Air Force Research Laboratory - VDL...Oct 10, 2012 · Why Spectral Inversion? • Science-grade...
Transcript of Air Force Research Laboratory - VDL...Oct 10, 2012 · Why Spectral Inversion? • Science-grade...
Air Force Research Laboratory
Integrity Service Excellence
Stuart Huston Boston College
For the Air Force Research Laboratory Space Vehicles Directorate
Kirtland Air Force Base, N.M.
10 October 2012
Spectral Inversion
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Outline
• Why spectral inversion? • How it works • PCA spectral model • Angular response
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Why Spectral Inversion?
• Science-grade instruments can measure directional, differential flux with good spectral and angular resolution
• Most detectors are not science-grade • wide field of view • small numbers of integral energy channels
• Spectral inversion allows us to determine energy spectra from integral-type detectors
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Spectral Inversion: How It Works
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Problem Formulation (1)
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0 0 02
900 0 0
sin ( ; ) ( ; , )
sin ( ; ) ( ; , )
i i
i
C dE d d A E j E
j dE d d A E F E
π π
π π
ϕ θ θ θ θ ϕ
ϕ θ θ θ θ ϕ
∞
∞
=
=
∫ ∫ ∫
∫ ∫ ∫
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the locally mirroring directional, differential particle flux
(e.g., in particles/cm -s-sr-MeV)( ; , ) angle- and energy-dependent particle angular distribution function
( ; ) the angle- ai
j
F EA Eθ ϕ
θ
=
==
th 2
nd energy-dependent effective area
for the i channel of the detector (e.g., in cm ), polar and azimuthal look directions in coordinates, polar and azimuthal look directions in
detectormagn
θ φ
θ ϕ
=
= coordinatesetic
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Problem Formulation (2)
• Recast as:
0( ) ( )y t G E f E dE bλ δ
∞≈ = +∫
a vector of observed counts a vector of expected counts integration time a vector of geometric factors (response functions)
( ) differential flux at energy a vector of background
y
tG
f E Eb
λδ
======
counts
• Solved by parameterizing and determining maximum likelihood value of • analytical (e.g., power-law, Maxwellian, …) • discrete (e.g., PCA)
( ) ( ; )f E f E q=
q
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Proton Analytical Spectral Inversion
CRRESPRO-A
Power law is a reasonable approximation between 10 – 100 MeV
Fit to exponential for E > MeV with fixed
e-folding rate determined from Selesnick, et al. model
( , , ) ( , ) ,nj E b Eθ ϕ θ ϕ −=Power law fit
CRRESPRO-Q
Power law like
Exponential like
From Selesnick, et al., Space Weather, 5, s04003, doi:10.1029/2006SW00275, 2007.
Selesnick model
( )
1 2
0
0
1 2
0
exp( ln ) ; ( )
exp ;
100 MeV345 MeVexp( ln )
exp
break
break break
break
breakbreak
break
q q E E Ej E Ej E E
E
EE
q q EjE E
− ≤ = − >
=
=
−=
−
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Selesnick PCA Model
• Selesnick model has fluxes at fixed values of M, K, L*
• Fluxes were interpolated to a uniform E grid, then gaps in K and L* were filled in
• Although energies extended as low as ~ 1 MeV, 10 MeV was used as a lower limit for PCA • Below this, not all K/L*
values are filled in, resulting in a bias towards higher fluxes
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10-1
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Energy (MeV)
flux
(#/c
m2 -
s-sr
-MeV
)
Selesnick Model, K=0
L=1.10L=1.15L=1.20L=1.25L=1.30L=1.35L=1.40L=1.50L=1.60L=1.70L=1.80L=1.90L=2.00L=2.20L=2.40
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10-1
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Energy (MeV)
flux
(#/c
m2 -
s-sr
-MeV
)
Selesnick Model, K=0.3
L=1.10L=1.15L=1.20L=1.25L=1.30L=1.35L=1.40L=1.50L=1.60L=1.70L=1.80L=1.90L=2.00L=2.20L=2.40
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Principal Components
• PCs well-behaved up through #5 (except near 1000 MeV)
• PC#4 and higher contribute very little to variance
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-1
0
1
2
3
4
5
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Energy (MeV)
Mea
n Lo
g Fl
ux
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-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
Energy (MeV)
PC#1PC#2PC#3PC#4PC#5
1 2 3 4 5 6 7 8 9 1010
-4
10-3
10-2
10-1
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102
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Principal Component
Var
ianc
e
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Inversion Results: Actual Counts vs. Expected Counts
• Comparisons with analytical inversions used 3 PCs • PCA inversion results in similar reconstruction of expected
counts (PCA may be a little better, at least at high count rates)
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Actual Counts
Exp
ecte
d C
ount
s
Analytical Inversion
D05D06T05T06T08T09
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Actual Counts
Exp
ecte
d C
ount
s
PCA Inversion
D05D06T05T06T08T09
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Flux Spectra – Analytical vs. PCA
• “Typical” spectra from analytical and PCA inversions
• PCA spectral shape is generally very close to analytical, except near Ebreak (in this example the reverse is true)
• Error bars for PCA are not always this bad
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Energy (MeV)
Inve
rted
Isot
ropi
c Fl
ux (#
/cm2 -
s-sr
-MeV
30-Jan-2005 04:00:40
PCA Analytical
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Angular Correction
• Much of our data come from wide-angle or omnidirectional detectors, which sample a fraction of the local omnidirectional flux
• Need a method to estimate j90 from this “semi-omnidirectional” flux • Particle angular distribution • Angular response of detector
• V1.0 used a correction after performing spectral inversion
• For V1.x, we plan to use combined energy/angular inversion as appropriate
“Typical” Pitch Angle Distribution
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Proton Angular Distribution Function
• Pitch angle distribution based on CRRESPRO model
( )( )0
0
0
;1
0;sin (equatorial pitch angle)
sin (equatorial loss cone angle), , , are fitting parameters
a bLC
LCaLC
LC
LC LC
LC
y y yj y y
j yy y
yya b j y
αα
−>
= −
≤=
=
• Azimuthal variation based on Lenchek-Singer, function of • Atmospheric scale height • Gyroradius • Magnetic Inclination
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Requirements
• Instrument response functions as function of energy (and angle) • at least threshold energy & geometric factor
• Prior knowledge of spectral shapes • PCA can provide a rational basis • For angular inversion, also need PAD
• Remember inversion is only valid for range of instrument response
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Implementation
• Spectral inversion routines have been implemented in invlib, a C- and MATLAB-callable library (code & documentation available on SourceForge) • many options for analytical spectral shapes, as well as
PCA • outputs include energy spectra, error bars, expected
counts • Used for TSX5/CEASE, HEO/dos, ICO/dos
• protons used Selesnick PCA model • electrons used PCA model based on CRRES MEA/HEEF
• New PCA models have been developed based on AP9 and AE9
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Questions?