Method and system for single-station carrier epoch difference real-time probing relative deformation

Abstract

The present application relates to the field of GNSS satellite navigation and positioning, and particularly relates to a method and system for single-station carrier epoch-difference real-time probing of relative deformation. The method comprises: obtaining an epoch-difference observation equation of a preset satellite based on an original carrier phase observation equation of the preset satellite; selecting a reference satellite to perform inter-satellite difference on the epoch-difference observation equation of the preset satellite to obtain a double-difference observation equation; judging whether a carrier phase observation value of a later epoch has a cycle slip; using a lambda algorithm to solve double-difference ambiguity, and substituting the double-difference ambiguity into the double-difference observation equation to solve the double-difference observation equation by using a least square method to obtain three-dimensional deformation variables of a receiver before and after an epoch. The present application can quickly solve a displacement vector under the condition of stable tracking without relying on a base station by using carrier phase double difference before and after an epoch of a receiver, and can meet the demand of precision and timeliness of disaster monitoring.

Description

Technical Field

[0001] This invention relates to the field of GNSS satellite navigation and positioning, specifically to a method and system for real-time differential detection of relative deformation between single-station carrier epochs. Background Technology

[0002] In geological disaster monitoring applications, GNSS positioning technology can capture the movement speed of corresponding vehicles, greatly assisting in geological disaster monitoring and early warning. GNSS positioning methods can be divided into differential positioning and point positioning based on operational methods and cost control. Differential positioning has two limitations: firstly, a base station is required; without a base station, differential technology cannot be implemented. For areas with a large service area, the density of base stations cannot be too low, otherwise the accuracy of differential positioning will be limited. Secondly, there is a certain distance limit between the rover and the base station. When the rover and the base station are far apart, the correlation of their common errors weakens, resulting in insufficient positioning accuracy, making it difficult to apply in areas with complex terrain, and also increasing costs by requiring the construction of a base station. Compared to differential positioning, GNSS point positioning does not require a base station; a single receiver can obtain the positioning result. However, point positioning generally only uses pseudorange observations of the current epoch, resulting in meter-level positioning accuracy. While precise point positioning technology can achieve centimeter-level positioning, it requires more external corrections and has poor real-time performance. Summary of the Invention

[0003] The technical problem to be solved by the present invention is to provide a method and system for real-time differential detection of relative deformation between single-station carrier epochs. This can solve the shortcomings of current geological disaster monitoring, which requires the construction of a reference station and long-term smoothing to achieve millimeter-level accuracy, and meet the requirements of geological disaster monitoring for accuracy and timeliness.

[0004] The technical solution of this invention to solve the above-mentioned technical problems is as follows: A method for real-time differential detection of relative deformation between single-station carrier epochs, comprising the following steps,

[0005] S1, using a receiver to obtain carrier phase observations and pseudorange observations for two adjacent epochs before and after from a preset satellite, as well as single-point solution coordinates for the next epoch;

[0006] S2, use the broadcast ephemeris to find the preset satellite position, and combine it with the single-point solution coordinates of the later epoch to calculate the true distance between the receiver and the preset satellite;

[0007] S3. Based on the original carrier phase observation equation of the preset satellite, the epoch difference is obtained by performing epoch difference based on the carrier phase observation value and pseudorange observation value of the two adjacent epochs and the real distance between the receiver and the preset satellite.

[0008] S4. Select a reference satellite and perform inter-satellite difference on the epoch difference observation equation of the preset satellite to obtain the double difference observation equation.

[0009] S5, determine whether a cycle slip has occurred in the carrier phase observation value of the later epoch; if yes, execute S6-S7; if no, execute S7.

[0010] S6. The double difference ambiguity is obtained by using the l ambda algorithm, and the double difference ambiguity is substituted into the double difference observation equation;

[0011] S7. The least squares method is used to solve the double-difference observation equation to obtain the three-dimensional deformation of the receiver before and after the epoch.

[0012] Based on the above-mentioned method for real-time differential detection of relative deformation between single-station carrier epochs, the present invention also provides a system for real-time differential detection of relative deformation between single-station carrier epochs.

[0013] A system for real-time differential detection of relative deformation between single-station carrier epochs includes the following modules:

[0014] The data acquisition module is used to acquire carrier phase observations and pseudorange observations of two adjacent epochs from a preset satellite using a receiver, as well as to acquire the single-point solution coordinates of the subsequent epoch.

