Precise orbit computations of Lageos for Wegener-Medlas

Printed in Great Britain. Allrights reserved. Copyright©1989 COSPAR

PRECISE ORBIT COMPUTATIONS OF

LAGEOS FOR WEGENER-MEDLAS

B. A. C. Ambrosius, H. Leenman, R. Noomen and K. F. Wakker

Delft University of Technology, Faculty of Aerospace Engineering, Kluyverweg 1, 2629 HS Deift, The Netherlands

ABSTRACT

The WEGENER-MEDLAS project was conceived to investigate crustal motions in the Eastern Mediterranean using highly accurate laser tracking of the LAGEOS satellite. The tracking data are processed to produce accurate point positions of the laser tracking systems. By repeating these position determinations at regular time intervals, changes in the relative positions may be observed,

providing quantitative data on the crustal motions which are expected to be of the order of 1 -

5

cm/year. Initial tracking started in January 1986. This paper discusses some of the data analysis techniques applied by the Section Orbital Mechanics of DelIt University of Technology. These techniques are based on one-week data arcs and on a data arc with a length of one year, and involve the accurate modeling of the trajectory of LAGEOS. The effects of various perturbations on the motion of this satellite are discussed including the earth’s gravity field, solar radiation pressure and

third-body perturbations. Recent improvements in the modeling of some of these effects are

highlighted and results are given for the evolution of some parameters recovered from one-week data arcs. To demonstrate the overall accuracy of the orbit modeling for one-week data periods, some examples of the results obtained for the relative motions between various global stations are presented. They are compared with results derived from independent geophysical data.

INTRODUCTION

The Section Orbital Mechanics (SOM) of the Faculty of Aerospace Engineering at Deift University of Technology (DUT) is actively involved in the WEGENER-MEDLAS project since January 1986.

This project is aimed at the determination of tectonic motions in the Mediterranean area, a region

where a large number of earthquakes occur each year. These motions will be determined from the analysis of observations on the geodetic satellite LAGEOS, obtained by mobile laser ranging systems being deployed at various Sites in this region during a number of years /1/. In 1986, a

German and a Dutch MTLRS mobile laser ranging system visited a total of 7 different locations in Italy and Greece. The 1987 operational campaign was even more successful, with the American transportable laser ranging system TLRS-1 supplementing the two MTLRS systems, resulting in the

occupation of 10 sites in Italy, Greece and Turkey. Six of these sites were also visited during the

first campaign of 1986. LAGEOS was launched in 1976 and orbits the earth in a nearly circular

orbit with a semi-major axis o~about12270 km. The spherical satellite has a mass of 407 kg and a

cross-sectional area of 0.283 m

The primary task of the Section Orbital Mechanics is the monitoring of the quality of the data taken by the systems participating in the WEGENER-MEDLAS laser observation campaigns. To this aim, the Section performs an extensive analysis of a sample of the measurements taken by these systems each week, and reports the analysis results to the project manager, the operations managers and other groups involved in the WEGENER-MEDLAS project. These samples of measurements are

(3)206 B. A.C.Ambrosius er al.

called quick-look data, and the Section is also known as the Quick-Look Data Analysis Center

(QLDAC).

Through the continuous analysis of LAGEOS data QLDAC has been able to keep track of the position of this satellite very accurately for a period of well over two years by now. In addition, the analysis has provided QLDAC with a unique series of recovered surface force scaling parameters. This paper deals with some aspects of the analysis of the quick-lock data. First, the accuracy of the modeling of the LAGEOS orbit in one-week data arcs will be discussed. These types of data arcs are the main element of the weekly analysis, and have proven to be a very powerful tool for the detection of possible data or other problems. Next, the results will be applied to address the

long-term evolution of the orbit of this satellite. The main objective of the WEGENER-MEDLAS project, the determination of tectonic motions, will be discussed in the final Section.

ONE-WEEK DATA ARCS

In order to perform its task in the WEGENER-MEDLAS project properly, QLDAC each week receives a batch of quick-look measurements, taken by both the mobile laser systems in the Mediterranean area and globally distributed fixed or mobile laser stations. The measurements are first passed through a screening and correcting process and converted into normal points. Subsequently these are analyzed using a dynamic approach in which an orbit is fitted to the data in

arcs of various lengths. Only the results obtained with one-week data arcs, which has proven to be

the most flexible analysis technique, will be addressed here.

Table I gives a summary of the computation model, denoted by QLDACMOD4, which is currently in use at QLDAC. In the weekly analyses, the state-vector at epoch, the solar reflectivity and the along-track acceleration parameter, the coordinates of the mobile laser systems and the earth rotation parameters at 5-day intervals are solved for. The weekly rms of the residuals of the quick-look

normal points and solutions for the solar reflectivity and the along-track acceleration parameter are

plotted in Figure 1.

