| Abstract: | Alekseev and Mikhailenko have developed a wavenumber-summation method which combines a finite integral transformation with a finite-difference calculation and involves no approximations other than numerical ones. However, numerical anisotropy causes velocity errors for shear waves which are unacceptable if Poisson's ratios are larger than 0.4 and unless the number of grid points per wavelength is chosen considerably higher than the value generally regarded as sufficient in finite-difference computations. To overcome this limitation in the applicability of the otherwise very powerful modelling scheme, the method is applied to the elastodynamic equations for the velocity vector. Thus, instead of solving a second-order hyperbolic system as in the case of the wave equation, solutions to a first-order hyperbolic system are computed. The finite-difference iteration is performed in a staggered grid. In addition to mastering the numerical difficulties in cases where the Poisson's ratio is unusually high, this approach results in a code which can be used for the modelling of liquid layers. With the new scheme, water reverberations are investigated in terms of normal modes. It is found that for realistic sea-bottom velocities the critical and supercritical cases exist only for P-waves. It means that compressional waves are trapped within the water layer but energy leaks into the substratum through converted shear waves. These leaky compressional normal modes attain properties similar to those of shear normal modes or Pseudo-Love waves. Due to their origin from conversion of dispersed multi-modal compressional waves the shear waves generated at the sea-bottom form a long complex wavetrain. They were found to mask the reflections from the target horizon in an offset-VSP field section. |
