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The adjustment of crustal motion parameters in satellite geodetic networks
JOURNAL OF GEODYNAMICS8, 221 234 (1987)
221
THE A D J U S T M E N T OF CRUSTAL MOTION PARAMETERS IN SATELLITE GEODETIC NETWORKS
R. DIETRICH
Academy o[ Sciences of the GDR. Central Institute for Physics of the Earth, Telegrafenberg A 17, 1561 Potsdam, G. D. R. (Accepted May 4, 1987)
ABSTRACT Dietrich, R., 1987. The adjustment of crustal motion parameters in satellite geodetic networks. In: I. Jo6 (Editor), Recent Crustal Movements in the Carpatho-Balkan Region. Journal of Geodynamics, 8: 221-234. Global satellite networks of the highest accuracy are needed for all geodynamic investigations. The investigation of possible changes in the inner geometry of the station network is necessary because such changes would influence the determination of polar coordinates, length of day and other parameters. The different possibilities for such investigations (epoch-related coordinate sets, adjustment of station motions, determination of global kinematic plate parameters) are discussed in detail. The estimable parameters describing crustal motions are pointed oat. Laser-ranging data from the LAGEOS satellite during the MERIT Campaign are used for practical computations. These computations demonstrate an accuracy for distance determinations below the decimeter level. The time span of the data (14 months) is not sufficient for the computation of significant station motions, but it is valuable for evaluation of the model's accuracy.
INTRODUCTION
All geodynamical investigations based on space techniques need a terrestrial reference system of the highest accuracy. In international campaigns like MERIT or possible future services using space techniques, we are faced with quasi-permanent observation series at each station. Methods are needed to check the consistency of the station network and to find out the magnitude and temporal behavior of possible changes in the geometry of the network. Such changes should be investigated very carefully, because they can influence the computed values of, e.g., polar motion and length of day. In the following, suitable ways for such investigations will be discussed, and practical computations using LAGEOS laser-ranging data of the MERIT campaign will illustrate the concepts. 0264-3707/87/$3.00 JOG 82-4 9
(~ 1987 Geophysical Press Ltd,
222
DIETRICH
SOME REMARKS ON THE USE OF SATELLITE METHODS FOR CRUSTAL MOTION INVESTIGATIONS
The suitable terrestrial coordinate system In classical geodetical computations we can find a subdivision into horizontal networks and vertical networks, and also horizontal and vertical crustal-motion parameters are usually determined independently. We decided, therefore, to use a spherical coordinate system with radius, longitude and latitude as the components of the station coordinates. This includes the possibility of investigating separately the horizontal and vertical positions of the stations (the sphere is a sufficient approximation for this purpose). The motion of tectonic plates can also easily be described on a sphere (see Minster et al., 1974). Because horizontal station motions are usually about one order of magnitude larger than vertical motions, in the following we shall especially investigate the horizontal ones. In comparing distances between the stations, we will consequently not compute three-dimensional vectors (baselines) but spherical distances (on a sphere with a mean Earth radius).
Estimability of crustal-motion parameters A careful analysis of the estimability of horizontal-motion parameters has been performed by Dermanis (1981). It holds also for global satellite networks. If station coordinates using satellite laser-ranging data are computed, the origin of the coordinate system used coincides with the Earth's center of mass. The scale of the network is a result of the used values of GM and the velocity of light. The three directions of the coordinate axis can, in principle, be choosen freely-there is a datum defect of the order 3. If station coordinates, horizontal station motions, polar coordinates and length of day are computed together, we have no control of changes of the orientation of the system with time-there exists an additional rank defect of the normal equation matrix of the order 3. This means that absolute station motions cannot be determined. We can compute -relative station motions, either with respect to a group of fixed stations or (in analogy to a free network adjustment) minimizing the (weighted) sum of all motions. -distance changes and resulting deformation parameters which are at least values of geophysical relevance.
THE ADJUSTMENT OF CRUSTAL MOTION
223
Model errors in satellite geodetic computations In dynamic satellite geodesy the remaining errors of the computational model cannot be neglected. A number of them were investigated and discussed in Gendt and Dietrich (1984). For instance, there exist errors: in the computed satellite positions due to errors in the force model used (Earth gravitational field, tidal forces, atmosphere) in the orientation of the Earth with respect to the quasi-inertial system which is used for orbit integration lprecesstion, nutation, pole coordinates, rotation phase) -
-
station positions (reduction for tidal displacement and tidal loading, station motions) -in the measurements and their reduction model (calibration uncertainties, refraction model). - i n
It is therefore not surprising that the use of formal error measures or error propagation for functions of unknown parameters represent values which are too optimistic. It is therefore necessary to use methods of external accuracy determinations, e.g., comparison of results using different data. For orbit computations of the satellite LAGEOS in the orbital model POTSDAM-5, we found a factor of about 3 between internal precision estimates and the accuracy determined by external comparisons (Dietrich and Gendt, 1984; Montag et al., 1985).
