Characterization and error analysis of an N×N unfolding 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 ill-posed 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}(E,t)—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 ∫S_{unfold}(E,t)dE. (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}(E,T) (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 ⟨S_{bb}⟩_{j} agreed to ∼20%, except where S_{bb}≪max⁡{S_{bb}}. 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}(E,T) 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)PRABFM1098-4402] 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.