In the construction of modern Celestial Reference Frames (CRFs) the overall rotational alignment is only weakly constrained by the data. Therefore, common practice has been to apply a 3-dimensional No-Net-Rotation (NNR) constraint in order to align an under-construction frame to the ICRF. We present evidence that correlations amongst source position parameters must be accounted for in order to properly align a CRF at the 5-10 (mu)as level of uncertainty found in current work. Failure to do so creates errors at the 10-40 (mu)as level

Heflin, M. B.

Jacobs, Chris S.

Lanyi, G. E.

Sovers, O. J.

Steppe, J. A.

NASA Technical Reports Server

C. S. Jacobs et al.: Rotational Alignment Altered by Source Position Correlations, IVS 2010 General Meeting
Proceedings, p.305–309
http://ivscc.gsfc.nasa.gov/publications/gm2010/jacobs2.pdf
Rotational Alignment Altered by Source Position Correlations
C. S. Jacobs, M. B. Heflin, G.E. Lanyi, O. J. Sovers, J. A. Steppe
Jet Propulsion Laboratory, California Institute of Technology
Contact author: C. S. Jacobs, e-mail: Chris.Jacobs@jpl.nasa.gov
Abstract
In the construction of modern Celestial Reference Frames (CRFs) the overall rotational alignment
is only weakly constrained by the data. Therefore, common practice has been to apply a 3-dimensional
No-Net-Rotation (NNR) constraint in order to align an under-construction frame to the ICRF. We
present evidence that correlations amongst source position parameters must be accounted for in order
to properly align a CRF at the 5–10 µas level of uncertainty found in current work. Failure to do so
creates errors at the 10–40 µas level.
1. Introduction to No-net-rotation Constraint Equations
Since the adoption of the ICRF [4] by the IAU in 1997 [1], the axes which define Right Ascension
and declination have been realized by an ensemble of quasar positions. Subsequent frames, most
notably the ICRF2 [5], have been aligned to the original ICRF by imposing a 3-dimensional No-
Net-Rotation (NNR) constraint in the form of the cross-product (see full derivation in appendix),
Nsrc∑
i=1
s0i × ∆si = 0 (1)
where for a set of Nsrc sources,
s0i = (cosα0i cos δ0i, sinα0i cos δ0i, sin δ0i) (2)
is the unit vector in the reference source direction for the ith source of coordinates (α0i, δ0i) and
∆si is the difference of the reference position from the estimated position.
s0i ×∆si = [− cosα0i sin δ0i cos δ0i ∆αi + sinα0i ∆δi], (3)
[− sinα0i sin δ0i cos δ0i ∆αi − cosα0i ∆δi],
[cos2 δ0i ∆αi]
The above triplet of constraints affects both the estimated set of αi and δi and their associated
covariance including the correlations amongst parameters.
The MODEST linear least squares fit software [7] requires the coefficient with respect to ∆αi
and ∆δi of each rotation constraint, Cj = (
∑
i s0i ×∆si)j where j goes from 1 to 3. The result is
∂C1/∂∆αi = − cosα0i sin δ0i cos δ0i, ∂C1/∂∆δi = sinα0i (4)
∂C2/∂∆αi = − sinα0i sin δ0i cos δ0i, ∂C2/∂∆δi = − cosα0i (5)
∂C3/∂∆αi = cos
2 δ0i, ∂C3/∂∆δi = 0 (6)
IVS 2010 General Meeting Proceedings 305
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C. S. Jacobs et al.: Correlations Alter CRF Alignment
Note that the constraint Eq. (1) is unweighted in the position parameters, s. The MODEST
software implements constraints by treating them as pseudo-observations with given uncertainties.
Specifying a vanishingly small uncertainty is equivalent to using an absolute constraint.
Using the preceding partials (Eqs. 4, 5, 6), each Cj was separately constrained to be zero to
within a tight constraint uncertainty of σj = 10
−10 radians.
2. Sensitivity of S/X, K, and X/Ka Frame Alignment to Correlations
Our procedure was first to estimate source positions under the influence of unweighted NNR
constraints (Eq. 1) to the ICRF2 defining sources. We used only sources with at least three
sessions, ten delay observations, and a formal error ellipse major axis smaller than 5 nrad, and
which were also in the ICRF2 defining list.
Second, we did an after-the-fact verification that the NNR constraints indeed kept the positions
from rotating with respect to the defining sources by estimating a 3-D rotation with respect to the
ICRF2 defining sources.
This second step was done using alternately diagonal-only parameter covariance and full covari-
