An Improved Empirical Harmonic Model of the Celestial Intermediate
Pole Offsets from a Global VLBI Solution
Santiago Belda
1,2,4
, Robert Heinkelmann
2
, José M. Ferrándiz
1
, Maria Karbon
2
, Tobias Nilsson
2
, and Harald Schuh
2,3
1
Department of Applied Mathematics, University of Alicante, Carretera San Vicente del Raspeig s/n,
2
Helmholtz Centre Potsdam, German Research Centre for Geosciences (GFZ), Telegrafenberg, A17, D-14473 Potsdam, Germany
3
Institute of Geodesy and Geoinformation Science, Technical University of Berlin, Straße des 17, Juni 135, 10623 Berlin, Germany
Received 2017 May 8; revised 2017 August 17; accepted 2017 August 22; published 2017 September 28
Abstract
Very Long Baseline Interferometry (VLBI)is the only space geodetic technique capable of measuring all the Earth
orientation parameters (EOP)accurately and simultaneously. Modeling the Earthʼs rotational motion in space within
the stringent consistency goals of the Global Geodetic Observing System (GGOS)makes VLBI observations essential
for constraining the rotation theories. However, the inaccuracy of early VLBI data and the outdated products could
cause non-compliance with these goals. In this paper, we perform a global VLBI analysis of sessions with different
processing settings to determine a new set of empirical corrections to the precession offsets and rates, and to the
amplitudes of a wide set of terms included in the IAU 2006/2000A precession-nutation theory. We discuss the results
in terms of consistency, systematic errors, and physics of the Earth. We find that the largest improvements w.r.t. the
values from IAU 2006/2000A precession-nutation theory are associated with the longest periods (e.g., 18.6-yr
nutation). A statistical analysis of the residuals shows that the provided corrections attain an error reduction at the level
of 15 μas. Additionally, including a Free Core Nutation (FCN)model into a priori Celestial Pole Offsets (CPOs)
provides the lowest Weighted Root Mean Square (WRMS)of residuals. We show that the CPO estimates are quite
insensitive to TRF choice, but slightly sensitive to the a priori EOP and the inclusion of different VLBI sessions.
Finally, the remaining residuals reveal two apparent retrograde signals with periods of nearly 2069and 1034 days.
Key words: astrometry –catalogs –reference systems –techniques: interferometric
1. Introduction
Due to gravitational attractions from the Moon, Sun, and
planets, the Earth rotation axis shows various periodical and
irregular motions w.r.t. its figure axis and w.r.t. an ideal inertial
reference system. The General Assemblies of the International
Astronomical Union (IAU)held in 2000 and 2006 defined the
transformation from the relevant Celestial Reference System
(CRS)to the Terrestrial Reference System (TRS)with the help
of the so-called Celestial Intermediate Reference Frame
(CIRF), e.g., Urban & Seidelman (2013). The precession-
nutation angles give the orientation of the CRF (usually the
International Celestial Reference Frame, or ICRF)w.r.t. the
CIRF. The position of the CIRF pole or Celestial Intermediate
Pole (CIP)in the CRF is defined by the precession-nutation
angles, the term precession being used when the long-term
motion is referred to, and nutation being applied to faster,
quasi-periodic variations with periods larger than two solar
days in the CRF according to the IAU resolutions; see, e.g., the
International Earth Rotation and Reference Systems Service
(IERS)Conventions (2010; Petit & Luzum 2010).
The transformation from CRS to TRS is completed through
the three rotation angles that connect the CIRF to the used
TRF, often designed as Earth rotation parameters (ERPs). Two
of them, xpand yp, give the relative position of the CIRF and
TRF poles and constitute the so-called polar motion (PM)
parameters, while the third one accounts for the irregular Earth
diurnal rotation, usually expressed by the parameter dUT1,
which is obtained from the difference between universal time
UT1 and UTC (universal time coordinated). The ensemble of
the two precession-nutation angles and the three ERPs is
usually designated as Earth orientation parameters (EOPs).
That is the terminology adopted in the IERS Conventions
(2010)and followed in this paper.
The IAU2000A nutation (Mathews et al. 2002)and IAU2006
precession models (Capitaine et al. 2003,2005)were adopted to
provide accurate approximations and predictions of the CIP.
However, they are not fully accurate and VLBI (Very Long
Baseline Interferometry)observations show that the CIP deviates
from the position resulting from the application of the IAU2006/
2000A model (see e.g., Petit & Luzum 2010). Those deviations
or offsets of the CIP are known as Celestial Pole Offsets (CPOs)
and are denoted as (dX,dY). Currently, accurate observations of
CPO can only be obtained by the VLBI technique. The observed
CPO can quantify the deficiencies of the IAU2006/2000A
precession-nutation model, including the astronomically forced
nutations and a component of nutation that is considered
unpredictable. The latter is mainly constituted by the free core
nutation (FCN), which is excited by angular momentum
exchanges between the Earthʼsmantleanditsfluid layers
(Toomre 1974;Smith1977;Wahr1981). It has a retrograde
long-period of about 430 days (with average amplitudes of about
100 μas)relative to the inertial frame (Krásná et al. 2013),ora
period slightly shorter than 1 day in the retrograde-diurnal band,
relative to the rotating terrestrial frame. The absence of a model
for FCN prediction is what causes the uncertainty of the IAU
The Astronomical Journal, 154:166 (14pp), 2017 October https://doi.org/10.3847/1538-3881/aa8869
© 2017. The American Astronomical Society. All rights reserved.
4
Corresponding author address and email: Department of Applied Mathe-
matics, University of Alicante, Carretera San Vicente del Raspeig s/n, San
Vicente del Raspeig, E-03690 Alicante, Spain
Original content from this work may be used under the terms
of the Creative Commons Attribution 3.0 licence. Any further
distribution of this work must maintain attribution to the author(s)and the title
of the work, journal citation and DOI.
