A sea single station single epoch model-aware regularized Vtec estimation method

Abstract

This invention discloses a model-aware regularized VTEC estimation method for a single-station, single-epoch model at sea, belonging to the field of ionospheric inversion. First, this invention utilizes non-differential, non-combined PPP to extract the oblique ionospheric total delay and combines it with external DCB products to correct the satellite DCB. Then, it constructs the ionospheric VTEC inversion observation equations and calculates the condition number of the normal equation matrix to quantify ill-conditioning. Further, it constructs a model-aware regularization method composed of a selectively structured matrix, an adaptive regularization factor, and a numerically stable term, dynamically adjusting the regularization strength according to the condition number. Finally, it superimposes the regularization terms onto a least-squares expression to solve for the robust VTEC model coefficients and calculates the VTEC value for the current epoch. This invention effectively suppresses the instability of VTEC inversion values ​​caused by small errors under ill-conditioning conditions, significantly improving the reliability and accuracy of ionospheric modeling in ocean observation-restricted areas.

Description

Technical Field

[0001] This invention relates to the field of ionospheric VTEC inversion, and particularly to a sensing-regularized VTEC estimation method for a single-station, single-epoch model at sea. Background Technology

[0002] The vertical total electron content (VTEC) of the marine ionosphere is one of the core parameters characterizing the ionospheric state, and its changes directly affect the propagation characteristics of Global Navigation Satellite System (GNSS) signals. High-precision, high-temporal-resolution VTEC information has significant research and application value in fields such as maritime radio communication, space-based radar imaging, and communication, navigation, and timing in deep-sea and ocean-going navigation.

[0003] Currently, there is a long-standing lack of fixed observation infrastructure in marine areas, with few and scattered GNSS stations. This situation results in insufficient constraints of existing global ionospheric models in marine regions, leading to significant uncertainties in ionospheric structural characteristics. Detailed characterization of the ionospheric state in marine areas has become a key bottleneck restricting the performance of related applications.

[0004] Given the limited conditions for marine observation, single-station ionospheric inversion has become an important technical approach for characterizing ionospheric changes in ocean regions. The core logic of ionospheric VTEC inversion is to estimate the VTEC model coefficients and then use these coefficients to reconstruct the VTEC sequence of the target region. Traditionally, the least squares method is often used to estimate the VTEC model coefficients. However, in single-station observation scenarios, the spatial distribution of ionospheric puncture points exhibits significant geometric degradation characteristics over a short period, causing the least squares design matrix to approach singularity and exhibit severe ill-conditioned behavior. This ill-conditioned behavior makes the estimated model coefficients extremely sensitive to observational noise and small perturbations, easily leading to numerical oscillations and non-physical anomalies. Due to the instability of the model coefficients, the traditional least squares method struggles to obtain stable and reliable single-station VTEC values.

[0005] Therefore, there is an urgent need for a method that can effectively suppress the ill-conditioned nature of the design matrix and improve the robustness of VTEC estimation under single-station and single-epoch conditions, so as to meet the requirements of high-precision ionospheric inversion in complex marine environments. Summary of the Invention

[0006] To address the aforementioned technical problems, this invention provides a sensing-regularized VTEC estimation method for a single-station, single-epoch model at sea. This method effectively suppresses the ill-conditioned nature of the least-squares design matrix and enhances the robustness and rationality of VTEC model coefficient estimation under single-station, single-epoch observation conditions, thereby achieving high-precision and high-reliability ionospheric VTEC inversion in complex marine environments.

[0007] To achieve the above objectives, the technical solution of the present invention is as follows:

[0008] A perception-regularized VTEC estimation method for a single-station, single-epoch model at sea includes the following steps:

[0009] Step 1: Using GNSS observation data, navigation ephemeris, and PPP-B2b or HAS correction information, extract and correct the total oblique ionospheric delay;

[0010] Step 2: Based on the projection function, the corrected total oblique ionospheric delay is mapped to VTEC, and a VTEC mathematical model is constructed;

[0011] Step 3: Establish the observation equation based on the VTEC mathematical model, extract the least squares design matrix, construct the normal equation matrix and calculate its condition number to evaluate the ill-conditioned nature of the matrix;

[0012] Step 4: Construct a model-aware regularization method consisting of a selectively structured matrix, a regularization factor, and a numerically stable term, and adaptively adjust the regularization strength based on the condition number of the normal equation matrix.

