Geodetic System Technical Report GS 1997/11

Transcription

1 Geodetic Sste Technical Report GS 1997/11 Docuent nae Recoended transforation paraeters fro WGS84 to NZGD49 Original uthor Merrin Pearse and Chris Crook Client Surveor General Distribution Unrestricted File reference GEO Docuent source L:\INFO\000MGT\REPORTS\Gs doc Revision date 22-ug :23 Print date 9-Mar :49 Pages 15 Docuent Histor uthor Date Description Suar Introduction Definition of New Zealand Geodetic Datu 1949 (NZGD49) Definition of World Geodetic Sste 1984 (WGS84) Data used to test transforation paraeters Transforation paraeters DM derived siilarit transforation paraeters : DM Multiple Regression Equations : Mackie derived siilarit paraeters Calculated latitude and longitude offset Calculated siilarit paraeters Results and discussion... 7 References... 9 ppendi : Transforation ethods Coordinate rotation ethods ursa-wolf odel Molodenskii-adekas odel Multiple Regression Equations (MRE) Method ppendi : Transforation paraeters between ITRS and WGS realisations... 15

2 Recoended transforation paraeters fro WGS84 to NZGD49 Page 2 of 15 Suar Most apping in New Zealand is based upon the NZGD49 datu. However positioning using GPS is based upon WGS84, a global datu. The NZGD49 and WGS84 datus differ due to the different ellipsoids (shape and position) used to odel the earth, observational ethods and the effects of earth deforation. These differences give a cobined apparent shift of approiatel 180 etres between WGS84 and NZGD49 coordinates, ie. if WGS84 coordinates are plotted on a ap based upon NZGD49 the position would have an apparent northing error of approiatel 180 etres. The WGS84 coordinate can be converted to a NZGD49 coordinate b several different ethods. The accurac achievable depends on the ethod selected. n accurac of approiatel 15 etres can be achieved b appling the nationwide id-point value of the range of differences in the latitude and longitude. That is adjust the WGS84 latitude 6.1 seconds southwards and the longitude 0.5 seconds westwards. better agreeent of approiatel 4 etres can be achieved using a 7 paraeter Molodenskii- adekas transforation. ppling this transforation involves converting the WGS84 latitude and longitude to Cartesian XZ coordinates relative to the centre of the ellipsoid, then appling a rotation, translation (shift), and scale change, and finall converting the resulting coordinates back to latitudes and longitudes on the NZGD49 ellipsoid. The recoended paraeters for this transforation are based upon calculations b Mackie in 1982 using doppler observations at 18 control stations. Mackie s calculations deterined the relationship between WGS72 and NZGD49. The paraeters to transfor fro WGS84 to NZGD49 are derived b appling the published relationship between WGS72 and WGS84 to Mackie s results. Recoended 7 Molodenskii-adekas paraeters for the transforation fro WGS84 to NZGD49 Ellipsoid paraeters Ellipsoid Sei ajor ais (etres) 1/flattening WGS NZGD Transforation paraeters: Translation (etres) T = T = 5.04 T = Rotation (seconds of arc) R = 0.47 R = R = Scale change (part per illion) s = These paraeters should onl be used to obtain NZGD49 (horiontal) coordinates and not heights. The following test point can be used to confir that the paraeters have been applied correctl: Datu Latitude Longitude WGS S E NZGD S E Note: converted coordinate accurac is approiatel ±4 etres: Recent GPS easureents of first order NZGD49 stations have enabled new Molodenskii-adekas transforation paraeters to be calculated. These new paraeters result in accurac of approiatel 3.5 etres which is considered an insignificant iproveent over the Mackie based paraeters (appro. 4 ) which have been inforall prooted b Land Inforation New Zealand (and its predecessor, the Departent of Surve and Land Inforation) since It is recoended that the Mackie based paraeters be officiall adopted b the departent to avoid the confusion which a different set of paraeters could introduce. For applications requiring a higher degree of accurac, eg. surveing, it is generall necessar to reference observations to local NZGD49 control stations rather than transfor WGS84 coordinates to NZGD49, as no siple national transforation can achieve an accurac significantl better than 4 etres. Transforing GPS derived heights is a separate proble which is beond the scope of this report.

