Phillips-Tikhonov Regularization Research Articles

Feasibility study of fast-ion velocity-space tomography in KSTAR via phantom tests

The fast-ion velocity distribution function is crucial for understanding fast-ion behavior and transport in future burning plasmas. However, direct measurements of this distribution are difficult due to its high-dimensional nature, necessitating inference from diagnostic data. To infer fast-ion velocity distributions in KSTAR experimental conditions, we explored the feasibility of using measurements from fast-ion Dα (FIDA) diagnostics. We assessed the reconstruction quality for two phantoms, representing a possible fast-ion distribution scenario and local velocity-space structures. We calculated the phase-space weight function of FIDA measurements, required for tomographic inversion, by modeling the measurements, and also developed a tomography code with Phillips–Tikhonov regularization. The phantom test results revealed limitations in the reconstruction capability of current FIDA systems in KSTAR, particularly near low-pitch regions. We also identified the influence of spatial bias of the weight function of the current FIDA systems. Introducing a new FIDA system to tomographic inversion process provided wider coverage in velocity space and the weight function with reduced spatial bias, thereby improving reconstruction capability, especially in low-pitch regions. We also scanned noise levels in the phantom tests and observed the benefits of using prior information to mitigate degradation of the reconstruction quality caused by noise.

Observational constraints on methane emissions from Polish coal mines using a ground-based remote sensing network

Abstract. Given its abundant coal mining activities, the Upper Silesian Coal Basin (USCB) in southern Poland is one of the largest sources of anthropogenic methane (CH4) emissions in Europe. Here, we report on CH4 emission estimates for coal mine ventilation facilities in the USCB. Our estimates are driven by pairwise upwind–downwind observations of the column-average dry-air mole fractions of CH4 (XCH4) by a network of four portable, ground-based, sun-viewing Fourier transform spectrometers of the type EM27/SUN operated during the CoMet campaign in May–June 2018. The EM27/SUN instruments were deployed in the four cardinal directions around the USCB approximately 50 km from the center of the basin. We report on six case studies for which we inferred emissions by evaluating the mismatch between the observed downwind enhancements and simulations based on trajectory calculations releasing particles out of the ventilation shafts using the Lagrangian particle dispersion model FLEXPART. The latter was driven by wind fields calculated by WRF (Weather Research and Forecasting model) under assimilation of vertical wind profile measurements of three co-deployed wind lidars. For emission estimation, we use a Phillips–Tikhonov regularization scheme with the L-curve criterion. Diagnosed by the emissions averaging kernels, we find that, depending on the catchment area of the downwind measurements, our ad hoc network can resolve individual facilities or groups of ventilation facilities but that inspecting the emissions averaging kernels is essential to detect correlated estimates. Generally, our instantaneous emission estimates range between 80 and 133 kt CH4 a−1 for the southeastern part of the USCB and between 414 and 790 kt CH4 a−1 for various larger parts of the basin, suggesting higher emissions than expected from the annual emissions reported by the E-PRTR (European Pollutant Release and Transfer Register). Uncertainties range between 23 % and 36 %, dominated by the error contribution from uncertain wind fields.

Three-dimensional tomographically reconstructed optical emission profiles of Hall thruster plasmas

A diagnostic system was developed for spectrally resolved, three-dimensional tomographic reconstruction of Hall thruster plasmas, and local intensity profiles of Xe I and Xe II emissions were reconstructed. In this diagnostic system, 28 virtual cameras were generated using a single, fixed charge-coupled device camera by rotating the Hall thruster to form a sufficient number of lines of sight. The Phillips–Tikhonov regularization algorithm was used to reconstruct local emission profiles from the line-integrated emission signals. The reconstruction performance was evaluated using both azimuthally symmetric and asymmetric synthetic phantom images including 5% Gaussian white noise, which resulted in a root-mean-square error of the reconstruction within an order of 10−3 even for a 1% difference in the azimuthal intensity distribution. Using the developed system, three-dimensional local profiles of Xe II emission (541.9 nm) from radiative decay of the excited state 5p4(3P2)6p2[3]°5/2 and Xe I emission (881.9 nm) from 5p5(2P°3/2)6p2[5/2]3 were obtained, and two different shapes were found depending on the wavelength and the distance from the thruster exit plane. In particular, a stretched central jet structure was distinctively observed in the Xe II emission profile beyond 10 mm from the thruster exit, while gradual broadening was found in the Xe I emission. Approximately 10% azimuthal nonuniformities were observed in the local Xe I and Xe II intensity profiles in the near-plume region (<10 mm), which could not be quantitatively distinguished by analysis of the frontal photographic image. Three-dimensional Xe I and Xe II intensity profiles were also obtained in the plume region, and the differences in the structures of both emissions were visually confirmed.

