GNGTS 2018 - 37° Convegno Nazionale

618 GNGTS 2018 S essione 3.2 since data are affected by larger uncertainties at greater distance. By using equations 1 and 2 we finally derive A 0 and A 1 for each single frequency. Thirdly, the local slope (or wavenumber) is retrieved. The developed algorithm takes the unwrapped (in offset) phase in a neighbourhood of each amplitude maximum i (Fig. 2), and computes a linear regression as in the classic MOPA approach (Strobbia and Foti, 2006). In formula: ∅ = G • M (7) where G is the data kernel matrix, containing the offset information, Φ the phase vector and M the vector containing the unknown polynomial coefficients, -k loci and φ 0i . Using a least-squares approach we obtain M as: M = G –g • ∅ (8) being G –g = ( G T G ) –1 • G T the pseudo-inverse of G . The value of the final local wavenumber k loc for each single frequency is then computed as the weighted (with respect to offset, as done for amplitude) average of all k loci values. This is possible since we are performing a 1D analysis (no lateral variations). Finally, dispersion curves for the two co-existing modes were calculated using equations 5 and 6. The application of the method to our synthetic dataset has given promising results. By overlapping the final dispersion curves on the normalized f-k spectrum we observe a perfect match between the retrieved dispersion curves and the peaks of maximal energy corresponding to the fundamental and first-order Rayleigh modes (Fig. 2). Results with real data. Multi-mode MOPA has been applied to a 2D real dataset with laterally uniform conditions. The site is located in Mirandola, Italy, and it is one of the three acquisition sites of the InterPACIFIC project (Garofalo et al. , 2016). Our analysis was focused on the 1 meter spaced 48 channels active acquisition and, in particular, on the first shot location (energized 10 times). Traces were recorded with 4.5 Hz natural frequency vertical geophones, with a length of 2 s and a sampling interval of 0.25 ms. From the observation of the f-k spectra we identified a possible frequency range for the analysis: between 10 Hz and 20 Hz the fundamental and the first higher modes are both present and quite energetic (Fig. 3). In this case, since we were using real data, amplitude distribution with offset was quite noisy: a light smoothing was necessary before fitting it with the periodic function. Uncertainties have also been estimated by propagating the error through the different steps of the analysis. Fig. 3: Example with real data: Left-upper panel: unwrapped phase at 19 Hz (blue stars) with values selected for each independent k loci computation (red circles). Left-bottom panel: Normalized amplitude at 19 Hz (blue stars), normalized smoothed amplitude after correction for attenuation (green line), fitting sinusoid (red line). Right panel: f-k spectrum overlapped by computed fundamental (blue stars) and higher order (red stars) dispersion curves. Error bars define uncertainty in the velocity estimation.