CN103163542A - Observation constant based method for detecting gross error in global navigation satellite system (GNSS) baseline solution - Google Patents

Abstract

The invention discloses an observation constant based method for detecting a gross error in global navigation system (GNSS) baseline solution. The method includes firstly calculating a difference value between a double-differenced carrier ionosphere-free combination value and an observation equation calculation value, then performing estimation by using difference values between double-differenced carrier ionosphere-free combination values and observation equation calculation values of prior two epochs to obtain a threshold value of a corresponding difference value of a current epoch, and finally judging whether a carrier observation value of the current epoch contains the gross error by comparing the threshold value with the observation value. According to the observation constant based method, problems that single gross error is difficult to position and continuous gross errors are difficult to detect in GNSS baseline solution are effectively solved.

Description

Gross error detection method based on observation constant in GNSS baseline solution

Technical Field

The invention relates to a gross error detection method in baseline solution, in particular to a gross error detection method based on a carrier observation constant.

Background

Baseline resolution is an important aspect of reference station integrity monitoring. The displacement correction quantity of each reference station is obtained by calculating the base line and carrying out net adjustment, and the coordinates of the reference stations are periodically corrected to ensure the accuracy of the coordinates of the reference stations and provide guarantee for quick and accurate positioning of users. Gross error detection is a research hotspot of baseline solution quality control, however, as least squares have no tolerance, once gross errors occur in observed values, even if the number is small, the final result is seriously influenced. Whether the gross error can be accurately detected directly influences the precision of the ambiguity floating solution and whether the ambiguity floating solution can be fixed correctly, and further influences the resolving quality of the baseline.

Most of the traditional gross error detection methods firstly solve ambiguity floating solutions by least squares and fix the ambiguity, and based on residual values generated after the ambiguity is fixed, a hypothesis test method is used for constructing statistical test quantities and detecting the gross error or a robust estimation model is used for detecting the gross error. However, due to least square residual statistic correlation, the inherent relation between the gross error and the residual error value is covered, so that the gross error cannot be correctly reflected to the corresponding residual error value, and thus, the gross error cannot be correctly detected by the statistic inspection quantity based on the residual error value, and because of the limitation of the number of satellites, the redundant observation in the double-difference carrier observation equation is less, so that the sensitivity of the robust estimation model to the gross error is greatly reduced. Therefore, it becomes quite difficult to detect a location based on the gross difference of the residual values.

Disclosure of Invention

Different from the conventional research thought, the rough error detection method based on the carrier observation constant is provided aiming at the defect of rough error detection based on the residual error value at present, and the problems of difficult positioning of single rough error and difficult continuous rough error detection are effectively solved.

The gross error judgment method is characterized in that a difference value between a double-difference carrier ionosphere-free combination value and an observation equation calculated value is constructed, and a threshold value of a corresponding difference value of a current epoch is estimated by using the difference value between the previous two epoch double-difference carrier ionosphere-free combination values and the observation equation calculated value, so that whether the current epoch carrier observed value contains gross error or not is judged. The method is realized by the following steps:

(1) constructing a double-difference carrier ionosphere-free combination value equation:

（1.1）

wherein,

a double difference operator; f. of₁And f₂Are respectively L₁、L₂(satellite carrier signals of two different frequencies) carrier frequency;

and

are respectively L₁、L₂A carrier phase observation; lambda [ alpha ]_WIs the widelane ambiguity; rho is a satellite-earth distance, and the precision in pseudo-range single-point positioning is meter level; o, M are error in orbit respectivelyDifferences and multipath effects, negligible in the double difference equation; t is tropospheric delay, corrected with the new university of unreal UNB3m tropospheric model; n is a radical of₁And N₂Are respectively L₁、L₂Carrier ambiguity. In equation (1.1), the equation holds when all parameters are exact values. The left side of the equal sign of the formula (1.1) is a double-difference carrier non-ionosphere combination value, and the right side of the equal sign is a double-difference carrier non-ionosphere combination calculated value. Due to L₁、L₂The precision of the carrier observed value is 0.01 week, so the left side of the equal sign can be regarded as the precise value, and the right side of the equal sign is subjected to double-difference distance between earth and earth

Double differential troposphere

The influence of parameter precision has certain deviation with the left-type double-difference carrier non-ionized layer combination value.

