A novel,inverse scattering-type,technique for solving Cauchy problems for integrable cellular automata will be presented for the case of the ultradiscrete KdV equation,defined over the real numbers.The action-angle variables that arise naturally in this approach turn out to be related in a rather simple way to those that can be obtained from a slightly modifed version of an algorithm proposed by Takagi for calculating the Kerov-Kirillov-Reshetikhin map in the context of rigged-configurations.