Where should we focus?
Seminar Room 1, Newton Institute
Seismic imaging or migration maps singly scattered data into the subsurface, providing an image of the interfaces between rock formations with different impedances. The corresponding linear inverse problem is the minimization of the least-squares error subject to the Born approximation of the acoustic wave equation.
Substantial preprocessing is usually required to remove data that do not obey the single scattering assumption. Also, an accurate background velocity is needed. Migration velocity analysis exploits the redundancy in the data to estimate the background velocity model. Data for different shot-receiver distances or offsets should provide the same image of the subsurface. Its implementation for the full wave equation invokes action at distance via a subsurface shift in space or time. Figure 1 shows a real-data example. The corresponding cost functional tries to focus energy at zero subsurface shift, thereby suppressing the unphysical action at distance.
Although removal of surface multiples is a common technique, interbed multiples as well as remnant surface multiples may still lead the focusing algorithms astray. Focusing in the data domain is a recent generalization that, in principle, should not suffer from the presence of surface and interbed multiples. Further development is, however, still required to mature the method.
Figure 1. Example of seismic velocity inversion with focusing based on horizontal shifts in the depth domain, starting from the best velocity model that increases linearly with depth. The left panel shows the extended image at a lateral position of x = 2 km, as a function of horizontal subsurface offset hx and depth z. The iteration count is displayed in the left upper corner. The central panel displays the migration image. The right one shows the reconstructed smooth background velocity model.