TABLE I. Summary of the computation model currently in use for the analysis of quick-look

measurements in one-week data arcs (QLDACMOD4). Dynamic and measurement model

PGS-1680 gravity field (ac = 6378.14411 km; GM = 398600.448 km3/s2); c = 299792.458 km/s;

solar and lunar attraction; solar radiation pressure; Wahr solid earth tide and Schwiderski ocean tide harmonic expansion models; DUT/SOM LSC 87.1 tracking station coordinates; tracking station dis-placement (h2= 0.6; 12= 0.075).

Estimated parameters

State-vector at epoch; solar reflectivity; along-track acceleration parameter, pole position and Al-.UTI difference at 5-day intervals; station coordinates of MTLRS-l, MTLRS-2 and TLRS-l and stations of opportunity.

Observations

20.0 0 0.0~ 1.00 :::::::::::::::::::::::~ 10.0 —20.0 660101 880101 TIME (TYMMOD)

Fig. I. The evolution of the weekly solutions of the rms of fit of the global quick-look normal points, the solar reflectivity and the along-track acceleration parameter in 1986 and 1987. The shaded areas in the middle plot represent the periods in which LAGEOS crossed the earth’s umbra

during its orbital revolutions.

The top plot clearly shows that the rms of fit averaged at about 10 cm throughout 1986 and the first months of 1987, but that it decreased significantly to approximately 7 cm since then. The improvement is a result of three changes in the computation model, effective since 8 March 1987. The first model improvement was the adoption of the DUT/SOM LSC 87.1 global set of laser station coordinates for the positions of the fi.xed laser stations. This solution was obtained from an analysis of full-rate LAGEOS data taken during the first three months of 1986 /2/. Before March 1987, the global stations were kept fixed at the positions according to the UT/CSR LSC 84.02 coordinate solution, a network solution based on tracking data taken in the period 1976 - 1984 /3/. It

(3)208 B. A. C. Ambrosius et at.

will be clear that the latter positions have changed considerably since then because of tectonic motions. The second basic improvement of the computation model was the introduction of the

Schwiderski ocean tide and the Wahr solid earth tide models after the MERIT standard /4/, supplementing the frequency-independent Love model that was used until March 1987. And, finally, the DE-96 planetary ephemerides were replaced by the improved DE- 118 ephemerides, which also

resulted in a better modeling of the earth’s nutation using the series by Wahr/5/.

The middle plot shows the solutions for the solar reflection coefficient C1~,obtained from the

one-week data arcs in 1986 and 1987. Also indicated are the periods during which LAGEOS crossed the

earth’s shadow during its orbital revolutions. The plot shows a number of very distinct features.

First, the scatter in the weekly solutions can be seen to decrease significantly with the adoption of

the new computation model: before March 1987, the scatter in the solutions amounts to about 0.08;

after this date, this value decreases to only0.05.

Secondly, there appears to be a significant increase in the scatter of the solutions, accompanied by larger standard deviations, during certain periods of time. A closer analysis revealed that these

events coincide with the periods when the earth-sun vector is approximately normal to the orbital

plane of LAGEOS, more-or-less halfway the periods indicated by the shaded areas. The first event

took place around September 1986, but the effect is partly obscured because of the use of the old computation model, resulting in a higher value for the “normal” scatter in the parameter solutions. The second time that the orbital plane and the sun were in this special constellation was in

May/June 1987, and this time the increase in the scatter is quite obvious. Furthermore, in 1986 the

minimum angle between the earth-sun vector and the normal to the orbital plane still amounted to about 28 deg, but in 1987 the earth-sun vector nearly became perfectly perpendicular to the orbital plane, the minimum angle being as small as 1.5 deg.

The larger scatter in the solutions for the solar radiation parameter can be explained by realizing that when the sun is perpendicular to the orbital plane, the solar radiation pressure exerts a constant force normal to the direction of the orbiting satellite’s motion. In this special case the effects of this force almost completely cancel out due to symmetry considerations, which means that there will hardly be any signal in the satellite positions and thus in the observative due to this perturbation force. This makes it difficult to recover the correct value of the force scaling parameter CR, resulting in a larger scatter of the weekly values during the period that the configuration evofves around this near-singularity. Further evidence for this phenomenon is provided by the relatively larger formal errors which accompany the increased scatter. The dependence of the recoverability of the solar reflectivity on the relative orientation of the sun and the orbital plane was already recognized almost 2 years ago /6/. At that time, however, the amount of data on which this conclusion was based was very small, but the conclusions made then are completely confirmed here.

Finally, the plot with CR solutions also suggests the existence of a periodic effect where it should be

realized that apparent thfferences in the amplitude before and after March 1987 may have been caused by the modifications of the computation model. Analysis of more data, including results for

1985 and 1988, not presented here, seems to indicate the existence of a period of precisely 1 year. Although this strongly suggests some error in the modeling of the solar radiation pressure (e.g. neglect of its dependence on the distance earth-sun), no evidence for this has yet been found.

