A conditioned backward fractional-derivative model for identifying super-diffusive pollutant source in aquatic systems with multiple observation data

Backward models for super-diffusion have been developed to identify pollutant source locations, but they are limited to a single observation and disregard field-measured concentrations. To overcome these limitations, this study derives the adjoint of the space-fractional advection–dispersion equation, incorporating measured concentrations from multiple observation data. Backward probabilities, such as the backward location probability density function (PDF), describe the likely source location(s) at a fixed time before sampling, offering a comprehensive modeling approach for source identification. By applying Bayes’ theorem, the individual PDFs from each observation and its corresponding concentration are combined into a joint PDF, enhancing both the information and reliability compared to the previous single PDF. Field applications show that the improved model enhances accuracy (with PDF peak locations closer to the actual source) and precision (with reduced variance) of backward PDFs for identifying point sources in a natural river and aquifer. The model’s performance is affected by observation count and measurement errors, with double peaks in the backward location PDF possible due to source mass uncertainty. Future refinements, such as incorporating backward travel time analysis and extending applications to reactive pollutants, could further enhance the utility of the conditioned backward fractional-derivative model developed in this study.