Abstract. Backward probabilities such as backward travel time probability density function for pollutants in natural aquifers/rivers had been used by hydrologists for decades in water-quality related applications. Reliable calculation of backward probabilities, however, has been challenged by non-Fickian pollutant transport dynamics and variability in the resolution of velocity at study sites. To address these two issues, we built an adjoint model by deriving a backward-in-time fractional-derivative transport equation subordinated to regional flow, developed a Lagrangian solver, and applied the model/solver to backtrack pollutant transport in various flow systems. The adjoint model applies subordination to a reversed regional flow field, converts forward-in-time boundaries to either absorbing or reflective boundaries, and reverses the tempered stable density to define backward mechanical dispersion. The corresponding Lagrangian solver is computationally efficient in projecting backward super-diffusive mechanical dispersion along streamlines. Field applications demonstrate that the adjoint subordination model can successfully recover release history, dated groundwater age, and spatial location(s) of pollutant source(s) for flow systems with either upscaled constant velocity, non-uniform divergent flow field, or fine-resolution velocities in a non-stationary, regional-scale aquifer, where non-Fickian transport significantly affects pollutant dynamics and backward probability characteristics. Caution is needed when identifying the phase-sensitive (aqueous versus absorbed) pollutant source in natural media. Possible extensions of the adjoint subordination model are also discussed and tested for quantifying backward probabilities of pollutants in more complex media, such as discrete fracture networks.;
