## 4 Field Tests of the GFZ-WISE
### Methodology of Surface-Wave Analysis
The phase velocity _c_(_o_) at each frequency is then estimated by a fitting procedure (_i.e._, an iterative grid-search procedure) between the experimental spatial correlation values for all the possible station pairs plotted as a function of distance \(r\), and theoretical values estimated by a Bessel function of the form:  
\[\phi(r,\omega_{o})=J_{0}\Bigg{(}\frac{\omega_{0}}{c(\omega_{0})}\,r\Bigg{)} \tag{2}\]  
The tentative phase velocity _c_(_o_)_ is generally varied over large intervals (e.g., between 100 and 3,000 m/s) in small steps (e.g.1 m/s). The best fit is achieved by minimizing the root mean square (RMS) of the differences between the values calculated using equations (1) and (2). Data points that differ by more than two standard deviations from the value obtained with the minimum-misfit velocity can be removed before the next iteration of the grid-search [7]. Concerning the H/V curve, [PERSON] and [PERSON] [36] suggested the normalization of the horizontal spectral amplitude with respect to the vertical one, with the aim of minimizing the source function effects in the noise spectra. In practice, the method for evaluating the H/V curves consists simply of (1) merging the two Fourier spectra of the two horizontal components, X_(o)_ and Y_(o)_, of motion to obtain a single combined horizontal, H_(_o_)_, component, and (2) computing the ratio between the H_(_o_)_ spectra with the one from the vertical component. We compute the modulus of the H_(_o_)_combined spectra by the root mean square average (RMS):  
\[H_{(\omega)}=\sqrt{\left(X_{(\omega)}^{\,2}+Y_{(\omega)}^{\,2}\right)/2} \tag{3}\]  
As discussed before, both the RWDC and H/V from the surface-wave analysis provides the necessary information for the S-wave velocity estimation. However, the relationship between the Rayleigh wave velocities and the H/V curve with the S-wave velocity and thickness of sediments is not linear. Therefore, analyses conducted to determine the subsoil S-wave velocity profile from surface-wave curves consists of solving a non-linear inverse problem. Unfortunately, non-linear inverse methods are generally time consuming, and thus not suitable for fieldwork uses. However, linearized procedures [37] might be used in the case some _a priori_ constraints are available, or assumptions can be made about the S-wave velocity _versus_ depth (_i.e._, a realistic 'input model' can be defined). For sake of simplicity, in this study we used this approach, which can be especially useful for real-time S-wave velocity estimates. In particular, the linearized inverse problem of surface-wave curves (_i.e._, RWDC or H/V) is solved using Singular Value Decomposition [SVD, 38] and by the generalized least-squares iterative minimization of the RMS of differences between observed and theoretical curves. Because of the non-linearity of the problem, the inversion is repeated, (1) until the RMS ceases to change significantly, and (2) starting from different input models. This inversion approach is particularly suitable for preliminary analysis during the field measurements, since it has the advantage of being computationally efficient, and thus, easy to implement in real-time for retrieving S-wave estimates. Of course, in the event that other information concerning the subsoil structure is not available, and the S-wave velocity structure is particularly complicated, the preliminary field model from the linearized inversion procedure should be validated by post-acquisition robust, but also time-consuming, non-linear inversion analysis [6, 39].