3 Recoended transforation paraeters fro WGS84 to NZGD49 Page 3 of 15 1 Introduction This report copares currentl available transforation paraeters for converting between NZGD49 (New Zealand Geodetic Datu 1949) and WGS84 (World Geodetic Sste 1984) coordinates. The need for transforation paraeters between WGS84 and NZGD49 coordinates coes fro the increasing use of the Navstar Global Positioning Sste (GPS) for easuring the spatial position of phsical features. The reference sste used b GPS is WGS84 which is a three diensional geocentric sste, unlike NZGD49 which is onl a two diensional (ie. horiontal) non-geocentric sste. This report will therefore onl copare the accurac of the transforation paraeters on the horiontal coordinates and not heights. 2 Definition of New Zealand Geodetic Datu 1949 (NZGD49) NZGD49 is a horiontal datu based on geodetic observations that were priaril observed between 1923 and detailed account of the establishent of NZGD49 is given in Lee (1978). The first order triangulation covered ost of the North Island but onl the eastern side of the South Island and coprised approiatel 290 stations, with a braced quadrilateral joining the networks across Cook Strait. The scale of the network was controlled b five North Island and three South Island baselines. With the aid of electro-echanical calculating achines the adjustent of the geodetic observations was undertaken in 8 figures using the ethod of condition equations. The reference ellipsoid is the International (Haford) ellipsoid which was positioned so as to approiate the geoid across New Zealand. Twelve Laplace stations were used to orientate the ellipsoid so the Z ais approiated the geocentric natural sstes Z ais (eg the Earth rotation ais). The adjusted first order station coordinates were assigned no foral errors so ust be considered to be error free. These first order stations have subsequentl been held fied when adjusting lower order networks. 3 Definition of World Geodetic Sste 1984 (WGS84) The United States Defense Mapping genc (DM) has been involved in the developent of World Geodetic Sstes since The World Geodetic Sste that was established in 1984 (WGS84) used Doppler data fro the US Nav Navigation Satellite Sste (NNSS). WGS84 was priaril developed to support the US DM s apping, charting and geodetic products. The WGS84 reference frae constitutes a ean or standard earth rotating at a constant rate around a ean pole of rotation fied in tie. Its origin is at the earth s centre of ass, and the aes are coincident with the Conventional Terrestrial Sste as defined b ureau International de l Heure (IH) for the epoch of The fundaental paraeters and reference ellipsoid of WGS84 are the sae as the International Union of Geodes and Geophsics (IUGG) sanctioned Geodetic Reference Sste 1980 (GRS80), as described b Morit (1980a), ecept for one paraeter, C 2. WGS84 defines the noralised second degree onal haronic coefficient of gravitational potential constant, C 2 (= -J 2 / 5 ), instead of the dnaical for factor, J 2, of GRS80. The indirect use of 5 thus introduced a truncated difference (after the eighth digit) in the flattening, f, for the WGS84 ellipsoid fro the f of GRS80 ellipsoid (Kuar, 1993). DM (1991) gives the full list of adopted constants for WGS84, while further inforation on GRS80 is given b Morit (1980a). The WGS84 reference frae, now over a decade old, was designed onl to have an accurac of 1-2 etres (1 siga), which is ore than adequate even for large scale apping (DM 1991). However recent geodetic requireents of the Departent of Defense (DoD) has required accurac at the decietre level. coparing the four defining paraeters (a, GM, J 2 and ω) of WGS84 with the ore recentl adopted scientific counit values of the International Earth Rotation Service (IERS) (McCarth, 1992), it was established that the GM value was the onl paraeter warranting revision.

