PAR Dilution of Precision Determination Method for Single-Epoch GNSS

Abstract

The present invention discloses a method for determining the PAR dilution of precision for single-epoch GNSS, which relates to the technical field of satellite navigation and positioning. It includes: dividing the pseudorange and carrier phase observations in the GNSS double-difference mathematical model into two groups, and constructing the functional relationship between the float solution #imgabs0# of the ambiguity vector and the variance-covariance matrix of the float solution #imgabs1# and the float solution #imgabs2#; constructing the single-frequency single-epoch single-system P-ADOP expression and establishing the analytical formula for the inverse transformation of the contraction factor k to obtain the approximate formula for the non-inverse transformation P-ADOP of single-frequency single-epoch single-system GNSS by simultaneous equations; and analogously deriving the above approximate formula for the non-inverse transformation P-ADOP of single-frequency single-epoch single-system GNSS to obtain the general approximate formula for the non-inverse transformation P-ADOP of single-frequency multi-system or multi-frequency single-epoch GNSS. The present invention has important theoretical value and practical guiding significance for the rapid selection of the optimal ambiguity subset of GNSS, positioning, and the analysis of the influence mechanism of the success rate of fixing some ambiguities.

Description

Technical Field

[0001] The present invention relates to the technical field of satellite navigation and positioning, and particularly to a method for determining the ambiguity dilution of precision factor for single-epoch GNSS. Background Art

[0002] The correct solution of the ambiguity is the key to high-precision relative positioning using the Global Navigation Satellite System (GNSS). The Ambiguity Dilution of Precision (ADOP) is an index factor for measuring the success rate of ambiguity fixing, and its approximate formula can accurately predict the ADOP value based on the observed quantities.

[0003] In terms of dimensions, ambiguity resolution can be divided into global ambiguity resolution and partial ambiguity resolution (PAR). Partial ambiguity resolution (PAR) is the optimal method for rapid GNSS positioning.

[0004] However, the current ADOP approximate formula is only applicable to global ambiguity resolution, and there is still a lack of an accurate estimation method for the ambiguity dilution of precision factor based on PAR (P-ADOP), that is, the existing formulas or methods cannot accurately predict or estimate the P-ADOP of the PAR ambiguity based on the observed quantities. Summary of the Invention

[0005] Based on this, it is necessary to provide a method for determining the ambiguity dilution of precision factor for single-epoch GNSS to address the above technical problems.

[0006] The embodiment of the present invention provides a method for determining the ambiguity dilution of precision factor for single-epoch GNSS, including:

[0007] Construct a single-epoch GNSS double-difference mathematical model, and divide the pseudorange observables and carrier observables in the model into two groups ( , ) and ( , ) to obtain the single-epoch GNSS double-difference mathematical model after splitting the observables, and the global ambiguity vector is correspondingly divided into two groups ;

[0008] Adopt the least squares method to construct from the observables , or from the observables , and The partial ambiguity vectors calculated by the single epoch GNSS double difference mathematical model after the observations are split Floating point solution The variance-covariance matrix of With the observed quantity and The partial ambiguity vectors calculated by the single epoch GNSS double difference mathematical model after the observations are split Floating point solution The variance-covariance matrix of The functional relationship between them; using the determinant reduction theorem and the partial fuzzy vector in the functional relationship Floating point solution The variance-covariance matrix of , construct the partial ambiguity vector Floating point solution The single-frequency single-epoch single-system P-ADOP expression;

[0009] Constructing the scaling factor when taking the first observed satellite as the reference satellite K middle The analytical formula for the inverse transformation; according to The de-inverse transform weighting coefficient matrix is determined by the de-inverse transform analytical formula, and the shrinkage factor representing the theoretical relationship between P-ADOP and ADOP is established. k The analytical expression of the inverse transformation of ;

[0010] Combined single-frequency single-epoch single-system P-ADOP expression and shrinkage factor k The de-inverse transform analytical expression is used to determine the PAR ambiguity precision factor P-ADOP.

[0011] Optionally, the single-epoch GNSS double-difference mathematical model after the observations are split is specifically:

[0012] ;

[0013] in, , " ” are , , B、 or ; , , and Respectively represent " The expectation and variance of is the pseudorange observation, is the carrier observation quantity, and are the baseline vector and the overall ambiguity vector, respectively, , is the carrier wavelength, is the -th order coefficient matrix and its rank is the cofactor matrix.

[0014] Optionally, construct the partial ambiguity vector float solution variance-covariance matrix and the partial ambiguity vector float solution variance-covariance matrix The specific functional relationship between them is:

[0015] ;

[0016] Wherein, and are respectively the partial ambiguity vectors , , , or the observables , , calculated from the observables float solution and the baseline vector b float solution variance-covariance matrix, is a positive definite or semi-positive definite matrix, , , is the variance-covariance matrix of the baseline vector and calculated only from b float solution , is the standard deviation of the undifferenced pseudorange observable;

[0017] If the partial ambiguity vector and the integer ambiguity vector correspond to t and carrier observables of frequencies respectively, then the double-difference cofactor matrix of the observables corresponding to the partial ambiguity vector is is the double-difference cofactor matrix of the observables corresponding to the -th j frequency ambiguity vector in the partial ambiguity vector

[0018] Optionally, construct the partial ambiguity vector float solution The single-frequency, single-epoch, single-system P-ADOP expression specifically includes:

[0019] According to the definition formula of the ambiguity dilution of precision ADOP, the single-frequency, single-epoch, single-system partial ambiguity vector float solution The P-ADOP formula is expressed as

[0020] ;

[0021] Among them, is the variance-covariance matrix of the partial ambiguity vector float solution calculated according to The partial ambiguity vector float solution ambiguity dilution of precision ADOP, , the scaling factor K is a positive number less than 1 and has the property of pseudorange precision; the single-frequency, single-epoch, single-system P-ADOP expression is used to characterize the relationship between the ADOP of the partial ambiguity vector float solution and the scaling factor K ;

[0022] The scaling factor K as the coefficient of the non-differenced pseudorange observation standard deviation , the contraction factor k is used to describe the non-differenced pseudorange observation standard deviation , , then the approximate expression of the P-ADOP of the single-frequency, single-epoch, single-system partial ambiguity vector float solution and the analytical formula of the contraction factor k are:

[0023] ;

[0024] Among them, k is a positive contraction factor, is the non-differenced carrier observation standard deviation, and are the coefficient matrix and cofactor matrix corresponding to the partial ambiguity vector , is order coefficient matrix and its rank , is the cofactor matrix, is the dimension of the partial ambiguity vector , is the number of observed satellites corresponding to the partial ambiguity vector .

[0025] Optionally, construct the de - inverse transformation analytical formula of the contraction factor k in , specifically including:

[0026] Let the double - difference coefficient matrix and the double - difference co - factor matrix when the first observation satellite is used as the reference satellite be and respectively, and the double - difference coefficient matrix and the double - difference co - factor matrix when the second observation satellite is used as the reference satellite be and respectively. The relationships between the double - difference coefficient matrices and between the double - difference co - factor matrices are specifically:

[0027] ;

[0028] wherein, is an invertible matrix with a determinant of 1, holds, has the property of being independent of the double - difference reference satellite;

[0029] Then the de - inverse transformation analytical formula of the contraction factor k in is specifically:

[0030] ;

[0031] wherein, and are the double - difference coefficient matrix and the double - difference co - factor matrix when the first observation satellite is used as the reference satellite respectively, is the weight of other observation satellites except the reference satellite, and the weight of the reference satellite is ;

[0032] According to the de - inverse transformation analytical formula, the de - inverse transformation analytical formula of the contraction factor k when the first observation satellite is used as the reference satellite is specifically:

[0033] ;

[0034] wherein, and are the de - inverse transformation weighted coefficient matrices for the partial ambiguity vector and the overall ambiguity vector respectively.

[0035] Optionally, the de - inverse transformation formula of when the first observation satellite is used as the reference satellite is specifically:

[0036] ;

[0037] Inverse Matrix and the adjoint matrix The functional relationship between them is:

[0038] ;

[0039] The analytical formula is:

[0040] ;

[0041] Among them, diag( ) is the vector " ” is a diagonal matrix with the main diagonal elements.

[0042] Optionally, when the pseudorange observations and carrier observations are single-frequency single-epoch single-system GNSS observations, the de-inverse transformation shrinkage factor is k In and Specifically:

[0043] ;

[0044] in, , is the imaginary unit of the complex number, , is the partial ambiguity vector The corresponding weighted double difference coefficient matrix, is the overall blur vector The corresponding weighted double difference coefficient matrix.

[0045] Optionally, combine the single-frequency single-epoch single-system P-ADOP expression and the shrinkage factor k The de-inverse transformation analytical formula is used to determine the PAR ambiguity precision factor P-ADOP, which includes:

[0046] Combined single-frequency single-epoch single-system P-ADOP expression and shrinkage factor k The analytical formula for the inverse transformation is used to obtain the single-frequency single-epoch single-system GNSS non-inverse transformation P-ADOP approximate formula; the single-frequency single-epoch single-system GNSS non-inverse transformation P-ADOP approximate formula is analogically derived to obtain the single-frequency multi-system or multi-frequency single-epoch GNSS non-inverse transformation P-ADOP general approximate formula; and the PAR ambiguity precision factor P-ADOP is determined by the single-frequency multi-system or multi-frequency single-epoch GNSS non-inverse transformation P-ADOP general approximate formula;

[0047] The single-frequency single-epoch single-system GNSS non-inverse transformation P-ADOP approximate formula is specifically:

[0048] ;

[0049] When the pseudorange observation and carrier observation are single-frequency multi-system or multi-frequency single-epoch GNSS observations, adjust the weighting coefficients and ambiguity dimensions in the non-inverse transformation P-ADOP approximate formula for single-frequency single-epoch single-system GNSS, multiply the SMRW factors of each frequency, and take the average of the wavelengths, non-differenced pseudorange standard deviations, and non-differenced carrier standard deviations of each frequency to obtain the general approximate formula for non-inverse transformation P-ADOP of single-frequency multi-system or multi-frequency single-epoch GNSS; the general approximate formula for non-inverse transformation P-ADOP of single-frequency multi-system or multi-frequency single-epoch single-baseline is specifically:

[0050] ;

[0051] Wherein, is the sum of weights and the ratio of the product of weights of the th j frequency in the partial ambiguity vector, N is the dimension of the partial ambiguity vector , and are the multi-frequency de-inverse transformation weighting coefficient matrices for the partial ambiguity vector and the overall ambiguity vector respectively, and are the average values of the pseudorange and carrier non-differenced standard deviations of the th t frequencies in the partial ambiguity vector is the average value of the carrier wavelengths of the th t frequencies in the partial ambiguity vector

[0052] The average value of the carrier wavelength , the average value of the carrier non-differenced standard deviation , the average value of the pseudorange non-differenced standard deviation , the sum of weights and the ratio of the product of weights of the th j frequency in the partial ambiguity vector , the dimension N of the partial ambiguity vector , the multi-frequency de-inverse transformation weighting coefficient matrix for the partial ambiguity vector and the multi-frequency de-inverse transformation weighting coefficient matrix for the overall ambiguity vector are specifically:

[0053] ;

[0054] in, and Corresponding to j Frequency and , , , and Respectively j The undifferenced standard deviation of the frequency pseudorange and carrier, For the j The carrier wavelength of the frequency.