1
2006/2000A precession-nutation model to be roughly of the
order of 0.2 mas (Dehant et al. 2003). Due to the variable
character of amplitudes and phases of the FCN effect, accurate
empirical models based on VLBI estimates are required.
Currently, different empirical FCN models, which were
determined using different adjustment strategies (Lambert &
Dehant 2007;Krásnáetal.2013; Malkin 2013a;Belda
et al. 2016), are available. The IERS Conventions recommend
the model of Lambert (2007), which was obtained by fitting the
FCN amplitude on a 2-year interval running by steps of one year.
Free inner core nutation (FICN)is one of the four free
rotational modes of the Earth considered in the theory of Earth
rotation, corresponding to the mode designated prograde free
core nutation (PFCN)in Mathews et al. (1991). Detecting this
signal in the observational data is a very important scientific
task that would allow for substantial improvement of the
knowledge about the Earthʼs interior and dynamics
(Malkin 2013b). Consequently, it could bring us significantly
closertomeetingthe accuracy goals pursued by the Global
Geodetic Observing System (GGOS)of the International
Association of Geodesy (IAG), i.e., 1 mm accuracy and
0.1 mm/year stability on global scales in terms of the ITRF
defining parameters (Plag & Pearlman 2009). Mathews et al.
(2002)were the first to discuss the existence of this resonance
in VLBI observations at a period of approximately 1035 days
supposedly caused by the FICN. According to Dehant et al.
(2005), the dynamics of the atmosphere and the oceans could
excite the FICN to amplitudes of a few tens of microarcse-
conds (μas).
Since the adoption of IAU2000, VLBI observations have
more than doubled in number and the quality of the data has
significantly improved. Empirically speaking, we assume that
evaluating the consistency and updating the precession-
nutation model could provide the necessary accuracy for
detecting signals of the aforementioned order of magnitude.
These issues were discussed by the past and more recent
studies done by Charlot et al. (1995)and Gattano et al. (2017),
respectively. In the former study, the IAU 1980 nutation theory
(Seidelmann 1982)was evaluated on the basis of combining
VLBI and Lunar Laser Ranging (LLR)data analyses, whereas
in the latter study different nutation series determined by
different VLBI Analysis Centers (AC)of the International
VLBI Service for Geodesy and Astrometry (IVS; Nothnagel
et al. 2015)were compared.
It is common practice in geodetic VLBI analysis to
distinguish between the so-called arc parameters and global
parameters. This terminology reflects the time epochs for which
the parameters are valid. Arc parameters are parameters that are
valid only during a particular observation session or parts of it.
These are, for example, the clock parameters, zenith wet delay,
or tropospheric gradients which customarily vary on a sub-
daily basis, whereas global parameters are valid for longer time
periods and not only for a single observing session. For
example, radio source coordinates, relativistic parameters, and
station coordinates and velocities can belong to the latter
category of parameters. The so-called global analysis uses a
large number of VLBI sessions and allows for solving for both
arc and global parameters. The computational strategy is based
on a separation of the normal equation (NEQ)system into two
parts. The first part contains the parameters that are estimated
and constitute the relevant output for a certain planned analysis.
And the second part corresponds to the remaining parameters,
which are “reduced”from the equations. Note that those
“reduced”parameters still belong to the functional model of
unknown parameters and are estimated implicitly from the
session-wise NEQ during the least-squares adjustment. One
possible approach is to accumulate reduced NEQ from single
sessions that no longer contain arc parameters, and then to
solve for the global parameters. Then, the arc parameters for
each session can be determined in a second step by substituting
the estimated global parameters (Haas 2004).
In this paper, we empirically evaluate the consistency,
systematics, and deviations of the IAU 2006/2000A preces-
sion-nutation model using several CPO time series derived
from the global analysis of VLBI sessions starting in 1990. To
reflect the impact of frames and other processing strategies on
the CPO estimates, we alternate several analysis settings
(Section 2). In Section 3, the various series are then used to
readjust the precession offset and rate, as well as the main
nutation amplitudes available in the IAU 2006/2000A
precession-nutation model for the sake of empirically improv-
ing the conventional values adopted by the IAU and the IAG as
published in the IERS Conventions (2010). The empirical FCN
model (Belda et al. 2016)is also included in the adjustment,
after a comparison with other empirical FCN models. In
Section 4, the remaining residuals of the fit are analyzed and
discussed by investigating possible geophysical signals in the
frequency band where the FICN is expected. Finally, conclud-
ing remarks are given in Section 5.
2. Methodology
2.1. Global VLBI Solution
The reassessment of the precession and nutation terms in
analogy to the IAU 2006/2000A precession-nutation model is
empirically done by performing different VLBI data analyses
with the GFZ version (Nilsson et al. 2015)of the Vienna VLBI
Software (VieVS)(Böhm et al. 2012). As detailed in
Section 2.2, we apply several VLBI processing options. In
order to achieve a high degree of consistency between the
VLBI data (and implicitly in the CPO estimates)with respect to
the ICRF2, we mainly focus on analyzing VLBI data globally,
processed at once in an accumulated normal equation system.
The number of sessions amounts to 2990, ranging from 1990 to
2010. We decided to exclude the sessions before 1990 and after
2010.0 because of the inaccuracy of VLBI data in the early
years (Malkin 2013a)and because the ICRF2 is based on VLBI
data until 2009 March only. The single-session analysis was
based on the IERS Conventions (2010). Before the global
solution/adjustment, we discarded VLBI sessions with
a posteriori sigmas of unit weights larger than 3. In addition,
sources observed less than 15 times over 2 years, which appear
in less than 3 VLBI sessions, and the coordinates of the so-
called ICRF2 special handling sources, were reduced in order
to refine the CPO measurements. Consequently, a total of 677
radio sources were estimated. The radio source velocities were
fixed to zero (no proper motion allowed), whereas station
positions along with the ERP (xp,yp,dUT1)were reduced from
the NEQ-system (in the sense explained in Section 1).