POSSIBILITIES PARAMETERS
OF
THE
DETERMINATION
OF
HORIZONTAL
CRUSTAL
MOT1ON
Epoch-related coordinate sets As in classical networks, we can compute a coordinate set and use the mean time of the observations as the epoch of the coordinate set. In such a way, observations of a longer period can be subdivided into data sets of smaller time intervals. The resulting coordinates are related to the mean date of the observations used. Due to unknown differences in the orientation of the coordinate sets, one can compare only relative coordinate changes. A good way to do this is to compute distances between the same stations from different data sets and to compare them. Adjustment o f station motions One can develop any temporal behaviour of the station coordinates q~ and 2 in a Taylor series:
224
DIETRICH
~o = ~o(t) = ~p(to) + 4(to). ,;~ = ,l(t) = X(to) + ;t(to).
(t - to) + (t - to) +
.
.
.
.
.
(1)
.
The new parameters ~b, ), can be included in the adjustment, and they are then related to the epoch to. It is quite conceivable that, depending on the density of information, also a more complicated motion model with determination of ~b, J,, ~, 2 .... can be used. In a first step, we decided to determine only changes of the station coordinates linear with time. The observation equations must be enlarged. The partial derivatives needed are determined in the following way:
81
3l 9q~
3l
=
9-(o =
if-
Cl 9--~=
3l C2 ~.2 92
CI 92 (t
,o
(2) to).
It is favourable to use the initial values ~b= 2 = 0. In this case, the value l0 computed as a function of all initial values of the problem will not change. It can be shown that in this case the normal equations for one pass (with observation epoch t) can be enlarged for the new unknows without return to observation equations. The new coefficients are computed by multiplying the coefficients for ~p and 2 by the factor ( t - t o ) . It is possible either to fix motion parameters for a number of stations or to perform a free adjustment; distances and distance changes can be computed as functions of the unknowns of the adjustment,
Adjustment of kinematic plate parameters About ten years ago, Minster et al., (1974) published numerical values for plate-tectonic motions. They used spreading rates, fracture zone trends and earthquake slip vectors and inverted these data in their computation. It was demonstrated by these authors that the relative motion of one plate on the sphere can be described by 3 parameters: The coordinates of the rotation pole (latitude q5 and longitude A) and the rotational velocity ~o. Drewes (1982) showed how these parameters can be determined from geodetically derived coordinate changes or distance changes, and presented the observation equations. In principle, it is possible to start with coordinate or distance changes determined as described in the previous section. But for both ways, also the variance-covariance matrix of these parameters must be used in the new adjustment. Therefore, it should be prefered to adjust kinematic plate parameters directly, together with the station coordinates.
THE A D J U S T M E N T O F CRUSTAL M O T I O N
225
In this case also, the observation equations must be enlarged for the new unknowns. We apply the following way for the determination of the partial derivatives: (?l
3l 8~o
8l 32
3l 3l 3O (?l (?2 3--~= (?~0 3A + c~fl 3A ?l (?co
3l &o (?~o 3co
(3)
3l 32 82 8co
The first factor in each summand exists already, and the second can be taken from formula (6) in Drewes (1982). Because of the non-linearity of the problem, we must use initial values q5o, Ao and coo which are not equal to zero. This means that the value l0 computed as a function of all initial values must be changed. We have found, however, a way to enlarge the normal equations for every pass without returning to the observation equations. The new coefficients are computed by multiplying the coefficients for ~0 and 2 with the abovementioned factors &o (?2 (?~' (?q5
or
c~0 ~72 (?A' ~?A
or
(?~0 32 (?co' &o"
We use the fact that the changes of q~ and 2 are linear functions of co. This leads to the trick of using, on the one hand, good initial values of qso, Ao, coo for the computation of the partial derivatives, but on the other hand of adjusting for the whole co = coo + do). In this way, no need for modification of lo arises• In practical computations every station must be related to one plate. In the adjustment, one plate with its stations can be fixed, or a free adjustment can be performed. We included the possibility of determining a reduced number of unknowns; this means, for instance, taking q:'t and Ai from any model (e.g., that of Minster and Jordan, 1978) without improvement, and only coi will be adjusted• This is one possible way to handle single stations on one plate. A combination of station motion and plate parameter determination is also possible in such a way that stations belonging to a plate and being representative for its motion~lecided also from a geophysical point of view-are used for the determination of plate parameters, whereas for other stations individual values for ~b and ), are adjusted•
