Basic Concepts of Optimization and Design of Geodetic Networks

Transcription

1 Basc oncepts of Optmzaton and Desgn of Geodetc Networks A. R. Amr-Smkooe, M.ASE ; J. Asgar ; F. Zangeneh-Nejad ; and S. Zamnpardaz 4 Downloaded from ascelbrary.org by Technsche Unverstet Delft on /9/. opyrght ASE. For personal use only; all rghts reserved. Abstract: Ths contrbuton revews a few basc concepts of optmzaton and desgn of a geodetc network. Proper assessment and analyss of networks s an mportant task n many geodetc-surveyng projects. Approprate qualty-control measures should be defned, and an optmal desgn should be sought. The qualty of a geodetc network s characterzed by precson, relablty, and cost. The am s to present a few case studes that have been desgned to meet optmal precson and relablty crtera. Though the case studes may be of nterest to the geodetc communty n ther own rght, the am s to gan nsght nto the general optmzaton problem of a geodetc network. Ths s also potentally of nterest for educatonal purposes. The case studes nclude a zeroth-order desgn to mprove the precson of the network ponts n a traverse network and a frst-order desgn to meet the hgh relablty and maxmum precson crtera n a geodetc network. It s shown that not only the confguraton of the network but also the type of the observatons used can affect the desgn crtera. For example, the case studes presented show that the optmal shape of the trlateraton network (ntersecton wth dstances) can result n a weak network n the sense of relablty and precson f the observatons are replaced by angles rather than dstances (trangulaton network). In close relaton to the optmzaton problem of a geodetc network, the global postonng system satellte confguraton s also optmzed for a partcular case that provdes the mnmum value of the geometrc dluton of precson. DOI:.6/(ASE)SU Amercan Socety of vl Engneers. E Database subject headngs: Optmzaton; Desgn; Networks; Qualty control; Geodetc surveys. Author keywords: Optmzaton and desgn of geodetc networks; Qualty control measures; Geometrc dluton of precson. Introducton Optmal desgn and optmzaton of a geodetc network s an mportant task n many geodetc applcatons. The qualty of a geodetc network s characterzed by ts precson, relablty, and cost (Seemkooe a, b). Repeatablty s another qualty-control crteron that s manly an ssue for satellte-based postonng technques, especally on very-long-baselne nterferometry, absolute global navgaton satellte system postonng, and contnuously operatng networks. In classcal geodetc networks, repeatablty s combned wth the precson and does not accomplsh any separate role. The precson of a geodetc network can be expressed by the covarance matrx of the parameters. In two- (or three-) dmensonal networks, error ellpses (or ellpsods) can be extracted from the elements of the covarance matrx. Also, as two extreme cases, the Assstant Professor, Dept. of Surveyng Engneerng, Faculty of Engneerng, Unv. of Isfahan, Hezar-Zerb Ave., Isfahan, Iran; and Acoustc Remote Sensng Group (ARS), Faculty of Aerospace Engneerng, Delft Unv. of Technology, Kluyverweg, 69 HS, Delft, Netherlands (correspondng author). E-mal: a.amrsmkooe@tudelft.nl Assstant Professor, Dept. of Surveyng Engneerng, Faculty of Engneerng, Unv. of Isfahan, Hezar-Zerb Ave., Isfahan, Iran. M.Sc. Student, Dept. of Surveyng Engneerng, Faculty of Engneerng, Unv. of Isfahan, Hezar-Zerb Ave., Isfahan, Iran. 4 M.Sc. Student, Dept. of Surveyng and Geomatcs Engneerng, Geodesy Dvson, Faculty of Engneerng, Unv. of Tehran, North-Kargar Ave., Amr-Abad, Tehran, Iran. Note. Ths manuscrpt was submtted on June, ; approved on January, ; publshed onlne on February,. Dscusson perod open untl Aprl, ; separate dscussons must be submtted for ndvdual papers. Ths paper s part of the Journal of Surveyng Engneerng, Vol. 8, No. 4, November,. ASE, ISSN 7-945//4-7e8/$5.. precson crteron can be expressed by ether the varance of the unknown parameters (dagonal entres) n the one-dmensonal case or the full structure of the covarance matrx of the parameters n the multdmensonal case. The relablty of a geodetc network, ntroduced by Baarda (968), s dvded nto the nternal and external relablty. Internal relablty refers to the ablty of a network to detect gross errors n observatons, and external relablty refers to the effect of the undetectable errors on the estmated parameters. The geometrcal strength analyss (robustness analyss) s another aspect of relablty (Vanícek et al. 99, ; Berber et al. 6). The relablty and geometrcal strength crtera have been shown to be hghly nterrelated (Seemkooe a, b). Hsu (4) and Hsu et al. (8) establshed the mathematcal foundaton for ths relaton. The geometrcal strength s not the subject of dscusson n the present contrbuton. Optmal desgn of a geodetc network nvolves desgnng an optmal confguraton for the network (e.g., selecton of the locaton and number of network ponts) and an optmal selecton of the type, number, and weght of the observatons. They need to be optmally selected to meet the desred crtera such as precson, relablty, and cost. Optmal desgn problem of a geodetc network orgnates from the work of Baarda (97) and Grafarend (974). The latter dentfed four orders of desgn: zeroth-order desgn (ZOD), desgnng a reference system (datum) for the network, frst-order desgn (FOD), desgnng an optmal confguraton for the network, second-order desgn (SOD), selectng the optmal weghts of the observatons, and thrd-order desgn (THOD), mprovng an exstng network. All of these order desgns are presented n detal by many authors n the reference book on optmzaton and desgn of geodetc networks (Grafarend and Sanso 985). The desgn problem can be solved usng two methods, namely, the analytcal method and the tral-and-error (heurstc) method. In the tral-and-error method, the user postulates a soluton upon whch the desgn crtera are computed. Should ether of the crtera 7 / JOURNAL OF SURVEYING ENGINEERING ASE / NOVEMBER J. Surv. Eng..8:7-8.

