GNGTS 2015 - Atti del 34° Convegno Nazionale

GNGTS 2015 S essione 3.3 157 constant. On the contrary, if we want a space step size of dx =27 m (the continuous green curve), it would be necessary that n ≈1.2. However, this is not possible because of the Nyquist theorem and consequently the solution will have a great numerical dispersion. If we place c min =1500 m/s and f max =15 Hz, (red curves in Fig. 1a) we obtain Therefore, if we want to use a space step size of dx =9 m, (the dashed red curve) it is sufficient to have n ≈11.1, which corresponds from Tab. 2 to ord s ≈4; a greater order of approximation will not produce significant improvements because the error remains almost constant hereafter. On the contrary, if we need to use a space step of size dx =27 m, we must set n ≈3.7, so it will be necessary a higher order ord s . As a final consideration the error, especially for low dx , seems to converge to a low constant value different from zero for each maximum frequency. This is due to the fact that the error related to time discretization takes over when the error related to space discretization decreases. This error is low for low maximum frequency f max , and increases with f max . In addition to this consideration about approximation error, it is important to evaluate the execution time. In Fig. 1b we represent the two execution time curves, in function of ord s , related to dx =[27 m, 9 m] with f max =15 Hz (for different maximum frequency the execution time did not change). We can note that execution time increases in function of dx and ord s . In particular the curve with dx =27 m is always below that with dx =9 m. This is due to the numerical method implemented, since we have where T is the execution time and nx * nz is the number of grid nodes. Then, the execution time increases quadratically as a function of dx and only linearly with ord s . Therefore it can be more convenient to increase ord s rather than decreasing dx , to obtain comparable error but with a lower execution time. As an example, the modeling with dx =9 m and ord s =4 has an error comparable with the modeling with dx =27 m and ord s =24. Moreover the first has an execution time of 11 s, while the second of only 5 s. A complex-velocity test. We simulated another seismic acquisition with the same parameters of acquisition as the previous one: one point source, the same numbers of receivers, a registration time of T =2 s and a time step dt =0.0005 s. We now assumed that the velocity varies as a function of the depth and length c = c ( x, z ) (Fig. 2a), with a range between 1500 m/s and about 4500 m/s. This velocity model is a readjustment of a portion of the Marmousi model Fig. 2 – Complex-velocity model (left) and its approximation error (right). a b

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