using&a&kalmanﬁlter toregularizethevlbi · 2018. 8. 21. · j. gispon – evga 2013 helsinki,...

J. Gispon – EVGA 2013 Helsinki, Finland

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.



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.



What is Nuta7on, anyway? (nuta7o)



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.



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.)



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



Kalman filtering -‐ Upda7ng

•  Predic2on is linear:

The A embodies the physics. noiseAAPP

AXXT

KKKK

KKKK

+=

=

−−−

−−−

1|11|

1|11|



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



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 →∞



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.



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.



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.



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



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



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




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


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




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.

using&a&kalmanﬁlter toregularizethevlbi · 2018. 8. 21. · j. gispon – evga 2013 helsinki,...

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