This section works on both code and phase observations, forming the so-called code-plus-phase PPP–RTK models. Considering different constraints on the atmospheric delays, we formulate the ionosphere-float, ionosphere-weighted, and ionosphere-fixed models. All models consider both CDMA and FDMA systems.

### Ionosphere-float models

In a larger-scale network, we parameterize the ionospheric delays without any constraints, forming the ionosphere-float models for both CDMA and FDMA systems.

#### CDMA model

Starting from CDMA observation equations in Eq. (1), we apply the S-basis theory to identify and eliminate the rank deficiencies in a step-by-step manner. To make this process straightforward, we introduce an identity in which a frequency-dependent quantity at the first two frequencies \(\left( . \right)_{,j = 1,2}\) can be decomposed as (Teunissen & Khodabandeh, 2015)

$$\left( . \right)_{,j = 1,2} = \left( . \right)_{{\rm ,IF}} + \mu_{j} \left( . \right)_{{\rm ,GF}}$$

(6)

where \(\left( . \right)_{{\rm ,IF}}\) and \(\left( . \right)_{{\rm ,GF}}\) are the ionosphere-free and geometry-free combinations, respectively, which are defined as.

$$\begin{aligned} \left( . \right)_{{\rm ,GF}} & = \frac{1}{{\mu_{2} - \mu_{1} }}\left[ {\left( . \right)_{,2} - \left( . \right)_{,1} } \right] \\ \left( . \right)_{{\rm ,IF}} & = \frac{{\mu_{2} }}{{\mu_{2} - \mu_{1} }}\left( . \right)_{,1} - \frac{{\mu_{1} }}{{\mu_{2} - \mu_{1} }}\left( . \right)_{,2} \\ \end{aligned}$$

(7)

Decomposing receiver and satellite code biases through Eq. (6), we rewrite Eq. (1) as

$$\begin{aligned} {\text{E}}\left[ {p_{r,j}^{s} } \right] &= m_{r}^{s} \tau_{r} + {\text{d}}t_{r} - {\text{d}}t^{s} \\ & \quad + \mu_{j} l_{r}^{s} + \left( {d_{{r,{\rm IF}}} + \mu_{j} d_{{r,{\rm GF}}} } \right) \\ &\quad - \left( {d_{{,{\rm IF}}}^{s} + \mu_{j} d_{{,{\rm GF}}}^{s} } \right) + \overline{d}_{r,j} - \overline{d}_{,j}^{s} \\ {\text{E}}\left[ {\phi_{r,j}^{s} } \right] &= m_{r}^{s} \tau_{r} + {\text{d}}t_{r} - {\text{d}}t^{s} \\ & \quad - \mu_{j} l_{r}^{s} + \delta_{r,j} - \delta_{,j}^{s} + \lambda_{j} z_{r,j}^{s} \\ \end{aligned}$$

(8)

with

$$\begin{gathered} \overline{d}_{r,j} = d_{r,j} - d_{{r,{\rm IF}}} - \mu_{j} d_{{r,{\rm GF}}} \, \left( {j > 2} \right) \hfill \\ \overline{d}_{,j}^{s} = d_{,j}^{s} - d_{{{\rm ,IF}}}^{{\rm s}} - \mu_{j} d_{{,{\rm GF}}}^{s} \, \left( {j > 2} \right) \hfill \\ \end{gathered}$$

(9)

being the recombined receiver and satellite biases. Of particulate note, although Eq. (9) holds for all frequencies and is equal to zero for \(j = 1,2\), we restrict them to \(j > 2\) for emphasizing that the recombined receiver and satellite biases exist only at the third frequency and above.

