## 3 Methodology
### Frequency-Wavenumber (f-k) Analysis  
The f-k method is an array technique based on the analysis of seismic noise. Assuming stationary noise in both time and space, the incoming waves can be represented as a function of the frequency and the vector velocities through the frequency-wavenumber spectral density function [8; 24]. Thus, the wave velocities and their respective directions of approach are derived as functions of the frequency.  
The power spectrum can be obtained through the maximum likelihood method (MLM) [8] or the beam-forming method (BFM) [24], which is the one used in this work. Additional information can be found elsewhere [47; 48; 49].  
The seismic noise array measurements were taken around the G\(\acute{\text{e}}\)ve\(\acute{\text{a}}\)r\(\acute{\text{e}}\)jar landslide at eight different sites (Figure 3). The implementation of the arrays was achieved with seismic refraction/reflection equipment consisting of twelve 4.5 Hz vertical geophones connected to a multichannel seismic recorder (SmartSeis ST, Geometrics, San Jose, CA, USA). A circular layout without a central station was deployed for the array layout (Figure 5a). For all the sites, the diameter of the arrays was configured to 20 m, which is the maximum possible aperture allowed by the cable length of the reflection/refraction equipment used. Taking into account that the wavelengths obtained by the f-k method can be several times greater than the array diameter [50; 51], this configuration would be enough for the soil classification purposes [52].  
The data acquisition consisted of approximately fifty records of 32 s without overlapping, sampled at 500 Hz and subsequently resampled at 100 Hz. Although the natural frequency of the geophones is 4.5 Hz, this equipment allows for estimating dispersion curves above approximately 2 Hz [53; 54].  
In Figure 5b, we show the theoretical response of the array, providing estimates of the aliasing (kmax) and the maximum resolution (kmin/2) limits, in terms of the wavenumber.  
The f-k analysis has also been accomplished using the GEOPSY software. The initial configuration requires the establishing of the grid of wavenumber (k) points to be analyzed, indicating the maximum value (grid size) as well as the minimum step (grid step). These values determine the aliasing limit and the maximum resolution, respectively. For the arrangement carried out in this work, the analysis of the theoretical array response provides a maximum resolution of 0.06 rad/m (kmin/4) and an aliasing limit of 0.64 rad/m (kmax/2) (see Figure 5b). The real array response depends on the number of sensors and the sensor geometry, but also on the slowness and the wavenumbers of the seismic phases observed within the array.  
With these considerations, a dispersion curve is first estimated for each analyzed 32 s window, which corresponds with the length of the recordings. Finally, a mean dispersion curve is computed by averaging the curves obtained from all the windows.  
Figure 5: (**a**) Array layout; (**b**) corresponding theoretical response. The k limits corresponding to the maximum resolution (kmin/2) and the aliasing limit (kmax) are also indicated.