Numerical evidence for the inconsistent separation

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Presentation transcript:

Numerical evidence for the inconsistent separation of the ITRF-ICRF transformation into precession-nutation, diurnal rotation and polar motion Athanasios Dermanis and Dimitrios Tsoulis Aristotle University of Thessaloniki IERS Workshop on Conventions, 20-21 September 2007, BIPM, Paris

A computation of the celestial pole direction as induced by geodetic observations and its comparison with the Celestial Intermediate Pole Athanasios Dermanis and Dimitrios Tsoulis Aristotle University of Thessaloniki IERS Workshop on Conventions, 20-21 September 2007, BIPM, Paris

A geodesist’s point of view Do not include astronomical / geophysical hypotheses in data analysis for the estimation of parameters which can be determined by geodetic observations in a hypothesis-free way Then data analysis provides theory-independent parameters appropriate for comparison with theoretical results Theory verification / Data validation

Comparing geodetic data with precession-nutation theory IAU2000 precession-nutation theory refers to the Celestial Intermediate Pole (CIP) The CIP is not “observable” (its position cannot be determined by observations) because it is defined by purely theoretical means in the framework of a particular solution and a particular mathematical representation The real observable is the 3-parameter rotation matrix R from the terrestrial to the celestial reference system From observed R it is possible to determine the direction (and modulus) of the instantaneous earth rotation vector and not the direction of the CIP

Attention !!! Updating a theory-provided rotation matrix R0 from the left (RL) and the right (RR) using geodetic data, does not provide an update to precession-nutation (RL) and update of LOD and estimates of polar motion (RR), respectively. ICRF ITRF not only an update of precession-nutation not only an update of LOD and an estimate of polar motion

Attention !!! Updating a theory-provided rotation matrix R0 from the left (RL) and the right (RR) using geodetic data, does not provide an update to precession-nutation (RL) and update of LOD and estimates of polar motion (RR), respectively. ICRF ITRF They both contribute to - precession-nutation, - LOD - polar motion PROOF: A simple exercise in matrix algebra

Attention !!! Updating a theory-provided rotation matrix R0 from the left (RL) and the right (RR) using geodetic data, does not provide an update to precession-nutation (RL) and update of LOD and estimates of polar motion (RR), respectively. ICRF ITRF

Attention !!! Updating a theory-provided rotation matrix R0 from the left (RL) and the right (RR) using geodetic data, does not provide an update to precession-nutation (RL) and update of LOD and estimates of polar motion (RR), respectively. ICRF ITRF They both contribute to - precession-nutation, - LOD - polar motion

Attention !!! Updating a theory-provided rotation matrix R0 from the left (RL) and the right (RR) using geodetic data, does not provide an update to precession-nutation (RL) and update of LOD and estimates of polar motion (RR), respectively. ICRF ITRF They both contribute to - precession-nutation, - LOD - polar motion Cannot be directly used for verifying precession-nutation theory e.g. small X, Y in Q = Q(X,Y) QIERS (IERS Conventions, Ch. 5) do not compare directly IAU200 precession-nutation

OUR APPROACH THEORY Theory of precession-nutation provides direction of instantaneous rotation axis “Removal” of selected precession-nutation theoretical components defines the Celestial Intemediate Pole (CIP) OBSERVATION Theory is updated by observational evidence to provide an “observed” rotation matrix R from terrestrial to celestial reference system mathematical compatibility provides COMPARISON an “observed” Compatible Celestial Pole (CCP) Computation of CCP – CIP differences

OUR APPROACH THEORY Theory of precession-nutation provides direction of instantaneous rotation axis “Removal” of selected precession-nutation theoretical components defines the Celestial Intemediate Pole (CIP) OBSERVATION Theory is updated by observational evidence to provide an “observed” rotation matrix R from terrestrial to celestial reference system mathematical compatibility provides COMPARISON an “observed” Compatible Celestial Pole (CCP) STOP Computation of CCP – CIP differences