[0015] The distance calculation module is used to determine the position of the preset satellite using the broadcast ephemeris and to calculate the actual distance between the receiver and the preset satellite by combining the single-point calculation coordinates of the later epochs.

[0016] The epoch difference module is used to perform epoch difference based on the original carrier phase observation equation of the preset satellite, according to the carrier phase observation value and pseudorange observation value of two adjacent epochs and the real distance between the receiver and the preset satellite, to obtain the epoch difference observation equation of the preset satellite.

[0017] The inter-satellite difference module is used to select a reference satellite and perform inter-satellite difference on the epoch difference observation equation of the preset satellite to obtain the double difference observation equation.

[0018] The cycle slip detection module is used to determine whether a cycle slip occurs in the carrier phase observation value of a later epoch;

[0019] The double-difference ambiguity resolution module is used to calculate the double-difference ambiguity using the lambda algorithm when a cycle slip occurs in the carrier phase observation value in the later epoch, and substitute the double-difference ambiguity into the double-difference observation equation;

[0020] The three-dimensional deformation calculation module is used to solve the double-difference observation equation using the least squares method to obtain the three-dimensional deformation of the receiver before and after the epoch.

[0021] The beneficial effects of this invention are as follows: In the method and system for real-time detection of relative deformation by differential measurement between carrier epochs of this invention, subtracting the carrier observations of adjacent epochs can eliminate ephemeris errors, tropospheric delay errors, and ionospheric delay errors. When cycle slips do not occur, it can also eliminate carrier phase integer ambiguity. Based on the differential measurement of consecutive epochs of a single satellite, selecting a reference satellite and performing inter-satellite differential measurement on the epoch difference results can eliminate the influence of receiver clock bias and reduce the number of unknowns. Only a three-dimensional vector needs to be solved, resulting in higher solution accuracy. This invention, through double differential measurement of carrier phase between consecutive epochs of the receiver, can quickly solve for the displacement vector without relying on a base station and under stable tracking conditions, meeting the accuracy and timeliness requirements of geological disaster monitoring. Furthermore, it does not rely on external reference station information, greatly saving costs. Attached Figure Description

[0022] Figure 1 This is a flowchart of a method for real-time differential detection of relative deformation between single-station carrier epochs according to the present invention;

[0023] Figure 2 This is a structural block diagram of a system for real-time differential detection of relative deformation between single-station carrier epochs according to the present invention. Detailed Implementation

[0024] The principles and features of the present invention are described below with reference to the accompanying drawings. The examples given are only for explaining the present invention and are not intended to limit the scope of the present invention.

[0025] like Figure 1 As shown, a method for real-time differential detection of relative deformation between single-station carrier epochs includes the following steps:

[0026] S1, using a receiver to obtain carrier phase observations and pseudorange observations for two adjacent epochs before and after from a preset satellite, as well as single-point solution coordinates for the next epoch;

[0027] S2, use the broadcast ephemeris to find the preset satellite position, and combine it with the single-point solution coordinates of the later epoch to calculate the true distance between the receiver and the preset satellite;

[0028] S3. Based on the original carrier phase observation equation of the preset satellite, the epoch difference is obtained by performing epoch difference based on the carrier phase observation value and pseudorange observation value of the two adjacent epochs and the real distance between the receiver and the preset satellite.

[0029] S4. Select a reference satellite and perform inter-satellite difference on the epoch difference observation equation of the preset satellite to obtain the double difference observation equation.

[0030] S5, determine whether a cycle slip has occurred in the carrier phase observation value of the later epoch; if yes, execute S6-S7; if no, execute S7.

[0031] S6. The double difference ambiguity is obtained by using the l ambda algorithm, and the double difference ambiguity is substituted into the double difference observation equation;

[0032] S7. The least squares method is used to solve the double-difference observation equation to obtain the three-dimensional deformation of the receiver before and after the epoch.

[0033] In the method for real-time differential detection of relative deformation between single-station carrier epochs of the present invention:

[0034] The original carrier phase observation equation for the preset satellite is:

[0035]

[0036] in, For the initial carrier wave observations of the pre-defined satellite, λ i For carrier wavelength, The carrier phase observation values ​​of the preset satellite, This represents the actual distance between the receiver and the preset satellite. To preset the integer ambiguity of the satellite, Let dt be the potential cycle slip of the pre-defined satellite, c be the speed of light in a vacuum, and dt be the potential cycle slip of the satellite. r For receiver clock bias, dt S To preset satellite clock bias, To preset the ionospheric delay of the satellite, To preset the tropospheric delay of the satellite, To pre-set the multipath error of the satellite, This represents the carrier phase observation noise of the preset satellite. Specifically, in the equation, the superscript S represents the preset satellite, the subscript r represents the receiver, and the subscript i represents the observation number.

[0037] In 1Hz solution processing, subtracting carrier observations from adjacent epochs can eliminate ephemeris errors, tropospheric delay errors, and ionospheric delay errors. In the absence of cycle slips, it can also eliminate carrier phase integer ambiguity. In 1Hz solution processing, the environment does not change significantly between adjacent times; therefore, multipath errors between adjacent epochs can be approximated as constant and eliminated by subtraction.

[0038] The epoch difference observation equation for the preset satellite is as follows:

[0039]

[0040] in, For the epoch difference observations of the preset satellite, To predetermine the original carrier observations of the satellite at epoch k, To pre-determine the original carrier observations of the satellite at epoch k-1, This represents the actual distance between the receiver and the preset satellite at epoch k. This represents the actual distance between the receiver and the preset satellite at the (k-1)th epoch. The receiver clock bias rate of change, To preset the satellite clock bias change rate, To preset the carrier phase observation noise of the satellite at epoch k, The carrier phase observation noise is preset for the satellite at epoch k-1.

[0041] In the epoch difference observation equation of the preset satellite,

[0042]

[0043]

[0044] in, Let be the unit vector between the receiver and the preset satellite at the k-th epoch. Let r be the unit vector between the receiver and the preset satellite at the (k-1)th epoch. S (k) represents the preset satellite position at the k-th epoch, r r (k) represents the receiver's position at the k-th epoch, r S (k-1) represents the preset satellite position at the (k-1)th epoch, r r (k-1) represents the receiver's position at the (k-1)th epoch;

[0045] The epoch difference observation equation of the preset satellite then simplifies to:

[0046]

[0047] in,

[0048]

[0049]

[0050] Specifically, This represents the Doppler effect caused by the pre-defined satellite motion. Δr represents the change in the geometric relationship between the preset satellite and the receiver, and Δr is the three-dimensional vector of the receiver before and after the epoch.

[0051] It is not hard to see that Noise residues belonging to the observed quantities are classified as residuals in the solution and are not considered unknowns. The above formula only contains four unknowns: the three-dimensional vectors at different epochs from the receiver and the clock bias rate of change. In this case, the three-dimensional vectors and clock bias rate of change can be obtained using least squares. Since most GNSS receivers use quartz clocks with relatively low stability, the receiver clock bias drifts over time, leading to a decrease in parameter estimation accuracy and consequently reducing the accuracy of the three-dimensional vector solution. By selecting a reference satellite and performing inter-satellite differencing on the epoch difference results from the single-satellite differencing, the influence of the receiver clock bias can be eliminated, and the number of unknowns can be reduced. Only the three-dimensional vector needs to be solved, resulting in higher solution accuracy.

[0052] The epoch difference observation equation for the reference satellite is as follows:

[0053]

[0054] in, For the epoch difference observations of the reference satellite, The Doppler effect represents the motion of the reference satellite. This represents the change in the geometric relationship between the reference satellite and the receiver. Let be the unit vector between the receiver and the reference satellite at epoch k. To reference the carrier phase observation noise of the satellite at epoch k, The carrier phase observation noise of the reference satellite at epoch k-1;

[0055] The double-difference observation equation is then given by:

[0056]

[0057] in, For double-difference observations, The difference in Doppler effect between the preset satellite and the reference satellite and The difference between the geometrical relationships between the preset satellite and the reference satellite and the receiver, and Specifically, the superscript Q in the equation represents the reference satellite.

[0058] The above discussion assumes stable carrier tracking and no cycle slips. When a cycle slip occurs in the observation, the unknowns also include double-difference ambiguity, which can be solved using the Lambda algorithm. Therefore, in real-time applications, cycle slip detection of the carrier observation is necessary every time a differential calculation is performed.

[0059] Based on the above-mentioned method for real-time differential detection of relative deformation between single-station carrier epochs, the present invention also provides a system for real-time differential detection of relative deformation between single-station carrier epochs.

