Author name used in this publication:  張國芹Author name used in this publication: SHI Wen-zhong2007-2008 > Academic research: refereed > Publication in refereed journalVersion of RecordPublishe

Shi, W

Zhang, GQ

Zhu, CQ

PolyU Institutional Repository

? 36 ?  ? 4 ? ?  ?  ?  ? Vol. 36, No. 4 2007 ? 11 ? ACTA GEODAETICA et CARTOGRAPHICA S INICA Nov. , 2007  ????: 1001-1595( 2007) 04-0463-05 ?????: P208     ?????: A2??????? ER??????????????朱长青1, 2, 张国芹3, 史文中4( 1. ????????????????????,?? ?? 210046; 2. ?????? ????,???? 450052;3. ?????????,?? ?? 450001; 4. ?????? ???????????,??)Algebraic Research on Geometrical Characteristic of Error Band for2D Linear Segment Uncertainty ER ModelZHU Chang-qing 1, 2, ZHANG Guo-qin3 , SHI Wen- zhong4(1. M ini st ry of Education K ey Labortory of VGE , N anj ing Universi ty, Nanj ing 210054; 2. I nst itut e of S urv eying and Map-p ing , I nf or mation Engineer ing Univer si ty , Zhengz hou 450052, China ; 3. I nst itut e of S ci ence, I nf ormat ion Engineering Uni-ver si ty , Zhengz hou 450001, China ; 4. Dep artment of L a nd Sur veying and Geo-Inf or mati cs , The Hong K ong Polyt ech nic Uni-ver si ty , H ong K ong, China)Abstract: The geometrical character istic of error band for 2D linear segment uncertainty ERmodel has been studied int his paper. T he monotony and extremum theor y of function are applied to prove the geometr ical character istic of er-ror band for 2D linear segment uncertainty ERmodel in theory. The result shows that the error band has the geomet-rical shape of / larg er ends and smaller middle0 or / one end lar ger and the ot her smaller0 . The minimum w idth andits ex act position of t he error band are calculated too . It is proved that the minimum width position of the er ror bandis near the end w ith t he smaller mean squar e error . Thus, the paper perfects the 2D linear segment uncertainty ERmodel in t heory and makes it more scientific, and also provides a new way to study the space data uncertainty inG IS.Key words: G IS; uncertainty; ER model; error band; geometrical characteristic?  ?:对地理信息系统( GIS)中 2维线元不确定性 ER模型的误差带的几何特征进行研究?运用函数单调性和极值理论,从理论上证明 ER模型的误差带的几何特征;得到线元误差带不仅具有/ 两端大?中间小0, 而且也可能具有/ 一端大?一端小0的几何形状; 给出了误差带的最小带宽及其位置; 论证了误差带的最小误差带宽的位置靠近中误差较小的一端?本研究完善了 2 维线元不确定性 ER模型,使其更具有严密性,同时为 GIS 空间数据不确定性的研究提供了新的方法?  ????: 2006-05-08; ????: 2007-01-29????: ?? 863????( 2006AA12Z223) ;??????????( 40501058)????: ???( 1962-) ,?,?????,??,?????,???? GIS?????????????????????????????E-mail: zcq88@ 263. net???: GIS; 不确定性; ER模型; 误差带;几何特征1  引  言? GIS?,????????????????????????????, GIS ?????????????????????????[ 1, 2]?? GIS????????, ?????????????, ????????????????????????????, ??????????????????????GIS ???????????????Perkal??? E-???