On the adjustment of combined GPS/levelling/geoid networks

Transcription

1 Journal of Geodesy (1999) 73: 412±421 On the adjustment of combned GPS/levellng/geod networks C. Kotsaks, M. G. Sders Department of Geomatcs Engneerng, Unversty of Calgary, 2500 Unversty Drve N.W., Calgary, Alberta, Canada T2N 1N4 e-mal: Tel.: ; Fax: Receved: 9 September 1998 / Accepted: 8 June 1999 Abstract. A detaled treatment of adjustment problems n combned global postonng system (GPS)/levellng/ geod networks s gven. The two man types of `unknowns' n ths knd of mult-data 1D networks are usually the gravmetrc geod accuracy and a 2D spatal eld that descrbes all the datum/systematc dstortons among the avalable heght data sets. An accurate knowledge of the latter becomes especally mportant when we consder employng GPS technques for levellng purposes wth respect to a local vertcal datum. Two modellng alternatves for the correcton eld are presented, namely a pure determnstc parametrc model, and a hybrd determnstc and stochastc model. The concept of varance component estmaton s also proposed as an mportant statstcal tool for assessng the actual gravmetrc geod nose level and/or testng a pror determned geod error models. Fnally, conclusons are drawn and recommendatons for further study are suggested. Key words. Geod evaluaton á GPS heght transformaton á Mult-data 1D networks á Adjustment 1 Introducton The combned use of global postonng system (GPS), levellng, and geod heght nformaton has been a key procedure n varous geodetc applcatons. Although these three types of heght nformaton are consderably d erent n terms of physcal meanng, reference surface de nton/realzaton, observatonal methods, accuracy, etc., they should ful ll the smple geometrcal relatonshp (Heskanen and Mortz 1967) h H N ˆ 0 Correspondence to: C. Kotsaks 1 where h are ellpsodal heghts obtaned from GPS observatons, H are orthometrc heghts derved from levellng methods, and N are geod heghts computed from a geod model. In practce, Eq. (1) s never sats ed due to the followng: (1) random nose n the values of h, H, N; (2) datum nconsstences and other possble systematc dstortons n the three heght data sets (e.g. long-wavelength systematc errors n N, dstortons n the vertcal datum due to an overconstraned adjustment of the levellng network, devaton between gravmetrc geod and reference surface of the levellng datum, etc.); (3) varous geodynamc e ects (post-glacal rebound, land subsdence, plate deformaton near subducton zones, mean sea level rse, monument nstabltes); and (4) theoretcal approxmatons n the computaton of ether H or N (e.g. mproper or non-exstent terran/ densty modellng n the geod soluton, mproper evaluaton of Helmert's formula for orthometrc heghts usng normal gravty values nstead of actual surface gravty observatons, neglgence of the sea surface topography (SST) at the tde gauges, error-free assumpton for the tde gauge observatons, etc.). The statstcal behavour and modellng of the msclosures of Eq. (1), computed n a network of levelled GPS benchmarks, have been the subject of many studes whch are often consderably d erent n terms of ther research objectves. The followng s a non-exhaustve lst of some of these objectves. The references provded are just representatve and are not the only mportant ones. 1. Testng the performance of global sphercal harmonc models for the Earth's gravty eld (Internatonal Geod Servce, IGeS 1997). 2. Testng the performance of local/regonal gravmetrc geod models and ther assocated computatonal technques (Manvlle et al. 1992; Sders et al. 1992). 3. Development of ntermedate corrector surfaces for optmal heght transformaton between geod surface and levellng datum surface (Manvlle et al. 1997; Smth and Mlbert 1996). 4. Development of corrector surfaces for long-wavelength gravmetrc geod errors (De Brujne et al.

