Realisation of celestial reference frames using the Allan variance classification. César Gattano1 ..... son of representatives for each group of celestial frame solu-.
Realisation of celestial reference frames using the Allan variance classification César Gattano1,2,* & Patrick Charlot1 : Laboratoire d’astrophysique de Bordeaux, Univ. Bordeaux, CNRS, B18N, allée Geoffroy Saint-Hilaire, 33615 Pessac, France. : funded by the Centre National d’Études Spatiales
abstract During the last IVS General meeting (and EVGA meeting), I presented a new classification of VLBI radio-sources on the basis of their astrometric stability revealed by Allan variance diagrams. Sources
are classified in three groups regarding the nature of noise contents of its astrometric time series. The global level of noise orders sources within each group. I present several strategy on the basis of this classification to realize celestial reference frames, i.e. on selecting the set of defining sources used to carry the fundamental axes of the frame. This set of sources is usually constrained in the data reduction by a no-net rotation constraint. Using two tools developed to determine the stability of realized frame, one that analyses the stability of the annual realisations of a given frame, another that analyses the coherence of random sub-frames, I determine the best usage of the classification. Performance given is then to be compared with the imminent release of the 3rd version of the ICRF.
Stability analysis of VLBI sources using Allan standard deviation
Flicker 1 The use of Allan standard deviation [Allan, 1966] enables to noise study the stability of a measurement series, such as the position monitoring of VLBI radio-sources. It discriminates 0.1 White Random between several types of noise contained in the data. Noise noise walk types are separated in two categories : (1) Noise types leading to a stable behaviour of the 0.01 data (grey background in Fig.1) : as the time scale increases, τ the estimated standard deviation decreases ; (2) Noise types leading to an unstable behaviour of the data (red background in Fig.1) : as the time scale Noise Slope Exponent p in Noise increases, the estimated standard deviation increases as well. type in log-log scale Sy (f ) ∝ f p color Additionally, we may evaluate the level of noise in the Random walk 0.5 -2 Red data. Taking into account the straight line that maximizes Flicker noise 0 -1 Pink the Allan standard deviation graph enables to quantify globally the level of noise in a way that we can White noise -0.5 0 White compare the VLBI sources. Fig.2 - Correspondence between the type of noise, associated
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We split the sources into three categories following the sequence of the dominating noise at each time scale, i.e. with respect to the behaviour of the data (see the frame 1) :
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AV0 : most stable astrometric behavior. AV0 sources are not dominated by unstable noise at any time scale ;
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with a color given by the exponent of the power law-type spectral density function, and the drift observed in the Allan standard deviation as a function of the data averaging time scale represented in logarithmic scale.
AV1 : intermediate astrometric stability. AV1 sources are dominated by unstable noise at some time scales but stable noise dominates on longest time scales appreciable regarded the observational history of the source ; Evalution of the source distribution within the classification with respect to the chosen rehabilitation threshold (more details on the text on right).
AV2 : least stable behavior. All sources whose longest time scales are dominated by an unstable noise.
complete study in Gattano et al. [2018] ; see associated web page http://www.cgattano-astrogeoresearch.fr/
The global level of noise order sources within each category.
Fig.3 -
Rehabilitation process : The developed statistical validation test aims to determine the probability that the detected slope (see section 1) results from a white noise process even if it is not -0.5 (due to the irregularity of the sampling or to non-uniform data standard errors). It is based on Monte-Carlo simulations of 1000 white noise processes distributed on the original sampling of the tested time series. The scatter of their corresponding Allan standard deviations provides an empirical error (see Fig.1).
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⇒ Each of the unstable sources (AV1/2) have therefore a certain probability to be in fact a stable source (AV0) offering the possibility to rehabilitate some sources whose probability is greater than a threshold chosen by a user. Fig. 3 gives the evolution of the classification as the threshold varies.
