We present a new technique for constructing the narrowest corridor containing all velocity profiles consistent with a finite collection of &tgr;(p) data and their statistical uncertainites. Earlier methods for constructing such bounds treat the confidence interval for each &tgr; datum as as trict interval within which the true value might lie with equal probability, but this interpretation is incompatible with the estimation procedure used on the original travel time observations. The new approach, based upon quardatic programming (QP), shares the advantages of the linear programming (LP) solution: it can invert &tgr;(p) and X(p) data concurrently; it permits the incorporation of constraints on the radial derivative of velocity for spherical earth models; and theoretical results about convergence and optimality can be obtained for the method. We compare P velocity bounds for the core obtained by QP and LP. The models produced by LP predict data values at the ends of the confidence intervals; these values are unlikely according to the proper statistical distribution of errors. For this reason the LP velocity bounds can be wider than those given by QP, which takes better account for the statistics. Sometimes, however, the LP bounds are more restrictive because LP never permits the predictions of the models to lie outside the conficence intervals even though occasional excursions are expected. The QP bounds grow narrower at lower levels of confidence, but the corridors at 95% and 99.9% are virtually indistinguishable: The data must be improved substantially to make a significant change in the velocity bounds. ¿ American Geophysical Union 1987