We discuss the formulation of a new nutation series to be used in the reduction of modern space geodetic data. The motivation for developing such a series is to develop a nutation series that has smaller short period errors than the IAU 1980 nutation series and to provide a series that can be used with techniques such as the Global Positioning System (GPS) that have sensitivity to nutations but can directly separate the effects of nutations from errors in the dynamical force models that effect the satellite orbits. A modern nutation series should allow the errors in the force models for GPS to be better understood. The series is constructed by convolving the Kinoshita and Souchay rigid Earth nutation series with an Earth response function whose parameters are partly based on geophysical models of the Earth and partly estimated from a long series (1979-1993) of very long baseline interferometry (VLBI) estimates of nutation angles. Secular rates of change of the nutation angles to represent corrections to the precession constant and a secular change of the obliquity of the ecliptic are included in the theory. Time dependent amplitudes of the Free Core Nutation (FCN) that is most likely excited by variations in atmospheric pressure are included when the geophysical parameters are estimated. The complex components of the prograde annual nutation are estimated simultaneously with the geophysical parameters because of the large contribution to the nutation from the S(sub 1) atmospheric tide. The weighted root mean square (WRMS) scatter of the nutation angle estimates about this new model are 0.32 mas and the largest correction to the series when the amplitudes of the ten largest nutations are estimated is 0.18 +/- 0.03 mas for the in phase component of the prograde 18. 6 year nutation

Herring, T. A.