[0055] Compared with the prior art, the above-mentioned method for determining the PAR ambiguity precision factor for single-epoch GNSS provided by the embodiment of the present invention has the following beneficial effects:

[0056] The present invention studies the influencing mechanism of the precision of partial ambiguity vector floating-point solution, derives the single-frequency single-epoch single-system P-ADOP expression, and demonstrates the independence of its double-difference reference satellite.

[0057] More importantly, the contraction factor that characterizes the theoretical relationship between P-ADOP and ADOP was constructed. k , analyze the shrinkage factor k The shrinkage and pseudorange accuracy properties of the shrinkage factor are used to establish the inverse transformation analytical formula of the shrinkage factor; the single-frequency single-epoch single-system P-ADOP expression and the shrinkage factor k By combining and analogically deducing the combined results, a general approximate formula for the non-inverse transformation P-ADOP of single-frequency multi-system or multi-frequency single-epoch GNSS is formed, which can improve the ambiguity precision dilution ADOP theory and has important theoretical value and practical guiding significance for the rapid selection and positioning of the optimal ambiguity subset of GNSS and the analysis of the mechanism affecting the success rate of partial ambiguity fixation. BRIEF DESCRIPTION OF THE DRAWINGS

[0058] Figure 1 The figure is a flow chart of a method for determining the PAR ambiguity precision factor for a single-epoch GNSS provided in an embodiment. DETAILED DESCRIPTION

[0059] In order to make the purpose, technical solution and advantages of the present invention more clearly understood, the present invention is further described in detail below in conjunction with the accompanying drawings and embodiments. It should be understood that the specific embodiments described herein are only used to explain the present invention and are not intended to limit the present invention.

[0060] In one embodiment, a method for determining PAR ambiguity precision factor for a single-epoch GNSS is provided, the method comprising:

[0061] Construct a single epoch GNSS double-difference mathematical model and transform the pseudo-range observations in the model and carrier observations Divided into two groups ( , )and( , ), and the single-epoch GNSS double-difference mathematical model after the observations are split is obtained, and the overall ambiguity vector Divide into two groups accordingly .

[0062] Using the least squares method, we construct , Or Observable , and The partial ambiguity vectors calculated by the single epoch GNSS double difference mathematical model after the observations are split Floating point solution The variance-covariance matrix of With the observed quantity and The partial ambiguity vectors calculated by the single epoch GNSS double difference mathematical model after the observations are split Floating point solution The variance-covariance matrix of The functional relationship between them.

[0063] According to the definition of ambiguity precision dilution ADOP, the determinant reduction theorem and the partial ambiguity vector in the functional relationship are used. Floating point solution The variance-covariance matrix of , construct the partial ambiguity vector Floating point solution The single-frequency single-epoch single-system P-ADOP expression is used to characterize the partial ambiguity vector Floating point solution ADOP and scaling factors K The relationship between.

[0064] Constructing the scaling factor when taking the first observed satellite as the reference satellite K middle The analytical formula for the inverse transformation is The de-inverse transform weighting coefficient matrix is determined by the de-inverse transform analytical formula, and the shrinkage factor representing the theoretical relationship between P-ADOP and ADOP is established. k The analytical expression for the inverse transformation of .

[0065] Combined single-frequency single-epoch single-system P-ADOP expression and shrinkage factor kThe inverse transformation analytical formula is used to obtain the approximate formula of the non-inverse transformation P-ADOP for single-frequency, single-epoch, single-system GNSS. By analogously deriving the approximate formula of the non-inverse transformation P-ADOP for single-frequency, single-epoch, single-system GNSS, the general approximate formula of the non-inverse transformation P-ADOP for single-frequency multi-system or multi-frequency single-epoch GNSS is obtained, and the precision factor P-ADOP of the PAR ambiguity is determined by the general approximate formula of the non-inverse transformation P-ADOP for single-frequency multi-system or multi-frequency single-epoch GNSS.

[0066] The specific implementation is as follows:

[0067] 1. Single-epoch GNSS positioning model

[0068] Assume that satellites are observed at the same time, then single-frequency double-difference observation (pseudorange and carrier) equations can be formed:

[0069] (1)

[0070] , (2)

[0071] In the formula, and represent the expectation and variance of " " respectively; is the pseudorange observation, is the carrier observation; and are the baseline vector and the overall ambiguity vector respectively; , is the carrier wavelength, is order coefficient matrix and its rank ; is the covariance matrix, , and are the covariance of the undifferenced observations of the reference satellite and the j th satellite respectively, w is the weight of the observed satellite; is the standard deviation of the undifferenced pseudorange observation, is the standard deviation of the undifferenced carrier observation. Among them, and and their inverse matrices are all symmetric positive definite matrices. Using the least squares method, the baseline vector and the overall ambiguity vector float solutions and and their variance-covariance matrices and are obtained as follows:

[0072] (3)

[0073] (4)

[0074] In the formula, and are both symmetric positive definite matrices. Using the LAMBDA method, the baseline vector and the overall ambiguity vector fixed solution and .

