Introduction

The Symmetrized Dot Pattern (SDP) technique, known also as Artificial Insymmetrized Pattern (AIP), have been introduced to help humans interpret underlying patterns in time series. Thanks to the fact that the data is mapped in a way that artificially induces symmetry in the dataset, humans can see small differences between virtually identical time series. The creation of patterns with symmetry elements makes the signal features more obvious to the human observer and can help in texture recognition.

The SDP method was originally created for sound analysis by C. A. Pickover [1, 2]. Therefore, at the beginning most of the work on this technique was in the field of speech. Some work was also dedicated on the analysis of cardiac sounds [3]. Later, SDP were employed for fault diagnosis of fans or hydraulic machines [4,5,6].

It is worth emphasizing that SDP requires very short computational time, which is an unquestionable advantage for fast data analysis. Potentially, it is possible to apply SDP for detection and characterization of the specific features of any sampled data.

In this paper, it is proposed for the first time to test the SDP technique in the field of plasma radiation, where monitoring the current situation and quick response time are really needed for plasma real-time control. In Sect. "Generalities on Symmetrized Dot Patterns (SDP)", a general description and typical usage examples of the SDP method are presented. Then, Sect. "SDP of Time Dynamics of Tokamak Plasma Emissivity" is devoted to the application of this approach to the temporal dynamics of the soft X-ray (SXR) line-integrated brightness, recorded during plasma discharges in WEST – Tungsten (W) Environment in Steady-state Tokamak [7], with a focus on the commonly encountered sawtooth instability. Finally, some conclusions and perspectives for future work are given in the last section.

Generalities on Symmetrized Dot Patterns (SDP)

Mathematical Description

Symmetrized Dot Patterns convert the sampled data into a dots representation that are subsequently reflected by straight lines passing through the center of the figure. Thanks to this, the pattern is often reminiscent of a snowflake with six-fold symmetry.

This method creates a scatter plot of neighboring amplitudes. The digitized time waveform is represented by a specific dot pattern by putting pairs of points onto a polar coordinate graph. SDP maps the normalized time series into symmetrical patterns on a polar plot in the following way. The point in the time series \(x\left( i \right)\) is attributed a radial component \(r\left( i \right)\), and the neighboring point \(x\left( {i + \tau } \right)\) is attributed two angular components \(\theta \left( i \right)\) and \(\phi \left( i \right)\), see Fig. 1. The polar transformation takes the form [2]:

$$ \begin{array}{*{20}l} {r\left( i \right) = \frac{{x\left( i \right) - x_{{min}} }}{{x_{{max}} - x_{{min}} }},} \hfill \\ {\theta \left( i \right) = \Omega + \frac{{x\left( {i + \tau } \right) - x_{{min}} }}{{x_{{max}} - x_{{min}} }}\zeta } \hfill \\ {\phi \left( i \right) = \Omega - \frac{{x\left( {i + \tau } \right) - x_{{min}} }}{{x_{{max}} - x_{{min}} }}\zeta ,} \hfill \\ \end{array} , $$
(1)

where \(\tau\) is the time delay expressed in units of the sampling period (corresponding to the index of the time series), \(\Omega = 2\pi /m\) the angle of rotation from the selected reference line with \(m\) the number of mirrors, \(\zeta\) the so-called increment of the graph (\(\zeta \le \Omega\)), and \(x_{{{\text{max}}}}\) and \(x_{{{\text{min}}}}\) are the highest and the lowest values of the time series. Based on Eq. (1), the correspondence between \(\theta \left( i \right)\) and \(\phi \left( i \right)\) is expressed as: \(\phi \left( i \right) + \theta \left( i \right) = 2\Omega\), i.e. there is symmetry with respect to the reference line. After this transformation, a basic pattern is obtained, which is then rotated \(m\) times to create a fully symmetrical diagram. Hereinafter, a number of mirrors \(m = 4\) is used.

