Proceedings of ITC/ISHW2007

Analysis of Density Fluctuation Data Measured by Microwave
Imaging Reflectometry
Z. B. Shi1), Y. Nagayama2), S. Yamaguchi2), Y. Hamada2)
1)

The Graduate University for Advanced Studies, Toki 509-5292, Japan2)
National Institute for Fusion Science, Toki 509-5292, Japan(Received 13 July, 2007 / Accepted 3 August, 2007)

The density fluctuation in the Large Helical Device (LHD) plasma has been investigated by using the microwave
imaging reflectometry (MIR) system. The statistical properties of the fluctuation spectral on MIR signals are quantified byusing the time-frequency analysis with ensemble averaging technique. Statistical analyses by using cross correlation
spectrum and coherence spectrum reveals characteristics of MHD modes, such as mode numbers, during high power NBI
heating.
Keywords: density fluctuation, Microwave Imaging Reflectometry, ensemble average

1. Introduction*

Turbulence and instabilities has been considered tocorrelate with the properties of confinement, so that thespatial structure of the fluctuation continues to be thebasic issue in the fluctuation study. The microwave
imaging Reflectometry (MIR) has been applied to LHD[1]. The MIR is expected to be one of the most powerful
diagnostics to investigate instabilities in plasmas, since itenables 2D/3D local measurement of density fluctuations[1-4]. This technology is based upon the reflection of
microwave at the density-dependent cutoff layer, and thefluctuating phase of the reflected wave is dominated by
the density fluctuation close to the cutoff layer. In fact,
the reflecting signal has rich physical phenomena, which
include the plasma turbulence and MHD instabilities.
Fluctuation signals often submerge in the strong
background noises such as electronic noise and thermal
noise, especially when the reflection surface is in the coreplasma region. After the onset of the turbulence thespectrum becomes broad, and the intermittently burst of
the large-scale turbulence eddy may cause the distortionof the spectrum. Therefore, it is hard to see somethingfrom the oscillation even in the frequency domain.
Many digital noise reduction methods have beendeveloped in previous studies [5-7]. These methods use
the statistical feature of the random noises that is its
power spectral density is similar in any frequency band.
The expected error rate of the ensemble average
decreases monotonically as a function of the number ofthe data sets in the ensemble average. Therefore, thestatistical analysis of a fluctuating quantity over a long
period of time may be useful to pickup fluctuating signals.
E-mail: shi.zhongbing@nifs.ac.jp

This work presents the methods to quantify thestatistical properties of the fluctuation spectra based on
MIR signals. The analysis methods and the effects ofensemble average on the noise reduction in the spectrum
are presented in section 2 and 3. In section 4, someexample of our analysis will be applied to MIR data in
LHD. Significant results are as follows: three types ofthe modes and turbulence are observed by using theensemble technique; the mode numbers are obtained by
the cross correlation technique. The turbulence shows anion drift characteristic during high power NBI heating.

2. Spectral analysis methods
Fourier analysis is the most broadly used signal
processing technique. In digital signal processing, weoften face the necessity to separate the weak signals froma serious noise contaminated time series. The oscillation
looks not very useful in many situations. However, if thetime series is transformed into the frequency domain, thefrequency spectrum will show the fluctuation power,
frequency and primary phase. The Fourier transformation
is given byX(f ) =+∞∫ w(t ) x(t )e−∞

− j 2πftdt(1)where x(t) is the time series, w(t) is the window function
which is used to reduce the leakage of the sideband. Here,hanning window is used. In general the spectrum ofdensity fluctuation changes with time, the short time FFTanalysis is used to show the time evolution of thespectrum.
The cross-power spectral analysis is used to identify
the two time series which have the similar spectral

Proceedings of ITC/ISHW2007

Gxy ( f ) = Y ( f ) X ( f ) *(2)here * denotes the complex conjugate. X(f) and Y(f) is thediscrete Fourier transforms of the time series x(t) and y(t),respectively. The auto-power spectrum is the same astaking the Fourier transform when use two identical timeseries, but the phase information is lost. The phase shift
between two time series is:[[]]⎧⎪ Im G xy ( f ) ⎫⎪⎬⎪⎩ Re G xy ( f ) ⎪⎭φ xy ( f ) = tan −1 ⎨(3)In order to obtain the phase shift whose value
corresponds to a high correlation in the frequency domain,
the coherence spectrum is introduced and it is defined by
the cross-power spectrum normalized by the total power,asγ xy ( f ) =|< Gxy ( f ) >|< Gxx ( f ) >< G yy ( f ) >(4)

where < > denotes ensemble average. The coherency isbounded between 0 and 1, and high value corresponds tohigh correlation, zero represents completely uncorrelated.
The statistical confident level of coherence spectrum isdetermined by the number of the independent time series
in the ensemble ( 1 / N ).
The phase-frequency spectrum has a prominent
advantage to show the dispersion relations of the MHD
mode and turbulence with a distinct phase shift andpropagation direction in a two dimensional figure. It canbe obtained by two-point cross-correlation method.

