and W5 are strongly correlated (cc = 0.73) during the three-year period (Figure 5). For the seasonal one-year period, there is a slight leading of E1 moisture relative to W5 (about 0.75 months) as calculated by the Cross Wavelet Transform (XWT) (see Materials and Methods Section 2.5). Indeed, the infiltration time of the meteoric water inside the clay layer at 1.2 m depth is lower in the E1 case, since the clay fraction of low permeability is shallower at EXT1 compared to EXT2 (Figure 3).

**Figure 5.** Comparison of three soil moistures at about 1.2 m depth (E1, E5, W5) to the Surface Soil Moisture SSM-ASC and SSM-DES using the SMOS ascending and descending track, respectively.

We use here the term Surface Soil Moisture (SSM) to refer to the volumetric soil moisture in the first few centimeters (0–5 cm) of the soil. We investigated also the SSM acquired by the ascending (SSM-ASC) and descending (SSM-DES) orbits of the SMOS satellite, showing rather positive correlation (cc = 0.48). For the one-year period, SSM-ASC and SSM-DES are both in-phase and there is a slight leading of SSM-ASC relative to SSM-DES for the 5-month period (about 0.9 months). Hence, the SSM-ASC dataset was considered more appropriate for calculating the phase differences to other time series in the discussion. For the one-year period, the phase lead of the surface soil moisture SSM-ASC relative to the in-ground soil moistures E1 and W5 calculated by XWT is about 0.16 yr and 0.22 yr, respectively.

#### **4. Discussion**

#### *4.1. Intercomparison of InSAR and In Situ Displacement Time Series*

Displacements measured by InSAR are in the direction of the Line-of-Sight (LoS) of the satellite, while displacements measured by extensometers depict motion in the vertical direction. However, the combination of ascending and descending InSAR measurements for the calculation of the actual vertical motion component (VERT-ASC-DESC) allows the direct inter-comparison of satellite and in situ observations.

When exposed to moisture, the magnitude of the vertical expansion of soils will depend on the amount of expansive clay minerals in the subsurface. In addition to the vertical displacement, the subsurface heterogeneities (i.e., clay depth and thickness and percentage of sand) may introduce spatial variability to the observed ground displacements. The above-mentioned factors need to be considered when examining the consistency of our measurements. In our case, the vertical displacements VERT-ASC-DESC from remote sensing are within the motion values of both extensometers, not only for linear displacement rates but also in terms of observed seasonality (see Section 3.1). This result is in agreement with the assumption that the effect of the spatial resolution of the remote sensing measurements (90 m by 90 m cell) represents an average of the vertical displacements within that resolution cell.

#### *4.2. Shrinking and Swelling Periods Using Fourier Power Spectra and Continuous Wavelet Transform*

We use first the Fourier spectra of vertical displacements to calculate the main frequency components of time series of both extensometers as well as at WP and East Point (EP) cells of the P-SBAS grid. The one-year seasonal period is found in the power spectrum of the four displacement time series at EXT1, EXT2, WP, and EP (Figure 6). In the frequency band between 2 and 4 yr−<sup>1</sup> (corresponding to a period between 0.5 and 0.25 yr), there are

three other frequencies which are noted F1, F21, and F22 (order of increasing frequency) and P1, P21, and P22 periods (order of decreasing period). It should be noted that the frequency values for all displacement time series are close to the first three sub-annual components of the Fourier transform of the soil moistures E1 and W5 at 1.2 m depth (Figure 6a) and the two ascending and descending time series of the surface soil moisture (SSM-ASC and SSM-DES) as measured by SMOS (Figure 6b).

**Figure 6.** In the top, the magnitude of the Fast Fourier Transform (FFT) of different time series between frequencies 0.5 and 5 year−<sup>1</sup> are shown: (**a**) vertical displacement of both extensometers (EXT1 and EXT2) and the Soil Moisture at E1 and W5 (about 1.2 m depth), (**b**) VERT-ASC-DES for vertical displacement using ascending and descending S-1A/1B tracks at West Point (WP) and East Point (EP), and SSM-ASC and SSM-DES for the Surface Soil Moisture using the ascending and descending SMOS track, respectively. In the bottom, the phase angle difference in degrees between the FFT of the SSM-ASC using SMOS ascending track and the FFT of the displacement of four time series are shown: (**c**) EXT1 and EXT2 and (**d**) VERT-ASC-DES at WP and EP.