4 Recoended transforation paraeters fro WGS84 to NZGD49 Page 4 of 15 The ain reason for updating the original WGS84 GM value was to reduce the 1.3 etre radial error (bias) in all DoD GPS orbit fits. The GM value of the IERS standards was adopted as the new WGS84 GM (Mals and Slater, 1994). t the tie of redefining the GM paraeter for WGS84, the coordinates of the five ir Force and five DM GPS onitor stations were updated. This was achieved b siultaneousl processing GPS data fro the ir Force, DM and selected International GPS Service for Geodnaics (IGS) sites during the 1992 global IGS capaign. The adjustent of these ir Force and DM sites was perfored while constraining a selection of the IGS sites to their ITRF91 (International Terrestrial Reference Frae 1992) values. This resulted in a new realisation of WGS84 through the adoption of new coordinates for the 10 DoD GPS tracking stations. The new realisation of WGS84 is reported b Swift (1994) to be coincident with ITRF91 at the order of 10 c. This refined WGS84 reference frae, along with the iproved GM value, have been given the designation WGS84 (G730), and was placed into DM s orbit processing fro the first da of GPS week 730 which corresponds to 2 Januar 1994 (Mals and Slater, 1994). 4 Data used to test transforation paraeters The Land Inforation New Zealand Datu 2000 project has reeasured all the NZGD49 first order stations, as well as an lower order NZGD49 stations, using GPS. This has allowed Land Inforation New Zealand to copare WGS84 coordinates against NZGD49 coordinates at 261 NZGD49 first order stations and 117 NZGD49 second order stations. These control points for the data set used to calculate and test transforations between WGS84 and NZGD49. The Datu 2000 GPS observations span a period fro 1994 to The have been reduced as baselines using Trible GPSurve software. The reference frae for the GPS data was introduced into the network using three fied ero order 2000 stations, Whangaparoa [1334], Heaph House [WELL] and the Universit of Otago Surveing School [OUSD]. Their coordinates were calculated in an adjustent of the ustralian regional network in ters of ITRF92 at the epoch of using the GRS80 ellipsoid (Morgan et al. 1996). This adjustent included the 1993 observations of 26 New Zealand stations. The ITRF92 reference frae differs fro WGS84 b less than 1 etre in New Zealand, which is better than the absolute accurac of the WGS84 realisation (see Table -1 for sie of transforation paraeters between WGS84 and ITRF92). The adjustent of the 1994 to 1996 GPS data did not take into account an earth deforation that occurred during the period of the surves. few observations were rejected as being inconsistent with the ajorit (in particular soe observations deonstrate probles with the station heights). 5 Transforation paraeters 5.1 DM derived siilarit transforation paraeters DM (1987a) lists four different sets of siilarit transforation paraeters for converting fro NZGD49 to WGS84, based on different cobinations of paraeters being solved for in the least squares solution. The values for the paraeters (Table 1) were derived fro 14 Doppler stations. No description of the location of these sites is contained in DM (1987a or b). The onl indication of the accurac of the transforation paraeters in Table 2 was given b DM (1987b, p. 10-9) for the three paraeter solution. The accuracies for the T X, T and T Z paraeters were ± 5, 3 and 5, respectivel, though the confidence interval was not specified. The siilarit transforation paraeters need to be applied to Cartesian coordinates. The local datu coordinates available to DM consisted of orthoetric heights rather than ellipsoidal heights. To convert orthoetric height to ellipsoidal height the geoid height (N) needs to be known. DM deterined N values in ters of NZGD49 b assuing the local geoid height at each Doppler station was ero. Then using the bridged Molodenskii Datu Transforation forula for H (DM,

5 Recoended transforation paraeters fro WGS84 to NZGD49 Page 5 of a, Table 7.8), deterined H values ( N). Then the N values were subtracted fro the WGS84 geoid height (degree and order 180) DM (1987a). Nuber of Paraeters solved for T X T T Z * ε ε ε s pp Table 1 : DM Siilarit Transforation Paraeters to convert fro WGS84 to NZGD49 (DM, 1987a, p. 7-47). *: s probabl printed with wrong sign (see Sections 5.3 and 5.5). 5.2: DM Multiple Regression Equations The DM (1987b, p ) MRE for the conversion of NZGD49 coordinates to WGS84 coordinates are in (1). The nuber of stations used to deterine the MRE is unclear as tests for deterining the accurac of the equations used either 14 points (DM, 1987a, p. 7-51) or 31 points (DM, 1987b, p ). where φ = U V UV U UV V U U 9 λ = U V UV U U 2 V U 3 V U 9 V 6 (1) paraeter definitions are the sae as (eqn 7) φ and λ units are arc seconds U = K (φ + 41), with φ being the geodetic latitude in decial degrees V = K (λ - 173), with λ being the geodetic longitude in decial degrees K = , the cobined scale factor and degree-to-radian conversion To convert fro WGS84 to NZGD49 the offsets φ and λ are calculated using WGS84 latitude and longitude in these forulae, and then subtracted fro the latitude and longitude. 5.3: Mackie derived siilarit paraeters Mackie (1982) deterined transforation paraeters between WGS72 and NZGD49 based on 18 Doppler stations that were well distributed across both the North and South Islands of New Zealand. Mackie coputed a geoid referenced to NZGD49, b integrating the coponents of the deviation of the vertical (ie. fro stro-geodetic levelling). This allowed the orthoetric heights to be converted to ellipsoidal heights and when cobined with the NZGD49 horiontal coordinates, provided a local three diensional set of coordinates that were copared with the Doppler coordinates to establish the transforation paraeters. Mackie solved for what he tered a 3-paraeter solution to the transforation vector and also a 7- paraeter solution. The seven paraeter solution for the transforation paraeters was obtained b a least squares solution based on the Molodenskii-adekas ethod (ppendi.3). The paraeters to transfor fro NZGD49 to WGS72 were stated b Mackie (1982, p. 22) as:

