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J Geod (2010) 84:31–50 DOI 10.1007/s00190-009-0342-1 ORIGINAL ARTICLE Global optimization of core station networks for space geodesy: application to the referencing of the SLR EOP with respect to ITRF David Coulot · Arnaud Pollet · Xavier Collilieux · Philippe Berio Received: 17 December 2008 / Accepted: 24 August 2009 / Published online: 10 September 2009 © Springer-Verlag 2009 Abstract We apply global optimization in order to optimize the referencing (and consequently the stability) of the Earth Orientation Parameters (EOP) with respect to ITRF2005. These EOP are derived at a daily sampling from SLR data, simultaneously with weekly station positions. The EOP ref- erencing is carried out with minimum constraints applied weekly to the three rotations and over core station networks. Our approach is based on a multi objective genetic algo- rithm, a particular stochastic global optimization method, the reference system effects being the objectives to minimize. We thus use rigorous criteria for the optimal weekly core sta- tion selection. The results evidence an improvement of 10% of the stability for Polar Motion (PM) series in comparison to the results obtained with the network specially designed for EOP referencing by the Analysis Working Group of the International Laser Ranging Service. This improvement of nearly 25 μas represents 50% of the current precision of the IERS 05 C04 PM reference series. We also test the possibil- ity of averaging the weekly networks provided by our algo- rithm (the Genetically Modified Networks—GMN) over the whole time period. Although the dynamical nature of the GMN is clearly a key point of their success, we can derive such a global mean core network, which could be useful Electronic supplementary material The online version of this article (doi:10.1007/s00190-009-0342-1) contains supplementary material, which is available to authorized users. D. Coulot (B ) · A. Pollet · X. Collilieux IGN/LAREG et ENSG, 6 et 8 Avenue Blaise Pascal, Cité Descartes, Champs-sur-Marne, 77455 Marne la Vallée Cedex 2, France e-mail: [email protected] P. Berio Observatoire de la Côte d’Azur, Boulevard de l’Observatoire, B.P. 4229, 06304 Nice Cedex 4, France for practical applications regarding EOP referencing. Using this latter core network moreover provides more stable EOP series than the conventional network does. Keywords Global optimization · Earth orientation parameters · Minimum constraints · Core station networks · Genetic algorithms · Satellite laser ranging 1 Introduction In this paper, we apply global optimization to find, each week and among all the available stations, the station networks over which apply Minimum Constraints (MC) in order to produce the best referenced Earth Orientation Parameter (EOP) series. We focus on a current space geodetic data processing provid- ing station positions and EOP. In the literature related to opti- mization, such a problem is a subset problem (Leguizamón and Michalewicz 1999). Improving the EOP referencing with respect to any given Terrestrial Reference Frame (TRF) aims to improve the stability and, consequently, the quality of EOP time series. In particular, we aim to guarantee the consistency of the estimated EOP among the different epochs at which the referencing is carried out, on the basis of MC. Indeed, some of the primary tasks of the International Earth Rotation and Ref- erence Systems Service (IERS) are (i) to ensure consistency between its three major reference products, the International Terrestrial Reference Frame (ITRF), the International Celes- tial Reference Frame (ICRF), and the EOP connecting these two frames and (ii) to guarantee the best feasible precisions and accuracies for these three reference products. Regarding the first goal, the ITRF2005 computation was recently a major step toward the consistency between the IERS products. Indeed, for the first time, the ITRF2005 com- bination provided the TRF together with consistent EOP time 123

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J Geod (2010) 84:31–50DOI 10.1007/s00190-009-0342-1

ORIGINAL ARTICLE

Global optimization of core station networks for space geodesy:application to the referencing of the SLR EOP with respectto ITRF

David Coulot · Arnaud Pollet · Xavier Collilieux ·Philippe Berio

Received: 17 December 2008 / Accepted: 24 August 2009 / Published online: 10 September 2009© Springer-Verlag 2009

Abstract We apply global optimization in order to optimizethe referencing (and consequently the stability) of the EarthOrientation Parameters (EOP) with respect to ITRF2005.These EOP are derived at a daily sampling from SLR data,simultaneously with weekly station positions. The EOP ref-erencing is carried out with minimum constraints appliedweekly to the three rotations and over core station networks.Our approach is based on a multi objective genetic algo-rithm, a particular stochastic global optimization method,the reference system effects being the objectives to minimize.We thus use rigorous criteria for the optimal weekly core sta-tion selection. The results evidence an improvement of 10%of the stability for Polar Motion (PM) series in comparisonto the results obtained with the network specially designedfor EOP referencing by the Analysis Working Group of theInternational Laser Ranging Service. This improvement ofnearly 25 µas represents 50% of the current precision of theIERS 05 C04 PM reference series. We also test the possibil-ity of averaging the weekly networks provided by our algo-rithm (the Genetically Modified Networks—GMN) over thewhole time period. Although the dynamical nature of theGMN is clearly a key point of their success, we can derivesuch a global mean core network, which could be useful

Electronic supplementary material The online version of thisarticle (doi:10.1007/s00190-009-0342-1) contains supplementarymaterial, which is available to authorized users.

D. Coulot (B) · A. Pollet · X. CollilieuxIGN/LAREG et ENSG, 6 et 8 Avenue Blaise Pascal,Cité Descartes, Champs-sur-Marne,77455 Marne la Vallée Cedex 2, Francee-mail: [email protected]

P. BerioObservatoire de la Côte d’Azur, Boulevard de l’Observatoire,B.P. 4229, 06304 Nice Cedex 4, France

for practical applications regarding EOP referencing. Usingthis latter core network moreover provides more stable EOPseries than the conventional network does.

Keywords Global optimization · Earth orientationparameters · Minimum constraints · Core station networks ·Genetic algorithms · Satellite laser ranging

1 Introduction

In this paper, we apply global optimization to find, each weekand among all the available stations, the station networks overwhich apply Minimum Constraints (MC) in order to producethe best referenced Earth Orientation Parameter (EOP) series.We focus on a current space geodetic data processing provid-ing station positions and EOP. In the literature related to opti-mization, such a problem is a subset problem (Leguizamónand Michalewicz 1999). Improving the EOP referencing withrespect to any given Terrestrial Reference Frame (TRF) aimsto improve the stability and, consequently, the quality of EOPtime series. In particular, we aim to guarantee the consistencyof the estimated EOP among the different epochs at which thereferencing is carried out, on the basis of MC. Indeed, some ofthe primary tasks of the International Earth Rotation and Ref-erence Systems Service (IERS) are (i) to ensure consistencybetween its three major reference products, the InternationalTerrestrial Reference Frame (ITRF), the International Celes-tial Reference Frame (ICRF), and the EOP connecting thesetwo frames and (ii) to guarantee the best feasible precisionsand accuracies for these three reference products.

Regarding the first goal, the ITRF2005 computation wasrecently a major step toward the consistency between theIERS products. Indeed, for the first time, the ITRF2005 com-bination provided the TRF together with consistent EOP time

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32 D. Coulot et al.

series, namely Polar Motion (PM) from VLBI and satellitetechniques and Universal Time (UT) and Length Of Day(LOD) from VLBI only (Altamimi et al. 2007). This resultled to the production of the new IERS EOP time seriesIERS 05 C04. The consistency of these EOP series and theITRF2005 is now regularly checked and is ensured at the levelof 20–30 µas (Altamimi et al. 2008). In comparison, Bizouardand Gambis (2009) assess the discrepancy between the pre-vious IERS C04 EOP time series and the ITRF2005 at thelevel of 300 µas. Concerning the IERS second task, the gen-eration of these new 05 C04 series was also the opportunityof implementing an upgraded computation procedure. Thisnew method induces a great improvement of the EOP accu-racy, now assessed at the level of 50 µas for PM (Bizouard andGambis 2009). Few years ago, Gambis (2004) still estimatedthis latter at approximately 100 µas. The natural sensitivitiesof the space geodetic techniques with respect to EOP (seeTable 1 of Gambis 2004) clearly affect the quality of theirdata-related EOP series; the way these EOP are referencedmay also have a significant influence.

There are currently several ways of rigorously referenc-ing EOP series with respect to any given TRF. The firstmethod consists in using a stacking approach like the oneapplied for ITRF2005 computation (Altamimi et al. 2007).Indeed, this latter was generated, for the first time in theITRF history, from time series (weekly from satellite tech-niques and 24-h session-wise from VLBI) of station posi-tions and daily EOP. This approach is mainly based on theHelmert transformation, a seven-parameter similarity—seefor instance Altamimi et al. (2002b), and on the equivalentrelations for EOP (Zhu and Mueller 1983). Schematically, theweekly seven-parameter transformations between the indi-vidual solutions and a stacked frame—a mean global secularframe realized in the TRF by the application of MC and/orinternal constraints—are estimated simultaneously with thisstacked frame and consistent EOP series. During the estima-tion process, the rotations are applied to the original dailyEOP to align them with respect to the orientation of thestacked frame. The consistency of EOP among the epochsof the original solution is thus only guaranteed by the factthat this computation method encompasses all the availablesolutions. The second method is similar to the previous onebut does not produce a stacked frame. More precisely, thisapproach consists in directly computing the seven-param-eter transformations between the weekly solutions and theTRF. The weekly rotations are then applied to the originaldaily EOP to align them with respect to the orientation ofthe TRF, in accordance with the relations derived in Zhuand Mueller (1983). Here, only an “optimal” estimation ofthe weekly rotations between the involved Terrestrial Frames(TF) and the TRF guarantees the EOP consistency among theepochs. The final approach directly references the daily EOPwith respect to the TRF by applying the appropriate MC at

the normal system level (Coulot et al. 2007). Indeed, MCwere mainly designed to adequately compensate for the rankdeficiencies of the normal matrices involved in any spacegeodetic data processing (Sillard and Boucher 2001); but,MC can also be used to nullify some of the transforma-tion parameters between the TF underlying the consideredsolutions and the TRF (Altamimi et al. 2002a). This lastapproach is the one used for the present study. The mainissue for this method is thus to guarantee the EOP consis-tency among epochs through the weekly application of MC,constraints which not originally aim at optimally defining aTF orientation. This latter point is more discussed later on,in Sect. 2.

