FERTIG, J. 1984, Shear Waves by an Explosive Point-Source: The Earth Surface as a Generator of Converted P-S Waves, Geophysical Prospecting 32, 1-17.The most common source of seismic energy is an explosion at some depth in a borehole.The radiated waves are reflected not only at the subsurface layers but also at the free surface.The earth's surface acts as a generator of both P-and S-waves.If the source depth is much less than the dominant wavelength the reflected waves resemble closely the waves generated by a single force. Theoretical seismograms were computed with different methods to look for the relevance of the surface-reflected waves. The numerical experiments show reflected shear waves even for small shotpoint-receiver distances. Due to their polarization these waves can be detected most easily on in-line horizontal geophones. The existence of these waves was examined during a conventional survey in Northern Germany. Conventional data analysis shows a large variability in the up/u, ratio. The method used here produced a shear-wave section with a rather good signal-to-noise ratio down to 4 s S-wave reflection time., I i
I N T R O D U C T I O NThe general formulation for the displacements in an elastic medium leads to very complicated formulae. The general solution of these equations is unknown. Only simplifications of the problem or approximations of the medium lead to equations which we can solve analytically, or at least approximately. We have the most complete knowledge for waves propagating in a homogeneous, infinite, isotropic, and ideally elastic medium. In a medium with so many contraints we have a wavepropagation of two independent kinds of waves. The most important one is the P-wave which is faster than the S-wave. Both propagation velocities depend on the elastic parameters of the medium. These velocities describe the kinematical aspects of wave-propagations. The causes of these waves can be described by single forces.
Phase‐shift migration techniques that attempt to account for lateral velocity variations make substantial use of the fast Fourier transform (FFT). Generally, the Hermitian symmetry of the complex‐valued Fourier transform causes computational redundancies in terms of the number of operations and memory requirements. In practice a combination of the FFT with the well‐known real‐to‐complex Fourier transform is often used to avoid such complications. As an alternative means to the Fourier transform, we introduce the inherently real‐valued, non‐symmetric Hartley transform into phase‐shift migration techniques. By this we automatically avoid the Hermitian symmetry resulting in an optimized algorithm that is comparable in efficiency to algorithms based on the real‐to‐complex FFT. We derive the phase‐shift operator in the Hartley domain for migration in two and three dimensions and formulate phase shift plus interpolation, split‐step migration, and split‐step double‐square‐root prestack migration in terms of the Hartley transform as examples. We test the Hartley phase‐shift operator for poststack and prestack migration using the SEG/EAGE salt model and the Marmousi data set, respectively.
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