# Earth Orientation Parameter Prediction (EOPP) Description

## May 2011

The National Geospatial-Intelligence Agency (NGA) is required to provide Earth Orientation Parameter Prediction (EOPP) coefficients and predictions to the Air Force GPS Master Control Station at Schriever Air Force Base and defense related customers on at least a weekly basis. The coefficients are unclassified and thus also available to non-military users. The cofficients are used in the summation equations below, to generate polar X (xp), polar Y (yp), and UT1-UTC prediction values for any number of days in the future. The NGA coefficients can be valuable in high precision satellite tracking in any Earth Centered-Earth fixed reference system, such as WGS84. The coefficients are computed daily by using updated xp, yp, and UT1-UTC values retrieved by NGA through the International Earth Rotation and Reference Systems Service (IERS) at the United State Naval Observatory. These updated values are then fit, in a least squares manner, to specified math models. The resulting coefficients, constants and variables used in the math models are output in a five (5) line (80 columns per line) format. By simply substituting the output values back into the math model it is possible to get parameter predictions for any day in the future. One must know the Modified Julian Day (MJD) of your day of interest. PLEASE NOTE HERE, that using the daily coefficients, provided on the NGA web site, in the following equations, may not reproduce the exact prediction values, also provided on the NGA web site. This is due to two reasons. First is the precision values used by NGA when computing the predictions (double precision) versus the formatted requirements of the listed web site coefficients. Second are the two RESTORATION methods to the predictions. Note the following two methods.NOTE:Before restoration, NGA removes a 41-term zonal model, with periods from 5.64 days to 34.85 days from the UT1-UTC data prior to computing the coefficients. 1)NGA EOPP BULLETIN WHICH APPROXIMATES ZONAL TIDES IN THE UT1-UTC PREDICTIONS ONLY.The web file named 'EOPP####.TXT', where #### is the bulletin number, has had the zonal tides restored to UT1-UTC predictions (only) in the EOPP bulletin. This is accomplished by using the unused UT1-UTC coefficients by fitting the two dominant periods, 27.56 days (lunar cycle) and 13.66 days (semi-lunar cycle), of the zonal tides. Zonal tide approximation utilizes the unused UT1-UTC coefficients, K1, K2, L1, L2 (all formerly set = 0.000000), as specified in the NGA bulletin. Zonal tide approximation requiresNO CHANGEto existing EOPP users that DO NOT restore zonal tides but may need better prediction accuracy in UT1-UTC. Also, using this method, the R1 and R2 values (both formerly set = 500.0) now equals 27.56 days and 13.66 days, respectively. 2)TOTAL RESTORATION TO THE POLAR X, POLAR Y, AND UT1-UTC PREDICTIONS.The web tabular file named 'USAF####.DAT', again where #### is the bulletin number, has had the 41-term, periods from 5.64 days to 34.85 days, zonal tide model applied to the UT1-UTC predictions along with the diurnals/sub-diurnals applied to the polar X, polar Y, and UT1-UTC. If these predictions are used, the K1, K2, L1, and L2 coefficients need to be ignored or reset (each) to zero. Please note below. This method gives the best accuracy of all three components when compared to the IERS Finals. The Air Force GPS Master Control Station uses these coefficients and predictions in their process. They no longer use the approximation method in 1) above. The following equations are the math models used in xp, yp, and UT1-UTC coefficient generation: A) 2 2 x(t) = A + B(t-ta) + Sum(Cj sin[(2pi(t-ta)/Pj]) + Sum(Dj cos[(2pi(t-ta)/Pj]) j=1 j=1 B) 2 2 y(t) = E + F(t-ta) + Sum(Gk sin[(2pi(t-ta)/Qk]) + Sum(Hk cos[(2pi(t-ta)/Qk]) k=1 K=1 C) 4 4 UT1-UTC(t) = I + J(t-tb) + Sum(Km sin[(2pi(t-tb)/Rm]) + Sum(Lm cos[(2pi(t-tb)/Rm]) m=1 m=1 The following is a list of the constants in the equations above: P1 and Q1 = 365.25 days (annual period) P2 and Q2 = 435 days (Chandler period) ----> R1, R2 = formerly set to 500.0 days...prior to EOPP502 (second week of 2005) <---- R1 = 27.56 days (lunar period) R2 = 13.66 days (semi-lunar period) R3 = 365.25 days (annual period) R4 = 