Appendix 13A Some remarks on the kinematical and dynamical aspects of multi-valued operators

When the velocity field of the subsurface contains strong lateral variations, traveltimes of reflection and/or transmission data can become multi-valued. Migrating these data with operators that contain first arrivals only causes migration artefacts, because energy contained in later arrivals is not accounted for (Geoltrain and Brac, 1993). The most simple solution to this problem is to account only for the most energetic event. More advanced techniques employ operators based on multi-valued Green’s functions [Nolan and Symes (1996), Ehinger et al. (1996), ten Kroode and Smit (1997), Van Barneveld and Verschuur (this chapter)]. Since multi-valued Green’s functions account for all arrivals at the acquisition surface, they handle the amplitudes better than single-valued operators. However, it is generally overlooked that even migration operators based on multi-valued Green’s functions ignore back-scattered energy (or to say it in another way, they do not compensate for transmission losses). Hence, although they improve the amplitudes of the migration output, they do not provide the final answer. In this appendix we will discuss this phenomenon in more detail and illustrate it with a numerical example. But first we review some kinematical aspects of multi-valued operators.