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We study a parameter estimation problem which arises from measurements with QUANTOM, a measurement facility utilizing Prompt-Gamma Neutron Activation Analysis (PGNAA) to analyze elemental masses in 200-l nuclear waste drums. The presence of neutron flux measurements in QUANTOM offers additional information for elemental composition reconstruction, motivating an alternative to the standard iterative approach. Therefore, the measurement is modelled using approximate models for neutron and photon transport. We introduce a mathematical formulation for solving the resulting parameter estimation problem of QUANTOM measurements and employ an adjoint-based method to determine the elemental composition. Uniqueness of solutions in this context is studied, and the results offer novel insights for PGNAA applications.