[ 3] , ???????? E????????????Blakemore? E-??????????????[ 4]???????????????????????: ¹ ????????, º ?????????[ 5]??????????????????????????????????/ G-?0 [ 1] , ????????????????????????????????/ G-?0????????????????[ 6] ,????/ G-?0???????????????????,??????????????????? R????????????????????[ 1, 5, 7] , ??????[ 8]???? ER????????????????????, ??????????????[ 1, 4, 6, 8] ,?????????????????????????????????, ?????????????????????????????????????????????, ???????????? ER???????????,????????????, ????????????????????????????,?????????,????????2  ER 模型误差带的表达式??,????? ER??????????,??????????????????????? Z1Z2 ???????? Z1 ( x 1,y 1) , Z2( x 2, y 2)?????-???????[ 6]D 1=R1x1Rx1y1Rx1y1R2y1,  D 2=R2x2Rx2y2Rx2y2R2y2,D 12=Rx1x2Rx1y2Ry1x2Ry1y2?? Z i ( x i , yi )??? Z 1Z2 ?????,? Z i? Z1, Z2 ?????[ 8, 9]x i= (1- r 1) x 1+ r 1 x 2yi= (1- r 1) y 1+ r 1 y 2(1)??, r 1= s 1i / s12, s 1i? Zi ??? Z1 ???, s12? Z1Z2 ??????, ? 0 [ r 1 [ 1??????????,??????? O )xcyc,????????? Oyc????????,?? 1???????????, ??????? O )xcyc?????????-?????( R ?????)D 1c= RD1R T  D2c= RD 2R T  D12c= RD 12R T??( 1) , ????-???????????????? Zi ??????????????,??????????????????R2xic= (1- r 1)2R2x1c+ r21R2x2c+ 2r 1(1- r 1) Rx 1cx 2c(2)? 1 ???????F ig. 1  The linear segment and coordinate systems???[ 8]??????????????Rxic?? 2??? ER?????Z1Z2 ??? Z i???????????( 2) , ???? R2xic??????R2xic= (1- r 1)2R2x1c+ r21R2x2c+ 2r 1(1- r 1) Rx 1cx 2cS ar 21+ 2br 1+ c (3)??a= R2x1c+ R2x2c- 2Rx1cx2c  b= - R2x1c+ Rx1cx2cc= R2x1c (4)??(3)??,???????????,??????? ER ??????????????Rxic?? r 1 ??, ????????????? Rxic? r 1(0 [ r 1 [ 1)???, ??????????????????? r 1?????? Rxic????????, ???????????????????????? Rxic> 0, ? Rxic????? R2xic?????????? (3)??, R2xic??????????????, ??????R2xic????,???? Rxic????, ?????? ER????????????3  基本引理?? 1: ??? Rx , Ry ???? Rxy?????Rx #Ry \Rxy???:?????????? Qxy = RxyRx#Ry?| Qxy | [ 1, ?? Rx#Ry \Rxy?464 ?  ?  ?  ?                   ? 36 ?????,????R2x R2y \R2xy (5)?? 2: ??(3)?? R2xic???a, b ? c,?ac \b2 (6)??:??(4)??( 5) ,????ac- b2= ( R2x1c+ R2x2c- 2Rx1cx2c) R2x1c-(- R2x1c+ Rx1cx2c)2= R2x2cR2x1c- R2x1cx2c \0?? 3: ??(4)???? a,? a \0???: ???-??????,?R2x1c+ R2x2c \2Rx1cRx2c???? 1??R2x1c+ R2x2c \2Rx 1cRx 2c \2Rx 1cx 2c????(4)???,??a= R2x1c+ R2x2c- 2Rx1cx2c \0?? 1: ???( 3)???? a = 0, ???Z1Z2 ? ER?????????????( 6)??(3)?????????? a=0??????????? 2???? 2  a= 0 ?????Fig . 2 T he error band for a= 0??? 1????, a = 0???????,??????????? a> 0?4  ER 模型误差带的几何特征2??? ER??????????,???????(3)???? r 1 ??? R2xic???????????? R2xic??,?dR2xicdr 1= 2ar 1+ 2 b (7)?????(7)??? r 1 ?????, ????? a> 0, ??? dR2xic / dr ????????? dR2xic/ dr , ???? dR2xic/ d r ?????r 0= - b/ a (8)?? dR2xic/ d r ????????? 