2 Downloaded from ascelbrary.org by Technsche Unverstet Delft on /9/. opyrght ASE. For personal use only; all rghts reserved. not be met, a new soluton s postulated, and the crtera are recomputed. Ths step s repeated untl a satsfactory network s acheved (ross 985). In the analytcal method, there s a unque seres of mathematcal steps that automatcally provde a network wth optmal qualty-control measures. A detaled descrpton of the analytcal methods was provded by Kuang (99, 996). Amr-Smkooe (4) and Amr-Smkooe and Sharf (4) proposed an analytcal SOD algorthm to make the relablty parameters as unform as possble. Some research s ongong n the feld of optmal desgn of geodetc networks usng the smulated annealng (SA), whch s a generc probablstc method for the global optmzaton problem. One may for nstance refer to Berné and Baselga (4) and Baselga (), who proposed the SA method for the classc FOD and SOD problems, respectvely. The SA was also used for robust estmaton methods by Baselga (7). The goal of the present contrbuton s to desgn and optmze a geodetc network. For ths purpose, dfferent desgn orders such as ZOD and FOD wll be consdered. The paper s organzed as follows. Frst, the qualty-control measures of a geodetc network are descrbed. The precson and relablty crtera are brefly explaned. Then, a few smple geodetc networks/problems are consdered. The analytcal and numercal ZOD and FOD problems are appled to the networks for an optmal desgn. Though the presented examples mght look ntutvely trval for some of the case studes presented, to our knowledge, they are unque n the sense of the analytcal dervatons gven. Also, the case studes provded can be of nterest n ther own rght to the geodetc communty for many surveyng applcatons lke cadastre, detal and land surveyng, and geographc nformaton system (GIS). In addton, they can also provde nsght nto the optmal desgn problem and can be nstructve from an educatonal pont of vew. Optmal Desgn rtera In ths secton, the qualty control measures precson and relablty of a geodetc network are brefly presented. onsder the lnear(zed) model of observaton equatons: EðyÞ 5 Ax; DðyÞ 5 Q y where A s the m n desgn matrx, Q y s the known m m covarance matrx of the observables, y s the m-vector of observables, x s the n-vector of unknown parameters, and E and D are the expectaton and dsperson operators, respectvely. The precson of the estmates ^x s expressed by the dagonal elements of the covarance matrx of the parameters, whch s of the form Q^x ¼ A T PA where P 5 Q y 5 weght matrx. Dfferent precson crtera can be defned on ths covarance matrx. For example, the dagonal elements of Q^x explan the varances of the parameters. When the unknown parameters are the coordnates of the network ponts, error ellpses (ellpsods) may be used. The dmensons and orentaton of the error ellpses are measures for the precson descrpton of the coordnates. Ths precson descrpton s also achevable at the desgn stage, because no observatons are requred for calculatng Q^x. The relablty matrx R, whch contans the redundancy numbers (r ) on ts man dagonal, has the followng form: ðþ ( # r # ). In an extreme case, when they are zeros, no gross errors, regardless of how large they are, can be detected, whle n the other extreme, when they are ones, all gross errors, regardless of how small they are, can be detected. For the former case, there s the same number of unknowns as observatons, and hence, A s an nvertble matrx resultng n R 5, whle the latter case occurs when measurng a known quantty (no unknown n the model) n whch the desgn matrx A s zero, and therefore, R 5 I. In ths case, all offdagonal entres of R are zeros, ndcatng that the resdual of an observable s not affected by the errors of other observables; t expresses only the error of that partcular observable. In the real case, however, t s desrable to have a network wth relatvely large and unform redundancy numbers, so that the ablty of the gross error detecton s dentcal n every part of the network. The nternal and external relablty crtera can therefore be of the form (Baarda 968) mnðr Þ max It should be mentoned that any of the prevously mentoned desgn orders can be used to reach a network wth hgh precson and/or maxmum relablty. The present contrbuton, through a few case studes, nvestgates how the ZOD and FOD problems can be used to acheve a network that fulflls these two optmal desgn crtera. For the precson crteron, ths can be acheved ether by mnmzng the absolute sze of the errors n the network ponts or by applyng homogenety n the precson of the network usng crcular errors rather than (elongated) ellpsodal errors. For the relablty crteron, one may seek an optmal network to reach unform (e.g., equal) redundancy numbers for the network observables. A Few Optmzaton ase Studes In the followng by means of a few analytcal examples the ZOD and FOD problems to reach networks that meet some precson and relablty crtera are presented. The case studes provded show how an optmzaton problem works, though some of the examples llustrated are of nterest n ther own rght n many geodetcsurveyng applcatons lke cadastre, detal and land surveyng, and GIS. ase : Open Traverse Traverse networks are wdely used n many surveyng engneerng projects. In an open traverse, the successve dstances and angles between ponts are measured (Fg. ). Gven the poston of the reference pont P, the am s to determne the poston of the last pont of the traverse. One also needs to know at least one azmuth (bearng) measured at pont, to determne the orentaton of the network. The goal of the optmzaton problem s to fnd the poston of such an azmuth (.e., to determne the ndex j) to reach a network n whch the poston error at pont s the smallest possble (mnmum). ðþ R ¼ I A A T PA A T P ðþ where I 5 dentty matrx of sze m. When the observables are uncorrelated, the redundancy numbers are between and Fg.. Open traverse network to be measured by successve dstances and angles JOURNAL OF SURVEYING ENGINEERING ASE / NOVEMBER / 7 J. Surv. Eng..8:7-8.

