ACOUSTIC MODELLING ON A GRID OF VERTICALLY VARYING SPACING¹

The Fourier spectral method and high‐order differencing have both been shown to be very accurate in computing spatial derivatives of the acoustic wave equation, requiring only two and three gridpoints per shortest wavelength respectively. In some cases, however, there is a lack of flexibility as both methods use a uniform grid. If these methods are applied to structures with high vertical velocity contrasts, very often most of the model is oversampled. If a complicated interface has to be covered by a fine grid for exact representation, both methods become less attractive as the homogeneous regions are sampled more finely than necessary.

In order avoid this limitation we present a differencing scheme in which the grid spacings can be extended or reduced by any integer factor at a given depth. This scheme adds more flexibility and efficiency to the acoustic modelling as the grid spacings can be changed according to the material properties and the model geometry. The time integration is carried out by the rapid expansion method. The spatial derivatives are computed using either the Fourier method or a high‐order finite‐difference operator in the x‐direction and a modified high‐order finite‐difference operator in the z‐direction. This combination leads to a very accurate and efficient modelling scheme. The only additional computation required is the interpolation of the pressure in a strip of the computational mesh where the grid spacing changes.