The PVT block is the last one in the GNSS-SDR flow graph. Hence, it acts as a signal sink, since the stream of data flowing along the receiver ends here.

The role of a PVT block is to compute navigation solutions and deliver information in adequate formats for further processing or data representation.

It follows a description of the available positioning algorithms and their parameters, the available output formats, and the description of the configuration options for this block.

Positioning modes

The positioning problem is generally stated as

where \(\mathbf{y}\) is the measurement vector (that is, the observables obtained from the GNSS signals of a set of \(m\) satellites), \(\mathbf{x}\) is the state vector to be estimated (at least, the position of the receiver’s antenna and the time), \(\mathbf{h}(\cdot)\) is the function that relates states with measurements, and \(\mathbf{n}\) models measurement noise. Depending on the models, assumptions, available measurements, and the availability of a priori or externally-provided information, many positioning strategies and algorithms can be devised. It follows a description of the positioning modes available at the RTKLIB_PVT implementation, mostly extracted from the excellent RTKLIB manual.

Single Point Positioning

The default positioning mode is PVT.positioning_mode=Single. In this mode, the vector of unknown states is defined as:

where \(\mathbf{r}_r\) is the receiver’s antenna position in an earth-centered, earth-fixed (ECEF) coordinate system (in meters), \(c\) is the speed of light, and \(dt_r\) is the receiver clock bias (in seconds).

The measurement vector is defi

As described in the Observables block, for a signal from satellite \(s\) in the i-th band, the pseudorange measurement \(P_{r,i}^{(s)}\) can be expressed as:

In the current implementation, if the receiver obtains pseudorange measurements from the same satellite in different frequency bands, only measurements in the L1 band are used.

Hence, the equation that relates pseudorange measurements to the vector of unknown states can be written as:

The geometric range \(\rho_r^{(s)}\) is defined as the physical distance between the satellite antenna phase center position and the receiver antenna phase center position in the inertial coordinates. For the expression in the ECEF coordinates, the earth rotation effect has to be incorporated. This is known as the Sagnac effect¹, and it can be approximated by:

\(\begin{equation} \rho_{r}^{(s)} \approx \left\| \mathbf{r}_r(t_r) - \mathbf{r}^{(s)}(t^{(s)}) \right\| + {\definecolor{dark-orange}{RGB}{255,165,0} \color{dark-orange} \frac{\omega_e}{c}\left(x^{(s)}y_r - y^{(s)}x_r \right)}~, \end{equation}\) where \(\omega_e\) is the Earth rotation angle velocity (in rad/s).

Geometric range and Earth rotation correction ²

Equation \(\mathbf{h}(\mathbf{x})\) is clearly nonlinear due to the presence of the Euclidean norm operator \(\left\| \cdot \right\|\). However, this term can be extended by using Taylor series around an initial parameter vector \(\mathbf{x}_0\) as \(\mathbf{h}(\mathbf{x}) = \mathbf{h}(\mathbf{x}_0) + \mathbf{H}(\mathbf{x}-\mathbf{x}_0) + ...\), where \(\mathbf{H} = \frac{\partial \mathbf{h}(\mathbf{x})}{\partial \mathbf{x}} \bigg\rvert_{\mathbf{x} = \mathbf{x}_{0} }\) is a partial derivatives matrix of \(\mathbf{h}(\mathbf{x})\) with respect to \(\mathbf{x}\) at \(\mathbf{x} = \mathbf{x}_{0}\). Assuming that the initial parameters are adequately near the true values and the second and further terms of the Taylor series can be neglected, equation \(\mathbf{y} = \mathbf{h}(\mathbf{x}) + \mathbf{n}\) can be approximated by \(\mathbf{y} \approx \mathbf{h}(\mathbf{x}_0) + \mathbf{H}(\mathbf{x}-\mathbf{x}_0) + \mathbf{n}\), and then we can obtain the following linear equation:

which can be solved by a standard iterative weighted least squares method.

Matrix \(\mathbf{H}\) can be written as:

and the weighted least squares estimator (LSE) of the unknown state vector is obtained as:

For the initial parameter vector \(\mathbf{x}_0\) for the iterated weighted LSE, just all \(0\) are used for the first epoch of the single point positioning. Once a solution obtained, the position is used for the next epoch initial receiver position. For the weight matrix \(\mathbf{W}\), the RTKLIB_PVT implementation uses:

where:

• \(F^{(s)}\) is the satellite system error factor. This parameter is set to \(F^{(s)} = 1\) for GPS and Galileo.

• \(R_r\) is the code/carrier‐phase error ratio. This value is set by default to \(R_r = 100\), and can be configured with the PVT.code_phase_error_ratio_l1 option.

• \(a_{\sigma}, b_{\sigma}\) is the carrier‐phase error factor \(a\) and \(b\) (in m). They are set by default to \(a_{\sigma} = b_{\sigma} = 0.003\) m, and can be configured with the PVT.carrier_phase_error_factor_a and PVT.carrier_phase_error_factor_b options, respectively.