[0013] Step 5: The selective structuring matrix, regularization factor and numerical stability term are superimposed on the least squares expression to obtain the VTEC model coefficients, and then the VTEC value of the current epoch is calculated.

[0014] In the above scheme, step 1 specifically includes the following sub-steps:

[0015] Step 1.1: Calculate the satellite position coordinates based on the navigation ephemeris data, and calculate the satellite elevation angle and azimuth angle based on the receiver position;

[0016] Step 1.2: The pseudorange and carrier phase observations of the current epoch are processed using non-differential, non-combined precise point positioning technology. The total oblique ionospheric delay, including satellite differential code bias and receiver differential code bias, is extracted and expressed as follows:

[0017] ;

[0018] In the formula, The total oblique ionospheric delay includes both satellite DCB and receiver DCB. The total electron content is slanted. , It is the coefficient of the difference in ionospheric delay between two frequencies. and These are the frequencies of satellite observations L1 and L2, respectively. r For receiver identification, c At the speed of light, For receiver DCB, For satellite DCB;

[0019] Step 1.3: Receive and decode BeiDou PPP-B2b or Galileo HAS data streams in real time to obtain satellite DCB correction numbers;

[0020] Step 1.4: Use the satellite DCB correction to eliminate the satellite DCB in the total oblique ionospheric delay, and obtain the corrected total oblique ionospheric delay:

[0021] ;

[0022] In the formula, This is the corrected total delay of the oblique ionosphere.

[0023] In the above scheme, step 2 specifically includes the following sub-steps:

[0024] Step 2.1: Calculate the zenith distance z at the receiver using the satellite elevation angle and the standard geometric transformation formula;

[0025] Step 2.2: Calculate the projection function value based on the improved ionospheric monolayer projection function model:

[0026] ;

[0027] ;

[0028] In the formula, Represents the projection function value. Let be the zenith distance at the IPP in the ionosphere, then , The average radius of the Earth The height of a single ionosphere. This is a correction factor for the single-layer projection function;

[0029] Step 2.3: Using spherical trigonometry formulas and coordinate transformation, calculate the geographic coordinates of the ionospheric puncture point based on the satellite azimuth, elevation angle, and receiver position. Then, transform these coordinates sequentially to the geomagnetic coordinate system and the solar-fixed geomagnetic coordinate system to obtain the solar-fixed geomagnetic latitude of the puncture point. And the geomagnetic longitude of the sun ;

[0030] Step 2.4: Substitute the sun-fixed geomagnetic coordinates into the selected VTEC mathematical model to construct a VTEC mathematical model suitable for a single station.

[0031] In the above scheme, the VTEC mathematical model is one of the following: a first-order planar linear model, an asymmetric quadratic polynomial model, a full quadratic polynomial model, or a spherical harmonic function model; the forms after substituting the coordinates are as follows:

[0032] First-order planar linear model:

[0033] ;

[0034] Asymmetric quadratic polynomial model:

[0035] ;

[0036] All quadratic polynomial model:

[0037] ;

[0038] Spherical harmonic function model:

[0039] ;

[0040] In the formula, Indicates the IPP location value, and These refer to the sun-fixed geomagnetic latitude and sun-fixed geomagnetic longitude of the ionospheric puncture point, respectively. and These are the maximum order and degree of the spherical harmonic function, respectively; For regularization of the Legendre function; and is the spherical harmonic coefficient.

[0041] In the above scheme, step 3 specifically includes the following sub-steps:

[0042] Step 3.1: Substitute the projection function value, VTEC mathematical model, and receiver DCB term into the corrected oblique ionospheric total delay formula to construct the ionospheric VTEC inversion model, as follows:

[0043] ;

[0044] In the formula, For the corrected total delay of the oblique ionosphere, Represents the improved projection function of a single ionospheric layer. Representing the VTEC mathematical model, For the receiver DCB;

[0045] This is further transformed into an observation equation:

[0046] ;

[0047] In the formula, These are physical constant coefficients, with values... ;

[0048] Step 3.2: Write the observation equations in matrix form. ,in This is a vector consisting of the total oblique ionospheric delay observations of all visible satellites at a single epoch. Design the least squares matrix. The parameter vector to be estimated includes VTEC model coefficients and receiver DCBs for each GNSS system;

[0049] Step 3.3: Construct the normal equation matrix using the least squares principle. ;

[0050] Step 3.4: Perform singular value decomposition on the normal equation matrix and calculate its maximum singular value. and minimum singular value This leads to the condition number. This is used to quantify the ill-conditioned nature of the design matrix;

[0051] .