(2) Estimating by using the difference value between the ionosphere-free combination value of the first two epoch double-difference carriers and the observation equation calculation value to obtain the threshold value of the corresponding difference value of the current epoch:

the coordinates of the reference station in the baseline solution are precisely known, so the double-difference range accuracy depends mainly on the rover range. The rover station satellite range can be obtained by the satellite coordinates and the rover station pseudorange single-point positioning coordinates:

$\widehat{\mathrm{\ρ}}=\sqrt{{\{X^i-(X_r+\mathrm{\ΔX})\}}^2+{\{Y^i-(Y_r+\mathrm{\ΔY})\}}^2+{\{Z^i-(Z_r+\mathrm{\ΔZ})\}}^2}---\left(1.2\right)$

wherein,

calculating the rover guard distance; xⁱ、Yⁱ、ZⁱRespectively the coordinates of the ith satellite; x_r、Y_r、Z_rThe r coordinates of the rover (whether the rover represents yes by r) are true values respectively; Δ X, Δ Y, and Δ Z are deviations of coordinates of the rover r (which is the same rover as the above rover) obtained from the pseudorange single-point positioning from real values of the rover coordinates, respectively.

As can be seen from the formula (2.2),

the magnitude of the change over time is only affected by the amount of satellite coordinate change. At adjacent time t_k、t_k+△kThe variation of the rover gauge to the ground is as follows:

$\mathrm{\Δ}\widehat{\mathrm{\ρ}}\left(\begin{array}{cc}{t}_k& t_{k+\mathrm{\Δk}}\end{array}\right)=\sqrt{\widehat{\mathrm{\ρ}}{\left(t_k\right)}^2+2\mathrm{\Δt}(V_X\·X+V_Y\·Y+V_Z\·Z)+{\mathrm{\Δ}}^2}-\widehat{\mathrm{\ρ}}\left(t_k\right)---\left(1.3\right)$

wherein, $\mathrm{\Δ}\widehat{\mathrm{\ρ}}\left(\begin{array}{cc}{t}_k& t_{k+\mathrm{\Δk}}\end{array}\right)$ is (t)_k,t_k+△k) The change of the sanitary distance of the flow station in the time period;

is t_kThe distance between the guards and the ground at the moment; Δ t = t_k+△k-t_k；V_X、V_Y、V_ZAre respectively satellite at (t)_k，t_k+△k) Average velocity along axis X, Y, Z over a period of time; x = Xⁱ-(X_r+△X)，Y=Yⁱ-(Y_r+△Y)，Z=Zⁱ-(Z_r+△Z),△²=△X²+△Y²+△Z²。

According to analysis, the variation of the rover satellite distance is related to the speed of the satellite along the coordinate axis direction and the offset of the rover coordinate. Since the velocity of the satellite in the coordinate axis direction is almost constant and the rover coordinate offset is constant in a short time, it can be considered that the rover satellite distance variation is approximately equal in a short time, that is, the rover satellite distance variation is approximately equal in a short time $\mathrm{\Δ}\widehat{\mathrm{\ρ}}\left(\begin{array}{cc}{t}_{k-\mathrm{\Δk}}& t_k\end{array}\right)\≈\mathrm{\Δ}\widehat{\mathrm{\ρ}}\left(\begin{array}{cc}{t}_k& t_{k+\mathrm{\Δk}}\end{array}\right).$ The influence of the rover coordinate offset on the double-difference carrier ionosphere-free combined calculated value is accumulative in time. Therefore, in the case of no gross error of the L1 and L2 carriers, the ionosphere-free combination value of the adjacent epoch double-difference carrier is approximately equal to the difference value of the calculation value of the observation equation, and the obvious linear characteristic is presented, that is: $\mathrm{\Δ}\▿P(j+1)-\mathrm{\Δ}\▿P\left(j\right)\≈\mathrm{\Δ}\▿P\left(j\right)-\mathrm{\Δ}\▿P(j-1)$ （1.4）

wherein,

and the difference value between the ionization layer-free combination value of the j +1 th epoch double-difference carrier and the calculation value of the observation equation is obtained.

