CN109270560B - Multi-dimensional gross error positioning and value fixing method for area elevation abnormal data - Google Patents

Multi-dimensional gross error positioning and value fixing method for area elevation abnormal data Download PDF

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CN109270560B
CN109270560B CN201811187757.0A CN201811187757A CN109270560B CN 109270560 B CN109270560 B CN 109270560B CN 201811187757 A CN201811187757 A CN 201811187757A CN 109270560 B CN109270560 B CN 109270560B
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张志伟
胡伍生
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Southeast 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
    • G01S19/38—Determining a navigation solution using signals transmitted by a satellite radio beacon positioning system
    • G01S19/39—Determining a navigation solution using signals transmitted by a satellite radio beacon positioning system the satellite radio beacon positioning system transmitting time-stamped messages, e.g. GPS [Global Positioning System], GLONASS [Global Orbiting Navigation Satellite System] or GALILEO
    • G01S19/40—Correcting position, velocity or attitude
    • 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
    • G01S19/38—Determining a navigation solution using signals transmitted by a satellite radio beacon positioning system
    • G01S19/39—Determining a navigation solution using signals transmitted by a satellite radio beacon positioning system the satellite radio beacon positioning system transmitting time-stamped messages, e.g. GPS [Global Positioning System], GLONASS [Global Orbiting Navigation Satellite System] or GALILEO
    • G01S19/42—Determining position

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Abstract

The invention discloses a multi-dimensional gross error positioning and value fixing method for regional elevation abnormal data, which organically combines the dimensional, positioning and value fixing according to a specified flow, and has accurate and efficient gross error positioning. Through analysis of results of a large number of example applications, when the coarse difference dimension reaches 1/6 of a total sample, and the maximum value of the coarse difference reaches 6 times of the medium error, the coarse difference can still be accurately positioned. Because the coarse difference value is obtained while the multi-dimensional coarse difference positioning is carried out, the invention has more flexible processing on the elevation abnormal value containing the coarse difference value, and particularly when the number of the control points is close to 3 times that of the parameters to be estimated.

Description

Multi-dimensional gross error positioning and value fixing method for area elevation abnormal data
Technical Field
The invention relates to the field of geodetic surveying, in particular to a multi-dimensional gross error positioning and value-fixing method for area elevation abnormal data.
Background
An area is a portion of the surface of the earth and an elevation anomaly is the difference between the earth's ellipsoid and the earth's quasi-geoid. Determining regional elevation anomalies is one of the key tasks in geodetic and engineering tasks. Currently, there are two main calculation methods for determining the area elevation anomaly: (1) gravity is like the calculation of the geodetic level. The elevation abnormity is determined according to the Molokinsky theory, and a two-time removal-recovery technical method is generally adopted in the calculation process. But the precision is lower, and the engineering requirement is difficult to meet. (2) And calculating the elevation anomaly by using the GPS level. The method is to obtain the GPS geodetic height and the precise level normal height of the point on the same point, the difference value of the GPS geodetic height and the precise level normal height is the elevation abnormity of the point, and the precision is higher than that of the first method. Moreover, if enough GPS level points are distributed in an area, an elevation anomaly calculation model of the area can be established by calculating elevation anomaly values of the points. The method (2) is a commonly used method for determining the elevation abnormality of the area (provincial and urban level). The accuracy of the elevation anomaly model depends on the data quality of elevation anomaly values, namely whether the elevation anomaly values contain gross errors or not, in addition to the accuracy of GPS geodetic determination and the accuracy of leveling measurement. If the gross error cannot be located and valued effectively (for correcting the gross error), it will have a serious impact on the estimation result of the parameters. The rough difference detection, positioning and setting of the elevation abnormal values are the premise for establishing an effective elevation abnormal model.
There are various methods for gross error testing, including the Bayes method; a green estimation-based gross error detection method; a standard detection method is simulated; a plurality of gross errors detection method based on variance expansion model; and distinguishing the multi-dimensional gross errors by applying a partial correlation coefficient, and carrying out overall significance detection and positioning on the multi-dimensional gross errors by using a complex correlation coefficient. In general, multi-dimensional gross error detection and localization are difficult, mainly the dimensionality of the gross error is difficult to determine. Although the one-dimensional gross error detection method is simple and feasible, the correlation of the gross error on the residual error cannot be considered, and the existence of a plurality of gross errors is judged successively by the method, so that the possibility of error judgment is high; when a plurality of gross errors are determined simultaneously by using a heuristic method, the combination number of the heuristic method is quite large and is difficult to realize when the data is more.