3 Downloaded from ascelbrary.org by Technsche Unverstet Delft on /9/. opyrght ASE. For personal use only; all rghts reserved. The coordnates of the pont P are assumed to be known up to a certan precson. It s also assumed that all angles have been measured wth the same precson s b 5 s b and that all dstances have been measured wth the same precson s l 5 s l. The measurements are assumed to be ndependent. Also, the only azmuth u j; j 5 a s measured wth precson s a. After a few smple algebrac operatons, the bearngs of the drectons before the pont are calculated as u j; j ¼ a u j; j ¼ a b j p u j; j ¼ a b j b j p «u ; ¼ a b j b jp whle the bearngs of the drectons after the pont j are u j; j ¼ a u j; j ¼ a b j p u j; j ¼ a b j b j p «u k; k ¼ a b j b k k j p The coordnates of the last pont are then computed as follows: and x k ¼ x Pj l sn u ; Pk l sn u ; ¼ y k ¼ y Pj l cos u ; Pk l cos u ; ¼ To obtan the standard error of the ndvdual coordnates, the error propagaton law s appled to Eqs. (4) and (5). From Eq. (5), t follows that ¼j ¼j s y k ¼ s yk y s a a Pk yk s l l Pk yk s b b ¼j Pj yk s By takng the partal dervatves and substtutng them n the precedng equaton, the varance n the y-component at pont k reads (Appendx) s y k ¼ s y ðx k x Þ s a s l s b " ðx x Þ Pk ¼j ðy y Þ l ðx k x Þ # fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} F whch s to be mnmzed. Note that the frst three terms n Eq. (7) are ndependent of the poston of the pont j. One may argue that the thrd term (the frst summaton) depends also on j. Although ths s ntutvely true, note that the expresson for the summand s dentcal b b ð4þ ð5þ ð6þ ð7þ for all values of rangng from to k. In other words, f one splts the summaton nto two parts (the frst from to j and the second from j tok), one would always obtan dentcal results (ndependent of j). The last term n Eq. (7) s thus the only one that should be mnmzed (.e.,f mn) over j. Ths mnmzaton problem s equvalent to the followng mnmzaton problem (Appendx): F ¼ Pj h ðx x Þ ðx k x Þ mn whch leads to determnng the ndex j usng the followng expresson: j ¼ arg mn t P t h ðx x Þ ðx k x Þ In a smlar manner, one obtans the followng formula for mnmzng the standard error n the x-drecton: j ¼ arg mn t P t h ðy y Þ ðy k y Þ ð8þ ð9þ ðþ To mnmze the standard error n the poston of the last pont, the ndex j s obtaned as j ¼ arg mn t P t l l k ðþ Analytcal evaluaton of the precedng mnmzaton problems s dffcult. In practce one has to be satsfed by the numercal evaluaton. For dfferent values of t (rangng from t 5 tok), one can evaluate the numercal values for any of Eqs. (9), (),or(). The t-value correspondng to the smallest possble (mnmum) value computed s set to be the optmum value for the j. Specal ase Now consder a specal case of the precedng equatons for whch an analytcal expresson can be obtaned. Assume that the dstances between successve statons are approxmately the same and also that the traverse contnues approxmately along a straght lne,.e., the angles are close to 8. In addton, assume that the pont P s located at the orgn. In ths case, one obtans x 5 x and y 5 y, wth x and y constant values. Eqs. (9)e() can then accordngly smplfy to j ¼ arg mn t P t The ndex j s then gven as (Appendx) j ¼ k h ðk Þ ðþ ðþ Ths ndcates that the best poston for measurng the azmuth n an open traverse s n the mddle of the network. Ths example can hence be consdered as an optmal ZOD problem n whch the orentaton of the datum s optmally chosen by an analytcal method. ase : Intersecton wth Two Dstances An ntersecton network s assumed (Fg. ). It s desred to estmate the poston of an unknown pont Pðx; yþ usng two dstances measured from two known ponts P and P. The two measurements 74 / JOURNAL OF SURVEYING ENGINEERING ASE / NOVEMBER J. Surv. Eng..8:7-8.

