## 2 Materials and Methods
### Methodology
#### 2.3.2 Model-Based Processing of TLS Data and Estimation of Oscillation Amplitude and Frequency  
Model-based processing of TLS data is widely used to determine with high precision the position of points materialized through known geometries. A typical example is given by the targets (spherical, cylindrical or plane) used for the registration of the scans carried out for the surveying of a 3D object [24; 25]. If a stable target having a known geometry is surveyed, the parameters of the fitting surface corresponding to this geometry can be estimated from the cloud of \(n\) points acquired during the scan, through a least-squares procedure. The RMS of surface fit residuals exhibits a linear trend as a function of sampling resolution, i.e., of the square root of the number \(n\) of sampled points [26]. The deviations between the fitting surfaces, obtained from different scans of the same target, are proportional to \(n^{-1/2}\) and, therefore, drastically lower than the single point precision of the instrument. The model-based processing, in our case, is based on the use of interpolation to reduce noise and increase the precision of displacement measurements [20]. This approach is particularly suited for linear structures such as wind towers. In [19] the authors showed that, using an interpolating curve, the precision achieved in measuring the deformation of a beam can be 20 times higher than that of a single point. These conclusions had been confirmed in [27; 28]. The knowledge of the geometry of the wind tower shown in Figure 3a is needed. A wind tower is a truncated cone with a circular section and a decreasing diameter upwards. Therefore, a generatrix is a straight line. The elastic line is certainly a continuous curve with a continuous derivative; in static conditions, for a load applied to the free end, it can be approximated by a third-order polynomial [29]. It is worth noting that the lines scanned by the VZ-1000 in line scanner mode do not belong to the same vertical plane, due to an imperfect orthogonality between the emitted laser beam and the rotation axis of the multi-facet mirror. This can also be inferred by observing the different horizontal angle associated with each scanned point. The laser beam would describe a conic section on a vertical surface instead of a straight line, as in the classic case of the the theodolite line-of-sight, not orthogonal to the secondary axis. Given the size of the misalignment, there is a maximum deviation of about 0.4 m at the top of the tower compared to  
Figure 4: TLS (**blue dots**) and GB-RAR (**red square**) measurement points along the wind tower. The range distance and resolution of each GB-RAR measurement point is given by Equation (2). the average generatrix. In any case, this effect does not affect the results obtained, since they are based on the differences between the scanned lines, which have the same deviation. To take into account these effects without making the calculation too heavy, the trace of the laser beam on the tower is approximated by a fourth-order polynomial in a (D,H)-plane, where D is the ground distance from the TLS and H is the height from the tower base. The following procedure is adopted for each line scan: (a) computation of coordinates of points of the TLS scanned line; (b) estimation of interpolating polynomial coefficients; (c) shift of the polynomial curve in order to make it pass through the base of the tower; (d) computation of the distance D for each selected height H along the tower. Figure 5 shows the polynomial fitting of the points of a single line scan along the tower. The procedure is repeated for all the TLS scanned lines. This provides the distance D(H,t), of a point P with a height H at the acquisition time t and hence the time series of the distance D of point P with respect to the TLS. Time series are used to obtain the amplitude of the oscillations. As a final step, the wind tower's vibration frequencies are obtained by a spectral analysis.