GNGTS 2015 - Atti del 34° Convegno Nazionale

158 GNGTS 2015 S essione 3.3 (Bourgeois et al. , 1991). We considered absorbing boundary conditions for lateral and bottom sides. On the contrary, for the topside we considered a reflective boundary condition (expressed by p ( x , 0, t )=0, A x, t ) to simulate the high contrast of velocity and density between air and water. For this modeling there is not an exact solution to be compared with the numerical ones. So as “exact” solution we use a numerical one with dx =1 m, dt =0.000125 s and ord s =4. To study the behavior of error and execution time, we consider again two different grid cell size dx =[27 m, 9 m], twelve different space orders of approximation ord s =2,4…24 and a wavelet Ricker with f s =10 Hz ( f max =30 Hz). The two execution time curves as a function of ord s are the same of Fig. 1b (execution time is not influenced by the velocity c ), while the two curves of error are represented in Fig. 2b. The behavior of these curves is similar to that of the previous test, but the error is higher. Indeed there are more seismic events in the seismogram for this test (reflected and refracted arrivals) than in the previous one. The curve of dx =27 m decreases slowly as a function of ord s , while the one of dx =9 m decreases fast until ord s =4, with the error remaining stable. We can note that, also in this case, it can be more convenient to use a space step of dx =27 m with high order ord s , rather then a space step of dx =9 m and low ord s . The errors we obtained in fact are comparable, but the execution time of the first case is lower (Fig. 1b). In Fig. 3 we reported a portion of three numerical seismograms we obtained. The first, on the left, is the seismogram obtained with dx =9 m and ord s =4, that corresponds to the better solution obtained with dx =9 m. The second, in the center, is the seismogram obtained with dx =9 and ord s =2. The third, on the right, is the seismogram with dx =27 and ord s =24, that corresponds to the best solution obtained with dx =27. All of them are confronted with the “exact solution” (the red seismograms on the graphics). We note that the third seismogram is better than the second one and, from Fig. 1b, it also has a lower execution time. On the other hand the first seismogram is better than the third one, even if the differences are not so pronounced, but it has a higher execution time. Therefore it can be more convenient to choose the modeling parameters used to compute the third seismogram instead of using the modeling parameters of the first one, especially if a huge number of forward modeling is required. Conclusions. Using a constant velocity model and a portion of the Marmousi model, we studied the 2D acoustic seismic wave equation and the parameters of modelling necessary to implement an efficient numerical solution as a function of approximation error and execution time. The approximation error depends on the stability condition and the grid dispersion relation. We found that under the stability condition, the approximation error is above all influenced by the space step size dx and the order ord s of the finite difference approximation of spatial Fig. 3 – Three portions of three different seismograms for different modelling parameters. On the left dx =9 m, ord s =4. In the center dx =9 m, ord s =2. On the right dx =27 m, ord s =24. All of them are confronted with the exact seismograms, in red.