[0075] 2. Single-epoch ADOP theory

[0076] The ambiguity dilution of precision (ADOP) is a scalar factor proposed by Professor Teunissen in 1997, which is easy to calculate and can be used to measure the estimation accuracy and solution success rate of the ambiguity vector. Its definition formula is as follows:

[0077] (5)

[0078] In the formula, represents the determinant of " ". The approximate formula of the ADOP for the geometry-based multi-frequency single-epoch single-baseline model can be written as:

[0079] (6)

[0080] In the formula, , is the carrier wavelength of the j th frequency, m - t is the dimension of the overall ambiguity vector, t is the number of frequencies (assuming that the same frequency of different systems selects reference satellites within their respective systems); and are the undifferenced standard deviations of the pseudorange and carrier of the j th frequency respectively, , . ADOP can also be used to measure the integer least squares success rate of the ambiguity vector:

[0081] (7)

[0082] In the formula, and are the success rate based on ADOP and the integer sequential rounding success rate respectively; is the standard normal distribution function. and All are monotonically decreasing functions with respect to ADOP.

[0083] 3.1 Influence mechanism of the accuracy of the partial ambiguity vector float solution

[0084] Construct a single-epoch GNSS double-difference mathematical model (double-difference function model and double-difference stochastic model), and divide the pseudorange observables and carrier observables in the model into two groups ( , ) and ( , ), to obtain the single-epoch GNSS double-difference mathematical model after splitting the observables. The coefficient matrix of the function model and the overall ambiguity vector are correspondingly divided into two groups, which are and .

[0085] If the observables in Equation (1) are divided into two groups, then Equation (1) and Equation (2) can be written as:

[0086] (8)

[0087] (9)

[0088] In the formula, , " " are respectively , , B、 or ; , , and respectively represent the expectation and variance of " ", is the pseudorange observable, is the carrier observable, and are the baseline vector and the overall ambiguity vector respectively, , is the carrier wavelength, is order coefficient matrix and its rank ; is the cofactor matrix.

[0089] Adopt the least squares method to construct from the observables , or the observables , and and the partial ambiguity vector calculated by the single-epoch GNSS double-difference mathematical model after splitting the observables float solution variance-covariance matrix and the variance-covariance matrix of the partial ambiguity vector calculated by the single-epoch GNSS double-difference mathematical model after splitting the observables and and the partial ambiguity vector calculated by the single-epoch GNSS double-difference mathematical model after splitting the observables float solution variance-covariance matrix The functional relationship between them: According to Equations (8) and (9), the variance-covariance matrix of the float solution of the partial ambiguity vector and the variance-covariance matrix of the float solution of the partial ambiguity vector is specifically as follows:

[0090] (10)

[0091] where and are the variance-covariance matrices of the partial ambiguity vector , , , or the observables , , calculated by the float solution of the partial ambiguity vector b and the baseline vector the float solution is a positive definite or semi-positive definite matrix, , , is the variance-covariance matrix of the baseline vector and calculated only by b the float solution of the baseline vector is the standard deviation of the undifferenced pseudorange observables, is the standard deviation of the undifferenced carrier observables.

[0092] If the partial ambiguity vector and the integer ambiguity vector correspond to t and carrier observables of respectively, then the double-difference cofactor matrix of the observables corresponding to the partial ambiguity vector is ​is the partial ambiguity vector in the j double-difference cofactor matrix of the observables corresponding to the frequency ambiguity vector.

[0093] Therefore, it can be seen from Equation (10) that the partial ambiguity vector float solution has a higher precision than the partial ambiguity vector float solution and the improvement in the precision of the partial ambiguity vector float solution is only related to the pseudorange observables and has nothing to do with the carrier observables .

[0094] 3.2, Single-Frequency Single-Epoch Single-System P-ADOP Approximation Formula

[0095] According to the definition of ADOP and Equation (10), with the help of the determinant reduction theorem (where A and R are both invertible square matrices), construct the single-frequency single-epoch single-system P-ADOP expression of the partial ambiguity vector float solution which is used to characterize the relationship between the ADOP of the partial ambiguity vector float solution and the scaling factor K :

[0096] (11)

[0097] In the formula, is the dimension of the partial ambiguity vector , is the number of observed satellites corresponding to the partial ambiguity vector , is the ambiguity dilution of precision ADOP of the partial ambiguity vector float solution calculated according to the variance-covariance matrix of the partial ambiguity vector float solution Taking into account and Equation (3), the scaling factor K can be written as:

[0098] (12)

[0099] The scaling factor K has the following two properties: (1) According to Weyl's theorem and the positive definite matrix ( = or ) and a positive definite or positive semi - definite matrix relationship , the relational expression holds, that is, the scaling factor K has contractivity; (2) According to the property that "compared with , the improvement in accuracy is only related to ", in equation (11), the scaling factor K can only be the contraction factor of the standard deviation of the non - differenced pseudorange observable, that is, the scaling factor K has the pseudorange accuracy property, and the single - frequency single - epoch single - system GNSS P - ADOP approximate formula is obtained.

