Comparison of Edge Turbulence Velocity
Analysis Techniques using Gas Puff Imaging Data on
Alcator C-Mod
by
Jennifer Marie Sierchio
B.S., University of Arizona (2011)
Submitted to the Department of Physics
in partial fulfillment of the requirements for the degree of
Master of Science in Physics
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@ Jennifer Marie Sierchio, MMXIV. All rights reserved.
The author hereby grants to MIT permission to reproduce and to
distribute publicly paper and electronic copies of this thesis document
in whole or in part in any medium now known or hereafter created.
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James L. Terry
Principal Research Scientist, Plasma Science and Fusion Center
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E
Comparison of Edge Turbulence Velocity
Analysis Techniques using Gas Puff Imaging Data on Alcator
C-Mod
by
Jennifer Marie Sierchio
Submitted to the Department of Physics
on May 9, 2014, in partial fulfillment of the
requirements for the degree of
Master of Science in Physics
Abstract
In the past, two methods for analyzing data from the Gas Puff Imaging diagnostic
on Alcator C-Mod have been used. One uses temporal and spatial Fourier analysis
to obtain wavenumber-frequency spectra, from which a phase velocity is computed
11, 2]. The other is based on time-delay cross-correlation of successive images used
to track the motion of discrete emission structures 13, 41. Several Gas-Puff-Imaging
experiments were conducted to obtain data taken using the GPI Phantom Camera.
The analysis of and results from these data are discussed in 131. The results showed
that the tracking time-delay-estimation technique found poloidal velocity magnitudes
in the 0.1-1.4 km/sec range. However, independent examination of these data using
the Fourier analysis yielded magnitudes up to a factor of 10 larger for the same data,
and sometimes even disagreed with the direction of motion found.
To understand the reasons for these discrepancies, we designed and generated synthetic data that mimics the real data. The user inputs the velocities, sizes, intensities,
and distributions of the synthetic emission structures. We have used the synthetic
data to test each code rigorously for strengths, weaknesses, and weighting. We have
found that the Fourier analysis perfectly returns the correct poloidal velocity when
there is no radial velocity component present. We have found that the tracking TDE
analysis weights low frequency, low wavenumber features most heavily since they are
typically the most intense, but systematically returns a smaller velocity than expected
due to issues associated with averaging. After ad.justing for these issues, the tracking
TDE code now returns the correct value of the poloidal and radial velocities to within
10% for synthetic data as long as there is only one velocity present in the synthetic
simulation.
We applied these corrections to the analysis of the real data, and found that
the measurements changed little in most cases. We then examined, in detail, the
Fourier-analysis-derived "conditional" spectra for each shot, and determined that the
likely causes for the discrepancies are due either to multiple velocities with emission
structures moving in opposite directions in the same field of view or to non-zero "dispersion" in which lower-frequency/lower-wavenumber features are moving with one
phase velocity and higher-frequency/higher-wavenumber features are moving with a
different phase velocity. In a couple of cases, there may be a radial component in
the actual images that may affect the poloidal velocity measurement for the Fourier
analysis. Accounting for these explanations, we believe that we have resolved the
discrepancies in many cases, and can explain it in the others.
Thesis Supervisor: James L. Terry
Title: Principal Research Scientist, Alcator C-Mod, MIT
Thesis Supervisor: Earl S. Marmar
Title: Senior Research Scientist, Department of Physics, MIT
Thesis Reader: Miklos Porkolab
Title: Professor of Physics, MIT
Acknowledgments
First and foremost, I would like to thank Jim Terry for being my supervisor these
past few years. His guidance, patience, support, as well as his expertise on GPI and
edge turbulence have greatly helped me in this thesis work. Next I wish to thank Earl
Marmar for being my formal advisor in the physics department, for his leadership of
Alcator C-Mod during the past couple of years, and for his comments and suggestions
which improved this thesis. Third, Miklos Porkolab deserves much gratitude for being
my thesis reader as well as being my academic advisor. His guidance and support
have proven invaluable. Fourth, I would like to thank Anne White for all of her help
and comments which improved this work.
Jim and Anne also deserve tremendous gratitude for the help and support they
have given me during my time in the physics department, and for their guidance as
I make the transition into a new program here at MIT.
My classmates: John, Ian, Mark, Leigh Ann, Jude, Evan, Silvia, Chuteng, Dan,
and Juan -
you guys are awesome. Thank you for the good times, especially when
the situation with C-Mod seemed really dire.
My fellow ARCers: Dennis Whyte, Paul Bonoli, Brandon, Justin, Tim, Chris,
Franco - while ARC is not the focus of this thesis, you all still deserve acknowledgment here. I have thoroughly enjoyed working with you all on the ARC design, and
am genuinely sad that we cannot do more with it.
To my family
thank you for constantly loving and supporting me all these years
(even though I'm 2500 miles away), making me laugh, and helping me to keep things
in perspective.
To Adam -
thank you for every single second I get to spend with you, and for
being incredibly understanding during the time I worked on this thesis.
This thesis work was partially supported by a National Science Foundation Graduate Research Fellowship, under grant No.
FC02-99ER54512 and DE-AC02-09CH11466.
1122374, and by USDoE awards DE-
6
Contents
1
Introduction
1.1
Fusion energy and tokamaks . . . . . . . . . . . . . . . . . . . . . . .
21
1.2
Turbulence at the edge of tokamak plasmas . . . . . . . . . . . . . . .
24
1.3
Gas-Puff-Imaging (GPI) hardware on Alcator C-Mod and detectors
.
25
1.4
2
3
21
1.3.1
Avalanche photodiodes (APDs)
. . . . . . . . . . . . . . . . .
27
1.3.2
Phantom Camera . . . . . . . . . . . . . . . . . . . . . . . . .
30
Organization of this thesis . . . . . . . . . . . . . . . . . . . . . . . .
30
Motivation
33
2.1
Comparison of two methods on recent C-Mod shots . . . . . . . . . .
34
2.2
A preliminary discussion of the observed discrepancies . . . . . . . . .
36
Fourier analysis of time-series images as a means of determining
poloidal phase velocity
39
3.1
Fourier representation of time series and the discrete Fourier transform
40
3.2
The discrete Fourier spatial transform . . . . . . . . . . . . . . . . . .
42
3.3
Calculating the average signal in time, frequency, and wavenumber
3.4
dom ains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
45
3.3.1
Time dom ain . . . . . . . . . . . . . . . . . . . . . . . . . . .
46
3.3.2
Frequency domain
. . . . . . . . . . . . . . . . . . . . . . . .
46
3.3.3
Wavenumber-frequency domain
. . . . . . . . . . . . . . . . .
46
Root-mean-square (RMS) value of the signal . . . . . . . . . . . . . .
46
. . . . . . . . . . . . . . . . . . . . . . . . . . .
47
3.4.1
Time domain
7
3.4.2
Frequency Domain
. . . . . . . . . . . . . . . . . . . . . . . .
47
3.4.3
Wavenumber-frequency domain . . . . . . . . . . . . . . . . .
47
Computation of fluctuations . . . . . . . . . . . . . . . . . . . . . . .
48
3.5.1
Epochs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
48
3.5.2
Oversampling the spatial Fourier transform . . . . . . . . . . .
49
3.5.3
Norm alization . . . . . . . . . . . . . . . . . . . . . . . . . . .
50
3.6
Conditional wavenumber-frequency spectrum . . . . . . . . . . . . . .
50
3.7
Velocity measurements with the Fourier technique . . . . . . . . . . .
52
3.8
Comparison of oversampled Fourier technique with the two-point method 53
3.9
Comparison of results between GPI Phantom Camera and APDs . . .
55
3.10 Limitations of the Fourier method . . . . . . . . . . . . . . . . . . . .
56
3.5
4
5
6
Tracking time-delay estimation technique for determining structure
velocities in time-series images
59
4.1
Cross-correlation function
. . . . . . . . . . . . . . . . . . . . . . . .
59
4.2
Im plementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
61
4.3
Known limitations of this method . . . . . . . . . . . . . . . . . . . .
63
Generation and use of synthetic data for testing analysis techniques 65
5.1
Generation of synthetic data . . . . . . . . . . . . . . . . . . . . . . .
65
5.2
Comparison of synthetic data to real data
. . . . . . . . . . . . . . .
67
5.3
Sample analysis on single-field and two-field shots . . . . . . . . . . .
69
5.3.1
Sample Fourier analysis
. . . . . . . . . . . . . . . . . . . . .
69
5.3.2
Sample tracking TDE analysis . . . . . . . . . . . . . . . . . .
70
Modifications needed for use of Fourier and tracking TDE techniques
73
6.1
Issues found when applying the Fourier analysis . . . . . . . . . . . .
73
6.1.1
Vertical velocity error when radial velocity is not small . . . .
73
6.1.2
Wavefront model to explain the observations from the Fourier
analysis applied in only one spatial dimension
8
. . . . . . . . .
74
6.2
7
Issues found when applying the Tracking TDE analysis . . . . . . . .
77
6.2.1
Effect of search box size on velocity measurement . . . . . . .
78
6.2.2
Requirement that there be at least four velocities lags in the
average . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
81
6.3
Effect of adding noise to the synthetic data . . . . . . . . . . . . . . .
81
6.4
Corrections to tracking TDE code, applied to real data . . . . . . . .
82
6.5
Corrections to Fourier analysis, applied to real data . . . . . . . . . .
83
Resolution of the velocity discrepancies
89
7.1
Revisiting the original problem and where we are so far . . . . . . . .
89
7.2
Analysis for dispersion in the real data . . . . . . . . . . . . . . . . .
91
7.3
Analysis for the effects of multiple lobes within the field of view
. . .
95
7.4
A resolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
98
7.5
Summary and conclusion . . . . . . . . . . . . . . . . . . . . . . . . .
105
109
A Codes to generate synthetic data
9
10
List of Figures
1-1
Schematic of the coils generating the toroidal and poloidal magnetic
fields. Graphic from CEA.15] . . . . . . . . . . . . . . . . . . . . . . .
1-2
22
Tokamak cross-section with minor (r)and major (R) radial directions,
and poloidal (0) and toroidal (#) angles labeled. The central axis is
the vertical line down the middle. The major radius, RO, goes from
the central axis to the center of the plasma.
1-3
. . . . . . . . . . . . . .
23
Image of a turbulence blob structure as seen with Alcator C-Mod's
gas puff imaging diagnostic. The separatrix is labeled as the dashed
line. The view is approximately along the local magnetic field. Since
the parallel wavelength of the turbulence is quite long, these images
actually show a cross-section of a blob filament.
. . . . . . . . . . . .
1-4
Diagram showing location of gas puffs and detectors in Alcator C-Mod.
1-5
Diagram illustrating how the gas puff line emission signal is transferred
25
27
from a telescope inside the vacuum vessel to the inner-wall APD detection system. The outer-wall viewing APDs as well as the Phantom
Camera (see below) work in a similar manner. . . . . . . . . . . . . .
1-6
28
Inner-wall APD electronic noise, measured with no light on the detectors. The measurements are from detectors whose fibers were coupled
to six radial views, hence the x-axis label. Typical signal-to-(electronic)
noise ratios range from ~50 (for views deepest into the plasma) to >
100 for other views.16]
. . . . . . . . . . . . . . . . . . . . . . . . . .
11
29
1-7
Power spectrum (square of the Fourier amplitude) per unit frequency
interval for C-Mod shots #1121002022 (plasma) and #1121002024 (no
plasma), as measured by the inner-wall APD system. The electronic
noise floor (measured on a shot for which there was no plasma light
present) is shown as the bottom trace. Both signals are normalized to
the DC component of the plasma signal.
2-1
. . . . . . . . . . . . . . . .
Vertical velocity vs.
rho for four Alcator C-Mod shots for the two
analysis techniques.
The red upright triangles are results using the
29
Fourier analysis, while the blue inverted triangles are the results using
tracking TDE. For the Fourier analysis, larger triangles indicate the
dominant velocity measurement at that location. The discrepancies
between the results from the Fourier analysis and the tracking TDE
can be as large as an order of magnitude. In some cases, they do not
predict the same direction. . . . . . . . . . . . . . . . . . . . . . . . .
3-1
35
Sample Phantom Camera frame showing the pixels included in one
poloidal transform. Most columns in the poloidal direction are nearly
vertical, and replace the simple column transform in Fourier analysis.
3-2
Wavenumber-frequency spectrum for C-Mod shot #1120815030,
44
p =
1.95 cm. The color scale is the log of the spectral power (the spectral
density squared).
3-3
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
51
Conditional spectrum for C-Mod shot #1120815030, at p = 1.95 cm.
(a) using the oversampled method with the oversampling factor = 4,
and (b) using the two-point method. The oversampled method found a
(hand-fitted) velocity of 3 km/s and a velocity using the max-integralcomputation of 3.13 km/s, while the two-point method failed to produce a spectrum clear enough to make a measurement.
3-4
. . . . . . . .
52
Conditional wavenumber-frequency spectrum for C-Mod shot #1120712027,
with hand-fit velocities shown. (a) Phantom Camera, (b) APDs. . . .
12
56
3-5
Conditional wavenumber-frequency spectrum for C-Mod shot
(top) Phantom Camera, and (bottom) APDs.
3-6
#1120224009.
. . . . . . . . . . . . .
57
Conditional spectra for shot #1110121019. The top plots use data from
the Phantom Camera, while the bottom plots use data from the APDs.
Results for each system are shown for two radial locations (p ~ -0.9
cm and ~0.09 cm). Qualitatively, the Phantom Camera and APDs detected the same features. However, there are differences in the poloidal
wavenumbers for the features, which lead to the inference of different
velocity measurements.
A cross-comparison of the Phantom-derived
and APD-derived feature velocities for one of the shots of Figure 2-1
is shown in Chapter 7 (as Figure 7-11).
Generally the agreement is
w ithin 20% . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4-1
58
Illustration of reference pixel inside its search box for the tracking
TDE analysis. Here, the peak of the cross-correlation function is found
within the box[7]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4-2
62
Example of a velocity map for 4ms of C-Mod data, with axis labels
representing pixel number. Velocity computations were attempted for
1/9 of the pixels in the frame. The maximum velocity in the field is
0.574 km /s.
5-1
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Example of synthetic blob field with noise added.
single blob is indicated by the arrow.
5-2
63
The motion of a
. . . . . . . . . . . . . . . . . .
67
Example signal in a time for a pixel viewing the plasma scrape-off-layer
(a), and example of a synthetic signal (b). Note that the time duration
shown for each is the same, i.e. 5 ms. . . . . . . . . . . . . . . . . . .
5-3
68
Normalized (to total fluctuations in signal) PDF for a real shot (left)
from a pixel viewing the far scrape-off-layer, and for a synthetic trial
where gamma=n9 (right). . . . . . . . . . . . . . . . . . . . . . . . . .
13
69
5-4
(left) Spectrogram of the log of the spectral power (Fourier amplitude squared) as a function of poloidal wavenumber and frequency
for a single-velocity field. (right) Conditional spectrum for the same
single-velocity field. Input vertical velocity = 1.01 km/sec and input
horizontal velocity = 0.0 km/sec. The red line through the lobe of the
conditional spectrum indicates a phase velocity of 1.0 km/sec, i.e. it
reproduces the input vertical velocity. The errors in the measurements
are ~5% . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5-5
Same as in Figure 5.4, but for a two-velocity field.
69
Input vertical
velocities were 0.39 km/s and 1.56 km/s, and input horizontal velocities
were 0.08 km/s and 0.39 km/s. The lines through the lobes indicate
that the Fourier analysis nearly reproduced the input vertical velocities.
Errors on the measurements are approximately 5%. Reasons for not
exactly reproducing the velocities will be explained in Chapter 6.
5-6
.
.
70
Velocity maps for single-velocity (left), and double-velocity (right) shots.
Axes are both in pixels, with the view as it actually is in the machine.
(left) Input vertical velocity = 1.01 km/s and input horizontal velocity
= 0 km/s, and the analysis yielded a vertical velocity of 0.8 km/s and a
horizontal velocity of 0.02 km/s. (right) Input vertical velocities = 0.39
km/s and 1.56 km/s and input horizontal velocities = 0.08 km/s and
0.39 km/s, while the analysis yielded a vertical velocity of 0.42 km/s
and a horizontal velocity of 0.04 km/s. Errors were approximately 5%
for each measurement.
6-1
. . . . . . . . . . . . . . . . . . . . . . . . . .
71
Ratio of output to input vertical velocity vs. ratio of input horizontal to
input vertical velocity for a series of synthetic shots. The blue dashed
line indicates what the output should have been based on the input
velocities.
6-2
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
74
Wavefront model with angle a shown. . . . . . . . . . . . . . . . . . .
75
14
6-3
Wavefront prediction for the output vertical Z velocity for the Fourier
analysis. The model prediction (Equation 6.3) is the green dashed line;
the synthetic shots are shown with error bars. . . . . . . . . . . . . .
6-4
For tracking TDE analysis:
(a) Output poloidal velocity vs.
76
input
poloidal velocity, (b) Output radial velocity vs. input radial velocity.
Both outputs are systematically smaller than the actual structure velocities, indicating that there are issues with the TDE method that
78
need to be understood.................................
6-5
Output poloidal velocity for tracking TDE vs. search box size in pixels.
The input (actual) poloidal velocity was 2.0 km/s. . . . . . . . . . . .
6-6
78
Cross-correlation function for a reference pixel, contoured for time lags
of -10 frames (upper left corner) through +10 frames (bottom right
corner), not including zero lag. The search box size is 17x17 pixels,
while the blobs have a FWHM of ~8 pixels. A well-defined peak in the
function that meets the threshold. is only in the search box for four
lags, yet the average velocity is calculated using values for 10 lags, as
listed in 6.1. The scale reflects the value of the cross-correlation with
the reference pixel for the given lag. . . . . . . . . . . . . . . . . . . .
6-7
79
Wavenumber-frequency spectrograms from the Fourier analysis for Elight/Enoise
-
2.2 at the breakdown level for the tracking TDE (a), and (b) the
conditional spectrum where a faint feature is still available for which
a valid velocity is obtained using the Fourier analysis. Eight/Eoise in
this case is 0.87. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6-8
Radial conditional spectra for varying input Vz/VR.
