GNGTS 2015 - Atti del 34° Convegno Nazionale

156 GNGTS 2015 S essione 3.3 numerical error: where temp =4000 is the number of time samples, nric =119 is the number of receivers, real i,j is the “exact” seismogram and sint i,j is the numerical one. Both seismograms are normalized to their maximum value. Fig. 1a shows the six curves (two for each frequency f s ) of the approximation error as a function of ord s . It is possible to see that the error increases with dx and f max , according to the relation of grid dispersion. We note also that the curves of approximation decrease, in general with ord s , but this behavior depends also on dx and f max . In particular, for frequency f max =45 Hz (the green curves), we note that the curve with dx =27 m slowly decreases as a function of ord s , while the one with dx =9 m decreases fast until ord s ≈8. Then we notice that the error remains stable around 5*10 -5 . Similarly, for frequency f max =30 Hz (the blue curves), the curve with dx =27 m slowly decreases as a function of ord s , while the one with dx =9 m decreases fast until ord s =6. Here the error remains stable around 2.5*10 -5 . Finally, for frequency f max =15 Hz (the red curves), the curve with dx =27 m decreases fast as a function of ord s until ord s =8 and then remains stable around an error of 2*10 -4 , while the curve with dx =9 m, decreases fast until ord s =6, it remains nearly constant until ord s =12 and finally increases slowly. Overall it remains stable around an error of 1*10 -5 . Therefore the whole behavior of the curves does not appear to be simple. However, for high frequencies, the better way to reduce the error is to decrease dx and slightly increase ord s . Increasing only the order of approximation ord s seems to bring only minor improvements. Instead, for low frequencies, there is no need to use short step sizes: dx =27 m with ord s ≈8 gives a sufficiently low approximation error, without the necessity to further increase ord s . For middle frequencies, the error appears to be more complex. However, using a short dx with a low ord s is a good compromise, while an equally valid solution is to use a high order of approximation ord s with higher space step sizes dx . The behavior of modeling as a function of dx , ord s and f max can be explained by the grid dispersion relation. If we place c min =1500 m/s and f max =45 Hz, (green curves in Fig. 1a) we obtain Therefore, if we want to use a space step size with dx =9 m (the dashed green curve in Fig. 1a) it is sufficient to have n ≈3.7, which corresponds from Tab. 2 to an ord s ≈8; greater order of approximation will not produce significant improvements, because the curve remains almost Fig. 1 – Approximation error (left) and execution time (right) for different modeling parameters. a b