Characterization and error analysis of anN×Nunfolding procedure applied to filtered, photoelectric x-ray detector arrays. I. Formulation and testing

Abstract

An algorithm for spectral reconstructions (unfolds) and spectrally integrated flux estimates from data obtained by a five-channel, filtered x-ray-detector array (XRD) is described in detail and characterized. This diagnostic is a broad-channel spectrometer, used primarily to measure time-dependent soft x-ray flux emitted by z-pinch plasmas at the Z pulsed-power accelerator (Sandia National Laboratories, Albuquerque, New Mexico, USA), and serves as both a plasma probe and a gauge of accelerator performance. The unfold method, suitable for online analysis, arises naturally from general assumptions about the x-ray source and spectral properties of the channel responses; a priori constraints control the illposed nature of the inversion. The unfolded spectrum is not assumed to be Planckian. This study is divided into two consecutive papers. This paper considers three major issues: (a) Formulation of the unfold method.-The mathematical background, assumptions, and procedures leading to the algorithm are described: the spectral reconstruction S unfold ðE; tÞ-five histogram x-ray bins j over the x-ray interval, 137 E 2300 eV at each time step t-depends on the shape and overlap of the calibrated channel responses and on the maximum electrical power delivered to the plasma. The x-ray flux F unfold is estimated as R S unfold ðE; tÞdE. (b) Validation with simulations.-Tests of the unfold algorithm with known static and time-varying spectra are described. These spectra included-but were not limited to-Planckian spectra S bb ðE; TÞ (25 T 250 eV), from which noise-free channel data were simulated and unfolded. For Planckian simulations with 125 T 250 eV and typical responses, the binwise unfold values S j and the corresponding binwise averages hS bb i j agreed to $20%, except where S bb ( maxfS bb g. Occasionally, unfold values S j & 0 (artifacts) were encountered. The algorithm recovered * 90% of the x-ray flux over the wider range, 75 T 250 eV. For lower T, the test and unfolded spectra increasingly diverged as larger fractions of S bb ðE; TÞ fell below the detection threshold ($ 137 eV) of the diagnostic. (c) Comparison with other analyses and diagnostics.-The results of the histogram algorithm are compared with other analyses, including a test with data acquired by the DANTE filtered-XRD array at the NOVA laser facility. Overall, the histogram algorithm is found to be most useful for x-ray flux estimates, as opposed to spectral details. The following companion paper [D. L. Fehl et al., Phys. Rev. ST Accel. Beams 13, 120403 (2010)] considers (a) uncertainties in S unfold and F unfold induced by both data noise and calibrational errors in the response functions; and (b) generalization of the algorithm to arbitrary spectra. These techniques apply to other diagnostics with analogous channel responses and supported by unfold algorithms of invertible matrix form.

An optimized online filter stack spectrometer

“…To propose a universal optimization method, we began with a formal model for encoding an X-ray spectrum S(E) into experimental channel D i for an N-channel FSS. The mathematical definitions and notations used in this paper are written as that Fehl used [44,45] . D i can be written as…”

“…These unfolding algorithms can hardly provide mathematically rigorous or realistic estimates of the error propagation relation of the spectral unfolding error ∆S and the data perturbation ∆D. But the error propagation relation of ∆S and ∆D is mainly determined by the response function R. For example, an upper-bound measure to ∆S can be estimated from ∆D and the condition number of RM cond(R) as [44] ||∆S||/||S|| ≤ cond(R)||∆D||/||D||,…”

Multichannel, triaxial, neutron time-of-flight diagnostic for experiments at the Z facility

“…Equation 3 [Solutions are not unique, if they exist at all, and are complicated by the presence of noise ε k ðtÞ and the need for appropriate, physical constraints-issues widely discussed in the literature [24][25][26][27].] In assessing nTOF data at NIF, Hatarik et al [14][15][16][17][18] have prescribed a more analytical model of G k ðtÞ with undetermined coefficients, based on the Brysk theory, which can be folded forward in R k * G k to yield a fitting function for D k ðtÞ and from which details of the plasma may be inferred.…”