Geophysical Unsupervised Signal Processing...SPSC - Geophysical Unsupervised Signal Processing...
Transcript of Geophysical Unsupervised Signal Processing...SPSC - Geophysical Unsupervised Signal Processing...
SPSC - Geophysical Unsupervised Signal Processing
Geophysical Unsupervised Signal Processing
Fuchs Anna & Pessentheiner Hannes
Signal Processing and Speech Communication Laboratory
Advanced Signal Processing 2
Fuchs Anna & Pessentheiner Hannes Advanced Signal Processing 2 page 1/50
SPSC - Geophysical Unsupervised Signal Processing
Outline
IntroductionModel Description and AssumptionsSummary
Approaches for Unsupervised ProcessingWiener Filter and Unsupervised ProcessingPredictive DeconvolutionBlind Source SeparationICA-Based Seismic DeconvolutionResultsSummary
Degenerate Unmixing Estimation Technique – DUET
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SPSC - Geophysical Unsupervised Signal Processing
Reflection Seismic
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SPSC - Geophysical Unsupervised Signal Processing
Reflectivity Function
→ Earth response to an ideal impulsive, seismic source
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SPSC - Geophysical Unsupervised Signal Processing
How?
Information on the subsurface of an area under analysis
Produce seismic waves (dynamite, air-guns,...)
Reflection on subsurface
Sensor grids located in the surface measure reflections
Why?
Exploration and monitoring of hydrocarbon reservoirs
Assessing sites for CO2 sequestration
Nuclear waste deposition
Information about structure of earth
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SPSC - Geophysical Unsupervised Signal Processing
Model DescriptionUnknown
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Source Signals Mixing Matrix AObservedSignals
s1s1
s2s2
s3
s3
sm
x1
x2
xm
a11
a12
a1m
a21
a22
a2m
an1
an2
amm
. . .
. . .
. . .
Different Goals
Finding s1(n) – Reflectivity Function
Finding A – Wavelets
Separation of si(n)
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SPSC - Geophysical Unsupervised Signal Processing
Problems:
Seismic source and subsurface propagation not ideal
Output is mixture of different waves which must be identifiedand separated
Hypothesis:
Linear distortion and mixtures
Convolution relationship – source signal and propagationenvironment are not ideal
Mixing – linear combination
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SPSC - Geophysical Unsupervised Signal Processing
Summary
Geophysical applications include Estimation of different sources Estimation of reflectivity function
Problems due to the not ideal environment conditions
Prior assumption of model to be able to use known algorithms
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SPSC - Geophysical Unsupervised Signal Processing
Approaches for Unsupervised Processing [1]
Wiener Filter and Unsupervised Processing
Geophysical SP - prominent role to develop unsupervisedmethods
Wieners theory to seismology (Enders A. Robinson 1954 [2])
SOS HOS
Blind Deconvolution
Unsupervised Processing
Blind Source Separation
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SPSC - Geophysical Unsupervised Signal Processing
Question
Additional information to the measured data about inputsources and the convolution/mixing system
If one of them is known – supervised methods
Assumption
No additional information available
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SPSC - Geophysical Unsupervised Signal Processing
Source Signature – known
Deterministic supervised deconvolution in seismology
e.g. estimate reflectivity function through linear filtering,using the Wiener-Levinson minimization of the MSE
Source Signature – unknown
Unsupervised task
Exploiting prior knowledge about structure of source signatureand the reflectivity
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SPSC - Geophysical Unsupervised Signal Processing
Task
Get reflectivity function from measured data usingdeconvolution
Predictive Deconvolution
Wiener theory on prediction and filtering could be used forpredictive, unsupervised deconvolution [2]
Two Hypothesis
1. The seismic wavelet is the impulse response of an all-pole,minimum phase system.
2. The impulse response of the layered earth model behaves likea decorrelated (white) signal, so that it has a flat frequencyspectrum.
