Modeling and quality control for reliable precise point positioning integer ambiguity resolution with GNSS modernization

Abstract

Recent research has demonstrated that the undifferenced integer ambiguities can be recovered using products from a network solution. The standard dual-frequency PPP integer ambiguity resolution consists of two aspects: Hatch-Melbourne-Wübbena wide-lane (WL) and ionosphere-free narrow-lane (NL) integer ambiguity resolution. A major issue affecting the performance of dual-frequency PPP applications is the time it takes to fix these two types of integer ambiguities, especially if the WL integer ambiguity resolution suffers from the noisy pseudorange measurements and strong multipath effects. With modernized Global Navigation Satellite Systems, triple-frequency measurements will be available to global users and an extra WL (EWL) model with very long wavelength can be formulated. Then, the easily resolved EWL integer ambiguities can be used to construct linear combinations to accelerate the PPP WL integer ambiguity resolution. Therefore, we propose a new reliable procedure for the modeling and quality control of triple-frequency PPP WL and NL integer ambiguity resolution. First, we analyze a WL integer ambiguity resolution model based on triple-frequency measurements. Then, an optimal pseudorange linear combination which is ionosphere-free and has minimum measurement noise is developed and used as constraint in the WL and the NL integer ambiguity resolution. Based on simulations, we have investigated the inefficiency of dual-frequency WL integer ambiguity resolution and the performance of EWL integer ambiguity resolution. Using almanacs of GPS, Galileo and BeiDou, the performances of the proposed triple-frequency WL and NL models have been evaluated in terms of success rate. Comparing with dual-frequency PPP, numerical results indicate that the proposed triple-frequency models can outperform the dual-frequency PPP WL and NL integer ambiguity resolution. With 1 s sampling rate, generally, only several minutes of data are required for reliable triple-frequency PPP WL and NL integer ambiguity resolution. Under benign observation situations and good geometries, the integer ambiguity can be reliably resolved even within 10 s.

Appendices

Appendix 1

Ignoring the UPDs in (11), the ambiguity term can be expressed as $x\lambda_{1} N_{1} + y\lambda_{2} N_{2} + z\lambda_{5} N_{5}$. Since our intention is to form the ambiguity term with both WL integer ambiguities and EWL integer ambiguities, we have:

$$\begin{aligned} x\lambda_{1} N_{1} + y\lambda_{2} N_{2} + z\lambda_{5} N_{5} & = x\lambda_{1} (N_{1} - N_{2} ) + (x\lambda_{1} + y\lambda_{2} )N_{2} + z\lambda_{5} N_{5} \\ & = x\lambda_{1} N_{\text{w}} - z\lambda_{5} N_{\text{ew}} + (x\lambda_{1} + y\lambda_{2} + z\lambda_{5} )N_{2} \\ \end{aligned}$$

(19)

Therefore, when the term $(x\lambda_{1} + y\lambda_{2} + z\lambda_{5} )$ equals to 0, the third condition and the objective model can be generated accordingly.

Appendix 2

With the conditions of (15a) and (15b), coefficients x and y can be expressed as $x = t_{1} + m_{1} z$ and $y = t_{2} + m_{2} z$, with $t_{1} = \frac{{f_{1}^{2} }}{{f_{1}^{2} - f_{2}^{2} }}$, $t_{2} = \frac{{ - f_{2}^{2} }}{{f_{1}^{2} - f_{2}^{2} }}$, $m_{1} = \frac{{ - f_{1}^{2} (f_{5}^{2} - f_{2}^{2} )}}{{f_{5}^{2} (f_{1}^{2} - f_{2}^{2} )}}$, $m_{2} = \frac{{ - f_{2}^{2} (f_{1}^{2} - f_{5}^{2} )}}{{f_{5}^{2} (f_{1}^{2} - f_{2}^{2} )}}$. The minimization of the third condition (15c) becomes:

$$(t_{1} + m_{1} z)^{2} + a^{2} (t_{2} + m_{2} z)^{2} + b^{2} z^{2} = \hbox{min}$$

(20)

Solving the above quadratic equation, we have:

$$\begin{gathered} \left( {m_{1}^{2} + a^{2} m_{2}^{2} + b^{2} } \right)\left( {z + \frac{{t_{1} m_{1} + a^{2} t_{2} m_{2} }}{{m_{1}^{2} + a^{2} m_{2}^{2} + b^{2} }}} \right)^{2} \hfill \\ \quad - \frac{{\left( {t_{1} m_{1} + a^{2} t_{2} m_{2} } \right)^{2} }}{{m_{1}^{2} + a^{2} m_{2}^{2} + b^{2} }} + t_{1}^{2} + t_{2}^{2} a^{2} = \hbox{min} \hfill \\ \end{gathered}$$

(21)

The minimization function can be obtained by setting the first term of (21) as 0 and x, y and z can be generated. Then it is observed that (21) is smaller than $t_{1}^{2} + t_{2}^{2} a^{2}$, which is the measurement noise variance of dual-frequency ionosphere-free model (9). In a similar manner, it can be proven that (21) is smaller than the measurement noises of ionosphere-free L1 and L5 model and L2 and L5 model.