The bottom plot of Figure 1 contains the weekly solutions for the along-track acceleration parameter

AT. This parameter is used to model an apparent drag force on LAGEOS for which a reasonable explanation has been found only recently

11/.

periodic signal, but the accuracy of the weekly solutions is too poor to resolve it, even after the model improvements in March 1987. The main conclusion that can be drawn from the plot is that

0.025 • I I — S • I SI.... S . .‘~• ~ ~ ~•• ~# ~ ~ ~. ~ —0.025 • I I 0.025 I S S S •• • U •S . • .• S.. • . • •. . . — I S •.~ •. • — I a ~ ~ s~. • h51 ‘. .•S • •~ ~ LU •, I. 1 ~ S •.S SI • .‘ —0.025 0.025 I I •.• — . S •.~ 5 I ~ • I LU • . •~ •s U, III •I •5 •• • S • .51s • ~‘ •~•.,•.• use. ~S.I.I ••, U S I S. U • U S S .• S S —0.025 I I 0,10 I S U LU I Cl, • U 5’ ~ S. ~ ~•.

~:

E

Fig. 2. The variation of the differences between the ending state-vector and the epoch state-vector solutions of connecting one-week quick-look data arcs of LAGEOS in 1986 and 1987. The results are given as differences in orbital elements.

After the fitting of an orbit through the range measurements, the LAGEOS ephemerides are stored each week and used to generate an initial slate-vector for next week’s 7-day arc analysis. This initial state-vector is therefore based on the previous week’s data and may be compared with the solution which results from the processing of next week’s data. The epoch of these state-vectors is always Sunday 00:00 GMT. A good impression of the orbital accuracy may be obtained by comparing these connecting orbits at each week’s epoch. Figure 2 shows the results for the years 1986 and 1987,

expressed as differences in orbital elements, i.e. the semi-major axis a, eccernricity e,inclination i

and argument oflatitude u, respectively. The right ascension ~l is not included in this figure, since its initial value on 5 January 1986 was only propagated in time and this parameter was not solved

for. As these results reflect the overall accuracy of the computation model, it is not surprising that the scatter decreases after 8 March 1987, when this model was drastically improved. Table 2 summarizes some statistics of the differences before and after this date, clearly underlining the

(3)2 10 B. A. C. Ambrosiuset a!.

improvements already demonstrated by the smaller values for the mis of fit in the top plot of

Figure 1. The scatter in all parameters decreases, with percentages ranging from 20 to 75 percent.

Some of the plots show indications of periodic variations in the differences during the period before 8 March 1987 (MJD 46862), when the model improvements were implemented. In particular, a 14-day period effect in the inclination plot may be observed, which is obviously related with the lunar

tide; after 8 March 1987, this signal has completely disappeared.

The mis values of all weekly differences between the initial and the adjusted epoch state-vectors,

listed in Table 3, may serve as an indication of the accuracy (consistency) of LAGEOS orbits of 7days length. For ease of interpretation, the differences have been transformed to cross-track, radial

and along-track position components. Again, the Large improvement due to the introduction of the new model is quite evident. If only the most recent results are considered, it seems that the current

accuracy is about 10, 7 and 71 cm in the three components, respectively, for LAGEOS orbits determined from quick-look data. The overall orbit accuracy may be even slightly better, however, since these results hold for the end-points of the weekly data arcs where secular and long-period

orbit errors are known to have the largest effects. Taking this into consideration, these results compare very favorably with similar data obtained from full-rate data analyses not presented here. Using a different technique in which the residuals are decomposed into apparent range and timing

biases, the full-rate results indicate a radial error of less than 5 cm and a 20 cm along-track position

error over the tracked portions of the orbit. Since the tracking is not continuous, the overall error

will be a little bit worse. it is therefore concluded that the true accuracy of the LAGEOS ephemerides for one-week data arcs is probably about 5 cm in the radial and 30 cm in the

cross-track and along-cross-track directions.

TABLE 2. The rms of the differences between weekly epoch state-vectors of LAGEOS computed from connecting one-week quick-look data arcs. The differences are given for the orbital elements semi-major axis a, eccentricity e, inclination i and argument of latitude ii. Period 1 ranges from 1 January 1986 to 8 March 1987, whereas period 2 starts on 8 March 1987 and ends on 31

December 1987. Period I Period 2 a (mn~ 7.4 4.8 e (10

)

TABLE 3. The mis of the differences between weekly epoch state-vectors of LAGEOS computed from connecting one-week quick-look data arcs. The differences are given in cross-track, radial and along-track position components. The two periods correspond to the periods mentioned in Table 2.

