The fixed point formulation for large sample PGNAA—Part 1: theory

The determination of large sample composition via prompt gamma measurements is examined as a non-linear inverse problem. We show that this non-linear problem can be formulated as a fixed point problem that always has a physically meaningful solution, even in the presence of significant contributions to photopeak area from gammas emitted by the surroundings. The formulation involves only ratios of measured photopeak areas, and, separately, ratios of modeled photopeak areas. It therefore does not require the absolute comparison of measured or modeled quantities. The proof of the existence of meaningful solutions relies on very simple and natural hypotheses of positivity and continuity. The natural fixed point iteration is examined, and certain physical limits where its global convergence can be guaranteed are examined. Several computational examples are presented.