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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.

    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 requires NO CHANGE to 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.

    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 

    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


    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


    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:

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.

Point of Contact: GPS Division
Phone Numbers:
  Com. (314)676-9140
  DSN 846-9140

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