(b) Vz/VR = 2. (c) Vz/V
(a) Vz/VR = 1.
= 5. (d) Vz/VR = 8. (e) VZ/VR = infinite.
15
82
84
6-9
Examples of spreads in lobes of poloidal conditional spectra for real
shots in the far scrape-off-layer (p > 0.85 cm).
(top) Two lobes on
opposite sides of the k-spectrum. The separation of the two lobes is
much too wide compared to the synthetic spectra where the separation
is due to a significant component in the perpendicular direction. (bottom) Spread in lobes (one dominant direction) which indicates that
there may be a radial component for this shot.
However, very few
real shots had significant width to the lobes in their poloidal spectra,
indicating small radial components.
. . . . . . . . . . . . . . . . . . .
6-10 Sample radial conditional spectrum for shot #1120224009.
7-1
. . . . . .
85
87
Poloidal velocity measurements for four Alcator C-Mod shots, using
Fourier analysis (red triangles) and tracking TDE (blue inverted trian-
gles). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7-2
90
Sample wavenumber-frequency spectrogram for a real shot (a) and a
synthetic one-velocity shot (b).
The color scale indicates the log of
the power (square of the absolute value of the Fourier coefficients) at
a given wavenumber and frequency. . . . . . . . . . . . . . . . . . . .
7-3
Poloidal conditional spectrum for C-Mod shot #1120224015.
92
The spec-
trum features a curved lobe indicating some dispersion in the plasma
at p = -0.65 cm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7-4
(a) Conditional spectrum for Shot #1120224024,
93
p = -0.1 cm. An
abrupt break in slope is clearly observed. Two slopes are shown: one
for a slower-moving component, and one for a fast moving component.
(b) Conditional spectrum for Shot #1120224009,
p = 1.0 cm, also
showing a clear break in slope, slower phase velocity (-0.6 km/s) for
the lower frequency/lower k features and a faster phase velocity (-3
km/s) for the higher frequency features. The solid white line is the
velocity obtained from maximum-in-the-integral-computation. ....
16
93
7-5
Conditional spectra for synthetic shots constructed in order to show
a break in slope. The red dashed line is the hand fit for frequencies
<
30 kHz and yields a velocity of -0.75 km/s. The dark green line
indicates the slope of the slower-moving feature (with a phase velocity
~ = -0.75 km/s) while the lighter green dashed line indicates the slope
of the faster-moving feature (-1.15 km s). Input values for these shots
are given in Table 7.1. (a) Uses only two fields, and (b) uses four fields.
7-6
94
Poloidal conditional spectra for four-field (a) and three-field (b and
c) synthetic shots.
The negative-k lobes appear blended.
Both a
maximum-integral computational fit (solid white line with the Vl
value) and hand fits (dashed red lines) were completed for these cases,
and are consistent with the input values given in Table 7.2, although
only 3 of the 4 velocities of four-field case are resolved.
. . . . . . . .
96
at p=-0.72 . . . . . . . .
98
7-7
Conditional spectrum for shot #1120224027,
7-8
Figures 2-1 and 7-1 modified after considering the presence of multiple
flows and non-zero dispersion. Again the blue triangles are using the
tracking TDE after applying the corrections discussed in Chapter 6.
The larger red triangles are the velocities from the Fourier analysis that
were hand-fitted to the lower frequency/lower k parts of the dominant
lobes in the conditional spectra. The smaller triangles are the handfitted velocities for lobes in the conditional spectra that were present
but not dominant or were present at the higher frequencies . . . . . .
7-9
Poloidal conditional spectra for shot // 1120224009.
99
Shown for each
point in Figure 7-8a are fits to the lobes in the spectra. (a-c) have two
lobes in the opposite directions while (d,e) have non-zero dispersion (a
slower-moving component at low k, low f with a faster component at
higher k and f). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
17
7-10 Poloidal conditional spectra for shot #1120224015.
Shown for each
point in Figure 7-8b are fits to the lobes in the spectra. Hand fits to
the velocity for
f
<30 kHz are shown in red, while a computational fit
is shown in white. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
102
7-11 Poloidal conditional spectra for shot #1120712027.
. . . . . . . . . .
103
7-12 Poloidal conditional spectra for shot #1120815018.
. . . . . . . . . .
104
7-13 Poloidal velocity vs. p for C-Mod shot ,if1120815018, comparing the
data from the APDs (blue) to the data from the Phantom Camera
(red), using the Fourier technique, as the tracking TDE code is not
designed for use with the APDs. The meaning of the small and large
triangles is the same as described in Figure 7-8. These measurements
are generally in agreement. . . . . . . . . . . . . . . . . . . . . . . . .
18
105
List of Tables
1.1
Typical values for key parameters for the Alcator C-Mod tokamak.18]
6.1
Lags (measured in ps) used for computation of poloidal and radial ve-
26
locities for a reference pixel in tracking TDE, for a synthetic shot with
a purely vertical input velocity (1 km/s), and the change in pixels Ai
and Aj for the horizontal and vertical directions, respectively. Only
lags whose maximum cross-correlations meet the 0.5 threshold are included in this table. After conversion from pixels/lag to km/s, the
output poloidal velocity for this shot was 0.82 km/s; the output radial
velocity was -0.1 km/s. . . . . . . . . . . . . . . . . . . . . . . . . . .
6.2
80
Results for same synthetic shot after requiring that the pixel at which
a maximum in the cross-correlation occurs be located in the search
box. Only four lags are used in the average. The new output poloidal
velocity for this shot was 0.93 km /s; the new output radial velocity
was -0.1 km /s. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.3
Original
80
13] (black entries) and new (bold entries) values of poloidal
velocity as computed with the tracking TDE analysis after applying
the improvements suggested in this chapter (km/s). For the data in
these shots, no measurements changed directions.
A few points are
not given because the data at those locations were too noisy to yield a
m easurem ent. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
19
83
6.4
Original [31 (black entries) and new (bold entries) values of radial velocity as computed with the tracking TDE analysis after applying the
improvements (km/s). For the data in these shots, no measurements
changed directions.
A few points are not given because the data at
those locations were too noisy to yield a measurement.
7.1
. . . . . . . .
83
Input values for two synthetic shots which show breaks in slope. Velocities are given in km/s, FWHM are given in cm, and max intensity
in arbitrary units.
The TDE analysis returns poloidal velocities of
-0.51 km/s and -0.61 km/s respectively for these synthetic cases, obviously weighting the lower-frequency components more heavily than
the higher ones. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.2
95
Input parameters for three multi-field synthetic shots. Velocities are
measured in km/s, FWHM in cm, and intensity in arbitrary units. The
input velocities are to be compared with the features and fits in the
conditional spectra of Figure 7.6 and the tracking TDE velocities listed
in Table 7.3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.3
96
Fourier and tracking TDE results for the synthetic shots given in Table
7.2. All velocities are in km/s. . . . . . . . . . . . . . . . . . . . . . .
20
97
Chapter 1
Introduction
This chapter will cover some topics important for the rest of the thesis. In the first
section, we will discuss the overall motivation for studying fusion, and describe the
main device we use in those studies as well as the typical geometry we work in. The
second section will introduce the reader to plasma turbulence and its characteristics.
The third section will describe one of the diagnostics we use to study plasma turbulence, as well as the source of all data for this work. And the fourth section will set
up the organization of the rest of the thesis.
1.1
Fusion energy and tokamaks
One of the major problems facing humanity today is to find an energy source which
is clean, efficient, renewable, and affordable. The current energy sources (fossil fuels,
wind, solar, nuclear fission) have their own set of advantages and disadvantages.
However in the 1930s it was discovered that stars, such as our Sun, have an internal
energy source completely different from anything currently used on Earth. Since then,
scientists have endeavored for almost 60 years to generate electricity here using the
same mechanism that allows the Sun to shine: thermonuclear fusion.
Fusion occurs when two lighter atoms "fuse" into a heavier atom which is lighter
than the sum of the original two. Because of this mass defect between the reactants
and the product, (kinetic) energy is released. The fusion of atoms requires extremely
21
high temperatures, on the order of tens of millions Kelvin at a minimum for hydrogen.
At these temperatures, the atoms are completely ionized, with the electrons normally
in the atoms no longer bound by the Coulombic attraction of their respective nuclei.
These free nuclei (ions) and electrons form what is called a plasma.[9, 101 In our Sun,
this plasma is confined by gravity; unfortunately on Earth we do not have the mass to
confine a plasma gravitationally so we must find other means of confining a plasma.
One way to confine plasmas is by using magnetic fields, since charged particles
gyrate around the field lines as they make their way around the torus.191 As one
might expect, not all configurations of magnetic field lines will successfully confine
the plasma. The most successful configuration, a shaped torus called a tokamak, has
magnetic fields both in the toroidal and poloidal directions, as shown in Figure 1-1.
We define the toroidal direction to be around the torus the long way; this direction
Primary winding
of the
transformer
Magnetic field
created by plasma
current
Toroidal
Plasma
current
Resultant helical
magnetic field
magnetic
field
Figure 1-1: Schematic of the coils generating the toroidal and poloidal magnetic fields.
Graphic from CEA.[51
circles around the device, when viewing the device from above. The poloidal direction
is the direction around the torus the short way (if we were to look at a cross section of
22
the torus, not as viewed from above but sideways, the poloidal direction would follow
the lines forming the cross section on one side of the torus). This forms a cross-section
as shown in Figure 1-2. Notice that the poloidal cross-sections need not be circular,
and often have a soft D shape. The poloidal field itself is created primarily by the
plasma current (with additional coils for shaping and stability), while the toroidal
field is created by coils outside the vacuum vessel holding the plasma.
Looking at the cross-section, the center of the plasma is located at a radius we
define to be the major radius RO, if we are measuring from the central axis. We can
also define a minor radius to be half the maximum horizontal length of the poloidal
cross-section.
two angles
#
So overall, this geometry defines two radial directions, R and r, and
and 0.
The radius of the tokamak at the outboard (outer) side is
R
r
Figure 1-2: Tokamak cross-section with minor (r)and major (R) radial directions,
and poloidal (0) and toroidal (#) angles labeled. The central axis is the vertical line
down the middle. The major radius, RO, goes from the central axis to the center of
the plasma.
greater, by definition, than it is on the inboard (inner) side. Drawing a horizontal
line from the central axis of the torus through the center of the plasma extending to
the outboard side, and then rotating it around the central axis defines what is known
as the midplane.
Even though the plasma is well-confined by the magnetic field configuration, the
23
confinement is not perfect, and it is crucial that the region of direct interaction between the plasma and the material surfaces that face it be moved further away from
the main plasma. The most common way to accomplish this is to use a divertor configuration, which splits the plasma into two regions: one with closed nested poloidal
field lines, and one with open poloidal field lines. These open lines intersect the vacuum vessel, usually at the divertor plates (see Figure 1-4 where the divertor plates
are located at the bottom of the vacuum vessel). The poloidal field exactly cancels
out at a location called the X-point. The flux surface that contains the X-point is
called the separatrix, or last-closed flux surface (LCFS). Outside the separatrix is the
region of open field lines, called the scrape-off layer (SOL). The scrape-off layer is
cooler than the core plasma, and is the region we will focus on in this thesis.
1.2
Turbulence at the edge of tokamak plasmas
Turbulence is one the major issues currently preventing the realization of fusion as a
viable energy source. It is the primary reason for plasma transport across the confining plasma (loss of plasma heat and particles, and thus poor plasma confinement).[111
Because this is such an important problem, we are interested in all properties of the
turbulent structures in the plasma: their dynamics (momentum), their fluctuation
magnitudes, how they perturb the plasma density and temperature profiles, and their
size/length scales. This thesis will be concerned most with the velocity and size scales
of the turbulent structures in the plasma boundary region near and outside the separatrix, the largest and brightest of which are colloquially referred to as blobs. Below
is an image taken with the gas puff imaging diagnostic (see below for description) of
one such blob structure (Figure 1-3)
These large and relatively bright turbulent structures have been measured to be
around 0.5 cm to 1 cm in diameter,[12] and are characterized by having a higher
density than the rest of the plasma. The magnitudes of the light intensity fluctuations, which are responding primarily to the plasma density fluctuations, vary from
being normally distributed inside the separatrix (the "edge" region) to being gamma-
24
(.-in)Radial Direction (out-.)
Figure 1-3: Image of a turbulence blob structure as seen with Alcator C-Mod's gas
puff imaging diagnostic. The separatrix is labeled as the dashed line. The view
is approximately along the local magnetic field. Since the parallel wavelength of the
turbulence is quite long, these images actually show a cross-section of a blob filament.
distributed in the far scrape-off layer.[13] The wavelength of the turbulence parallel to
the toroidal magnetic field is quite long, forming filaments along the field lines. 114, 15]
What is actually being measured in the GPI images are cross-sections of the turbulent structures, perpendicular to the toroidal field. These cross-sections need not be
circular. Rather, away from the outboard midplane (poloidally), they are elliptical,
with their ellipticity and angle of orientation varying depending on the location of
measurement. 116]
1.3
Gas-Puff-Imaging (GPI) hardware on Alcator CMod and detectors
Alcator C-Mod 18] is a currently operational tokamak located on MIT's campus. In
Table 1.1, key parameters for C-Mod's plasmas as well as relevant dimensions are
given. At a major radius of 0.68 m, it is one of the smaller of the major tokamaks in
operation, but its magnetic field on axis can be as high as 8 T.
25
Parameter
Major radius
Minor radius
Plasma current
Toroidal magnetic field
Density
Central temperature
Value for Alcator C-Mod
0.68 m
0.22 m
~1 MA
up to 8 T
2 x 1019 - 8 x 1020 / m 3
-5 keV
Table 1.1: Typical values for key parameters for the Alcator C-Mod tokamak.181
One of the diagnostics on C-Mod to measure properties of plasma turbulence is
Gas Puff Imaging (GPI).117] The GPI system works by puffing a neutral gas such
as deuterium or helium into the plasma. The neutrals are not constrained to follow
the magnetic field lines, and they will emit line radiation from their interactions with
the plasma. The light from the puffed-gas cloud is typically filtered so that only one
emission line of the particular neutral is detected (D, at 656 nm or He-I at 587 nm).
Figure 1-4 shows the arrangement in Alcator C-Mod of the two gas puffs, both located
near the midplane, one on the inboard side and one on the outboard side. There are
currently two different types of detectors used to detect the light signals: avalanche
photodiodes (APDs) and a commercially-available fast-framing (400,000 frames/sec)
Phantom Camera. The view of the Phantom Camera is aligned along the magnetic
field, while the view of the APDs is aligned toroidally'.
Each can measure the same quantity: the intensity of light signal received in each
view or pixel. Since the light signal is dependent upon the neutral density as well as
the local electron density and temperature (I ~ non'T ),[18, 191 only the normalized
intensity fluctuation level I is useful for our purposes. With this parameterization, I
is related to the normalized density and temperature fluctuations as derived in [1:
I
+
= a
<I>
< n>
__
where a and
13
__
< >(1.1)
< T>
are coefficients dependent on the type of gas puffed into the plasma,
and I, ii, and t are the signal, density, and temperature fluctuations respectively.
'Not being perfectly aligned with the magnetic field (i.e. at an angle of ~ 80) should, in principle,
result in only a 1% smearing of field-aligned structures as detected by the APD system.
26
2D an"
ggTwo
Inboard
m =n
gm-4.
mpeaisnuo
-A---s
WnersW.r--4x4
Figure 1-4: Diagram showing location of gas puffs and detectors in Alcator C-Mod.
We take the RMS value as a measure of the fluctuation.
1.3.1
Avalanche photodiodes (APDs)
Arrays of APD detectors are used for detection of interference-filtered gas puff emission from both the inboard and the outboard side. Figure 1-5 shows schematically
how the detector system is set up for the inboard side; the outboard side is set up
in an analogous fashion.[11 There is an in-vessel telescope, which images the gas puff
into 54 unbroken fibers. Each fiber sees a different location in space. The fibers are
brought through the vacuum interface and run to a "breakout box" where each fiber
can be accessed via an SMA termination. From the "breakout box" the fibers are run
individually to SMA connectors that couple the light to individual APD detectors on
a 4x8 detector array. The fibers are proximity coupled though a thin (2 mm) interference filter (D, or He I). The amplified detector outputs are digitized at 2 MHz. Each
APD array coupler only allows for 30 views; thus only 30 of the 54 are currently used
at any one time. The outboard side utilizes three APD arrays, with 30 views each
for a total of 90 views. The inner and outer APD systems share a common water
chiller, which cools the components to 15 C in order to keep the APD gain constant
27
and maintain a low noise level.
Schematic of Innerwall Imaging System
af
55
iOkhI
ed
"$MIA
Figure 1-5: Diagram illustrating how the gas puff line emission signal is transferred
from a telescope inside the vacuum vessel to the inner-wall APD detection system.
The outer-wall viewing APIs as well as the Phantom Camera (see below) work in a
similar manner.
Each APD array is structured with two cathodes, each biased with a separate
high-voltage.
Each cathode connects to 16 detectors, 15 of which are illuminated
with fibers viewing the gas puff, while one is kept dark and is used to monitor the
dark current.
Since each fiber is accurately registered to illuminate a given ARD
detector, there is almost no cross-talk between the cathodes on a typical APD chip
(0.3%). Each APD cathode is controlled by a remotely-programmed high voltage.
At a high voltage of approximately 420V, a gain of 50 is achieved4[6j These detectors
have very high quantum efficiency (QE > 0.8 at both wavelengths of interest), and
when combined with their signal amplifiers, have very low electronic noise, as shown
in Figure 1-6. To examine the typical signal-to-noise as a function of frequency for
the inner-wall APDs, we show in Figure 1-7 power spectra (the square of the Fourier
amplitude per unit frequency interval) for both a signal without plasma light and with
plasma light normalized to the DC component of the plasma shot. As can be seen,
28
Z=-O.D679782
-
Electronic Noise
0.0025
0.0020
-
0.0015
-
0.0010
-
AA
A
C
0
0
6~
44.5
45.0
46.5
46.0
45.5
Rodius (cm)
Figure 1-6: Inner-wall APD electronic noise, measured with no light on the detectors.
The measurements are from detectors whose fibers were coupled to six radial views,
hence the x-axis label. Typical signal-to-(electronic) noise ratios range from ~50 (for
views deepest into the plasma) to > 100 for other views.[6
the electronic level is below the signal at all frequencies.
In this example, beyond
about 300 kHz the noise is due to photon statistical noise.
Shot 1121002022 (plasma) & 024 (no plasma)
10-
p = 0.22+/-0.19cm
Rmaj= 4 4 .90 cm
N
106
V 10-7
3 10-8
0
2
no light noise floor
a10-9i
10
100
10 100
Frequency (kHz)
Figure 1-7: Power spectrum (square of the Fourier amplitude) per unit frequency
interval for C-Mod shots #1121002022
(plasma) and #1121002024
(no plasma), as
measured by the inner-wall APD system. The electronic noise floor (measured on a
shot for which there was no plasma light present) is shown as the bottom trace. Both
signals are normalized to the DC component of the plasma signal.
29
1.3.2
Phantom Camera
The Phantom 710 Camera is a commercial fast-framing camera used to view the
filtered line emission from the gas puff, from which we can study turbulence at the
edge of the plasma. The 6 cm by 6 cm image of the gas puff is transferred to the
camera through a coherent fiber bundle, and then re-imaged onto the camera's CCD
detector, which has a 64x64 array of pixels. It has a maximum frame rate of 391,000
frames per second, thus a time per frame of approximately 2.5 pus, and a typical
exposure time of 2.1 ps at this frame rate. The gas puff is centered ~3 cm below the
midplane on the outboard side and roughly at the typical radial location of the LCFS;
the viewing geometry is shown schematically in Figure 1-3. The gas-puff imaging and
image transfer work in a similar fashion as those for the APDs: a telescope inside the
vessel images the emission onto a coherent fiber optic bundle, which then brings the
images to lenses which reimage the light through an interference-filter for line emission
from deuterium or helium, and then onto a camera CCD. The actual chordal spatial
resolution for the system estimated to be roughly 2-3 mm at the midplane.[3]
The remainder of this thesis will focus primarily on data taken with the Phantom
Camera, the reasons for which will be explained in Chapter 2. Since the Phantom
Camera is less sensitive that the APD detector arrays, it has a lower signal-to-noise
ratio than the APDs. In some cases, the data taken with the APDs for the same
C-Mod shots will be used for verification purposes.
1.4
Organization of this thesis
This thesis is organized as follows. Chapter 2 will describe the motivation for the
rest of this work, discussing some recent published data and the analysis discrepancies discovered, and introduce the reader to our findings.
Chapters 3 and 4 will
go into detail about the two analysis techniques: Fourier analysis, and a time-delayestimation/pattern tracking hybrid, respectively. Chapter 5 will present how we chose
to investigate the discrepancies found using the two analysis techniques, i.e. through
the creation and use of synthetic data. Chapter 6 will explain some of the initial
30
issues encountered when using the analysis methods on the synthetic data, and offer
modifications to the codes to mitigate these issues. And Chapter 7 will provide evidence that suggests a resolution to the discrepancies that motivate this work and will
also conclude this thesis.
31
32
Chapter 2
Motivation
As mentioned in Chapter 1, this thesis will focus on the velocities, size scales, and
intensities of the emission fluctuations as measured by GPI. Therefore, we wish to
understand how these quantities are measured given the data from the GPI diagnostic. Many methods for analyzing GPI data have been developed;[1, 2, 3, 4, 13, 20, 21]
we will consider only two methods here: one using spatial and temporal Fourier analysis, and another using a hybrid between pattern tracking and time-delay estimation
(TDE). The two methods will be discussed in detail, in Chapters 3 and 4, respectively,
so only a brief description will be provided here.
As mentioned previously, the GPI takes time signals for each pixel/view. This
produces a three-dimensional set of data: two spatial dimensions and one temporal.
The first analysis method,[1, 21 Fourier decomposition, involves both temporally and
spatially Fourier transforming the time series of images to produce a wavenumberfrequency spectrum (defined in Chapter 3). Using this spectrum, a phase velocity of
fluctuating emission structures can be easily obtained as 27rf /k, where f is the frequency and k is the wavenumber, as long as the Fourier amplitudes used are maximal
for each frequency at a wavenumber such that f/k is approximately constant. Because the Fourier analysis decomposes the signal, it can detect multiple components
moving at different phase velocities.
The second method, tracking TDE,[3] involves detecting a maximum in the timedelayed cross-correlation function between time histories of signals from image pixels
33
(defined in Chapter 4) in some search box of pixels surrounding a reference pixel.
This is completed for several time lags. Since it searches for maxima in the crosscorrelations, it favors bright, large features. If each cross-correlation function for the
set of time lags meets a threshold, then a velocity is computed for that time lag as
the distance between the location of the peak and the reference pixel divided by the
time lag. If the threshold is met for more than one lag in the set, then the associated
velocities are averaged. Thus, the tracking TDE analysis can only output one velocity
for the region, rather than multiple velocities for multiple flows. It is important to
note that the tracking TDE does not compute the phase velocity.
2.1
Comparison of two methods on recent C-Mod
shots
As will be investigated, each method has its strengths and weaknesses. Ideally, we
would like to use one method to validate the other, and vice-versa to converge on a
common estimate for the velocities of the emission structures. As a starting point,
we present some recently published
13] GPI Phantom Camera results for a sampling
of Alcator C-Mod shots from a specific implementation of the tracking TDE method,
as well as the results from using an implementation of the Fourier analysis technique
on the same data, as shown in Figure 2-11. The plots show the calculated vertical
velocity (Z-direction) of the emission structures vs. p, where p is the distance outside
the separatrix for view locations that have been mapped magnetically to the outboard
midplane. Negative p indicates locations on closed field lines and flux surfaces, while
positive p indicates we are looking at the scrape-off layer (p=O is the separatrix). The
red upright triangles are the results from the Fourier analysis, while the blue inverted
triangles are the results from the tracking TDE analysis. There are some instances
in which the Fourier analysis outputted two results for the same location; this is due
1[3] actually has the locations in rho 1.0 cm greater than shown in the figure. This is due to the
fact that during a vacuum break in 2011 the Phantom Camera alignment, but was only recently
accounted for. The figures and data in this thesis reflect that recent correction.
34
to the fact that (as will be shown in Chapter 3), the Fourier analysis examines both
directions (up and down) for feature propagation, and in many instances finds features
propagating in each direction.
Thus, both a positive and a negative velocity are
assigned to the same p location. As shown in Figure 2-1, the Fourier analysis and the
Shot #1120224009
Shot #1120224015
4
2
(b)A
I
(a)
A
A
-
A
a
E
E
V
VV
0
V
-1
A
A
-1
0
A
-0
5
.A
D5
A
-2
-
0.0
Rho (Cm)
3
-05
0.0
Rho (cm)
1.0
D5
Shot #1120815018
4
&
A
A
-1.0
-15
1.0
Shot #1120712027
4
A
A
-2
(d)
-(0)
3
2
E
E
2
A
A
A
1
a
A
AL
AA
-1
5
AA
-V- -- -- -- -- -- -- --.
V
T
-1
0
-0.5
Rho
A
A
-
D0
0.5
-1
(cm)
5
-1.0
-0.5
A A
A
00
Rho (cm)
0.5
A
1.0
Figure 2-1: Vertical velocity vs. rho for four Alcator C-Mod shots for the two analysis
techniques. The red upright triangles are results using the Fourier analysis, while the
blue inverted triangles are the results using tracking TDE. For the Fourier analysis,
larger triangles indicate the dominant velocity measurement at that location. The
discrepancies between the results from the Fourier analysis and the tracking TDE can
be as large as an order of magnitude. In some cases, they do not predict the same
direction.
tracking TDE output velocities do not agree. The computed velocities differ by up to
an order of magnitude and even direction (if there is only one direction detected with
the Fourier analysis). This poses serious issues with the published results using either
method. First, this implies that neither method can be used to validate the other,
and second, it is certainly possible that the results previously obtained with either
code could be incorrect. The purpose of this thesis is to understand the reasons for
35
these differences, and make possible adjustments to the methods and implementations
by which these results were obtained to allow for validation of future results.
2.2
A preliminary discussion of the observed discrepancies
The first limitation in these measurements is due to the gas puff itself. Because a gas
is being puffed into the plasma, and interacting with it to induce line emission, there
is a risk that the puff perturbs the plasma to the point of affecting the measurement.
Estimates have been made for this effect; it is predicted that the energy loss due to
the puff should be small, but the particle increase due to the puff could be significant
enough to change the density in the midplane scrape-off layer
13] and thus affect the
intrinsic turbulence. However, this effect is difficult to quantify without other independent measurements. Its effect on the velocity measurements from these methods
is also unknown, but is likely not significant.
It can now be mentioned that both methods themselves have limitations for computing the velocities of the emission structures which could affect the outputs and
need to be stated. First, since GPI cannot yield any information about the structures'
motion in and out of the image plane, neither method can determine velocities parallel
to the toroidal magnetic field. Second, if there is dispersion in the plasma, where the
phase velocity mentioned above is not equal to the group velocity of the structures
(f/k is not constant), then neither method will produce the correct measurement.
Limitations for the individual methods will be discussed in Chapters 3 and 4.
Most importantly, we mentioned above that the Fourier analysis can yield multiple
measurements if there are multiple flows while the tracking TDE method cannot.131
As we will see in Chapter 7, this, in combination with dispersion that causes abrupt
changes in the phase velocity as a function of wavenumber, ultimately plays a significant role in explaining the discrepancies. Multiple flows and dispersion are fairly
common occurrences in the real data. Once we account for these reasons, along with
36
the fact that the tracking TDE favors bright features, we will be explain to explain
most of the discrepancies and possibly recover them for some shots.
37
38
Chapter 3
Fourier analysis of time-series images
as a means of determining poloidal
phase velocity
In this chapter, we describe in detail the first of the two methods for analyzing
GPI data we wish to compare, the Fourier analysis method, as implemented by I.
Cziegler.[1, 2J The general idea of this method is to temporally and spatially Fourier
transform the time series from the images into a wavenumber-frequency spectrum,
from which a phase velocity is derived. We will first describe the mathematics behind
the Fourier method, as well as explain how to compute the fluctuation amplitudes.
We then discuss the specific implementation of this method and how to compute the
phase velocity from the conditional spectrum. We end by comparing our method to
the well-known two-point method, and by comparing the Fourier-analyzed Phantom
Camera data to the Fourier-analyzed APD data.
39
3.1
Fourier representation of time series and the discrete Fourier transform
Let us begin by assuming we have obtained a signal in time from the GPI diagnostic.
The GPI diagnostic measures fluctuations of visible line emission for He I or D, due
to interaction of the plasma with the atoms from the local gas puff. We assume that
we can represent the signal from a pixel in the Phantom Camera images as g(t). This
signal g(t) will include a slow-variation component from the plasma background and
gas puff as well as a fast-variation from density and temperature fluctuations. We
can represent g(t) as a Fourier series [221 of period N, in which the signal exists for
only one period and N is the total number of time-points (the total number of data
points digitized from a single view of the plasma):
27rnt
(an
g(t) = ao +
a, cos N
n=1
27rnt
+ bn sin N
N
(3.1)
where the Fourier amplitudes ao, an, and bn are obtained from Euler's formulas. We
can manipulate the above to write g(t) as a cosine function only, with a phase shift:
g(t) = ao +
ancos
/-
N
(3.2)
n=1
where the coefficients and phase shifts are given by:
2+b2
tan On =
(3.4)
b
Alternatively we can write g(t) as a complex series using another of Euler's formulas: 1231
00
Cn3
g(t) =
?=-00
40
)
(3.5)
where cn is given by: 1231
0
2
Cn -
1
(an - ib,)
2
1
-(an + ib!.)
2
if n = 0,
(3.6a)
if n > 0,
(3.6b)
if n < 0
(3.6c)
Inputting the equations for the coefficients back into the series representation of
g(t) and changing variables leads to the Fourier integral transform representation of
g(t). Of course, for our purposes, g(t) is not a continuous function; it is discrete with
the intensity measured at a given set of times. We can write the complex coefficients,
and hence the discrete frequency spectrum, as the following:
N-1
gn = Nan =
(-
i
l)
(3.7)
t=0
This transforms g(t) into and these coefficients tell us which frequencies dominate
in the signal. The Fourier amplitudes for the sine/cosine or cosine-alone decomposition can be obtained from the complex Fourier transform coefficients:
Cn + c*,
(3.8)
b- =c* - Crj
(3.9)
an
a, = 2|cn 1
tan On
- img(C)
real(c,,)
(3.10)
(3.11)
where we have taken advantage of the fact the g(t) is a real-valued function, so
c-n = c*.
If the signal arises from multiple independent processes then we can
individually transform the functions representing each process. This arises from the
fact that the integral of two separable functions is the sum of the integrals over
each function individually. For example, we can use GPI to characterize the quasicoherent mode (QCM). GPI returns the total signal of the QCM and the broadband
41
background.
In this case, the two processes can be separated and represented as
different functions, g(t) =
fQcM(t) +hBB(t), whose transform can be decomposed: 122]
g
3.2
= fQcM
+ hBB
(3.12)
The discrete Fourier spatial transform
The Fourier analysis technique we use calls for not only transforming in time, but
in space as well for two-dimensional images. Thus, we will now describe what the
general 2D spatial transformation looks like. It is important to note that while transforming in time gives frequency (in units of s-1), transforming in space will give,
in principle, wavenumbers in both spatial directions (in units of cm- 1 ). If we have
a two-dimensional image with coordinates r and z, the 2D spatial transformation is
given by:1221
R-1 Z-1
H[kz, kR] = R
1
h[z, r]e-2i(R+
(3.13)
)
r=O z=O
where kR and kz are the wavenumbers in the r and z directions, respectively, and R
and Z are the total number of spatial points, respectively. And the inverse transform
is given by:
R-1 Z-1
h[z, r] = RZ
LH[kz, kRe
R
(3.14)
z)
r=O z=O
The time-to-frequency transformation involves only one dimension. However, this
involves two dimension and thus introduces subtleties when actually implementing
this. We can think of the image as a matrix of values, and in performing the transform,
we are operating on that matrix. To begin, we rewrite Equation 3.13 in a slightly
different, but equivalent form:
IRH[kR, kz] = RE
11
r=0
Z-1
zEh[z,
-2
kZ
r]e-2i z
-,ik
-2e
R
(3.15)
z=0
First we perform a column transformation on the component in brackets. The summation is with respect to the row z, and column r is ignored. We can define the Z by
42
Z column transform operator, which will act on our image as a matrix multiplication
from the left:
1
e-2
7r zk7
z
(3.16)
H' = Kh[z, r]
(3.17)
K =
z
and the column transformation is:
Where the bold font indicates that K and H' are matrices operating on h. We define
the R, by R row transformation operator which will act on our column-transformed
image from the right, and will perform a summation over column r, leaving kz free:
L=
R
Re2R
(3.18)
So our final result is the following:
H[kz, kR] = Kh[z, r]L
(3.19)
To summarize, we have defined for a spatial 2D Fourier transform what the column and row transforms are. These will play a role in Chapter 6 when issues with this
method will be discussed. There are two caveats to this formulation. First, because
we are interested in the poloidal velocity rather than in the perfectly vertical one, the
transform we perform are actually not perfectly vertical in space, but rather along
flux surfaces of constant p (distance from separatrix).
For the poloidal transform
the differences from a simple transformation of pixels along adjacent vertical column
pixels are shown in Figure 3-1. Second, in our particular implementation, only the
poloidal transform is performed using the signals from pixels viewing along flux surfaces with a similar p coordinate, giving a wavenumber in the poloidal direction. We
do not perform the radial transform using pixels viewing similar Z coordinates, and
thus do not find a radial wavenumber. The magnetic flux surfaces are approximately
parallel to the poloidal direction, and perpendicular to the radial direction. Thus for
43
the poloidal transform, the columns over which the process is completed are almost
parallel to the magnetic flux surfaces, while the radial transform would attempt to
complete the Fourier process over rows which are perpendicular to them. The transform in the radial dimension is problematic since the separatrix is typically in the
field of view and radial propagation of turbulence inside and outside of the separatrix
is observed to be quite different.
2
.
. . . .
.
.
i
.
.
. 4
0
0E
-2
-4-
-61
84
86
R
88
(cm)
90
92
Figure 3-1: Sample Phantom Camera frame showing the pixels included in one
poloidal transform. Most columns in the poloidal direction are nearly vertical, and
replace the simple column transform in Fourier analysis.
Since both the GPI APDs and the Phantom camera are two-dimensional arrays of
views, we add a spatial dimension to the time-Fourier analysis. This adds important
information about the spatial characteristics of the turbulence, which we would like to
determine. We accomplish this by following the Fourier temporal transform (Equation
3.7) by a spatial Fourier transform that transforms the time/frequency analysis from
multiple spatial points into frequency, wavenumber spectra
44
(f) --
(f,k).
The
frequency dependence remains intact while we have added a new dependence on
wavenumber. Because the 2D spatial information is available, both the time series
and the frequency spectrum are
g(t) = g(,t), h(f) = (, f)
(3.20)
We perform the spatial transformation on the time-transformed coefficients from the
appropriate set of views.
Let us now rewrite the above formulation in a slightly
different manner, allowing the user to choose the length of the poloidal wavenumber
vector, rather than in typical Fourier analysis in which the length of the wavenumber
vector is the same as the number of spatial points. Because the spatial locations of the
signals and wavenumbers are vectors, we use the tensor and outer product operators
to perform the spatial transforms. In general, the spatially transformed spectrum is
given the equivalent of Equation 3.17:
g(k, f) = C( , k) O® j(2, f)
(3.21)
where 9t is the tensor product operator and CQx, k) are the spatial Fourier coefficients
given by:
C = efo
(3.22)
Here, 00 represents the outer product operator of the x and k vectors. Again, for our
purposes, we transform in the poloidal direction only, so we are left with the poloidal
component of the wavenumber vector.
