CN104502935B - A kind of network RTK Ambiguity Solution Methods based on the non-combined model of non-difference - Google Patents

Abstract

The invention discloses a network RTK ambiguity resolution method based on a non-difference and non-combination model. The undifferenced uncombined model is used to estimate the undifferenced ambiguity of the whole network, and the satellite clock is used as an unknown parameter, and the real-time satellite clock difference product is added as a pseudo-observation. According to the user's location, extract the non-difference ambiguity of the user's adjacent sites, select one of the adjacent sites as the reference site, and use the difference operator to restore the integer characteristics of the double-difference ambiguity, and obtain the double-difference with integer characteristics in a single epoch Blur results. Using the method proposed by the present invention can effectively improve the calculation performance in a large-scale reference station network, and at the same time weaken the influence of atmospheric errors and lengths on the fixed double-difference ambiguity, reduce the ill-conditionedness of the double-difference model, and improve the model's ability to resist errors and Ambiguity fixed success rate.

Description

A network RTK ambiguity resolution method based on non-difference and non-combination model

technical field

The invention relates to the field of global navigation satellite system (GNSS) satellite positioning, in particular to GNSS non-difference and non-combined precise single point positioning.

Background technique

With the gradual increase in the number of reference stations and the compatibility of multi-system and multi-frequency signals, a large-scale CORS (Continuous Operational Reference System: Continuous Operational Reference System) system has been widely constructed and has gradually become an important national infrastructure to support high-precision positioning applications. Users can use the multi-reference station network to obtain centimeter-level positioning accuracy in real time.

In the current network RTK application, it is necessary to ensure the correct calculation of the double difference ambiguity, so that the difference correction information of the whole network and the VRS observation value can be generated, so as to ensure the positioning effect. Therefore, most CORS system software adopts the baseline solution method to achieve network-wide ambiguity resolution. Although the baseline solution method reduces the number of unknown parameters and solves the rank deficiency problem to a certain extent, it usually has complex and difficult to solve correlation problems. At the same time, due to the influence of atmospheric error and satellite orbit error, the baseline length is limited during the baseline calculation process, and a new atmospheric error estimation model has to be considered to expand the baseline length. Moreover, as the number of reference stations increases, the number of baselines to be solved increases exponentially. Therefore, the network-wide ambiguity resolution based on the baseline solution method is not conducive to the construction and operation of the distributed architecture of the large-scale CORS system, and limits the application of the large-scale CORS system.

Meanwhile, with the successful application of non-differenced PPP, an innovative PPP-RTK method has been extensively studied. In the uncombined observation equation, the receiver and satellite carrier pseudo-range bias items are absorbed by the ambiguity, which makes the undifferenced ambiguity lose the integer property. But using fixed double-difference ambiguity and correct reference ambiguity, the integer characteristics of non-difference ambiguity can be restored, and it has numerical equivalence with double-difference equation, but it does not need to face the correlation problem. Therefore, the ambiguity fixation of the whole network is worthy of in-depth study, and will greatly benefit the development of large-scale CORS systems.

Contents of the invention

Purpose of the invention: Aiming at the above existing technologies, a network RTK ambiguity resolution method based on the non-difference and non-combination model is proposed, which can solve the problems that the current network RTK is affected by the correlation problem and the baseline length and quantity are limited, and will greatly benefit Conducive to the development of a wide range of CORS systems.

Technical solution: A network RTK ambiguity resolution method based on the non-difference and non-combination model. Firstly, the non-difference and non-combination model is used to estimate the single-station non-difference ambiguity of the whole network. The satellite clock is used as an unknown parameter, and the real-time satellite The clock difference product is added as a pseudo-observation; then according to the user's location, the non-difference ambiguity of the user's adjacent sites is extracted, one of the adjacent sites is selected as the reference site, and the integer characteristics of the double-difference ambiguity are obtained by using the difference operator to restore, Single epochs obtain double-differenced ambiguity results with integer properties.

Further, the described network RTK ambiguity resolution method based on the non-difference non-combination model includes the following specific steps:

Step 1), use the original observations of each station to estimate the single-station indifferent ambiguity of each station, and estimate the satellite clock error as an unknown parameter, as shown in formula (1.1):

Among them: s represents the satellite, k represents the receiver of the base station, j represents the frequency of each observation value, and j=1, 2; Indicates the pseudo-range observations at frequency j between satellite s and station k, Indicates the phase observation at frequency j between satellite s and station k, Indicates the geometric distance from the satellite to the receiver antenna phase center, δt _k indicates the receiver clock error, δt ^s indicates the satellite clock error, represents the tropospheric delay, Indicates the ionospheric delay; α _j is the frequency ratio, Indicates the value of frequency j; Denote the hardware delay of the receiver and the satellite on frequency j, respectively; are other modelable errors; Respectively represent the phase fractional deviation of the receiver and satellite on frequency j; λ _j is the carrier wavelength on frequency j, and N _j is the phase ambiguity on frequency j; is the pseudorange observation value and phase observation value noise at frequency j; c represents the speed of light;

According to the formula (1.1), use the Kalman filter to estimate the unknown parameters to obtain the non-difference floating-point ambiguity, including the following specific steps:

