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Pixon maps of the same flares as Figures 1 and 2. Again, we use 2 arcsec pixel −1 , 10 – 100% contour levels, and a 128 × 128 field of view.
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We have used Ramaty High Energy Solar Spectroscopic Imager (RHESSI) modulation profiles in the 25 – 300keV range to construct
high-fidelity visibilities of 25 flares having at least two components. These hard X-ray visibilities, which are mathematically
identical to the visibilities of radio imaging, were input to software developed for mapping sol...
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Context 1
... of component sizes clearly shows the "super resolution" characteristic of MEM algorithms (Cornwell and Evans, 1985). Figure 3 is an array of maps for the same 25 flares, this time using the RHESSI Pixon algorithm. Comparison with Figure 1 shows that the Pixon components, with few excep- tions (flares #2 and #22), are similar in size to the MEM components. ...
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... Native measurements in both radio astronomy (Thompson et al. 2017) and modern hard-X-ray solar imaging (Piana et al. 2022) are sets of spatial Fourier components of the incoming source flux, named visibilities, measured at specific spatial frequency samples, named (u, v) points. In both radio astronomy and hard-X-ray solar imaging, CLEAN (Högbom 1974;Schmahl et al. 2007) is a nonlinear image reconstruction algorithm that iteratively deconvolves the instrumental pointspread function (PSF) from the so-called dirty map, i.e., the discretized inverse Fourier transform of the experimental visibility set. More specifically, the CLEAN algorithm is made of a CLEAN loop, which generates the following: a set of CLEAN components located at the points of the solar disk from where most of the source emission propagates; an estimate of the background; the convolution of the CLEAN components map with an idealized PSF, named the CLEAN beam; and, eventually, the CLEANed map, i.e., the sum of the outcome of this convolution step with the convolved background residuals. ...
... Instead, Step 4 of the pipeline, i.e., the construction of the CLEANed map, is clearly ambiguous and mostly biased by the user's decision about the shape of the CLEAN beam K C . In the version of the CLEAN code originally developed for RHESSI (Schmahl et al. 2007), this convolution kernel is modeled by a two-dimensional Gaussian function whose FWHM is chosen by the user according to heuristic rules of thumb. This convolution product is the main reason for the low photometric reliability of the CLEANed map, while conservative choices for FWHM typically lead to underresolved reconstructions, with correspondingly high χ 2 values. ...
CLEAN is an iterative deconvolution method for radio and hard-X-ray solar imaging. In a specific step of its pipeline, CLEAN requires the convolution between an idealized version of the instrumental point-spread function (PSF), and a map collecting point sources located at positions from where most of the flaring radiation is emitted. This step has highly heuristic motivations and the shape of the idealized PSF, which depends on the user’s choice, impacts the shape of the reconstruction. This study introduces a user-independent release of CLEAN for image reconstruction from observations recorded by the Spectrometer/Telescope for Imaging X-rays (STIX) on board Solar Orbiter. Specifically, we show here that this unbiased release of CLEAN outperforms the standard version of the algorithm, with reconstructions in line with the ones offered by other imaging methods developed in the STIX framework.
... However, this constraint is needed for simplifying the optimization problem. MEM_GE represents a computational upgrade of the MEM_NJIT algorithm implemented for the Reuven Ramaty High Energy Solar Spectroscopic Imager (RHESSI) (Bong et al., 2006;Schmahl et al., 2007), since it involves a mathematically sound optimization problem and a robust optimization technique. • Back projection (see, e.g., Hurford et al., 2002), which realizes the discrete Fourier transform inversion of STIX visibilities and is equivalent to the dirty map in radio interferometry. ...
... An estimate of the retrieved parameter uncertainty is obtained by perturbing 20 times the data with Gaussian noise, by forward-fitting the perturbed data and computing the standard deviation of the set of optimized parameters. • EM (Massa et al., 2019), which is the Expectation Maximization algorithm, also known as the Richardson-Lucy algorithm when applied to image deconvolution problems (Richardson, 1972;Lucy, 1974). EM takes as input the measured STIX counts instead of the corresponding visibilities. ...