2 ), and general gravmetrc geod re nement strateges (Jang and Duquenne 1996). 5. Evaluaton of the achevable accuracy for `levellng by GPS' surveys (Forsberg and Madsen 1990). 6. Montorng, testng, and/or mprovng (strengthenng) of already-exstng vertcal datums (Hen 1986; Kearsley et al. 1993). The above lst can be further extended f we substtute n Eq. (1) the GPS heght h wth altmetrc observatons, and the orthometrc heght H wth the SST. A study for such marne applcatons s ncluded n De Brujne et al. (1997). In vew of the many d erent uses for such multdata 1D networks, the purpose of ths paper s to present some general adjustment and modellng schemes that can be employed for an optmal analyss of the msclosures of Eq. (1). In partcular, we are manly nterested n applcatons of the types (1), (2), or (3) from the prevous lst. In Sect. 2, a general overvew s gven for varous adjustment schemes that have already been appled n practce. Some general modellng consderatons and the ntal formulaton of our models are presented n Sect. 3, and the general adjustment model s fully developed n Sect. 4. Two d erent cases of ths model are examned n detal n Sects. 5 and 6, and nally some conclusons and recommendatons are gven. 2 Overvew of varous adjustment/modellng schemes A bref revew of varous adjustment and modellng schemes that have already been appled for the applcatons mentoned n the prevous secton wll be gven here. Some general aspects of adjustment problems wth combned heght data sets can be found n Pelzer (1986). 2.1 Geod evaluaton Most of the geod evaluaton studes, based on comparsons wth GPS/levellng data, make use of the followng basc model: h H N ˆ a T x v 2 where x s an n 1 vector of unknown parameters, a s an n 1 vector of known coe cents, and v denotes a resdual random nose term. The parametrc part a T x s supposed to descrbe all possble datum nconsstences and other systematc e ects n the data sets. In practce, for these studes, the usual four-parameter model s often used,.e. a T x ˆ x o x 1 cos u cos k x 2 cos u sn k x 3 sn u 3 and rarely ts ve-parameter extenson (see e.g. Duquenne et al. 1995). a T x ˆ x o x 1 cos u cos k x 2 cos u sn k x 3 sn u x 4 sn 2 u 4 has also been employed. Both Eqs. (3) and (4) correspond to the followng datum transformaton model for the geod undulaton N, whch s descrbed extensvely n Heskanen and Mortz (1967, Sect. 5±9): DN ˆ Da DX o cos u cos k DY o cos u sn k DZ o sn u adf sn 2 u 5 where DX o, DY o, and DZ o are the shft parameters between two `parallel' datums and Df, Da are the changes n attenng and sem-major axs of the correspondng ellpsods. In our case, the two d erent datums wll correspond to (1) the GPS datum and (2) the datum used for the development of the global sphercal harmonc model that supports the gravmetrc geod, and for the computaton of the gravty anomaly data Dg. The model of Eq. (2) s appled to all network ponts and a least-squares (LS) adjustment s performed to estmate the resduals v, whch are tradtonally taken as the nal external ndcaton of the geod accuracy. The man problem under ths approach s that the v terms wll contan a combned amount of GPS, levellng, and geod random error that needs to be separated nto ts ndvdual components for a more relable geod assessment. Furthermore, an optmal adjustment n a statstcal sense would requre the proper weghtng of the resduals, whch s hardly appled n practce. Fnally, the use of such oversmpl ed parametrc models as Eqs. (3) or (4), combned wth mproper weghtng of the resduals v, creates mportant problems n terms of the `separablty' of the varous random and systematc e ects between the two unknown components a T x and v. 2.2 Corrector surface for GPS/levellng The development of corrector surfaces ams bascally at provdng GPS users wth an optmal transformaton model between ellpsodal heghts h and orthometrc heghts H wth respect to a gven levellng datum. For a general dscusson regardng theoretcal and practcal aspects of ths problem, see Featherstone (1998). Two such developments have been reported n North Amerca, n the US by the Natonal Geodetc Survey (NGS; Smth and Mlbert 1996) and n Canada by the Geodetc Survey Dvson (GSD; Manvlle et al. 1997). Both studes followed a smlar methodology, usng ntally the basc model of Eq. (2) wth ts parametrc part gven by Eq. (3). The obtaned adjusted values for the resduals v were then spatally modelled n a grd form usng an nterpolaton procedure. In the GSD study a mnmum-curvature nterpolaton algorthm was used, whereas the NGS tted an sotropc Gaussan covarance functon to the statstcs of the rregularly dstrbuted values v and then used smple collocaton formulas for the grddng. From the combnaton of the grdded values for the resduals and the adjusted values for the parameters x, a corrector surface to the gravmetrc geod was nally computed. Some comments regardng the `drawbacks' of these modellng approaches wll be gven later on n the present paper.

