Knowing the travel-time distributions from infiltrating rivers to pumping wells is important in the management of alluvial aquifers. Commonly, travel-time distributions are determined by releasing a tracer pulse into the river and measuring the breakthrough curve in the wells. As an alternative, one may measure signals of a time-varying natural tracer in the river and in adjacent wells and infer the travel-time distributions by deconvolution. Traditionally this is done by fitting a parametric function such as the solution of the one-dimensional advection-dispersion equation to the data. By choosing a certain parameterization, it is impossible to determine features of the travel-time distribution that do not follow the general shape of the parameterization, i.e., multiple peaks. We present a method to determine travel-time distributions by nonparametric deconvolution of electric-conductivity time series. Smoothness of the inferred transfer function is achieved by a geostatistical approach, in which the transfer function is assumed as a second-order intrinsic random time variable. Nonnegativity is enforced by the method of Lagrange multipliers. We present an approach to directly compute the best nonnegative estimate and to generate sets of plausible solutions. We show how the smoothness of the transfer function can be estimated from the data. The approach is applied to electric-conductivity measurements taken at River Thur, Switzerland, and five wells in the adjacent aquifer, but the method can also be applied to other time-varying natural tracers such as temperature. At our field site, electric-conductivity fluctuations appear to be an excellent natural tracer.