An Overview of Existing Algorithms for Resolving the 180° Ambiguity in Vector Magnetic Fields: Quantitative Tests with Synthetic Data
TL;DR: In this article, the authors report on the state-of-the-art in algorithms used for resolving the 180° ambiguity in solar vector magnetic field measurements and compare them quantitatively and seek to understand where each succeeds, where it fails and why.
Abstract: We report here on the present state-of-the-art in algorithms used for resolving the 180° ambiguity in solar vector magnetic field measurements. With present observations and techniques, some assumption must be made about the solar magnetic field in order to resolve this ambiguity. Our focus is the application of numerous existing algorithms to test data for which the correct answer is known. In this context, we compare the algorithms quantitatively and seek to understand where each succeeds, where it fails, and why. We have considered five basic approaches: comparing the observed field to a reference field or direction, minimizing the vertical gradient of the magnetic pressure, minimizing the vertical current density, minimizing some approximation to the total current density, and minimizing some approximation to the field's divergence. Of the automated methods requiring no human intervention, those which minimize the square of the vertical current density in conjunction with an approximation for the vanishing divergence of the magnetic field show the most promise.
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TL;DR: The Helioseismic and Magnetic Imager (HMI) instrument and investigation as a part of the NASA Solar Dynamics Observatory (SDO) is designed to study convection-zone dynamics and the solar dynamo, the origin and evolution of sunspots, active regions, and complexes of activity, the sources and drivers of solar magnetic activity and disturbances as mentioned in this paper.
Abstract: The Helioseismic and Magnetic Imager (HMI) instrument and investigation as a part of the NASA Solar Dynamics Observatory (SDO) is designed to study convection-zone dynamics and the solar dynamo, the origin and evolution of sunspots, active regions, and complexes of activity, the sources and drivers of solar magnetic activity and disturbances, links between the internal processes and dynamics of the corona and heliosphere, and precursors of solar disturbances for space-weather forecasts. A brief overview of the instrument, investigation objectives, and standard data products is presented.
2,242 citations
Cites methods from "An Overview of Existing Algorithms ..."
...A further product is computed as the final vector field by applying a field direction disambiguation technique, since the Stokes parameters cannot distinguish the polarity of the component of the field normal to the line-of-sight (Metcalf, 1994; Metcalf et al., 2006; Leka et al., 2009)....
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TL;DR: In this article, the Helioseismic and Magnetic Imager (HMI) active region patches (HARPs) are used to track the location and shape of magnetic regions throughout their lifetime.
Abstract: The Helioseismic and Magnetic Imager (HMI) began near-continuous full-disk solar measurements on 1 May 2010 from the Solar Dynamics Observatory (SDO). An automated processing pipeline keeps pace with observations to produce observable quantities, including the photospheric vector magnetic field, from sequences of filtergrams. The basic vector-field frame list cadence is 135 seconds, but to reduce noise the filtergrams are combined to derive data products every 720 seconds. The primary 720 s observables were released in mid-2010, including Stokes polarization parameters measured at six wavelengths, as well as intensity, Doppler velocity, and the line-of-sight magnetic field. More advanced products, including the full vector magnetic field, are now available. Automatically identified HMI Active Region Patches (HARPs) track the location and shape of magnetic regions throughout their lifetime. The vector field is computed using the Very Fast Inversion of the Stokes Vector (VFISV) code optimized for the HMI pipeline; the remaining 180∘ azimuth ambiguity is resolved with the Minimum Energy (ME0) code. The Milne–Eddington inversion is performed on all full-disk HMI observations. The disambiguation, until recently run only on HARP regions, is now implemented for the full disk. Vector and scalar quantities in the patches are used to derive active region indices potentially useful for forecasting; the data maps and indices are collected in the SHARP data series, hmi.sharp_720s. Definitive SHARP processing is completed only after the region rotates off the visible disk; quick-look products are produced in near real time. Patches are provided in both CCD and heliographic coordinates. HMI provides continuous coverage of the vector field, but has modest spatial, spectral, and temporal resolution. Coupled with limitations of the analysis and interpretation techniques, effects of the orbital velocity, and instrument performance, the resulting measurements have a certain dynamic range and sensitivity and are subject to systematic errors and uncertainties that are characterized in this report.
599 citations
Cites background or methods from "An Overview of Existing Algorithms ..."
