Robust estimates of the ratio between S- and P-wave velocity anomalies in the Earth's mantle using normal modes

Seismic tomography allows us to image the interior of the Earth. In general, to determine the nature of seismic anomalies, constraints on more than one seismic parameter are required. For example, the ratio $R$ between perturbations in $v_s$ and $v_p$ ($\mathrm{dln}{v}_s$ and $\mathrm{dln}{v}_p$, respectively) is studied extensively in the lowermost mantle and interpreted in terms of thermal and/or chemical anomalies. However, to jointly interpret tomographic models of variations in $v_s$ and $v_p$ or their ratio $R$, it is essential for them to share the same local resolution. Most existing models do not provide resolution information, and thus cannot guarantee to honour this condition. In addition, uncertainties are typically not provided, making it difficult to robustly interpret the ratio $R=\mathrm{dln}{v}_s/\mathrm{dln}{v}_p$. To overcome these issues, we utilise the recently developed SOLA tomographic method, a variant of the linear Backus–Gilbert inversion scheme. SOLA retrieves local-average model estimates, together with information on their uncertainties, whilst it also provides direct control on model resolution through target kernels. In this contribution, we apply SOLA to normal-mode data with sensitivity to both $v_s$ and $v_p$, as well as density throughout the mantle. Specifically, we aim to develop models of both $v_s$ and $v_p$ with the same local resolution. We test our methodology and approach using synthetic tests for various noise cases (random noise, data noise or also additional 3D Earth noise due to variations in other physical parameters than the one of interest). We find that the addition of the 3D noise increases the uncertainties in our model estimates significantly, only allowing us to find model estimates in six or four layers for $v_s$ and $v_p$, respectively. While the synthetic tests indicate that no satisfactory density models can be obtained, we easily manage to construct models of $\mathrm{dln}{v}_s$ and $\mathrm{dln}{v}_p$ with almost identical resolution, from which the ratio $R$ can be robustly inferred. The obtained values of $R$ in our synthetic experiments significantly depend on the noise case considered and the method used to calculate it, with the addition of 3D noise always leading to an overestimate of $R$. When applying our approach to real data, we obtain values of $R$ in the range of 2.5–4.0 in the lowest 600 km of the mantle, which are consistent with previous studies. Our model estimates with related resolving kernels and uncertainties can be used to test geodynamic model predictions to provide further insights into the temperature and composition of the mantle.