A 4D tomographic ionospheric model to support PPP-RTK

Abstract

Successful implementation of integer ambiguity resolution enabled precise point positioning (aka PPP-RTK) algorithms is inextricably linked to the ability of a user to perform near real-time positioning by quickly and reliably resolving the integer carrier-phase ambiguities. In the PPP-RTK technique, a major barrier to successful ambiguity resolution is the unmodelled impact of the ionosphere. We present a 4D ionospheric tomographic model that computes in real time the ionospheric electron density as a linear combination of basis functions, namely B-splines. The results show that when the ionospheric estimates are provided as atmospheric corrections for a PPP-RTK end-user, the time to fix its horizontal position below 10 cm is around 20 epochs (the sample rate is 30 s) at the $90\%$ of the cumulative distribution function (CDF), as opposed to the time it takes when no external corrections are provided, which is around 80 epochs at $90\%$ of the CDF.

Appendix

Equation (9) can be rearranged into one single equation as follows:

$$\begin{aligned} \left[ \begin{array}{l} \mathbf{y}_k \\ \mathbf{f}\left( {\hat{\mathbf{x}}}_{k-1}\right) \end{array} \right] = \left[ \begin{array}{l} \mathbf{H}_k \\ \mathbf {I} \end{array} \right] \,\mathbf{x}_k + \left[ \begin{array}{l} {\varvec{\epsilon }}_k \\ {\varvec{\delta }}_k \end{array} \right] . \end{aligned}$$

(16)

The optimal solution (in a LLS sense) for Eq. (16) corresponds to the argument $\mathbf{x}_k$ that minimizes the error, i.e.

$$\begin{aligned} {\hat{\mathbf{x}}}_k=\hbox {arg\,min}_{\mathbf{x}_k} ||\mathbf{z}_k-{{\varvec{{\Omega }}}}_k \mathbf{x}_k||^2_{{{\varvec{{\Sigma }}}}_k}, \end{aligned}$$

(17)

whose solution is as follows

$$\begin{aligned} {\hat{\mathbf{x}}}_k=\left[ {{\varvec{{\Omega }}}}_k^\mathrm{T} {{\varvec{{\Sigma }}}}_k^{-1} {{\varvec{{\Omega }}}}_k \right] ^{-1} \left[ {{\varvec{{\Omega }}}}_k^\mathrm{T} {{\varvec{{\Sigma }}}}_k^{-1} \mathbf{z}_k \right] , \end{aligned}$$

(18)

where $\mathbf{z}_k\,\equiv [\mathbf{y}_k^\mathrm{T}, \,\, \mathbf{f}^\mathrm{T}\left( {\hat{\mathbf{x}}}_{k-1}\right) ]^\mathrm{T}$; ${{\varvec{{\Omega }}}}_k$ is the matrix that maps the state-space estimates onto the observations and onto the a priori state-space estimates,

$$\begin{aligned} {{\varvec{{\Omega }}}}_k= \left[ \begin{array}{c} \mathbf{H}_k \\ \mathbf {I} \end{array} \right] , \end{aligned}$$

(19)

which is always full rank due to the presence of the identity matrix.

The covariance of the state-space estimates is

$$\begin{aligned} {{\varvec{{\Sigma }}}}_k \equiv \left[ \begin{array}{lc} \mathbf{E}_k &{} \quad \mathbf{0}\\ \mathbf{0} &{} \quad \mathbf{Q}_{\mathbf{f}\left( {\hat{\mathbf{x}}}_{k-1}\right) } \end{array} \right] , \end{aligned}$$

(20)

where $\mathbf{E}_k$ is the covariance of the measurements $\mathbf{y}_k$, and $\mathbf{Q}_{\mathbf{f}\left( {\hat{\mathbf{x}}}_{k-1}\right) }$ is the covariance of the stochastic dynamic model $\mathbf{f}\left( {\hat{\mathbf{x}}}_{k-1}\right)$.

