It is common practice to fit mathematical models to radionuclide activity-depth profiles in soils in order to quantify rates of vertical migration through the soil profile. We have fitted six such models to 21 different activity-depth profiles of radiocaesium (137Cs) derived from Chernobyl and determined relations between the models and the values of their parameters. The advection and dispersion parameters obtained using four solutions to the advection-dispersion equation (each based on different initial and boundary conditions or different simplifications) are in good agreement. We further develop a relation between parameter values obtained using the advection-dispersion models and those determined by a simpler exponential function of the form Aexp(-Bt) where t is the time and A and B are parameters to be estimated. One of the advection-dispersion models proved to be significantly better than the others in terms of goodness-of-fit, versatility and ease of use. A simple model, using calculations based on measured characteristics of the activity-depth profile, was shown to accord well with parameters derived from more complex models based on statistical curve fitting. We have also evaluated the 'residence time' or 'compartmental' model approach to characterizing radionuclide activity-depth profiles. We relate such models to a numerical solution of a simple advection equation, and we show that apparent dispersion in compartmental models is an artefact of numerical dispersion, which can be quantified by the Courant condition. For activity profiles that have a significant advection component, using solutions to the advection-dispersion equation, we have observed a strong positive correlation between advection and dispersion in the profile.