1 2 4 8 τ : time scale [years]
Fig.1 -
(Left) Astrometric offset with respect to the mean position computed for each VLBI session. (Right) Allan standard deviation of the regularized time series over time scale τ . The log-log diagram is plotted in black solid line with its uncertainties at 90% in black dashed line. Colored background indicates the behaviour of the noise at each time scale (more details in the text above). The blue straight line is the lowest one that maximizes the diagram down to τ = 1 yr. It leads to the global noise level of the source (more details in the text above). The grey solid lines are the dispersion of the Monte-Carlo test (more details on the section 2). How much the black diagram remains within the dispersion lines leads to the written probabilities. They are compared with the threshold of the rehabilitation process.
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Only a subset of Nsubset sources among the NDS defining sources is observed during a VLBI session and the frame orientation within the session is determined to a certain extent by this subframe.
The Allan standard deviation analysis reveals that only few sources have a stable behaviour. All the others show a perceptible variability affecting their astrometric position. Consequently : how far can we state the non-rotation of the frame ?
How much the subframe represents statistically the same orientation as for the complete frame ?
The difficulty is that we do not have the capability to measure the real stability of the frame because we cannot observe frequently the whole subset of defining sources.
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Fig.4 -
Comparison of the partial annual stability between representatives of each solution group (see section 3). The comparison criterion is set on the maximum between the 3 maximum differences within each rotation angle annual values Ai (t) (i = x for rotation around (Ox) axis, i = y around (Oy ) and i = z around (Oz)). The maximum is computed on a time interval delimited by a lower limit and the study date. The value of this lower limit is given in abscissa and the related comparison criterion in ordinate in µas.
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Results and discussion
No. N subset
Strategies to select defining sources
Prehab σAV 0 σAV 1 σAV 2 prior [%] [mas] [mas] [mas] GROUP 1 : Only AV0 sources 1 100 50 10 0 0 0 2 100 25 10 0 0 0 3 200 25 10 0 0 0 4 100 0 10 0 0 0 5 200 0 10 0 0 0 6 300 0 10 0 0 0 7 400 0 10 0 0 0 8 500 0 10 0 0 0 GROUP 2 : First AV0, then AV1, no AV2 9 200 50 10 10 0 0 10 300 50 10 10 0 0 11 300 25 10 10 0 0 GROUP 3 : smallest level of noise 12 100 10 10 10 1 13 200 10 10 10 1 14 300 10 10 10 1 15 400 10 10 10 1 16 500 10 10 10 1
GROUP 4 : smallest level 27 100 50 10 28 200 50 10 29 300 50 10 30 100 25 10 31 200 25 10 32 300 25 10 33 100 0 10 34 200 0 10 35 300 0 10 36 400 0 10 37 500 0 10 Tab.1 -
of noise but 10 0 10 0 10 0 10 0 10 0 10 0 10 0 10 0 10 0 10 0 10 0
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the chosen rehabilitation threshold (see section 2) ; σAV 0 , σAV 1 , σAV 2 0.1 0
exclude all sources) ; prior = 0 gives the priority on the source class (data behaviour) followed by the noise level. prior = 1 is reversed. In light blue, we emphasize the representative of each group, selected for their most stable quality within the group and with a number N not too large so the representatives are quite different with each other.
Selection process : First, we rehabilitate AV1/2 sources into AV0 category regarding the chosen Prehab threshold. Then, we exclude sources in each category for which their level of noise is greater than chosen σAVi . Then, if prior = 0, we select the N first sources in remaining AV0 sources. If AV0 is used up, we complete with remaining AV1 sources and then remaining AV2 sources. If prior = 1, we first gather all remaining sources and order them with respect to their level of noise. Then, we select the first N ones.