3???? 3  ??dR2xicd r 1????Fig . 3  The monotone ofdR2xicd r 1??, ? r 1 [ r 0 ?, ?dR2xicd r 1[ 0, ?? R2xic??????,?? Rxic??????????,?r 1 \r 0?,?dR2xicd r 1\0,?? R2xic??????,?? Rxic???????????????, R2xic(? Rxic)? r 0= - b/ a??????????????? 2??? 2: ¹ ? r 1 [ r 0 ?, Rxic????; º ?r 1 \r 0 ?, Rxic????; » Rxic? r 0= - b / a ??????????? r 0 ???[ 0, 1]?????,?? 0 [ r 0 [ 1, ?? 0 [ - ba[ 1, ?b [ 0,  a+ b \0 (9)???(4)??,?Rx1cx2c [ R2x1c,  Rx1cx2c [ R2x2c??, ????????, ? Rx1cx2c < m in{ R2x1c, R2x2c} ,? 0< r 0< 1???? (8) , ? (9)?? (4)??, ? Rx1cx2c \ R2x1c?,? r 0 [ 0; ? Rx1cx2c \R2x2c?, r 0 \1?????????? 2????????????????? Rxic????, ???????????? ER???????????? 3??? 3: ¹ ?? ER?????????? r 0= - b/ a ???, ?? 3???º ? Rx1cx2c< m in{ R2x1c , R2x2c} ,?? ER?????????? r 0= - b/ a ??????,????????, ? 0< r 0< 1???, ?????/???, ???0?????,?? 4???465? 4 ?         ????: 2 ??????? ER??????????????» ? Rx1cx2c \R2x1c, ?? ER ????????????????????,?????/???,???0?????, ?? 5???¼? Rx1cx2c \R2x2c, ?? ER ????????????????,?????/???,???0?????,?? 6???? 4 / ???,???0????F ig. 4 The error band with / larger ends and smaller middle0? 5 / ???,???0????F ig. 5  The error band w ith / smaller left end and larg erright end0? 6 / ???,???0????F ig. 6  The error band w ith / larger left end and smallerright end05  实验与分析?????????,??? 3?????? ER????????????????????? 1 ????????: Z1 ( 250, 280) ,Z2( 400, 210) ; ?? 2 ????? ???:Z1( 500, 400) , Z 2( 600, 550) ; ?? 3 ????????: Z1(500, 100) , Z2(600, 152) , ?????3???????????-????????2xy1=1. 681 2  0. 267 0. 026 7  0. 036 50. 267  0. 646 2 0. 040 3 - 0. 065 40. 026 7  0. 036 5 0. 463 2  0. 082 50. 036 5 - 0. 065 4 0. 082 5  0. 232 3,2xy2=0. 871  0. 322  0. 099  0. 0780. 322  1. 090 - 0. 034 - 0. 0670. 099 - 0. 034  1. 919  0. 6870. 078  0. 067  0. 687  0. 971,2xy3= 1. 024 5  - 0. 181    0. 054 5  - 0. 035 5- 0. 181    0. 763   - 0. 038 3   0. 067 7 0. 054 5  - 0. 038 3   1. 024 5  - 0. 181- 0. 035 5  - 0. 065 4  - 0. 181    0. 763???? 1???????, ?? 2???????, ?? 3????????? 1 ???????(1) , ?(2) ,?( 4)??(9)???? 3?????????????????????????? a, b ????? 1??, ???( 1) ,? (2) , ? (4)??(8) ,?????? E0 ???? r 0 ?( x 0, y 0)???????????????????????????????????, ????????????????, ????????????????????????????????,?? 1?????????,????????????????????????,???????????????????????????,????? 1  ?????????????Tab. 1  The minimum width and its position of the linear error band??? a b r 0 E0 x 0 y 01 0. 961 7 - 0. 676 0 0. 702 9 0. 348 6    383. 557 9    329. 205 22 1. 496 6 - 0. 572 3 0. 382 4 0. 649 8 538. 240 1 457. 360 13 1. 743 5 - 0. 871 7 0. 5 0. 728 7 550. 00 126. 00  ?? 4: ¹ ?? Rx1c< Rx2c, ? r 0< 1/ 2; º ? ? Rx1c> Rx2c, ? r 0> 1/ 2??????????466 ?  ?  ?  ?                   ? 36 ???????????????????? 3?» , ¼????????????,???????6  结  论1. ???????????????????,?????????????? ER??????????, ??????????????ER??, ?????????2. ?????????????????????????, ?????????????,?? GIS ?????????????????????, ???????????????3. ???????????????????????????4. ????????????, ? GIS ?????????????????????????????5. ??????? ER??????????????,??????????????????????????????参考文献:[ 1 ]  SHI Wen-zhong, LIU Wen-bao. A Stochast ic Process-basedModel for Posit ional Error of Line Segments in GIS [ J] . Inter-nat ional Journal of Geographical Information S cience, 2000,14: 51-66.[ 2]  SH I Wen-zhong, Cheung C K, ZHU Chang-qing. Model ing Er-ror Propagat ion of Buf fer S patial Analysis in a Vector-basedGIS[ J] . International Journal of Geographic Information Sc-ien ce, 2003, 17: 251-271.[3]  PERKAL J. On Epsilon Length [ J] . Bulletin de i. AcademiePolonaise des Sciences. 1956, 4: 399-403.[ 4]  BLAKEMORE M. Generalizat ion and Error in S patial Data-bases[ J] . Cartographica, 1984, 21: 131-139.[ 5]  SH I Wen-zhong.M odeling Posit ional and Themat ic Uncertain-t ies in Integrat ion of Rem ote S ensing and Geographic Informa-t ion System[ M] . Enschede: IT C Publicat ion, 1994.[ 6]  LIU Wen-bao, DAI Hong-lei, et al . The Analyt ic Expression ofGeomet ric Figure on Planar Lines Error Band[ J] . Acta Geo-dat ica et Cartographica Sinica, 1998, 27: 231-237. (???,???.????????????????.????,1998, 27: 231-237. )[7]  CASPARY W, SCHEURING R. Error-band as Measurers ofGeographic Accuracy [ A] . Processing of EGIS. 92 [ C] . M u-nich: Springer Netherlands Press, 1992. 226-233.[8 ]  L IU Da-jie, SHI Wen- zhong, et al . Accuracy Analysis andQuality Cont rol of S patial Data in GIS [M ] . S hanghai: Shang-hai Press of Scient ifie and Technical Literature, 1999. (???,???,???, ?. GIS ??????????????.??:???????????, 1999. )[9]  ZHU Chang-qing. Numerical Calculat ion M ethod and It s Ap-plicat ions[ M ] . Beijing: Science Press, 2006. (???. ??????????.??:?????, 2006. )(责任编辑:丛树平)52007年版中国期刊引证报告6发布???????????? 2007 ? 11 ? 15 ?????????????????, ???2006?????????????????????52007??????????(???)6?2006 ???????????????????????? ????? ?????? ???? ???? ?? ???? ???????? 149 9 - 0. 70 0. 234 9 - 0. 62???? 468 5 - 0. 06 0. 749 4  0. 23???? 630 3  0. 27 0. 299 8 - 0. 51???? 673 2  0. 35 0. 789 3  0. 29?????????? 503 4  0. 01 0. 792 2  0. 30?????? 425 7 - 0. 15 0. 667 5  0. 09??????? 458 6 - 0. 08 0. 436 7 - 0. 29???? 285 8 - 0. 43 0. 529 6 - 0. 13???? 894 1  0. 80 1. 005 1  0. 64??? 498 0. 611467? 4 ?         ????: 2 ??????? ER??????????????

2維線元不確定性ε[sub σ]模型誤差帶幾何特征的代數研究

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