Wave field extrapolation techniques in seismic migration, a tutorial

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Wave field extrapolation techniques in seismic migration, a tutorial

A. J. Berkhout

https://doi.org/10.1190/1.1441172

The objective of this paper is to provide a general view on methods of wave field extrapolation as used in seismic modeling and seismic migration, i.e., the Kirchhoff‐summation approach, the plane‐wave method (k–f method), and the finite‐difference technique. Particular emphasis is given to the relationship between the different methods. By formulating the problem in the space‐frequency domain (x, y, ω‐domain), a systems approach can be adopted which results in simple and concise expressions. These expressions clearly show that forward extrapolation is described by a spatial convolution procedure and inverse extrapolation is described by a spatial deconvolution procedure. In the situation of lateral velocity variations, the (de)convolution procedure becomes space‐variant. The space‐frequency domain is most suitable for recursive depth migration. In addition, frequency dependent properties such as absorption, dispersion, and spatial bandwidth can be handled easily. It is shown that all extrapolation methods are based on two equations: Taylor series and wave equation. In the Kirchhoff‐summation approach all terms of the Taylor series are summed to an exact analytical expression—the Kirchhoff‐integral for plane surfaces. It formulates the extrapolation procedure in terms of a spatial convolution integral which must be discretized in practical applications. The Fourier‐transformed version of the Kirchhoff‐integral is used in the plane wave method (k–f method). This actually means that spatial (de)convolution in the x, y, ω‐domain is translated into multiplication in the kx, ky, ω‐domain. Of course, this is not allowed if the extrapolation operators are space‐variant. In explicit finite‐difference techniques a truncated version of the Taylor series is used with some optimum adjustments of the coefficients. For only one or two terms in the Taylor series, a spatial low‐pass filter must be applied to compensate for the amplitude errors at high tilt angles. Explicit methods are simple and most suitable for three‐dimensional (3-D) applications. In implicit finite‐difference schemes the wave field extrapolator is written in terms of an explicit forward extrapolator and an explicit inverse extrapolator. Properly designed implicit schemes do not show amplitude errors and, therefore, amplitude correction filters need not be applied. In comparison with explicit schemes, implicit schemes are more sensitive to improper boundary conditions at both ends of the data file. It is shown that the forward seismic model can be elegantly described by a matrix equation, using separate operators for downward and upward traveling waves. Using this model, inverse extrapolation involves one matrix inversion procedure to compensate for the downward propagation effects and one matrix inversion procedure to conpensate for the upward propagation effects.