A note on frame transformations with applications to geodetic datums

Abstract

Rigorous equations in compact symbolic matrix notation are introduced to transform coordinates and velocities between ITRF frames and modern GPS-based geocentric geodetic datums. The theory is general but, after neglecting higher than second-order terms, it is shown that the equations revert to the formulation currently applied in most major continental datums. We discuss several examples: the North American Datum of 1983 (NAD83), the European Terrestrial Reference System of 1989 (ETRS89), the Geodetic Datum of Australia of 1994 (GDA94), and the South American Geocentric Reference System (SIRGAS).

Appendix

Part A

A counterclockwise active rotation of vectors (body rotation) by an angle $\dot \Omega$ about a line of direction cosines$\left\{ {\ell _x } \right\}$ can be written in compact matrix notation as

$$ {\bf{R}}_\ell \left( {\dot \Omega } \right) = \left[ I \right] + \sin \dot \Omega \,\underline {\left[ \ell \right]} + \left( {1 - \cos \dot \Omega } \right)\,\underline {\left[ \ell \right]} ^2 $$

(33)

This equation can be reduced by assuming a differential rotation $\dot \Omega$, substituting $\sin \dot \Omega \approx \dot \Omega$, $\cos \dot \Omega \approx 1$, and taking into consideration Eq. (14). The result is

$$ {\bf{R}}\delta _\ell \left( {\dot \Omega } \right) = \left[ I \right] + \underline {\left[ {\dot \Omega } \right]} $$

(34)

It can be easily proved by simple geometric arguments that an active counterclockwise rotation of vector coordinates is equivalent to a passive clockwise rotation of frames. However, if we assume that all rotations (active and passive) are positive when rotated in the counterclockwise sense, a passive counterclockwise rotation by an angle θ around the three frames i=1,2,3, using the transpose of Eq. (33) we can write:

$$R_i \left( \theta \right) = \left[ I \right] + \sin \theta \,\underline {\left[ {\ell _i } \right]} ^t + \left( {1 - \cos \theta } \right)\,\underline {\left[ {\ell _i } \right]} ^2 $$

(35)

The fact that $\underline {\left[ {\ell _i } \right]} ^2$ is a symmetric matrix was taken into consideration to write Eq. (35). The symbols $\ell _i ;\;\,i = 1,2,3$ correspond to the three direction cosines of the three Cartesian axes, i.e., $\ell _1 = \left\{ {1\;\;\;0\;\;\;0} \right\}^t$; $\ell _2 = \left\{ {0\;\;\;1\;\;\;0} \right\}^t$; and $\ell _3 = \left\{ {0\;\;\;0\;\;\;1} \right\}^t$. For example, if we want to obtain the standard counterclockwise rotation about the first axis denoted as R ₁ (θ), it follows from Eq. (35) that

$$R_1 \left( \theta \right) = \left[ {\matrix{ 1 & 0 & 0 \cr 0 & 1 & 0 \cr 0 & 0 & 1 \cr } } \right] + \sin \theta \left[ {\matrix{ 0 & 0 & 0 \cr 0 & 0 & 1 \cr 0 & { - 1} & 0 \cr } } \right] + \left( {1 - \cos \theta } \right)\left[ {\matrix{ 0 & 0 & 0 \cr 0 & { - 1} & 0 \cr 0 & 0 & { - 1} \cr } } \right] = \left[ {\matrix{ 1 & 0 & 0 \cr 0 & {\cos \theta } & {\sin \theta } \cr 0 & { - \sin \theta } & {\cos \theta } \cr } } \right]$$

(36)

which reduces to the well-known definitions of counterclockwise rotation around coordinate axes R _i (θ), i=1,2,3 prevalent on the geodetic literature (Kaula 1966, p. 13; Mueller 1969, p. 43). The reader should be reminded that the counterclockwise positive sense for rotations and the basic rotation matrices described above are adopted by most scientists, and it is even assumed in the formulation presented in the set of IERS conventions when discussing rotation of frames due to precession, nutation, and polar motion.

Part B

The following partial derivatives based on Eqs. (8) and (10) are rigorous (no high-order terms are neglected). This new set of partial derivatives should replace the original ones previously given in Soler and Marshall (2002) in order to have a consistent set of transformation equations, and accompanying partial derivatives, totally independent of any approximations. In our notation $\Im$ is associated with the functional relationship of Eq. (8); similarly, ${\cal A}$ is associated with Eq. (10).