Using the first two known epochsAnd interpolating the value to obtain the estimated value of the current epoch:

$\mathrm{\Δ}\▿\tilde{P}(j+1)\≈2\·\mathrm{\Δ}\▿P\left(j\right)-\mathrm{\Δ}\▿P(j-1)---\left(1.5\right)$

wherein,

is the j +1 th epoch

An estimate of (2);

and

j and j-1 epochs respectively

True value of (1). If the (j + 1) th epoch is in gross error, then use

And

interpolating an estimate of the j +2 th epoch

And so on.

(3) Judging whether the current epoch carrier observed value contains gross error:

due to the fact that

From the formula (1.1), when L is₁、L₂When the gross error occurs in the carrier wave (the satellite carrier wave signals with two different frequencies), the gross error is respectively reduced by 0.562 times and 0.438 times to be reflected in

In (1). In order to detect a gross error of 0.1 week or more, the threshold is set to

When in use

If so, the observed value is considered to contain no gross error, otherwise, the observed value is considered to contain gross error.

By the observation constant-based gross error detection method in GNSS baseline solution, the problems that a single gross error cannot be positioned and continuous gross error detection is difficult in GNSS baseline solution are effectively solved.

Drawings

FIG. 1 is a flow chart of gross error detection based on observed constants in baseline solution.

FIG. 2 shows the P346 station after adding a single gross error

The true value and its threshold.

FIG. 3 shows the P346 station after adding a single gross error

The case where the true value exceeds the upper threshold value is represented by the difference between the upper threshold value and the true value.

FIG. 4 shows P346 station after adding successive gross errors

The true value and its threshold.

FIG. 5 shows P346 station after adding successive gross errors

The case where the true value exceeds the lower threshold is represented by the difference between the lower threshold and the true value.

Detailed Description

The gross error detection method based on the observation constant is specifically realized as follows:

(1) pseudo-range point positioning is carried out to obtain the approximate coordinates of the rover station;

(2) correcting the convection layer by using a UNB3m model;

(3) substituting the approximate coordinates of the rover station and the troposphere delay into the right side of the equation (1.1) with equal sign, neglecting the orbit error and the multipath effect, and obtaining a calculated value of the double-difference carrier ionosphere-free combination;

(4) substituting the observed values of the L1 and L2 carriers into the left side of a formula (1.1) to obtain a double-difference carrier ionosphere-free combined value;

(5) calculating the difference value between the combined value without the ionized layer of the double-difference carrier and the calculated value of the observation equation

Using the first two epochs

Linear interpolation of the values to obtain the current epoch

Threshold value of

(6) Calculating the current epoch

Value and threshold

And comparing to judge whether the gross errors exist.

Example one

To demonstrate the effectiveness of the gross error detection method based on observed constants, a baseline of 200km length was tested using a combination of the CMBB station of the CORS network (university of Columbia, Inc., Calif.) and the P346 station (Laporter, Inc., Indiana, Laporte, Calif.) at 2011, 15 minutes at 23 months and 15 minutes at 23 days, to establish a baseline, where CMBB (university of Columbia, Inc., Calif.) is the base station and P346 is the rover. And selecting a G03 satellite as a research object by taking a G08 satellite as a reference satellite.