3.3
Calculating the average signal in time, frequency,
and wavenumber domains
We calculate the average value of the signal, since only normalized intensity fluctuations between GPI views yield meaningful comparisons due to the spatial variation of
the gas-puff emission, differences in sensitivity for each view, and differences in gain.
45
The average value can be computed for the time series after computing the temporal
and spatial transforms.
3.3.1
Time domain
In the time domain, the average signal is simply computed at the average over all
time-points:
<I>
o9(t)
(3.23)
N
where N is again the total number of digitized points in the time series.
3.3.2
Frequency domain
If we refer back to Equation 3.5, we note that g(t) in Fourier series representation has
a constant "DC offset" ao. This offset is the average value of the signal in the time
domain and is the
f
= 0 component of the frequency spectrum:
N-1
1
2ir(n=O)t
( g(t)e
t=o
N
(3.24)
which reduces to Equation 3.23 for n = 0.
3.3.3
Wavenumber-frequency domain
Since the spatial transform has added a dependence on wavenumber to the frequency
spectrum, we can think of
Therefore, to obtain
(k,
f)
as having split
(0) we must sum
.4(f) into its various k-components.
(k, 0) over the k-spectrum:
=
Z() (k, 0)
(3.25)
k
3.4
Root-mean-square (RMS) value of the signal
If we wish to characterize the intensity fluctuations through the AC RMS value of the
signal, we show in this section how this is accomplished in the time, frequency, and
46
wavenumber/frequency domains. To do this, we introduce Parseval's theorem:124, 25]
2
N-1
N-1
ig(t)
2
f)
=(k,
A2
E
N f=0
t=O
f=0
(3.26)
k
Parseval's theorem states that power is conserved after performing a Fourier transform. This theorem also ensures that we obtain the same RMS value in each domain.
3.4.1
Time domain
The AC component of the RMS value for a discrete time series is defined as follows:
N-1
IRMS =
3.4.2
(3.27)
N2 t=O g(t)- < I > |2
Frequency Domain
We can simply input Equation 3.26 into Equation 3.27 to obtain the RMS in the
frequency domain:
IN-1
IRMs =
(3.28)
5(f)|2
2E
f=1
Since we want the AC component only, we begin the sum with
f
1 instead of
f
0.
3.4.3
Wavenumber-frequency domain
For the wavenumber-frequency domain, we must first sum over the k-resolved coefficients, yielding:
IRMS
1N-i
N2
Z
f=1
47
kt
2
(k, f)
(3.29)
3.5
Computation of fluctuations
Here, we will briefly discuss some important points about the codes used to compute
the fluctuations and perform the Fourier analysis.
The basic method involves producing the discrete Fourier frequency spectrum as
a function of view, frequency, and time. The user must specify the time series and
corresponding times, along with a sampling frequency. The time series is split into
a coarser set of time epochs of a length determined by the sampling frequency. An
epoch is simply a block of time or time window; the total number of epochs multiplied
by the number of time points in each epoch gives the total time sampled. Thus, if we
have one second of data and each epoch is 0.0056 seconds, then we have 178 epochs;
the leftover time is simply cut out. The RMS can be computed over a single epoch or
multiple epochs. The user can also choose to normalize the signal, apply a hanning
filter, and overlap blocks of signal to reduce error.
3.5.1
Epochs
As mentioned above, we can take the RMS for a single epoch or for multiple epochs
individually. If we want a single value for the RMS but accounting for all epochs,
there are two ways to accomplish this. The first is to simply take the RMS for each
epoch and then average those values over epochs:
1T-1
IRS
=
T E
(3.30)
IRMS,t
i=O
where IRMS,t is the RMS computed for each epoch t using one of Equations 3.27, 3.28,
or 3.29. The second way first computes the spectrum for each epoch, then averages
over the epochs to produce an average spectrum, and then calculates the RMS based
on that average spectrum. In the frequency domain, this looks like:
N-1
RMS
I~s
=
f=1
48
T-1(
(f,It)|
T t=0
(3.31)
and in the wavenumber-frequency domain this becomes:
,t)
IRMs =(k,
t=0-
f=1
2
(3.32)
k
It is important to note that averaging in the first method (Equation 3.30) removes
dependence on any slow variation, while the second method will leave slower-varying
components if the epochs are long enough. For our purposes, however, we are not
concerned about slow variation in the signal, which is the background and is not
characteristic of the turbulence. We want the RMS average to account for fast variation in the plasma due to smaller-scale fluctuations and not due to slow changes
in the plasma or gas-puff. The Fourier coefficients .(k,
f,
t) for individual epochs
vary far more than the time-averaged Fourier coefficients (i.e. averaged over epochs).
Normally this is not an issue and the first method will lead to a reasonable result.
However, sometimes it can be an issue if we want, for example, to fit the spectra to
a curve. If the variation is not averaged out (as in the second method), this extra
variation leads to difficulty when using fitting programs.
3.5.2
Oversampling the spatial Fourier transform
Oversampling allows the number of points in the k spectrum to be increased beyond the number of spatial points included the transform. It allows extension the
wavenumber spectrum to wavenumbers smaller than that defined by Ikmin = 27/L,
where L is the linear dimension of the view (~6 cm for the Phantom and ~3.9 cm
for the APD system in the poloidal dimension). Ikmin| can be reduced by "oversampling," which assumes that any large features in the images vary smoothly across the
included views. Of course the resolution of a single-k component will be limited by
the linear dimension of the view (Ak = 27/L), but the spectra will be made smoother
by this oversampling. For these analyses, we oversample by about a factor of 4, with
Ikmin
~ 27/(4L). The maximum k cannot be increased above the Nyquist limit
(kNyquist
= i/pixel-spacing), so that
ko,Nyquist =
49
33.5 cm-
1
for the Phantom camera
images and
ko,Nyquist
= 8-1 cm' for the APD-based system). In other words, instead
of setting the number of elements in the wavenumber array to the number of spatial
points (as would be the case in an FFT). we have increased that number, effectively
oversampling. It will be shown later in this chapter that the oversampling provides
a justifiable wavenumber resolution. Our method of spatially Fourier transforming
(Equations 3.20-3.22) allows the user to define the oversampling rate, and thus the
oversampled wavenumber vector with which to perform the spatial transform. To
avoid aliasing, km, is limited to 0.85*kNyquist for these analyses.
3.5.3
Normalization
Each APD/Phantom view can in principle have a different gain and is in a different
location relative to the gas puff nozzle, and thus each raw signal can have a different
average value. Since we need to be able to compare different channels, we divide
each frequency and wavenumber-frequency spectrum by the DC offset (i.e. the co
component) in each epoch. We do not just subtract the DC offset since we have
already subtracted it out in finding the AC component of the RMS value. Since the
co component is divided by itself,
IRMS/ <
I> is unaffected by the normalization.
No correction to IRMs/ < I > is needed.
3.6
Conditional wavenumber-frequency spectrum
The numerical Fourier analysis gives the poloidal wavenumber-frequency spectrum
for a signal in time from some set of views or pixels. This spectrum is a function
of poloidal wavenumber, frequency, and time bin and is computed using the process
described above. Note that only the poloidal wavenumber is computed, not the radial
wavenumber.
This means that only the spatial poloidal transform is performed.
We call the values of the spectrum the spectral density S(k, w, t).
This spectral
density can be plotted as a contour over all frequencies and poloidal wavenumbers.
An example using experimental data from a C-Mod shot is shown in Figure 3-2, where
we have averaged over epochs.
50
Shot 1120815030
t
-
1.2596 thru 1.2636 afc
15-1
-2
-4
-6
-4
-2
0
2
4
6
Iko (cm'1)
Figure 3-2: Wavenumber-frequency spectrum for C-Mod shot #1120815030, p = 1.95
cm. The color scale is the log of the spectral power (the spectral density squared).
The wavenumber-frequency spectrum automatically favors the brightest features
in a time signal. Thus, to be able to pick out all features in a signal, we can normalize the spectral density at each frequency to produce what is called the conditional
spectrum:1261
s(klw)
-
S(k,w)
S(w)
(3.33)
Where s(klw) is the conditional spectral density, S(k, w) is the spectral density as
a function of both wavenumber and frequency, and S(w) is the spectral density integrated over the wavenumbers at each frequency. This gives equal weight to each
frequency in evaluating the relationship between poloidal wavenumber and frequency
and does not weight it according to fluctuation power. The conditional wavenumberfrequency spectrum calculated for the spectrum in Figure 3-2 is shown in Figure 3-3a.
We will use the conditional spectrum to perform the turbulence velocity analysis.
51
Shot 1120615030
150
iso
50
-6
-4
-2
ke
0
2
4
6
-10
0
-5
k
(Cm~')
5
'0
ev"
Figure 3-3: Conditional spectrum for C-Mod shot #1120815030, at p =1.95 cm. (a)
using the oversamrpled method with the oversampling factor = 4, and (b) using tile
two-point method. Tile oversampled method found a (hand-fitted) velocity of 3 km/s
and a velocity using the max-integral-comiputation of 3.13 km/s, while the two-point
method failed to produce a spectrum clear enough to make a measurement.
3.7
Velocity measurements with the Fourier technique
There are multiple characteristics of the turbulence which call be studied with tile
Fourier technique. The most important characteristic for this thesis is the velocity of
the turbulence structures in the plasma. We call measure the velocity by using either
the square of the spectral density or the conditional wavenumber-frequency spectrum,
but most often the conditional spectrum. There are two different velocities which call
be measured: the phase velocity and thle group velocity. The phase velocity is defined
as:
Vphase =-(3.34)
k
where w is the frequency (=27rf) and k is the wavenumber. Typically U) is a function
of k; this function is called the dispersion relation. There call be a phase velocity
associated with both the radial and the poloidal directions by simply using the radial
52
and poloidal wavenumbers, respectively. The group velocity is defined as:
dk
dw
Vqru
(3.35)
We consider the plasma turbulence to be dispersionless when both the derivative
of w with respect to k is a constant and w(k = 0) = 0.
When this occurs, the
group and phase velocities are equal. In the implementation of this technique, we
typically assume no dispersion, and thus take the poloidal phase velocity to be the
slope along the maxima in the conditional wavenumber-frequency spectrum. If there
were non-zero dispersion, a curve could be fit, to the maxima, and the derivative could
be taken.
There are two different ways to fit the dispersionless velocity. The first involves
handfitting a line by eye, i.e. by defining by eye a "best fit" line passing through
the maxima of the s(klw) values and through f = 0, k = 0; the slope of the line
is the phase velocity (shown in Figure 3-3a). The second method is computational.
It takes a set of f/k=constant lines (passing through f = 0, k = 0) and integrates
the s(klw) values along each line.
The maximum integral determines which line
gives the best fit. The slope of this line is then chosen as the phase velocity. This
method only gives one output per direction (positive or negative wavenumbers). This
method was used to determine the Fourier analysis points in Figure 2-1. If a lobe,
i.e. a roughly straight positive-k or negative-k extended feature in the conditional
spectrum, indicates dispersion, handfitting should be used, and several points along
the lobe should be chosen to compute a velocity for each part of the lobe.
3.8
Comparison of oversampled Fourier technique with
the two-point method
The now well-known two-point method for measuring the wavenumber-frequency
spectrum with fixed probe pairs was introduced by [261. To summarize the method,
we imagine two measuring points located at x, and
53
Xb.
The local wavenumber spec-
trum is essentially the difference between the phases of the signals at each point
divided by their spatial separation:
0(Xb, W) - 0(a, W)
K(x, w) ~(.6
where x is the average location between xa and
Xb.
(3.36)
To find the phases, we must
first find the locally Fourier transformed signal, for a time signal g(x, t) to give and,
following Equation 3.11, the phase angle will then be:
9(x, w) = arctan -img
xW)
real (x, w)
Kth_ phantom
(3.37)
twopoint.pro is an implementation of this method, which I wrote.
To compute the conditional spectrum, normally five sets of two points are used to
compute a local wavenumber and then averaged. An example conditional spectrum
for shot #1120815030,
p = 1.95 cm is shown in Figure 3-3b, where this averaging of
actually ten sets of two-point evaluations failed to produce a spectrum clear enough
to make a phase velocity determination.
Thus we conclude that the Fourier transform analyses, described in Sections 3.13.7 and used to produce the conditional spectrum shown in Figure 3-2, provide much
more sensitivity and accuracy than the two point method.[27] The reasons are as
follows.
We can consider the two-point method to essentially be oversampling by
a factor of 50-100. Such a large oversampling rate introduces error. As mentioned
previously the multi-point spatial Fourier-transform also oversamples, but only by a
factor of -4, which results in smoother spectra and is allowed as long as any small-k
features are coherent across the array view. When the oversampling is 50-100, this
assumption is far less good and loss of accuracy results. Using more spatial points
reduces the error and provides greater wavenumber resolution. Thus the multi-point
spatial transform, with an oversampling rate of ~4, provides a reasonable tradeoff
between wavenumber range and mathematical rigor.
54
3.9
Comparison of results between GPI Phantom
Camera and APDs
As discussed in Chapter 1, we have two different detector systems for the GPI: the
avalanche photodiodes (APDs) and the Phantom Camera. Because the APDs have
significantly better signal to noise while the Phantom Camera has slightly better
special resolution, we want to compare both the qualitative and the quantitative
similarities and differences between the two detector systems. We can use the APDs
to validate (or not), the results of the Phantom Camera. To do this, we will use the
conditional spectra of a few actual shots. Qualitatively, both the Phantom Camera
and the APDs should find the same features; quantitatively these features should
have the same location in k -
.f space, giving the same velocity.
The first shot we will investigate is #/1120712027, an Ohmic H-Mode
131 with a
clearly visible quasi-coherent mode structure visible. Figure 3.4 shows the conditional
spectra from the Phantom Camera and the APDs averaged over 10 msec of data.
Both the Phantom Camera and the APDs have detected the QCM, the feature at
80-120 kHz and k ~ 2cm7 1 . It is apparent that the ko resolution for the Phantom is
better, due a 6 cm vertical view compared with a 3.9 cm for the APD-based system
and six times the number of vertical points in a column (recall that Akin = 27r/L).
However, the two systems find values of k for the QCM, that differ by ~20%, with k0
- 1.8 cm-
1
(Phantom) and at 2.2 cm-
1
(APDs). The reason for this difference in the
detected ko for essentially the same turbulence structures is still not understood. It is
also apparent that the APD system detects more relative fluctuation in the 150-200
kHz band and in the 25-60 kHz band. This is due to the higher sensitivity of the APD
system compared to the Phantom camera. More importantly though, the differences
in poloidal wavenumber between the data for the two detectors causes differences in
the measured phase velocities, a ~20% difference for the QCM phase velocity and a
~50% difference for the k < 0 lobe (see Figure 3-4).
55
Sot 1120712027
0
50
-6
Figure
t - 1.4400 thru 1.4431 SUe
ShAt #1120712=7
t - 1,4400 thrU 1.4434 sft
3-4:
-4
-2
0
k# (emW')
2
Conditional
4
5
-6
wavenumber-frequency
-4
-2
0
ko(m'
spectrum
2
for
4
6
C-Mod
shot
#1120712027, with hand-fit velocities shown. (a) Phantom Camera, (b) APDs.
The next shot will we analyze is #1120224009,
for 15 msec of data, during an
L-mode phase.[3 Figure 3-5 shows that that the Phantom Camera and the APDs
find qualitatively similar ko,
f
spectra, with features moving upward. However, the
differences remain. The Phantom Camera is less sensitive and therefore shows no
features above the noise at frequencies higher than -150 kHz.
Let us now consider a third shot, #1110121019.
Figure 3-6 (a-d) show results
from both the Phantom Camera and the APDs for two different locations, along with
the hand-fitted velocities through some of the main features.
Note that while the
frequencies of the features seem to match, the poloidal wavenumbers differ, causing a
difference in the poloidal phase velocity. The reasons for this difference are currently
unknown.
3.10
Limitations of the Fourier method
To summarize, there are a few limitations on the Fourier method for velocity analysis,
which have arisen in this discussion. The first is that due to the distinct behavior
change of the plasma at the separatrix, we cannot complete a radial Fourier transform
56
Shot #I I
I
-6
$ot
-4
-2
J1 120224009
0
k# (cm')
t
2
4
-6
6
-4
-2
0
2
4
6
k (CM-')
ShOt I 120224009
- 0.7010 thm 0.7153 sec
t - 0.7010 thm 0.7153 sWc
1150
o100
-6
Figure 3-5:
#1120224009.
-4
-2
0
kO(.m')
2
4
6
-6
-4
-2
0
k
2
4
1
(c)
Conditional wavenumber-frequency spectrum
(top) Phantom Camera, and (bottom) APDs.
for
C-Mod
shot
using the full radial coverage of the views (this will be discussed further in Chapter
6).
The second limitation is that the formalism for performing a spatial Fourier
transform was designed for a rectilinear coordinate system in which the directions
are at perfect right angles. While the poloidal direction is almost vertical, and the
radial direction is nearly perpendicular to it, some (small) error has been introduced.
Third, as has been shown in the previous section, the Fourier method applied to
data from both GPI detectors (APD array and Phantom Camera) do not show the
same features with exactly the same wavenumber-frequency spectrum; this affects the
velocity measurements. The reasons for this are unknown, and thus we cannot say
57
ShWr
1110121019
t
-
1.2200 thni 1.2498 sec
150
150
-
-4
-2
0
2
ko (0m')
4
6
Figure 3-6: Conditional spectra for shot
-6
-4
-2
0
k
2
4
6
(Cm-')
#1110121019. The top plots use data from
the Phantom Camera, while the bottom plots use data from the APDs. Results
for each system are shown for two radial locations (p ~ -0.9 cm and ~0.09 cm).
Qualitatively, the Phantom Camera and APDs detected the same features. However,
there are differences in the poloidal wavenumbers for the features, which lead to the
inference of different velocity measurements. A cross-comparison of the Phantomderived and APD-derived feature velocities for one of the shots of Figure 2-1 is shown
in Chapter 7 (as Figure 7-11). Generally the agreement is within 20%.
with total confidence that the data from one detector perfectly validates the data from
the other. However, as was shown, qualitatively they agree well, and the differences
in quantitative measurements are generally not large.