[0052] In the above scheme, step 4 specifically includes the following sub-steps:

[0053] Step 4.1, Construct a selective structured matrix This matrix is ​​a diagonal matrix, whose diagonal elements impose no constraints on the constant terms but impose unit constraints on the higher-order parameters, specifically defined as follows:

[0054] ;

[0055] In the formula, for diagonal matrix, This represents the diagonalization operator, used to diagonalize an input vector. The mapping is a diagonal matrix, where the diagonal elements are the components of the input vector and the off-diagonal elements are 0. This represents the total number of valid observations at that epoch. For the elements on the main diagonal, the piecewise function is defined as:

[0056] ;

[0057] In the formula, Let the parameter number be the parameter to be estimated; Step 4.2, construct the numerical stability term. :

[0058] ;

[0059] In the formula, For numerically stable terms, The total number of unknowns to be solved in the observation equation. It is a very small constant. The trace operator;

[0060] Step 4.3: Determine the regularization factor using a sub-strategy approach based on the type of the selected VTEC mathematical model. ,Establish condition number The mapping relationship is as follows: the regularization strength increases when the condition number increases, and decreases when the condition number decreases.

[0061] Step 4.4, selectively structure the matrix Regularization factor Sum of numerical stability terms Combining these terms forms an adaptive regularization term, which is used to correct the original least squares problem.

[0062] In the above scheme, the regularization factor The calculation formula is:

[0063] ;

[0064] In the formula, This represents taking the median. For matrix The set of singular values, The total number of unknowns to be solved in the observation equation. For the maximum singular value, It is the second largest singular value. The minimum singular value is ; LP, SH, AQ, and FQ models are respectively a first-order planar linear model, a spherical harmonic function model, an asymmetric quadratic polynomial model, and a full quadratic polynomial model.

[0065] In the above scheme, step 5 specifically includes the following sub-steps:

[0066] Step 5.1, selectively structure the matrix Regularization factor Sum of numerical stability terms Superimposed onto the original least squares normal equations, the final solution expression for the perception-regularized VTEC estimation method of the single-station, single-epoch model at sea is constructed:

[0067] ;

[0068] In the formula, for An identity matrix of order 1;

[0069] Step 5.2: Solve the above expression to obtain the vector of parameters to be estimated. This includes VTEC model coefficients and receiver DCBs for each GNSS system;

[0070] Step 5.3: Obtain the geographic coordinates of the receiver's location using precise point positioning technology, and convert them to latitude in the sun-fixed geomagnetic coordinate system. and longitude ;

[0071] Step 5.4: Substitute the obtained VTEC model coefficients and the solar-fixed geomagnetic coordinates at the receiver into the selected VTEC mathematical model to calculate the VTEC value at the receiver at the current epoch.

[0072] In the above scheme, the non-differential non-combined precise single-point positioning technology uses GNSS pseudorange and carrier phase observations to calculate the receiver position, clock error, tropospheric delay, and oblique ionospheric total delay including hardware delay epoch-by-epoch through Kalman filtering or least squares filtering.

[0073] Through the above technical solution, the present invention provides a sensing-regularized VTEC estimation method for a single-station, single-epoch model at sea, which has the following beneficial effects:

[0074] 1. Effectively suppresses the ill-conditioned nature of the design matrix and improves the stability of parameter estimation.

[0075] This invention addresses the ill-conditioned problem of least squares design matrices caused by geometric degradation in single-station marine observations by proposing a model-aware regularized VTEC estimation method. By constructing a selectively structured matrix, constraints are imposed only on higher-order spatially varying parameters while the constant term remains unpenalized. Combined with an adaptive regularization factor, the regularization strength can be dynamically adjusted based on the condition number of the normal equation matrix. Experiments show that under higher-order models (such as fully quadratic polynomial models), the RMS curve of the traditional least squares method exhibits severe oscillations (exceeding 10 TECUs), while the method of this invention stably controls the error within the low range of 1–5.5 TECUs, significantly suppressing numerical divergence caused by ill-conditioned phenomena.

[0076] 2. Adaptive regularization strategy, balancing accuracy and stability.

