GNGTS 2015 - Atti del 34° Convegno Nazionale

GNGTS 2015 S essione 3.3 153 Optimal parameters for finite difference modeling of 2D seismic wave equation B. Galuzzi 1 , E. Stucchi 1 , E. Zampieri 2 1 Dipartimento di Scienze della Terra “A. Desio”, Università degli Studi di Milano, Italy 2 Dipartimento di Matematica “F. Enriques”, Università degli Studi di Milano, Italy Introduction. Full waveform inversion is a classical contest of data inversion in which the numerical solution of the wave equation is compared with one or more seismograms to obtain information on the Earth’s subsurface (Tarantola, 1986), �������� ��� ������� ������ (Virieux and Operto, 2009). Consequently, accurate and efficient numerical implementation of the wave equation is still an active research field and involves sampling quantities such as time, space and physical properties of the subsurface, along with choosing an appropriate numerical method of resolution and writing an efficient resolution code. Approximation error and execution time determine the effectiveness of the implementation. An effective code exhibits the right balance between these two factors because the use of high-resolution parameters to decrease the approximation error causes a large execution time, which, for seismic inversion applications, should remain in the order of a few seconds or lower. For example it may be necessary to use about ten thousand or one hundred thousand synthetic seismograms to resolve a problem of seismic inversion by global optimization algorithms ������� (Sajeva et al. , 2014). In this work we study the relationship between these two factors in the contest of the numerical solution of the 2D acoustic wave equation. The numerical solution is obtained from finite difference software, written at the University of Milan, in which the implementation parameters can be set in order to get an efficient solution. At the beginning we derive the 2D acoustic seismic wave equation and explain the numerical implementation and the parameters of modeling used. Then, we analyze which parameters cause the highest approximation error and find that they are the space step size and the order of approximation of space derivatives. We study their behavior in a simple constant-velocity model as a function of the maximum frequency of the source signal and we analyze the relation with the execution time. Finally, we apply these considerations on a complex-velocity model and find the right parameters of modeling to get the optimum trade-off between the approximation error and the execution time. The 2D acoustic seismic equation. The seismic wave propagation in a geological medium is often modeled by the acoustic 3D equation ��������� ����� (Fichner, 2010) with p acoustic pressure of the wave, f seismic source and c acoustic wave speed. A realistic range for wave speed can be between 1500 m/s (water) and 7000 m/s (granite). Since the seismic source has a space dimension much smaller than the geological medium, it can be approximated by a point source in space where s ( t ) is the seismic wavelet, describing the variation of seismic source in time. One important aspect of the modeling is the maximum frequency f max of the wavelet. Since many source-receiver geometries are often confined to a plane (for example y =0), it is possible to use the acoustic 2.5-D equation (Bleinstein, 1986) which differs from the 3D equation only for the fact that c varies only as a function of the depth z and the length x . Finally, because of the large computational cost of 3D modeling, we consider the 2D acoustic wave equation