...This method has been shown to be among the best performing on a wide range of tests on synthetic data for which the answer is known (Metcalf et al., 2006; Leka et al., 2009)....
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...A plethora of methods have been proposed to resolve this ambiguity (for an overview, see Metcalf et al., 2006)....
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...A neighboring-pixel acute-angle algorithm based on the University of Hawai’i Iterative method (UHIM; Canfield et al., 1993; Metcalf et al., 2006) that locally maximizes the sum of the dot product of the field at one pixel with the field at its neighbors is used....
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TL;DR: In this paper, the authors discuss NLFFF modeling of NOAA Active Region 10953 using Hinode/SOT-SP, Hinode-XRT, STEREO/SECCHI-EUVI, and SOHO/MDI observations, and in the process illustrate three such issues.
Abstract: Nonlinear force-free field (NLFFF) models are thought to be viable tools for investigating the structure, dynamics, and evolution of the coronae of solar active regions. In a series of NLFFF modeling studies, we have found that NLFFF models are successful in application to analytic test cases, and relatively successful when applied to numerically constructed Sun-like test cases, but they are less successful in application to real solar data. Different NLFFF models have been found to have markedly different field line configurations and to provide widely varying estimates of the magnetic free energy in the coronal volume, when applied to solar data. NLFFF models require consistent, force-free vector magnetic boundary data. However, vector magnetogram observations sampling the photosphere, which is dynamic and contains significant Lorentz and buoyancy forces, do not satisfy this requirement, thus creating several major problems for force-free coronal modeling efforts. In this paper, we discuss NLFFF modeling of NOAA Active Region 10953 using Hinode/SOT-SP, Hinode/XRT, STEREO/SECCHI-EUVI, and SOHO/MDI observations, and in the process illustrate three such issues we judge to be critical to the success of NLFFF modeling: (1) vector magnetic field data covering larger areas are needed so that more electric currents associated with the full active regions of interest are measured, (2) the modeling algorithms need a way to accommodate the various uncertainties in the boundary data, and (3) a more realistic physical model is needed to approximate the photosphere-to-corona interface in order to better transform the forced photospheric magnetograms into adequate approximations of nearly force-free fields at the base of the corona. We make recommendations for future modeling efforts to overcome these as yet unsolved problems.
443 citations
Cites methods from "An Overview of Existing Algorithms ..."
...…algorithms to do this, we chose as an alternative to AZAM to employ the automated University of Hawai‘i Iterative Method (UHIM) (Canfield et al. 1993) because it has been used extensively in the literature and also scored highly amongst other ambiguity resolution algorithms (Metcalf et al. 2006)....
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...We choose to perform the disambiguation using the interactive Azimuthal Ambiguity Method (AZAM), which is one of several methods have been devised and tested to resolve this ambiguity (see Metcalf et al. 2006, and references therein)....
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TL;DR: In this article, the authors report the evolution of magnetic field and its energy in NOAA active region 11158 over 5 days based on a vector magnetogram series from the Helioseismic and Magnetic Imager (HMI) on board the Solar Dynamic Observatory (SDO).
Abstract: We report the evolution of magnetic field and its energy in NOAA active region 11158 over 5 days based on a vector magnetogram series from the Helioseismic and Magnetic Imager (HMI) on board the Solar Dynamic Observatory (SDO). Fast flux emergence and strong shearing motion led to a quadrupolar sunspot complex that produced several major eruptions, including the first X-class flare of Solar Cycle 24. Extrapolated non-linear force-free coronal fields show substantial electric current and free energy increase during early flux emergence near a low-lying sigmoidal filament with sheared kilogauss field in the filament channel. The computed magnetic free energy reaches a maximum of ∼2.6 × 10 32 erg, about 50% of which is stored below 6 Mm. It decreases by ∼0.3 × 10 32 erg within 1 hour of the X-class flare, which is likely an underestimation of the actual energy loss. During the flare, the photospheric field changed rapidly: horizontal field was enhanced by 28% in the core region, becoming more inclined and more parallel to the polarity inversion line. Such change is consistent with the conjectured coronal field “implosion”, and is supported by the coronal loop retraction observed by the Atmospheric Imaging Assembly (AIA). The extrapolated field becomes more “compact” after the flare, with shorter loops in the core region, probably because of reconnection. The coronal field becomes slightly more sheared in the lowest layer, relaxes faster with height, and is overall less energetic.