MC and core station networks are key issues of thesethree approaches. Indeed, the simultaneous estimation ofITRF2005 and its joined EOP provides EOP series consistentwith the orientation of this TRF. This orientation is definedin order to get null rotation parameters at epoch 2000.0 andnull rotation rates between the ITRF2005 and the ITRF2000.These two conditions are fulfilled by means of MC appliedover a core set of 70 stations located in 55 sites, see Fig. 9 ofAltamimi et al. (2007). This empirical core set of stations wasderived to avoid any artifact, related to any frame misdefi-nition, in the so-computed EOP time series. Moreover, forboth first approaches, the seven-parameter transformationcomputation requires realistic station position variance–covariance matrices for the least-squares estimation. Suchvariance–covariance matrices can be deduced from MCapplied on appropriate networks (Altamimi et al. 2002b).Finally, the importance of MC and, especially, of the net-work over which these latter are applied is clearly illustratedlater on, in Sect. 3.

The present study consists in finding the optimal SLRweekly core networks over which apply MC to guarantee thebest achievable referencing, by means of these MC, of SLRderived EOP with respect to ITRF2005. The SLR weeklycore networks are derived with the help of a particular sto-chastic global optimization method, the Genetic Algorithms(GA). These weekly networks are thus called “GeneticallyModified Networks” (GMN). We decided to first carry outthis study for the SLR technique as its tracking network isboth sparsely distributed and restricted concerning the num-ber of stations. The first feature is a challenge regarding theuse of MC, whereas the second one preserves the applicationof our algorithm based on GA from being too time consum-ing over a long data span. The methodology developed here,despite of its necessary tuning for SLR solutions, is generaland can be applied to other techniques.

Our aim here is in fact twofold. On one hand, our algo-rithm is used to test whether the stability of the SLR EOPseries could be improved, on the basis of MC, with the helpof appropriate weekly core networks. On the other hand, andin the same spirit than the core set of stations used for the

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Global optimization of core station networks 33

ITRF2005 datum realization, we also question the possibilityof making a mean reference network emerge by averagingthe weekly GMN. Indeed, such a core network may be veryuseful from a practical point of view. These two major goalsare supplemented by other results provided by our analy-sis. These latter are linked to the choice of the criteria tooptimize, to the possible incommensurability between theobjectives optimized by our algorithm, and finally to the pos-sible influence of some geometrical parameters on the EOPreferencing.

The first section, after a brief description of the SLR solu-tions used for this study, discusses the definition of a TForientation and evidences the influence, on the EOP referenc-ing, of the network over which MC are applied. The secondsection then describes three fixed reference networks cur-rently usable for EOP referencing and the results providedby these three networks. The third section is fully dedicatedto global optimization: we present the choice of the objec-tives to optimize to ensure the best EOP referencing, theprocedure developed for our issue on the basis of an existingGA, and the inherent numerical tests. The results providedby our algorithm are then detailed and discussed. We partic-ularly emphasize the stability of the EOP series so producedand the issue concerning the possible emergence of a globalfixed core network from the weekly GMN. Finally, we pro-vide some conclusions and prospects.

2 Definition of a terrestrial frame orientation

In this section, we first briefly describe the data used for thepresent study and the inherent features of the SLR network.The Reference System Effect (RSE) and MC concepts arethen presented. Finally, the influence of MC on the EOP ref-erencing is illustrated with numerical examples.

2.1 Data used

2.1.1 ASI solution

To carry out this study, we use the official solution (v10 ver-sion) computed by the International Laser Ranging Service(ILRS) (Pearlman et al. 2002) analysis centre hosted at theItalian Spatial Agency (analysis centre named ASI). Over aparticular week, a solution consists of daily EOP (PM andLOD), station positions for the whole available SLR track-ing network, and possible range biases, simultaneously esti-mated from a week of data on both LAGEOS satellites andon both ETALON satellites. These solutions are availablein the form of SINEX files. The corresponding time spanis nearly 15 years, from 12/27/1992 to 04/05/2008 inclusive.This corresponds to exactly 797 weeks.

The strategy adopted for the computation of these solu-tions is the one which was discussed and accepted during the

ILRS Analysis Working Group (AWG) meeting in Grasse,France, in September 2007. This strategy is detailed in thecorresponding minutes.1

2.1.2 SLR network features

A world-wide network of 77 stations is involved over theconsidered data span of nearly 15 years but the number ofstations available each week varies. On the whole, there isa minimum number of seven stations, a maximum numberof 26 stations, and an average number of 19 stations eachweek, the standard deviation being three. This points out thedynamic nature of the involved weekly SLR station networks.

This feature concerns not only the number of available sta-tions each week, but also the geographical distribution andthe presence of these stations. Figure 1 shows the SLR track-ing network, classifying the stations with respect to their pres-ence duration in the weekly solutions over the whole periodof time. Among the 77 stations, only seven are availablebetween 75 and 100% of the time (black dots on Fig. 1): thereare three stations located in the US, three stations located inEurope, and one station located in the West of Australia.

Finally, the quality of the SLR tracking station data is alsoheterogeneous. Indeed, not all the SLR stations are equiva-lent regarding the observation capability or the data accuracy.This quality is moreover regularly checked and reported bythe ILRS. It is thus quite challenging to get a SLR refer-ence network over which applying the MC. Nevertheless,such networks exist and two of them are described later on.Next Subsections discuss the TF orientation and illustrate theinfluence of MC on EOP.

2.2 Terrestrial frame orientation

The definition of any TF orientation in fact amounts to solvea free-network adjustment problem and must consequentlybe apprehended in this general framework. The followingpart is mainly based on the review of algebraic constraintsfor TF datum definition by Sillard and Boucher (2001).

2.2.1 Free-network adjustment

The normal systems related to station positions involved inany space geodesy data processing are singular. Indeed, bynature, the space geodesy measurements do not carry all thenecessary information to define the TF underlying the stationpositions. For instance, VLBI is not sensitive to the Earth’scentre of mass whereas satellite techniques are. On one hand,the estimation of station positions requires the definition of acoordinate system, but, on another hand, the measurements

1 Available at http://ilrs.gsfc.nasa.gov/docs/AWG_GRASSE_minutes_24.09.2007.pdf.

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34 D. Coulot et al.

180°

180°

240°

240°

300°

300°

60°

60°

120°

120°

180°

180°

-80° -80°

-40° -40°

0° 0°

40° 40°

80° 80°

Fig. 1 Station network named “ASI network” in text. Each week, thisnetwork is made up by all the stations available in the ASI solution. Theblack (respectively, green, blue, and red) dots correspond to the stations

available between 75 and 100 (respectively 50 and 75, 25 and 50, and0 and 25)% of the whole data time span

are invariant with respect to some modifications in this coor-dinate system (Grafarend and Schaffrin 1976). For the pres-ent case study, as it will be numerically demonstrated later on,the geometrical invariants are the three rotations related to theorientation of the weekly TF underlying the ASI solutions.

The problem consisting in estimating, in a least-squaresense, station positions from space geodesy measurementshas thus an infinite number of solutions (Dermanis 1994)(with different parameter values and associated variance–covariance matrices), and any solution of the involved sin-gular normal system is called a free-network solution. Sucha solution belongs to the order zero, among the four ordersof network design proposed by Grafarend (1974).

The foundations of free-network adjustment were first for-mulated by Blaha (1971) and a wealth of literature has beenpublished since, both on the free network concept and onthe optimal methods to compute a solution for such a sin-gular normal equation. These mathematical methods mostlyrely on the generalized matrix inverses (Bjerhammar 1973).On the basis of these latter, Koch (1987) provided the generalform of a free-network adjustment solution; he also dem-onstrated that the solution of minimum norm was the onebased on the Moore–Penrose pseudo-inverse (Penrose 1956).This particular solution is also the one with the maximumprecision.

Blaha (1982) found the same minimum norm andmaximum precision solution but with a set of minimum

constraints, in the sense defined by Schaffrin (1985); this“minimum” characteristic is guaranteed by a relation betweenthe constraint matrix (then called the “matrix of inner con-straint adjustment”) and the design matrix of the involvednormal system. The term “constraint” is here quite ambigu-ous as, in fact, Blaha solved a conditional system.

These methods based on generalized matrix inverses mayprovide biased estimations of the station positions (Grafarendand Schaffrin 1974). As the previous equivalence shows,these methods in fact amount to add a regularization con-dition to the involved normal system. But, if this conditiondoes not rely on the TF geometrical invariants, it may lead toestimations fully dissociated from their physics. It is the rea-son why Sillard and Boucher (2001) developed the concept ofRSE and MC under the form recommended by the ITRF prod-uct centre since the ITRF2000 realization (Altamimi et al.2002a). Both next parts briefly present these concepts andthe transformation between two TF.

