Global assessment of tomographic resolution and uncertainty with the SOLA method

Interpretations of seismic tomography and applications of the resulting tomographic images, e.g. for estimating present-day mantle temperatures, require information on their resolution and uncertainty. Assessing these model properties is often difficult due to the large size of tomographic systems on global scales. In consequence, there have been only few attempts to consistently analyse the spatially variable quality of tomographic images of deep mantle structure. For linear problems, both resolution and uncertainty can be quantified with the tools provided by classic Backus–Gilbert (B–G) inversion. In this theory, averaging kernels define the local resolving power at each model parameter, while uncertainties represent the propagation of data errors into the model values. By using a more efficient variant of B–G inversion, the method of 'Subtractive Optimally Localized Averages' (SOLA), global tomography can be performed with complete information for model appraisal.

Based on the SOLA framework, we present a concept for the assessment of the 3-D resolution information contained in a global set of averaging kernels. It is based on the rigorous estimation of resolution lengths from a 3-D Gaussian parametrization of the averaging kernels, together with a test for the robustness of this approximation. This is a necessary step because a perfectly bell-shaped or delta-like behaviour of resolution can not always be guaranteed in global tomography due to the inhomogeneous data coverage. Therefore, we also develop a classification scheme, which enables a basic identification of those averaging kernels that are too complex to be sufficiently described by the chosen definition of resolution length. We note that this approach is more generally applicable, i.e. it can be used with any explicitly available set of averaging kernels or point-spread functions, but also with alternative parametrizations.

In the context of the SOLA method, our resolution analysis can be further used to locally calibrate the inversion parameters. This involves on the one hand the specification of a target (resolution) kernel. On the other hand, a trade-off parameter needs to be selected that regulates the fit of the averaging to the target kernel, and the conversely affected propagation of data errors.

To this end, we apply our concept for robust resolution estimation to different sets of averaging kernels from SOLA inversions with varying parameter combinations. Most notably, we systematically increase the spatial extent of the target kernels (taken as 3-D Gaussian functions here as well). The final maps of global (and classified) resolution and uncertainty can be viewed together for a complete picture of the model quality. They reveal where, and for which target size and amount of error propagation, resolution lengths are meaningful and model values can be interpreted appropriately.