8/16/2019 29 Brouwer

1/192

8/16/2019 29 Brouwer

2/192

8/16/2019 29 Brouwer

3/192

8/16/2019 29 Brouwer

4/192

8/16/2019 29 Brouwer

5/192

8/16/2019 29 Brouwer

6/192

8/16/2019 29 Brouwer

7/192

8/16/2019 29 Brouwer

8/192

8/16/2019 29 Brouwer

9/192

8/16/2019 29 Brouwer

10/192

8/16/2019 29 Brouwer

11/192

8/16/2019 29 Brouwer

12/192

8/16/2019 29 Brouwer

13/192

8/16/2019 29 Brouwer

14/192

8/16/2019 29 Brouwer

15/192

8/16/2019 29 Brouwer

16/192

8/16/2019 29 Brouwer

17/192

8/16/2019 29 Brouwer

18/192

8/16/2019 29 Brouwer

19/192

8/16/2019 29 Brouwer

20/192

8/16/2019 29 Brouwer

21/192

8/16/2019 29 Brouwer

22/192

8/16/2019 29 Brouwer

23/192

8/16/2019 29 Brouwer

24/192

8/16/2019 29 Brouwer

25/192

8/16/2019 29 Brouwer

26/192

8/16/2019 29 Brouwer

27/192

8/16/2019 29 Brouwer

28/192

8/16/2019 29 Brouwer

29/192

8/16/2019 29 Brouwer

30/192

8/16/2019 29 Brouwer

31/192

8/16/2019 29 Brouwer

32/192

8/16/2019 29 Brouwer

33/192

8/16/2019 29 Brouwer

34/192

8/16/2019 29 Brouwer

35/192

8/16/2019 29 Brouwer

36/192

8/16/2019 29 Brouwer

37/192

8/16/2019 29 Brouwer

38/192

8/16/2019 29 Brouwer

39/192

8/16/2019 29 Brouwer

40/192

8/16/2019 29 Brouwer

41/192

8/16/2019 29 Brouwer

42/192

8/16/2019 29 Brouwer

43/192

8/16/2019 29 Brouwer

44/192

8/16/2019 29 Brouwer

45/192

8/16/2019 29 Brouwer

46/192

8/16/2019 29 Brouwer

47/192

8/16/2019 29 Brouwer

48/192

8/16/2019 29 Brouwer

49/192

8/16/2019 29 Brouwer

50/192

8/16/2019 29 Brouwer

51/192

8/16/2019 29 Brouwer

52/192

8/16/2019 29 Brouwer

53/192

8/16/2019 29 Brouwer

54/192

8/16/2019 29 Brouwer

55/192

8/16/2019 29 Brouwer

56/192

8/16/2019 29 Brouwer

57/192

8/16/2019 29 Brouwer

58/192

8/16/2019 29 Brouwer

59/192

8/16/2019 29 Brouwer

60/192

8/16/2019 29 Brouwer

61/192

8/16/2019 29 Brouwer

62/192

8/16/2019 29 Brouwer

63/192

8/16/2019 29 Brouwer

64/192

8/16/2019 29 Brouwer

65/192

8/16/2019 29 Brouwer

66/192

2.7.6 Oce an Load ing

The changing loads of ocea n masses on the continental shelves due totides generate a tidal motion of the continent s. The relative phase lagof the ocean tide to the corresponding Earth tide and the height dis-placement ampl itud e can be computed for the main tides using e.g. the[Sc hwi de rsk i,l 978 ] odel. As yet , this is not implemented in DEGRIAS.

Discuss on

For MERIT, the ocea n loading displacements due to the 9 main tides arecomputed for 25 stat ions all over the world [Melbourne et a1.,19831 .For the majority of these stations (including Westerbork/Kootwijk) thedisplacement is less than 1 cm. Only for stations like Bermuda has amaximum displacement of cm been computed. On the average, therefore acont ribu tion to the total error budget of DEGRIAS of one centimetre inregions near the oceans can be expected.

2.7.7 Miscellaneous

To con clude, a number of geophysical phenomena are mentioned influencingsite positions , notably: groundwat er, atmospheric loading and crustaldynamics.

Discussion

Changing groundwater storage due to seasonal variations and changes inthe atmospher ic masses (atm ospheric load ing) may vary the positions oftelescopes by one centim etre in height [Larden,l 980]. It will, however,be hard to model this reliably.Furthermo re, there exists some speculation about a secular expansion ofthe Earth [Dic ke,1 969] . The possible rate of this expansion is esti-mated to be 10-100 mm/century.Fina lly, the effect of crustal movement s and plate tectonics is men-tioned, which is well-accepted but st ill a hypothesis to be confirmed bydirect meas urements ; cf. the objectives of geodetic VLBI in 51.2. Theaverage relative motions of the major plates are shown in Figure 17.Typically the plates move at rates of a few cm per year , with a maximumof nearly 20 cm per year along the Pacific-Nazca boundary. A directmeasure of these motions will shed light on the rheology of the crust

and upper mantle and the responsible (conve ctio n) mechanism, ultimatelyenabling the prediction of earthquake activity.

2.8 ASSESSMENT F ACCURACIES

In the preceding sect ions an idea is conveyed of the various phenomenacomposing the real world for geodetic VLBI. Algorithms have been dis-cussed composing the model worl d in DEGR IAS, together with an indica-tion of their level of accuracy. In this section an assessmen t will bemade of these accu raci es, and compared to bottom line results thatwill be possible in the near future.

8/16/2019 29 Brouwer

67/192

Figure 17: Motions of Tecton ic Plates

In this assess ment a careful distinc tion must be made between relativ eand absolu te accura cy. Relati ve is defined here as: influencing therelati ve positi ons of statio ns, wherea s absolu te is defined as: in-fluencing the position of the polyhedron formed by all stations withrespect to the adopted frame of reference. It may be clear that thesedefini tions are direct ly related to the primary goal of geodetic VLBI(51.2): world- wide positi oning of statio ns. It may also be clear thatthis refers to the definition of form-elements [Baarda, 19661; see51.4. An exampl e may illustr ate the differ ence.In S2.4.2 the differ ence betwee n the 1950.0 precessional consta nt andthe 52000 one was given as about 1.1 arcsec per centur y. Betwee n DEG-RIAS coordinates and results of other software packages using theJ2000-ephemeris, a rotation is therefore present; this can be regardedas absolute accuracy. If, however, from the coordinates (presuming thatall other algori thms are the same) the baseline length s and the anglesbetween baselines are comput ed, the result s will be ( almost ) identical:relative accuracy.

This is also shown by a computational example. With DEGRIAS simulated

observ ations are generated for the MERIT-SC schedu le (§3.7), during 48hours on two baselines: Effelsberg (Bonn, FRG ) OVRO (Cal., USA) andEffelsberg-Haystack (Mass., USA). Next, two adjustments are made, oneusing the 1950.0-system preces sional constan t and one using the interpo-lated new preces sional constant. From the result s under item 2 in Ta-bles 2 and 3 it is clear that an apparen t rotation of the network occursof 0.317 arcsec (which equals 7 metres) but that baseline lengths staythe same. In the same way 1 1 other examples have been computed pertain-ing e.g. to gravitation al deflec tion and nutati on, as fol lows:

1 standard simulation fit, using full DEGRIAS model;delays onl y; a priori standard devia tion 0.2 ns

2 using 1950.0 instead of interpolated new preces sional consta nt3 no nutati on included

8/16/2019 29 Brouwer

68/192

4 = no E-terms in aberration included5 no gravitational deflection correction applied6 = tropospheric refraction without weather data7 no ionospheric refraction correction applied8 changing all polar motion Xp-components by 0.01 arcsec9 changing Xp as in 8, but only for one day

10 with a 10% change in Earth tide Love numbers11 no Earth tides included12 with a 5% error in retarded baseline correction13 1 cm axis offset introduced in Haystack

Table 2.

Simulated Distortions of EFF-OVRO Baseline

- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

I RMS I dX dY I dZ dL dAzim.I I (ns) I (cm) I (cm) I (cm) I (cm) l II----+--------+--------+--------+--------+---------+-------l) 1 0 . 1 7 1 1 1 1 1 1

with sigma: I 11 1 24 9 11 7 1(----+--------+--------+--------+--------+---------+-------

1 2 ( 0 . 1 7 ( - 7 6 4 1 - 9 9 1 1 0 0 31711 3 1 0 . 9 8 1 552 1 733 18 1 8 - 2 3 3 11 4 1 0 . 1 7 1 64 1 -83 1 0 I 0 2 7 1( 5 0 . 2 4 1 - 1 2 0 1 - 1 5 7 1 9 1 -1 1 5 0 11 6 1 0 . 2 1 -14 -21 1 8 1 0 I 7 1( 7 1 0 . 2 5 1 -55 -28 1 1 26 1 1 4 1

8 1 0.17 1 5 1 0 - 3 1 1 0 - 1 11 9 0.18 3 -2 1 - 1 1 1 -2 1 0110 1 0.17 0 I 2 1 1 1 1 1 0 I111 1 0.22 1 4 -16 1 6 1 -12 3 1( 1 2 1 0 . 2 3 ) 1 2 1 2 6 ) 15 1 5 - 7 1113 0.17 0 O I 1 I 1 1 0

----------------------------------------------------------

A few remarks about the results of these simulations can be made.Firstly, t is immediately clear that the introduction of an error in

the adjustment model affects the relative positions of stations farless than their absolute ones. This means that most phenomena haveabout the same influence on all baselines. A good example is nutation.Adjustment 3 of the tables shows that completely neglecting nutation inthe adjustment increases RMS residuals by a factor of five and changesstation positions by several metres, but the rotation of both baselinesaround the Z-axis (azimuth) is the same and the change in length is ap-proximately proportional to baseline length (EFF-OVRO 8200 km and EFF-HAY 5600 km). This is of course correct, as nutation is to a first or-der approximation a pure rotation and deviations from this are mainlyseen as residuals and only partly as a change in the estimated parame-ters. For instance, ionospheric refraction (example 7) and Earth tides(example 11) can obviously not be described by a rotation and/or achange in scale alone.

8/16/2019 29 Brouwer

69/192

Table 3.

