Toward quantifying uncertainty in travel time tomography using the null-space shuttle

The solution of large linear tomographic inverse problems is fundamentally non-unique. We suggest to explore the non-uniqueness explicitly by examining the null-space of the forward operator. We show that with the null-space shuttle it is possible to assess robustness in tomographic models, and we illustrate the concept for the global P-wave model MIT-P08. We found a broad range of acceptable solutions compatible with the travel time data. The root mean square (RMS) velocity perturbations vary from 0.2 to 0.6% in the lowermost mantle and from 0.3 to 1.3% in the upper mantle. Such large variations in average amplitudes prohibit meaningful inferences on temperature or chemical variations in the Earth from tomographic models alone. On a global scale much short wavelength structure resides in the null-space of the forward operator, suggesting that the data do not everywhere resolve structure on the smallest length scale (<200 km) allowed by the (block) parameterization used in MIT-P08 and similar models. This indicates that great care should be taken when interpreting such structure. As a practical measure, we suggest that only those structures for which the wave speed perturbations do not change sign within the range of models permitted by the data should be considered robust. With this criterion, the model null-space analysis shows that the high velocity anomalies in the lower mantle, which are often interpreted as remnants of slabs of subducted lithosphere, are required by the seismic data. Low-velocity anomalies underneath, for instance, Hawaii, Iceland, and Africa show varying degrees of robustness.