Double-filter high-resolution soft x-ray tomographic diagnostic for investigating electron acceleration in TS-6 reconnection merging experiments.

An innovative tangential-view soft x-ray (SXR) tomographic imaging measurement was developed on the TS-6 spherical tokamak merging device as a key diagnostic for investigating the mechanism of electron acceleration. In order to measure SXR with different energy ranges, two micro-channel plates (MCPs) are, respectively, installed in two vacuum chambers, which are equipped with different filters. Especially designed lenses and fiber bundles serve as an optical system to transfer images from phosphor plates of MCPs to a high speed imaging system. This design also enables us to simultaneously measure two images appearing on phosphor plates of MCPs by just one high speed imaging system. The temporal and spatial resolution of this diagnostic can be up to 5 µs and 4mm, respectively, at present. The tomographic method based on the Phillips-Tikhonov regularization is employed to reconstruct line-integrated images into the local emissivity of SXR, which reflects the spatial distribution of high-energy electrons. Owing to this diagnostic, we successfully measured SXR emitted from the downstream region of magnetic reconnection in TS-6 merging experiments for the first time. The energy range of SXR turned out to be higher than 100eV but lower than 400eV.

Soft x-ray tomographic reconstruction of Heliotron J plasma for the study of magnetohydrodynamic equilibrium and stability

The preparation for an experimental soft x-ray tomography study on the Heliotron J (H-J) machine is carried out, with the objectives of evaluating the capability of the current soft x-ray tomographic system in terms of the identification of different mode structures and their poloidal rotation, and the axis shift with different plasma and machine parameters, and fixing the physics goals for the experimental study. These preparations were carried out via a simulated soft x-ray data set arising from different plasma conditions, such as magnetic islands, low beta and high beta. Soft x-ray tomography (SXT) is performed by the discrete pixel method including singular value decomposition and Phillips–Tikhonov regularization, to obtain clear and smooth images. The H-J soft x-ray tomography results from simulated soft x-rays for the equilibrium H-J plasma sensed the magnetic axis shift clearly and an estimate was also achieved. Successful reconstruction for mode structure m = 1/n = 1 was obtained along with the realization of the poloidal rotation of the structure. The reconstruction for the m = 2/n = 1 mode was not very clear for the current soft x-ray diagnostic design. Effective mode identification was not possible due to the lack of measurements. The SXT from the current soft x-ray diagnostic on H-J, the magnetic axis shift can be estimated and the m = 1/n = 1 mode can be studied. Study of higher poloidal modes is difficult with the current design.

Sparse data recovery of tomographic diagnostics for ultra-large-area plasmas

As the size of plasma systems reaches multiple square meters, several phenomena lead to non-uniform plasma production; as a result, the requirement for real-time monitoring of plasma uniformity is increasing, particularly for industrial plasmas. Although non-intrusive diagnostics (e.g. optical emission spectroscopy) are preferred to monitor such plasmas, line (or volume)-integrated measurement is indispensable. Thus, special attention has been paid to the tomographic reconstruction technique; however, limitations remain for tomographic diagnostics for ultra-large-area plasmas. Here, we report an inversion technique that remarkably improves the performance of tomographic reconstruction for large-area industrial plasmas. A computational tool for the tomographic reconstruction with Phillips–Tikhonov regularization was prepared, and reconstruction performance tests were conducted using artificial (phantom) images (namely, expected plasma emission distributions). While very sparse line-of-sight and projection data result in poor reconstruction performance, two suggested operators, which constrain the reconstruction solution, noticeably improve the reconstruction performance and reduce the overall reconstruction error. Although the projection data are contaminated by random noise (<5% of data), the reconstruction error was improved by 43.8% and 40.5% for flat- and hollow-shaped phantoms, respectively. The computational analysis clearly demonstrates that tomographic plasma diagnostics with the proposed inversion technique can be used as a real-time tool for detecting arcing and monitoring plasma spatial uniformity in large-area industrial plasmas.

