Fundamentals

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PPP (Precise Point Positioning) is a technology that uses external precise products (e.g., satellite orbit/clock), comprehensively considers and meticulously models various errors, processes single GNSS (Global Navigation Satellites System) receiver’s observation by undifferenced calculation. It was put forward to reduce the huge computing burden of GNSS network solutions due to massive data, it has ushered in rapid development and application. Compared to the relative positioning, the popularity of PPP is that no nearby reference stations are required, the user can achieve high accuracy positioning with only a single receiver. Besides, compared to the SPP (Standard Point Positioning) based on broadcast ephemeris and pseudo-range, PPP takes advantage of utilizing both pseudo-range observations and carrier phase observations, and more precise satellite-related parameters.

PPP integrates the advantage of GNSS SPP and GNSS relative positioning and overcomes their disadvantages to some extent. However, PPP does not eliminate or weaken the influence of various observation errors by difference, so all error terms must be finely considered and corrected. And the number of parameters to be solved is so large that external files need to be introduced. Moreover, the phase bias caused by hardware delay from satellite ends and receiver ends will be absorbed in the ambiguity, the corresponding ambiguity will not be an integer. Therefore, the difficulty of PPP is to separate the phase bias from ambiguity to achieve ambiguity resolution (AR).

Mathematical model of PPP

For GNSS dual-frequency observations from station r to satellite s, the raw observation equation for original pseud-orange and carrier-phase of the i-th frequency (i=1,2) in the unit of length is

$$ \begin{cases} P_{r,i}^s=\rho_r^s+c\left(\delta t_r-\delta t^s\right)+\frac{A}{f_i^2}+d_{r,i}-d_i^s\\ L_{r,i}^s=\rho_r^s+c\left(\delta t_r-\delta t^s\right)-\frac{A}{f_i^2}+\lambda_iN_{r,i}^s+b_{r,i}-b_i^s \tag{1} \end{cases} $$

where, $P_{r,i}^s$ are pseudo-range observations; $L_{r,i}^s$ are carrier-phase observations; $\rho_r^s$ is the station-satellite geometric distance; $c$ is the speed of light in vacuum; δtr and δts are the receiver and satellite clock errors, respectively; $\frac{A}{f_i^2}$ denotes the impact of the first-order ionosphere delays; $f_1$ and $f_2$ are the frequencies of $L_1$ and $L_2$; $\lambda_1$ and $\lambda_2$ are the corresponding wavelength; $N_1$ and $N_2$ are integer ambiguities; $d_{r,i}$ and $d_i^s$ denote the pseudo-range bias of the i-th frequency, which is caused by the hardware delay of the receiver and the satellite; $b_{r,i}$ and $b_i^s$ denote the phase bias of the i-th frequency, which is caused by the hardware delay of the receiver and the satellite; for simplicity, high-order ionospheric delay, tropospheric delay, multipath effect and random noise are omitted.

The geometric distance from the satellite to the receiver in equation $(1)$ can be expressed as

$$ \begin{cases} \rho_r^s=\left|\it{X}^s(t_s)-\it{X}_r(t_r)\right| \tag{2} \end{cases} $$

where, $t_s$ and $t_r$ are signal transmission time and signal reception time respectively; ${X}^s(t_s)$ and ${X}_r(t_r)$ is the satellite coordinate vector at the signal transmitting time and the receiver coordinate vector at the signal receiving time respectively; |·| represents the vector module length.

The first-order ionospheric delay in equation $(1)$ can be eliminated by the difference between the product of the dual-frequency observations and their frequency squares, while the difference between the two frequency squares is divided by the above equation to keep the geometric distance constant. The ionosphere-free observation equation is then formed

$$ \begin{cases} P_{r,0}^s&=\alpha P_{r,1}^s-\beta P_{r,2}^s\\ &=\rho_r^s+c\left(\delta t_r-\delta t^s\right)+d_{r,0}-d_0^s\\ P_{r,0}^s&=\alpha L_{r,1}^s-\beta L_{r,2}^s\\ &=\rho_r^s+c\left(\delta t_r-\delta t^s\right)+\alpha\lambda_1N_{r,1}^s-\beta\lambda_2N_{r,2}^s+b_{r,0}-b_0^s \tag{3} \end{cases} $$

where

$$ \begin{cases} \alpha=\frac{f_1^2}{f_1^2-f_2^2}\\ \beta=\frac{f_2^2}{f_1^2-f_2^2}\\ \alpha-\beta=1 \end{cases} $$

and

$$ \begin{cases} d_{r,0}&=\alpha d_{r,1}-\beta d_{r,2}\\ d_0^s&=\alpha d_1^s-\beta d_2^s\\ b_{r,0}&=\alpha b_{r,1}-\beta b_{r,2}\\ b_0^s&=\alpha b_1^s-\beta b_2^s \end{cases} $$

