Direct Projection An Efficient Approach for Datum Transformation of Plane Co-ordinates

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1 Drect Projecton An Effcent Approach for Datum Transformaton of Plane Co-ordnates Lars E. ENGBERG and Mkael LILJE, Sweden Key words: datum transformaton, map-grd co-ordnates. SUMMARY Lantmäteret has decded that SWEREF 99 shall be the Swedsh offcal reference frame and replace RT 90 for surveyng and mappng purposes. Transformaton between global and local reference frames often leads to lengthy computatons nvolvng several consecutve transformaton steps. The concept of the drect projecton s to project the geodetc (global) system drectly to the local system. The result s very good; the resduals are normally around 5 centmetres when local systems are transformed and a few more centmetres for the natonal system RT 90. Comparsons between drect projecton and 7-parameter transformaton show that t could be used especally when only 2D-poston (map-data) s to be transformed. 1/8

2 Drect Projecton An Effcent Approach for Datum Transformaton of Plane Co-ordnates Lars E. ENGBERG and Mkael LILJE, Sweden 1 BACKGROUND Implementaton of a globally adapted reference frame as a natonal standard wll affect all those who rely on co-ordnate data. Transformaton between global and local reference frames often leads to lengthy computatons nvolvng several consecutve transformaton steps. A typcal sequence of transformatons mght look lke ( ϕ, λ) global (X,Y,Z) global (X,Y,Z) natonal ( ϕ, λ) natonal (x,y) natonal (x,y) local (1) In 1997 B-G Ret proposed a method (Ret, 1997) that n most cases makes t possble to shorten the above sequence. Ths approach s based on the assumpton that gven a geodetc datum A and a plane rectangular system of another datum B, t s possble to fnd a set of projecton parameters (usng the same projecton as used for the gven plane coordnates of datum B) to defne a plane system of datum A, whch approxmates the plane system of datum B. In strct mathematcal sense the two systems wll not be concdent, but the dfferences may be acceptable for some applcatons. If the dfferences between the plane systems are acceptable, the transformaton sequence (1) can be reduced to the smple expresson ( ϕ, λ) (x,y global ) local (2) At the Geodetc Department of Lantmäteret (the Natonal Land Survey of Sweden), B-G Ret has developed an algorthm and also mplemented t n a software that can be used for computaton of the parameters of the projecton. 2 THE CHANGE OF REFERENCE SYSTEM 2.1 Introducton of the ETRS 89 realsaton SWEREF 99 Lantmäteret has decded that SWEREF 99 shall be the Swedsh offcal reference frame and replace RT 90 for surveyng and mappng purposes. A formal decson regardng map projectons for natonal mappng purposes as well as for local surveyng was taken n 2003 (Lantmäteret 2003), all the projectons are of Transverse Mercator type. The work wth the ntroducton of SWEREF 99 among other authortes n Sweden, such as local authortes, s n progress. Approxmately 70 of the 290 Swedsh muncpaltes have started the process to replace ther old reference frames wth SWEREF 99 and 11 have so far fnalsed the replacement. 2/8

3 To rectfy dstorted geometres of local reference frames, correcton models (Kempe et al, 2006) can be used by the muncpaltes together wth the transformaton parameters from the RIX 95-project 1. The models are based on resduals exstng after transformaton and the rectfcaton s done by a so-called rubber sheetng algorthm. The result s a homogenous network n SWEREF 99 and geographcal data wth less deformaton. 2.2 Old Systems Natonally and Locally The Swedsh old system was ntroduced for trangulaton works n the southern part of the country n the frst decade of the 20 th century. Lattude and longtude on Bessel s ellpsod are projected wth the Gauss-Krüger (Transverse Mercator) projecton. A system of sx zones wth 2.5 gon (2 15') between central merdans was also ntroduced for cadastral works n rural areas. The towns mostly had ther own local systems. As a result of the thrd natonal trangulaton there are also 12 regonal systems mostly used n cadastral works. Therefore, the stuaton s rather complcated wth several hundreds of local systems n the muncpaltes and dozens and dozens of natonal systems. 3 TRANSFORMATION STRATEGIES Tradtonally, the transformaton between two reference systems (dfferent geodetc datums) s performed wth three-dmensonal smlarty transformaton, so-called 7-parameter transformaton. One complcaton n that respect s that even though only plane co-ordnates wll be transformed nformaton about the alttude has to be provded. In the transformaton sequence (1), the step between (φ,λ) and (X,Y,Z) requres the heght above the ellpsod. Thus, the transformed plane co-ordnates are dependent on heght nformaton. Of course, there s a possblty to always put n zero as heght value and stll use ths 7-parameter transformaton. However, many local systems do not have a rgorous geodetc defnton, whch means that there s no way to calculate lattude and longtude from plane co-ordnates. In these cases a straghtforward applcaton of the 7-parameter transformaton s not possble. 3.1 The Drect Projecton Approach The concept of the drect projecton s to project the geodetc (global) system drectly to the local system. The approach s as follows (from Ret, 2003): Gven n ponts wth co-ordnates known n both the geodetc system and the local system, (ϕ,λ) G and (x,y) L, respectvely. Estmate values of the Transverse Mercator projecton parameters λ 0, k 0, F N and F E by a least squares ft whch mnmse the sum of the squares of the quanttes v x and v y of the equatons y = TM ( ϕ, λ, λ,k,f,f,a, f ) + v y 0 0 N E y (3a) 1 RIX 95 s a natonal project, whch ams at creatng hgh qualty connectons between local, natonal and global reference frames. 3/8