58
Chapter 4
Tracking time-delay estimation
technique for determining structure
velocities in time-series images
As mentioned in Chapter 2, we are considering two different methods for analyzing
GPI measurements and trying to understand why they can disagree for the same data.
The first, Fourier analysis, was discussed in detail in the previous chapter. In this
chapter, we will discuss the second method, tracking time-delay-estimation.[3, 41 The
general idea for this method is a combination of time-delay-estimation (TDE) and
pattern tracking within some defined search box. We will begin by defining some of
the important mathematics necessary as well as the specific implementation used to
perform the analysis of data in this thesis. We will end by discussing some limitations
of this method.
4.1
Cross-correlation function
There are several different ways to implement TDE.[281 We will focus on a method using a direct cross-correlation between two discrete time signals. The cross-correlation
59
function is defined for a continuous signal as:
(f * g) (t) =
where
f
(4.1)
f*(t)g(t +,T)dt
and g are two functions of time t, and
T
is a finite time interval called the
time lag.[29, 301 For a discrete signal, we can use a summation instead of an integral:
00
f*[n]g[n + T]
(f * g) [r] =
71
(4.2)
=-00
where n is now a discrete time instead of continuous time t. This function is a measure
of how much a signal's behavior varies in time relative to another signal's behavior.
We consider different time lags because changes in one signal could be seen at one
time, but correlated changes in the other could take place a little earlier (negative
-r; the first signal lags the second) or later (positive
T;
the second signal lags the
first). When two signals are perfectly correlated (i.e. the changes in the behavior of
one signal match changes in the behavior of the other), their cross-correlation is +1.
Autocorrelation is the cross-correlation of a signal with itself as a function of time
delay or lag. When two signals are perfectly anti-correlated, their cross-correlation is
-1.
For time delay estimation, we define the time delay in the cross-correlation as:
Tdelay =
arg max((f * g) (T))
(4.3)
which is just the argument of the maximum in the cross-correlation function, where
we are convolving g with the complex conjugate of
f
as a function of t.
The phantom camera produces a set of two-dimensional frames in time.
The
pixels in the camera each have their own sensitivities to signal, which could affect
the value of the cross-correlation function. Thus we wish to normalize out the pixelto-pixel variation in the response. We do this by subtracting the mean signal and
then dividing by the square root of the product of the standard deviation in the two
60
signals. This is implemented numerically using the following formula:
N-T
(f-k+r|-
E-
Zkt
I) (9k
-
0
(4.4)
r > 0
(4.5)
N-11-
1 (fk- f
(
-
,
where N is the number of points in the time series of the signals f and g.
4.2
Implementation
As described in
13], Phantom Camera images taken over roughly three to five millisec-
onds duration were analyzed. This length of time was chosen to ensure that enough
data were selected to obtain an accurate answer while ensuring efficiency of the code
(longer time series takes longer to process). Selecting shorter time series would result
in not including lower frequency correlations. In principle, a time series of any length
could be used. Each frame in the series is first spatially smoothed and normalized
to reduce noise. Then a reference pixel is selected and a search box of ±8 pixels in
each direction surrounding it is designated, for a total of 17x17 or 289 pixels. The
cross-correlation function for the reference pixel and every other pixel in the search
box is computed for lags up to ±10 frames, with the time between frames being 2.5
ps. The maximum in the cross-correlation function for each time lag is found, and Ai
pixels and Aj pixels are computed, where Ai and Aj represent the distance in radial
and vertical pixel-space, respectively, between the reference pixel and the pixel for
which the maximum cross-correlation was found (for that time lag). This is shown
for a time lag of +1 frame in Figure 4-1.
Once the cross-correlation maxima vs. time lag are found, with the corresponding
Ai and Aj in pixels, a horizontal (vertical) velocity is computed for each time lag,
by dividing the Ai (Aj) by the lag, and converting to km/s. As long as the crosscorrelation function was at least 0.5, and as long as there are at least four time lags
61
One
Frame
t
25 is time series
t
Figure 4-1: Illustration of reference pixel inside its search box for the tracking TDE
analysis. Here, the peak of the cross-correlation function is found within the box[7].
(of the 20 total) meeting that criterion, the final velocity in that location is computed
by averaging over the velocities meeting those criteria. This process is computed for
reference pixels that are not within 8 pixels of the edge of the frame. The user has
the choice of further restricting the region of reference pixels. Because of the search
box size and the requirement that four lags yield cross-correlation maxima above the
threshold, this particular implementation detects velocities only between -2.0 km/s
and +2.0 km/s, in both the Z and the R directions.
There are two outputs of the code. First are poloidal and radial velocity profiles
as a function of rho, where the velocities for reference pixels within 0.5 cm in rho are
averaged together, and reported as the velocity in the median of the range considered
for rho. They are evaluated at p (cm) = -1.25, -0.75, -0.25, 0.25, and 0.75, with error
bars computed as the standard deviation of the measurement. This output is meant
to be the "mean" flow at particular locations near the separatrix. The second output
is a velocity map, in which the velocity vectors are represented as arrows (Figure 4-2).
The arrows point in the direction of motion at each location, and the lengths of the
arrows are relative to that of the maximum velocity in the field (which has a fixed
length). Not every vector is shown, for two reasons. First, the user can choose the
62
percentage of pixels for which a velocity computation is attempted in order to reduce
computation time (e.g. every third pixel in every third row so that 1/9 of the total
number of pixels are used). Second, if the computed lag-velocities for that pixel do
not meet the threshold criteria stated above, then no velocity is found for that pixel.
Shot 1 120712027
60
50
t
40
30
20
10
0
0
10
20
30
4D
50
60
Figure 4-2: Example of a velocity map for 4ms of C-Mod data, with axis labels
representing pixel number. Velocity computations were attempted for 1/9 of the
pixels in the frame. The maximum velocity in the field is 0.574 km/s.
4.3
Known limitations of this method
As we did in Chapter 3 for the Fourier analysis, we will describe the known uncertainties and limitations with the tracking TDE analysis method. The most obvious
is that this particular implementation is limited to finding velocities no larger than
2.0 km/s. Certain flows are known to move at faster speeds (e.g. when using RF
heating
121). These flows will not be detected by this particular implementation of
the tracking TDE method. By the virtue of searching for the maxima in the cross-
63
correlation function, it cannot find multiple flows in the same region; it can only
find the dominant motion caused by the largest structures and the most intense. At
most it will find some combination of any multiple flows, so that the output should
be interpreted as some average velocity in the region. It makes the assumption that
the individual structures do not change in size or intensity; we know in the real data
that the blobs characteristics may change in time. This method is subject to what
is called the aperture effect, in which a change in intensity for the reference pixel is
necessary to detect the motion.141
64
Chapter 5
Generation and use of synthetic data
for testing analysis techniques
We have chosen to use synthetic data as a means of testing for the strengths and
weaknesses of the implementations of the Fourier analysis technique [1
(described in
Chapter 3) and the tracking TDE technique [3] (described in Chapter 4). Currently,
neither code can be used to benchmark the other, and this motivates the need to
generate data for which we know the relevant input parameters such as velocity, size,
distribution, and intensity beforehand so that the accuracy and fidelity of each code
and method can be tested rigorously. This will provide insight on the quantities that
are "weighted" most heavily by each analysis technique. In the end we wish to understand why the two techniques differ in their conclusions about turbulence velocities.
To this end, the challenge is to generate synthetic data that mimics as closely as possible the real data. We will describe how the synthetic data are generated, compare
the synthetic data to real, and provide a sample analysis.
5.1
Generation of synthetic data
The basic principle behind the generation was to assume that the GPI turbulence
structures can be treated as "blobs" which have a given shape, velocity, and distribution of sizes and intensities. The large blobs detected by the Phantom Camera
65
system appear to be roughly circular in shape when viewed parallel to the local field.
The APD system detects blob shapes that are more elliptical than circular for reasons
that are still unclear.[1] We have chosen the synthetic blobs to be circular. The user
is allowed to input both a horizontal (R) and vertical velocity (Z) for the blob field.
This means that all blobs in a single generated field will be moving with the same
velocity at all times and do not change shape as they move. The user can specify a
statistical distribution from which to assign a size and intensity to individual blobs in
the field. Currently, the options are a normal distribution and a gamma distribution
(where gamma can be one up to nine in value). The user can then define a maximum
intensity and maximum full width at half of the intensity maximum (FWHM) for the
blob distribution. Note that, as implemented, a given blob intensity and FWHM are
determined by the same randomly generated value.
Three other important options for the user are the number of fields, the number of
time steps, and whether or not to add noise to the synthetic signal. First, the user may
want to be able to have blobs moving at different speeds within the same field of view.
The user can choose how many blob fields to generate, and the signals in each pixel
or view will be added for each field generated. Thus it is possible to choose different
fields to have different velocities, distributions, and different maximum intensities
and sizes. Second, rather than fixing the length of time for which the signals will be
generated, the length of time is variable to the fidelity of the codes in terms of the
length of the time series. Third, the real data are subject to noise in the Phantom
Camera. Thus, the user has the option to add noise from a no-plasma C-Mod shot
(Shot #1120712007).
An example of the end result is shown in Figure 5-1.
There are two methods for generating the blob fields. The first method, developed
by J. Terry, involves creating a large static field of blobs, and then reading out the
intensity values in each pixel at a defined rate to simulate a moving field. This has
the advantage that once the field is generated it can be read multiple times to obtain
a signal from multiple fields moving at different speeds. It also has the advantage
of not limiting the size of the blobs generated.
It has the disadvantage of being
unable to choose different distributions, maximum sizes, and maximum intensities for
66
Figure 5-1: Example of synthetic blob field with noise added. The motion of a single
blob is indicated by the arrow.
a single generated field. However, the user can generate multiple static fields, scan
each, and then add the signals from each. The second method, which I developed,
generates the blobs as the signals for each view are saved to the C-Mod data archive.
The blobs are generated outside the field of view, and move into the field of view
after so many time steps depending on the chosen velocity. This also allows the user
to generate multiple fields with completely different characteristics for an individual
shot. It has the disadvantage of limiting blob size because of the fixed size of the
field over which the blobs are generated before moving into the field of view. It is
also slightly more computationally intensive since the signals must be computed at
each time step. However, both methods lead to the same results for the same input
parameters. Included in Appendix A is part of the code which implements the second
method, as well as definitions for the input variables.
5.2
Comparison of synthetic data to real data
We wish to mimic the real data as much as possible. As such, it is important to
compare certain statistical properties of the synthetic data to those of the real. First,
67
both a real and a synthetic signal are shown in Figure 5-2.
Nhot i1209i1IM&Rft
%ne
Nnb
a
4
10
1.190
1.11
1.10
1.105
Tk"W"TW
1.1"4
0.100
1.105
on*nees
0.01
.0
0.03
0.004
Figure 5-2: Example signal in a time for a pixel viewing the plasma scrape-off-layer
(a), and example of a synthetic signal (b). Note that the time duration shown for
each is the same, i.e. 5 ms.
Unless the signal is simply too low, or there is no signal in a pixel, the absolute
magnitude of the signal is unimportant since the signals are normalized during the
analysis.
Thus, to compare the synthetic data to the real data, we compare their
statistical properties. One such property is the probability density function (PDF),
which is just the probability distribution as a function of fluctuation magnitude, which
is defined as follows:
~I- <I>
I =
0'(I)
(5.1)
where a(I) is the standard deviation of the signal. Examples of PDFs for both a real
shot and a synthetic trial are given in Figure 5-3 a and b, respectively. The synthetic
data were specifically designed to output any gamma (or normal) distribution of blob
sizes and intensities. This is to mimic how blob intensity varies from being (approximately) normally-distributed inside and just outside the last closed flux surface to
being gamma distributed (gamnma-9) in the far scrape off layer. 1131 Since we are most
concerned with the turbulence in the scrape off layer, for most trial shots, gamma-9
was chosen. No initial dependence of distribution on the velocity measurement was
found (as seen in Figure 5-4).
Thus, to first order, we believe our synthetic data
properly mimic the real data well enough to be used for this study.
68
Shot 1120815021
U_
LF.M
Gunmma4
**6 000'
I
iL0.0100:
'
0.001
O010 -
OMWO~l
0.0001
---.-- -1
-2
0
,
a.M1
1
2
3
4
-2
.
.
-1
0
1
,
,
2
3
4
(0sp0YOM(
Figure 5-3: Normalized (to total fluctuations in signal) PDF for a real shot (left)
from a pixel viewing the far scrape-off-layer, and for a synthetic trial where gamma-9
(right).
t
Shot I I 1161123
-6
-4
-2
0.9000 thru 0.9046 sec
0
ke (Cm')
2
4
SW
-6
6
t
111181123
-4
-2
-
0.90
0
thr.u 0.9046 Wec
2
4
6
kS (em')
Figure 5-4: (left) Spectrogram of the log of the spectral power (Fourier amplitude
squared) as a function of poloidal wavenumber and frequency for a single-velocity field.
(right) Conditional spectrum for the same single-velocity field. Input vertical velocity
= 1.01 km/sec and input horizontal velocity = 0.0 km/sec. The red line through the
lobe of the conditional spectrum indicates a phase velocity of 1.0 km/sec, i.e. it
reproduces the input vertical velocity. The errors in the measurements are ~5%.
5.3
5.3.1
Sample analysis on single-field and two-field shots
Sample Fourier analysis
When developing the synthetic data, we expected the Fourier analysis for a singlevelocity field to be straightforward and easy to predict. If the input poloidal velocity
69
is positive, then the blobs move in the electron diamagnetic drift direction (positive
wavenumber for Alcator C-Mod's normal toroidal field direction), and vice-versa for
a negative input poloidal velocity. Both the wavenumber-frequency spectrum and the
conditional spectrum were expected to have a single lobe. The slope of the lobe is the
measured velocity. This was indeed demonstrated for several trials (see Figure 5-4).
A two-velocity shot was expected to be more complicated, but similar in output.
We expected there to be two lobes, with the velocity of each field being the slope
measured for each lobe. In several trials, this was demonstrated provided that the
intensity of the second field was large enough to be detected (see Figure 5-5).
The
effect of intensity of the blob fields will be discussed in Chapter 7.
Shot I11161081
-6
-4
t
-2
-
0.9000 Uwu 0.9046 sec
0
2
4
Shot
6
111161081
-6
-4
t
-2
- 0.900G
0
thru 0.9048 sc
2
4
6
Figure 5-5: Same as in Figure 5.4, but for a two-velocity field. Input vertical velocities
were 0.39 km/s and 1.56 km/s, and input horizontal velocities were 0.08 km/s and 0.39
km/s. The lines through the lobes indicate that the Fourier analysis nearly reproduced
the input vertical velocities. Errors on the measurements are approximately 5%.
Reasons for not exactly reproducing the velocities will be explained in Chapter 6.
5.3.2
Sample tracking TDE analysis
We expected the tracking TDE analysis to produce velocity maps which were uniform in direction for the synthetic data since the synthetic data would have uniform,
isotropic fields of blobs all moving in the same direction, or in the case of different
70
directions (two or more fields overlapping), the differences would average out over a
long enough period of time. This was indeed the case for most trials, especially when
there was one field, or two fields with the same sign in poloidal/radial velocity. See
Figure 5-6 for examples of velocity maps for the same single-velocity (input vertical
velocity = 1.01 km/s and input horizontal velocity = 0.0 km/s) and double-velocity
shots (input vertical velocities = 0.39 km/s and 1.56 km/s and input horizontal velocities = 0.08 km/s and 0.39 km/s). For the single-velocity synthetic shot, the velocities
found by the analysis were Vert = 0.8 km/s and Va, = 0.02 km/s; for the two-velocity
shot, t he velocities found were Vert = 0.42 km/s and Vho, =0.04 km/s.
..
SO
Shot 111161081
Shot 111161123
........
LL~
L*
L
j
j
60
L
50
50 1
pol
L
40
40
30
30
I
20
r
10
10
I
SI I
A .-
S i .. jI..j
0
0
10
20
30 40 50
60
0
10
20
30
40
I
I
50
60
Figure 5-6: Velocity maps for single-velocity (left), and double-velocity (right) shots.
Axes are both in pixels, with the view as it actually is in the machine. (left) Input
vertical velocity = 1.01 km/s and input horizontal velocity = 0 km/s, and the analysis
yielded a vertical velocity of 0.8 km/s and a horizontal velocity of 0.02 km/s. (right)
Input vertical velocities = 0.39 km/s and 1.56 km/s and input horizontal velocities
= 0.08 km/s and 0.39 km/s, while the analysis yielded a vertical velocity of 0.42
km/s and a horizontal velocity of 0.04 km/s. Errors were approximately 5% for each
measurement.
71
72
Chapter 6
Modifications needed for use of
Fourier and tracking TDE techniques
After generating synthetic data, we tested each code rigorously for strengths and
weaknesses. In this chapter, we describe some of the weaknesses of each code that
could possibly affect the results of the real data, and offer ways to correct these issues.
The first section will cover issues we found with the Fourier analysis code; the second
section will cover issues we found with the tracking TDE analysis code.
Ve also tested
the effect of adding noise to the synthetic data; this is described in the third section.
The last two sections apply the corrections to analysis for real data. Ultimately we
found that these corrections only explain a small portion of the original discrepancies.
6.1
Issues found when applying the Fourier analysis
6.1.1
Vertical velocity error when radial velocity is not small
A velocity scan using synthetic images with single (known) vertical and radial structure (blob) velocities was completed. Single velocity fields were generated, each with
the same maximum intensity and FWHM, but varying the radial and vertical input
velocities. Figure 6-1 shows the relationship between the output vertical velocity and
the input vertical velocity, using the Fourier analysis. Increasing the radial veloc-
73
ity, relative to the vertical, increases artificially the vertical velocity found from the
analysis above the actual vertical velocity. As seen in Figure 6-1, when the ratio of
the actual radial to actual vertical velocities is less than -0.5, the returned vertical
velocity is within 20% of the input value. However, when the ratio is 1, the returned
vertical velocity is wrong by a factor of ~2, with the error increasing steeply as the
ratio increases beyond 1.
