Linear programming inversions of seismic data may include constraints of the second derivative of the delay time, d2&tgr;/dp2=-dX/dp= -X'. Geometric ray theory predicts that the second derivative of the delay time is proportional to the inverse of the (complex) seismic amplitude squared, so that the use of this additional constraint constitutes the incorporation of amplitude information in linear inversions. The realizability constraint that velocity be a single valued function of depth must be included explicitly. For higher-order derivatives of delay time than the second, the realizability constraint is nonlinear and cannot be included in a linear scheme. Given that the data are adequate to allow identification of the major prograde and retrograde branches and the intervening caustics, useful bounds on X' can be established. The function X' is calibrated with respect to measured amplitudes by requiring that the T(X) curve obtained by successively integrating the X' curve satisfy the observed travel times. This calibration may be cast as a linear least squares problem. Applications to data from the ROSE (Rivera Ocean Seismic Experiment) area reveal a well-defined layer 2-layer 3 transition, an apparently isotropic and nearly homogeneous layer 3, and a Moho transition that is quite gradual. This transition appears to be more rapid in the fast Pn direction.