$P_{r,0}^s$ and $P_{r,0}^s$ are the ionosphere-free pseudo-range observation and carrier-phase observation, respectively. Correspondingly, $d_{r,0}$ and $d_0^s$ are the ionosphere-free pseudo-range bias at the receiver end and the satellite end, $b_{r,0}$ and $b_0^s$ are the ionosphere-free combination phase bias at the receiver end and the satellite end respectively.

Error correction of PPP

As mentioned above, PPP uses undifferenced data processing and does not eliminate or weaken the impact of various observation errors through difference. Hence all error terms must be considered finely and corrected as much as possible. Usually, there are two types of error correction: (1) model correction is used for errors that can be finely modeled, such as the correction of satellite antenna PCO/PCV (Phase Center Offset/Variation); (2) for the errors that cannot be accurately modeled, they can be estimated as parameters or eliminated by using combined observations. For example the tropospheric delay after model correction still needs to be estimated by adding parameters, and the low-order term of ionospheric delay error can be eliminated by using dual-frequency combined observations.

In PPP, the main error sources can be divided into three categories: (1) errors related to satellites, (2) errors related to signal propagation paths, and (3) errors related to receivers and stations.

Errors related to satellites

(1) Satellite ephemeris error and clock error

Satellite ephemeris error refers to the discrepancy between the orbit represented by the satellite ephemeris and the real orbit. For the satellite coordinate vector ${X}^s(t_s)$ in equation $(2)$, the nominal accuracy of the post precise ephemeris product of IGS (International GNSS Service) is better than 2.5cm. The user can use Lagrange interpolation to calculate the satellite coordinates at the time of signal transmission. The calculation formula of signal transmission time is:

$$ \begin{cases} t_s=t^r+\delta t_r-\tau \tag{4} \end{cases} $$

where $\tau$ is the signal propagation time. It can be calculated by the geometric distance between the satellite and the observation station after correcting various errors, and the geometric distance is related to the satellite coordinates, so iterative calculation is required in this process.

Satellite clock error can be eliminated or weakened by utilizing precise satellite clock error products, that is, it can be substituted into the observation equation as a known value. At present, the precision of IGS legacy clock error products has reached 75ps, which can fully meet the needs of PPP.

(2) Earth rotation correction

Because the earth-fixed coordinate system is rotating with the rotation of the earth, the earth-fixed coordinate system corresponding to the satellite signal transmitting time and the receiver signal receiving time is different. Therefore, it is necessary to consider this correction to calculate the geometric distance from the satellite to the receiver in the earth-fixed coordinate. Set $\omega$ as the earth rotation angular velocity, and the resulting satellite coordinate change is

$$ \begin{cases} \it{X}^{s\prime}=\it{R}\cdot\it{X}^s \tag{5} \end{cases} $$

where, $\it{R}$ is the rotation matrix

$$ \it{R}=\begin{bmatrix} cos\omega\tau & sin\omega\tau & 0 \\ -sin\omega\tau & -sin\omega\tau & 0\\ 0 & 0 & 1 \end{bmatrix} $$

The correction of corresponding geometric distance is

$$ \begin{cases} \mathrm{\Delta\rho}=\frac{\omega}{c}\left[\it{Y}^s(\it{X}_r-\it{X}^s)-\it{X}^s(\it{Y}_r-\it{Y}^s)\right] \tag{6} \end{cases} $$

(3) Relativistic effects

The relativistic effect is caused by the different states (motion speed and gravity potential) of the satellite clock and the receiver clock. The change of clock frequency caused by different velocities is called the special relativity effect, and the change of clock frequency caused by different gravity potentials is called the general relativity effect. Under the combined influence of the special relativity effect and the general relativity effect, the relative clock error occurs between the satellite clock and the receiver clock, and the satellite clock moves faster than the receiver clock. Its constant part can reduce its standard frequency when producing satellite clock. However, the frequency difference between the satellite clock and the receiver clock is related to the operating speed of the satellite and its distance from the earth center, so there are still residuals after the above correction, which can be corrected by the following formula:

$$ \begin{cases} \mathrm{\Delta}\rho_{rel}=-\frac{2}{c}\it{X}^s·\dot{\it{X}^s} \tag{7} \end{cases} $$

where $\it{X}^s$ is the coordinate vector of the satellite and $\dot{\it{X}^s}$ is the velocity vector of the satellite. In addition to the frequency drift of satellite clock, the effect of general relativity also includes the delay of the geometric distance caused by the earth's gravitational field, which is called gravitational delay. The corresponding correction can refer to relevant paper.

(4) PCO/PCV for satellite

PCO of satellite antenna refers to the deviation between satellite center of mass and satellite antenna phase center. The satellite orbit products used in PPP are based on the satellite center of mass and the signal observations are ranging from the phase center of the satellite antenna. For a satellite, PCO can be regarded as a fixed deviation vector.

Because the phase center changes with time during the actual transmission and reception of signals, there is a deviation compared with the average phase center, which is called PCV. It is necessary to correct the change of phase center in high-precision applications.

(5) Phase wind-up

GNSS satellite signal adopts polarization wave. When the satellite antenna or receiver antenna rotates around its longitudinal axis, the carrier phase observation value will change, and its value can be up to one cycle. When the relative rotation occurs between the transmitting antenna and the receiver antenna, the carrier phase observation value will include error. In positioning, after the antenna pointing of the receiver changes, its error will be automatically absorbed into the receiver clock error, so there is no need to consider it. Since the solar panel on the satellite needs to be always aligned with the sun, the satellite antenna will rotate slowly. After entering the eclipse period, the satellite will accelerate the rotation, resulting in the error of carrier phase observation. The influence of phase wind-up on PPP is very obvious, and this error must be taken into account.

Errors related to signal propagation path

(1) Ionospheric delay

The ionosphere is a dispersive medium, mainly located in the atmospheric area about 70km to 1000km above the earth's surface. In this region, some neutral gas molecules are ionized, producing a large number of electrons and positive ions, thus forming an ionized region. In dispersive media, the propagation velocity of wave is a function of wave frequency. The phase velocity of electromagnetic wave propagation in the ionosphere (the phase velocity of electromagnetic wave with single frequency) will exceed the group velocity (the propagation velocity of a group of electromagnetic wave signals with different frequencies as a whole). Therefore, in the GNSS signal, the pseudo-range code is delayed and the carrier phase is advanced.

As mentioned above, to eliminate and weaken the influence of ionospheric delay, ionospheric correction models and ionospheric grid models can be adopted. In addition, dual-frequency correction can be adopted to eliminate ionospheric delay error through linear combination of observations. After using the dual-frequency observations to eliminate the first-order ionospheric influence, the influence of the remaining high-order terms is very small and can be ignored.

(2) Tropospheric delay

The troposphere is the lower part of the atmosphere and is non-dispersive at frequencies above 15 GHz. Tropospheric delay can be divided into dry component and wet component. The common method of tropospheric delay correction in PPP is to correct the tropospheric delay by using the model as a priori value, estimate the residual tropospheric delay as piecewise constant or random walk noise, and map it to the direction of satellite signal propagation path through mapping function. Tropospheric delay can be expressed as:

$$ \begin{cases} \mathrm{\Delta}\rho_{trop}=ZTD_{dry}·M_{dry}+ZTD_{wet}·M_{wet} \tag{8} \end{cases} $$

where, $ZTD_{dry/wet}$ is the zenith tropospheric dry/wet component delay and $M_{dry/wet}$ is the dry/wet component mapping function.

(3) Multi-path effect

The multi-path effect means that if the satellite signal (reflected wave) reflected by the reflector near the measured station enters the receiver antenna, it will interfere with the signal (direct wave) directly from the satellite, to make the observed value deviate from the true value. Multi-path errors vary greatly, depending on the receiver environment, satellite elevation angle, receiver signal processing method, antenna gain type, and signal characteristics.