Disclosure of Invention
The purpose of the invention is as follows: the invention aims to provide a multi-dimensional gross error positioning and value fixing method for area elevation abnormal data, which can accurately position gross errors.
The technical scheme is as follows: in order to achieve the purpose, the invention adopts the following technical scheme:
the invention relates to a multi-dimensional gross error positioning and value fixing method for elevation abnormal data of an area, which comprises the following steps:
s1: obtaining elevation abnormal values of n GPS points to establish a model as follows:
Figure BDA0001826683410000021
in the formula (1), LiIs the elevation anomaly value of the ith GPS point, Li＝HGi-H0i，HGiGeodetic height, H, for the ith GPS point0iNormal high for the ith GPS point, i ═ 1, …, n; xjJ is 0, 1, …, 5, which is the undetermined parameter of the quadric surface model; x is the ordinate of the GPS point on the Gaussian plane, and y is the abscissa of the GPS point on the Gaussian plane;
writing equation (1) in matrix form:
L＝AX+Δ (2)
in the formula (2), the reaction mixture is,
Figure BDA0001826683410000022
is an elevation outlier vector;
Figure BDA0001826683410000023
is a parameter vector to be determined;
Figure BDA0001826683410000024
is a matrix of coefficients for the unknown parameters,
Figure BDA0001826683410000025
is a random error vector;
s2: obtaining a least squares estimate of X according to equation (3)
Figure BDA0001826683410000026
Obtaining the Unit weight variance according to equation (4)
Figure BDA0001826683410000027
Least squares estimation of
Figure BDA0001826683410000028
Figure BDA0001826683410000029
Figure BDA00018266834100000210
Wherein N is ATPA and P are weight arrays; v is the least square estimation of the random error vector, obtained according to equation (5);
Figure BDA00018266834100000211
in formula (5), H ═ A (A)TPA)-1ATP；
S3: obtaining Δ t (i) according to formula (6):
ΔT(i)＝T(i)-Ta (6)
in the formula (6), T (i) is the influence function statistic of the ith GPS point, TaIs the arithmetic mean of T (1), …, T (n), obtained according to formula (7), TaObtained according to formula (8);
Figure BDA00018266834100000212
in the formula (7), the reaction mixture is,
Figure BDA00018266834100000213
performing least square estimation of regression X from the remaining n-1 groups of data after deleting the ith group of data;
Figure BDA00018266834100000214
strong points of influence are determined by the following method: when the delta T (i) is more than 0, the ith point is a strong influence point;
making the sum of the number of all strong influence points be m;
s4: step S4 specifically includes the following steps:
s4.1: the test statistic F is calculated according to equation (9):
Figure BDA0001826683410000031
in the formula (9), the reaction mixture is,
Figure BDA0001826683410000032
obtained according to the formula (10),
Figure BDA0001826683410000033
obtained according to formula (11), omega obtained according to formula (12), F(α，m，n-6-m)The method is characterized in that the method is used for testing an F quantile value with the level of alpha and the degree of freedom of (m, n-6-m);
Figure BDA0001826683410000034
Figure BDA0001826683410000035
Figure BDA0001826683410000036
in the formula (12)
Figure BDA0001826683410000037
Obtained according to the formula (13),
Figure BDA0001826683410000038
wherein B ═ E1，E2，…，Em) Is an n multiplied by m matrix; ekIs an n × 1 vector, k is 1, …, m, where the i-th row element is 1 and the remaining elements are zero;
s4.2: gross error location was performed by the following method: judging that F is larger than FαWhether or not: if yes, judging that at least one coarse difference point exists, and performing step S4.3; otherwise, the coarse difference point does not exist, the following steps are not carried out, and the calculation is directly finished;
s4.3: obtaining test statistic F of the u strong influence point according to equation (14)u：
Figure BDA0001826683410000039
In the formula (14), F(α，1，n-6-m)To examine the F quantile value with level alpha and degree of freedom (1, n-6-m),
Figure BDA00018266834100000310
is composed of
Figure BDA00018266834100000311
The u-th main diagonal element of (1);
when F is presentu＞FαThe u-th point is a coarse difference point;
s4.4: counting the number m of coarse difference points according to the calculation1；
S5: let B1＝(E1，E2，…，E m1) Is n × m1Matrix, EhIs n × 1 vector, h is 1, …, m1Wherein the ith row element is 1 and the rest elements are zero; solving according to equation (15) to obtain a coarse difference value
Figure BDA00018266834100000312
Figure BDA0001826683410000041
In the formula (15), the reaction mixture is,
Figure BDA0001826683410000042
s6: if m is in step S41Not equal to 0, there are two options to choose one:
the first method comprises the following steps: removing m1Data containing coarse difference points to obtain n1＝n-m1Grouping the elevation abnormal values, and circulating the steps S1-S4 until no gross error points exist;
and the second method comprises the following steps: calculating the elevation abnormal value after gross error correction according to the formula (16), and circulating the steps S1-S4 until no gross error point exists;
Figure BDA0001826683410000043
in the formula (16), the compound represented by the formula,
Figure BDA0001826683410000044
the elevation abnormal value of the ith GPS point after rough error correction,
Figure BDA0001826683410000045
is the gross error value of the ith GPS point.
