EPSG Dataset :: v12.003

Time-dependent Position Vector tfm (geocentric)

Coordinate Operation Method Details [VALID]

Name:

Time-dependent Position Vector tfm (geocentric)

Code:

1053

Operation is Reversible:

Yes

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

Method
Parameters:

Parameter Name	Parameter Code	Sign reversal	Parameter Description
X-axis translation	8605	Yes	The difference between the X values of a point in the target and source coordinate reference systems.
Y-axis translation	8606	Yes	The difference between the Y values of a point in the target and source coordinate reference systems.
Z-axis translation	8607	Yes	The difference between the Z values of a point in the target and source coordinate reference systems.
X-axis rotation	8608	Yes	The angular difference between the Y and Z axes directions of target and source coordinate reference systems. This is a rotation about the X axis as viewed from the origin looking along that axis. The particular method defines which direction is positive, and what is being rotated (point or axis).
Y-axis rotation	8609	Yes	The angular difference between the X and Z axes directions of target and source coordinate reference systems. This is a rotation about theY axis as viewed from the origin looking along that axis. The particular method defines which direction is positive, and what is being rotated (point or axis).
Z-axis rotation	8610	Yes	The angular difference between the X and Y axes directions of target and source coordinate reference systems. This is a rotation about the Z axis as viewed from the origin looking along that axis. The particular method defines which direction is positive, and what is being rotated (point or axis).
Scale difference	8611	Yes	Scale factor for source CRS axes minus one, where the scale factor is a multiplication factor to be applied to coordinates in the source CRS to obtain the correct scale in the target CRS. dS = (M – 1). If a distance of 100 km in the source CRS translates into a distance of 100.001 km in the target CRS, the scale difference is 1 ppm (the ratio being 1.000001). When the scale difference dS is expressed in parts per million (ppm), M = (1 + dS10-6). When the scale difference dS is expressed in parts per billion (ppb), M = (1 + dS10-9).
Rate of change of X-axis translation	1040	Yes	Time first derivative of X-axis translation.
Rate of change of Y-axis translation	1041	Yes	Time first derivative of Y-axis translation.
Rate of change of Z-axis translation	1042	Yes	Time first derivative of Z-axis translation.
Rate of change of X-axis rotation	1043	Yes	Time first derivative of X-axis rotation.
Rate of change of Y-axis rotation	1044	Yes	Time first derivative of Y-axis rotation.
Rate of change of Z-axis rotation	1045	Yes	Time first derivative of Z-axis rotation.
Rate of change of scale difference	1046	Yes	Time first derivative of scale difference.
Parameter reference epoch	1047	No	The reference epoch for the parameters of a time-dependent transformation.

Meta Data

Remarks:

Note the analogy with the Time-dependent Coordinate Frame rotation (code 1056) but beware of the differences! The Position Vector convention is used by IAG. See method codes 1054 and 1055 for similar methods operating between other CRS types.

Information Source:

EPSG guidance note #7-2, http://www.epsg.org

Data Source:

EPSG

Revision Date:

February 21, 2019

Change ID:

[2013.021] [2013.065] [2014.068] [2018.001] [2019.006]

Alias:

Alias	Naming System	Remarks
Time-dependent Position Vector transformation (geocentric domain)	EPSG alias

Wildcard	Example
% or *	Any wildcard characters
? or _	One wildcard character
[ - ]	[ - ] Range e.g. N[a-c] will return all geodetic objects that contains a string starting with the letter 'N' or 'n' followed by any of the letters contained within the range specified within the square bracket, e.g. letters a, b and c.
\|	Or e.g. E[a\|e]M will return all geodetic objects that contains a string starting with the letter E(e), ends with the letter M(m) and also contain either the letter a or letter e.