Seismic migration is used to image recorded reflection and scatter events based on wave theoretical approaches, by de-propagating them to their true subsurface positions. Wave propagation and de-propagation is based on perturbation theory. Nowadays, prestack depth migration (PSDM) has become a most effective tool for imaging complex geological structures. Assuming the macro-velocity field is accurate enough, the migration operators or wave propagators should be competent for accurately expressing wave propagation in media with severe lateral velocity variations.
Stoffa et al. (1990) introduced a first-order split-step correction, which is accurate for flat reflectors. However, the accuracy of the migration operator for depicting wave propagation decreases as reflectors become steeper and the differences between the reference and migration velocities increase. Therefore, Ristow and Ruhl (1994) put forward the Fourier finite-difference (FFD) operator, and Huang et al. (1999) and Rousseau and de Hoop (2001) introduced the generalized-screen propagator GSP, to compensate for high-order terms neglected when imaging steeply dipping reflectors with the Split-Step Fourier SSF operator. The FFD operator deals with the high-order terms in the frequency-space domain; the GSP processes them in the frequency-wavenumber domain. The FFD operator requires that the reference velocity is lower than the minimum velocity in an extrapolation layer, because the sign of the coefficients in the finite-difference equation can not be changed at different lateral points in a depth layer. Otherwise, calculation instability will occur. Therefore, even if a set of suitable reference velocities is given, the FFD operator cannot give a very good image.
Biondi (2002) modified the general FFD operator by introducing the interpolation of two wavefields: the first wavefield is obtained by applying the FFD correction, starting from a reference velocity lower than the medium velocity; the second wavefield is obtained by applying the FFD correction starting from a reference velocity higher than the medium velocity.
In fact, the high-order terms in wave-propagation perturbation theory are generated by the velocity perturbation between the true velocity and the reference velocity. If the reference velocity is chosen as close as possible to the true velocity, the SSF, FFD and GSP operators can accurately characterize the wave propagation in media with severe lateral velocity variations. Futhermore, the imaging quality can be improved with such propagators.
Assuming that the macro-velocity model for imaging is accurate enough, the main reason for most imaging errors is that the one-way wave equation can not accurately characterize the wave propagation in the case of severe lateral velocity variations. In such cases, steeply dipping reflectors cannot be clearly imaged. We can decrease these errors either by using an optimized, but complex propagator, or by choosing a set of reasonable reference velocities and using a simple propagator. We think that the latter is much more flexible and cost-effective. Of course, both approaches can be combined together.
Clapp (2004) proposed that the reference velocity can be selected by a generalized Lioyd method. The basic idea behind Lioyd's method is to iteratively improve the quantization of a function by looking at the velocity statistics of each region (such as mean, median, and variance), and then changing the boundaries of the regions at each iteration to find the solution which is optimal based on some criterion.
Now, we introduce a self-adaptive strategy for selecting a reasonable reference velocity in the presence of lateral velocity variations. The main steps include setting a threshold for the ratio between two adjoining reference velocities, sorting the velocity slice into an array, and detecting the edges of velocity regions using velocity averages and variances in the different regions. The resulting reference velocity field and the imaging results produced with it indicate that our approach is correct and effective. Meanwhile, the method is flexible in use, computationally efficient, and easy to program in either 2D or 3D. The approach can also be used for 2D or 3D image edge detecting. Numerical tests demenstrate that the method is effective.