3 Gravmetrc geod re nement In De Brujne et al. (1997) a 28-parameter surface model was estmated to correct the gravmetrcally derved geod n the North Sea area for ts long-wavelength errors. TOPEX altmetrc data (h) and gravmetrc geod heghts (N) were only used n the general observaton equaton, Eq. (2), snce the SST was neglected n ths study. The parametrc model a T x was comprsed of a smple blnear part wth four parameters (one bas, two tlts, one torson), and a more complcated part of trgonometrc polynomals wth 24 coe cents. For the optmal estmaton of ths correcton model, only the external altmetrc data were properly weghted, accordng to ther precomputed standard devatons. Extensve statstcal testng was also appled to valdate the nal adjustment results. For the re nement of land gravmetrc geod models usng GPS/levellng data, Jang and Duquenne (1996) proposed the dvson of the entre test area nto smaller adjacent networks, n order to better model the hgher frequency geod dstortons due to the nsu cent local gravty data coverage and the errors n the used Dgtal Terran Model (DTM). 2.4 Vertcal datum testng/re nement For such applcatons the analyzed network usually contans a combnaton of some, or all, of the followng data: (1) relatve DH from conventonal levellng; (2) relatve Dh from local GPS surveys; (3) N or DN from a geod model; (4) absolute H and SST values at tde gauge statons; and (5) absolute h from Satellte Laser Rangng (SLR) or global GPS campagns. The above data con guraton was proposed by Kearsley et al. (1993). In ther extensve study, nvestgatng the qualty of sample subsets of the Australan Heght Datum (AHD), they used the followng general mathematcal model for known observatons and unknown parameters: Dh j ˆ h j h v Dhj DH j ˆ H j H v DHj DN j ˆ N j N v DNj h j H j N j ˆ 0 6a 6b 6c 6d All avalable observatons Dh j ; DH j ; DN j, along wth ther a pror accuracy estmates, were smultaneously adjusted, usng Eq. (6d) as a geometrcal constrant for the unknown parameters at each staton pont j. For the unknown parameters, addtonal a pror nformaton can also be ncorporated n the adjustment algorthm (Bayesan estmaton), n the form of ndependent pont measurements wth ther assocated varances and possble co-varances (e.g. measurement of H at tde gauge stes). The above methodology suggests a powerful adjustment tool that can be used for vertcal datum re nement/rede nton, where both geometrcal (GPS, SLR) and physcal (levellng, geod, mean sea level) quanttes are optmally combned n a un ed fashon (see also Vancek 1991). Among the crtcal ssues exstng n ths approach (as well as n the prevously overvewed applcatons) s the estmaton of the a pror covarance matrces for the d erent data sets. Snce these types of weghtng measures are only used to descrbe the behavour of the random errors n the measurements, some augmentaton of the observaton equatons, Eqs. (6), by addtonal auxlary parametrc models, descrbng possble systematc/datum o sets n the avalable data sets, should also be consdered. 3 General modellng consderatons In general, Eq. (1) does not hold exactly, due not only to the presence of zero-mean random errors n the heght data, but also due to a number of other drect or ndrect systematc e ects. Snce there are not usually avalable a pror correctons for many of these e ects, they should be ndvdually modelled and estmated durng an adjustment process. In ths way, the followng three general equatons can be wrtten for each pont P n a combned GPS/levellng/geod (GLG) network: h ˆ h a f h v h ; H ˆ H a f H v H ; N ˆ N a f N v N 7 where h, H, and N denote the avalable `observed' values for the GPS, orthometrc and geod heghts, respectvely. The superscrpt a denotes true values wth respect to a un ed geodetc datum, such that the followng equaton holds: h a H a N a ˆ 0 8 The f terms correspond to all the necessary reductons that need to be appled to the orgnal data n order to elmnate the datum nconsstences and other systematc errors. Fnally, the v terms descrbe zero-mean random errors, for whch a second-order stochastc model s avalable: E v h v T h ˆ Ch ; E v H v T H ˆ CH ; E v N v T N ˆ CN 9 For the orthometrc heghts, the covarance (CV) matrx C H s known from the adjustment of the levellng network. In the same way, C h can be computed from the adjustment of the GPS surveys performed at the levelled benchmarks. In the gravmetrc geod case, the covarance matrx C N s computed by smple error propagaton from the orgnal nosy data used n the geod soluton; for detaled formulas, see L and Sders (1994). For a more realstc stochastc error model, full knowledge of the CV matrces should not be assumed. Ths s especally true for the geod heghts where the often vaguely known nose level of the nput data (geopotental model coe cents, gravty, DTM), and the always necessary statonary nose assumpton when fast spectral technques are employed for the computatons, may cause C N to devate consderably from realty. Hence,