226
DIETRICH
PRACTICAL C O M P U T A T I O N S USING LAGEOS LASER-RANGING DATA OF THE MERIT CAMPAIGN
As a designated analysis centre for the satellite laser-ranging method in the MERIT project, the Central Institute for Physics of the Earth, Potsdam, analysed all LAGEOS laser data. Altogether, the data consists of about 4500 satellite passes with approximately 45.000 normal points from September 1983 up to October 1984. The coordinates of the 32 participating stations and Earth rotation parameters were presented by Montag et al.,
(1985). The orbital program system POTSDAM-5 (Gendt, 1984) was used for all numerical investigations; the program part "SOLVE" for the adjustment, based on passwise-stored normal equations, was enlarged for additional station-motion parameter determinations in the way described before. In the program, all constants and parameters published in the MERITstandards (Melbourne et al., 1983) were implemented. Five-day arcs were analysed, the orbital fit was on the order of 5 . . - 1 0 c m for most of the TABLE I
Stations used for practical computations (OF: orbital fit in the computations with POTSDAM-5) Day of Number
first
last pass
Station Number
Lon. ()
Lat. ()
of passes
OF (cm)
1
2
3
4
5
6
7
50.7 52.0 49.0
329 71 320
6.7 12.7 7.6
45611 .45601 45583
46004 46004 46004
Herstmonc. Kootwijk Wettzell
7840 7833 7834
0.3 5.8 12.9
(MJD)
Potsdam
1181
13.1
52.2
115
18.9
,J,5583
46004
Graz
7839 7939 7090 7838
15.5 16.7 115.4 135.9
46.9 40.5 28.9 33.4
182 339 271 229
6,0 8.5 6.6 8.9
45587 45583 45639 45586
46003 46004 46004 46004
7210 7121 7109 7110 7122 7112 7086 7105 7907
203.7 209.0 239.0 243.6 253.5 255.3 256.0 283.2 288.5
20.6 - 16.6 39.8 32.7 23.2 40.0 30.5 38.8 - 16.4
379 134 445 384 180 136 124 231 415
6.1 8.0 5.(1 4.8 8.3 7.7 8.8 5.5 9.0
45585 45583 45584 45614 45603 45583 45586 45592 45582
46000 46004 45999 46004 46004 45985 45990 46(10(I 46002
Matera Yaragadec Simosato Maui
Huahine Quincy Mon. Peak
Mazatlan Plattevil. Ft. Davis
Greenbelt Arequipa
THE ADJUSTMENT OF CRUSTAL MOTION
227
stations. The adjusted parameters include orbital elements, pole coordinates, length of day and station coordinates as a standard. Additionally, horizontal-motion parameters were determined. In the following computations, all those stations were included which were observing regularly during the whole campaign (sufficient time span and density of observations). Table I presents the stations used, with additional information.
Epoch-related coordinate sets In these computations the data of the whole MERIT campaign were subdivided into 4 intervals with an interval duration of about 100 days. The resulting station coordinates were used to compute the spherical distances of all station pairs. Examples of these results are presented in Tables II, Ill and IV. The precision of the distances computed from the program is on the order of 1 ... 2 cm in most cases. If we neglect possible station motions, we can take the four values presented for each distance as an independent determination. The differences in these determinations give the possibility of computing an external accuracy measure. It is possible to conclude that the distances are determined with an accuracy of 1 decimeter or even better. There seems to be no dependence with respect to the absolute value of the distance. Also laser instruments of the 2nd generation (station 1181) give reliable results in this computation variant. This means that if there is, for example, a permanent change of a distance on the order of 5 centimeters, one can be sure of detecting this effect using satellite laser-ranging data after about 5 years of observations.
Station-motion adjustment It can be expected that data with a duration of 14 months are not sufficient for a station-motion determination. But they can be used advantageously for an estimation of reachable accuracies. For this purpose, horizontal station motion parameters ,~ and 4, were adjusted. For the interpretation of the results, the following standpoint was used: The real station motions during the 14 months are neglectable small, all computed motion values reflect remaining model errors and can therefore be used to determine r.m.s, values. In Table V the r.m.s, values of the computed station motions in dependence of the time interval (200, 300, 400 days) are presented, One can expect decreasing r.m.s, values of ~b and )~ with increasing time, reaching + 9 cm/year for ). and + 6 cm/year for ~b for a data span of 14 months.