Tomography-based spatial uniformity diagnostics for meter-sized plasmas

Optical emission spectral tomography diagnostics have been developed to measure the spatial uniformity of meter-sized processing plasmas. The lines of sight and detector location are selected based on tomographic reconstruction tests using synthetic (phantom) images. The developed collection optics system is composed of slits, a plano-convex lens, optical interference filters and photodiode arrays. Using the collection optics, the line-integrated emission intensity is acquired from a meter-sized rectangular argon plasma (inductively coupled plasma at a frequency of 13.56 MHz), which was developed for display panel manufacturing. From the measured intensity, two-dimensional (2D) spatial distributions of argon atomic line intensity (675.3 nm and 852.1 nm) are obtained via the tomographic reconstruction technique based on the Phillips–Tikhonov regularization. In addition, 2D profiles of the electron excitation temperature (Texc) are obtained from those of Ar intensity at the wavelengths 675.3 nm and 852.1 nm via two line method (in essence, a Boltzmann plot). The 2D argon plasma emission and Texc profiles match well with the shape of the electrode and etch rate profile.

Tomographic reconstruction for a tangentially viewing visible light imaging diagnostic on EAST

A two-dimensional tomographic reconstruction method based on the Phillips-Tikhonov regularization is proposed in this study, for a tangentially viewing visible light imaging diagnostic on EAST. Phantom tests are demonstrated to evaluate the performance of this method. The purpose of this method is to reconstruct the local emission from experimental images in various parts of the visible spectrum, which can be employed to reflect spatial distributions of certain particles (Li-II in this study) in plasmas. As a potential application of the Li-II distribution, the measurement of plasma boundary displacements in the presence of resonant magnetic perturbations is tried. Though the result of this attempt is limited by the computational precision and conditions of diagnostic system at present, it provides valuable references for further works.

Combined neural network/Phillips–Tikhonov approach to aerosol retrievals over land from the NASA Research Scanning Polarimeter

Abstract. In this paper, an algorithm for the retrieval of aerosol and land surface properties from airborne spectropolarimetric measurements – combining neural networks and an iterative scheme based on Phillips–Tikhonov regularization – is described. The algorithm – which is an extension of a scheme previously designed for ground-based retrievals – is applied to measurements from the Research Scanning Polarimeter (RSP) on board the NASA ER-2 aircraft. A neural network, trained on a large data set of synthetic measurements, is applied to perform aerosol retrievals from real RSP data, and the neural network retrievals are subsequently used as a first guess for the Phillips–Tikhonov retrieval. The resulting algorithm appears capable of accurately retrieving aerosol optical thickness, fine-mode effective radius and aerosol layer height from RSP data. Among the advantages of using a neural network as initial guess for an iterative algorithm are a decrease in processing time and an increase in the number of converging retrievals.

Development of bolometric tomography technique for high-contrast radiation distribution in toroidal devices.

An extended Computed Tomography (CT) technique with a priori information based on Phillips-Tikhonov regularization has been developed to handle a high-contrast radiation distribution, which can result in large reconstruction errors in the region where radiation intensity is low, with few line of sight (LOS) data. Reference profiles generated from LOS data for every time slice are employed as the a priori information. In the extended technique, the weighting parameter for the reference profile is automatically determined from the LOS data, to avoid an inappropriate reference. The extended technique has been examined with a reconstruction test with the previously designed CT system using an infrared imaging video bolometer in JT-60SA. In the reconstructed profile of the high-contrast radiation distribution, the extended technique shows improvement of the weak radiation region, such as the scrape-off layer, without the deterioration of the strong radiation region, such as around the divertor. The results indicate that the extended CT technique can help to handle a high-contrast radiation distribution with few LOS data.

Algorithm and software to automatically identify latency and amplitude features of local field potentials recorded in electrophysiological investigation

BackgroundLocal field potentials (LFPs) evoked by sensory stimulation are particularly useful in electrophysiological research. For instance, spike timing and current transmembrane current flow estimated from LFPs recorded in the barrel cortex in rats and mice are exploited to investigate how the brain represents sensory stimuli. Recent improvements in microelectrodes technology enable neuroscientists to acquire a great amount of LFPs during the same experimental session, calling for algorithms for their quantitative automatic analysis. Several computer tools were proposed for LFP analysis, but many of them incorporate algorithms that are not open to inspection or modification/personalization. We present a MATLAB software to automatically detect some important LFP features (latency, amplitude, time-derivative value in the inflection-point) for a quantitative analysis. The software features can be customized by the user according to his/her personal research needs. The incorporated algorithm is based on Phillips-Tikhonov regularization to deal with noise amplification due to ill-conditioning. In particular, its accuracy in the estimation of the features of interest is assessed in a Monte Carlo simulation mimicking the acquisition of LFPs in different SNR (signal-to-noise-ratio) conditions. Then, the algorithm is tested by analyzing a real set of 2500 LFPs recorded in rat after whisker stimulation at different depths in the primary somatosensory (S1) cortex, i.e., the region involved in the cortical representation of touch in mammals.ResultsAutomatic identification of LFP features by the presented software is easy and fast. As far as accuracy is concerned, error indices from simulated data suggest that the algorithm provides reliable estimates . Indeed, results obtained from LFPs recorded in rat after whisker stimulation are in line with the known sequential activation of the microcircuits of the S1 cortex.ConclusionA MATLAB software implementing an algorithm to automatically detect the main LFPs features was presented. Simulated and real case studies showed that the employed algorithm is accurate and robust against measurement noise. The available code can be used as it is, but the reported description of the algorithms allows users to easily modify the code to cope with specific requirements.