• \(El_r^{(s)}\) is the elevation angle of satellite direction (in rad).

• \(\sigma_{bclock,s}\) is the standard deviation of the broadcast ephemeris and clock error (in m). This parameter is estimated internally from URA (User Range Accuracy) or or similar indicators.

• \(\sigma_{ion,s}\) is the standard deviation of ionosphere correction model error (in m). This parameter is set to \(\sigma_{ion} = 5\) m by default (PVT.iono_model=OFF) and \(\sigma_{ion} = 0.5 \cdot I_{r,i}^{(s)}\) m when the option PVT.iono_model=Broadcast is set in the configuration file.

• \(\sigma_{trop,s}\) is the standard deviation of troposphere correction model error (in m). This parameter is set to \(\sigma_{trop} = 3\) m by default (PVT.trop_model=OFF) and \(\sigma_{trop} = 0.3 / \left(\sin(El_r^{(s)}) + 0.1\right)\) m when the option PVT.trop_model=Saastamoinen, PVT.trop_model=Estimate_ZTD or PVT.trop_model=Estimate_ZTD_Grad is set in the configuration file.

• \(\sigma_{cbias}\) is the standard deviation of code bias error (in m). This parameter is set to \(\sigma_{cbias} = 0.3\) m.

The estimated receiver clock bias \(dt_r\) is not explicitly output, but incorporated in the solution time‐tag. That means the solution time‐tag indicates not the receiver time‐tag but the true signal reception time measured in GPS Time.

Solution validation

The estimated receiver positions described in (\(\ref{eq:lse}\)) might include invalid solutions due to unmodeled measurement errors. To test whether the solution is valid or not, and to reject the invalid solutions, the RTKLIB_PVT applies the following validation tests after obtaining the receiver’s position estimate:

1) Residuals Test

Defining the residuals vector \(\boldsymbol{\nu} = \left( \nu_1, \nu_2, \nu_3, ..., \nu_m \right)^T\) with:

the residuals test is defined as:

where \(n\) is the number of estimated parameters, \(m\) is the number of measurements, \(\chi_{\alpha}^2(n)\) is the chi‐square distribution of degree of freedom \(n\), and with a significance level of \(\alpha=0.001\) (that is, \(prob > 0.001\)).

2) GDOP Test

The Geometric Dilution of Precision, defined as \(\text{GDOP} = \sqrt{\sigma_{r_{x}}^2 + \sigma_{r_{y}}^2 + \sigma_{r_{z}}^2 + \sigma_{c \cdot dt}^2 }\), must be better (that is, lower) than a certain threshold:

The threshold value is set by default to \(\text{GDOP}_{\text{threshold}} = 30\), and it can be configured via the option PVT.threshold_reject_GDOP in the configuration file.

If any of the validation fails, the solution is rejected as an outlier (that is, no solution is provided).

Receiver Autonomous Integrity Monitoring (RAIM)

In addition to the solution validation described above, RAIM (receiver autonomous integrity monitoring) FDE (fault detection and exclusion) function can be activated. If the chi-squared test described above fails and the option PVT.raim_fde is set to \(1\), the implementation retries the estimation by excluding one by one of the visible satellites. After all of the retries, the estimated receiver position with the minimum normalized squared residuals \(\boldsymbol{\nu}^T \boldsymbol{\nu}\) is selected as the final solution. In such a scheme, an invalid measurement, which might be due to satellite malfunction, receiver fault, or large multipath, is excluded as an outlier. Note that this feature is not effective with two or more invalid measurements. It also needs two redundant visible satellites, which means at least 6 visible satellites are necessary to obtain the final solution.

Precise Point Positioning

When the PVT.positioning_mode option is set to PPP_Static or PPP_Kinematic in the configuration file, a Precise Point Positioning algorithm is used to solve the positioning problem. In this positioning mode, the state vector to be estimated is defined as:

where \(Z_r\) is ZTD (zenith total delay), \(G_{N_r}\) and \(G_{E_r}\) are the north and east components of tropospheric gradients (see the tropospheric model below) and \(\mathbf{B}_{LC} = \left( B_{r,LC}^{(1)}, B_{r,LC}^{(2)}, B_{r,LC}^{(3)}, ..., B_{r,LC}^{(m)} \right)^T\) is the ionosphere‐free linear combination of zero‐differenced carrier‐phase biases (in m), defined below in Equation (\(\ref{eq:bias-lc}\)).

