EXAFS Phase Retrieval Solution Tracking for Complex Multi-Component System: Synthesized Topological Inverse Computation

Using the FEFF kernel A(k,r), we describe the inverse computation from χ(k)-data to g(r)-solution in terms of a singularity regularization method based on complete Bayesian statistics process. In this work, we topologically decompose the system-matched invariant projection operators into two distinct types, (A+AA+A) and (AA+AA+), and achieved Synthesized Topological Inversion Computation (STIC), by employing a 12-operator-closed-loop emulator of the symplectic transformation. This leads to a numerically self-consistent solution as the optimal near-singular regularization parameters are sought, dramatically suppressing instability problems connected with finite precision arithmetic in ill-posed systems. By statistically correlating a pair of measured data, it was feasible to compute an optimal EXAFS phase retrieval solution expressed in terms of the complex-valued χ(k), and this approach was successfully used to determine the optimal g(r) for a complex multi-component system.