Additionally, the datum definition of the TRF was constrained
by applying no-net-translation (NNT)and no-net-rotation
(NNR)conditions at the session level, referring to the station
coordinates reported in the respective catalogs. Although CPOs
are usually determined as arc parameters from single VLBI
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The Astronomical Journal, 154:166 (14pp), 2017 October Belda et al.
sessions, in our approach the CPOs and source coordinates
were estimated as global parameters by imposing no-net-
rotation conditions with respect to the ICRF2 defining sources
(Fey et al. 2015). We had two motives. First, the comparison/
update of precession-nutation model terms is expected to gain
consistency and accuracy when using a global solution, since
determining those parameters from individual VLBI sessions
would add additional noise, e.g., rotation of sub-frames or other
datum inconsistencies (Belda et al. 2017). Second, global
parameters allow direct calculation of formal errors, whereas
the time series only allow for an empirical step-wise error
assessment. Finally, we repeated the VLBI analysis when
extending the period until 2015.0, with a total of 3594 sessions,
to study the impact of including the most recent data. In this
case, 744 radio sources were included in the solutions.
2.2. General Design of the Different Testing Approaches
In an ideal case, the estimated values of the CPO should be
the same, independent of the a priori values. However, highly
accurate estimation of the full set of EOP is not simple from
either a mathematical or physical perspective, and the
possibility of having effects derived from the choice of the
initial solution should not be discarded (Belda et al. 2017).
Following up on this basic idea, multiple VLBI analyses were
carried out with different a priori EOP series (IERS 08 C04
(Bizouard & Gambis 2009), United States Naval Observatory
(USNO)finals), and TRFs (ITRF2014 (Altamimi et al. 2016),
ITRF2008 (Altamimi et al. 2011), VTRF2008 (Böckmann
et al. 2010)) to determine the effect of changing those inputs on
the CPO, the precession constant and rate, and the main
nutation terms of the 2006/2000A precession-nutation model.
The former and latter approaches, designated “A”or “B”
respectively in Table 1, are explained in the next paragraphs.
2.2.1. Test A: Different a Priori EOP Series
Using ITRF2014 and ICRF2, several global VLBI solutions
were estimated from various a priori EOP series (USNO finals
and IERS 08 C04)and different settings. In all cases, we
performed a global analysis of all the VLBI data in the selected
period of time that treats all the sessions jointly, to derive a
series of differences dX,dY, between the XVLBI,
Y
VLBI,
coordinates of the CIP estimated from VLBI data minus the
a priori EOP values that define the case.
1. Case A1.A priori EOPvalues are taken from USNO
finals (case A1.a)or IERS 08 C04 (case A1.b):
XX dX
YY dY.1
VLBI IERS USNO apriori
VLBI IERS USNO apriori
=+
=+
()
() ()
This case serves to compare the new global solution to well-
known local solutions such as the conventional IERS C04 and
USNO finals.
2. Case A2.A priori EOP values are: (1)the X,Y, predicted
by the IAU2006/2000A model:
XX dX
YY dY;2
VLBI IAU2006 2000A apriori
VLBI IAU2006 2000A apriori
=+
=+
()
() ()
and (2)the ERP from USNO finals in A2.a and from IERS 08
C04 in A2.b. This choice is similar to Belda et al. (2016). This
case is the closest to compare the new global solution to the
conventional definition of CPO, based on IAU2000A.
3. Case A3.A priori EOP values are: (1)the sum of the X,
Y, predicted by the IAU2006/2000A model and the free
nutation provided by one of the empirical FCN models by
Belda et al. (2016; hereafter B16)either derived from
USNO finals (case A3.a)or from IERS 08 C04 (case
A3.b):
XX X dX
YY Y dY;3
VLBI IAU2006 2000A FCN apriori
VLBI IAU2006 2000A FCN apriori
=++
=++
()
()()
and (2)ERP like in Case A2. This approach is useful for
assessing the joint predictive capability of the a priori
combined model, regarding the global solution as target.
For each of the aforementioned cases, the Celestial Pole
Coordinates (CPCs, XVLBI and
Y
VLBI)were calculated using the
adjustments obtained by VLBI (dX,dY). As an example,
Figure 1represents the CPCs of case A1.a using USNO finals
as a priori values.
That test can provide an indication of the predicting
capabilities of each a priori CPC series: the more accurate a
starting EOP set is, the less variability the CPO has. Note that
only the dX,dYresulting from case A2 can be strictly
Table 1
Overview of the Different Applied Approaches (A and B)
Choice of the a Priori Data
TRF EOP Series EOP ERP and IAU2006/2000A ERP and IAU2006/2000A +FCN B16
ITRF2014 USNO finals A1.a and B1 A2.a A3.a
IERS 08 C04 A1.b A2.b A3.b
VTRF2008 USNO finals B2 LL
ITRF2008 USNO finals B3 LL
Figure 1. Celestial Pole Coordinates estimated from case A1.a using USNO
finals as a priori values. Precession and obliquity rates were removed between
1990 and 2015.
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The Astronomical Journal, 154:166 (14pp), 2017 October Belda et al.
denominated as a CPO according to IAU resolutions, whereas
the dX,dY, from cases A1 and A3 are more properly denoted as
CPO-like parameters. However, for the sake of concision we
denote of all them as CPO from now onward, taking into
account the fact that the dispersion of each parameter provides
an indication of the accuracy of its prediction by means of the
assumed a priori solution. One remarkable feature is that case
A3 produces the lowest WRMS for both coordinates (150.7
μas in Xand 153.1 μas in Yas displayed in Figure 2, bottom),
although one could reasonably expect that A1 would cause
a priori values closer to our global VLBI solution, since both
series (from IERS and USNO solutions)contain actual CPOs
derived from a combination of VLBI observations.