Fig. 1
figure 1

Basic scheme of the SDP mapping method

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It follows from the above that any given point \(x\left( i \right)\) is mapped into radial component and the adjacent point \(x\left( {i + \tau } \right)\) is mapped into angular component. The correlations between points in this pairs will determine the structure of the pattern. And so, for example in case of white noise, a uniform distribution of points is obtained, while for Gaussian noise the contours in the diagram are blurred and the points are concentrated in the centers of the patterns and appear less frequently on the edges. However, for data coming from processes containing deterministic components, points accumulate in specific areas and the diagrams are characterized by a small dispersion of points, as presented in Fig. 2.

Fig. 2
figure 2

SDP plots for: a white noise, b normally distributed noise and c sinusoid signal

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Typical Usage Examples

As mentioned previously, SDP were originally invented for graphic displays of speech waveforms. Due to the fact that SDP is sensitive to frequency variations, differences in the animal vocalization contrast with each other and SDP from a rooster, a dolphin, a frog and a cat could be clearly distinguished, see [1]. Some global similarity of the sound “oo” was also clearly visible when several people were pronouncing the word “boot” as reported in [2].

Another application for SDP is the representation of cardiac sounds. Thanks to this method, the difference between normal and pathological heart sound waveforms (with mild mitral stenosis or with mitral regurgitation) could be easily recognized [3].

Later, SDP started to be used for other purposes, for example Xu et al. took advantage of this method to visualize fans defects [4]. SDP can discriminate between different working conditions of the fans and also indicate a failure, allowing an easy visual identification. For example for an unbalanced rotor, the patterns show thinner lobes with a large region without dots in its center.

The above method was also used during Hall Effect Thruster data analyzing. Namely, SDP of discharge current and ion current waveforms measured simultaneously for different engine operating conditions were plotted. As it can be seen in Fig. 3, the pattern changes with discharge voltage \({U}_{d}\), and the greatest qualitative change (i.e. filling of the interior of the structures) occurs after the thruster switches from the so-called global (\(U_{d}\) = 300 and 450 V) to the local (\(U_{d}\) = 580 and 600 V) operating mode [8].

Fig. 3
figure 3

SDP of ion current (top) and discharge current (bottom) of a Hall Effect Thruster for different discharge voltages [8]. Although the signal for the ion current is noisier, the corresponding time series exhibit similar patterns

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The examples presented here demonstrate the variety of fields in which the SDP method can be used with success. With the development of Artificial Intelligence (AI) and neural network technologies [9], the number of applications can be expected to increase.

SDP of Time Dynamics of Tokamak Plasma Emissivity

Motivation

One of the main challenges to ensure fusion plasma stability and performance is the appearance of plasma instabilities. A commonly encountered instability in tokamak plasmas is related to the sawtooth-like (ST) oscillations [10], mostly observed in the temporal dynamics of the electron temperature and soft X-ray (SXR) emissivity. The phenomenon is characterized by a regular increase of the central electron temperature, until a sudden crash leads to a reorganization of the plasma core and a transient flattening of the radial profile, i.e. a decrease in the very core and an increase at mid-radius. Such ST have been proven to impact the core transport, for example in helping to flush heavy impurities out of the plasma core [11]. Therefore, studying the time dynamics of SXR emissivity with the help of SDP representation could be valuable to identify different phase of the plasma state, e.g. change in the heating power mix, appearance of specific instabilities or radiative collapse potentially associated with a disruption [12]. In this context, WEST – Tungsten (W) Environment in Steady-state Tokamak, recently upgraded from Tore Supra [7], can be an excellent opportunity to test the SDP approach.

Selection of WEST Data

Two WEST discharges #54,765 and #54,802 recently analysed in [13] and [14] respectively, were considered for the analysis. SXR measurements were provided with a time resolution of 4 ms by the horizontal camera of the DTOMOX diagnostic [15] in the photon energy range 2–15 keV, as this range is especially adapted to observe the dynamics of the tokamak plasma core [16]. Two channels were used to test the SDP approach: a “core” channel for which the line-of-sight (LoS) goes through the center of the plasma, and a lower “mid-radius” channel for which the LoS reaches the \(\rho_{norm} = 0.5\) magnetic surface, as shown in Fig. 4 (left). The corresponding line-integrated brightness for WEST #54,802, averaged over t = 3.5–7.0 s is presented in Fig. 4 (right).