4 Downloaded from ascelbrary.org by Technsche Unverstet Delft on /9/. opyrght ASE. For personal use only; all rghts reserved. are assumed to be uncorrelated wth the same precson. Achevng an optmal precson descrpton for the unknown pont s requested. Ths optmalty s provded f the (absolute) error ellpse at the pont P becomes a crcle. It s requred to determne the geometrc locus of such postons n whch ths optmalty crteron s fulflled. Therefore, an optmum FOD problem needs to be solved. To acheve ths goal, one needs to compute the covarance matrx Q^x 5 ða T Qy AÞ. The observaton equatons are p ffffffffffffffffffffffffffffffff l ¼ x y q l ¼ ffffffffffffffffffffffffffffffffffffffffffffffffffffff ðx sþ y from whch the desgn (coeffcent) matrx A s obtaned as x y A ¼ B l x s y A l l ð4þ ð5þ where l and l are gven n Eq. (4) and s s a known quantty. Wthout loss of generalty, one may n addton assume that the weght matrx s an dentty matrx,.e., P 5 Q y 5 I. The covarance matrx of the unknown parameters s then Q x ¼ l l 4s y 6 4 y l y l xy ðx sþy l l xy ðx sþy l l 7 5 x l ðx sþ l ð6þ n whch the dagonal entres are the varances of x and y and the offdagonal entry s the covarance between x and y. The error ellpse, representng the precson descrpton at pont P, becomes an error crcle f () the dagonal entres of the covarance matrx are the same (.e., s x 5 s y ), and () the off-dagonal entry becomes zero (s xy 5 ). Under these condtons, one has y l l l 4s y y l ¼ x l " xy l ðx sþ l # ðx sþy l ¼ ð7þ ð8þ Eq. (7) wll hold f ether of the followng two relatons are fulflled (Appendx): y x sx ¼ ðx sþ y ¼ s ð9þ ðþ In a smlar manner, Eq. (8) wll hold f ether of the followng two relatons are fulflled (Appendx): x ¼ s ðx sþ y ¼ s ðþ ðþ The prevous two sets of relatons obtaned from Eqs. (7) and (8) ndcate that the common term [.e., Eqs. () and ()] wll fulfll both equatons. Therefore, the error ellpse becomes an error crcle f ðx sþ y ¼ s ðþ whch s an equaton for a crcle centered at (s, ) wth the radus s. Ths means that all (unknown) ponts on ths crcle (.e., ntersecton wth rght angle, 9 ) have an error crcle that descrbes the homogenety of ther precson. Note that the radus of the error crcle s the smallest possble (mnmum) compared wth the semmajor axes of all ponts ether nsde or outsde of the crcle n Eq. (). Ths example shows how the FOD problem can be solved to reach a network that meets a proper precson descrpton for the unknown pont P. Ths optmalty crteron was met by ntroducng an error crcle nstead of an error ellpse. ase : Intersecton/Resecton wth Three Ponts Three cases wll be consdered: () ntersecton usng three dstances, () ntersecton usng three azmuths, and () resecton usng three angles. Intersecton Usng Three Dstances An ntersecton network s assumed (Fg. ). It s desred to estmate the poston of an unknown pont Pðx; yþ usng three dstances measured from three known ponts P, P, and P. The three ponts are assumed to make an equlateral trangle n whch all three sdes are equal. Wthout loss of generalty, one may assume that the center of mass of the trangle s the orgn (, ). The three measurements are assumed to be uncorrelated wth the same precson. Achevng an optmal precson descrpton for the unknown pont s requested. Ths optmalty s provded f the (absolute) error ellpse at the pont P becomes a crcle. Fg.. Intersecton network usng two dstances measured from two known ponts to an unknown pont; s s a known quantty Fg.. Intersecton network usng three dstances measured from three known ponts to an unknown pont JOURNAL OF SURVEYING ENGINEERING ASE / NOVEMBER / 75 J. Surv. Eng..8:7-8.

5 Downloaded from ascelbrary.org by Technsche Unverstet Delft on /9/. opyrght ASE. For personal use only; all rghts reserved. It s requred to determne the geometrc locus of such postons n whch ths optmalty crteron s fulflled. Therefore, an optmal FOD problem needs to be solved. To acheve ths goal, one needs to compute the covarance matrx Q^x 5 ða T Qy AÞ. The observaton equatons are qffffffffffffffffffffffffffffffffffffffffffffffffffffff l ¼ x ðysþ sffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff l ¼ x s y s ð4þ sffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff l ¼ x s y s from whch the desgn (coeffcent) matrx A s obtaned as Q x ¼ s l 6 4 xðy sþ l x x s l l x s y s l x s l x pffff s y s n whch the dagonal entres are the varances and the off-dagonal entry s the covarance. The error ellpse, representng the precson descrpton at pont P, becomes an error crcle f () the dagonal entres of the covarance matrx are the same (.e., s x 5 s y ) and () the off-dagonal entry becomes zero (s xy 5 ). Because the covarance matrx Q x s dagonal under these crcumstances, there s no need to nvert Eq. (6). One then has xðy sþ l x l pffff x s ¼ l ðy sþ l x y s l pffff x s y s l s l y s x l l ð7þ s y s ¼ l ð8þ After some rather lengthy algebrac and mathematcal operatons, one can conclude that Eq. (7) wll hold f any of the followng three relatons are fulflled: ys y 4 x 4 ¼ x y s ¼ x ¼ y ¼ ð9þ ðþ ðþ x l pffff x s A ¼ l pffff B s l y s l y s l s y A l ð5þ where l, l, and l are gven n Eq. (4). Wthout loss of generalty, one may n addton assume that the weght matrx s a scaled dentty matrx,.e., P 5 Q y 5 s l I. The covarance matrx of the unknown parameters s as follows: xðy sþ l x s y s l ðy sþ y s l l x y s l s y s l 7 5 ð6þ In a smlar manner, Eq. (8) wll hold f any of the followng three relatons are fulflled: x s y x y ¼ x y s ¼ x ¼ y ¼ ðþ ðþ ð4þ The prevous two sets of relatons obtaned from Eqs. (7) and (8) ndcate that the common terms [.e., Eqs. () and () and Eqs. () and (4)] wll fulfll both equatons. Therefore, one has x y ¼ s ð5þ whch s the equaton of a crcle centered at (, ) wth radus s. Ths means that all (unknown) ponts on ths crcle (.e., ntersecton of 6 ) have an error crcle whch descrbes the homogenety of ther precson. Note that the radus of the error crcle s the smallest possble (mnmum) compared wth the semmajor axes of all (but one) ponts nsde or outsde of the crcle n Eq. (5). The other possblty [Eqs. () and (4)]stheorgn n whch x 5 y 5, and the error ellpse also has a crcular shape. In all of these cases, the radus of the error ellpse equals the semmajor (a) pand semmnor (b) axesoftheerrorellpse,and hence a 5 b 5 ffffffffffffff = sl,wheres l s the precson of the observables. One can now do a SOD problem to determne the precson of the observatons. Inp the 95% confdence regon, ths reads as a max 5 b max 5 :447 ffffffffffffff = sl, whch mght become smaller than the desred precson crteron from whch s l can be obtaned. (Here, a max 5 :447a and b max 5 :447b are, respectvely, the semmajor and semmnor axes n the 95% confdence regon.) 76 / JOURNAL OF SURVEYING ENGINEERING ASE / NOVEMBER J. Surv. Eng..8:7-8.