2.2.2 Transformation between two terrestrial frames

The standard relation of transformation between two TF is anEuclidian similarity based on seven parameters (the Helmertparameters), three translations tx , ty , and tz , a scale factoroffset d with respect to the unit scale, and three rotationsεx , εy , and εz (Altamimi et al. 2002b). The transformationof the a priori position vector of station i, xi

0, expressed in

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Global optimization of core station networks 35

ITRF2005, into its corresponding estimated position vectorxi

0 + δxi provided by any SLR data processing, is given by

δxi = t + d · xi0 + Rxi

0 (1)

with

δxi =⎛⎝

δxi

δyi

δzi

⎞⎠, t=

⎛⎝

txty

tz

⎞⎠, and R=

⎡⎣

0 εz −εy

−εz 0 εx

εy −εx 0

⎤⎦

The corresponding transformation between the daily a pri-ori EOP set eopj

0 (equal to the IERS 05 C04 EOP time series)

and the corresponding estimated set eopj0 + δeopj provided

by the considered SLR data processing is deduced from therotations as follows (Zhu and Mueller 1983):

δeopj =(

δx jp

δy jp

)and

{δx j

p = −εy

δy jp = −εx

(2)

The previous relations (1) and (2) can be matricially exp-ressed as:

δxi = Aiθ and δeopj = Djρ (3)

with

θT = (tx , ty, tz, d, εx , εy, εz

)and ρT = (

εx , εy)

and

Ai =⎡⎣

1 0 0 xi0 0 −zi

0 yi0

0 1 0 yi0 zi

0 0 −xi0

0 0 1 zi0 −yi

0 xi0 0

⎤⎦, Dj =

[0 −1

−1 0

]

From the relations (3), we can deduce the least-squares esti-mation of the vectors θ andρ, based respectively on a networkof n stations and on a vector of m EOP sets (in the current con-figuration, the station network corresponds to a given weeklyestimation and the EOP vector corresponds to the seven dailyestimations of the EOP sets over the considered week),{

θ = (AT�−1A

)−1 AT�−1δx

ρ = (DT�−1D

)−1 DT�−1δeop(4)

with

δx =⎛⎜⎝

δx1

...

δxn

⎞⎟⎠, A =

⎡⎢⎣

A1

...

An

⎤⎥⎦, var(δx) = �,

δeop =⎛⎜⎝

δeop1

...

δeopm

⎞⎟⎠, D =

⎡⎢⎣

D1

...

Dm

⎤⎥⎦, and var(δeop) = �

2.2.3 Reference system effects and minimum constraints

The RSE concept was introduced by Sillard and Boucher(2001) for TRF, namely sets of station positions andvelocities. The main goal of this concept is to identify the

geometrical invariants of any TF,2 with respect to any givenmeasurement set. This identification aims to add the onlynecessary constraints to the considered singular normal sys-tem, without spoiling the underlying physics. With the helpof these constraints, a solution is then estimated in the “free-network adjustment” framework. The formulation of RSE ismainly based on the S-transformations, first introduced byBaarda (1973). This statistical concept was generalized toEOP by Coulot et al. (2007), independently from any equiv-alent MC concept.

From a numerical point of view, to get these quantities,we need to compute loosely constrained solutions, i.e. to addloose constraints3 to the normal systems involved in orderto make them invertible. The ASI weekly solutions pres-ently used are loosely constrained solutions with a prioristandard deviations of 100 m for station positions and ofequivalent values for EOP (PM and LOD). The relations link-ing the RSE and the variance–covariance matrices � and �

of, respectively, the station positions and the EOP are (Sillardand Boucher 2001; Coulot et al. 2007):{

� = �inner + A.�θ .AT, �θ = (AT�−1A

)−1

� = �inner + D.�θ .DT,�θ = (DT�−1D

)−1 (5)

The matrices �θ and �θ are the RSE, respectively,deduced from the station positions and the EOP. The matri-ces A and D are defined in previous Eqs. (3) and (4). Goingback to Eq. (4), we see that these effects are just the variance–covariance matrices of the estimations of the transformationparameters between the loosely constrained and the a pri-ori solutions. The fuzzy defined global parameters (the geo-metrical invariants for the measurements) are then the onesevidencing large formal errors (root squares of the diagonalelements of the matrices �θ and �θ ). Moreover, the matrices�inner and�inner are commonly called the “inner noises” anddirectly correspond to the real quality of space geodesy tech-niques in providing station positions and EOP. Regarding thestation positions, this name directly comes from the work ofMeissel (1965) on inner error theory for geodetic networks.We finally must specify that the two decompositions (5) areunique.

According to Sillard and Boucher (2001), once the fuzzydefined global parameters are identified with the help ofRSE, there are several ways to access to the inner precision�inner (and thus to the minimum norm and maximum preci-sion free-network solution). One of these (quite equivalent)approaches is the application of MC. The MC equation canbe easily derived, thanks to the expression of the transforma-tion between ITRF2005, xi

0 for station i , and the estimated

2 More precisely, in the present study, one of the seven global parame-ters tx , ty, tz, d, εx , εy , and εz , see previous Subsect. 2.2.2.3 Equality constraints to zero for some parameters with large standarddeviations.

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36 D. Coulot et al.

positions xi0 + δxi. By restricting the Eq. (4) to the station

positions and the estimation of the three rotations (the fuzzydefined global parameters in the present case), we get:

θε =(

BT�−1B)−1

BT�−1δx (6)

with

θεT = (

εx , εy, εz), δx =

⎛⎜⎝

δx1

...

δxn

⎞⎟⎠, B =

⎡⎢⎣

B1

...

Bn

⎤⎥⎦,

Bi =⎡⎣

0 −zi0 yi

0zi

0 0 −xi0

−yi0 xi

0 0

⎤⎦, and var(δx) = �

The vector θε is an estimation of the three rotationsbetween the estimated TF and the a priori TRF. We can,for instance, fix θε to zero. In fact, it corresponds to aligningthe estimated TF with respect to ITRF2005 in orientation.The corresponding MC equation is, �mc being the variance–covariance matrix associated to the constraint,(

BT�−1B)−1

BT�−1δx = 0 (�mc) (7)

This equation is to be added, in its “normal” form, to thenormal system involved in the considered data processing.Doing so thus defines the three rotations of the TF underly-ing the estimated station positions and EOP (Altamimi et al.2002a).

Consequently, although the primary aim of MC is to find aminimum norm and maximum precision solution among aninfinite set of possible ones, these constraints can be twistedto define the orientation of a given TF underlying any esti-mated station positions and EOP. We use them in that wayin the present work. To conclude these theoretical consider-ations, we cite the following sentence, in relation with rela-tions (5) and the inner noise matrix, from Sillard and Boucher(2001): it may occur that the projection over some subset ofthe original set instead of the whole set of coordinates givesbetter results (i.e. smaller trace of the resulting variance–covariance matrix).

2.3 Influence of minimum constraints on EOP referencing

After providing some numerical examples related to RSE andMC, we demonstrate here the influence of the network overwhich MC are applied on the EOP referencing so obtained.

2.3.1 Numerical illustrations

Table 1 provides the RSE (translated into parameter formalerrors; it will always be the case in the following) deducedfrom the relations (5) for the loosely constrained solution(original ASI solution) of the GPS week 1,031. We remark

a slight effect for the tz translation, directly linked to thesparse distribution of the SLR network. The three rotationsclearly appear as the fuzzy defined global parameters of theTF. And the RSE values for the two first rotations are consis-tent for station positions and EOP. Furthermore, for this par-ticular week, the median value of the spherical errors4 of theestimated loosely constrained station positions is 270.7 mm.This is clearly not the precision of the SLR positioning.The median value of the formal errors of the estimated x p

(respectively yp) pole coordinates is 5,667 (respectively5,821) µas. All these numerical values indicate that the RSEdominate the variance–covariance matrix of the loosely con-strained solution. It is worth noting that the effects deducedfrom the EOP variance–covariance matrices are larger thanthose obtained for station positions.

We then compute a minimally constrained solution (solu-tion computed with the application of MC at the normalsystem level, after the withdrawal of the original loose con-straints) for the same GPS week 1,031. MC are applied inaccordance with relation (7) and over the ASI network(Fig. 1). Regarding the station positions, the median valueof the spherical errors is now 6.7 mm (to compare to the pre-vious 270.7 mm value). The median value of the estimatedx p (respectively yp) formal errors is now 132 (respectively102) µas. The application of MC clearly reduces the exist-ing RSE, as shown in Table 1. The values of the effects onthe three rotations are clearly lower than those provided inthe same Table for the loose constrained solution. And theeffects on the translations and the scale factor have been leftunchanged. Moreover, the RSE values are no more consistentfor station positions and EOP variance–covariance matrices.And, once again, the effects deduced from the EOP variance–covariance matrices are larger than those obtained for stationpositions. There are thus residual RSE in the EOP variance–covariance matrices of the solutions computed with MC. Thisfact was already evidenced by Coulot et al. (2007). Similarresults, regarding both loosely and minimally constrainedweekly solutions, are observed over the whole time periodof nearly 15 years.