Constructing a double-difference carrier ionosphere-free combination value equation:

（1.1）

wherein,

a double difference operator; f. of₁And f₂Are respectively L₁、L₂(satellite carrier signals of two different frequencies) carrier frequency;and

are respectively L₁、L₂A carrier phase observation; lambda [ alpha ]_WIs the widelane ambiguity; rho is a satellite-earth distance, and the precision in pseudo-range single-point positioning is meter level; o, M are orbit errors and multipath effects, respectively, which can be ignored in the double difference equation; t is tropospheric delay, corrected with the new university of unreal UNB3m tropospheric model; n is a radical of₁And N₂Are respectively L₁、L₂Carrier ambiguity. In equation (1.1), the equation holds when all parameters are exact values. The left side of the equal sign of the formula (1.1) is a double-difference carrier non-ionosphere combination value, and the right side of the equal sign is a double-difference carrier non-ionosphere combination calculated value. Due to L₁、L₂The precision of the carrier observed value is 0.01 week, so the left side of the equal sign can be regarded as the precise value, and the right side of the equal sign is subjected to double-difference distance between earth and earth

Double differential troposphere

The influence of parameter precision has with the left type double difference carrier non-ionized layer combination valueA certain deviation.

(2) Estimating by using the difference value between the ionosphere-free combination value of the first two epoch double-difference carriers and the observation equation calculation value to obtain the threshold value of the corresponding difference value of the current epoch:

the coordinates of the reference station in the baseline solution are precisely known, so the double-difference range accuracy depends mainly on the rover range. The rover station satellite range can be obtained by the satellite coordinates and the rover station pseudorange single-point positioning coordinates:

$\widehat{\mathrm{\ρ}}=\sqrt{{\{X^i-(X_r+\mathrm{\ΔX})\}}^2+{\{Y^i-(Y_r+\mathrm{\ΔY})\}}^2+{\{Z^i-(Z_r+\mathrm{\ΔZ})\}}^2}---\left(1.2\right)$

wherein,

calculating the rover guard distance; xⁱ、Yⁱ、ZⁱRespectively the coordinates of the ith satellite; x_r、Y_r、Z_rThe r coordinates of the rover (whether the rover represents yes by r) are true values respectively; Δ X, Δ Y, and Δ Z are deviations of coordinates of the rover r (which is the same rover as the above rover) obtained from the pseudorange single-point positioning from real values of the rover coordinates, respectively.

As can be seen from the formula (2.2),

the magnitude of the change over time is only affected by the amount of satellite coordinate change. At adjacent time t_k、t_k+△kThe variation of the rover gauge to the ground is as follows:

$\mathrm{\Δ}\widehat{\mathrm{\ρ}}\left(\begin{array}{cc}{t}_k& t_{k+\mathrm{\Δk}}\end{array}\right)=\sqrt{\widehat{\mathrm{\ρ}}{\left(t_k\right)}^2+2\mathrm{\Δt}(V_X\·X+V_Y\·Y+V_Z\·Z)+{\mathrm{\Δ}}^2}-\widehat{\mathrm{\ρ}}\left(t_k\right)---\left(1.3\right)$

wherein, $\mathrm{\Δ}\widehat{\mathrm{\ρ}}\left(\begin{array}{cc}{t}_k& t_{k+\mathrm{\Δk}}\end{array}\right)$ is (t)_k,t_k+△k) The change of the sanitary distance of the flow station in the time period;

is t_kThe distance between the guards and the ground at the moment; Δ t = t_k+△k-t_k；V_X、V_Y、V_ZAre respectively satellite at (t)_k，t_k+△k) Average velocity along axis X, Y, Z over a period of time; x = Xⁱ-(X_r+△X)，Y=Yⁱ-(Y_r+△Y)，Z=Zⁱ-(Z_r+△Z),△²=△X²+△Y²+△Z²。