Has the advantages that: the invention discloses a multi-dimensional gross error positioning and value fixing method for area elevation abnormal data, which has the following beneficial effects compared with the prior art:
1) the gross error positioning is accurate and efficient; through analysis of a large number of example application results, when the coarse difference dimension reaches 1/6 of a total sample, and the maximum value of the coarse difference reaches 6 times of the medium error, the coarse difference can still be accurately positioned;
2) because the coarse difference value is obtained while the multi-dimensional coarse difference positioning is carried out, the invention has more flexible processing on the elevation abnormal value containing the coarse difference value, and particularly when the number of the control points is close to 3 times that of the parameters to be estimated.
Detailed Description
The technical solution of the present invention will be further described with reference to the following embodiments.
The specific embodiment discloses a multi-dimensional gross error positioning and value-fixing method for area elevation abnormal data, and the method of the specific embodiment is described in detail below by combining a specific engineering example.
Obtaining elevation abnormal values of n-36 GPS points to establish the following model:
Figure BDA0001826683410000046
in the formula (1), LiIs the elevation anomaly value of the ith GPS point, Li＝HGi-H0i，HGiGeodetic height, H, for the ith GPS point0iNormal high for the ith GPS point, i ═ 1, …, n; xjJ is 0, 1, …, 5, which is the undetermined parameter of the quadric surface model; x is the ordinate of the GPS point on the Gaussian plane, and y is the abscissa of the GPS point on the Gaussian plane;
writing equation (1) in matrix form:
L＝AX+Δ (2)
in the formula (2), the reaction mixture is,
Figure BDA0001826683410000051
is an elevation outlier vector;
Figure BDA0001826683410000052
is a parameter vector to be determined;
Figure BDA0001826683410000053
is a matrix of coefficients for the unknown parameters,
Figure BDA0001826683410000054
is a random error vector.
In order to obtain the elevation abnormal values of 36 GPS points, two measurement tasks, namely GPS measurement and leveling measurement, are required to be carried out on the measurement control points.
GPS measurement: according to the requirements of national measurement specifications, all control points are subjected to corresponding-grade GPS measurement to obtain the Gaussian plane coordinates (x) of each control pointi，yi) And ground height HGi。
Leveling: according to the requirements of national measurement specifications, all control points are subjected to level measurement of corresponding levels to obtain the normal height H of each control point0iAnd calculating the elevation abnormity of each control point.
