Investigating uncertainties in travel time tomography using the null space shuttle

Abstract

Due to the underdetermined nature of large tomographic inverse problems, a sizable null space exists. It is therefore important to investigate the uncertainties in tomographic models produced by inverse problems with multiple solutions. Conventional methods to analyze model quality all have shortcomings, which we discuss in this report. As an alternative, we use the the null space shuttle technique (Deal and Nolet, 1996). This method has been designed to exploit components of the model null space, along with a priori information or a physical model, in order to improve or enhance the original solution. We generalize the null space shuttle technique to analyze the robustness of a global P wave speed perturbation mantle model PMIT08 produced by a classical study of travel time tomography (Li et al., 2008) and examine a range of models that is consistent with the travel time data. We find that the RMS amplitude of velocity perturbations within this set of solutions ranges from 0.2 to 0.6 % in the lowermost mantle and 0.3 to 1.3 % in the upper mantle and deduce that the travel time data provide little constraint on the amplitudes of the solution. Furthermore, solutions exist that contain structures different in geometry than those imaged in PMIT08. Upper mantle portions of slab-like anomalies are altered or removed from the tomogram in some regions, while the vertical extent of low-velocity anomalies in PMIT08 appears to be poorly constrained. On the contrary, most slab-like anomalies in the lower mantle seem to be robust. We strongly advise against physical interpretations of a single solution of an underdetermined tomographic problem. The null space shuttle technique is straightforward to implement and can be utilized in an efficient way to analyze the solution robustness. We suggest that this technique should be routinely applied before physical interpretation of tomographic images are made.