If we compare the CPO adjustments (dX,dY)computed from
cases A1 and A2 (Figure 2)we clearly find that a low-
frequency part of about 18.6 yr of periodicity remains with
approximately the same magnitude and opposite signs for dX
and dY, respectively. Other authors already observed this signal
and applied several methods to delete the small curvature, e.g.,
Capitaine et al. (2009)fitted a parabola plus a term of 18.6 year
periodicity to the CPO adjustments. In case A3 this curvature is
not visible in the adjustments resulting from the consideration
of the FCN model fitted a priori, where the authors, in addition
to accounting for the FCN signal, also accounted for such low-
frequency signals in order to avoid contamination of the FCN
signal determination (Belda et al. 2016).
To better compare the pairs of time series of CPO estimates,
we calculated the weighted mean (WM)of the differences and
the weighted root mean square (WRMS)differences between
each pair of them by means of the equation provided by Belda
et al. (2017). In case A1.a, formal errors of dX range in the
interval [7.5, 169.2]μas with a median of 60.7 μas; for dY the
figures are similar, with a range [7.6, 169.6]μas and a median
of 60.8 μas. There are no significant differences to the other
cases. The orders of magnitude of the WM between all the
cases are similar at about up to 2 μas, which is well below the
current accuracy of the CRF axes at the level of 10 μas,
providing evidence of the high consistency level of VLBI. This
can be seen in Figure 3(top and middle of the panel). However,
it is important to note that the choice of a priori values (cases
A1, A2, and A3)generates a noticeable scatter of about 50 μas
and 70 μas from the USNO finals and IERS 08 C04,
respectively. On the other hand, the smallest WRMS
differences (around 10 μas)can be found when comparing
case A2 to case A3. Moreover, note that case A1.a w.r.t case
A1.b exhibits a WRMS difference of about 70 μas for both
CPO components.
2.2.2. Test B: Different a Priori TRFs
This test is similar to case A1 but now CPOs were estimated
making use of different terrestrial reference frames, i.e., B1:
ITRF2014, B2: VTRF2008, and B3: ITRF2008 (Table 1), with
USNO finals as a priori EOP values. It can assess the extent of
the effect of the mutual consistency of EOP solutions and
TRFs, indicated by the “repeatability”in terms of scatter
or WRMS.
Figure 2. Adjustments to three CPO modeling approaches (blue dots). Top: case A1.a; middle: case A2.a; bottom: case A3.a. A priori data: USNO finals and
ITRF2014. Left: dX. Right: dY. Orange line: 18.6-yr periodic term.
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The Astronomical Journal, 154:166 (14pp), 2017 October Belda et al.
Once again, a low-frequency signal with a period of about
18.6 yr remains in the adjustments, with an amplitude
comparable to the A1.a and A1.b approaches. CPO adjustments
from cases B2 and B3 show practically identical scatter (of
about 152 μas and 164 μas for Xand Y, respectively). However,
CPO estimates from ITRF2014 (case B1)produce a slight
deterioration of the accuracy (157.6 μas in Xand 166 μas in Y).
The CPO differences between pairs of the different approaches
(B1, B2, B3)are close to each other (Figure 4), showing only
small WM (<1.1 μas)and WRMS differences (Figure 3,
bottom). ITRF2008 w.r.t. VTRF2008 presents the smallest
WRMS, about 5 μas, which was expected due to their mutual
consistency: VTRF2008 presents the VLBI input to ITRF2008.
On the other hand, ITRF2008 and VTRF2008 w.r.t ITRF2014
exhibit a scatter three times larger, which may be attributable to
consistency issues w.r.t. the a priori EOP values and the
different time spans of input data.
3. Least-squares Adjustment: Toward Refining the IAU
2006/2000A Precession-Nutation Models
The refinement of the VLBI-derived amplitudes of nutation
terms presents a number of difficulties. Since some of the
astronomical nutations have periods sufficiently close to the
FCN, their amplitudes can be magnified by resonance. The
most notable example is the retrograde annual nutation. On
occasion, separation of periods is problematic and some
periods need to be constrained to their theoretical amplitude
values to properly fit the nutation series parameters—e.g.,
Herring et al. (2002)fixed the out-of-phase 386.0-day period
amplitude to the MBH2000 a priori value.
In this study, to reduce the inconveniences derived from the
aforementioned resonance effect, a preliminary FCN signal was
subtracted from the CPC before fitting them to the main
nutation amplitudes. The preliminary model was estimated
from the CPO adjustments observed by VLBI w.r.t. the IAU
2006/2000A precession/nutation theory (the CIP coordinates)
and following the sliding window length of 400 days’approach
as outlined in Belda et al. (2016). The equations used to
estimate the FCN model are:
X A tA tX
Y A tA tY
cos sin
cos sin , 4
cs
sc
FCN FCN FCN 0
FCN FCN FCN 0
ss
ss
=-+
=++
() ()
() () ()
where P2
FCN
s
p=is the frequency of FCN in the Celestial
Reference System (CRS),
AC
and
A
Sdenote the amplitudes as
Figure 3. WM (left)and WRMS (right)differences between the EOP estimated with different a priori EOPs and different TRFs (legend: see Table 1). Units: μas. For
an explanation of the various approaches see Table 1and the text.
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The Astronomical Journal, 154:166 (14pp), 2017 October Belda et al.
Figure 4. CPO differences (μas)between CPOs estimated with different TRFs: ITRF2014, ITRF2008, and VTRF2008. A priori EOPs are from USNO finals. Left: dX.
Right: dY.
