EPSG guidance note #7, http://www.epsg.org
Note: These formulas have been transcribed from EPSG Guidance Note #7. Users are encouraged to use that document rather than the text which follows as reference because limitations in the transcription will be avoided. XT = XT0 + XS . k . dSX . cos q + YS . k . dSY . sin q YT = YT0 – XS . k . dSX . sin q + YS . k . dSY . cos q where: XT0 ,YT0 = the coordinates of the origin point of the source coordinate reference system, expressed in the target coordinate reference system; dSX , dSY = the length of one unit of the source axis, expressed in units of the target axis, for the X axes and the Y- axes respectively; k = point scale factor of the target coordinate reference system in a chosen reference point; q = the angle through which the source coordinate reference system axes must be rotated to coincide with the target coordinate refderence system axes (counter-clockwise is positive). Alternatively, the bearing (clockwise positive) of the source coordinate reference system Y-axis measured relative to target coordinate reference system north.
Source coordinate system: imaginary 3D seismic acquisition bin grid. The two axes are orthogonal, but the unit on the I-axis is 25 metres, whilst the unit on the J-axis is 12.5 metres. The target projected coordinate system is WGS 84 / UTM Zone 31N and the origin of the bin grid (centre of bin 0,0) is defined at E = 456781.0, N = 5836723.0. The projected coordinate system point scale factor at the bin grid origin is 0.99984. The map grid bearing of the I and J axes are 110* and 20* respectively. Thus the angle through which both the positive I and J axes need to be rotated to coincide with the positive Easting axis and Northing axis respectively is +20 degrees. Hence: XT0 , = 456 781.0 m YT0 = 5 836 723.0 m dSX = 25 dSY = 12.5 k = 0.99984 q = +20 degrees Forward calculation for centre of bin with coordinates: I = 300, J = 247: XT = Easting = XT0 + XS . k . dSX . cos q + YS . k . dSY . sin q = 464 855.62 m. YT = Northing = YT0 – XS . k . dSX . sin q + YS . k . dSY . cos q = 5 837 055.90 m Reverse calculation for this point: XS = [( XT – XT0) . cos qY – (YT – YT0) . sin qY ] / [k . dSX . cos (qX – qY)] = 230 bins YS = [(XT – XT0) . sin qX + (YT – YT0) . cos qX ] / [k . dSY . cos (qX – qY)] = 162 bins