Four types of rank deficiencies then become clear: (1) between \({\text{d}}t_{r}\), \(d_{{r,{\rm IF}}}\), and \(\delta_{r,j}\) of size *n*; (2) between \({\text{d}}t_{r}\), \(d_{{,{\rm IF}}}^{s}\), and \(\delta_{,j}^{s}\) of size *m*; (3) between \(l_{r}^{s}\), \(d_{{r,{\rm GF}}}\), and \(\delta_{r,j}\) of size *n*; and (4) between \(l_{r}^{s}\), \(d_{{,{\rm GF}}}^{s}\), and \(\delta_{,j}^{s}\) of size *m*. We select \(d_{{r,{\rm IF}}}\), \(d_{{,{\rm IF}}}^{s}\), \(d_{{r,{\rm GF}}}\), and \(d_{{,{\rm GF}}}^{s}\) as the S-basis to eliminate these rank deficiencies, yielding

$$\begin{aligned} {\text{E}}\left[ {p_{r,j}^{s} } \right] &= m_{r}^{s} \tau_{r} + {\text{d}}\overline{t}_{r} - {\text{d}}\overline{t}^{s} \\ & \quad + \mu_{j} \overline{l}_{r}^{s} + \overline{d}_{r,j} - \overline{d}_{,j}^{s} \\ {\text{E}}\left[ {\phi_{r,j}^{s} } \right] &= m_{r}^{s} \tau_{r} + {\text{d}}\overline{t}_{r} - {\text{d}}\overline{t}^{s} \\ & \quad - \mu_{j} \overline{l}_{r}^{s} + \overline{\delta }_{r,j} - \overline{\delta }_{,j}^{s} + \lambda_{j} z_{r,j}^{s} \\ \end{aligned}$$

(10)

with

$$\begin{aligned} & {\text{d}}\overline{t}_{r} = {\text{d}}t_{r} + d_{{r,{\rm IF}}} \\ & {\text{d}}\overline{t}^{s} = {\text{d}}t^{s} + d_{{,{\rm IF}}}^{s} \\ & \overline{l}_{r}^{s} = l_{r}^{s} - d_{{,{\rm GF}}}^{s} + d_{{r,{\rm GF}}} \\ & \overline{\delta }_{r,j} = \delta_{r,j} - d_{{r,{\rm IF}}} + \mu_{j} d_{{r,{\rm GF}}} \\ & \overline{\delta }_{,j}^{s} = \delta_{,j}^{s} - d_{{,{\rm IF}}}^{s} + \mu_{j} d_{{,{\rm GF}}}^{s} \\ \end{aligned}$$

(11)

being the recombined parameters. Except for the recombined ionospheric delay \(\overline{l}_{r}^{s}\), other parameters are still inestimable.

We then consider two other types of rank deficiencies: (5) between \({\text{d}}\overline{t}_{r}\) and \({\text{d}}\overline{t}^{s}\) of size 1; and (6) between \(\overline{d}_{r,j}\) and \(\overline{d}_{,j}^{s}\) of size \(f - 2\). Selecting \({\text{d}}\overline{t}_{1}\) and \(\overline{d}_{1,j > 2}\) as the S-basis reforms the equations as

$$\begin{aligned} {\text{E}}\left[ {p_{r,j}^{s} } \right] &= m_{r}^{s} \tau_{r} + {\text{d}}\tilde{t}_{r} - {\text{d}}\tilde{t}^{s} \\ & \quad + \mu_{j} \overline{l}_{r}^{s} + \tilde{d}_{r,j} - \tilde{d}_{,j}^{s} \\ {\text{E}}\left[ {\phi_{r,j}^{s} } \right] &= m_{r}^{s} \tau_{r} + {\text{d}}\tilde{t}_{r} - {\text{d}}\tilde{t}^{s} \\ & \quad - \mu_{j} \overline{l}_{r}^{s} + \overline{\delta }_{r,j} - \overline{\delta }_{,j}^{s} + \lambda_{j} z_{r,j}^{s} \\ \end{aligned}$$

(12)

with

$$\begin{aligned} {\text{d}}\tilde{t}_{r} & = {\text{d}}\overline{t}_{r} - {\text{d}}\overline{t}_{1} \quad \left( {r > 1} \right) \\ {\text{d}}\tilde{t}^{s} & = {\text{d}}\overline{t}^{s} - {\text{d}}\overline{t}_{1} \\ \tilde{d}_{r,j} & = \overline{d}_{r,j} - \overline{d}_{1,j} \quad \left( {r > 1,j > 2} \right) \\ \tilde{d}_{,j}^{s} & = \overline{d}_{,j}^{s} - \overline{d}_{1,j} \quad \left( {j > 2} \right) \\ \end{aligned}$$

(13)

being the estimable receiver clock error, satellite clock error, receiver code bias, and satellite code bias, respectively.