Table 2
Reassessment of Precession Offset b and Rate p
Data Period A Priori Values
Before Fit-
ting FCN
a
After Fit-
ting FCN
a
TRF EOP bxsbyspxspys
r
x
r
y
r
x
r
y
USNO finals
1990–2010 ITRF2014 CPO 38.8±2.1 −67.9±2.1 1.4±48.6 −144.2±49.2 180.7 188.6 147.6 158.4
IAU 2006/2000A 40.9±1.8 −71.8±1.9 2.9±42.9 −83.1±43.4 173.5 173.4 139.7 140.2
IAU 2006/2000A +FCN 40.7±1.8 −71.8±1.9 −0.2±42.7 −87.8±43.1 172.4 173.0 138.5 140.0
IERS 08 C04
ITRF2014 CPO 39.5±2.3 −70.2±2.3 −66.6±53.9 −167.5±54.5 193.7 194.9 163.7 166.7
IAU 2006/2000A 40.5±1.8 −71.9±1.8 8.7±42.6 −86.9±43.0 173.1 173.0 139.1 140.3
IAU 2006/2000A +FCN 40.5±1.8 −72.0±1.8 9.0±42.4 −87.4±42.9 173.1 172.9 139.0 140.2
USNO finals
ITRF2008 CPO 39.5±2.1 −69.1±2.1 −11.5±47.9 −124.1±48.5 177.4 187.0 144.1 156.6
VTRF2008 CPO 39.6±2.1 −68.9±2.1 −10.8±48.0 −128.8±48.5 177.9 187.3 144.7 156.7
USNO finals
1990–2015 ITRF2014 CPO 38.6±2.1 −68.9±2.1 178.1±27.1 −208.3±27.3 187.9 194.0 144.2 154.0
IAU 2006/2000A 41.0±1.9 −73.2±1.9 162.0±24.0 −177.6±24.2 184.9 183.2 138.6 140.1
Note. Formal errors for each parameter are provided following the±sign. Units: μas or μas/century.
a
Root mean squares rof the residuals after fitting the amplitudes of the main nutation terms of the IAU 2006/2000A precession-nutation model when considering or
not considering the FCN effect through the estimated empirical models.
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The Astronomical Journal, 154:166 (14pp), 2017 October Belda et al.
represented in the two components, tis the time relative to
J2000.0, Pis the FCN period, and X0and
Y0
are offsets. The
amplitudes and offsets were estimated for each fitting interval.
The offsets absorb the low-frequency part of the residual signal.
Therefore, the contribution of the FCN to the CIP offsets
(CPO)was computed using Equation (4)after removing the
offsets X0and
Y0
—which are thus ignored.
The estimated CPCs, which should now be almost free of the
FCN signal, were used to recalculate and model the precession
offset and rate, as well as the main nutation amplitudes
corresponding to the periods included in the IAU 2006/2000A
precession-nutation model. Our fitting method is based on the
least-squares (LS)method, with weights taken as the inverse of
the squared errors given by the VLBI estimates (e.g., Gattano
et al. 2017). After LS fitting, the preliminary FCN model was
added back to the remaining residuals in order to recalculate a
final empirical FCN model using the same method as before.
3.1. Corrections to the Precession Offset and Rate
Obtained by Various Approaches
Table 2displays the corrections to the precession offsets and
rates resulting from the LS adjustments. First, note that the
order of magnitude of the correction to the precession offset is
similar for all the approaches, which shows that this parameter
is not sensitive to the a priori values. The corrections w.r.t. the
IAU 2006/2000A precession-nutation model show mean
values close to 40 μas and −70 μas for
b
xand
b
y, respectively,
with formal errors of about 2 μas. And second, the precession
rates exhibit significantly different values depending on the
approach, especially when compared to case A1, i.e., the
a priori CPOs taken from IERS 08 C04 or USNO finals. Note
that the precession rate py, which corresponds to the obliquity
rate, presents the largest deviations (differences near 70 μas/
century—small effect), with formal errors of 50 μas/century
for both components (pxand py). The magnitudes of these
errors are completely in accordance with Liu & Capitaine
(2017)and Gattano et al. (2017).
The inclusion of our FCN model into a priori CPO
coordinates (approach A3)provides the smallest residuals
(rms of 138.5. μas in dX and 140.0 μas in dY after removing the
FCN oscillations). In addition, it causes an rms reduction of
about 15 μas w.r.t. case A1. Analyzing the different TRFs, the
most comparable results are reached between ITRF2008 and
VTRF2008 (approaches B2 and B3)since they are the most
consistent with ICRF2—at least regarding the years of data
used in their realization. On the other hand, the poorest
accuracy (rms)was found with ITRF2014 and USNO finals or
IERS 08 C04 as a priori EOPs, mainly because these EOP
series are not consistent with this frame. Note that the inclusion
of additional more recent VLBI data up to 2015.0 causes a
significant reduction of the formal error in precession rates, of
about 50%. However, the estimated rates using different time
spans (1990–2015 versus 1990–2010)show significant differ-
ences of about 170 μas/cy and 100 μas/cy for pxand py,
respectively.
3.2. Corrections to the Main Nutation Amplitudes
In this section, we use our previous global VLBI solutions to
fit a set of corrections to the amplitudes given in IAU2000A.
That theory was fitted to observations in 2000 using only 21
terms (Herring et al. 2002)and their WRMS were noticeably
larger than the WRMS attainable at present. We start from a set
of 197 terms that correspond to the lunisolar nutations with
amplitudes larger than 4 μas, and are hence smaller than the set
used by Petrov (2007)for fitting an empirical harmonic model
for the Earth’s three Euler angles. When two close frequencies
do not meet the separation criterion, the frequency with larger
amplitude in IAU2000 was kept and the other one was removed
from the fitting. The criterion is the standard one, which was
successfully applied by Petrov before: the frequency separation
was considered unfeasible when the difference was smaller
than 2π/ΔT, where ΔTdenotes the interval length of
observations, here 20 years; the resulting difference in
frequencies is less than 10 8
»-rad
s
1-
. In this way, the total
number of constituents used in the fit was reduced to 178. The
corrections to the remaining amplitudes were derived for each
of the global solutions described in the previous sections,
together with their formal errors that stayed below 3 μas—all
but the one for the 18.6-yr period.
Table 3contains a summary of the most significant
amplitude deviations, ordered by decreasing absolute values
of the period. Only the terms with amplitude corrections larger
than three times the median error are displayed, using a row for
each CPO. The arguments for each period are the same that
appeared in IAU2000A and are thus not displayed for the sake
of concision. Columns headed by As and Ac show the
Table 3
Main Corrections to the IAU2000A Nutation Amplitudes
Median Range
Median
Error
Period (days)CPO As Ac As Ac As Ac
6798.383 dX −6.7 63.7 2.0 10.4 2.9 3.7
dY 20.8 −63.0 4.3 6.3 2.9 3.7
3399.192 dX −7.1 5.6 2.1 1.0 3.0 2.8
dY −6.3 −11.4 4.6 2.6 3.0 2.8
1615.748 dX −1.4 −6.8 3.4 1.7 2.7 2.8
dY 4.0 −8.7 3.5 2.9 2.7 2.8
1305.479 dX 0.1 5.6 1.5 1.4 2.8 2.8
dY 7.2 12.2 1.6 1.1 2.8 2.8
1095.175 dX 2.7 −15.1 3.0 0.6 2.7 2.8
dY 9.3 2.1 2.5 1.8 2.7 2.8
182.621 dX 10.6 −12.7 4.3 5.5 2.7 2.8
dY 24.6 10.2 1.6 2.9 2.7 2.8
169.002 dX 8.6 0.4 3.3 2.3 2.7 2.7
dY 4.2 4.6 5.0 2.5 2.7 2.7
29.531 dX 4.0 −0.3 4.2 2.7 2.8 2.8
dY 6.4 8.7 4.2 1.8 2.8 2.8
27.555 dX −3.5 −16.4 3.0 3.6 2.7 2.7
dY 10.3 −5.9 6.1 4.7 2.7 2.7
27.093 dX 3.2 −6.0 3.0 2.0 2.7 2.7
dY 3.9 11.9 3.4 1.9 2.7 2.7
26.985 dX −3.1 4.3 1.9 3.3 2.8 2.8
dY 1.9 −9.9 4.2 5.4 2.8 2.8
25.325 dX −2.1 −2.9 3.2 1.6 2.7 2.8
dY 8.3 −0.4 2.9 2.8 2.7 2.8
13.749 dX 1.6 3.0 3.0 2.2 2.6 2.7
dY 0.4 9.2 4.6 1.2 2.6 2.7
13.661 dX −22.4 −12.2 12.7 3.8 2.7 2.7
dY −0.9 5.9 7.2 6.7 2.7 2.7
Note. Corrections are estimated from the different VLBI global solutions (see
Table 1)and feature the median amplitude, range and the median of the formal
errors for each term. As and Ac correspond to sine and cosine components,
respectively. Units: μas.
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The Astronomical Journal, 154:166 (14pp), 2017 October Belda et al.
corrections to the amplitudes of the sine and cosine terms for
the relevant argument, respectively. The values given in the
table are obtained as the median of the corresponding
corrections provided by each of the different approaches.
Table 3also displays the range of amplitude corrections (i.e.,
the maximum difference between the maximum and minimum
values for all the solutions)and the median of the formal errors.
The comparison of both parameters is useful to get more insight
into the magnitude of the actual errors and repeatability.
Mainly, the most significant deviations occur at the longest
periods, particularly the 18.6-yr nutation.
4. Detailed Study of the Remaining Residuals
After the estimation of the selected set of nutation
amplitudes, including the free motion associated with the
FCN, a detailed study of the remaining residuals is presented.
To ensure the “cleanness”of the residual time series we
discarded the rather noisy data before 1993—a fact also pointed
out by Chao & Hsieh (2015). Consequently, our study is solely
based on post-1993 data, for better quality. Figure 5displays
the FCN model that was calculated with a sliding window
length of 400 days and a step size of 50 days using a priori
CPOs from USNO finals (approach A1.a), along with the
Fourier spectra estimated from XFCN and
Y
FCN(Equation (4)),
and the complex quantity XFCN +i
Y
FCN. The FCN signal, with
its dominant retrograde period that is somewhat longer than a
year, is well captured across the Fourier spectra. Note that some
more long-periodic signals appear in the retrograde nutation
frequency band (Figure 5bottom right); we will investigate
these below.
In the time series of the parameters named X0and
Y0
in
Equation (4), estimated together with the FCN corrections, which
absorb the low-frequency signals and suppress the high-
frequency signals, we clearly detect two long-periodic oscilla-
tions by means of Fourier analysis performed independently for
both the real variables X0and
Y0
and the complex notation X0+i
Y0
(Figure 6). First, a clear signal at about 2069 days (≈5.7 years)
Figure 5. Upper plot: CPOs (blue dots)and FCN model (red line). Lower plots: Fourier spectra estimated from XFCN and YFCN (left)and XFCN +iYFCN (right)using
different window lengths and different step sizes between subsequent fits.
8
The Astronomical Journal, 154:166 (14pp), 2017 October Belda et al.
Figure 6. Fourier spectra of the parameters X
0
and Y0(left), and X
0
+iY0(right), estimated along with the FCN, with different window lengths and step sizes. The 1σ
and 3σconfidence levels are displayed as horizontal black and red lines, respectively. Green vertical lines are inserted at the periods that exhibit significant signals:
1034 and 2069 days.
Figure 7. Left: constant offsets X
0
estimated usingEquation (4), with different sliding window lengths and step sizes. Right: fit of the precession constant X
0
based on
the residuals after the fit of the main nutation amplitudes and the FCN amplitudes.
Figure 8. Comparison of the offsets X
0
and Y0estimated in this study, provided by Malkin (2013a)(http://www.gao.spb.ru/english/as/persac/), and based on the
residuals after empirical correction of the main nutation amplitudes and the FCN amplitudes using a sliding window length of 400 days and a step size of 100 days.
9
The Astronomical Journal, 154:166 (14pp), 2017 October Belda et al.
can be easily identified in the remaining residuals, predominantly
in the Xcomponent. While adopting a fixed value for the FCN
period, its time-variable amplitudes and phases were estimated
based on various sliding window lengths (400, 500, and 600
days)displaced by different step sizes (50, 100, and 200 days).