English

NASA Technical Reports Server

NASA-CR-200040
AN A PRIORI MODEL FOR THE REDUCTION OF
NUTATION OBSERVATIONS: A'SVi 994.3 NUTATION SERIES
T. A. H E R R I N G
Massachusetts Institute of Technology
77 Massachusetts Avenue,
Cambridge, MA. 02139, USA.
r •Abst.ra.ct! We discuss the formulation of a new nutation series to be used
'in the reduction of modern space geodetic data. The motivation for devel-
: oping such a series is to develop a nutation series that has smaller short
. period errors than the IAU 1980 nutat ion series and to provide a series that
can be used wi th techniques such as the Global Positioning System (GPS)
eo • that have sensitivity to nutations but can directly separate the effects of
o M ' nutations from errors in the dynamical force models that effect the satellite
^ to ,$• orbits. A modern nuta t ion series should allow the errors in the force models
^ *o <? . f o r GPS to be bet ter understood. The series is constructed by convolving
2 £ § the Kinoshi ta and Souchay rigid Earth nuta t ion series wi th an Earth re-
sponse funct ion whose parameters are par t ly based on geophysical models
^ 't of the Earth and partly estimated from a long series (1979-1993) of very
x. long baseline interferometry (VLBI) estimates of nutation angles. Secular
2 rates of change of the nutation angles to represent corrections to the pre-
cession constant and a secular change of the obliquity of the ecliptic are
included in the theory. Time dependent amplitudes of the Free Core Nuta-
tion (FCN) that is most likely excited by variations in atmospheric pressure
are included when the geophysical parameters are estimated. The complex
components of the prograde annual nutation are estimated simultaneously
with the geophysical parameters because of the large contribution to the
nutation from the 5i atmospheric tide. The weighted root mean square
(WRMS) scatter of the nutation angle estimates about this new model are
0.32 mas and the largest correction to the series when the amplitudes of
the ten largest nutations are estimated is 0.17 ± 0.03 mas for the in phase
component of the prograde 18.6 year nutation.
222
/. Appenteiler led.). Highlights of Astronomy. Vol. 10. 222-227.
© 1995 IAU. Printed in the Netherlands.
Z O CO 01
x^ U. O W
https://ntrs.nasa.gov/search.jsp?R=19960011482 2020-06-16T05:42:12+00:00Z
223
1. Introduction
The formulation of modern nutat ion series such as the IAU 1980 nutat ions
series (VVahr, 198 L) are based on the (almost) linear response of the Earth
to torques applied to the mantle of the Earth. These torques primarily
arise from the luni-solar forces acting on the equatorial bulge of the Earth
but there are significant contributions from the planets and from torques
associated with the motions of the fluid parts of the Earth (atmosphere,
oceans and fluid inner core). There have also been proposed contributions
from the solid inner core as well (Mathews et a/., 1991a. de Vries and Wahr,
1991). The linear nature of the problem is used to separate the formation of
the nutation series in to two dis t inct parts: (a) the derivation of the nu ta t ions
of a rigid Earth and (b) the derivation of the response function of the Earth
which gives the ratio of the nutations of a "real" Earth to those of the rigid
Earth. Much of the error in the IAU1980 nutations is due to imperfect
knowledge of the geophysical parameters that effect the response function
of the "real" Earth. The IAU 1980 nutations also suffers from the effects of
truncation to 0.1 mas and the neglect of several second-order terms in the
derivation of the rigid series.
In conventional formulations of the transformation between an Earth
fixed frame and inertia! frame, the only nearly diurnal motions are associ-
ated with nutations. However, recent analyses of space geodetic data have
shown that other non-nutation related motions of the Earth rotation axis
and the rate of rotation exist in the diurnal and semidiurnal bands. These
motions arise from ocean tidal effects and tend to be dominated by ti dally
driven currents (Severs et a/.. 1993. Herring and Dong, 1994. Wat kins and
Eanes. 1994). For a complete representation of the transformation from
Earth fixed coordinates to inertial space these additional motions whose
amplitudes can reach 0.5 mas need also to be included. The data set used
here include these other motions in the analysis of the space geodetic data.
A general representation of a nutation series is given by Mathews et al.
(I991a) (see also Wahr (1981). de Vries and Wahr (1991)). The amplitude
of the nutation at frequency v is given by
where CH3td(°") is the nutation amplitude for the rigid Earth with the
same flattening as the real Earth; rjm(a) is the response of the Earth; and
£,Eo.Tth.(0) is tne nutation of the real Earth. The function, r jm(ff) . depends
on many geophysical parameters of the Earth but is most sensitive to the
elastic properties of the Earth, and the flattening of the Earth and its fluid
core. At this time geophysical models of the Earth are not accurate enough
to reliably determine rjm(cr) mainly because of non-hydrostatic forces in
224
the Earth, the effects of mantle anelasticity, and the effects of ocean tides.
However, the form that Tjm(a} should take is probably well represented by
geophysical theory. The derivation of the nutation series discussed here is
based on estimating the parameters in T7m(cr) that are not well known or
are effected by ocean tides and mantle anelasticity.
2. Series derivation
The series derived here is based on the Kinoshita and Souchay (1990) rigid
Earth nutations series, with some slightly motivations discussed below, and
the NASA Goddard Space Flight Center (GSFC) estimates of nutation
angles from the analysis of VLBI data between 1979 and 1993. We refer to
this series as A' 5 ^ 994. (The specific version we discuss here is the A'5V1994 3
series i.e., the third of a series of different types of analyses whose details
are given below).
Before using the Kinoshita and Souchay (1990) rigid Earth nutation
series we scaled the series by 0.9999404 to account for the rate of change
of the nutation in longitude observed in VLBI and Lunar Laser Ranging
( L L R ) data (Herr ing et a/., 1991. Dickey ei a/., 1994) of approximately
-0.3 "/century that we interpret as arising from luni-solar precession and
thus an error in the dynamical ellipticity of the Earth. We also combined
duplicated 'terms in the series (each arising from different sources but wi th
the same fundamental arguments) and we removed semidiurnal nutations
due to the triaxiality of the Earth since these are implicitly included in the
prograde diurnal polar motions terms in the VLBI analysis (see Herring