6 Recoended transforation paraeters fro WGS84 to NZGD49 Page 6 of 15 T = ±0.27 T = ±0.27 T = ±0.27 s = ± (2) ε = ± = ±0.27 ε = ± = ±0.22 ε = ± = ±0.28 Note that the sign of s as given b Mackie in (2) has the opposite sign to the s given b DM (Table 1). It would appear fro a coparison of the DM (Table 1), Mackie (2) and calculated (Table 4) 7 paraeters that the DM value for s has been printed with the incorrect sign. The values that were used for the centroid ( X Z ) when deriving the paraeters in (2) were not specificall stated. However if one assues that the centroid was calculated using Mackie s equation 5 (1982, p. 22), which is equivalent to that given in (eqn 5), the coordinates of the centroid can be calculated fro the Cartesian coordinates given in Mackie s table 1 (ibid., p. 22) to be: X = = (3) Z = To copare the paraeters derived b Mackie with paraeters contained in Section 5.5, the paraeters need to be converted fro the Molodenskii-adekas ethod (ppendi.3) to the ursa- Wolf ethod (ppendi.2). This can be achieved b cobining equations (1), (4) and (6) to give the ursa-wolf translation paraeters: T T T X Z X = Z s F 1 ε ε ε 1 ε ε ε 1 X Z T + T T Solving (4) using the values in (2) and (3) results in (the sense, fro NZGD49 to WGS72): T X = T = (5) T Z = The scale and rotation paraeters reain unchanged (see ppendi.3). Having now converted Mackie s Molodenskii-adekas based paraeters to ursa-wolf paraeters it is now possible to convert these paraeters so that the for transforing fro WGS84 to NZGD49, rather than transforing fro NZGD49 to WGS72. This is achieved, due to the sall rotation angles, b adding algebraicall the paraeters for converting WGS84 to WGS72 (Table -1), with the rotation and scale paraeters fro (2) and the translation paraeters fro (5) (Table 2). (4)

7 Recoended transforation paraeters fro WGS84 to NZGD49 Page 7 of 15 Fro To T X T T Z s ε ε ε 10-6 WGS84 WGS WGS72 NZGD WGS84 NZGD Table 2 : Seven paraeter siilarit transforation paraeters derived fro Mackie (1982) for conversion fro WGS84 to NZGD Calculated latitude and longitude offset ver siple approach to converting WGS84 latitude and longitude to NZGD49 is to deterine an offset to appl to the latitude and longitude. One ethod of calculating the offset is to find the value which iniies the aiu reaining error after the offset is applied. This is equivalent to the ean of the iniu and aiu observed offsets. The offset calculated fro the test data is 6.1 seconds south and 0.5 seconds west (to be applied to WGS84 latitudes and longitudes). 5.5 Calculated siilarit paraeters Siilarit transforation paraeters have been calculated using the control points in the test data set. These points onl provide horiontal control, since NZGD49 is onl a two diensional coordinate sste. In order to adequatel constrain the solution, Mackie s approach has been followed. This approach endeavours to obtain a geoid height of approiatel ero at the locations of the baselines easured for the 1949 adjustent. The EGM96 global geopotential odel has been used to calculate the geoid heights relative to WGS84 at the baseline stations. This is the WGS84 ellipsoidal height at which the orthoetric height should be ero (to within the accurac of the geoid odel - of the order of 1 etre). ecause the geoid height relative to NZGD49 is constrained to be ero, it follows that the NZGD49 ellipsoidal height of these points is close to ero. Two sets of paraeters have been deterined based upon these criteria - one odel coprises onl translations (3 paraeters), and the other is a full 7 paraeter odel. The paraeters are: Nuber of Paraeters solved for T X T T Z ε ε ε s pp Table 3 : Calculated siilarit paraeters to convert fro WGS84 to NZGD49 6 Results and discussion In order to test the proposed transforations the WGS84 coordinates of the control stations have been converted to NZGD49 using each transforation. These and the NZGD49 coordinates for the points have been converted to the NZMG ap projection. The residual horiontal error in the transforation is the distance between the coordinates derived fro WGS84 and those fro NZGD49.

