CN113985457B - GBAS protection level calculation method - Google Patents

GBAS protection level calculation method Download PDF

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CN113985457B
CN113985457B CN202111291369.9A CN202111291369A CN113985457B CN 113985457 B CN113985457 B CN 113985457B CN 202111291369 A CN202111291369 A CN 202111291369A CN 113985457 B CN113985457 B CN 113985457B
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protection level
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薛瑞
赵萌宇
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Beihang University
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    • G—PHYSICS
    • G01—MEASURING; TESTING
    • G01S—RADIO DIRECTION-FINDING; RADIO NAVIGATION; DETERMINING DISTANCE OR VELOCITY BY USE OF RADIO WAVES; LOCATING OR PRESENCE-DETECTING BY USE OF THE REFLECTION OR RERADIATION OF RADIO WAVES; ANALOGOUS ARRANGEMENTS USING OTHER WAVES
    • G01S19/00—Satellite radio beacon positioning systems; Determining position, velocity or attitude using signals transmitted by such systems
    • G01S19/01—Satellite radio beacon positioning systems transmitting time-stamped messages, e.g. GPS [Global Positioning System], GLONASS [Global Orbiting Navigation Satellite System] or GALILEO
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Abstract

The invention relates to a GBAS protection level calculation method, which comprises the following steps: calculating a positioning domain error sequence of each satellite observed by the airborne receiver during positioning; performing correlation analysis on the positioning domain error sequence to obtain correlation characteristics; screening data according to the relevant characteristics, and establishing a distribution model of a positioning domain error sequence; fitting a distribution function of the positioning domain error sequence according to the distribution model; and calculating the distribution position corresponding to the fault-free missed detection probability according to the distribution function, and determining the corresponding protection level. The invention considers the relevance of the positioning domain error sequence, so that the protection level calculation result is more accurate, and the continuity and the usability index of the whole navigation system are improved.

Description

GBAS protection level calculation method
Technical Field
The invention relates to the technical field of satellite navigation, in particular to a GBAS protection level calculation method based on a multivariate extreme value model.
Background
As Global Navigation Satellite Systems (GNSS) enter a modernized development environment, civil requirements for satellite navigation are becoming increasingly important directions of application. However, applying satellite navigation systems to civil aviation, which must first be made to meet the civil aviation's performance requirements for navigation systems, including 4 aspects of accuracy, integrity, continuity and availability, GBAS is considered to be the GNSS augmentation system that has the greatest potential to meet the near-three demands for precision in civil aviation.
The GBAS applied to civil aviation broadcasts differential correction information through a ground subsystem, and the airplane receives the information and then carries out differential positioning and protection level calculation. When the protection level calculation is carried out, in the traditional calculation method, a zero-mean Gaussian model is established to carry out modeling on pseudo-range errors, the standard deviation relation of errors in a pseudo-range domain and a positioning domain is found through projection matrix weighting, and then the single-constellation protection level is obtained through the standard deviation by using a quantile method.
The protection level calculation method is based on the premise that pseudo-range errors are zero mean and symmetrical, and pseudo-range errors among satellites in a single constellation are independently and uniformly distributed. However, the errors are enveloped by the independent and identically distributed zero-mean Gaussian distribution, the errors are amplified to a certain extent, and when the final protection level PL envelops and positions the errors with the probability of 1-p, the compactness of the envelops is influenced, so that the calculation accuracy of the actual protection level is influenced.
Disclosure of Invention
In view of the above analysis, the present invention aims to provide a GBAS protection level calculation method based on a multivariate extreme value model, so as to solve the problem of protection level calculation accuracy in the prior art.
The technical scheme provided by the invention is as follows:
the invention discloses a GBAS protection level calculation method, which comprises the following steps:
the positioning domain error sequence of each satellite observed by the computer-mounted receiver during positioning;
performing correlation analysis on the positioning domain error sequence to obtain correlation characteristics; screening data according to the relevant characteristics, and establishing a distribution model of a positioning domain error sequence;
fitting a distribution function of the positioning domain error sequence according to the distribution model;
and calculating the distribution position corresponding to the fault-free missed detection probability according to the distribution function, and determining the corresponding protection level.