182.625 days (semi-annual period) ----> K1, K2, L1, L2 = formerly set to zero (0)...prior to EOPP502 <---- K1, K2, L1, L2 = computed variables fitted to the two dominant (new) R1 and R2 periods (note above)...values are in seconds K3 = -0.022 seconds K3, K4, L3, and L4 are seasonal K4 = 0.006 seconds variation coefficients found L3 = 0.012 seconds through astronomical observations. L4 = -0.007 seconds (They are assumed to be constant.) B, F = 0 (currently...may be used/filled in future upgrades/improvements) pi = 3.1415926535... The following is a list of the variables in the equations above: A, E - offset terms (in arcseconds) Cj, Dj, Gk, Hk - polar position coefficients (in arcseconds) I - offset term (in seconds) J - time drift coefficient (in seconds per day) The following is a list of the time variables in the equations above: t - the MJD of the day that predictions are desired (prediction or effectivity dates) ta - the MJD of the first day of data used as input (435 days, Chandler Period, prior to the generation date) tb - the MJD of the day before January 1st of the current year...this is the MJD of December 31st of the previous year...the UT1-UTC equation needs this date along with the seasonal variation coefficients to account for a 'correct phase' of the seasonal effects during the current year. eg. Jan. 1st, 2005 = MJD 53371 Dec. 31st, 2004 = MJD 53370 1) WEEKLY COMPUTATIONS: A) The following is an example of the five record output prior to EOPP502, i.e., EOPP501 (with no changes to K1, K2, L1, L2, R1, and R2): 52951.00 .048890 .000000 -.054723 -.088378 .028945 .109437365.25 435.00 .347596 .000000 -.017649 -.116192 -.046661 -.095657365.25435.00 53370.00 -.510031 -.000276 .000000 .000000 -.022000 .006000 .000000 .000000 .012000 -.007000 500.0000 500.0000 365.2500 182.6250 32 501 53372 53368 B) The following is an example of the five record output starting with EOPP502 (with added changes to K1, K2, L1, L2, R1, and R2): 52958.00 .054272 .000000 -.056994 -.103402 .014470 .106800365.25 435.00 .348539 .000000 -.011996 -.106209 -.049085 -.106233365.25435.00 53370.00 -.510015 -.000337 .000615 -.000667 -.022000 .006000 .000829 .001090 .012000 -.007000 27.5600 13.6600 365.2500 182.6250 32 502 53379 53376 2) DAILY COMPUTATIONS: A) The following is an example of the five record output starting with daily computations (with added changes to K1, K2, L1, L2, R1, and R2): 55756.00 .086462 .000000 .143942 -.028860 .025332 -.047571365.25 435.00 .338797 .000000 -.021268 .049319 .123890 -.018568365.25435.00 55926.00 .619928 -.000991 .000623 -.000636 -.022000 .006000 -.000400 .000006 .012000 -.007000 27.5600 13.6600 365.2500 182.6250 35 2249 56175 56174 The record output format is: RECORD NUMBER COLUMN START FORMAT VALUE 1 1 F10.2 ta 11 F10.6 A 21 F10.6 B 31 F10.6 C1 41 F10.6 C2 51 F10.6 D1 61 F10.6 D2 71 F6.2 P1 77 4X Fill 2 1 F6.2 P2 7 F10.6 E 17 F10.6 F 27 F10.6 G1 37 F10.6 G2 47 F10.6 H1 57 F10.6 H2 67 F6.2 Q1 73 F6.2 Q2 79 2X Fill 3 1 F10.2 tb 11 F10.6 I 21 F10.6 J 31 F10.6 K1 41 F10.6 K2 51 F10.6 K3 (*) 61 F10.6 K4 (*) 71 10X Fill 4 1 F10.6 L1 11 F10.6 L2 21 F10.6 L3 (*) 31 F10.6 L4 (*) 41 F9.4 R1 50 F9.4 R2 59 F9.4 R3 68 F9.4 R4 77 4X Fill 5 1 I4 TAI-UTC 5 I5 Bulletin Number (EOPP week/day) 10 I6 t - Effectivity Date 16 1X Fill 17 A18 Generation Date / Info 35 46X Fill (*) - values computed by I. I. Mueller, at Ohio State University, in the 1960's ... note the text by Moritz, H. and I.I. Mueller, Earth Rotation: Theory and Observations, 1987, Ungar, New York. An analysis of NGA's polar parameter predictions show that over the week the coefficients are in effect, xp and yp should each have an RMS error of under 0.003 arcseconds (10 cm. at the equator) with the IERS Final Values. The RMS error for UT1-UTC should be found to be below 0.8 milliseconds with the Final Values. It is important to realize that the accuracy of the coefficients degrade with time. Therefore, when using the NGA coefficients one should always use the most recent set available. The coefficients are recomputed every day at NGA and sent to the users after a quality control check. They are labeled to go into effect on the following day.

**GPS Division**

Phone Numbers:

Com. (314)676-9140

DSN 846-9140

Document last modified Monday, 05-May-2014 16:30:34 EDT