A multiscale approach to the
analysis of anisotropic plasma
turbulence
(Or studying turbulence wearing wavelet spectacles)
Khurom Kiyani
the aim
to compute theoretically relevant
statistical quantities from a turbulent
(stochastic) time-series
background guide field
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background guide field
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background guide field
background guide field
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question 1
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we some how define a background
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which takes all intermediate scales
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question 1
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Can
we some how define a background
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field
which takes all intermediate scales
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into account, self-consistently?
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extreme time locality (Haar filter)
increments
y(t, τ ) = x(t + τ ) − x(t)
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τ
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extreme time locality (Haar filter)
increments
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question 2
Can we obtain the best of both worlds
and live ‘well’ in the time and frequency
domains?
time (space)
domain
better reference to
detailed physics and
structure
Fourier
domain
much theory; normally
coarse-grained picture
question 2
Can we obtain the best of both worlds
and live ‘well’ in the time and frequency
domains?
time
domain
Fourier
domain
outline
•Discrete wavelets as band-pass digital filters
•Picking and designing an appropriate wavelet/transform
•Some results on separating anisotropic signatures in
plasma turbulence in the solar wind
adaptive local scale-dependent fluctuations and
background guide field
background field
fluctuations
small
scales
large
scales
local scale-dependent fluctuations and background
field using low/high pass filters
f1
spectral power
b1
fluctuations
background field
frequency
local scale-dependent fluctuations and background
field using low/high pass filters
spectral power
b1
b2
f1
f2
fluctuations
background field
filter bank
implementation
using quadrature
mirror filters
b3 f3
frequency
discrete wavelet transform (dwt)
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discrete wavelet transform (dwt)
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DWT steps:
Leave filter as it is
Decimate signal by 2
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2
0
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log
10
PSD ((signal units)2Hzï1)
let’s see it working
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log10 Frequency (Hz)
2
log10 PSD ((signal units)2Hzï1)
let’s see it working
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Frequency (Hz)
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log10 PSD ((signal units) Hz )
let’s see it working
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Frequency (Hz)
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log10 PSD ((signal units)2Hzï1)
let’s see it working
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log10 Frequency (Hz)
2
importance of zero moments
nr of levels = 21 wavelet is haar
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PSD (signal units)2Hzï1
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frequency (Hz)
4
importance of zero moments
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x 10
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importance of zero moments
P SD(f ) ∼ f
1
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−β
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n > (β − 1)/2
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�
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n
x ψ(x)dx = 0
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importance of zero moments
nr of levels = 19 wavelet is db2
log10 PSD (signal units)2Hzï1
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2
log10 frequency (Hz)
4
importance of zero moments
2
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importance of zero moments
nr of levels = 18 wavelet is db4
log10 PSD (signal units)2Hzï1
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log10 frequency (Hz)
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importance of zero moments
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db4 response
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importance of zero moments
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x 10
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x 10
choice of wavelet transform & filters
1. Self-consistent high/low pass filters [Quadrature Mirror
Filters], full reconstruction -- can’t use continuous
wavelets.
2. Time event preservation -- can’t use normal discrete
wavelets.
3. Zero or linear phase (symmetric filters) -- for exact time
localisation of events.
4. Sharp Fourier frequency resolution for studying
spectra -- need higher order wavelets.
5. Time domain implementation with compact digital
filters to minimise edge effects, increase time locality
and reduce computational expense -- wavelet order
can’t be too high; again can’t use continuous wavelets.