The Precise Point Positioning measurement model is based on the fact that, according to the phase and code ionospheric refraction, the first order ionospheric effects on code and carrier-phase measurements depend (99.9 %) on the inverse of squared signal frequency \(f_i\). Thence, dual-frequency receivers can eliminate their effect through a linear combination of pseudorange \(P_{r,i}^{(s)}\) and phase-range \(\Phi_{r,i}^{(s)}\) measurements (where the definitions at Observables apply):

with \(C_i = \frac{f_i^2}{f_i^2 - f_j^2}\) and \(C_j = \frac{-f_j^2}{f_i^2 - f_j^2}\), where \(f_i\) and \(f_j\) are the frequencies (in Hz) of \(L_i\) and \(L_j\) measurements. Explicitly:

with

In the current implementation, satellites and receiver antennas offset and variation are not applied, so \(\mathbf{d}_{r,pco,i} = \mathbf{d}_{pco,i}^{(s)} = \mathbf{0}\) and \(d_{r,pcv,i} = d_{pcv,j}^{(s)} = 0\). The correction terms for the Earth tide³ \(\mathbf{d}_{r,disp}\) and the phase windup effect⁴ \(\phi_{pw}\) are deactivated by default, and can be activated through the PVT.earth_tide and PVT.phwindup options, respectively.

The measurement vector is then defined as:

where \(\boldsymbol{\Phi}_{LC} = \left(\Phi_{r,LC}^{(1)}, \Phi_{r,LC}^{(2)}, \Phi_{r,LC}^{(3)}, ..., \Phi_{r,LC}^{(m)} \right)^T\) and \(\mathbf{P}_{LC} = \left( P_{r,LC}^{(1)}, P_{r,LC}^{(2)}, P_{r,LC}^{(3)}, ..., P_{r,LC}^{(m)} \right)^T\).

In the current implementation, if the receiver obtains pseudorange measurements from the same satellite in different frequency bands, only measurements in the L1 band are used.

The equation \(\mathbf{h}(\mathbf{x})\) that relates measurements and states is:

where:

This is again a nonlinear equation that could be solved with the iterative weighted least squares estimator as in the case of the Single Point Positioning case. However, here we want to incorporate some a priori information, such as a basic dynamic model for the receiver, and some statistical knowledge about the status of the troposphere. The Extended Kalman Filter offers a suitable framework for that.

The partial derivatives matrix \(\mathbf{H} = \frac{\partial \mathbf{h}(\mathbf{x})}{\partial \mathbf{x}} \bigg\rvert_{\mathbf{x} = \mathbf{x}_{0} }\) can be written as:

where \(\mathbf{D} = \left( \begin{array}{ccccc} 1 & -1 & 0 & \cdots & 0 \\ 1 & 0 & -1 & \cdots & 0 \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ 1 & 0 & 0 & \cdots & -1 \end{array} \right)\) is known as the single‐differencing matrix, \(\mathbf{E} = \left( \mathbf{e}_{r}^{(1)}, \mathbf{e}_{r}^{(2)}, \mathbf{e}_{r}^{(3)}, ..., \mathbf{e}_{r}^{(m)} \right)^T\) with \(\mathbf{e}_{r}^{(s)}\) defined as in equation (\(\ref{eq:H-single}\)), and

\(\scriptstyle \begin{equation} \!\!\!\!\!\!\!\!\!\!\! \mathbf{M}_T \; = \; \left( \begin{array}{ccc} m_{WG,r}^{(1)} \left( El_r^{(1)} \right) & m_{W,r}^{(1)} \left( El_r^{(1)} \right) \cot \left( El_r^{(1)} \right) \cos \left( Az_r^{(1)} \right) & m_{W,r}^{(1)} \left( El_r^{(1)} \right) \cot \left( El_r^{(1)} \right) \sin \left( Az_r^{(1)} \right) \\ m_{WG,r}^{(2)} \left( El_r^{(2)} \right) & m_{W,r}^{(2)} \left( El_r^{(2)} \right) \cot \left( El_r^{(2)} \right) \cos \left( Az_r^{(2)} \right) & m_{W,r}^{(2)} \left( El_r^{(2)} \right) \cot \left( El_r^{(2)} \right) \sin \left( Az_r^{(2)} \right) \\ m_{WG,r}^{(3)} \left( El_r^{(3)} \right) & m_{W,r}^{(3)} \left( El_r^{(3)} \right) \cot \left( El_r^{(3)} \right) \cos \left( Az_r^{(3)} \right) & m_{W,r}^{(3)} \left( El_r^{(3)} \right) \cot \left( El_r^{(3)} \right) \sin \left( Az_r^{(3)} \right) \\ \vdots \\ m_{WG,r}^{(m)} \left( El_r^{(m)} \right) & m_{W,r}^{(m)} \left( El_r^{(m)} \right) \cot \left( El_r^{(m)} \right) \cos \left( Az_r^{(m)} \right) & m_{W,r}^{(m)} \left( El_r^{(m)} \right) \cot \left( El_r^{(m)} \right) \sin \left( Az_r^{(m)} \right) \end{array} \right) \end{equation}\) is a matrix related to the tropospheric model (see below).

With all those definitions, the Precise Point Positioning solution is computed as follows:

The transition matrix \(\mathbf{F}_k\) models the receiver movement:

• If PVT.positioning_mode=PPP_Static:

• If PVT.positioning_mode=PPP_Kinematic:

where \(\Delta_k = t_{k+1} - t_k\) is the time between GNSS measurements, in s.