(S (φ , f ) = | Gxy ( f ) | δ φ xy ( f ) − φ)(5)

In the calculation the delta function is replaced by arectangular window, and the width of the window isdependent on the number of the discrete sections in thevalue range of φxy(f). From phase shift, we can obtain themode speed and the mode number by estimating thedistance between two detecting points.

3. Ensemble averages
In many situations the signal from plasma is strongly
contaminated by random noise. Sometimes its amplitudein the frequency domain is higher than the signal that weare interested in, causing difficult to get the useful
information. Fortunately, by using the ensemble
averaging technique in the frequency domain, theamplitude of the random jumps becomes an average
power level in all frequency range and the peaks of thenoise can be removed. The averaging has less influenceon the mode whose amplitude doesn’t change in theensemble time. To show how the effect of the noise on
signal in the frequency spectrum, a program with a test
parameter composed of a sinusoid and a random function
is developed. By changing the amplitude ratio of therandom noise to signal (N/S), the ratio of the power
spectra between the noise and the signal can be obtained.
Figure 1 shows the ratio of the FFT amplitudes as afunction of N/S. Here, the red star denotes the ratio of theamplitude spectra without ensemble and the black
diamond is with ensemble. The ratio of FFT amplitudes
changes as a linear function of N/S. The discriminating
rate of the N/S is defined as it in the amplitude spectrum
has decreased 1/e. With the ensemble technique the N/Sincrease to about 25, while it is only about 10 withoutensemble. Therefore, the ensemble technique is aneffective way to reduce noise. It requires the lifetime of
the mode should be longer than the time window of FFT.
If not, the signal might be distorted by averaging and new
analysis method which has both high time and highfrequency ability should be used, for example wavelet
transforms [8-9].
Ratio of FFT Amp.

properties. The cross-power spectrum between two timeseries x(t) and y(t) is defined as

0.8
0.6
0.4
0.2
01020
N/S

3040Fig. 1 The effects of the noise to signal (N/S) on thepower spectrum. (Red star: without ensemble, Blackdiamond: with ensemble, dotted line: 1 − e −1 )4. Analysis of MIR signals in LHD
At present MIR system on LHD has three antennas
with 8.4cm spacing in toroidal and poloidal direction, andthree probe beams with frequencies of 53, 66 and 69GHz
in either O-mode or X-mode to illuminate the plasma [1].
The fluctuation signals are measured by the heterodyne
receivers with the sampling frequency of 1 MHz. In thispaper, we will present an analysis of one shot (75414)
with the toroidal magnetic field of 1.5 T and the major
radius of 3.6 m, heated by the co-injected NBI with thepower of 2.5 MW and counter-injected NBI with thepower of 1 MW between t=0.3 and 2.3 s, and the ECHwith the total power of 1.2 MW is injected between t=1.4
and 2.0 s. X-mode is used for this shot. Therefore, theProceedings of ITC/ISHW2007

432106

precession frequency of the resonant trapped fast ions
[10].
When turning on the ECH power, a high frequency
mode (~55 kHz) appears. This is in the range of Alfvénfrequency. The mode frequency increases with time and it
is up to 70 kHz at t=2.2 s. At t=2.0 s the turning-off ofthe ECH power causes no obviously effect on this mode.
That means this mode may be related to the energetic ion
mode but it is induced by the energetic electrons.
(a)

ECRH
0204060

NBI808042060

02040f(kHz)6080

Fig. 2 Power spectrum at 1.6s (top: with ensemble
averaging, 50 data sections with the time scale of 2mseach are used. bottom: without averaging and 4ms
window is used.)f (kHz)P(a.u.)

P(a.u.)

cutoff layer is determined by both the toroidal filed andthe electron density. The electron density is obtained by
the Thomson scattering with calibrating to the microwave
interferometer. The normalized radius of the cutoff layer
is about 0.1-0.3 during NBI heating.

40

20

0
0.5

1.0

1.5

2.0

2.5

t(s)

P(a.u.)