6 Downloaded from ascelbrary.org by Technsche Unverstet Delft on /9/. opyrght ASE. For personal use only; all rghts reserved. Fg. 4. Measurements from three known ponts to an unknown pont: measurng (a) and (b) three dstances, (c) and (d) three azmuths, and (e) and (f) three angles; ndcated n the plots are also qualty-control measures; rato of (a), (c), and (e) semmajor axs to semmnor axs and (b), (d), and (f) mnmum redundancy number JOURNAL OF SURVEYING ENGINEERING ASE / NOVEMBER / 77 J. Surv. Eng..8:7-8.

7 Downloaded from ascelbrary.org by Technsche Unverstet Delft on /9/. opyrght ASE. For personal use only; all rghts reserved. The problem has also been solved numercally. The rato of the semmajor axs to the semmnor axs has been plotted n Fg. 4(a). As can be seen, the crcle passng through the ponts P, P, and P along wth the orgn (, ) s the geometrc locus where ths rato becomes one. As a measure for the relablty, the mnmum redundancy number s also maxmzed. Ideally, the mnmum redundancy number should be the mean redundancy number, whch s r 5 = for ths example. Fg. 4(b) shows the mnmum redundancy numbers for the entre area. The maxmum value of ths measure s acheved on the crcle x y 5 s and at the orgn. Therefore, ths geometrc locus not only has the optmal precson descrpton by means of crcular error ellpses but also has the maxmum relablty by means of unform redundancy numbers. Note that these conclusons were made because the observatons are equally weghted and that the ntersectons of the three dstances are at an angle of 6 (on a crcle and/or at the orgn). Intersecton Usng Three Azmuths Another stuaton occurs when three azmuths (or bearngs) are measured ether from the unknown pont to the known ponts or from the known ponts to the unknown pont. Smlar analyss to what was done before usng the three dstances can also be performed when measurng three azmuths. Fg. 4(c and d) llustrates the best confguratons usng these measurements, assumng that the observables are uncorrelated wth dentcal precson. Theoretcally, there exst four dstnct ponts n whch both the precson and relablty crtera are fulflled. Therefore, the geometrcal locus wll reduce to these four ponts. Ths s n contrast to the prevous stuaton, where the geometrcal locus was a crcle. Resecton Usng Three Angles The thrd stuaton occurs when one measures three angles from the unknown pont to the known ponts P, P,andP, whch s called resecton. ontrary to the prevous stuatons n whch the best confguraton, n the sense of hgh precson, was obtaned on a crcle passng through the ponts P, P,andP, the results ndcate that ths stuaton should be reversed [Fgs. 4(e and f)]. Ths crcle n fact gves the worst stuaton for a resecton usng three angles. Except for the central pont, the best stuaton s acheved when the unknown pont s far enough from ths crcle. The relablty crteron s, however, ndependent of the poston of the unknown pont n ths case. The mnmum redundancy number s the maxmum value of one-thrd over the entre area. ase 4: Lnear Regresson In ths case study, the smplest problem of lnear regresson s consdered. Although ths cannot be consdered to be a geodetc network, as t s a redundant and hence nconsstent system of equatons, t s conceptually very smlar to a geodetc network. Let us assume that the parameters of the lne y 5 ax b are unknown (.e., the offset b and slope a). They are to be estmated usng the observables y ; 5 ;...; m, measured at the fxed postons on the x-axs as x 5 ; 5 ;...; m (Fg. 5). Further, assume that the observables are uncorrelated wth the same precson s. The goal s to fnd the most relable observable y j (ndex j) for whch the maxmum redundancy number s obtaned. The am s also to gve some nterpretatons and relate them to the prevous work n the feld of geodetc networks. The desgn matrx for ths lnear regresson model s of the form x x A ¼ ««A ¼ ««A x m m ð6þ The redundancy numbers are the dagonal entres of R n Eq. (), whch wth P 5 Q y 5 I smplfes to R 5 I AðAT AÞ A T. After a few smple algebra operatons, one obtans A T A ¼ m Pm P m m Pm ¼ The redundancy number of the jth observable s r j ¼ ½RŠ jj ¼ j A T A j whch smplfes to r j ¼ 6 4 m Pm P m mj j Pm Pm ¼ P m ¼ A ð7þ ð8þ Pm 7 5 ð9þ The redundancy number r j s a functon of the ndex j. Tofnd the maxmum redundancy number, one needs to take the dervatve of r j wth respect to j and set the result equal to zero,.e., dr j dj Fg. 5. Lnear regresson on an evenly spaced x-axs ¼ m Pm whch smplfes to P m mj Pm ¼ ð4þ j ¼ m ð4þ Note that the second dervatve of r j wth respect to j s negatve, meanng that Eq. (4) s n fact the soluton for a local maxmum. Therefore, t was proved that the maxmum redundancy number belongs to an observaton n the mddle of the network. Ths ndcates that the blunder detectablty of an observaton n the mddle s hgher compared wth observatons at the edges. In other words, the nternal 78 / JOURNAL OF SURVEYING ENGINEERING ASE / NOVEMBER J. Surv. Eng..8:7-8.