2.3.2 Importance of the referencing network

The MC (7) aim not only to reduce the RSE evidenced for thethree rotations (and, as a consequence, to get realistic vari-ance–covariance matrices for station positions and EOP), butalso to align the orientation of the weekly TF with respect toITRF2005. As a consequence, and in accordance with Eq. (2),the estimated EOP time series are referenced in the referenceunderlying the IERS 05 C04 series. Figure 2 shows, overthe period 2000.0–2001.5, the differences, with the IERS

4 σsph =√

σ 2x + σ 2

y + σ 2z + 2σxy + 2σxz + 2σyz

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Global optimization of core station networks 37

Table 1 Reference system effects deduced from the variance–covariance matrices of the estimated station positions (lines “Stations”) and EOP(lines “EOP”)

Parameters

tx ty tz d εx εy εz

Loose constrained solution

Stations 0.5 0.5 1.5 0.1 5,594 5,553 9,685

EOP − − − − 5,768 5,640 −Solution with minimum constraints

Stations 0.5 0.5 1.5 0.1 21 17 7

EOP − − − − 90 117 −The effects are provided for a solution computed with loose constraints (original ASI solution) and for a solution computed with minimum constraints(applied to the three rotations, with respect to ITRF2005, over the ASI network, Fig. 1). Both solutions correspond to the GPS week 1,031. Onlyeffects regarding both first rotations can be deduced from the estimated EOP variance–covariance matrix. The effects are translated into parameterformal errors and are given in cm for the three translations tx , ty , and tz , in ppb for the scale factor offset d, and in µas for the three rotations εx , εy ,and εz

2000 2000.5 2001 2001.5 −3

−2

−1

0

1

2

3

4

5

6

7

x 10 4

Year

x p (

µ ) s a

Fig. 2 Differences between the x p pole coordinate daily series com-puted with loose constraints (original ASI solution, dashed line) andthe IERS 05 C04 series versus the similar differences for the x p dailyseries computed with minimum constraints (solid line), in 104 µas, overthe period 2000.0–2001.5. The minimum constraints are applied to thethree rotations, with respect to ITRF2005, over the ASI network, Fig. 1

05 C04 series, of the estimated x p pole coordinate dailyseries. These daily series are respectively provided by theloose constrained solution (the original ASI solution) andby the solution computed with MC applied over the ASI net-work. The differences for the loose constrained series appearto be successions of steps, these latter corresponding to theweeks of estimation (similar results are observed for the yp

pole coordinate). Over the whole time period, the weightedmean and standard deviation values of these difference seriesare, respectively, 53,432 and 129,451 µas. On the other hand,the minimally constrained solution is clearly more stable.Indeed, over the whole time period, the weighted mean andstandard deviation values of the corresponding differenceseries are, respectively, 3 and 324 µas.

Figure 3 shows, over the period 2000.0–2001.5, the differ-ences between the estimated yp pole coordinate daily seriesprovided by solutions computed with MC. These MC arerespectively applied over the ASI network (Fig. 1) and thenetwork recommended by the ILRS AWG for EOP refer-encing. As on Fig. 2 for the loose constrained series, thedifference time series are successions of weekly steps. Thedifference between both time series thus mainly comes fromthe referencing carried out each week of estimation (similarresults are observed for the x p pole coordinate). Over thewhole time period, the weighted mean and standard devia-tion values of these differences are, respectively, −43 and228 µas. It is worth noting that the Weighted Root MeanSquare (WRMS5) of the post fit PM residuals resulting fromthe ITRF2005 combination is approximately 130 µas for theILRS SLR combined solution (Altamimi et al. 2008).

The reference network used for the application of MCis thus of great importance for the referencing and, conse-quently, for the stability of the daily EOP series estimatedwith the weekly station position series. The application ofglobal optimization aims to improve this stability by pro-viding, each week, the optimal reference network for theapplication of MC.

Before going through this issue, we present hereafter, inthe next section, the results produced with three networks:the ASI, IGN, and ILRS networks. Both latter networks canbe used for EOP referencing. In the following, these resultswill be used as reference ones for comparison with thoseproduced with our algorithm.

5 Considering a random set (xi )1≤i≤n with associated standard devia-

tions (σi )1≤i≤n , the WMRS is defined as

√∑i

x2i

σ 2i

/√∑i

1σ 2

i.

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38 D. Coulot et al.

2000 2000.5 2001 2001.5−800

−600

−400

−200

0

200

400

600

800

Year

y p( sec

nereffid

µ)sa

Fig. 3 Differences between the yp pole coordinate daily solution com-puted with minimum constraints applied over the ASI network (Fig. 1)and the yp daily solution computed with minimum constraints appliedover the network recommended by the ILRS analysis working group forEOP referencing, in µas, over the period 2000.0–2001.5. The minimumconstraints are applied to the three rotations, with respect to ITRF2005

3 State-of-the-art: usual core station networks

This section presents the three ASI, ILRS, and IGN networksand the results produced with these networks regarding theEOP referencing and the station position time series.

3.1 Reference networks and quality criteria

The three reference networks used for carrying out somecomparisons with the networks provided by our algorithm

are called the ASI, ILRS, and IGN networks. They are respec-tively shown on Figs. 1, 4 and 5.

As previously mentioned in Sect. 2.1, the ASI networkis made up by all the stations available in the ASI solutioneach week. Consequently, it cannot really be considered as areference network designed to optimally apply MC. Never-theless, as it is the most obvious usable network for the MCapplication, we keep it for comparisons.

The IGN network is based on 16 stations. This network isnot particularly designed for EOP referencing but is routinelyused by the ITRF product centre, at the Institut Géographi-que National (IGN), to apply MC during any SLR solutionstacking with CATREF software (Altamimi Z, private com-munication, 2008).

Finally, the third network (the ILRS network) is based on21 stations. But, among these 21 stations, only 13 stationscan be considered over the whole time period (black dots onFig. 5), the other eight stations (green dots) being only usableover given time periods. This network was designed by theILRS AWG for EOP referencing and is currently named “theILRS list of core sites”. It can thus be considered as the cur-rent conventional core station network to use for SLR EOPreferencing. Rigorously speaking, the station 7810 (Zimmer-wald, Switzerland) can only be considered as a core stationregarding its measurements carried out in the blue wave-length. We do not work here directly at the observation level,and, as this station is one of the most effective of the ILRStracking network, we decided to consider this station as acore one over the period recommended by the ILRS AWG.

180°

180°

240°

240°

300°

300°

60°

60°

120°

120°

180°

180°

-80° -80°

-40° -40°

0° 0°

40° 40°

80° 80°

Fig. 4 Station network named “IGN network” in text. This network is based on 16 stations. Each week, it is made up by all the stations, takenfrom these 16, available in the ASI solution

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Global optimization of core station networks 39

180°

180°

240°

240°

300°

300°

60°

60°

120°

120°

180°

180°

-80° -80°

-40° -40°

0° 0°

40° 40°

80° 80°

Fig. 5 Station network named “ILRS network” in text. This networkis based on 21 stations among which only 13 stations (black dots) canbe considered over the whole time span. The other eight stations (green

dots) can only be considered over given periods of time. Each week, thisnetwork is made up by all the stations, taken from these 21, availablein the ASI solution

Both IGN and ILRS networks sound like fixed networksbut it is not the case. Indeed, each week, due to the dynamicnature of the SLR network, the corresponding networks arein fact made up by all the stations, taken from the fixed listsof stations, available in the ASI solution.

As the geographical distribution of the stations is obvi-ously a primary quality criterion of any reference network, wedevelop a distribution quality criterion. To assess the distri-bution of the stations composing a given network, we use thecoordinates of the Centre of Network (CN), i.e. the isobary-centre of the network, expressed in ITRF2005. The more min-imal the absolute values of these coordinates are, the morereliable is the distribution of the corresponding SLR network.This assertion should of course be moderated regarding densenetworks. In the present case, this is true only because morethan 80% of the SLR stations are located at mid-latitude posi-tions (see Fig. 1).

This criterion can be summed up by a unique quantity, thedistance from this CN to the origin of ITRF2005 dCN, which,for a network of n stations, is expressed as:

∀c ∈ {x, y, z}, c = 1

n

n∑i=1

ci , dCN =√

x2 + y2 + z2

(8)

xi , yi , zi designating the cartesian coordinates of the stationi in the ITRF2005 solution, computed at the 2000.0 refer-ence epoch. Table 2 gives the mean and standard deviation

values of the weekly numbers of stations and of the CN coor-dinates for the three reference networks. It is worth notingthat the numbers of stations are both lower and more sta-ble for the IGN and ILRS networks. Moreover, no networksimultaneously optimizes the three CN coordinates, aboveall the ASI network that minimizes none of them. The largemean values obtained for the CN z coordinates obviouslycome from the heterogeneous distribution of the SLR net-work between both hemispheres.

In order to test the stability of a given weekly network, overa long time period, we finally introduce the “mean presence”criterion defined as follows. We first define the presence of astation over the considered data time span by:

presence = occurrence in the ASI network

797× 100 (9)

For a given weekly network, made up of m stations, the meanpresence of the network is just the mean value of all individ-ual indicators presencei ,

1m

∑mi=1 presencei , expressed in %.

A high mean presence value thus shows that the consideredweekly network is made up by stations regularly present inthe ASI solution.

3.2 Results produced with the reference networks

As noticed in Gambis (2004), the precision of any productis directly linked to the stability or the reproductibility ofa given data set whereas the accuracy mainly corresponds

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40 D. Coulot et al.

Table 2 Weekly numbers of available stations and Centre of Network (CN) coordinates (mean ± standard deviation values) for the three referencenetworks ASI, IGN, and ILRS

Network

ASI IGN ILRS

Numbers of stations 19 ± 3 11 ± 2 12 ± 2

CN x coordinates 1.88 ± 0.58 1.25 ± 0.66 1.13 ± 0.50

CN y coordinates 0.36 ± 0.66 −0.07 ± 0.64 −0.27 ± 0.93

CN z coordinates 2.83 ± 0.42 3.13 ± 0.32 2.69 ± 0.38

The CN coordinates are expressed in 1,000 km and the corresponding minimum statistical values are evidenced in bold

to an external estimation of the quality of the consideredproduct with respect to the “true” value of this latter. Thus,biases, drifts, and WRMS with respect to reference series arecommonly used to assess the quality of EOP series (Gambis2004; Bizouard and Gambis 2009). The WRMS is particu-larly used as an accuracy or, more precisely, as an inaccuracyassessment quantity. For the present study, we decided to usethe WRMS of the differences between the computed seriesand the IERS 05 C04 series, which is dominated by the GPSPM. We consider it as a mixed precision/inaccuracy assess-ment quantity, the precision (with the meaning of stability)being more rigorously provided by the weighted standarddeviation.

