Many depth migration methods use one-way frequency-space depth extrapolation methods. These methods are generally considered to be expensive, so it is important to find the most efficient way of implementing them. This usually means making spatial convolution operators that are as short as possible. Applying the extrapolation operators in a recursive way, using small depth steps, also demands that the operators do not amplify the wavefield at every depth step.Weighted least squares is an appropriate method to use for designing extrapolation operators that are accurate and efficient and that remain stable in a recursive algorithm. The extrapolated wavefields calculated with these operators are comparable with the extrapolation results obtained with other known operator design techniques as the Remez exchange method and nonlinear optimization. In this paper, the weighted leastsquares technique is refined by using different model functions. By smoothing the phase and amplitude transition at the evanescent cutoff, we can stabilize the resulting operators.The accuracy of the operators is shown in zero-offset migration impulse responses in 2D and 3D media. The Sigsbee2A data set is used to illustrate the quality of the extrapolation operators in prestack depth migration in a complex medium.
SummaryA spatial convolution operator is designed so that it is stable and accurately matches the exact operator in the wavenumber domain in a desired wavenumber range. This can be done by a weighted least-squares optimization. Several applications require the use of an asymmetric operator. By introducing an asymmetric weight function, the properties of the operator can be in uenced to reach an optimal design.
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