The Spectrometer/Telescope for Imaging X-rays (STIX) is one of six remote sensing instruments on-board Solar Orbiter. The telescope applies an indirect imaging technique that uses the measurement of 30 visibilities, i.e., angular Fourier components of the solar flare X-ray source. Hence, the imaging problem for STIX consists of the Fourier inversion of the data measured by the instrument. In this work, we show that the visibility amplitude and phase calibration of 24 out of 30 STIX sub-collimators has reached a satisfactory level for scientific data exploitation and that a set of imaging methods is able to provide the first hard X-ray images of solar flares from Solar Orbiter. Four visibility-based image reconstruction methods and one count-based are applied to calibrated STIX observations of six events with GOES class between C4 and M4 that occurred in May 2021. The resulting reconstructions are compared to those provided by an optimization algorithm used for fitting the amplitudes of STIX visibilities. We show that the five imaging methods produce results morphologically consistent with the ones provided by the Atmospheric Imaging Assembly on board the Solar Dynamic Observatory (SDO/AIA) in UV wavelengths. The $\chi ^{2}$ χ 2 values and the parameters of the reconstructed sources are comparable between methods, thus confirming their robustness.
... Among count-based methods, SSW includes Back Projection , Clean (Högbom 1974), Forward Fit (Aschwanden et al. 2002), Pixon (Metcalf et al. 1996), and Expectation Maximization (Benvenuto et al. 2013). Among visibility-based methods, SSW includes MEM_NJIT (Bong et al. 2006;Schmahl et al. 2007), a Maximum Entropy method; VIS_FWDFIT (Schmahl et al. 2007), which selects pre-defined source shapes based on their best fitting of visibilities; uv_smooth (Massone et al. 2009), an interpolation/extrapolation method in the Fourier domain; VIS_CS (Felix et al. 2017), a catalog-based compressed sensing algorithm; and VIS_WV ), a wavelet-based compressed sensing algorithm. Although each one of these algorithms combines specific values with applicability limitations and specific flaws, a critical comparison of the maps of a given flaring event obtained by the application of all (or most) of these algorithms provides a good picture of what a reliable image of the event could be. ...
... Among count-based methods, SSW includes Back Projection , Clean (Högbom 1974), Forward Fit (Aschwanden et al. 2002), Pixon (Metcalf et al. 1996), and Expectation Maximization (Benvenuto et al. 2013). Among visibility-based methods, SSW includes MEM_NJIT (Bong et al. 2006;Schmahl et al. 2007), a Maximum Entropy method; VIS_FWDFIT (Schmahl et al. 2007), which selects pre-defined source shapes based on their best fitting of visibilities; uv_smooth (Massone et al. 2009), an interpolation/extrapolation method in the Fourier domain; VIS_CS (Felix et al. 2017), a catalog-based compressed sensing algorithm; and VIS_WV ), a wavelet-based compressed sensing algorithm. Although each one of these algorithms combines specific values with applicability limitations and specific flaws, a critical comparison of the maps of a given flaring event obtained by the application of all (or most) of these algorithms provides a good picture of what a reliable image of the event could be. ...
Maximum Entropy is an image reconstruction method conceived to image a sparsely occupied field of view, therefore it is particularly effective at achieving super-resolution effects. Although widely used in image deconvolution, this method has been formulated in radio astronomy for the analysis of observations in the spatial frequency domain, and an Interactive Data Language code has been implemented for image reconstruction from solar X-ray Fourier data. However, this code relies on a non-convex formulation of the constrained optimization problem addressed by the Maximum Entropy approach, and this sometimes results in unreliable reconstructions characterized by unphysical shrinking effects. This paper introduces a new approach to Maximum Entropy based on the constrained minimization of a convex functional. In the case of observations recorded by the Reuven Ramaty High Energy Solar Spectroscopic Imager (RHESSI), the resulting code provides the same super-resolution effects of the previous algorithm, while also working properly when that code produces unphysical reconstructions. We also obtain results via testing the algorithm with synthetic data simulating observations of the Spectrometer/Telescope for Imaging X-rays (STIX) in Solar Orbiter. The new code is available in the HESSI folder of the Solar SoftWare (SSW) tree.
