Erncos C. PAVLIS, Ivan 1. MUELLER Department of Geodetic Science and Surveying,

Ohio State University Columbus, Ohio 43210.

T H E E F F E C T OF E A R T H O R I E N T A T I O N E R R O R S

IN BASELINE D E T E R M I N A T I O N

Abstract

The determination o f baseline lengths from certain space techniques is based on the derived coordinates o f the terminal stations. As such, the estimated baselines are susceptible to systematic errors that affect the relative coordinates. One source of error is in the set o f parameters which describes the continuously changing relative orientation o f the Conventional Terrestrial (CI'S) and Inertial (CIS) Reference frames. Due to these errors, the coordinates o f each terminal station may in fact refer to a slightly different coordinate system, and, therefore, when used for computing the length between the stations, errors will result. The expected magnitudes of such errors and their possible presence in current solutions are investigated.

In conclusion, we find that the present level of accuracy and stability o f the available parameters connecting the CTS and CIS (e.g., the ERP series} is unsatisfactory for centimeter level baseline length determinations. The available options are either the use o f strictly simultaneous SLR data sets (similar to the VLBI data sets} or the improvement o f the parameters connecting the CTS and the CIS. The first long-range step in the latter direction is the support o f the IAG / IA U Joint Working Group COTES proposal [CSTG Bulletin, 1982], endorsed by both the IAG and IAU in various resolutions [ [AU, 1983 ; l A G , 1982] .

The determination of baseline lengths from Satellite Laser Ranging (S LR) as well as from some other space techniques is based on the derived coordinates of the terminal stations, i.e., i t is an indirect determination. As such, the estimated baselines are susceptible to systematic errors that affect the relative coordinates. If the coordinates of the terminal stations are not determined from "simultaneous" observations, one source of error is in the set of parameters which describes the continuously changing relative orientation of the Conventional Terrestrial (CTS) and Inertial (CIS) Reference frames. Due to these errors, the nonsimultaneously determined coordinates of each terminal station may in fact refer to a slightly different reference frame, and, therefore, when used for computing the length between the stations, errors wil l result. In the fol lowing we investigate the expected magnitudes of such errors and their possible presence in current solutions.

Analysis of the Lageos data distribution has shown that even during periods of Presented at the Annual Fall Meeting of the American Geophysical Union, San Francisco, December 7-15, 1982, and at the Fifth Annual NASA Geodynamics Program Conference and Crustal Dynamics Project Review, Washington, D.C., January 24--28, 1983. Bull GJod. 57 (]983)pp. 273-282.

273

E.C. PAVLIS, I.I. MUELLER

intensive observational campaigns (such as the MER IT Short Campaign, August-October, 1980), very few stations collected ranging data simultaneously. In several cases, therefore, a baseline is estimated from the terminal stations ' coordinates which were derived from data collected over disjunctive time intervals.

Assume that the coordinates of two stations ( i and j ) have been determined at two different epochs ( ] and 2 ) . Assuming that the orbit is perfectly known, the only source of systematic error (apart from the observations themselves) is the set of parameters used in the transformation between the CIS and CTS. Let the two sets of

, 0)1 and (• , y , 0)2 . Since we are interested only in the effectof parameters, constant biases are of no concern. Let & x , Ay ,and variation in the rotational parameters between the two epochs. Then,

rotations be ( x , y variations in these A 0 denote the real

X2 = Xl + A X

Y2 = Yl +Ay

0= = 01 + AO

(1)

Due to the usually unaccountable part of these real variations or to oti~er errors , each station coordinate set determined at a different epoch is referred to a coordinate system which can differ from the other by as much as 0".010 in each of three orthogonal directions [Mue~er et aL, 1982 ; Feissel, 1982]. Obviously, when one computes the baseline lengths from coordinates referenced to different frames of reference, an error is committed which to the f i rst-order is proportional to the orientation errors, and its value depends on the particular station combination.

