Extrapolation of Seismic Waveforms by Fourier Methods

The problem of constructing a cross section of reflectivity from the wave field recorded at the surface of the medium is discussed with particular reference to migration of seismic records. The numerical procedures are formulated in the frequency and wavenumber domain. The operations are defined in a fixed coordinate system, whereas finite difference methods require a downward-moving reference frame. The numerical algorithms in the frequency wavenumber domain are simpler and give more accurate results than finite difference methods. This is particularly true when the lateral velocity variation in the medium can be neglected. In this case the downward wave extrapolation is accomplished by implementing a phase change in the Fourier coefficients. Numerically, this is equivalent to the multiplication by a complex number of unit modulus. There is no stability condition associated with this operation. This means that the source and recorder positions can be lowered by any amount within one computational step.