Further, by establishing a projection matrix from the pseudo-range domain to the positioning domain, projecting the pseudo-range error of each satellite to the positioning domain to obtain a positioning domain error sequence of each satellite;
wherein the positioning domain error of the jth satellite is Sν,jεj；Sν,jFor the projection matrix of the pseudorange domain to the location domain, εjIs pseudo range error; j is 1, …, and N is the total number of satellites that can be tracked by the receiver; pseudorange error ejAnd processing the pseudo-range measured value read in the message and then performing difference operation on the processed pseudo-range measured value and the user position actually calculated through satellite ephemeris to obtain the user position.
Further, the projection matrix S of the pseudo-range domain to the positioning domainν,j＝(GTWG)-1GTW; wherein G is an orientation cosine matrix and W is a weighting matrix; the orientation cosine matrix is obtained by a least square differential positioning solution of the airborne receiver.
Further, the weighting matrix
Figure BDA0003332177030000021
Wherein the content of the first and second substances,
Figure BDA0003332177030000022
is the pseudorange error standard deviation for the jth satellite.
Further, the pseudo range error standard deviation of the jth satellite
Figure BDA0003332177030000023
Wherein the content of the first and second substances,
Figure BDA0003332177030000024
is the pseudorange error standard deviation of the ground station,
Figure BDA0003332177030000031
is the pseudorange error standard deviation of the airborne receiver,
Figure BDA0003332177030000032
is the standard deviation of the residual ionospheric error,
Figure BDA0003332177030000033
residual tropospheric error standard deviation.
Further, the tail correlation characteristic of the positioning domain error sequence is obtained through correlation analysis, data screening is carried out through a data screening method including a block method or a super-threshold method according to the tail correlation characteristic, and an extreme value model of the positioning domain error sequence is established.
Further, in the block method for data screening, pseudo-range error values obtained by resolving the same satellite at different ground stations are selected as modeling data, and the maximum value in each interval is selected according to a certain step length for data screening.
Furthermore, in the data screening by the above-threshold method, a threshold is set firstly, all observed data are processed integrally, modeling is performed by using the part exceeding the threshold, and a proper threshold is selected for the pseudo-range error value to screen data.
Further, fitting out an expression of multivariate extremum distribution of the localization domain error sequence according to the extremum model is as follows:
Figure BDA0003332177030000034
wherein x isjAnd delta is a correlation parameter of the positioning domain error extreme value sequence screened out for the jth satellite.
Further, the parameter δ is estimated by moment estimation or maximum likelihood estimation.
The invention has the beneficial effects that:
the GBAS protection level calculation method can obtain more accurate protection level calculation results, and compared with the traditional method, the GBAS protection level calculation method uses more information, so that more accurate results can be obtained. Because the protection level envelops the positioning error with the probability of 1-p, the more accurate calculation result of the protection level can obviously improve the continuity and the usability index of the whole navigation system.
Drawings
The drawings are only for purposes of illustrating particular embodiments and are not to be construed as limiting the invention, wherein like reference numerals are used to designate like parts throughout.
Fig. 1 is a flowchart of a GBAS protection level calculation method according to an embodiment of the present invention.
Detailed Description
The preferred embodiments of the present invention will now be described in detail with reference to the accompanying drawings, which form a part hereof, and which together with the embodiments of the invention serve to explain the principles of the invention.
As shown in fig. 1, a method for calculating a GBAS protection level disclosed in this embodiment includes the following steps:
step S1, calculating a positioning domain error sequence of each satellite observed by the airborne receiver during positioning;
step S2, carrying out correlation analysis on the positioning domain error sequence to obtain correlation characteristics; screening data according to the relevant characteristics, and establishing a distribution model of a positioning domain error sequence;
step S3, fitting a distribution function of the positioning domain error sequence according to the distribution model;
and step S4, calculating the distribution position corresponding to the fault-free missed detection probability according to the distribution function, and determining the corresponding protection level.