Comparison of two regularization methods for soft x-ray tomography at Tore Supra

Soft x-ray (SXR) emission in the range 0.1–20 keV is widely used to obtain valuable information on tokamak plasma physics, such as particle transport, magnetic configuration or magnetohydrodynamic activity. In particular, 2D tomography is the usual plasma diagnostic to access the local SXR emissivity. The tomographic inversion is traditionally performed from line-integrated measurements of two or more cameras viewing the plasma in a poloidal cross-section, like at Tore Supra (TS). Unfortunately, due to the limited number of measured projections and presence of noise, the tomographic reconstruction of SXR emissivity is a mathematical ill-posed problem. Thus, obtaining reliable results of the tomographic inversion is a very challenging task. In order to perform the reconstruction, inversion algorithms implemented in present tokamaks use a priori information as additional constraints imposed on the plasma SXR emissivity. Among several potential inversion methods, some of them have been identified as well suited to tokamak plasmas. The purpose of this work is to compare two promising inversion methods, i.e. the minimum fisher information method already used at TS and planned for WEST configuration, and the alternative 2nd order Phillips–Tikhonov regularization with smoothness constraints imposed on the second derivative norm. Respective accuracy of both reconstruction methods as well as overall robustness and computational time are studied, using several synthetic SXR emissivity profiles. Finally, a real case is studied through tomographic reconstruction from TS SXR database.

Phillips-Tikhonov regularization with a priori information for neutron emission tomographic reconstruction on Joint European Torus.

A method of tomographic reconstruction of the neutron emissivity in the poloidal cross section of the Joint European Torus (JET, Culham, UK) tokamak was developed. Due to very limited data set (two projection angles, 19 lines of sight only) provided by the neutron emission profile monitor (KN3 neutron camera), the reconstruction is an ill-posed inverse problem. The aim of this work consists in making a contribution to the development of reliable plasma tomography reconstruction methods that could be routinely used at JET tokamak. The proposed method is based on Phillips-Tikhonov regularization and incorporates a priori knowledge of the shape of normalized neutron emissivity profile. For the purpose of the optimal selection of the regularization parameters, the shape of normalized neutron emissivity profile is approximated by the shape of normalized electron density profile measured by LIDAR or high resolution Thomson scattering JET diagnostics. In contrast with some previously developed methods of ill-posed plasma tomography reconstruction problem, the developed algorithms do not include any post-processing of the obtained solution and the physical constrains on the solution are imposed during the regularization process. The accuracy of the method is at first evaluated by several tests with synthetic data based on various plasma neutron emissivity models (phantoms). Then, the method is applied to the neutron emissivity reconstruction for JET D plasma discharge #85100. It is demonstrated that this method shows good performance and reliability and it can be routinely used for plasma neutron emissivity reconstruction on JET.

Automated analysis of local field potentials evoked by mechanical whisker stimulation in rat barrel cortex.

Local field potentials (LFPs) recorded in the barrel cortex in rats and mice are important to investigate somatosensory systems, the final aim being to start to understand mechanisms of brain representation of sensory stimuli in humans. Parameters extracted from LFP of particular interest include spike timing and transmembrane current flow. Recent improvements in microelectrodes technology have enabled neuroscientists to acquire a great amount of LFP signals during the same experimental session, calling for the development of algorithms for their quantitative automatic analysis. In the present work, an algorithm based on Phillips-Tikhonov regularization is presented to automatically detect the main features (in terms of amplitude and latency) of LFP waveforms recorded after whisker stimulation in rat. The accuracy of the algorithm is first assessed in a Monte Carlo simulation mimicking the acquisition of LFP in three different conditions of SNR. Then, the algorithm is tested by analyzing a set of 100 LFP recorded in the primary somatosensory (S1) cortex, i.e., the region involved in the cortical representation of touch in mammals.