This test allows us to assess possible impacts due to the
assumption of certain numerical values avoiding misleading
conclusions caused by possible hidden mathematical artifacts. As
canbeseenintheFourierspectra(Figure 6)and the X0parameter
(Figure 7), essentially the same behavior and peaks at 2069 days
wereobtainedwhen using different window lengths and step
sizes. This indicates that this predominant retrograde signal is of
physical nature.
Next, X0and
Y0
, the window-dependent constant offsets
estimated before fitting the amplitudes of the main nutation
terms of the IAU 2006/2000A model, were compared to the
similar parameters determined by Malkin (2013a), which were
computed using 430 day long groups of data and a step size of
Figure 9. SOI (bottom)and its Fourier spectra considering different periods: 1993 to 2010, corresponding to the time span of the VLBI data (left), and 1987 to 2015
(right).
Figure 10. Fourier spectra of the offsets X
0
and Y0(left), and X
0
+iY0(right)estimated using Equation (4)after removing the 2069-day signal, for a selection of
different window lengths and step sizes. The 1σand 3σconfidence levels are displayed as horizontal black and red lines, respectively. Green vertical lines are inserted
at the periods of 1034 days and 2069 days discussed in the text.
10
The Astronomical Journal, 154:166 (14pp), 2017 October Belda et al.
1 day. The results for Malkinʼs solution and ours with a
window length of 400 days and advancing each 100 days, are
shown in Figure 8. The two curves have a similar shape, which
indicates positive validation of our new solution. A third
solution is displayed in the figure that was obtained as follows:
the signal composed by the corrections to the IAU2000
amplitudes is subtracted from the residuals and then the FCN
time-varying amplitude is re-fitted too, again by means of a
sliding window method with a 400-day window length and a
100-day step forward, which provides a new time series (X0,
Y0
)
Figure 11. Upper row: comparison of the offsets X
0
and Y0estimated in this study in comparison to the values provided by Malkin (2013a)(http://www.gao.spb.ru/
english/as/persac/), as well as the fitofX
0
and Y0based on the residuals after empirical correction of the main nutation amplitudes and the FCN amplitudes using a
sliding window length of 400 days and a step size of 100 days. Middle: offsets (X
0
,Y0)estimated using the FCN equations with different window width strategies
shown in the legend. Bottom: fit of the offsets using the remaining residuals. Left: Xcomponent, right: Ycomponent.
11
The Astronomical Journal, 154:166 (14pp), 2017 October Belda et al.
showing less variability than the other two. The sinusoidal
signal is superimposed on the third solution and looks to be
quite in agreement with it (left plot).
That apparent signal with a period of about 5.7 yr shows
amplitudes of about 25 μas and 10 μas in Xand Y, respectively,
which could be taken into consideration for future improve-
ments and could be attributed to a variety of potential
candidates that share similar periodicity. It could be caused/
mimicked by, e.g., (1)strong ENSO (El-Nino Southern
Oscillation)events; (2)geomagnetic jerk (GMJ)events as
Figure 12. Morlet–Wavelet energy spectrum of the constant offsets X
0
(top)and Y0(bottom)estimated using Equation (4). The x-axis shows the time domain (years)
and the y-axis shows the frequency domain as given in periods (days). Constant offsets are estimated with a sliding window length of 400 days and a step size of 100
days. The green horizontal dashed lines denote the period of 1035 days where the FICN is expected (Mathews et al. 2002). Approach A1. Units: power.
Figure 13. Coefficients
A
c(top)and
A
s(bottom)of different empirical FCN models computed with a sliding window length of 400 days and a step size of 50 days
using USNO finals as a priori EOPs, along with the values published in the IERS Conventions (2010)and the coefficients estimated by Malkin (2013a), during the
interval 1990–2010.
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The Astronomical Journal, 154:166 (14pp), 2017 October Belda et al.
pointed out by Malkin (2013a)and Shirai et al. (2005);(3)
errors in the nearby 2120.652 day nutation term determination;
(4)a modulation phenomenon associated with the proximity of
the FCN and nearly annual frequencies (see, e.g., Gattano et al.
2016);or(5)the 5.8±0.8 yr oscillation present in the length
of day (LOD)attributed to gravitational coupling between the
mantle and inner core (Mound & Buffet 2006), among other
potential factors. The investigation of the origin of that signal is
out of the scope of our paper, which focuses on improving CPO
modeling with an empirical approach. Nevertheless, the ENSO
possibility (1)was roughly investigated by performing a
spectral analysis of the time series representative of the
Southern Oscillation index—SOI (Figure 9). The results were
not conclusive since they lack robustness: analyzing the period
1993–2010, the highest spectral peak is located around 1500
days, but extending the time span from 1987 to 2015 the
maximum power moves to the neighborhood of 2000 days.
After pre-whitening the residuals from the 5.7 yr signal, the
remaining residuals were again reexamined using the offsets X0
and
Y0
- determined from Equation (4). Now, the harmonic
constituent with maximal amplitude is exactly located at 1034
days (2.8 years)for both Xand Ycomponents in all the tested
scenarios (Figure 10, left). That period is almost equal to
the 1035±52 days that Mathews et al. (2002)assigned to the
FICN in the celestial frame.
5
The possibility of associating the
detected 1034-day periodic term to the FICN was assessed by
means of Fourier analysis estimated from the complex quantity
XYi
00
+. The obtained Fourier spectra (Figure 10, right)
provide clear evidence of a signal in the retrograde nutation
frequency band (negative period)at about 1034 days.
Figure 11 is similar to Figure 8, but the third solution has
been changed by a new one derived from the residuals after
cleaning the 5.7-yr oscillation. It can be seen in the upper plots
of Figure 11 that the slow variation patterns of X0and
Y0
associated with the 18.6-yr and 5.7 yr periods have disappeared
after applying the empirical corrections derived so far in the
paper. The periodic signal near 1034 days that was mentioned
before is now dominant and can be easily identified and
recognized in the time series offsets (Figure 11, upper row);
this signal is implicit in the remaining residuals (Figure 11,
bottom row)and the robustness of its determination is
illustrated in (Figure 11, middle row). In order to study the
temporal variation of this signal, we determined the distribution
of energy within the data by using a Morlet–Wavelet power
spectrum (Figure 12).