The statistics of the results produced with the referencenetworks ASI, IGN, and ILRS, are provided in Table 3.For each of these three networks, weekly station positionsolutions are computed with the application of MC to thethree rotations, with respect to ITRF2005, over the consid-ered network. Daily EOP (PM and LOD) series are computedtogether with these station positions.

First, Table 3 provides the statistics of the results linkedto the EOP time series: the formal errors of these series,the residual RSE for both rotations εx and εy deduced fromthe second relation of Eq. (5), Sect. 2.2, the rotations esti-mated with respect to the IERS 05 C04 series according to thesecond relation of Eq. (4), and the bias, drift, weighted stan-dard deviation, and WRMS values of the differences betweenthese series and the IERS 05 C04 series.

According to all these results, the ILRS network appearsto be the best network for the application of MC. Indeed, itsuse induces the largest reduction of the residual RSE (firstlines of Table 3). Moreover, it provides the most stable EOPtime series, concerning both residual rotations with respect tothe IERS 05 C04 series and the weighted standard deviationsof the difference series (last lines of Table 3).

Regarding the bias values, we note a significant bias withrespect to the IERS 05 C04 series for the yp pole coordinatefor all the computed series. According to Gambis (2004), theEOP series provided by the different space geodetic tech-niques are heterogeneous, these inconsistencies being mainly

modeled by biases and drifts. Moreover, the same can be saidfor different solutions computed from the data of a given tech-nique, according to Table 8 of the article above-mentioned.Indeed, for the three SLR solutions studied in this article,the x p and yp biases range from −366 to 110 µas. The biasevidenced in our yp series is most probably due to any incon-sistency intrinsic to the ASI solution used here. And there isno reason to see it disappear, even if it seems that it canbe reduced, according to the results obtained with the ASInetwork. We do not consider neither the mean values of thedifferences of the EOP series (which are quite equivalent tothe weekly rotation mean values), nor the drift values, as realquality indicators.

Independently from the study conducted with EOP, Table 4provides the statistics of the results linked to the station posi-tion time series: the spherical errors of the series (previouslydefined in Sect. 2.3), the residual RSE of the three rotationsεx , εy , and εz , deduced from the first relation of Eq. (5), therotations estimated with respect to ITRF2005 according tothe first relation of Eq. (4), the CN distances dCN (8), andthe mean presences. Regarding the estimation of the threerotations, it must be specified that all the seven parametersare estimated but only the results regarding the three rota-tions are reported here, the results for the translations andthe scale factors being unchanged. Furthermore, no stationposition rejection process is carried out for these computa-tions.

From all the results concerning both the residual rotationRSE and the residual rotations with respect to ITRF2005, theASI network now appears as the best network. Indeed, it isthe network for which the RSE are the most reduced. It alsoprovides the most stable residual rotation series, accordingto their standard deviation values. It is worth noting that, inthis case, the residual rotation mean values are quality indi-cators; indeed, the MC consists in aligning the orientation ofthe weekly TF on the orientation of ITRF2005.

Regarding the mean presence indicator, the IGN networkappears as the most stable network over time. Indeed, this net-work is partly designed with this requirement (Altamimi Z,private communication, 2008). In the opposite, the ASI

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Global optimization of core station networks 41

Table 3 Statistics of the results produced with the differences between the daily EOP series, simultaneously estimated with the weekly stationposition series, and the IERS 05 C04 series

Reference network

ASI IGN ILRS

x p yp x p yp x p yp

EOP series formal errors

Median value 114 118 92 92 89 87

Bound valuea 396 492 262 294 237 255

EOP series differences

Biasb −2 88 −26 120 −17 120

Bias formal error 4 4 4 3 4 3

Drift 5 0 9 8 7 7

Drift formal error 1 1 1 1 1 1

Weighted standard deviation 324 301 294 257 287 246

WRMS 324 314 295 286 287 275

εx εy εx εy εx εy

Weekly rotation reference system effects

Median value 94 87 58 58 53 54

Bound valuea 427 346 235 182 194 170

Weekly rotationsc

Mean value −80 8 −124 30 −126 21

Standard deviation 336 317 250 238 216 221

The weekly solutions are computed with minimum constraints for the three rotations, applied with respect to ITRF2005, over the three referencenetworks ASI, IGN, and ILRS. All values are provided in µas or in µas/a (drift values). The minimum values are evidenced in bold for the quantitieschosen as quality indicatorsaValue val such as 95% of the values are in the interval [0, val]bThe bias value is simultaneously estimated with the drift value at the mid epoch of the whole time span (08/16/2000)cThe rotations are estimated with respect to the IERS 05 C04 time series, using the second relation of Eq. (4)

network, made up each week by all the available stations,evidences the lowest mean value for this criterion.

From all these results, we can conclude that the networkused for the application of the MC clearly influences theEOP weekly referencing and, as a consequence, the stabilityof the computed EOP series. Furthermore, it seems that thenetwork providing the best EOP referencing is not neces-sarily the best one for the station position referencing. Thiswill be discussed later on. The rest of this paper is dedicatedto the application of global optimization to ensure the bestEOP weekly referencing on the basis of the MC. The nextsection describes the approach used to achieve this optimalreferencing.

4 Global optimization

Global optimization methods, and, in particular, heuristics,have been recently applied to geodetic problems. Forinstance, Dare and Saleh (2000) applied optimal (for small

networks) and near-optimal (for larger networks) optimiza-tion methods, such as simulated annealing, for GPS networkdesign problems. Simulated annealing was also used for thefirst-order design of geodetic networks by Berné and Baselga(2004). In the present study, we use GA. Recently, Saleh andChelouah (2004) used GA for designing GPS surveying net-works, from a topometric and observational point of view.Baselga and García-Asenjo (2008) also applied GA to solvethe GNSS double difference positioning model with robustestimation.

GA are Evolutionary Algorithms (EA), i.e. stochasticalgorithms that emulate the evolution theory by using somegenetic operators such as chromosome selection, crossover(recombination), and gene mutation and the rule of survivalof the fittest in probabilistic terms. The main idea of the GAis to make subsets, called populations, of possible solutionsof a given optimization problem evolve in order to obtainthe global optimum for the problem. During the successivegenerations, the individuals of the population are evaluatedwith respect to their ability of optimizing the considered

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42 D. Coulot et al.

Table 4 Statistics of the results produced with the weekly station position time series

Reference network

ASI IGN ILRS

Weekly station position spherical errors

Median value 7 6 6

Bound valuea 56 52 51

Weekly CN distances dCN

Mean value 3.11 3.50 3.11

Standard deviation 0.47 0.33 0.42

Weekly mean presence

Mean value 58.7 76.3 71.3

Standard deviation 4.8 5.2 3.8

εx εy εz εx εy εz εx εy εz

Weekly rotation reference system effects

Median value 14 17 5 15 21 5 14 19 6

Bound valuea 29 34 10 33 37 11 41 36 12

Weekly rotationsb

Mean value 32 −19 3 55 −1 −1 50 4 2

Standard deviation 116 122 38 131 150 45 139 139 55

These series are computed, simultaneously with EOP series, with minimum constraints for the three rotations, applied with respect to ITRF2005,over the three reference networks ASI, IGN, and ILRS. All values concerning the three rotations are provided in µas. The spherical error statisticsare provided in mm, the statistics related to the Centre of Network (CN) distances dC N , Eq. (8), are provided in 1,000 km, and the statistics for themean presences are given in %. The optimum values are evidenced in boldaValue val such as 95% of the values are in the interval [0, val]bThe rotations are estimated with respect to ITRF2005, using the first relation of Eq. (4)

objective(s) and the best individuals are favored in order tomake better solutions emerge. Goldberg (1989) wrote thefirst book popularizing GA. A wealth of literature has beenpublished since and GA have become an active research field.

Here, we first justify the choice of the objectives to opti-mize. We then briefly present our algorithm and the approachwe retained for the choice of the final optimal solutionsamong the sets of non-dominated solutions provided by ourprocedure based on a Multi Objective GA (MOGA). Text1,electronic supplementary material, is fully dedicated to thepresentation of this procedure and of the major concepts onwhich it relies; it moreover provides a concise review ofmethods for handling preferences in MOGA (in other words,for choosing the final optimal solution) and related numericaltests.

4.1 Multiple objectives to optimize

We aim here at guaranteeing the best referencing of dailyEOP series produced together with weekly station positionsolutions. This referencing is achieved with MC restricted tothe three rotations of the weekly TF with respect toITRF2005. These MC are completely defined through the Bmatrix of Eq. (7) and, especially, through the station network

used for designing this latter. In order to assess the qualityof the derived EOP and, more particularly, their stability, wehave to choose objectives directly linked to this referencing.

As shown by the results provided in Tables 3 and 4, manyquantities depend on the network over which MC are applied:formal errors of the EOP and station position time series,standard deviations (scatters) of the weekly rotations deducedfrom the EOP estimates and of the EOP time series them-selves, as well as mean and standard deviation values of rota-tions deduced from the station positions estimates, etc. Thus,the choice spectrum is broad. But, in essence, the RSE are thebest statistical quantities to assess the MC handling. Indeed,they are global criteria, which furthermore assess the a prioriquality of the estimations of the seven Helmert parameters.And, finally, the MC are intrinsically based on these globalparameters.

