Transformations and OSGM02 user guide - Leica Geosystems

The OSi/OSNI polynomial transformationTo some extent distortions within traditional triangulation networks are inevitable. Within the triangulation network of theRepublic of Ireland and Northern Ireland these distortions are not generally significant; however, regional distortions dooccur. A 3 rd order polynomial transformation has been developed to model these distortions.A polynomial expression was fitted to the coordinate differences of a number of points in the different coordinatereference systems. This is a one-dimensional fitting method that is applied to the geographical coordinate, requiringindependent parameters to be computed for both latitude and longitude.In general, the polynomial model can be expressed as:∆φ = ∑∑ Α ij(φ−φ m) i (λ−λ m) j∆ λ=∑∑ Β ij(φ−φ m) i (λ−λ m) jThe fully expanded forms of the 3 rd order polynomial are as follows:∆φ = [A 00 + A 10 U + A 01 V + A 11 UV + A 20 U 2 + A 02 V 2 + A 21 U 2 V + A 12 UV 2 + A 22 U 2 V 2 + A 30 U 3 + A 03 V 3+ A 31 U 3 V + A 13 UV 3 + A 32 U 3 V 2 + A 23 U 2 V 3 + A 33 U 3 V 3 ] / 3600∆λ = [B 00 + B 10 U + B 01 V + B 11 UV + B 20 U 2 + B 02 V 2 + B 21 U 2 V + B 12 UV 2 + B 22 U 2 V 2 + A 30 U 3 + B 03 V 3 +B 31 U 3 V + B 13 UV 3 + B 32 U 3 V 2 + B 23 U 2 V 3 + B 33 U 3 V 3 ] / 3600Where A ijand B ijare the computed parameters, and U and V are the normalised coordinates calculated as follows:U = k 0 (φ - φ m ) and V = k 0 (λ - λ m )Where φ m and λ m are the coordinates of the approximate centre of the region. The parameters A ij, B ij,K 0 , φ m and λ m aregiven in table 2 below. The transformed geographical coordinates are then obtained as follows:λ ETRS = λ IG + ∆λ and φ ETRS = φ IG + ∆φThe reverse transformation from ETRS89 to Irish Grid cannot be calculated directly and requires iteration.Conversions between geographical and grid coordinates are computed using standard Transverse Mercator projectionformulae in association with the published Irish Grid parameters.Transformations and OSGM02 user guide chapter 3v2.1 9/2002 © crown copyrightPage 23