A method for designing weakly cross-correlation observation matrices for tropospheric tomography models

Abstract

This invention discloses a method for designing a weakly cross-correlated observation matrix for a tropospheric tomography model, comprising the following steps: First, dividing the tomographic grid and constructing an observation matrix based on satellite signal directions; Second, selecting random matrices such as Gaussian, Rademacher, and Bernoulli, dividing them into left and right modules, and concatenating them into a single matrix, using the new random matrix to correct the observation matrix; Third, using a cross-correlation function to measure the degree of cross-correlation between the observation matrix and the adaptive dictionary matrix, determining convergence. If convergence fails, reconstructing the random matrix to correct the observation matrix, and re-evaluating the cross-correlation until both matrices converge. This invention addresses the problem of severe ill-posedness in tomographic models due to numerous parameters to be estimated and strong correlations in tomographic observations. By designing an observation matrix with weak cross-correlation, it not only ensures a unique solution for sparse coefficients but also guarantees stable solutions for compressed sensing tropospheric tomography models.

Description

Technical Field

[0001] This invention relates to the field of radio navigation, and in particular to a method for designing a weakly cross-correlation observation matrix for a tropospheric tomography model. Background Technology

[0002] The troposphere is a crucial component of Earth's near-Earth space environment and is closely related to human life. Accurately acquiring high-precision tropospheric information can provide data support for early warning of severe weather events such as rainstorms and typhoons, and has significant scientific and practical value for studying climate change and improving weather forecasting.

[0003] The tropospheric delay effect is also a major bottleneck restricting the positioning, navigation and timing (PNT) of my country's BeiDou Navigation Satellite System (BDS) and Global Navigation Satellite System (GNSS), and is of great significance to the research of space geodesy technology.

[0004] High-precision tropospheric grid information broadcast from ground-based systems provides data support for PNT enhancement and extreme weather warnings, but the large number of model parameters presents a challenge for practical applications. GNSS tropospheric delay modeling mainly includes empirical tropospheric models, regional / global 2D models, and tropospheric tomography models. Empirical tropospheric models are easy to use, but their lower accuracy cannot meet the needs of high-precision PNT applications and weather warning services. Regional / global 2D models can provide high-precision tropospheric correction information in a two-dimensional plane, but their large data volume is not conducive to real-time broadcasting. The large number of model parameters and other issues pose greater challenges to satellite-based broadcasting applications.

[0005] To meet the needs of high-precision PNT applications and weather warning services, it is necessary to improve the accuracy of empirical tropospheric models. Although some progress has been made in atmospheric inversion research based on compressed sensing at home and abroad, there are still problems to be solved in terms of data sparsity, adaptive dictionary, and cross-correlation calculation of observation matrices.

[0006] To address the problem of severe ill-posedness in tomographic models due to the large number of parameters to be estimated and the strong correlation of observations in the tomographic region, it is urgent to design an observation matrix with weak cross-correlation. This not only ensures a unique solution for the sparse coefficients but also guarantees the stable solution of the compressed sensing tropospheric tomographic model. Summary of the Invention

[0007] To address the shortcomings of existing technologies, the purpose of this invention is to provide a design method for a weakly cross-correlation observation matrix in a tropospheric tomography model. In order to reduce the influence of ill-posed problems in the tomographic equation model, a compressed sensing tropospheric tomography model is established. Data such as radiosonde and ECMWF are used to evaluate the compressed sensing tomography results from multiple aspects. A cross-correlation coefficient is designed to measure the correlation strength of the observation matrix, and the functional relationship between the cross-correlation coefficient of the observation matrix and the accuracy of tomographic modeling is explored.

[0008] The above-mentioned technical objective of this invention is achieved through the following technical solution: a method for designing a weak cross-correlation observation matrix for a tropospheric tomography model, comprising the following steps:

[0009] Step 1: Analyze the tomographic grid and construct the observation matrix based on the satellite signal direction. ;

[0010] Step 2: Select random matrices from Gaussian, Rademacher, Bernoulli, etc., divide them into left and right modules, mix and concatenate them into a new matrix. Correcting the observation matrix yields ;

[0011] Step 3: Use the cross-correlation function to measure the observation matrix With adaptive dictionary matrix The degree of cross-correlation between the two matrices is used to determine whether convergence has occurred. If convergence has not occurred, a new random matrix is ​​constructed to correct the observation matrix, and the degree of cross-correlation is determined again until the two matrices converge.

[0012] In a preferred embodiment, the present invention can be further configured to: construct the observation matrix in the first step. In this process, the tomographic mesh is divided, including the following steps:

[0013] S1. Divide the chromatography region in the horizontal and vertical directions to obtain several uniformly distributed small cuboids, i.e., voxel blocks.

[0014] S2. Treat the water vapor density of each grid at the center point as a parameter to be estimated. Assuming that the water vapor density within the grid is uniform and constant in any given time period, the water vapor content along the satellite signal path can be discretized into the following form:

[0015] ;

[0016] in, This represents the oblique path water vapor observation value corresponding to the GNSS signal. and Indicates the GNSS signal at the 1st Layer, First line, number The intercept information and water vapor density value in the voxel block are used if the GNSS signal does not pass through the voxel block. =0;

[0017] Representing all rays within the study area in the above form, we obtain the observation equation:

[0018] ;

[0019] in, This is a column vector composed of delay information on rays emanating from the top of the study area. For the observation matrix, It is a column matrix composed of unknown parameters such as water vapor density.

[0020] In a preferred embodiment, the present invention can be further configured to include the following steps in the construction of the random matrix in the second step:

[0021] S1. Determine the size of the matrix dimension to ensure that the dimension of the random matrix matches the dimension of the three-dimensional tomographic observation matrix, so as to ensure that effective corrections can be made and the randomness intensity of the random matrix can be controlled.

[0022] S2. Based on the determined parameters, select appropriate random matrices for block splicing and mixing, mainly considering random matrices such as Gaussian, Rademacher, and Bernoulli.

[0023] S3, random matrix With observation matrix The corrected observation matrix is ​​obtained by performing multiplication. , .

[0024] In a preferred embodiment, the present invention can be further configured as follows: after correcting the observation matrix with a random matrix in the third step, a cross-correlation function is used to measure the corrected observation matrix. With adaptive dictionary matrix The degree of cross-correlation between them includes the following steps:

[0025] S1. Calculate the cross-correlation coefficient between the corrected observation matrix and the adaptive dictionary matrix, and determine whether they converge. The cross-correlation coefficient is defined as follows:

[0026] ;

[0027] in, and These are the observation matrices. and dictionary matrix The row vectors and column vectors, and They are respectively The number of rows and columns, This is called the Welch lower bound;

[0028] , A smaller cross-correlation value indicates a weaker cross-correlation, resulting in higher accuracy in calculating the sparse coefficients. Cross-correlation is considered to be excellent when it is less than a certain threshold.

[0029] S2. If the obtained cross-correlation coefficient does not converge, reconstruct the random matrix and correct the observation matrix again. Calculate the cross-correlation coefficient between the corrected observation matrix and the dictionary matrix to determine whether it converges. If it does not converge, continue to repeat the above steps until it converges.

[0030] By adopting the above technical solution,

[0031] In summary, the present invention has the following beneficial effects: to address the problem that the tomographic model suffers from severe ill-posedness due to the large number of parameters to be estimated and the strong correlation of observations in the tomographic region, an observation matrix with weak cross-correlation is designed. This not only ensures that the solution for the sparse coefficients has a unique solution, but also guarantees the stable solution of the compressed sensing tropospheric tomographic model. Attached Figure Description

[0032] Figure 1 These are the design steps for a weakly cross-correlated observation matrix;

[0033] Figure 2 The distribution characteristics of certain observation matrices, Gaussian random matrices, and discrete cosine (DCT) dictionary values ​​are shown.

[0034] Figure 3 A technical approach to reduce the cross-correlation of observation matrices by constructing random matrices. Detailed Implementation

[0035] The present invention will be further described in detail below with reference to the accompanying drawings.

[0036] like Figure 1 , Figure 2 , Figure 3 As shown, a method for designing a weakly cross-correlation observation matrix for a tropospheric tomography model includes the following steps:

[0037] Step 1: Analyze the tomographic grid and construct the observation matrix based on the satellite signal direction. .

[0038] The first step involves constructing the observation matrix. In this process, the tomographic mesh is divided, including the following steps:

[0039] S1. Divide the chromatography region in the horizontal and vertical directions to obtain several uniformly distributed small cuboids, i.e., voxel blocks.

[0040] S2. Treat the water vapor density of each grid at the center point as a parameter to be estimated. Assuming that the water vapor density within the grid is uniform and constant in any given time period, the water vapor content along the satellite signal path can be discretized into the following form:

[0041] ;

[0042] in, This represents the oblique path water vapor observation value corresponding to the GNSS signal. and Indicates the GNSS signal at the 1st Layer, First line, number The intercept information and water vapor density value in the voxel block are used if the GNSS signal does not pass through the voxel block. It is 0.

[0043] Representing all rays within the study area in the above form, we obtain the observation equation:

[0044] ;

[0045] in, This is a column vector composed of delay information on rays emanating from the top of the study area. For the observation matrix, It is a column matrix composed of unknown parameters such as water vapor density.

[0046] Step 2: Select random matrices from Gaussian, Rademacher, Bernoulli, etc., divide them into left and right modules, mix and concatenate them into a new matrix. Correcting the observation matrix yields Gaussian, Rademacher, and Bernoulli are Gauss, Rademacher, and Bernoulli, respectively.

[0047] The construction of the random matrix in the second step includes the following steps:

[0048] S1. Determine the matrix dimension to ensure that the dimension of the random matrix matches the dimension of the three-dimensional tomographic observation matrix, so as to ensure effective correction and control of the randomness intensity of the random matrix.

[0049] S2. Based on the determined parameters, select appropriate random matrices for block splicing and mixing, mainly considering random matrices such as Gaussian, Rademacher, and Bernoulli.

[0050] S3, random matrix With observation matrix The corrected observation matrix is ​​obtained by performing multiplication. , .

[0051] Step 3: Use the cross-correlation function to measure the observation matrix With adaptive dictionary matrix The degree of cross-correlation between the two matrices is used to determine whether convergence has occurred. If convergence has not occurred, a new random matrix is ​​constructed to correct the observation matrix, and the degree of cross-correlation is determined again until the two matrices converge.

[0052] After correcting the observation matrix with a random matrix in the third step, the cross-correlation function is used to measure the corrected observation matrix. Between and adaptive dictionary matrix The degree of cross-correlation includes the following steps:

[0053] S1. Calculate the cross-correlation coefficient between the corrected observation matrix and the adaptive dictionary matrix, and determine whether they converge. The cross-correlation coefficient is defined as follows:

[0054] ;

[0055] in, and These are the observation matrices. and dictionary matrix The row vectors and column vectors, and They are respectively The number of rows and columns, This is called the Welch lower bound.

[0056] , A smaller cross-correlation value indicates a weaker cross-correlation, resulting in higher accuracy in calculating the sparse coefficients. Cross-correlation is considered to be good if it is less than a certain threshold.

[0057] S2. If the obtained cross-correlation coefficient does not converge, reconstruct the random matrix and correct the observation matrix again. Calculate the cross-correlation coefficient between the corrected observation matrix and the dictionary matrix to determine whether it converges. If it does not converge, continue to repeat the above steps until it converges.

[0058] The specific embodiments are merely illustrative of the present invention and are not intended to limit the invention. After reading this specification, those skilled in the art can make modifications to these embodiments without contributing any inventive step, but such modifications are protected by patent law as long as they are within the scope of the claims of the present invention.

Claims

1. A method for designing a weakly cross-correlation observation matrix for a tropospheric tomography model, characterized in that: Includes the following steps: Step 1: Analyze the tomographic grid and construct the observation matrix based on the satellite signal direction. ; Step 2: Select Gaussian, Rademacher, and Bernoulli random matrices, divide them into left and right modules, and then combine them to form a new random matrix. G New random matrix G Correcting the observation matrix yields ; The construction of the random matrix in the second step includes the following steps: S1. Determine the size of the matrix dimension to ensure that the dimension of the random matrix matches the dimension of the three-dimensional tomographic observation matrix, so as to ensure that effective corrections can be made and the randomness intensity of the random matrix can be controlled. S2. Based on the determined parameters, select random matrices from Gaussian, Rademacher, and Bernoulli, and perform block concatenation and mixing to obtain a new random matrix. G ; S3, random matrix With observation matrix The corrected observation matrix is ​​obtained by performing multiplication. , ; Step 3: Use the cross-correlation function to measure the observation matrix With adaptive dictionary matrix The degree of cross-correlation between the two matrices is used to determine whether convergence has occurred. If convergence has not occurred, a new random matrix is ​​constructed to correct the observation matrix, and the degree of cross-correlation is determined again until the two matrices converge. After correcting the observation matrix with a random matrix in the third step, the cross-correlation function is used to measure the corrected observation matrix. With adaptive dictionary matrix The degree of cross-correlation between them includes the following steps: S1. Calculate the cross-correlation coefficient between the corrected observation matrix and the adaptive dictionary matrix, and determine whether they converge. The cross-correlation coefficient is defined as follows: ； in, and These are the observation matrices. and dictionary matrix The row vectors and column vectors, and They are respectively The number of rows and columns, This is called the Welch lower bound; , A smaller cross-correlation value indicates a weaker cross-correlation, resulting in higher accuracy in calculating the sparse coefficients. Cross-correlation is considered to be excellent when it is less than a certain threshold. S2. If the obtained cross-correlation coefficient does not converge, reconstruct the random matrix and correct the observation matrix again. Calculate the cross-correlation coefficient between the corrected observation matrix and the dictionary matrix to determine whether it converges. If it does not converge, continue to repeat the above steps until it converges.

2. The method for designing a weakly cross-correlation observation matrix for a tropospheric tomography model according to claim 1, characterized in that: In the first step, construct the observation matrix. In this process, the tomographic mesh is divided, including the following steps: S1. Divide the chromatography region in the horizontal and vertical directions to obtain several uniformly distributed small cuboids, i.e., voxel blocks. S2. Treat the water vapor density of each grid at the center point as a parameter to be estimated. Assuming that the water vapor density within the grid is uniform and constant in any given time period, the water vapor content along the satellite signal path can be discretized into the following form: ; in, This represents the oblique path water vapor observation value corresponding to the GNSS signal. and The GNSS signal is shown in the first Layer, First line, number The intercept information and water vapor density value in the voxel block are used if the GNSS signal does not pass through the voxel block. =0; Representing all rays within the study area in the above form, we obtain the observation equation: ； in, This is a column vector composed of delay information on rays emanating from the top of the study area. For the observation matrix, It is a column matrix composed of unknown parameters such as water vapor density.

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