using&a&kalmanfilter toregularizethevlbi · 2018. 8. 21. · j. gispon – evga 2013 helsinki,...
TRANSCRIPT
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J. Gispon – EVGA 2013 Helsinki, Finland
Using a Kalman filter to regularize the VLBI nuta7on 7me series
Using a Kalman filter to regularize the VLBI nuta7on 7me series
J. Gipson, K. Le Bail, S. Bolo2n
NVI, Inc. At GSFC/NASA, Greenbelt, U.-‐S.
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J. Gispon – EVGA 2013 Helsinki, Finland
Using a Kalman filter to regularize the VLBI nuta7on 7me series
Contents • Goal: State-‐of-‐the-‐art: VLBI nuta2on 2me series. – Importance of the VLBI nuta2on 2me series; – Why regularizing them to daily 2me series?
• A Kalman filter to regularize the VLBI nuta2on 2me series: nutkal2012.f . – How to use it? Control file and op2ons.
• Applica2ons: – Op2mal choice of appropriate parameters; – Es2mator of the quality of the es2mate: the Goodness Of Fit.
• Conclusions and perspec2ves: – nutkal2012.f for predic2on.
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J. Gispon – EVGA 2013 Helsinki, Finland
Using a Kalman filter to regularize the VLBI nuta7on 7me series
What is Nuta7on, anyway? (nuta7o)
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J. Gispon – EVGA 2013 Helsinki, Finland
Using a Kalman filter to regularize the VLBI nuta7on 7me series
Why Regularize? • Nuta2on es2mated by VLBI at irregular intervals. • Time-‐tag depends on epoch of the experiment • For many purposes nice to have regularly spaced data.
• We (IVS) are closest to the data. We should be able to produce the best results.
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J. Gispon – EVGA 2013 Helsinki, Finland
Using a Kalman filter to regularize the VLBI nuta7on 7me series
Our Approach 25 years ago we (GSFC) faced a similar problem for UT1 and PM. Needed a good a-‐priori ERP model. Developed eopkal (EOP Kalman filter) s2ll in use. 10 years later developed nutkal. This report is about nutkal2012. (An updated version of nutkal.)
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J. Gispon – EVGA 2013 Helsinki, Finland
Using a Kalman filter to regularize the VLBI nuta7on 7me series
Kalman filtering -‐ No7ons (1)
XK-‐1|K-‐1 PK-‐1|K-‐1
Prior knowledge
XK|K-‐1 PK|K-‐1
XK|K PK|K
Predic2on
Add measurement Z PZ
New predic2on
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J. Gispon – EVGA 2013 Helsinki, Finland
Using a Kalman filter to regularize the VLBI nuta7on 7me series
Kalman filtering -‐ Upda7ng
• Predic2on is linear:
The A embodies the physics. noiseAAPP
AXXT
KKKK
KKKK
+=
=
−−−
−−−
1|11|
1|11|
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J. Gispon – EVGA 2013 Helsinki, Finland
Using a Kalman filter to regularize the VLBI nuta7on 7me series
Kalman filtering – Upda7ng Example • Example: Linear mo7on. – State:
– Update
€
XK −1VK −1
⎛
⎝ ⎜
⎞
⎠ ⎟
⎟⎟⎠
⎞⎜⎜⎝
⎛ Δ+→⎟⎟⎠
⎞⎜⎜⎝
⎛
−
−−
−
−
1
11
1
1
K
KK
K
K
VVtX
VX €
A =1 Δt0 1⎛
⎝ ⎜
⎞
⎠ ⎟
€
P =σX2 CXV
CX σV2
⎛
⎝ ⎜
⎞
⎠ ⎟
⎟⎟⎠
⎞⎜⎜⎝
⎛
Δ+
Δ+Δ+Δ+=→ 2
222 2
VXV
XVVXVX
tVCtVCttC
PPσ
σσ
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J. Gispon – EVGA 2013 Helsinki, Finland
Using a Kalman filter to regularize the VLBI nuta7on 7me series
Kalman filtering – Adding measurement • Add measurement:
Schema2cally
Note that : we are just combining according to sigmas.
€
XK |K =PK |K −1−1 XK |K −1 + PZ
−1ZPK |K −1−1 + PZ
−1
€
P −1 ≈1σ2
€
XK |K = XK |K−1 − XK |K−1 +PK |K−1−1 XK |K−1 +PZ
−1ZPK |K−1−1 +PZ
−1
⎛
⎝ ⎜
⎞
⎠ ⎟
• Let: . So:
€
XK |K−1 =(PK |K−1
−1 +PZ−1)XK |K −1
PK |K−1−1 +PZ
−1
€
XK |K = XK |K −1 +PZ−1(Z − XK |K −1)PK |K −1−1 + PZ
−1
Predicted
Measurement
Predic2on Correc2on • Note: As , give more importance to data. As , give more importance to predic2on.
