Post on 05-Mar-2021
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
2
Outline
• Why spectral inversion? • How it works • PCA spectral model • Angular response
3
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
4
Spectral Inversion: How It Works
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Problem Formulation (1)
2
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
∞
∞
=
=
∫ ∫ ∫
∫ ∫ ∫
902
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
6
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
101
102
103
10-1
100
101
102
103
104
105
106
107
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
101
102
103
10-1
100
101
102
103
104
105
106
107
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
9
Principal Components
• PCs well-behaved up through #5 (except near 1000 MeV)
• PC#4 and higher contribute very little to variance
101
102
103
-1
0
1
2
3
4
5
6
7
Energy (MeV)
Mea
n Lo
g Fl
ux
101
102
103
-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
100
101
102
103
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)
100
101
102
103
104
105
100
101
102
103
104
105
Actual Counts
Exp
ecte
d C
ount
s
Analytical Inversion
D05D06T05T06T08T09
100
101
102
103
104
105
100
101
102
103
104
105
Actual Counts
Exp
ecte
d C
ount
s
PCA Inversion
D05D06T05T06T08T09
11
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
101
102
100
101
102
103
Energy (MeV)
Inve
rted
Isot
ropi
c Fl
ux (#
/cm2 -
s-sr
-MeV
30-Jan-2005 04:00:40
PCA Analytical
12
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
13
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
14
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
15
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
16
Questions?