Design of one‐way wavefield extrapolation operators, using smooth functions in WLSQ optimization

Abstract

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 least‐squares 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.