$$\left[ {\delta \Re } \right] = \left[ I \right] + \underline {\left[ {\varepsilon \left( {t_k } \right)} \right]} ^t $$

$${{\partial \Im } \mathord{\left/ {\vphantom {{\partial \Im } {\partial \left\{ x \right\}}}} \right. \kern-\nulldelimiterspace} {\partial \left\{ x \right\}}} = \left( {1 + s\left( {t_k } \right)} \right)\left[ {\delta \Re } \right] + \left( {t_D - t_k } \right)\left[\kern-0.15em\left[ {\left( {1 + s\left( {t_k } \right)} \right)\underline {\left[ {\dot \varepsilon } \right]} ^t + \dot s\left[ {\delta \Re } \right]} \right]\kern-0.15em\right] + \left( {t_D - t} \right)^2 \dot s\underline {\left[ {\dot \varepsilon } \right]} ^t = \left[ {\partial x} \right]$$

$${{\partial \Im } \mathord{\left/ {\vphantom {{\partial \Im } {\partial \left\{ {v_x } \right\}}}} \right. \kern-\nulldelimiterspace} {\partial \left\{ {v_x } \right\}}} = \left( {t_D - t} \right)\left[\kern-0.15em\left[ {\left( {1 + s\left( {t_k } \right)} \right)\left[ {\delta \Re } \right] + \left( {t_D - t_k } \right)\left[\kern-0.15em\left[ {\left( {1 + s\left( {t_k } \right)} \right)\underline {\left[ {\dot \varepsilon } \right]} ^t + \dot s\left[ {\delta \Re } \right]} \right]\kern-0.15em\right] + \left( {t_D - t} \right)^2 \dot s\underline {\left[ {\dot \varepsilon } \right]} ^t } \right]\kern-0.15em\right] = \left[ {\partial v_x } \right]$$

$${{\partial \Im } \mathord{\left/ {\vphantom {{\partial \Im } {\partial \left\{ {T_x \left( {t_k } \right)} \right\}}}} \right. \kern-\nulldelimiterspace} {\partial \left\{ {T_x \left( {t_k } \right)} \right\}}} = \left[ I \right]$$

$${{\partial \Im } \mathord{\left/ {\vphantom {{\partial \Im } {\partial \left\{ {\varepsilon \left( {t_k } \right)} \right\}}}} \right. \kern-\nulldelimiterspace} {\partial \left\{ {\varepsilon \left( {t_k } \right)} \right\}}} = \left( {\left( {1 + s\left( {t_k } \right)} \right) + \left( {t_D - t_k } \right)\dot s} \right)\left[\kern-0.15em\left[ {\underline {\left[ x \right]} + \left( {t_D - t} \right)\underline {\left[ {v_x } \right]} } \right]\kern-0.15em\right] = \left[ {\partial \varepsilon } \right]$$

$${{\partial \Im } \mathord{\left/ {\vphantom {{\partial \Im } {\partial s\left( {t_k } \right)}}} \right. \kern-\nulldelimiterspace} {\partial s\left( {t_k } \right)}} = \left[\kern-0.15em\left[ {\left[ {\delta \Re } \right] + \left( {t_D - t_k } \right)\underline {\left[ {\dot \varepsilon } \right]} ^t } \right]\kern-0.15em\right]\left\{ {\left\{ x \right\} + \left( {t_D - t} \right)\left\{ {v_x } \right\}} \right\} = \left\{ {\partial s} \right\}$$

$${{\partial \Im } \mathord{\left/ {\vphantom {{\partial \Im } {\partial \left\{ {\dot T_x } \right\}}}} \right. \kern-\nulldelimiterspace} {\partial \left\{ {\dot T_x } \right\}}} = \left( {t_D - t} \right)\left[ I \right]$$

$${{\partial \Im } \mathord{\left/ {\vphantom {{\partial \Im } {\partial \{ \dot \varepsilon \} }}} \right. \kern-\nulldelimiterspace} {\partial \{ \dot \varepsilon \} }} = \left( {\left( {t_D - t_k } \right)\left( {1 + s\left( {t_k } \right)} \right) + \left( {t_D - t_k } \right)^2 \dot s} \right)\left[\kern-0.15em\left[ {\underline {\left[ x \right]} + \left( {t_D - t} \right)\underline {\left[ {v_x } \right]} } \right]\kern-0.15em\right] = \left[ {\partial \dot \varepsilon } \right]$$

$${{\partial \Im } \mathord{\left/ {\vphantom {{\partial \Im } {\partial \dot s}}} \right. \kern-\nulldelimiterspace} {\partial \dot s}} = \left[\kern-0.15em\left[ {\left( {t_D - t_k } \right)\left[ {\delta \Re } \right] + \left( {t_D - t} \right)^2 \underline {\left[ {\dot \varepsilon } \right]} ^t } \right]\kern-0.15em\right]\left\{ {\left\{ x \right\} + \left( {t_D - t} \right)\left\{ {v_x } \right\}} \right\} = \left\{ {\partial \dot s} \right\}$$