We notice in Sect. 3 the dualism between the RSE deducedfrom EOP and from station position variance–covariancematrices. But, according to the results given in Table 3, thenetwork providing the minimum rotation residual RSEdeduced from the EOP variance–covariance matrices(namely, the ILRS network) is the one providing the most sta-ble EOP time series. Thus the rotation residual RSE [Eq. (5),second relation] deduced from the EOP variance–covariance

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Global optimization of core station networks 43

matrices (denoted σεeopx and σε

eopy , in the following) appear

as the best suited objectives regarding the EOP referencing.From a strictly geometrical point of view, we must also

keep the residual RSE on the third rotation [σεstaz , see Eq. (5),

first relation] provided by the station position variance-covariance matrices as a possible objective to optimize.Indeed, the SLR loose constrained solutions evidence resid-ual RSE for the three rotations (see Table 1) and not only forboth rotations εx and εy . And these solutions do not provideUT but its time derivative LOD, which is not directly linkedto the third rotation εz . Consequently, this third rotation resid-ual RSE, σεsta

z , is only accessible through the station positionvariance-covariance matrices.

We have three objectives to minimize: σεeopx , σε

eopy , and

σεstaz . When dealing with multiple objectives, a classical

approach, in the EA framework, consists in aggregating allthese objectives in one single scalarizing objective func-tion, such as a weighted sum of the objectives for example.But, serious drawbacks have been proven for these methodsand, regarding Multi Objective Optimization (MOO), the EAcommunity mostly favors the approaches relying on Paretodominance relation (Knowles and Corne 2004). Consideringa minimization problem with m objective functions fi , thisdominance relation � is defined by (Coello Coello 2005)

x � y ⇐⇒{∀i ∈ {1, . . . , m} fi (x) ≤ fi (y)

∃ j ∈ {1, . . . , m} / f j (x) < f j (y)(10)

Such an approach does not provide the user with a singleoptimal solution, but with a set of non-dominated solutions,called the Pareto optimal set (Konak et al. 2006). Moreover,this optimal set can help the user, not only to study the rela-tionships among the objectives, but also to gain insights aboutthe inherent structure of the problem at hand (Ulrich et al.2008). For all these reasons, we decided to design an algo-rithm on the basis of an existing MOGA and, consequently,to consider independently the three objectives σε

eopx , σε

eopy ,

and σεstaz . This moreover avoids the necessity of specifying

weights, normalizing objectives, etc., as noted by Knowlesand Corne (2004).

4.2 Algorithm used

Konak et al. (2006) provide an overview and a tutorial on GAspecifically developed for problems with multiple objectives,the so-called MOGA. These authors made an inventory of 13famous existing MOGA and, among them, we chose the spe-cific algorithm NSGA-II (Deb et al. 2002). Indeed, NSGA-IIis one of the most popular MOGA of the so-called secondgeneration, which has moreover become a landmark (CoelloCoello 2006).

Each week, on the basis of NSGA-II, our algorithmsearches, among all the possible networks usable to build theB matrix of Eq. (7) with at least three stations, those which

provide the best compromises between the three objectivesσε

eopx , σε

eopy , and σεsta

z . In order to improve the convergenceof our approach, our algorithm in fact relies, each week, ontwo runs of NSGA-II. The first run aims at providing a dis-tance boundary in the objective space. This distance is thenused in the second run for concentrating the search on themost interesting solutions, regarding EOP referencing. But,for the weekly solutions comprising a number of availablestations between seven and 11,6 the Pareto optimal sets aredirectly found among all possible solutions on the basis of(10). Thus, using GA is not necessary for those cases. For theother weeks, our algorithm based on NSGA-II is used, withsettings tuned with respect to the number of available stationsin the network. Finally, we use 15 particular test weeks (eachweek corresponding to a given number of stations between12 and 26) for carrying convergence tests to check the effi-ciency of our algorithm. Interested readers can refer to Text1,where all these points are detailed.

4.3 Choice of the optimal solution

As previously noted, our algorithm supplies a set of non-dominated solutions, the so-called Pareto optimal set. Thisthus raises the issue of the choice of the final optimal solu-tion in this Pareto optimal set. Paradoxally, little attention hasbeen paid to the decision making process required to selectthis optimal solution (Coello Coello 2000), even if the situa-tion seems to have recently evolved, at least regarding the apriori methods (Cvetkovic and Coello Coello 2004).

The Pareto optimal sets provided each week by both runsof our two-step procedure permitted us to carry out numeroustests regarding this issue of the final optimal solution selec-tion. Some of them are described in Text1 (in particular, seeTables 8, 9, 10, 11). We just report here the most importantresults.

First, choosing the optimal solution (in the Pareto optimalset) as the one optimizing one particular objective σε

eopx ,

σεeopy , or σεsta

z , proves the incommensurability of σεstaz with

both σεeopx and σε

eopy criteria. Indeed, the WRMS of the dif-

ferences between the so-obtained EOP and the IERS 05 C04series are 302 and 294 µas, respectively, for x p and yp, whenσεsta

z is the choice criterion, against 277 (respectively 266)and 243 (respectively 251) µas, when σε

eopx (respectively

σεeopy ) is the choice criterion.Second, the results are not drastically modified when the

optimal solution selection relies on the minimization of theCN distance dCN (8), the maximization of the number of sta-tions involved in the network, or the maximization of theweekly mean presence (9). This certainly proves that the

6 These particular solutions only represent 1.25% of the 797 consideredweeks.

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44 D. Coulot et al.

Table 5 Weekly numbers of available stations and Centre of Network (CN) coordinates (mean ± standard deviation values) for the geneticallymodified networks corresponding to the solutions GMNopt

eop, GMNneteop, and GMNopt

sta

Solution

GMNopteop GMNnet

eop GMNoptsta Best

Numbers of stations 8 ± 2 8 ± 2 6 ± 2 –

CN x coordinates 0.29 ± 0.54 0.47 ± 0.50 −0.04 ± 0.15 1.13 ± 0.50

CN y coordinates −0.03 ± 0.54 −0.21 ± 0.50 −0.02 ± 0.18 −0.07 ± 0.64

CN z coordinates 1.87 ± 0.89 2.18 ± 0.81 0.34 ± 0.57 2.69 ± 0.32

The CN coordinates are expressed in 1,000 km and the corresponding minimum statistical values, among the three GMN results, are evidenced inbold. The best values obtained with the three reference networks ASI, IGN, and ILRS (Table 2) are recalled in the last column for comparison

optimization of the chosen three objectives σεeopx , σε

eopy , and

σεstaz implicitly optimizes also these station related criteria.Finally, we tested the most commonly used a posteriori

approach, namely the weighted sum, one of the most recentapproaches, the evaluation line (Vergidis et al. 2008), and thefollowing intuitive geometric criterion,

dσ =√

σεeop 2x + σε

eop 2y + σεsta 2

z (11)

to select the final optimal solutions among the Pareto opti-mal sets. According to the simple preference method of deCaritat Condorcet (1785),7 the choice criterion providing thebest results (especially concerning the EOP referencing) isthe one relying on the dσ distance (11) in the objective space.This criterion is thus the one used in the following. The nextsection is dedicated to the results provided by our globaloptimization approach.

5 Results

This section presents the results provided by our procedure.We first detail the results obtained from three different GMNsolutions. Then, to test whether our algorithm would permitto deduce a mean core network usable for the MC applicationover a long period of time, we precisely study the networkshidden behind the GMN.

5.1 Weekly genetically modified networks

In addition to the 10 GMN solutions computed for this studyand reported in Text1, we computed two other solutions.We thus report here the results obtained with the three fol-lowing GMN solutions: GMNopt

eop, GMNneteop, and GMNopt

sta .

GMNopteop is the solution for which the distance dσ (11)

is used for the selection of the final optimal solutions.

7 Alternative A is favored over B if the number of criteria where A isbetter than B is greater then the number of criteria where B is betterthan A (Cvetkovic and Coello Coello 2004).

This choice criterion favors both objectives σεeopx and σε

eopy ,

compared to σεstaz . It thus aims to optimize the EOP

referencing.GMNnet

eop is based on the results of our algorithm with

respect to the three objectives σεeopx , σε

eopy , and σεsta

z , butwith an optimum solution choice criterion different fromthe one of solution GMNopt

eop. Here, for each solution of thePareto optimal set, we compute the dCN distance (8), the meanpresence, and the number of stations nbrsta. On the basisof these new three objectives and with the Pareto dominancerelation (10), we compute a new Pareto optimal set. Finally,among this new optimal subset, the optimal solution is chosenon the basis of the evaluation line approach (Vergidis et al.2008) carried out with respective weights of 1 for the oppo-site of the mean presence and 2 for both dCN and −nbrstavalues. This solution thus aims at making the most stableglobal core network emerge from the weekly GMN, withreasonable geographical distribution and number of stations.

Finally, GMNoptsta is equivalent to the solution GMNopt

eop, butbased on the three objectives σεsta

x , σεstay , and σεsta

z and theequivalent distance dσ computed with these latter objectives.This last solution aims at comparing the results obtained withthose produced by the GMNopt

eop solution.The statistics of the results provided by these three GMN

solutions are provided in Tables 5 and 6.

5.1.1 EOP indicators

All the indicators linked to EOP are improved with the useof the GMNopt

eop solution in comparison with the best valuesobtained with the three reference networks ASI, IGN, andILRS (see Tables 3, 4).

More precisely, using the GMN corresponding to this solu-tion induces a great improvement of the EOP indicators incomparison to those related to the ILRS network. Indeed, thegains are: 24 and 23% for the median values of the formalerrors of the x p and yp series, 43% for both rotations εx andεy regarding the median values of the weekly residual RSE,and 31 and 25% for the standard deviation values of both

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Global optimization of core station networks 45

Table 6 The upper part provides the statistics of the results producedwith the differences between the daily EOP series, simultaneously esti-mated with the weekly station position series, and the IERS 05 C04series. The weekly solutions are computed with minimum constraintsfor the three rotations, applied with respect to ITRF2005, over the threegenetically modified networks corresponding to the solutions GMNopt

eop,

GMNneteop, and GMNopt

sta . All values are provided in µas or in µas/a (driftvalues). The lower part provides the statistics of the results producedwith the weekly station position series. All values concerning the three

rotations are provided in µas. The spherical error statistics are providedin mm, the statistics related to the Centre of Network (CN) distancesdC N , Eq. (8), are provided in 1000 km, and the statistics for the meanpresences are given in %. The optimum values, among the three GMNresults, are evidenced in bold for the quantities chosen as quality indi-cators. The best values obtained with the three reference networks ASI,IGN, and ILRS (Tables 3 and 4) are recalled in the last column forcomparison

opteop

neteop

optsta

a

a

b

c

d

a Value val such as 95% of the values are in the interval [0, val]b Rotations estimated with respect to the IERS 05 C04 series, using the second relation of (4)c Bias simultaneously estimated with drift at the mid epoch ofthe whole time span (08/16/2000)d The values not evidenced in italics correspond to the values obtained with the ILRS network

opteop

neteop

optsta

e

e

f

e Value val such as 95% of the values are in the interval [0, val]f Rotations estimated with respect to ITRF2005, using the first relation of (4)

rotations εx and εy . Moreover, the WRMS of the differencesbetween the EOP series computed with the networks corre-sponding to the GMNopt

eop solution and the IERS 05 C04 seriesare respectively reduced by 8 (22 µas) and 10 (28 µas) % forx p and yp with respect to the WRMS of the series producedwith the ILRS network. These reductions of about 25 µascorrespond to 50% of the present IERS 05 C04 series accu-racy (Bizouard and Gambis 2009). Reductions of the samelevel are also noted for the standard deviation values of these

EOP differences. Finally, even if they are not really consid-ered as quality indicators, the bias values of the EOP serieswith respect to the IERS 05 C04 series are slightly reducedin comparison with those of the EOP series computed withthe ILRS network. They are contrariwise still larger than thebias values of the EOP series computed with the ASI network(Table 3). In light of these results, it is thus possible to reducethe influence of the network used to apply MC on the EOPreferencing so obtained.

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46 D. Coulot et al.

Regarding the GMNneteop solution, the results obtained are

consistent with those produced by the GMNopteop solution.

Indeed, even if they are slightly worse than these latter, theresults provided by this solution are still better than thoseobtained with the ILRS network.

On the contrary, a clear corruption of the results obtainedappears regarding the GMNopt

sta solution. Indeed, in compari-son with the values produced with the ILRS network, the val-ues obtained are increased by 20–30% for the median valueof the EOP formal errors, by 40–70% for the median value ofthe weekly RSE deduced from the EOP variance–covariancematrices, by 80–90% for the standard deviation values of theweekly rotation time series, and by nearly 20–30% for theWRMS of the differences between the computed EOP seriesand the IERS 05 C04 series.

Finally, concerning the drift values of the differencesbetween the EOP series computed with all the three GMNsolutions and the IERS 05 C04 series, they are at the max-imum level of six µas/a; this corresponds to a variation ofabout three mm at the Earth’s surface after 15 years. Thesevalues are thus negligible as are the values obtained with thethree reference networks ASI, IGN, and ILRS (see Table 3).

5.1.2 Station positions indicators

A great majority of indicators linked to the station positiontime series is improved with the use of the GMNopt

sta solutionin comparison with the best values obtained with the threereference networks ASI, IGN, and ILRS. It is particularly truefor the weekly RSE deduced from the station position var-iance–covariance matrices, the weekly estimated rotationswith respect to ITRF2005, and the mean values of the CNcoordinates. These latter indicators (as well as the dCN indica-tor in Table 6) show the importance of the network geometryfor the reduction of the RSE derived from the station positionvariance–covariance matrices. This geometry is clearly notessential for the reduction of the RSE deduced from the EOPvariance-covariance matrices, as shown in Table 5.

Comparing the three GMN solutions, it is moreover worthnoting that, even if the values obtained with both solutionsGMNopt

eop and GMNneteop are not the best achievable station

positions indicators, they are on the whole at the level orsometimes better than the best values produced with the threereference networks ASI, IGN, and ILRS. Regarding the meanpresence indicator, we note that the GMNnet

eop solution, espe-cially designed with this requirement, provides the maximummean value. Much attention is paid on this particular solutionin next Sect. 5.2.

Finally, regarding the weekly spherical errors, a quite sur-prising result appears: The networks which provide the bestresults regarding the RSE computed from the station positionvariance-covariance matrices are also the networks which

produce the worst results regarding the station position spher-ical errors in comparison with those provided by both GMNsolutions GMNopt

eop and GMNneteop. The same fact occurs with

the three reference networks (see ASI network vs. IGN andILRS networks in Table 4). A possible explanation is pre-sented in Section 2 of Text2. The GMNopt

sta solution choosesmore often non-core stations or at least stations not oftenpresent in the ASI solution with the consequence of glob-ally spoiling the spherical errors. This is not the case of bothsolutions GMNopt

eop and GMNneteop which more rely on core sta-

tions. Considerations about the stations selected by the GMNsolutions are provided in next Sect. 5.2.

All the results in Tables 5 and 6, concerning EOP and sta-tion positions indicators, show that, even if the primary roleof MC is to reduce the reference system noise contained in thestation position variance-covariance matrices, it is not possi-ble to only stick to these uncertainties to search for an optimalnetwork to reference EOP series. The same can be said forthe network geometry. Indeed, the global optimizations car-ried out on the basis of the three objectives σε

eopx , σε

eopy ,

and σεstaz , provide an optimal EOP referencing without dis-

torting the underlying weekly TF. Furthermore, they do notcorrupt any of the indicators linked to the station positions.On the contrary, an optimization approach only based on cri-teria deduced from the station position variance–covariancematrices clearly corrupts the EOP time series.

5.2 Toward long-term core station networks?

As such a network would be easier to use, at least from apurely operational point of view, we question here the feasi-bility of deducing, from the GMN, a mean core network forthe application of the MC. To reach this goal, we have thus tohighlight a mean network underlying the GMN, if this latterexists. We use two criteria that we cross to make such meannetworks emerge from the GMN: the presence criterion, pre-viously defined in Eq. (9), and the choice criterion definedby:

choice = occurrence in the GMN

occurrence in the ASI network× 100 (12)

5.2.1 Nature of the selected stations

On the basis of these criteria, Table 7 provides the mean net-works corresponding to each of the three GMN solutions andto given intervals regarding the choice and the presence val-ues. The stations corresponding to ILRS core stations (Fig. 5)are underlined in the lists provided in this Table.

Both solutions GMNopteop and GMNnet

eop clearly favor theseILRS core stations. It is a proof of the efficiency of ourapproach as these stations are classified as the best onesby the ILRS AWG. More precisely, the GMN correspond-

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Global optimization of core station networks 47

Table 7 Mean networks underlying the GMN evidenced with bothcriteria choice (12) and presence (9), for the three solutions GMNopt

eop,

GMNneteop, and GMNopt

sta

Solution

GMNopteop GMNnet

eop GMNoptsta

75% ≤ choice 7090 7090 7090

75% ≤ presence 7110 7110

7839

7840

75% ≤ choice 7210

50% ≤ presence < 75%

75% ≤ choice 7849 7403

25% ≤ presence < 50% 7501

7843

7849

50% ≤ choice < 75% 7501 7080

25% ≤ presence < 50% 7810 7105

7832 7210

7839 7810

7840 7832

7843 7843

8834 7849

8834

25% ≤ choice < 50% 7080 7403 7080

25% ≤ presence < 50% 7105 7501 7110

7403 7838 7210

7835 7837 7824

7837 7832

7939 7835

7941 7939

presence < 25% 7109 7109 7097

50% ≤ choice 7119 7119 7119

7122 7122 7122

7405 7406 7123

7406 7502 7124

7502 7520 7404

7520 7541 7405

7541 7825 7406

7825 7410

7883 7411

7502

7520

7525

7530

ing to the GMNopteop (respectively GMNnet

eop) solution makesuse of 90% (respectively 81%) of these ILRS core stations.The GMNopt

sta solution makes use of only 52% of the ILRScore stations. This substantiates again the idea of the optimi-

Table 7 continued

Solution

GMNopteop GMNnet

eop GMNoptsta

7545

7548

7825

7847

7882

7883

Stations always missing 1831 1831 1831

1864 1864 1885

1873 1873 7231

1884 1884 7295

1885 1885 7328

1893 1893 7335

1953 1953 7337

7123 7123 7339

7237 7237 7541

7249 7249 7805

7358 7404 7848

7404 7548

7410 7805

7548 7847

7805 7848

7847

7848

The systematically missing stations are also listed. The underlined sta-tion codes correspond to ILRS core stations (Fig. 5). For the lists ofthe stations always missing, the station codes evidenced in bold corre-spond to stations present during at least 25% of the time period in theASI network, Fig. 1

zation of mixed objectives and not only of objectives deducedfrom station position variance–covariance matrices. We notealso that, on average, the GMNnet

eop solution really promotes

the most present stations in comparison to the GMNopteop solu-

tion, even if the mean networks deduced from these two solu-tions are quite similar.

Furthermore, it is worth noting that a significant numberof stations is systematically missing in the GMN; some ofthem (evidenced in bold in Table 7) are yet present during atleast 25% of the time period. The number of these missingstations even reaches 22% of the total number of possiblestations for the GMNopt

eop solution.If we consider all the mean networks provided in Table 7

(including the three GMN solutions) and only keep the sta-tions present during at least 5% of the time period, our algo-rithm mostly chooses, by a majority, 26 stations. These lat-ter are listed in Table 1 of Text2. These 26 stations onlyrepresent 34% of the total number of possible stations, but57% of the total number of stations present during at least

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48 D. Coulot et al.

