Numerical evidence for the inconsistent separation of the ITRF-ICRF transformation into precession-nutation, diurnal rotation and polar motion

Transcription

1 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, September 2007, BIM, aris

2 A computation of the celestial pole direction as induced by geodetic observations and its comparison with the Celestial Intermediate ole Athanasios Dermanis and Dimitrios Tsoulis Aristotle University of Thessaloniki IERS Workshop on Conventions, September 2007, BIM, aris

3 A geodesist s point of 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

4 Comparing geodetic data with precession-nutation theory IAU2000 precession-nutation theory refers to the Celestial Intermediate ole (CI The CI 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 CI

5 Attention!!!!!! Updating a theory-provided rotation matrix R 0 from the left (δr L and the right (δr R using geodetic data, does not provide an update to precession-nutation (δr L and update of LOD and estimates of polar motion (δr R, respectively. ICRF R = δ R R δ R L 0 R ITRF not only an update of precession-nutation not only an update of LOD and an estimate of polar motion

6 Attention!!!!!! Updating a theory-provided rotation matrix R 0 from the left (δr L and the right (δr R using geodetic data, does not provide an update to precession-nutation (δr L and update of LOD and estimates of polar motion (δr R, respectively. ICRF R = δ R R δ R L 0 R ITRF They both contribute to - precession-nutation, -LOD - polar motion ROOF: A simple exercise in matrix algebra

7 Attention!!!!!! Updating a theory-provided rotation matrix R 0 from the left (δr L and the right (δr R using geodetic data, does not provide an update to precession-nutation (δr L and update of LOD and estimates of polar motion (δr R, respectively. ICRF R = δ R R δ R L 0 R ITRF R = δq( δx, δy Q ( X, Y R ( ψ R ( δψ R ( x R ( y

8 Attention!!!!!! Updating a theory-provided rotation matrix R 0 from the left (δr L and the right (δr R using geodetic data, does not provide an update to precession-nutation (δr L and update of LOD and estimates of polar motion (δr R, respectively. ICRF R = δ R R δ R L 0 R ITRF They both contribute to - precession-nutation, -LOD - polar motion

9 Attention!!!!!! Updating a theory-provided rotation matrix R 0 from the left (δr L and the right (δr R using geodetic data, does not provide an update to precession-nutation (δr L and update of LOD and estimates of polar motion (δr R, respectively. ICRF R = δ R R δ R L 0 R ITRF They both contribute to - precession-nutation, -LOD - polar motion Cannot be directly used for verifying precession-nutation theory e.g. small δx, δy in Q = δq(δx,δy Q IERS (IERS Conventions, Ch. 5 do not compare directly IAU200 precession-nutation

10 OUR AROACH THEORY Theory of precession-nutation provides direction of instantaneous rotation axis Removal of selected precession-nutation theoretical components defines the Celestial Intemediate ole (CI OBSERVATION Theory is updated by observational evidence to provide an observed rotation matrix R from terrestrial to celestial reference system mathematical compatibility provides COMARISON an observed Compatible Celestial ole (CC Computation of CC CI differences

11 OUR AROACH THEORY Theory of precession-nutation provides direction of instantaneous rotation axis Removal of selected precession-nutation theoretical components defines the Celestial Intemediate ole (CI OBSERVATION Theory is updated by observational evidence to provide an observed rotation matrix R from terrestrial to celestial reference system mathematical compatibility provides COMARISON an observed Compatible Celestial ole (CC Computation of CC CI differences STO

12 EARTH ROTATION COMONENTS recession-nutation Diurnal Rotation olar motion Q( X, Y R ( s ( θ 3 3 ( s ( x ( y R R3 R2 R1 3 C r 3 ω r T ω r ω 2 T X Y 2 C x y ψ 1 C ψ = s+ θ + s 1 T celestial reference system 1 C, 2 C, 3 C terrestrial reference system 1 T, 2 T, 3 T IERS earth rotation representation: R = δq( δx, δy Q ( X, Y R ( ψ R ( x R ( y Separation by NRO conditions ψ = sxy (, + θ + s ( x, y

13 THE CELESTIAL INTERMEDIATE OLE R ( ψ 3 IERS Representation: Diurnal rotation around the Celestial Intermediate ole (CI CI = Direction provided by theoretical earth rotation after removal of particular frequency terms Q ( X, Y 0 THE COMATIBLE CELESTIAL OLE 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 r ω by mathematical compatibility. r ω Compatible Celestial ole (CC = direction of the rotation vector mathematically compatible with the IERS provided rotation matrix R COMATIBLE EARTH ROTATION RERESENTATION R ( ψ = R ( s θ + s r ω θ dθ / dt = ω = r ω Diurnal rotation 3 3 takes place around and diurnal rotation angle satisfies: (compatibility in direction and magnitude

14 ω r Mathematical separation of of the the rotation matrix matrix R into into precession-nutation, diurnal diurnal motion motion (LOD (LOD and and polar polar motion motion = rotation vector, with components ω (celestial and ω (terrestrial C T d T [ ωc ] = R R [ ωt ] = dt R T dr dt The mathematically induced Compatible Celestial ole (CC has components celestial terrestrial n C X% 1 = ωc = Y% ω X% Y% n x% 1 = ω = y% ω 1 x% y% T T 2 2 ω = ωω = ωω T T C C T T

15 COMATIBLE EARTH ROTATION RERESENTATION R = Q( XY %, % R ( ψ % R ( x % R ( y % where ψ% = s% ( X %, Y % % θ s% ( x%, y% % θ (% τ = A+ B UT1 = A+ B(UTC % τ % τ = UTC UT1 COMUTATIONS R ( ψ% = Q( XY %, % RR ( y% R ( x% NRO conditions ψ% s% ( X%, Y% s% ( x%, y% % θ = s% ( X%, Y% s% ( x%, y% ψ% % θ θτ ( A τ = UTC % % % % τ = UTC UT1 B

16 Comparison of the CC with the Celestial Intermediate ole (CI recession-nutation components X % X X % X T 1 T 1 T 2 Units = meters on the earth surface (30 m 1 arcsec Two dominant components with periods: T 1 = days T 2 = 13.6 days

17 Comparison of the CC with the Celestial Intermediate ole (CI Y% Y recession-nutation components Y% Y T 1 T 1 T 2 Units = meters on the earth surface (30 m 1 arcsec Two dominant components with periods: T 1 = days T 2 = 13.6 days

18 Comparison of the CC with the Celestial Intermediate ole (CI x% x olar motion components x% x T 1 T 1 T 2 Units = meters on the earth surface (30 m 1 arcsec Two dominant components with periods: T 1 = days T 2 = 14.2 days

19 Comparison of the CC with the Celestial Intermediate ole (CI y% y olar motion components y% y T 1 /2 T 1 T 2 Units = meters on the earth surface (30 m 1 arcsec Two dominant components with periods: T 1 = days T 2 = 14.2 days

20 VALIDATION OF RESULTS ART 1 Computation with 4 different methods from original IERS data: R = R( p p = { XY,, δ X, δy, τ, x, y} 1 NUMERICAL 2 ANALYTICAL p R numerical differentiation R & p numerical differentiation p& ω (, C = ωc RR & ω (, C = ωc pp& ω = ω ( RR, & ω = ω ( pp&, T T T T

21 Separation in components R = [ δqq ] R ( ψ [ R ( x R ( y ] = QD [ ω Q ] = & T [ ω ] QQ D = & T [ ω ] DD = & T ωc = ωq + QωD + QDω ω T = T T T DQωC 3 4 NUMERICAL BY COMONENTS ANALYTICAL BY COMONENTS p QD,, numerical differentiation QD &, &, & ωq, ωd, ω ω, ω C T p numerical differentiation ω = ω ( XYXY,, & Q Q, &, δ X, δy, δx &, δy & ωd = ω D( ψ, ψ& ω = ω ( x, y, x&, y& p& ω C, ω T

22 VALIDATION OF RESULTS ART 2 Stability of numerical differentiation df Determination of derivative fi = ( ti dt from equidistant values: f f( t i = i i 1 t = + t +Δ t i Use of 2k+1 values: f,..., f, f, f,..., f i k i 1 i i+ 1 i+ k Moving polynomial interpolation: ( t = a + at a t ik, 0 1 2k 2k ik, ( tm = fm, m= i k,..., i+ k f = i dik. ( ti dt Various choices of k give essentially identical results!

23 VALIDATION OF OF RESULTS ART ART 3 Effect Effect of of data data noise noise High frequencies in data errors may create large error values in computed derivatives Treatment: Data smoothing by moving averages Simple moving average: f% i = k 1 2k + 1 m= k f i+ m Effect on final results: Somewhat smaller amplitudes for larger k in computed differences between CC & CI parameters. But 2 basic frequencies remain dominant!

24 SECTRA OF DIFFERENCES BETWEEN CC & CI X% X Y% Y % x y% y x

25 CONCLUSIONS Differences between the position of the Compatible Celestial ole (CC and the position of the Celestial Intermediate ole (CI are significant. The respective parameters referring to the celestial (X,Y and the terrestrial reference system (polar motion x, y demonstrate differences which vary in time with two dominant terms: % T T = 13.6 days 1 = days 2 X Y X % T T = 13.6 days 1 = days 2 Y % T 1 = days T 2 = 14.2 days x x % T 1 = days T 2 = 14.2 days y y

26 FUTURE WORK Investigate theoretically the effect of biases & systematic errors in the rotation matrix R, on the CC coordinates XY %, % Investigate theoretically the effect of aliasing on data with diurnal resolution. Higher resolution data available? BEFORE Comparing with CI Instantaneous Celestial ole separation as defined by astronomical theory.