The covariance of the estimates ${\hat{\mathbf{x}}}_k$ is as follows:

$$\begin{aligned} \mathbf{C}_k = \left[ {{\varvec{{\Omega }}}}_k^\mathrm{T} {{\varvec{{\Sigma }}}}_k^{-1} {{\varvec{{\Omega }}}}_k \right] ^{-1}, \end{aligned}$$

(21)

which, after developing the right-hand side, yields

$$\begin{aligned} \mathbf{C}_k = \left[ \left( \mathbf{Q}_{\mathbf{f}\left( {\hat{\mathbf{x}}}_{k-1}\right) }\right) ^{-1} + \mathbf{H}_k^\mathrm{T} \mathbf{E}_k^{-1} \mathbf{H}_k\right] ^{-1}. \end{aligned}$$

(22)

The matrix inversion lemma, e.g. Stengel (1994), states that

$$\begin{aligned}&\left[ \left( \mathbf{Q}_{\mathbf{f}\left( {\hat{\mathbf{x}}}_{k-1}\right) }\right) ^{-1} + \mathbf{H}_k^\mathrm{T} \mathbf{E}_k^{-1} \mathbf{H}_k\right] ^{-1} \nonumber \\&\quad = \mathbf{Q}_{\mathbf{f}\left( {\hat{\mathbf{x}}}_{k-1}\right) } - \mathbf{Q}_{\mathbf{f}\left( {\hat{\mathbf{x}}}_{k-1}\right) }{} \mathbf{H}_k^\mathrm{T}\left( \mathbf{H}_k \mathbf{Q}_{\mathbf{f}\left( {\hat{\mathbf{x}}}_{k-1}\right) } \mathbf{H}_k^\mathrm{T} + \mathbf{E}_k\right) ^{-1} \mathbf{H}_k \mathbf{Q}_{\mathbf{f}\left( {\hat{\mathbf{x}}}_{k-1}\right) } \nonumber \\ \end{aligned}$$

(23)

thence, after replacing Eq. (23) in Eq. (22), it leads to the following expression

$$\begin{aligned} \mathbf{C}_k = \left( {\mathbf {I}} - \mathbf{K}_k\mathbf{H}_k \right) \mathbf{Q}_{\mathbf{f}\left( {\hat{\mathbf{x}}}_{k-1}\right) }, \end{aligned}$$

(24)

where $\mathbf{K}_k$, the gain matrix, is defined as follows

$$\begin{aligned} \mathbf{K}_k\equiv \mathbf{Q}_{\mathbf{f}\left( {\hat{\mathbf{x}}}_{k-1}\right) } \mathbf{H}_k^\mathrm{T} \left( \mathbf{H}_k \mathbf{Q}_{\mathbf{f}\left( {\hat{\mathbf{x}}}_{k-1}\right) } \mathbf{H}_k^\mathrm{T}+\mathbf{E}_k\right) ^{-1}. \end{aligned}$$

(25)

Finally, by using Eq. (24) in Eq. (18), and noting that

$$\begin{aligned} {\mathbf {I}}=\left( \mathbf{H}_k \mathbf{Q}_{\mathbf{f}\left( {\hat{\mathbf{x}}}_{k-1}\right) } \mathbf{H}_k^\mathrm{T} + \mathbf{E}_k\right) ^{-1} \left( \mathbf{H}_k \mathbf{Q}_{\mathbf{f}\left( {\hat{\mathbf{x}}}_{k-1}\right) } \mathbf{H}_k^\mathrm{T} + \mathbf{E}_k\right) , \end{aligned}$$

(26)

the following expression is obtained:

$$\begin{aligned} \hat{\mathbf{x}}_k = \mathbf{f}\left( {\hat{\mathbf{x}}}_{k-1}\right) +\mathbf{K}_k\left[ \mathbf{y}_k - \mathbf{H}_k\mathbf{f}\left( {\hat{\mathbf{x}}}_{k-1}\right) \right] \, \end{aligned}$$

(27)

which is Eq. (10) presented in Sect. 2.