Simulated Distortions of EFF-HAY Baseline

I I RMS dX I dY dZ I dL I d ~ z i m .I I (ns) I (cm) l (cm) l (cm) I (cm) I II----+--------+--------+--------+--------+---------+-------l( 1 0 . 2 0 1 I I II w i t h s i g m a : ( 13 1 1 3 3 7 1 6----+--------+--------+--------+--------+---------+-------l

1 2 1 0 . 2 0 760 391 0 I 0 3171 3 0.66 -554 -293 -7 6 -2331 4 1 0 . 2 0 1 6 4 ) 3 2 1 0 0 2 7 11 5 1 0 . 2 4 121 61 -9 0 5 0 )1 6 1 0 . 2 4 14 5 1 4 1 2 1 61 7 ( 0 . 3 2 42 6 0 1 1 4 1 1 5 1( 8 0.20 -3 1 0 1 12 0 1 - 1 1

9 0.21 -1 I 2 1 4 - 2 I 0 l1 1 0 1 0 . 2 1 1 - 1 1 - 1 1 0 I 0 I 0 I111 1 0 . 2 7 2 9 0 1 - 6 2112 0.25 -15 11 -11 -18 -3(13 0.20 0 O o O l 0 I

- - - - - - - - - - - - - - - - - - - -- - - - - - - - - - - - - - - - - - - - - -- - - - - - - - - - - - - - - -

In the above examples some effects are completely neglected, but in thesame way this discussion holds if an error in the model is present ofe.g. 10 percent in the total magnitude of the corresponding correction.The considerable rotation over 1.5 metres due to neglect of gravita-

tional deflection is caused by the position of the reference source3C273-B, which was at an angular separation of only 6 degrees from theSun at the time of the MERIT Short Campaign.The other examples speak for themselves.

With the above in mind, Table 4 is composed as an assessment of the dis-cussions of sections 2.3 to 2.7 for the delay observable. In the firstcolumn, the estimated absolute accuracy of DEGRIAS is mentioned whereappropriate; in the second column the relativeI1 DEGRIAS accuracy isshown and in the last column the expected [Trask et a1.,19821 bottomline results for VLBI on the basis of one-day datasets are given, all

for a 5000 km baseline.From the table it appears that the accumulated error budget of DEGRIASmodelling is 7.1 cm. To avoid optimism, it is preferable to state theconclusion that:

----------------------------------------------------------------------

I DEGRIAS model accuracy at its present stage is one decimetre Ifor intercontinental baselines.

----------------------------------------------------------------------

I

For shorter baselines, this accuracy increases to few centimetres.

8/16/2019 29 Brouwer

70/192

Table 4

Assessment of DEGRIAS and VLBI Accuracy

- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

I I DEGRI S I Iunits = cm I modelling accuracy I Bottom I

I I I lineI phenomenon I absolute l re1ative J resultsI-------------------------+---------- ---------- ----------lI 1. clock behaviour I 2.0 0.61 2 . antenna structure I 3.0 0.7

3. source structure I 0.5 0.34. precession 200.0 0.2 I5. nutation 50.0 4.0 0.9 16. aberration 20.0 3.0 I I7. grav. deflection I 0.5 I8. (special) relativity I 0.7 I

9. dry troposphere 1 1.5 0.5(10. wet troposphere I 3.0 1.5111. ionosphere I 0.2 0.2112. Earth rotation 20.0 1.0 I I113. retarded baseline 0.2 I114. polar motion 20.0 0.2 I115. solid Earth tides 2.0 I116. ocean loading I I 1.0 I -117. miscellaneous I I 1.0 II------------------------- ---------- ---------- ----------

II

I TOTAL 210.0 7.1 1.9-------------------------+----------+----------+----------lI ESTIMATED RESULTS I I II baseline length (cm) I 7.1 1.9 II pole position (mas) 19 7.0 2.0

UT1-UTC (ms) 1.3 0.5 0.06I source positions (mas) 70 7.0 2.0

- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

I

The last column of Table 4 (cf. also [Schaffer,l984]) clearly demon-strates that the wet tropospheric refraction component is the majorVLBI error source. In addition, the dry component is a major error

source at low elevations (0.5 cm is for elevations above 10 degrees).If a significant improvement (1.5 cm includes WVR-measurements) can bereached in their determination, the way is open to an approach usingphase-delay observables [Campbell,l979a] instead of the present group-delays. This could bring VLBI accuracy to the level of Connected Ele-ment Radio Interferometry (S1.1), meaning millimetre precision.

As the computed observation of the delay rate observable is found by anumerical differentiation using computed delay observations (S3.1), thelevel of accuracy of this type of observation is in principle the sameas for delay, noting, however, the inherent limitation of delay-rate:its insensitivity to the Z-component of the baseline.

8/16/2019 29 Brouwer

71/192

Chapter 3

D E G R I A S S O F T W A R E P A C K A G E

Summary: In the preceding chapt er the real world for geodetic VLBI hasbeen described. In 53.1 all relevant phenomena will be com-bined to arrive at the standard model world (formula system )for the DEGRIA S software package (DElft Geodetic Radio Inter-ferometry Adjustment System). For this so-called kinem aticmodel the linearized obser vatio n equati ons for delay and delayrate observ ables are derived. Thi s is followed by a brief de-scrip tion of the idiosyncrasies of DEGRI AS and a discus sionabout rank defici encie s of the system of normal equat ions of

Least Squar es adjustment. The description concludes with anindication of some envisaged improvements of DE GRIAS (53.5).As an example of the c apabilities of this software package, theresults of analysis of two multi-station geodetic VLBI experi-ments are prese nted in sec tio ns 3.6 and 3.7, viz.: ERIDO C (Eu-ropean Radio Inte rferometry and DOppler Cam paign ) and a part ofthe MERIT Short Campaign (to Monitor Earth Rotation and to In-tercompare the Techniques of observation and analysis). Forthe latter, the differences in estimated results are also stud-ied when several choice s are made for the set of observationsor model parameters.

3.1 FUL L COMPU TING MODEL OF DEGRI AS

In 52.2 the sim ple computing model (22.3) and the related non-linear ob-servat ion equati on (22.5 ) have been presented. Now the full compu tingmodel (22.7) as implemented in DEGRIA S will be described. This comput -ing model is called kinema tic model, si nce it is based o n the rotationalmotio ns of the Earth with respect to the fixed sourc es, or - in otherwords sin ce the posit ions with respect to the Earth-fixed interferome-ter of a g iven sourc e at se veral insta nts of obser vatio n are related viathe (pr esuma bly) known algorithms for precession, nutation, Earth rota-

tion, etc..The following models and quantities for chapter 2 are applied:

( W = Rx( YP) Ry(XP) polar motion matrix (27.9)( S = R,(GAST), the Ea rt h' s spin (27.3)

p = precession matrix (24.2)(NI = nutat ion matrix (24.7)(A ) aberration vector (24.15)( G ) = gravitational deflection vector (25.3 )X,Y,Z = coord inates of stat ions a and b in the CIO-system( B = (Xb-Xa, Yb-Ya, Zb-Za)

= (D Xl DY 1D Z) , he baseline vector( U = the source position vector in the 1950.0 sys tem ,

expressed by (right ascen sion) and (declination)

8/16/2019 29 Brouwer

72/192

cos cos6= [sin a cos6 (31.1)

sin 6TCLO = the clock function for stations a,b (23.1)TANT = antenna motion correction (23.3) for stations a,bTRTB = retarded baseline correction (27.6)TION ionosp heric refract ion correct ion for a,b (26.13 or 26.15)

TAMB BWS-am biguity correc tion (52.3.3)TTRO tropos pheric refract ion correct ion for a,b (26.2,26.7,26.11)TTID = solid Earth tides correction for a,b (27.14)

In DEGRI AS ;he following parame ters are estimable in the LS Q fit:

X,Y,Z = station coord inates in the CIO-systema ,6 = right ascensio n and decl inatio n of a

source in the 1950.0 systemTO, . . 5 = six coefficie nts of the clock funct ion (23.1)Xp, Yp = X- and Y-component (as one-day's averages)

of polar motion (27.9)UT1 = UT1-UTC value, a s one-day's averages (27.3)

- Ttz = troposp heric zenith delay, the sum of (26.1) and(26.11)= velocity of light in vacuo.

Differ entiati ng the full comput ing model (22.7) with respect to theseestimable parameters (using their approximate values) yields the follow-ing linearized observ ation equation (being one row of the design matrixfor the LSQ fit ) for statio ns a and b and source i (GHAi is theactual Greenwich hour angle, 6i is the apparen t declin ation, both cor-rected for gravitational deflection).

d T = + cos(GHAi) cos 6i / c

+ sin(GHAi) cos 6i / c

+ sin 6i / c

cos(GHAi) cos 6i / c

sin(GHAi) cos 6i / c

sin 6i / c

8/16/2019 29 Brouwer

73/192

+ cos GHAi) cos 6i DZ-sin 6i DX) / c d ~ p

+ sin GHAi) cos 6i DZ-sin 6i DY) / c dYp

+ cos 6i sin GHAi)*DXcos GHAi)*DY) / c dUTl

cos 6i sin GHAi)*DXcos GHAi)*DY) / c d a

For computing the model value of the delay rate observa ble two possiblemethods exist: 1) a direct formula via the differentiated formula

22.8) and 2) a numerical differentiation of 22.7) during execution

time of the input module of DEGRIAS. The disadva ntage of 1) is that itis a tedi ous task to completely implement all the chain-r ule differenti-ations . Therefo re, in DEGRIAS only an approxi mate direct model for

22.8) is im plement ed, in which only the time-derivatives of clock func-tion 23.2), of precess ion 24.3), of nutation 24.10), of aberrat ion

24.16), of tropospheri c refrac tion 26.4) and of retarded baseline cor-rectio n 27.8) are present , so that the time-dependence of the followi ngphenom ena is neglected: gravit ational deflec tion, ionospheric refrac-tion, solid Earth ti des, antenna motion correct ion and equati on of equi-noxes in GAST.The numerica l differe ntiati on of option 2) is computed as:

8/16/2019 29 Brouwer

74/192

with dt having a value of 0.5 or 5 seconds, depending on the baselinelength. It is clea r that the latter approach requires more computertime since precession, nut ation, etc. are computed for three time-points(t,t +dt and t-dt). The advantage is that all phenomena are automati-cally incl uded in the computed obser vati on for delay rate.The coe fficients of the linearized observation equation for delay ratewhich need not be so accu rate can be computed by directly differenti-ating the delay coef fici ents , using the approximati on that changes re-sult only from Earth rotation. From ( 27.3 ) one finds:

R dGAST/dt = 2 7 ~ (mean sidereal day + dUTl + dEQE) (31.4)

This yields:

sin(GHAi) cos 6i R / cI

+ cos(GHAi) cos 6i R / c

+ sin(GHAi) cos 6i R / cI

cos(GHAi) cos 6i R / c

lsin(GHAi) cos 6 i R DZ / c

8/16/2019 29 Brouwer

75/192

- f / c dc

+ cos i (-cos(GHAi *DX-sin(GHAi)*DY) / c d a i

+ sin i (-sin(GHAi)*DX+cos(GHAi)*DY) / c

No effort has been made here to express all quantities in appropriateunits; therefore multiplicat ion by 2~ etc. has been omitted. The ac-tual units used in DEGR IAS to scale the system of normal equations are

(for the most important quantities): station coordinates in Mm, ob-served del ays in ms and source coordinates in radians.