8 Downloaded from ascelbrary.org by Technsche Unverstet Delft on /9/. opyrght ASE. For personal use only; all rghts reserved. and external relablty of an observaton on the permeter of the network s lower compared wth an observaton n the mddle. Smlar ssues have already been reported n the real and smulated geodetc networks by Seemkooe (a, b). He showed the redundancy numbers of the observatons n the mddle of a network are, on average, larger than those at the edges. Ths s because of the stronger observaton tes for observatons located n the mddle rather than those located on the permeter of the network. ase 5: Postonng wth Global Postonng System The last case study s a global postonng system (GPS) example. Though ths s drectly not a geodetc network, t s conceptually comparable to the classcal optmzaton problem of a geodetc network, because the dluton-of-precson (DOP) values have the same nterpretaton as the precson crteron n a classcal network. It s also smlar to the geodetc network, as t may be compared wth a threedmensonal trlateraton network. The GPS satellte confguraton s optmzed for a partcular case to meet the smallest value for geometrc DOP (GDOP). For postonng usng GPS, one needs at least four satelltes. The relatve confguraton of the recever and satelltes may play an mportant role for such a postonng method. For four satelltes, the coeffcent (desgn) matrx s of the form A ¼ X X o r o X X o r o X X o r o X4 X o r 4 o Y Y o r o Y Y o r o Y Y o r o Y4 Y o r 4 o Z Z o r o Z Z o r o Z Z o r o Z4 Z o r 4 o A ð4þ In a local apparent (LA) system, the desgn matrx A smplfes to cos E cos a cos E sn a sn E cos E cos a cos E sn a sn E A ¼ cos E cos a cos E sn a sn E A cos E 4 cos a 4 cos E 4 sn a 4 sn E 4 ð4þ To exemplfy the precedng desgn matrx, one may assume that the frst three satelltes are located at an equal elevaton angle of E and azmuths of,, and 4, respectvely. The am now s to fnd the optmal poston (azmuth and elevaton) of a fourth satellte such that the mnmum DOP s obtaned for that confguraton. The mnmum DOP corresponds to the maxmum precson of the postonng. Under the prevously mentoned condtons, the desgn n Eq. (4) reads as cos E sn E p ffffff :5 cos E cos E sn E A ¼ :5 cos E cos E sn E A cos E cos a cos E sn a sn E The covarance matrx of the estmates, wth P 5 I, reads as ð44þ Q^x ¼ A T A The GDOP follows from the precedng equaton as GDOP ¼ p ffffffffffffffffffffffffffffffffffffff traceðq^x Þ ¼ rffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff trace hða T AÞ ð45þ ð46þ Because the desgn matrx A s square and nvertble, the precedng equaton yelds GDOP ¼ trace A A T ¼ P 4 4 P j¼ a ; j ð47þ where a ; j s the entry of the nverse of matrx A at row and column j (.e., t represents A n ndex notaton). After some mathematcal and algebrac operatons, one has GDOP ¼ 8sn E sn E 4 cos E cos E cos E cos 4 E 8 cos E ðcos E sn E sn E cos E Þ ð48þ If the partal dervatve of Eq. (48) s taken wth respect to E and the dervatve s set to zero, the optmal soluton n whch ths equaton gets t mnmum value can be obtaned. It then follows that the optmal soluton s obtaned at 8p p >< E ¼ arc tan >: arc tan 6 sn 4 E sn E ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff 4sn 4 E sn E 9sn 8 E 5sn 6 E 6! 6 sn 4 E sn E ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff 4sn 4 E sn E 9sn 8 E 5sn 6 E 6! ð49þ Among these solutons, the only acceptable soluton s the frst, whch mnmzes the functon. The second s not acceptable, because the fourth satellte s located under the horzon plane and wll not be vsble. (It s at ts nadr.) The last two solutons cannot be accepted, because they result n magnary numbers for values # E # 9. The only acceptable soluton s then at E 5 9, when the satellte s at ts zenth. Ths s ntutvely what one would expect, but ths ssue has been shown analytcally. The practcal applcaton of ths result s that, when dealng wth four satelltes, t s optmal to have the fourth satellte far away from the plane passng through the other three satelltes. Further explanaton s gven as follows. Sngular Models Some numercal results on the behavor of Eq. (48) are now consdered. Fg. 6 shows the GDOP values versus E at E 5. The GDOP gets nfntely ncreased at E 5. Ths means that when the four satelltes are smultaneously located n a plane, the desgn matrx becomes sngular, and the geometrcal confguraton becomes extremely weak. Another sngular case occurs when the azmuths of the satelltes are dentcal. In ths case, agan, all satelltes are located n a plane. The desgn matrces for the cases that JOURNAL OF SURVEYING ENGINEERING ASE / NOVEMBER / 79 J. Surv. Eng..8:7-8.

