Within ray theory, travel times sample a finite set of points on the core-mantle boundary (CMB). Only if the CMB is smooth can we estimate its shape from travel times. The assumed degree of smoothness affects the uncertainty of the estimate. We find lower bounds on the uncertainty of the CMB shape by assuming (1) travel time errors are independent and identically distributed with zero mean and known variance, (2) linearization does not compromise accuracy, (3) event locations are exact, (4) radial velocity follows the IASP91 model, and (5) the only three-dimensional structure is a low-degree spherical harmonic perturbation to the CMB. We compute confidence intervals and simultaneous confidence envelopes for the CMB with a new technique based on reproducing kernels. Simultaneous confidence envelopes have a specified probability of containing the entire CMB, rather than just one point on the CMB, and thus allow inferences about the CMB shape. Suppose the standard deviation of travel time errors is 0.25 s (it is probably closer to 1.0 s), and that the CMB has a degree 4 spherical harmonic expansion.

Then the width of a 95% confidence envelope around the least squares estimate of the CMB using 776,958 PKP phases and 84,947 PcP phases ranges from 0.32 km under Europe and Tonga to a maximum of 1.06 km under the mid-Pacific, with a median width of 0.58 km. If the CMB has structure up to degree 20, the envelope width ranges from 8.14 km to 245.0 km. It is wider than 40.5 km for 50% of the CMB, and wider than 67.3 km over 25% of the CMB. We construct a degree 20 CMB model extending almost 93 km into the lower mantle beneath the mid-Pacific that changes travel times by only 0.1 s rms. ¿ American Geophysical Union 1993