[0077] This invention establishes a mapping relationship between the regularization factor and the condition number of the normal equation matrix. When the condition number is small, the regularization factor automatically approaches zero, and the method degenerates into the standard least squares method, preserving the original observation accuracy. When the condition number is large, the regularization strength is adaptively increased to suppress noise amplification. In a first-order planar linear model, the RMS curve of the method in this invention almost completely coincides with that of the traditional least squares method, indicating that no additional bias is introduced under well-conditioned conditions, achieving a balance between accuracy and stability.

[0078] 3. Enhance the reliability of VTEC inversion under extreme observation conditions

[0079] In a seven-day field study using data from marine buoys, the traditional least squares method exhibited significant random oscillations under both asymmetric and full quadratic polynomial models, resulting in a severe deviation of the VTEC estimate from the reference benchmark. In contrast, the method of this invention successfully reconstructed a smooth curve that closely matches the GIM reference benchmark, demonstrating a clear daily trend, stable performance at night, and precise daytime peaks. This fully proves that even in extreme environments with a limited number of observed satellites and poor geometry, the method of this invention maintains high robustness and accuracy.

[0080] 4. Compatible with multiple VTEC mathematical models, highly applicable.

[0081] This invention is applicable to various commonly used VTEC mathematical models, such as first-order planar linear models, asymmetric quadratic polynomial models, full quadratic polynomial models, and spherical harmonic function models. It adopts a regularization factor determination method with different strategies for different models, which has good scalability and adaptability and can be flexibly applied to different ionospheric inversion scenarios at a single station and a single epoch at sea.

[0082] 5. Enhance the ability to model the ionosphere in marine areas

[0083] This invention solves the technical problem of unstable model coefficients caused by geometric degradation of single-station observations at sea, significantly improves the reliability of VTEC inversion in the ionosphere in limited marine observation areas, and provides key technical support for high-precision positioning, navigation and timing in marine radio communication, space-based radar imaging, and deep-sea ocean navigation. Attached Figure Description

[0084] To more clearly illustrate the technical solutions in the embodiments of the present invention or the prior art, the accompanying drawings used in the description of the embodiments or the prior art will be briefly introduced below.

[0085] Figure 1 This is a schematic diagram of the process of a sensing regularized VTEC estimation method for a single-station, single-epoch model at sea, as disclosed in an embodiment of the present invention.

[0086] Figure 2 The following are comparison charts showing the RMS evolution of the least squares algorithm derived from multi-station averaging of IGS data and the method of this invention at a 30-second epoch interval and four VTEC mathematical models. Among them, (a) is a comparison chart of the RMS evolution of the first-order planar linear model, (b) is a comparison chart of the RMS evolution of the spherical harmonic function model, (c) is a comparison chart of the RMS evolution of the asymmetric quadratic polynomial model, and (d) is a comparison chart of the RMS evolution of the full quadratic polynomial model.

[0087] Figure 3The figures show a comparison of the VTEC inversion time series of the least squares algorithm and the method of this invention over a continuous seven-day period of measured data from marine buoys, using four VTEC mathematical models at a 30-second epoch interval. Among them, (a) is a comparison of the VTEC inversion time series of the first-order planar linear model, (b) is a comparison of the VTEC inversion time series of the spherical harmonic function model, (c) is a comparison of the VTEC inversion time series of the asymmetric quadratic polynomial model, and (d) is a comparison of the VTEC inversion time series of the full quadratic polynomial model. Detailed Implementation

[0088] The technical solutions of the present invention will be clearly and completely described below with reference to the accompanying drawings in the embodiments of the present invention.

[0089] This invention provides a sensing-regularized VTEC estimation method for a single-station, single-epoch model at sea, such as... Figure 1 As shown, it includes the following steps:

[0090] Step 1: Using GNSS observation data, navigation ephemeris, and PPP-B2b or HAS correction information, extract and correct the total oblique ionospheric delay.

[0091] Specifically, it includes the following sub-steps:

[0092] Step 1.1: Calculate the satellite position coordinates based on the navigation ephemeris data, and calculate the satellite elevation angle and azimuth angle based on the receiver position.

[0093] In this embodiment, a marine GNSS receiver is installed on a buoy platform to collect pseudorange and carrier phase observation data from GPS, BeiDou, Galileo, and other systems in real time. It also receives navigation ephemeris data and BeiDou PPP-B2b or Galileo HAS correction data streams. The data sampling interval is 30 seconds.

