GNGTS 2015 - Atti del 34° Convegno Nazionale

GNGTS 2015 S essione 3.3 155 time step as a function of the space step size dx and the maximum velocity c max . The second is the numerical dispersion (Alford et al. , 1974) where n = n ( ord s ) is the number of points per wavelength. Grid dispersion limits the maximum possible space step by the minimum velocity cmin and the maximum frequency f max of the source signal s ( t ).We consider the function cos , where l denotes the wavelength, to estimate the values of n. If we calculate the second derivative of this function analytically at x =0, and we set l =1, we obtain -4π 2 . The numerical solution can be obtained by sampling the function with different sampling intervals dx = l/n , where n is the number of points per wavelength and by using different 2n-order operators. Tab. 2 lists the values of n , as a function of the order of approximation ord s , to obtain an error below 1%. Numerical stability is a necessary condition to implement any explicit finite difference method. Because of the order of magnitude of the wave speed in rocks, the stability condition implies that the maximum possible time step must be approximately three order of magnitude at least lower than the minimum possible space step size dt max ≈10 -3 dx min ; if this condition is met, the error of approximation is more sensitive to spatial parameters ( dx , ord s ). For this reason, we study the error of approximation and execution time as a function of dx and ord s . Aconstant-velocity test. To simulate a simple seismic acquisition, we consider a rectangular region with dimensions X=3240 m and Z=1620 m, with a constant velocity of c =1500 m/s (water velocity). We choose a seismic source located in ( x 0 , z 0 )=(27 m, 27 m), characterized by a Ricker wavelet with a =π 2 f 0 2 and t 0 =0.2 s . The choice of this wavelet is so because it is simple to control its maximum frequency, that is f max ≈3 f 0 . The recording spread is composed of 119 receivers, equally spaced by 27 m, with a depth of 27 m and also an offset of 27 m to the first receiver. In the case of constant velocity model without boundary, the solution of the equation is (Aki and Richards, 2002) with t s that depends on the duration of the wavelet, usually much smaller than the duration of registration T . If we use a quadrature formula to approximate the integral, we have a solution of the problem whose accuracy is independent of distance and time, but depends only on the accuracy of the quadrature formula. This procedure allows building an “exact” solution that can be compared with our numerical solutions (we can consider our model as unbounded using a large absorbing boundary condition). A numerical implementation of this problem was made with a fixed time step of dt =0.0005 s and a recording length of T =2 s . To study the behavior of the approximation error and execution time, we consider two different grid cell size dx =[27 m, 9 m], twelve different space orders of approximation ord s =[2,4…24] and three different wavelets with f s =[5 hz, 10 hz, 15 hz] (which correspond to f max =[15 hz, 30 hz, 45 hz]). To compare the numerical solution with the “exact” solution, we use the following measure of Tab. 2 - Values of n as a function of ords, to obtain an error below 1%. ord s 2 4 6 8 10 12 14 16 18 20 22 24 n 18 6.3 4.5 3.75 3.5 3.25 3 2.9 2.8 2.7 2.6 2.5