2.1 Initial ambiguity resolution process

There are more available satellites for multi-GNSS positioning. It is difficult to fix ambiguities for all the satellites, however, because of multipath and atmospheric residuals. The partial ambiguity resolution method is developed to solve this problem. But its solution is unreliable when few satellites are available for partial ambiguity resolution. This section presents the application of an iterative partial ambiguity resolution method to obtain an approximate solution like Site B in Figure 1.

Figure 1. Schematic diagram of the IIAR method

GNSS observation equation is expressed as follows:

(1)\begin{equation}Y = AX + Bb + \varepsilon \end{equation}

where $Y \in {R^m}$ denotes the vector of the pseudo-range and carrier phase observation; $A \in {R^{m\ast n}}$ denotes the matrix of designed coefficients; $X \in {R^n}$ denotes the vector of the estimated coordinates; $B \in {R^{m\ast p}}$ denotes the matrix of carrier phase length; $b \in {Z^p}$ denotes the vector of the estimated integer ambiguity; $\varepsilon \in {R^m}$ denotes the noise vector, which is assumed to follow a zero-mean Gaussian distribution. The float solution of X and b and their variance matrices are expressed as follows:

(2)\begin{equation}\left[ {\begin{matrix} {{X_f}}\\ {{b_f}} \end{matrix}} \right]\quad \left[ {\begin{matrix} {{Q_{11}}}&{{Q_{12}}}\\ {{Q_{21}}}&{{Q_{22}}} \end{matrix}} \right]\end{equation}

where ${b_f} \in {R^p}$ denotes the float solution vector of b; ${X_f} \in {R^n}$ denotes the float solution vector of X; ${Q_{11}} \in {R^{n\ast n}}$ denotes the variance submatrix of coordinates; ${Q_{22}} \in {R^{q\ast q}}$ denotes the variance submatrix of ambiguity; ${Q_{12}} \in {R^{n\ast q}}$ and ${Q_{21}} \in {R^{q\ast n}}$ denote the covariance submatrices.

Integer ambiguity resolution is essential for RTK positioning. It is difficult to fix ambiguities for all the satellites in a complex environment, so partial ambiguity resolution is implemented (Teunissen, Reference Teunissen1999, Reference Teunissen2001; Teunissen et al., Reference Teunissen, Joosten and Tiberius1999). The key for partial ambiguity resolution is to select a feasible ambiguity subset. An ambiguity subset is fixed, with a bootstrapped success rate greater than or equal to a minimum required value ${P_0}$, to select the feasible subset such that:

(3)\begin{equation}F(b )= \mathop \prod \limits_{i = k}^p \left( {2\mathrm{\Phi }\left( {\frac{1}{{2{\sigma_{{b_{i|I}}}}}} - 1} \right)} \right) \ge {P_0}\end{equation}

where $\mathrm{\Phi }(x )$ denotes the cumulative normal distribution; ${\sigma _{{b_{i|I}}}}$ denotes the conditional variance element; ${b_{i|I}} \in {R^{p - i + 1}}$ denotes the feasible ambiguity subset. Equation (3) denotes that only the last $p - k + 1$ ambiguities are in the feasible subset. The ambiguity resolution is then applied for the selected subset.

Many studies have been conducted to resolve the integer ambiguity, such as integer least-squares (Teunissen, Reference Teunissen1999), integer bootstrapping (Teunissen, Reference Teunissen1998) and integer rounding (Jonge and Tiberius, Reference Jonge and Tiberius1998). The formula of integer least-squares ambiguity resolution is expressed as follows:

(4)\begin{equation}f({\nabla \Delta b} )= \min {(\nabla \Delta b - \nabla \Delta {b_f})^T}{Q_{22}}({\nabla \Delta b - \nabla \Delta {b_f}} )\end{equation}

where $\nabla \Delta$ denotes the double-difference symbol. This estimator is known to be optimal, which means the probability of the correct integer estimation is maximised (Teunissen, Reference Teunissen1999). In this study, the least-squares ambiguity decorrelation adjustment (LAMBDA) method is used to resolve ambiguities for the selected satellites (Teunissen, Reference Teunissen1995). The initial ambiguity resolution process is an iterative partial ambiguity resolution method. The iterative model guarantees the correctness of the virtual station. Equation (4) is the criterion of the partial ambiguity resolution. The iterative model will end if the adjacent iterations have the same ambiguity subsets.

The initial ambiguity resolution is to obtain an approximate solution, which is set as the virtual fixed solution of the virtual station. Previous studies apply the iterative partial ambiguity resolution method to obtain the solution, but the solution is unreliable when the selected satellites have small redundancy. This study applied an iterative partial ambiguity resolution to obtain the virtual fixed solution, and the iteration will end when the adjacent iterations have the same ambiguity subsets. The virtual fixed solution is unreliable for high-precision positioning, and the IIAR process is essential.

2.2 IIAR process

The initial ambiguity resolution process applies an iterative partial ambiguity resolution method to obtain the virtual fixed solution, which omits the ambiguity resolution for poorly-observed satellites. It is hypothesised that the virtual station and the estimated rover station have the same integer ambiguities in RTK positioning. There is no need to resolve integer ambiguities for other satellites with the virtual fixed solution. The IIAR process transforms all of the adjustment equations to a new form. More formula derivations are given in the following section.

Figure 1 illustrates a double-difference for site B and site C in. According to Equation (1), the double-difference ambiguity formula for each observation equation is expressed as follows:

(5)\begin{equation}\nabla \Delta {b_i} = \textrm{Round}\left( {\nabla \Delta {\varphi_i} - \frac{{\nabla \Delta {\rho_i}}}{{{\lambda_i}}}} \right)\end{equation}

where i denotes the frequency i; $\nabla \Delta {b_i}$ denotes the double-difference ambiguity element; $\textrm{Round}({\cdot} )$ denotes the rounding function; $\nabla \Delta {\varphi _i}$ denotes the double-difference carrier phase observation element; $\nabla \Delta {\rho _i}$ denotes the double-difference pseudo-range observation element; ${\lambda _i}$ denotes the carrier phase length element of frequency i. Zhang et al. (Reference Zhang, Liu and Li2003) applied Equation (5) for short baseline monitoring. They obtained the estimated solution based on the hypothesis that the current epoch had the same integer ambiguities as the initial epoch. In the presetn study, the equation is applied to substitute the integer ambiguities based on the initial ambiguity resolution process. The selected observations in the initial ambiguity resolution process and the remaining observations are used to make double differences for Equation (5). The integer bootstrapping estimator is a sequential least-squares adjustment. It starts with the more precise ambiguity, here assumed to be the last ambiguity, and rounds its value to the nearest integer. By contrast, the proposed model divides the ambiguities into two parts. The selected observations in the initial ambiguity resolution process obtain the virtual fixed solution. This solution is unreliable with a small redundancy. The remaining observations eliminate the ambiguities by Equation (5) to obtain a larger redundancy.

Equation (5) is aimed at the code division multiple access model. As for the frequency division multiple access model, the double-difference ambiguity of GLObal NAvigation Satellite System (GLONASS) is not an integer. To solve this problem, the formula of Equation (5) is expressed as follows:

(6)\begin{equation}{\lambda _i}\Delta {b_i} - {\lambda _j}\Delta {b_j} = {\lambda _i}({\Delta {b_i} - \Delta {b_j}} )+ \Delta {b_j}({{\lambda_i} - {\lambda_j}} )= {\lambda _i}\nabla \Delta {b_{ij}} + \Delta {\lambda _{ij}}\Delta {b_j}\end{equation}

where j denotes the frequency j; $\Delta {b_i}$ denotes the single-difference ambiguity element of frequency i; $\Delta {b_j}$ denotes the single-difference ambiguity element of frequency j; ${\lambda _j}$ denotes the carrier phase length element of frequency j; $\nabla \Delta {b_{ij}}$ denotes the double-difference element of frequency i and j; $\Delta {\lambda _{ij}}$ denotes the single-difference element of frequency i and j. To solve the problem of the satellite geometry, Equations (5) and (6) are aimed at all of the observation equations. Teunissen (2019) proposed a new GLONASS ambiguity resolution model. The authors will apply this model to the proposed method in future studies.

Equation (6) has a new residual $\Delta {\lambda _{ij}}\Delta {b_j}$, which is influenced by the single-difference carrier phase length $\Delta {\lambda _{ij}}$ and single-difference integer ambiguity $\Delta {b_j}$. The new residual $\Delta {\lambda _{ij}}\Delta {b_j}$ is less than 0⋅1 cycle, in the condition that $\Delta {\lambda _{ij}}$ is minimal or $\Delta {b_j}$ is more than several cycles (Zhang et al., Reference Zhang, Xu, Wang and Zhang2001), and $\Delta {\lambda _{ij}}\Delta {b_j}$ is absorbed into double-difference noise $\nabla \Delta \varepsilon$. According to Equations (1), (5) and (6), the formula for each double-difference observation equation is as follows:

(7)\begin{equation}{A_i}\Delta X = {\lambda _i}\nabla \Delta {\varphi _i} - \nabla \Delta {\rho _i} - {\lambda _i}Round\left( {\nabla \Delta {\varphi_i} - \frac{{\nabla \Delta {\rho_i}}}{{{\lambda_i}}}} \right) - \nabla \Delta {\varepsilon _i}\end{equation}

where ${A_i} \in {R^{m\ast n}}$ denotes the designed matrix for Equations (5) and (6); $\mathrm{\Delta }X \in {R^n}$ denotes the difference vector between the virtual station and the estimated station. The coordinate formula is then as follows:

(8)\begin{equation}{X_{cs}} = {X_{\textit{vfs}}} + \Delta X\end{equation}

where ${X_{cs}} \in {R^n}$ denotes the current solution vector. The IIAR process obtains the difference vector $\Delta X$, and then the current solution vector ${X_c}$ will be output if $\Delta X$ meets the requirement. The proposed method divides the observations into two parts. The selected observations in the initial ambiguity resolution process obtain a range solution, while the IIAR process obtains a high-precision solution with larger redundancy. It is hypothesised that the virtual fixed solution and the final solution have the same double-difference ambiguities. The selected observations are used to eliminate the integer ambiguity parameters of the remaining observations. The biases in the remaining observations will be decreased or eliminated by Equations (5) and (6).

Different from the previous studies, this method transforms the integer ambiguity into a new form. It is hypothesised that the virtual station and the estimated rover station have the same integer ambiguities in RTK positioning. With the virtual fixed solution, the integer ambiguity vector is substituted by the pseudo-range and carrier phase observation. The IIAR process resolves the adjustment equations for all the satellites, which solves the problem of poor satellite geometry.

2.3 Threshold of the difference vector $\Delta X$

Equations (5) and (8) indicate that $\Delta X$ should be within a certain range. It is hypothesised that the virtual station and the estimated rover station have the same integer ambiguities in RTK positioning. That means their difference is less than half wavelength. The formula is expressed as follows:

(9)\begin{equation}|{{A_i}\Delta X} |\le 0 \cdot 5\lambda \end{equation}

Supposing that maximum displacements for E (east), N (north) and U (up) are equal, Equation (8) is derived as follows (Liu, Reference Liu2014):

(10)\begin{equation}|{\Delta X} |\le \frac{{\sqrt 3 \cdot 0 \cdot 5{\lambda _i}}}{{MAX({{{|{{A_i}} |}^2}} )}} = \frac{{\sqrt 3 \cdot 0 \cdot 5{\lambda _i}}}{{\sqrt 2 }} = 0 \cdot 6{\lambda _i}\end{equation}

where $|{{A_i}} |$ denotes the determinant value of ${A_i}$. According to Equations (8) and (10), the formula is expressed as follows:

(11)\begin{equation}|{\Delta X} |= |{X - {X_{\textit{pre}}}} |< 0 \cdot 6{\lambda _i}\end{equation}

where ${X_{\textit{pre}}}$ denotes the solution in the previous iteration. The difference between X and ${X_{\textit{pre}}}$ should satisfy Equation (11). But the SPP float solution ${X_f}$ cannot meet this requirement. To obtain a more accurate solution, an iterative partial ambiguity resolution method is applied for the initial ambiguity resolution process, and its solution substitutes ${X_v}$ for ${X_f}$. The proposed IIAR method obtains an approximate solution in the initial ambiguity process, while the irrespective of integer ambiguity process resolves the adjustment equations to obtain the high-precision estimated coordinates. It is hypothesised that the virtual station and the estimated rover station have the same double-difference ambiguities in RTK positioning. So the virtual fixed solution should meet the requirements in this section. Otherwise, it indicates that the iterative partial ambiguity resolution obtains a low-precision solution. the experiment in section 3.2 shows that the proposed model cannot guarantee a high-precision solution when the virtual fixed solution cannot meet Equation (11). For clarity, the algorithm schematic diagram is presented in Figure 1.

The detailed steps of the proposed method are as follows, where site A is the base station; site C is the rover station; sites p and q are the satellite p and q:

1. 1. Select feasible satellites for the initial ambiguity resolution process. The site B in Figure 1 is set as the virtual station and the initial ambiguity resolution process obtains its solution ${X_{init}}$.

2. 2. Update the float solution ${X_f}$ and the corrections of the virtual station by ${X_{init}}$.

3. 3. Set the current solution ${X_{init}}$ as the virtual fixed solution ${X_{\textit{vfs}}}$, if the adjacent iterations have the same ambiguity subsets. Otherwise, back to step 1.

4. 4. Substitute the combination of carrier phase and pseudo-range observations for the integer ambiguity as Equation (5).

5. 5. Obtain the difference solution $\Delta X$ as Equation (7), and update the current solution ${X_{cs}}$. Now the current solution ${X_{cs}}$ is resolved by all the observed satellites.

6. 6. Set the current solution ${X_{\textit{cs}}}$ as the final solution X. Go to step 7, if it meets the requirement of Equation (11). Otherwise, back to step 1.

7. 7. Output X as the fixed solution, if $|{X - {X_{init}}} |< 0.6\lambda$. Otherwise, output ${X_{\textit{pre}}}$ as the float solution.