Wave-equation imaging

The previous two operators have in common the description of the image in the time domain. However, there are several limitations describing a converted-wave image in the time domain. NMO requires knowledge of two parameters and although the two parameters can be estimated very accurately, NMO is still an approximation and will not produce an accurate image in geologically complex areas. DMO also presents limitations, as for example the transformation from CMP to CRP. There are several other operators that will also provide a better image of converted-wave data, such as, poststack and prestack time migration. One of the main ideas behind this thesis is to prove that the ideal domain to describe and interpret converted-wave images is in depth.

Recent advances in computer power make it practical to use prestack depth migration as an imaging operator. This operator directly transforms our data, which is in data-midpoint position, data-offset, and time coordinates [(m_D,h_D,t)], into an image that is in image-midpoint location, image-offset, and depth coordinates [($m_{\xi},h_{\xi},z_{\xi}$)].

Prestack depth migration can be achieved using Kirchhoff migration methods, which are based on a high-frequency approximation of the wave-equation. Kirchhoff methods require the summation of the data over complex multivalued surfaces, and cannot correctly handle complex subsurfaces. The high-frequency approximation breaks down for very complex subsurface structures.

Wave-equation-based methods provide a solution to the wave-equation along the entire range of frequencies, wave-equation-based methods can handle complex subsurface structures such as overthrusts, salt bodies, and others. There are several ways to implement wave-equation migration, which differ mainly in propagation of the wavefield and the imaging condition. The next section describes the basics of prestack wave-equation imaging, Its main purpose is to describe the algorithm I use to transform the prestack data into the prestack image space, and describe the elements of this space.

 

• Downward-continuation migration