[0100] The P - ADOP formula for the floating - point solution of the single - frequency single - epoch single - system partial ambiguity vector is expressed as:

[0101] (13)

[0102] where is the ambiguity dilution of precision ADOP of the floating - point solution of the partial ambiguity vector calculated according to the variance - covariance matrix of the floating - point solution of the partial ambiguity vector , the scaling factor is a positive number less than 1 and has the pseudorange accuracy property; K ,

[0103] The scaling factor K as the coefficient of the standard deviation of the non - differenced pseudorange observable, the contraction factor is used to describe the standard deviation of the non - differenced pseudorange observable , , then the P - ADOP approximate expression of the floating - point solution of the single - frequency single - epoch single - system partial ambiguity vector and the analytical formula of the contraction factor k are: (14)

[0104]

[0105] where k is a positive - valued contraction factor, is the standard deviation of the non - differenced pseudorange observable, is the standard deviation of the non - differenced carrier observable,​​ and are the coefficient matrix and the cofactor matrix corresponding to the partial ambiguity vector , is order coefficient matrix and its rank , is the cofactor matrix, is the partial ambiguity vector dimension, is the partial ambiguity vector corresponding number of observed satellites.

[0106] Therefore, compared with the partial ambiguity vector float solution , (1) the partial ambiguity vector float solution has a smaller ADOP value, that is, its ambiguity fixing success rate is higher; (2) in terms of improving the accuracy of the partial ambiguity vector float solution or improving its ADOP value, increasing the pseudorange observation has the same effect as improving the accuracy of the original pseudorange observation . This property further reveals the influence mechanism of the accuracy of the partial ambiguity vector float solution.

[0107] 3.3. P-ADOP Double-Difference Reference Satellite Correlation Analysis

[0108] Let the baseline vector and the variance-covariance matrix of the double-difference ambiguity vector when the first observed satellite is used as the reference satellite be and respectively. Let the baseline vector and the variance-covariance matrix of the double-difference ambiguity vector when the second observed satellite is used as the reference satellite be and respectively. and hold, that is , and P-ADOP are independent of the double-difference reference satellite. For the convenience of calculation and to improve the calculation efficiency, it is stipulated to construct the double-difference function model and the double-difference stochastic model with the first observed satellite as the reference satellite.

[0109] The ADOP value is independent of the reference satellite, that is, it is independent of the choice of the reference satellite. As an extended formula of ADOP, P-ADOP should be consistent with ADOP in this property. Therefore, the correlation between P-ADOP and the reference satellite is demonstrated by analyzing the relationship between the shrinkage factor k and the reference satellite.

[0110] If the double-difference observation equation is formed with the first observed satellite as the reference satellite, then in equations (3) and (4) and can be respectively expressed as and , that is and , where , is the order coefficient matrix composed of the non-differenced unit direction vectors between the receiver and the satellite; , , is a column vector with all elements being 1, is order identity matrix, is a diagonal matrix. Similarly, let and be respectively and when the second observed satellite is used as the reference star, then:

[0111] (15)

[0112] In the formula, and are respectively the double-difference coefficient matrix and the covariance factor matrix of the observables when the second satellite is used as the reference star; is an invertible matrix with a determinant of 1.

[0113] 3.4 Shrinkage factor k Analytical formula for the inverse transformation

[0114] Construct the analytical formula for the inverse transformation of the scaling factor K in when the first observed satellite is used as the reference satellite; Determine the weighted coefficient matrix of the inverse transformation according to the analytical formula for the inverse transformation of , and establish the analytical formula for the inverse transformation of the shrinkage factor k characterizing the theoretical relationship between P-ADOP and ADOP.

[0115] Assume that both Equation (1) and Equation (2) are models with the first observed satellite as the reference star. The specific formula for constructing the inverse transformation formula of when the first observed satellite is used as the reference satellite is:

[0116] (16)

[0117] Among them, the functional relationship between the inverse matrix and the adjoint matrix is:

[0118] (17)

[0119] The analytical formula is:

[0120] (18)

[0121] where diag( ) is a diagonal matrix with the vector " " as the main diagonal elements.

[0122] Let the double-difference coefficient matrix and the double-difference cofactor matrix when the first observation satellite is used as the reference satellite be and respectively, and the double-difference coefficient matrix and the double-difference cofactor matrix when the second observation satellite is used as the reference satellite be and respectively. The relationships between the double-difference coefficient matrices and between the double-difference cofactor matrices are specifically:

[0123] (19)

[0124] where is an invertible matrix with a determinant of 1, holds, has the property of being independent of the double-difference reference satellite.

[0125] The analytical formula for the inverse transformation of the shrinkage factor k in when the first observation satellite is used as the reference satellite is specifically:

[0126] (20)

[0127] where and are the double-difference coefficient matrix and the double-difference cofactor matrix respectively when the first observation satellite is used as the reference satellite, ω is the weight of the other observation satellites except the reference satellite, and the weight of the reference satellite is .

[0128] According to the inverse transformation analytical formula, the analytical formula for the inverse transformation of the shrinkage factor k of single-frequency single-epoch single-system GNSS when the first observation satellite is used as the reference satellite is specifically:

[0129] (21)

[0130] where and are the inverse transformation weighted coefficient matrices for the partial ambiguity vector and the overall ambiguity vector respectively.