3.2 FUNC TION AL DESCRIPTION DEGRIAS

The DEGR IAS package is integrated in the SCAN-I1 (System for Computer-ized Adjustment of Networks) modular software system of the ComputingCentre of the Delft Department of Geodesy which is written in FORTRAN-IVand developed by J.J.Kok. It is running on the AMDAHL 470-V7-B compute rof the Computing Centre of the Delft University of Technology. An out-line of DEGRIA S is presented in Figure 18.

DEGR IAS starts with the input of a number of steering parameters. Thes einclude such items as: number of stations and observed sources, parame-ters for statistical testing and an option list indicating the choicefor the computing model (included subroutines and formula system).DEGR IAS uses a special format for the input of the observation s. Oneobservation card contains the following data: the stations of the in-terferometer , the observed source, the day and time of observation andthe observed delay and delay rate with their respective standard devia-tions. Thes e card s can be in any sequence: per time of observatio n orper baseline in obser ving sequence, or just randomly. Observation s

which ar e not to be included in the LSQ fit can be indicated by thecharacter X in the first column of the card and will automaticall y beskipped. If dual-frequen cy observation s are used to eliminate ionos-pheric refr action, two files are read simultaneously. If one of the twoobservatio ns is missing, the observation is also skipped. A third inputfile i s used when weather data a re available, for a tro pospheric refrac-tion correction according to (26.7). If required, this file can be cre-ated by separa te software , called METEOINT (53.5) which interpolat es ob-served values for tem perature, pressure and relative humidity (e.g. fromweather map data if no on-site values are available) for every instantof observation.

8/16/2019 29 Brouwer

76/192

INPUT MODULEread steering parametersread approximate values of) unknowns

read tim e, value and standarddeviation of observation

compute C- 0 value of the observation and itsdesign matrix coefficients according to thechosen model with subroutines for:

clock model grav. deflec tionantenna motion troposph. refractionprecess ion ionosph. refract ionnutation Earth rotationaberrat ion polar motionsolid Earth tides retarded baseline

compute a priori weight matrix. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

CENTRAL MODULE

compute adjusted unknown parameterscompute var/covar matrix of unknown parameter scompute post-fit residualsdo outlier testing and compute reliabilityiterate for outliers

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

OUTPUT MODULEprint adjusted values of unknown parameters,their standard deviationsand correlation matrixprint residua ls of observation s and RMS-valuesprint mar ginall y detectable errors andexternal reliability parametersprint results of statistical testingprint plots of the residuals

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Figure 18: Outlin e of DEGRIA S

As a consequence of the correla tion process, the a priori weight matrixof the observa tions is always chosen to be diagona l [Bock,1 983], al-though DEGRIAS allows a full weight matrix. This would be especi allyuseful for simulation studies to investigate the effect of the introduc-tion of off-di agonal element s caused by common factors such as tropos-pheric refraction for baselines forming a closed triangle. The correla-tion process yields a standard deviation based on the signal-to-noise-ratio SNR) of the observation which depends mainly on source strength

8/16/2019 29 Brouwer

77/192

and length of the coherent correlation interval [Brouwer&Visser,l978].For stronger sources, even using Mk-I1 BWS, the standard deviat ion fordelay typically s catter s around 0.01 ns, which equals 3 mm. Th is is ev-idently too optimi stic, for the SNR is not the only item of interest,but als o the behaviour of the real world during the entire experiment.In DEGRIA S it is therefore possible to change this SNR standard devia-tion by a multip licatio n factor or by adding an ad hoc number. In par-

ticular the latter accoun ts in an acceptable way for error sources thatar e not merely a function of signal strength (correlated flu x) and pro-vides a more equal weighting of the observations. In additi on, it ispossible to divide the standard deviations by the sine of the elevation,to suppress the influence of low elevation observations.

The centra l module applies Choleski decomposition using spars e matrixtechniques [Kok,1981]. This computer storag e and CPU-time saving ap-proach a llows a large number of observations and estimable parameters t obe included. In the standard version of DEGRIAS therefore the followingdimens ions are valid for the estimable parameters:

number of stations 2number of sources 4number of clock polynomials 5number of tropospheric parameters 50number of polar motion parameters 10number of UT1 parameters 10speed of light 1

Th is amount s to a total of 521 unknowns with a maximum of 2500 observa-tions. Any of these unknowns can be constrained in the LSQ adjustment.The number of observations can easily be extended to 10000, as computerstorag e is no major limitation.

Th e statis tical testing is done in two steps following the B-method oftestin g [Baarda,l9681. The first step makes use of the overal l Fisher-test to test the zero hypothesis Ho, which consists of: (a ) mathematicalor computing model (desig n matrix and linearization), (b ) probabilisticmodel (norma l distribution and a priori variance/covariance matrix ) and(C) the validity of the vector of observations with respect to these twomodels:

with

a posteriori variance factorC i v iz / ux i 2 ) / (m-b)

a priori variance factorcritic al value of F-testlevel of significance of the test (54.3.3)total number of observationsnumber of estimated parametersthe correct ion (or post-fit residual) of the observationthe standard deviation of the observation.

8/16/2019 29 Brouwer

78/192

If the above test fail s, the normalised test values "wl' are computed forall observations. They are related to the alternative hypothesis thatonl y on e obser vatio n at a time is erroneous. The process of computingthis serie s of test values is denoted by "data-snooping" [Baarda,1968].The general formula for "W" is valid for correlated observati ons (43.7),but in the case of a diagonal a priori weight matrix o ne finds:

with i= the standard deviation of the correction of the i-th ob-servation and F the crit ical value and a o the level of significance.

T o enabl e faster operat ion, it is possible to iterate the LSQ solutiono n the basis of this W-test. It may e.g. be that a comparatively smallnumber of obser vatio ns are r ejected, for instance due to a low elevationor due to a n inappropriate SNR-sigma. If the optio n for iteration is

cho sen , DEGRI AS automatically downweights all rejected observati ons by afactor of four and comput es a new LSQ solution and new test-values.Thi s process may be repeated several times. The approach is of courseonly allowed if some observa tions are rejected more or less randomly andtherefore no systematic trends are present in the residuals. Appendix Dshows the validity of this process for the case of VLBI. As the LSQ so-lution ( =t he estimated parameters) hardly changes during the iterations(N.B. o ne should of cour se check this ), the main advanta ge of this ap-proach is the smoother appearance of the residuals (32.5) and the test-values.

DEG RIA S enabl es one als o to compute the marginally detectable error

m.4.g.) connected with the W-test as a measure for the so-called in-ter nal reliability. Th e m.d.e. is the size of error in an observ ationwhich cause s the W-test to just fail. In additi on, the measure iscomput ed for the so-called external reliability [Baarda,1972]. This isthe normalized length of the vector of deviations in adjusted parametersdu e to the abo ve error with size m.d.e. [Brouwer,l982a]. These quant i-ties are of special interest for design studies (chapter 4). For thispurpo se a random obser vatio n generator is present a s well.

In addit ion to the data-snooping which tests for individual error s, anumber of special alternat ive hypotheses are also formulated to test for

syste matic trends. The following three types are used:

a constant bias is present in all delay observations including onespecific stationa constant bias is present in all delay obser'vations of on e specificbaselinea constant bias is present in all delay observations made at on e spe-cific source.

For the mathematical formulation of these tests one is referred toS4.3.2 and App end ix G.

8/16/2019 29 Brouwer

79/192

The results of the output module are clear. Due to a reordering proc-ess, the res ults can be printed either per baseline or in time sequence .A plot of dela y and del ay rate residua ls is produced per baseline.Two RMS (Root Mean Squared)-values are computed, an unweighted and aweighted one, defined as:

RMSU sqrt(C (v iz ) (ml*(m-b)/m)) (32.4)

with m1 number of relevant observ ations (per baseli ne, delay or delayrate observables).

3.3 ESTIMABLE PARAMETERS IN DEGRIAS

In 53.1 the parame ters have been mentioned that serve in DEGRIA S as un-knowns in the LSQ fit. It should be recalled that every stat ion may havesevera l clock polynomials , depending on clock jumps or re-starts of themeasurements. Furthermore, tropospheric zenith delays can be chosen forsevera l time interv als; it may e.g. be favour able to split the measur e-ments into night and day sectio ns with separat e tropos pheric parameters.In addition, t is recalled that polar motion and UT1 parame ters arecomputed as at least one day averages.It is obv iou s that not all these parame ters are estimab le at the sametime: in genera l the system of normal equati ons of the LSQ fit will besingular. Thi s can be due to two reasons:

a) rank deficiencies due to coordinate system definitionb) rank deficiencies due to a critical configuration.

An example of the latter is the single baseline experi ment with source sat only one declination. In this situation clock offset and verticalbaseli ne compon ent become insepa rable; see e.g. [Schut ,19831. Thesecritic al configurat ions, however , will be dealt with in $4.4.1.