choice of wavelet transform & filters
Solution: Undecimated discrete wavelet transform
undecimated discrete wavelet transform (udwt)
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undecimated discrete wavelet transform (udwt)
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2
UDWT steps:
Leave signal as it is
Upsample filter by 2
0
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choice of wavelet transform & filters
Solution: Undecimated discrete wavelet transform
scaling
(low-pass)
filter
wavelet
(high-pass)
filter
[Coiflet-2 wavelet and scaling (digital) time-domain filters ]
choice of wavelet transform & filters
Solution: Undecimated discrete wavelet transform
scaling
function
wavelet
function
[Coiflet-2 wavelet and scaling (digital) time-domain filters ]
choice of wavelet transform & filters
Solution: Undecimated discrete wavelet transform
[Coiflet-2 wavelet and scaling (digital) time-domain filters ]
choice of wavelet transform & filters
Solution: Undecimated discrete wavelet transform
[Coiflet-2 wavelet and scaling (digital) time-domain filters ]
choice of wavelet transform & filters
Solution: Undecimated discrete wavelet transform
Phase (radians)
0
Coif2 Phase Response
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0
qmf(Lo D,0) [Walden]
linear
0.2
0.4
0.6
0.8
Normalized Frequency (×/ rad/sample)
1
0.1
residuals
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0
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0.6
0.8
1
choice of wavelet transform & filters
Solution: Undecimated discrete wavelet transform
Rice wavelet toolbox for Matlab (C mex files)
https://github.com/ricedsp/rwt
http://dsp.rice.edu/software/rice-wavelet-toolbox
PyWavelets (Cython)
http://www.pybytes.com/pywavelets/
Anisotropic plasma turbulence applications
local scale-dependent fluctuations and background
field using low/high pass filters
f1
spectral power
b1
fluctuations
background field
frequency
local scale-dependent fluctuations and background
field using low/high pass filters
spectral power
b1
b2
f1
f2
fluctuations
background field
filter bank
implementation
using quadrature
mirror filters
b3 f3
frequency
adaptive local scale-dependent fluctuations and
background guide field
background field
fluctuations
small
scales
large
scales
2
||
PSD /PSD
−2
2
−4
0
−6
−2
−8
−4
−0.4
−0.6
−6
−0.8
−1
−1.2
−8
⊥
log
2
−1
PSDlog
/PSDPSD [nT2 Hz−1log
PSD
[nT
Hz
]
]
10
|| 10 ⊥
10
0
power spectral
density
−0.4
−0.6
−0.8
−1
kρi � 1
parallel
transverse
search coil noise
kρe � 1
isotropy
parallel
transverse
search coil noise
Cluster FGM and
STAFF search-coil
−2
0
1
isotropy −1
log10 spacecraft frame frequency [Hz]
Kiyani et al. ApJ (2013)
angles of measurement w.r.t. B
2πfsc
k=
VSW
Taylor frozen-in approx,
for low-frequency
phenomena
-VSW
VA
<1
MHD V
SW
else
Vφ
<1
VSW
Anisotropy (angle between B and V)
4
9
x 10
k oblique
8
k oblique
k Perp
7
Histogram count
6
DR
IR
5
4
3
2
1
0
20
30
40
50
60
70
80
B.V angle (degrees)
90
100
110
120
higher order scaling
(a i.)
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B4;)2+,;
#
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B4;)2+,;
note: Inertial range
data is from ACE
Also see:
K. H. Kiyani et al.
Phys. Rev. Lett.
103, 075006 (2009)
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�m
N �
�
�
�
δB
(t
,
τ
)
1
j
�(⊥)
m
�
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√
S�(⊥) (τ ) =
�
�
N j=1
τ
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%!
(a ii.)
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Structure Functions
#
standardised probability densities
0
−1
parallel
transverse
0
Inertial
Range
log10 Ps(δ Bs)[σ nT−1]
s
−1
log10 Ps(δ B )[σ nT ]
1
−2
−3
−4
−5
−6
−20
−10
0
10
δ Bs [nT/σ]
parallel
transverse
Dissipation
Range
−3
−4
−5
−6
−20
0
δ Bs [nT/σ]
20
1
isotropy
)[σ nT−1]
Parallel
Fluctuations
s
−1
)[σ nT ]
s
−2
−7
20
1
0 dissipation
range
−1
inertial
−2
range
−1
0 dissipation
range
−1
inertial
−2
range
Kiyani et al.
ApJ (2013)
Transverse
Fluctuations
magnetic compressibility
N
2
�
δB
1
� (tj , f )
C� (f ) =
2 (t , f )
N j=1 δB�2 (tj , f ) + δB⊥
j
0
Magnetic Compressibility C||
10
isotropy
ï1
10
local field 55º-65º
local field 65º-75º
local field 75º-85º
local field 85º-95º
global field
ï2
10 ï2
10
ï1
0
1
10
10
10
Normalised wavenumber kli
Kiyani et al.
ApJ (2013)
take home message
•Wavelets are excellent tools for studying turbulence and
anisotropy
•They are especially useful for both simultaneous event
localisation as well as frequency localisation
•Tried extending this to m-band wavelet packets to get
better frequency resolution; but generally this is not
possible. Recent work on over/undersampling is more
promising.
guides and literature
• M. Farge et al., Wavelets and Turbulence, Proc. IEEE, 84(4), 639, (1996)
• A. Walden & A. Cristan, Proc. R. Soc. Lond. A, 454, 2243 (1998)
• K. Kiyani et al, ApJ, (2013)
Essential Wavelets for
Statistical Applications and
Data Analysis
T. Ogden
Ten Lectures on Wavelets
I. Daubechies
Wavelet Methods for Time
Series Analaysis
D. Percival & A. Walden
A Wavelet Tour of Signal
Processing
S. Mallat