METHOD OF NETWORK RELIABILITY ANALYSIS BASED ON ACCURACY CHARACTERISTICS

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1 METHOD OF NETWOK ELIABILITY ANALYI BAED ON ACCUACY CHAACTEITIC ławomr Łapńsk hd tudent Faculty of Geodesy and Cartography Warsaw Unversty of Technology ABTACT Measurements of structures must be precse and accurate as they affect the qualty of the fnal result of geodetc tasks. In some cases there s a rsk of commttng a gross error. Dagnostc analyses may allow such an observaton n to the adustment and estmaton of unknown parameters. The paper presents results of network deformaton and robustness analyss based on the characterstcs of network accuracy and relablty. The maxmum undetectable error that may occur n an ndvdual observaton consttutes the source of deformaton. The method tested n the present paper s proposed n rószyńsk and arzyńsk (2009). The paper evaluates theoretcal relatonshps and ther utlty for analyses of large geodetc networks. Addtonally, the paper attempts to transfer the presented relatonshps to the case of free adustment, whch was not ncluded n theoretcal dervatons n the paper mentoned above. The research has been conducted usng a computer generated 2D lnear-angular network. The analyss focuses on the results of network deformaton, dependng on the assumed reference system. Moreover, an attempt has been made to modfy the proposed method of network relablty analyss for the case of free adustment. Key words: gross error, nternal relablty, external relablty, relablty analyss, accuracy analyss 1. BAIC FOMULAE AND DEFINITION Usng the model Ax = y + v; C y (1)

2 266 the formula descrbng node dsplacement caused by the gross error (gross errors) (Vancek et al. 2001) can be shown as follows: T T ΔXˆ 1 = ( A A ) A Δl (2) where Δ s the (nx1) vector of standardzed gross errors. l The source of network deformaton s the occurrence of the maxmum undetectable gross error (standardzed gross error) (Baarda 1968) of the magntude shown by the followng formula G c = (3) where c a constant whch s the functon of test parameters (α 0, β 0 ); relablty ndex for the -th observaton. 2. ELATIONHI BETWEEN ELIABILITY INDICE AND ACCUACY CHAACTEITIC In rószyńsk and arzyńsk (2009), the Authors present a concept of network robustness measures based on responses to undetectable observaton errors. These measures are formulated on the bass of network deformaton and robustness analyss. Moreover, the Authors have found propertes connectng accuracy characterstcs to relablty ndces. It has been proved that the end of the vector of node dsplacement Δ Xˆ ( ) of the -th node les ether on or nsde an ellpse analogous to that of covarance ellpse for the -th node, wth the followng scale factor k c 1 or =, (3) = 1 to k G (4) The graph of the above relatonshp k = f c, ) for c = 4. 1 s shown n Fg. 1. ( Fg. 1. Graph of the relatonshp k = f ( c, ) wth c = 4. 1 (rószyńsk and arzyńsk 2009).

3 267 Ths property enables one to specfy the upper bound for the vectors of dsplacement of the -th node owng to the maxmum undetectable error, whch may occur n all the ndvdual observatons. The bound for the -th node, marked by U( Δ X ) can be wrtten as follows U ( Δ X ) = k E (5) where: k a coeffcent formulated on the bass of mn ; E - s a covarance ellpse of the -th node; - a symbol for the operaton of magnfyng the ellpse sem-axes. 3. TET NETWOK ELIABILITY ANALYI In order to analyze the network relablty I have used a 30-pont horzontal geodetc network (Fg. 2) to conduct analyses for varous defntons of the reference system. Fg. 2. Outlne of the test network. The network conssts of sxty-nne lnear and one hundred and thrty-eght angular observatons. The redundancy of the observaton system s f =140 (f=n-u+d; n=207, u=60, d=3) wth the condtons defnng the reference system as w=d ( free and pontlne type). If w>d then redundancy s f=n-u+w. For the condtons specfyng the pont-pont type reference system w=4, the redundancy of such a system s f=141. The network s dsplays vared nternal relablty ndces rangng from mn =0.32 to max =0.92. A pror standard devaton have been employed to standardze the observatons 10 cc for angles and 2mm + 3ppm for lengths. The network has been adusted upon the assumpton of the followng condtons defnng the reference system: a) free type condtons set on all ponts; b) pont-lne type condtons (two varants: pont 27, lne 27-21; pont 14, lne 14-27); c) pont-pont type condtons(one varant: ponts 1 and 11). elected results of the network relablty and deformaton analyss have been presented n the Fgures 3 and 4. The vectors of node dsplacement that result from the