As for item a) short discuss ion is necess ary here:Since Euclidean coordinates in 3-D are defined as distances with respect

to certai n (o rthogo nal) planes, they are not estima ble as such[Baarda,19731. In a Euclidean space the coordinate system definitionrequir es 7 conven tions , because the defini tion is related to the num-ber of parameters in the similarity transf ormati on (seven for 3-D). Themost commo n way to define the coordi nate system is to constr ain 7 well-chosen parameters in the LSQ adjustment. The more or less international

standa rd choice for these seven parame ters for VLBI is:

the X, Y and Z coordi nate of one statio n, to define the origin (3translations),the epoch (1950.0 in DEGRIA S) epheme ris pole position to define theequato rial plane (2 rotations),

8/16/2019 29 Brouwer

80/192

the right ascension of one source, to define the orientation in theequatorial plane 1 rotation),the velocity of light in vacuo, as a sca le parameter.

The Car tes ian coordi nate system in use in DEGRIA S is left-handed, geo-centr ic and Earth fixed, as descri bed in S2.2. The origin of this sys-tem is operat ional ly defined by some arbit rary coordi nates adopted forthe point of intersection of the azimut h and eleva tion axes of one an-tenna, mostly Haysta ck, if included e.g. MERIT Short Cam pai gn; §3.7),as derived from , for instance, spacecraft tracking. The origin of rightascens ion is taken as 12h 26m 33.246s for the radio sourc e 3C273-B atepoch 1950.0 ell ipt ic aberr ation remove d) which is the quasar with thehighest radio intensity and therefore practically alway s included ingeodet ic VLBI experiments. The value for the speed of light used toconver t measured d elays to distances is 299,792.458 km/s.The intr oduct ion of precession and nutation in the compu ting model de-fines the ephemeris pole position at epoch time. Consequently, the def-inition of the Z-axis is tacitly included in the precession/nutation

formul ae s ee 24.1): R,, etc.) and no unknown parameters have tobe constrained explicitly.

In additi on, a rank deficiency occurs as a combined effect of coordinatesystem defini tion and a criti cal configuration. The unknowns for polarmotion and UT1 yield a singular system of normal equat ions if they arenot held fixed to a priori values for at least one moment =o ne obser va-tion: in DEGRIAS thus one day). Thi s is due to the fact that these par-amete rs describe a rotation of the Earth-fixed frame with respect to thestellar f ram e, and evidently only chan ges in these rotations can be mon-itored. It is probably more precise to state that with the kinema ticmodel for VLBI one has to defin e two coord inate frames: a stellar on ewith only orientations thus three directions fixed: ephemeris pole andorigi n of right ascen sion) and a terrestrial one with the standard sev enparameters: three stati on coord inates , one scale factor and polar motionand UT1 as three orientations.Howev er, other conventions for the coordinate system definition may bechosen ; for example:

a) Constr ain seven coordinates of VLBI-stations plu s the additionalthree values for polar motion and UT1). As yet, this is not possiblewith DEGRI AS for the kinema tic model since the epheme ris pole posi-tion is alread y implicitly fixed in the precession formulae. If on e

therefore fixed seven station coordinates in DEGRIAS, the LSQ fitwould be overconstrained.

b) From a) it follows that a valid choice for DEGRIAS would be: to con-strain five station coordinates. With the implicit assump tion ofpole position and again the three Earth orientation parameters) thiswould defin e the coord inate frame completely and make e.g. the rightascens ion of 3C273-B estimable.

c ) The use of a minimum norm solution: this yields a minimal trace forthe variance/covariance matrix so that smoother looking variances ap-pear cf. the larger sigma ellip ses in plane geodetic networ ks, awayfrom the two constrained points). This solution is not directly pos-

8/16/2019 29 Brouwer

81/192

sibl e in DEGR IAS because the inversion routine cannot handle a gen-eral inverse, but requir es an explicit indica tion of the constrainedunknowns.

It is stre ssed , however, that in principle these solut ions are the sa me,because the real worl d does not chan ge when describing it with the aidof a different coordinate frame. Therefore, a relation exists betweenthese solutions. [Baa rda, 1973 ] called this relation S-transformat ion.For the background to this, one is referred to chap ter and Appendix E.

In addition it is mentioned that for the clock polynomial the same situ-ation occurs as for the station coordinates. Time is a one dimensionalphenomenon. Therefore origin ( zero point of time) and scale (thelength of the secon d) must be defined. For the orig in this can be doneby assuming the clock offset at one of the station s to be zero. By put-ting the clock drift of this station also equal to zero , the unit oftime is defined as the length of the second of this clock at any give nmoment. The same holds for other terms in the clock function (23.1).

This is because only time differences are measured.

An interesting interdependen ce, however, can be observed between clock-drift and speed of light. Consider the situation of Figure 19. Thetimes of arriv al of the signals are recorded as ta and tb. Let therate s with whic h the two cloc ks are running with respect to real timebe: la and lb. For rea l time one can think of the time fra me belo ng-ing to the adopted speed of light (coo rdin ate time).

Figure 19: Clock Time-scales

0 ? 2 3 ta

Then the following r elat ions hold:

I I

t b0 2

I

I

0 1 1 1 2 4 5 6I I I I I I

ta = (Ta TaO) / latb = (Tb TbO) lb

Time A x i s Station a

3I Time A x i s Station b

I Coordinate Time

The delay resulting from the correlation process is (22.1): c = tb-ta.The time diff eren ce, howe ver, that belongs to the coordinate time frameis cg = Tb-Ta, where c g is the geometric delay, or the innerproduct (22.2) of ( B ) and (U). Now the following relation exists:

TbOTaO Tb Ta

Z

8/16/2019 29 Brouwer

82/192

( tb - ta) (Tb/lb - Ta/la) - (TbO/lb - TaO/la)

so that

One immediately sees the scaling effect of the clock drift at stationl b

. The speed of light and the clock drift have therefore the samebackground and consequently they have also an equal impact on the solu-tion. Const rainin g the speed of light in the LSQ adjustment thus makesthe clock drift at one station ( b ) estimable.Du e to the term (la-lb)/lb*ta, the clock drift of stati on a is inprinciple als o estimable. Howev er, because of the very small coeffi-cient - (la -lb ) is of order 10-l4 this is not possi ble in prac-tice. Hen ce not all clock drifts are estim able if the speed of light isconstrained, or , put the other way, constraining one clock drift doesnot yield an estimable speed of light.

3.4 D T ANALYSIS PROCEDURE

Thi s secti on is included here to present an idea of how the data han-dling for ERIDOC and MERIT-SC was done and to serve as a guide to oth-ers.

The analysis starts with a tape containing approximate values for sourceand station coord inate s plus observations with timing and correspondingformal errors. DEGRI AS input requires a special format for the observa-tions (S3.2). There fore, an interface exists to transform the observa-tion file from the commonly used NGS-format [Cart er,l9 81] to the DEGRI-AS-format. Th is programme is called NGSFORMAT; see S3.5. Possibleother forma ts can , via some minor actions also be transferred to theDEGRIAS-format. Four files are neede d, containing: steering parametersand station/source coordinates, observations, observations at a secondfrequency (for ionospheric refraction correction) and meteo data respec-tively, of which the last two ar e optional.Then the analysis is continued with the following steps:

1. Delete observa tions via the character X (S3.2) with an obviously poorformal error, since a formal error considerably above the avera ge

noise level of the observations usually indicates that there is some-thing wrong with the corresponding observation, rather than that thisobservation is simply of lower precision.

2. Make an LSQ fit of the first 2-3 hours of data of the experiment wit hclock offset and clock drift as the only unknown parameters. Apply -if required - the automatic ambiquity elimination procedure here(S2.3.3). At the same tim e, the C- 0 values ca n be used to eli min atelarge outliers. Thi s should be done for n-l baselines, if n is thenumber of stati ons, to establish values of clock offset and driftparameters for all stations. Note that dual-frequency obser vatio nsfor the determination of ionospheric refraction cannot be applied be-fore all ambig uitie s are eliminated. In particular, when ionospheric

8/16/2019 29 Brouwer

83/192

corrections have almost the same magnitude as the ambigu ity spacing,the appli cation of the ionoso nde model is helpful 52.6.3).

3. Now one has some estimates for clock offset and drift , iterate st ep 2for larger portions of observ ation s to find better estimates for theclock parameters and st ation coordinates and to eliminate the ambigu-ities on all other baselines as well. Do not hesitate to introduce

clock breaks in the clock fun ction of a statio n to speed up this it-eration procedure by separating the observations into sections.It is advis able to exclude observations taken at elevations below 10degrees in this procedure, because of the uncertainties in the tro-pospheric refraction correction.

Note: It is probably advisable to eliminate obser vation s takenat elevations below 10 degree s completely, even in the finaladj ust men t; see 52.6.1 and Appendix B.

4. The triangle-condition, commo nly known as closure phase, ca n be used

to eliminate any remaining ambiguities in the delay observations andas only single baselines have been considered so far any confu-sion between clock offset and ambiguity spacing. Since the closur ephase relation is implicitly included in the adjus tment model 31.2),special software exists to compute the closures in the triangles.The input to this TRIANG LE programme is the observ ation file. Theseobser vation s are then corrected for retarded baseline 27.6), becauseotherwise the triangle will not close. The output consists of a listof all closu res and a list of the observations not included in anytriangle condition. The latter observations are somewhat suspectsince they are less well-checked in the LSQ adjustment and err ors maypass undetected. Also o n the basis of these closur es, observ ationsca n be excluded.

5. Make an improved LSQ adjustment for all single baselines with moreunknown parameters. This offers the opportunity of eliminating badobser vation s on the basis of the W-test 53.2). In the present ap-proach, only at this stage are delay rate observations included inthe adjustment since the following rule, although not strictly neces-sar y, is applie d: if the delay observ able is excluded in the LSQfit , the delay rate obser vable is excluded as well; the oppos ite doesnot hold.

6. Compu te the overa ll multi-station LSQ solution with all possible es-timable param eters using, where required, the iteration procedure andapplying the special altern ative hypothesis testing 53.2) to assessthe final accuracy.

For a single baseline experiment the analogous procedure is followedwith DEGRIA S, without step 4, of course.