4 415 we wll adopt the followng stochastc model for the random nose e ects n the three heght data sets: E v h v T h ˆ r 2 h Q h ; E v H v T H ˆ r 2 H Q H ; 10 E v N v T N ˆ r 2 N Q N where the cofactor matrces Q h, Q H, and Q N are assumed known from the sources prevously ndcated, and the three varance components are treated as unknown parameters controllng the valdty of the a pror random error models. One could also extend the above stochastc model a lttle bt more, by decomposng the covarance matrx C N nto two d erent CV matrces wth assocated unknown varance components, whch would correspond to the two man geod random error sources (nosy geopotental coe cents, nosy gravty anomaly data). In ths paper, the set of observaton Eqs. (7) and ther assocated stochastc model n Eq. (10) represent the basc framework upon whch all the dervatons n the followng sectons wll be based. 4 A general adjustment model Let us assume that, at each pont P of a test network wth m ponts, we have a trplet of heght observatons (h, H, N ), or equvalently one `synthetc' observaton l ˆ h H N. By combnng Eqs. (7) and (8), we obtan the followng observaton equaton for each network pont: l ˆ h H N ˆ f h f H or, n a more compact form f N v h v H v N 11 l ˆ f v h v H v N 12 If the man objectve for usng such a test network s to evaluate the gravmetrc geod accuracy, then we are naturally nterested n the estmaton of the v N terms. Snce there s a stochastc model [Eq. (10)] that has been assocated wth these terms, the values of v N are supposed to re ect all the geod random error sources that were taken nto account for the computaton of the CV matrx Q N. Furthermore, the ablty to estmate the unknown parameter r 2 N accordng to some varance component estmaton algorthm (see e.g. Rao 1971, 1997), provdes probably the most powerful statstcal tool for a relable estmate of the actual geod nose, and a useful means of testng all the assumptons that were ncorporated n the constructon of the prelmnary geod error model Q N. There s stll, however, an amount of geod error whch s not ncluded n the v N terms, and for whch no a pror nformaton s avalable n general: alasng e ects, mproper (or omtted) terran and densty modellng, varous bases n the coe cents of the geopotental model, etc. Such geod errors, whch do not follow a zero-mean random behavour, wll be absorbed n the f correcton term along wth many other systematc e ects n the GPS and levellng data. In the absence of any pror statstcal and/or determnstc nformaton for these error sources, lterng them out and estmatng ther magntude ndvdually s mpossble. If, on the other hand, ths test network s to be used for the determnaton of an optmal corrector surface for future GPS/levellng applcatons, then the values f have to be estmated and spatally modelled n the best possble way. The random nose terms v h, vh, v N should be left out of the modellng for such a correcton surface. Ths can be easly realzed by lookng at the form of the basc observaton equaton n a future orthometrc heght network whch wll utlze GPS/geod nformaton, as well as the computed corrector surface from our orgnal test network,.e. h N c ˆ H v 13 where the term c represents the reducton e ect of the computed correcton surface model. A system of equatons, created by takng the d erences of Eq. (13) between the GPS survey ponts, has now to be adjusted for the optmal estmaton of the orthometrc heght d erences DH wth respect to the local levellng datum. Correctng, pror to ths adjustment, the GPS/levellng observatons for ther random nose e ects [whch s the case f the terms v h and v N from Eq. (12) are ncluded n the modellng of the corrector surface term c] makes no sense statstcally. Furthermore, f the resdual values v H from Eq. (12) are ncluded n the modellng of the corrector surface, then the avalable orgnal observatons n Eq. (13) wll be `corrected' for an error source whch does not even exst n them! Let us now return to our ntal observaton model of Eq. (12). The correcton term f ˆ f P represents a 2D spatal eld of values, and t can be further decomposed n the general form f ˆ a T x s 14 where a s an (n 1) vector of known coe cents, and x s an (n 1) vector of unknown determnstc parameters. The term s denotes some `resdual correcton', the nature of whch (determnstc or stochastc) s left unspec ed for now. The nal observaton equaton for each pont n the test network wll therefore have the followng form: l ˆ a T x s v h v H v N 15 and by usng matrx notaton n order to combne all the network ponts, we obtan l ˆ Ax s Bv where 16 l ˆ l 1 l l m Š T ; s ˆ s 1 s s m Š T ; 17a v ˆ v T h v T H v T T N T; v # ˆ v # 1 v # v # m # : h; H; N 17b