Moment discretization for ill-posed problems with discrete weakly bounded noise

We study moment discretization for compact operator equations in Hilbert space with discrete noisy data. Instead of assuming that the error in the data converges strongly to 0, we only assume weak convergence of the noise as introduced by Eggermont et al. (Inverse Probl 25:115018, 2009). A specific instance would be random noise. Under the usual source conditions, we derive optimal convergence rates for Phillips-Tikhonov regularization. The analysis is based on the comparison of the discrete problem with a semi-discrete version of the problem, which is made possible by virtue of a quadrature result in a suitable reproducing kernel Hilbert space. Some numerical results using strong and weak discrepancy principles for the selection of the regularization parameter are presented.

Soft X-Ray Imaging Design and Analysis Methods on DIII-D

A new tangential 2D Soft X-Ray Imaging System (SXRIS) is being designed to examine the edge magnetic island structure in the lower X-point region of DIII-D. A synthetic diagnostic calculation coupled to 3D emissivity estimates is used to generate phantom images. Phillips-Tikhonov regularization is used to invert the phantom images for comparison to the original emissivity model. Noise level, island size, and equilibrium accuracy are scanned to assess the feasibility of detecting edge island structures. Models of typical DIII-D discharges indicate integration times > 1 ms with accurate equilibrium reconstruction are needed for small island (< 3 cm) detection.

First results of the resistive bolometers on KSTAR

The resistive bolometers have been successfully installed in the midplane of L-port in Korea Superconducting Tokamak Advanced Research (KSTAR) device. The spatial and temporal resolutions, 4.5 cm and ∼1 kHz, respectively, enable us to measure the radial profile of the total radiated power from magnetically confined plasma at a high temperature through radiation and neutral particles. The radiated power was measured at all shots. Even at low plasma current, the bolometer signal was detectable. The electron cyclotron resonance heating (ECH) has been used in tokamak for ECH assisted start-up and plasma control by local heating and current drive. The detectors of resistive bolometer, near the antenna of ECH, are affected by electron cyclotron wave. The tomographic reconstruction, using the Phillips-Tikhonov regularization method, will be carried out for a major radial profile of the radiation emissivity of the circular cross-section plasma.

On weakly bounded noise in ill-posed problems

We study compact operator equations with noisy data in Hilbert space. Instead of assuming that the error in the data converges strongly to zero, we only assume a type of weak convergence. Under the source conditions that are usually assumed in the presence of convex constraints, we derive optimal convergence rates for convexly constrained Phillips–Tikhonov regularization. We also discuss a version of the Lepskii method for selecting the regularization parameter.

Modified Phillips–Tikhonov regularization for plasma tomography

The tomography has been used as a powerful diagnostic method for visualizing cross-sectional images in plasma research. Without any biasing data, the Phillips–Tikhonov (P–T) regularization method was attempted in this work to reconstruct cross-sectional phantom images of the plasma by minimizing the gradient between adjacent pixel data. A comparison of the tomographic reconstructions of the cross-sectional images similar to the tokamak plasmas by the P–T method and the maximum entropy (ME) method showed that the P–T method produced more accurate results. In addition, the P–T method was modified by adding an iteration procedure with a second-order correction to improve the accuracy of the reconstruction for noisy line-integrated data. Tomographic reconstructions in the presence of Gaussian-distributed random noise demonstrated that the modified P–T method with only several iterations significantly reduced the mean reconstruction error by (30–50)% of the original root-mean-square error caused by the random noise.

Analysis of the solution of the elastic light scattering inverse problem for polymeric emulsions

We consider the problem of inversion of elastic light scattering (ELS) measurements from polymeric emulsions, to obtain its particle size distribution (PSD) and its refractive index. The mathematical formulation results in a nonlinear inverse problem. A Fredholm integral equation of the first kind appears with an unknown parameter in its kernel. We discuss the existence, uniqueness, and stability of the generalized solutions of the problem when it is stated as a minimization problem with a least square functional. First, we assume that the PSD is known, and for this case we prove that the solution exists and is unique as long as the relation between the measurements and the parameter is by an injective function. Then, we use this result to state sufficient conditions for the complete problem. The analysis of existence and uniqueness of the solution for the problem in hand is supported by numerical simulation. The Phillips–Tikhonov regularization method is proposed to stabilize the problem when noisy-data is available.