8/16/2019 29 Brouwer

84/192

3.5 DEGRIAS SYSTEM SUMMAR Y AND IMPROVE MENTS

The complete data-flow through the DEGRIAS system is shown in Figure 20.The tasks of the input, output and central module of DEGRIAS are pres-ented in detail in Fi gur e 18.As yet, DEGRIAS is a prototype softwar e package: all relevant items arepresent , but both the coding and the internal organiz ation are not ye t)optimal for use in a product ion manner. The envisaged improvements arementioned here:

the change of programming language from standar d FORTRAN -IV to stand-ard FORTRAN-77,the introduction of the improved IAU 520 00 system in the comput ingmodel, plus some other minor) changes in this model,an automatic iteration with improved estimates for unknown parameters,further optimiz ation of use of CPU time and compute r storage in viewof the large number of observa tions compar ed to the number of estima-ble parameters,

the integration of NGSFORM AT, METEOIN T, TRIANGL E and SORTOBS in theINPUT and OUTPUT modules of DEGRIAS,better plot facilities for post-fit residuals,additio nal non-con vention al hypothe ses, e.g. tests for errors in cer-tain correcti ons, or the introduction of multi-variate linear hypothe-sis tests [K okIl984 ],the introdu ction of more estimab le paramete rs. Good candida tes are:the precessional constant, the aberration constant, the Love numbersand phase lag of solid Earth tides and the gravit ational bending par-ameter 25.2),the improvement of the numeric al differe ntiatio n for delay rate. Com-

putatio ns are in princip le only required for two time points t+d t andt-dt) if the data for the observa tion equatio n are determined via in-terpolation as well.

3.6 EXAMPLE L: ERIDOC VLBI CAMPAIGN

3.6.1 Genera l Informa tion

The fourth objecti ve S1.4) of the research project which led to thispublication was to cooperate in the organization of VLBI experiments andto carry out t he analysis of the data obtained.In 1980 a programme of four geodetic VLBI experiments was started withinthe European network of radio astronomy observatories. This programmewas denoted by the acronym WEJO, for the initials of the four major par-ticipants. The joint responsibility lay with the Geodeti c Institu te ofthe Univers ity of Bonn D) and the Department of Geodesy of the DelftUniversity of Technology. The programme was designed to gain experiencewith geodet ic VLBI measure ments on a European scale and to finally servethe objectives of S1.2.

The first campaig n took place in October 1980 but due to a number oftechnical failures only the measurements on the baseline Effelsberg D)

to Metsahovi SF ) could be correlated. Baselin e results were presentedin [Beyer et a1.,19821. WEJO-3 was a complete failure in December 1981.

8/16/2019 29 Brouwer

85/192

* (NGS-FORMAT) TAPE * * RAW DATA ** _ station coord's * * meteo + ionosonde ** - source coord's * * - axis offsets ** observations * * - ambiguity spacing ** meteo data * * - UTl/polar motion ** - standard deviat's * * - clock intervals *. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

I lI . . . . . . . . . . . . . . . . . . . . . . . . Ill l

NGSFORMAT METEOINT1' I

l

* observ's * observ's * * meteo data ** freq. 1 * freq. 2 * * *. . . . . . . . . . . . . . . . . . . . . . . * * * * * * * * * * * * * * * *

I ITRIANGLE SORTOBS I

I ll

* options ** *

I

DEGRIAS-INPUT

DEGRIAS-CENTRAL

I

DEGRIAS-OUTPUT

l) _ _ _ _ _ _ - _ _ _ - _ _ _ _ _ _ _ _ _ _

l

SORTOBS

l) _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _

READY

Figure 2 : Data Flow in DEGRIAS Software Package

8/16/2019 29 Brouwer

86/192

Snowfall and high winds prevented any observations at Onsala S); therecording equipment was out of order at Jodrell Bank GB) and a wrongfrequency setting hampered observations at Westerbork NL). As Effels-berg was the only station left, no correlation could be done. The lastcampaign with the same four observatories December 1982) has been suc-cessfully correlated for the most part, but no baseline results are re-ported.

The secon d, most successful experiment took place in April 1981 and wascalled: Project ERIDOC Europ ean Radio Interferometry and DOppler Cam-paign). In the project, simultaneous VLBI and Doppler satellite posi-tioning campaigns were condacted between those European radio observato-ries that were or may in the future be equipped with VLBI recording sys-tems. In this way, the performance of the VLBI system on a Europeanscale could be tested and compared with the Doppler system; in addition,precise Doppler positions would be available for as yet non-operationalVLBI observatories.CO-located VLBI and Doppler satellite observations were thus made at

five radio observatories: Effelsberg D), Westerbork NL), ChilboltonGB), Jodrell Bank GB) and Onsala S); see Table 5. Thirteen otherDoppler statio ns also took part in the campaign, five of them located atsatellite laser observatories, to provide a connection to the laser net-work as well. Those stations were: Dwingeloo NL), Roble do E), Weil-heim D), Wettzell D), Bologna I), Graz A), Leeuwar den NL), Herst-monceux GB), Kootwijk NL), Dionysos GR), Florence I), Brussels B)and Metsahovi SF); see Figure 21.

Table 5

ERIDOC VLBI Stations

I I I l II Antenna I Mount Axis Clock I

I Station diameter I type I offset I systemI l m I l m) I lI I I I II-------------------- ------------ --------- -------- -----------

l

l II Westerbork dish B)[ 2

I Effelsberg 100

I I l IEQ 4.95 Rubid ium1

I AltAz 1 0.0 I H-Maser II Jodre ll Bank M ~ I I ) ~ 26*38 AltAz 0.47 Rubid iumOnsal a OS 0 25.6) 25.6 EQ 2.15 H-Ma ser

I Chilbolton 2 AltAz 0.31 H-Maser II l l I

- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

I

The VLBI session lasted from April 12, 18h UT to April 13, 18h UT andthe Doppler observations from April 7 , Oh UT to April 16, 24h UT. Bythis simultaneity of the observations uncertainties in the chang es oforientation of the reference frames due to precession and nutation onthe one hand and polar motion and UT1 on the other , do not enter intothe comparison results.

8/16/2019 29 Brouwer

87/192

W TTZ LL

W ILH IM

G R Z

Figure 21: ERIDOC Network

The data were analysed independently at the two computin g centres Bonnand Delft, and a report is presented in [B rouwer et a1.,19831. Althoughmost of the results are already available in this paper, a short reviewof the achievements is given here, because the data are connected to oneof the main objectiv es of the present researc h project 51.4) and theresults of the updated VLBI solutio n presented here with th e aid of DEG-RIA S are used in chapter to illustra te the necessity of the use ofrigorous comparison techniques.

8/16/2019 29 Brouwer

88/192

At the time of ERIDOC, not all observatories were equipped with a Mk-I11recording system. Due to the accuracy limitations of the Mk-I1 system ,a sequential bandwidth synthesis (BW S) technique was used in order toimprove the delay resolution of the 6-cm observations (52.3.3). On theEuropean scale a two channel approach was considered to be the bestchoi ce, because it mini mize s the loss in sensitivity and was relativelyeasy to implement. This 40 MHz two channel approach (odd seconds on

4990 MHz; even seconds on 5030 MHz) yields a theoretical precision ofabout 0.25 ns in the BWS-delay, but leaves an ambiguity spacing of 25ns. With the relatively high signal-to-noi se-ratio (S NR ) provided bythe rather sensitive European station configuratio ns, the non-ambiguoussingle channel BSA-delays (51.3.4) tend to be fairly accurate (betwee n1.5 and 5 ns) and can thus be included in the LSQ solu tion.

A relatively simple observing schedule was used with fixed 15 minutescans, including a varying portion of time for slewing. This schedulewas compiled by the SCH ED module of DE GRIA S (54.3).

3.6.2 ERIDOC multi-station solu tion

The ERIDOC VLBI measurements were not completely successful fro m a tech-nical point of view due to a number of factors. Onsa la, for instance,could only observe during the second half of the campaign due to a re-corder malfunction. The BWS-delays on all baselines involving eitherWesterbork or Jodrell Bank were corrupted by strong sinusoidal phasevariations during the integration interval. The exact cause of thesevariations has not been established, but is connected to insufficientstability of the rubidium standards or the two synthesizers. At Chil-bolton, the H-maser was not operational during 12 hours of the sessionand also showed a somewhat lower stability performance than usual forthe remainder of the campaign. More over , its 6 cm receiver had a rela-tively high system temperature.In this way, the BWS-delays on the Effelsberg-Onsala baseline proved tobe the only ones fully consistent with the two channel 40 MHz BWS-accu-racy level expected from theory.

For the ERIDOC anal ysis , the standard coordinate system definition of53.3 was used, with for the Effelsberg co ordinates as the arbitra ry val-ues of Tabl e 6. For the LSQ adjustment the kinematic model of DEGR IASwas chosen with the following features:

fixed s ource positions as published in [Fanselow et a1.,19811,precession defined by the 5200 0 precessional constant transformed tothe 1950.0-system,second order polynomials a s clock function where required,tropospheric zenith path delays computed from interpolated weatherdata based o n readings every 3 or 6 hours,the ionosonde ionospheric correction model, based on data observed atthe KNMI in De Bilt (The Netherlands )

The latter three items in particular, are included as improvements inthe present solution as compared to that presented in [Brouwer eta1.,19831.

8/16/2019 29 Brouwer

89/192

In the ERIDOC ca mpaign 10 baselines are present, ea ch with two BSA chan-nels and one BWS channel . Due to the technical problems descr ibedabove, only the BWS data of the EFF-ONS baseline were fu l ly usable sothat a simultaneous LSQ fit was made to the observation s of one BWS and20 BSA baselines. After removing bad obse rvat ions with the W-te st(32.3), a total number of 1029 group delay observatio ns and 877 delay-rate observation s were used in the adjustment , each with a weight based

on its SNR.

Table 6.

ERIDOC VLBI Coordinates

- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

l l1 S TAT I O N ( m ) Y ( m ) ( m )

l

Effelsbg 4033911.00 1 -487046.00 4900424.00/ - l - 0 l - 0 - 0

West erbk 3828565.81 -443931.86 5064915.40/ - 0.07 0.04 0.13

Chi lb l tn 4008274.97 100595.26 4943788.22I / - I 0.17 0.18 0.41I Jo d- Ba nk 3822812.23 153747.33 5086279.44

/ - 0.10 1 0.08 0.20Ons ala 3370929.24 -711520.42 1 5349657.18

/ - 0.02 1 0.02 0.04l

- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

In the LS Q fit 58 unknowns were estimated, comprising: coordinate s offour stations and 18 clock polynomials. The latter is due to a numberof clock breaks. Second-order terms were only introduced at those sta-t ions where the Googe number (S4.2.1) i ndicated that the parameter wouldbe estimable. The multi-dimensional F-test (32.1) yielded a result of0.795 with crit ical value 1.000 with = 5 , indicating that the esti-mated formal error is a l i t t le pessimistic. The estimated coordinate s

and their ( form al) standard deviations are presented in Tabl e 6.In Table 7 the computed baseline lengths ar e shown, together with theirformal erro rs and the weighted RMS-values of the post-fi t residuals(32.5). In gene ral it can be stated that the over all precision of theERIDOC Mk-I1 BWS VLBI results is around 2 0 cm.The major c oncl usio n of this campaign, however, was that i t proved to befeasible to perform geodetic VLBI on a European scale at the cm level,even using a t wo channel 40 M z Mk-I1 BWS-scheme if hydrogen masers wereto become avai labl e at all stations.