Period 1 Period 2 cross-track (cm) 38.0 10.0

radial (cm) 11.5 7.1

ONE-YEAR DATA ARCS

It has been shown in the previous Section that during the operational activities of the WEGENER-MEDLAS QLDAC, the position of LAGEOS has been determined in consecutive one-week data arcs with an accuracy of a few decimeters throughout 1986 and 1987. It was realized, that this series

of ephemerides which resulted from these efforts might serve as an excellent reference for studies into the long-term effects of errors in the computation model. These errors may be caused, for

instance, by inadequate modeling of long-periodic variations in the earth’s gravity field or by

deficiencies in the modeling of the surface forces acting on LAGEOS. If they are not neutralized by

the inclusion of suitable extra solve-for parameters, as is the usual practice, they will seriously affect

the overall accuracy of a long-arc solution. Their effects may be as large as tens of meters for arc-lengths of up to a year. To investigate the problems of the long data arcs, a data period of about one

year was selected as the result of a trade-off between the required computation time and the

expected yield of the analysis. Eventually the year 1986 was chosen because all its data, notably the full-rate normal points, were first available.

To establish a particular trajectory of LAGEOS during this one-year period, a single orbit was fitted to a sample of the full-rate normal points. The state-vector at epoch (5 January 1986) and a single solar reflectivity and along-track acceleration parameter were solved for. A major excursion from the QLDACMOD4 model was the adoption of a new set of station coordinates, the DUT/SOM LSC 87.4 global network solution /8/, derived from a multi one-week arc analysis of all 1986 full-rate data. These coordinates were held fixed in the long-arc analyses.

The original goal of the analyses of one-year data arcs was twofold. In the first place, the absolute

accuracy of the LAGEOS ephemerides in terms of mis of fit and apparent biases was to be

established, and, secondly, the effect of specific elements of the computation model (gravity field, tides, nutation) was to be investigated.

To start with, the rms of the residuals was found to be of the order of 13 - 15 m, and hardly

changed due to substitutions of different elements of the computation model. This result was a little bit unexpected and slightly disappointing. It will be further addressed later on in this Section. However, it is also possible to compare the weekly solutions of the epoch state-vectors mentioned above and a single one-year ephemeris. Figure 3 presents the results of such a comparison in terms of orbital element differences, which may also be considered as orbital element residuals. In the first place it may be noted that these differences are at least one order of magnitude larger than the results presented in the previous Section. Furthermore there are significant long-periodic trends while the inclination differences also exhibit a typical “short-period” signal with a period of about 2 weeks. It should be pointed out that there were some differences in the dynamic models applied in the one-week and the long-arc solutions. Notably the tide models were different, a simple Love model being used for the short arcs versus the Wahr and Schwiderski models for the long arc. Also, different versions of the planetary ephemerides were used, DE-96 for the week-arcs and DE-200 for the one-year arc, primarily resulting in different nut.ation series being applied (Woolard versus Wahr). Both model differences may be responsible for the short-period signal in the inclination residuals, but it seems unlikely that they are responsible for the large systematic variations in the other orbital elements, except for the semi-major axis, which are of the order of tens to hundreds of meters. When in addition the large residual mis of the long arc, mentioned before, is also

considered, all this can only lead to the conclusion that the dynamic model applied for the long-arc

solution was not adequate. Statistics of the orbital element differences plotted in Figure 3 are summarized in the first column of Table 4.

(3)2 12 B. A. C. Ambrosiuser a!. 0.25 S . E • • • ••• S •••S••SS•••I•••••• • •S •S • S S • —0.25 0.25~ I S.. 5 . S — •I, ..5••, S... 5•. S Li • •..S —0.25 0.0 I • • S • • • • S . • S. •.5• •5I S • • • • S S — • •. . .5 (_) SS . . S u_i • I • Cr) S 5 C...) .5 I • • —0.1 I U_i Cr) •S~. . • .•S.

E

Fig. 3. Orbital residual histories computed from the differences between the epoch state-vector

solutions of one-week quick-look data arcs and the corresponding results from a single one-year arc

of LAGEOS data in 1986. The one-year solution was obtained using the PGS-1680 gravity field, the

Wahr and Schwiderski tide models and the DE-200 planetary ephemerides. Solar eclipses by the moon were not modeled.

TABLE 4. Statistics of the orbital residuals of LAGEOS in 1986, computed from the differences

between the epoch state-vector solutions of one-week data arcs and the corresponding results from a single one-year data arc. Listed are the mis values of the differences in orbital elements.