The dynamics model noise covariance matrix \(\mathbf{Q}_k\) is set to:

with:

• \(\mathbf{Q}_r = \mathbf{E}_r^T \text{diag} \left( \sigma_{re}^2 , \sigma_{rn}^2 , \sigma_{ru}^2 \right) \mathbf{E}_r\), where \(\mathbf{E}_r\) is the coordinates rotation matrix from ECEF to local coordinates at the receiver antenna position (defined below), and \(\sigma_{re}\), \(\sigma_{rn}\) and \(\sigma_{ru}\) are the standard deviations of east, north, and up components of the receiver position model noises (in m).
  • If the positioning mode is set to PVT.positioning_mode=PPP_Static, these values are initialized to \(\sigma_{re} = \sigma_{rn} = \sigma_{ru} = 100\) m in the first epoch and then set to \(0\) in the following time updates.
  • If the positioning mode is set to PVT.positioning_mode=PPP_Kinematic, these values are set to \(\sigma_{re} = \sigma_{rn} = \sigma_{ru} = 100\) m for all time updates.
• \(\mathbf{Q}_v = \mathbf{E}_r^T \text{diag} \left( \sigma_{ve}^2 \Delta_k , \sigma_{vn}^2 \Delta_k, \sigma_{vu}^2 \Delta_k \right) \mathbf{E}_r\), where \(\sigma_{ve}\), \(\sigma_{vn}\) and \(\sigma_{vu}\) are the standard deviations of east, north, and up components of the receiver velocity model noises (in m/s/\(\sqrt{s}\)). In the current implementation, those parameters are set to \(\sigma_{ve} = \sigma_{vn} = \sigma_{vu} = 0\).
• \(\sigma_{c \cdot dt_{r}}\) is the standard deviation of the receiver clock offset (in m). This value is set to \(\sigma_{c \cdot dt_{r}} = 100\) m.
• \(\mathbf{Q}_{T} = \text{diag} \left( \sigma_{Z}^2 \Delta_k, \sigma_{G_{N}}^2 \Delta_k, \sigma_{G_{E}}^2 \Delta_k \right)\) is the noise covariance matrix of the troposphere terms. These values are set to \(\sigma_{Z} = 0.0001\), and \(\sigma_{G_{N}} = \sigma_{G_{E}}\) are initialized to \(\sigma_{G_{N}} = \sigma_{G_{E}} = 0.001\) m/\(\sqrt{s}\) in the first epoch and then set to \(\sigma_{G_{N}} = \sigma_{G_{E}} = 0.1 \cdot \sigma_{Z}\) in the following time updates. The default value of \(\sigma_{Z} = 0.0001\) m/\(\sqrt{s}\) can be configured with the PVT.sigma_trop option.
• \(\sigma_{bias}\) is the standard deviation of the ionosphere-free carrier-phase bias measurements, in m/\(\sqrt{s}\). This value is initialized at the first epoch and after a cycle slip to \(\sigma_{bias} = 100\) m/\(\sqrt{s}\), and then is set to a default value of \(\sigma_{bias} = 0.0001\) m/\(\sqrt{s}\) in the following time updates. This value and can be configured with the option PVT.sigma_bias.
• \(\mathbf{E}_r = \left( \begin{array}{ccc} - \sin(\theta_r) & \cos (\theta_r) & 0 \\ -\sin (\psi_r) \cos(\theta_r) & - \sin (\psi_r)\sin(\theta_r) & \cos (\psi_r) \\ \cos(\psi_r)\cos(\theta_r) & \cos(\psi_r)\sin(\theta_r) & \sin(\psi_r)\end{array} \right)\) is the rotation matrix of the ECEF coordinates to the local coordinates, where \(\psi_r\) and \(\theta_r\) are the geodetic latitude and the longitude of the receiver position.

The measurement model noise covariance matrix \(\mathbf{R}_k\) is defined as:

where:

in which \(\sigma_{\Phi,1}^{(s)}\) is the standard deviation of L1 phase‐range measurement error (in m), and \(\sigma_{P,1}^{(s)}\) is the standard deviation of L1 pseudorange measurement error (in m). These quantities are estimated as:

• \({\sigma_{\Phi,1}^{(s)}}^2 = a_{\sigma}^2 + \frac{b_{\sigma}^2}{\sin(E_r^{(s)})^2} + \sigma_{ion,s}^2 + \sigma_{bclock}^2 + \sigma_{trop,s}^2\), where:
  • \(a_{\sigma} = 0.003\) and \(b_{\sigma} = 0.003\) are the carrier phase error factors (configurable via PVT.carrier_phase_error_factor_a and PVT.carrier_phase_error_factor_b),
  • \(\sigma_{ion,s}\) is the standard deviation of ionosphere correction model error (in m). This parameter is set to \(\sigma_{ion} = 5\) m by default (PVT.iono_model=OFF) and \(\sigma_{ion} = 0.5 \cdot I_{r,i}^{(s)}\) m when the option PVT.iono_model=Broadcast is set in the configuration file.
  • \(\sigma_{bclock} = 30\) m is the standard deviation of the broadcast clock,
  • \(\sigma_{trop}\) is the standard deviation of the troposphere correction model error (in m). This parameter is set to \(\sigma_{trop} = 3\) m when PVT.trop_model=OFF and \(\sigma_{trop,s} = \frac{0.3}{\sin(El_r^{(s)}) + 0.1}\) m when PVT.trop_model=Saastamoinen, PVT.trop_model=Estimate_ZTD or PVT.trop_model=Estimate_ZTD_Grad.
• \({\sigma_{P,1}^{(s)}}^2 = R_r \cdot \left( a_{\sigma}^2 + \frac{b_{\sigma}^2}{\sin(E_r^{(s)})^2} \right) + \sigma_{ion,s}^2 + \sigma_{bclock}^2 + \sigma_{trop,s}^2 + \sigma_{cbias}^2\), where:
  • \(R_r = 100\) (configurable via PVT.code_phase_error_ratio_l1),
  • \(a_{\sigma} = b_{\sigma} = 0.003\) m (configurable via PVT.carrier_phase_error_factor_a and PVT.carrier_phase_error_factor_b),
  • \(\sigma_{ion,s}\), \(\sigma_{bclock}\) and \(\sigma_{trop,s}\) defined as above.
  • \(\sigma_{cbias}\) is the standard deviation of code bias error (in m). This parameter is set to \(\sigma_{cbias} = 0.3\) m.

Outlier rejection

In each of the executions of the Extended Kalman Filter defined in (\(\ref{eq:state-update}\))-(\(\ref{eq:meas-cov-update}\)), if the absolute value of a residual \(\nu_s = \frac{P_r^{(s)} - \left( \hat{\rho}_r^{(s)} +c \hat{dt}_r - c \cdot dT^{(s)} + I_r^{(s)} + T_r^{(s)} \right)}{\sigma_s}\) for a satellite \(s\) is above a certain threshold, that observation is rejected as an outlier. The default threshold is set to \(30\) m and can be configured via the option PVT.threshold_reject_innovation.

Ionospheric Model

The ionosphere is a region of Earth’s upper atmosphere, from about 60 km to 1,000 km altitude, surrounding the planet with a shell of electrons and electrically charged atoms and molecules. This part of the atmosphere is ionized by ultraviolet, X-ray and shorter wavelengths of solar radiation, and this affects GNSS signals’ propagation speed.

The propagation speed of the GNSS electromagnetic signals through the ionosphere depends on its electron density, which is typically driven by two main processes: during the day, sun radiation causes ionization of neutral atoms producing free electrons and ions. During the night, the recombination process prevails, where free electrons are recombined with ions to produce neutral particles, which leads to a reduction in the electron density.

The frequency dependence of the ionospheric effect (in m) is described by the following expression:

where STEC is the Slant Total Electron Content, which describes the number of free electrons present within one square meter between the receiver and satellite \(s\). It is often reported in multiples of the so-called TEC unit, defined as \(\text{TECU} = 10^{16}\) el/m\(^2\). Ionospheric effects on the phase and code measurements have the opposite signs and have approximately the same amount. It causes a positive delay on code measurements (so it is included with a positive sign in the pseudorange measurement model) and a negative delay, or phase advance, in phase measurements (so it is included with a negative sign in the phase-range measurement model).

This dispersive nature (i.e., the ionospheric delay is proportional to the squared inverse of \(f_i\)) allows users to remove its effect up to more than 99.9% using two frequency measurements (as in the see ionosphere-free combination for dual-frequency receivers shown in the Precise Point Positioning algorithm described above), but single-frequency receivers have to apply an ionospheric prediction model to remove (as much as possible) this effect, that can reach up to several tens of meters.

Broadcast

For ionosphere correction for single-frequency GNSS users, GPS navigation data include the following broadcast ionospheric parameters:

By using these ionospheric parameters, the L1 ionospheric delay \(I_{r,1}^{(s)}\) (in m) can be derived by the following procedure⁵ (this model is often called as the Klobuchar model⁶):

This correction is activated when PVT.iono_model is set to Broadcast.

Tropospheric Model

The troposphere is the lowest portion of Earth’s atmosphere, and contains 99% of the total mass of water vapor. The average depths of the troposphere are 20 km in the tropics, 17 km in the mid-latitudes, and 7 km in the polar regions in winter. The chemical composition of the troposphere is essentially uniform, with the notable exception of water vapor, which can vary widely. The effect of the troposphere on the GNSS signals appears as an extra delay in the measurement of the signal traveling time from the satellite to the receiver. This delay depends on the temperature, pressure, humidity as well as the transmitter and receiver antennas location, and it is related to air refractivity, which in turn can be divided into hydrostatic, i.e., dry gases (mainly \(N_2\) and \(O_2\)), and wet, i.e., water vapor, components:

• Hydrostatic component delay: Its effect varies with local temperature and atmospheric pressure in quite a predictable manner, besides its variation is less than the 1% in a few hours. The error caused by this component is about \(2.3\) meters in the zenith direction and \(10\) meters for lower elevations (\(10^{o}\) approximately).