We finally identify the remaining two types of rank deficiencies: (7) between \(\overline{\delta }_{,j}^{s}\) and \(z_{r,j}^{s}\) of size \(fm\); and (8) between \(\overline{\delta }_{r,j}\) and \(z_{r,j}^{s}\) of size \(fn\). Here, the choice of the S-basis must keep the integer nature of the ambiguities. To this end, we select \(z_{1,j}^{s} ,z_{r \ne 1,j}^{1} + \overline{\delta }_{1,j}\) as the S-basis, yielding the full-rank model

$$\begin{aligned} {\text{E}}\left[ {p_{r,j}^{s} } \right] & = m_{r}^{s} \tau_{r} + {\text{d}}\tilde{t}_{r} - {\text{d}}\tilde{t}^{s} \\ & \quad + \mu_{j} \overline{l}_{r}^{s} + \tilde{d}_{r,j} - \tilde{d}_{,j}^{s} \\ {\text{E}}\left[ {\phi_{r,j}^{s} } \right] & = m_{r}^{s} \tau_{r} + {\text{d}}\tilde{t}_{r} - {\text{d}}\tilde{t}^{s} - \mu_{j} \overline{l}_{r}^{s} \\ & \quad + \tilde{\delta }_{r,j} - \tilde{\delta }_{,j}^{s} + \lambda_{j} \tilde{z}_{r,j}^{s} \\ \end{aligned}$$

(14)

with

$$\begin{aligned} \tilde{\delta }_{r,j} & = \overline{\delta }_{r,j} - \overline{\delta }_{1,j} + \lambda_{j} (z_{r,j}^{1} - z_{1,j}^{1} )\quad (r > 1) \\ \tilde{\delta }_{,j}^{s} & = \overline{\delta }_{,j}^{s} - \overline{\delta }_{1,j} - \lambda_{j} z_{1,j}^{s} \\ \tilde{z}_{r,j}^{s} & = (z_{r,j}^{s} - z_{1,j}^{s} ) - (z_{r,j}^{1} - z_{1,j}^{1} )\quad (r > 1,s > 1) \\ \end{aligned}$$

(15)

being the estimable receiver phase bias, satellite phase bias, and double-differenced ambiguities, respectively.

Equation (14) represents the full-rank model on the network side. Providing users with the satellite clock corrections, satellite biases, and optionally, the atmospheric delays, one can form the user model parameterizing integer ambiguities. For brevity, we do not correct the atmospheric delays but parameterize them as unknowns, thereby forming the full-rank user model as

$$\begin{aligned} {\text{E}}\left[ {p_{u,j}^{s} + {\text{d}}\tilde{t}^{s} + \tilde{d}_{,j}^{s} } \right] &= {\varvec{e}}_{u}^{s} \Delta {\varvec{x}} + m_{u}^{s} \tau_{u} \\ & \quad + {\text{d}}\tilde{t}_{u} + \mu_{j} \overline{l}_{u}^{s} + \tilde{d}_{r,j} \\ {\text{E}}\left[ {\phi_{u,j}^{s} + {\text{d}}\tilde{t}^{s} + \tilde{\delta }_{,j}^{s} } \right] &= {\varvec{e}}_{u}^{s} \Delta {\varvec{x}} + m_{u}^{s} \tau_{u} \\ & \quad + {\text{d}}\tilde{t}_{u} - \mu_{j} \overline{l}_{u}^{s} + \tilde{\delta }_{u,j} + \lambda_{j} \tilde{z}_{u,j}^{s} \\ \end{aligned}$$

(16)

where \(\Delta {\varvec{x}}\) denotes the coordinate vector of a user receiver multiplied by the line-of-sight vector \({\varvec{e}}_{u}^{s}\). The definitions of other parameters are consistent with those on the network side, replacing the network receiver index ‘*r*’ with a user receiver index ‘*u*’. Equation (16) (the user model) is equivalent to Eq. (14) (the network model) if Eq. (16) moves the satellite clock errors and satellite biases in Eq. (14) from the right side to the left side and parameterizes the user coordinates.

#### FDMA model

The FDMA model adopts the same S-basis as the CDMA model, thereby forming the equations as

$$\begin{aligned} {\text{E}}\left[ {p_{r,j}^{s} } \right] &= m_{r}^{s} \tau_{r} + {\text{d}}\tilde{t}_{r} - {\text{d}}\tilde{t}^{s} \\ & \quad + \mu_{j} \overline{l}_{r}^{s} + \tilde{d}_{r,j} - \tilde{d}_{,j}^{s} \\ {\text{E}}\left[ {\phi_{r,j}^{s} } \right] &= m_{r}^{s} \tau_{r} + {\text{d}}\tilde{t}_{r} - {\text{d}}\tilde{t}^{s} - \mu_{j} \overline{l}_{r}^{s} \\ & \quad + \tilde{\delta }_{r,j} - \tilde{\delta }_{,j}^{s} + \left( {\lambda_{j}^{s} \overline{z}_{r,j}^{s} - \lambda_{j}^{1} \overline{z}_{r,j}^{1} } \right) \\ \end{aligned}$$

(17)

where \(\overline{z}_{r,j}^{s} = z_{r,j}^{s} - z_{1,j}^{s}\) is the between-receiver single-differenced ambiguity. Since the estimable receiver biases (\(\tilde{d}_{r > 1,j}\) and \(\tilde{\delta }_{r > 1,j}\)) are the bias differences between two receivers, we ignore the between-receiver single-differenced IFB, making estimable receiver biases free of satellite identifiers. This is reasonable in the network where the receivers and the connected antennas and firmware are homogenous (Wanninger, 2012).

Ambiguities in Eq. (17) are still inestimable. Teunissen (2019) proposed to parameterize the ambiguities as

$$\begin{aligned} {\text{E}}\left[ {p_{r,j}^{s} } \right] &= m_{r}^{s} \tau_{r} + {\text{d}}\tilde{t}_{r} - {\text{d}}\tilde{t}^{s} \\ & \quad + \mu_{j} \overline{l}_{r}^{s} + \tilde{d}_{r,j} - \tilde{d}_{,j}^{s} \\ {\text{E}}\left[ {\phi_{r,j}^{s} } \right] &= m_{r}^{s} \tau_{r} + {\text{d}}\tilde{t}_{r} - {\text{d}}\tilde{t}^{s} \\ & \quad - \mu_{j} \overline{l}_{r}^{s} + \tilde{\delta }_{r,j} - \tilde{\delta }_{,j}^{s} + \frac{{2848\lambda_{j}^{0} }}{{a_{1} a_{s} }}\tilde{\tilde{z}}_{r,j}^{s} \\ \end{aligned}$$

(18)

where \(\lambda_{j}^{0}\) denotes the wavelength of the GLONASS center frequency. \(a_{s} = 2848 + \kappa^{s}\) with \(\kappa^{s} \in \left[ { - 7, + 6} \right]\) being the channel number of GLONASS satellites; \(\tilde{\tilde{z}}_{r,j}^{s} { = }a_{1} (z_{r,j}^{s} - z_{1,j}^{s} ) - a_{s} (z_{r,j}^{1} - z_{1,j}^{1} )\) is the estimable GLONASS ambiguity.

Although ambiguities in Eq. (18) are estimable and integers, the integer nature of the original ambiguities cannot be guaranteed. For rigorous ambiguity resolution, we must transform the estimable ambiguities to integer-estimable ones, whose definition is in Teunissen (2019).

Along a similar line with Eq. (16), we form the FDMA user model as

$$\begin{aligned} {\text{E}}\left[ {p_{u,j}^{s} + {\text{d}}\tilde{t}^{s} + \tilde{d}_{,j}^{s} } \right] &= {\varvec{e}}_{u}^{s} \Delta {\varvec{x}} \\ & \quad + m_{u}^{s} \tau_{u} + {\text{d}}\tilde{t}_{u} + \mu_{j} \overline{l}_{u}^{s} + \tilde{d}_{r,j} \\ {\text{E}}\left[ {\phi_{u,j}^{s} + {\text{d}}\tilde{t}^{s} + \tilde{\delta }_{,j}^{s} } \right] &= {\varvec{e}}_{u}^{s} \Delta {\varvec{x}} + m_{u}^{s} \tau_{u} \\ & \quad + {\text{d}}\tilde{t}_{u} - \mu_{j} \overline{l}_{u}^{s} + \tilde{\delta }_{u,j} + \frac{{2848\lambda_{j}^{0} }}{{a_{1} a_{s} }}\tilde{\tilde{z}}_{r,j}^{s} \\ \end{aligned}$$

(19)

where the estimable ambiguities must be transformed to integer-estimable ambiguities.

### Ionosphere-weighted models

In medium-scale networks, we consider the spatial correlation between ionospheric delays by introducing a zero-mean constraint to between-receiver single-differenced ionospheric delays

$${\text{E}}\left[ {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{l}_{r}^{s} } \right] = l_{r}^{s} - l_{1}^{s}$$

(20)

where \(\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{l}_{r}^{s}\) denotes the pseudo ionospheric observations set as zero. Its uncertainty is modeled by a stochastic model, for example, the stochastic model in Zha et al. (2021).

With the introduction of weighted ionospheric constraints, some rank deficiencies change. Additionally, one new rank deficiency between the satellite clock error and tropospheric delay occurs since tropospheric mapping functions are almost identical in medium-scale networks. This section clarifies these rank deficiencies and forms the ionosphere-weighted models for both CDMA and FDMA systems.

#### CDMA model

Combining pseudo ionospheric observations with GNSS code and phase observations, the size of the rank deficiency between \(l_{r}^{s}\), \(d_{{r,{\rm GF}}}\), and \(\delta_{r,j}\) is no longer *n*, but only one. We select the geometry-free code bias of the first receiver \(d_{{1,{\rm GF}}}\) as the S-basis instead of the code biases of all receivers \(d_{{r,{\rm GF}}}\). Keeping other S-basis unchanged, we form the ionospheric-weighted model as

$$\begin{aligned} {\text{E}}\left[ {p_{r,j}^{s} } \right] & = m_{r}^{s} \tau_{r} + {\text{d}}\tilde{t}_{r} - {\text{d}}\tilde{t}^{s} + \mu_{j} \tilde{l}_{r}^{s} \\ & \quad + \mu_{j} \tilde{d}_{{r,{\rm GF}}} + \tilde{d}_{r,j} - \tilde{d}_{,j}^{s} \\ {\text{E}}\left[ {\phi_{r,j}^{s} } \right] & = m_{r}^{s} \tau_{r} + {\text{d}}\tilde{t}_{r} - {\text{d}}\tilde{t}^{s} - \mu_{j} \tilde{l}_{r}^{s} \\ & \quad + \tilde{\tilde{\delta }}_{r,j} - \tilde{\delta }_{,j}^{s} + \lambda_{j} \tilde{z}_{r,j}^{s} \\ {\text{E}}\left[ {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{l}_{r}^{s} } \right] & = \tilde{l}_{r}^{s} - \tilde{l}_{1}^{s} \\ \end{aligned}$$

(21)

with

$$\begin{aligned} \tilde{l}_{r}^{s} &= l_{r}^{s} - d_{{,{\rm GF}}}^{s} + d_{{1,{\rm GF}}} \\ \tilde{d}_{{r,{\rm GF}}} &= d_{{{\rm r,GF}}} - d_{{1,{\rm GF}}} \, \left( {r > 1} \right) \\ \tilde{\tilde{\delta }}_{r,j} &= \delta_{r,j} - d_{{r,{\rm IF}}} - \delta_{1,j} + d_{{1,{\rm IF}}} \\ & \quad + \lambda_{j} (z_{r,j}^{1} - z_{1,j}^{1} ) \, \left( {r > 1} \right) \\ \end{aligned}$$

(22)

being the estimable ionospheric delay, receiver code bias, and receiver phase bias, respectively, which differ from the parameters of the ionosphere-float model. Of particular note, although the estimable ionospheric delays \(\tilde{l}_{r}^{s}\) are not the original ionospheric delays \(l_{r}^{s}\), constraints on the estimable ionospheric delays in Eq. (21) are equivalent to the constraints on the original ionospheric delays in Eq. (20), as we can verify that \({\text{E}}\left[ {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{l}_{r}^{s} } \right] = \tilde{l}_{r}^{s} - \tilde{l}_{1}^{s} = l_{r}^{s} - l_{1}^{s}\).

Since the tropospheric mapping functions are almost identical in a medium-scale network, an additional rank deficiency between the tropospheric delay and satellite clock error occurs. We select the tropospheric delay of the first receiver as the S-basis, yielding the full-rank ionosphere-weighted model

$$\begin{aligned} {\text{E}}\left[ {p_{r,j}^{s} } \right] &= m_{r}^{s} \tilde{\tau }_{r} + {\text{d}}\tilde{t}_{r} - {\text{d}}\tilde{\tilde{t}}^{s} \\ & \quad + \mu_{j} \tilde{l}_{r}^{s} + \mu_{j} \tilde{d}_{{r,{\rm GF}}} + \tilde{d}_{r,j} - \tilde{d}_{,j}^{s} \\ {\text{E}}\left[ {\phi_{r,j}^{s} } \right] &= m_{r}^{s} \tilde{\tau }_{r} + {\text{d}}\tilde{t}_{r} - {\text{d}}\tilde{\tilde{t}}^{s} \\ & \quad - \mu_{j} \tilde{l}_{r}^{s} + \tilde{\tilde{\delta }}_{r,j} - \tilde{\delta }_{,j}^{s} + \lambda_{j} \tilde{z}_{r,j}^{s} \\ {\text{E}}\left[ {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{l}_{r}^{s} } \right] &= \tilde{l}_{r}^{s} - \tilde{l}_{1}^{s} \\ \end{aligned}$$

(23)

with

$$\begin{gathered} \tilde{\tau }_{r} = \tau_{r} - \tau_{1} \, \left( {r > 1} \right) \hfill \\ {\text{d}}\tilde{\tilde{t}}^{s} = {\text{d}}\overline{t}^{s} - {\text{d}}\overline{t}_{1} - m_{1}^{s} \tilde{\tau }_{1} \hfill \\ \end{gathered}$$

(24)

being the estimable ionospheric delay and satellite clock error, respectively.

On the user side, we consider, among others, the atmospheric products, forming the full-rank user model as

$$\begin{aligned} {\text{E}}\left[ {p_{u,j}^{s} + {\text{d}}\tilde{\tilde{t}}^{s} + \tilde{d}_{,j}^{s} - m_{u}^{s} \tilde{\tau }_{u} } \right] &= {\varvec{e}}_{u}^{s} \Delta {\varvec{x}} \\ & \quad + {\text{d}}\tilde{t}_{u} + \mu_{j} \tilde{l}_{u}^{s} + \mu_{j} \tilde{d}_{{u,{\rm GF}}} + \tilde{d}_{u,j} \\ {\text{E}}\left[ {\phi_{u,j}^{s} + {\text{d}}\tilde{\tilde{t}}^{s} + \tilde{\delta }_{,j}^{s} - m_{u}^{s} \tilde{\tau }_{u} } \right] &= {\varvec{e}}_{u}^{s} \Delta {\varvec{x}} \\ & \quad + {\text{d}}\tilde{t}_{u} - \mu_{j} \tilde{l}_{u}^{s} + \tilde{\tilde{\delta }}_{u,j} + \lambda_{j} \tilde{z}_{u,j}^{s} \\ {\text{E}}\left[ {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{l}_{u}^{s} } \right] &= \tilde{l}_{u}^{s} \\ \end{aligned}$$

(25)

where \(\tilde{\tau }_{u}\) and \(\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{l}_{u}^{s}\) are the tropospheric and ionospheric delays, respectively, interpolated at the user station.