228
DIETRICH
I I I
+++
+ I l l + + +
Z
A
t
0
¢b
o
H
C~
II
THE ADJUSTMENT
OF CRUSTAL
MOTION
I
o
Z
oo
~a e'-I
[...,
©
-,'-t
©
N E
g "~_~
~o°o b-
229
+
+
+
230
DIETRICH
"r--
+
;
Z
.7
oo
P~
r',h,
©
"4
0
©
©
~4
-
I
I
+
+
t
"1"
~
oc
+
+
+
+
~4
THE ADJUSTMENT OF CRUSTAL MOTION
231
TABLE V r.m.s, values for 2 and ~b (in cm/year} for time intervals of 200, 300 and 400 days with observations time interval 200 days
300 days
400 days
r.m.s. ,~
+18
_+14
-+9
r.m.s. ~
+12
±7
4:6
Table VI shows the results of another computational variant: All 5 day arcs starting from the beginning of the campaign were given an integer number NARc in ascending order. Three data groups were selected. This first group contained all arcs with NARc= 1 rood 3, the second all arcs with NARc = 2 rood 3, and the third all arcs with N A R c = 0 mod 3. The table presents also the resulting station coordinate changes for every group. The datum difference between the three solutions seems to be small enough for first interpretations. In about 50% of all cases of computed motion values there still exist different signs among the 3 data groups, only some stations show stable trends. Larger differences for station 7833 and 1181 may be explained by the ranging accuracy, but for station 7086 for instance, they may be an indication of data problems (calibration), even taking into consideration the smaller number of observed passes. The motion values are used to compute distance changes with time. Tables II, III and IV show the results for a selected number of distances (column 11). For all presented results the r.m.s, of the ,~-value is +_ 11 cm/year.
Kinematic plate parameter determination The formalism described in the previous chapter was used to compute kinematic plate parameters. One fundamental assumption in this model is the non-existence of intraplate deformations. The results of the computations in Tables II, III, IV and V already indicate that this assumption seems to be a bad approximation of reality. The results of our computations show that a plate-motion model cannot be found which gives a good fit to the data. Even a reduced parameter set [4~,, A~ fixed on the values of RM2 of Minster and Jordan, 1978) with the determination of ~oi alone ( the r.m.s, values of the ~oi are in the order of
232
DIETRICH
G~
<
O~
I
I
I
I
I
I
I
< I
,
d
I
r
I
1
I
r
I
I
I
J
[
I
I
r
L rl
I I I A I I L
dJ
~ l f l ~
Pr
I
I
i I
< J
o
o -~, c~
.,,-.
,..a
I
I
[
r
l
F o
o
o
rl
IF
IF
~
J i
Z Z Z '
k.
THE ADJUSTMENT OF CRUSTAL MOTION
233
4-0.1-.-0.2 deg/mill, years) gave only a partial similarity to the co-values of RM2. But this is a problem of the data used and not of the applied algorithm. We will continue our research.
CONCLUSIONS
For monitoring the consistency of a conventional on satellite laser-ranging data and the detection station coordinates, different practicable methods adjustment and analysis were presented. These are -computation of epoch-related subsets of station
terrestrial system based of possible changes of in the process of data
coordinates
-adjustment of linear shifts of station coordinates -adjustment of kinematic plate parameters. The main purpose of the paper was discussion of the adjustment variants. Practical computations using LAGEOS laser-ranging data of the MERIT campaign show: - A subdecimeter level in distance determinations is reached for good stations. - F u r t h e r investigations should be performed to increase the significance of the results. - A number of problems should be discussed with geophysicists (e.g., the problem of intraplate deformations or regions which can be regarded as representative with their motion for the entire plate). - A longer time span with data is needed for significant motion determinations. REFERENCES Dermanis, A., 1981. Geodetic estimability of crustal deformation parameters. Quatern. Geod. Thessaloniki 2: pp. 159-169. Dietrich, R. and Gendt, G , 1985. An attempt to detect geometric tidal information using LAGEOS laser ranging data. Proceedings 5th Intern. Syrup. on Geod. and Physics of the Earth, Magdeburg, GDR, Sept. 23-29, 1984, Publ. ZIPE, No. 81: part 1, pp. 77-81, Potsdam. Drewes, H., 1982. A geodetic approach for the recovery of global kinematic plate parameters. Bull. Geod., Paris 56: I, pp. 70-79. Gendt, G,, 1984, Further improvements of the orbital program system POTSDAM-5 and their utilization in geodetic-geodynamic investigations. Nabl. isk. sputn, zemli, Praha 23: pp. 421-428. Gendt, G. and Dietrich, R., 1984. On homogenious global satellite network determinations for tectonic motion investigations. Int. Syrup. on Space Techniques for Geodynamics, Sopron, Hungary, Juli 9-13, 1984. Proceedings 2: pp. 110-118.