This effect is caused by the fact that we are only performing the column transform (vertical in viewing space) during the analysis, rather than performing both the
column (vertical) and row (radial) transforms.
12
p100.
>
42>2
0.0
0.5
1.0
1.5
2.0
VRAVz (input)
2.5
3.0
Figure 6-1: Ratio of output to input vertical velocity vs. ratio of input horizontal to
input vertical velocity for a series of synthetic shots. The blue dashed line indicates
what the output should have been based on the input velocities.
6.1.2
Wavefront model to explain the observations from the
Fourier analysis applied in only one spatial dimension
We will now derive a model in which we treat the blobs as wavefronts, and will show
that this wavefront model explains these results. Let us begin by assuming that in
the R-Z plane, the wavefronts are moving so that the normal to the fronts is at an
angle a to the horizontal (see Figure 6-2). The normal to the wavefronts points in the
direction of the total actual velocity of the blobs. Each wave has a wavelength A, the
74
distance between successive wavefronts. We assert that the spatial Fourier analysis
breaks up the wavelength into its R or Z component, when performing only a row
transform or only a column transform, respectively. Simple geometry dictates that:
A'
n Ax/=
A
sin a
cos a
(6.1)
R
Figure 6-2: Wavefront model with angle a shown.
Thus, there are wavenumbers in both the Z and R directions. These are given by:
k'
27r
k'
Z A'zR
27r
-
,
A/
(6.2)
As mentioned in Chapter 3, the phase velocity is given by 27rf/k. Thus, there is an
associated phase velocity with each direction, given by:
' = 27rf
Vz = 27rf
k' V6.3I
k,
Z
(6.3)
R
The frequency f of the wave does not change with direction. Recognizing that the
total phase velocity can also be written as
f A,
and each directional phase velocity
component is just the product of the frequency and the directional wavelength, we
75
can write the directional velocities in terms of the total velocity:
Ij =
(6.4)
.V , V = I
sina
cosa
where V is the input velocity of the wave. From the diagram, we can write sine and
cosine in terms of the Z and R components of the actual velocity V. Simplifying our
expression then, we can write the effective phase velocities in the Z and R directions,
obtained by examining only the projections of the wavelengths with the frequencies,
and normalized by the actual components of the phase velocity, as:
J
Vj
-=1+
VR)
V'
Vz
VZ
VR
(Vz
(-Z)
2
(6.5)
VR
This implies, for example, that if the input VR is large compared to the input Vz, the
Vz found by the Fourier analysis performed in the Z direction will be large compared
to the actual Vz component. Of course, the same will be true of the VR found by
the radial Fourier analysis when Vz is large compared to VR. When Equation 6.5 is
plotted against the data found from the poloidal Fourier analysis results shown in
Figure 6-1, it is an excellent fit (Figure 6-3).
8
4
-------------------
0.0
0.5
1.0
1,5
2.0
2.5
v'/v' (input)
Figure 6-3: Wavefront prediction for the output vertical Z velocity for the Fourier
analysis. The model prediction (Equation 6.3) is the green dashed line; the synthetic
shots are shown with error bars.
76
We can solve for the input velocities individually in terms of the outputs. In Equation 6.5, dividing the equation for V5/Vz by the equation for Vj'/VR, we eventually
arrive at a cubic equation for the ratio of Vz/V,
Z3
q
as a function of Vz/VR:
V
Z
VZ2+
+
=0
(6.6)
There is one real solution to this equation:
-=
- 11(6.7)
VR
Vz
We can substitute this back into Equation 6.5 to find the original velocities as a
function of the outputs:
V,z2(6.8)
1+
6.2
!V ),'
=R
+()
Issues found when applying the Tracking TDE
analysis
We performed the tracking TDE analysis on the same trials used for the velocity
scan for the Fourier analysis described above. We predict that there should not be
any dependence on the input horizontal velocity VR for the output Vz measurement,
or vice-versa, since the change in pixels in each direction corresponding to the peak
in the cross-correlation function are computed independently. This proved to be the
case; however, the tracking TDE analysis systematically underestimates both outputs
by 20-30% (Figure 6-4).
Obviously the biggest concern is with the limit in maximum velocity that can be
detected by the code. If one has knowledge that there is a fast flow in the plasma
boundary (V > ~ 4 km/s), it is simply recommended to not use the TDE analysis.
77
2.0I
1.6 --
lp lA
A/
02.0
0
'
Q .0
0.0
0.5
1.0
I.
2.0
0.0
33
2.5
0.6
np. Paboid Vbcly (ns)
1.0
1.6
2.0
6WuRdWVdoty "m)
2.5
3.0
Figure 6-4: For tracking TDE analysis: (a) Output poloidal velocity vs. input poloidal
velocity, (b) Output radial velocity vs. input radial velocity. Both outputs are systematically smaller than the actual structure velocities, indicating that there are issues
with the TDE method that need to be understood.
6.2.1
Effect of search box size on velocity measurement
Recalling from Chapter 4 that the cross-correlations are examined only within search
boxes of size t8 pixels with respect to the reference pixel, we can understand that the
size of the search box limits the detectable velocity. A search box size of ±8 pixels
theoretically allows a maximum velocity measurement of t
2.8 km/s [3] as long as
only one time lag with a cross-correlation value above the threshold is required. A
scan of the relationship between search box size and output poloidal velocity was
completed for an input vertical velocity of 2.0 km/s and the results are shown in
Figure 6-5.
2.0
E
1,9
A
1.7
CL
16
18
20
22
Vbaf size (pixels)
24
26
Figure 6-5: Output poloidal velocity for tracking TDE vs. search box size in pixels.
The input (actual) poloidal velocity was 2.0 km/s.
78
40
4D
.5
6
is
a
0.6
0.4
0.2
0.0
Figure 6-6: Cross-correlation function for a reference pixel, contoured for time lags of 10 frames (upper left corner) through +10 frames (bottom right corner), not including
zero lag. The search box size is 17x17 pixels, while the blobs have a FWHM of ~8
pixels. A well-defined peak in the function that meets the threshold, is only in the
search box for four lags, yet the average velocity is calculated using values for 10 lags,
as listed in 6.1. The scale reflects the value of the cross-correlation with the reference
pixel for the given lag.
Increasing the size of the search box improves the results. There are two reasons
for this. First, typical blob sizes are roughly 1 cm in diameter. Each pixel is slightly
less than 0.1 cm across, so most blobs fill most of a search box (11 pixels across). The
second reason is that because for larger velocities the peak of the cross correlation
function for a reference pixel could be outside the search box (Figure 6-6). Nonetheless, as long as the threshold cross-correlation value is obtained there, the code treats
the peak as being at the edge of the box; for a given time lag, this decreases the change
79
in the number of pixels (Ai and Aj ) thereby decreasing the velocity computed for
that lag (Table 6.1).
Lag
Max CC
Ai
Aj
-12.5
0.70
-1
-8
-10
0.91
-1
-8
-7.5
0.99
0
-8
-5
0.99
0
-5
-2.5
0.99
0
-3
2.5
0.99
0
3
5
0.99
0
5
7.5
0.99
0
8
10
0.92
0
8
12.5
0.72
1
8
Table 6.1: Lags (measured in ps) used for computation of poloidal and radial velocities
for a reference pixel in tracking TDE, for a synthetic shot with a purely vertical input
velocity (1 km/s), and the change in pixels Ai and Aj for the horizontal and vertical
directions, respectively. Only lags whose maximum cross-correlations meet the 0.5
threshold are included in this table. After conversion from pixels/lag to km/s, the
output poloidal velocity for this shot was 0.82 krn/s; the output radial velocity was
-0.1 km/s.
There are two ways to correct this issue. The first entails increasing the search
box size.
However, as can be seen from Figure 6-5, the search box would have to
be larger than it is currently (17x17 pixels). This creates a second issue of how to
meaningfully count the change in pixels from one frame to the next for a given time
lag. The second way to correct avoids this issue by simply requiring that the peak
in the cross correlation function be within the search box for a given lag, in order for
the velocity corresponding to that lag to be included in the average. We have done
this for the shot in Table 6.1, and the new results are listed in Table 6.2.
Lag
Max CC
Ai
Aj
-5
0.99
-0.35
-5.2
-2.5
0.99
-0.20
-2.65
2.5
0.99
0
2.5
5
0.99
0.25
5.15
Table 6.2: Results for same synthetic shot after requiring that the pixel at which
a maximum in the cross-correlation occurs be located in the search box. Only four
lags are used in the average. The new output poloidal velocity for this shot was 0.93
km/s; the new output radial velocity was -0.1 km/s.
80
6.2.2
Requirement that there be at least four velocities lags
in the average
The second major issue in the tracking TDE code is that in order to evaluate the
velocity for a location in the Phantom Camera images, the velocities computed from
at least four time lags must meet the threshold. If that is the case, then those values
are averaged. This reduces the maximum detectable velocity from ~0.5x the searchbox dimension/minimum-lag-time to half that. To correct this, we have modified the
code to eliminate this requirement. For example, if only two velocities computed from
the time lags meet the threshold, then only those two get averaged.
6.3
Effect of adding noise to the synthetic data
A separate issue possibly affecting both methods/codes was that of the noise inherent
in the real data. Noise could possibly affect the measurements if the pixel to pixel
variation has a gradient or some pattern to it, especially for the tracking TDE analysis.
Thus, it is important to quantify this effect by comparing trials with and without
noise, while keeping all other inputs constant (same velocities, intensities, sizes, and
distributions).
To do this, a series of single-velocity shots was created each with the same level of
absolute real noise, varying the absolute average signal. A sampling of the results is
shown in Figure 6-7. The Fourier analysis code finds the expected result (accounting
for the systematic errors described above) for all levels of signal, although quality was
significantly reduced at low signal-to-noise (as seen in Figure 6-7b). The tracking TDE
breaks down for a ratio of the standard deviations of light to noise of approximately
2.2. In other words, it fails to find a velocity measurement. At larger Elight/Znoise,
the tracking TDE analysis yielded expected values (after accounting for systematic
errors).
Thus, we conclude that noise does not inherently invalidate the velocity
measurements.
81
Sho. 11311220
I
-
0.90CM thru 0,9097 sec
1.
0,0 F~~~
-0.$
-2.0
-20
-10
0
10
20
-20
-10
g(Cfwr)
0
ke (CM 1)
0
20
Figure 6-7: Wavenumber-frequency spectrograms from the Fourier analysis for
-oise
= 2.2 at the breakdown level for the tracking TDE (a), and (b) the
Eight/E
conditional spectrum where a faint feature is still available for which a valid velocity
is obtained using the Fourier analysis. Elight/Enoise in this case is 0.87.
6.4
Corrections to tracking TDE code, applied to
real data
We wish to see whether or not the improvements we have made the tracking TDE
code will help resolve the initial discrepancies we saw in Chapter 2. We will discuss
the wavefront issue for the Fourier analysis in the next section.
Table 6.3 has the original and new values for six shots, using the improvements
we have proposed here, for the rho-profiles. Table 6.4 displays similar data for the
radial velocities. In general, the larger percentage changes were for small velocities
(< 0.2 km/s in magnitude).
As discussed in Chapter 2, the discrepancies between
the velocities obtained with the Fourier analysis and those obtained with the tracking
TDE were usually at least a factor of 2, or more than 100% in terms of percentages.
Therefore, we can confidently say that these improvements in the tracking TDE code
only account for a small portion of the total discrepancy.
82
Rho (cm)
224009
-1.25
-0.75
-0.25
0.25
0.75
Shot number (1120+)
224015 224022 224023 224024
224027
-
-1.369
-0.744
-0.596
-0.295
-1.084
-
-
-0.844
-0.699
-0.320
-1.177
-0.288
-1.238
-0.504
-0.436
-0.278
-0.205
-0.289
-1.38
-0.652
-0.559
-0.415
0.013
0.143
-0.889
-0.681
-0.874
-0.452
0.538
0.072
-1.124
-0.733
-1.022
-0.390
0.623
0.532
-0.837
-0.131
-0.063
0.038
0.571
0.043
0.603
0.550
-0.844
-0.080
-0.202
0.561
-0.304
0.203
0.057
0.088
0.479
0.523
0.062
0.229
0.053
0.081
0.457
Table 6.3: Original [31 (black entries) and new (bold entries) values of poloidal velocity as computed with the tracking TDE analysis after applying the improvements
suggested in this chapter (km/s). For the data in these shots, no measurements
changed directions. A few points are not given because the data at those locations
were too noisy to yield a measurement.
Rho (cm)
-1.25
-0.75
-0.25
0.25
0.75
Shot number (1120+)
224009
-
224015
-0.054
224022
-0.146
224023
0.024
224024
0.017
224027
-0.107
-
-
-0.039
0.007
-0.019
-0.101
-0.108
0.006
0.038
0.062
0.06
0.019
-0.120
0.02
0.052
0.062
0.067
0.041
0.001
0.069
0.052
0.096
0.102
0.068
-0.005
0.039
0.046
0.057
0.106
0.099
0.046
0.074
0.088
0.096
0.135
0.131
0.057
0.092
0.084
0.101
0.137
0.128
0.071
0.12
0.088
0.17
0.15
0.187
0.049
0.195
0.096
0.157
0.150
0.155
Table 6.4: Original 131 (black entries) and new (bold entries) values of radial velocity
as computed with the tracking TDE analysis after applying the improvements (km/s).
For the data in these shots, no measurements changed directions. A few points are
not given because the data at those locations were too noisy to yield a measurement.
6.5
Corrections to Fourier analysis, applied to real
data
We will now investigate how much of a correction, if any is needed to the poloidal
velocities in the real data due to any radial component, as according to the wavefront
83
model in Figure 6-3. To test this, we must first understand exactly what happens
to the conditional spectrum when there is a significant radial component if we are
trying to measure the poloidal velocity, and vice-versa. Figure 6-8 illustrates what
happens when we increase the ratio of input Vz/VR when analyzing for VR (which
is equivalent to analyzing Vz and increasing input VR/Vz). As is seen, the region in
Shot
t - 0.9000 theu 0.9046 et
116 1137
Shot #111161138
t
0.9000 theu 0.9046 "ec
-
ISO
100
100
50
50
-6
S"#t
-4
-2
0
k, (CM-)
2
6
4
-6
-2
k'
t - V9WhU 0.S46 WCe
111t161135
-4
Shot
#111161134
t
-
0
(Oen-)
0.9000
2
4
6
thru 0.9046 see
160
160
ISO
100
so
-6
-4
-2
0
2
6
4
Shot 0111161123
-6
t - 0.9000
-4
thcu
0.9046
2
4
-2
2
4
6
ec
100
Sc
-6
-4
-2
0
k'
6
(.m)
Figure 6-8: Radial conditional spectra for varying input Vz/VR. (a) Vz/VR
Vz/VR = 2. (c) Vz/VR = 5. (d) Vz/VR = 8. (e) Vz/VR - infinite.
84
1. (b)
which the Fourier amplitudes peak (lobe), broadens. If the velocity is supposed to be
positive, the larger the ratio, the more the lobe spills into the negative k region of the
graph (it would be just the opposite if the velocity were supposed to be negative).
This continues until the lobe is divided in half, when the ratio is infinite (i.e. Vz is
finite, and VR
-
0, analyzing for VR, and vice-versa).
Shot 1120224015
Shot 1120224027
150
150
>1 100
>100
LA_
50
50
-6
Sh ot
-4
0 2
-2
k. (cm~')
4
6
-6
1120224022
Shct
150
150
0> 100
C
100
50
50
-6
-4
-2
0 2
ke (cm-')
4
6
-4
-2
0
2
k8 (cm~')
4
6
1120224022
L
-6
-4
-2
0 2
ke (cm~')
4
6
Figure 6-9: Examples of spreads in lobes of poloidal conditional spectra for real
shots in the far scrape-off-layer (p > 0.85 cm). (top) Two lobes on opposite sides
of the k-spectrum. The separation of the two lobes is much too wide compared to
the synthetic spectra where the separation is due to a significant component in the
perpendicular direction. (bottom) Spread in lobes (one dominant direction) which
indicates that there may be a radial component for this shot. However, very few real
shots had significant width to the lobes in their poloidal spectra, indicating small
radial components.
85
While the tracking TDE finds the radial velocities of real shots to be small compared to the poloidal velocities, additional assurance would be provided if we could
find another independent way of determining an upper limit on the radial velocity
in order to evaluate how much of an effect any radial component will have on the
Fourier measurements of the poloidal velocities.
Using what was shown in Figure
6.8, we have examined the measured poloidal velocities of real shots from,[3 looking
for instances in which there are broadened lobes, and lobes split between negative
and positive k values in the conditional k-f spectra from the Fourier analysis. Some
examples are shown in Figure 6-9.
There were very few instances in which there were broad lobes. There were several
instances in which there appeared to be lobes split between positive and negative
wavenumbers.
However, in these cases, if we compare the separation of the split
lobes in the real data to the separation of the split lobes in the synthetic data, the
split lobes in the real data are much further apart (Figure 6-9). This indicates that
for these cases the lobes in opposite directions are not likely due to a significant radial
component, but are actually two separate lobes.
We have also independently analyzed for the radial velocities in the real data, by
modifying the Fourier code to handle the horizontal direction. The wavefront model
still applies, but in this case applying the radial form of Equation 6.5. We have done
this solely for the SOL; the region inside the separatrix in the Phantom Camera field
of view for each shot was not included in the Fourier spatial transform. This ensures
that there is no issue with the results due to crossing the separatrix when performing
the summation in the transform. In almost all cases examined, the radial conditional
spectrum yielded no measurement (Figure 6-10).
This is due to the fact that the
radial component is most likely too weak to be detected above the noise level.
These analyses strongly support our belief that the radial phase velocities, in
general, are small compared to the poloidal phase velocities, and that little, if any,
correction is necessary for the Fourier analysis. As was demonstrated in the previous
section, the corrections to the tracking TDE code, made only a slight difference in the
poloidal velocity measurement in a few cases, and essentially no difference in most
86
Shot
1120224009
150
100
50
-6
-4
-2
0
2
kr (cm~')
4
6
Figure 6-10: Sample radial conditional spectrum for shot #1120224009.
cases. Thus, we conclude that the reasons for the discrepancies in results between
these two techniques must be due to some other causes. We explore these other causes
in the next chapter.