8 Recoended transforation paraeters fro WGS84 to NZGD49 Page 8 of 15 The residuals are suarised in Table 4. Seven transforation odels are ordered fro the sallest to largest aiu residual horiontal error. Model Residual Horiontal error (etres) verage RMS 95 Percentile Maiu Calculated 7 paraeter Mackie 7 paraeter Calculated 3 paraeter DM MRE forula DM 3 paraeter Latitude/longitude offset No transforation Table 4 : Coparison of the residual horiontal errors due to appling different transforation odels. The ost appropriate ethod to use depends upon the accurac required and the software available. There is little to choose between the first three ethods in ters of accurac - Mackie's paraeters are recoended as the are alread in use. The MRE ethod in principle should be able to give a better result since it can take account of distortion. However we have not recalculated these paraeters using the new control points as the transforation tends to be unstable awa fro the control points and is not widel supported in software. Indeed, the DM (now NIM) have not pursued this ethod since the id 1980 s (G.Stent, pers co). The DM 3 paraeter transforation has little to recoend it copared to the alternative transforation paraeters. For an navigation applications appling the latitude and longitude offset will be ore than adequate if the equipent and software do not support a 3 or 7 paraeter siilarit transforation. This accurac is well within that of stand-alone (non differential) GPS navigation. Figure 1 shows the residual errors at the control points after the Mackie 7 paraeter transforation has been applied to the WGS84 coordinates. s there are significant local trends, ore accurate transforations could be deterined for these local areas but this is beond the scope of this paper. Figure 1 also iplies that surves referenced to a local base station with known NZGD49 coordinates can obtain a significantl better accurac. For eaple it should be straightforward to position to within 0.3 (coonl required in utilit apping) provided suitable equipent and procedures are used. Note that this assues that relative consistenc of NZGD49 is preserved in the breakdown fro second order stations into the local cadastre. The irregular distribution of error across the countr also iplies that siple transforations cannot provide a uch better overall accurac since the cannot adequatel odel the distortion in NZGD49.

9 Recoended transforation paraeters fro WGS84 to NZGD49 Page 9 of 15 4 etres Figure 1: Shows the reaining difference between WGS84 and NZGD49 after the Mackie 7 paraeter transforation has been applied to the WGS84 coordinates. References ppelbau, L.T., Geodetic datu transforation b ultiple regression equations. Proceedings of the Third International Geodetic Sposiu on Satellite Doppler Positioning, New Meico State Universit, Phsical Science Laborator, Las Cruces, New Meico, US, 8-12 Februar. adekas, J., Investigations Related to the Establishent of a World Geodetic Sste. Report No. 124, Departent of Geodetic Science and Surveing, Ohio State Universit, US. oucher, C. and Z. ltaii, Developent of a Conventional Terrestrial Reference Frae, in D.E. Sith and D.L. Turcotte (Eds), Contribution of Space Geodes to Geodnaics: Earth