234
DIETRICH
Melbourne, W., Anderle, R., Feissel, M., King, R., McCarthy, D., Smith, D., Tapley, B. and Vicente, R., 1983. Project MERIT standards. United States Naval Observatory Circular No. 167, Washington. Minster, J. B, Jordan, T. H., Molnar, P. and Haines, E., 1974. Numerical modelling of instantaneous plate tectonics. Geophys. J. R. astr. Soc. 36: pp. 541--576. Minster, J. B. and Jordan, T. H., 1978. Present-day plate motions. J. of. Geoph. Res. 83: B l l, pp. 5331-5353. Montag, H., Gendt, G., Dietrich, R. and Kurth, K., 1985. Investigation of polar motion and the length of day by means of SLR data of the MERIT Campaign. Paper Int. MERIT/COTES Conference on Earth rotation and the terrestrial reference system, Columbus/Ohio, USA, July-August 1985. Publication of the Central Institute for Physics of the Earth No. 1472.
221
THE A D J U S T M E N T OF CRUSTAL MOTION PARAMETERS IN SATELLITE GEODETIC NETWORKS
R. DIETRICH
Academy o[ Sciences of the GDR. Central Institute for Physics of the Earth, Telegrafenberg A 17, 1561 Potsdam, G. D. R. (Accepted May 4, 1987)
ABSTRACT Dietrich, R., 1987. The adjustment of crustal motion parameters in satellite geodetic networks. In: I. Jo6 (Editor), Recent Crustal Movements in the Carpatho-Balkan Region. Journal of Geodynamics, 8: 221-234. Global satellite networks of the highest accuracy are needed for all geodynamic investigations. The investigation of possible changes in the inner geometry of the station network is necessary because such changes would influence the determination of polar coordinates, length of day and other parameters. The different possibilities for such investigations (epoch-related coordinate sets, adjustment of station motions, determination of global kinematic plate parameters) are discussed in detail. The estimable parameters describing crustal motions are pointed oat. Laser-ranging data from the LAGEOS satellite during the MERIT Campaign are used for practical computations. These computations demonstrate an accuracy for distance determinations below the decimeter level. The time span of the data (14 months) is not sufficient for the computation of significant station motions, but it is valuable for evaluation of the model's accuracy.
INTRODUCTION
All geodynamical investigations based on space techniques need a terrestrial reference system of the highest accuracy. In international campaigns like MERIT or possible future services using space techniques, we are faced with quasi-permanent observation series at each station. Methods are needed to check the consistency of the station network and to find out the magnitude and temporal behavior of possible changes in the geometry of the network. Such changes should be investigated very carefully, because they can influence the computed values of, e.g., polar motion and length of day. In the following, suitable ways for such investigations will be discussed, and practical computations using LAGEOS laser-ranging data of the MERIT campaign will illustrate the concepts. 0264-3707/87/$3.00 JOG 82-4 9
(~ 1987 Geophysical Press Ltd,
222
DIETRICH
SOME REMARKS ON THE USE OF SATELLITE METHODS FOR CRUSTAL MOTION INVESTIGATIONS
The suitable terrestrial coordinate system In classical geodetical computations we can find a subdivision into horizontal networks and vertical networks, and also horizontal and vertical crustal-motion parameters are usually determined independently. We decided, therefore, to use a spherical coordinate system with radius, longitude and latitude as the components of the station coordinates. This includes the possibility of investigating separately the horizontal and vertical positions of the stations (the sphere is a sufficient approximation for this purpose). The motion of tectonic plates can also easily be described on a sphere (see Minster et al., 1974). Because horizontal station motions are usually about one order of magnitude larger than vertical motions, in the following we shall especially investigate the horizontal ones. In comparing distances between the stations, we will consequently not compute three-dimensional vectors (baselines) but spherical distances (on a sphere with a mean Earth radius).
Estimability of crustal-motion parameters A careful analysis of the estimability of horizontal-motion parameters has been performed by Dermanis (1981). It holds also for global satellite networks. If station coordinates using satellite laser-ranging data are computed, the origin of the coordinate system used coincides with the Earth's center of mass. The scale of the network is a result of the used values of GM and the velocity of light. The three directions of the coordinate axis can, in principle, be choosen freely-there is a datum defect of the order 3. If station coordinates, horizontal station motions, polar coordinates and length of day are computed together, we have no control of changes of the orientation of the system with time-there exists an additional rank defect of the normal equation matrix of the order 3. This means that absolute station motions cannot be determined. We can compute -relative station motions, either with respect to a group of fixed stations or (in analogy to a free network adjustment) minimizing the (weighted) sum of all motions. -distance changes and resulting deformation parameters which are at least values of geophysical relevance.