In one embodiment, the area of the region is about 200km2The elevation anomaly model of (2) is used for distributing 36 measurement control points. And C-level GPS measurement and third-level leveling measurement are carried out on all control points according to the requirements of national measurement specifications, and GPS elevation abnormal values without gross errors are obtained. Now, the difference of the error of 2.5 to 6 times, specifically (0.030m, 0.035m, -0.025m, 0.040m, 0.060m, -0.040m) is applied to spots No. 5, 10, 15, 20, 25 and 30T. Data containing gross errors are obtained as shown in table 1:
TABLE 1 GPS elevation anomaly table with gross error
Figure BDA0001826683410000055
S2: obtaining a least squares estimate of X according to equation (3)
Figure BDA0001826683410000056
Obtaining the Unit weight variance according to equation (4)
Figure BDA0001826683410000057
Least squares estimation of
Figure BDA0001826683410000061
Figure BDA0001826683410000062
Figure BDA0001826683410000063
Wherein N is ATPA and P are weight arrays; v is the least square estimation of the random error vector, obtained according to equation (5);
Figure BDA0001826683410000068
in formula (5), H ═ A (A)TPA)-1ATP；
S3: obtaining Δ t (i) according to formula (6):
ΔT(i)＝T(i)-Ta (6)
in the formula (6), t (i) is the influence function statistic of the ith GPS point, as shown in table 2; t isaIs the arithmetic mean of T (1), …, T (n), obtained according to formula (7), TaObtained according to formula (8);
TABLE 2T (i) (dimensionless) TABLE
Figure BDA0001826683410000064
Figure BDA0001826683410000065
In the formula (7), the reaction mixture is,
Figure BDA0001826683410000066
performing least square estimation of regression X from the remaining n-1 groups of data after deleting the ith group of data;
Figure BDA0001826683410000067
strong points of influence are determined by the following method: when the delta T (i) is more than 0, the ith point is a strong influence point; it can be seen that Δ t (i) > 0 at 1, 4, 5, 10, 15, 16, 20, 25 and 30 points, so it is a strong point of influence;
making the sum of the number of all strong influence points be m, and then m is equal to 9;
s4: step S4 specifically includes the following steps:
s4.1: the test statistic F is calculated according to equation (9):
Figure BDA0001826683410000078
in the formula (9), the reaction mixture is,
Figure BDA0001826683410000071
obtained according to the formula (10),
Figure BDA0001826683410000072
obtained according to formula (11), omega obtained according to formula (12), F(α，m，n-6-m)F quantile values with test level α ═ 0.05, degree of freedom (m, n-6-m);
Figure BDA0001826683410000073
Figure BDA0001826683410000074
Figure BDA0001826683410000075
in the formula (12)
Figure BDA0001826683410000076
Obtained according to the formula (13),
Figure BDA0001826683410000077
wherein B ═ E1，E2，…，Em) Is an n multiplied by m matrix; ekIs an n × 1 vector, k is 1, …M, where the element in row i is 1 and the remaining elements are zero;
Figure BDA0001826683410000081
s4.2: gross error location was performed by the following method: judging that F is larger than FαWhether or not: if yes, judging that at least one coarse difference point exists, and performing step S4.3; otherwise, the coarse difference point does not exist, the following steps are not carried out, and the calculation is directly finished;
the positioning result is shown in table 3, and all the 6 gross error points are positioned correctly;
TABLE 3 gross error positioning results table
Figure BDA0001826683410000082
S4.3: obtaining test statistic F of the u strong influence point according to equation (14)u：
Figure BDA0001826683410000091
In the formula (14), F(α，1，n-6-m)To examine the F quantile value with level alpha and degree of freedom (1, n-6-m),
Figure BDA0001826683410000092
is composed of
Figure BDA0001826683410000093
The u-th main diagonal element of (1);
when F is presentu＞FαThe u-th point is a coarse difference point;
s4.4: counting the number m of coarse difference points according to the calculation1＝6；
S5: let B1＝(E1，E2，…，E m1) Is n × m1Matrix, EhIs n × 1 vector, h is 1, …, m1Wherein the ith row element is 1 and the other elements areZero; solving according to equation (15) to obtain a coarse difference value
Figure BDA0001826683410000094
Figure BDA0001826683410000095
In the formula (15), the reaction mixture is,
Figure BDA0001826683410000096
s6: if m is in step S41Not equal to 0, there are two options to choose one:
the first method comprises the following steps: removing m1Data containing coarse difference points to obtain n1＝n-m1Grouping the elevation abnormal values, and circulating the steps S1-S4 until no gross error points exist;
and the second method comprises the following steps: calculating the elevation abnormal value after gross error correction according to the formula (16), and circulating the steps S1-S4 until no gross error point exists;
Figure BDA0001826683410000097
in the formula (16), the compound represented by the formula,
Figure BDA0001826683410000098
the elevation abnormal value of the ith GPS point after rough error correction,
Figure BDA0001826683410000099
the rough difference value of the ith GPS point; the coarse difference (0.0315m, 0.0358m, -0.0251m, 0.0395m, 0.0617m, -0.0375m) is obtained according to formula (16)TThe relative error of the solution is 6% at the maximum.

