# Application of the spectral analysis for the mathematical modelling

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```							Application of the spectral analysis
for the mathematical modelling of
the rigid Earth rotation
V.V.Pashkevich

Central (Pulkovo) Astronomical Observatory
St.Petersburg
Space Research Centre of Polish Academy of Sciences
Warszawa

2004
The aim of the investigation:
Construction of a new high-precision series for the rigid Earth
rotation, dynamically consistent with DE404/LE404 ephemeris
and based on the SMART97 developments.

A L G O R I T H M:
1.   Numerical solutions of the rigid Earth rotation are constructed.
Discrepancies of the comparison between our numerical
solutions and the SMART97 ones are obtained in Euler angles.
2.   Investigation of the discrepancies was carried out by the least
squares (LSQ) and by the spectral analysis (SA) methods. The
secular and periodic terms were determined from the
discrepancies.
3.   New precession and nutation series for the rigid Earth,
dynamically consistent with DE404/LE404 ephemeris, were
constructed.
Numerical            Discrepancies:
Initial conditions      integration of           Numerical
from SMART97          the differential       Solutions minus
equations             SMART97

Precession             LSQ method              6-th degree
terms of            calculate secular      Polinomial of time
SMART97                    terms

Compute new           Remove the secular        SA method
precession                 trend                calculate
parameters           from discreapancies     periodical terms

New precession
and nutation          Construct a new
series              nutation series
The calculations on Parsytec computer with a quadruple precision.
Fig.1. Difference between our numerical solution and SMART97
a) in the longitude.
Kinematical case                              Dynamical case

Secular terms of…                            Secular terms of…
smart97 (as)             - d (as)         smart97(as)              - d (as)
7.00                                           6.89
50384564881.3693 T           - 206.50 T       50403763708.8052 T           - 206.90 T
2                    2                        2                    2
- 107194853.5817 T          - 3451.30 T       - 107245239.9143 T          - 3180.80 T
3                    3                        3                    3
- 1143646.1500 T           1125.00 T          - 1144400.2282 T            1048.00 T
4                    4                         4                   4
1328317.7356 T            - 788.00 T           1329512.8261 T           - 306.00 T
5                    5                        5                    5
- 9396.2895 T            - 57.50 T             - 9404.3004 T           - 65.50 T
6                                             6
- 3415.00 T                                   - 3421.00 T
Fig.1. Difference between our numerical solution and SMART97
b) in the proper rotation.
Kinematical case                                   Dynamical case

Secular terms of…                               Secular terms of…
smart97 (as)               d(as)               smart97 (as)               d (as)
1009658226149.3691             6.58                1009658226149.3691             6.53
474660027824506304.0000 T        99598.30 T        474660027824506304.0000 T        97991.40 T
2                 2                                2                 2
- 98437693.3264 T        - 7182.30 T                 98382922.2808 T        - 6934.40 T
3                 3                               3                  3
- 1217008.3291 T        1066.80 T                  -1216206.2888 T         1004.00 T
4              4                                4                  4
1409526.4062 T         - 750.00 T                  1408224.6897 T         - 226.00 T
5                 5                                  5                5
- 9175.8967 T         - 30.30 T                    - 9168.0461 T         - 37.80 T
6                                                   6
- 3676.00 T                                        - 3682.00 T
Fig.1. Difference between our numerical solution and SMART97
c) in the inclination.
Kinematical case                             Dynamical case

Secular terms of…                            Secular terms of…
 smart97(as)              d(as)           smart97(as)               d(as)
84381409000.0000               1.42          84381409000.0000                1.39
- 265011.2586 T           - 96.61 T          - 265001.7085 T           - 96.73 T
2                 2                          2                   2
5127634.2488 T          - 353.10 T          5129588.3567 T           - 595.90 T
3                   3                        3                   3
- 7727159.4229 T           771.50 T          - 7731881.2221 T          - 945.10 T
4                   4                        4                   4
- 4916.7335 T           - 84.50 T            - 4930.2027 T            - 76.50 T
5                     5                        5                   5
33292.5474 T           - 86.00 T            33330.6301 T             - 70.00 T
6                                            6
- 247.50 T                                   - 247.80 T
Fig.2. Difference between our numerical solution and SMART97
after formal removal of secular trends.
Kinematical case                Dynamical case
Fig.3. Spectra of discrepancies between our numerical solution and
SMART97 for proper rotation angle.
Kinematical case                  Dynamical case
Fig.3. Spectra of discrepancies between our numerical solution and
SMART97 for proper rotation angle.
Kinematical case                 Dynamical case

3

1   2
Fig.3. Spectra of discrepancies between our numerical solution and
SMART97 for proper rotation angle. DETAIL 1
Kinematical case                  Dynamical case
Fig.3. Spectra of discrepancies between our numerical solution and
SMART97 for proper rotation angle. DETAIL 1
Kinematical case                  Dynamical case

B

A
Fig.3. Spectra of discrepancies between our numerical solution and
SMART97 for proper rotation angle. DETAIL 1 A
Kinematical case                  Dynamical case
Fig.3. Spectra of discrepancies between our numerical solution and
SMART97 for proper rotation angle. DETAIL 1 A
Kinematical case                 Dynamical case