THE ADJUSTMENT OF CRUSTAL MOTION
223
Model errors in satellite geodetic computations In dynamic satellite geodesy the remaining errors of the computational model cannot be neglected. A number of them were investigated and discussed in Gendt and Dietrich (1984). For instance, there exist errors: in the computed satellite positions due to errors in the force model used (Earth gravitational field, tidal forces, atmosphere) in the orientation of the Earth with respect to the quasi-inertial system which is used for orbit integration lprecesstion, nutation, pole coordinates, rotation phase) -
-
station positions (reduction for tidal displacement and tidal loading, station motions) -in the measurements and their reduction model (calibration uncertainties, refraction model). - i n
It is therefore not surprising that the use of formal error measures or error propagation for functions of unknown parameters represent values which are too optimistic. It is therefore necessary to use methods of external accuracy determinations, e.g., comparison of results using different data. For orbit computations of the satellite LAGEOS in the orbital model POTSDAM-5, we found a factor of about 3 between internal precision estimates and the accuracy determined by external comparisons (Dietrich and Gendt, 1984; Montag et al., 1985).
POSSIBILITIES PARAMETERS
OF
THE
DETERMINATION
OF
HORIZONTAL
CRUSTAL
MOT1ON
Epoch-related coordinate sets As in classical networks, we can compute a coordinate set and use the mean time of the observations as the epoch of the coordinate set. In such a way, observations of a longer period can be subdivided into data sets of smaller time intervals. The resulting coordinates are related to the mean date of the observations used. Due to unknown differences in the orientation of the coordinate sets, one can compare only relative coordinate changes. A good way to do this is to compute distances between the same stations from different data sets and to compare them. Adjustment o f station motions One can develop any temporal behaviour of the station coordinates q~ and 2 in a Taylor series:
224
DIETRICH
~o = ~o(t) = ~p(to) + 4(to). ,;~ = ,l(t) = X(to) + ;t(to).
(t - to) + (t - to) +
.
.
.
.
.
(1)
.
The new parameters ~b, ), can be included in the adjustment, and they are then related to the epoch to. It is quite conceivable that, depending on the density of information, also a more complicated motion model with determination of ~b, J,, ~, 2 .... can be used. In a first step, we decided to determine only changes of the station coordinates linear with time. The observation equations must be enlarged. The partial derivatives needed are determined in the following way:
81
3l 9q~
3l
=
9-(o =
if-
Cl 9--~=
3l C2 ~.2 92
CI 92 (t
,o
(2) to).
It is favourable to use the initial values ~b= 2 = 0. In this case, the value l0 computed as a function of all initial values of the problem will not change. It can be shown that in this case the normal equations for one pass (with observation epoch t) can be enlarged for the new unknows without return to observation equations. The new coefficients are computed by multiplying the coefficients for ~p and 2 by the factor ( t - t o ) . It is possible either to fix motion parameters for a number of stations or to perform a free adjustment; distances and distance changes can be computed as functions of the unknowns of the adjustment,
Adjustment of kinematic plate parameters About ten years ago, Minster et al., (1974) published numerical values for plate-tectonic motions. They used spreading rates, fracture zone trends and earthquake slip vectors and inverted these data in their computation. It was demonstrated by these authors that the relative motion of one plate on the sphere can be described by 3 parameters: The coordinates of the rotation pole (latitude q5 and longitude A) and the rotational velocity ~o. Drewes (1982) showed how these parameters can be determined from geodetically derived coordinate changes or distance changes, and presented the observation equations. In principle, it is possible to start with coordinate or distance changes determined as described in the previous section. But for both ways, also the variance-covariance matrix of these parameters must be used in the new adjustment. Therefore, it should be prefered to adjust kinematic plate parameters directly, together with the station coordinates.