... Some of these methods apply directly to RHESSI counts while others have been conceived to process RHESSI visibilities, i.e. calibrated samples of the Fourier transform of the incoming photon flux, generated via a data stacking process. Among count-based methods, SSW includes Back Projection , Clean (Högbom 1974), Forward Fit (Aschwanden et al. 2002), Pixon (Metcalf et al. 1996), and Expectation Maximization (Benvenuto et al. 2013); among visibility-based methods, SSW includes MEM NJIT (Bong et al. 2006;Schmahl et al. 2007), a Maximum Entropy method; VIS FWDFIT (Schmahl et al. 2007), which selects pre-defined source shapes based on their best fitting of visibilities; uv smooth (Massone et al. 2009), an interpolation/extrapolation method in the Fourier domain; VIS CS (Felix et al. 2017), a catalogue-based compressed sensing algorithm; and VIS WV ), a wavelet-based compressed sensing algorithm. Although each one of these algorithms combines specific values with applicability limitations and specific flaws, a critical comparison of the maps of a given flaring event obtained by the application of all (or most) of these algorithms provides a good picture of what a reliable image of the event could be. ...
... Some of these methods apply directly to RHESSI counts while others have been conceived to process RHESSI visibilities, i.e. calibrated samples of the Fourier transform of the incoming photon flux, generated via a data stacking process. Among count-based methods, SSW includes Back Projection , Clean (Högbom 1974), Forward Fit (Aschwanden et al. 2002), Pixon (Metcalf et al. 1996), and Expectation Maximization (Benvenuto et al. 2013); among visibility-based methods, SSW includes MEM NJIT (Bong et al. 2006;Schmahl et al. 2007), a Maximum Entropy method; VIS FWDFIT (Schmahl et al. 2007), which selects pre-defined source shapes based on their best fitting of visibilities; uv smooth (Massone et al. 2009), an interpolation/extrapolation method in the Fourier domain; VIS CS (Felix et al. 2017), a catalogue-based compressed sensing algorithm; and VIS WV ), a wavelet-based compressed sensing algorithm. Although each one of these algorithms combines specific values with applicability limitations and specific flaws, a critical comparison of the maps of a given flaring event obtained by the application of all (or most) of these algorithms provides a good picture of what a reliable image of the event could be. ...
Maximum Entropy is an image reconstruction method conceived to image a sparsely occupied field of view and therefore particularly appropriate to achieve super-resolution effects. Although widely used in image deconvolution, this method has been formulated in radio astronomy for the analysis of observations in the spatial frequency domain, and an Interactive Data Language (IDL) code has been implemented for image reconstruction from solar X-ray Fourier data. However, this code relies on a non-convex formulation of the constrained optimization problem addressed by the Maximum Entropy approach and this sometimes results in unreliable reconstructions characterized by unphysical shrinking effects. This paper introduces a new approach to Maximum Entropy based on the constrained minimization of a convex functional. In the case of observations recorded by the Reuven Ramaty High Energy Solar Spectroscopic Imager (RHESSI), the resulting code provides the same super-resolution effects of the previous algorithm, while working properly also when that code produces unphysical reconstructions. Results are also provided of testing the algorithm with synthetic data simulating observations of the Spectrometer/Telescope for Imaging X-rays (STIX) in Solar Orbiter. The new code is available in the {\em{HESSI}} folder of the Solar SoftWare (SSW)tree.
... Volume Estimates from RHESSI Imaging Calculating the volume from RHESSI imaging can be difficult because of the partial Fourier coverage of the instrument. Different imaging algorithms will give different estimates, and the CLEAN algorithm is known to overestimate the source size Schmahl et al. 2007;Kontar et al. 2010). In this study, we used the estimates derived from the Visibility Forward Fit algorithm, which provide a measure of the uncertainty of the size of the source. ...
We present the statistical analysis of 33 flare-related coronal jets, and discuss the link between the jet and the flare properties in these events. We selected jets that were observed between 2010 and 2016 by the Atmospheric Imaging Assembly (AIA) on board the Solar Dynamic Observatory ( SDO ) that are temporally and spatially associated with flares observed by the Reuven Ramaty High Energy Solar Spectrometric Imager ( RHESSI ). For each jet, we calculated the jet duration and projected velocity in the plane of sky. The jet duration distribution has a median of 18.8 minutes. The projected velocities are between 31 and 456 km s ⁻¹ , with a median at 210 km s ⁻¹ . For each associated flare, we performed X-ray imaging and spectroscopy and identify nonthermal emission. Nonthermal emission was detected in only 1/4 of the events considered. We did not find a clear correlation between the flare thermal energy or soft X-ray (SXR) peak flux and the jet velocity or jet duration. There is no preferential time delay between the flare and the jet. The X-ray emission is generally located at the base of the jet. The analysis presented in this paper suggests that the flare and jet are part of the same explosive event, that the jet is driven by the propagation of an Alfvénic perturbation, and that the energy partition between flare and jets varies substantially from one event to another.