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SPSC - Geophysical Unsupervised Signal Processing
Predictive Deconvolution
Deconvolution using prediction-error filter uses only SOS –simplification of task BUT
Prediction-error filter acts as a whitening filter – ideallyrecovers uncorrelated signals
Problem if seismic wavelet cannot be modeled as an all-pole,minimum phase filter
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SPSC - Geophysical Unsupervised Signal Processing
Limitation of linear predictive deconvolution
Next step – from SOS to HOS
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SPSC - Geophysical Unsupervised Signal Processing
Blind Source Separation
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SPSC - Geophysical Unsupervised Signal Processing
Blind Source Separation
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Source Signals Mixing Matrix AObservedSignals
s1s1
s2s2
s3
s3
sm
x1
x2
xm
a11
a12
a1m
a21
a22
a2m
an1
an2
amm
. . .
. . .
. . .
x(n) = [x1(n) x2(n) ... xM (n)]T
s(n) = [s1(n) s2(n) ... sN (n)]T
x = A · s → s = W · x with W = A−1
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SPSC - Geophysical Unsupervised Signal Processing
Goal: Estimate all source signals
W can recover sources if mixing system is linear, memory-lessand time-invariant
SOS: W is adjusted so as to decorrelate the recovered signalss (using PCA)
HOS: W is adjusted so that recovered signals becomemutually independent
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SPSC - Geophysical Unsupervised Signal Processing
Blind Source Separation using SOS and HOS
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SPSC - Geophysical Unsupervised Signal Processing
ICA-Based seismic Deconvolution
ICA – Independent Component Analysis
Computational model for separating multiple sources
Produces additive subcomponents
Assumption:
- Mutual statistical independence- Source signals are non-Gaussian and i.i.d (– satisfied byreflectivity)
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SPSC - Geophysical Unsupervised Signal Processing
Modelx = As
x(0)x(1)
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.x(N − 1)
︸ ︷︷ ︸
Trace
= A
ρ(0)ρ(1)
.
.
.ρ(N − 1)
︸ ︷︷ ︸
Reflectivity
A =
h(0) 0 . . . 0 0 . . . 0h(1) h(0) . . . 0 0 . . . 0
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.... 0
h(Nh − 1)
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.... h(0) 0
... 0
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...
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0
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.... 0
0 0 . . . h(Nh − 1) h(Nh − 2) . . . h(0)
→ Single Snapshot → delayed versions of x(n) and ρ(n) – N
snapshots
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SPSC - Geophysical Unsupervised Signal Processing
Step 1 – Rearrangement
Obtaining of M < N mixtures
Improvement of statistical properties of the mixture matrix
X =
0 0 . . . 0 x(0) . . . x(N −M)...
.... . .
. . ....
. . ....
0 x(0) . . . . . . x(M − 2) . . . x(N − 2)x(0) x(1) . . . . . . x(M − 1) . . . x(N − 1)
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SPSC - Geophysical Unsupervised Signal Processing
Step 2 – Data Whitening
Decorrelation of the signals
Equalize power
z(n) = WSOSx(n)
WSOS obtained with SOS
Step 3 – Dimension Reduction
Linear transformation – new set of Nh mixtures
x(n) = NTkW
TSOSz(n)
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SPSC - Geophysical Unsupervised Signal Processing
Step 4 – ICA
s(n) = WHOSx(n)
WHOS = QWSOS with Q given by ICA
Step 5 – Wavelet and Reflectivity Estimation
si = hTi x(n)
Optimal Pair (s∗(n),h∗) gives original reflectivity and wavelet
hi from WHOS
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SPSC - Geophysical Unsupervised Signal Processing
Results I – Synthetic Data
(a) Noiseless synthetic trace; (b) Synthetic reflectivity function; (c)Linear prediction error filter; (d) Estimated reflectivity based onestimated wavelet
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SPSC - Geophysical Unsupervised Signal Processing
Results II
(a) 35-points Berlage wavelet; (b) estimated wavelet by B-ICA; (c)zero-pole plot for the original wavelet; (d) zero-pole plot forestimated wavelet
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SPSC - Geophysical Unsupervised Signal Processing
Summary
Differences between supervised and unsupervised processing
Predictive deconvolution to estimate reflectivity function
Limitations of predictive deconvolution – go to HOS
Blind source separation to estimate sources
Combination of deconvolution and BSS – ICA-Based seismicDeconvolution
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SPSC - Geophysical Unsupervised Signal Processing
Volcano Hazards
many villages close to volcanos
predict eruptions, tremors, earthquakes to avoid casualties
Problem:lack of prediction knowledge
- number of sources?- types of sources?