4(b)
20
1.0
0.8
0.6
0.4
0.2
0.0
4
20-2-40

Coherence

00PhaseTo extract the fluctuation information, FFT analysis
with ensemble averaging technique is used. Figure 2
shows the power spectrum with/without ensemble
averaging at 1.6 s. Without ensemble averaging, theMHD modes are concealed in the strong backgroundfluctuation, leading to difficult to decide the mode
frequency. With the ensemble averaging, the MHD
modes clearly appear and the turbulence exhibits a broad
spectrum with high coherency. Figure 3(a) shows thetime-frequency spectrum. Three types of fluctuation
appear in the MIR signals.
In the low frequency range, the density fluctuation has
a fundamental frequency of 2.3 kHz and its higher
harmonics. The frequency of this low frequency mode ismuch lower than that of the Alfvén eigen modes, andabout 3 times higher than the electron diamagnetic
frequency. It appears when turning on the NBI power anddisappears after turning off the NBI power. It seems as if
the onset of this mode depends strongly on the power ofneutral beam. For the tangentially NBI heating, it is noteasy to destabilize the fishbone instability. Indeed, thetypical fluctuation of fishbone instability was notobserved in magnetic probe signal. Therefore this mode is
not the fishbone instability.
At t=0.9 s a mid-frequency mode (~23 kHz) appears
when the plasma temperature increase to flat top. When
turning on the ECH power, this frequency increases to 26
kHz and it disappears after turning off the ECH power.
The frequency of this mode is close to the toroidal

20406080

Confidence level20406080

2040f(kHz)6080Fig. 3 (a) Time-frequency plot of FFT spectrum, (b)Cross-power spectrum (black: cross-spectrum, red andgreen: auto-spectrum), coherence and phase shift in thepoloidal direction at t=1.6s.
When these modes appear, the cross-power spectrum
is peaked and coherence become high, and the phase shift
shows less jumps, as shown in figure 3(b). Whencalculating the cross-power and coherence spectra, FFT isProceedings of ITC/ISHW2007

f (kHz)done at every 200 data sections with the time period of 4
ms each. The overlap between the neighboring sections is
the half of the time window. Before calculating thespectrum, the mean value and the linear trend have been
removed from every time series.
Figure 4 shows the contour plot of the phasefrequency spectrum in the toroidal and poloidal direction
at t=1.55 s. The high light color corresponds to the large
amplitude. When the mode appears, it shows a wide
spectral profile vs. the phase shift, maybe, this is becauseof the strong effects of the turbulence which may cause
the distorted distribution of the spectrum. However, theexact phase shift can be obtained from the statisticalprofile of the phase spectrum. It should be the same as thephase shift in Fig. 3(b). The fluctuation is dominated by
the frequency lower than 70 kHz and the turbulence
propagates along the ion drift direction. The mode
number of the 26 kHz is m=2/n=6, and the mode numberof 56 kHz is m/n=4/7. The errors of mode numbers areabout ±1.

60

60

40

40

202000-1.0 -0.5 0.0 0.5 1.0 -1.0 -0.5 0.0 0.5 1.0φ/π
θ/πFig. 4 The phase-frequency spectra in poloidal (left) andtoroidal (right) directions at t=1.55s.
5. Summary
In summary, the analysis of plasma density fluctuation
on LHD has been carried out based on MIR signals. Theensemble technique has been developed to reduce thenoise effect in the spectrum analysis. This technique
improves accuracy better than single data set whenobtaining the statistical property of the fluctuation. Novel
MHD modes and turbulence are observed during highpower NBI heating. The mode numbers are obtained by
the cross correlation technique. The turbulence shows anion drift characteristic during high power NBI and ECH
heating.

Acknowledgments:
This work is supported by National Institute of NaturalSciences (NIFS07KEIN0021) and by National Institutefor Fusion Science (NIFS07ULPP525).
Reference:[1] S. Yamaguchi, et al., Rev. Sci. Instrum. 77, 10E930(2006).
[2] R. Sabot, et al., Plasma Phys. Control. Fusion. 48, B421
(2006)[3] H. Park, et al., Rev. Sci. Instrum. 74, 4239 (2003).
[4] E. Mazzucato, Rev. Sci. Instrum. 69, 2201(1998).
[5] E. B. Hooper, Plasma Physics, 13, 1 (1971)[6] Ch. P. Ritz, et al., Rev. Sci. Instrum. 59, 1739 (1988).
[7] G. S. Xu, et al., Physics of Plasmas. 13, 102509 (2006)[8] A. Grinsted, J. C. Moore, and S. Jevrejeva, Nonlinear
Processes in Geophysics 11,561(2004)[9] V. P. Budaev, I. M. Pankratov, S. Takamura, etal., Nucl.
Fusion 46, S175 (2006)[10] T. Kass, H. S. Bosch, F. Hoenen, et al., Nucl. Fusion 38,807 (1998)