EARTH ROTATION COMPONENTS Precession-Nutation Diurnal Rotation Polar motion 3C 3T 2T 2C 1C 1T celestial reference system 1C, 2C, 3C terrestrial reference system 1T, 2T, 3T IERS earth rotation representation: Separation by NRO conditions

THE CELESTIAL INTERMEDIATE POLE IERS Representation: Diurnal rotation around the Celestial Intermediate Pole (CIP) CIP = Direction provided by theoretical earth rotation after removal of particular frequency terms THE COMPATIBLE CELESTIAL POLE IERS provided rotation matrix R, as updated by observations, defines an estimate of the complete earth rotation and thus also a corresponding rotation vector estimate by mathematical compatibility. Compatible Celestial Pole (CCP) = direction of the rotation vector mathematically compatible with the IERS provided rotation matrix R COMPATIBLE EARTH ROTATION REPRESENTATION Diurnal rotation takes place around and diurnal rotation angle satisfies: (compatibility in direction and magnitude)

Mathematical separation of the rotation matrix R into precession-nutation, diurnal motion (LOD) and polar motion = rotation vector, with components (celestial) and (terrestrial) The mathematically induced Compatible Celestial Pole (CCP) has components celestial terrestrial

COMPATIBLE EARTH ROTATION REPRESENTATION where COMPUTATIONS NRO conditions

Comparison of the CCP with the Celestial Intermediate Pole (CIP) Precession-nutation components T1 T1 T2 Units = meters on the earth surface (30 m  1 arcsec) Two dominant components with periods: T1 = 186.1 days T2 = 13.6 days

Comparison of the CCP with the Celestial Intermediate Pole (CIP) Precession-nutation components T1 T1 T2 Units = meters on the earth surface (30 m  1 arcsec) Two dominant components with periods: T1 = 186.2 days T2 = 13.6 days

Comparison of the CCP with the Celestial Intermediate Pole (CIP) Polar motion components T1 T1 T2 Units = meters on the earth surface (30 m  1 arcsec) Two dominant components with periods: T1 = 341.2 days T2 = 14.2 days

Comparison of the CCP with the Celestial Intermediate Pole (CIP) Polar motion components T1/2 T1 T2 Units = meters on the earth surface (30 m  1 arcsec) Two dominant components with periods: T1 = 341.2 days T2 = 14.2 days

1 2 VALIDATION OF RESULTS – PART 1 Computation with 4 different methods from original IERS data: 1 2 NUMERICAL ANALYTICAL numerical differentiation numerical differentiation

3 4 Separation in components NUMERICAL BY COMPONENTS ANALYTICAL BY COMPONENTS numerical differentiation numerical differentiation

VALIDATION OF RESULTS – PART 2 Stability of numerical differentiation Determination of derivative from equidistant values: Use of 2k+1 values: “Moving” polynomial interpolation: Various choices of k give essentially identical results!

VALIDATION OF RESULTS – PART 3 Effect of data noise High frequencies in data errors may create large error values in computed derivatives Treatment: Data smoothing by moving averages Simple moving average: Effect on final results: Somewhat smaller amplitudes for larger k in computed differences between CCP & CIP parameters. But 2 basic frequencies remain dominant !

SPECTRA OF DIFFERENCES BETWEEN CCP & CIP 13.6 186.1 13.6 186.2 14.2 341.2 14.2 341.2

CONCLUSIONS Differences between the position of the Compatible Celestial Pole (CCP) and the position of the Celestial Intermediate Pole (CIP) are significant. The respective parameters referring to the celestial (X,Y) and the terrestrial reference system (polar motion xP, yP) demonstrate differences which vary in time with two dominant terms: T1 = 186.1 days T2 = 13.6 days T1 = 186.2 days T2 = 13.6 days T1 = 341.2 days T2 = 14.2 days T1 = 341.2 days T2 = 14.2 days

FUTURE WORK Investigate theoretically the effect of biases & systematic errors in the rotation matrix R, on the CCP coordinates Investigate theoretically the effect of aliasing on data with diurnal resolution. Higher resolution data available? BEFORE Comparing with CIP – Instantaneous Celestial Pole separation as defined by astronomical theory.