8/16/2019 29 Brouwer

90/192

Table 7.

ERIDOC Baseline Lengths

BASELINE I length (m ) I formal error1 RMS (ns)l I (m) II----------+---------------+-------------+-----------lI l I I l

EFF-JOD 699,800.71 0.09 1.70EFF-CH1 589,796.51 0.18 2.01

I EFF-WES 266,613.77 0.09 I 1.56I EFF-ONS 831,711.51 0.02 0.20

JOD-ONS 1,011,065.95 0.09 I 3.54CHI-ONS 1~109,266.01 0.22 13.01

WES-ONS 601,758.04 0.07 3.59I JOD-WES 598,088.57 0.09 3.86

CHI-WES 586,069.08 0.20 I 5.32

3.7 EXAMPLE 11: MERIT SHORT CAMPAIGN

3.7.1 General Information

Project MERIT is the cooperative effort of IAU and IUGG, sponsored by

COSPAR, to Monitor Earth Rotation and to Intercompare the Techniques of

observation and analysis . The main objective of this project is to

measure polar motion and UT1-UTC as accurately and intensively as possi-

ble with all available techniques (optical astrometry, Satelite LaserRanging (SLR), Lunar Laser Ranging (LLR), Satellite Doppler, CERI (51.1)

and, naturally, VLBI), to intercompare the results of the individual

techniques and thus to have a better understanding of the phenomena thatexcite the regular and sudden changes in Earth rotation and polar motion

[Wilkins,l980]. The main measurement campaign of MERIT was held from

1983 September 1 to 1984 October 31, thus spanning a complete period ofthe Chandler Wobble.

To provide realistic tests of the types of operational arrangements that

were required during the main campaign and to provide high-quality ob-

servational data for scientific analysis and for use in a preliminaryintercomparison of the techniques, a test was held during a three-month

period (1980 August 1 to October 31), called: the MERIT Short Campaign,here denoted by: MERIT-SC. One part of this MERIT-SC consisted of a se-

ries of fourteen sessions of 24 hour VLBI observations at the following

observatories (not all stations observed for the whole series); see Fig-

ure 22:

1. Haystack Radio Observatory, USA

2. Harvard Radio Astronomy Observatory, USA

3. Owens Valley Radio Observatory, USA

4. Onsala Space Observatory, Sweden

5. Chilbolton Radio Observatory, United Kingdom

6. Effelsberg Radio Observatory, Federal Republic of Germany

8/16/2019 29 Brouwer

91/192

Fig ure 22: MERIT-S C Network

The observa tions were made using the Mk-I11 recording system with ap-proxim ately 140 schedu led delay and delay rate S- and X-band v alues perbaseline per day. Each value represents an observation (integration) ofabout 110 seconds. For the elimination of ionospheric refraction and forincreasing the precision of the observatio ns by means of the BWS-techni-qu e, eight X-band channel s (near 8.3 GHz ) and six S-band channels (ne ar2.2 GHz) were recorded. The ambiguity spacing was 20 ns for X-band and40 ns for S-band, and the oretic ally, depending on the sensitivity of thetelesc opes, a delay accura cy of about 0.2 to 0.3 ns could be expec ted.

Here an analysis is made of the observations during the first 48 hoursof this MERIT-SC (Septe mber 26, 21h UT to Septem ber 28, 20h UT). Thecooper ating observ atorie s during this period were the above statio ns ex-cept Chilbo lton. Inform ation about the MERIT-SC for Earth rotati on de-termin ation can be found in [Robertson Carter, l9811.

The obj ective of the present analysis is twofold. In the first place, itprovides a validation of the DEGRIAS software package by comparing theresults with previously reported data, especially [Carter,1983]. Sec-ondly, some studie s are done in S3.7.3 about deform ations of the net-work when slight change s in the computing model or in the number of ob-servat ions are made. The latter analys is is compar able to the one inS2.8, but now a network is considered with actually observed data.

8/16/2019 29 Brouwer

92/192

Table 8 .

Number of Scans per MERIT-SC Baseline

- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

I I I l lI BASE- SCHEDULED ~ C O R R E L AT E D ~ CCEPTED I

I LINE I SCANS I SCANS I DELAYS II I I no.I-----------+----------+------------+-----------I HAY-ONS 240 231 89 39 1

HAY-OVR 240 194 57 29I HAY-HRA 240 183 56 31I HAY-EFF 240 209 90 43I EFF-ONS 240 226 179 79I EFF-OVR 240 160 95 60I EFF-HRA 240 151 108 72I ONS-OVR 240 180 88 49I ONS-HRA 240 173 85 49I OVR-HRA 240 185 136 74 1)-----------+----------+----------+-------------l

100% 79% 41%

3.7.2 MERIT-SC: Refe renc e Fit

The "reference fit" is a DEGRIA S LSQ fit to the data of th e part of MER-IT-SC described above, which includes all relevant options. The phrase"optimal solution" is deliberately not used, a s this thesis is concerned

with the principles , assumptio ns and methods of geodetic VLBI. The ob-jective was therefore not to deliver an optimal solution for the MERIT-SC , but to assess a general validation of DE GRIAS and geodetic VLBI.

This reference fit is taken as the on e with delays only. The analysisof the observed data was carried out on the basis of the procedure of53.4. From Tabl e 8 the large drop-out of observatio ns on all Haystackbaselines is immediately obvious. The reason for this is that due to a nerror in the steering computer, all pointings of the Haystack ante nnatowards the West were wrong and the observations therefore useless; com-pare the percentages: about 30% for the baselines to HRAS and OVR O(West) and about 40% to Effelsberg and Onsala (East) Of cour se thishas some impact on the reliability and comparability of the resultspresented here. For the other baselines, the figures in Table 8 aretypical for a global VLBI-experiment. Reasons for the difference s be-tween the number of scheduled scans and successfully correlated scan sare primarily visibility (some sources are not observable by the longes tbaselines and the telescopes were idle at that moment) and in addi tionequipment failures, bad spots on the tape, etc.. The additional de-crea se in the number of accepted observations is partly due to the sameproblems and partly to poor environment al conditions (e.g. low eleva-tions) s o that the observations do not fit the computing model. Th ey ar eexcluded from the LSQ fit either on the basis of the closed triangle

condition (53.4) or o n the ba si s of the W-test (32.3).

8/16/2019 29 Brouwer

93/192

- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - +

l I20.0 I19.1 I18.3 I17.4 I16.5 I15.7 l14.813.9

II

13.1 112.2 I11.3 I10.4 I

9.6 l8.7 l7.8 I7.0 l

6.1 l5.2 I4.4 l3.5 I2.6

. . . . . . . . . . . . . . . . . . . . . .I

1.7. . . . . . . . . . . . . . . . . . . . . . . . . . . .

l0.9

I 0.0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . II - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - II (ns) 0.032 0.105 0.179 0.252 0.325 0.399 0.472 +.5I II MEANVALUE: 0.105 ns

MIN. VALUE: 0.0 nsI

l lI MAX VALUE: 0.866 ns

STANDARD DEVIATION: 0.110 nsl

I- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

I+

Figure 23: Closures of MERIT-SC Triangles

A histogram with the closures of 768 triangles before the final editingof the observed data was done, is shown in Figure 23. From this histo-gram t follows that the maximum closure is 0.866 ns. The mean value,

however, is only 0.105 ns. Hence, a few observations still have to beexcluded To compute these closures, the observations are corrected forthe retarded baseline effect. Without this correction, closures will befound with a mean value of 4.6 ns and a maximum value of up to 22.8 ns,which equals about 7 metres.The distribution per source of the 983 delays which were finally ac-cepted is given in Table 9. Note that relatively few observations arepresent for low declination sources.

As regards the LSQ fit, the coordinate system definition is the same asfor ERIDOC, but with adopted arbitrary coordinates for Haystack (Table10). For the computing model the kinematic model was chosen for epoch1950.0 with the 52000 precessional constant; the corrections for tropos-

8/16/2019 29 Brouwer

94/192

Table 9.

Sour ces Used in MERIT-SC

OBJECT No. RIGHT ASCENSION DECLINATION SPECIAL

pheric zenith delay were based on surf ace meteo readings and a clockfunction was chosen which was valid for two days with five unknowns:off set, linear drift and sine wave. All source positio ns were estimatedexcept those of the low declina tion sources 010 6t01 3, 2134t00 and3C273-B which were scarcel y observed. The pole position andUT1-para meter were determin ed for the 28th September. The a prioristandard dev iati ons, based on the SNR of the correlated sig nals , rangedfrom 0.09 to 0.35 ns (dependi ng on baseline and observed so ur ce) with anaverage value of 0.19 ns.

In this way, 6 2 unknowns were estimated in the LSQ fit. The most impor-tant results of this adjustm ent are shown in Table 10: the coor dina tes

of the sta tion s and their correspondi ng formal errors.The multi-d imension al F-test (32.1) with a level of signifi cance of 5 ,yielded a value of 1.5, indicating that some improve ments in the modelor updating of the observed materia l would be possible . Only four indi-vidual obse rvat ions were left rejected as they had a W-value just abov ethe critica l value. As reg ards the special alternat ive hypothes is test-ing (Table g , it became clear that there are indeed some problems withthe low declinat ion sour ces , because both 0106+ 013 and 3C273-B are re-jected with a W-va lue of about 5 8 (critic al value 3.3). The marginal lydetectab le error (54.2.3) for these tests was about 0.18 ns, so that aconstant bias of more than 0.3 ns for the corresp onding obs erva tion s ispresent. Th is indicat es the influence of refract ion, because all theobservations were made at low elevations.

