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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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Can we some how define a background field which takes all intermediate scales
into account, self-consistently?
question 1
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Can we some how define a background field which takes all intermediate scales
into account, self-consistently?
extreme time locality (Haar filter)
! " # $ % & ' ( ) * "!
+,"!&
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!%!!
!#!!
!
#!!
%!!
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)!!
-./0
12-3 τ
incrementsy(t, τ) = x(t + τ)− x(t)
0.5 1 1.5 2 2.50.8
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extreme time locality (Haar filter)
! " # $ % & ' ( ) * "!
+,"!&
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12-3 τ
incrementsy(t, τ) = x(t + τ)− x(t)
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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
Fourierdomain
much theory; normally coarse-grained picture
better reference to detailed physics and
structure
question 2
Can we obtain the best of both worlds and live ‘well’ in the time and frequency
domains?
timedomain
Fourierdomain
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
smallscales
largescales
fluctuationsbackground field
frequency
spec
tral
pow
erlocal scale-dependent fluctuations and background field using low/high pass filters
f1b1
frequency
spec
tral
pow
er
b2 f2
local scale-dependent fluctuations and background field using low/high pass filters
f1b1
b3 f3
filter bank implementation
using quadrature mirror filters
fluctuationsbackground field
discrete wavelet transform (dwt)
1 2 3 4 5 6 7 80.2
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0
0.1
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0.8
0 10 20 30 40 50 60 70 80 90 10014
12
10
8
6
4
2
0
2
4
discrete wavelet transform (dwt)
1 2 3 4 5 6 7 80.2
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0 10 20 30 40 50 60 70 80 90 10014
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6
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2
0
2
4
DWT steps:Leave filter as it isDecimate signal by 2
0 5 10 15 20 25 30 35 40 45 5012
10
8
6
4
2
0
2
4
let’s see it working
2 1 0 1 2
8
6
4
2
0
2
log 10
PSD
((si
gnal
uni
ts)2 H
z1 )
log10 Frequency (Hz)
let’s see it working
2 1 0 1 215
10
5
0
log 10
PSD
((si
gnal
uni
ts)2 H
z1 )
log10 Frequency (Hz)
let’s see it working
2 1 0 1 2
15
10
5
0
log 10
PSD
((si
gnal
uni
ts)2 H
z1 )
log10 Frequency (Hz)
let’s see it working
2 1 0 1 2
15
10
5
0
log 10
PSD
((si
gnal
uni
ts)2 H
z1 )
log10 Frequency (Hz)
importance of zero moments
4 2 0 2 410
5
0
5lo
g 10 P
SD (s
igna
l uni
ts)2 H
z1
log10 frequency (Hz)
nr of levels = 21 wavelet is haar
importance of zero moments
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5x 10 3
0
0.1
0.2
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0.9
1
0 500 1000 1500 2000 2500 3000 3500 4000 45001.5
1
0.5
0
0.5
1
1.5
importance of zero moments
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5x 10 3
0
0.1
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1
n > (β − 1)/2
PSD(f) ∼ f−β
�xnψ(x)dx = 0
importance of zero moments
4 2 0 2 410
5
0
5lo
g 10 P
SD (s
igna
l uni
ts)2 H
z1
log10 frequency (Hz)
nr of levels = 19 wavelet is db2
importance of zero moments
0 0.5 1 1.5 2 2.5 3 3.5x 10 3
0
0.1
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1
0 2000 4000 6000 8000 10000 12000 140001.5
1
0.5
0
0.5
1
1.5
2
importance of zero moments
3 2 1 0 1 2 310
5
0
5lo
g 10 P
SD (s
igna
l uni
ts)2 H
z1
log10 frequency (Hz)
nr of levels = 18 wavelet is db4
importance of zero moments
db4 response
0 0.5 1 1.5 2x 10 3
0
0.1
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0.8
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1
0 0.5 1 1.5 2 2.5 3x 104
1
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0
0.5
1
1.5
importance of zero moments
2 0 2 4 6 8 10 12 14 16x 10 4
0
0.2
0.4
0.6
0.8
1
0 2 4 6 8 10 12 14 16x 104
0.8
0.6
0.4
0.2
0
0.2
0.4
0.6
0.8
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)
1 2 3 4 5 6 7 80.2
0.1
0
0.1
0.2
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0.5
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0.7
0.8
0 10 20 30 40 50 60 70 80 90 10014
12
10
8
6
4
2
0
2
4
undecimated discrete wavelet transform (udwt)
1 2 3 4 5 6 7 80.2
0.1
0
0.1
0.2
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0.5
0.6
0.7