To derive the analytical expression for the errors, assume that between the two epochs ] and 2, the differentially small errors in the three rotations are 8 x , 5 y , and 8 8. One can view these quantities as errors in the real variations in eq. (1) and also as the orientation error between the two coordinate systems to which the station coordinates refer. Without loss of generality we take the epoch 1 determination as the standard. The coordinates of the epoch 2 determination are related to the frame at epoch ] through the following differential transformation :

X 1 80 8x X[

Y ~ - 5 0 1 - 5 y Y[ (2)

Z i - S x 8 y 1 2

.the subscripts indicating the epoch of the frame to which the vectors refer. In other words, the errors in the coordinates are

8X

8Y

8Z 1 - 2

o z I - Z 0

Y - X 0 2

8 y

8x

60

(3)

Now compute the baseline length for stations i and j , where the first was determined in the epoch ! frame, the latter in the epoch 2 frame. For station coordinates in the same frame, the length is

d = [ ( X i - X j ) T (X i - X j ) ] 1/2 (4)

274

T H E E F F E C T OF E A R T H O R I E N T A T I O N E R R O R S .....

where

Xi = [Xi Yi Zi ]T (5)

X'j = [X j Yj Zj ]T (6)

are the station coordinates.

Because of the reference frame difference, first apply eq. (3) to the coordinates of station j (at epoch 2 ) to bring them in the same frame as those of station i (at epoch 1) . If this is not done, the error committed is

8d1-2 d i 8

8 1 - 2

(7)

which can be obtained by differentiating eq. (4) wi th respect to X j . From eq. (7), by substitution of eq. (3), after simplification, one arrives at

8d = (xi -- XJ'~ ZJ - ( Z i - Zj)Xj 8x 1--2 d

+ ( Z i - Z j ) Y J - ( Y i - Y j ) Z J 8y d

+ ( X i - X j ) Y j + ( Y i - Y j ) X j 80 (8) d

Expression (8) gives the total error in the ( i - j ) baseline length determination due to the unaccounted for changes or errors in the reference frame orientatton between epochs 1 and 2.

Using the above equation to get a feel for the magnitude of things, we have computed the errors in all baselines between some of the more important SLR stations for an error of 8 x = 8 y = 80 = +0''.001 . Table 1shows the results for the worst possible case when the absolute values of the three error components are added together. Inspection shows that even in the case of the extremely optimistic orientation errors of one milliarcsecond ( ~ 3 cm on the equator), the corresponding errors in the baselines can reach 5 cm. Current actual errors in the orientation parameters could cause baseline errors which are larger than this value by at least a factor of three.

Such a level of uncertainty is unacceptable when baselines are to be used for the determination of centimeter level crustal or tectonic motions. Unless the rotational variations can be improved to better than a milliarcsecond, the baselines must be based on coordinates determined simultaneously, preferably from observations to the same satellite passes. If simultaneity becomes a rigid constraint for data collection, one can further improve the results by analyzing the SLR range data as "simultaneous range-- differences (SRD)," a technique studied and proposed in [Pavtis, 1983 ]. These methods of analysis wil l not only eliminate the reference frame errors as discussed here, but will also minimize the effect of orbital model errors and others which have been ignored in

275

E . C , P A V L I S , I . I . M U E L L E R

Table 1

Effect of Differential Rotations on Baseline Lengths

n u ~ s 1 8 8 l e s s ~ H r a i l = . ~ m ILal4WW. ~ - - ~J e t e ~ m s ~ i . ~ s m I ~ , B ~ aS m m m w ~ U S ~ 8 m ~ mR mine a m

F ~ d � 9 t e a m f f r A T A R G a B O B R HAW N C ~ HAY T ~ 2 7 0 9 0 T I I I ~ T I ~ ~ T O g i

7 0 9 0 O. ?

x + y 2 . 0 O . , 7 1 1 t i x- t .~ '~ �9 4 . 3 1 . 8

ORK x + y , . 8 2 . 3 1 . | T 9 4 ~ x + y + � 9 1 . 6 4 . T 2 . T

H A , z + y I . T 1 . 8 1 .9 T I 2 0 x + T . ~ � 9 8 . ? S . O 4 . 8

HCD x + y 2 . ? , . g 2 . T T 0 8 6 x~ 'T+O 5 . 0 1 . 2 | . !