and Dong, 1994) for discussion). The planetary nutation contributions to
the nutation angles from Kinoshita and Souchay (1990) were removed from
the observed nutation angles before they were used in the analysis.
Two other contributions were removed from observed nutations angles
before they are analyzed: the annual modulation of the geodetic precession
discussed in Fukushima (1991) (annual nutation in longitude 0.15 mas).
and corrections to the 18.6 and 9.3 year nutations for planetary tilt effects
(Williams, 1994) (largest correction 0.14 mas for nutation in longitude). In
the analysis presented here no corrections were applied for ocean tides or
mantle anelasticity. Instead, it was assumed that these corrections could be
absorbed into the parameters of r)m(a).
The form of rjm(cr) used here is based on that of Mathews et al. (1991a)
with a small modification to better represent the effects of ocean tidal
loading. rjm{a) is written as
= R + R'(Sl + a)
225
where fi is the rotation rate of the Earth and we take a to be the frequency
of the nutation as seen on the rotating Earth; R and R' are constants de-
rived from geophysical theory; and the sum over a represents the effects
of resonances in the rotation of the Earth wi th strengths Ra and reso-
nance frequencies aa. In the Mathews et al. (1991a) theory there were four
resonances: the Chandler wobble, denoted CW; the retrograde free core
nutation. RFCN; and two resonances associated with the ,;olid inner core
denoted by PFCN and ICW. The CW and ICW have resona.nce frequencies
far from the diurnal band and have little effect on the nutations. The RFCN
has a marked effect, and the PFCN which has a much smaller strength than
the RFCN could have observable effects. After extensive searching for the
PFCN, we could find no detectable evidence for this resonance. The fit to
the VLBI data when the resonance was included were worse then when
its strength was set to zero, thus indicating that resonance parameters are
sufficiently far from the geophysically determined values, that we have been
unsuccessful in isolating this resonance. In the analysis here we have set the
strengths for the inner core resonances to zero.
The parameters, treated as complex values, estimated from the VLBI
data were R. RRFCK- and PRFCW- Trail analyses were performed with
R' estimated, but we found slightly better fits to the VLBI data if more
explicitly included the effects of loading phenomena. To include the load
effects from ocean tides we modified the nuta t ional response of the Earth
to the following form:
+ "-ir-u"-
where the complex quan t i t y v represents the effects of load (see for example
Wahr and Sasao. 1981). Our analyses shows that the estimates of i> is highly
correlated with estimates of R' and simultaneous estimation of these two
complex parameters tends to be unstable and we have therefore estimated
v and keep R' fixed at its geophysically determined value. The values we
used are from Mathews et al. (1991b).
The formulation above assumes that the forcing of the nutations is either
proportional to the rigid Earth nuta t ion or the torque applied to the Earth
and that the response is modified by the elastic and anelastic parameters
and the resonances of the Earth wi th the primary resonance being that due
to the fluid core. There is however one nutat ion frequency for which these
assumptions are probably not valid. The prograde annual nutation is likely
to be affected by the thermal S\ atmospheric tide which is much larger then
the amplitude of the amplitude of the simple tidal forcing (see for example
Chapman and Lindzen , 1970). We have therefore explicitly estimated the
ampli tude of this nuta t ion while estimating the parameters of the resonance
forcing.
226
The primary role of the RFCN resonance is to modify the amplitudes of
the forced nutat ions but the resonance itself can also be excited by quasi-
random excitations most likely from the atmospheric pressure variations
(Sasao and Wahr, 1981). The amplitude and phase of this freely excited
mode will vary with time (in much the same way that the Chandler wobble
amplitude and phase does) and we therefore included estimates of the freely
excited RFCN in two years intervals after 1984 wi th one estimate for the
1979-1984 interval because of the sparseness and lower quality of the VLBI
data during this time.
The VLBI data set analyzed was the NASA/GFSC analysis of the data
from 1979 to 1993 and consisted of 1840 pairs of estimates of nutation
angles in obliquity and longitude. We root sum square (RSS) added 0.25
and 0.60 mas to the standard deviat ions given in the GSFC analysis to
bring the x2 per degree of freedom of the postfit nutat ion angle residuals
to approximately unity (see Herring et ai, 1991). From this data set we
estimated 26 parameters comprised of 6 complex components for the time
dependent RFCN amplitude, offsets and rates for the nutations in longitude
and obliquity, the amplitude of the prograde annual, and the four complex
parameters R. RRFCN^ <7RFC,\'^ and i>. The VVRMS scatter of the postfit
nutation angle residuals was 0.32 mas.
3. Results and Discussion
The fit to the VLBI nutation estimates was almost the same when only the
26 parameters listed above were estimated compared to a trial analysis in
which corrections to the coefficients to the nuta t ion series were estimated
(0.321 mas versus 0.318 mas) ind ica t ing that the above formulation is
adequate for representing the VLBI nutat ion angles and is likely to be a
good predictor of future nutation angles.
The estimated rates of change of the nutat ions in longitude (interpreted
as a change to the IAU 1976 precession constant) and in obliquity were
-0.298 ± 0.01 "/cent and -0.024 ± 0.005 "/cent. The latter value is in very
good agreement with that recently determined theoretically by Williams
(1994).
The correction to the prograde annual nutation (relative to that value
predicted by the new series) was — 0.01 —tO.09 ± 0.01 mas. The amplitude of
the RFCN varied from 0.34 ± 0.06 mas (1979-1984 interval) to 0.11 ± 0.02
mas (1992-1994 interval) and showed a steady decrease in amplitude over
the 14 year interval of the VLBI data. When corrections to the coefficients of
the series were estimated, all amplitude corrections were less than 0.016 mas
except for the 18.6 year nutation for which the corrections were -0.17+iO.ll
mas and 0.08 — tO.13 mas with uncertainties of 0.10 mas in both cases.
227
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An a priori model for the reduction of nutation observations: KSV(1994.3) nutation series

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