0.8
0 10 20 30 40 50 60 70 80 90 10014
12
10
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6
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2
0
2
4
UDWT steps:Leave signal as it isUpsample filter by 2
0 2 4 6 8 10 12 14 160.2
0.1
0
0.1
0.2
0.3
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0.5
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0.7
0.8
choice of wavelet transform & filters
Solution: Undecimated discrete wavelet transform
[Coiflet-2 wavelet and scaling (digital) time-domain filters ]
scaling (low-pass)
filterwavelet
(high-pass) filter
choice of wavelet transform & filters
Solution: Undecimated discrete wavelet transform
[Coiflet-2 wavelet and scaling (digital) time-domain filters ]
scaling function
wavelet function
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
0 0.2 0.4 0.6 0.8 10.1
0.05
0
0.05
0.1residuals
0 0.2 0.4 0.6 0.8 1
10
5
0Coif2 Phase Response
Normalized Frequency (× rad/sample)
Phas
e (ra
dian
s)
qmf(Lo D,0) [Walden] linear
choice of wavelet transform & filters
Solution: Undecimated discrete wavelet transform
Rice wavelet toolbox for Matlab (C mex files)
https://github.com/ricedsp/rwthttp://dsp.rice.edu/software/rice-wavelet-toolbox
PyWavelets (Cython)
http://www.pybytes.com/pywavelets/
Anisotropic plasma turbulence applications
fluctuationsbackground field
frequency
spec
tral
pow
erlocal scale-dependent fluctuations and background field using low/high pass filters
f1b1
frequency
spec
tral
pow
er
b2 f2
local scale-dependent fluctuations and background field using low/high pass filters
f1b1
b3 f3
filter bank implementation
using quadrature mirror filters
fluctuationsbackground field
adaptive local scale-dependent fluctuations and background guide field
background field fluctuations
smallscales
largescales
power spectral density
!8
!6
!4
!2
0
2
log10
PSD
[nT2 H
z!1 ]
!2 !1 0 1
!1.2!1
!0.8!0.6!0.4
log10 spacecraft frame frequency [Hz]
log10
PSD
||/PSD
!
paralleltransversesearch coil noise
isotropy
kρe � 1
kρi � 1
Kiyani et al. ApJ (2013)
!8
!6
!4
!2
0
2
log10
PSD
[nT2 H
z!1 ]
!2 !1 0 1
!1.2!1
!0.8!0.6!0.4
log10 spacecraft frame frequency [Hz]
log10
PSD
||/PSD
!
paralleltransversesearch coil noise
isotropy
Cluster FGM and STAFF search-coil
angles of measurement w.r.t. B
-VSW
VA
VSW< 1MHD else
Vφ
VSW< 1
k =2πfsc
VSW
Taylor frozen-in approx, for low-frequency
phenomena
Anisotropy (angle between B and V)
20 30 40 50 60 70 80 90 100 110 1200
1
2
3
4
5
6
7
8
9x 10
4
B.V angle (degrees)
His
tog
ram
cou
nt
DR
IR
k Perpk oblique k oblique
higher order scaling
! " #!$
!
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%!
%$
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<'.'**)*
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!="
!=#
!=>
!=?
%
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%=#@2).+3'*,.'25),-0'*325,)A<42)2+-
B4;)2+,;
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<'.'**)*
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,
! " #!
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"
C
#
$F3--3<'+342,.'25),-0'*325,)A<42)2+-
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,
<'.'**)*GH!=?I
+.'2-().-)GH!=J>
(a i.)
(a ii.)
(b i.)
(b ii.)note: Inertial range data is from ACE
Also see:K. H. Kiyani et al. Phys. Rev. Lett.
103, 075006 (2009)
Sm�(⊥)(τ) =
1N
N�
j=1
����δB�(⊥)(tj , τ)
√τ
����m
Structure Functions
standardised probability densities
!20 0 20!7
!6
!5
!4
!3
!2
!1
0
! Bs [nT/"]log
10 P
s(! Bs )["
nT!
1 ] DissipationRange
paralleltransverse
!20 !10 0 10 20!6
!5
!4
!3
!2
!1
0
1
! Bs [nT/"]
log10
Ps(!
Bs )[" n
T!1 ]
InertialRangeparallel
transverse
!20 !10 0 10 20!7!6!5!4!3!2!1
01
! Bs [nT/"]
log10
Ps(!
Bs )[" n
T!1 ] Parallel
Fluctuationsdissipation
rangeinertialrange
!20 !10 0 10 20!7!6!5!4!3!2!1
01
! Bs [nT/"]
log10
Ps(!
Bs )[" n
T!1 ] dissipation
rangeinertialrange
TransverseFluctuations
isotropyKiyani et al.ApJ (2013)
magnetic compressibility
10 2 10 1 100 10110 2
10 1
100
Normalised wavenumber k i
Mag
netic
Com
pres
sibi
lity
C||
local field 55º-65ºlocal field 65º-75ºlocal field 75º-85ºlocal field 85º-95ºglobal field
isotropy
C�(f) =1
N
N�
j=1
δB2�(tj , f)
δB2�(tj , f) + δB2
⊥(tj , f)
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
Ten Lectures on WaveletsI. Daubechies
Essential Wavelets for Statistical Applications and
Data Analysis T. Ogden
A Wavelet Tour of Signal ProcessingS. Mallat
• 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)
Wavelet Methods for Time Series Analaysis
D. Percival & A. Walden