[0060] like Figure 2 As shown, a system for real-time differential detection of relative deformation between single-station carrier epochs includes the following modules:

[0061] The data acquisition module is used to acquire carrier phase observations and pseudorange observations of two adjacent epochs from a preset satellite using a receiver, as well as to acquire the single-point solution coordinates of the subsequent epoch.

[0062] The distance calculation module is used to determine the position of the preset satellite using the broadcast ephemeris and to calculate the actual distance between the receiver and the preset satellite by combining the single-point calculation coordinates of the later epochs.

[0063] The epoch difference module is used to perform epoch difference based on the original carrier phase observation equation of the preset satellite, according to the carrier phase observation value and pseudorange observation value of two adjacent epochs and the real distance between the receiver and the preset satellite, to obtain the epoch difference observation equation of the preset satellite.

[0064] The inter-satellite difference module is used to select a reference satellite and perform inter-satellite difference on the epoch difference observation equation of the preset satellite to obtain the double difference observation equation.

[0065] The cycle slip detection module is used to determine whether a cycle slip occurs in the carrier phase observation value of a later epoch;

[0066] The double-difference ambiguity resolution module is used to calculate the double-difference ambiguity using the lambda algorithm when a cycle slip occurs in the carrier phase observation value in the later epoch, and substitute the double-difference ambiguity into the double-difference observation equation;

[0067] The three-dimensional deformation calculation module is used to solve the double-difference observation equation using the least squares method to obtain the three-dimensional deformation of the receiver before and after the epoch.

[0068] In a system for real-time differential detection of relative deformation between single-station carrier epochs according to the present invention:

[0069] The original carrier phase observation equation for the preset satellite is:

[0070]

[0071] in, For the initial carrier wave observations of the pre-defined satellite, λ i For carrier wavelength, The carrier phase observation values ​​of the preset satellite, This represents the actual distance between the receiver and the preset satellite. To preset the integer ambiguity of the satellite, Let dt be the potential cycle slip of the pre-defined satellite, c be the speed of light in a vacuum, and dt be the potential cycle slip of the satellite.r For receiver clock bias, dt S To preset satellite clock bias, To preset the ionospheric delay of the satellite, To preset the tropospheric delay of the satellite, To pre-set the multipath error of the satellite, The carrier phase observation noise of the preset satellite is used.

[0072] The epoch difference observation equation for the preset satellite is as follows:

[0073]

[0074] in, For the epoch difference observations of the preset satellite, To predetermine the original carrier observations of the satellite at epoch k, To pre-determine the original carrier observations of the satellite at epoch k-1, This represents the actual distance between the receiver and the preset satellite at epoch k. This represents the actual distance between the receiver and the preset satellite at the (k-1)th epoch. The receiver clock bias rate of change, To preset the satellite clock bias change rate, To preset the carrier phase observation noise of the satellite at epoch k, The carrier phase observation noise is preset for the satellite at epoch k-1.

[0075] In the epoch difference observation equation of the preset satellite,

[0076]

[0077]

[0078] in, Let be the unit vector between the receiver and the preset satellite at the k-th epoch. Let r be the unit vector between the receiver and the preset satellite at the (k-1)th epoch. S (k) represents the preset satellite position at the k-th epoch, r r (k) represents the receiver's position at the k-th epoch, r S (k-1) represents the preset satellite position at the (k-1)th epoch, r r (k-1) represents the receiver's position at the (k-1)th epoch;

[0079] The epoch difference observation equation of the preset satellite then simplifies to:

[0080]

[0081] in,

[0082]

[0083]

[0084] Specifically, This represents the Doppler effect caused by the pre-defined satellite motion. Δr represents the change in the geometric relationship between the preset satellite and the receiver, and Δr is the three-dimensional vector of the receiver before and after the epoch.

[0085] The epoch difference observation equation for the reference satellite is as follows:

[0086]

[0087] in, For the epoch difference observations of the reference satellite, The Doppler effect represents the motion of the reference satellite. This represents the change in the geometric relationship between the reference satellite and the receiver. Let be the unit vector between the receiver and the reference satellite at epoch k. To reference the carrier phase observation noise of the satellite at epoch k, The carrier phase observation noise of the reference satellite at epoch k-1;

[0088] The double-difference observation equation is then given by:

[0089]

[0090] in, For double-difference observations, The difference in Doppler effect between the preset satellite and the reference satellite and The difference between the geometrical relationships between the preset satellite and the reference satellite and the receiver, and

[0091] In the method and system for real-time detection of relative deformation by differential measurement between carrier epochs of this invention, subtracting the carrier observations of adjacent epochs can eliminate ephemeris errors, tropospheric delay errors, and ionospheric delay errors. When cycle slips do not occur, it can also eliminate carrier phase integer ambiguity. Based on the differential measurement of consecutive epochs of a single satellite, selecting a reference satellite and performing inter-satellite differential measurement on the epoch difference results can eliminate the influence of receiver clock bias and reduce the number of unknowns. Only a three-dimensional vector needs to be solved, resulting in higher solution accuracy. This invention, through double differential measurement of carrier phase between consecutive epochs of the receiver, can quickly solve for the displacement vector without relying on a base station and under stable tracking conditions, meeting the accuracy and timeliness requirements of geological disaster monitoring. Furthermore, it does not rely on external reference station information, significantly saving costs.

[0092] The above description is only a preferred embodiment of the present invention and is not intended to limit the present invention. Any modifications, equivalent substitutions, improvements, etc., made within the spirit and principles of the present invention should be included within the protection scope of the present invention.

Claims

1. A method of single-station carrier epoch-difference differential real-time probing of relative deformation, characterized by: Includes the following steps, S1, using a receiver to obtain carrier phase observations and pseudorange observations for two adjacent epochs before and after from a preset satellite, as well as single-point solution coordinates for the next epoch; S2, use the broadcast ephemeris to find the preset satellite position, and combine it with the single-point solution coordinates of the later epoch to calculate the true distance between the receiver and the preset satellite; S3. Based on the original carrier phase observation equation of the preset satellite, the epoch difference is obtained by performing epoch difference based on the carrier phase observation value and pseudorange observation value of the two adjacent epochs and the real distance between the receiver and the preset satellite. S4. Select a reference satellite and perform inter-satellite difference on the epoch difference observation equation of the preset satellite to obtain the double difference observation equation. S5, determine whether a cycle slip has occurred in the carrier phase observation value of the later epoch; if yes, execute S6-S7; if no, execute S7. S6. Use the lambda algorithm to find the double-difference ambiguity and substitute the double-difference ambiguity into the double-difference observation equation; S7. The least squares method is used to solve the double-difference observation equation to obtain the three-dimensional deformation of the receiver before and after the epoch. The original carrier phase observation equation for the preset satellite is: ； in, For the initial carrier wave observations of the preset satellite, For carrier wavelength, The carrier phase observation values ​​of the preset satellite, This represents the actual distance between the receiver and the preset satellite. To preset the integer ambiguity of the satellite, To pre-set the potential cycle slip of the satellite, The speed of light in a vacuum. For receiver clock bias, To preset satellite clock bias, To preset the ionospheric delay of the satellite, To preset the tropospheric delay of the satellite, To pre-set the multipath error of the satellite, Preset the carrier phase observation noise for the satellite; The epoch difference observation equation for the preset satellite is as follows: ； in, For the epoch difference observations of the preset satellite, For the preset satellite number The original carrier observations of the epoch, For the preset satellite number The original carrier observations of the epoch, For the first The actual distance between the receiver and the preset satellite at the epoch. For the first The actual distance between the receiver and the preset satellite at the epoch. The receiver clock bias rate of change, To preset the satellite clock bias change rate, For the preset satellite number Carrier phase observation noise at epoch, For the preset satellite number Carrier phase observation noise at epoch; In the epoch difference observation equation of the preset satellite, ， ； in, For the first The unit vector between the receiver and the preset satellite at the epoch. For the first The unit vector between the receiver and the preset satellite at the epoch. For the first The position of the satellite is preset at the epoch. For the first The position of the receiver at the epoch. For the first The position of the satellite is preset at the epoch. For the first The receiver's position at the epoch; The epoch difference observation equation of the preset satellite then simplifies to: ； in, ； ； Specifically, This represents the Doppler effect caused by the pre-defined satellite motion. This represents the change in the geometric relationship between the preset satellite and the receiver. This is a three-dimensional vector representing the epochs before and after the receiver.

2. The method for real-time differential detection of relative deformation between single-station carrier epochs according to claim 1, characterized in that: The epoch difference observation equation for the reference satellite is as follows: ； in, For the epoch difference observations of the reference satellite, The Doppler effect represents the motion of the reference satellite. This represents the change in the geometric relationship between the reference satellite and the receiver. For the first The unit vector between the receiver and the reference satellite at the epoch. For reference satellite number Carrier phase observation noise at epoch, For reference satellite number Carrier phase observation noise at epoch; The double-difference observation equation is then given by: ； in, For double-difference observations, The difference in Doppler effect between the preset satellite and the reference satellite and , The difference between the geometrical relationships between the preset satellite and the reference satellite and the receiver, and .

3. A system for real-time differential detection of relative deformation between single-station carrier epochs, characterized in that: Includes the following modules, The data acquisition module is used to acquire carrier phase observations and pseudorange observations of two adjacent epochs from a preset satellite using a receiver, as well as to acquire the single-point solution coordinates of the subsequent epoch. The distance calculation module is used to determine the position of the preset satellite using the broadcast ephemeris and to calculate the actual distance between the receiver and the preset satellite by combining the single-point calculation coordinates of the later epochs. The epoch difference module is used to perform epoch difference based on the original carrier phase observation equation of the preset satellite, according to the carrier phase observation value and pseudorange observation value of two adjacent epochs and the real distance between the receiver and the preset satellite, to obtain the epoch difference observation equation of the preset satellite. The inter-satellite difference module is used to select a reference satellite and perform inter-satellite difference on the epoch difference observation equation of the preset satellite to obtain the double difference observation equation. The cycle slip detection module is used to determine whether a cycle slip occurs in the carrier phase observation value of a later epoch; The double-difference ambiguity resolution module is used to calculate the double-difference ambiguity using the lambda algorithm when a cycle slip occurs in the carrier phase observation value in the later epoch, and substitute the double-difference ambiguity into the double-difference observation equation; The three-dimensional deformation calculation module is used to solve the double-difference observation equation using the least squares method to obtain the three-dimensional deformation of the receiver before and after the epoch. The original carrier phase observation equation for the preset satellite is: ； in, For the initial carrier wave observations of the preset satellite, For carrier wavelength, The carrier phase observation values ​​of the preset satellite, This represents the actual distance between the receiver and the preset satellite. To preset the integer ambiguity of the satellite, To pre-set the potential cycle slip of the satellite, The speed of light in a vacuum. For receiver clock bias, To preset satellite clock bias, To preset the ionospheric delay of the satellite, To preset the tropospheric delay of the satellite, To pre-set the multipath error of the satellite, Preset the carrier phase observation noise for the satellite; The epoch difference observation equation for the preset satellite is as follows: ； in, For the epoch difference observations of the preset satellite, For the preset satellite number The original carrier observations of the epoch, For the preset satellite number The original carrier observations of the epoch, For the first The actual distance between the receiver and the preset satellite at the epoch. For the first The actual distance between the receiver and the preset satellite at the epoch. The receiver clock bias rate of change, To preset the satellite clock bias change rate, For the preset satellite number Carrier phase observation noise at epoch, For the preset satellite number Carrier phase observation noise at epoch; In the epoch difference observation equation of the preset satellite, ， ； in, For the first The unit vector between the receiver and the preset satellite at the epoch. For the first The unit vector between the receiver and the preset satellite at the epoch. For the first The position of the satellite is preset at the epoch. For the first The position of the receiver at the epoch. For the first The position of the satellite is preset at the epoch. For the first The receiver's position at the epoch; The epoch difference observation equation of the preset satellite then simplifies to: ； in, ； ； Specifically, This represents the Doppler effect caused by the pre-defined satellite motion. This represents the change in the geometric relationship between the preset satellite and the receiver. This is a three-dimensional vector representing the epochs before and after the receiver.

4. The system for real-time differential detection of relative deformation between single-station carrier epochs according to claim 3, characterized in that: The epoch difference observation equation for the reference satellite is as follows: ； in, For the epoch difference observations of the reference satellite, The Doppler effect represents the motion of the reference satellite. This represents the change in the geometric relationship between the reference satellite and the receiver. For the first The unit vector between the receiver and the reference satellite at the epoch. For reference satellite number Carrier phase observation noise at epoch, For reference satellite number Carrier phase observation noise at epoch; The double-difference observation equation is then given by: ； in, For double-difference observations, The difference in Doppler effect between the preset satellite and the reference satellite and , The difference between the geometrical relationships between the preset satellite and the reference satellite and the receiver, and .

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