GNGTS 2018 - 37° Convegno Nazionale

616 GNGTS 2018 S essione 3.2 The method. In order to highlight the main concepts underlying the proposed approach, let us consider a simple theoretical case: the interference between two sinusoidal signals, a and b , with identical frequency and different amplitude, and yet propagating with different apparent velocities. Note that no specific reference is here made to surface waves, but only to the combination of two monochromatic waves. The amplitude presents a regular oscillating pattern with offset, due to positive and negative interference between the two waves (beats). Conversely, the phase presents regular smooth jumps in conjunction with amplitude minima, at the positions of phase opposition, better visible with a linear-move-out (LMO) correction applied (Fig. 1). The example above clear show, to no surprise, that the combination of two different monochromatic signals having the same frequency produces observable patterns on seismograms. In particular: 1. Amplitude maxima and minima give information about the sum (constructive or in-phase interference) and the difference (destructive or out-of-phase interference) of the individual amplitudes of the two interfering waves. Conversely, we can infer the amplitudes of the single sinusoids, A a and A b , from the observed A max and A min : (1) (2) 2. The oscillating spatial period of amplitude and phase, let us call it ΔX , is linked to the difference of wavenumber (i.e. to the difference in velocity) between the two signals. In formula: (3) where k a and k b are the individual wavenumbers of the two sinusoids. 3. The local slope of the phase versus offset, k loc , corresponding to the position of amplitude maxima, is equal to the weighted mean of the two individual slopes k a and k b , where the weights are given by the sinusoid amplitudes: (4) Once we compute individual wave amplitudes, A a and A b , from equations 1 and 2, we can retrieve the phase slopes, or wavenumbers, from the solution of the system of equations 3 and 4. We have: (5) (6) The concepts above, illustrated in the simple case of interference between twomonochromatic waves propagating (in the same direction) with different velocities, can be easily extended to surface waves and, specifically, to the interference between two different modes. In fact, when using MOPA the surface wave analysis is performed independently for each frequency. Therefore, given a 2D seismic record with laterally homogeneous subsurface conditions (a vertical 1D model), it is possible to extract, for each frequency, the wavenumber information of two interfering modes, which consists in retrieving the multi-modal dispersion curves for that