8/16/2019 29 Brouwer

95/192

Table 10.

MERIT-SC Station Coordinates

l II STATION I

]----------+-------------+-------------+-------------

I l I IHaystack 1492406.691 4457267.330 4296882.102

l / - I - 0 I - 0 I - 0 II OVRO 1-2409598.948 4478357.281 3838603.839I l- .029 .062 .060I HRAS 1-1324207.728 5332028.819 3232118.815I / - I .030 .097 .069

Onsala 3370598.846 -711919.776 5349830.519I / - I .060 .047 .084I

Effelsbg 4033940.609 -486993.888 4900430.445I / - I .059 .048 .079I I I l

- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

Since hardly any baseline components can be found in the literature, for

reference purposes the baseline lengths as computed from these coordi-nates are listed in Table 11, together with the differences with respect

to the results of [Carter,19831 and the corresponding formal errors and

weighted RMS-values and special alternative hypothesis test values

(critical value 3.3). From these tests t is clear that error sources

as a function of source are more significant than biases per baseline.The error detectable by these tests was about 0.1 ns.

As was to be expected, the estimated baseline lengths are not the sameas the ones of [Carter,1983]; exact similarity would even have been sus-

picious, e.g. because in this reference fit no delay rate observations

are included and because of the problems with the Haystack pointings.

It is obvious from the differences (column 3 of Table 11) that the over-

all scale of the network of this DEGRIAS solution is smaller: the four

longest baselines are all about 20 cm shorter. In addition, t can bestated that Haystack moves to the East, because HAY-EFF and HAY-ONS

are both 10 cm shorter and HAY-HRAS and HAY-OVRO both about 10 cmlonger. This leads to the conclusion that Haystack also moves inheight. The explanation is of course that the lack of observations to-

wards the West means that the estimated tropospheric zenith delay param-

eter is poorly determinable. Consequently, via the correlation, the

local height coordinate of the station becomes biased. With this in

view, the correspondence between the DEGRIAS results and those of

[Carter,1983] can be considered as within the error limits.

At any rater for the Haystack-Onsala baseline e.g., the following re-

sults can be found as adjustment results (in metres):

5,599r714.63 [Shapiro et a1.,1979]5r599,714.69 [Herring et a1.,19811

5,599,714.54 [Robertson&Carter,l981]

8/16/2019 29 Brouwer

96/192

Table 11.

MERIT-SC Baseline Lengths

- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

I I I DEGRIAS I I baseline I I one IBASELINE DEGRIAS I minus formall RMS I bias I(observ.1

I I length (m ) [Carter,831 sigma 1 (ns) (hypothesi s(l ore II I l (cm) l (cm) l l test l l (cm) I----------+-------------+---------+-------+-------+----------++-------]

I I I I I II II EFF-ONS 832210.535 1 0 I 1 ( 0 . 1 2 1 4.52 1 0 1( E F F - O V R 1 8 2 0 3 7 4 2 . 4 3 8 1 - 2 0 1 5 1 0 . 2 8 1 4.85 1 ) +6 11 HRA-ONS 17940732.094 1 - 1 3 1 8 1 0 . 2 7 1 1 - 8 0 1 1+l9I HRA-OVR 1508195.371 1 0 1 2 1 0 . 1 9 1 0.91 1 + 4 II EFF-HRA 18084184.714 1 - 3 0 1 8 1 0 . 2 5 1 0.18 1 1+l91 OVR-ONS 7914130.941 1 - 16 1 5 1 0.22 1 0.78 l l +6 1

HAY-ONS 5599714.414 1 - 11 1 3 1 0.24 1 3.44 l l +l I

I HAY-HRA 1 3135641.130 1 10 1 3 1 0.16 0.17 1 +7HAY-OVR 3928881.745 9 1 3 0.21 2.44 1 3HAY EFF 1 5591903.458 - 16 1 3 1 0.28 1 1.10 1 +l

II

I I l l I l I I- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

l

5,599,714.53 [Carter,198315,599,714.4 3 [Ro ger s et a1.,198315,599,714.51 [Lundqvist,l98415,599,714.41 [present publ icat ion]

The most recent information about a series of intercontinental baselinedeterminations can be found in [Lundqvist, l9841; cf. also [ ~ h o m a s ta1.,19831.

3.7.3 MERIT-SC: Alte rnat ive Fit s and Stab ilit y

3.7.3.1 One obs erv ati on more

To demonstrate the possible influence of one single observation on theLSQ esti mates, the results are reproduced here of an alt ernative fit for

MERIT-SC. The diff eren ce is just that one additional obse rvat ion ofsource 2134 +00 by the baseline Effelsberg -HRAS is included. The obse rv-ing elevations are 32.6 and 11.1 degrees respectiv ely. At the time ofthe observation (day 270, 22h 58m UT) the local zenith angles of the Sunwere 131 and 6 8 degrees.The differences in estimated baseline lengths with the reference fit areshow n in the last colu mn of Table 11. From these numbers t followsthat a difference of only one observation yields a maximum change inbaseline length of 19 cm Inspection of the test results reveals thatthe W-test for this obse rvat ion has a value of 3.95 with a crit icalvalue of 3.29, so that the observation is only narrowly rej ected, with apost-fit residual of 0.75 ns. The marginally detectable error (43.8)is, in addi tion , about twice the average value for this baseline and the

8/16/2019 29 Brouwer

97/192

external reliability parameter 43.11) has a value 5.45, whereas the av-erage value is one. Possi ble error s in this observation therefore havea large impact on the estimated parameters.Hence , this obser vatio n has a very poor reliability, and when an erroris present, the geometry of the solution is severely corrupted. Th emain reason for such an error in this obser vatio n is of course the lowelevation. Th e effect is so large because the sourc e was not observed

often , twelve times in total and only once by this baseline. In addi-tion, it should be noted that the results of the special hypothesis testremain about the same: 0.23 vs. 0.10 Table 11) with a critical value of3.29. It is therefore obvious that just this single obser vatio n is er-roneous. Even taking into account the non-optimal DEGRIA S computingmodel of whi ch the siz es of the m.d.e. and X however, are independ-ent), it follo ws from this example that for many obse rvatio ns it is als onecessary to inspect all observations individually.

3.7.3.2 Exclud ing sub set s of obs erv ati ons

To analy se the stability of the multi-station fit furth er, in Table 12the differenc es with respect to the reference fit are show n for the fol-lowing cases:

1. No EFF-OVR baseline observations included2. No HRA-ONS baseline observations included3. No HRA-OVR baseline observations included4. No EFF-HRA ba selin e obser vation s included5. No OVR-ONS baseline observations included6. No HAY-ONS baseline observations included7. No HAY-OVR baseline observations included8 No HAY-HRA baseline observations included9. No EFF-ONS baseline observations included

10. No HAY-EFF baseline observations included11. All baselines with station OVRO excluded12. All baselines with station HRAS excluded13. All ba selines with station EFF excluded14. All baselines with station HAY excluded15. All baselines with station ONS excluded16. All observ ations of source 0106t013 excluded

Run 16 was included because of the results of the special alter nativ e

hypothesis testing of S3.7.2. The analo gous elimination of obser vatio nsof all sourc es other than 0106t013 yielded changes in baseline lengthbelow the 2 cm threshold, which a re not reproduced here.As a main impression from these results, it can be stated in the firstplace that the stability of the network solution is not very good. Se-veral examples can be found where changes with a magnitude of about thestandard devia tion of the estimated baseline length ar e caused by ex-cluding a number of observations. That this is purely a matter of poorreliability is indicated by the fact that no significant differences arefound between the multi-dimensional F-test values 32.1). Since possi-ble error s in the set of obser vatio ns or in the computing model cannottherefore be detected by statistical testing of the main LSQ fit alo ne,

it is recommended that such a compari son of a series of fit s should al-

8/16/2019 29 Brouwer

98/192

Table 12.

Comparison of MERIT-SC Results I (cm)

+---- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

l R U N 1 1 / 2 1 3 1 4 1 5 1 6 1 7 / 8 1I-------+-----+-----+-----+-----+-----+-----+-----+-----l

I E F - O N ) 0 1 0 1 0 1 0 1 0 1 0 1 0 1 - 1 1( E F - O V I - 1 0 ) 3 1 - 1 1 1 1 1 1 - 1 1 0 1I H R - O N ) - 4 1 - 1 - 6 1 - 2 1 0 1 0 1 1 1 6 1I H R - O V I - 1 1 0 1 - 1 - 1 1 0 ) 0 1 - 1 1 2 1I E F - H R I - 5 1 0 1 - 5 1 - 1 - 1 1 0 1 0 1 6 )I O V - O N I - 3 1 0 1 2 1 - 1 1 1 1 1 0 1 - 1 1I H A - O N ( - 2 1 1 1 2 1 1 1 0 ) 1 0 1 1 1\ H A - H R I - 2 1 0 1 - 1 1 0 1 0 1 0 1 - 1 1 1I H A - O V I - 1 1 0 1 3 1 0 1 0 1 0 1 1 3 1I H A - E F I - 2 1 1 1 3 1 1 1 2 1 - 1 1 - 1 1 2 )

- - - - - - -+--- - -+--- - -+--- - -+--- - -+--- - -+--- - -+--- - -+--- - - II-------+-----+-----+-----+-----+-----+-----+-----+-----lI RUN 1 9 1 1 0 1 1 1 1 1 2 1 1 3 1 1 4 1 1 5 1 1 6 1I-------+-----+-----+-----+-----+-----+-----+-----+-----I E F - O N 1 1 0 1 0 1 0 1 1 0 1 - 1 - 2 1I E F - O V 1 8 ) 5 1 - 1 0 1 - 1 1 3 1 7 1 - 7 1I H R - O N I - 7 1 6 1 - 1 1 1 - 1 2 3 1 - 1 - 1 0 1I H R - O V I - 1 1 1 1 1 - 1 0 1 0 1 - 1 1 - 2 1I E F - H R ) 7 1 6 1 - 1 3 1 1 1 2 3 1 5 1 - 9 1I O V - O N I - 7 1 5 1 - 1 - 1 1 - 1 1 1 4 1 - 1 - 8 )

H A - O N ( - 5 ) 0 ) 1 7 1 - 4 1 1 - 1 - 4 1I H A - H R ( 0 ) 0 1 - 6 1 - 1 - 2 1 1 0 1 - 3 1( H A - O V I 0 1 0 1 - 1 6 1 - 2 1 1 0 1 - 2 1I H A - E F ) 4 1 - 1 - 2 1 7 1 - 1 1 4 1 - 3 1

+--- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

ways be performed when adjusting a set of VLBI observations, as thisgives a better indication of the true accuracy (preci sion reliability)of the experiment than the quoted formal errors of the fin al LS Q fit.