9 Downloaded from ascelbrary.org by Technsche Unverstet Delft on /9/. opyrght ASE. For personal use only; all rghts reserved. the satelltes have the same elevaton and the same azmuth are, respectvely, and cosðeþ cos a cosðeþ sn a snðeþ cosðeþ cos a cosðeþ sn a snðeþ A ¼ cosðeþ cos a cosðeþ sn a snðeþ A cosðeþ cos a 4 cosðeþ sn a 4 snðeþ ð5þ cosðe Þ cos a cosðe Þ sn a snðe Þ cosðe Þ cos a cosðe Þ sn a snðe Þ A ¼ cosðe Þ cos a cosðe Þ sn a snðe Þ A cosðe 4 Þ cos a cosðe 4 Þ sn a snðe 4 Þ ð5þ The former s sngular, because the thrd and fourth columns are lnearly dependent, whle the latter s sngular, because the frst and second columns are lnearly dependent. The prevous concluson can be generalzed. Assume that all satelltes are n a plane and on a sphere centered at the locaton of nterest; they are at an dentcal dstance (r) from the orgn. In ths case, the desgn matrx A can be wrtten as A ¼. r X Y Z X Y Z X Y Z X ffl{zffl} 4 Y 4 Z ffl{zffl} ffl{zffl} {z} V V V V 4 A ð5þ It s not dffcult to show that ths matrx s sngular. Assume that the equaton of the plane passng through all satelltes s ax by cz d 5. Multply the column vectors V ; V ; V ; and V 4 wth the coeffcents a, b, c, and d, respectvely. It s clear that Fg. 6. The GDOP for, E, 9 and E 5 these four vectors fulfll the plane equaton, and hence, the columns of matrx A are lnearly dependent. oncludng Remarks In ths contrbuton, basc concepts of optmzaton and desgn of geodetc networks were consdered. A few case studes were presented. Although the examples consdered may be of nterest for many geodetc applcatons n ther own rght, the am was also to gan nsght nto the general optmzaton problem n geodetc networks. It may therefore be potentally of nterest to the geodetcsurveyng communty and also for educatonal purposes. In close relaton to the classcal optmzaton problem of a geodetc network, the GPS satellte confguraton was optmzed for a partcular case to meet the smallest value for the GDOP. Two qualty-control measures, namely, the precson and relablty, were consdered as crtera for whch the optmal desgn needs to be sought. Among the crtera, the precson requrements can be acheved at the ZOD and the SOD stages. The ZOD was nvestgated n a traverse by choosng an approprate poston for measurng an azmuth n the network. The FOD problem n whch the optmal confguraton of the network was sought was nvestgated. The FOD s manly responsble for the geometrcal strength crteron and hence the relablty of the network. It was also shown that the precson and relablty of a network s a functon of not only the confguraton but also the type of observatons carred out. For example, a trlateraton network wth hgh geometrcal strength does not necessarly result n the same confguraton f the network becomes a trangulaton one. The type and number of observatons along wth ther precson can also play an mportant role n the precson and relablty of a geodetc network. Appendx. Dervaton of Equatons Dervaton of Eq. (7) To derve Eq. (7), take 8 / JOURNAL OF SURVEYING ENGINEERING ASE / NOVEMBER J. Surv. Eng..8:7-8.

10 Downloaded from ascelbrary.org by Technsche Unverstet Delft on /9/. opyrght ASE. For personal use only; all rghts reserved. a ¼ l sn u ; lj sn u j;j l j snðaþ l k sn u k;k ¼ x x k 8 ¼ l sn u ; ¼ x x b y >< k ¼ l sn u ; l sn u ; lj sn u j;j ¼ xj x b j >: b ¼ l sn u ; l sn u ; ¼ x x «b j ¼ l j sn u j;j ¼ xk x j «and ¼ l j sn u j;j lj sn u j;j b k l k sn u k;k ¼ xk x k 8 l >< «Dervaton of Eq. (8) To derve Eq. (8), take Pk ¼ cos u ; ¼ ðy y Þ l >: ¼ cos ðy u k;k ¼ k y k Þ l k ðx x Þ Pk ¼j x x x x x jx x ¼j l k ðx x k Þ mn x x x k x k mn x k j x k x k x x k x k k j x k jx x x x k k P ¼j Pk ¼j x mn x mn x k j x k jx x x x x F ¼ x x or ¼j x x k ¼j x k x kx x x ¼j x x k x x k x x fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} F Pj ¼j x x x x x mn x x k x ¼j x x k x mn x jx k jx mn [ F mn x x x Pj x k Pj x x k x fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} G G ¼ Pj x x k x kx x x ¼ Pj x x k x x x Pj x k mn [ G mn x k x x x mn F ¼ Pj h ðx x Þ ðx k x Þ mn Dervaton of Eq. () To derve Eq. (), take j ¼ arg mn t P t The expresson for F can be wrtten as F ¼ Pj Pj h ðk Þ F ¼ Pj h ðk Þ k Pj k Pj ¼ Pj k Pj F ¼ jk jð j Þ k ¼ jk kj j k x k j x k jx x x x k ¼j x mn F j ¼ F j ¼ k jk k ¼ JOURNAL OF SURVEYING ENGINEERING ASE / NOVEMBER / 8 J. Surv. Eng..8:7-8.