#### FDMA model

Along the similar line, one can form the FDMA model by replacing the CDMA estimable ambiguities with the FDMA ones. In this way, we write the FDMA network model as

$$\begin{aligned} {\text{E}}\left[ {p_{r,j}^{s} } \right] & = m_{r}^{s} \tilde{\tau }_{r} + {\text{d}}\tilde{t}_{r} - {\text{d}}\tilde{\tilde{t}}^{s} + \mu_{j} \tilde{l}_{r}^{s} \\ & \quad + \mu_{j} \tilde{d}_{{r,{\rm GF}}} + \tilde{d}_{r,j} - \tilde{d}_{,j}^{s} \\ {\text{E}}\left[ {\phi_{r,j}^{s} } \right] & = m_{r}^{s} \tilde{\tau }_{r} + {\text{d}}\tilde{t}_{r} - {\text{d}}\tilde{\tilde{t}}^{s} - \mu_{j} \tilde{l}_{r}^{s} \\ & \quad + \tilde{\tilde{\delta }}_{r,j} - \tilde{\delta }_{,j}^{s} + \frac{{2848\lambda_{j}^{0} }}{{a_{1} a_{s} }}\tilde{\tilde{z}}_{r,j}^{s} \\ {\text{E}}\left[ {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{l}_{r}^{s} } \right] & = \tilde{l}_{r}^{s} - \tilde{l}_{1}^{s} \\ \end{aligned}$$

(26)

while the user mode reads

$$\begin{aligned} & {\text{E}}\left[ {p_{u,j}^{s} + {\text{d}}\tilde{\tilde{t}}^{s} + \tilde{d}_{,j}^{s} - m_{u}^{s} \tilde{\tau }_{u} } \right] = {\varvec{e}}_{u}^{s} \Delta {\varvec{x}} + {\text{d}}\tilde{t}_{u} \\ & \quad + \mu_{j} \tilde{l}_{u}^{s} + \mu_{j} \tilde{d}_{{u,{\rm GF}}} + \tilde{d}_{u,j} \\ & {\text{E}}\left[ {\phi_{u,j}^{s} + {\text{d}}\tilde{\tilde{t}}^{s} + \tilde{\delta }_{,j}^{s} - m_{u}^{s} \tilde{\tau }_{u} } \right] = {\varvec{e}}_{u}^{s} \Delta {\varvec{x}} + {\text{d}}\tilde{t}_{u} \\ & \quad - \mu_{j} \tilde{l}_{u}^{s} + \tilde{\tilde{\delta }}_{u,j} + \frac{{2848\lambda_{j}^{0} }}{{a_{1} a_{s} }}\tilde{\tilde{z}}_{u,j}^{s} \\ & {\text{E}}\left[ {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{l}_{u}^{s} } \right] = \tilde{l}_{u}^{s} \\ \end{aligned}$$

(27)

where the estimable ambiguities must be transformed to integer-estimable ambiguities for rigorous ambiguity resolution.

### Ionosphere-fixed models

In small-scale networks, we assume that atmospheric delays at all stations are identical. Based on this assumption, we investigate how the rank deficiencies change and form the full-rank ionosphere-fixed models for both CDMA and FDMA systems.

#### CDMA model

Assuming that \(m_{1}^{s} \tau_{1} = \cdots = m_{n}^{s} \tau_{n} = \tau^{s}\) and \(l_{1}^{s} = \ldots l_{n}^{s} = l^{s}\), the tropospheric delay \(\tau^{s}\) can be directly absorbed into satellite clock errors, while the rank deficiencies related to the ionospheric delays are the same as those in the ionosphere-weighted model. We follow the S-basis selected for the ionosphere-weighted model and write the ionosphere-fixed model as

$$\begin{aligned} {\text{E}}\left[ {p_{r,j}^{s} } \right] & = {\text{d}}\tilde{t}_{r} - {\text{d}}\overline{\overline{t}}^{s} + \mu_{j} \overline{\overline{l}}^{s} + \mu_{j} \tilde{d}_{{r,{\rm GF}}} + \tilde{d}_{r,j} - \tilde{d}_{,j}^{s} \\ {\text{E}}\left[ {\phi_{r,j}^{s} } \right] & = {\text{d}}\tilde{t}_{r} - {\text{d}}\overline{\overline{t}}^{s} - \mu_{j} \overline{\overline{l}}^{s} + \tilde{\tilde{\delta }}_{r,j} - \tilde{\delta }_{,j}^{s} + \lambda_{j} \tilde{z}_{r,j}^{s} \\ \end{aligned}$$

(28)

with

$$\begin{aligned} & {\text{d}}\overline{\overline{t}}^{s} = {\text{d}}t^{s} + d_{{,{\rm IF}}}^{s} - \tau^{s} - {\text{d}}t_{1} - d_{{1,{\rm IF}}} \\ & \overline{\overline{l}}^{s} = l^{s} - d_{{,{\rm GF}}}^{s} + d_{{{\rm 1,GF}}} \\ \end{aligned}$$

(29)

being the estimable satellite clock error and ionospheric delay, respectively.