• Wet component delay: It is caused by the water vapor and condensed water in form of clouds and, thence, it depends on weather conditions. The excess delay is small in this case, only some tens of centimetres, but this component varies faster than the hydrostatic component and in a quite random way, being very difficult to model.

The troposphere is a non-dispersive media with respect to electromagnetic waves up to 15 GHz, so the tropospheric effects are not frequency-dependent for the GNSS signals. Thence, the carrier phase and code measurements are affected by the same delay, and this effect can not be removed by combinations of dual-frequency measurements.

Saastamoinen

The standard atmosphere can be expressed as:⁷

where \(p\) is the total pressure (in hPa), \(T\) is the absolute temperature (in K) of the air, \(h\) is the geodetic height above MSL (mean sea level), \(e\) is the partial pressure (in hPa) of water vapor and \(h_{rel}\) is the relative humidity. The tropospheric delay \(T_{r}^{(s)}\) (in m) is expressed by the Saastamoinen model with \(p\), \(T\) and \(e\) derived from the standard atmosphere:

where \(z^{(s)}\) is the zenith angle (rad) as \(z^{(s)} = \frac{\pi}{2} - El_{r}^{(s)}\), where \(El_{r}^{(s)}\) is elevation angle of satellite direction (rad).

The standard atmosphere and the Saastamoinen model are applied in the case that the processing option PVT.trop_model is set to Saastamoinen, where the geodetic height is approximated by the ellipsoidal height and the relative humidity is fixed to 70%.

Estimate the tropospheric zenith total delay

If the processing option PVT.trop_model is set to Estimate_ZTD, a more precise troposphere model is applied with strict mapping functions as:

where \(Z_{T,t}\) is the tropospheric zenith total delay (in meters), \(Z_{H,r}\) is the tropospheric zenith hydro‐static delay (in meters), \(m_{H}\left(El_{r}^{(s)}\right)\) is the hydro‐static mapping function and \(m_{W}\left(El_{r}^{(s)}\right)\) is the wet mapping function. The tropospheric zenith hydro‐static delay is given by Saastamoinen model described above with the zenith angle \(z = 0\) and relative humidity \(h_{rel} = 0\). For the mapping function, the software employs the Niell mapping function⁸. The zenith total delay \(Z_{T,r}\) is estimated as an unknown parameter in the parameter estimation process.

Estimate the tropospheric zenith total delay and gradient

If the processing option trop_model is set to Estimate_ZTD_Grad, a more precise troposphere model is applied with strict mapping functions as⁹:

where \(Az_{r}^{(s)}\) is the azimuth angle of satellite direction (rad), and \(G_{E,r}\) and \(G_{N,r}\) are the east and north components of the tropospheric gradient, respectively. The zenith total delay \(Z_{T,r}\) and the gradient parameters \(G_{E,r}\) and \(G_{N,r}\) are estimated as unknown parameters in the parameter estimation process.

A de-noising Kalman filter for the PVT solution

The PVT block can apply a simple Kalman filter to the computed PVT solutions. This filter can be enabled by setting PVT.enable_pvt_kf=true in the configuration file. The structure of this filter is as follows:

• State model: \(\begin{equation} \mathbf{x} = \left[ x, y, z, v_x, v_y, v_z \right]^{T} \end{equation}\)

  \[\begin{equation} \mathbf{x}_k = \mathbf{F} \mathbf{x}_{k-1} + \mathbf{v}_k~, \quad \mathbf{v}_k \sim \mathcal{N}(\mathbf{0},\mathbf{Q}) \end{equation}\] \[\begin{equation} \textbf{F} = \left[ \begin{array}{cccccc} 1 & 0 & 0 & T & 0 & 0 \\ 0 & 1 & 0 & 0 & T & 0 \\ 0 & 0 & 1 & 0 & 0 & T \\ 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 \end{array} \right] \end{equation}\] \[\begin{equation} \textbf{Q} = \begin{bmatrix} \sigma_{s\_pos}^{2} & 0 & 0 & 0 & 0 & 0 \\ 0 & \sigma_{s\_pos}^{2} & 0 & 0 & 0 & 0 \\ 0 & 0 & \sigma_{s\_pos}^{2} & 0 & 0 & 0 \\ 0 & 0 & 0 & \sigma_{s\_vel}^{2} & 0 & 0 \\ 0 & 0 & 0 & 0 & \sigma_{s\_vel}^{2} & 0 \\ 0 & 0 & 0 & 0 & 0 & \sigma_{s\_vel}^{2} \end{bmatrix} \end{equation}\]

