On the nonuniqueness of the coupled origin time-velocity tomography problem

When earthquakes are used as sources in velocity tomography, the unknown velocity structure and the unknown hypocentral parameters (that is, source location and origin time) must be simultaneously estimated during the imaging process. This coupling allows the two sets of unknowns to trade off, and increases the degree of nonuniqueness of the resulting tomographic image above what would have been present had the hypocentral parameters been precisely known. We analyze, in detail, the nonuniqueness associated with unknown origin time, which we argue is often a more important source of nonuniqueness than is unknown location. While this type of nonuniqueness has long been understood to be a problem in teleseimic tomography, we show here that it is of equal importance in all coupled problems. We provide a practical method for calculating null solutions and calculate them for several commonly encountered experimental geometries. We also show that the attenuation tomography possesses a mathematically identical nonuniqueness, with unknown source amplitude being the analogue to unknown origin time.