Introduction

With the opening of the five-frequency BDS-3 system, the technology of multi-frequency and multi-constellation conditions is the general trend in GNSS applications. The ambiguity resolution (AR) is the prerequisite of high-precision positioning and navigation, such as real-time kinematic positioning (RTK). In order to improve the success rate and efficiency of AR, the error terms of code and phase observations should be processed appropriately. However, unmodeled errors such as atmospheric delays and multipath usually cannot be ignored (Li et al. 2018).

Earlier research mainly focuses on obtaining the ambiguities with high efficiency and high reliability by using single-frequency or dual-frequency data. The criterion is systematically studied, including rounding, bootstrapping, or integer least-squares (ILS) criterion (Dong and Bock 1989; Teunissen 1993, 1998). The rounding strategy is simple, but the correlations of float ambiguities are ignored, and the bootstrapping method only takes the correlations between adjacent ambiguities into account. The ILS criterion then accounts for all the correlations by finding the minimum norm fixed solutions according to the covariance matrix of the ambiguities (Teunissen et al. 2002). The decorrelation method is then used in ILS (Teunissen 1995). Later, the emergence of three-frequency GNSS brings great benefits to AR. Forssell et al. (1997) and Vollath et al. (1998) proposed the concept of three-frequency carrier ambiguity resolution (TCAR). Typically, the TCAR method means that the ambiguities of the selected linear phase combinations are fixed by rounding directly according to a certain order. The success rate and efficiency of AR are improved significantly by the TCAR method (Cocard et al. 2008). As usual, the TCAR method is based on the geometry-free (GF) model. The geometry-based (GB) model which has better geometric strength, can also be used (Feng 2008; Zhang and He 2016). More recently, since the BDS-3 and Galileo satellites can broadcast four or five frequencies, four-frequency carrier ambiguity resolution (FCAR) or five-frequency carrier ambiguity (FiCAR) became possible, improving the success rate and efficiency of AR even further (Zhang et al. 2020).

In multi-frequency carrier ambiguity resolution (MCAR), including TCAR, FCAR, and FiCAR, mitigating or even avoiding the impacts of unmodeled errors is one of the most important issues since it will deteriorate the accuracy of float ambiguities. The tropospheric and ionospheric delays can be largely eliminated or mitigated via observation combination, model correction, or parameterization (Wang and Rothacher 2013; Li 2018). Li et al. (2010) proposed a distance-independent AR method using a GF and ionospheric-free (IF) semi-generated signal. Zhao et al. (2015) applied a modified TCAR method to mitigate the residual ionospheric delays in three-frequency linear combinations. Since the BDS system has three types of satellites, the BDS-2 or BDS-3 AR is also investigated (Tang et al. 2014; Li et al. 2020). Because multipath cannot be treated by the above traditional methods (Luo et al. 2014; Leick et al. 2015; Zhang et al. 2019), using additional data processing methods is an indispensable approach, especially in canyon environment, low-cost receiver, and long baseline. As usual, the signals with longer wavelengths are more affected by multipath (Lima Filho and Moraes 2020). Based on the satellite period, the sidereal filtering is used, where the undifferenced (UD), single-differenced (SD), or double-differenced (DD) strategy in either the coordinate (Bock et al. 2000; Ragheb et al. 2007) or observation (Zhong et al. 2010; Ye et al. 2015) domain can be conducted. Thanks to the spatial repeatability of satellites, in a static or relative static condition, a so-called hemispherical map providing multipath corrections can also be applied (Moore et al. 2014; Fuhrmann et al. 2015); for the relative and standalone modes, see Dong et al. (2016) and Zheng et al. (2019) respectively. In addition, the other strategies can also be applied, such as the front-end signal processing (McGraw and Braasch 1999; Henkel et al. 2016), advanced tracking method (Yang et al. 2020), improved antenna array (Daneshmand et al. 2013), wavelet analysis (Pugliano et al. 2016), ray-tracing method (Lau and Cross 2007), support vector regression (Phan et al. 2013), and Vondrak filtering (Zheng et al. 2005).

However, currently, there are still some limitations of multipath processing, of course including in the area of MCAR. First, as aforementioned, the traditional multipath processing methods mainly focus on the non-parametric way, where the errors cannot be fully captured or eliminated. Limited research focuses on multipath mitigation in the MCAR (Moradi et al. 2015; Chen et al. 2016; Wang et al. 2018). This topic is crucial since the multipath effects are amplified in the MCAR, which will be proved in the following text. In essence, most of these methods are like the model correction in the time or space field. At present, there is no universal observation-domain parameterized method for code and phase observations. In theory, multipath parameterization may be the best choice since the temporal and spatial properties can be fully considered. Unfortunately, how to parameterize the multipath is still an urgent problem to be solved. Second, traditional multipath mitigation methods cannot work well in real-time or kinematic cases. If coupled with complex conditions, the multipath may become a dominant error source of unmodeled effects. The multipath always has apparent adverse effects when we conduct the MCAR, where the baseline length is usually long. Unfortunately, the multipath is always ignored in the MCAR.

We propose a new code and phase multipath parameterized mitigation method based on the observation domain. Specifically, the multi-frequency code multipath (MCM) and multi-frequency phase multipath (MPM) combinations are formed and preprocessed first in five or least four frequencies cases. After that, the multipath errors of code and phase observations are estimated by the least-squares (LS) criterion.

The main contribution of this research is twofold. First, the general models, methods of the MCAR, especially the FiCAR considering the multipath, are systematically studied. Second, a new code and phase multipath mitigation method is proposed and fully considered in the FiCAR.

According to the wavelength, the linear combinations can be defined as extra-wide lane (EWL, \(\uplambda \ge 2.93\mathrm{ m}\)), wide lane (WL, \(0.75\mathrm{ m}\le\uplambda <2.93\mathrm{ m}\)), medium lane (ML, \(0.19\mathrm{ m}\le\uplambda <0.75\mathrm{ m}\)), and narrow lane (NL, \(0.10\mathrm{ m}\le\uplambda <0.19\mathrm{ m}\)). The baseline length can be divided into three categories, including medium-to-short baseline (\(l\le 100\mathrm{km}\)), medium-to-long baseline (\(100\mathrm{ km}<l\le 200\mathrm{ km}\)) and long baseline (\(l>200\mathrm{ km}\)). The symbols and operators used are described as follows. The symbols ‘\(\Delta\)’ and ‘\(\nabla\)’ denote the between-receiver and between-satellite SD operators, ‘\({\left[.\right]}_{\mathrm{round}}\)’ denotes the rounding operator, ‘diag’ denotes the operator of a diagonal matrix, ‘\({\mathbb{D}}\)’ denotes the dispersion operator, and \({\varvec{I}}\) denotes the identity matrix.

Five-frequency carrier ambiguity resolution

As an important part of the MCAR, the optimal linear combinations are studied comprehensively in this section. Then different FiCAR methods, including the GF FiCAR and GB FiCAR are discussed.

Five-frequency GNSS linear combination

If there are \(k\) (\(k=5\) in this study) frequencies satisfying \({f}_{1}>{f}_{2}>\cdots >{f}_{k}\) in multi-frequency GNSS, the frequency, wavelength, and DD ambiguities of a certain linear combination read

$$f_{\left[ k \right]} = f_{{\left( {i_{1} ,i_{2} , \cdots ,i_{k} } \right)}} = i_{1} \cdot f_{1} + i_{2} \cdot f_{2} + \cdots + i_{k} \cdot f_{k}$$
(1)
$$\lambda_{\left[ k \right]} = \lambda_{{\left( {i_{1} ,i_{2} , \cdots ,i_{k} } \right)}} = c/f_{\left[ k \right]}$$
(2)
$$\nabla \Delta N_{\left[ k \right]} = \nabla \Delta N_{{\left( {i_{1} ,i_{2} , \cdots ,i_{k} } \right)}} = i_{1} \cdot \nabla \Delta N_{1} + i_{2} \cdot \nabla \Delta N_{2} + \cdots + i_{k} \cdot \nabla \Delta N_{k}$$
(3)

where \({i}_{1},{i}_{2},\cdots ,{i}_{k}\) are the integer coefficients, \(\uplambda\) denotes the wavelength, \(c\) denotes the light speed in a vacuum, and \(N\) denotes the ambiguity. Accordingly, the linear code and phase combinations read

$$\nabla \Delta P_{\left[ k \right]} = \nabla \cdot P_{{\left( {i_{1} ,i_{2} , \cdots ,i_{k} } \right)}} = \left( {i_{1} \cdot f_{1} \cdot \nabla \Delta P_{1} + i_{2} \cdot f_{2} \cdot \nabla \Delta P_{2} + \cdots + i_{k} \cdot f_{k} \cdot \nabla \Delta P_{k} } \right)/f_{\left[ k \right]}$$
(4)
$$\nabla \Delta \Phi_{\left[ k \right]} = \nabla \Delta \Phi_{{\left( {i_{1} ,i_{2} , \cdots ,i_{k} } \right)}} = \left( {i_{1} \cdot f_{1} \cdot \nabla \Delta \Phi_{1} + i_{2} \cdot f_{2} \cdot \nabla \Delta \Phi_{2} + \cdots + i_{k} \cdot f_{k} \cdot \nabla \Delta \Phi_{k} } \right)/f_{\left[ k \right]}$$
(5)

where \(P\) and \(\phi\) denote the code and phase observations, respectively. Then the observation models of linear code and phase combinations can be formed

$$\nabla \Delta P_{\left[ k \right]} = \nabla \Delta \rho + \beta_{\left[ k \right]} \nabla \Delta I_{1} + \nabla \Delta T + \nabla \Delta M_{\left[ k \right]} + \nabla \Delta \varepsilon_{\left[ k \right]}$$
(6)
$$\nabla \Delta \Phi_{\left[ k \right]} = \nabla \Delta \rho + \lambda_{\left[ k \right]} \nabla \Delta N_{\left[ k \right]} - \beta_{\left[ k \right]} \nabla \Delta I_{1} + \nabla \Delta T + \nabla \Delta m_{\left[ k \right]} + \nabla \Delta e_{\left[ k \right]}$$
(7)

with the first-order ionospheric scale factor \({\beta }_{[k]}={f}_{1}^{2}\left({i}_{1}/{f}_{1}+{i}_{2}/{f}_{2}+\cdots +{i}_{k}/{f}_{k}\right)/{f}_{[k]}\), where \(\rho\) denotes the receiver-to-satellite range, and \(I\) and \(T\) denote the ionospheric and tropospheric delays. \(M\) and \(m\) denote the code and phase multipath, and \(\varepsilon\) and \(e\) denote the code and phase observation noise.

In the case of the five-frequency BDS-3 system, detailed information is listed in Table 1. It is worth noting that the code chipping rates of different signals are not the same. Hence the code precisions from different signals are also different. To obtain the optimal linear combination, the code and phase total noise levels (TNLs) are used. Since the baseline length is usually relatively long in MCAR, the second-order ionospheric delay and the orbit error should be better considered. Unlike the other traditional studies, here, the multipath is also considered in code and phase TNLs. The main reason is that the multipath may play an important role under complex conditions. Besides, the multipath is amplified in MCAR, which will be mentioned in the following test. The code TNL \({\sigma }_{{TN}_{P}}\) and phase TNL \({\sigma }_{{TN}_{\Phi }}\) read

$$\sigma_{{TN_{P} }} = \sqrt {\beta_{\left[ k \right]}^{2} \sigma_{{\nabla \Delta I_{1} }}^{2} + \theta_{\left[ k \right]}^{2} \sigma_{{\nabla \Delta I_{2} }}^{2} + \sigma_{\nabla \Delta T}^{2} + \mu_{\left[ k \right]}^{2} \sigma_{\nabla \Delta M}^{2} + \sigma_{\nabla \Delta O}^{2} + \eta_{\left[ k \right]}^{2} \sigma_{{\varepsilon_{\nabla \Delta P} }}^{2} }$$
(8)
$$\sigma_{{TN_{\Phi } }} = \sqrt {\beta_{\left[ k \right]}^{2} \sigma_{{\nabla \Delta I_{1} }}^{2} + \theta_{\left[ k \right]}^{2} \sigma_{{\nabla \Delta I_{2} }}^{2} + \sigma_{\nabla \Delta T}^{2} + \mu_{\left[ k \right]}^{2} \sigma_{\nabla \Delta m}^{2} + \sigma_{\nabla \Delta O}^{2} + \mu_{\left[ k \right]}^{2} \sigma_{{\varepsilon_{\nabla \Delta \Phi } }}^{2} } /\lambda_{\left[ k \right]}$$
(9)

with the second-order ionospheric scale factor \({\theta }_{[k]}={f}_{1}^{3}\left({i}_{1}/{f}_{1}^{2}+{i}_{2}/{f}_{2}^{2}+\cdots +{i}_{k}/{f}_{k}^{2}\right)/{f}_{[k]}\), the multipath and phase noise amplitude factor \({\mu }_{\left[k\right]}^{2}=\left[{\left({i}_{1}\bullet {f}_{1}\right)}^{2}+{\left({i}_{2}\bullet {f}_{2}\right)}^{2}+\cdots +{\left({i}_{k}\bullet {f}_{k}\right)}^{2}\right]/{f}_{\left[k\right]}^{2}\), the code noise amplitude factor \({\eta }_{\left[k\right]}^{2}=\left[{\left({i}_{1}\bullet {f}_{1}\right)}^{2}+{\left({\alpha }_{12}\bullet {i}_{2}\bullet {f}_{2}\right)}^{2}+\cdots +{\left({\alpha }_{1k}\bullet {i}_{k}\bullet {f}_{k}\right)}^{2}\right]/{f}_{\left[k\right]}^{2}\) where \({\alpha }_{1k}\) denotes the scale factor between the frequencies 1 and \(k\). \({\sigma }_{\nabla \Delta {I}_{1}}\) and \({\sigma }_{\nabla \Delta {I}_{2}}\) denote the standard deviations (STDs) of the DD first-order and second-order ionospheric delays, \(\nabla \Delta T\) denotes the STD of the DD tropospheric delay, \(\nabla \Delta O\) denotes the STD of the DD orbit error, \({\sigma }_{\nabla \Delta M}\) and \({\sigma }_{\nabla \Delta m}\) denote the STDs of the DD code and phase multipath, and \({\sigma }_{{\varepsilon }_{\nabla \Delta P}}\) and \({\sigma }_{{\varepsilon }_{\nabla \Delta \Phi }}\) denote the STDs of the DD code and phase noise. According to the code chipping rates listed in Table 1, without loss of generality, there has the relationship \({\sigma }_{{\varepsilon }_{\nabla \Delta P}}={\sigma }_{{\varepsilon }_{{\nabla \Delta P}_{1}}}={{\alpha }_{12}\sigma }_{{\varepsilon }_{{\nabla \Delta P}_{2}}}={{\alpha }_{13}\sigma }_{{\varepsilon }_{{\nabla \Delta P}_{3}}}={\alpha }_{14}{\sigma }_{{\varepsilon }_{{\nabla \Delta P}_{4}}}={\alpha }_{15}{\sigma }_{{\varepsilon }_{{\nabla \Delta P}_{5}}}\) with empirical values \({{\alpha }}_{12}=0.5\), \({{\alpha }}_{13}=0.1\), \({{\alpha }}_{14}=0.1\), and \({{\alpha }}_{15}=0.1\); \({\sigma }_{{\varepsilon }_{\nabla \Delta \Phi }}={\sigma }_{{\varepsilon }_{{\nabla \Delta \Phi }_{1}}}={\sigma }_{{\varepsilon }_{{\nabla \Delta \Phi }_{2}}}={\sigma }_{{\varepsilon }_{{\nabla \Delta \Phi }_{3}}}={\sigma }_{{\varepsilon }_{{\nabla \Delta \Phi }_{4}}}={\sigma }_{{\varepsilon }_{{\nabla \Delta \Phi }_{5}}}\). Similar processing approaches can be found in other research such as Tang et al. (2014) and Zhang et al. (2020).

Table 1 Information of the five-frequency BDS-3 signals
Full size table

In order to determine the optimal linear combinations, Table 2 presents the error budgets of different DD code and phase error types for different baseline lengths. The empirical values are widely adopted in other research (Li et al. 2010; Zhang et al. 2020). Unlike other related studies, the code and phase multipath effects are considered here. Then according to Tables 1 and 2, the wavelength of linear phase combinations can be computed by (1) and (2). Five independent combinations need to be determined in advance. Since only four independent combinations can be found under the conditions of \({i}_{1}+{i}_{2}+{i}_{3}+{i}_{4}+{i}_{5}=0\), the raw signal satisfying \({i}_{1}+{i}_{2}+{i}_{3}+{i}_{4}+{i}_{5}=1\) is treated as the fifth independent signal. It is worth noting that the correlations between the selected optimal liner combinations are often ignored in MCAR. The reason is that the EWL or WL is easy to be fixed directly due to their relatively low noise level and long wavelength. The four independent EWL/WL optimal linear phase combinations and NL optimal linear code combinations under different conditions can be found according to the TNLs. Tables 3 and 4 list the detailed information of the optimal linear phase and code combinations for the five-frequency BDS-3 system in this study.

Table 2 DD code and phase error budgets for the medium-to-short, medium-to-long, and long baselines (cm)
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Table 3 Optimal linear phase combinations of five-frequency BDS-3 observations under the circumstances of medium-to-short, medium-to-long, and long baselines
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Table 4 Optimal linear code combinations of five-frequency BDS-3 observations under the circumstances of medium-to-short, medium-to-long, and long baselines
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FiCAR method

There are two main approaches in FiCAR. Specifically, one is the GF-FiCAR method, and the other one is the GB-FiCAR method. For the GF-FiCAR method, one can compute two main specific types

$$\nabla \Delta \check{N}_{{{\text{LC}}1}} = \left[ {\left( {\nabla \Delta P_{\left[ k \right]} - \nabla \Delta \Phi_{{{\text{LC}}1}} } \right)/\lambda_{{{\text{LC}}1}} } \right]_{{{\text{round}}}}$$
(10)
$$\nabla \Delta \check {N}_{{{\text{LC}}2}} = \left[ {\left( {\nabla \Delta \Phi_{{{\text{LC}}1}} - \lambda_{{{\text{LC}}1}} \cdot \nabla \Delta \mathop N\limits_{{{\text{LC}}1}} - \nabla \Delta \Phi_{{{\text{LC}}2}} } \right)/\lambda_{{{\text{LC}}2}} } \right]_{{{\text{round}}}}$$
(11)

where the subscripts \(LC1\) and \(LC2\) denote the first and second linear phase combinations, and \(\nabla \Delta \mathop N\limits\) denotes the fixed DD integer ambiguities. The ambiguities of chosen linear combinations are fixed one by one according to the values of TNL or wavelength. In real applications, one can choose either (10) or (11) according to the precisions of \({\nabla \Delta P}_{[k]}\) and \({\nabla \Delta \Phi }_{LC1}\).

In the GB-FiCAR method, the three-dimensional coordinates are resolved together with the ambiguities. There have two main types

$$\left[ {\begin{array}{*{20}c} {{\varvec{v}}_{{P_{\left[ k \right]} }} } \\ {{\varvec{v}}_{LC1} } \\ \end{array} } \right] = \left[ {\begin{array}{*{20}c} {\varvec{A}} & 0 \\ {\varvec{A}} & {{\varvec{I}} \cdot \lambda_{LC1} } \\ \end{array} } \right]\left[ {\begin{array}{*{20}c} {\varvec{x}} \\ {\nabla \Delta {\varvec{N}}_{LC1} } \\ \end{array} } \right] - \left[ {\begin{array}{*{20}c} {{\varvec{l}}_{{P_{\left[ k \right]} }} } \\ {{\varvec{l}}_{LC1} } \\ \end{array} } \right]$$
(12)
$$\left[ {\begin{array}{*{20}c} {{\varvec{v}}_{LC1}^{ {\prime }} } \\ {{\varvec{v}}_{LC2} } \\ \end{array} } \right] = \left[ {\begin{array}{*{20}c} {\varvec{A}} & 0 \\ {\varvec{A}} & {{\varvec{I}} \cdot \lambda_{LC2} } \\ \end{array} } \right]\left[ {\begin{array}{*{20}c} {\varvec{x}} \\ {\nabla \Delta {\varvec{N}}_{LC2} } \\ \end{array} } \right] - \left[ {\begin{array}{*{20}c} {{\varvec{l}}_{LC1}^{^{\prime}} } \\ {{\varvec{l}}_{LC2} } \\ \end{array} } \right]$$
(13)

where the \({\varvec{v}}\) and \({{\varvec{v}}}^{\boldsymbol{^{\prime}}}\) denote the residual and fixed residual vectors. \({\varvec{l}}\) and \({{\varvec{l}}}^{\boldsymbol{^{\prime}}}\) denote the observation and fixed observation vectors, and \({\varvec{A}}\) and \({\varvec{x}}\) denote the design matrix and baseline component. Here, the ambiguities of chosen linear combinations are also fixed one by one through a certain sequence. Similarly, if the precision of \({{\varvec{l}}}_{LC1}^{^{\prime}}\) is higher than that of \({{\varvec{l}}}_{P}\), once can choose (13) to fix the ambiguities. It is also worth noting that more code or linear code combination observations can be used, thus making more redundant observations. Besides, the high-precision fixed NL can also be used in (13).

Multipath mitigation in multi-frequency code and phase observations

In this section, the amplification effects of code and phase multipath in five-frequency linear combinations are discussed. Then the theory and method of the new multipath parameterized mitigation method are proposed.

Multipath effects in five-frequency linear combinations

The code and phase multipath in multi-frequency linear combinations can be derived

$$\nabla \Delta M_{\left[ k \right]} = \left( {i_{1} \cdot f_{1} \cdot \nabla \Delta M_{1} + i_{2} \cdot f_{2} \cdot \nabla \Delta M_{2} + \cdots + i_{k} \cdot f_{k} \cdot \nabla \Delta M_{k} } \right)/f_{\left[ k \right]}$$
(14)
$$\nabla \Delta m_{\left[ k \right]} = \left( {i_{1} \cdot f_{1} \cdot \nabla \Delta m_{1} + i_{2} \cdot f_{2} \cdot \nabla \Delta m_{2} + \cdots + i_{k} \cdot f_{k} \cdot \nabla \Delta m_{k} } \right)/f_{\left[ k \right]}$$
(15)

Then the multipath factor can be obtained

$$\nu = \left( {\left| {i_{1} } \right| \cdot f_{1} + \left| {i_{2} } \right| \cdot f_{2} + \cdots + \left| {i_{k} } \right| \cdot f_{k} } \right)/f_{\left[ k \right]}$$
(16)

The multipath factors and the wavelength magnifications for the wavelength of the optimal combination corresponding to the wavelength of each raw observation are presented in Table 5. It can be seen that the multipath factor is rather large, which can be up to 488.5. We can also find that the multipath factor of each optimal linear phase combination is larger than the wavelength magnifications. Specifically, the multipath factors are approximately 2.0, 3.4, and 5.0 times of wavelength magnifications under the circumstances of medium-to-short, medium-to-long, and long baselines, respectively. This value can be up to 8.6 times, where the multipath cannot be ignored apparently. It indicates that the multipath will be more significant in optimal linear phase combinations, especially in long baselines. Unfortunately, the FiCAR at this time usually cannot work best. Therefore, it is highly urgent to mitigate the multipath in FiCAR.

Table 5 Multipath factors and the wavelength magnifications for the wavelength of the optimal combination corresponding to the wavelength of each raw observation under the circumstances of medium-to-short, medium-to-long, and long baselines
Full size table

A new parameterization method for mitigation code and phase multipath

To mitigate the code and phase multipath by observation-domain parameterization, starting from the UD and uncombined observation equations, the between-receiver SD MCM and SD MPM combinations are formulated, which eliminate the clock offset and hardware delay of the satellite. In addition, since the SD MCM and SD MPM combinations are the GF and IF combinations, there only leave the receiver-dependent code and phase hardware delays, the code and phase multipath, and the observation noise. The ambiguities are also existent in SD MPM combinations. The between-receiver SD MCM and SD MPM combinations read

$$\begin{aligned} \Delta {\text{MCM}}_{{{\text{ijk}}}} & = \left( {\lambda_{k}^{2} - \lambda_{j}^{2} } \right)\Delta P_{i} + \left( {\lambda_{i}^{2} - \lambda_{k}^{2} } \right)\Delta P_{j} + \left( {\lambda_{j}^{2} - \lambda_{i}^{2} } \right)\Delta P_{k} = \left( {\lambda_{k}^{2} - \lambda_{j}^{2} } \right)\Delta \xi_{i} + \left( {\lambda_{i}^{2} - \lambda_{k}^{2} } \right)\Delta \xi_{j} \\ & \quad + \left( {\lambda_{j}^{2} - \lambda_{i}^{2} } \right)\Delta \xi_{k} + \left( {\lambda_{k}^{2} - \lambda_{j}^{2} } \right)\Delta M_{i} + \left( {\lambda_{i}^{2} - \lambda_{k}^{2} } \right)\Delta M_{j} + \left( {\lambda_{j}^{2} - \lambda_{i}^{2} } \right)\Delta M_{k} + \varepsilon_{{\Delta {\text{MCM}}_{{{\text{ijk}}}} }} \\ \end{aligned}$$
(17)
$$\begin{aligned} \Delta {\text{MPM}}_{{{\text{ijk}}}} & = \left( {\lambda_{k}^{2} - \lambda_{j}^{2} } \right)\Delta \phi_{i} + \left( {\lambda_{i}^{2} - \lambda_{k}^{2} } \right)\Delta \phi_{j} + \left( {\lambda_{j}^{2} - \lambda_{i}^{2} } \right)\Delta \phi_{k} = \left( {\lambda_{k}^{2} - \lambda_{j}^{2} } \right)\lambda_{i} \Delta N_{i} + \left( {\lambda_{i}^{2} - \lambda_{k}^{2} } \right)\lambda_{j} \Delta N_{j} \\ & \quad + \left( {\lambda_{j}^{2} - \lambda_{i}^{2} } \right)\lambda_{k} \Delta N_{k} + \left( {\lambda_{k}^{2} - \lambda_{j}^{2} } \right)\Delta \zeta_{i} + \left( {\lambda_{i}^{2} - \lambda_{k}^{2} } \right)\Delta \zeta_{j} + \left( {\lambda_{j}^{2} - \lambda_{i}^{2} } \right)\Delta \zeta_{k} + \left( {\lambda_{k}^{2} - \lambda_{j}^{2} } \right)\Delta m_{i} \\ & \quad + \left( {\lambda_{i}^{2} - \lambda_{k}^{2} } \right)\Delta m_{j} + \left( {\lambda_{j}^{2} - \lambda_{i}^{2} } \right)\Delta m_{k} + e_{{\Delta {\text{MPM}}_{{{\text{ijk}}}} }} \\ \end{aligned}$$
(18)

where the subscripts \(i\), \(j\) and \(k\) denote the frequencies,\(\xi\) and \(\zeta\) denote the code and phase hardware delays. \({\varepsilon }_{{\Delta \mathrm{MCM}}_{ijk}}\) and \({e}_{\Delta {\mathrm{MPM}}_{ijk}}\) denote the noise of \({\Delta \mathrm{MCM}}_{ijk}\) and \({\Delta \mathrm{MPM}}_{ijk}\). The SD code and phase hardware delays and the SD ambiguities need to be removed to estimate the SD code and phase multipath in (17) and (18). Since the hardware delays and the ambiguities can be regarded as a constant if there are no cycle slips within a certain period (de Bakker et al. 2012; Cai et al. 2016), the method of removing the averages over a certain period is used. Therefore, the preprocessed SD MCM (\(\Delta {\mathrm{MCM}}_{ijk}^{^{\prime}}\)) and SD MPM (\(\Delta {\mathrm{MPM}}_{ijk}^{^{\prime}}\)) combinations can be computed

$$\begin{aligned} \Delta {\text{MCM}}_{{{\text{ijk}}}}^{ {\prime }} & = \left( {\lambda_{k}^{2} - \lambda_{j}^{2} } \right)\Delta P_{i} + \left( {\lambda_{i}^{2} - \lambda_{k}^{2} } \right)\Delta P_{j} + \left( {\lambda_{j}^{2} - \lambda_{i}^{2} } \right)\Delta P_{k} - \frac{1}{n}\mathop \sum \limits_{t = 1}^{n} \left[ {\Delta {\text{MCM}}_{{{\text{ijk}}}} \left( t \right)} \right] \\ & = \left( {\lambda_{k}^{2} - \lambda_{j}^{2} } \right)\Delta M_{i} + \left( {\lambda_{i}^{2} - \lambda_{k}^{2} } \right)\Delta M_{j} + \left( {\lambda_{j}^{2} - \lambda_{i}^{2} } \right)\Delta M_{k} + \varepsilon_{{\Delta {\text{MCM}}_{{{\text{ijk}}}} }} \\ \end{aligned}$$
(19)
$$\begin{aligned} \Delta {\text{MPM}}_{{{\text{ijk}}}}^{ {\prime }} & = \left( {\lambda_{k}^{2} - \lambda_{j}^{2} } \right)\Delta \phi_{i} + \left( {\lambda_{i}^{2} - \lambda_{k}^{2} } \right)\Delta \phi_{j} + \left( {\lambda_{j}^{2} - \lambda_{i}^{2} } \right)\Delta \phi_{k} - \frac{1}{n}\mathop \sum \limits_{t = 1}^{n} \left[ {\Delta {\text{MPM}}_{{{\text{ijk}}}} \left( t \right)} \right] \\ & = \left( {\lambda_{k}^{2} - \lambda_{j}^{2} } \right)\Delta m_{i} + \left( {\lambda_{i}^{2} - \lambda_{k}^{2} } \right)\Delta m_{j} + \left( {\lambda_{j}^{2} - \lambda_{i}^{2} } \right)\Delta m_{k} + e_{{\Delta {\text{MPM}}_{{{\text{ijk}}}} }} \\ \end{aligned}$$
(20)

In the case of five frequencies, 10 different SD MCM and SD MPM combinations can be formed with different frequency combinations in a single epoch. Since there are only 5 SD code and phase multipaths to be determined for a certain satellite, these SD code and phase multipath can be parameterized and estimated by the LS criterion. It is worth noting that 4 different SD MCM and SD MPM combinations can be formed when there are four frequencies. Accordingly, four unknown SD code and phase multipath can be uniquely determined. That is, the proposed procedure can be used in at least four-frequency situation. It is worth noting that the UD or DD MCM and MPM multipath combinations can also be formed and estimated in a similar way. The reason why the SD MCM and MPM combinations are used here is that they are more suitable for the relative mode and are not affected by the reference satellite.

Taking the five-frequency case as an example, the specific steps of multipath parameterization are as follows. First, the code and phase functional models of UD, SD or DD multipath parameterization can be derived

$${\mathbf{MCM}} = {\mathbf{BM}} + {\varvec{\varepsilon}}$$
(21)
$${\mathbf{MPM}} = {\mathbf{Bm}} + {\varvec{e}}$$
(22)

where \(\mathbf{M}\mathbf{C}\mathbf{M}\) and \(\mathbf{M}\mathbf{P}\mathbf{M}\) denote the preprocessed MCM and MPM observation vectors, respectively. \({\varvec{B}}\) denotes the design matrix, \({\varvec{M}}\) and \({\varvec{m}}\) denote the code and phase multipath vectors to be estimated. \({\varvec{\varepsilon}}\) and \({\varvec{e}}\) denote the noise vectors of \(\mathbf{M}\mathbf{C}\mathbf{M}\) and \(\mathbf{M}\mathbf{P}\mathbf{M}\). Then the (21) and (22) can be expanded to

$$\left[ {\begin{array}{*{20}c} {{\text{MCM}}_{123}^{ {\prime }} } \\ {{\text{MCM}}_{124}^{ {\prime }} } \\ {{\text{MCM}}_{125}^{ {\prime }} } \\ {{\text{MCM}}_{134}^{ {\prime }} } \\ {{\text{MCM}}_{135}^{ {\prime }} } \\ {{\text{MCM}}_{145}^{ {\prime }} } \\ {{\text{MCM}}_{234}^{ {\prime }} } \\ {{\text{MCM}}_{235}^{ {\prime }} } \\ {{\text{MCM}}_{245}^{^{\prime}} } \\ {{\text{MCM}}_{345}^{^{\prime}} } \\ \end{array} } \right] = \left[ {\begin{array}{*{20}c} {\alpha_{32} } & {\alpha_{13} } & {\alpha_{21} } & 0 & 0 \\ {\alpha_{42} } & {\alpha_{14} } & 0 & {\alpha_{21} } & 0 \\ {\alpha_{52} } & {\alpha_{15} } & 0 & 0 & {\alpha_{21} } \\ {\alpha_{43} } & 0 & {\alpha_{14} } & {\alpha_{31} } & 0 \\ {\alpha_{53} } & 0 & {\alpha_{15} } & 0 & {\alpha_{31} } \\ {\alpha_{54} } & 0 & 0 & {\alpha_{15} } & {\alpha_{41} } \\ 0 & {\alpha_{43} } & {\alpha_{24} } & {\alpha_{32} } & 0 \\ 0 & {\alpha_{53} } & {\alpha_{25} } & 0 & {\alpha_{32} } \\ 0 & {\alpha_{54} } & 0 & {\alpha_{25} } & {\alpha_{42} } \\ 0 & 0 & {\alpha_{54} } & {\alpha_{35} } & {\alpha_{43} } \\ \end{array} } \right]\left[ {\begin{array}{*{20}c} {M_{1} } \\ {M_{2} } \\ {M_{3} } \\ {M_{4} } \\ {M_{5} } \\ \end{array} } \right] + \left[ {\begin{array}{*{20}c} {\varepsilon_{{{\text{MCM}}_{123} }} } \\ {\varepsilon_{{{\text{MCM}}_{124} }} } \\ {\varepsilon_{{{\text{MCM}}_{125} }} } \\ {\varepsilon_{{{\text{MCM}}_{134} }} } \\ {\varepsilon_{{{\text{MCM}}_{135} }} } \\ {\varepsilon_{{{\text{MCM}}_{145} }} } \\ {\varepsilon_{{{\text{MCM}}_{234} }} } \\ {\varepsilon_{{{\text{MCM}}_{235} }} } \\ {\varepsilon_{{{\text{MCM}}_{245} }} } \\ {\varepsilon_{{{\text{MCM}}_{345} }} } \\ \end{array} } \right]$$
(23)
$$\left[ {\begin{array}{*{20}c} {{\text{MPM}}_{123}^{ {\prime }} } \\ {{\text{MPM}}_{124}^{ {\prime }} } \\ {{\text{MPM}}_{125}^{ {\prime }} } \\ {{\text{MPM}}_{134}^{ {\prime }} } \\ {{\text{MPM}}_{135}^{ {\prime }} } \\ {{\text{MPM}}_{145}^{ {\prime }} } \\ {{\text{MPM}}_{234}^{ {\prime }} } \\ {{\text{MPM}}_{235}^{ {\prime }} } \\ {{\text{MPM}}_{245}^{ {\prime }} } \\ {{\text{MPM}}_{345}^{ {\prime }} } \\ \end{array} } \right] = \left[ {\begin{array}{*{20}c} {\alpha_{32} } & {\alpha_{13} } & {\alpha_{21} } & 0 & 0 \\ {\alpha_{42} } & {\alpha_{14} } & 0 & {\alpha_{21} } & 0 \\ {\alpha_{52} } & {\alpha_{15} } & 0 & 0 & {\alpha_{21} } \\ {\alpha_{43} } & 0 & {\alpha_{14} } & {\alpha_{31} } & 0 \\ {\alpha_{53} } & 0 & {\alpha_{15} } & 0 & {\alpha_{31} } \\ {\alpha_{54} } & 0 & 0 & {\alpha_{15} } & {\alpha_{41} } \\ 0 & {\alpha_{43} } & {\alpha_{24} } & {\alpha_{32} } & 0 \\ 0 & {\alpha_{53} } & {\alpha_{25} } & 0 & {\alpha_{32} } \\ 0 & {\alpha_{54} } & 0 & {\alpha_{25} } & {\alpha_{42} } \\ 0 & 0 & {\alpha_{54} } & {\alpha_{35} } & {\alpha_{43} } \\ \end{array} } \right]\left[ {\begin{array}{*{20}c} {m_{1} } \\ {m_{2} } \\ {m_{3} } \\ {m_{4} } \\ {m_{5} } \\ \end{array} } \right] + \left[ {\begin{array}{*{20}c} {e_{{{\text{MPM}}_{123} }} } \\ {e_{{{\text{MPM}}_{124} }} } \\ {e_{{{\text{MPM}}_{125} }} } \\ {e_{{{\text{MPM}}_{134} }} } \\ {e_{{{\text{MPM}}_{135} }} } \\ {e_{{{\text{MPM}}_{145} }} } \\ {e_{{{\text{MPM}}_{234} }} } \\ {e_{{{\text{MPM}}_{235} }} } \\ {e_{{{\text{MPM}}_{245} }} } \\ {e_{{{\text{MPM}}_{345} }} } \\ \end{array} } \right]$$
(24)

with \({\alpha }_{ij}={\lambda }_{i}^{2}-{\lambda }_{j}^{2}\) (α21 = 0 in this study). According to the law of covariance propagation, the corresponding stochastic models read

$${\mathbb{D}}\left( {{\mathbf{MCM}}} \right) = \sigma_{\Delta P}^{2} {\text{diag}}\left( {\gamma_{123} ,\gamma_{124} , \cdots ,\gamma_{345} } \right)$$
(25)
$${\mathbb{D}}\left( {{\mathbf{MPM}}} \right) = \sigma_{\Delta \Phi }^{2} {\text{diag}}\left( {\gamma_{123} ,\gamma_{124} , \cdots ,\gamma_{345} } \right)$$
(26)

with \({\gamma }_{ijk}={\left({\lambda }_{k}^{2}-{\lambda }_{j}^{2}\right)}^{2}+{\left({\lambda }_{i}^{2}-{\lambda }_{k}^{2}\right)}^{2}+{\left({\lambda }_{j}^{2}-{\lambda }_{i}^{2}\right)}^{2}\). It is worth noting that a fully populated stochastic model can also be used here if necessary. At last, the code and phase multipath can be estimated based on the LS criterion

$${\varvec{M}} = \left[ {{\varvec{B}}^{{\text{T}}} \left[ {{\mathbb{D}}\left( {{\mathbf{MCM}}} \right)} \right]^{ - 1} {\varvec{B}}} \right]^{ - 1} {\varvec{B}}^{{\text{T}}} \left[ {{\mathbb{D}}\left( {{\mathbf{MCM}}} \right)} \right]^{ - 1} {\mathbf{MCM}}$$
(27)
$${\varvec{m}} = \left[ {{\varvec{B}}^{{\text{T}}} \left[ {{\mathbb{D}}\left( {{\mathbf{MPM}}} \right)} \right]^{ - 1} {\varvec{B}}} \right]^{ - 1} {\varvec{B}}^{{\text{T}}} \left[ {{\mathbb{D}}\left( {{\mathbf{MPM}}} \right)} \right]^{ - 1} {\mathbf{MPM}}$$
(28)

After the multipath corrections are estimated, the multipath-reduced observations can be obtained. Theoretically, the multipath effects will be mitigated to a great extent in the FiCAR when using the multipath-reduced DD observations. Compared with other traditional methods, the parameterization method can be conducted based on the UD, SD, and DD modes in real-time or kinematic situations.

Test on real observations and result analysis

To evaluate and verify the performance of the proposed method, two baselines with different lengths were tested. The Trimble receivers were used as the reference and rover stations, where the 1-Hz five-frequency BDS-3 code and phase observations were collected. The high-end antennas with chokes are adopted in the reference stations and placed on the roof, whereas the low-cost antenna without a choke is placed in relatively obstructed places. Hence, the multipath effects are existent. Table 6 lists the detailed information of the baselines. It indicates that the two different baselines can represent two typical baseline lengths in single-baseline RTK or network RTK.