422 citations
Advanced Technology Center1, National Center for Atmospheric Research2, University of California, Berkeley3, Max Planck Society4, University of Sydney5, École Polytechnique6, Paris Diderot University7, University of Potsdam8, Japan Agency for Marine-Earth Science and Technology9, University of St Andrews10
TL;DR: In this article, a nonlinear force-free field (NLFFF) model was applied to the highest resolution and quality vector-magnetographic data observed by the recently launched Hinode satellite on NOAA AR 10930 around the time of a powerful X3.4 flare.
Abstract: Solar flares and coronal mass ejections are associated with rapid changes in field connectivity and are powered by the partial dissipation of electrical currents in the solar atmosphere. A critical unanswered question is whether the currents involved are induced by the motion of preexisting atmospheric magnetic flux subject to surface plasma flows or whether these currents are associated with the emergence of flux from within the solar convective zone. We address this problem by applying state-of-the-art nonlinear force-free field (NLFFF) modeling to the highest resolution and quality vector-magnetographic data observed by the recently launched Hinode satellite on NOAA AR 10930 around the time of a powerful X3.4 flare. We compute 14 NLFFF models with four different codes and a variety of boundary conditions. We find that the model fields differ markedly in geometry, energy content, and force-freeness. We discuss the relative merits of these models in a general critique of present abilities to model the coronal magnetic field based on surface vector field measurements. For our application in particular, we find a fair agreement of the best-fit model field with the observed coronal configuration, and argue (1) that strong electrical currents emerge together with magnetic flux preceding the flare, (2) that these currents are carried in an ensemble of thin strands, (3) that the global pattern of these currents and of field lines are compatible with a large-scale twisted flux rope topology, and (4) that the ~1032 erg change in energy associated with the coronal electrical currents suffices to power the flare and its associated coronal mass ejection.
343 citations
Cites methods from "An Overview of Existing Algorithms ..."
...The algorithm applied includes the enhancements described in Metcalf et al. (2006) and was the top-performing automated method among those compared in that same study....
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References
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TL;DR: There is a deep and useful connection between statistical mechanics and multivariate or combinatorial optimization (finding the minimum of a given function depending on many parameters), and a detailed analogy with annealing in solids provides a framework for optimization of very large and complex systems.
Abstract: There is a deep and useful connection between statistical mechanics (the behavior of systems with many degrees of freedom in thermal equilibrium at a finite temperature) and multivariate or combinatorial optimization (finding the minimum of a given function depending on many parameters). A detailed analogy with annealing in solids provides a framework for optimization of the properties of very large and complex systems. This connection to statistical mechanics exposes new information and provides an unfamiliar perspective on traditional optimization problems and methods.
41,772 citations
TL;DR: In this article, a modified Monte Carlo integration over configuration space is used to investigate the properties of a two-dimensional rigid-sphere system with a set of interacting individual molecules, and the results are compared to free volume equations of state and a four-term virial coefficient expansion.
Abstract: A general method, suitable for fast computing machines, for investigating such properties as equations of state for substances consisting of interacting individual molecules is described. The method consists of a modified Monte Carlo integration over configuration space. Results for the two‐dimensional rigid‐sphere system have been obtained on the Los Alamos MANIAC and are presented here. These results are compared to the free volume equation of state and to a four‐term virial coefficient expansion.
35,161 citations
"An Overview of Existing Algorithms ..." refers methods in this paper
...Hence, the “simulated annealing” algorithm (Metropolis et al., 1953; Kirkpatrick et al., 1983) is used to find the global minimum, an extremely robust approach when faced with a large, discrete problem (there are two and only two possibilities at each pixel) with many local minima (Metcalf, 1994)....
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TL;DR: The Michelson Doppler Imager (MDI) as mentioned in this paper was used to probe the interior of the Sun by measuring the photospheric manifestations of solar oscillations, revealing the static and dynamic properties of the convection zone and core.
