Imaging by Wavefield Extrapolation
In Chapter 2, we have seen that the wavefi eld on a horizontal plane can be derived from the wavefi eld observed on another horizontal plane given that the space lying between the two planes is homogeneous and free from any sources. This operation is called extrapolation of wavefi eld; forward extrapolation is when we go away from the sources and backward extrapolation is when we go toward the sources (Figure 10.1). Wavefi eld extrapolation enables us to map the source distribution provided it is known that all sources are confi ned to a layer. This problem is known as an inverse source problem . A slightly different situation arises in the scattering problem. An external source induces a fi eld on the surface of a scatterer, which in turn will radiate wavefi eld, known as scattered fi eld, back into the space. This scattered fi eld contains information about the scatterers. The inverse scattering problem pertains to extraction of information about the scatterers from the scattered fi eld. The tomographic imaging covered in Chapter 7 falls in the realm of the inverse scattering problem. In the present chapter, we seek a means of reconstructing a layered (but not necessarily horizontally layered) medium using the refl ected wavefi eld. This problem is of great signifi cance in seismic exploration, where it is commonly known as migration. An image of subsurface refl ectors can also be achieved through focused beamformation, which gives an estimate of the refl ected energy received from a subsurface point. The focused beamformation is based on a ray theoretic description of the wavefi eld, as in optics, but the migration is based on diffraction properties of the wavefi eld. Both approaches lead to similar results. For imaging, an essential input is the wave speed, which, fortunately, has to be estimated from the observed wavefi eld only.