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Description: Traditionally P- and S-wave tomography has been based on the inversion of data that are sensitive to the desired Earth structure, and model covariance is estimated from imperfect resolution and data error propagation. This analysis ignores the usually large null-spaces, and hence the significant non-uniqueness of the solution encountered in seismic tomography problems. Here we perform a model space search for P- and S-velocity structure to find acceptable fits to recent normal-mode splitting and fundamental-mode phase velocity data. The survey of the model space employs the neighbourhood algorithm of Sambridge, which preferentially samples the good data-fitting regions. A Bayesian approach is used subsequently to extract robust information from the ensemble of models.We particularly focus on posterior marginal probability density functions and covariances for the various model parameters. The covariance matrix obtained is very useful in providing insights into the trade-offs between the different variables and the uncertainties associated with them. We stay within the framework of perturbation theory, meaning that our emphasis is on the null-space of the linear inverse problem rather than the neglected non-linearity. The whole model space (including the nullspace) is sampled within reasonable parameter bounds, and hence the error bars are determined by all fitting models rather than subjective prior information. We estimated P and S models for spherical harmonic degree two only. The uncertainties are quite large and corresponding relative errors can exceed 100 per cent in the mid-mantle for Vp. We find a good correlation of our most likely S model with previous models but some small changes in amplitude. Our most likely P model differs quite strongly from the recent P model SB10L18 and the correlation between our most likely P and S models is small. Among all the good data-fitting models, there are, however, many that have a significant Vp −Vs correlation. We compute dlnVs/dlnVp from the models that correlate significantly.We find an increase with depth in the top 1500 km. Deeper in the mantle, normal-mode data prefer modest values compared with traveltime data.