BAY x ~ , 8 . 8 O . * 2 . 4 7 0 9 1 x + y + � 9 I . � 9 � 9 4 . 6

OVR x + y 2 . , 0 . 9 $ . 1 T I i 4 x + ~ � 9 4 . 8 ! . 9 4 . 1

2 . 0 8 . 6

1 . 8 1 . 8 2 . s 4 . 2

O . T ] . T 1 . 4 8 . 1

I . S 2 . 0 2 . T 4 . 2

2 . 9 e . l

2 . 8 2 . e J . � 9 4 . 4

J o t e a $ ( 1 ) / b D t a t l o m m i n x , y a n d �9 : A 4 2 N I i m o s e h .

( 2 ) ~ o o i t m e o r r o r e I n e e m t l i t o r ~ u .

( 3 ) " x + y " �9 C o m b i n e d e r r o r f r o m �9 a i d y .

( 4 ) e x + l r + O e �9 T o t a l e r r o r I n b a n � 9 d u e t o t h e e r r � 9 i n x , y e n d �9 .

( S ) A b e � 9 v n l u o m o f e r r � 9 u n o d - ~q*rmt esuwe p o e a t b l e .

( 6 ) � 9 4 o e i l D m t i o m m a r e a e f o l l e u u :

8 T A - 8 T A L A a , G r e e n b e l t , HD. O V R - 05mnm V a l l e y , C A . - - . . , . . , .

~'IAA~ ~la;n~t&'~ld~Le,HAAJm- O I U t - O r r o r a l , A m m t r a l l a , ~ n a - G o l d � 9 CA.

( 7 ) l h a e l l n e e f r o m w k i e k d a t a ~ r o u e o d i n t h i 8 e t u d y :

�9 B e e o l t ~ L e n l t k ( k i d

I 8 T A - ORR 1 2 1 1 8 . 5 4 2 H~D - O n ! 1 1 K 3 8 . 1 6 3 Y A R - OAR 3 1 9 4 1 . ~ �9 H A y - OILK 1 2 2 4 4 ) . 6 0 8 e T A - Y ~ 1 ~ 4 ~ . 9 g 6 ] S C D - Y A R 1 2 1 8 9 . 0 1 T H A y - Y k R 1 2 6 3 8 . 1 6 8 O V R - CD8 2 ~ J . . ~ ) 9 l P ~ u ORR 7 8 8 0 , 9 9

l e H A W - YAR 9 4 ~ 6 . 4 6 I I O V R - YAR 1 1 7 6 8 . 6 3 1 2 f r / ' k - R A Y 6 4 ~ . 0 8

this paper. In any case, what should be done at least as to select the observations in such a way that their mean epoch (within a few days) should be the same for both terminal stations.

To test for the presence of the suspected errors described above, we analyzed some of the NASA/GSFC preliminary SL5 monthly baseline variations over the 1979--1981 time interval [Chfistodou]idis and Smith, 1982' Chfistodoulidis, private communication, 1983]. The analysis is based on the previous error equations, using them as observation equations in monthly least squares adjustments, in which through minimizing the variations of all baselines, with respect to independently determined annual (1980) values, we searched each month for the unaccounted for rotations.

276

THE E F F E C T O F EARTH O R I E N T A T I O N ERRORS .....

011 U3 ~ ~ I n ! i

0

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,0ST

~ ~ "S

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1 ~ ~ ~ , .0~

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2 7 7

E.C. PAVLIS, I.I. MUELLER

The expectation was that the rotational errors thus monthly removed would not only reduce the monthly variations of each baseline with respect to its annual value and about the mean, but also would bring the mean nearer to the annual value. Since information to determine the exact epochs when the terminals of each baseline were observing was unavailable, we assumed that within each month one end of each baseline was observed at a certain common epoch, while the other ends at another but also common epoch.