According to analysis, the variation of the rover satellite distance is related to the speed of the satellite along the coordinate axis direction and the offset of the rover coordinate. Since the velocity of the satellite in the coordinate axis direction is almost constant and the rover coordinate offset is constant in a short time, it can be considered that the rover satellite distance variation is approximately equal in a short time, that is, the rover satellite distance variation is approximately equal in a short time $\mathrm{\Δ}\widehat{\mathrm{\ρ}}\left(\begin{array}{cc}{t}_{k-\mathrm{\Δk}}& t_k\end{array}\right)\≈\mathrm{\Δ}\widehat{\mathrm{\ρ}}\left(\begin{array}{cc}{t}_k& t_{k+\mathrm{\Δk}}\end{array}\right).$ Shadow of rover coordinate offset on double-difference carrier ionosphere-free combined calculated valueThe responses are additive in time. Therefore, in the case of no gross error of the L1 and L2 carriers, the ionosphere-free combination value of the adjacent epoch double-difference carrier is approximately equal to the difference value of the calculation value of the observation equation, and the obvious linear characteristic is presented, that is: $\mathrm{\Δ}\▿P(j+1)-\mathrm{\Δ}\▿P\left(j\right)\≈\mathrm{\Δ}\▿P\left(j\right)-\mathrm{\Δ}\▿P(j-1)$ （14）

wherein,

and the difference value between the ionization layer-free combination value of the j +1 th epoch double-difference carrier and the calculation value of the observation equation is obtained.

Using the first two known epochs

And interpolating the value to obtain the estimated value of the current epoch:

$\mathrm{\Δ}\▿\tilde{P}(j+1)\≈2\·\mathrm{\Δ}\▿P\left(j\right)-\mathrm{\Δ}\▿P(j-1)---\left(1.5\right)$

wherein,is the j +1 th epoch

An estimate of (2);

and

j and j-1 epochs respectivelyTrue value of (1). If the (j + 1) th epoch is in gross error, then use

Andinterpolating an estimate of the j +2 th epoch

And so on.

(3) Judging whether the current epoch carrier observed value contains gross error:

due to the fact that

From the formula (1.1), when L is₁、L₂When the coarse difference appears in the carrier wave, the coarse difference is respectively reduced by 0.562 times and 0.438 times to be reflected inIn (1). In order to detect a gross error of 0.1 week or more, the threshold is set to

When in use $|\mathrm{\&Delta;}\&dtri;\tilde{P}(j+1)-\mathrm{\&Delta;}\&dtri;P(j+1)|<0.04$ If so, the observed value is considered to contain no gross error, otherwise, the observed value is considered to contain gross error.

The pseudorange single-point positioning equation is constructed to obtain the approximate coordinates of the P346 station, and the tropospheric delay values of the CMBB station and the P346 station are obtained by using the UNB3m model. Constructing a double-difference ionosphere-free carrier combination value (formula 1.1), and calculating the difference value between the double-difference ionosphere-free carrier combination value and the calculation value of an observation equation

Using the first two epochs

Andinterpolation is carried out to obtain the estimated value of the difference value between the current epoch double-difference carrier ionosphere-free combination value and the observation equation calculated value

(equation 1.5), setting a threshold valueAnd judging whether the current epoch carrier observed value contains a gross error.

To prove that the gross error detection method based on the observation constant has good effect on detecting single gross error and continuous gross error, the experiment is divided into two parts:

experiment one: coarse differences of 0.1, 0.2 and 0.3 weeks are added to the 11 th epoch, the 21 st epoch and the 31 st epoch respectively at the P346 station, and the threshold value and the true value of the difference value between the double-difference carrier ionosphere-free combination value and the calculated value of the observation equation of the corresponding epoch are calculated as shown in fig. 1, and fig. 2 shows the case that the true value of the difference value between the double-difference carrier ionosphere-free combination value and the calculated value of the observation equation exceeds the upper limit (lower limit) of the threshold value.

