Short note: analysis of the non-uniqueness of seismic travel-times through brute-force counting

Kristopher A. Innanen

The questions of how "hard" seismic inversion is, and of how good a solution we can expect, both rest on the uniqueness of a seismic datum (such as a travel time), and the degree to which multiple independent data of the same type can mitigate it. Since large numbers of unknowns are involved, one might think to use the methods of statistical mechanics to characterize this non-uniqueness. Anyway, I did. Statistical analyses usually start with a counting of possibilities (e.g., the number of states of a many-particle system with the same energy). In our case the counting would be of the number of the possible models which produce the same traveltime (or amplitude, or waveform component, etc.). How precisely can we count the number of discrete slowness models which produce a given traveltime?Here we will set out an approach in a short note. Once we have a method for counting, we may be able to glean interesting facts about the inverse problem in a range of experiments and seismic source/receiver configurations.