*************************************************************************** Gazette IERS Gazette IERS Gazette IERS Gazette IERS Gazette _____________________________________ No 13, 30 January 1997 / ____________________________________/ Contact: iers@obspm.fr ftp: hpvlbi.obspm.fr (145.238.100.7) WWW: ftp://hpvlbi.obspm.fr/iers/ierscb.html *************************************************************************** Title: INTERPOLATING IERS EARTH ORIENTATION DATA Authors: Dennis McCarthy, Daniel Gambis INTRODUCTION IERS Earth Orientation Parameters (EOP) are produced at regular intervals (daily or longer) with the effects of semidiurnal and diurnal variations removed. Users who require high accuracy information may want to interpolate the published data and include the semidiurnal/diurnal variations. This gazette provides a recommended procedure to follow in order to determine the most accurate Earth orientation for a given instant. Currently all analysis centers contributing to the IERS employ a procedure using a priori values of the EOP which are then corrected through the analysis of observations. These a priori values are best estimates based on the standard knowledge of the Earth's orientation, usually the interpolated tabular values plus a diurnal/semidiurnal model plus lower frequency tides (if appropriate). The corrections to the a priori estimates are determined from the analyses of data taken over some period of time, ranging from minutes to days. Thus, they represent the mean of the unmodelled variations over the length of the analyzed period of time. When the reported estimates (a priori + mean estimated correction) are based on distinct time intervals, they may be referred to as normal points. The IERS provides to users polar motion and UT1 based on the combination of the analysis centers data, using either smoothed estimates in order to reduce observational errors (Bulletins A and B, EOP 97 C 04, see Explanatory Supplement to IERS Bulletins A and B), or normal points (EOP 97 C 01, 02, 03, see IERS 1995 Annual Report, part II.4). These results are provided without the effects of the diurnal/semidiurnal tides. The IERS Conventions (McCarthy 1996) for transformations between terrestrial and celestial frames, however, imply that IERS EOP data are exact estimates at the instant in time reported. SEMIDIURNAL/DIURNAL VARIATIONS The existence of diurnal and semidiurnal variations caused by ocean tides is well known (Eubanks 1993; Sovers et al. 1993; Herring & Dong 1994; Ray et al. 1994). In recent years, through continuous, high-precision VLBI experiments (e.g. ERDE, EPOCH92, CONT93, etc.), analytical models have been derived for the prominent diurnal and semidiurnal tides (Herring and Dong 1991; Brosche et al. 1991; Herring 1993; Herring and Dong 1994; Gipson 1996). This in turn has prompted a refinement in the theoretical models (Brosche et al. 1989; Seiler 1991; Dickman 1989, 1990, 1991, 1993; Gross 1993; Ray et al. 1994; Ray 1995; Seiler and Wunsch 1995). As the observational data have improved in accuracy, the models (both empirical and theoretical) have quickly converged. Although there are differences between recent models (e.g. Ray et al. 1994; Ray 1995; Seiler and Wunsch 1995; Gipson 1996), these differences