4 268 occurrence of the gross error n each observaton are marked by lnes startng from the same -th pont. The smaller ellpse presents the standard error ellpse, whereas the larger one demonstrates the ellpse whch has been magnfed by the scale factor n accordance wth formula (4). Fg. 3. The results of the robustness analyss - free type condton set on all ponts. Fg. 4. The results of the robustness analyss pont-lne condton (14,14-27).

5 AN ATTEMT OF MODIFICATION OF THE OBUTNE ANALYI FO FEE ADJUTMENT On the bass of the analyss results presented n the prevous paragraph, there may be observed an overestmaton of the U( Δ X ) areas for the maorty of the ponts, whch decreases the confdence as regard the method relablty analyss. An attempt has been made to modfy the robustness analyss, whch ams to ndvdualze the computaton of the k value (the k value s the global scale factor). As a result of the modfcaton, the U( ΔX ) areas proportons wll be reduced n order to show relable node dsplacement areas. The analyses n ths secton apply exclusvely to the case of large geodetc network adustment where the reference system s of the free type set on all the ponts. For ths type of reference, none of the dsplacement vectors reaches the boundary determned by the U( Δ X ) area for each pont. Addtonally, t s necessary to note that the propertes formulated n rószyńsk and arzyńsk (2009) do not nclude the free adustment case. The ndvdualzaton of the computaton of k value (.e. local scale of the standard error ellpse) ams at reducng propagaton of the nfluence of the observaton of a low nternal relablty ndex around the entre area of the network. On the bass of the formula 4, the value of the scale factor for each pont goes as follows: where k 1 = c, or k = G 1 s the modfed value of the nternal relablty ndex. Accordng to Otrębsk Theorem, the redundancy of observatons n a network makes ther a posteror accuracy hgher. It may be wrtten as follows: u z = (7) n where: n the number of observatons n a network; u the number of unknowns n a network. On the bass of a graphc-analytc analyss, the followng modfcaton of the nternal relablty ndex has been proposed: a) for external ponts of the network mn = (8) (1 z) b) for nternal ponts of the network = mn (1 + z) (9) where mn s the mnmum value of the nternal relablty ndex for an observaton conducted drectly on the pont under consderaton. (6)

6 270 Fgure 5 shows the results of the modfcaton. The notatons are the same as those n the prevous paragraph. The thrd ellpse has been added (located between the standard error ellpss and the large ntal ellpss). Fg. 5. esults of the modfcaton of the network robustness analyss free type condton set on all ponts. 5. CONCLUDING EMAK The relatonshpss between the relablty theory and the accuracy characterstcs have been proved to be correct. In the case of free adustment, none of the node dsplacement vectors caused by the undetectable gross error exceeds the desgnated area. It can be stated that ths way of research s correct and worth further theoretcal formulatons. In the case where large networks are analyzed, the above approach s not useful as t reduces the actual relablty level of a gven network. The areas of the standard error ellpsss are sgnfcantly larger than the actual area of the node dsplacement. The modfcaton, n the case of free adustment, was based on the graphc-analytc analyss. The results showed areas that much better reflected the actual of network reactons to undetectable errors. All the conclusons are the result analyses of specfc test networks. In order to make them more general and relable, t s necessary to conduct further research ncludng a larger number of dverse networks. Fnally, to prove the correctness of some of the conclusons, further theoretcal formulatons are necessary. EFEENCE Baarda W. (1968). A testng procedure for use n geodetc netwerks, ublcatons on Geodesy, New eres, Netherlands Geodetc Commsson, vol. 2, No. 5, Delft. rószyńsk W., arzyńsk Z. (2009). Network robustness measures based on responses to undetectable observaton errors, Warsaw: eports on Geodesy, No. 2(87), pp Vanček., Craymer M., Krakwsky EJ. (2001) obustness analyss of geodetc horzontal networks, Journal of Geodesy 75: