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Single Point Positioning Basics

Table of Contents

Estimators

Measurement model

Used observations

  • $ \tilde{p}_r^s $ Pseudorange measurements [m]
  • $ \tilde{d}_r^s $ Doppler/Pseudorange rate measurements [m/s]

Measurement innovation

Innovation vector

\begin{equation} \label{eq:eq-SppBasics-measurementModel-innovation} \delta\mathbf{z}(\boldsymbol{x}) = \mathbf{z} - \mathbf{h}(\mathbf{x}) \end{equation}

Measurement vector

\begin{equation} \label{eq:eq-SppBasics-measurementModel-innovation-measurementVector} \mathbf{z} = \left\{ \begin{array}{c:c} \tilde{p}_r^1,\quad \tilde{p}_r^2,\quad \dots,\quad \tilde{p}_r^m & \tilde{d}_r^1,\quad \tilde{d}_r^2,\quad \dots,\quad \tilde{d}_r^m \end{array} \right\} \end{equation}

Estimates vector

\begin{equation} \label{eq:eq-SppBasics-measurementModel-innovation-estimatesVector} \mathbf{h}(\mathbf{x}) = \left\{ \begin{array}{c:c} \hat{p}_r^1,\quad \hat{p}_r^2,\quad \dots,\quad \hat{p}_r^m & \hat{d}_r^1,\quad \hat{d}_r^2,\quad \dots,\quad \hat{d}_r^m \end{array} \right\} \end{equation}

  • Pseudorange estimates

    \begin{equation} \label{eq:eq-SppBasics-measurementModel-innovation-estimates-pseudorange} \hat{p}_r^s = \hat{\rho}_r^s + c [d\hat{t}_r - d\hat{t}^s + d\hat{t}_{ISB}] + \hat{I}_r^s + \hat{T}_r^s + \delta\hat{\rho}_r^s \end{equation}

    where

    • $ \hat{\rho}_r^s $ Receiver-satellite range [m]
    • $ d\hat{t}_r $ Receiver clock error [s]
    • $ d\hat{t}^s $ Satellite clock error [s]
    • $ d\hat{t}_{ISB} $ Inter-system bias clock error [s]
    • $ \hat{I}_r^s $ Ionospheric delay [m]
    • $ \hat{T}_r^s $ Tropospheric delay [m]
    • $ \delta\hat{\rho}_r^s $ Sagnac correction [m]

    ([45] Springer Handbook, ch. 21.1.1, eq. 21.1, p. 606)

  • Pseudorange-rate estimates

    \begin{equation} \label{eq:eq-SppBasics-measurementModel-innovation-estimates-pseudorangeRate} \hat{d}_r^s = {\mathbf{u}_{as}^e}^T [\mathbf{\hat{v}}^e_s - \mathbf{\hat{v}}^e_r] + c [d\hat{\dotup{t}}_r - d\hat{\dotup{t}}^s + d\hat{\dotup{t}}_{ISB}] - \delta\hat{\dotup{\rho}}_r^s \end{equation}

    where

    • $ \mathbf{u}_{rs}^e $ Line-of-Sight vector
    • $ \mathbf{\hat{v}}^e_s $ Satellite velocity [m/s]
    • $ \mathbf{\hat{v}}^e_r $ Receiver velocity [m/s]
    • $ d\hat{\dotup{t}}_r $ Receiver clock drift [s/s]
    • $ d\hat{\dotup{t}}^s $ Satellite clock drift [s/s]
    • $ d\hat{\dotup{t}}_{ISB} $ Inter-system bias clock drift [s/s]
    • $ \delta\hat{\dotup{\rho}}_r^s $ Range-rate Sagnac correction [m/s]

    ([17] Groves, ch. 9.4.1, eq. 9.142, p. 412 (Sagnac correction different sign))

Measurement estimates

Receiver-satellite range

\begin{equation} \label{eq:eq-SppBasics-measurementModel-innovation-estimates-rho} \hat{\rho}_r^s = \mid(\mathbf{x}^s\left(t_{\mathrm{E}}\right) - \mathbf{x}_r\left(t_{\mathrm{A}}\right))\mid \end{equation}

Ionospheric delay
See Ionosphere-Model-Klobuchar

Tropospheric delay
See Troposphere-Model-Saastamoinen

Sagnac correction

\begin{equation} \label{eq:eq-SppBasics-measurementModel-innovation-estimates-sagnac} \delta\hat{\rho}_r^s = \frac{1}{c}\left(\boldsymbol{x}_{r}\left(t_{\mathrm{A}}\right)-\boldsymbol{x}^{s}\left(t_{\mathrm{E}}\right)\right) \cdot \left(\boldsymbol{\omega}_{ie} \times \boldsymbol{x}_{r}\left(t_{\mathrm{A}}\right)\right) \end{equation}

([45] Springer Handbook, ch. 19.1.1, eq. 19.7, p. 562)

Range-rate Sagnac correction

\begin{equation} \label{eq:eq-SppBasics-measurementModel-innovation-estimates-sagnac-rate} \delta\hat{\dotup{\rho}}_r^s = \frac{\boldsymbol{\omega}_{ie}}{c}\left(v_y^s \cdot x_r + y^s \cdot v_{x,r} - v_x^s \cdot y_r - x^s \cdot v_{y,r}\right) \end{equation}

([17] Groves, ch. 8.5.3, eq. 8.46, p. 342)

Design matrix / Measurement sensitivity matrix

\begin{equation} \label{eq:eq-SppBasics-measurementModel-sensitivityMatrix-general} \mathbf{H}_k = \left.\frac{\delta\mathbf{h}(\mathbf{x}, t_k)}{\delta\mathbf{x}}\right|_{x=\hat{x}_k^-} \end{equation}

Measurement error models

See NAV::GnssMeasurementErrorModel