are less than 10%. The effects of these tides are on the order of 0.1 milliarcseconds (mas) in the polar motion coordinates x and y and 10**-6 s in UT1. The inclusion of any of these models in high-precision, Earth orientation analysis software clearly improves the solution. If the data do not correctly model the diurnal/semidiurnal tide, the errors can reach up to +/-0.1 milliarcseconds (mas). Systematic errors of this magnitude may be unacceptable to high-accuracy users. Possible ambiguities caused by unclear procedures concerning subdiurnal EOPs and the epoch of observation can be eliminated. Although the problems are only on the fringes of detectability, they may cause significant systematic errors. It is expected that eventually subdiurnal observations of EOPs will be possible routinely from a reduction standpoint. RECOMMENDATION The following software is recommended to interpolate the IERS polar motion and Universal Time products and account for the semidiurnal/diurnal variations in the Earths orientation. This procedure makes use of a Lagrangian interpolation scheme and applies the Ray model of the semidiurnal/diurnal variations in the Earth's orientation as recommended in the IERS Conventions (McCarthy, 1996). Any equivalent interpolation scheme could, of course, be substituted. This software can be obtained in machine readable form by anonymous ftp to maia.usno.navy.mil, cd to dist, and get interp.f. or hpvlbi.obspm.fr; cd iers/model; get interp.f (interp.note is the above text) ------------ SUBROUTINE INTERP (RJD,X,Y,T,N,RJDINT,XINT,YINT,TINT) C C THIS SUBROUTINE TAKES A SERIES OF X, Y, AND UT1-UTC VALUES C AND INTERPOLATES THEM TO AN EPOCH OF CHOICE. THIS ROUTINE C ASSUMES THAT THE VALUES OF X AND Y ARE IN SECONDS OF C ARC AND THAT UT1-UTC IS IN SECONDS OF TIME. AT LEAST C ONE POINT BEFORE AND ONE POINT AFTER THE EPOCH OF THE C INTERPOLATION POINT ARE NECESSARY IN ORDER FOR THE C INTERPOLATION SCHEME TO WORK. C C PARAMETERS ARE : C RJD - ARRAY OF THE EPOCHS OF DATA (GIVEN IN MJD) C X - ARRAY OF X POLAR MOTION (ARCSEC) C Y - ARRAY OF Y POLAR MOTION (ARCSEC) C T - ARRAY OF UT1-UTC (SEC) C N - NUMBER OF POINTS IN ARRAYS C RJDINT- EPOCH FOR THE INTERPOLATED VALUE C XINT - INTERPOLATED VALUE OF X C YINT - INTERPOLATED VALUE OF Y C TINT - INTERPOLATED VALUE OF UT1-UTC C DOUBLE PRECISION RJD(N), X(N), Y(N), T(N), . RJDINT, XINT, YINT, TINT, CORX, CORY, CORT CALL LAGINT (RJD,X,N,RJDINT,XINT) CALL LAGINT (RJD,Y,N,RJDINT,YINT) CALL LAGINT (RJD,T,N,RJDINT,TINT) CALL RAY (RJDINT,CORX,CORY,CORT) XINT = XINT + CORX YINT = YINT + CORY TINT = TINT + CORT RETURN END C C---------------------------------------------------------------- C SUBROUTINE LAGINT (X,Y,N,XINT,YOUT) C C THIS SUBROUTINE PERFORMS LAGRANGIAN INTERPOLATION C WITHIN A SET OF (X,Y) PAIRS TO GIVE THE Y C VALUE CORRESPONDING TO XINT. THIS PROGRAM USES A C WINDOW OF 4 DATA POINTS TO PERFORM THE INTERPOLATION. C IF THE WINDOW SIZE NEEDS TO BE CHANGED, THIS CAN BE C DONE BY CHANGING THE INDICES IN THE DO LOOPS FOR C VARIABLES M AND J. C C PARAMETERS ARE : C X - ARRAY OF VALUES OF THE INDEPENDENT VARIABLE C Y - ARRAY OF FUNCTION VALUES CORRESPONDING TO X C N - NUMBER OF POINTS C XINT - THE X-VALUE FOR WHICH ESTIMATE OF Y IS DESIRED C YOUT - THE Y VALUE RETURNED