[0094] First, calculate the position coordinates of each satellite based on navigation ephemeris data. Assuming the approximate position of the receiver is known (which can be obtained through stepwise convergence via PPP), calculate the elevation and azimuth angles of each satellite using the spatial geometric relationship between the satellite and the receiver.

[0095] Step 1.2: The pseudorange and carrier phase observations of the current epoch are processed using non-differential, non-combined precise point positioning technology. The total oblique ionospheric delay, including satellite differential code bias and receiver differential code bias, is extracted and expressed as follows:

[0096] ;

[0097] In the formula, The total oblique ionospheric delay includes both satellite DCB and receiver DCB. The total electron content is slanted. , It is the coefficient of the difference in ionospheric delay between two frequencies. and These are the frequencies of satellite observations L1 and L2, respectively. r For receiver identification, c At the speed of light, For receiver DCB, For satellite DCB;

[0098] The aforementioned non-differential, non-combined precise single-point positioning technology utilizes GNSS pseudorange and carrier phase observations, and calculates receiver position, clock error, tropospheric delay, and total oblique ionospheric delay including hardware delay epoch-by-epoch through Kalman filtering or least squares filtering.

[0099] Step 1.3: Receive and decode BeiDou PPP-B2b or Galileo HAS data streams in real time to obtain satellite DCB correction numbers;

[0100] Step 1.4: Use the satellite DCB correction to eliminate the satellite DCB in the total oblique ionospheric delay, and obtain the corrected total oblique ionospheric delay:

[0101] ;

[0102] In the formula, This is the corrected total delay of the oblique ionosphere.

[0103] Step 2: Based on the projection function, the corrected total oblique ionospheric delay is mapped to VTEC, and a VTEC mathematical model is constructed.

[0104] Specifically, it includes the following sub-steps:

[0105] Step 2.1: Calculate the zenith distance at the receiver using the satellite elevation angle and the standard geometric transformation formula. ;

[0106] Step 2.2: Calculate the projection function value based on the improved ionospheric monolayer projection function model:

[0107] ;

[0108] ;

[0109] In the formula, Represents the projection function value. Let be the zenith distance at the IPP in the ionosphere, then , The average radius of the Earth The height of a single layer of the ionosphere is typically taken as 450 km. This is the correction factor for the single-layer projection function, typically taken as 0.9782;

[0110] Step 2.3: Using spherical trigonometry formulas and coordinate transformation, calculate the geographic coordinates of the ionospheric pierce point (IPP) based on the satellite azimuth, elevation angle, and receiver position. Then, transform these coordinates sequentially to the geomagnetic coordinate system and the sun-fixed geomagnetic coordinate system to obtain the sun-fixed geomagnetic latitude of the pierce point. And the geomagnetic longitude of the sun ;

[0111] Step 2.4: Substitute the sun-fixed geomagnetic coordinates into the selected VTEC mathematical model to construct a VTEC mathematical model suitable for a single station.

[0112] In single-station ionospheric VTEC inversion modeling, commonly used mathematical models include polynomial function models and spherical harmonic function models. The VTEC mathematical model selected in this invention is one of the following: a first-order planar linear model, an asymmetric quadratic polynomial model, a full quadratic polynomial model, or a spherical harmonic function model. The forms after substituting the coordinates are as follows:

[0113] First-order planar linear model:

[0114] ;

[0115] Asymmetric quadratic polynomial model:

[0116] ;

[0117] All quadratic polynomial model:

[0118] ;

[0119] Spherical harmonic function model:

[0120] ;

[0121] In the formula, Indicates the IPP location value, and These refer to the sun-fixed geomagnetic latitude and sun-fixed geomagnetic longitude of the ionospheric puncture point, respectively. and These are the maximum order and degree of the spherical harmonic function, respectively; For regularization of the Legendre function; and is the spherical harmonic coefficient.

[0122] Step 3: Establish the observation equations based on the VTEC mathematical model, extract the least squares design matrix, construct the normal equation matrix and calculate its condition number to evaluate the ill-conditioned nature of the matrix.