Gravity field PGS-1680 PGS-l680 GEM-TI

Tides: solid-earth Wahr Wahr Wahr

ocean Schwiderski Schwiderski GEM-Ti

Ephemerides DE-200 DE-118 DE-118

Solar eclipses modeled no yes yes

a (mn~ 44.7 52.9 52.4

e (10

)

(marcsec) 29.1 87.4 107.0

(marcsec) 719.4 430.7 422.8

u (marcsec) 985.3 1447.3 963.3

TABLE 5. Overview of the solar eclipses by the moon as seen from LAGEOS, that took place in

1986. For each eclipse, the length and the maximum occultation factor are listed.

Start Length Max. Occultation

(minutes) (percent) April 9, 03:33 GMT 66.7 87.9 May 8. 23:58 GMT 11.7 5.6 October 3, 16:08 GMT 20.0 85.0 October 3, 22:29 GMT 10.0 10.7 November 2, 08:03 GMT 23.3 42.0

A potential candidate for the model deficiencies was the lack of modeling of solar eclipses by the moon in the long-arc solution. It was known from experience that the temporary reduction of the

solar radiation force on the satellite due to partial obscuration of the sun by the moon can cause a

significant perturbation of the satellite’s orbit. In 1986, a total of five of such eclipses took place, the first of which (dated April 9) was unusually large and lasted extremely long (Table 5). Actually, it was this particular eclipse which resulted in the recognition by the authors of the significance of

these phenomena since it seriously affected the results of the operational analyses of QLDAC in that

period /6/. In Figures 4 and 5, the disturbing effects of eclipses on the along-track position of LAGEOS and on the semi-major axis are depicted. Both figures show the differences between

one-year LAGEOS orbits derived with and without modeling these eclipses. The impact of the eclipse of

9 April 1986, on the along-track position of LAGEOS is clearly visible in Figure 4. In the analysis in which the eclipses are not modeled, the resulting errors are partly absorbed by the solved-for semi-major axis and along-track acceleration parameter. This is demonstrated by Figure 5 which clearly shows that the semi-major axis changes abruptly by about 3 cm during the 67 mm eclipse. Since the sun was in or near LAGEOS’ orbital plane around this date, the non-modeling of the eclipse had no effect on the evolution of neither the inclination nor the ascending node.

(3)2 14 B. A. C. Ambrosius et a!. S 11.0 I _I I S I S C_I S U_i I Cl) LI •, S S

E

Fig. 4. The effect of solar eclipses by the moon on the evolution of the argument of latitude of the LAGEOS orbit during the year 1986. The variation is primarily the result of a major eclipse on April 9. 0.2 S _~ — ~ SI. ~5 ~ - •~••~~ ‘~5•, ~ c.~... c. .‘~._ ~ PS.~ .I..SS..ImS~ •~I. •.SSS. ~ ~ 55 . .••s•..•:..•:..•’•.5S....•,~ ISI •~ 5555555

1~fflf,t,,f~

Fig. 5. The effect of solar eclipses by the moon on the evolution of the semi-major axis of the LAGEOS orbit during the year 1986. Only the variation during the period 8-10 April 1986 is shown. Because of the large effect the eclipse of April 9 had on the along-track position of LAGEOS

(Figure 4), the analysis of the one-year data arc was repeated with all eclipses now properly

modeled. The software that was modified to enable the modeling of the eclipses still uses the 1950 astronomical reference system, however, instead of the J2000 system. Therefore the DE-200 planetary ephemerides could no longer be used and had to be replaced by the DE-1 18 series. It is understood that these are similar, though, the only difference being the reference system. Figure 6, now, shows the differences in the inclination and the argument of latitude between this new solution and the “true” LAGEOS positions, according to the weekly epoch state-vector solutions. The pattern in the inclination residuals has not changed at all, but the entire envelope has shifted over some 0.06 arcsec due to the new overall fit. The variation of the along-track position of LAGEOS has changed considerably in the sense that the periodic signal has become even more pronounced, while the amplitude is still about the same. This means that the eclipses were not responsible for the rather large along-track residuals. A summary of the statistics of the residuals of all orbital elements is given in the middle column of Table 4.

—0.05 I I I I .• • S • • (_) • . .~ •• S • 5 SI I. •I 5 55 CI~ I LI • •. • _I i~, — SI I ~ —0.15W I I 11.0 I I I S C-i) I. •5 C_I • 555 55 5S II.. S .‘ S•S5•1 —2.0 I ‘161100 116500 ‘16600 116700 ‘46800 TIME (MJOJ

Fig. 6. Orbital residual histories for the inclination and the argument of latitude of LAGEOS, computed from the differences between the epoch stale-vector solutions of one-week quick-look data arcs and the corresponding results from a single one-year arc of data. The latter was obtained using the PGS-1680 gravity field, the Wahr and Schwiderski tide models and the DE-i18 planetary ephemerides. In addition, solar eclipses by the moon were modeled.

—0.05 I I U) Li • Cr) . I I• LI I ~ 1 5,0 S •5 • I55Ss5~ II — ~• IS• — S S I.. —0.15- I 3.0 I I I 6 I~ .SSSSSIISSI.S Li • •• Cl) ._{I I S} S._{I •I} S S_I •I5• •~ I.~ S —3.0 I I I ‘161100 t16500 ‘46600 ‘46700 ‘16800 TIME (MJO)

Fig. 7. Orbital residual histories for the inclination and the argument of latitude of LAGEOS, computed from the differences between the epoch state-vector solutions of one-week quick-look data arcs and the corresponding results from a single one-year arc of data. The latter was obtained using the GEM-Ti gravity field, the GEM-Ti tide models and the DE-118 planetary ephemerides. In addition, solar eclipses by the moon were modeled.