€
PK |K −1 →∞ PK |K −1−1 →0
€
PK |K −1 →0 PK |K −1−1 →∞
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J. Gispon – EVGA 2013 Helsinki, Finland
Using a Kalman filter to regularize the VLBI nuta7on 7me series
Kalman filtering -‐ No7ons (5) • Our model for nuta2on is: Integrated Random walk + N harmonic terms. Each term has associated with it some noise. The harmonics also have associated a width or Q factor.
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J. Gispon – EVGA 2013 Helsinki, Finland
Using a Kalman filter to regularize the VLBI nuta7on 7me series
A Kalman filter to regularize VLBI nuta7on 7me series: nutkal2012.f
• The program reads in a series of nuta2on values from the snoop nuta2on files…
• Control file.
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J. Gispon – EVGA 2013 Helsinki, Finland
Using a Kalman filter to regularize the VLBI nuta7on 7me series
Op7mal choice for the Kalman filter parameters (1)
• Snoop file: GSC 2011a solu2on. • Which periodic signals? – From the FCN (Lambert XX) = -‐430.21 days; – From our study in 2012 (IVS GM) R1 and R4 sessions weekly series = 450, 510 and 385 days;
– From PSD = 470.25 days.
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J. Gispon – EVGA 2013 Helsinki, Finland
Using a Kalman filter to regularize the VLBI nuta7on 7me series
Op7mal choice for the Kalman filter parameters (2)
Linear Harmonic Goodness Of Fit
Yes/No Process noise
Period days Width days Process noise
Psi Epsilon
Test 1 No -‐ 430 0 0.001 4.140 2.177
Test 2 No -‐ 470.25 0 0.001 4.140 2.177
Test 3 No -‐ 470.25 0 0.01 1.968 1.067
Test 4 No -‐ 450 510 385
0 0 0
0.01 0.01 0.01
1.387 0.744
Test 5 Yes 0.01 450 510 385
0 0 0
0.01 0.01 0.01
1.282 0.663
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J. Gispon – EVGA 2013 Helsinki, Finland
Using a Kalman filter to regularize the VLBI nuta7on 7me series
Op7mal choice for the Kalman filter parameters (3) .prn files: observa2ons and es2mated
values at the date of the observa2ons.
5 5.1 5.2 5.3 5.4 5.5x 104
−4
−2
0
2
4
Test
4Lo
ngitu
de a
ngle
in m
as
Observationsnutkal2012 estimated values
5 5.1 5.2 5.3 5.4 5.5x 104
−2
−1
0
1
2
Test
4O
bliq
uity
ang
le in
mas
Observationsnutkal2012 estimated values
5 5.1 5.2 5.3 5.4 5.5x 104
−4
−2
0
2
4
Test
4 −
Diff
eren
ces
Long
itude
ang
le in
mas
5 5.1 5.2 5.3 5.4 5.5x 104
−2
−1
0
1
2Te
st 4
− D
iffer
ence
sO
bliq
uity
ang
le in
mas
Observations − nutkal2012 estimated values
Observations − nutkal2012 estimated values
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J. Gispon – EVGA 2013 Helsinki, Finland
Using a Kalman filter to regularize the VLBI nuta7on 7me series
Op7mal choice for the Kalman filter parameters (4) Snoop file and nutkal2012 daily series.
5 5.1 5.2 5.3 5.4 5.5x 104
−4
−2
0
2
4
Test
4Lo
ngitu
de a
ngle
in m
as
5 5.1 5.2 5.3 5.4 5.5x 104
−2
−1
0
1
2
Test
4O
bliq
uity
ang
le in
mas
nutkal2012 daily seriessnoop file
nutkal2012 daily seriessnoop file
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J. Gispon – EVGA 2013 Helsinki, Finland
Using a Kalman filter to regularize the VLBI nuta7on 7me series
Problem: Filter runs away Nutkal2012.f for predic2on
5.56 5.561 5.562 5.563 5.564 5.565x 104
−20
0
20
40
60
80
100
Test
4 −
Pre
dict
ion
1 m
onth
Long
itude
ang
le in
mas
nutkal2012 daily seriessnoop file
5.56 5.561 5.562 5.563 5.564 5.565x 104
−50
−40
−30
−20
−10
0
10
Test
4 −
Pre
dict
ion
1 m
onth
Obl
iqui
ty a
ngle
in m
as
nutkal2012 daily seriessnoop file
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J. Gispon – EVGA 2013 Helsinki, Finland
Using a Kalman filter to regularize the VLBI nuta7on 7me series
Conclusions and perspec7ves Goddard is developing a nuta2on Kalman filter. Goal is to use the Kalman filter to regularize the nuta2on series.
This series can be used: • As an a priori model for VLBI data analysis. • For geophysical inves2ga2ons. S2ll a work in progress.