ance for the Celestial Reference Frame (CRF) being constrained. The reference, ICRF2, always
used diagonal covariances because the off-diagonal correlations are not published.
The results for three data sets are given in Table 1 with descriptions of the sets as follows:
• The S/X data set included observations from October 1978 to September 2009 totalling 5.3M
observations from 1709 sources. The NNR constraint used 267 ICRF2 defining sources.
• The K-band data set [3] covers 2002 to 2009 with over 100,000 observations of 275 sources.
The NNR constraint used 125 ICRF2 defining sources.
• The X/Ka-band data set [2] covers 2005 to 2010 with over 10,000 observations of 387 sources.
The NNR constraint used 153 ICRF2 defining sources.
Table 1. 3-D Rotations (µas) estimated with vs. without correlations.
CRF NDef R1 σR1 R2 σR2 R3 σR3
S/X with corr. 267 1.4 ±5.4 2.4 ±5.6 2.0 ±4.4
without corr. 267 −25.8 ±4.9 9.4 ±4.9 −1.6 ±4.3
K with corr. 125 0.1 ± 6.8 1.2 ± 7.4 −0.1 ±5.2
without corr. 125 −20.5 ±11.5 −17.7 ±11.9 −8.2 ±7.9
X/Ka with corr. 153 0.3 ± 6.3 2.9 ± 6.6 − 0.1 ± 4.9
without corr. 153 −35.1 ±17.9 31.1 ±18.3 38.6 ±11.2
3. Discussion
NNR Constraint Weighting: Given that the NNR constraint to the ICRF2 was unweighted,
there may be problems caused by using a diagonal weighting for our after-the-fact check of rota-
tional alignment. Our alignment check used the sum of the covariance of the frame under study
and the diagonal ICRF2 covariance (note that the ICRF2 off-diagonal terms have not been made
306 IVS 2010 General Meeting Proceedings
C. S. Jacobs et al.: Correlations Alter CRF Alignment
available). For the three cases studied above, the X/Ka frame has the largest uncertainties, which
are expected to dominate the combined X/Ka + ICRF2 covariance. The K-band uncertainties are
somewhat larger than the ICRF2 and will partially dominate the combined covariance. However,
because the S/X frame comes from almost the same data as the ICRF2, its uncertainties are com-
parable to those of the un-inflated ICRF2. Once the ICRF2 diagonal covariances are inflated by
a scale factor of 1.5 and then have a 40 µas noise floor RSS’ed into each position coordinate, the
ICRF2 covariance dominates the diagonal uncertainties.
These remaining nuances in weighting may explain why—even with full covariance—our after-
the-fact check shows a few µas residual non-alignment. We have not yet studied these issues closely
and can only say that the subject deserves more careful investigation.
Nutation constraints: One important lesson learned in the course of our comparisons is that
other constraints can distort the desired effect of the NNR constraint. In the case of our K-band
solution, our a priori nutation model was the IAU 2000 standard of Matthews, Herring, Buffet
(MHB) [6]. Because there was evidence external to our study that the MHB model had 100–150
µas errors, we at first estimated nutation corrections with “weak” 200 µas a priori uncertainties
on the nutation angles for each of twelve day-long sessions. Noting that a priori constraints act as
pseudo-measurements, this effectively added 12 observations of the nutation pole each with 200 µas
uncertainty. Thus, in retrospect, we realized that we had added a global constraint on the nutation
pole of 200 µas /
√
12 = 58 µas which was strong enough to bias the 3-D rotational alignment of
our celestial reference frame. This example is shared as a reminder that celestial frame alignment
at the level of 10 µas requires careful attention to many details.
4. Conclusions
The celestial frame No-Net-Rotation constraint,
∑
s0i×∆si = 0, alters the α-δ full covariance
produced by a CRF solution. The correlations contain significant information which is needed to
correctly estimate 3-D rotations and associated sigmas. At the 5–10 µas levels of axial alignment
precision found in current CRF work, these correlations must be accounted for in order to avoid
statistically significant misalignment of a given CRF.
5. Acknowledgements
The research described herein was performed at the Jet Propulsion Laboratory of the California
Institute of Technology, under a contract with the National Aeronautics and Space Administration.
Copyright 2010 c© California Institute of Technology. Government sponsorship acknowledged.
References
[1] IAU General Assembly XXIII, Resolution B2-d, Kyoto, Japan, August 1997.