... Another way of showing the modulation in the different detectors is to plot the amplitude of the visibility vectors ( Schwartz et al. 2002;Schmahl et al. 2007) for each detector as a function of the position angle defined as the spatial direction of each grid response referenced to solar north. This is essentially the same as plotting just the amplitudes of the oscillations seen in Figure 4 but here plotted for only a half rotation from 0° to 180° with the second half rotation assumed Figure 1. ...
... because no "noise" is allowed outside the range of the assumed sources. 2. MEM_NJIT 6 ( Schmahl et al. 2007) is a Maximum Entropy Method (MEM) algorithm based on visibilities that was originally developed at New Jersey Institute of Technology (NJIT). It provides the best fit to the visibility amplitudes especially for the subcollimators with the finest pitch as indicated in Figure 5. ...
The unusually narrow X-ray source imaged with the Ramaty High Energy Solar Spectroscopic Imager ( RHESSI ) during an impulsive spike lasting for ∼10 s during the Geostationary Operational Environmental Satellite C7.9 flare on 2011 September 25 (SOL2011-09-25T03:32) was only ∼2″ wide and ∼10″ long. Comparison with Helioseismsic and Magnetic Imager magnetograms and Atmospheric Imaging Assembly images at 1700 Å shows that the X-ray emission was primarily from a long ribbon in the region of positive polarity with little if any emission from the negative polarity ribbon. However, a thermal plasma source density of ∼10 ¹² cm ⁻³ estimated from the RHESSI -derived emission measure and source area showed that this could best be interpreted as a coronal hard X-ray source in which the accelerated electrons with energies less than ∼50 keV were stopped by Coulomb collisions in the corona, thus explaining the lack of the more usual bright X-ray footpoints. Analysis of RHESSI spectra shows greater consistency with a multi-temperature distribution and a low-energy cutoff to the accelerated electron spectrum of 22 keV compared to 12 keV if a single-temperature distribution is assumed. This leads to a change in the lower limit on the total energy in electrons by an order of magnitude, given the steepness of the best-fit electron spectrum with a power-law index of ∼6.
... Another way of showing the modulation in the different detectors is to plot the amplitude of the visibility vectors Schmahl et al. 2007) for each detector as a function of the position angle defined as the spatial direction of each grid response referenced to solar north. This is essentially the same as plotting just the amplitudes of the oscillations seen in Figure 4 but here plotted for only a half rotation from 0°to 180°with the second half rotation assumed Figure 1. ...
... because no "noise" is allowed outside the range of the assumed sources. 2. MEM_NJIT 6 (Schmahl et al. 2007) is a Maximum Entropy Method (MEM) algorithm based on visibilities that was originally developed at New Jersey Institute of Technology (NJIT). It provides the best fit to the visibility amplitudes especially for the subcollimators with the finest pitch as indicated in Figure 5. ...
The unusually narrow X-ray source imaged with RHESSI during an impulsive spike lasting for $\sim$10~s during the GOES C7.9 flare on 25 September 2011 (SOL2011-09-25T03:32) was only $\sim$2~ arcsec wide and $\sim$10~arcsec long. Comparison with HMI magnetograms and AIA images at 1700~\AA~shows that the X-ray emission was primarily from a long ribbon in the region of positive polarity with little if any emission from the negative polarity ribbon. However, a thermal plasma source density of $\sim$10$^{12}~cm^{-3}$ estimated from the RHESSI-derived emission measure and source area showed that this could best be interpreted as a coronal hard X-ray source in which the accelerated electrons with energies lass than $\sim$50~keV were stopped by Coulomb collisions in the corona, thus explaining the lack of the more usual bright X-ray footpoints. Analysis of RHESSI spectra shows greater consistency with a multi-temperature distribution and a low energy cutoff to the accelerated electron spectrum of 22 keV compared to 12 keV if a single temperature distribution is assumed. This leads to a change in the lower limit on the total energy in electrons by an order of magnitude given the steepness of the best-fit electron spectrum with a power-law index of $\sim$6.
... [ For SOL2013-05-13T02:12, the volume was estimated by applying the imaging algorithm Visibility Forward Fitting (VIS FWDFIT; Schmahl et al. 2007). Using VIS FWDFIT, the volume of the coronal source varies with energy (see Jeffrey et al. 2015), so we determined the mean volume over the energy range of 10-25 keV (Figure 3), yielding a value of V = (0.86 ± 0.20) × 10 27 cm 3 ≈ (0.9 ± 0.2) × 10 27 cm 3 . ...