Abstract: The Solar Oscillations Investigation (SOI) uses the Michelson Doppler Imager (MDI) instrument to probe the interior of the Sun by measuring the photospheric manifestations of solar oscillations. Characteristics of the modes reveal the static and dynamic properties of the convection zone and core. Knowledge of these properties will improve our understanding of the solar cycle and of stellar evolution. Other photospheric observations will contribute to our knowledge of the solar magnetic field and surface motions. The investigation consists of coordinated efforts by several teams pursuing specific scientific objectives. The instrument images the Sun on a 10242 CCD camera through a series of increasingly narrow spectral filters. The final elements, a pair of tunable Michelson interferometers, enable MDI to record filtergrams with a FWHM bandwidth of 94 mA. Normally 20 images centered at 5 wavelengths near the Ni I 6768 spectral line are recorded each minute. MDI calculates velocity and continuum intensity from the filtergrams with a resolution of 4″ over the whole disk. An extensive calibration program has verified the end-to-end performance of the instrument. To provide continuous observations of the longest-lived modes that reveal the internal structure of the Sun, a carefully-selected set of spatial averages are computed and downlinked at all times. About half the time MDI will also be able to downlink complete velocity and intensity images each minute. This high rate telemetry (HRT) coverage is available for at least a continuous 60-day interval each year and for 8 hours each day during the rest of the year. During the 8-hour HRT intervals, 10 of the exposures each minute can be programmed for other observations, such as measurements in MDI's higher resolution (1.25″) field centered about 160″ north of the equator; meanwhile, the continuous structure program proceeds during the other half minute. Several times each day, polarizers will be inserted to measure the line-of-sight magnetic field. MDI operations will be scheduled well in advance and will vary only during the daily 8-hour campaigns. Quick-look and summary data, including magnetograms, will be processed immediately. Most high-rate data will be delivered only by mail to the SOI Science Support Center (SSSC) at Stanford, where a processing pipeline will produce 3 Terabytes of calibrated data products each year. These data products will be analyzed using the SSSC and the distributed resources of the co-investigators. The data will be available for collaborative investigations.
2,154 citations
01 Dec 1995
TL;DR: The Solar Oscillations Investigation (SOI) as mentioned in this paper uses the Michelson Doppler Imager (MDI) instrument to probe the interior of the Sun by measuring the photospheric manifestations of solar oscillations.
Abstract: The Solar Oscillations Investigation (SOI) uses the Michelson Doppler Imager (MDI) instrument to probe the interior of the Sun by measuring the photospheric manifestations of solar oscillations. Characteristics of the modes reveal the static and dynamic properties of the convection zone and core. Knowledge of these properties will improve our understanding of the solar cycle and of stellar evolution. Other photospheric observations will contribute to our knowledge of the solar magnetic field and surface motions. The investigation consists of coordinated efforts by several teams pursuing specific scientific objectives.
1,910 citations
"An Overview of Existing Algorithms ..." refers methods in this paper
...…condition can be alleviated by padding the observed line-of-sight field with non-vector data from an instrument with a large field-of-view (e.g. SOHO/MDI, Scherrer et al., 1995), padding the boundary data with zeroes, or by assuming the walls of the extrapolation box are perfectly conducting....
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TL;DR: In this article, the average magnetic helicity of 69 diverse active regions, adopting the linear force-free field parameter alpha as a measure, was determined by minimizing the differences between the computed constant-alpha force free and observed horizontal magnetic fields.
Abstract: Using a 1988-1994 data set of original photospheric vector magnetograms as well as published data, we have studied the average magnetic helicity of 69 diverse active regions, adopting the linear force-free field parameter alpha as a measure. This average value was determined by minimizing the differences between the computed constant-alpha force-free and observed horizontal magnetic fields. The average magnetic helicity shows a sign difference at the 2 sigma level in opposite hemispheres. In our data set, 76% of the active regions in the northern hemisphere have negative helicity, and 69% in the southern hemisphere, positive. Although the data show considerable variation from one active region to the next, the data set as a whole suggest that the magnitude of the average helicity increases with solar latitude, starting at zero near the equator, reaches a maximum near 15 deg - 25 deg in both hemispheres, and drops back toward smaller values avove 35 deg - 40 deg. Qualitative comparison with published models shows that such latitudinal variation of the average magnetic helicity may result from either turbulent convective motions or differential rotation, although our studies of rotating sunspots lead us to favor the former.
568 citations
"An Overview of Existing Algorithms ..." refers methods in this paper
...The code is part of a code-tree which includes a program that computes a best-fit α , selected by minimizing the difference between the transverse components of the observed field and the linear force-free field (Leka and Skumanich, 1999; Pevtsov et al., 1995)....
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