[0131] De-inverse transformation shrinkage factor k In and Specifically:

[0132] (twenty two)

[0133] in, , is the imaginary unit of the complex number, , is the partial ambiguity vector The corresponding weighted double difference coefficient matrix, is the overall blur vector The corresponding weighted double difference coefficient matrix. Equations (14) and (21) are only applicable to the estimation of P-ADOP values of single-frequency, single-epoch and single-system.

[0134] 3.5 General approximate formula for single-frequency multi-system or multi-frequency single-epoch GNSS non-inverting transformation P-ADOP

[0135] Combined single-frequency single-epoch single-system P-ADOP expression and shrinkage factor k The analytical formula for the inverse transformation is used to obtain the approximate formula for the single-frequency, single-epoch, single-system GNSS non-inverse transformation P-ADOP. The approximate formula for the single-frequency, single-epoch, single-system GNSS non-inverse transformation P-ADOP is derived by analogy to obtain the general approximate formula for the single-frequency, multi-system or multi-frequency, single-epoch GNSS non-inverse transformation P-ADOP.

[0136] The single-frequency single-epoch single-system GNSS non-inverse transformation P-ADOP approximate formula is as follows:

[0137] (twenty three)

[0138] When the pseudorange observations and carrier observations are single-frequency multi-system or multi-frequency single-epoch GNSS observations, adjust the weighting coefficients and ambiguity dimensions in the single-frequency single-epoch single-system GNSS non-inverse transform P-ADOP approximate formula, multiply the SMRW factors of each frequency, and average the wavelength, undifferenced pseudorange standard deviation, and undifferenced carrier standard deviation of each frequency to obtain the single-frequency multi-system or multi-frequency single-epoch GNSS non-inverse transform P-ADOP general approximate formula; the single-frequency multi-system or multi-frequency single-epoch single baseline non-inverse transform P-ADOP general approximate formula is:

[0139] (twenty four)

[0140] in, is the partial ambiguity vector Middle jThe ratio of the sum of the weights of the frequencies to the product of the weights, where N is the partial ambiguity vector is the dimension of, and are the multi-frequency de-inversion transformation weighting coefficient matrices for the partial ambiguity vector and the overall ambiguity vector respectively, and are the average values of the pseudo-range and carrier non-differenced standard deviations of the in the t frequencies in the partial ambiguity vector, is the average value of the carrier wavelengths of the in the t frequencies in the partial ambiguity vector.

[0141] The average value of the carrier wavelength in the single-frequency multi-system or multi-frequency single-epoch GNSS non-inversion transformation P-ADOP general approximation formula , the average value of the carrier non-differenced standard deviation , the average value of the pseudo-range non-differenced standard deviation , the partial ambiguity vector in the j ratio of the sum of the weights to the product of the weights of the frequency, the dimension N of the partial ambiguity vector , the multi-frequency de-inversion transformation weighting coefficient matrix for the partial ambiguity vector and the multi-frequency de-inversion transformation weighting coefficient matrix for the overall ambiguity vector Specifically:

[0142] (25)

[0143] Among them, , , is the dimension of the partial ambiguity vector , is the number of observed satellites corresponding to the in the j frequency ambiguity vector in the partial ambiguity vector; and are the average values of the pseudo-range and carrier non-differenced standard deviations of the in the t frequencies in the partial ambiguity vector, which have the same meaning as and in Equation (6), that is and ; and are the non-differenced standard deviations of the pseudo-range and carrier of the j frequency respectively, and correspond to the j frequency of and respectively. , , and are respectively the non-differenced standard deviations of the j frequency pseudorange and carrier, is the j carrier wavelength of the

[0144] The embodiments described above only represent several implementation manners of the present invention. The description is relatively specific and detailed, but it should not be construed as a limitation on the scope of the invention patent. It should be noted that for those of ordinary skill in the art, without departing from the concept of the present invention, several modifications and improvements can be made, and these all belong to the protection scope of the present invention.

Claims

1. Method for determining PAR dilution of precision for single-epoch GNSS, characterized in that, Including: Construct a single-epoch GNSS double-difference mathematical model, and divide the pseudorange observable p and carrier observable in the model into two groups (p1, p2) and obtain the single-epoch GNSS double-difference mathematical model after splitting the observables, and the overall ambiguity vector a is correspondingly divided into two groups (a1, a2); Using the least squares method, construct the variance-covariance matrix of the float solution a1 of the partial ambiguity vector calculated by the single-epoch GNSS double-difference mathematical model after splitting the observables p, or the observables p1, p2 and and the float solution a1 of the partial ambiguity vector calculated by the single-epoch GNSS double-difference mathematical model after splitting the observables; Construct the function relationship between the variance-covariance matrix and the variance-covariance matrix of the float solution a1 of the partial ambiguity vector calculated by the single-epoch GNSS double-difference mathematical model after splitting the observables p1 and and the float solution a1 of the partial ambiguity vector calculated by the single-epoch GNSS double-difference mathematical model after splitting the observables; Construct the single-frequency single-epoch single-system P-ADOP expression of the float solution a1 of the partial ambiguity vector by using the determinant reduction theorem and the variance-covariance matrix of the float solution a1 of the partial ambiguity vector in the function relationship; Construct the variance-covariance matrix of the float solution a1 of the partial ambiguity vector; Construct the single-frequency single-epoch single-system P-ADOP expression of the float solution a1 of the partial ambiguity vector; Constructing B in the scaling factor K when the first observation satellite is used as the reference satellite T Q -1 The analytical formula for the inverse transformation of B; According to B T Q - 1 Determine the weighted coefficient matrix of the inverse transformation according to the analytical formula for the inverse transformation of B, and establish the analytical formula for the inverse transformation of the contraction factor k characterizing the theoretical relationship between P-ADOP and ADOP; where B T Q -1 B is the coefficient matrix of the baseline vector normal equation; Combining the single-frequency, single-epoch, single-system P-ADOP expression and the analytical formula of the inverse transformation of the shrinkage factor k to determine the PAR ambiguity dilution of precision P-ADOP.

