Seismic wave synthesis by Gaussian beam summation: A comparison with finite differences

We apply Gaussian beam summation to the calculation of seismic reflections from complex interfaces, introducing several modifications of the original method. First, we use local geographical coordinates for the representation of paraxial rays in the vicinity of the recording surface. In this way we avoid the time‐consuming evaluation of the ray‐centered coordinates of the observation points. Second, we propose a method for selecting the beams that ensures numerical stability of the synthetic seismograms. Third, we introduce a simple source wave packet that simplifies and stabilizes the calculations of inverse Fourier transforms. We compare reflection seismograms computed using the Gaussian beam‐summation method with those calculated by finite differences. Two simple models are used. The first is a continuous curved interface separating an elastic layer from a free half‐space. A double caustic, or degenerate focal point, appears due to the crossing of reflected rays. In this instance the finite‐difference simulation and the Gaussian beam summation are in excellent agreement. Both phase and amplitude are modeled correctly for both the direct and reverse branches. When compared to geometrical ray theory, Gaussian beam summation provides a good approximation of the field near the caustics while geometrical ray theory does not. The second, more complex, model we consider is a trapezoidal dome with sharp corners in the interface. The corners of the dome in this model produce rather strong diffractions. Also, creeping head waves propagate along the interface. The results compare well with the finite‐difference simulation except for the diffracted branches, where the traveltime of diffracted waves is poorly approximated by the Gaussian beam‐summation method.