Experiment two: the cycle gross errors of-0.1, -0.2 and-0.3 are added into 10 th to 12 th, 20 th to 22 th and 30 th to 32 th epochs of the P346 station respectively, the threshold value and the true value of the difference value between the double-difference carrier ionosphere-free combination value and the calculation value of the observation equation of the corresponding epoch are calculated as shown in figure 3, and figure 4 shows the condition that the true value of the difference value between the double-difference carrier ionosphere-free combination value and the calculation value of the observation equation exceeds the upper limit (lower limit) of the threshold value.

Through the first experiment and the second experiment, the gross error detection method based on the observation constant has good effect on detecting single gross error and continuous gross error.

The above description is only of the preferred embodiments of the present invention, and it should be noted that: it will be apparent to those skilled in the art that several contemplated modifications and adaptations can be made without departing from the principles of the invention and these are intended to be included within the scope of the invention.

Claims (3)

1. A method for detecting gross error based on observation constant in global satellite navigation system GNSS baseline solution is characterized in that:

(1) pseudo-range point positioning is carried out to obtain the approximate coordinates of the rover station;

(2) correcting the convection layer by using a UNB3m model;

(3) according to the approximate coordinates of the rover station and the troposphere delay, and neglecting the orbit error and the multipath effect, obtaining a calculated value of the double-difference carrier ionosphere-free combination;

(4) obtaining a double-difference carrier ionosphere-free combination value according to L1 and L2 carrier observation values, wherein L1 and L2 are satellite carrier signals with two different frequencies;

(5) the difference value of the combined value without the ionized layer of the double-difference carrier and the calculated value of the observation equation is obtainedUsing the first two epochs

Linear interpolation of the values to obtain the current epoch

Threshold value of

(6) Will be the current epoch

Value and threshold

And comparing to judge whether the gross error is contained.

2. The method of claim 1, wherein the equation for the double-difference carrier ionosphere-free combined value is:

（1.1）

wherein,a double difference operator; f. of₁And f₂Are respectively L₁、L₂Carrier frequency, wherein L1 and L2 are satellite carrier signals of two different frequencies;

and

are respectively L₁、L₂A carrier phase observation; lambda [ alpha ]_WIs the widelane ambiguity; rho is a satellite-earth distance, and the precision in pseudo-range single-point positioning is meter level; o, M are orbit errors and multipath effects, respectively, which can be ignored in the double difference equation; t is tropospheric delay, corrected with the new university of unreal UNB3m tropospheric model; n is a radical of₁And N₂Are respectively L₁、L₂Carrier ambiguity, in equation (1.1), when all parameters are accurate values, the equation holds;

in the formula (1.1), the left side of the equal sign is a double-difference carrier non-ionized layer combination value, and the right side of the equal sign is a double-difference carrier non-ionized layer combination calculation value because L₁、L₂The precision of the carrier observed value is 0.01 week, so the left side of the equal sign can be regarded as the precise value, and the double-difference distance between the earth and the ground is received

Double differential troposphere

The influence of parameter precision, equal sign right and left formula side double difference carrier do not have the ionosphere composite value and have certain deviation.

3. The GNSS baseline solution of claim 1, wherein the difference between the ionosphere-free combination of the double difference carriers and the calculated value of the observation equation is calculated by

Using the first two epochs

Linear interpolation of the values to obtain the current epoch

Threshold value of

Comprises the following steps:

the coordinates of the reference station in the baseline solution are accurately known, so the precision of the double-difference range mainly depends on the rover range, and the rover range can be obtained by the satellite coordinates and the rover pseudorange single-point positioning coordinates:

$\widehat{\mathrm{\&rho;}}=\sqrt{{\{X^i-(X_r+\mathrm{\&Delta;X})\}}^2+{\{Y^i-(Y_r+\mathrm{\&Delta;Y})\}}^2+{\{Z^i-(Z_r+\mathrm{\&Delta;Z})\}}^2}---\left(1.2\right)$

wherein,calculating the rover guard distance; xⁱ、Yⁱ、ZⁱRespectively the coordinates of the ith satellite; x_r、Y_r、Z_rRespectively are r coordinate true values of the rover station; DeltaX, DeltaY and DeltaZ are respectively the deviation of the coordinate of the rover r obtained by pseudo-range single-point positioning compared with the true value of the coordinate of the rover,

as can be seen from the formula (1.2),

the magnitude of the change over time is only affected by the amount of satellite coordinate change. At adjacent time t_k、t_k+△kThe variation of the rover gauge to the ground is as follows:

$\mathrm{\&Delta;}\widehat{\mathrm{\&rho;}}\left(\begin{array}{cc}{t}_k& t_{k+\mathrm{\&Delta;k}}\end{array}\right)=\sqrt{\widehat{\mathrm{\&rho;}}{\left(t_k\right)}^2+2\mathrm{\&Delta;t}(V_X\&CenterDot;X+V_Y\&CenterDot;Y+V_Z\&CenterDot;Z)+{\mathrm{\&Delta;}}^2}-\widehat{\mathrm{\&rho;}}\left(t_k\right)---\left(1.3\right)$

wherein, $\mathrm{\&Delta;}\widehat{\mathrm{\&rho;}}\left(\begin{array}{cc}{t}_k& t_{k+\mathrm{\&Delta;k}}\end{array}\right)$ is (t)_k,t_k+△k) The change of the sanitary distance of the flow station in the time period;

is t_kThe distance between the guards and the ground at the moment; Δ t = t_k+△k-t_k；V_X、V_Y、V_ZAre respectively satellite at (t)_k，t_k+△k) Average velocity along axis X, Y, Z over a period of time; x = Xⁱ-(X_r+△X)，Y=Yⁱ-(Y_r+△Y)，Z=Zⁱ-(Z_r+△Z)，△²=△X²+△Y²+△Z²；

As can be seen from the analysis, the rover satellite distance variation is related to the speed of the satellite in the coordinate axis direction and the rover coordinate offset, and since the speed of the satellite in the coordinate axis direction is almost constant and the rover coordinate offset is constant in a short time, the rover distance variation is considered to be approximately equal in a short time, that is, the rover distance variation is considered to be approximately equal in a short time $\mathrm{\&Delta;}\widehat{\mathrm{\&rho;}}\left(\begin{array}{cc}{t}_{k-\mathrm{\&Delta;k}}& t_k\end{array}\right)\&ap;\mathrm{\&Delta;}\widehat{\mathrm{\&rho;}}\left(\begin{array}{cc}{t}_k& t_{k+\mathrm{\&Delta;k}}\end{array}\right),$ The influence of the rover coordinate offset on the double-difference carrier ionosphere-free combination calculated value is accumulative in time, so that under the condition that the L1 and the L2 carriers have no gross error, the difference value of the adjacent epoch double-difference carrier ionosphere-free combination value and the observation equation calculated value is approximately equal, and the obvious linear characteristic is presented, namely:

$\mathrm{\&Delta;}\&dtri;P(j+1)-\mathrm{\&Delta;}\&dtri;P\left(j\right)\&ap;\mathrm{\&Delta;}\&dtri;P\left(j\right)-\mathrm{\&Delta;}\&dtri;P(j-1)---\left(1.4\right)$

wherein,

the difference value between the combined value of the j +1 th epoch double-difference carrier wave without the ionized layer and the calculation value of the observation equation is obtained;

using the first two known epochs

And interpolating the value to obtain the estimated value of the current epoch:

$\mathrm{\&Delta;}\&dtri;\tilde{P}(j+1)\&ap;2\&CenterDot;\mathrm{\&Delta;}\&dtri;P\left(j\right)-\mathrm{\&Delta;}\&dtri;P(j-1)---\left(1.5\right)$

wherein,

is the j +1 th epoch

An estimate of (2);

andj and j-1 epochs respectivelyTrue value of (1). If the (j + 1) th epoch is in gross error, then useAnd

interpolating an estimate of the j +2 th epoch

And so on.

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