[0123] Specifically, it includes the following sub-steps:

[0124] Step 3.1: Substitute the projection function value, VTEC mathematical model, and receiver DCB term into the corrected oblique ionospheric total delay formula to construct the ionospheric VTEC inversion model, as follows:

[0125] ;

[0126] In the formula, For the corrected total delay of the oblique ionosphere, Represents the improved projection function of a single ionospheric layer. Representing the VTEC mathematical model, For the receiver DCB;

[0127] This is further transformed into an observation equation:

[0128] ;

[0129] In the formula, These are physical constant coefficients, with values... ;

[0130] Step 3.2: Write the observation equations in matrix form. ,in This is a vector consisting of the total oblique ionospheric delay observations of all visible satellites at a single epoch. Design the least squares matrix. The vector of parameters to be estimated includes VTEC model coefficients and receiver DCBs for each GNSS system.

[0131] Taking the asymmetric quadratic polynomial model among the four models as an example, we substitute it into the ionospheric VTEC inversion model to complete the matrix reconstruction of the observation equations. Considering A combined observation scenario of n GNSS systems, let the nth GNSS system be... The number of satellites in each system is The VTEC inversion observation equation for the ionosphere is obtained as follows:

[0132] ;

[0133] The known oblique ionospheric total delay observation vector is derived from the oblique ionospheric total delay observations of all visible satellites in each GNSS system at a single epoch. Composition, in which, Indicates the first The system is numbered as follows The corresponding ionospheric delayed observations of the satellite; The least squares design matrix has the following form: ;in,

[0134] , Representing the first The first system The satellite's composite projection frequency factor and receiver bias mapping coefficient, and The first System No. The solar-fixed geomagnetic longitude and solar-fixed geomagnetic latitude parameters of the puncture point corresponding to each satellite; The coefficients to be estimated are... , , , These are the coefficients of the VTEC model for the ionosphere. The first one to be estimated The receiver DCB corresponding to each GNSS system.

[0135] Step 3.3: Construct the normal equation matrix using the least squares principle. ;

[0136] Step 3.4: Perform singular value decomposition on the normal equation matrix and calculate its maximum singular value. and minimum singular value This leads to the condition number. This is used to quantify the ill-conditioned nature of the design matrix;

[0137] .

[0138] Step 4: Construct a model-aware regularization method consisting of a selective structuring matrix, a regularization factor, and a numerical stability term, and adaptively adjust the regularization strength based on the condition number of the normal equation matrix.

[0139] To solve the condition number of the normal equation matrix To address the problem of excessively large least-squares design matrices exhibiting severe ill-conditioned behavior, a perceptually regularized VTEC estimation method for a single-station, single-epoch model at sea is constructed. This method includes selectively structured matrices. Regularization factor Numerical stability term The calculation.

[0140] Specifically, it includes the following sub-steps:

[0141] Step 4.1, Construct a selective structured matrix This matrix is ​​a diagonal matrix, whose diagonal elements impose no constraints on the constant terms but impose unit constraints on the higher-order parameters, specifically defined as follows:

[0142] ;

[0143] In the formula, for diagonal matrix, This represents the diagonalization operator, used to diagonalize an input vector. The mapping is a diagonal matrix, where the diagonal elements are the components of the input vector and the off-diagonal elements are 0. This represents the total number of valid observations at that epoch. For the elements on the main diagonal, the piecewise function is defined as:

[0144] ;

[0145] In the formula, The parameter number to be estimated;

[0146] Step 4.2: Considering that the number of observable satellites per epoch may decrease drastically under the extremely harsh observation environment at sea, causing the normal equation matrix to fall into a rank deficiency state, the method further introduces a minimal numerical stability term. :

[0147] ;

[0148] In the formula, For numerically stable terms, The total number of unknowns to be solved in the observation equation. It is a very small constant. The trace operator;

[0149] Step 4.3: Determine the regularization factor using a sub-strategy approach based on the type of the selected VTEC mathematical model. ,Establish condition number The mapping relationship is as follows: Regularization strength increases when the condition number increases, and decreases when it decreases; regularization factor The calculation formula is:

[0150] ;

[0151] In the formula, This represents taking the median. For matrix The set of singular values, The total number of unknowns to be solved in the observation equation. For the maximum singular value, It is the second largest singular value. The minimum singular value is ; LP, SH, AQ, and FQ models are respectively a first-order planar linear model, a spherical harmonic function model, an asymmetric quadratic polynomial model, and a full quadratic polynomial model.

[0152] Step 4.4, selectively structure the matrix Regularization factor Sum of numerical stability terms Combining these terms forms an adaptive regularization term, which is used to correct the original least squares problem.