Specifically, in step S1, the pseudorange error of each satellite is projected to the positioning domain by establishing a projection matrix from the pseudorange domain to the positioning domain, so as to obtain a positioning domain error sequence of each satellite;
wherein the positioning domain error of the jth satellite is Sν,jεj；Sν,jFor the projection matrix of the pseudorange domain into the positioning domain, εjIs pseudo range error; j is 1, …, and N is the total number of satellites that can be tracked by the receiver;
pseudorange error ejAnd processing the pseudo-range measured value read in the message and then performing difference operation on the processed pseudo-range measured value and the user position actually calculated through the satellite ephemeris to obtain the pseudo-range positioning method.
More specifically, the projection matrix S of the pseudorange domain to the positioning domainν,j＝(GTWG)-1GTW; g is an orientation cosine matrix, W is a weighting matrix; the orientation cosine matrix is obtained by a least square differential positioning solution of the airborne receiver.
Wherein the weighting matrix
Figure BDA0003332177030000051
Wherein, the first and the second end of the pipe are connected with each other,
Figure BDA0003332177030000052
the pseudo range error standard deviation of the jth satellite is taken as the standard deviation of the pseudo range error of the jth satellite;
preferably, the pseudorange error of the ground station, the pseudorange error of the airborne receiver, the residual ionosphere error and the residual troposphere error are considered, and the standard deviation of the pseudorange error of the jth satellite is
Figure BDA0003332177030000053
Wherein the content of the first and second substances,
Figure BDA0003332177030000054
is the pseudorange error standard deviation of the ground station,
Figure BDA0003332177030000055
is the pseudorange error standard deviation of the airborne receiver,
Figure BDA0003332177030000056
as a residual standard deviation of ionospheric error,
Figure BDA0003332177030000057
residual tropospheric error standard deviation.
Wherein the pseudorange error standard deviation of the ground station
Figure BDA0003332177030000058
Pseudorange error standard deviation to airborne receiver
Figure BDA0003332177030000059
Can be obtained by the navigation data processing of the ground station and the airborne receiver.
Standard deviation of residual ionospheric error σiono,j＝Fppσvert_iono_grad(xair+2τvair) (ii) a Wherein, FppIs a tilt factor, σvert_iono_gradIs the standard deviation of the residual vertical ionospheric error, xairIs the horizontal distance, v, between the airborne receiver and the ground stationairAnd tau is the smoothing time interval for the horizontal flight speed of the airplane.
More specifically, the tilt factor
Figure BDA00033321770300000510
Wherein R iseIs the radius of the earth, hiThe shell height of the ionized layer is generally 350km, eliIs the satellite altitude at the current moment.
Residual vertical ionospheric error standard deviation σvert_iono_gradTaking 2-4mm/km, the smoothing interval τ is typically taken as 100 s.
Standard deviation of residual tropospheric error
Figure BDA00033321770300000511
Wherein σnIs refractive index uncertainty, eliAs satellites at the present timeHeight angle, h07600 is generally taken for tropospheric elevation, and Δ h is the height of the airborne end relative to the reference point of the GBAS.
Specifically, the correlation analysis in step S2 may employ a linear correlation coefficient, a correlation degree analysis, and a tail correlation.
Wherein the linear correlation coefficient
Figure BDA0003332177030000061
Two satellites are selected, observed quantities at the same receiver and the same moment are substituted for XY in the formula, and then the correlation between the observed quantities at the current moment can be obtained.
Wherein, the relevance analysis formula F (X, Y) ═ P (X > X, Y > Y) > P (X > X) P (Y > Y); selecting the observed quantities of two satellites, the same receiver and the same time, substituting the observed quantities into XY by taking the observed quantities larger than the average value as a large-value standard (namely small XY in a formula), and calculating corresponding probability to obtain the association degree between the variables.
The tail correlation refers to the probability that when one variable takes an extreme value, the other variable also takes an extreme value; is given by the formula
Figure BDA0003332177030000062
Similar to the calculation of the correlation degree, since the tail is calculated, u represents a quantile, a certain quantile (for example, 95%, which is close to a double standard difference digit of a standard normal distribution) is selected as a large-value standard, u is substituted, and the observed quantity is substituted into X and Y, so that the tail correlation between variables can be calculated.
The linear correlation coefficient uses the covariance between variables and the variance of the variables, but for data with the distribution of pseudo-range errors conforming to the characteristic of peak thick tail, the variance does not exist, so the linear correlation coefficient loses meaning and needs to be quantitatively described by other methods. However, considering the limited data amount, the variance of the sampled data may exist, so that the linear correlation coefficient may be used to assist in determining the existence of the correlation between the pseudo-range errors.