• Measurement model: \(\begin{equation} \mathbf{z} = \left[ x , y , z , v_{x}, v_{y}, v_{z} \right]^{T} \end{equation}\)

  \[\begin{equation} \mathbf{z}_k = \mathbf{H}\mathbf{x}_k + \mathbf{w}_k , \quad \mathbf{w}_k \sim \mathcal{N}(\mathbf{0},\mathbf{R}) \end{equation}\] \[\begin{equation} \textbf{H} = \begin{bmatrix} 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 \end{bmatrix} \end{equation}\] \[\begin{equation} \textbf{R} = \begin{bmatrix} \sigma_{m\_pos}^{2} & 0 & 0 & 0 & 0 & 0 \\ 0 & \sigma_{m\_pos}^{2} & 0 & 0 & 0 & 0 \\ 0 & 0 & \sigma_{m\_pos}^{2} & 0 & 0 & 0 \\ 0 & 0 & 0 & \sigma_{m\_vel}^{2} & 0 & 0 \\ 0 & 0 & 0 & 0 & \sigma_{m\_vel}^{2} & 0 \\ 0 & 0 & 0 & 0 & 0 & \sigma_{m\_vel}^{2} \end{bmatrix} \end{equation}\]

• Initialization: \(\begin{equation} \mathbf{x}_{0|0} = \left[ \begin{array}{cccc} x_{0} & y_{0} & z_{0} & v_{x_{0}} & v_{y_{0}} & v_{z_{0}} \end{array} \right]^T \end{equation}\)

  \[\begin{equation} \mathbf{P}_{0|0} = \begin{bmatrix} \sigma_{s\_pos}^{2} & 0 & 0 & 0 & 0 & 0 \\ 0 & \sigma_{s\_pos}^{2} & 0 & 0 & 0 & 0 \\ 0 & 0 & \sigma_{s\_pos}^{2} & 0 & 0 & 0 \\ 0 & 0 & 0 & \sigma_{s\_vel}^{2} & 0 & 0 \\ 0 & 0 & 0 & 0 & \sigma_{s\_vel}^{2} & 0 \\ 0 & 0 & 0 & 0 & 0 & \sigma_{s\_vel}^{2} \end{bmatrix} \end{equation}\]

• Prediction: \(\begin{equation} \hat{\mathbf{x}}_{k|k-1} = \mathbf{F} \hat{\mathbf{x}}_{k-1|k-1} \end{equation}\)

  \[\begin{equation} \mathbf{P}_{k|k-1} = \mathbf{F} \mathbf{P}_{k-1|k-1} \mathbf{F}^T + \mathbf{Q} \end{equation}\]

• Update: \(\begin{equation} \mathbf{K}_k = \mathbf{P}_{k|k-1} \mathbf{H}^T \left( \mathbf{H}\mathbf{P}_{k|k-1} \mathbf{H}^T + \mathbf{R} \right)^{-1} \end{equation}\)

  \[\begin{equation} \hat{\mathbf{x}}_{k|k} = \hat{\mathbf{x}}_{k|k-1} + \mathbf{K}_k \left( \mathbf{z}_k - \mathbf{H}_k\hat{\mathbf{x}}_{k|k-1} \right) \end{equation}\] \[\begin{equation} \mathbf{P}_{k|k} = \left( \mathbf{I} - \mathbf{K}_{k} \mathbf{H} \right)\mathbf{P}_{k|k-1} \end{equation}\]

The following parameters are exposed in the configuration, here with their defaut values:

• \[\sigma_{m\_pos} = \text{PVT.kf_measures_ecef_pos_sd_m} = 1.0 \text {, in [m].}\]
• \[\sigma_{m\_vel} = \text{PVT.kf_measures_ecef_vel_sd_ms} = 0.1 \text {, in [m/s].}\]
• \[\sigma_{s\_pos} = \text{PVT.kf_system_ecef_pos_sd_m} = 0.01 \text {, in [m].}\]
• \[\sigma_{s\_vel} = \text{PVT.kf_system_ecef_vel_sd_ms} = 0.001 \text {, in [m/s].}\]

Output formats

Depending on the specific application or service that is exploiting the information provided by GNSS-SDR, different internal data will be required. The software provides such output data in standard formats:

KML, GeoJSON, GPX

For Geographic Information Systems, map representation and Earth browsers: KML, GeoJSON, and GPX files are generated by default, upon the computation of the first position fix.

RINEX

For post-processing applications: RINEX 2.11 and 3.02. Version 3.02 is generated by default, and version 2.11 can be requested by setting PVT.rinex_version=2 in the configuration file.

IMPORTANT: In order to get well-formatted GeoJSON, KML, GPX, and RINEX files, always terminate gnss-sdr execution by pressing key ‘q’ and then key ‘ENTER’. Those files will be automatically deleted if no position fix has been obtained during the execution of the software receiver.

NMEA-0183

For sensor integration: NMEA-0183. A text file containing NMEA messages is stored with a default name of gnss_sdr_pvt.nmea, configurable via PVT.nmea_dump_filename. In addition, NMEA messages can be forwarded to a serial port by setting PVT.flag_nmea_tty_port=true. The default port is /dev/tty1, and can be configured via PVT.nmea_dump_devname.