Table 8 Validity time periods for the 16 stations mostly used by thesolution GMNnet

eop (start date–end date)

Network

GMNneteop ILRS

7080 Period Period

7090 Period Period

7105 49011–54442 Period

7110 Period Period

7210 48983–53126 49384–53161

7403 49025–51873 49025–51873

7501 49172–49206 –

7501 51762–54309 51762–54554

7810 Period 50814–54491

7832 Period Period

7835 49032–52664 48997–53560

7837 49046–53448 50449–53476

7839 Period Period

7840 Period Period

7843 49095–51103 non ILRS core

7849 Period Period

8834 Period 52504–54561

The time periods are also provided for the ILRS network, when the con-sidered station belongs to this core network. Dates are given in modifiedJulian days. Regarding the ILRS network, the original validity time peri-ods are crossed with the presence time periods in the considered ASIsolution. Finally, the term “Period” indicates that the considered stationcan be used as a reference station over its whole presence time periodin the ASI solution

5% of the time period. Our algorithm thus clearly selects thestations used. And this selection is dynamical as illustratedby the intersection of the intervals “presence < 25%” and“50% ≤ choice” in Table 7. Indeed, a significant number ofnon-core stations is also selected in the GMN. The dynam-ical feature of the GMN obviously comes partly from theintrinsic dynamical nature of the SLR network (see Fig. 1;Sect. 2.1). But this does not fully explain this characteristicwhich appears to be a strength of the GMN. Indeed, whena stable core station is missing for a given week, the GMNprobably switch to another station which is not forcely a pri-ori classified as a core one.

5.2.2 Mean station core networks

Despite this dynamical nature of the GMN, it is possible, onthe basis of the five first categories in Table 7, to evidence amean core network for the dedicated solution GMNnet

eop. Thismean core network is listed in Table 8, together with theinherent validity periods in the GMN solution and the ILRSnetwork.

The case of the station 7249 (station of Beijing in China,DOMES number 21601S004, with a presence of 32.5%) is

interesting. Indeed, the ILRS AWG recommends to considerthis station as a core one before 1999.0. On the contrary,the GMN never use it. This station may not be well suitedfor EOP referencing. This was moreover pointed out duringthe ILRS AWG meeting of April 12, 2008. Indeed, as writ-ten in the corresponding minutes (Pavlis EC, private com-munication, 2008), Beijing site (7249) should be removedfrom the core site list. In the opposite, the station 7843 (sta-tion of Orroral in Australia, DOMES number 50103S007,with a presence of 29.1%, and which is no more active) isused by GMN. This station should probably be consideredas a core station. The case of both ILRS core stations 7939and 7941 (stations of Matera in Italy, respective DOMESnumbers 12734S001 and 12734S008, respective presencevalues of 34.9 and 30.5%, 7939 is no more active) ques-tions also. Indeed, these stations are not mainly usedin the GMNnet

eop solution, even if they are not alwaysmissing.

Other cases are noticeable. First, the three stations 7210(station of Haleakala in Hawaii, DOMES number40445M001, presence of 59.1%, no more active), 7810 (sta-tion of Zimmerwald in Switzerland, DOMES numbers14001S001 and 14001S007, presence of 65.5%), and 8834(station of Wettzell in Germany, DOMES number14201S018, presence of 84.8%) are used over their wholepresence periods by the GMN whereas their use is only rec-ommended over restricted time periods by the ILRS AWG.Second, the validity time periods deduced from the GMNare different than the ones established for the ILRS net-work regarding both stations 7501 (station of Hartebeesthoekin South Africa, DOMES number 30302M003, presence of39.3%) and 7835 (SLR station of Grasse, France, DOMESnumber 10002S001, presence of 60.2%). Finally, we cannotice that, for the station 7403 (station of Arequipa in Peru,DOMES number 42202M003, presence of 47.7%), theGMNnet

eop solution evidences the same validity time periodsthan the one recommended in ILRS network, the end datecorresponding to an Earthquake.

As a test, we merge the 16 stations of the mean networkcorresponding to the GMNnet

eop solution (especially designedto reach this particular goal) into a core network (as the IGNand ILRS ones), taking into account the respective valid-ity time periods deduced from the GMN. The differencesbetween the so-computed PM time series and the IERS 05C04 series provide WRMS of 285 and 267 µas, for x p and yp

respectively. These results are better than those provided bythe ILRS network (Table 3) but they are worse than resultsobtained with any of both solutions GMNopt

eop and GMNneteop

(Table 6). However, they prove the possibility of evidencing acore network providing a satisfactory EOP referencing fromthese weekly GMN. This core network moreover questionsthe ILRS network, regarding the stations chosen as well astheir validity time periods.

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Global optimization of core station networks 49

6 Discussion and conclusions

In this article, we propose a new convention for EOP ref-erencing, based on the weekly selection of station core net-works for MC application with respect to a given TRF. Thisnew approach leads to an improvement of 10% of the stabilityof the PM time series with respect to the IERS 05 C04 series,in comparison to the ILRS network which was especiallydesigned for EOP referencing. Indeed, the WRMS of thedifferences are reduced from 287 to 265 µas for x p, and from275 to 247 µas for yp. These reductions of about 20–25 µasrepresent 50% of the current assessed accuracy of the IERS05 C04 series (Bizouard and Gambis 2009). For some part,the accuracy of the PM series may be thus corrupted by ref-erencing problems and it is essential to ensure an optimal ref-erencing. Our method does so. Moreover, we obtained theseresults on the basis of rigorous criteria for the optimizationof core station selection. These rigorous criteria (namely, themixed RSE deduced from EOP and station position variance-covariance matrices) support the fundamental followingrequirement: although the considered problem falls into thefree-network adjustment framework, the best core networkfor EOP referencing (carried out on the basis of MC appli-cation) should be more designed on its adequacy to best rep-resent the real EOP, by preserving their consistency amongthe involved epochs, rather than on its adequacy to reducethe station positions uncertainties, and even on its geometry.On this latter point, it is yet worth noting that our optimizedcriteria do not distort the underlying weekly TF.

The dynamical nature of the weekly GMN appears as akey point of their success for the MC application. It was thusessential to work on a weekly basis to evidence this fact.This feature is another fundamental difference between ourmethod and the current approaches based on (more or lessempirical) mean core networks. Nevertheless, we designeda specific GMN solution to make such a mean core networkemerge. This latter, even if it does not provide results at thelevel of those produced with the weekly GMN, improvesthe EOP accuracy in comparison to the ILRS core network.Moreover, it questions this latter network regarding both sta-tions 7249 and 7843 and the validity time periods of otherones.

The multi objective approach, which is the heart of ouralgorithm, permitted us to show the incommensurabilitybetween the considered objectives, to test the possible influ-ence of geographical indicators (such as the mean presenceor the distance dCN), and more particularly to compute thespecific GMNnet

eop solution for producing a mean core net-work. Nevertheless, in light of the results provided by theGMNopt

eop solution, and of all the numerical tests reported inText1, Section 3, permitted by this multi objective approach,it seems that, for this particular application to the SLR tech-nique, a single objective approach (based on the distance

dσ , for instance) should provide optimal solutions. This wasconfirmed by numerical tests carried out for the 15 test GPSweeks we used for assessing our algorithm results.

Our approach is general and can consequently be usedfor other case studies. Regarding SLR technique, it wouldbe interesting to carry out a similar study, using other solu-tions or even the ILRS official combined solution, whichis currently under investigation for the next ITRF generation(ITRF2008) at the time of writing of this article. Similar stud-ies for the DORIS, VLBI, and GPS techniques would alsobe profitable to strengthen the present conclusions. To ourmind, we should also pay attention to the issue of designinga specific global optimization approach for the selection of amean fixed core network, valid over a given time period.

Finally, regarding the consistency of EOP and TRF or TF,Gambis (2004) concludes that there are presently two waysof improvement: using the ITRF stacking approach or car-rying out a combination of space geodetic techniques at themeasurement level. For this latter, that can be carried outon a weekly basis, we know that referencing problems arestill key issues (Coulot et al. 2007) and that the applicationof MC is essential. Such a study, based on global optimiza-tion, may solve some of these problems or, at least, guideus towards possible ways of improvement. For the stackingapproach, which must encompass all available solutions topreserve consistency among epochs and which also mainlyrelies on MC (Altamimi et al. 2007), it would probably bebeneficial to design a similar global optimization method.In this context, regarding the study carried out by Ray andAltamimi (2005), such an approach could also help for theevaluation (and even the selection) of the co-location ties.

Acknowledgments This study was funded by the Institut Géographi-que National (IGN), France. We also acknowledge the Centre Nationald’Études Spatiales (CNES), France, for recent financial support. We aregrateful to the International Laser Ranging Service (ILRS) (Pearlmanet al. 2002) and, more especially, to the analysis center ASI (Agen-zia Spaziale Italiane, Italian Spatial Agency, Italy) for the productionand the distribution of the solution used for this research. We thankthe professor K. Deb (Kanpur GA Laboratory, KanGAL, India) andhis colleagues for the design, the programming, the testing, and themaintenance of NSGA-II software. We have also a friendly thoughtfor our colleague H. Duquenne (IGN/LAREG, France) who found theso present French expression “Réseau Génétiquement Modifié” (thatwe have translated into “Genetically Modified Network” in English)after a preliminary presentation of this work at IGN/LAREG. Finally,we acknowledge the anonymous referees who reviewed the submittedand revised versions of this article: Their remarks and comments havehelped us to make this paper more precise and comprehensive.

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