In this embodiment, the positioning domain error sequence of each satellite is subjected to correlation analysis to obtain that the positioning domain error sequence has tail correlation. Therefore, tail correlation analysis is introduced, data screening is determined by a data screening method including a block method or a super-threshold method, and an extreme value model of a positioning domain error sequence is established;
specifically, in the block method for data screening, pseudo-range error values obtained by resolving the same satellite at different ground stations are selected as modeling data, and the maximum value in each interval is selected according to a certain step length for data screening.
The mathematical expression of the block maximum model is:
Figure BDA0003332177030000071
pseudo-range error values obtained by resolving the same satellite at different ground stations are selected as modeling data, and the maximum value in each interval is selected according to a certain step length (block length), so that the data can be screened out.
For example, each satellite has 3600 data points for each ground station per second, and 360 data points can be selected within one hour of data according to the step size of 10.
The block method is used for grouping data and then selecting the maximum value of each group for mathematical modeling, and when the data size is not large, the method has limitation and can cause that extreme value information in the data cannot be fully utilized. So the super threshold method can also be employed.
Specifically, in the data screening by the super-threshold method, a threshold is set, all observed data are processed integrally, modeling is performed by using the part exceeding the threshold, and a proper threshold is selected for a pseudo-range error value to perform data screening. This method has low data volume requirements.
For a sufficiently large threshold u, the superthresholding method is that the gradual distribution of the excess X-u becomes a generalized Pareto distribution under the condition that X > u.
The threshold selection in the above-threshold method of this embodiment is aided by the use of an average excess function, whose formula is E (u) ═ E { x-u | x > u }; an average of the portions of the sample that are greater than a given threshold value exceeding the threshold value may be obtained.
Similarly, for example, each satellite may analyze 3600 data per hour for each ground station, set a certain threshold (assuming that the threshold is selected as the average value of the data set), and select the part exceeding the threshold, i.e., perform modeling.
In the present embodiment, the granule method and the super threshold method can be flexibly selected according to the size of the data amount. And a block method can be adopted for assistance, and a corresponding data is selected by using a super-threshold model for modeling.
Specifically, since the distribution model is a multivariate extremum model, in step S3, parameters of the multivariate extremum distribution function need to be estimated.
In this embodiment, a logistic model is used, and the specific form of the multivariate extreme value distribution function is
Figure BDA0003332177030000081
Wherein x isjAnd delta is a correlation parameter of the positioning domain error extreme value sequence screened out for the jth satellite.
The estimation of the parameter δ can be done by moment estimation and maximum likelihood estimation. And approximating the fitted curve to the extreme value model data distribution by estimating the parameter delta.
In step S4, according to the multivariate extreme value distribution function F (x)1,...,xj,...,xN(ii) a Delta) calculating the distribution position corresponding to the fault-free missed detection probability, and determining the corresponding protection level.
Specifically, in this embodiment, the multivariate extreme value distribution function F (x) is continuously applied1,...,xj,...,xN(ii) a Delta) performing mathematical derivation;
since, the estimated correlation parameter 0 < δ < 1; thus, a multivariate extremum distribution function
Figure BDA0003332177030000082
Continuously deducing the above formula;
to find
Figure BDA0003332177030000083
For different xjEdge distribution and edge probability density function fj。
From the edge probability density function, get
Figure BDA0003332177030000084
Probability density function of the distribution of (a):
Figure BDA0003332177030000085
wherein, frGet f1,f2...fjAny one of them.
Distribution function
Figure BDA0003332177030000091
The distribution function model used is calculated for the final protection level. And calculating the distribution position corresponding to the fault-free missed detection probability according to the distribution function, namely determining the corresponding protection level.
In summary, the GBAS protection level calculation method of this embodiment may obtain a more accurate protection level calculation result based on the multivariate extreme value model, and compared with the conventional method, more information is used, so that a more accurate result may be obtained. Because the protection level envelops the positioning error with the probability of 1-p, the more accurate calculation result of the protection level can obviously improve the continuity and the usability index of the whole navigation system.