87
88
Chapter 7
Resolution of the velocity
discrepancies
7.1
Revisiting the original problem and where we are
so far
We began this thesis by presenting recently published data [31 in which the poloidal
velocities of turbulent filaments in the boundary of C-Mod plasmas were presented
as a function of distance from the separatrix. Using a method which we have called
tracking Time-Delay-Estimation, the Zweben paper analyzed data from the Gas Puff
Imaging system which employs the Phantom Camera as a detector. As described
in Chapter 4, this method finds the maxima in the cross-correlation function for an
array of time lags between the time series for a reference location in the Phantom
Camera image and the time series from other locations in a search box surrounding
the reference location.
The velocities computed for each time lag are then aver-
aged to obtain a final velocity for the emission structures, at that reference location.
We contrasted the velocities computed in this way with those computed using another method, which uses temporal and spatial Fourier analysis, from which we produced conditional wavenumber-frequency spectrograms as described in Chapter 3.
The Fourier analysis finds poloidal phase velocities of emission structures using those
89
spectrograms. As shown in Figure 2-1 (repeated here as Figure 7-1) and emphasized
in Chapter 2, these velocities do not agree, with discrepancies of up to an order of
magnitude for the same data. The goal of this thesis has been to understand and
resolve these discrepancies.
Shot #1 12G224009
Shot #1120224015
4 -A
(a)
A
2
A
AN.
E
M
b
1
(3
E
V
-V
V
ai
V
V
A
-1
0
-2
A
0.0
-05
AL
D5
1.
-1 S
-05
-1.0
05
0.0
1.0
PhD (-cm)
Shot # 1120712027
A
AL
A
hD (cm)2
Shot #1120515016
4
A
(d)
3
3
2
E
E
a
-it
A
AL
-2
--
---
-1 5
-10
D D
A
A
A
A
0.5
1.0
A
-1
A
-0.5
A
A
-- -- -- -- ---AA
A
2
-U
0.5
A A
5
-1.0
-0.5
00
RhD
RhD (Cm)
(cm)
Figure 7-1: Poloidal velocity measurements for four Alcator C-Mod shots, using
Fourier analysis (red triangles) and tracking TDE (blue inverted triangles).
In Chapter 5, we presented our method for studying this issue, i.e. the use of
synthetic data for which we know the velocities, sizes, intensities, and distributions
of the emission structures ("blobs") before analysis.
This allowed us to test each
method and code for their individual strengths and weaknesses, as well as systematic
issues. The systematic issues that were found are described in Chapter 6. Overall
we found that the Fourier analyses correctly returned the poloidal velocities of the
structures in the synthetic image sequences as long as the ratio of radial to poloidal
velocity <0.7. This condition appears to be satisfied in the experimental data, based
90
on the very small values of the TDE-determined radial velocities and based on radialvelocity Fourier analysis for the shots. Affecting the tracking TDE determinations
were two significant issues. First, if the maximum in the cross-correlation function is
at the edge of the search box for a reference pixel (meaning that the center of a blob
has already passed out of the box), it was still used in the average over time lags.
Second, cross-correlations above the threshold value at four different time lags were
required for inclusion in computing the detected velocity, thus limiting the maximum
detectable velocity to a value well below the quoted maximum of ±2.8 km/s.
To
address these two issues, we modified the analysis code so that time lags for which
the maximum in the cross-correlation function is at the edge of the search box are not
included, and so that the cross-correlation above the threshold is required at only one
time-lag for inclusion. With these changes the TDE analysis returns velocities that
are within
10% of the velocities in the synthetic image sequences as long as there
is only one velocity field and as long as the maximum velocity does not exceed -2.8
km/s. However, even with these changes to the tracking TDE analysis, most of the
discrepancy between the results of the two analyses remains. Thus, accounting for
these systematic effects was not able to explain the original discrepancies.
With the knowledge that analysis methods do return the appropriate velocities
under the restricted conditions discussed above, we are forced to examine the details
of the Fourier spectrograms in search of a resolution for the remaining discrepancies.
Using the spectrograms we will explore the evidence and impact in the analyses of
multiple velocities existing within the field-of-view, including non-zero dispersion in
w(k). These were mentioned previously (Chapters 2, 3, and 4) as possible problems.
7.2
Analysis for dispersion in the real data
The first issue is dispersion in the plasma.
have dispersion when w/k
Mathematically, we can detect if we
# dw/dk; the phase velocity does not equal the group
velocity. This can manifest itself in different ways. First, dw/dk could be a continuous
function in which the group velocity changes gradually.
91
Second, dw/dk could be
discontinuous, meaning that the group velocity changes abruptly. We will treat the
first case immediately and discuss the second case later in this section. Physically,
dispersion in this case would mean that different size structures move at different
velocities. As mentioned before, the presence of dispersion leads both codes to output
incorrect values of the poloidal velocities of the emission structures since there is no
single velocity for all features. For the Fourier analysis, this is obvious since it relies
on the principle that a straight line can be fit to the peaks of the conditional spectrum
(through zero), and that the slope of the line is the phase velocity. A changing slope
implies dispersion. In the tracking TDE code, the presence of continuous dispersion is
more subtle, because the tracking TDE method relies on finding the maximum in the
cross-correlation function and heavily weights the most intense features. Typically,
the most intense features are in the low-frequency, low-wavenumber range. This can
be seen by looking at a sample wavenumber-frequency spectrogram, such as the one
shown in Figure 3-1 (reproduced here as Figure 7-2). Note that the color scale is the
log of the spectral power.
$hc4t
1 251$03i
I
1,2.8" thrW
~.
Z .4
I .263& 04.A
1111811
4-~~
8
4
23
t
-2
-
OUQO
0
thru
2
O0044 s-
4
6
Figure 7-2: Sample wavenumber-frequency spectrogram for a real shot (a) and a
synthetic one-velocity shot (b). The color scale indicates the log of the power (square
of the absolute value of the Fourier coefficients) at a given wavenumber and frequency.
The simplest way to check for dispersion is to examine the conditional spectra
from the Fourier analysis for curved lobes. This, we have done. Figure 7-3 shows an
example of a clear curved lobe (dw/dk continuous) in the sample of shots published
92
in the Zweben paper.
150
-4
-6
-2
0
2
6
4
k. (cm)
Figure 7-3: Poloidal conditional spectrum for C-Mod shot #1120224015. The spectrum features a curved lobe indicating some dispersion in the plasma at p = -0.65
cm.
S
it 120224024
t-
1.13W0t
1.134
..
Sht 1120224009
c
.0
1 00
C
(D
100
50
i
U.
-6
-4
-2
0
k* (cm')
-4
-2
0
2
4
6
ko (CMr')
2
Figure 7-4: (a) Conditional spectrum for Shot #1120224024, p = -0.1 cm. An abrupt
break in slope is clearly observed. Two slopes are shown: one for a slower-moving
component, and one for a fast moving component. (b) Conditional spectrum for Shot
#1120224009, p = 1.0 cm, also showing a clear break in slope, slower phase velocity
(~0.6 km/s) for the lower frequency/lower k features and a faster phase velocity (~.3
km/s) for the higher frequency features. The solid white line is the velocity obtained
from maximum-in-the-integral-computation.
Now let us consider the second kind of dispersion, in which the slope of the lobe in
93
a conditional spectrum abruptly changes. There are instances of this kind of dispersion in the real data, as well. Examples are shown in Figure 7-4 for shots #1120224024
and #1120224009,
with the fitted velocities included. Note that the fitted line for
the higher-frequency, higher-wavenumber component does not go through the origin.
Producing a clearly curved lobe in the synthetic data was extremely difficult. However, we were able to produce two synthetic shots in which the velocity appears to
change quickly ("split-slope") as a function of frequency. The conditional spectra for
these shots are shown in Figure 7-5. The input velocities for these synthetic shots
S~W
-10
111161119
-5
t-
0.9000 thru 0,9046 ftc
0
5
I111161118
SW
-10
10
-5
t
-0.9000
0
thru 0,9046 sc
5
10
k. (cm-!
ko (Cm')
Figure 7-5: Conditional spectra for synthetic shots constructed in order to show a
break in slope. The red dashed line is the hand fit for frequencies < 30 kHz and yields
a velocity of -0.75 km/s. The dark green line indicates the slope of the slower-moving
feature (with a phase velocity - = -0.75 km/s) while the lighter green dashed line
indicates the slope of the faster-moving feature (-1.15 km/s). Input values for these
shots are given in Table 7.1. (a) Uses only two fields, and (b) uses four fields.
are listed in Table 7.1. The important result of the test analysis is that the tracking
TDE analysis returns poloidal velocities of -0.51 km/s and -0.61 km/s, respectively
for these synthetic cases, obviously weighting the lower frequency more heavily than
the higher frequency ones.
94
Field
Field
Field
Field
Field
Vertical Velocity
1
2
3
4
Field 5
Field 6
Horizontal Velocity
FWHM
Synthetic shot #111161118: Four-field shot
-0.59
0.31
0.8
-0.82
0.27
0.8
-1.09
0.23
0.4
-1.21
0.27
0.4
Synthetic shot #111161119: Two-field shot
-0.59
0.31
0.8
-1.13
0.27
0.4
Max Intensity
3.0
2.7
1.1
1.1
3.0
0.7
Table 7.1: Input values for two synthetic shots which show breaks in slope. Velocities
are given in km/s, FWHM are given in cm, and max intensity in arbitrary units. The
TDE analysis returns poloidal velocities of -0.51 km/s and -0.61 km/s respectively
for these synthetic cases, obviously weighting the lower-frequency components more
heavily than the higher ones.
7.3
Analysis for the effects of multiple lobes within
the field of view
The second issue we mentioned previously (Chapters 2 and 4) is that of multiple flows
in the same region. It is already known that the tracking TDE cannot distinguish
between multiple flows in the same region; the best it can do is provide a weighted
"average" velocity for the region. Because the Fourier analysis breaks up the signal
into its individual components and does not only select the brightest part of the
signal, it can detect multiple flows in a region. We have already shown this for two
flows with the same sign in velocity (Figure 5-5).
However, it is certainly possible
that these flows can "blend" together to appear as though they are the same one.
Figure 7.6 shows three more synthetic examples in which there are multiple flows in
the same region.
See Table 7.2 for the inputs. These inputs were chosen so that
the poloidal conditional spectra would produce one or two blended lobes while the
number of input fields was actually three or four.
We obtained the poloidal velocity as measured from both the Fourier and the
tracking TDE analysis. The results are listed in Table 7.3. If the four-field shot is
treated as a single-lobe by the Fourier analysis, and we only return a single velocity
95
Sy
U.-L
--
c"
g
k4()4
4-20246-6
-4
0
-2
2
4
6
Synthetic shot 1161117
1150
~
~
00
tikPO
0
k
(cm)k")(m'
-2
0
1
~50£
LL
-8
-4
2
kPO1 (Cnf 1 )
4
8
Figure 7-6: Poloidal conditional spectra for four-field (a) and three-field (b and c)
synthetic shots. The negative-k lobes appear blended. Both a maximum-integral
computational fit (solid white line with the Vp, value) and hand fits (dashed red
lines) were completed for these cases, and are consistent with the input values given
in Table 7.2, although only 3 of the 4 velocities of four-field case are resolved.
Field
Field
Field
Field
Field
1
2
3
4
Vertical Velocity Horizontal Velocity FWHM Max Intensity
Synthetic shot #111161115: Four-field shot
-1.95
-1.37
-0.98
-0.82
0.39
0.39
0.39
0.39
Synthetic shot #111161116:
Field 1
Field 2
Field 3
-1.95
-1.29
0.19
0.39
0.39
0.08
Synthetic shot #111161117:
Field 1
Field 2
Field 3
-1.95
-1.29
0.78
0.39
0.39
0.08
0.4
0.4
1.0
1.0
2.3
2.2
2.7
3.0
Three-field shot
0.4
0.6
1.0
2.5
1.9
1.3
Three-field shot
0.4
0.6
1.0
2.5
1.9
1.3
Table 7.2: Input parameters for three multi-field synthetic shots. Velocities are measured in km/s, FWHM in cm, and intensity in arbitrary units. The input velocities
are to be compared with the features and fits in the conditional spectra of Figure 7.6
and the tracking TDE velocities listed in Table 7.3.
96
using the maximum-integral computational fit, then the returned velocity is -1.4 km/s
as listed.
Each three-field shot has two negative-k lobes and one positive-k lobe.
Again, if the Fourier analysis is used to return only a single velocity for each direction,
then the two found velocities are as listed.
Vp,, (Fourier)
Vp,, (TDE)
111161115
-1
-0.86
111161116
0.2
-1.9
0.13
111161117
1.9
-0.11
-
0.88
-
Synthetic shot
-
Table 7.3: Fourier and tracking TDE results for the synthetic shots given in Table
7.2. All velocities are in km/s.
The last case shown in Figure 7-6c and listed in Table 7.3 demonstrates how
discrepancies of up to an order of magnitude and even direction might arise from
a multi-field signal, moving mainly in the poloidal direction.
The Fourier analysis
yields a maximum-integral-computational velocity of -1.9 km/s, with obvious additional lobes of -1.36 and 0.88 km/s. These correspond quantitatively with the input
velocities of -1.95, -1.29, and 0.78 km/s respectively.
However, the tracking TDE
analysis yields a poloidal velocity of -0.11 km/s. Thus, in this case, the disagreement
is by a factor of more than 10 for the negative-k lobes, and in the opposite direction for the positive-k lobe. Its magnitude is a factor of 9 smaller than the actual
positive-k input velocity.
After thoroughly examining the conditional spectra for the shots published in
131,
there are many instances in which there are multiple flows in the real data, many
with both positive and negative-k lobes. One is shown in Figure 7-7.
97
Shot 91120224027
t a 1.1440 thru 1.1474 sec
150
-4
-2
0
k* (cm')
2
4
Figure 7-7: Conditional spectrum for shot #1120224027, at p--0.72
7.4
A resolution
We have modified Figures 2-1 and 7-1 in light of this more detailed analysis of the
Fourier conditional spectrograms. The modified presentation is shown in Figure 7-8
a-d. We have observed that the multiple flows are manifested in two different ways.
In some cases there appear to be flows moving in opposite directions. For other cases
(which we have examined as non-zero dispersion cases), there appears to be a single
lobe actually composed of two or more velocities. We have designed a synthetic data
set, whose conditional spectra are shown in Figure 7-4, to have the rough appearance
of the non-zero dispersion case, as an example. In this synthetic case and in those
cases of actual data with a break-in-slope lobe, we have found that the TDE analysis
returns a velocity that is much more heavily weighted to the lower-f/lower-k part of
the spectrum. To arrive at the results of Figure 7-8, we considered each shot carefully.
First for shot #1120224009,
Figure 7-9 (a-e) are the conditional spectra from the
Phantom Camera for Figure 7.8a. As can be seen in (a-d), the Fourier analysis yields
two lobes in opposite directions at each location in p (-0.81, -0.35, 0.1, and 0.55). In
addition, the positive lobe in each appears to have two relatively distinct slopes. For
the spectra at p
-
0.55 and 1 cm (Figures 7.9d-e), the lobes with positive wavenumbers
98
Shot #1120224009
Shot #1120224015
1.5
4
1.0
A
2
U
A
A
0.5
A
A
AV
(b)
0.G ------------------- --
9'
A VA
c
#6
-0.5
VA
A
-2
-10
A_
-0
5
Shot
A
0.0
VA
A
D5
A
1.0
i
-0.5
-1G
0.0
1 0
G15
rho (Cm)
rho (Cm)
#1120712027
Shot #1120815018
4
:6
A
(d)
e-
.2
A
E
A
A
2
A
A
0.5
1.0
A
A
-. 9
-1
[
A A
A
5
-1.0
-05
D0
-1 1
-15
0.5
-1.0
-0.5
rho (Cm)
00
rho (cm)
Figure 7-8: Figures 2-1 and 7-1 modified after considering the presence of multiple
flows and non-zero dispersion. Again the blue triangles are using the tracking TDE
after applying the corrections discussed in Chapter 6. The larger red triangles are the
velocities from the Fourier analysis that were hand-fitted to the lower frequency/lower
k parts of the dominant lobes in the conditional spectra. The smaller triangles are
the hand-fitted velocities for lobes in the conditional spectra that were present but
not dominant or were present at the higher frequencies.
are clearly dominant. However, these dominant lobes have a clear break-in-slope: one
representing slower motion (for low
f
and k), and one at higher
f
and k indicating
faster motion. These cases are examples of non-zero dispersion. The slower-moving
component has larger Fourier coefficients and is dominant.
Looking back at Figure 7-8a, we can now understand the discrepancy for this
shot.
First, the tracking TDE code is finding some weighted average velocity for
the region. For the first three points (p = -0.81, -0.36, 0.10 cm), this "average" is
over the two components moving in opposite directions, as well as the split slopes.
The tracking TDE poloidal velocity measurement moves from negative to positive
(while still staying around zero magnitude) as the lobes in the conditional spectra
99
Shot 1120224009
Shot 1120224009
N150
N150
)00
2100
0)
so0
2,)50
LL
WA.
-6
-4
-2
0
2
kPW (cm1 )
4
6
-6
-4
-2
0
2
4
6
4
6
kPW (cm')
Shot 1120224009
Shot 1120224009
150
%.0
100
90
U.5
U_
-6
-4
-2
0
2
4
6
-6
-4
kW (cm~')
-2
0
2
k,,. (cm')
Shot 1120224009
N150
100
50
UL
-6
-4
-2
0
2
4
6
ka (cm-')
Figure 7-9: Poloidal conditional spectra for shot #1120224009. Shown for each point
in Figure 7-8a are fits to the lobes in the spectra. (a-c) have two lobes in the opposite
directions while (d,e) have non-zero dispersion (a slower-moving component at low k,
low f with a faster component at higher k and f).
move from being more dominant in negative wavenumber to being more dominant
in positive wavenumber.