4 x = TM ( ϕ, λ, λ,k,f,f,a, f ) + v x 0 0 N E x (3b) where TM x and TM y are the Transverse Mercator functons that maps the surface of the ellpsod to a plane. The equatons are lnearsed by Taylor expanson around approxmate values of the parameters λ 0, k 0, F N and F E. The soluton s terated untl the correctons to the parameters are neglgble. Normally and provded the system covers a reasonably wde area, a sutable projecton of the global (lattudes, longtudes) wll produce co-ordnates n good agreement wth the local system Networks wth bad orentaton If the local co-ordnate systems are rotated compared to the global system, the projecton must be done n combnaton wth a transformaton. There are bascally two dfferent methods that have been used due to the fact that dfferent software s can not use the same type of transformatons see fgure 1. These two methods gve almost the same accuracy but not the same co-ordnates. Lantmäteret calculates both combnatons and the fnal decson on whch combnaton that should be the offcal one s done by the local authortes n the muncpalty. Trans. Mercator-projecton 7-parameter-transformaton + + Helmert-transformaton Trans. Mercator-projecton Fgure 1: Dfferent transformaton combnatons to handle rotated co-ordnate systems 4 APPLICATIONS OF THE DIRECT PROJECTION METHOD Snce the drect projecton method was ntroduced t has been appled wthn Sweden. In the begnnng t was seen as a completon to the 7-parameter transformaton but n course of tme t became more or less the man method n all cases. 4.1 Transformaton of Local Systems Changng from a local system nto a SWEREF 99 system needs a transformaton that ncludes a datum shft. Snce on the one hand most of the control ponts n the local trangulaton networks do not have heghts and on the other many of the local systems do not have a rgorous geodetc defnton t was obvous to use the drect projecton approach. The result s very good, the RMS value for the resduals s normally between 0.01 and 0.08 m, the maxmum resdual s normally less than twce these values. Takng n account that the nternal geometrc qualty n these local networks s of the same order we can not expect better results. 4/8

4.2 Transformaton of the Natonal System RT 90 No transformaton between RT 90 and SWEREF 99 s perfect snce the system dfferences can not be modelled n analytcal form.

5 4.2 Transformaton of the Natonal System RT 90 No transformaton between RT 90 and SWEREF 99 s perfect snce the system dfferences can not be modelled n analytcal form. Tradtonally, the transformaton between two reference systems (dfferent geodetc datum) s performed wth 7-parameter transformaton. Usng that method two problems arose, the need of heght nformaton and the absence of 7-parameter transformaton module n some software. Tests of the drect projecton approach proved that t was as good as the 7-parameter transformaton. The two methods wll gve almost the same accuracy n the transformaton; a mean error not qute 0.07 m and a maxmal error slghtly more than 0.2 m. The dfference between plane coordnates obtaned by drect projecton and co-ordnates transformed wth 7-parameter transformaton s generally less than 0.1 m but n certan areas come to 0.3 m. Fgure 4a and 4b wll show the resduals n the fundamental ponts n the network of our permanent GPS statons (SWEPOS ). Fgure 2a: 7-parameter transformaton 5 DIRECT PROJECTION VS 7-PARAMETER TRANSFORMATION Fgure 2b: Drect projecton The experences from our work wth transformaton of local as well as natonal systems wth the drect projecton put up some questons. Is t possble to use drect projecton elsewhere or are the condtons n Sweden specal? A procedure was created to calculate dfferences between drect projecton and 7-parameter transformaton. Usng parameters from the CRS EU web-ste coordnates n dfferent natonal systems were calculated from lattudes and longtudes n the European reference frame ETRS 89. After that we calculated parameters for a drect projecton wth our least squares ft procedure. In table 1 result from some European countres all of whch have a TM projecton are lsted. 5/8

