Time-dependent Position Vector tfm (geocentric)

Formula

Note: These formulas have been transcribed from EPSG Guidance Note #7-2. Users are encouraged to use that document rather than the text which follows as reference because limitations in the transcription will be avoided.

The transformation between source and target CRS geocentric coordinates is usually described as a simplified 7-parameter Helmert transformation, expressed in matrix form with 7 parameters:

(Xt)        (  1     -rZ'  +rY')   (Xs)   (tX')
(Yt) = M' * ( +rZ'    1    -rX') * (Ys) + (tY')
(Zt)        ( -rY'   +rX'    1 )   (Zs)   (tZ')

where 
tX' = tX + dtX (t – t0)
tY' = tY + dtY (t – t0)
tZ' = tZ + dtZ (t – t0)
rX' = rX + drX (t – t0)
rY' = rY + drY (t – t0)
rZ' = rZ + drZ (t – t0)
dS' = dS + ddS (t – t0)
M' = 1 + dS'

and the parameters are defined as:
(tX', tY', tZ'): Translation vector, to be added to the point's position vector in the source coordinate reference system in order to transform from source system to target system; also: the coordinates of the origin of the source coordinate reference system in the target coordinate reference system.

(rX', rY', rZ'): Rotations to be applied to the point's vector.  The sign convention is such that a positive rotation about an axis is defined as a clockwise rotation of the position vector when viewed from the origin of the Cartesian coordinate reference system in the positive direction of that axis; e.g. a positive rotation about the Z-axis only from source system to target system will result in a larger longitude value for the point in the target system.  Although rotation angles may be quoted in any angular unit of measure, the formula as given here requires the angles to be provided in radians.

M': The time-adjusted scale correction to be made to the position vector in the source coordinate reference system in order to obtain the correct scale in the target coordinate reference system. M' = (1 + dS'). When the time-adjusted scale difference dS' is expressed in parts per million, M' = (1 + dS'*10^-6). When the time-adjusted scale difference dS' is expressed in parts per billion, M' = (1 + dS'*10^-9).

Example

Transformation from ITRF2008 to GDA94 at epoch 2013.90.

Input point coordinate reference system: ITRF2008 at epoch 2013.90 (geocentric Cartesian coordinates):

Xs = -3789470.710 m
Ys = 4841770.404 m
Zs = -1690893.952 m
t = 2013.90

Transformation parameter values:

tX = –84.68 mm
dtX = +1.42 mm/yr
tY = –19.42 mm
dtY = +1.34 mm/yr
tZ = +32.01 mm
dtZ = +0.90 mm/yr
rX = +0.4254 msec
drX = –1.5461 msec/yr
rY = –2.2578 msec
drY = –1.1820 msec/yr
rZ = –2.4015 msec
drZ = –1.1551 msec/yr
dS = +0.00971 ppm
ddS = +0.000109 ppm/yr
t0 = 1994.00 
				
First calculate the correction due to rate of change to each of the 7 transformation parameters for the period (t-t0), taking care to convert the translations to the same units as the source CRS (in this case metres) and the rotations to radians:
tX' = –84.68 + [+1.42 * (2013.90-1994.00)] = –56.42 mm = –0.056 m
tY' = –19.42 + [+1.34 * (2013.90-1994.00)] = +7.25 mm = +0.007 m
tZ' = +32.01 + [+0.90 * (2013.90-1994.00)] = +49.92 mm = +0.050 m
rX' = 0.4254 + [–1.5461 * (2013.90-1994.00)] = –30.34 msec = –1.471021E-07 rad
rY' = –2.2578 + [–1.1820 * (2013.90-1994.00)] = –25.78 msec = –1.249830E-07 rad
rZ' = –2.4015 + [–1.1551 * (2013.90-1994.00)] = –25.39 msec = –1.230844E-07 rad
dS' = +0.00971 + [+0.000109 * (2013.90-1994.00)] = +0.01188 ppm

Then M' = 1.00000001188

Using these parameter values, application of the 7 parameter Position Vector transformation to the given (source) ITRF2008 coordinates results in:
Xt = -3789470.004m
Yt =  4841770.686 m
Zt = -1690895.108 m
on the GDA94 (target) geocentric coordinate reference system.

For the reverse transformation from GDA94 coordinates to ITRF08 coordinates at epoch 2013.90, the signs of all parameters need to be reversed except for the reference epoch. Then:
tX' = +84.68 + [(–1.42 * (2013.90 –1994.00)] = +56.42 mm = +0.056 m 
and similarly for the other six parameters. Hence tY' = –0.007 m, tZ' = –0.050 m, rX' = 1.471021E-07 rad, rY' = 1.249830E-07 rad, rZ' = 1.230844E-07 rad and dS' = –0.01188 ppm.

Using these time-adjusted parameters values, M' = 0.99999998812 and then applying the 7 parameter Position Vector transformation fomula to the GDA94 coordinates of 
Xs = –3789470.004 m
Ys = 4841770.686 m
Zs = –1690895.108 m 

results in ITRF08 coordinates at epoch 2013.90 of:

Xt = –3789470.710 m
Yt = 4841770.404 m
Zt = –1690893.952 m

EPSG:1053

Time-dependent Position Vector tfm (geocentric)

Attributes

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Formula

Example