On the user side, we directly use the ionospheric delay estimated on the network side to correct the ionospheric delay at a user station. Along with other corrections, we form the user model as

$$\begin{aligned} {\text{E}}\left[ {p_{r,j}^{s} + {\text{d}}\overline{\overline{t}}^{s} - \mu_{j} \overline{\overline{l}}^{s} + \tilde{d}_{,j}^{s} } \right] & = {\varvec{e}}_{u}^{s} \Delta {\varvec{x}} + {\text{d}}\tilde{t}_{r} + \mu_{j} \tilde{d}_{{r,{\rm GF}}} + \tilde{d}_{r,j} \\ {\text{E}}\left[ {\phi_{r,j}^{s} + {\text{d}}\overline{\overline{t}}^{s} + \mu_{j} \overline{\overline{l}}^{s} + \tilde{\delta }_{,j}^{s} } \right] & = {\varvec{e}}_{u}^{s} \Delta {\varvec{x}} + {\text{d}}\tilde{t}_{r} + \tilde{\tilde{\delta }}_{r,j} + \lambda_{j} \tilde{z}_{r,j}^{s} \\ \end{aligned}$$

(30)

where the tropospheric delays are actually corrected since they are absorbed in the satellite clock errors.

#### FDMA model

It is now straightforward to extend the CDMA models to the FDMA models: the network model

$$\begin{aligned} {\text{E}}\left[ {p_{r,j}^{s} } \right] &= {\text{d}}\tilde{t}_{r} - {\text{d}}\overline{\overline{t}}^{s} + \mu_{j} \overline{\overline{l}}^{s} \\ & \quad + \mu_{j} \tilde{d}_{{r,{\rm GF}}} + \tilde{d}_{r,j} - \tilde{d}_{,j}^{s} \\ {\text{E}}\left[ {\phi_{r,j}^{s} } \right] &= {\text{d}}\tilde{t}_{r} - {\text{d}}\overline{\overline{t}}^{s} - \mu_{j} \overline{\overline{l}}^{s} \\ & \quad + \tilde{\tilde{\delta }}_{r,j} - \tilde{\delta }_{,j}^{s} + \frac{{2848\lambda_{j}^{0} }}{{a_{1} a_{s} }}\tilde{\tilde{z}}_{r,j}^{s} \\ \end{aligned}$$

(31)

and the user model

$$\begin{aligned} & {\text{E}}\left[ {p_{r,j}^{s} + {\text{d}}\overline{\overline{t}}^{s} - \mu_{j} \overline{\overline{l}}^{s} + \tilde{d}_{,j}^{s} } \right] = {\varvec{e}}_{u}^{s} \Delta {\varvec{x}} \\ & \quad + {\text{d}}\tilde{t}_{r} + \mu_{j} \tilde{d}_{{r,{\rm GF}}} + \tilde{d}_{r,j} \\ & {\text{E}}\left[ {\phi_{r,j}^{s} + {\text{d}}\overline{\overline{t}}^{s} + \mu_{j} \overline{\overline{l}}^{s} + \tilde{\delta }_{,j}^{s} } \right] = {\varvec{e}}_{u}^{s} \Delta {\varvec{x}} \\ & \quad + {\text{d}}\tilde{t}_{r} + \tilde{\tilde{\delta }}_{r,j} + \frac{{2848\lambda_{j}^{0} }}{{a_{1} a_{s} }}\tilde{\tilde{z}}_{u,j}^{s} \\ \end{aligned}$$

(32)

where we emphasize again the necessity to transform the estimable ambiguities to integer-estimable ambiguities.