Solar flare hard X-ray (HXR) spectroscopy serves as a key diagnostic of the accelerated electron spectrum. However, the standard approach using the collisional cold thick-target model poorly constrains the lower-energy part of the accelerated electron spectrum, hence the overall energetics of the accelerated electrons and consequently the flare energetics are typically constrained only to within one or two orders of magnitude. I will discuss the development and application of a physically self-consistent, warm-target approach that involves the use of both HXR spectroscopy and imaging data. The approach allows an accurate determination of the electron distribution low-energy cutoff, and hence the electron acceleration rate and the contribution of accelerated electrons to the total energy released, by constraining the coronal plasma parameters. Using a solar flare observed in X-rays by RHESSI, we demonstrate that using the standard cold-target methodology, the low-energy cutoff (hence the energy content in electrons) is essentially undetermined. However, the warm-target methodology can determine the low-energy electron cutoff with ˜7% uncertainty at the 3σ level, hence it permits an accurate quantitative study of the importance of accelerated electrons in solar flare energetics.
... B. APPENDIX: VOLUME ESTIMATES FROM RHESSI IMAGING Calculating the volume from RHESSI imaging can be difficult, because of the partial Fourier coverage of the instrument. Different imaging algorithms will give different estimates, and the CLEAN algorithm is known to sur-estimate the source size Schmahl et al. 2007;Kontar et al. 2010). In this study, we used the estimates derived from the visibility forward fit algorithm, which provide a measure of the uncertainty of the size of the source. ...
We present the statistical analysis of 33 flare-related coronal jets, and discuss the link between the jet and the flare properties in these events. We selected jets that were observed between 2010 and 2016 by the Atmospheric Imaging Assembly (AIA) on board the Solar Dynamic Observatory (SDO) and are temporally and spatially associated to flares observed by the Reuven Ramaty High Energy Solar Spectrometric Imager (RHESSI). For each jet, we calculated the jet duration and projected velocity in the plane of sky. The jet duration distribution has a median of 18.8 minutes. The projected velocities are between 31 km/s and 456 km/s with a median at 210 km/s. For each associated flare, we performed X-ray imaging and spectroscopy and identify non-thermal emission. Non-thermal emission was detected in only 1/4 of the event considered. We did not find a clear correlation between the flare thermal energy or SXR peak flux and the jet velocity. A moderate anti-correlation was found between the jet duration and the flare SXR peak flux. There is no preferential time delay between the flare and the jet. The X-ray emission is generally located at the base of the jet. The analysis presented in this paper suggests that the flare and jet are part of the same explosive event, that the jet is driven by the propagation of an Alfvenic perturbation, and that the energy partition between flare and jets varies substantially from one event to another.
... [ For SOL2013-05-13T02:12, the volume was estimated by applying the imaging algorithm Visibility Forward Fitting (VIS FWDFIT; Schmahl et al. 2007). Using VIS FWDFIT, the volume of the coronal source varies with energy (see Jeffrey et al. 2015), so we determined the mean volume over the energy range of 10-25 keV (Figure 3), yielding a value of V = (0.86 ± 0.20) × 10 27 cm 3 ≈ (0.9 ± 0.2) × 10 27 cm 3 . ...
Solar flare hard X-ray (HXR) spectroscopy serves as a key diagnostic of the accelerated electron spectrum. However, the standard approach using the collisional cold thick-target model poorly constrains the lower-energy part of the accelerated electron spectrum, hence the overall energetics of the accelerated electrons are typically constrained only to within one or two orders of magnitude. Here, we develop and apply a physically self-consistent, warm-target approach that involves the use of both HXR spectroscopy and imaging data. This approach allows an accurate determination of the electron distribution low-energy cutoff, and hence the electron acceleration rate and the contribution of accelerated electrons to the total energy released, by constraining the coronal plasma parameters. Using a solar flare observed in X-rays by RHESSI, we demonstrate that using the standard cold-target methodology, the low-energy cutoff (hence the energy content in electrons) is essentially undetermined. However, the warm-target methodology can determine the low-energy electron cutoff with ∼7% uncertainty at the 3σ level, hence it permits an accurate quantitative study of the importance of accelerated electrons in solar flare energetics.