Table 6 Details of the baselines used in the test
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The traditional and improved FiCAR methods are both applied and compared. Four different AR modes are used in each type of FiCAR method, i.e., single-epoch GF FiCAR, single-epoch GB FiCAR, multi-epoch GF FiCAR, and multi-epoch GB FiCAR. The fixing strategy is rounding. The modified Hopfield model and ionospheric-fixed model are used. The cycle slips have been detected and repaired (Zhang and Li 2020). The true ambiguities are validated by comparing the fixed baseline solutions with the precise coordinates in advance. According to the total noise level, the first to fourth signals to be fixed are \({\nabla \Delta \Phi }_{\left(\mathrm{0,0},\mathrm{0,1},-1\right)}\), \({\nabla \Delta \Phi }_{\left(\mathrm{0,0},1,-\mathrm{1,0}\right)}\), \({\nabla \Delta \Phi }_{\left(1,-\mathrm{1,0},\mathrm{0,0}\right)}\) and \({\nabla \Delta \Phi }_{\left(\mathrm{0,1},-\mathrm{1,0},0\right)}\), then the fifth independent signal is \({\nabla \Delta \Phi }_{\left(\mathrm{0,0},\mathrm{0,0},1\right)}\). The optimal linear code combination \({\nabla \Delta P}_{\left(\mathrm{0,0},\mathrm{1,1},0\right)}\) is used as the precise code observation. The fixed \({\nabla \Delta \Phi }_{\left(\mathrm{0,0},1,-\mathrm{1,0}\right)}\) is used as another precise signal. To further improve the success rate, in the single-epoch mode, the AR of the fifth signal may rely on the high-precision fixed NL \({\nabla \Delta \Phi }_{\left(\mathrm{4,4},-3,-3,-2\right)}\) of which the wavelength and precision are as small as 0.1084 and 0.1420 m, respectively. Here the ambiguities of \({\nabla \Delta \Phi }_{\left(\mathrm{4,4},-3,-3,-2\right)}\) can be computed directly with the above four EWLs. In the multi-epoch mode, the AR is set to reinitialize every 5 min. To avoid the potential sudden changes of the float solutions caused by the NL \({\nabla \Delta \Phi }_{\left(\mathrm{4,4},-3,-3,-2\right)}\), the AR of the fifth signal in multi-epoch mode is only assisted by the fixed \({\nabla \Delta \Phi }_{\left(\mathrm{0,0},1,-\mathrm{1,0}\right)}\).

As an example, \({\Delta \mathrm{MCM}}_{123}\) and \({\Delta \mathrm{MPM}}_{123}\) as well as the elevation for PRN 36 are illustrated in Figs. 1 and 2, respectively. It can be found that the SD MCM and SD MPM combinations are rather stable during this period. Hence, it is feasible to avoid the impacts of the hardware delays and the ambiguities by removing the average in a certain period, e.g., 3 h in this study. The SD MCM or the SD MPM combinations from different baselines are highly consistent. It demonstrates that the distance-independent SD code and phase multipath effects are apparently existent and even dominant here.

Fig. 1
figure 1

SD MCM combination (blue) and elevation (red) of B1C, B1I, and B3I for PRN 36, where the left and right panels denote the results of baselines No.1 and 2, respectively

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Fig. 2
figure 2

SD MPM combination (blue) and elevation (red) of B1C, B1I, and B3I for PRN 36, where the left and right panels denote the results of baselines No.1 and 2, respectively

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Figures 3, 4, 5 and 6 show the \(\Delta {\mathrm{MCM}}_{123}^{^{\prime}}\), \(\Delta {\mathrm{MCM}}_{145}^{^{\prime}}\), \(\Delta {\mathrm{MPM}}_{123}^{^{\prime}}\) and \(\Delta {\mathrm{MPM}}_{145}^{^{\prime}}\), hence the information of all five frequencies is included. Each color denotes one satellite in each panel. It can be clearly seen that all the preprocessed SD MCM combinations fluctuate between −0.16 and 0.16 m regardless of the satellite and baseline length. Similar conclusions can be found in the preprocessed SD MPM combinations, of which the values fluctuate between −0.16 and 0.16 cm.

Fig. 3
figure 3

Preprocessed SD MCM combination of B1C, B1I, and B3I, where the left and right panels denote the results of baselines No.1 and 2, respectively. Each color denotes one satellite

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Fig. 4
figure 4

Preprocessed SD MCM combination of B1C, B2b, and B2a, where the left and right panels denote the results of baselines No.1 and 2, respectively. Each color denotes one satellite

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Fig. 5
figure 5

Preprocessed SD MPM combination of B1C, B1I, and B3I, where the left and right panels denote the results of baselines No.1 and 2, respectively. Each color denotes one satellite

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Fig. 6
figure 6

Preprocessed SD MPM combination of B1C, B2b, and B2a, where the left and right panels denote the results of baselines No.1 and 2, respectively. Each color denotes one satellite

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Figures 7 and 8 illustrate the SD code and phase multipath of the two baselines. We can find that all the code and phase multipath effects from different frequencies and baselines fluctuate between −4 and 4 m and −4 and 4 cm, respectively. Once again, since there are no significant differences between the two baselines, the site-specific multipath effects are indeed estimated.

Fig. 7
figure 7

SD code (up) and phase (bottom) multipath of each frequency for baseline No.1, where the left to right panels denote the results of B1C, B1I, B3I, B2b, and B2a, respectively. Each color denotes one satellite

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Fig. 8
figure 8

SD code (up) and phase (bottom) multipath of each frequency for baseline No.2, where the left to right panels denote the results of B1C, B1I, B3I, B2b, and B2a, respectively. Each color denotes one satellite

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Figure 9 illustrates the float solutions of the first four EWLs for baseline No.2 using traditional and improved methods in the single-epoch GF-FiCAR mode. The panels in the same row denote one satellite pair. Similar conclusions can be found in baseline No.1 or GB-FiCAR mode. First, it can be clearly seen that all the ambiguities of the first three EWLs can be easily fixed by using either the traditional or improved methods. This is because the first three optimal phase liner combinations have enough small TNLs (i.e., only 0.0815, 0.0892, and 0.1047 cycles, respectively), and can weaken the influences of the multipath effects to a great extent. The proposed method has a better performance. For the fourth EWL, the proposed method can still fix the ambiguities easily, whereas the traditional method cannot. It can also be found that after the fixed EWL \({\nabla \Delta \Phi }_{\left(\mathrm{0,0},1,-\mathrm{1,0}\right)}\) is used as the precise observation in the third and fourth signals, the float solutions of the proposed method have much better behaviors than those of the traditional method. It is reasonable that the multipath effects are mitigated to a great extent in the ionospheric-reduced signals. The results also indicate that since the proposed method can work much better than the traditional method, the proposed method can be used under complex conditions.

Fig. 9
figure 9

Float solutions of the traditional (blue) and improved (red) methods for baseline No.2, where the left to right panels denote the results of \({\nabla \Delta \Phi }_{\left(\mathrm{0,0},\mathrm{0,1},-1\right)}\), \({\nabla \Delta \Phi }_{\left(\mathrm{0,0},1,-\mathrm{1,0}\right)}\), \({\nabla \Delta \Phi }_{\left(1,-\mathrm{1,0},\mathrm{0,0}\right)}\), and \({\nabla \Delta \Phi }_{\left(\mathrm{0,1},-\mathrm{1,0},0\right)}\). Four panels in the same row denote one satellite pair

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Figure 10 shows the results of the fifth signal for baseline No.2 where the methods without NL assistance and with NL assistance are both tested. Once again, the float solutions of the proposed method are much better than the ones of the traditional method. Specifically, when there is no NL assistance, the fluctuations of the proposed method are much smaller than those of the traditional method. If the NL is used in FiCAR, the impacts of the multipath effects can be reduced. The main reason is that, as aforementioned, the multipath effects are significantly increased in EWL/WL. However, the shortcoming of the NL assistance is that it highly relies on the accuracy of the previously fixed ambiguities. Otherwise, there may exist sudden changes in the float solutions. Therefore, only the proposed method can simultaneously obtain a better AR performance without the potential problem of ambiguity jump.

Fig. 10
figure 10

Float solutions of the traditional (blue) and improved (red) methods for baseline No.2, where the left to right panels denote the results of the methods without NL assistance and with NL assistance. Two panels in the same row denote one satellite pair

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Tables 7 and 8 present the root mean square (RMS) values between the float and true ambiguities for all the satellite pairs. The numbers 1–5 denote the first to fifth signals to be fixed. It can be found that the RMS values of the first three signals are all smaller than 0.07 cycles. Hence, it indicates that the ambiguities can be fixed with high reliability by both the traditional and improved methods. Compared with the RMS of the traditional method, where the mean value is approximately 0.0439 cycles, and the one of the proposed methods is only 0.0158 cycles. The improvement is as much as 64.0%. For the fourth signal, the RMS values can also be reduced significantly, where the mean values of the traditional and improved methods are 0.1602 and 0.0292 cycles, respectively. The results are consistent with the analysis mentioned above. When fixing the fifth signal, it can be found that the traditional method cannot fix the ambiguities well. The traditional GF-FiCAR method is the worst among all these methods, and the improved GB-FiCAR method is the best. In addition, the method with NL assistance is generally better than the method without NL assistance, except for the traditional GB-FiCAR method. It is because that the traditional GB-FiCAR method can resist the impacts of unmodeled errors, including the multipath, to a great extent.

Table 7 RMS between the float and true ambiguities for all the satellite pairs by the traditional and improved methods, including the single-epoch GF-FiCAR and GB-FiCAR modes for baseline No.1 (cycle)
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Table 8 RMS between the float and true ambiguities for all the satellite pairs by the traditional and improved methods, including the single-epoch GF-FiCAR and GB-FiCAR modes for baseline No.2 (cycle)
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In the single-epoch process strategy, the indicator AR success rate \(P=\left({n}_{suc}/{n}_{tot}\right)\times 100\%\) is used with \({n}_{suc}\) and \({n}_{tot}\) denoting the epoch numbers of successful AR and the total epochs, respectively. The AR success rates by the traditional and improved methods are shown in Tables 9 and 10. From these tables, all the AR success rates of the first to third signals are 100%. For the fourth signal, the AR success rates of the proposed method can still reach 100%, whereas the traditional method cannot. The mean AR success rates of the traditional GF-FiCAR and GB-FiCAR methods are approximately 84.65% and 93.75%, respectively. Then for the fifth signal, the improved method can still slightly increase AR success rates, especially in the GF mode without NL assistance, where the maximum improvement can reach 50.7%. It again proves the effectiveness of the proposed method, which is especially useful for the last two signals.

Table 9 Single-epoch AR success rates by the traditional and improved methods including the GF and GB modes for baseline No.1
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Table 10 Single-epoch AR success rates by the traditional and improved methods including the GF and GB modes for baseline No.2
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If the ambiguities cannot be fixed instantaneously, the multi-epoch processing strategy may be used. The traditional and improved methods, including GF and GB modes for baselines No.1 and 2 are all tested. The time to first fix (TTFF) is used, where the critical thresholds of fraction and STD are set to \(\pm 0.2\) and 0.1 cycles, respectively (Zhang et al. 2020). Table 11 presents the mean TTFF of all the satellite pairs calculated from each liner combination. We can find that the TTFF of the first to fourth signals by using the improved method is 1.00 s, whereas the corresponding TTFF by using the traditional method ranges from 1.00 to 74.66 s. For the fifth signal, the mean TTFF values of the traditional and improved methods are approximately 193.0 and 144.3 s, respectively. In conclusion, the new multipath mitigation method is indeed effective.

Table 11 TTFF by the traditional and improved methods including GF and GB modes for baselines No.1 and 2 (s)
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Concluding remarks

We first propose a code and phase multipath mitigation method based on the observation-domain parameterization, which can be used in the FCAR and FiCAR. First, the basic theory and method of the MCAR, especially the FiCAR, are given. Second, the multipath effects in FiCAR are discussed. We can find that the presence of multipath may lead to difficulty in MCAR. Finally, the methodology of the multipath mitigation method is proposed. Specifically, different MCM and MPM from different frequencies are formed and preprocessed appropriately. Then the code and phase multipath are estimated based on the LS criterion.

To validate the effectiveness of the proposed method, the real five-frequency BDS-3 observations are applied. The single-epoch GF-FiCAR and GB-FiCAR, and multi-epoch GF-FiCAR and GB-FiCAR modes are all tested. According to the results, it can be concluded that the proposed method can indeed improve AR performance. First, the RMS values of the signals to be fixed by the proposed method are all smaller than the traditional method. Second, for the first four signals, only the AR success rates of the proposed method are 100%. The mean improvement of AR success rate is approximately 19.4% when fixing the fifth signal. Third, the TTFF for the first four signals of the proposed method is 1.00 s, which is significantly shorter than the one of the traditional method. For the fifth signal, approximately 49 s of TTFF can be shortened. In conclusion, the success rate and efficiency of AR can be significantly improved when using the proposed method.

In the future, the UD or DD multipath can also be parameterized in a similar way. Optimal multipath parameterization according to the user demands can be extended. Also, the proposed method can be used in other GNSS applications.