*2.3. Methodology*

This section describes the methodology proposed in this paper for the accurate characterization of dynamical behavior of tall structures. The methodology goes through the following steps:


The comparison and integration of TLS-based vibration frequencies with those provided by GB-RAR makes it possible to identify instrument artifacts and to extend measurements to higher oscillation modes, with an amplitude smaller than the enhanced TLS precision attained at point (2).

Below, we will examine the previous points to describe in detail each step of the proposed methodology applied to a wind tower.

#### 2.3.1. Co-Location of TLS and GB-RAR Instruments and Co-Registration of Their Measurements

The co-location of TLS and GB-RAR is carried out by a TS topographic survey providing the coordinates of the TLS and GB-RAR, as well as of the base of the wind tower in the same local reference system. Figure 2 shows the layout of the TS survey (a) and a sketch of the local reference system with the positions of the TS, TLS, GB-RAR and wind tower (b). The slant ranges RTS,TLS and RTS,GB-RAR of TLS and GB-RAR with respect to the TS are indicated, along with the slant distances RTS,WT-B and RTS,WT-T of the base and top of the wind tower. The TS survey makes it possible to compute the coordinates of each point P of the tower, in the same local reference system, and hence their slant ranges RP,TLS and RP,GB-RAR with respect to the TLS and GB-RAR. The topographic survey is not necessary if the entire structure can be surveyed by the TLS, which in this case allows us to obtain, along with the mutual positioning of the structure and GB-RAR, a 3D model of the structure and of the surrounding environment.

Figure 3 displays a high-resolution 3D digital model of the tower provided by the TLS with the slant ranges RP,GB-RAR and RP,TLS. As a result, the spatial co-registration of TLS and GB-RAR measurements is obtained in a seamless way, taking into consideration the different spatial resolutions of TLS and GB-RAR measurements. Figure 4 shows the TLS measurements points mapping the shape of the wind tower and the points corresponding to the center of each of 0.75 m-resolution cell of the GB-RAR. For each range cell of GB-RAR data, the corresponding TLS measurements are easily identified and compared or merged with GB-RAR data. The co-registration of the data in time is provided by a synchronization of TLS and GB-RAR acquisitions.

**Figure 2.** The TS survey layout with slant distances of TLS, GB-RAR, top and bottom of wind tower (**a**); The local reference system (TLS centered) used to co-locate TLS and GB-RAR measurements (**b**).

**Figure 3.** 3D digital surface model of the wind tower (**a**); The distances RP, GB-RAR and RP, TLS of point P on the tower with respect to TLS and GB-RAR (**b**).

**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).

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*<sup>−</sup>1/<sup>2</sup> 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 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

the average generatrix. In any case, this e ffect does not a ffect the results obtained, since they are based on the di fferences between the scanned lines, which have the same deviation. To take into account these e ffects 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 coe fficients; (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.

**Figure 5.** Polynomial fitting of points along the wind tower of a single line scan.

2.3.3. Interferometric Processing of GB-RAR Data and 2D Visualization of Oscillation Frequency Spectra

The stack of complex-valued GB-RAR acquisitions is processed by means of the interferometric radar technique by computing the phase di fferences of each pixel with respect to the first GB-RAR acquisition. The phase is first unwrapped and then mapped to propagation delay. Hence, the contribution of delay due to propagation in the atmosphere is modelled and removed, resulting in the measurement of the range RP,GB-RAR between the GB-RAR location and a target on the wind tower. A spectral analysis of the stack of range vectors {RP,GB-RAR(t)}PTOWER, results in the 2D spectrum of the vibration frequencies. This procedure has been described in [18] and applied to the monitoring of bell-towers and monuments in [30]. An important step of the adaption of the estimation of the GB-RAR estimation of vibration frequencies to the study of the characterization of the dynamical properties of a wind tower is the co-registration of TLS and GB-RAR spectra as sketched in Figure 6. After co-registering TLS and GB-RAR measurements, it is possible to compare vibration spectra obtained by TLS and GB-RAR by a spectral analysis of the time series of distances. It is worth noting that due to the di fferent spatial resolutions of TLS and GB-RAR data, more TLS spectra correspond to the a given frequency spectrum provided by GB-RAR interferometry. In fact, a GB-RAR range resolution of 0.75 m corresponds to a coarser spatial resolution along the wind tower, depending on the radar looking angle and the antenna irradiation pattern (Figure 4).

**Figure 6.** Sketch of proposed methodology to estimate vibration frequencies by the joint use of TLS and GB-RAR techniques.
