QSER Decomposition for Forward and Inverse Modeling of Radon Transport in Lithosphere and Atmosphere

The Source-Environment-Response (SER) framework introduced a decomposition of radon transport into source-driven and environmental components, but treated the environment as a passive correction rather than an active memory accumulator. We present QSER (Generalized Source-Environment-Response), which extends SER by making the memory kernel explicit through the environmental accumulator equation. The QSER framework separates the governing transport operator into conser-vative (advective) and dissipative (diffusive and decay) parts, revealing the observed concentration as the difference between a conservative ghost field S and an environmental memory field E that accu-mulates the history of dissipative interactions. We derive the full three-dimensional time-dependent solution for atmospheric radon transport, including generalizations for stochastic velocity fields and shape-changing plumes while preserving strict mass conservation. The memory kernel “Q”, which is the Green’s function of the system, captures how past emissions influence present concentrations. The QSER decomposition solution successfully recovers industry-standard atmospheric dispersion models (Gaussian plume, RESRAD-OFFSITE, CAP88-PC, ISCLT3, SCREEN3, ALOHA/CAMEO) as special cases. Validation against the Nazaroff solution for soil radon transport demonstrates exact decomposition and recovery of physical parameters (F, v, a, D). Four internal consistency tests pro-vide built-in verification of the decomposition and Physical consistency of results. The framework transforms ill-posed inverse problems into a manageable sequential optimization over conservative source parameters, enabling robust parameter estimation for radon monitoring applications in earth-quake precursor studies, uranium exploration, and atmospheric transport. All implementation code is available at “https://github.com/1030ahmad1030/QSER”.