6 Table 1: Relatve comparson between 7-parameter transformaton and drect projecton Country Number of ponts It seems from these examples that drect projecton could be used also n other areas than Sweden. These comparsons just reflect the dfference between 7-parameter transformaton and drect projecton but are the only way to do t wthout real data (co-ordnates n both systems). For Sweden and Fnland also we have compared results from 7-parameter transformaton and drect projecton. In table 2 results from these tests and tests n Australa by Featherstone (Featherstone & Ret, 1998) are lsted. Table 2: Statstcs of the dfferences between transformed and projected co-ordnates, the number of ponts n Sweden was 185, n Fnland 90 and n Australa 82. From table 2 t s obvous that the dfference between the two methods relatve the old coordnates s not so bg. It means that the accuracy s almost the same but the fgures are dfferent. 6 CONCLUDING REMARKS The drect projecton offers a smple and effcent method to transform co-ordnates between the exstng and new reference frame (datum) and map projecton. The accuracy s commensurable wth the 7-parameter transformaton and much more easy to use. RMS (m) Max resdual (m) Mn lattude Mn longtude Max lattude Max longtude Austra, west Austra, mddle Austra, east Croata, west Croata, east Denmark Fnland Great Brtan Ireland Italy, west Italy, east Luxembourg Slovena Sweden Sweden Fnland Australa 7-par transf. Dr. proj. 7-par transf. Dr. proj. 7-par transf. Dr. proj. RMS (m) Max (m) Mn (m) Mean (m) Std (m) /8

7 When usng some GIS software that have mplemented the 7-parameter transformaton for transformaton of the 2D-poston (horzontal) also the heghts wll be altered and specal arrangements have to be naugurated. Usng drect projecton no such problem wll occur. REFERENCES Featherstone, W. E. and Ret, B.-G. (1998) Modfed Map Projecton: Transformng Map Grd Coordnates Between the Australan Geodetc Datum and Geocentrc Datum of Australa, Surveyng Australa, Vol. 20, No. 3, pp Kempe Ch., Alfredsson A., Engberg L. E., Llje M. (2006) Correcton Model To Rectfy Dstorted Co-Ordnate Systems, Munch, Germany, October 8-13, Ret, B.-G. (1997) A smple way of ntroducng a global reference frame n surveyng and mappng. Survey Revew, 34(264) 1997 pp Ret, B.-G. (2003) Connectng a local system to a geocentrc reference frame. Internal paper, Lantmäteret, BIOGRAPHICAL NOTES Lars E Engberg Mr Engberg obtaned hs masters degree from the Royal Insttute of Technology n Stockholm He has been workng as a lecturer n geodesy at the School of Surveyng for many years. Between 1989 and 1996 he was at the Cty Surveyng Department n Stockholm and responsble for the establshment of an mproved reference network n Greater Stockholm. Snce 1996 he s workng at the Geodetc Research Department at Lantmäteret. At present, he s nvolved n a natonal project amng to mplement the new reference frame SWEREF 99 as a natonal standard. He s also engaged as an nternatonal advser. Mr Engberg s a member of the Swedsh Assocaton of Chartered Surveyors, the Nordc Geodetc Commsson as well as the Swedsh Cartographc Socety. Mkael Llje Mr Llje graduated n 1993 from the Royal Insttute of Technology as a Land Surveyor wth emphass on Geodesy and Photogrammetry. He s workng at Lantmäteret snce 1994 wth varous topcs, manly at the Geodetc Research Department. Currently he s the head of a group workng wth reference frame and co-ordnate system questons. Mr Llje s char of the Swedsh Map and Measurng Technque Socety and char of the FIG Commsson 5 Workng Group on Reference Frame n Practce. Mr Llje was also secretary for FIG Commsson 5 durng the perod /8

8 CONTACTS Lars E. Engberg Geodetc Research Department Lantmäterverket Gävle SWEDEN Tel Fax Emal: Web ste: http// 8/8

Proper Choice of Data Used for the Estimation of Datum Transformation Parameters

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