At present, there is no more effective solution to the multipath effect. The main measures to weaken the multipath error are: selecting an appropriate station site, equipping the receiver with a diameter suppression plate or circle, appropriately prolonging the observation time, estimating additional parameters, etc. Because the satellite signal with low elevation is more likely to produce multi-path effect, the cut-off elevation can also be set during data preprocessing, and the impact of multi-path effect on precise point positioning can be weakened through long-time observation and smoothing.

At present, the main algorithms for compensating multipath delay are sidereal filtering (SF) and multipath hemispherical map (MHM). The former utilizes the repeatability of satellite orbits to filter observations in the time domain, which requires a longer observation time and requires the establishment of separate models for each satellite, resulting in more complex calculations. The MHM also utilizes the repeatability of satellite orbits, but considers that the multipath delay of satellite signals from the same direction and frequency should be the same. A model is established with altitude and azimuth as independent variables, and satellites of the same frequency can use the same MHM model, which is simpler to calculate and more suitable for application.

Errors related to receivers and stations

(1) Receiver clock error

Because the receiver generally adopts quartz clock, its stability is worse than satellite clock, so the polynomial fitting method is generally not applicable. Instead, the receiver clock of each observation epoch is treated as an unknown parameter. In the process of processing, the receiver clock error is usually regarded as a group of white noise. It should be noted that unlike the calculation of satellite position, the receiver clock error in equation $(1)$ of the original observation method needs to be closely estimated, because in the calculation of satellite position, the measurement error is multiplied by the satellite operating speed of 3.9km/s, and the influence of measurement error on geometric distance needs to be multiplied by the vacuum speed of light.

(2) Tidal correction

Under the gravitational action of the moon and the sun, the elastic earth surface will produce periodic changes, which is called solid tide. It lengthens the earth in the connecting direction between the earth's center and the celestial body, and tends to be flat in the vertical direction. The influence of earth tide on stations includes long-term migration related to latitude and short-term term mainly composed of daily period and sub-daily period. For the daily solution of PPP, although the periodic error can be basically eliminated, the residual effect can reach 5cm in the horizontal direction and 12cm in the vertical direction.

Ocean loading results from the load of the ocean tides on the underlying crust. The displacement due to the ocean loading is one order of magnitude smaller than the earth tide. In the daily solution of PPP, the impact is mm, when the station is more than 1000km away from the coastline, the impact is negligible. The influence on a single epoch can reach 5cm.

(3) PCO/PCV for receiver

When GNSS receiver is used for measurement, the measured position of antenna phase center, and the antenna height is generally measured to the position of ARP (Antenna Reference Point). These two points generally do not coincide. This deviation is called receiver antenna PCO, and the PCO is also inconsistent for signals of different frequencies. It must be considered in PPP data processing.

The phase center of the receiver antenna is not fixed, and its instantaneous phase center changes with the elevation angle, azimuth angle and signal strength of the received signal. Similarly, the difference between the instantaneous phase center and the average phase center of the receiver antenna is called the antenna phase center change, correction is also required, just like the satellite-side PCV.

Undifferenced ambiguity resolution

The hardware delay term in equation $(1)$ includes two parts which are the time-invariant part and the time-varying part, i.e.,

$$ \begin{cases} d_{r,}&=\mathrm{\Delta}d_{r,\ast}+\delta d_{r,\ast}\\ d_\ast^s&=\mathrm{\Delta}d_\ast^s-\delta d_\ast^s\\ b_{r,\ast}&=\mathrm{\Delta}b_{r,\ast}-\delta b_{r,\ast}\\ b_\ast^s&=\mathrm{\Delta}b_\ast^s-\delta b_\ast^s \tag{9} \end{cases} $$

where, $\ast$ is a wildcard character representing observations of different frequencies and their combination.

Another commonly used combined observation in PPP is Melbourne-Wübbena combination.

$$ \begin{cases} L_{r,m}^s&=\lambda_w\left(\frac{L_{r,1}^s}{\lambda_1}-\frac{L_{r,2}^s}{\lambda_2}\right)-\lambda_n\left(\frac{P_{r,1}^s}{\lambda_1}-\frac{P_{r,2}^s}{\lambda_2}\right)\\ &=\lambda_w\left(N_{r,w}^s+\frac{b_{r,1}-b_1^s}{\lambda_1}-\frac{b_{r,2}-b_2^s}{\lambda_2}\right)-\lambda_n\left(\frac{d_{r,1}-d_1^s}{\lambda_1}+\frac{d_{r,2}-d_2^s}{\lambda_2}\right) \tag{10} \end{cases} $$