The Continuous Wavelet Transform (CWT) tool permits the recognition of power in time-frequency space, along with assessing confidence levels against red noise backgrounds (see Materials and Methods section). We use now the continuous wavelet power spectrum in order to unravel the intermittent physical mechanisms of the shrink/swell process at EXT2 and WP cell during 3.3 years (September 2016–December 2020):


**Figure 7.** The continuous wavelet transform (CWT) during 3.3 years (September 2016–December 2020) are shown for two vertical displacements: (**a**) EXT2 and (**b**) VERT-ASC-DES at WP. The thick contour designates the 5% significant level against red noise (see Materials and Methods section). The Cone of Influence (COI) where edge effects might distort the picture is shown as a lighter shadow.

In conclusion, the CWT analysis highlights two sub-annual periods that reflect both average ground shrinking (P1) and swelling timeframes (P21 and P22). A similar behavior is observed at EXT1 extensometer and within EP cell (data not shown).

### *4.3. Estimating Variations of Expansive Clay Depth and Thickness Using the Time Series Phase Difference*

The phase difference between the Surface Soil Moisture (SSM) and InSAR displacement time series indicates the time lag between the cause and the effect in the shrink/swell process. We test the assumption that depth and thickness of expansive clays are linked to the phase difference between both time series for the shrinkage and swelling periods, respectively. We use and compare the Fourier and XWT analyses to calculate this phase difference (see Materials and Methods section). The XWT spectra will be high in the time-frequency areas where both CWTs display high values, so this helps identify common time patterns in the two data sets. The XWT permits also the recognition of relative phase lags in time-frequency space (noted ΔΦ). While P1 and P21 are the shrinking and swelling periods found by the Fourier method (as described above), P1\* and P21\* are the same similar periods we found using the thick contours of XWT that designate the 5% significant level against red noise (Figure 8). XWT tool is used as well to calculate the circular standard deviation of this phase difference ΔΦ (see Materials and Methods section) and all these values are reported in Table 2.

No significant difference is found between ΔΦ values of both extensometers EXT1 and EXT2 for the shrinking period P1\* and the swelling period P21\* (Table 2). This result is consistent with the tomography results which show that both depth and thickness of the clay is indeed comparable at the location of the extensometers (Figure 3b). For the shrinking period P1\*, the phase difference in degrees is three times higher at EXT2 (ΔΦ = 107◦) than at WP (ΔΦ = 34◦) (Table 2). The same phase difference for the shrinking period is observed between EXT1 and EP (Table 2 and Figure 9). This result is further consistent with the tomographic findings indicating clays lenses in the first 0.5 m subsurface layer of the three 28m-wide profiles in the WP cell (Figures 3 and 4). Conversely, ΔΦ of the swelling periods is higher at EP cell (ΔΦ = 110◦) in comparison to the WP cell (ΔΦ = 72◦) (Table 2 and Figure 9). A higher clay thickness in the EP cell in comparison to the WP cell may be an explanation of this result (there is a need of electrical tomography data to check this assumption).

**Figure 8.** In the top, the cross wavelet transform (XWT) between SSM-ASC and two times series is shown: (**a**) EXT2 and (**b**) VERT-ASC-DES at WP. The relative phase relationship is shown as arrows, with in-phase pointing right and anti-phase pointing left and the Surface Soil Moisture leading by 90◦ pointing straight down. The Cone of Influence (COI) where edge effects might distort the picture is shown as a lighter shadow. In the bottom, the phase lags ΔΦ of SSM-ASC relative to the same times series are shown: (**c**) EXT2 and (**d**) VERT-ASC-DES at WP. The shrinking (P1\*) and swelling (P21\* and P22\*) periods are found using the thick contours of XWT that designate the 5% significant level against red noise (see Materials and Methods section).


**Table 2.** Comparison of Fourier FFT and cross-wavelet XWT analysis of phase angles and time lags of the surface soil moisture (SSM-ASC) relative to the vertical displacement at EXT1, EXT2, West Point (WP), and East Point (EP) for P0, P1, P21, and P22 periods.

**Figure 9.** In the top, the cross wavelet transform (XWT) between SSM-ASC using SMOS ascending track and two times series is shown: (**a**) EXT1 and (**b**) VERT-ASC-DES at East Point (EP). The relative phase relationship is shown as arrows, with in-phase pointing right and anti-phase pointing left and the Surface Soil Moisture leading by 90◦ pointing straight down. The Cone of Influence (COI) where edge effects might distort the picture is shown as a lighter shadow. In the bottom, the phase lags ΔΦ of SSM-ASC relative to the same times series are shown: (**c**) EXT1 and (**d**) VERT-ASC-DES at EP. The shrinking (P1\*) and swelling (P21\* and P22\*) periods are found using the thick contours of XWT that designate the 5% significant level against red noise (see Materials and Methods section).
