A new technique has been developed to overcome difficulties which arise when body-wave data in the form of T-delta is reduced to the tau-p form in the presence of cusps. The technique uses sophisticated constrained spline fits to allow for constraints on the derivatives and thus eliminates the possibility of points of inflexion occurring as well as making sure that the curvature is in the right direction. Having arrived at a good fit to the T-delta data with a single spline, the tau-p curve can be derived which has taken the cusp properly into consideration. The tau-p curve is then inverted with a new linear inversion method, which again takes into consideration whether a particular point lies on a prograde or retrograde branch.

The technique is applied to the lunar seismic data, where at least one triplication is presumed to occur for the P-wave travel-time curve. It is shown that the present analysis is not accurate enough to address itself to the problem of a possible lunar core, since there are insufficient data points for events close to the antipode of the center of the lunar network.

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