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

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 ${S}_{\mathrm{unfold}}(E,t)$---five histogram x-ray bins $j$ over the x-ray interval, $137\ensuremath{\le}E\ensuremath{\le}2300\text{ }\text{ }\mathrm{eV}$ 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 ${\mathcal{F}}_{\mathrm{unfold}}$ is estimated as $\ensuremath{\int}{S}_{\mathrm{unfold}}(E,t)dE$. (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 ${S}_{bb}(E,T)$ ($25\ensuremath{\le}T\ensuremath{\le}250\text{ }\text{ }\mathrm{eV}$), from which noise-free channel data were simulated and unfolded. For Planckian simulations with $125\ensuremath{\le}T\ensuremath{\le}250\text{ }\text{ }\mathrm{eV}$ and typical responses, the binwise unfold values ${S}_{j}$ and the corresponding binwise averages $⟨{S}_{bb}{⟩}_{j}$ agreed to $\ensuremath{\sim}20%$, except where ${S}_{bb}\ensuremath{\ll}\mathrm{max}﻿{{S}_{bb}}$. Occasionally, unfold values ${S}_{j}\ensuremath{\lesssim}0$ (artifacts) were encountered. The algorithm recovered $\ensuremath{\gtrsim}90%$ of the x-ray flux over the wider range, $75\ensuremath{\le}T\ensuremath{\le}250\text{ }\text{ }\mathrm{eV}$. For lower $T$, the test and unfolded spectra increasingly diverged as larger fractions of ${S}_{bb}(E,T)$ fell below the detection threshold ($\ensuremath{\sim}137\text{ }\text{ }\mathrm{eV}$) 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 ${S}_{\mathrm{unfold}}$ and ${\mathcal{F}}_{\mathrm{unfold}}$ 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.