Solution:use blind source separationfor data-mining
Landscape after ashfall.
How to improve knowledge of volcanic behaviour?
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SPSC - Geophysical Unsupervised Signal Processing
DUET: Degenerate Unmixing Estimation Technique
Features
for blind separation of multiple sources
- Volcano Tectonic (VT) events- Long Period (LP) events
based on pair of seismograph sensors (stations)
second sensor enables use of a time-frequency–lattice (τ, ω)
stations separated by less than half a wavelength of interest
performs best if there’s no signal overlap between sources
robust behaviour in echoic mixtures
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SPSC - Geophysical Unsupervised Signal Processing
DUET: Degenerate Unmixing Estimation Technique
Sensor 2
Sensor 1
δ
Seismic sources & absorbing sensors.
Are there any assumptions?Fuchs Anna & Pessentheiner Hannes Advanced Signal Processing 2 page 29/50
SPSC - Geophysical Unsupervised Signal Processing
Assumptions
Anechoic Mixing Model
W-Disjoint Orthogonality
Sensor 2
Sensor 1
Anechoic Mixing Model.
Source 1
Source 2
t
f
1 Hz
5 Hz
W-Disjoint Orthogonality.
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SPSC - Geophysical Unsupervised Signal Processing
Assumptions: Anechoic Mixing Model
no realistic model
direct paths only (non-echoic signals)
stations provide attenuation & delay parameters of waves
Sensor 2
Sensor 1
Anechoic (Ideal) Mixing Model.
Sensor 2
Sensor 1
Echoic (Realistic) Mixing Model.
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SPSC - Geophysical Unsupervised Signal Processing
Assumptions: Anechoic Mixing Model cont’d
model
xk(t) :=
N∑
j=1
ak,j · sj(t− δk,j) k = 1, 2
where
xk(t) . . . mixture of k-th sensor
sj(t) . . . source signal
ak,j . . . attenuation coefficient
δk,j . . . time delay
k . . . sensor index
j . . . source index
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SPSC - Geophysical Unsupervised Signal Processing
Assumptions: W-Disjoint Orthogonality (WDO)
disjoint supports of signals’ windowed Fourier transforms (F)
every (τ, ω)-point in a mixture xj(t) dominated bycontribution of at most one source
Ω1 ∩ Ω2 = 0
Ω1
Ω2
Ω
Disjoint sets Ω1 & Ω2.
t
f
S1
! !"
S2
S1
S2 S1
S2
frame-size
WDO and non-WDO signals.
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SPSC - Geophysical Unsupervised Signal Processing
Assumptions: W-Disjoint Orthogonality (WDO) cont’d
given window function w(t) & source signals sj(t), sk(t)
given windowed F of sj(τ, ω), sk(τ, ω)
sj(τ, ω) :=1√2π
∫ +∞
−∞
w(t− τ)sj(t)e−iwtdt
where ω is the angular frequency and τ is the window shift
WDO if supports of sj(τ, ω), sk(τ, ω) are disjoint
sj(τ, ω) · sk(τ, ω) = 0 ∀τ, ω, ∀j 6= k
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SPSC - Geophysical Unsupervised Signal Processing
Assumptions: W-Disjoint Orthogonality (WDO) cont’d
if w(t) = 1- simply F of whole mixture- signals have to be frequency-disjoint to satisfy WDO-condition(frequency-division multiplexed signals)
if w(t) = δ(t)- signals have to be time-disjoint to satisfy WDO-condition(time-division multiplexed signals)
t
f
S2
S1 S1 S1
S2 S2
Frequency-Division Multiplexed Signals.
t
f
S2S1 S1
Time-Division Multiplexed Signals.
How does the algorithm work?