In addit ion, attention is drawn to the fact that for run 14 (wher e allHaystack observations are excluded) a value of about two decim etre s isfound equal to the number by which the DEGRIAS baselines of the refer-

ence fit are shorter than the ones presented in [Carter,1983]. Thisjustifies the conclusion that the DEGRIAS computing model accuracy isindeed at the 10 cm level and that the main deviations for these MERIT-SC data ar e caused by the Haystack pointing error bias.

3.7.3.3 Cha nge s in the compu ting mode l

Analogous to the computations in 52.8, this section describes the influ-ence of some changes in the computing model on the behaviour of the LSQfit.Column 1 of Table 13 shows the differences in baseline lengths with re-spect to the refe rence fit , due to a change in the a priori standard de-

8/16/2019 29 Brouwer

99/192

viation: now a fixed value of 0.2 ns is chos en for all observations. Bycomparison with the reference fit , this means that the observatio ns ofEFF-ONS and HRA-OVR are down-weighted and those of EFF-HRA and ONS-HRAup-weighted . In view of the standard devi atio ns of the estimated base-line lengths of Table 11, no significant changes can be found, althoughthere exis ts a trend that baseline lengths increase proportion al totheir length. The coordinates of Onsala and of Effelsberg change some-

what more than their standard deviation: in the X- and Y-di rect ions by-7 cm and in the Z-directio n by -17 cm. The network shape , howe ver, re-mains constant in view of the above.The results of such statistical testing as data-snoop ing, multi-dimen-sional F-test, special alternative hypothesis tests and RMS-values arealso in complete agreement.

Table 13.

Compariso n of MERIT-SC Results I1 cm)

EF-ONEF-OVHP.-ON1-m-ovEF-h?

OV-ONHA-ONH A - mHA-ov

In the second example, all standard deviations of the observatio ns aremultiplied by 1/ 2*cos1); this is a down-weight ing as a functi on of ze-nith angle. The facto r of 2 is applied to arri ve at about the same for-mal errors for the estimated unknowns. Although the changes in baselinelength are somewhat larger and more random than in the previous example,no surprising effects are found. The considerab le changes in estimatedcoordinates should be noted again, with a maximum for X,F and Z of EFFof t18, +9 and t28 cm respectively. Here one sees again that the shapeof the network is better defined than its orie ntat ion or its scale. Inthese fits all observations below 10 degrees elevation had already beenexcluded, otherwi se the changes would have been considera bly greater.

As regards the introduction of delay rate observation s in run 3, onlythe chan ges in the length of the intercontine ntal baselines to HRAS are

8/16/2019 29 Brouwer

100/192

worth noting. Further inspection of the adjustment results shows thatthe estimated tropospheric zenith delay of HRAS changes by about 10 cmas a result of the introduction of delay rates. Its value becomes 2.16metres , a more realistic number in view of the height of HRAS above sealevel. Th e conclusion is therefore that with delay obser vatio ns only acorrel ation between zenith delay and local height-coordinate for theHRA S stati on exists for this experi ment, which is too high (N.B. the

shift in EIRAS coordi nates compa red t o the referenc e fit is exactly 1 4 cmin height direction ) and that the additi onal delay rate obser vatio nsremedy this situation.

In the 4th example the correctio n for ionospheric refraction is omitted.It can be concluded from Appendix C that this includes correct ions of upto 2.2 ns. It is immediately obviou s from Table 13 that the chang es inbaseline length are more or less proportional to baseline length, whichmeans that the scale is ill-determined. Such errors cannot therefore bedetected easily. Thi s follow s als o from the multi-dimensional F-test,which yields 2.1 (compared to the 1.53 for the reference fit) for a 5%level of significance. Thi s is probably als o the main reason for ef-fects such as the large differences between published lengths for HAY

ONS (53.7.2).

In the last example, in view of fit 12 of 52.8 and the resu lts of Figu re23 , a 5% error in the retarded baseline correction is introduced, whichhas a smaller magnitude than the error for the previous example. Thedifferences are immediately obvious: the multi-dimensional test-variateF rises to 2.22, compared to 1.53 for the reference fit. As regards thespeci al hypoth esis, the tests for source biases remain about the same,but almost all baseline bias tests are rejected and also one stationbias test. Th e largest test value is 11.0 (with criti cal value 3.3) for

a baseline bias in EFF-OVR. Th e problem is als o indicated by the in-crease of the RMS value of the residuals from 0.28 to 0.44 ns for thesame baseline. In addit ion, the number of rejected observ ation s by thedata-snooping is seven times as large. All these facto rs indicate thatthe geometry of the network (espec ially the closed triangles ) andtherefore of the LSQ fit is severely disturbed. Consequently, such anerror, unlike that of the preceding example, can be easily detected.The changes in estimated station coordinates are als o considerable:about 0.5 metres. In geographical coordinates this means a rotation ofabout 0.015 arcsec to the East for the European stations and a change ofabo ut -65 cm in height. OV RO and EIRAS cha nge mainly in height: by +60

and +37 cm respectively.

3.7.4 Conclusions

Th e first conclusion of these MERIT-SC computations is that the claimedDEGRIA S computing model accuracy of 52.8 is indeed achieved and may ev enbe so mewhat better than the quoted number of 10 cm.

Furthe rmore , it can be stated that one should be careful in claim ing acerta in precision for the estimated station coordinates or baselinelengths, on the basis of formal errors and RMS-values only. It is shownthat the "stability" of the estimated MERIT Short Campai gn baseline

8/16/2019 29 Brouwer

101/192

lengths is significa ntly poorer than indicated by the formal errors.Perh aps the MERIT-SC is a comparat ively bad exampl e due to the Haystackpointing err or, but i t is stated here that such f acto rs will al ways ham-per a complete ly successfu l observing session. Mention should be madeof the following points: the ionosphere can suddenly become very active,parts of some obse rvat ions will lack a second frequency or an error inthe clock system lasting for several hours may occur . Thes e factors de-

termine for the most part the final accuracy of the experi ment , as theycause sit uations like that of 53.7.3.1, where o ne obse rvat ion has a tre-mendous influence on the LSQ estimates.

Even though the recommend ation is based on the example of the MERIT-SCwhich has some very significa nt problems, and noting that DEGRIAS doesnot have a computin g model accuracy of 1 cm, i t is to be recommende dthat computat ions such as those described above (d eleting all observa-tions of one bas eline, of one source, etc.) should alway s be carried outso that the final solu tion is accompan ied with a better indicatio n ofits accuracy . One thus has a reliable estimate of the internal consist-

ency or repeatability. The reproducibility (indepen dent observati ons,correlation, geodetic analysis software) will then be about a factor oftwo worse.

Another conc lusi on is that a marked differenc e exists between errorsources that corrupt the geometry of a closed triangle of VLBI baselinesand those that do no t, with respect to the possibili ty of detectingthose sources of error. This is because the former type of error sourcetends to increase the shifting variate and therefo re can be detected bystatistical testing, whereas the latter kind of error source leadsmainly to changes in the scale and orientation of the network.An exampl e of the first kind of error source is an error in the computa-tion of the retarded baseline correctio n, which results in considerableclosing errors for the triangles. An example of the second error sourceis the neglect of the ionospheric refraction correcti on.This leads to the conc lusi on that in general the relative positions ofVLBI observato ries can be determined to one order of magnitude betterthan the posi tion , orie ntat ion and scale of the entire network. This isdue to the coor dina te system definiti on (53. 3) and the fact that errorsin certain correcti ons or observati ons yield a considerable common ef-fect on all baselines and therefore on the position, orientat ion andscale of the netwo rk.

Th e last conc lusi on is that troposph eric refractio n, both dry at lowelevatio ns and wet , is the most prominent sour ce of erro r, especiall ybecause i t is of the second type mentioned above. To avoid prob lems , noobservati ons below 10 degr ees elevation should be included in the expe-riment. In addi tion , care should be taken with sources which have notbeen observed on many occasions . For the Northern hemispher e tele-scopes, these are mostly low declination sources, observed in a smallsection of the sky. The esti mate of the troposphe ric zenith delay par-ameter then becomes less well determined due to corr elat ion, and alsothe situation of observati ons with a poor reliability may occur. Thesource position then becomes biased as well and affe cts the estimatedstation locations.

8/16/2019 29 Brouwer

102/192

Chapter 4

D E S I G N O F V L B I E X P E R I M E N T

Summary: In this chapter first some general consid eration s relevant tothe design of a geodeti c VLBI experiment are discussed. Next,two items of the quality control of a geodeti c network , i.e.precisi on and r eliabi lity, are discussed in S4.2 with a view onthe special Delft approac h for point positioning. The quan-tities of this approach for describing precision and reliabil-ity are derive d and discus sed. In connect ion with this, thecharacteristics of the SCHED-module of the DEGRIAS softwarepackage are briefly summarized. This module generates a sched-

ule for a geodetic VLBI campaign starting from visibility con-siderat ions for the sources and optimiz ing for telesco pe slew-ing times. Thi s is defined as costs , the third constit uentof quality control.In S4.4 the above is applied to some simulat ion studies for op-timizing the quality of a geodetic VLBI network. Examples arethe network studie s previously performed in [Derma nis,l97 7],the MERIT-SC config uration and a Europe an geodyna mics network.

4.1 INTRODUCTION

General ly speaki ng, every geodet ic VLBI experi ment is conduc ted for thedetermination of station coordinates and Earth orientation parameters ata specific moment. The quality control [Teunissen8l984a1 of such ageodeti c network demands a design which is precis e and reliabl e enoughat minimal costs. Precisi on is defined here as a measure of the charac-teristi cs of the network in propagating random errors. Reliabi lity de-scribes the ability of the redundant observations to check model errors.Cost s of geodetic VLBI are mainly determined by the number of telescopesinvolved (especiall y via correlator time) and the durati on of the cam-paign.

First of al l, choice