Claims (1)

1. The multi-dimensional gross error positioning and value fixing method for the area elevation abnormal data is characterized by comprising the following steps of: the method comprises the following steps:
s1: obtaining elevation abnormal values of n GPS points to establish a model as follows:
Figure FDA0001826683400000011
in the formula (1), LiIs the elevation anomaly value of the ith GPS point, Li＝HGi-H0i，HGiGeodetic height, H, for the ith GPS point0iNormal high for the ith GPS point, i ═ 1, …, n; xjJ is 0, 1, …, 5, which is the undetermined parameter of the quadric surface model; x is the ordinate of the GPS point on the Gaussian plane, and y is the abscissa of the GPS point on the Gaussian plane;
writing equation (1) in matrix form:
L＝AX+Δ (2)
in the formula (2), the reaction mixture is,
Figure FDA0001826683400000012
is an elevation outlier vector;
Figure FDA0001826683400000013
is a parameter vector to be determined;
Figure FDA0001826683400000014
is a matrix of coefficients for the unknown parameters,
Figure FDA0001826683400000015
is a random error vector;
s2: obtaining a least squares estimate of X according to equation (3)
Figure FDA0001826683400000016
Obtaining the Unit weight variance according to equation (4)
Figure FDA0001826683400000017
Least squares estimation of
Figure FDA0001826683400000018
Figure FDA0001826683400000019
Figure FDA00018266834000000110
Wherein N is ATPA and P are weight arrays; v is the least square estimation of the random error vector, obtained according to equation (5);
Figure FDA00018266834000000111
in formula (5), H ═ A (A)TPA)-1ATP；
S3: obtaining Δ t (i) according to formula (6):
ΔT(i)＝T(i)-Ta (6)
in the formula (6), T (i) is the influence function statistic of the ith GPS point, TaIs the arithmetic mean of T (1), …, T (n), obtained according to formula (7), TaObtained according to formula (8);
Figure FDA00018266834000000112
in the formula (7), the reaction mixture is,
Figure FDA00018266834000000113
performing least square estimation of regression X from the remaining n-1 groups of data after deleting the ith group of data;
Figure FDA0001826683400000021
strong points of influence are determined by the following method: when the delta T (i) is more than 0, the ith point is a strong influence point;
making the sum of the number of all strong influence points be m;
s4: step S4 specifically includes the following steps:
s4.1: the test statistic F is calculated according to equation (9):
Figure FDA0001826683400000022
in the formula (9), the reaction mixture is,
Figure FDA00018266834000000210
obtained according to the formula (10),
Figure FDA0001826683400000023
obtained according to formula (11), omega obtained according to formula (12), F(α，m，n-6-m)The method is characterized in that the method is used for testing an F quantile value with the level of alpha and the degree of freedom of (m, n-6-m);
Figure FDA00018266834000000211
Figure FDA0001826683400000024
Figure FDA0001826683400000025
in the formula (12)
Figure FDA0001826683400000026
Obtained according to the formula (13),
Figure FDA0001826683400000027
wherein B ═ E1，E2，…，Em) Is an n multiplied by m matrix; ekIs an n × 1 vector, k is 1, …, m, where the i-th row element is 1 and the remaining elements are zero;
s4.2: gross error location was performed by the following method: judging that F is larger than FαWhether or not: if yes, judging that at least one coarse difference point exists, and performing step S4.3; otherwise, the coarse difference point does not exist, the following steps are not carried out, and the calculation is directly finished;
s4.3: obtaining test statistic F of the u strong influence point according to equation (14)u：
Figure FDA0001826683400000028
In the formula (14), F(α，1，n-6-m)To examine the F quantile value with level alpha and degree of freedom (1, n-6-m),
Figure FDA0001826683400000029
is composed of
Figure FDA00018266834000000212
The u-th main diagonal element of (1);
when F is presentu＞FαThe u-th point is a coarse difference point;
s4.4: counting the number m of coarse difference points according to the calculation1；
S5: let B1＝(E1，E2，…，E m1) Is n × m1Matrix, EhIs n × 1 vector, h is 1, …, m1Wherein the ith row element is 1 and the rest elements are zero; solving according to equation (15) to obtain a coarse difference value
Figure FDA0001826683400000031
Figure FDA0001826683400000032
In the formula (15), the reaction mixture is,
Figure FDA0001826683400000036
s6: if m is in step S41Not equal to 0, there are two options to choose one:
the first method comprises the following steps: removing m1Data containing coarse difference points to obtain n1＝n-m1Grouping the elevation abnormal values, and circulating the steps S1-S4 until no gross error points exist;
and the second method comprises the following steps: calculating the elevation abnormal value after gross error correction according to the formula (16), and circulating the steps S1-S4 until no gross error point exists;
Figure FDA0001826683400000033
in the formula (16), the compound represented by the formula,
Figure FDA0001826683400000034
the elevation abnormal value of the ith GPS point after rough error correction,
Figure FDA0001826683400000035
is the gross error value of the ith GPS point.
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