$\lambda_w=\frac{c}{f_1-f_2}$ and $\lambda_n=\frac{c}{f_1+f_2}$ are the wide-lane wavelength and narrow-lane wavelength respectively; $N_{r,w}^s=N_{r,1}^s-N_{r,2}^s$ is the ambiguity of wide-lane. M-W combination eliminates ionospheric delay, geometric distance from satellite to receiver, satellite clock and receiver clock. It is only affected by multipath effect, measurement noise and hardware delay. Because the wide-lane wavelength $\lambda_w$ is up to 86cm, it is easy to determine its integer ambiguity, that is, the wide lane ambiguity is solved through M-W combination $L_{r,m}^s$. The corresponding receiver phase deviation and satellite phase deviation are

$$ \begin{cases} b_{r,w}&=\lambda_w\left(\frac{b_{r,1}}{\lambda_1}-\frac{b_{r,2}}{\lambda_2}\right)-\lambda_n\left(\frac{d_{r,1}}{\lambda_1}+\frac{d_{r,2}}{\lambda_2}\right)\\ b_w^s&=\lambda_w\left(\frac{b_1^s}{\lambda_1}-\frac{b_2^s}{\lambda_2}\right)-\lambda_n\left(\frac{d_1^s}{\lambda_1}+\frac{d_2^s}{\lambda_2}\right) \tag{11} \end{cases} $$

After the ambiguity of wide-lane is resolved through M-W combination, we can substitute $N_{r,2}^s=N_{r,1}^s-\check{N}_{r,w}^s$ into ionosphere-free combination equation $(3)$ which can then be transformed into

$$ \begin{cases} P_{r,0}^s&=\alpha P_{r,1}^s-\beta P_{r,2}^s\\ &=\rho_r^s+c\left(\delta t_r-\delta t^s\right)+d_{r,0}-d_0^s\\ \bar L_{r,0}^s&=L_{r,0}^s-\beta\lambda_2 \check N_{r,w}^s\\ &=\rho_r^s+c\left(\delta t_r-\delta t^s\right)+\lambda_nN_{r,1}^s+b_{r,0}-b_0^s \tag{12} \end{cases} $$

where, $\check N_{r,w}^s$ denotes the resolved wide-lane ambiguity; and the $N_{r,1}^s$ in this formula is also called narrow-lane ambiguity; $\bar L_{r,0}^s$ is the ionosphere-free combined carrier-phase observation after correcting the wide-lane ambiguity.

In the process of GNSS data, the ambiguity in the continuous arc is generally constrained as a constant, and the clock error is generally estimated as white noise. In this way, the constant part of the hardware delay is absorbed by the ambiguity parameter, and the time-varying part is absorbed by the clock parameter. Therefore, whether the hardware delay is constant or varies with time, the effect on the ambiguity is to introduce a constant deviation. The key to fixing the un- differenced ambiguity is to separate the constant bias from the integer ambiguity.

There are several methods to fix the undifferenced ambiguity: integer clock model, decoupled clock model, UPD (uncalibrated phase delay) model and phase clock/bias model.

(1) Integer clock model and decoupled clock model

The basic idea of the integer clock model is to assume that the wide-lane phase bias remains stable in a single day, estimate the wide-lane ambiguity through M-W combination, extract its fractional part from the wide-lane ambiguity estimation as the wide-lane phase bias, and the integer part is the wide-lane integer ambiguity. Then the fixed wide-lane ambiguity is brought into the ionosphere-free combination to solve the narrow-lane ambiguity. By rounding the resolved narrow-lane ambiguity, the corresponding narrow-lane phase bias is absorbed into the clock parameters. The decoupled clock model is similar to the integer clock model, except that the wide lane phase bias is estimated epoch by epoch. The positioning accuracy of integer clock model is high, but the satellite clock product is incompatible with IGS legacy clock product and DCB (Differential Code bias) product. The decoupled clock model needs to estimate two sets of clock products, which is rarely used by analysis centers and scientific research institutions.

(2) UPD model

In UPD model, the processing of wide-lane phase bias is the same as that of integer clock model, and the calculation process of narrow-lane phase deviation is consistent with that of wide-lane phase bias. The UPD model directly uses the IGS legacy clock product, but its narrow-lane phase bias is not stable, it needs to be estimated every ten minutes empirically, and the positioning accuracy is lower than the integer clock model.

