Fast error analysis of continuous GNSS observations with missing data

Abstract

One of the most widely used method for the time-series analysis of continuous Global Navigation Satellite System (GNSS) observations is Maximum Likelihood Estimation (MLE) which in most implementations requires $\mathcal{O }(n^3)$ operations for $n$ observations. Previous research by the authors has shown that this amount of operations can be reduced to $\mathcal{O }(n^2)$ for observations without missing data. In the current research we present a reformulation of the equations that preserves this low amount of operations, even in the common situation of having some missing data.Our reformulation assumes that the noise is stationary to ensure a Toeplitz covariance matrix. However, most GNSS time-series exhibit power-law noise which is weakly non-stationary. To overcome this problem, we present a Toeplitz covariance matrix that provides an approximation for power-law noise that is accurate for most GNSS time-series.Numerical results are given for a set of synthetic data and a set of International GNSS Service (IGS) stations, demonstrating a reduction in computation time of a factor of 10–100 compared to the standard MLE method, depending on the length of the time-series and the amount of missing data.

Appendix: Derivation of key equations

In this appendix we give the derivations of Eqs. (7) and (8) that were presented in Sect. 2. These derivation make use of Jacobi’s determinant identity and the formula for the inverse of a block matrix (Brualdi and Schneider 1983). To start, assume that our square covariance matrix $\mathbf C$, size $n\times n$, can be partitioned into four sub-matrices:

$$\begin{aligned} \mathbf{C } = \begin{pmatrix} \mathbf{C }_{oo}&\mathbf{C }_{om} \\ \mathbf{C }_{mo}&\mathbf{C }_{mm} \end{pmatrix} \end{aligned}$$

(33)

where the subscripts $m$ and $o$ represent the rows and columns of missing and observed data respectively. Consequently, $\mathbf{C }_{mm}$ is a square $m\times m$ matrix.

Using Gaussian elimination we can reduce $\mathbf{C }_{mo}$ to a zero matrix:

$$\begin{aligned} \begin{pmatrix} \mathbf{I }^{n-m}&\mathbf{0 } \\ -\mathbf{C }_{mo}\mathbf{C }_{oo}^{-1}&\mathbf{I }^{m} \end{pmatrix} \begin{pmatrix} \mathbf{C }_{oo}&\mathbf{C }_{om} \\ \mathbf{C }_{mo}&\mathbf{C }_{mm} \end{pmatrix} = \begin{pmatrix} \mathbf{C }_{oo}&\mathbf{C }_{om} \\ \mathbf{0 }&\mathbf{C }^{\prime }_{mm} \end{pmatrix} \end{aligned}$$

(34)

where $\mathbf{C }^{\prime }_{mm}=\mathbf{C }_{mm}-\mathbf{C }_{mo}\mathbf{C }_{oo}^{-1} \mathbf{C }_{om}$ which is called the Schur complement of $\mathbf{C }_{oo}$ in $\mathbf C$. When we take the determinant of Eq. (34), we obtain:

$$\begin{aligned} \text{ det}(\mathbf{C })=\text{ det}(\mathbf{C }_{oo})\,\text{ det}(\mathbf{C }^{\prime }_{mm}) \end{aligned}$$

(35)

Using the matrices $\breve{\mathbf{C }}$ (=$\mathbf{C }_{oo}$) and $\mathbf F$ that were introduced in Sect. 2, we can rewrite Eq. (35) as:

$$\begin{aligned} \text{ det}(\mathbf{C })=\text{ det}(\breve{\mathbf{C }})\, \text{ det}\left((\mathbf{F }^\mathrm{T}\mathbf{C }^{-1}\mathbf{F })^{-1}\right) \end{aligned}$$

(36)

The last term of Eq. (36) can be obtained by realising that the two multiplications with matrix $\mathbf F$ do nothing more than selecting a submatrix of matrix $\mathbf{C }^{-1}$. The relation with $\mathbf{C }^{\prime }_{mm}$ can be found by looking at the formula for the matrix inverse for a block matrix (Brualdi and Schneider 1983):

$$\begin{aligned}&\mathbf{C }&^{-1}=\begin{pmatrix} \mathbf{C }_{oo}^{-1} \!+\! \mathbf{C }_{oo}^{-1} \mathbf{C }_{om}(\mathbf{C }^{\prime }_{mm})^{-1}\mathbf{C }_{mo}\mathbf{C }_{oo}^{-1}&\!-\!\mathbf{C }_{oo}^{-1}\mathbf{C }_{om}(\mathbf{C }^{\prime }_{mm})^{-1}\nonumber \\ \!-\!(\mathbf{C }^{\prime }_{mm})^{-1}\mathbf{C }_{mo}\mathbf{C }_{oo}^{-1}&(\mathbf{C }^{\prime }_{mm})^{-1} \end{pmatrix}\\ \end{aligned}$$