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SPSC - Geophysical Unsupervised Signal Processing
Signal Representation
mixture signals can be decomposed into weighted & shiftedsource signals in time & frequency domain
xk(t) :=
N∑
j=1
ak,j · sj(t− δk,j)
xk(τ, ω) :=
N∑
j=1
ak,j · sj(τ, ω) · e−iωδk,j
δ1
δ2
Sources, sensors, and delays.
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SPSC - Geophysical Unsupervised Signal Processing
Signal Representation cont’d
without loss of generality
a1,j = 1 δ1,j = 0 a2,j = aj δ2,j = δj
interested in the differences of both captured signals
vectorized mixture signals (k = 1, 2)
[x1(τ, ω)x2(τ, ω)
]
︸ ︷︷ ︸mixture signals
=
[1 . . . 1
a1e−iωδ1 . . . aNe−iωδN
]
︸ ︷︷ ︸relative propagation matrix
s1(τ, ω). . .
sN (τ, ω)
︸ ︷︷ ︸source signals
How to decompose the mixture signals?
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SPSC - Geophysical Unsupervised Signal Processing
Binary Mask
given mixture signal xk(t)
given F of xk(τ, ω)
xk(τ, ω) :=1√2π
∫ +∞
−∞
N∑
j=1
ak,j · sj(t− δk,j)e−iwtdt
xk(τ, ω) := a1 · s1(τ, ω) · e−iωδ1
+ a2 · s2(τ, ω) · e−iωδ2
+ . . .
+ aN · sN (τ, ω) · e−iωδN
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SPSC - Geophysical Unsupervised Signal Processing
Binary Mask cont’d
source separation in case of WDO using an ideal binary mask
sj(τ, ω) := Mj(τ, ω) · xk(τ, ω), ∀τ, ω
where
Mj(τ, ω) :=
1 sj(τ, ω) 6= 0
0 otherwise
and
Mj(τ, ω) ·Mk(τ, ω) = 0 ∀τ, ω ∀j 6= k
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SPSC - Geophysical Unsupervised Signal Processing
Binary Mask cont’d
400
600
800
1,000
1,200
1,400
1,600
1,800
2,000
2,200
Samples
Frequency (Hz) Binary Mask Source 2
1
2
3
Binary mask of a mixture signal for source 2.
How do we obtain coefficients aj & δj in case of WDO-signals?
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SPSC - Geophysical Unsupervised Signal Processing
Parameter Estimation
WDO- & Anechoic Source Signals
ratio of (τ, ω)-representations of mixtures depends on activesource component mixing parameters only
R(τ, ω) :=x2(τ, ω)
x1(τ, ω)
assume set of (τ, ω) on Ωj := (τ, ω) : Mj(τ, ω) = 1
R(τ, ω) := aj · e−iωδj
aj := |R(τ, ω)| δj := − 1
ωargR(τ, ω)
Do we have ideal conditions in real world?
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SPSC - Geophysical Unsupervised Signal Processing
Parameter Estimation cont’d
Approximated WDO- & Echoic Source Signals
in field scenarios perfect WDO never given
high degree of WDO approximation per frame if
- small overlap of source spectra in frequency- small overlap of source spectra in time
increase degree of WDO approximation by
- changing sensor position- changing window type/size (ao)
In case of non-WDO, why is the DUET still suitable?
Volcano Tectonic and Long Period events ...
- hardly occur simultaneously and, thus,- exhibit high degree of WDO approximation
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SPSC - Geophysical Unsupervised Signal Processing
Parameter Estimation cont’d
accurate estimation of true coefficients required
α = aj −1
aj
αj =
∑(τ,ω)∈Ωj
|x1(τ, ω) · x2(τ, ω)|2 · α∑(τ,ω)∈Ωj
|x1(τ, ω) · x2(τ, ω)|2
δj =
∑(τ,ω)∈Ωj
|x1(τ, ω) · x2(τ, ω)|2 · δ∑(τ,ω)∈Ωj
|x1(τ, ω) · x2(τ, ω)|2
How do we obtain mask Mj to separate source signals?