2. The method for determining the PAR ambiguity dilution of precision for single-epoch GNSS according to claim 1, wherein The specific single-epoch GNSS double-difference mathematical model after splitting the observables is as follows: Among them, "*” are p, B, a, or E[*] and D[*] represent the expectation and variance of "*” respectively, p is the pseudo-range observable, is the carrier observable, b and a are the baseline vector and the overall ambiguity vector respectively, B λ = λ·I n×n , λ is the carrier wavelength, B is an n×3 coefficient matrix and its rank rank(B) = 3, Q is the cofactor matrix.

3. The method for determining the PAR ambiguity dilution of precision for single-epoch GNSS according to claim 2, wherein The construction is composed of the observation quantity p, or the observation quantities p1, p2, and the float solution of a part of the ambiguity vector a1 calculated by the single-epoch GNSS double-difference mathematical model after splitting the observation quantity of the variance-covariance matrix and the function relationship between the variance-covariance matrix of the observation quantity p1 and the float solution of a part of the ambiguity vector a1 calculated by the single-epoch GNSS double-difference mathematical model after splitting the observation quantity is as follows: Among them, and are respectively the partial ambiguity vector a1 floating-point solution or the partial ambiguity vector a1 floating-point solution calculated from the observables p1, p2, and the variance-covariance matrix of the baseline vector b floating-point solution , is a positive definite or semi-positive definite matrix, is the variance-covariance matrix of the baseline vector b floating-point solution calculated only from p1 and , σ p is the standard deviation of the non-differenced pseudorange observable, is the standard deviation of the non-differenced carrier observable; If the partial ambiguity vector a1 and the overall ambiguity vector correspond to the carrier observations of t and τ frequencies respectively, then the double-difference cofactor matrix of the observations corresponding to the partial ambiguity vector a1 is is the double-difference cofactor matrix of the observations corresponding to the j-th frequency ambiguity vector in the partial ambiguity vector a1.

4. The method for determining the PAR ambiguity dilution of precision factor for single-epoch GNSS according to claim 1, wherein The floating-point solution of the constructed partial ambiguity vector a1 The single-frequency, single-epoch, single-system P-ADOP expression specifically includes: According to the definition formula of the Ambiguity Dilution of Precision (ADOP), the floating-point solution of the partial ambiguity vector a1 for single-frequency, single-epoch, and single-system is expressed by the P-ADOP formula as follows: Among them, is the floating-point solution of the partial ambiguity vector a1 of the variance-covariance matrix The floating-point solution of the partial ambiguity vector a1 calculated of the ambiguity dilution of precision ADOP, The scaling factor K is a positive number less than 1 and has the property of pseudorange precision; the single-frequency, single-epoch, single-system P-ADOP expression is used to characterize the relationship between the ADOP of the floating-point solution of the partial ambiguity vector a1 and the scaling factor K; The scaling factor K serves as the coefficient of the undifferenced pseudorange observation standard deviation σ p and the contraction factor k is used to describe the undifferenced pseudorange observation standard deviation σ p , Then the P-ADOP approximate expression of the floating solution of the single-frequency single-epoch single-system partial ambiguity vector a1 and the analytical formula of the contraction factor k are as follows: Among them, k is a contraction factor with a positive value, and σ p is the standard deviation of the undifferenced pseudorange observation, is the standard deviation of the undifferenced carrier observation, B1 and Q1 are the coefficient matrix and the cofactor matrix corresponding to the partial ambiguity vector a1, B is an n×3 order coefficient matrix and its rank rank(B) = 3, Q is the cofactor matrix, is the dimension of the partial ambiguity vector a1, is the number of observed satellites corresponding to the partial ambiguity vector a1.