Fig. 4
figure 4

Left: WEST vertical cross-section, showing the magnetic equilibrium for the WEST discharge #54,802 at t = 5.0 s, DTOMOX lines-of-sight geometry and the “core” (in red) and lower “mid-radius” (in blue) channels used for the analysis. Right: SXR profile averaged over t = 3.5–7.0 s for #54,802

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The time evolution of the two considered SXR channels is presented in the upper plot of Fig. 5 and Fig. 6, together with the plasma current Ip and the plasma stored energy WMHD, while in the lower plot, it can be seen that the main sources of plasma heating are ohmic heating (inductive) and Lower Hybrid Current Drive (LHCD) heating (non-inductive). As the latter heating method deposit most of its power to the electrons, switching for ohmic to LHCD heating leads to a significant increase of the core electron temperature and therefore of the associated SXR emissivity, accompanied with an increase of the plasma stored energy WMHD and a decrease of the loop voltage Vloop, LHCD being a very efficient current drive system decreasing by such the request in terms of inductive current. The total radiated power Prad represents a high fraction of the heating power (\(\gtrsim \) 60%) due to the significant amount of W impurities present in WEST plasmas. The line-averaged electron density ne,l is obtained by interferopolarimetry.

Fig. 5
figure 5

Time evolution of different measured quantities for WEST #54,765 – Top: SXR core and mid-radius channels brightness (see Fig. 4), plasma current Ip, stored energy WMHD. Bottom: Ohmic power Pohm, Lower Hybrid Current Drive heating power PLHCD, radiated power Prad, loop voltage Vloop and line-averaged electron density ne,l

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Fig. 6
figure 6

Time evolution of different measured quantities for WEST #54,802 – Top: SXR core and mid-radius channels brightness (see Fig. 4), plasma current Ip, stored energy WMHD. Bottom: Ohmic power Pohm, Lower Hybrid Current Drive heating power PLHCD, radiated power Prad, loop voltage Vloop and line-averaged electron density ne,l

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Results

For the SDP analysis, SXR time series of each discharge have been separated in six time intervals of 0.5 s—corresponding to about 120 data points, as presented in the right column of Fig. 7 and Fig. 8. The left column displays the SDP representation of the core channel time dynamics, while the middle column depicts the mid-radius channel. The SXR time traces and associated SDP exhibits similar characteristic for both studied WEST discharges.

Fig. 7
figure 7

SDP representation of the time evolution of SXR core (left) and mid-radius (middle) channels brightness for WEST #54,765 in the interval t = 3.5–7.0 s (the interval 5.0–5.5 s is omitted due to its redundancy with neighbouring intervals). The corresponding times series are displayed on the right column to facilitate the interpretation of SDP plots

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Fig. 8
figure 8

SDP representation of the time evolution of SXR core (left) and mid-radius (middle) channels brightness for WEST #54,802 in the interval t = 3.5–6.5 s. The corresponding times series are displayed on the right column to facilitate the interpretation of SDP plots

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During the stationary ohmic phase (t = 3.5–4.0 s), small and fast oscillations are present for the core channel, while the signal of the mid-radius channel is more noisy. This can mean that the radial extension of the ST is contained within \(\rho_{{{\text{norm}}}} \; < \;0.5\). For the core channel, the ST crashes are characterized in the SDP by singular points surrounding the main lobe structure—which comes from the rising signal during the recovery phase of the ST, each point representing one crash. The position of such point on the SDP is determined by the initial emissivity before the crash and by the crash amplitude, according to Eq. (1).

The transition from ohmic to LHCD heating (t = 4.0–4.5 s) exhibit a strong increase of SXR signal, as shown by the radical change of SDP for the mid-radius channel and the dominant main SDP sinusoidal-like lobe, cf. Figure 2, for both channels.

