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Figure S1: Schematic of a portion of one typical kernel (column) of R for a block or node at the centre of layer n for a typical teleseismic tomography study (grey lines and dots) showing the relative magnitudes of diagonal (black dot) and off-diagonal (grey dots) elements of R. These dots are (horizontally) at block centres and vertically proportional to the amplitude of R in that element (relative scale at left). Such representations can be thought of as the “impulse response” to the presence of a single-block anomaly at the centre of layer n, for the filter comprising a particular dataset and (to a lesser degree) the particular inversion method used.

Figure S2: Typical but idealized two-dimensional ray set for teleseismic tomography. Arbitrary scaling; no vertical exaggeration (after Evans and Achauer, 1993).

Figure S3: Two-dimensional synthetic-data tests of resolution for (a) a best-case and (b) more typical ray sets, with similar numbers of rays in each. The models used to create synthetic travel times are in (c), (f), (i), and (l). Inversion results using the best-case ray set are in (d), (g), (j), and (m) and those for the more typical ray set are in (e), (h), (k), and (n). Green ovals in (a) and (b) are estimated minimum-resolvable objects in various locations (from Yanovskaya, 1997).

Figure S4: Synthetic wave-speed model for the Ethiopian rift (left) consisting of high wave speed (ΔvP = 5%) rift flanks. A relative arrival-time dataset is computed for the same station-earthquake pairs used in the study of Bastow et al. (2008). The resulting tomographic model is characterized not only by high-wave-speed flanks but also by a low-wave-speed zone beneath the rift valley. In reality, in Ethiopia, P- and S-wave arrival times are ubiquitously late compared with the global mean, with the implication that the “high” wave-speed (blue) regions are low wave-speed compared with the global mean (Bastow et al., 2008; Bastow, 2012). A model result with high and low wave-speed structure is the inevitable consequence of relative arrival-time inversions.

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