II

I
Fig.3. Spectra of discrepancies between our numerical solution and
SMART97 for proper rotation angle. DETAIL 1 A-I
Kinematical case                  Dynamical case
Fig.3. Spectra of discrepancies between our numerical solution and
SMART97 for proper rotation angle. DETAIL 1 A-II
Kinematical case                  Dynamical case
Fig.3. Spectra of discrepancies between our numerical solution and
SMART97 for proper rotation angle. DETAIL 1 A-II
Kinematical case                  Dynamical case
Fig.3. Spectra of discrepancies between our numerical solution and
SMART97 for proper rotation angle. DETAIL 1 A-II (zoom)
Kinematical case                  Dynamical case
Fig.3. Spectra of discrepancies between our numerical solution and
SMART97 for proper rotation angle. DETAIL 1 B
Kinematical case                  Dynamical case
Fig.3. Spectra of discrepancies between our numerical solution and
SMART97 for proper rotation angle. DETAIL 1 B
Kinematical case                  Dynamical case
Fig.3. Spectra of discrepancies between our numerical solution and
SMART97 for proper rotation angle. DETAIL 1 B (zoom)
Kinematical case                  Dynamical case
Fig.3. Spectra of discrepancies between our numerical solution and
SMART97 for proper rotation angle. DETAIL 1 B (zoom)
Kinematical case                  Dynamical case
Fig.3. Spectra of discrepancies between our numerical solution and
SMART97 for proper rotation angle. DETAIL 1 B (zoom2)
Kinematical case                  Dynamical case
Fig.3. Spectra of discrepancies between our numerical solution and
SMART97 for proper rotation angle. DETAIL 2
Kinematical case                  Dynamical case
Fig.3. Spectra of discrepancies between our numerical solution and
SMART97 for proper rotation angle. DETAIL 3
Kinematical case                  Dynamical case
Fig.4. Difference between our numerical solution and SMART97 after
formal removal the secular trends and 9000 periodical harmonics.
Kinematical case                 Dynamical case
Fig.5. Repeated Numerical Solution minus New Series.

Kinematical case                 Dynamical case
Fig.6. Numerical solution minus New Series after formal removal
secular trends in the proper rotation angle.
Kinematical case                  Dynamical case
Fig.6. Numerical solution minus New Series after formal removal
secular trends in the proper rotation angle. (zoom)
Kinematical case                  Dynamical case
The calculations on PC with a double precision.
Fig.7. Numerical solution minus New Series after formal removal
secular trends in the proper rotation angle.

Kinematical case                  Dynamical case
Fig.8. Sub diurnal and diurnal spectra of discrepancies between our
numerical solution and SMART97 for proper rotation angle.
Kinematical case                  Dynamical case
Fig.9. Numerical solution minus New Series including sub diurnal and
diurnal periodical terms after formal removal secular trends in the proper
rotation angle.
Kinematical case                     Dynamical case
CONCLUSION

• Spectral analysis of discrepancies of the numerical solutions and
SMART97 solutions of the rigid Earth rotation was carried out for the
kinematical and dynamical cases over the time interval of 2000 years.
• Construction of a new series of the rigid Earth rotation, dynamically
consistent with DE404/LE404 ephemeris, were performed for
dynamical and kinematical cases.
• The power spectra of the residuals for the dynamical and kinematical
cases are similar.
• The secular trend in proper rotation found in the difference between
the numerical solutions and new series is considerably smaller than
that found in the difference between the numerical solutions and
SMART97.
ACKNOWLEDGMENTS

The investigation was carried out at the
Central (Pulkovo) Astronomical Observatory of Russian Academy of Science
and the
Space Research Centre of Polish Academy of Science,
under a financial support of the
Cooperation between Polish and Russian Academies of Sciences,
Theme No 25 and of the
Russian Foundation for Fundamental Research,
Grant No 02-02-17611.
http://www.csa.ru/

Center for supercomputing applications
The massive-parallel computer system Parsytec CCe20
•   Parsytec CCe20 is a supercomputer of
massive-parallel architecture with separated
memory. It is intended for fulfilment of
high-performance parallel calculations.

Hardware:
• 20 computing nodes with processors PowerPC
604e (300MHz);
• 2 nodes of input-output;
Massive-parallel supercomputers Parsytec is
designed by Parsytec GmbH, Germany,          • The main memory:
using Cognitive Computer technology.              o 32 Mb on computing nodes;
o 64 Mb on nodes of an input / conclusion;
The system approach is based on using of     • disk space 27 Gb;
PC technology and RISC processors
• tape controller DAT;
PowerPC which are ones of the most
powerful processor platforms available today • CD-ROM device;
and are clearly outstanding in price /       • network interface Ethernet (10/100 Mbs);
performance.                                 • communication interface HighSpeed Link (HS-
There are 5 Parsytec computers in CSA now .
SA method for cleaning the discrepancies calculated periodical terms

Discrepancies after removal            Set of nutation terms of
SMART97
the secular trend

LSQ method compute amplitude
of power spectrum of discrepancies

LSQ method determine amplitudes
and phases of the largest rest harmonic       Nutation terms
of SMART97
if |Am| > ||      Compute a new
nutation term                 Construct
No               Yes
a new
Until the end        Remove this harmonic from      nutation
of specta            discrepancy and Spectra        series
Quadruple precision corresponding to 32- decimal
representation of real numbers.

Double precision corresponding to 16- decimal representation
of real numbers.
Fig.5 Repeated Numerical Solution minus New Series.

Kinematical case                 Dynamical case

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