TO CALLER C REAL*8 X(N),Y(N),XINT,YOUT,TERM INTEGER N,I,J C YOUT = 0.0D0 DO 5 I = 1,N-1 IF ( XINT .GE. X(I) .AND. XINT .LT. X(I+1) ) K = I 5 CONTINUE IF ( K .LT. 2 ) K = 2 IF ( K .GT. N-2 ) K = N-2 DO 20 M = K-1,K+2 TERM = Y(M) DO 10 J = K-1,K+2 IF ( M .NE. J ) THEN TERM = TERM * (XINT - X(J))/(X(M) - X(J)) END IF 10 CONTINUE YOUT = YOUT + TERM 20 CONTINUE RETURN END C C---------------------------------------------------------------- C SUBROUTINE RAY (RJD,CORX,CORY,CORT) C C THIS SUBROUTINE IMPLEMENTS THE RAY MODEL FOR C DIURNAL/SUBDIURNAL TIDES. IT USES THE SIMON ET AL. C FUNDAMENTAL ARGUMENTS. THE CORRECTIONS IN X AND Y ARE IN C UNITS OF SEC. OF ARC AND UT1-UTC IN SEC. OF TIME. THESE C CORRECTIONS SHOULD BE ADDED TO "AVERAGE" EOP VALUES TO GET C ESTIMATES OF THE INSTANTANEOUS VALUES. C C PARAMETERS ARE : C RJD - EPOCH OF INTEREST GIVEN IN MJD C CORX - TIDAL CORRECTION IN X (SEC. OF ARC) C CORY - TIDAL CORRECTION IN Y (SEC. OF ARC) C CORT - TIDAL CORRECTION IN UT1-UTC (SEC. OF TIME) C IMPLICIT DOUBLE PRECISION (A-H,O-Z) DOUBLE PRECISION . L, LPRIME HALFPI = 1.5707963267948966d0 T = (RJD - 51544.5D0)/36525.0D0 L = -0.00024470d0*T**4 + 0.051635d0*T**3 + 31.8792d0*T**2 . + 1717915923.2178d0*T + 485868.249036d0 L = DMOD(L,1296000d0) LPRIME = -0.00001149d0*T**4 - 0.000136d0*T**3 . - 0.5532d0*T**2 . + 129596581.0481d0*T + 1287104.79305d0 LPRIME = DMOD(LPRIME,1296000d0) CAPF = 0.00000417d0*T**4 - 0.001037d0*T**3 - 12.7512d0*T**2 . + 1739527262.8478d0*T + 335779.526232d0 CAPF = DMOD(CAPF,1296000d0) CAPD = -0.00003169d0*T**4 + 0.006593d0*T**3 - 6.3706d0*T**2 . + 1602961601.2090d0*T + 1072260.70369d0 CAPD = DMOD(CAPD,1296000d0) OMEGA = -0.00005939d0*T**4 + 0.007702d0*T**3 . + 7.4722d0*T**2 . - 6962890.2665d0*T + 450160.398036d0 OMEGA = DMOD(OMEGA,1296000d0) THETA = (67310.54841d0 + . (876600d0*3600d0 + 8640184.812866d0)*T + . 0.093104d0*T**2 - . 6.2d-6*T**3)*15.0d0 + 648000.0d0 ARG7 = DMOD((-L - 2.0D0*CAPF - 2.0D0*OMEGA + THETA) . * 3.14159265D0/648000.0D0,6.28318530718D0) - HALFPI ARG1 = DMOD((-2.0d0*CAPF - 2.0d0*OMEGA + THETA) . * 3.14159265D0/648000.0D0,6.28318530718D0) - HALFPI ARG2 = DMOD((-2.0d0*CAPF + 2.0d0*CAPD - 2.0d0*OMEGA . + THETA) . * 3.14159265D0/648000.0D0,6.28318530718D0) - HALFPI ARG3 = DMOD(THETA * . 3.14159265D0/648000.0D0,6.28318530718D0) . + HALFPI ARG4 = DMOD((-L - 2.0d0*CAPF - 2.0D0*OMEGA + 2.0d0*THETA) . * 3.14159265D0/648000.0D0,6.28318530718D0) ARG5 = DMOD((-2.0D0*CAPF - 2.0D0*OMEGA + 2.0d0*THETA) . * 3.14159265D0/648000.0D0,6.28318530718D0) ARG6 = DMOD((-2.0d0*CAPF + 2.0d0*CAPD - 2.0d0*OMEGA . + 2.0d0*THETA) . * 3.14159265D0/648000.0D0,6.28318530718D0) ARG8 = DMOD((2.0d0*THETA) . * 3.14159265D0/648000.0D0,6.28318530718D0) CORX = - 0.026D0*DSIN(ARG7) + 0.006D0*DCOS(ARG7) . - 0.133D0*DSIN(ARG1) + 0.049D0*DCOS(ARG1) . - 0.050D0*DSIN(ARG2) + 0.025D0*DCOS(ARG2) . - 0.152D0*DSIN(ARG3) + 0.078D0*DCOS(ARG3) . - 0.057D0*DSIN(ARG4) - 0.013D0*DCOS(ARG4) . - 0.330D0*DSIN(ARG5) - 0.028D0*DCOS(ARG5) . - 0.145D0*DSIN(ARG6) + 0.064D0*DCOS(ARG6) . - 0.036D0*DSIN(ARG8) + 0.017D0*DCOS(ARG8) CORY = - 0.006D0*DSIN(ARG7) - 0.026D0*DCOS(ARG7) . - 0.049D0*DSIN(ARG1) - 0.133D0*DCOS(ARG1) . - 0.025D0*DSIN(ARG2) - 0.050D0*DCOS(ARG2) . - 0.078D0*DSIN(ARG3) - 0.152D0*DCOS(ARG3) . + 0.011D0*DSIN(ARG4) + 0.033D0*DCOS(ARG4) . + 