RTCM-104

For real-time, possibly networked processing: RTCM-104 messages, v3.2. A TCP/IP server of RTCM messages can be enabled by setting PVT.flag_rtcm_server=true in the configuration file, and will be active during the execution of the software receiver. By default, the server will operate on port 2101 (which is the recommended port for RTCM services according to the Internet Assigned Numbers Authority, IANA), and will identify the Reference Station with ID=1234. These values can be changed with PVT.rtcm_tcp_port and PVT.rtcm_station_id. The rate of the generated RTCM messages can be tuned with the options PVT.rtcm_MT1045_rate_ms (it defaults to 5000 ms), PVT.rtcm_MT1019_rate_ms (it defaults to 5000 ms), PVT.rtcm_MSM_rate_ms (it defaults to 1000 ms). The RTCM messages can also be forwarded to the serial port PVT.rtcm_dump_devname (it defaults to /dev/pts/1) by setting PVT.flag_rtcm_tty_port=true in the configuration file.

Custom streaming

In addition to the standard output formats, the PVT block offers a custom mechanism for streaming its internal data members to local or remote clients over UDP through a monitoring port which can be enabled by setting PVT.enable_monitor=true in the configuration file. This feature is very useful for real-time monitoring of the PVT block and its outputs. By default, the data is streamed to the localhost address on port 1234 UDP. These settings can be changed with PVT.monitor_client_addresses and PVT.monitor_udp_port. The streamed data members (28 in total) are serialized via Protocol Buffers into a format defined at monitor_pvt.proto. This allows other applications to easily read those messages, either using C++, Java, Python, C#, Dart, Go, or Ruby, among other languages, hence enhancing Interoperability.

The following table shows the complete list of streamed parameters:

The PVT monitor can also stream GPS and Galileo ephemeris data by setting PVT.enable_monitor_ephemeris=true in the configuration file. The streamed data members are serialized via Protocol Buffers into formats defined at gps_ephemeris.proto and galileo_ephemeris.proto, prepended by character G for GPS data and by character E for Galileo data. By default, data are streamed to the localhost address on port 1234 UDP. These settings can be changed with PVT.monitor_ephemeris_client_addresses and PVT.monitor_ephemeris_udp_port parameters in the configuration file.

Read more about standard output formats on our Interoperability page.

Implementation: RTKLIB_PVT

This implementation makes use of the positioning libraries of RTKLIB, a well-known open-source program package for standard and precise positioning. It accepts the following parameters:

PVT implementation: RTKLIB_PVT.

Example:

;######### PVT CONFIG ############
PVT.implementation=RTKLIB_PVT
PVT.positioning_mode=PPP_Static
PVT.output_rate_ms=100
PVT.display_rate_ms=500
PVT.iono_model=Broadcast
PVT.trop_model=Saastamoinen
PVT.flag_rtcm_server=true
PVT.flag_rtcm_tty_port=false
PVT.rtcm_dump_devname=/dev/pts/1
PVT.rtcm_tcp_port=2101
PVT.rtcm_MT1019_rate_ms=5000
PVT.rtcm_MT1045_rate_ms=5000
PVT.rtcm_MT1097_rate_ms=1000
PVT.rtcm_MT1077_rate_ms=1000
PVT.rinex_version=2

The generation of output files is controlled by the parameter output_enabled. If set to true (which is its default value), RINEX, XML, GPX, KML, GeoJSON, NMEA and binary RTCM files will be generated. You can turn off the generation of such files by setting output_enabled=false, and then select, for instance, rinex_output_enabled=true or kml_output_enabled=true. Files are stored in the path indicated in output_path, which by default is the current folder (that is, the folder from which GNSS-SDR was called). This can be changed for all outputs (for instance, output_path=gnss-products or output_path=/home/user/Documents/gnss-products/day1), or it can be defined per type of output (e.g., rinex_output_path=gnss-products/rinex, gpx_output_path=gnss-products/gpx, geojson_output_path=gnss-products/geojson, etc.).

Example:

PVT.output_enabled=false
PVT.rtcm_output_file_enabled=false
PVT.gpx_output_enabled=true
PVT.geojson_output_enabled=true
PVT.kml_output_enabled=true
PVT.xml_output_enabled=true
PVT.rinex_output_enabled=true
PVT.nmea_output_file_enabled=false
PVT.output_path=gnss-products/others
PVT.gpx_output_path=gnss-products/gpx
PVT.kml_output_path=./
PVT.xml_output_path=./
PVT.rinex_output_path=gnss-products/rinex

This will create in your current directory:

.
├── PVT_181028_093651.kml
├── gnss-products
│   ├── gpx
│   │   └── PVT_181028_093651.gpx
│   ├── others
│   │   └── PVT_181028_093651.geojson
│   └── rinex
│       ├── GSDR301j36.18N
│       └── GSDR301j36.18O
└── gps_ephemeris.xml

In order to shut down the generation of output files, you can just include in your configuration file the line:

PVT.output_enabled=false

Please note that this only concerns the generation of mentioned file formats, and it does not affect the generation of dump files activated in the configuration of each processing block. If the RTCM server is activated with flag_rtcm_server=true, it will still work even if the binary RTCM file is deactivated with rtcm_output_file_enabled=false.

References