5 416 preselected determnstc parametrc form. In order to avod any rank de cency problems, the total number of selected parameters should be always smaller than the number of network ponts. In ths case, the adjustment model of Fg. 1 wll be reduced to the form Fg. 1. A general model for GPS/levellng/geod network adjustment A ˆ a 1 a a m Š T ; 17c B ˆ I m I m I m Š; I m : m m unt matrx Ths nal adjustment model s summarzed n Fg. 1. The assocated stochastc model follows from the one ntroduced n Eq. (10). Such adjustment problems where, apart from the unknown determnstc parameters x and the zero-mean random errors v, there appear also some quanttes s that depend on an underlyng unknown functon (the corrector surface n our case), are very common n geodetc applcatons. When the emphass s placed on the estmaton of the functonals s, t s tradtonally called a LS collocaton problem wth unknown parameters (Mortz 1980). In the case where the man nterest s on the parameters x, t s vewed as a smple LS adjustment problem `n the presence of sgnals' (Dermans 1978, 1984). Both approaches are of course equvalent, wth an mmedate relaton to the classc mxed lnear models of statstcal theory (see e.g. Koch 1987). The crucal pont for the soluton of the adjustment model n Fg. 1 s how to treat the sgnals s. In a rst smple determnstc approach these sgnals can be treated just as addtonal dscrete unknown parameters, and ther mplct relaton wth the underlyng unknown functon s completely gnored (see e.g. Dermans 1984). Ths approach, however, s not applcable to our spec c case of Eq. (16), because the resultng matrx of the normal equatons, under the mnmzaton prncple v T Pv ˆ v T h Q 1 h v h v T H Q 1 H v H v T N Q 1 N v N ˆ mn 18 wth the weght matrx beng 2 3 Q 1 h 0 0 P ˆ 4 0 Q 1 H Q 1 N wll always be sngular. In order to obtan a unque soluton, therefore, some addtonal constrants need to be mposed on the resdual systematc correctons s. Two d erent cases wll now be dent ed for applyng these necessary constrants. 5 A purely determnstc approach One easy way to solve the general adjustment model n Fg. 1 s to neglect the presence of the resdual correcton sgnals s. Essentally, ths means that the corrector surface wll be exclusvely modelled by a l ˆ Ax Bv 20 where A s some approprate desgn matrx wth full column rank. The nal soluton of Eq. (20), under the mnmzaton prncple of Eq. (18), wll be gven by the equatons W ˆ I m A A T Q h Q H Q N 1 A 1 A T Q h Q H Q N 1 ^x ˆ A T Q h Q H Q N 1 A 1 A T Q h Q H Q N 1 l ^v h ˆ Q h Q h Q H Q N 1 Wl ^v H ˆ Q H Q h Q H Q N 1 Wl ^v N ˆ Q N Q h Q H Q N 1 Wl ^v total ˆ B^v ˆ ^v h ^v H ^v N ˆ Wl 21a 21b 21c 21d 21e 21f In the case where a full CV matrx for the heght data nose s not avalable, but only some gross estmates for the mean heght accuracy are known, a much smpler verson of the above equatons occurs. If we denote by q 2 h, q2 H, and q2 N the a pror unform accuracy estmates for the ellpsodal, orthometrc, and geod heghts, respectvely, then we obtan the followng soluton: W ˆ I m A A T 1A T A 22a ^x ˆ A T 1A A T l 22b q 2 h ^v h ˆ q 2 h q2 H Wl q2 N q 2 H ^v H ˆ q 2 h q2 H Wl q2 N q 2 N 22c 22d ^v N ˆ q 2 h q2 H Wl 22e q2 N From the last three equatons [also from Eqs. (21c)± (21e)] the crucal role of the stochastc model for the random nose n the heght data s obvous. It o ers the means of applyng an optmal lterng to the total resduals B^v ˆ Wl of the adjustment by separatng the nose comng from each ndvdual heght component. It s rather nterestng, though a hghly unrealstc case,

6 417 that when statonary whte nose has been assumed for all heght data types, the estmates for the unknown parameters ^x and the total resduals B^v wll not depend at all on the three d erent nose levels q 2 h, q2 H, and q2 N. By applyng covarance propagaton to the above results, the CV matrx C^x of the adjusted model parameters can be also determned, whch should always be used to evaluate the qualty of the parametrc corrector surface for future GPS/levellng applcatons. Another useful matrx s the cross-cv matrx between the adjusted model parameters and the adjusted resduals for the varous heght data sets, from whch mportant nformaton can be extracted regardng the correlaton of the corrector surface wth the avalable data. The relablty of the prevous results depends on (1) the sutablty of the parametrc model Ax to descrbe e ectvely all the systematc e ects n the heght data sets, and (2) the correctness of the stochastc model for the observatonal nose (Q h, Q H, Q N ). It s therefore necessary to estmate also the three unknown varance components (see Fg. 1). The method of varance component estmaton tradtonally used n geodesy s Rao's Mnmum Norm Quadratc Unbased Estmaton ± MINQUE (Rao 1971). In the geodetc lterature ths problem has been solved ndependently, for