0.037D0*DSIN(ARG5) + 0.196D0*DCOS(ARG5) . + 0.059D0*DSIN(ARG6) + 0.087D0*DCOS(ARG6) . + 0.018D0*DSIN(ARG8) + 0.022D0*DCOS(ARG8) CORT = + 0.0245D0*DSIN(ARG7) + 0.0503D0*DCOS(ARG7) . + 0.1210D0*DSIN(ARG1) + 0.1605D0*DCOS(ARG1) . + 0.0286D0*DSIN(ARG2) + 0.0516D0*DCOS(ARG2) . + 0.0864D0*DSIN(ARG3) + 0.1771D0*DCOS(ARG3) . - 0.0380D0*DSIN(ARG4) - 0.0154D0*DCOS(ARG4) . - 0.1617D0*DSIN(ARG5) - 0.0720D0*DCOS(ARG5) . - 0.0759D0*DSIN(ARG6) - 0.0004D0*DCOS(ARG6) . - 0.0196D0*DSIN(ARG8) - 0.0038D0*DCOS(ARG8) CORX = CORX * 1.0d-3 CORY = CORY * 1.0d-3 CORT = CORT * 0.1d-3 RETURN END ---------------- REFERENCES Brosche, P., Seiler, U., Sundermann, J., and Wunsch, J., 1989, "Periodic Changes in Earth's Rotation due to Oceanic Tides," Astron. Astrophys., 220, pp. 318-320. Brosche, P., Wunsch, J., Campbell, J., and Schuh, H., 1991, "Ocean Tide Effects in Universal Time detected by VLBI," Astron. Astrophys., 245, pp. 676-682. Dickman, S. R., 1989, "A Complete Spherical Harmonic Approach to Luni-Solar Tides," Geophys. J., 99, pp. 457-468. Dickman, S. R., 1990, "Experiments in Tidal Mass Conservation," (research note), Geophys. J., 102, pp. 257-262. Dickman, S. R., 1991, "Ocean Tides for Satellite Geodesy," Mar. Geod., 14, pp. 21-56. Dickman, S. R., 1993, "Dynamic ocean-tide effect on Earth's rotation," Geophys. J. Int., 112, pp. 448-470. Eubanks, T. M., 1993, "Variations in the Orientation of the Earth," in Contributions of Space Geodesy to Geodynamics: Earth Dynamics Geodynamics, American Geophysical Union, Washington, DC, 24, pp. 1-54. Gipson, J., 1996, "VLBI Determination of Neglected Tidal Terms in High-Frequency Earth Orientation Variation," submitted to J. Geophys. Res. Gross, R. S., 1993, "The Effect of Ocean Tides on he Earth's Rotation as Predicted by the Results of an Ocean Tide Model," Geophys. Res. Lett., 20, pp. 293-296. Herring, T. A., 1993, "Diurnal and semi-diurnal variations in Earth rotation," in The Orientation of the Planet Earth as Observed by Modern Space Techniques, M. Feissel (ed.), Pergamon Press, in press. Herring, T. A. and Dong, D., 1991, "Current and future accuracy of Earth rotation measurements," in Proceedings of the Chapman conference on Geodetic VLBI: Monitoring Global Change, NOAA Technical Report NOS 137 NGS 49, pp. 306-324. Herring, T. A. and Dong, D., 1994, "Measurement of Diurnal and Semidiurnal Rotational Variations and Tidal Parameters of Earth," J. Geophys. Res., 99, pp. 18051-18071. IERS Annual Report, Available from the Central Bureau of the IERS, Paris Observatory, Paris. McCarthy, D. D., 1996, IERS Conventions, IERS Technical Note, 21. Ray, R., Steinberg, D. J., Chao, B. F., and Cartwright, D. E., 1994, " Diurnal and Semidiurnal Variations in the Earth's Rotation Rate Induced by Oceanic Tides," Science, 264, pp. 830-832. Ray, R., 1995, Personal Communication. Seiler, U., 1991, "Periodic Changes of the Angular Momentum Budget due to the Tides of the World Ocean," J. Geophys. Res., 96, pp. 10287-10300. Seiler, U. and Wunsch, J. 1995, "A refined model for the influence of ocean tides on UT1 and polar motion," Astron. Nachr., 316, pp. 419-423. Sovers, O. J., Jacobs, C. S., and Gross, R. S., 1993, "Measuring rapid ocean tidal Earth orientation variations with VLBI, J. Geophys. Res., 98, 19959-19971.