Step a), setting the unknown parameter vector X _i , as shown in formula (1.2):

Among them, the unknown parameter X _i is divided into two parts: time-varying and time-invariant, and the time-varying part includes the zenith wet delay ZTD _w,k ; the absorbing receiver has no ionospheric hardware delay The receiver clock error δt _k ′, in n-Dimensional Ionospheric Slope Delay of Absorbing Receiver and Satellite Frequency Dependent Hardware Delay Part in n is the number of satellites observed in this epoch, and the absorbing satellite has no ionospheric hardware delay The n-dimensional satellite clock error δt ^s ′, The time-invariant part includes n-dimensional non-differenced initial N1 integer ambiguities and n-dimensional non-differenced initial N2 integer ambiguities

Step b), set the Kalman filter design matrix B _i and observation vector L _i , as shown in formula (1.3):

Wherein, the observation vector L _i contains four non-difference observation values and a real-time satellite clock product result As pseudo observations, represent the phase observations between satellite s and station k at frequencies 1 and 2 respectively, represent the pseudo-range observations between satellite s and station k at frequencies 1 and 2, respectively;

n represents the number of observed satellites in this epoch, θ ⁿ represents the satellite altitude angle of the nth satellite in this epoch, MF _w (θ) is the mapping function related to the zenith wet delay and satellite altitude angle, c represents the speed of light, λ ₁ , λ ₂ represents the wavelength of frequency 1, 2 respectively;

According to the unknown parameter vector X _i and the design matrix B _i and observation vector L _i , Kalman filtering is performed to obtain the non-difference floating-point ambiguity

Step 2), according to the distance between the user's location and the site, extract the non-difference ambiguity of at least three user adjacent sites, and the non-difference ambiguity of each user's adjacent site As shown in formula (1.5):

in, are the non-difference basic ambiguities N1 of the 1st to nth satellites corresponding to adjacent stations; are the non-difference basic ambiguities N2 of the 1st to nth satellites corresponding to adjacent stations;

Step 3), first select a site in each user's adjacent sites as a reference site, and other adjacent sites as non-reference sites; then set the inter-station difference operator C _kd and reference The inter-satellite difference operator C ^sd of the satellite and the non-reference satellite, according to the difference operator C _kd and C ^sd restores the double-difference ambiguity with integer characteristics between the stations, and the restoration steps are shown in formula (1.6) :

Among them, k1 and k2 are the reference station and non-reference station respectively, r is the reference satellite, s is the non-reference satellite, is the inter-satellite double-difference ambiguity, are the undifferenced ambiguities of sites k1 and k2, respectively, Respectively, double-difference basic ambiguity N1, N2;

Step 4), use the LAMBDA algorithm to search and fix the ambiguities, specifically: first convert the double-difference basic ambiguities N1, N2 into wide-lane ambiguities and N1 ambiguity part Then use the LAMBDA algorithm to obtain a fixed integer wide-lane ambiguity, and finally use the wide-lane ambiguity as a known value to determine the integer value of N1 ambiguity.

Beneficial effects: a network RTK ambiguity resolution method based on the non-difference and non-combination model proposed by the present invention incorporates real-time satellite clock difference products as pseudo observations, which improves the real-time PPP solution effect and improves the robustness of the model degree. The difference operator is used to restore the non-difference ambiguity to obtain the double-difference integer ambiguity. In the process, only the time-invariant ambiguity parameters are introduced for the whole network adjustment, which reduces the influence of the baseline length and atmospheric delay on the results, and improves the calculation efficiency. Efficiency is conducive to the development of large-scale CORS systems.

Description of drawings

Fig. 1 is a flow chart of the method of the present invention;

Fig. 2 is the distribution diagram of embodiment benchmark site network;

Fig. 3 is embodiment real-time satellite clock difference product accuracy statistical diagram;

Figure 4 is the satellite clock deviation diagram of the PRN8 satellite and the PRN14 satellite;

Fig. 5 is the change diagram of the condition number under different satellite product clock precisions;

Figure 6 is a graph of ADOP variation under different satellite product clock precisions;

Fig. 7 is the observation value residual diagram of the ionosphere-free combination model, the non-combination model, and the non-combination model of the estimated satellite clock;

Fig. 8 is the change diagram of the condition number of the double-difference ambiguity obtained by adopting the non-difference method and the double-difference method;

Fig. 9 is the floating-point deviation of the double-difference wide-lane ambiguity obtained by using the non-difference method and the double-difference method;

Fig. 10 shows the floating-point deviation of the double-difference N1 ambiguity obtained by using the non-difference method and the double-difference method.

detailed description

The present invention will be further described below in conjunction with the accompanying drawings.