The partials of the functional relationship (10) with respect to the 14 transformation parameters are:

$${{\partial {\cal A}} \mathord{\left/ {\vphantom {{\partial {\cal A}} {\partial \left\{ x \right\}}}} \right. \kern-\nulldelimiterspace} {\partial \left\{ x \right\}}} = {{\partial \left[ {\partial x} \right]} \mathord{\left/ {\vphantom {{\partial \left[ {\partial x} \right]} {\partial t_D }}} \right. \kern-\nulldelimiterspace} {\partial t_D }} = \left( {1 + s\left( {t_k } \right)} \right)\underline {\left[ {\dot \varepsilon } \right]} ^t + \dot s\left[ {\delta \Re } \right] + 2\left( {t_D - t_k } \right)\dot s\underline {\left[ {\dot \varepsilon } \right]} ^t = \left[ {\overline \partial x} \right]$$

$$ \matrix{ {{{\partial {\cal A}} \mathord{\left/ {\vphantom {{\partial {\cal A}} {\partial \left\{ {v_x } \right\}}}} \right. \kern-\nulldelimiterspace} {\partial \left\{ {v_x } \right\}}}} \hfill & { = {{\partial \left[ {\partial v_x } \right]} \mathord{\left/ {\vphantom {{\partial \left[ {\partial v_x } \right]} {\partial t_D }}} \right. \kern-\nulldelimiterspace} {\partial t_D }} = \left( {1 + s\left( {t_k } \right)} \right)\left[ {\delta \Re } \right] + \left( {2t_D - \left( {t_k + t} \right)} \right)\left[\kern-0.15em\left[ {\left( {1 + s\left( {t_k } \right)} \right)\underline {\left[ {\dot \varepsilon } \right]} ^t + \dot s\left[ {\delta \Re } \right]} \right]\kern-0.15em\right]} \hfill \cr {} \hfill & { + \left( {3\left( {t_D^2 + t_k^2 } \right) - 4t_D t_k + 2t\left( {t_k - t_D } \right)} \right)\dot s\underline {\left[ {\dot \varepsilon } \right]} ^t = \left[ {\bar \partial v_x } \right]} \hfill \cr } $$

$$\partial {\cal A}/\partial \left\{ {T_x \left( {t_k } \right)} \right\} = \left[ 0 \right]$$

$$ \partial {\cal A}/\partial \left\{ {\varepsilon \left( {t_k } \right)} \right\} = \partial \left[ {\partial \varepsilon } \right]/\partial t_D = \left( {1 + s\left( {t_k } \right)} \right)\left[ {v_x } \right] + \dot s\underline {\left[ x \right]} + \left( {2t_D - \left( {t_k + t} \right)} \right)\,\underline {\left[ {v_x } \right]} = \left[ {\overline \partial \varepsilon } \right] $$

$$ \partial {\cal A}/\partial s\left( {t_k } \right) = \partial \left\{ {\partial s} \right\}/\partial t_D = \left[ {\delta \Re } \right]\left\{ {v_x } \right\} + \underline {\left[ {\dot \varepsilon } \right]} ^t \left\{ {\left\{ x \right\} + \left( {2t_D - \left( {t_k + t} \right)} \right)\left\{ {v_x } \right\}} \right\}\, = \left\{ {\bar \partial s} \right\} $$

$$ \partial {\cal A}/\partial \left\{ {\dot T_x } \right\} = \left[ I \right] $$

$$\matrix{ {\partial {\cal A}/\partial \left\{ {\dot \varepsilon } \right\} = \partial \left[ {\partial \dot \varepsilon } \right]/\partial t_D } \hfill & { = \left( {1 + s\left( {t_k } \right)} \right)\underline {\left[ x \right]} + \left( {2t_D - \left( {t_k - t} \right)} \right)\,\underline {\left[ {v_x } \right]} } \hfill \cr {} \hfill & { + \dot s2\left( {t_D - t_k } \right)\,\underline {\left[ x \right]} + \left( {3t_D^2 + t_k^2 - 4t_D t_k + 2t\left( {t_k - t_D } \right)} \right)\,\underline {\left[ {v_x } \right]} = \left[ {\bar \partial \dot \varepsilon } \right]} \hfill \cr } $$

$$\matrix{ {\partial {\cal A}/\partial \dot s = \partial \{ \partial \dot s\} /\partial t_D } \hfill & { = \left[ {\delta \Re } \right]\left\{ {\left\{ x \right\} + \left( {2t_D - \left( {t_k + t} \right)} \right)\left\{ {v_x } \right\}} \right\}} \hfill \cr {} \hfill & { + \underline {\left[ {\dot \varepsilon } \right]} ^t \left\{ {2\left( {t_D - t_k } \right)\left\{ x \right\} + \left( {3t_D^2 + t_k^2 - 4t_D t_k + 2t\left( {t_k - t_D } \right)} \right)\left\{ {v_x } \right\}} \right\} = \left\{ {\bar \partial \dot s} \right\}} \hfill \cr } $$