(37)

one will note the equivalence of this submatrix with the inverse of the Schur complement. Thus, we have:

$$\begin{aligned} \mathbf{C }^{\prime }_{mm} = \left(\mathbf{F }^\mathrm{T}\mathbf{C }^{-1}\mathbf{F }\right)^{-1} \end{aligned}$$

(38)

By taking the logarithm of Eq. (36), and using the relation $\text{ det}(\mathbf{C })=1/\text{ det}(\mathbf{C }^{-1})$, Eq. (8) of Sect. 2 is obtained.

Of course the missing data will normally not occur after all observations have been made. Determinants have the property that rows and columns of the matrix can be interchanged without changing the value except for a change in sign. However, when rows $i$ and $j$ are swapped of the covariance matrix, the columns $i$ and $j$ are swapped at the same time which means that the sign does not change. Therefore, Eq. (8) is valid for any sequence of missing data.

Next, using Eq. (38) we have:

$$\begin{aligned} \mathbf{F }(\mathbf{F }^\mathrm{T}\mathbf{C }^{-1}\mathbf{F })^{-1}\mathbf{F }^\mathrm{T} = \begin{pmatrix} \mathbf{0 }&\mathbf{0 } \\ \mathbf{0 }&\mathbf{C }^{\prime }_{mm} \end{pmatrix} \end{aligned}$$

(39)

Using Eq. (37), this leads to the following result:

$$\begin{aligned}&\mathbf{C }^{-1}\mathbf{F }(\mathbf{F }^\mathrm{T}\mathbf{C }^{-1}\mathbf{F })^{-1}\mathbf{F }^\mathrm{T}\mathbf{C }^{-1} \nonumber \\&\quad = \begin{pmatrix} \mathbf{C }_{oo}^{-1}\mathbf{C }_{om}(\mathbf{C }^{\prime }_{mm})^{-1} \mathbf{C }_{mo}\mathbf{C }_{oo}^{-1}&-\mathbf{C }_{oo}^{-1}\mathbf{C }_{om}(\mathbf{C }^{\prime }_{mm})^{-1}\\ -(\mathbf{C }^{\prime }_{mm})^{-1}\mathbf{C }_{mo}\mathbf{C }_{oo}^{-1}&(\mathbf{C }^{\prime }_{mm})^{-1} \end{pmatrix}\nonumber \\ \end{aligned}$$

(40)

Changing the sign of Eq. (40) and adding $\mathbf{C }^{-1}$ finally gives us:

$$\begin{aligned} \mathbf{C }^{-1} - \mathbf{C }^{-1}\mathbf{F }(\mathbf{F }^\mathrm{T}\mathbf{C }^{-1}\mathbf{F })^{-1} \mathbf{F }^\mathrm{T}\mathbf{C }^{-1} = \begin{pmatrix} \mathbf{C }_{oo}^{-1}&\mathbf{0 } \\ \mathbf{0 }&\mathbf{0 } \end{pmatrix} \end{aligned}$$

(41)

This provides us the proof for Eq. (7) in Sect. 2:

$$\begin{aligned} {\breve{\mathbf{r }}}^\mathrm{T} {\breve{\mathbf{C }}}^{-1}{\breve{\mathbf{r }}} = \mathbf{r }_o^\mathrm{T} \left( {\mathbf{C }^{-1} - \mathbf{C }^{-1} \mathbf{F } (\mathbf{F }^\mathrm{T}\mathbf{C }^{-1} \mathbf{F })^{-1} \mathbf{F }^\mathrm{T} \mathbf{C }^{-1} }\right) \mathbf{r }_o \end{aligned}$$

(42)

Swapping rows $i$ and $j$ of the covariance matrix $\mathbf C$ will cause a swap of columns $i$ and $j$ of matrix $\mathbf{C }^{-1}$. Swapping the columns of $\mathbf C$ will cause a swap of the rows of $\mathbf{C }^{-1}$ in a similar way. Thus, we can reshuffle our set of observations and missing data, while at the same time adjusting matrix $\mathbf F$, to obtain the form of Eq. (34) which proves the general validity of Eq. (42) for any sequence of missing data.