For the Fourier points at p = 0.55 and 1.0 cm), we have
split up the measurement we made for Figure 2-1a (7.1a) into two separate velocities
100
to reflect the non-zero dispersion. The slower-moving component in this region now
agrees with the measurement made by the tracking TDE (p
-
0.75). This confirms
that the tracking TDE method is finding the dominant component, which is usually
the component with lower
f
and lower k.
Figure 7-10 (a-e) show the conditional spectra for shot #1120224015.
At p
-
-1.11, -0.65, -0.20, and 0.25 cm, there appears to be only one negative-k lobe. We
have fitted by hand the velocities of these lobes at frequencies <30 kHz, where the
Fourier amplitudes are largest. We can see that these hand-fits at low frequency yield
somewhat smaller velocities than the maximum-integral-computational ones that were
plotted in the original Figure 2-1 and 7-1. Looking back at Figure 7-8b, we can see that
with this fitting criterion the discrepancy is now mostly resolved between the Fourier
analysis and the tracking TDE. The corrections in Chapter 6 made non-significant
changes to the outputs from the tracking TDE; those are reflected in Figure 7-8b as
well. Moving to the outermost scrape-off layer point at p
-
0.7 cm (Figure 7-10e), we
can see that there is now a second lobe in addition to the first, moving in the opposite
direction. Indeed at this location the Fourier and the TDE results no longer agree,
and we conclude that for this location, the discrepancy can be resolved by recognizing
that the tracking TDE code is taking a weighted average of the two features. The
tracking TDE analysis finds a small positive velocity in this case, even though the
negative-k lobe appears to dominate the Fourier spectrum.
For shot #1120712027, at p - -1.37 cm and -0.92 cm, we note the presence of the
quasi-coherent mode (at k
-
2 cm- 1 and
downward (split-lobe with two slopes).
f
= 100 kHz), and another feature moving
The QCM has a fast phase velocity that
is beyond the detection limit of the TDE analysis. Moving towards the separatrix
and into the scrape-off-layer, the QCM disappears, leaving only the feature moving
downward, which still appears to be split into two slopes. This feature now appears
more spread out, in a similar fashion to the lobes shown in Figure 6.9, indicating
a possible radial component to its motion. We attempted to verify that there is a
radial component and were unable to do so because the Phantom Camera frames
did not include enough of the SOL layer to obtain a meaningful measurement of the
101
Shot 1120224015
Shot 1120224015
N150
N150
100
100
0)D
950
LL
50
U
-6
-4
-2
0
2
4
6
-6
-4
ko (cm")
-2
k
Shot 1120224015
0
2
4
6
2
4
6
(cm')
Shot 1120224015
N150
N150
100
00
a)a
950
50
6
-4
-2
0
2
ka (cm1 )
4
6
-6
-4
-2
k
0
(cm")
Shot 1120224015
150
'100
950
h6
4
-2
0
2
kM (cm' 1)
4
6
Figure 7-10: Poloidal conditional spectra for shot #1120224015. Shown for each point
in Figure 7-8b are fits to the lobes in the spectra. Hand fits to the velocity for f ,30
kHz are shown in red, while a computational fit is shown in white.
radial phase velocity in that region. Comparing the spectra in Figure 7-11 to those
in Figure 6-9, it is certainly possible that if there is a radial component, it is likely to
be comparable in magnitude to the poloidal component, if not slightly larger. This
radial component would increase the magnitude of the poloidal phase velocity by a
102
ISO.-
150 .
50
-6
-4
-2
ho
S"o 01120712027
0
(2)
2
6
4
-6
k . 1.44M0 11W 1.4434 2ft
-4
-2
S"o P 120712027
0
2
k7 (CM4)
4
6
t - 1.4400thna 1.4434 *ft
150
IDO
50*
-6
-4
-2
0
k ()
2
SWo
6
4
-4
-6
-2
0
2
4
5
I 1.4434 sec
#1120712027
150
-6
-4
-2
kO
0
(cm')
2
4
6
Figure 7-11: Poloidal conditional spectra for shot #1120712027.
factor of -2.
The tracking TDE (corrected) found velocities slightly smaller than the
smaller of the two velocities associated with the two slopes, in all cases. If we assume
that there is a radial component for the first three Fourier points, the correction to
the Fourier velocities would account for the difference.
103
Considering the last of the four shots, #1120815034, we can see immediately the
origin of the discrepancy (Figure 7-12). At each location, there are lobes indicative
Sho 1120615018
1.2734 sec
t - 1.2700 thr
Sho
120815018
-
1.2700 tw
1.2734 sec
100
50
-6
-4
Sho #1120815018
-2
0
k# (-"')
2
4
1 0 1.2700 tI
-6
6
1,2734 se
-4
-2
0
k# (c-')
t
Shot #1120815018
2
4
6
12700 thn 1.2734 sec
ISO
200
50
-6
-4
-2
0
4
2
6
k. (Oi2M-)
Shot #1120815018
-6
t
-
1.2'
k9
0
2
(cm-1
-2
-4
0
k. (cmt*)
2
4
5
2SO
100
50
-6
-4
-2
4
6
Figure 7-12: Poloidal conditional spectra for shot #1120815018.
of motion in opposite directions. At p
-
-0.82 and -0.37 cm, we can see the QCM,
which has a fast poloidal phase velocity, as well as a slower moving lobe indicating
104
downward motion. Eventually the QCM is replaced by another positive lobe moving
somewhat slower. Looking back at Figure 7-8d, the tracking TDE code is taking some
average of these lobes, but favoring the positive ones since they are somewhat more
intense.
Finally, we consider APD measurements for one of the shots we showed in Figures
7-8, #1120815018.
In Chapter 3, we discussed that while the APDs and Phantom
Camera showed qualitatively the same features, in some cases the measured wavenumbers and phase velocities were not exactly the same. Figure 7-13 is a profile of the
poloidal velocity versus p, derived using the Fourier analysis on both the APD and the
Phantom camera data. As shown, while the velocity measurements are not exactly
the same, they are in good agreement.
Shot #1120815018
4
V
E2
A V
-1.0
-0.5
0.O
0.5
1.0
rho (Cm)
Figure 7-13: Poloidal velocity vs. p for C-Mod shot #1120815018, comparing the
data from the APDs (blue) to the data from the Phantom Camera (red), using the
Fourier technique, as the tracking TDE code is not designed for use with the APDs.
The meaning of the small and large triangles is the same as described in Figure 7-8.
These measurements are generally in agreement.
7.5
Summary and conclusion
The focus of this thesis was to understand why two different analysis codes for determining the poloidal velocities of emission structures captured with GPI yield results
105
differing up to an order of magnitude for the same data. Chapter 1 introduced the
Gas Puff Imaging diagnostic as well as the two detectors we use to collect the data
from it: the Phantom Camera and the avalanche photodiodes (APDs). The data
considered in this thesis were taken mainly with the Phantom Camera. Chapter 2
provided the motivation for this work, as illustrated in Figure 2.1 which shows the
discrepancies in the results from the two codes. Chapter 3 discussed in detail the first
code, [1, 2] which uses temporal and spatial Fourier transforms to create (among other
things) conditional wavenumber-frequency spectra from the time signals for which the
poloidal phase velocities of the emission structures can be extracted. Chapter 4 then
described the second code, [3, 4] which employs a hybrid of pattern tracking and
time-delay-estimation.
It works by finding time-delayed cross-correlations between
spatially separated locations in the images. In Chapter 5, we described how we chose
to address this issue: by generating synthetic data that mimics the real data. We used
the synthetic data to test each code rigorously. As part of those tests, we identified
some issues which could lead to errors in the determinations of the desired poloidal
velocities; these issues are described in Chapter 6.
However, after accounting for
these issues, either by modifying the analysis (in the tracking TDE case) or by determining that the assumption Vz >>
VR
is justified (in the Fourier case), it was
shown that these issues did not resolve the original discrepancies. In this Chapter,
we have examined the Fourier spectra in detail, considered the "weighting" that each
analysis uses in its calculation, and concluded that the discrepancies are mostly due
to two reasons. First is the presence of multiple velocities in the same field of view
that are either "averaged" in some way by the tracking TDE analysis or weighted
differently by each analysis, so that the velocity that dominates the tracking TDE
analysis result is not necessarily dominating the Fourier-derived result. And second
is the presence of non-zero dispersion in the propagation of the emission structures
and again a difference in weighting by the two analyses under these conditions.
We will leave the reader with a few words on the use of these codes. These two
codes are looking at different properties of the emission structures.
The tracking
TDE code returns a single velocity that is associated with the largest and typically
106
brightest features, while the Fourier analysis allows examination of multiple-size and
time-scale features. It is appropriate to use the tracking TDE method when one wants
to study how the largest features move. It is not appropriate to use when one wants
to study multiple size- and time- scales of the turbulence; one can use the Fourier
method for this. The most information will be gained when using both methods on
the same data with caution and with understanding of what is being measured by
each.
107
108
Appendix A
Codes to generate synthetic data
Construct__fakeblobs.pro - generates synthetic data for APDs
Construct _fakeblobs_ phantom.pro - generates synthetic data for Phantom Camera
/home/terry/idl-lib/gaussian-lD.pro
0/home/sierchio/readcol.pro
@/home/sierchio/remchar.pro
0/home/sierchio/gettok.pro
0/home/sierchio/strnumber.pro
@/home/terry/idl-lib/cmreplicate.pro
@/usr/local/cmod/codes/spectroscopy/gpi/write-phantom-t-hists.pro
@/home/terry/gpi/phantom/get-phantom-data.pro
pro construct-fake-blobs-phantom,normal=normal,tree-write=tree-write,
addnoise=addnoise
input variables and options
normal = 1 allows the user to select from a normal distribution
instead of a gamma distribution
tree-write allows user to write the shot to the tree for future use
addnoise allows the user to add a real noise background to each
109
frame using a non-plasma C-Mod shot, #1120712007
steps = the number of the time steps the user wishes to run for.
2000 steps will give roughly 5 ms of data.
num-fields = number of fields the user wishes to include in the
total field, essentially the number of sets of parameters the user
wishes to include. This must be decided before running.
variables for each field
Vp = input vertical velocity, measured in km/s
Vr = input horizontal velocity, measured in km/s
intensity = maximum intensity a single blob is allowed to have
(usually 3.0 is adequate, especially if running multiple fields)
FWHM = full width half max of the blobs, measured in cm. For an
intensity of 3.0, the max FWHM for gamma=9 is 1.2 cm.
gamma = the gamma parameter for the gamma distribution
print,'enter time steps and number of fields'
read,steps,num-fields
fields-arr=fltarr(num-fields,64,64,steps)
keydefault,normal,0
for w=0,num-fields-1 do begin
start:
frames1=fltarr(64,64,steps)
print,'enter phVp (km/s), phVr (km/s), intensity, FWHM,gamma'
print,'Do NOT input both Vp=0 AND Vr=0'
read,Vp,Vr,intensity,fwhm,gamma
around 70 blobs will cover the surface for an intensity of 3.0
and FWHM of 0.8 cm. To get the best coverage and signal for a
single field, it is recommended that the user run a single field
110
at least twice. This will ensure that as few pixels as possible
will have zero signal in them at any given time. Double or multiple
fields in the same shot only need to be run once for each set of
parameters.
parameters necessary for writing to the tree, such as trigger
time, frame rate, clock, time, and segment size are taken from
shot #1120911021. The trigger time is currently 0.9 s.
vratio=Vp/Vr
x=sqrt(Vp^2+Vr^2)
counter=1
counter2=3
mdsopen,'spectroscopy',1120911021
r_arr=mdsvalue('\SPECTROSCOPY::TOP.GPI.PHANTOM.IMAGEPOS:R-ARR')
z_arr=mdsvalue('\SPECTROSCOPY::TOP.GPI.PHANTOM.IMAGEPOS:Z-ARR')
time=mdsvalue('dim-of(\SPECTROSCOPY::TOP.GPI.PHANTOM:THISTS)')
segsizeo=mdsvalue('getsegmentinfo(\SPECTROSCOPY::TOP.GPI.PHANTOM:$
frames,0,*,_dimct,_dims,_nrow),_nrow')
exp=mdsvalue('\SPECTROSCOPY::TOP.GPI.PHANTOM.SETTINGS:EXPOS')
frame.rate=mdsvalue('\SPECTROSCOPY::TOP.GPI.PHANTOM.SETTINGS:$
FRAMERATE')
clock=mdsvalue('\SPECTROSCOPY::TOP.GPI.PHANTOM.SETTINGS:$
FRAMECLOCK')
numframes=mdsvalue('\SPECTROSCOPY::TOP.GPI.PHANTOM.SETTINGS:$
NUMFRAMES')
ncols=mdsvalue('\SPECTROSCOPY::TOP.GPI.PHANTOM.SETTINGS:XPIX')
n-rows=mdsvalue('\SPECTROSCOPY::TOP.GPI.PHANTOM.SETTINGS:YPIX')
trig-time=mdsvalue('\SPECTROSCOPY::TOP.GPI.PHANTOM.SETTINGS:$
111
TRIGTIME')
mdsclose
back.filename='/home/terry/gpi/phantom/$
phantom-background-exp_2us.sav'
restore,file=backfilename,/verb
back-shot-num=1120712007
Vp=Vp*100000.0 ;converts from km/s to cm/s
Vr=Vr*100000.0
delta-t=1.0/frame-rate ; sec
pixstep=delta-t*10000000.0
blob-params=fltarr(30000,5)
;for each blob, stores r,z coordinates, max intensity, radius
Height.pix=1907
Width-pix=1877
field=fltarr(1548,1578)
z_field=-11.87+0.01*indgen(1578)
r-field=98.47-0.01*indgen(1548)
tot-field=fltarr(width-pix,height-pix)
blob-params[*,0]=fwhm
blob-params[*,11=150.0
blob-params[*,21=-60.0
if normal then begin
blob-params[*,31= (3.+RANDOMN(SEED,30000,/normal) > 0.01)
siggauss=(blob-params[*,3]/3.)*blob-params[*,0]/$
sqrt(8.*0.693)
blob.params[*,4]=3*siggauss
endif else begin
blobparams[*,3]= intensity/gamma*RANDOMN(SEED,30000,$
112
gamma=gamma)
sig-gauss=(blob-params[*,3]*blob-params[*,0]/intensity)/$
sqrt(8.*0.693)
blob-params[*,4]=3*sig-gauss
endelse
z-dum=([fltarr(1548)+1.1#zjfield)
r-dum=r-field#[fltarr(1578)+1.1
ttl=systime(1)
new=0
ttl=systime(1)
mdsopen,'spectroscopy',1120911021L
r-arr=mdsvalue('\SPECTROSCOPY::TOP.GPI.PHANTOM.IMAGEPOS:$
R_ARR')
z_arr=mdsvalue('\SPECTROSCOPY::TOP.GPI.PHANTOM.IMAGEPOS:$
Z_ARR')
mdsclose
r-index=where(r-arr GT 0)
z-index=where(z-arr LT 0)
huh=array-indices(r-arr,r-index)
time=fltarr(steps)
view=lonarr(4096)
FMT='L'
fname='/home/sierchio/viewrad-phantom.txt'
readcol,fname,view,F=FMT,/sil
t=0
t=float(t)
113
fillfactor=fltarr(steps)
loadct,45,/sil
This is just an example to show what the code looks like
for a particular chosen direction. The other three directions
are included in the code in /home/sierchio
if (Vp GE 0.0) AND (Vr GE 0.0) then begin
tot-field[0:1547,0:15771=field
newz=-7.56+0.01*indgen(756)
new-r=93.16-0.01*indgen(756)
for t=0.0,steps-1 do begin
time[t]=t*deltat+trig-time
;loop to add up signal in pixels and store in frames
for i=0,4095 do begin
fields-arr(w,huh[0,i] ,huh[1,i],nint(t))=field(view[i])
endfor
totfield[nint(pixstep*Vr/100000.0):1547+nint(pixstep*Vr/$
100000.0),nint(pixstep*Vp/100000.0):1577+nint(pixstep*Vp/$
100000.0)1=field
field=0
blob.params[*,1]=blob-params[*,1]-Vr*delta_t
blob-params[*,21=blob-params[*,2]+Vp*delta_t
tot-field[1548:*,*]=0
tot-field[*,1578:*]=0
field=tot-field[0:1547,0:15771
if (t MOD counter EQ 0) then begin
blob-params[new,11=93.9 Ad
A6
A
;-(3-blobparams[new,41)
blob.params[new,2]=new-z(nint(755* RANDOMN(SEED,$
/uniform)))
114
smallrdum=sqrt((r-dum-blob-params[new,11)^2+(z-dum-$
blob-params [new ,21) ^2)
r_ind=where(smallr-dum lt
blob-params[new,41)
intens=gaussian_1D(smallr-dum(r-ind),[blob-params[$
new,31,0,blob-params[new,41/3.1)
field(r-ind)=field(r-ind)+intens
new=new+1
blob-params[new,1]=new-r(nint(755* RANDOMN(SEED,/uniform)))
blob-params[new,2]=-7.1 ;+(3-blob-params[new,4])
smallr-dum=sqrt((r-dum-blob-params[new,1])^2+(z-dum$
-blob-params[new,21)^2)
r-ind=where(smallr-dum lt
blob-params[new,41)
intens=gaussianiD(smallr-dum(r-ind),
new,3],0,blob-params[new,41/3.1)
field(r-ind)=field(r-ind)+intens
new=new+1
endif
totfield[0:1547,0:15771=field
endfor
endif
endfor
tree1:
stop
frames=total(fields-arr,1)
test-shot=OL
ans=OL
maxf=max(frames)
tlen=n-elements(time)
if keyword-set(addnoise) then begin
115
[blob-params[$
get-phantom-data, back_shotnum,back-frames,back-timesexp,$
nsegl=O,nseg2=40,/silent
frames=UINT((frames/max-f*275)+back-frames(*,*,O:tlen-1))
endif else frames=UINT(frames/max-f*1000+cmreplicate(back,tlen))
THE REVERSE IS IMPORTANT BECAUSE THE IMAGES ARE STORED MIRRORED
ABOUT A VERTICAL AXIS
make the max of the signal 1000 counts plus the "offset" or
backgnd level
if keyword-set(tree-write) then begin
the rest of the code writes to the tree and is included in the
version in /home/sierchio
116
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