10 Recoended transforation paraeters fro WGS84 to NZGD49 Page 10 of 15 Dnaics. Geodnaical series Vol. 24, erican Geophsical Union ooks oard, Washington, US. ursa, M., Fundaentals of the Theor of Geoetric Satellite Geodes. Travau de l institut Geophsique de l acadeie Teheco-slovaque des Sciences, Vol DM, 1987a. Suppleent to Departent of Defense World Geodetic Sste 1984 Technical Report: Part I - Methods, Techniques, and Data used in WGS84 developent. DM TR , first edition, Deceber 1. DM, 1987b. Suppleent to Departent of Defense World Geodetic Sste 1984 Technical Report: Part II - Paraeters, Forulas, and Graphics for practical application of WGS84. DM TR , first edition, Deceber 1. DM, Departent of Defense World Geodetic Sste 1984: Its definition and Relationship with Local Geodetic Sstes. DM TR , second edition, Septeber 1. Harve,.R., Transforation of 3D co-ordinates. The ustralian Surveor, June, Vol. 33, No. 2, pp Higgins, M.., Transforation fro WGS84 to GD84 - n Interi Solution. Internal report of the Departent of Mapping and Surveing, Queensland, ustralia. Kuar, M., World geodetic sste 1984: reference frae for global apping, charting and geodetic applications. Surveing and Land Inforation Sstes, Vol. 53, No. 1, pp Lee, L.P., First-Order Geodetic Triangulation of New Zealand and Technical Series No. 1, Dept. Lands and Surve, New Zealand. Mackie, J.., The relationship between the WGS72 doppler satellite datu and the New Zealand Geodetic Datu Report No. 178, Geophsics Division, Departent of Scientific and Industrial Research, New Zealand Mals, S. and J. Slater, Maintenance and enhanceent of the world geodetic sste Proceedings of ION GPS-94, Salt Lake Cit, Utah, US, Septeber. McCarth, D.D. (Ed.), IERS Standards (1992). IERS Technical Note 13, Observatoire de Paris, Paris, France. Morgan, P.,. ock, R. Colean, P. Feng, D. Garrard, G. Johnston, G. Luton,. McDowall, M. Pearse, C. Rios and R. Tiesler, Zero Order GPS Network for the ustralian Region. UNISURV S-46, School of Geoatic Engineering, The Universit of New South Wales, Sdne, ustralia. Morit, H, 1980a. Geodetic reference sste ulletin Geodesique, Vol. 54, No. 3, pp Steed, J., practical approach to transforation between coonl used reference sstes. The ustralian Surveor, Vol. 35, No. 3, pp Swift, E., Iproved WGS84 coordinates for the DM and ir Force GPS tracking sites. Proceedings of ION GPS-94, Salt Lake Cit, Utah, US, Septeber, pp Wolf, H., Geoetric connection and re-orientation of three-diensional triangulation nets. ulletin Geodesique, Vol. 68, pp

11 Recoended transforation paraeters fro WGS84 to NZGD49 Page 11 of 15 ppendi : Transforation ethods There are a nuber of was of defining the relationship between one reference sste and another. The choice of the ost appropriate network transforation odel is influenced b: i) the etents of the area for which it is to be applied ii) iii) iv) the presence of distortion in either of the reference sstes the diensions of the reference sstes (two- or three-diensional), and the accurac requireents. This section focuses on those ethods which have been applied to the New Zealand situation of converting between NZGD49 coordinates and global reference sstes. It is noted that other theoretical transforation ethods are available but paraeters have not been developed (or at least widel disseinated) for New Zealand. One of the ost coonl used transforation ethods in surveing is the siilarit transforation, which preserves the shape, so angles are not changed, but lengths of lines and the position of points a be changed. It assues that there are no ssteatic distortions within either network. The general siilarit transforation is given b: where X Z X = s R F Z X,, Z X,, Z T X, T, T Z R s F TX + T TZ Cartesian coordinates in coordinate sste Cartesian coordinates in coordinate sste translations ters, are the coordinates of the origin of the XZ coordinate sste in the XZ coordinate sste, respectivel 3 3 orthogonal rotation atri (Section.1) scale factor = 1 + s, where s is the differential scale There are seven paraeters which are usuall associated with a siilarit transforation; three rotation angles, three translation coponents and one scale factor. If the rotations are sall, as is epected when both coordinate sstes refer to the sae CTRS, then (1) is approiatel linear and the order of the rotations is uniportant. The siilarit transforation is popular due to: i) the sall nuber of paraeters involved ii) iii) the siplicit of the odel, which is ore easil ipleented into software, and the fact that it is adequate for relating two coordinate sstes which are hoogeneous (no local distortion in scale or orientation). s a result siilarit transforation paraeters have been published to allow the conversion between coordinate sstes used in New Zealand (Section 5). It is therefore necessar to outline soe of the different odels that are available for deterining siilarit transforations, especiall those used for establishing New Zealand transforation paraeters. One disadvantage of the seven paraeter siilarit transforation ethod is that both networks are assued to have onl linear distortions (ecluding shear coponents). Often older terrestrial networks do have non-linear distortions because of the adjustent and surve ethodologies eploed. (1)

