A method for treating, in a joint manner, both distance and delay time data, including uncertainties, is presented. Full advantage is taken of the linear relationship between kinematic data and velocity structure. The inversion is computed using linear programing, a technique developed to treat economic optimization problems. Because of the linearity of the inverse problem and the ability to constrain solutions to be physically realizable the method will accurately predict achievable velocities within the earth. In general, after choosing a convenient basis in which to expand the velocity-depth function, the technique will provide the tightest possible constraints consistent with a kinematic data set composed of upper and lower bounds on &tgr;(p) and &khgr;(p). The effects of two nonlinear constraints caused by a possible lack of knowledge of velocities within a low-velocity zone and/or the surface velocity are examined.