The original monthly baseline variations with respect to the 1980 annual solution are given in Table2. Three solutions were performed : one solving for all three rotation variations (Table 3) , one solving only for the two polar-mot ion- type parameters, and finally one only for a rotation about the z-axis.

Due to the limited number of available variations ("observations"), the parameters are not strongly determined, but this is irrelevant since the purpose of these tests is not to improve the parameters, i.e., the reference frame orientation, but to demonstrate the very existence of the problem. In all three adjustments, the variation of each monthly baseline was in general reduced over the entire time span both with respect to its value based on the independent 1980 annual solution and also with respect to their own mean. The results are be.~t for the three rotation solutions mentioned above, showing an average rms reduction by 38 %, while the one-parameter (z-axis rotation) solution shows the lowest average rms reduction by only 11 ~. The results of this solution in a graphical form are also shown in Figure2. Note that for all baselines the mean of the monthly values after the rotations is much nearer to the 1980 annual solution than the value before the rotational effect had been removed.

In conclusion then, we find that the present level of accuracy and stability of the available parameters connecting the CTS and CIS (e.g., the ERP series) is unsatisfactory for centimeter level short-term baseline length determinations. Even if some of the rotational errors detected are not due to the reference frame, identification of the cause cannot be made without first eliminating the reference frame inconsistencies. The available options are either the use of strictly simultaneous SLR data sets (similar to the VLBI data sets) or the improvement of the parameters connecting the CTS and the CIS. The first long range step in this latter direction is the support of the IAG/ IAU COTES proposal [CSTG Bulletin, 1982], endorsed by both the lAG and IAU in various resolutions [IAU, 1983 ; lAG, 19821 This option would place less stringent requirements on the schedules of the SLR stations in the long run, and it thus may seem to be the more general and logical opti.on. In the short run, strict simultaneity must be the remedy.

Acknowledgement.

This work has been supported by NASA Cont rac t NAS5- -25888 ,

0

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278

T H E E F F E C T OF E A R T H O R I E N T A T I O N ERRORS .....

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E.C. PAVLIS, I.I. MUELLER

REFERENCES

D.C. CHRISTODOULIDIS and D.E. SMITH (1982) : "SL5 Geodetic Solution," Error Sources in SLR Baseline Determinations. NASA/Goddard Space Flight Center. Third Crustal Dynamics Working Group Meeting, Greenbelt, Maryland, October.

CSTG Bulletin (1982) : "Reference Frame Requirements and the MERIT Campaign -- Proposal for Extra Observations," June 9 (avail. Dept. of Geodetic Science and Surveying, Ohio State Univ., Columbus).

M. FEISSEL (1982) : "Combined Estimates of the Earth Rotation Parameters-BIH Report," Project MERIT, G .A. Wilkins and M. Feissel, eds., Royal Greenwich Observatory, Herstmonceaux, England.

lAG (1982) : "Resolutions Adopted by the General Meeting of the International Association of Geodesy, Tokyo, May 7--10," Bulletin G#od~sique, 56, 4,396-397.

IAU (1983) : "The XVII I th General Assembly of the International Astronomical Union, Patras, Greece, August, 1982, Resolution R5 : The MERIT Campaign," IAU Information Bulletin 49, January, 1983, p. 9,

I.I. MUELLER, B.S. RAJAL and Y.S. ZHU (1982) : "Comparison of Polar Motion Data from the 1980 Project MERIT Short Campaign," High Precision Earth Rotation and Earth-Moon Dynamics, Proc. of IAU Colloq. 63, Grasse, France, May, t981, O. Calame, ed., Reidel.

E.C. PAVLIS (1983) : "On the Geodetic Applications of Simultaneous Range--Differencing to Lageos," Dept. of Geodetic Science and Surveying Rep. 338, Ohio State Univ., Columbus.

.... j

R e c e i v e d : 1 0 . 1 1 . 1 9 8 2

A c c e p t e d : 2 6 . 0 5 . 1 9 8 3

282