The wavelet analysis (Figure 12)reveals a temporally relatively
stable oscillation with a period of about 1034 days at the level of
about 10 μas, especially in the Xcomponent between 1993 and
2010, and in the Ycomponent between 1995 and 2006. The
strong energy distribution is less clear and smaller in the Y
component in early and late years, which is probably caused by
an aliasing effect with other frequencies (e.g., 1500 days),but
wavelet scalograms should not be seriously interpreted at the data
borders due to possible edge effects. These pictures are similar for
all the analyzed series in this study and for all the various
approaches (Figure 11,middle). Therefore, the same conclusions
can be drawn when using different sliding window lengths and
step sizes for the fits.
Finally, in Figure 13 we show a comparison of the different
estimated empirical FCN models using different a priori CPO
coordinates (cases A1, A2, and A3)w.r.t. current models (IERS,
i.e., Lambert 2007, and Malkin 2013a)in order to ensure the
consistency of our results and the followed methodology during
the study. Note that all the amplitudes
A
cand
As
are reasonably
close to each other along the whole period.
5. Conclusions
In this paper, we obtain corrections to the precession offsets
and rates, and to the amplitudes of a wide set of terms of the IAU
2006/2000A precession-nutation model. These corrections are
derived from several solutions of the celestial pole coordinates
determined from a global analysis of VLBI sessions in the period
1990–2010 applying different a priori data, such as EOPs and
TRFs. We find that the tested TRFs do not have a considerable
influence on the CPO estimates in our approach. However, using
Table 4
Solution Difference Statistics/Statistical Analysis of the Residuals
A Priori Values Before re-fitting After re-fitting Removing Signals Removing
IAU 2006/2000A IAU 2006/2000A 5.7 years/2.8 years FCN Model
TRF EOP
r
x
r
y
r
x
r
y
r
x
r
y
r
x
r
y
USNO finals
ITRF2014 CPO 183.5 196.2 170.5 179.9 169.5 179.6 133.1 146.7
IAU 2006/2000A 175.4 180.7 162.8 164.9 161.7 164.6 124.3 128.2
IAU 2006/2000A +FCN 174.2 180.2 161.9 164.3 160.7 164.0 123.2 127.8
IERS 08 C04
ITRF2014 CPO 196.8 200.1 181.9 183.8 181.3 183.5 148.5 152.6
IAU 2006/2000A 175.2 179.7 162.7 164.0 161.6 163.7 123.6 127.7
IAU 2006/2000A +FCN 174.5 179.0 162.6 163.9 161.5 163.6 123.5 127.6
USNO finals
ITRF2008 CPO 179.7 194.4 167.0 177.9 165.9 177.6 129.0 144.3
VTRF2008 CPO 180.1 194.6 167.5 178.1 166.4 177.8 129.7 144.4
Note. Comparison of root mean squares rof the residuals before and after fitting the main nutation amplitudes of the IAU 2006/2000A precession-nutation models,
considering observed periodic signals of about 2.8 yr and 5.7 yr, plus the FCN. Units: μas.
5
As pointed out by the cited study, that period is retrograde-diurnal w.r.t the
terrestrial frame and prograde long-periodic w.r.t the celestial frame.
13
The Astronomical Journal, 154:166 (14pp), 2017 October Belda et al.
different a priori CPO results in slight deviations, particularly to
the precession rate and the scatter of the residuals. The error
analyses, displayed in Tables 2and 4, show that the most accurate
results (minimum rms)are always obtained when including the
IAU 2006/2000A model plus the FCN model published by
Belda et al. (2016)as a priori CPC (case A3).
Including the VLBI sessions after 2010 leads to increasingly
different precession rate estimates. The reason is not obvious;
although a plausible explanation could be that the ICRF2 is only
fully consistent with data before 2009 March. After this date, the
nutation estimates by VLBI could absorb the limitations of the
CRF. This means that the ICRF2 needs to be updated as already
done for the current conventional EOP series and TRF, in order to
improve the consistency with the more recent VLBI data. A new
release, “ICRF3,”is expected to be presented at the upcoming
IAU General Assembly in 2018.
A statistical analysis of the residuals (Table 4)between the
reported time series (approaches A and B)before and after re-
fitting the main nutation terms demonstrates that the empirical
corrections (Table 3)attain an error reduction by almost 15 μas
(≈0.5 millimeters)for both the Xand Ycomponents. Further-
more, using the most consistent VLBI data w.r.t ICRF2 revealed
two apparent predominant retrograde signals, one at about 2069
days and one at about 1034 days, the former being clearly
statistically significant. Considering these long-term periodic
signals results in a modest mean improvement of about 1 μas
and 0.3 μas in the Xand Ydirections, respectively, as can be seen
by comparing the pairs of columns 5/7and6/8inTable4.
Finally, the rms of the residuals after FCN amplitudes have been
removed show mean values close to 130 μas (Table 4).
For further improvement of the estimates in the future, we think
that the following points should be addressed, among others: (i)
put more stringent constraints on the theoretical estimates; (ii)
identify inconsistencies between TRFs and CRFs; (iii)consider
further unmodeled geophysical signals; (iv)update the ICRF2 in
terms of consistency with the involved TRF; and (v)combine
different space geodetic techniques (i.e., VLBI and LLR).
This work was funded and realized in the framework of the
project AYA2016-79775-P (AEI/FEDER, UE)and APOSTD/
2026/079. The authors also acknowledge the IVS and all its
components for providing VLBI data (Nothnagel et al. 2015).
We thank the anonymous referee for valuable comments and
suggestions.
ORCID iDs
Santiago Belda https://orcid.org/0000-0003-3739-6056
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