While the invention has been described with reference to specific preferred embodiments, it will be understood by those skilled in the art that various changes and modifications may be made without departing from the spirit and scope of the invention as defined in the following claims.

Claims (8)

1. A GBAS protection level calculation method is characterized by comprising the following steps:
calculating a positioning domain error sequence of each satellite observed by the airborne receiver during positioning;
projecting the pseudo-range error of each satellite to a positioning domain by establishing a projection matrix from the pseudo-range domain to the positioning domain to obtain a positioning domain error sequence of each satellite;
wherein the positioning domain error of the jth satellite is Sν,jεj；Sν,jFor the projection matrix of the pseudorange domain to the location domain, εjIs pseudo range error; j is 1, …, N is the total number of satellites that can be tracked by the receiver; pseudorange error ejProcessing the pseudo-range measured value read in the message and then carrying out difference operation on the processed pseudo-range measured value and the user position actually calculated through satellite ephemeris to obtain a difference value;
performing correlation analysis on the positioning domain error sequence to obtain correlation characteristics; screening data according to the relevant characteristics, and establishing a distribution model of the error sequence of the localization domain;
fitting a distribution function of the error sequence of the positioning domain according to the distribution model;
fitting an expression of the multivariate extremum distribution of the positioning domain error sequence according to the extremum model, wherein the expression comprises the following steps:
Figure FDA0003633399560000011
wherein x isjThe positioning domain error extreme value sequence screened out for the jth satellite is delta is a correlation parameter; calculating the distribution position corresponding to the fault-free missed detection probability according to the distribution function, and determining the corresponding protection level;
to find
Figure FDA0003633399560000012
For different xjEdge distribution and edge probability density function fj；
According to the edge probability density function fjIs obtained as to
Figure FDA0003633399560000013
Probability density function of the distribution of (a):
Figure FDA0003633399560000014
wherein, frTake f1,f2...fjAny one of the above;
distribution function
Figure FDA0003633399560000021
Calculating a distribution function model for the final protection level; and calculating the distribution position corresponding to the fault-free missed detection probability according to the distribution function, and determining the corresponding protection level.
2. Method for GBAS protection class calculation according to claim 1, characterized in that said pseudorange domain to positioning domain projection matrix Sν,j＝(GTWG)-1GTW; wherein G is an orientation cosine matrix and W is a weighting matrix; the orientation cosine matrix is obtained by a least square differential positioning solution of the airborne receiver.
3. The GBAS protection level calculation method of claim 2,
the weighting matrix
Figure FDA0003633399560000022
Wherein, the first and the second end of the pipe are connected with each other,
Figure FDA0003633399560000023
is the pseudorange error standard deviation for the jth satellite.
4. The method for calculating the level of protection for GBAS as in claim 3 wherein the pseudorange error standard deviation for the jth satellite
Figure FDA0003633399560000024
Wherein, the first and the second end of the pipe are connected with each other,
Figure FDA0003633399560000025
is the pseudorange error standard deviation of the ground station,
Figure FDA0003633399560000026
is the pseudorange error standard deviation of the airborne receiver,
Figure FDA0003633399560000027
is the standard deviation of the residual ionospheric error,
Figure FDA0003633399560000028
residual tropospheric error standard deviation.
5. The GBAS protection level calculation method of any one of claims 1 to 4, wherein a tail correlation characteristic of the localization domain error sequence is obtained by correlation analysis, and an extreme value model of the localization domain error sequence is established by performing data screening according to the tail correlation characteristic by a data screening method including a block method or a super-threshold method.
6. The GBAS protection level calculation method of claim 5,
and selecting pseudo-range error values obtained by resolving the same satellite at different ground stations as modeling data in the data screening by the block method, and selecting the maximum value in each interval according to a certain step length to screen the data.
7. The GBAS protection level calculation method of claim 5,
in the data screening by the super-threshold method, a threshold is set firstly, all observed data are processed integrally, modeling is carried out by using the part exceeding the threshold, and a proper threshold is selected for pseudo-range error values to carry out data screening.
8. The GBAS protection level calculation method of claim 1,
the parameter δ is estimated by a moment estimate or a maximum likelihood estimate.
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