Rank defect analysis and the realization of proper singularity in normal equations of geodetic networks

Abstract

The singularity of input normal equations (NEQ) is a crucial element for their optimal handling in the context of terrestrial reference frame (TRF) estimation under the minimal-constraint framework. However, this element is often missing in the recovered NEQ from SINEX files after the usual deconstraining based on the stated information for the stored solutions. The same setback also occurs with the original NEQ that are formed by the least-squares processing of space geodetic data due to the datum information which is carried by various modeling choices and/or software-dependent procedures. In the absence of this datum-related singularity, it is not possible to obtain genuine minimally constrained solutions because of the interference between the input NEQ’s content and the external datum conditions, a fact that may alter the geometrical information of the original measurements and can cause unwanted distortions in the estimated solution. The main goal of this paper is the formulation of a filtering scheme to enforce the proper (or desired) singularity in the input NEQ with regard to datum parameters that will be handled by the minimal-constraint setting in TRF estimation problems. The importance of this task is extensively discussed and justified with the help of several numerical examples in different GNSS networks.

Appendix

The equivalency between the CDR filtering that was presented in Sect. 4.1 and the related methodology for deriving datum-free NEQ according to Bloßfeld (2015, p. 31) will be demonstrated here.

Let us consider a normal system ${\varvec{N}}\left( {{\varvec{X}}-{\varvec{X}}_o } \right) ={\varvec{u}}$ that is obtained from the analysis of space geodetic observations for estimating the station positions ${\varvec{X}}$ in a geodetic network with respect to a target frame. It is assumed that this system has full rank due to datum information which is present in the form of (unknown) minimal constraints. This setting can cover two different cases of “input NEQ” with practical relevance for TRF estimation problems, namely:

• 1. minimally constrained NEQ recovered from SINEX files which do not report the information for the applied constraints in the stored solutions; or

• 2. deconstrained NEQ recovered from SINEX files which report the information for the applied constraints in the stored solutions.

The former case is considered in Bloßfeld (2015) and Rebischung (2014), yet the latter is also of interest as it could require the implementation of an additional datum removal scheme (see the discussion and examples in Sect. 2).

The normal system ${\varvec{N}}\left( {{\varvec{X}}-{\varvec{X}}_o } \right) ={\varvec{u}}$ can be algebraically associated with a “fictitious” system of observation equations, in the sense that

$$\begin{aligned} {\varvec{N}}={\varvec{A}}^{T}{\varvec{PA}}, \quad {\varvec{u}}={\varvec{A}}^{T}{\varvec{Pb}} \end{aligned}$$

(19)

where ${\varvec{A}}$, ${\varvec{P}}$ and ${\varvec{b}}$ stem from a full-rank linear Gauss–Markov model

$$\begin{aligned} {\varvec{b}}={\varvec{A}}\left( {{\varvec{X}}-{\varvec{X}}_o } \right) +{\varvec{v}}, \quad {\varvec{v}}{ \sim }(\mathbf{0},\sigma ^{2}{\varvec{P}}^{-1}) \end{aligned}$$

(20)

The selective removal of datum information from the normal system can be implemented by introducing an artificial frame-related rank defect in the above system of observation equations. This is achieved via a simple re-parameterization using the Helmert transformation model

$$\begin{aligned} {\varvec{X}}={{\varvec{X}}}'+{\varvec{E}}^{T}{\varvec{\theta }} \end{aligned}$$

(21)

where the elements of ${\varvec{\theta }}$ (and the rows of the transformation matrix ${\varvec{E}})$ correspond to the datum parameters that we wish to filter out. By substituting Eq. (21) into Eq. (20), we obtain the extended system of observation equations

$$\begin{aligned} {\varvec{b}}={\varvec{A}}\left( {{{\varvec{X}}}'-{\varvec{X}}_o } \right) +{\varvec{AE}}^{T}{\varvec{\theta }} +{\varvec{v}}, \quad {\varvec{v}}{ \sim }(\mathbf{0},\sigma ^{2}{\varvec{P}}^{-1}) \end{aligned}$$

(22)

which, in turn, is linked with the augmented normal system

$$\begin{aligned} \left[ {{\begin{array}{cc} {\varvec{N}}&{} {{\varvec{NE}}^{T}} \\ {{\varvec{EN}}}&{} {{\varvec{ENE}}^{T}} \\ \end{array} }} \right] \left[ {{\begin{array}{c} {{{\varvec{X}}}'-{\varvec{X}}_o } \\ {\varvec{\theta }} \\ \end{array} }} \right] =\left[ {{\begin{array}{c} {\varvec{u}} \\ {{\varvec{Eu}}} \\ \end{array} }} \right] \end{aligned}$$

(23)

Obviously the above system retains the same information about the network’s geometrical characteristics as the original system ${\varvec{N}}\left( {{\varvec{X}}-{\varvec{X}}_o } \right) ={\varvec{u}}$, yet it is singular since the parameter vectors ${{\varvec{X}}}'$ and ${\varvec{\theta }}$ cannot be separately estimated from the same observations.

If we reduce the (unknown) datum parameters from Eq. (23), then the following “filtered” normal system is derived

$$\begin{aligned}&\left( {{\varvec{N}}-{\varvec{NE}}^{T}({\varvec{ENE}}^{T})^{-1}{\varvec{EN}}} \right) \left( {{{\varvec{X}}}'-{\varvec{X}}_o } \right) \nonumber \\&\quad ={\varvec{u}}-{\varvec{NE}}^{T}({\varvec{ENE}}^{T})^{-1}{\varvec{Eu}} \end{aligned}$$

(24)

or, equivalently

$$\begin{aligned}&\left( {{\varvec{I}}-{\varvec{NE}}^{T}({\varvec{ENE}}^{T})^{-1}{\varvec{E}}} \right) {\varvec{N}}\left( {{{\varvec{X}}}'-{\varvec{X}}_o } \right) \nonumber \\&\quad =\left( {{\varvec{I}}-{\varvec{NE}}^{T}({\varvec{ENE}}^{T})^{-1}{\varvec{E}}} \right) {\varvec{u}} \end{aligned}$$

(25)

The last equation is identical to the reconstructed singular NEQ according to Eqs. (9) and (12) of the present paper.