iau.org/static/resolutions/IAU1997 French.pdf
[2] Jacobs, C. & O. Sovers, ‘X/Ka Global Astrometry,’ Proc. 19th EVGA, Bordeaux, France, 24 Mar 2009.
www.u-bordeaux1.fr/vlbi2009/index.php?numpage=13&nompage=Proceedings
[3] Lanyi, G.E., et al, ‘The Celestial Frame at 24 and 43 GHz I. Astrometry,’ AJ, 139, 5, p. 1695, 2010.
doi: 10.1088/0004-6256/139/5/1695
iopscience.iop.org/1538-3881/139/5/1695
IVS 2010 General Meeting Proceedings 307
C. S. Jacobs et al.: Correlations Alter CRF Alignment
[4] Ma, Arias, Eubanks, Fey, Gontier, Jacobs, Sovers, et al, ‘ICRF as Realized by VLBI,’ AJ, 116, 1998.
doi: 10.1086/300408
iopscience.iop.org/1538-3881/116/1/516/
[5] Ma, C., et al, ‘The Second Realization of the ICRF by Very Long Baseline Interferometry,’
IERS Tech. Note 35, editors: A. L. Fey, D. Gordon, & C. S. Jacobs, Frankfurt, Germany, Oct. 2009.
www.iers.org/nn 11216/IERS/EN/Publications/TechnicalNotes/tn35.html
[6] Matthews, P. M., T. A. Herring, and B. A. Buffet, ‘Modeling of Nutation and Precession: New
nutation series for Nonrigid Earth and Insights into the Earth’s Interior,’ JGR, vol. 107, no. B4,
10.1029/2001JB000390, Apr. 2002.
www.agu.org/journals/jb/jb0204/2001JB000390/
[7] Sovers, O. J., J. L. Fanselow, and C. S. Jacobs, ‘Astrometry and Geodesy with Radio Interferometry:
Experiments, Models, Results’, Rev. Mod. Phys., vol. 70, no. 4, pp. 1393–1454, Oct. 1998.
doi: 10.1103/RevModPhys.70.1393
rmp.aps.org/abstract/RMP/v70/i4/p1393 1
Figure 1. K-band (24 GHz) 12-session distri-
bution of α-α correlations vs. arclength for all
combinations of two sources. The correlations
are systematically positive for short arcs and
become negative for the longest arcs. These
trends alter both the estimated rotations and
their uncertainties. Color coding of 22.5◦ dec.
bands: low to high (orange, red, green, blue,
purple, black).
Figure 2. X/Ka (8.4/32 GHz) CRF’s distribu-
tion of δ-δ correlations vs. arclength. The cor-
relations remain systematically positive over
the full range of arc lengths, thus altering both
the estimated rotations and their uncertain-
ties. Color coding is the same as in the pre-
ceding figure. Note the tendency for low dec-
lination to be more strongly correlated.
6. Appendix: Derivation of NNR Constraint Equation
Given two sets of sources forming two frames, one frame is a priori assigned the role of the
absolute reference and the other frame we wish to estimate from our data. Next, we establish
the mathematical formula which describes the angular rotational offset between these two frames.
Then we estimate the positions from data by least squares such that the rotational offset is zero.
308 IVS 2010 General Meeting Proceedings
C. S. Jacobs et al.: Correlations Alter CRF Alignment
Let s0i be the reference position vector of the i
th source, and let si be the estimated position.
Denote the difference as ∆si = s0i − si. Suppose an approximately infinitesimal rotation, R(~ǫ), is
applied to si. Then the rotated position is
R(~ǫ) si = si + ~ǫ× si (7)
We want the rotation R(~ǫ) that minimizes the mean square angular position difference:
(∆θ)2 =
1
Nsrc
Nsrc∑
i=1
(∆si − ~ǫ× si) · (∆si − ~ǫ× si) (8)
We minimize with respect to variations in ǫj by setting the derivative, ∂ (∆θ)
2/ ∂ǫj = 0
Nsrc∑
i=1
[
∂
∂ǫj
(∆si − ~ǫ× si)] · (∆si − ~ǫ× si) + (∆si − ~ǫ× si) · [
∂
∂ǫj
(∆si − ~ǫ× si)] = 0 (9)
Nsrc∑
i=1
(−(
∂
∂ǫj
~ǫ)× si) · (∆si − ~ǫ× si) + (∆si − ~ǫ× si) · (−(
∂
∂ǫj
~ǫ)× si) = 0 (10)
Since dot products commute, this can be re-written:
Nsrc∑
i=1
(−2)(∆si − ~ǫ× si) · ((
∂
∂ǫj
~ǫ)× si) = 0 (11)
We want the above derivative to be zero, ∂(∆θ)2/∂ǫj = 0 when evaluated at ~ǫ = 0, thus
Nsrc∑
i=1
(∆si) · ((
∂
∂ǫj
~ǫ)× si) = 0 for j = 1, 2, 3 (12)
Using the general vector property v1 · (v2 × v3) = v2 · (v3 × v1) we have:
Nsrc∑
i=1
( ∂
∂ǫj
~ǫ) · (si ×∆si) = 0 for j = 1, 2, 3 (13)
Since ∂
∂ǫj
~ǫ is just the elementary unit vector on the jth axis, this represents three equations which
can be combined into the vector equation:
Nsrc∑
i=1
si × ∆si = 0 (14)
Noting that si×∆si = (s0i−∆si)×∆si = s0i×∆si−∆si×∆si = s0i×∆si since for any vector,
v, v × v = 0, we have:
Nsrc∑
i=1
s0i × ∆si = 0 (15)
IVS 2010 General Meeting Proceedings 309


Rotational Alignment Altered by Source Position Correlations

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