Nowadays, with the dramatic increase in computational ability, the discrete wavenumber integration method (DWIM) (see, e.g., Bouchon and Aki, 1977; Bouchon, 1979, 1981) has been one of the most favorable techniques of computing the synthetic seismograms for a layered half-space because of its simplicity, accuracy, and fair efficiency for some cases, particularly for the case of the near field. However, it becomes less efficient for the case of far field, that is, at large epicentral distances, and the larger the epicentral distance is, the less efficient DWIM will be. In this study, we propose an efficient numerical wavenumber integration method, the self-adaptive Filon's integration method (SAFIM), to compute efficiently the dynamic Green's functions for a layered half-space at large epicentral distances. This new integration technique is build upon the particular fifth-order Filon's integration scheme (Apsel, 1979; Apsel and Luco, 1983) and the principle of the self-adaptive Simpson integration technique. By using numerical examples, we demonstrate that SAFIM is not only accurate but also very efficient for large epicentral distances. According to our study, we find that at a relatively short epicentral distance (r < 500 km), the classical DWIM is more efficient than SAFIM; at a medium range of epicentral distance (500 km < r < 1200 km), both methods have similar efficiency; at large epicentral distance (r > 1200 km), however, SAFIM is significantly more efficient than DWIM, and the larger the epicentral distance is, the more efficient SAFIM will be. For instance, when r = 2000 km, SAFIM only needs about 1/3 of the computation time of DWIM. Therefore, this new integration method is expected to be very useful in computing synthetic seismograms at large epicentral distances.

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