GNGTS 2015 - Atti del 34° Convegno Nazionale

84 GNGTS 2015 S essione 3.2 where f is the frequency, φ( f ) is the phase lag at the frequency f and d is the distance between the two receivers. This cloud of points, which is just a suitable transformation of the original data, still retains the locality information and to this level, the information regarding different modes of propagation, although mixed. These frequency-dependent phase lags would be suitable to build an objective function to be used during inversion, for example, in a Full Waveform Inversion style (Virieux and Operto, 2009; Masoni et al. , 2014; Groos et al. , 2014), assuming a suitable forward model is available. In classical SASW, however, this information is usually inverted using a parallel- layered based forward model. In order to avoid introducing such approximation, we proceed without inversion. We rather express the COP in the wavelength-velocity domain (λ-V R ) which is then associated to the subsurface between the pair. 2-D approach. In the two dimensional approach, source and receivers are all placed along the same line. The advantage of this approach is basically that starting from the same data we can compute the frequency-velocity spectrum and use its amplitude to filter out the points in the COP corresponding to harmonics that do not carry a meaningful amount of energy (say less than 5%). To do so, we require E ( f , V R ) p > 0.05 E max ( f ), (2) where E ( f , V R ) p is the energy carried by the harmonic wave represented by point p , having frequency f and traveling at speed V R , and E max ( f ) is the maximum energy transferred by any harmonic wave at the same frequency. An example of the points calculated for a whole linear array of 24, 4.5 Hz proper frequency geophones is shown in Fig. 1. The blue points are the points that are discarded once equation 2 is taken into account, while the valid points are drawn in black. The valid points agree quite well with the frequency-velocity transform obtained for the same data and shown in the background. The remaining points are then expressed in the wavelength-velocity domain (λ-V R ). The COP obtained from all receiver pairs, for all the source points and for all the shots (when multiple shots are performed), can now be assembled into one pseudo-section by means of a suitable weighted average algorithm. We refer to this result as a “��������������� ������� ��� ������� pseudo-section� ������� ��� ������� ” because the maximum depth to which each (λ,V R ) point brings its contribution is associated to a suitable fraction β of Fig. 1 – An example of the SASW-like cloud of points in the f-VR domain. In this example, the blue points are filtered out while the black points are those which comply with Eq. 2. For sake of comparison, the corresponding frequency-velocity transform is shown both clean (above) and with superimposed points (below) to show how the filtered COP adapts to the propagating modes.