The Herglotz‐Wiechert integral for the direct inversion of ray parameter versus distance curves can be manipulated to find the envelope of all possible models consistent with geometrical body wave observations (travel time and ray parameter versus distance). Such an extremal inversion approach has been used to find the uncertainty bounds for the velocity structure in the mantle and core. We find, for example, that there is an uncertainty of ±40 km in the radius of the inner core boundary, ±18 km at the core‐mantle boundary, and ±35 km at the 435‐km transition zone. The velocity uncertainty is about ±0.08 km/sec forPandSwaves in the lower mantle and about ±0.20 km/sec in the core. Experiments with various combinations of ray types in the core indicate that rather crude observations ofSKKS‐SKStravel times confine the range of possible models far more dramatically than do the most precise estimates ofPmKPtravel times. Comparisons of results from extremal inversion and linearized perturbation inversions indicate that body wave behavior is too strongly nonlinear for linearized schemes to be effective for predicting