In groundwater, hydraulic heads and solute arrival times depend primarily on the hydraulic conductivity field and hydraulic boundary conditions. The spread of breakthrough curves, in contrast, depends also on longitudinal and transverse dispersion coefficients. The shape of point-like measured breakthrough curves can be reproduced by simulations only when appropriate dispersivities are applied. The values and spatial distributions of dispersivities depend on the resolution of the underlying hydraulic conductivity field. We present a geostatistical method for the joint estimation of log conductivity and log dispersivities from measurements of hydraulic heads and temporal moments of local breakthrough curves. The parameter fields are considered random space variables, and they are conditioned on the measurements by Bayesian inference. The estimated longitudinal and transverse dispersivities are to be applied in conjunction with the estimated conductivity field. We apply our technique to data of a technical-scale tracer experiment. In the particular application, the amount and quality of measured data are sufficient to infer the conductivity distribution at a spatial resolution at which the spread of locally obtained breakthrough curves is dominated by pore-scale transverse dispersion.