Jörg Dalkolmo, Wolfgang Friederich
Gemini II is a program package to calculate exact Green's functions for the elastic wave equation in one-dimensional, isotropic or transversely isotropic, depth dependent media. Applications of the code range from high-frequency, small-scale wave propagation problems like ultrasonic waves, seam waves and shallow seismics to continental-scale seismic waves from earthquakes.
Gemini II is derived from an older version (Gemini I) originally designed to solve the elastic wave propagation problem for a spherically symmetric, isotropic or transversely isotropic earth model. A description of the underlying mathematical approach can be found in the paper by Friederich and Dalkolmo, Geophysical Journal International, 122, 537-550, 1995. <p> The basic solution method is a transformation of the elastic wave equation into the frequency domain followed by a separation of variables leading to Helmholtz equations describing the angular dependencies of the displacement field and systems of ordinary differential equations (ODE) describing the vertical dependence. The spherical Helmholtz equations are solved by spherical harmonics. The vertical systems of equations are solved numerically by a Bulirsch-Stoer adaptive step size approach with Richardson extrapolation. To avoid numerical instabilities, the vertical ODEs are transformed to new variables which represent second order minors of the relevant basis solutions. Attenuation is included in an exact manner by using complex elastic constants. Once the vertical ODEs are solved, the displacement can be computed as a spherical harmonics expansion. This representation of the solution is implemented in the earlier version Gemini I. It can be obtained from the SPICE website (www.spice-rtn.org). The method can also be used for regional wave propagation, shallow seismics or even ultrasonics, with only slight modifications. The numerical integration in vertical direction remains unchanged whereas the spherical harmonics are replaced by an asympotic expression in terms of Bessel functions. They can be stably and efficiently calculated up to high frequencies and wavenumbers. The drawback is that they are no longer valid for large epicentral distances in case of global wave propagation. This approach is implemented in Gemini II together with a massive reorganization of the entire program.