It should be noted that the IGS legacy clock product is defined as dual-frequency ionosphere-free combined clock. Therefore, in its legacy products, in addition to the real satellite clock, it also includes the hardware bias part of ionosphere-free combination. Considering the weight difference between pseudo-range observation and carrier-phase observation, the clock includes the time-invariant part of pseudo-range bias and the time-varying part of phase bias. Its theoretical form is

$$ \begin{cases} \delta t_{r,F}&=\delta t_r+\frac{\mathrm{\Delta}d_{r,0}+\delta b_{r,0}}{c}\\ \delta t_F^s&=\delta t^s+\frac{\mathrm{\Delta}d_0^s+\delta b_0^s}{c} \tag{13} \end{cases} $$

(3) Phase clock/bias model

The instability of narrow-lane phase bias in UPD model is considered to be due to the influence of satellite orbit/clock error and residual atmospheric error. Based on integer clock model and UPD model, phase bias/clock calculates the mean value of narrow lane ambiguity in UPD model in a single day and fixes it in subsequent data processing, then re-estimates the clock parameter, and absorbs the residual narrow-lane phase bias relative to the mean value of narrow-lane ambiguity into the clock error parameter. Therefore, the required integer ambiguity and its bias of narrow-lane are the integer part and fractional part of the mean value of narrow-lane ambiguity respectively. The re-estimated clock is the phase clock in the model. In the UPD model, the narrow-lane phase bias between the receiver and the satellite is

$$ \begin{cases} b_{r,n}&=\mathrm{\Delta}b_{i,0}-\mathrm{\Delta}d_{i,0}\\ b_n^s&=\mathrm{\Delta}b_0^s-\mathrm{\Delta}d_0^s \tag{14} \end{cases} $$

After calculating the daily mean value of narrow-lane ambiguity and its phase bias based on UPD model, taking into account equation $(13)$ , the clock is re-estimated in the ionosphere-free combination (equation (12)) , i.e.

$$ \begin{cases} P_{r,0}^s=\rho_r^s+c\left(\delta t_{r,F}-\delta t_F^s\right)+(\delta d_{r,0}-\delta d_0^k-\delta b_{i,0}+\delta b_0^k)\\ \bar L_{r,0}^s-\lambda_n\check N_{r,1}^s+\check b_n^s=\rho_r^s+c(\delta t_{r,F}-\delta t_F^s)+b_{r,n} \tag{15} \end{cases} $$

where, $\delta t_{r,F}$ and ${\delta t}_F^s$ are receiver clock error and satellite clock error to be estimated, respectively; $(\delta d_{r,0}-\delta d_0^k-\delta b_{i,0}+\delta b_0^k)$ is the residual term, which will be absorbed into the pseudo-range residual and it can be ignored; the narrow lane phase bias at the receiver end $b_{r,n}$ will be absorbed by the receiver clock error $\delta t_{r,F}$.

Accordingly, the user's mathematical model for PPP-AR using phase bias/clock model is as follows:

$$ \begin{cases} L_{r,m}^s+\hat b_w^s=\lambda_wN_{r,w}^s+b_{r,w}\\ P_{r,0}^s+c\hat t_F^s\approx\rho_r^s+ct_{r,F}\\ L_{r,0}^s+c\hat t_F^s-\beta\lambda_2\check N_{r,w}^s+\hat b_n^s=\rho_r^s+ct_{r,F}+\lambda nN_{r,1}^s+b_{r,n} \tag{16} \end{cases} $$

where, $\hat b_w^s$ and ${\hat b}_n^s$ are the phase bias products of wide-lane and narrow-lane at the satellite end; $\hat t_F^s$ is the satellite clock product; in equation $(15)$, the $(\delta d_{r,0}-\delta d_0^k-\delta b_{i,0}+\delta b_0^k)$ in the ionosphere-free combined pseudo-range observations is ignored here. The narrow-lane phase bias $b_{r,n}$ at the receiver end will be absorbed by the receiver clock $\delta t_{r,F}$.

In the data processing, first fix the wide-lane ambiguity by M-W combination, and then wide-lane integer ambiguity, satellite clock and narrow-lane phase bias are brought into the ionosphere-free combination to fix the narrow-lane ambiguity.