As shown in Figure 1, a network RTK ambiguity resolution method based on the non-difference and non-combination model first uses the non-difference and non-combination model to estimate the single-station non-difference ambiguity of the entire network, and the satellite clock is used as an unknown parameter , the real-time satellite clock difference product is added as a pseudo-observation; then according to the user's position, extract the non-difference ambiguity of the user's adjacent sites, select one of the adjacent sites as the reference site, and use the difference operator to restore the double-difference ambiguity Integer properties, single epoch Obtain double-differenced ambiguity results with integer properties. The inventive method comprises the following concrete steps:

Step 1), use the original observation values of each station to estimate the single-station undifference ambiguity of each station, and estimate the satellite clock error as an unknown parameter; the non-combined observation equation of each station is as shown in formula (1.1) Show:

Among them: s represents the satellite, k represents the receiver of the base station, j represents the frequency of each observation value, and j=1, 2; Indicates the pseudo-range observations at frequency j between satellite s and station k, Indicates the phase observation at frequency j between satellite s and station k, Indicates the geometric distance from the satellite to the receiver antenna phase center, δt _k indicates the receiver clock error, δt ^s indicates the satellite clock error, represents the tropospheric delay, Indicates the ionospheric delay; α _j is the frequency ratio, Indicates the value of frequency j; Denote the hardware delay of the receiver and the satellite on frequency j, respectively; are other modelable errors, such as tidal corrections, pole shift errors; Respectively represent the phase fractional deviation of the receiver and satellite on frequency j; λ _j is the carrier wavelength on frequency j, and N _j is the phase ambiguity on frequency j; is the pseudorange observation value and phase observation value noise at frequency j; c represents the speed of light;

According to the formula (1.1), use the Kalman filter to estimate the unknown parameters to obtain the non-difference floating-point ambiguity, including the following specific steps:

Step a), setting the unknown parameter vector X _j , as shown in formula (1.2):

Among them, the unknown parameter X _i is divided into two parts: time-varying and time-invariant, and the time-varying part includes the zenith wet delay ZTD _w,k ; the absorbing receiver has no ionospheric hardware delay The receiver clock error δt _k ′, in n-Dimensional Ionospheric Slope Delay of Absorbing Receiver and Satellite Frequency Dependent Hardware Delay Part in n is the number of satellites observed in this epoch, and the absorbing satellite has no ionospheric hardware delay The n-dimensional satellite clock error δt ^s ′, The time-invariant part includes n-dimensional non-differenced initial N1 integer ambiguities and n-dimensional non-differenced initial N2 integer ambiguities

Step b), set the Kalman filter design matrix B _i and observation vector L _i , as shown in formula (1.3):

Wherein, the observation vector L _i contains four non-difference observation values and a real-time satellite clock product result As pseudo observations, represent the phase observations between satellite s and station k at frequencies 1 and 2 respectively, Represent the pseudo-range observations between satellite s and station k at frequencies 1 and 2, respectively. It should be pointed out that since the real-time satellite clock error product is a pseudo-observation value, which is significantly affected by the interpolation accuracy and product accuracy, the satellite clock error is estimated as an unknown parameter, and the addition of the pseudo-observation equation can reduce the ill-conditionedness of the equation. in:

n represents the number of observed satellites in this epoch, θ ⁿ represents the satellite altitude angle of the nth satellite in this epoch, MF _w (θ) is the mapping function related to the zenith wet delay and satellite altitude angle, c represents the speed of light, λ ₁ , λ ₂ represents the wavelength of frequency 1, 2 respectively;

According to the unknown parameter vector X _i and the design matrix B _i and observation vector L _i , Kalman filtering is performed to obtain the non-difference floating-point ambiguity Among them, the form of Kalman filtering is shown in formula (1.5):

In the formula, Φ _{i, i-1} is the state transition matrix of (4n+2)×(4n+2), as shown in formula (1.6):

Q _i is a (4n+2)×(4n+2) dynamic noise covariance matrix, expressed as follows:

in, are the relative times of the corresponding stochastic process, respectively, E _n is an n×n unit matrix.

In Equation 1.7, the dynamic models for zenith wet delay, receiver, satellite clock bias and ionospheric tilt delay depend on their respective dynamic characteristics. For the zenith wet delay parameter, it can be considered as a first-order Markov process: q _ZWD =1~9cm ² /h, Δt is the time interval of epoch. White noise can simply and effectively describe the random process of receiver and satellite clock error. For the ionospheric tilt delay parameter, it can also be considered as a random walk related to the zenith angle z′ of the position of the ionospheric puncture point: Ambiguity is considered as a time-invariant parameter: represents the Kronecker product.

Step c), verify the ambiguity resolution model from the three indicators of ambiguity covariance condition number, ADOP value and observation value residual, the specific steps are as follows:

The condition number of ambiguity covariance reflects the internal relationship between ambiguities. The smaller the condition number is, the stronger the tolerance is and the more stable the model structure is. The accuracy of the pseudo-observation equation will affect the result of the condition number. To evaluate the model method equation, since the ambiguity method equation is a symmetric positive definite matrix, the condition number is used as the measurement index, and the calculation is shown in formula (1.8):

in, is the covariance matrix of the floating-point solution of the ambiguity; ||·|| represents the norm symbol; λ _max and λ _min are the maximum and minimum values of the eigenvalues, respectively.

The ADOP value is used to evaluate the internal model strength of the ambiguity resolution success rate, similar to the PDOP used to determine the influence of the geometric position of the satellite receiver on the positioning result, and the ADOP is used to evaluate the accuracy value of the ambiguity filter solution itself. It is affected by various indicators to obtain comprehensive results. In this method, in addition to being affected by the number of epochs and satellites, it is also affected by the accuracy of pseudo-observation equations and the number of parameters to be estimated. Its calculation is shown in formula (1.9):

Observation residuals directly reflect the posterior error of the model, which can intuitively reflect the influence of observation noise on the results under different methods, and at the same time, the characteristics of wave noise at different altitude angles need to be considered.

Step 2), according to the distance between the user's location and the site, extract the non-difference ambiguity of at least three user adjacent sites, and the non-difference ambiguity of each user's adjacent site and its covariance matrix As shown in formula (1.10):

in, are the non-difference basic ambiguities N1 of the 1st to nth satellites corresponding to adjacent stations; are the non-difference basic ambiguities N2 of the 1st to nth satellites corresponding to adjacent stations;

Step 3), first select a site in each user's adjacent sites as a reference site, and other adjacent sites as non-reference sites; then set the inter-station difference operator C _kd and reference The inter-satellite difference operator C ^sd between the satellite and the non-reference satellite is shown in formula (1.11):

In the inter-satellite difference operator, the position of the "-1" column is determined by the sorting position of the reference satellite in the observation satellite;

According to the difference operators C _kd and C ^sd restore the double-difference ambiguity with integer characteristics between stations and stars, the restoration steps are shown in formulas (1.12) and (1.13):

Among them, k1 and k2 are reference sites and non-reference sites respectively, r is a reference satellite, and s is a non-reference satellite; is the double-difference ambiguity between stations and satellites, and the corresponding covariance matrix is are the undifferenced ambiguities of sites k1 and k2, respectively, Respectively, double-difference basic ambiguity N1, N2;

Step 4), use the LAMBDA algorithm to search and fix the ambiguity, including the following steps:

a), converting the double-difference basic ambiguities N1, N2 into wide-lane ambiguities and N1 ambiguity part Used to convert the observation array and covariance matrix The linear transformation matrix C _linear is shown in formula (1.14):

in, respectively with The corresponding covariance matrix.

b) Use the LAMBDA algorithm to obtain a fixed integer wide-lane ambiguity, use the wide-lane ambiguity as a known value, and determine the integer value of the N1 ambiguity, as shown in formula (1.15):

In formula (1.15), are the integer values of N1 ambiguity and widelane ambiguity respectively, for The corresponding covariance matrix.

The double-difference ambiguities of integer characteristics between adjacent sites of the user's location are obtained, and the step-by-step method is used to reduce the number of ambiguities, which improves the network-wide search efficiency of the LAMBDA algorithm.

This embodiment adopts the 24-hour RINEX observation file and navigation file of CORS in the United States, and the IGS real-time product matching the observation file. In the experiment, the whole network includes nine reference stations, and the distribution of reference stations is shown in Figure 2.

In order to simulate the sparse network RTK, station p332 is set as the reference station, and eight baselines with an average baseline length of about 150 km are formed, as shown in Table 1:

Table 1 Eight baselines composed of nine base stations in the US CORS network

It can be seen from Figure 3 that the mean square error of most satellite clock errors is within 0.25s, but there are still some satellites with large errors. Therefore, in the non-difference and non-combination model, the satellite clock bias needs to be estimated to ensure the time-invariant property of the ambiguity. Figure 5-7 shows the comparison of the condition number, ADOP value and observation residual under different estimation strategies for the satellite clock error. It can be seen that although the estimation of the satellite clock error needs to sacrifice more time to ensure the ambiguity resolution Reliability, but the condition number and phase residual are reduced to ensure the stability and tolerance of the model. It can be seen from Figure 8 that the non-difference method restores the double-difference ambiguity, and has a better estimation effect on the floating-point ambiguity than the double-difference method. From Figures 9 and 10, it can be seen that compared with the traditional MW method, the non-difference and non-combination method has significantly improved the fixing success rate of wide-lane ambiguity and N1 ambiguity.

The above is only a preferred embodiment of the present invention, and it should be pointed out that for those of ordinary skill in the art, some improvements and modifications can be made without departing from the principle of the present invention. It should be regarded as the protection scope of the present invention.

Claims (2)

1. A network RTK ambiguity resolution method based on the non-difference and non-combination model, characterized in that, firstly, the non-difference and non-combination model is used to estimate the single-station non-difference ambiguity of the whole network, wherein the satellite clock error is used as the unknown Parameters, real-time satellite clock difference products are added as pseudo observations; then according to the user's location, extract the non-difference ambiguity of the user's adjacent sites, select one of the adjacent sites as the reference site, and use the difference operator to restore the double-difference ambiguity Integer properties of , a double-differenced ambiguity result with integer properties is obtained for a single epoch. 2. the network RTK ambiguity resolution method based on non-difference non-combination model according to claim 1, is characterized in that, comprises following specific steps: Step 1), use the original observations of each station to estimate the single-station indifferent ambiguity of each station, and estimate the satellite clock error as an unknown parameter, as shown in formula (1.1): $\begin{array}{c}{\mathrm{<span\; class="notranslate">\; P</span>}}_{\mathrm{<span\; class="notranslate">\; j</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; k</span>}}^{\mathrm{<span\; class="notranslate">\; the\; s</span>}}<span\; class="notranslate">\; =</span>{\mathrm{<span\; class="notranslate">\; \&rho;</span>}}_{\mathrm{<span\; class="notranslate">\; k</span>}}^{\mathrm{<span\; class="notranslate">\; the\; s</span>}}<span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; c\&delta;t</span>}}_{\mathrm{<span\; class="notranslate">\; k</span>}}<span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; c\&delta;t</span>}}^{\mathrm{<span\; class="notranslate">\; the\; s</span>}}<span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; T</span>}}_{\mathrm{<span\; class="notranslate">\; k</span>}}^{\mathrm{<span\; class="notranslate">\; the\; s</span>}}<span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; \&alpha;</span>}}_{\mathrm{<span\; class="notranslate">\; j</span>}}{\mathrm{<span\; class="notranslate">\; I</span>}}_{\mathrm{<span\; class="notranslate">\; k</span>}}^{\mathrm{<span\; class="notranslate">\; the\; s</span>}}<span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; d</span>}}_{\mathrm{<span\; class="notranslate">\; k</span>}<span\; class="notranslate">\; ,</span>{\mathrm{<span\; class="notranslate">\; P</span>}}_{\mathrm{<span\; class="notranslate">\; j</span>}}}<span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; d</span>}}_{{\mathrm{<span\; class="notranslate">\; P</span>}}_{\mathrm{<span\; class="notranslate">\; j</span>}}}^{\mathrm{<span\; class="notranslate">\; the\; s</span>}}<span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; \&epsiv;</span>}}_{\mathrm{<span\; class="notranslate">\; k</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; o</span>}\mathrm{<span\; class="notranslate">\; t</span>}\mathrm{<span\; class="notranslate">\; h</span>}\mathrm{<span\; class="notranslate">\; e</span>}\mathrm{<span\; class="notranslate">\; r</span>}\mathrm{<span\; class="notranslate">\; the\; s</span>}}^{\mathrm{<span\; class="notranslate">\; the\; s</span>}}<span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; \&epsiv;</span>}}_{\mathrm{<span\; class="notranslate">\; k</span>}<span\; class="notranslate">\; ,</span>{\mathrm{<span\; class="notranslate">\; P</span>}}_{\mathrm{<span\; class="notranslate">\; j</span>}}}^{\mathrm{<span\; class="notranslate">\; the\; s</span>}}\\ {\mathrm{<span\; class="notranslate">\; \&Phi;</span>}}_{\mathrm{<span\; class="notranslate">\; j</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; k</span>}}^{\mathrm{<span\; class="notranslate">\; the\; s</span>}}<span\; class="notranslate">\; =</span>{\mathrm{<span\; class="notranslate">\; \&rho;</span>}}_{\mathrm{<span\; class="notranslate">\; k</span>}}^{\mathrm{<span\; class="notranslate">\; the\; s</span>}}<span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; c\&delta;t</span>}}_{\mathrm{<span\; class="notranslate">\; k</span>}}<span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; c\&delta;t</span>}}^{\mathrm{<span\; class="notranslate">\; the\; s</span>}}<span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; T</span>}}_{\mathrm{<span\; class="notranslate">\; k</span>}}^{\mathrm{<span\; class="notranslate">\; the\; s</span>}}<span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; \&alpha;</span>}}_{\mathrm{<span\; class="notranslate">\; j</span>}}{\mathrm{<span\; class="notranslate">\; I</span>}}_{\mathrm{<span\; class="notranslate">\; k</span>}}^{\mathrm{<span\; class="notranslate">\; the\; s</span>}}<span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; b</span>}}_{\mathrm{<span\; class="notranslate">\; k</span>}<span\; class="notranslate">\; ,</span>{\mathrm{<span\; class="notranslate">\; \&Phi;</span>}}_{\mathrm{<span\; class="notranslate">\; j</span>}}}<span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; b</span>}}_{{\mathrm{<span\; class="notranslate">\; \&Phi;</span>}}_{\mathrm{<span\; class="notranslate">\; j</span>}}}^{\mathrm{<span\; class="notranslate">\; the\; s</span>}}<span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; \&epsiv;</span>}}_{\mathrm{<span\; class="notranslate">\; k</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; o</span>}\mathrm{<span\; class="notranslate">\; t</span>}\mathrm{<span\; class="notranslate">\; h</span>}\mathrm{<span\; class="notranslate">\; e</span>}\mathrm{<span\; class="notranslate">\; r</span>}\mathrm{<span\; class="notranslate">\; the\; s</span>}}^{\mathrm{<span\; class="notranslate">\; the\; s</span>}}<span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; \&lambda;</span>}}_{\mathrm{<span\; class="notranslate">\; j</span>}}{\mathrm{<span\; class="notranslate">\; N</span>}}_{\mathrm{<span\; class="notranslate">\; j</span>}}<span\; class="notranslate">\; +</span>{\mathrm{<span\; class="notranslate">\; \&epsiv;</span>}}_{\mathrm{<span\; class="notranslate">\; k</span>}<span\; class="notranslate">\; ,</span>{\mathrm{<span\; class="notranslate">\; \&Phi;</span>}}_{\mathrm{<span\; class="notranslate">\; j</span>}}}^{\mathrm{<span\; class="notranslate">\; the\; s</span>}}\end{array}<span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 1.1</span>}<span\; class="notranslate">\; )</span>$ Among them: s represents the satellite, k represents the receiver of the base station, j represents the frequency of each observation value, and j=1, 2; Indicates the pseudo-range observations at frequency j between satellite s and station k, Indicates the phase observation at frequency j between satellite s and station k, Indicates the geometric distance from the satellite to the receiver antenna phase center, δt _k indicates the receiver clock error, δt ^s indicates the satellite clock error, represents the tropospheric delay, Indicates the ionospheric delay; α _j is the frequency ratio, f _j represents the value of frequency j; Denote the hardware delay of the receiver and the satellite on frequency j, respectively; are other modelable errors; Respectively represent the phase fractional deviation of the receiver and satellite on frequency j; λ _j is the carrier wavelength on frequency j, and N _j is the phase ambiguity on frequency j; is the pseudorange observation value and phase observation value noise at frequency j; c represents the speed of light; According to formula (1.1), use Kalman filter to estimate the unknown parameters, and obtain the non-difference floating-point ambiguity, including the following specific steps: Step a), setting the unknown parameter vector X _i , as shown in formula (1.2): ${\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}<span\; class="notranslate">\; =</span>{\left[\begin{array}{cccccc}{\mathrm{<span\; class="notranslate">\; ZTD</span>}}_{\mathrm{<span\; class="notranslate">\; w</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; k</span>}}& {{\mathrm{<span\; class="notranslate">\; \&delta;t</span>}}_{\mathrm{<span\; class="notranslate">\; k</span>}}}^{<span\; class="notranslate">\; \&prime;</span>}& {{\mathrm{<span\; class="notranslate">\; I</span>}}_{\mathrm{<span\; class="notranslate">\; k</span>}}^{\mathrm{<span\; class="notranslate">\; the\; s</span>}}}^{<span\; class="notranslate">\; \&prime;</span>}& {{\mathrm{<span\; class="notranslate">\; \&delta;t</span>}}^{\mathrm{<span\; class="notranslate">\; the\; s</span>}}}^{<span\; class="notranslate">\; \&prime;</span>}& {\mathrm{<span\; class="notranslate">\; N</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}^{\mathrm{<span\; class="notranslate">\; the\; s</span>}}& {\mathrm{<span\; class="notranslate">\; N</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}^{\mathrm{<span\; class="notranslate">\; the\; s</span>}}\end{array}\right]}^{\mathrm{<span\; class="notranslate">\; T</span>}}<span\; class="notranslate">\; ,</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; the\; s</span>}<span\; class="notranslate">\; =</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; ...</span>\mathrm{<span\; class="notranslate">\; no</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 1.2</span>}<span\; class="notranslate">\; )</span>$ Among them, the unknown parameter X _i is divided into two parts: time-varying and time-invariant, and the time-varying part includes the zenith wet delay ZTD _w,k ; the absorbing receiver has no ionospheric hardware delay The receiver clock error δt _k ', in n-Dimensional Ionospheric Slope Delay of Absorbing Receiver and Satellite Frequency Dependent Hardware Delay Part in n is the number of satellites observed in this epoch, and the absorbing satellite has no ionospheric hardware delay The n-dimensional satellite clock error δt ^s ', The time-invariant part includes n-dimensional non-differenced initial N1 integer ambiguities and n-dimensional non-differenced initial N2 integer ambiguities Step b), set the Kalman filter design matrix B _i and observation vector L _i , as shown in formula (1.3): ${\mathrm{<span\; class="notranslate">\; L</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}<span\; class="notranslate">\; =</span>\left[\begin{array}{c}{\mathrm{<span\; class="notranslate">\; \&Phi;</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; k</span>}}^{\mathrm{<span\; class="notranslate">\; the\; s</span>}}\\ {\mathrm{<span\; class="notranslate">\; P</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; k</span>}}^{\mathrm{<span\; class="notranslate">\; the\; s</span>}}\\ {\mathrm{<span\; class="notranslate">\; \&Phi;</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; k</span>}}^{\mathrm{<span\; class="notranslate">\; the\; s</span>}}\\ {\mathrm{<span\; class="notranslate">\; P</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; k</span>}}^{\mathrm{<span\; class="notranslate">\; the\; s</span>}}\\ {\mathrm{<span\; class="notranslate">\; \&delta;t</span>}}_{\mathrm{<span\; class="notranslate">\; I</span>}\mathrm{<span\; class="notranslate">\; G</span>}\mathrm{<span\; class="notranslate">\; S</span>}}^{\mathrm{<span\; class="notranslate">\; the\; s</span>}}\end{array}\right]<span\; class="notranslate">\; ,</span>{\mathrm{<span\; class="notranslate">\; B</span>}}_{\mathrm{<span\; class="notranslate">\; i</span>}}<span\; class="notranslate">\; =</span>\left[\begin{array}{cccccc}{\mathrm{<span\; class="notranslate">\; B</span>}}_{{\mathrm{<span\; class="notranslate">\; MF</span>}}_{\mathrm{<span\; class="notranslate">\; w</span>}}}& {\mathrm{<span\; class="notranslate">\; B</span>}}_{{{\mathrm{<span\; class="notranslate">\; \&delta;t</span>}}_{\mathrm{<span\; class="notranslate">\; k</span>}}}^{<span\; class="notranslate">\; \&prime;</span>}}& {\mathrm{<span\; class="notranslate">\; B</span>}}_{\mathrm{<span\; class="notranslate">\; I</span>}}& <span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; B</span>}}_{{{\mathrm{<span\; class="notranslate">\; \&delta;t</span>}}^{\mathrm{<span\; class="notranslate">\; the\; s</span>}}}^{<span\; class="notranslate">\; \&prime;</span>}}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}\\ {\mathrm{<span\; class="notranslate">\; B</span>}}_{{\mathrm{<span\; class="notranslate">\; MF</span>}}_{\mathrm{<span\; class="notranslate">\; w</span>}}}& {\mathrm{<span\; class="notranslate">\; B</span>}}_{{{\mathrm{<span\; class="notranslate">\; \&delta;t</span>}}_{\mathrm{<span\; class="notranslate">\; k</span>}}}^{<span\; class="notranslate">\; \&prime;</span>}}& \frac{{\mathrm{<span\; class="notranslate">\; f</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}^{\mathrm{<span\; class="notranslate">\; 2</span>}}}{{\mathrm{<span\; class="notranslate">\; f</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}^{\mathrm{<span\; class="notranslate">\; 2</span>}}}{\mathrm{<span\; class="notranslate">\; B</span>}}_{\mathrm{<span\; class="notranslate">\; I</span>}}& <span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; B</span>}}_{{{\mathrm{<span\; class="notranslate">\; \&delta;t</span>}}^{\mathrm{<span\; class="notranslate">\; the\; s</span>}}}^{<span\; class="notranslate">\; \&prime;</span>}}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}\\ {\mathrm{<span\; class="notranslate">\; B</span>}}_{{\mathrm{<span\; class="notranslate">\; MF</span>}}_{\mathrm{<span\; class="notranslate">\; w</span>}}}& {\mathrm{<span\; class="notranslate">\; B</span>}}_{{{\mathrm{<span\; class="notranslate">\; \&delta;t</span>}}_{\mathrm{<span\; class="notranslate">\; k</span>}}}^{<span\; class="notranslate">\; \&prime;</span>}}& <span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; B</span>}}_{\mathrm{<span\; class="notranslate">\; I</span>}}& <span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; B</span>}}_{{{\mathrm{<span\; class="notranslate">\; \&delta;t</span>}}^{\mathrm{<span\; class="notranslate">\; the\; s</span>}}}^{<span\; class="notranslate">\; \&prime;</span>}}& {\mathrm{<span\; class="notranslate">\; B</span>}}_{{\mathrm{<span\; class="notranslate">\; N</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}}& \mathrm{<span\; class="notranslate">\; 0</span>}\\ {\mathrm{<span\; class="notranslate">\; B</span>}}_{{\mathrm{<span\; class="notranslate">\; MF</span>}}_{\mathrm{<span\; class="notranslate">\; w</span>}}}& {\mathrm{<span\; class="notranslate">\; B</span>}}_{{{\mathrm{<span\; class="notranslate">\; \&delta;t</span>}}_{\mathrm{<span\; class="notranslate">\; k</span>}}}^{<span\; class="notranslate">\; \&prime;</span>}}& <span\; class="notranslate">\; -</span>\frac{{\mathrm{<span\; class="notranslate">\; f</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}^{\mathrm{<span\; class="notranslate">\; 2</span>}}}{{\mathrm{<span\; class="notranslate">\; f</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}^{\mathrm{<span\; class="notranslate">\; 2</span>}}}{\mathrm{<span\; class="notranslate">\; B</span>}}_{\mathrm{<span\; class="notranslate">\; I</span>}}& <span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; B</span>}}_{{{\mathrm{<span\; class="notranslate">\; \&delta;t</span>}}^{\mathrm{<span\; class="notranslate">\; the\; s</span>}}}^{<span\; class="notranslate">\; \&prime;</span>}}& \mathrm{<span\; class="notranslate">\; 0</span>}& {\mathrm{<span\; class="notranslate">\; B</span>}}_{{\mathrm{<span\; class="notranslate">\; N</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}}}\\ \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}& {\mathrm{<span\; class="notranslate">\; B</span>}}_{{{\mathrm{<span\; class="notranslate">\; \&delta;t</span>}}^{\mathrm{<span\; class="notranslate">\; the\; s</span>}}}^{<span\; class="notranslate">\; \&prime;</span>}}& \mathrm{<span\; class="notranslate">\; 0</span>}& \mathrm{<span\; class="notranslate">\; 0</span>}\end{array}\right]<span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 1.3</span>}<span\; class="notranslate">\; )</span>$ Wherein, the observation vector L _i contains four non-difference observation values and a real-time satellite clock product result As pseudo observations, represent the phase observations between satellite s and station k at frequencies 1 and 2 respectively, represent the pseudo-range observations between satellite s and station k at frequencies 1 and 2, respectively; n represents the number of observed satellites in this epoch, θ ⁿ represents the satellite elevation angle of the nth satellite in this epoch, MF _w (θ) is the mapping function related to the zenith wet delay and satellite elevation angle, c represents the speed of light, λ ₁ , λ ₂ represents the wavelength of frequency 1, 2 