11 Downloaded from ascelbrary.org by Technsche Unverstet Delft on /9/. opyrght ASE. For personal use only; all rghts reserved. or Soluton to Eq. (7) The soluton to Eq. (7) s or y l y l j ¼ k ¼ x l ðx sþ l y l l x l ðx sþ l ¼ y h x y ðx sþ y x h ðx sþ y ðx sþ x y ¼ y x y 4 4s y 4sxy x 4 8s x Soluton to Eq. (8) The soluton to Eq. (8) s 8sx y x 4s y 4sxy ¼ y 4 x 4 4s x 4sx ¼ y 4 x x 4s 4sx ¼ y 4 x ðx sþ ¼ y x x s y x x s ¼ ( y x x s ¼ l l sy y x x s ¼ y x sx ¼ ðx sþ y ¼ s " xy l # ðx sþy l ¼ x ðx sþ l l ¼ xl ðxsþl ¼ h x ðx sþ y ðxsþ x y ¼ x x 4s 4sx y x xy sx sy ¼ x 4s x 6sx xy sy ¼ or x s x sx fflfflffl{zfflfflffl} sx sx xy sy ¼ x ðx sþ y ðx sþ sxðx sþ ¼ ðx sþ x y sx ¼ x ¼ s ðx sþ y ¼ s Explanng Eq. (4) The poston vector! r n the LA coordnate system can be wrtten as follows: cos E cos a X X! ro ¼ o cos E sn a A B Y Y A sn E Z Z By substtutng the precedng equaton nto Eq. (4), Eq. (4) wll be obtaned. Acknowledgments We would lke to acknowledge the valuable comments of the edtorn-chef and two anonymous revewers, whch mproved the presentaton of ths paper. References Amr-Smkooe, A. R. (4). A new method for second order desgn of geodetc networks: Amng at hgh relablty. Surv. Rev., 7(9), 55e56. Amr-Smkooe, A. R., and Sharf, M. A. (4). Approach for equvalent accuracy desgn of dfferent types of observatons. J. Surv. Eng., (), e5. Baarda, W. (968). A testng procedure for use n geodetc networks, Netherland Geodetc ommsson, Delft, Netherlands. Baarda, W. (97). S-transformaton and crteron matrces, Netherland Geodetc ommsson, Delft, Netherlands. Baselga, S. (7). Global optmzaton soluton of robust estmaton. J. Surv. Eng., (), e8. Baselga, S. (). Second order desgn of geodetc networks by the smulated annealng method. J. Surv. Eng., 7(4), 67e7. Berber, M., Dare, P., and Vanícek, P. (6). Robustness analyss of twodmensonal networks. J. Surv. Eng., (4), 68e75. Berné, J. L., and Baselga, S. (4). Frst-order desgn of geodetc networks usng the smulated annealng method. J. Geod., 78(-), 47e54. ross, P. A. (985). Numercal methods n network desgn. Optmzaton and desgn of geodetc networks, E. W. Grafarend and F. Sanso, eds., Sprnger, Berln, 49e45. Grafarend, E. W. (974). Optmzaton of geodetc networks. Boll. Geod. Sc. Aff., (4), 5e46. Grafarend, E. W., and Sanso, F., eds. (985). Optmzaton and desgn of geodetc networks, Sprnger, Berln. Hsu, R., Lee, H.., and Kao, S. P. (8). Three-dmensonal networks are horzontally superor n robustness: A mathematcal reasonng. J. Surv. Eng., 4(), 6e65. 8 / JOURNAL OF SURVEYING ENGINEERING ASE / NOVEMBER J. Surv. Eng..8:7-8.

12 Downloaded from ascelbrary.org by Technsche Unverstet Delft on /9/. opyrght ASE. For personal use only; all rghts reserved. Hsu, R., and L, S. (4). Decomposton of deformaton prmtves of horzontal geodetc networks: Applcaton to Tawan s GPS network. J. Geod., 78(4-5), 5e6. Kuang, S. L. (99). Optmzaton and desgn of deformatons mentorng schemes. Ph.D. dssertaton, Tech. Rep. 57, Dept. of Surveyng Engneerng, Unv. of New Brunswck, Fredercton, NB, anada. Kuang, S. L. (996). Geodetc network analyss and optmal desgn: oncepts and applcaton, Ann Arbor Press, helsea, MI. Seemkooe, A. A. (a). omparson of relablty and geometrcal strength crtera n geodetc networks. J. Geod., 75(4), 7e. Seemkooe, A. A. (b). Strategy for desgnng geodetc network wth hgh relablty and geometrcal strength crtera. J. Surv. Eng., 7(), 4e7. Vanícek, P., raymer, M. R., and Krakwsky, E. J. (). Robustness analyss of geodetc horzontal networks. J. Geod., 75(4), 99e9. Vanícek,P.,Krakwsky,E.J.,raymer,M.R.,Gao,Y.,andOng,P.S. (99). Robustness analyss. Tech. Rep. 56, Dept of Surveyng Engneerng, Unv. of New Brunswck, Fredercton, NB, anada. JOURNAL OF SURVEYING ENGINEERING ASE / NOVEMBER / 8 J. Surv. Eng..8:7-8.