(3)216 B. A. C. Ambrosius et a!.

To investigate the effects of recent improvements in the models of the earth’s gravity field and the ocean tides, the one-year data arc was finally processed one more time using the latest GEM-Ti solutions for these models ,W. The resulting ephemeris was compared again with the one-week arc solutions of the state-vector. The last column of Table 4 lists the statistics of the differences as orbital element residuals. The variation of the inclination and the argument of latitude residuals is plotted in Figure 7. Although there seems to be some improvement in the along-track direction, the overall character of the fit has not changed. The long-term trends in the residuals persist from which it is concluded that the dynamic model needs some further refinement for long-arc applications. As mentioned, in the long-arc solutions only one along-track acceleration parameter was adjusted. It is well known, however, that the drag force on LAGEOS, whatever its cause may be, exhibits periodic variations with various frequencies 17/. It seems quite likely that these are responsible for the major part of the periodic signal in the along-track residuals. The signals in the residuals of the other elements like the eccentricity and the inclination are probably caused by more subtle effects. Investigators of the Centre for Space Research at the University of Austin, Texas, recently found evidence for seasonal variations in the earth’s gravity field due to atmospheric effects like the distribution of air-mass and precipitation over the globe. It may therefore be concluded that very long arcs of LAGEOS data are very useful to provide information for further dynamic model development. For geodesy, which was the primary purpose of the LAGEOS mission, it is preferred to use shorter data arcs to eliminate these model errors.

TECTONIC MOTIONS

Obviously, the computation of very precise LAGEOS orbits is not a goal in itself. The requirements are primarily dictated by the geodetic applications. The main goal is to measure crustal motions due to plate tectonics which are of the order of 0.1 to 10 cm per year. This requires that the position of many points on earth has to be repeatedly measured with an accuracy of better than 1 cm. One way, therefore, to judge the accuracy of the LAGEOS orbit is to see how well the overall LAGEOS data processing performs in terms of accuracy of the recovered station positions. In this Section, some of the results that were obtained by DUT/SOM from the analysis of full-rate data t2,8/ will be discussed. It is emphasized that the analyses were performed in support of the WEGENER-MEDLAS project.

The way in which the full-rate data are analyzed at DUT/SOM is very similar to the analysis

method applied for the quick-look observations. The analysis model used for the full-rate analysis deviates from QLDACMOD4 (Table 1) in several respects:

- much more-accurate full-rate normal points have replaced the quick-look normal points, essentially reducing the noise contribution of the measurements by a factor of 10;

- instead of using values from USNO predictions, a priori values for the earth rotation parameters are taken from the BIH Circular D;

- the coordinates of all laser stations are solved for, instead of those of only a few selected ones; - the coordinate solutions are based on the simultaneous processing of many one-week data arcs.

For the analyses, the year 1986 was divided into five data intervals of approximately equal length (Table 6). The selection of these time spans was based on the deployment periods of the WEGENER-MEDLAS mobile laser systems. The data intervals are sufficiently long to guarantee that all stations have sufficient tracking coverage, but on the other hand are small enough to prevent the unmodeled tectonic motions to become a signthcant error source. Each of the resulting five batches of data were analyzed in a multi-arc approach, solving for a state-vector, a solar reflectivity and an along-track acceleration parameter per one-week arc, and global station coordinates and earth rotation parameters at 5-day intervals for each data period. Since the WEGENER-MEDLAS mobile systems occupied different sites during each of the five data periods, no repeated solutions were obtained for either of these sites in 1986. However, most of the 20 stationary global laser systems were taking measurements during all of the five data periods, which resulted in multiple solutions of their coordinates.

As a first testof theaccuracy of the recovered station coordinates, the five solutions were compared with the DUT/SOM LSC 87.4 solution. To eliminate systematic errors, the solutions were first transformed to the reference system defined by the DUT/SOM LSC 87.4 coordinate set, while in

addition possible scale differences were eliminated. Subsequently, the differences between the coordinates of the common stations in the reference solution and in each of the five individual solutions were computed. Table 7 lists the mis of these differences in the three coordinate directions for each data period. As can be seen, the scatter averages at about 2.5 cm for all solutions. From the formal statistics it is known that the error due to data noise is less than 1 cm. This means that all other error sources contribute about 2 cm to the uncertainty in the station position coordinates. These error sources include orbit errors and systematic measurement errors.

