Characterization and error analysis of an $N\times N$ unfolding procedure applied to filtered, photoelectric x-ray detector arrays. II. Error analysis and generalization

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 $\int S_{\mathrm{unfold}}(E,t)dE$, 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 $\nu$ 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 ${\mathcal{F}}_{\mathrm{unfold}}$ ($\nu \equiv NS{R}_{\mathrm{output}}/NS{R}_{\mathrm{input}}$): for $S_{\mathrm{unfold}}$, $1\lesssim {\nu }_j\lesssim 30$, an outcome that is strongly spectrally dependent; for ${\mathcal{F}}_{\mathrm{unfold}}$, $0.6\lesssim {\nu }_{\mathcal{F}}\lesssim 1$, 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 $\sim 15\%$. (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.

Figure 1

Noise amplification in the unfolding process for uniform relative data errors. Part (a) gives the binwise noise factor (NF) ${\nu }_j[T]$—i.e., $NSR(S_j)/NSR(D_i)$—for Planckian spectra, $T=150–250\mathrm{eV}$. (The colored, broken lines indicate the envelope of these calculations.) Part (b) shows the corresponding NF’s ${\nu }_{\mathcal{F}}[T]$—i.e., $NSR({\mathcal{F}}_{\mathrm{unfold}})/NSR(D_i)$ and $NSR({\mathcal{F}}_{LS})/NSR(D_i)$—for the unfolded flux estimate ${\mathcal{F}}_{\mathrm{unfold}}$ (red curve) and for a different flux estimate ${\mathcal{F}}_{LS}$ [Ref. 8] (Planckian spectra, $T=25–250\mathrm{eV}$). The sequence of numbers $m^{-(1/2)}$ ($m=1,\dots ,5$) on the left-hand scale represent hypothetical degrees of freedom for noise averaging.

Figure 2

Estimated experimental uncertainties for the $Z$ diagnostic, and fluctuations for repeated experiments. Part (a) shows the unfolded peak-power spectrum for $Z$-shot 165 (cf. Pt. 1: Figs. 3 and 4). Here, for clarity, $S_{\mathrm{unfold}}(E,t_{\mathrm{peak}})$ is denoted by solid (red) points at the midpoint of each unfold bin (Pt. 1: Table II), with horizontal bars indicating the bin widths; the ordinates [$\mathrm{GW}{\mathrm{sr}}^{-1}{\mathrm{eV}}^{-1}$] are the unfold coefficients, $S_j$. Vertical error bars give the combined, propagated uncertainties, $\pm \sigma (\Delta S_j)$. The blue curve is a 200-eV Planckian spectrum [$\mathrm{GW}{\mathrm{sr}}^{-1}{\mathrm{eV}}^{-1}{\mathrm{cm}}^{-2}$] scaled by integrated flux to the same units. Part (b) compares unfold coefficients and flux for two independent, filtered-XRD arrays (denoted “A” and “B”), which were simultaneously fielded on five, nominally identical, tungsten $z$-pinch shots at the $Z$ facility ($Z$-shots 163-167). The solid points are the binwise unfold ratios, ${}_{\mathrm{A}}S_j/{}_{\mathrm{B}}S_j$, for each shot at peak power (left scale); the additional points at the right side of the figure are the corresponding unfold flux ratios, ${}_{\mathrm{A}}\mathcal{F}_{\mathrm{unfold}}/{}_{\mathrm{B}}\mathcal{F}_{\mathrm{unfold}}$ (right axis, same scale). The notation “2X” indicates two nearly identical ratios; and the red “$+$” symbols denote the sample ratio mean for each bin $j$ and the flux estimate.

Figure 3

Binwise passband functions, ${\rho }_{A,j}(E)$ and ${\rho }_{U,j}(E)$, for the averaging operator $\mathcal{A}$ and the unfold operator $\mathcal{U}={\mathcal{M}}_{\mathsf{B}\mathsf{D}}^{-1}\mathcal{M}$, respectively. In both figures, the baselines are displaced for clarity, dashed vertical lines denote the unfold interval $[\Delta E]$, and arrowheads indicate the unfold bin boundaries. Part (a) graphs averaging passband functions ${\rho }_{A,j}(E)$, each of constant value $(\Delta E_j{)}^{-1}$ within the $j$th unfold bin, but zero elsewhere (Pt. 1: Table II). Part (b) shows the corresponding unfold passband functions ${\rho }_{U,j}(E)$, which exhibit x-ray absorption-edge features in the responses (Pt. 1: Fig. 1) but not the high-energy humps of $R_1$, $R_2$, $R_3$ (Pt. 1: Figs. 1 and 2). The ${\rho }_{U,j}(E)$’s can be nonzero, and even negative valued, outside bin $[\Delta E_j]$.

Figure 4

Passband functions $H_{[\Delta E]}(E)$ and $H_{\mathrm{unfold}}(E)$ for the incident flux ${\mathcal{F}}_{[\Delta E]}$ and unfolded flux estimate ${\mathcal{F}}_{\mathrm{unfold}}$, respectively. The encircled numbers indicate roughly where specific response functions (Pt. 1: Fig. 1) dominate $H_{\mathrm{unfold}}(E)$.

Figure 5

Products of $\Delta H(E)$ with various Planckian spectra. Part (a) plots $\Delta H(E)$ alone, while (b) and (c) show the products, $\Delta H(E)S_{bb}(E,250\mathrm{eV})$ and $\Delta H(E)S_{bb}(E,25\mathrm{eV})$, respectively.