a varety of adjustment models, by many researchers; an extensve revew wth further references to the relevant lterature s gven n Grafarend (1985). The followng algorthm follows the MINQUE crteron and computes optmal estmates for the unknown varance components of the ellpsodal heghts (^r 2 h ), orthometrc heghts (^r2 H ), and geod heghts (^r 2 N ): ^r ˆ J 1 k 23a ^r ˆ ^r 2 h ^r 2 H ^r 2 T N 23b k ˆ ^v T Q 1 ^v ; j : h; H; N 23c J j ˆ tr Q h Q H Q N 1 WQ Q h Q H Q N 1 WQ j 23d There are occasons, however, where the use of algorthm (23) may lead to negatve estmates for the unknown varance components. In such cases, a mod caton of the MINQUE method s requred (see e.g. Sjoberg 1984; Rao 1997). A number of statstcal tests and subsequent teratons are nally needed n order to valdate the adjustment results. An overvew of the whole adjustment procedure descrbed n ths secton s gven n the owchart of Fg A `collocaton' approach The man dsadvantage of the prevous adjustment approach s the d culty of ndng a good parametrc model Ax to descrbe all the possble systematc nconsstences n the heght data sets (too many e ects to model). Ths, n turn, causes problems wth respect to how relable the results would be for the GPS/levellng/ geod nose resduals and for the correspondng varance components. The use of classcal statstcal testng may help to dentfy, to some degree, the weakness of a spec c parametrc model, but t cannot provde the means for model mprovement. It should be noted that a good parametrc model does not necessarly mply small values for the estmated resduals B^v, snce the nose level n the orgnal heght data (Q h, Q H, Q N ) may allow relatvely large values. It s the accuracy C^x of the adjusted model parameters that should determne how good a model s, and how e ectvely t can be further used n future GPS/levellng applcatons. Fg. 2. Flowchart for the adjustment procedure n the determnstc approach

7 418 Although the parameterzaton of the dstorton effects n combned GLG networks s a very nterestng topc on ts own, t may be more approprate not to try puttng all the systematc errors n a preselected parametrc form. For small networks, n partcular, ths should be the gudng rule, snce n such cases only a small number of unknown determnstc parameters can be ntroduced n order to keep the degrees of freedom relatvely large and the relablty of the adjustment results relatvely hgh. Such a `compact' model s of course unable to fully descrbe the complexty of the varous systematc e ects and t should be accompaned by addtonal resdual correctons, whch were prevously ntroduced n the form of sgnals s [see Eq. (16)]. The soluton of the general adjustment model n Fg. 1 requres the ncorporaton of the sgnals s n the mnmzaton prncple, whch now takes the form s T Q 1 s s v T h Q 1 h v h v T H Q 1 H v H v T N Q 1 N v N ˆ mn 24 wth Q 1 s beng an approprate weght matrx for the unknown correcton sgnals. Although the soluton obtaned by usng Eq. (24) does not necessarly have to admt a stochastc nterpretaton for the sgnal part, t s useful to consder the sgnals as addtonal stochastc parameters, just lke the zero-mean random errors v; an excellent dscusson on ths aspect can be found n Dermans (1984). The stochastcty of s s actually necessary n the case where statstcal tests related to the valdty of ther weght (covarance) model Q s are to be appled. One of the man d cultes n ths approach s that the mean value m s ˆ Efsg of the stochastc sgnals wll not necessarly be zero, due to the systematc behavour that s supposed to exst n ther values. As a result, m s should appear n the nal estmaton formulas f we are seekng unbased estmators (.e. equvalence between the LS prncple of Eq. (24) and the best lnear unbased estmaton (BLUE) for E{s} 6ˆ 0; for detaled formulas see e.g. Dermans 1987). In order to avod such computatonally useless estmates, we can ntally solve the system l ˆ Ax s Bv usng Eq. (24) wth a unt sgnal weght matrx. The ntal soluton for the sgnal part, W ˆ I m A A T Q h Q H Q N I m 1 A 1 A T Q h Q H Q N I m 1 ^s nt ˆ Q h Q H Q N I m 1 Wl 25a 25b can be vewed as the `smoothest' resdual correcton eld that best ts the avalable observatons l, the selected parametrc model Ax, and the assocated stochastc model for the random nose e ects (Q h, Q H, Q N ). Now, we can easly compute the overall trend n the sgnals s by ttng some smooth surface to the ^s nt values. If we evaluate ths tted surface at the test network ponts, we obtan n general some values ^m s 6ˆ ^s nt. We can then create the followng `reduced' observatons and sgnals: l r ˆ l ^m s s r ˆ s ^m s 26a 26b It s now safe to assume that the reduced sgnals s r have zero mean,.e. E{s r }=0. Furthermore, the numercal values ^s nt ^m s can be used for an emprcal determnaton of a covarance functon model descrbng the average