respectively; According to the unknown parameter vector X _i , design matrix B _i and observation vector L _i , perform Kalman filtering to obtain n-dimensional non-difference initial N1, N2 integer ambiguities Step 2), according to the distance between the user's location and the site, extract the non-difference ambiguity of at least three user adjacent sites, and the non-difference ambiguity of each user's adjacent site As shown in formula (1.5): $\underset{\mathrm{<span\; class="notranslate">\; 2</span>}\mathrm{<span\; class="notranslate">\; no</span>}<span\; class="notranslate">\; \&times;</span>\mathrm{<span\; class="notranslate">\; 1</span>}}{{\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; k</span>}}^{\mathrm{<span\; class="notranslate">\; the\; s</span>}}}<span\; class="notranslate">\; =</span>{\left[\begin{array}{cccccc}{\mathrm{<span\; class="notranslate">\; N</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; k</span>}}^{\mathrm{<span\; class="notranslate">\; 1</span>}}& <span\; class="notranslate">\; ...</span>& {\mathrm{<span\; class="notranslate">\; N</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; k</span>}}^{\mathrm{<span\; class="notranslate">\; no</span>}}& {\mathrm{<span\; class="notranslate">\; N</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; k</span>}}^{\mathrm{<span\; class="notranslate">\; 1</span>}}& <span\; class="notranslate">\; ...</span>& {\mathrm{<span\; class="notranslate">\; N</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; k</span>}}^{\mathrm{<span\; class="notranslate">\; no</span>}}\end{array}\right]}^{\mathrm{<span\; class="notranslate">\; T</span>}}<span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 1.5</span>}<span\; class="notranslate">\; )</span>$ in, are the non-difference basic ambiguities N1 of the 1st to nth satellites corresponding to adjacent stations; are the non-difference basic ambiguities N2 of the 1st to nth satellites corresponding to adjacent stations; Step 3), first select a site in each user's adjacent sites as a reference site, and other adjacent sites as non-reference sites; then set the inter-station difference operator C _kd and reference The inter-satellite difference operator C ^sd of the satellite and the non-reference satellite, according to the difference operator C _kd and C ^sd restores the double-difference ambiguity with integer characteristics between the stations, and the restoration steps are shown in formula (1.6) : $\underset{\mathrm{<span\; class="notranslate">\; 2</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; no</span>}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; )</span><span\; class="notranslate">\; \&times;</span>\mathrm{<span\; class="notranslate">\; 1</span>}}{{\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; k</span>}\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; k</span>}\mathrm{<span\; class="notranslate">\; 2</span>}}^{\mathrm{<span\; class="notranslate">\; the\; s</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; r</span>}}}<span\; class="notranslate">\; =</span>{\mathrm{<span\; class="notranslate">\; C</span>}}^{\mathrm{<span\; class="notranslate">\; the\; s</span>}\mathrm{<span\; class="notranslate">\; d</span>}}<span\; class="notranslate">\; \&Center\; Dot;</span>{\mathrm{<span\; class="notranslate">\; C</span>}}_{\mathrm{<span\; class="notranslate">\; k</span>}\mathrm{<span\; class="notranslate">\; d</span>}}<span\; class="notranslate">\; \&Center\; Dot;</span>\left[\begin{array}{c}{\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; k</span>}\mathrm{<span\; class="notranslate">\; 1</span>}}^{\mathrm{<span\; class="notranslate">\; the\; s</span>}}\\ {\mathrm{<span\; class="notranslate">\; x</span>}}_{\mathrm{<span\; class="notranslate">\; k</span>}\mathrm{<span\; class="notranslate">\; 2</span>}}^{\mathrm{<span\; class="notranslate">\; the\; s</span>}}\end{array}\right]<span\; class="notranslate">\; =</span>{\left[\begin{array}{cc}{\mathrm{<span\; class="notranslate">\; N</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; k</span>}\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; k</span>}\mathrm{<span\; class="notranslate">\; 2</span>}}^{\mathrm{<span\; class="notranslate">\; the\; s</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; r</span>}}& {\mathrm{<span\; class="notranslate">\; N</span>}}_{\mathrm{<span\; class="notranslate">\; 2</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; k</span>}\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; k</span>}\mathrm{<span\; class="notranslate">\; 2</span>}}^{\mathrm{<span\; class="notranslate">\; the\; s</span>}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; r</span>}}\end{array}\right]}^{\mathrm{<span\; class="notranslate">\; T</span>}}<span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 1.6</span>}<span\; class="notranslate">\; )</span>$ Among them, k1 and k2 are the reference station and non-reference station respectively, r is the reference satellite, s is the non-reference satellite, is the inter-satellite double-difference ambiguity, are the undifferenced ambiguities of sites k1 and k2, respectively, Respectively, double-difference basic ambiguity N1, N2; Step 4), use the LAMBDA algorithm to search and fix the ambiguities, specifically: first convert the double-difference basic ambiguities N1, N2 into wide-lane ambiguities and N1 ambiguity part Then use the LAMBDA algorithm to obtain a fixed integer wide-lane ambiguity, and finally use the wide-lane ambiguity as a known value to determine the integer value of N1 ambiguity.