Velocity Inversion

The next step is to invert the seismic trace for the sonic velocities. This is done by matching candidate traces to the synthetic trace obtained from the log (right panel in Figure 1). I used a micro-genetic algorithm since it proved to be superior to a standard genetic algorithm for this problem Alvarez (2002). To perform the inversion, the micro-genetic algorithm starts with a small random population of synthetic sonic logs from which trial solutions, in the form of synthetic seismic traces, are generated using the procedure described in the previous section. Each of these traces is matched to the synthetic trace obtained from the real log and a figure of merit (fitness value) assigned according to the ``goodness'' of the match. Two possible choices of fitness functions are described below. The genetic algorithm ``evolves'' the solution according to the rules of evolution and survival of the fittest Goldberg (1989) until a satisfactory match is achieved between the synthetic seismic trace from the real log and a trial synthetic trace. Once this match is achieved, the corresponding well log will hopefully be a good match to the real one (given the restrictions of no multiple reflections and no attenuation).

This is the ideal situation but in reality we must deal with the important issue of the impossibility of directly recovering the general velocty-depth trend present in the sonic log. This happens because the seismic data lack the very low frequencies. This issue is obviously very important and will be addressed in some detail below.