5. The method for determining the PAR ambiguity dilution of precision for single-epoch GNSS according to claim 4, wherein When the first observation satellite is used as the reference satellite in the construction, B in the scaling factor K T Q -1 The analytical formula for the inverse transformation of B; According to B T Q - 1 Determine the inverse transformation weighted coefficient matrix according to the analytical formula for the inverse transformation of B, and establish the analytical formula for the inverse transformation of the contraction factor k that characterizes the theoretical relationship between P-ADOP and ADOP, specifically including: Let the double-difference coefficient matrix and the double-difference cofactor matrix when the first observation satellite is used as the reference satellite be B 1 and Q 1 , and let the double-difference coefficient matrix and the double-difference cofactor matrix when the second observation satellite is used as the reference satellite be B 2 and Q 2 . The relationships between the double-difference coefficient matrices and between the double-difference cofactor matrices are specifically as follows: B 2 = Z T B 1 , Q 2 = Z T Q 1 Z; Among them, is an invertible matrix with a determinant of 1, (B 2 ) T (Q 2 ) -1 B 2 =(Z T B 1 ) T (Z T Q 1 Z) -1 Z T B 1 =(B 1 ) T (Q 1 ) -1 B 1 holds, and B T Q -1 has double-difference reference satellite independence; Build B in the scaling factor K when taking the first observation satellite as the reference satellite T Q -1 The analytical formula for the inverse transformation of B is specifically as follows: Among them, B 1 and Q 1 are the double-difference coefficient matrix and the double-difference cofactor matrix respectively when the first observation satellite is used as the reference satellite. ω = [w2,..., w m is the weight of other observation satellites except the reference satellite, and the weight of the reference satellite is w1; According to the de-inverse transformation analytical formula of B in the scaling factor K when the first observation satellite is used as the reference satellite, determine the de-inverse transformation weighted coefficient matrix, and establish the de-inverse transformation analytical formula of the contraction factor k that characterizes the theoretical relationship between P-ADOP and ADOP: T Q -1 ​ Among them, and are the inverse transformation weighted coefficient matrices for the partial fuzzy vector a1 and the overall ambiguity vector a respectively, and B T Q -1 B is the coefficient matrix of the baseline vector normal equation.

6. The method for determining the PAR ambiguity dilution of precision for single-epoch GNSS according to claim 5, wherein When the first observation satellite is used as the reference satellite for the construction (Q 1 ) -1 The specific inverse transformation formula is as follows: Inverse matrix Q -1 and adjoint matrix Q * The functional relationship between them is as follows: Q -1 = Q * / |Q|; The analytical formula of |Q| is: where diag(*) is a diagonal matrix with the vector "*" as the main diagonal elements.

7. The method for determining the PAR ambiguity dilution of precision for single-epoch GNSS according to claim 5, wherein When the pseudorange observation and carrier observation are single-frequency, single-epoch, single-system GNSS observations, the and in the inverse transformation shrinkage factor k are specifically as follows: Among them, i is the imaginary unit of a complex number, i 2 = -1, is the weighted double-difference coefficient matrix corresponding to the partial ambiguity vector a1, is the weighted double-difference coefficient matrix corresponding to the overall ambiguity vector a.

8. The method for determining the PAR ambiguity dilution of precision for single-epoch GNSS according to claim 7, wherein The process of combining the single-frequency, single-epoch, single-system P-ADOP expression and the analytical formula of the inverse transformation of the shrinkage factor k to determine the PAR ambiguity dilution of precision P-ADOP specifically includes: Combining the single-frequency, single-epoch, single-system P-ADOP expression and the analytical formula of the inverse transformation of the shrinkage factor k to obtain the approximate formula of the non-inverse transformation P-ADOP of single-frequency, single-epoch, single-system GNSS; performing analogical derivation on the approximate formula of the non-inverse transformation P-ADOP of single-frequency, single-epoch, single-system GNSS to obtain the general approximate formula of the non-inverse transformation P-ADOP of single-frequency multi-system or multi-frequency single-epoch GNSS; and determining the PAR ambiguity dilution of precision P-ADOP through the general approximate formula of the non-inverse transformation P-ADOP of single-frequency multi-system or multi-frequency single-epoch GNSS; The approximate formula of the non-inverse transformation P-ADOP of single-frequency, single-epoch, single-system GNSS is specifically: When the pseudorange observables and carrier observables are single-frequency multi-system or multi-frequency single-epoch GNSS observables, adjust the weighting coefficients and ambiguity dimensions in the approximate formula of the non-inverse transformation P-ADOP of single-frequency, single-epoch, single-system GNSS, multiply the SMRW factors of each frequency, and take the average of the wavelengths, undifferenced pseudorange standard deviations, and undifferenced carrier standard deviations of each frequency to obtain the general approximate formula of the non-inverse transformation P-ADOP of single-frequency multi-system or multi-frequency single-epoch GNSS; The general approximate formula of the non-inverse transformation P-ADOP of single-frequency multi-system or multi-frequency single-epoch single-baseline is specifically: Among them, SMRW j is the ratio of the sum of the weights to the product of the weights of the j-th frequency in the partial ambiguity vector a1, N is the dimension of the partial ambiguity vector a1, and are the multi-frequency inverse transformation weighting coefficient matrices for the partial ambiguity vector a1 and the overall ambiguity vector a respectively, and are the average values of the pseudo-range and carrier non-differential standard deviations of t frequencies in the partial ambiguity vector a1 respectively, is the average value of the carrier wavelengths of t frequencies in the partial ambiguity vector a1; Average carrier wavelength in the general approximation formula of single-frequency multi-system or multi-frequency single-epoch GNSS non-inverse transformation P-ADOP Average non-differenced carrier standard deviation Average non-differenced pseudorange standard deviation Ratio SMRW of the sum to the product of weights of the j-th frequency in the partial ambiguity vector a1 j , dimension N of the partial ambiguity vector a1, multi-frequency de-inverse transformation weighted coefficient matrix for the partial ambiguity vector a1 and multi-frequency de-inverse transformation weighted coefficient matrix for the overall ambiguity vector a Specifically: Among them, and correspond to and at the j-th frequency respectively, where j = 1, 2, ..., t, ..., τ, τ ≥ t, σ pj and σ φj are the undifferenced standard deviations of the pseudo-range and carrier at the j-th frequency respectively, and λ j is the carrier wavelength at the j-th frequency.

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