An approach to recovering an initial tsunami waveform with its application to the 16 September 2015 Illapel tsunami is proposed. The approach is based on inverting the remote measurements of water-level data from the deep-ocean DART buoys without a priori information on a source except for the common information about its spatial localization. The ill-posed inverse problem in question is regularized by means of a least-squares inversion using the truncated singular value decomposition method and the r-solution method. The method proposed suppresses the instability of the numerical solution of the ill-posed problem under consideration. The computational algorithm allows one to find the way to improve the inversion by selecting the most informative set of available observation stations. The method proposed was successfully applied to the 16 September 2015 Illapel tsunami.