12 Recoended transforation paraeters fro WGS84 to NZGD49 Page 12 of 15 Multiple Regression Equations (MRE) are one ethod that attepts to account for non-linear distortions in either of the networks and is outlined in Section.4..1 Coordinate rotation ethods The Cardanian rotation atri is the ost coonl applied ethod of rotating a coordinate sste. If the rotation angles are sall, the order of appling the Cardanian angles to their respective aes does not influence the result. One of the si possible cobinations is that contained in (2). where and R = R Z (ε ) * R (ε ) * R X (ε ) R X, R, R Z ε ε ε cosε sin ε 0 R Z (ε ) = sin ε cos ε R X (ε ) = 0 cos ε sin ε 0 sin ε cos ε rotation atrices about the X,, Z aes respectivel. rotation angles in radians about the X,, Z aes respectivel. Positive rotations are clockwise rotations as viewed looking fro the origin to the positive end of the ais in a right handed coordinate sste. cos ε 0 sin ε R (ε ) = sin ε 0 cos ε (2) Therefore R can now be written as: R = cosε cosε cos ε sin ε sin ε + sin ε cosε sin ε sin ε cos ε sin ε cos ε sin ε cosε cos ε cos ε sin ε sin ε sin ε sin ε sin ε cos ε + cos ε sin ε sin ε cos ε sin ε cos ε cosε (3) For sall rotation angles the rotation atri (3) can be approiated b R 1 ε ε ε 1 ε ε ε 1 (4).2 ursa-wolf odel This odel, presented b ursa (1965) and Wolf (1963), solves for a seven paraeter transforation - a scale factor, three rotation angles and three translation coponents. The ursa-wolf odel is also known in Geodes as the Seven Paraeter Siilarit odel and takes the sae for as the general siilarit transforation of (1). One proble with the ursa-wolf odel is that the adjusted paraeters are highl correlated when the network of points used to deterine the paraeters covers onl a sall portion of the earth.

13 Recoended transforation paraeters fro WGS84 to NZGD49 Page 13 of 15.3 Molodenskii-adekas odel The Molodenskii-adekas odel (adekas, 1969) reoves the high correlation between paraeters b relating the paraeters to the centroid of the network. where X Z X = Z T + T + s T F R X - X - Z - Z X = X i / n = centroid X coordinate for the points in coordinate sste = i / n = centroid coordinate for the points in coordinate sste Z = Z i / n = centroid Z coordinate for the points in coordinate sste T T T Molodenskii-adekas translations ters reaining ters are as defined for the ursa-wolf odel (1) (5) The adjusted coordinates, baseline lengths, scale factor, rotation angles, their Variance Covariance (VCV) atrices and the a posteriori variance factor coputed b this odel are the sae as those fro the corresponding ursa-wolf solution. However, the translations are different and their precisions are generall an order of agnitude saller (Harve, 1986). The difference between the translation ters of the ursa-wolf and Molodenskii-adekas odels is due to the different scaling and rotating of the centroid of the network. This can be seen clearl b epanding (5) to give (6), where P c is a constant ter for all points and obviousl affects the translation ters. X Z T X X X = + T + s Pc F R where Pc = s F R T Z Z Z When transforation paraeters fro the Molodenskii-adekas odel are to be applied to transfor coordinates of points, it is essential to know what values were used for the centroid ( X Z ) when deriving the paraeters. However, in the past the have not alwas been published with the transforation paraeters (eg. Mackie, 1982). It should be noted that when working with a global network of points the Molodenskii-adekas odel has centroid coordinates that equal the centre of the ellipsoid (ie. X = = Z = 0) and therefore reduces to the ursa-wolf odel. (6).4 Multiple Regression Equations (MRE) Method Multiple Regression Equations ethod has the advantage over both the ursa-wolf and Molodenskii- adekas odels of being able to account for non-linear distortion in either of the networks. The ain disadvantage of the MRE ethod is that outside the area of the control points used to deterine the MRE the results can be etreel unreliable. Therefore the control points need to etend to the boundaries of the datu for which the transforation is to appl. There are various fors that an MRE can take, but onl the for used b DM(1987a) will be presented, as this for has been used to deterine transforation paraeters between WGS84 and NZGD49 (Section 5.2). For each coordinate coponent a difference between datu values ( φ, λ,

