GNGTS 2013 - Atti del 32° Convegno Nazionale

In general the following properties are of interest: in the rage –1 ≤ u ≤ 1 and m > 0, all the polynomials oscillate about zero even- or odd- symmetrically (for m even or odd respectively) with maximum amplitude of the oscillations equal to 1. For u > 1, T m (u) increases monotonically; for u < -1, T m (u) increases monotonically if m is even and decreases monotonically if m is odd. For all the polynomials: T m (1) = 1 . Given the acquisition parameters used in the field, we exploit these properties for determining the weights of the source and receiver arrays and the maximum rejection ratio R possible. Here below we briefly outline the procedure for the source array weights computation. The polynomial variable u is set to where x is given by with D 0 the distance between two consecutive elements of the array (1.25 m) and the wavenumber at which the first notch of the array will occur. In our case with 9 elements of the array ( ), we choose , slightly softening the indications suggested by the ground roll characteristics in the test but achieving a higher rejection ratio R . This gives and . The array coefficients can now be computed by means of the following formula (Holzman, 1963 eq. 35): (2) where: [] is the floor function, , and if s=0 otherwise , giving for the a k : 0.3682, 0.5261, 0.7610, 0.9355, 1.0000, 0.9355, 0.7610, 0.5261, 0.3682 for k=1...9. Array simulations. The redundant number of single blow gathers with closely spaced sources and receivers acquired, allow now to perform source and/or receiver array simulations in the processing lab that achieve the desired spatial filter responses. As an example, the blue line in Fig. 2 illustrates the spectrum of the simulated array composed by 9 evenly spaced and equally weighted sources. It is evident the performance of the array in attenuating the wavenumber components above 0.1 [cycles/m], which are mainly due to the surface waves noise and air wave. However, as shown by the lobe amplitudes in the stop-band region, the array response is not uniform, suggesting that the equal weights array performance can be improved in the lab. Indeed, an even more effective noise attenuation is achieved by the application of a simulated array composed by 9 evenly spaced but weighted blows, where the weights are computed using the procedure described in the previous paragraph based on the Chebyshev polynomials. The corresponding filter response is shown by the red line in Fig. 2. Note the uniform maximum amplitude of the lobes in the stop-band region at the expense of a slight higher position of the first notch due to the used. Results. The results obtained filtering the data with equally and optimized (Chebyshev) weighted spatial filters are checked on shot gathers and on stacked data. On shot gathers the surface waves noise attenuation is appreciable both in offset-time (x,t) and in frequency (f,k) domain, even if at short offsets residuals still remain. Concerning the stacked data, the close-ups Fig. 2 – Comparison between the spatial response of the simulated array composed by 9 evenly spaced and equally weighted sources (blue) and by the optimal Chebyshev weighted sources (red). 68 GNGTS 2013 S essione 3.1