THE A D J U S T M E N T O F CRUSTAL M O T I O N
225
In this case also, the observation equations must be enlarged for the new unknowns. We apply the following way for the determination of the partial derivatives: (?l
3l 8~o
8l 32
3l 3l 3O (?l (?2 3--~= (?~0 3A + c~fl 3A ?l (?co
3l &o (?~o 3co
(3)
3l 32 82 8co
The first factor in each summand exists already, and the second can be taken from formula (6) in Drewes (1982). Because of the non-linearity of the problem, we must use initial values q5o, Ao and coo which are not equal to zero. This means that the value l0 computed as a function of all initial values must be changed. We have found, however, a way to enlarge the normal equations for every pass without returning to the observation equations. The new coefficients are computed by multiplying the coefficients for ~0 and 2 with the abovementioned factors &o (?2 (?~' (?q5
or
c~0 ~72 (?A' ~?A
or
(?~0 32 (?co' &o"
We use the fact that the changes of q~ and 2 are linear functions of co. This leads to the trick of using, on the one hand, good initial values of qso, Ao, coo for the computation of the partial derivatives, but on the other hand of adjusting for the whole co = coo + do). In this way, no need for modification of lo arises• In practical computations every station must be related to one plate. In the adjustment, one plate with its stations can be fixed, or a free adjustment can be performed. We included the possibility of determining a reduced number of unknowns; this means, for instance, taking q:'t and Ai from any model (e.g., that of Minster and Jordan, 1978) without improvement, and only coi will be adjusted• This is one possible way to handle single stations on one plate. A combination of station motion and plate parameter determination is also possible in such a way that stations belonging to a plate and being representative for its motion~lecided also from a geophysical point of view-are used for the determination of plate parameters, whereas for other stations individual values for ~b and ), are adjusted•
226
DIETRICH
PRACTICAL C O M P U T A T I O N S USING LAGEOS LASER-RANGING DATA OF THE MERIT CAMPAIGN
As a designated analysis centre for the satellite laser-ranging method in the MERIT project, the Central Institute for Physics of the Earth, Potsdam, analysed all LAGEOS laser data. Altogether, the data consists of about 4500 satellite passes with approximately 45.000 normal points from September 1983 up to October 1984. The coordinates of the 32 participating stations and Earth rotation parameters were presented by Montag et al.,
(1985). The orbital program system POTSDAM-5 (Gendt, 1984) was used for all numerical investigations; the program part "SOLVE" for the adjustment, based on passwise-stored normal equations, was enlarged for additional station-motion parameter determinations in the way described before. In the program, all constants and parameters published in the MERITstandards (Melbourne et al., 1983) were implemented. Five-day arcs were analysed, the orbital fit was on the order of 5 . . - 1 0 c m for most of the TABLE I
Stations used for practical computations (OF: orbital fit in the computations with POTSDAM-5) Day of Number
first
last pass
Station Number
Lon. ()
Lat. ()
of passes
OF (cm)
1
2
3
4
5
6
7
50.7 52.0 49.0
329 71 320
6.7 12.7 7.6
45611 .45601 45583
46004 46004 46004
Herstmonc. Kootwijk Wettzell
7840 7833 7834
0.3 5.8 12.9
(MJD)
Potsdam
1181
13.1
52.2
115
18.9
,J,5583
46004
Graz
7839 7939 7090 7838
15.5 16.7 115.4 135.9
46.9 40.5 28.9 33.4
182 339 271 229
6,0 8.5 6.6 8.9
45587 45583 45639 45586
46003 46004 46004 46004
7210 7121 7109 7110 7122 7112 7086 7105 7907
203.7 209.0 239.0 243.6 253.5 255.3 256.0 283.2 288.5
20.6 - 16.6 39.8 32.7 23.2 40.0 30.5 38.8 - 16.4
379 134 445 384 180 136 124 231 415
6.1 8.0 5.(1 4.8 8.3 7.7 8.8 5.5 9.0
45585 45583 45584 45614 45603 45583 45586 45592 45582
46000 46004 45999 46004 46004 45985 45990 46(10(I 46002
Matera Yaragadec Simosato Maui
Huahine Quincy Mon. Peak
Mazatlan Plattevil. Ft. Davis
Greenbelt Arequipa
THE ADJUSTMENT OF CRUSTAL MOTION
227
stations. The adjusted parameters include orbital elements, pole coordinates, length of day and station coordinates as a standard. Additionally, horizontal-motion parameters were determined. In the following computations, all those stations were included which were observing regularly during the whole campaign (sufficient time span and density of observations). Table I presents the stations used, with additional information.
Epoch-related coordinate sets In these computations the data of the whole MERIT campaign were subdivided into 4 intervals with an interval duration of about 100 days. The resulting station coordinates were used to compute the spherical distances of all station pairs. Examples of these results are presented in Tables II, Ill and IV. The precision of the distances computed from the program is on the order of 1 ... 2 cm in most cases. If we neglect possible station motions, we can take the four values presented for each distance as an independent determination. The differences in these determinations give the possibility of computing an external accuracy measure. It is possible to conclude that the distances are determined with an accuracy of 1 decimeter or even better. There seems to be no dependence with respect to the absolute value of the distance. Also laser instruments of the 2nd generation (station 1181) give reliable results in this computation variant. This means that if there is, for example, a permanent change of a distance on the order of 5 centimeters, one can be sure of detecting this effect using satellite laser-ranging data after about 5 years of observations.
Station-motion adjustment It can be expected that data with a duration of 14 months are not sufficient for a station-motion determination. But they can be used advantageously for an estimation of reachable accuracies. For this purpose, horizontal station motion parameters ,~ and 4, were adjusted. For the interpretation of the results, the following standpoint was used: The real station motions during the 14 months are neglectable small, all computed motion values reflect remaining model errors and can therefore be used to determine r.m.s, values. In Table V the r.m.s, values of the computed station motions in dependence of the time interval (200, 300, 400 days) are presented, One can expect decreasing r.m.s, values of ~b and )~ with increasing time, reaching + 9 cm/year for ). and + 6 cm/year for ~b for a data span of 14 months.