Another way to assess the accuracy of the station coordinate solutions is to study the consistency of the relative positions of the stations. Since station coordinate solutions are used to determine crustal motions, which are primarily horizontal motions, a suitable quantity to test is the geodesic distance between pairs of stations. By definition this is the smallest distance measured over the surface of an ellipsoid between the projections of two station positions on this ellipsoid. Of course, the common stations in the five solutions share about 200 geodesic distances between each other. From these, three were selected to demonstrate the accuracy of the relative station positions. In Figure 8 the five solutions of these three geodesic distances are plotted as a function of time. The midpoint of each individual time interval was chosen as the epoch of the corresponding solution.

TABLE 6. Overview of the data periods in which the 1986 full-rate data were subdivided for their analysis. Listed are the start and stop times of the data periods, the number of one-week data arcs that they comprised and the overall mis of fit for each period.

Data period Number of arcs Rms of fit

(cm) January 5 - March 16 10 4.6 Marchl6-May25 10 5.0 May 25 - August 31 14 4.7 August 31 - October 26 8 4.9 October 26 - December 28 9 5.2

TABLE 7. Overview of the differences between the independent global network solutions and the DUT/SOM LSC 87.4 coordinate solution. Listed are the start and stop times of the data periods, the number of common stations and the mis differences of the x-, y- and i-coordinates. The latter values were obtained after solving for Helmert transformation parameters.

Number of Rms x/y/z coordinate

Data period common stations difference

(cm)

January 5-March16 17 4.1 /3.4/3.2

March16- May25 14 1.6/2.5/1.4

May 25 - August31 i2 2.3 / 2.2 / i.8

August 31 - October 26 12 2.2 / 2.0 / 1.5

(3)218 B. A. C. Ambrosiuset at. 719728.qO a, ‘I, a, -~---~

I

Fig. 8. The histories of five independent solutions of the geodesic distances between three pairs of stations, derived from full-rate LAGEOS laser ranging observations in 1986. From top to bottom, the results are for Graz - Matera, Quincy - Monument Peak and Yarragadee - Simosato, respectively. The solid lines represent straight line fits to the individual solutions, while the dashed lines indicate the rateof change according tothe Minster-Jordan AM1-2 model.

The top plot shows the solutions for the geodesicdistance betweem the European stations Graz in Austria (station number 7839)andMatera in Italy (7939), which is about 720 km. The figure clearly displays the good repeatability of the solutions for this particular pair of stations: the scatter is less than 1 cm, which is well within the envelope of the formal standard deviations. The latter are indicated by the error bars and were scaled to a measurement noise-level of 5 cm. The solid line is a weighted least squares fit through the five solutions, the slope of which represents a rate of change of no more than 0.7 cm per year. For comparison also the predicted rate of change according to the Minster-Jordan AMI-2 model /10/, derived from geological information, is indicated by the dashed line. In this model, both stations are assumed to be on the same Eurasian plate from which it follows that the rate should be zero. The actual situation may be a little bit more complicated, however, but in view of the overall uncertainty, the agreement is considered to be quite good.

The second case for which the results are presented is the station pair Quincy (7109) and Monument Peak (7110), both in the Western United States. The distance between them is also relatively short,

about 884 km. but they are located on either side of the San Andreas Fault. This is where the so-called Pacific plate slides along the North American plate with a rate of about 5 cm per year. Furthermore, the relative location is such that the line connecting the two sites is more-or-less parallel to the direction of the (transform) fault, which means that tectonic motions will be largely reflected in distance variations between these two stations. This particular interstation baseline has been the subject of intensive study using satellite laser ranging (SLR) for almost two decades so that the rate of change has been established quite accurately by now. From the last 6 years of LAGEOS tracking a decrease of 2 to 3 cm per year has been recovered /11/. In the middle plot of Figure 8 the results obtained at DUT/SOM from the analysis of 1986 data only are presented. The decreasing trend in the successive solutions for the geodesic distance is quite evident, and fitting a straight line to the data yields a slope of -2.3 cm per year while the remaining scatter is less than 0.3 cm. This is in excellent agreement with the result obtained from the 6 years of tracking mentioned above. An interesting detail, finally, is that these values disagree with the Minster-Jordan model (dashed line), which predicts a rate of -5.3 cm/year. When the sites were selected it was thought that they were rigidly located on either plate. Recent investigations /12/ seem to indicate, however, that this is not the case for Monument Peak.

The final result that is presented here is for a much longer geodesic distance of about 7260 km between Yarragadee in Australia (7090) and Simosato in Japan (7838). These two sites are also located on different tectonic plates which move considerably with respect to each other. Again, this trend is very distinct in the bottom plot of Figure 8, but a straight line fit (a rate of change of 12.5cm/year) leaves a scatter of only 0.7 cm. Currently there is no satisfactory explanation for the significant difference with the Minster-Jordan model (dashed line). Important, however, is that the dynamic modeling of the LAGEOS orbit seems to have worked quite well to connect two points on earth which are very far apart.