spatal behavour of the reduced sgnals s r. In ths way, we can repeat the adjustment of the model n Fg. 1, usng a new `mproved' verson for the stochastc model of the correcton sgnals: l r ˆ Ax s r Bv Efs r g ˆ 0; 27a E s r s T r ˆ Csr ˆ r 2 s r Q sr 27b The elements of the cofactor matrx Q sr are computed from the emprcal CV model estmated at the prevous step. An unknown varance component has been also ntroduced n order to dagnose any problems related to the valdty of the emprcal sgnal covarance functon. The soluton of the adjustment model n Eq. (27) wll be gven by the followng unbased estmators: W ˆ I m A A T Q h Q H Q N Q sr 1 A 1 A T Q h Q H Q N Q sr 1 ^x ˆ A T Q h Q H Q N Q sr 1 A 1 A T Q h Q H Q N Q sr 1 l r ^v h ˆ Q h Q h Q H Q N Q sr 1 Wl r ^v H ˆ Q H Q h Q H Q N Q sr 1 Wl r ^v N ˆ Q N Q h Q H Q N Q sr 1 Wl r ^s r ˆ Q s Q h Q H Q N Q sr 1 Wl r 28a 28b 28c 28d 28e 28f The nal soluton equatons are smlar to the ones obtaned under the determnstc approach, wth the only d erences beng (1) the use of reduced observatons l r nstead of the orgnal l, and (2) the ncorporaton of the sgnal covarance matrx Q sr. In the specal case of statonary whte nose n all three heght data sets, no sgn cant smpl caton of the above formulas occurs due to the appearance of the matrx Q sr. The estmaton of the four unknown varance components ^r 2 h, ^r2 H, ^r2 N, ^r2 s r follows a straghtforward extenson of the MINQUE algorthm of Eq. (23) and t s omtted. As t was mentoned n Sect. 5, varous statstcal tests and teratve solutons can be performed n order to - nally valdate the adjustment results. For statstcal testng procedures n extended adjustment models wth sgnals, see Dermans and Rosskopoulos (1991) and the references gven theren. In any case, a complete answer

8 419 for the estmated corrector surface should nclude: (1) the estmated parametrc model A^x, (2) the parameters descrbng the non-zero mean sgnal trend ( ^m s are just the values of ths trend at the test network ponts); (3) the estmated values for the reduced zero-mean sgnals ^s r at the network ponts; and (4) a covarance model for the zero-mean sgnals s r. A combned use of (3) and (4), n a collocaton-type predcton formula, s requred for the nterpolaton of the zero-mean part of the correcton sgnal at other non-levelled ponts. A general owchart for the whole computatonal procedure descrbed n ths secton s gven n Fg. 3a, b. 7 Summary and conclusons The use of combned GPS/levellng/geod networks de ntely provdes a very attractve evaluaton scheme for the accuracy of gravmetrc geod models. At the same tme, GLG networks consttute the skeleton of `common ponts' n the attempt to nd optmal transformaton models between GPS and orthometrc heghts. These are two d erent problems whch, nevertheless, can be attacked smultaneously through a un ed adjustment settng. As far as the geod evaluaton problem s concerned, a GLG network adjustment can essentally be used for testng the relablty of prelmnary geod error models, whch have been derved va nternal error propagaton from the source data and ther nose used n the gravmetrc soluton. Varance component estmaton has been proposed as a useful statstcal tool for computng and testng the actual geod nose level. The mportant role played by the stochastc nose model of the other two heght components was also demonstrated through the derved lterng equatons for the total nose resduals of the adjustment. Ths general approach also allows us to check ndvdually varous addtve geod error models (see comment at the end of Sect. 3). In the absence of any pror geod error model, we can stll use a unt weght matrx and obtan an estmate for the a posteror unt weght varance for the geod nose. For the problem of modellng a corrector surface to the gravmetrc geod for GPS-to-orthometrc heght transformaton, t s mportant to lter out all the zeromean random nose e ects comng from the trplet of the Fg. 3. a Flowchart for the adjustment procedure n the `collocaton' approach. b Flowchart for the adjustment procedure n the `collocaton' approach (contnued)

9 420 Fg. 3b heght data. Ths s the man weakness of some of the presently avalable attempts for such a modellng. Ths correcton surface wll also absorb a part of the geod long-wavelength error whch does not necessarly follow a zero-mean random behavour, and whch s not generally possble to be explctly solated. Two modellng alternatves have been presented for the descrpton of the systematc correcton eld. These are (1) purely dscrete determnstc modellng and (2) hybrd determnstc and `stochastc' modellng. Agan, the tool of varance component estmaton provdes the statstcal means to test the admssblty of the correcton eld's CV model when (2) s employed. The problem of statstcal testng for varous hypotheses, regardng the a pror accuracy nformaton and the