Based on the two statistical tools ( see sections 4 and 5), we note that the “sol 3”, representative of the solution group “Only AV0 sources” is the least stable celestial frame. Note that AV0 sources, although having a stable behaviour, show in general a higher level of noise. On the opposite, “sol 14”, as the representative of the solution group where only the level of noise of the sources is taken into account to select defining sources, is the most stable celestial frame. This means that, for the realisation of a celestial frame at a given time, the dominant information is the noise level of sources and the data behaviour is not important. The hierarchy between representative solutions goes in the same sense : first, “sol 11” better than “sol 3” shows the advantage of taking AV1 sources into account, and “sol 14” better than “sol 35” shows the disadvantage of excluding AV2 sources. Additionally, “sol 5” better than “sol 3”, “sol 11” better than “sol 10” and “sol 35” better of its group show the advantage to lower the rehabilitation threshold Prehab to its minimum value, 0%. By doing so, we largely increase the population of AV0 sources with the consequence of reducing the average noise level of the pool in which we select defining sources. In the spirit of updating the official celestial reference frame, the ICRF, it is not important to focus on the data behaviour. Nevertheless, it is probably when it is question of the evolution of a realized celestial frame that the data behaviour could have its importance. “How can the Allan variance classification help to anticipate the degradation of a realized celestial frame stability with time” will be the subject of future investigations.
sol 35
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N is the number of selected defining sources ; Prehab is
are upper limits for the noise level in each category (10mas enables to
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We randomly draw a thousand times Nsubset sources among the total set of NDS defining sources of a celestial frame solution (see section 3). Hence, we get a thousand random subframes of identical size. We compute their statistical relative differences of orientation and retrieve the standard deviation on each orientation parameter Ax , Ay and Az . We repeat the process with different values of ratio Nsubset/NDS and draw graphs such as those in Fig. 4 where is showed the comparison of representatives for each group of celestial frame solutions (see section 3). The most stable one is that with the lowest dispersion between random subframes.
MAX[ σA ] [µas]
Through our observational history of VLBI radio-sources, we access partially the total stability. We quantify this part on an annual time scale by computing annual versions of celestial frame solutions (see section 3) by means of the annual averaged position of the observed sources. Then, we compute the relative differences between annual frames on three parameters Ax (t), Ay (t) and Az (t) for each rotation around the frame axis. Finally, the stability is quantified by computing the maximum difference on each parameter values on a time-interval down-limited by a given epoch. In Fig. 4, we show the largest of three depending on the chosen lower time limit. For lisibility, we only plot representatives of each group of solution. sol 6
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Stability by sub-frame orientation dispersion
Partial annual stability
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VLBI sources classification on the basis of Allan standard deviation
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Fig.5 -
Comparison of the subframe orientation dispersion between representatives of each solution group (see section 3). The comparison criterion is set on the maximum between the 3 standard deviations σ Ai (i = x for rotation around (Ox) axis, i = y around (Oy ) and i = z around (Oz)) of the thousand random subframes of size Nsubset ∈ [0 : NDS ], NDS being the total number of defining sources of the complete frame.
references
• D. W. Allan. (1966) Statistics of atomic frequency standards. IEEE Proceedings, 54 : 221-230 • C. Gattano, S. B. Lambert, and K. Le Bail. (2018) Extragalactic radio source stability and [...]. submitted in A&A.
On the topic of realizing the celestial reference frame with VLBI, see also : • P. Charlot and the ICRF3 Working group, various communications • A. L. Fey, D. Gordon, C. S. Jacobs et al. (2015) The Second Realization of the International Celestial Reference Frame [...]. AJ 150 : 58 • M. Karbon, R. Heinkelmann, J. Mora-Diaz et al. (2017) The extension of the parametrization of the radio source coordinates in geodetic VLBI [...]. Journal of Geodesy, 91 : 755-765 • K. Le Bail, D. Gordon, C. Ma, Selecting sources that define a stable celestial reference frame with the Allan variance. (2016) IVS General Meeting Proceedings 288-291 • N. Liu, J.-C. Liu, Z. Zhu. (2017) Test of source selection for cinstructing a more stable and uniforme celestial reference frame. MNRAS, 466 : 1567-1574. • B. Soja, R. Gross, C. Jacobs, et al. (2017) A Kalman filter approach for the determination of celestial reference frames. EGU General Assembly
