GNGTS 2015 - Atti del 34° Convegno Nazionale

154 GNGTS 2015 S essione 3.3 In general a 2D modeling of wave propagation cannot be used in general to make a direct quantitative comparison, including amplitude information, with seismic data acquired along a line and assumed to be 2.5-D, but there are many strategies that make the passage from 2D to 2.5-D possible ������� ����� ���������� ��� ������ ����� ���� ��� ����������� ������ (Liner, 1991; Williamson and Pratt, 1995; Song and Williamson, 1995)�. Modeling of the acoustic seismic equation. In our numerical implementation of the acoustic wave equation we use an explicit finite difference method ������� ������ ���� ������� ���� (Cohen, 2002), with uniform time step and uniform space step (with the same step for depth and length). In order to approximate the time and space derivatives we use different finite difference operators. We implement a second order operator to approximate the time-derivative and implement a 2n-order space operator to approximate the space derivatives where the c i are the coefficients for the 2n-order of approximation of derivatives (Cohen, 2002). Tab. 1 lists the values of c i , obtained as a function of the order of approximation ord s =2 n , for some values of n . Tab. 1 - Values of the coefficients for different order of approximation of spatial derivative. ords c1 c2 c3 c4 c5 c6 2 1 0 0 0 0 0 4 4/3 -1/12 0 0 0 0 6 3/2 -3/20 1/60 0 0 0 8 8/5 -1/5 8/135 -1/560 0 0 10 5/3 -5/21 5/126 -5/1008 1/3150 0 12 12/7 -15/56 10/189 -1/112 2/1925 -1/16632 In the interest of computational efficiency, the limitation of the computational domain to only a part of the true physical domain introduces reflecting boundaries that do not exist. We use the Gaussian taper method (Cerjan et al. , 1985) to suppress the undesired reflections, in which we introduced a thin absorbing region along the artificial boundary where the wave field is attenuated. Approximation error of the implementation. There are four main parameters that influence the approximation error of the implementation: the time step dt , the space step dx , the order of approximation of the space operator ord s =2 n and the size of the absorbing region, expressed as the number of grid nodes. To focus the attention only on the first three parameters, we can choose an absorbing region so large as to make irrelevant the error introduced by the boundaries of the model. There are two main relations between these three parameters. The first is numerical stability (Courant et al. , 1967) with λ = λ ( ord s ) ∈ [0.5,1], that is the Courant number. This relation limits the maximum possible