Characterization and error analysis of anN×Nunfolding procedure applied to filtered, photoelectric x-ray detector arrays. II. Error analysis and generalization

Abstract

A five-channel, filtered-x-ray-detector (XRD) array has been used 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). The preceding, companion paper [D. L. Fehl et al., Phys. Rev. ST Accel. Beams 13, 120402 (2010)] describes an algorithm for spectral reconstructions (unfolds) and spectrally integrated flux estimates from data obtained by this instrument. The unfolded spectrum ${S}_{\mathrm{unfold}}(E,t)$ is based on ($N=5$) first-order B-splines (histograms) in contiguous unfold bins $j=1,\dots{},N$; the recovered x-ray flux ${\mathcal{F}}_{\mathrm{unfold}}(t)$ is estimated as $\ensuremath{\int}{S}_{\mathrm{unfold}}(E,t)dE$, where $E$ is x-ray energy and $t$ is time. This paper adds two major improvements to the preceding unfold analysis: (a) Error analysis.---Both data noise and response-function uncertainties are propagated into ${S}_{\mathrm{unfold}}(E,t)$ and ${\mathcal{F}}_{\mathrm{unfold}}(t)$. Noise factors $\ensuremath{\nu}$ are derived from simulations to quantify algorithm-induced changes in the noise-to-signal ratio (NSR) for ${S}_{\mathrm{unfold}}$ in each unfold bin $j$ and for ${\mathcal{F}}_{\mathrm{unfold}}$ ($\ensuremath{\nu}\ensuremath{\equiv}NS{R}_{\mathrm{output}}/NS{R}_{\mathrm{input}}$): for ${S}_{\mathrm{unfold}}$, $1\ensuremath{\lesssim}{\ensuremath{\nu}}_{j}\ensuremath{\lesssim}30$, an outcome that is strongly spectrally dependent; for ${\mathcal{F}}_{\mathrm{unfold}}$, $0.6\ensuremath{\lesssim}{\ensuremath{\nu}}_{\mathcal{F}}\ensuremath{\lesssim}1$, a result that is less spectrally sensitive and corroborated independently. For nominal $z$-pinch experiments, the combined uncertainty (noise and calibrations) in ${\mathcal{F}}_{\mathrm{unfold}}(t)$ at peak is estimated to be $\ensuremath{\sim}15%$. (b) Generalization of the unfold method.---Spectral sensitivities (called here passband functions) are constructed for ${S}_{\mathrm{unfold}}$ and ${\mathcal{F}}_{\mathrm{unfold}}$. Predicting how the unfold algorithm reconstructs arbitrary spectra is thereby reduced to quadratures. These tools allow one to understand and quantitatively predict algorithmic distortions (including negative artifacts), to identify potentially troublesome spectra, and to design more useful response functions.