14 Recoended transforation paraeters fro WGS84 to NZGD49 Page 14 of 15 h) is deterined b an MRE, and this is then applied to the known datu coordinate coponent to obtain the unknown datu coordinate using: and φ = φ + φ λ = λ + λ h = h + h (φλh) known curvilinear coordinates of a station in ters of datu (φλh) unknown datu curvilinear coordinates of the sae station The general for of the difference between the two datu, using an MRE, for the latitude coponent is (DM, 1987a, eqn. 7-14): where φ = U + 2 V + 3 U UV + 5 V V U 9 V + 56 U 8 V U 9 V U 8 V U 9 V U 8 V U 9 V 9 (7) 0, 1,..., 99 U = K (φ - φ ) V = K (λ - λ ) K = 100 possible coefficients deterined in a stepwise ultiple regression procedure with U and V each liited to single digit eponents = noralised geodetic latitude of the coputation point = noralised geodetic longitude of the coputation point = scale factor and degree-to-radian conversion φ, λ = local geodetic latitude and longitude, respectivel, of the coputation point (in degrees) φ, λ = id-latitude and id-longitude values, respectivel, of the local geodetic datu area (in degrees). Siilar equations are obtained for λ and h b replacing φ in the left hand side of (7) b λ and h, respectivel. The coefficients for the MRE were coputed b DM (1987a, p. 7-18) using the following approach. Prior to beginning the developent process, individual φ, λ, h coordinate differences are fored for each station within the datu area that has coordinates in ters of both datu. The ultiple regression procedure of ppelbau (1982) is then initiated to develop separate equations to fit the φ, λ and h coordinate differences. The first step of the procedure produces a constant and a variable. The variable will either be a function of φ or λ, or both. The procedure then sequentiall adds one variable at a tie to the equation. fter a variable is added, all variables previousl incorporated into the equation are tested and, if one is no longer statisticall significant, it is reoved. This stepwise addition or reoval of variables ensures that onl significant variables are retained in the final equation. In keeping with (7), each variable consists of products of powers of noralised geodetic latitude (U), or noralised geodetic longitude (V), or both (ie. U 3 V 4 is a single variable). The stepwise regression procedure continues until the precision desired for the equation is obtained. The DM derived MRE for transforing NZGD49 coordinates to WGS84 coordinates are contained in Section 5.2 for which the desired precision was that the rs difference be approiatel less than 1.5 (DM, 1987a, p. 7-19).

15 Recoended transforation paraeters fro WGS84 to NZGD49 Page 15 of 15 ppendi : Transforation paraeters between ITRS and WGS realisations The transforation paraeters used in this report for converting between different realisations of ITRS and WGS are suarised in Table -1. The application of these paraeters was perfored using the seven-paraeter siilarit transforation (ursa-wolf) forula given in (1) using the siplified rotation atri of (4). Fro To Reference T X WGS84 WGS72 preprint of DM 1987a T T Z s 10-8 ε as ε as ε as WGS84 WGS72 DM 1987a WGS84 WGS84 (G730) WGS84 (G730) Swift ITRF92 Swift ITRF91 ITRF92 oucher et al Table - 1 : Transforation paraeters between ITRS and WGS realisations. It is worth noting that the value given b DM (1987a) for s, to convert WGS84 to WGS72 coordinates, is However, the preprint of DM (1987a and b) stated that s = , which is the value quoted in ustralia (eg. Higgins, 1987 and Steed, 1990) and used within DOSLI to odif Mackie s paraeters (Section 5.3). This difference of approiatel 0.04 at the Earth s surface can be considered insignificant when copared to the accurac obtainable b the Doppler ethod used to establish WGS72 and WGS84.