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TABLE V r.m.s, values for 2 and ~b (in cm/year} for time intervals of 200, 300 and 400 days with observations time interval 200 days
300 days
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Table VI shows the results of another computational variant: All 5 day arcs starting from the beginning of the campaign were given an integer number NARc in ascending order. Three data groups were selected. This first group contained all arcs with NARc= 1 rood 3, the second all arcs with NARc = 2 rood 3, and the third all arcs with N A R c = 0 mod 3. The table presents also the resulting station coordinate changes for every group. The datum difference between the three solutions seems to be small enough for first interpretations. In about 50% of all cases of computed motion values there still exist different signs among the 3 data groups, only some stations show stable trends. Larger differences for station 7833 and 1181 may be explained by the ranging accuracy, but for station 7086 for instance, they may be an indication of data problems (calibration), even taking into consideration the smaller number of observed passes. The motion values are used to compute distance changes with time. Tables II, III and IV show the results for a selected number of distances (column 11). For all presented results the r.m.s, of the ,~-value is +_ 11 cm/year.
Kinematic plate parameter determination The formalism described in the previous chapter was used to compute kinematic plate parameters. One fundamental assumption in this model is the non-existence of intraplate deformations. The results of the computations in Tables II, III, IV and V already indicate that this assumption seems to be a bad approximation of reality. The results of our computations show that a plate-motion model cannot be found which gives a good fit to the data. Even a reduced parameter set [4~,, A~ fixed on the values of RM2 of Minster and Jordan, 1978) with the determination of ~oi alone ( the r.m.s, values of the ~oi are in the order of
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4-0.1-.-0.2 deg/mill, years) gave only a partial similarity to the co-values of RM2. But this is a problem of the data used and not of the applied algorithm. We will continue our research.
CONCLUSIONS
For monitoring the consistency of a conventional on satellite laser-ranging data and the detection station coordinates, different practicable methods adjustment and analysis were presented. These are -computation of epoch-related subsets of station
terrestrial system based of possible changes of in the process of data
coordinates
-adjustment of linear shifts of station coordinates -adjustment of kinematic plate parameters. The main purpose of the paper was discussion of the adjustment variants. Practical computations using LAGEOS laser-ranging data of the MERIT campaign show: - A subdecimeter level in distance determinations is reached for good stations. - F u r t h e r investigations should be performed to increase the significance of the results. - A number of problems should be discussed with geophysicists (e.g., the problem of intraplate deformations or regions which can be regarded as representative with their motion for the entire plate). - A longer time span with data is needed for significant motion determinations. REFERENCES Dermanis, A., 1981. Geodetic estimability of crustal deformation parameters. Quatern. Geod. Thessaloniki 2: pp. 159-169. Dietrich, R. and Gendt, G , 1985. An attempt to detect geometric tidal information using LAGEOS laser ranging data. Proceedings 5th Intern. Syrup. on Geod. and Physics of the Earth, Magdeburg, GDR, Sept. 23-29, 1984, Publ. ZIPE, No. 81: part 1, pp. 77-81, Potsdam. Drewes, H., 1982. A geodetic approach for the recovery of global kinematic plate parameters. Bull. Geod., Paris 56: I, pp. 70-79. Gendt, G,, 1984, Further improvements of the orbital program system POTSDAM-5 and their utilization in geodetic-geodynamic investigations. Nabl. isk. sputn, zemli, Praha 23: pp. 421-428. Gendt, G. and Dietrich, R., 1984. On homogenious global satellite network determinations for tectonic motion investigations. Int. Syrup. on Space Techniques for Geodynamics, Sopron, Hungary, Juli 9-13, 1984. Proceedings 2: pp. 110-118.
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DIETRICH
Melbourne, W., Anderle, R., Feissel, M., King, R., McCarthy, D., Smith, D., Tapley, B. and Vicente, R., 1983. Project MERIT standards. United States Naval Observatory Circular No. 167, Washington. Minster, J. B, Jordan, T. H., Molnar, P. and Haines, E., 1974. Numerical modelling of instantaneous plate tectonics. Geophys. J. R. astr. Soc. 36: pp. 541--576. Minster, J. B. and Jordan, T. H., 1978. Present-day plate motions. J. of. Geoph. Res. 83: B l l, pp. 5331-5353. Montag, H., Gendt, G., Dietrich, R. and Kurth, K., 1985. Investigation of polar motion and the length of day by means of SLR data of the MERIT Campaign. Paper Int. MERIT/COTES Conference on Earth rotation and the terrestrial reference system, Columbus/Ohio, USA, July-August 1985. Publication of the Central Institute for Physics of the Earth No. 1472.