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SPSC - Geophysical Unsupervised Signal Processing
Parameter Estimation & Demixing cont’d
1. construct (τ, ω)-representations x1(τ, ω) & x2(τ, ω)
2. label each non-zero (τ, ω)-point with computed pair
(α , δ)
3. weight each pair with energy of corresponding (τ, ω)-point
4. generate histogram of these labels
5. find peaks in histogram for each source
6. construct masks Mj to determine Ωj for each source
7. apply masks to mixture
sj(τ, ω) = Mj(τ, ω) · x1(τ, ω)
8. convert each sj(τ, ω) back into time domain
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SPSC - Geophysical Unsupervised Signal Processing
Results (with both parameters α and δ)
(α, δ)-lattice for one (top left) and six (top right) sources. Original, mixed, andseparated signals (bottom).
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SPSC - Geophysical Unsupervised Signal Processing
Results (with single parameter δ)
0 1,000 2,000 3,000 4,000 5,000 6,000−0.02
−0.01
0
0.01
0.02
Samples
0 1,000 2,000 3,000 4,000 5,000 6,000−0.02
−0.01
0
0.01
0.02
Samples
(a)
(b)
0 1,000 2,000 3,000 4,000 5,000 6,000−0.01
−0.005
0
0.005
0.01
Samples
0 1,000 2,000 3,000 4,000 5,000 6,000−0.01
−0.005
0
0.005
0.01
Samples(a)
Mixtures of Long Period (LP) and Volcano Tectonic (VT) events at bothsensors (top/bottom left). Band-pass filtered mixtures (top/bottom right).
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SPSC - Geophysical Unsupervised Signal Processing
Results (with single parameter δ) cont’d
0 1,000 2,000 3,000 4,000 5,000 6,000−0.01
−0.005
0
0.005
0.01
Samples
0 1,000 2,000 3,000 4,000 5,000 6,000−0.02
−0.01
0
0.01
0.02
Samples
−30 −20 −10 0 10 20 300
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
Delay in Samples
Energy
SamplesFrequency (Hz)
0 1,000 2,000 3,000 4,000 5,000 6,000
2
4
6
8
Samples
Frequency (Hz)
0 1,000 2,000 3,000 4,000 5,000 6,000
2
4
6
8
Amplitude (dB)
−5
−4
−3
−2
−1
Amplitude (dB)
−4
−3
−2
−1
(b)
Band-pass filtered mixtures (top left) and corresponding delay histogram of VTand LP events (top right). Separated signals of VT (bottom top) and LP
(bottom bottom).
Fuchs Anna & Pessentheiner Hannes Advanced Signal Processing 2 page 47/50
SPSC - Geophysical Unsupervised Signal Processing
Conclusion
DUET applied to geophysical data
for blind source separation of VT and LP events
results show that DUET ...
- is a powerful tool- for identifying, separating, and locating VT and LP events- to better understand the source mechanism behind tremors,earthquakes, and eruptions caused by volcanic activities
Fuchs Anna & Pessentheiner Hannes Advanced Signal Processing 2 page 48/50
SPSC - Geophysical Unsupervised Signal Processing
References
Andre K. Takahata and Everton Z. Nadalin and Rafael Ferrari and Leonardo Tomazeli Duarte and Ricardo
Suyama and Renato R. Lopes and Joao Marcos Travassos Romano and Martin Tygel,“Unsupervised Processing of Geophysical Signals: A Review of Some Key Aspects of Blind Deconvolutionand Blind Source Separation,”IEEE Signal Process. Mag., vol. 29, pp. 27–35, 2012.
E. A. Robinson,
“Predictive decomposition of time series with applications to seismic exploration”,Ph.D. dissertation,Dept. Geol. Geophys., MIT, Cambridge, 1954.
Yilmaz, O., Rickard, S.,
“Blind Separation of Speech Mixtures via Time-Frequency Masking”,IEEE Transactions on Signal Processing, Vol. 52, No. 7, July, 2004.
Moni, A., Bean, C. J., Lokmer, I., Rickard, S.,
“Source Separation on Seismic Data: Application in a geophysical setting”,IEEE Signal Processing Magazine, May, 2012.
Moni, A., Bean, C. J., Lokmer, I., Rickard, S.,
“Seismic Signal Source Separation”,ISSC 2011, Trinity College Dublin, June 2011.
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