Then, during the stationary phase with LHCD heating and stronger ST oscillations (t = 4.5–6.0 s for #54,765 and t = 4.7–5.5 s for #54,802), the regular spikes following ST crashes in the mid-radius channel indicate the wider radial extension of the ST. In this case, the SDP of the mid-radius channel are characterized by open lines for the main lobe. Regarding the core channel, ST crashes are still characterized in the SDP by singular points around the main lobe. The position of these points is determined by the initial emissivity before the crash and by the crash amplitude, as in the ohmic phase.

While the SDP of the mid-radius channel drawn for #54,802, t = 4.5–5.0 s in Fig. 8 might look quite different than the other SDP of this channel at first sight, it should be noted that its basic structure is actually very similar to the one of the next time frame or of the other discharge—a main lobe with open legs. However, there is a rescaling due to one of the sawtooth spikes being particularly sharp (around t = 4.8 s), such that the corresponding point in the SDP, at the edge of the figure, remains far away from the rest of the signal.

Finally, a global collapse of SXR emissivity due e.g. backward transition from LHCD to ohmic heating is characterized by discontinuities in the SDP, see for example the time interval (t = 6.5–7.0 s) for WEST #54,765.

In summary, these results show that different types of dynamics in the SXR time series can be identified in a single plot, thanks to the SDP representation:

  1. (1)

    stationary signals with noise fluctuations are governed by the typical noise-like structures presented in Fig. 2,

  2. (2)

    continuously increasing or decreasing signals are characterized by a sinusoidal-like main lobe,

  3. (3)

    sudden and temporary signal crashes are identified by singular points surrounding the main lobe (e.g. ST crash for the core channel),

  4. (4)

    sharp spikes in the signal are characterized by a main lobe with open legs (e.g. ST crash for the mid-radius channel),

  5. (5)

    global and rapid collapse of the signal is associated with discontinuities in the SDP.

An interesting feature is that the intensity of each ST crash may be extracted from the SDP, based on the relative position of the corresponding individual point with respect to the main lobe. Additionally, the ST frequency may be calculated by counting the number of such points in each plot – dividing by the number of mirrors \(m\) of the SDP. Besides, the ST radial extent may also be estimated by comparing the SDP of different channels, the first channel (starting from the core) exhibiting lobes with open legs, i.e. spikes instead of crashes, setting the upper limit for the ST radius. Further work will be necessary to test such approach and compare it with more conventional methods employing line-integrated X-ray emissivity and tomographic tools [16, 17].

Conclusions and Perspectives

It is difficult to compare and characterize the time dynamics of diagnostic data in a continuous way, which motivates to create convenient graphical representations of time series. The SDP method is a simple recipe for taking points of amplitude-time waveform and computing an understandable pattern. Moreover SDP is able to visualize information which are missing in Fourier spectra and other popular methods.

SDP creates output easy to detect and memorize with previous patterns for human analyst. Thanks to this, it allows examining the current and predicting future states of the examined process.

In this paper, a preliminary attempt of using SDP representation to simply and quickly capture, in one plot, the dynamics of plasma radiation in a fusion device is presented for the first time. It is demonstrated in particular that SDP of SXR time series contain rich information about the characteristics of sawtooth oscillations (intensity, radial extension, frequency) in the plasma core. It should be noted that other tokamak plasma instabilities, such as Edge Localized Modes (ELMs) that exhibit periodic and abrupt crashes of the pedestal pressure [18] as seen for example by Thomson scattering diagnostics, can represent good candidates for studies of their temporal dynamics with the SDP approach. Although further work will be necessary to prove the reliability of the method in a more systematic manner, this first analysis opens valuable perspectives for facilitating the rapid identification of the plasma state, either for the purpose of post-processing and physics studies, or for real-time control of the discharge [19, 20]. In the case of the sawtooth instability, mitigation measures can include the application of localized electron cyclotron current drive [21]. Machine learning methods, for example based on neural networks which have proven to be efficient for image identification and pattern recognition, could be used to process the SDP from numerous diagnostic channels in a automatized way [8, 22,23,24].