Summarizing, the results from the absolute and relative station positioning analyses seem to indicate that the current accuracy of the models used in the processing of LAGEOS laser ranging data is adequate to support the detection and monitoring of global tectonic motions at the 1 cm/year level. This is achieved by keeping the length of the data arcs relatively short (one week), to prevent the dynamic model errors from building up and propagating into the coordinate solutions. To ensure a sufficient tracking coverage of all stations in a global network, many one-week data arcs can be processed simultaneously to generate an accurate solution from data covering an extended period of time.

CONCLUSIONS

In this paper, it has been shown that the state-of-the-art of orbit computations for the geodetic satellite LAGEOS has reached a level of accuracy which is sufficient to support its basic geodynamics mission. It was demonstrated that changes in the relative positions of laser ranging Stations due to global tectonic motions with rates of the order of 1 cm/year may be detected from just one year of observations of LAGEOS. This was achieved by breaking up the one-year data period into many data arcs with a length of one week. Thus the accuracy of the orbit could be maintained at a level of 5 cm in the radial direction and 30 cm in both the cross-track and along-track directions.

The long-term evolution of the orbit, it was found, can be modeled with an overall accuracy of about 100 m for a one-year arc using the best general dynamic and kinematic models currently available. The largest component of this error is, of course, in the along-track direction which is primarily caused by unmodeled variations of the observed drag force acting on LAGEOS. Furthermore, the improvements in the latest models for the gravity field and the ocean tides were verified. The GEM-Ti model yields a better fit in the along-track direction which is probably mainly due to the improved tidal model. Finally, it was shown that such rare events like solar eclipses by the moon may have significant long-term effects on the orbit of LAGEOS, and should be modeled properly. The remaining errors are thought to be caused by recently discovered seasonal variations

(3)220 B. A.C.Ambrosiuseta!.

in the earth’s gravity field. This is another demonstration of the valuable contributions of the study

of very accurate satellite orbits to our understanding of geophysics.

REFERENCES

1. P. Wilson, E. Reinhart and M. Pearlman (Eds.), WEGENER-MEDLAS project plan, Institut für

Angewandte Geod~sie,Frankfurt, draft, 1985.

2. B.A.C. Ambrosius, R. Noomen, K.F. Wakker and E. Papazissi, Accuracy aspects of recent laser

station positioning experiments, Annales Geophysicae, Vol SB, No. 6, p. 523 (1987).

3. B.D. Tapley, B.E. Schutz and RJ. Eanes, Station coordinates, baselines, and earth rotation from LAGEOS laser ranging: 1976-1984,J.Geophys. Res., Vol. 90, No. Bil, p. 9235 (1985).

4. W. Melbourne, R. Anderle, M. Feissel, R. King, D. McCarthy, D. Smith, B. Tapley and R. Vicente, Project MERIT standards, USNO Circ. No. 167,27 December 1983.

5. J.M. Wahr, The forced nut.ation of an elliptical, rotating, elastic and oceanless earth, Geophys. J

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Roy. Astr. Soc., Vol. 64, p. 705 (1981).

6. R. Noomen, B.A.C. Ambrosius, K.F. Wakker and H.J.D. Piersma, European laser station positioning from LAGEOS laser ranging, in: Proceedings of the Second International Symposium on Spacecraft Right Dynamics, Darmstadt, 20-23 October 1986, ESA SP-255, p. 215 (1986). 7. D.P. Rubincam, LAGEOS orbit decay due to infrared radiation from earth, i. Geophys. Res.,

Vol. 92, No. B2, p. 1287 (1987).

8. B.A.C. Ambrosius, R. Noomen, H. Leenman and K.F. Wakker, Analysis of 1986 and 1987 global LAGEOS laser ranging data, paper presented at the NASA Crustal Dynamics Working Group Meeting, Jet Propulsion Laboratory, Pasadena, 22-24 March 1988.

9. J.G. Marsh, FJ. Larch, B.H. Putney, D.C. Christodoulidis, TI. Felsentreger, B.V. Sanchez, D.E. Smith, S.M. Kiosko, T.V. Martin, E.C. Pavlis, J.W. Robbins, R.G. Williamson, O.L. Colombo, N.L. Chandler, K.E. Rachlin, G.B. Patel, S. Bhati and D.S. Chinn, An improved model of the earth’s gravitational field: GEM-Ti, NASA Technical Memorandum 4019, Goddard Space Right Center, Greenbelt (1987).

10. J.B. Minster and T.H. Jordan, Present-day plate motions, J. Geophys. Res., Vol. 83, No. BIl, p. 5331 (1978).

11. D.E. Smith, R. Kolenkiewicz, Pi. Dunn, M.H. Torrence, E.C. Pavlis, J.W. Robbins, R.G. Williamson, S.M. Klosko, L. Carpenter and S.K. Fricke, Global laser solution: SL7.l, paper

presented at the NASA Crustal Dynamics Meeting, Goddard Space Right Center, Greenbelt,

21—23 October 1987.

12. J.B. Minster and T.H. Jordan, Vector constraints on western U.S. deformation from space