A method for detecting gross errors in satellite gravity gradients based on tensor invariants and Dixon's statistical theory

Abstract

The present invention discloses a method for detecting gross errors in satellite gravity gradients based on tensor invariants and Dixon statistical theory, which relates to the technical field of geodesy. This method first establishes a tensor invariant system for satellite gravity gradient observations, calculates the time series of the first invariant and the Dixon statistic, and then globally detects gross errors in the main diagonal components of the satellite gravity gradient observation tensor. At the same time, by introducing a priori gravity field models, the residual sequences of the second and third invariants are calculated, and the Dixon statistic is calculated, so as to globally detect gross errors in all six tensor components of the satellite gravity gradient observation tensor. The method of the present invention is based on the tensor invariant theory, combines the tensor invariant characteristics of satellite gravity gradient observation data with the Dixon statistical theory, realizes the reliable detection of global gross errors in the satellite gravity gradient observation tensor, improves the efficiency and reliability of gross error detection, and provides an effective means for the accurate determination of the short-wave spectral structure in the Earth's gravity field.

Description

Technical Field

[0001] The present invention belongs to the technical field of geodetic surveying, and particularly relates to a method for detecting gross errors in satellite gravitational gradients based on tensor invariants and Dixon statistical theory. Background Technique

[0002] Gravity gradient satellites, as high-precision and sophisticated earth observation satellites, are crucial for determining the short-wave fine structure in the earth's gravity field. Changes in the on-orbit space physical environment of the satellite cause unstable changes in the measurement state of the gravity gradiometer carried by the satellite, resulting in inevitable gross errors in the satellite gravitational gradient observation data. The existence of gross errors in the observed values affects the mathematical statistical characteristics of the observed values, brings the inapplicability of data processing methods such as least squares adjustment and the ill-condition of the solution process, makes the usability of satellite gravitational gradient observation data poor, and restricts the improvement of the short-wave frequency spectrum accuracy in the earth's gravity field. Therefore, gross error detection and rejection are required for satellite gravitational gradient observation data.

[0003] The methods for detecting gross errors in satellite gravitational gradient observation data are mostly based on a certain component of the satellite gravitational gradient observation tensor and use various mathematical statistics methods. This processing process does not consider the mathematical relationships existing among the components of the satellite gravitational gradient observation tensor. The detection process is relatively cumbersome and has a certain degree of repetition. It is impossible to conduct an overall detection of the satellite gravitational gradient tensor, and it is impossible to achieve the mutual comparison and verification between the overall detection of the observation tensor and the detection results of gross errors in each component, resulting in poor gross error detection efficiency and reliability, indirectly bringing poor performance in the use of satellite data, and affecting the accurate determination of the short-wave frequency spectrum structure in the earth's gravity field. Summary of the Invention

[0004] The purpose of the present invention is to provide a method for detecting gross errors in satellite gravitational gradients based on tensor invariants and Dixon statistical theory. By combining the Dixon statistical theory with the tensor invariant mathematical characteristics of the satellite gravitational gradient tensor, it solves the problem that most of the current satellite gravitational gradient observation data is carried out based on a certain observation component, resulting in poor quality or even unusability of the satellite gravitational gradient observation data due to low gross error detection efficiency.

[0005] To solve the above technical problems, the present invention is realized through the following technical solutions:

[0006] The present invention is a method for detecting gross errors in satellite gravitational gradients based on tensor invariants and Dixon statistical theory, including the following steps:

[0007] Step 1: Overall detection of gross errors in the main diagonal components of the satellite gravitational gradient observation tensor;

[0008] Step 2: Overall detection of gross errors in all six tensor components of the satellite gravitational gradient observation tensor.

[0009] As a preferred technical solution of the present invention, step 1 specifically includes:

[0010] Step 101: Establish a tensor invariant system {I1, I2, I3} of satellite gravity gradient observation values;

[0011] Step 102: Calculate the time series of the first invariant of satellite gravity gradient observation values;

[0012] Step 103: Calculate the Dixon statistic of the first invariant sequence of satellite gravity gradient observation values

[0013] Step 104: According to the comparison result of the Dixon statistic with the critical value, overall detect the gross error of the main diagonal components of the satellite gravity gradient observation tensor.

[0014] As a preferred technical solution of the present invention, the expression of the tensor invariant system {I1, I2, I3} in step 101 is:

[0015] I1 = V 11 + V 22 + V 33 (1)

[0016]

[0017] As a preferred technical solution of the present invention, step 104 specifically includes:

[0018] Given a significance level α and a window size K;

[0019] Select statistic;

[0020] If the Dixon statistic of the first invariant of the satellite gravity gradient observation value one of which is greater than the critical value D of the Dixon test α , it is considered that the observation epoch where the Kth element of this window length is located is abnormal data, and a gross error mark is made;

[0021] If the Kth element is abnormal data, perform the detection step operation described in step 104 on the first K - 1 elements of this time series until no abnormal data can be detected.

[0022] As a preferred technical solution of the present invention, step 2 specifically includes:

[0023] Step 201: Establish the same tensor invariant system {I1, I2, I3} of satellite gravity gradient observation values as in step 101;

[0024] Step 202 calculates the residual sequences of the second invariant and the third invariant of the satellite gravity gradient.

[0025] Step 203 globally detects gross errors in all six tensor components of the satellite gravity gradient observation tensor. As a preferred technical solution of the present invention, the specific steps of Step 202 include:

[0026] Step 2021 calculates the prior model values of the gravity gradient tensor in the local north-referenced coordinate system using a prior gravity field model, converts it to the satellite gradiometer coordinate system, and then calculates the second invariant and the third invariant of the prior model values of the satellite gravity gradient.

[0027] Step 2022 calculates the second invariant and the third invariant of the satellite gravity gradient observation values in the satellite gradiometer coordinate system.

[0028] Step 2023 calculates the residual sequences of the second invariant and the third invariant of the satellite gravity gradient observation values.

[0029] As a preferred technical solution of the present invention, the specific steps of Step 203 include:

[0030] Step 2031 calculates the Dixon statistics of the residual sequences of the second invariant and the third invariant of the satellite gravity gradient observation values and

[0031] Step 2032 globally detects gross errors in all six tensor components of the satellite gravity gradient observation tensor based on the comparison results of the Dixon statistics and with the critical values.

[0032] As a preferred technical solution of the present invention, the specific steps of Step 2032 include:

[0033] Given a significance level α and a window size K, select and statistics;

[0034] If one of the Dixon statistics and of the second invariant and the third variable of the satellite gravity gradient observation values is greater than the critical value D α of the Dixon test, it is considered that the observation epoch where the Kth element of this window length is located is abnormal data;

[0035] If the Kth element is abnormal data, after performing a gross error mark, the detection step operation of Step 2031 is performed on the first K - 1 elements of this time series until no abnormal data can be detected.

[0036] The present invention has the following beneficial effects:

[0037] The overall detection method for gross errors of satellite gravity gradients based on tensor invariants and Dixon statistical theory disclosed by the present invention is based on the tensor invariant theory, combines the tensor invariant characteristics of satellite gravity gradient observation data with Dixon statistical theory, and applies it to the overall detection of gross errors in satellite gravity gradient observation data, realizing the reliable detection of overall gross errors of satellite gravity gradient observation tensors; by introducing a prior gravity field model, the overall detection of gross errors of the main diagonal components and all six tensor components of the gravity gradient tensor can be independently achieved.

[0038] Of course, it is not necessary for any product implementing the present invention to simultaneously achieve all the above-mentioned advantages. Description of the Drawings

[0039] To more clearly illustrate the technical solutions of the embodiments of the present invention, the accompanying drawings required for describing the embodiments will be briefly introduced below. Obviously, the accompanying drawings in the following description are only some embodiments of the present invention. For those of ordinary skill in the art, other accompanying drawings can be obtained based on these drawings without creative efforts.

[0040] Figure 1 is the flowchart of the satellite gravity gradient gross error detection method based on tensor invariants and Dixon statistical theory;

[0041] Figure 2 is the overall detection flowchart of gross errors of the main diagonal components of the satellite gravity gradient observation tensor;

[0042] Figure 3 is the overall detection flowchart of gross errors of all six tensor components of the satellite gravity gradient observation tensor. Detailed Embodiments

[0043] The technical solutions in the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings in the embodiments of the present invention. Obviously, the described embodiments are only a part of the embodiments of the present invention, rather than all the embodiments. All other embodiments obtained by those of ordinary skill in the art without creative efforts based on the embodiments of the present invention belong to the scope of protection of the present invention.

[0044] A specific application of the present invention is:

[0045] As Figure 1 shown, the satellite gravity gradient gross error detection method based on tensor invariants and Dixon statistical theory includes:

[0046] Step 1: Overall detection of gross errors of the main diagonal components of the satellite gravity gradient observation tensor;

[0047] Step 2 globally detects gross errors in all six tensor components of the satellite gravity gradient observation tensor.

[0048] As Figure 2 shown, the global detection of gross errors in the main diagonal components of the satellite gravity gradient observation tensor in the above Step 1 specifically includes the following steps:

[0049] Step 101 establishes a tensor invariant system {I1, I2, I3} of the satellite gravity gradient observation values;

[0050] The expression of the tensor invariant system {I1, I2, I3} of the satellite gravity gradient observation values is:

[0051] I1 = V 11 + V 22 + V 33 (1)

[0052]

[0053] In the formula: I1 is the first invariant of the satellite gravity gradient tensor invariant system; I2 is the second invariant of the satellite gravity gradient tensor invariant system; I3 is the third invariant of the satellite gravity gradient tensor invariant system; V 11 is the component of the satellite gravity gradient observation value in the xx direction of the satellite gradiometer coordinate system or the XX direction of the local north - pointing coordinate system; V 12 is the component of the satellite gravity gradient observation value in the xy direction of the satellite gradiometer coordinate system or the XY direction of the local north - pointing coordinate system; V 13 is the component of the satellite gravity gradient observation value in the xz direction of the satellite gradiometer coordinate system or the XZ direction of the local north - pointing coordinate system; V 22 is the component of the satellite gravity gradient observation value in the yy direction of the satellite gradiometer coordinate system or the YY direction of the local north - pointing coordinate system; V 23 is the component of the satellite gravity gradient observation value in the yz direction of the satellite gradiometer coordinate system or the YZ direction of the local north - pointing coordinate system; V 33 is the component of the satellite gravity gradient observation value in the zz direction of the satellite gradiometer coordinate system or the ZZ direction of the local north - pointing coordinate system.

[0054] It should be noted here that the global detection of gross errors in the main diagonal components of the satellite gravity gradient observation tensor proposed in the present invention only involves the first invariant of the satellite gravity gradient.

[0055] Step 102 calculates the time series of the first invariant of the satellite gravity gradient observation values.

[0056] The main diagonal components of the satellite gravity gradient observation values at different observation epochs k in the satellite gradiometer coordinate system Where \(i = x,y,z\), substitute it into the calculation formula (1) of the tensor invariant system to obtain the first invariant of the satellite gravitational gradient observation value at different observation epochs \(k\):

[0057]

[0058] Select the window length \(K\) of the satellite gravitational gradient observation value for the Dixon test, and for the first invariant of the satellite gravitational gradient observation value at different observation epochs \(k\) After taking the absolute value, sort them in ascending order to form a time series:

[0059]

[0060] Step 103 Calculate the Dixon statistic of the first invariant sequence of the satellite gravitational gradient observation value.

[0061] Introduce the Dixon test theory and use the pseudo-range difference and to estimate the standard deviation of the time series, and obtain the Dixon statistic of the first invariant of the satellite gravitational gradient observation value:

[0062]

[0063] Step 104 Detect the gross error of the main diagonal components of the satellite gravitational gradient observation tensor as a whole.

[0064] Given the significance level \(\alpha\) and the window size \(K\), select the statistic. If one of the Dixon statistics of the first invariant of the satellite gravitational gradient observation value is greater than the critical value \(D\) α of the Dixon test, then the observation epoch where the \(K\)th element of this window length is located is considered as abnormal data; if the \(K\)th element is abnormal data, after marking the gross error, perform the same detection step operation (Step 104) on the first \(K - 1\) elements of this time series until no abnormal data is detected. The \(D\) α values corresponding to different significance levels \(\alpha\) are shown in Table 1.

[0065] Thus, the overall detection of the gross error of the main diagonal components of the satellite gravitational gradient observation tensor is completed.

[0066] Table 1 Critical values of the Dixon test

[0067]

[0068] As Figure 3 shown, the above Step 2 specifically includes:

[0069] Step 201 Establish a tensor invariant system \(\{I_1, I_2, I_3\}\) of the satellite gravitational gradient observation value;

[0070] Step 202 calculates the residual sequences of the second invariant and the third invariant of the satellite gravity gradient.

[0071] Step 203 globally detects the gross errors of all six tensor components of the satellite gravity gradient observation tensor.

[0072] The said Step 201 is the same as Step 101.

[0073] The said Step 202 specifically includes:

[0074] Step 2021 calculates the prior model value of the satellite gravity gradient tensor in the local north-pointing coordinate system .

[0075] Calculate the prior model value of the gravity gradient tensor in the local north-pointing coordinate system using the prior gravity field model The calculation formula is:

[0076]

[0077] In the formula: GM is the geocentric gravitational constant, r, θ, and λ are the geocentric radius vector, geocentric co-latitude, and geocentric longitude respectively, R is the average radius of the Earth, n and m are the degree and order of the spherical harmonic expansion of the Earth's gravity field, and N is the highest degree of the adopted prior gravity field model. λ ij , are the gravity gradient tensor coefficients, and their expressions are shown in Table 2, where is the fully normalized spherical harmonic coefficient of the gravitational potential of the prior gravity field model, and are the fully normalized associated Legendre functions and their first and second derivatives with respect to the geocentric co-latitude θ respectively.

[0078] Table 2 Expressions of gravity gradient components in the local north-pointing coordinate system

[0079]

[0080]

[0081] Substitute the geocentric radius vector, geocentric co-latitude, geocentric longitude, and prior gravity field model coefficients corresponding to a certain observation epoch k into (6), and the prior model value of the satellite gravity gradient tensor in the local north-pointing coordinate system at the observation epoch k can be obtained

[0082] Calculation of the prior model value of the satellite gravity gradient tensor in the satellite gradiometer coordinate system .

[0083] Convert the prior model value of the satellite gravitational gradient tensor in the local north-pointing coordinate system to the satellite gradiometer coordinate system through the following conversion processes: local north-pointing coordinate system → Earth-fixed coordinate system → inertial coordinate system → gradiometer coordinate system. The conversion formula is:

[0084]

[0085] In the formula: and are the satellite gravitational gradient tensors before and after coordinate transformation respectively, and R is the coordinate transformation matrix.

[0086] Substitute the prior model value of the satellite gravitational gradient tensor in the local north-pointing coordinates at the observation epoch k k and the coordinate transformation matrix R at this moment

[0087] into (7), and the prior model value of the satellite gravitational gradient tensor in the satellite gradiometer coordinate system at the observation epoch k can be obtained

[0088] Calculation of the second invariant of the prior model value of the satellite gravitational gradient and the third invariant of the prior model value of the satellite gravitational gradient in the satellite gradiometer coordinate system. Substitute the prior model value of the satellite gravitational gradient tensor

[0089]

[0090] in the satellite gradiometer coordinate system at different observation epochs k

[0091] into the calculation formulas (2) and (3) of the tensor invariant system, and the second invariant of the prior model value of the satellite gravitational gradient and the third invariant of the prior model value of the satellite gravitational gradient at different observation epochs k are obtained:

[0092]

[0093] Calculation of the second invariant and the third invariant of the satellite gravitational gradient observation value in the satellite gradiometer coordinate system in step 2022.

[0094] Substitute the component of the satellite gravitational gradient observation value in the satellite gradiometer coordinate system at different observation epochs k

[0092]

[0093] into the calculation formulas (2) and (3) of the tensor invariant system, and the second invariant of the satellite gravitational gradient observation value and the third invariant of the satellite gravitational gradient observation value at different observation epochs k are obtained:

[0094] Calculation of the second invariant residual sequence of the satellite gravitational gradient observation value and the third invariant residual sequence of the satellite gravitational gradient observation value in the satellite gradiometer coordinate system in step 2023.

[0094] The differences are respectively calculated between the second invariant of the prior model value of the satellite gravity gradient and the third invariant of the prior model value of the satellite gravity gradient, as well as between the second invariant of the observed value of the satellite gravity gradient and the third invariant of the observed value of the satellite gravity gradient at different observation epochs k, to obtain the residual sequence of the second invariant of the observed value of the satellite gravity gradient and the residual sequence of the third invariant of the observed value of the satellite gravity gradient:

[0095]

[0096] Select the window length K of the observed value of the satellite gravity gradient for the Dixon test, and for the residual sequence of the second invariant of the observed value of the satellite gravity gradient at different observation epochs k and the residual sequence of the third invariant of the observed value of the satellite gravity gradient After taking the absolute values, they are sorted in ascending order to respectively form time series:

[0097]

[0098] The specific steps of step 203 include:

[0099] Step 203 globally detects gross errors in all six tensor components of the observed satellite gravity gradient tensor.

[0100] Step 2031 calculates the Dixon statistics of the residual sequence of the second invariant of the observed value of the satellite gravity gradient and the residual sequence of the third invariant of the observed value of the satellite gravity gradient.

[0101] Introduce the Dixon test theory and use the pseudo-range and to estimate the standard deviation of the time series, and obtain the Dixon statistic of the second invariant of the observed value of the satellite gravity gradient:

[0102]

[0103] Use the pseudo-range and to estimate the standard deviation of the time series, and obtain the Dixon statistic of the third invariant of the observed value of the satellite gravity gradient:

[0104]

[0105] Step 2032 globally detects gross errors in all tensor components of the observed satellite gravity gradient tensor.

[0106] Given the significance level α and the window size K, select the statistic. If one of the Dixon statistics of the second invariant of the observed value of the satellite gravity gradient is greater than the critical value D of the Dixon test α, it is considered that the observation epoch where the K-th element of the window length is an outlier. If the K-th element is an outlier, after performing a gross error mark, the same detection step operation (step 2031) is performed on the first K - 1 elements of the time series until no outlier can be detected. Similarly, given a significance level α and a window size K, the statistic is selected. If the Dixon statistic of the third invariant of the satellite gravitational gradient observation value is greater than the critical value D of the Dixon test α , it is considered that the observation epoch where the K-th element of the window length is an outlier. If the K-th element is an outlier, after performing a gross error mark, the same detection step operation (step 2031) is performed on the first K - 1 elements of the time series until no outlier can be detected. The D α values corresponding to different significance levels α are shown in Table 1.

[0107] Thus, the overall detection of gross errors for all six tensor components of the satellite gravitational gradient observation tensor is completed.

[0108] In the description of this specification, the descriptions referring to terms such as "one embodiment", "example", "specific example", etc. mean that the specific features, structures, materials or characteristics described in connection with the embodiment or example are included in at least one embodiment or example of the present invention. In this specification, the schematic representations of the above terms do not necessarily refer to the same embodiment or example. Moreover, the specific features, structures, materials or characteristics described can be combined in any one or more embodiments or examples in a suitable manner.

[0109] The preferred embodiments of the present invention disclosed above are only used to help explain the present invention. The preferred embodiments do not elaborate on all the details, nor do they limit the invention to the specific embodiments described. Obviously, many modifications and variations can be made according to the content of this specification. These embodiments are selected and specifically described in this specification to better explain the principles and practical applications of the present invention, so that those skilled in the art in the relevant technical field can well understand and utilize the present invention. The present invention is only limited by the claims and their full scope and equivalents.

Claims

1. A method for detecting gross errors in satellite gravity gradients based on tensor invariants and Dixon statistical theory, characterized in that It includes the following steps: Step 1: Overall detection of gross errors in the main diagonal components of the satellite gravity gradient observation tensor; The specific content of Step 1 includes: Step 101: Establish a tensor invariant system of satellite gravity gradient observation values ; Step 102 Calculate the time series of the first invariant of the satellite gravity gradient observation values; Step 103 calculates the Dixon statistic of the time series of the first invariant of the satellite gravity gradient observations ; Step 104 determines the gross error of the main diagonal components of the satellite gravity gradient observation tensor as a whole according to the comparison result between the Dixon statistic and the critical value; Step 2: Overall detection of gross errors in all six tensor components of the satellite gravity gradient observation tensor.

2. A satellite gravity gradient gross error detection method based on tensor invariance and Dixon statistical theory according to claim 1, characterized in that The tensor invariant system in step 101 has the following expression: （1） （2） （3） Where: I1 is the first invariant of the satellite gravity gradient tensor invariant system; I2 is the second invariant of the satellite gravity gradient tensor invariant system; I3 is the third invariant of the satellite gravity gradient tensor invariant system; V11 is the component of the satellite gravity gradient observation value in the xx direction of the satellite gradiometer coordinate system or the XX direction of the local north-pointing coordinate system; V12 is the component of the satellite gravity gradient observation value in the xy direction of the satellite gradiometer coordinate system or the XY direction of the local north-pointing coordinate system; V13 is the component of the satellite gravity gradient observation value in the xz direction of the satellite gradiometer coordinate system or the XZ direction of the local north-pointing coordinate system; V22 is the component of the satellite gravity gradient observation value in the yy direction of the satellite gradiometer coordinate system or the YY direction of the local north-pointing coordinate system; V23 is the component of the satellite gravity gradient observation value in the yz direction of the satellite gradiometer coordinate system or the YZ direction of the local north-pointing coordinate system; V33 is the component of the satellite gravity gradient observation value in the zz direction of the satellite gradiometer coordinate system or the ZZ direction of the local north-pointing coordinate system.

3. A satellite gravity gradient gross error detection method based on tensor invariance and Dixon statistical theory according to claim 1, characterized in that, The specific content of Step 104 includes: Given significance level and window size K ; Select statistic; If the Dixon statistic of the first invariant of the satellite gravity gradient observation value is greater than the critical value of the Dixon test , it is considered that the observation epoch where the Kth element of the window length is located is abnormal data, and a gross error mark is made; If the Kth element is abnormal data, perform the detection step operation described in Step 104 on the first K - 1 elements of this time series until no abnormal data can be detected.

4. A satellite gravity gradient gross error detection method based on tensor invariants and Dixon statistical theory according to claim 2, characterized in that The specific content of Step 2 includes: Step 201: Establish a tensor invariant system of satellite gravity gradient observations identical to that in Step 101 ; Step 202 Calculate the residual sequences of the second invariant and the third invariant of the satellite gravity gradient; Step 203 Overall detection of gross errors in all six tensor components of the satellite gravity gradient observation tensor.

5. A method for detecting gross errors in satellite gravity gradients based on tensor invariants and Dixon statistical theory according to claim 4, characterized in that The specific content of Step 202 includes: Step 2021 Use the prior gravity field model to calculate the prior model values of the gravity gradient tensor in the local north-pointing coordinate system, and convert them to the satellite gradiometer coordinate system, and then calculate the second invariant and the third invariant of the prior model values of the satellite gravity gradient; Step 2022 Calculate the second invariant and the third invariant of the satellite gravity gradient observation values in the satellite gradiometer coordinate system; Step 2023 Calculate the residual sequences of the second invariant and the third invariant of the satellite gravity gradient observation values.

6. A method for detecting gross errors in satellite gravity gradients based on tensor invariants and Dixon statistical theory according to claim 5, characterized in that The specific content of Step 203 includes: Step 2031 calculates the Dixon statistics of the second invariant residual sequence and the third invariant residual sequence of the satellite gravity gradient observation values and ; Step 2032, according to the Dixon statistic and Based on the comparison result with the critical value, globally detect gross errors in all six tensor components of the satellite gravity gradient observation tensor and all six tensor components of the gross error observation tensor.

7. A method for detecting gross errors in satellite gravity gradients based on tensor invariants and Dixon statistical theory according to claim 6, characterized in that The specific steps of Step 2032 include: Given significance level and window size K , select and statistics; If the Dixon statistic of the second invariant and the third variable of the satellite gravity gradient observation value is greater than the critical value of the Dixon test , it is considered that the observation epoch where the K th element of the window length is abnormal data; The K element is abnormal data. After performing a gross error mark, the detection step operation of step 2031 is performed on the first K -1 elements of the time series until no abnormal data can be detected.

Images