modellng choces n GLG networks needs to be addressed n more detal, especally n vew of the many d erent levels of accuracy desred by GPS/levellng users. In ths drecton, the problem of optmzaton and desgn of GLG networks s another mportant and nterestng topc that certanly needs to be explored. Although the present paper was restrcted to a theoretcal framework only, numercal work usng the proposed methodologes s well underway and t wll be presented n a future paper. Acknowledgements. The authors would lke to thank Prof. Dr. Erk Grafarend, who kndly presented the materal of ths paper at the IV Hotne±Maruss Symposum on Mathematcal Geodesy, Trento, Italy, 14±17 September Fnancal support was provded through a contract wth the Geodetc Survey Dvson of Geomatcs Canada. References De Brujne AJT, Haagmans RHN, de Mn EJ (1997) A prelmnary North Sea Geod Model GEONZ97. Project report, Drectoraat-Generaal Rjkswaterstaat, Meetkundge Denst Dermans A (1978) Adjustment of geodetc observatons n the presence of sgnals. Int School of Advanced Geodesy, 2nd course: ÔSpace-tme geodesy, d erental geodesy, and geodesy n the large', Lecture notes, Erce, Italy, 18 May ±2 June Dermans A (1984) Sgnals n geodetc networks. Int School of Advanced Geodesy, 3nd course: ÔDesgn and optmzaton of geodetc networks', Lecture notes, Erce, Italy, 25 Aprl±10 May Dermans A (1987) Geodetc applcatons of nterpolaton and predcton. Int School of Geodesy A. Maruss, 4th course:

10 421 ÔAppled and basc geodesy: present and future trends', Lecture notes, Erce, Italy, 15±25 June Dermans A, Rosskopoulos D (1991) Statstcal nference n ntegrated geodesy. Paper presented at IUGG XXth General Assembly, Venna, 11±21 August Duquenne H, Jang Z, Lemare C (1995) Geod determnaton and levellng by GPS: some experments on a test network. IAG Symposa Gravty and Geod, No 113, pp 559±568, Sprnger- Verlag Featherstone W (1998) Do we need a gravmetrc geod or a model of the Australan heght datum to transform GPS heghts n Australa? Austr Surv 43(4): 273±280 Forsberg R, Madsen F (1990) Hgh-precson geod heghts for GPS levellng. Proc 2nd Int Symp Precse Postonng wth the Global Postonng System, Ottawa 3±7 September, pp 1060±1074 Grafarend E (1985) Varance±Covarance estmaton, theoretcal results and geodetc applcatons. Statst Decsons, Suppl Iss 2: 407±441 Hen GW (1986) Heght determnaton and montorng wth tme usng GPS observatons and gravty data. In: Pelzer H, Nemeer W (eds) Determnaton of heghts and heght changes. Contrbutons to the Symposum on Heght Determnaton and Recent Vertcal Crustal Movements n Western Europe, Hannover, 15±19 September pp 349±360 Heskanen WA, Mortz H (1967) Physcal geodesy. WH Freeman, San Francsco Internatonal Geod Servce (1997) The Earth gravty model EGM96: testng procedures at IGeS. Sp Iss, Bull n.6, DIIAR, Poltechnco d Mlano, Italy Jang Z, Duquenne H (1996) On the combned adjustment of gravmetrcally determned geod and GPS levellng statons. J Geod 70: 505±514 Kearsley AHW, Ahmad Z, Chan A (1993) Natonal heght datums, levellng, GPS heghts and geods. Aust J Geod Photogram Surv 59: 53±88 Koch K-R (1987) Parameter estmaton and hypothess testng n lnear models. Sprnger, Berln Hedelberg New York L YC, Sders MG (1994) Mnmzaton and estmaton of geod undulaton errors. Bull Geod 68: 201±219 Manvlle A, Forsberg R, Sders MG (1992) Global postonng system testng of geods computed from geopotental models and local gravty data: a case study. J Geophy Res 97 (B7): 11, 137±11, 147 Manvlle A, Craymer M, Blacke S (1997) The GPS heght transformaton 1997, an ellpsodal±orthometrc heght transformaton for use wth GPS n Canada. Report of Geodetc Survey Dvson, Geomatcs Canada, Ottawa Mortz H (1980) Advanced physcal geodesy. Herbert Wchmann, Karlsruhe Pelzer H (1986) Heght determnaton ± adjustment models for combned data sets. In: Pelzer H, Nemeer W (eds) Determnaton of heghts and heght changes. Contrbutons to the Symposum on Heght Determnaton and Recent Vertcal Crustal Movements n Western Europe, Hannover, 15±19 September pp 327±340 Rao CR (1971) Estmaton of varance components ± MINQUE theory. J Multvar Statst 1: 257±275 Rao PSRS (1997) Varance components estmaton: mxed models, methodologes, and applcatons. Monographs on statstcs and appled probablty, vol 78. Chapman & Hall, London Sders MG, Manvlle A, Forsberg R (1992) Geod testng usng GPS and levellng (or GPS testng usng levellng and the geod?). Aust J Geod Photogram Surv 57: 62±77 Sjoberg L (1984) Non-negatve varance component estmaton n the Gauss±Helmert adjustment model. Manuscr Geod 9: 247± 280 Smth DA, Mlbert DG (1996) The GEOID96 hgh resoluton geod heght model for the Unted States. Natonal Oceanc and Atmospherc Admnstraton (NOAA), US Natonal Geodetc Survey, Slver Sprngs, MD VancÏ ek P (1991) Vertcal datum and NAVD88. Surv Land Inf Syst 51(2): 83±86