[0153] Step 5: Superimpose the selective structuring matrix, regularization factor, and numerical stability term into the least squares expression to obtain the VTEC model coefficients, and then calculate the VTEC value of the current epoch.

[0154] Specifically, it includes the following sub-steps:

[0155] Step 5.1, selectively structure the matrix Regularization factor Sum of numerical stability terms Superimposed on the normal equation matrix of the original least squares In this paper, the final solution expression of the perception-regularized VTEC estimation method for a single-station, single-epoch model at sea is constructed:

[0156] ;

[0157] In the formula, for An identity matrix of order 1;

[0158] Step 5.2: Solve the above expression to obtain the vector of parameters to be estimated. This includes VTEC model coefficients and receiver DCBs for each GNSS system;

[0159] Step 5.3: Obtain the geographic coordinates of the receiver's location using precise point positioning technology, and convert them to latitude in the sun-fixed geomagnetic coordinate system. and longitude ;

[0160] Step 5.4: Substitute the obtained VTEC model coefficients and the solar-fixed geomagnetic coordinates at the receiver into the selected VTEC mathematical model to calculate the VTEC value at the receiver at the current epoch.

[0161] Taking the asymmetric quadratic polynomial model among the four models as an example, the solar-fixed geomagnetic latitude at the receiver is... With the sun's magnetic longitude Substitute it into the VTEC mathematical model:

[0162] ;

[0163] This allows you to obtain the high-precision VTEC value at the receiver.

[0164] Seventeen European stations were selected as average experimental examples, and VTEC obtained by interpolation using GIM products provided by IGS was used as an external reference benchmark. A seven-day VTEC monitoring experiment was conducted using buoys in a certain sea area. This invention uses the least squares method and a single-station, single-epoch model-based perception-regularized VTEC estimation method to estimate the parameters of the VTEC mathematical model. The RMS evolution curves of four different models at 30-second epoch intervals within a day for the seventeen European stations were compared as core evaluation indicators. Furthermore, the VTEC inversion time series of the four different models at 30-second epoch intervals were compared for seven consecutive days using buoys in the same sea area to evaluate the robustness of the single-station, single-epoch model-based perception-regularized VTEC estimation method in dealing with the ill-conditioned nature of the least squares design matrix. TECU is the unit of VTEC, 1 TECU = 10 electrons / m 2 .

[0165] Figure 2 In the diagram, the blue line represents the result calculated using the traditional least squares method, while the red line represents the result calculated using the sensing-regularized VTEC estimation method based on a single-station, single-epoch model at sea. Figure 2 As shown in (a) (first-order planar linear model), due to its extremely low order and minimal parameters, its normal equation matrix exhibits strong stability and rarely suffers from ill-conditioned problems. It can be seen that the RMS curves of the blue and red lines almost completely overlap. This indicates that the sensing-regularized VTEC estimation method for the single-station, single-epoch model at sea automatically reduces the regularization factor to near zero when the matrix conditions are favorable, preserving the original observation accuracy. Figure 2 As shown in (b) (spherical harmonic function model), applying the global spherical harmonic basis to a single-station local scenario can lead to extreme structural rank deficiency. The blue line in the figure shows global divergence and oscillation, while the red line remains relatively stable throughout the day. Figure 2 As shown in (c) (asymmetric quadratic polynomial model), with the increase of model order, the traditional least squares method begins to exhibit a matrix ill-conditioned trend in certain periods, manifested as local rises and oscillations in the curve. It can be seen that the red line is mostly lower than the blue line. This indicates that the algorithm successfully suppresses the ill-conditioned design matrix of the least squares method. Figure 2As shown in (d) (all quadratic polynomial model), when a model with higher parameters is used, the condition number of the normal equations increases sharply. The traditional least squares method completely fails, and its RMS curve exhibits violent oscillations, significantly exceeding 10 TECU. However, the red line still robustly controls the error within the low range of 1-5.5 TECU. This proves that in extremely ill-conditioned high-order models, the perception-regularized VTEC estimation method for single-station, single-epoch models at sea can effectively suppress ill-conditioned problems and significantly improve the stability and modeling accuracy of parameter estimation. In summary, through comprehensive analysis of data from seventeen stations, it is found that the evolution of epoch accuracy of this method over time has higher accuracy and stronger stability compared to the least squares method, with a relatively lower root mean square error and a stable RMS.

[0166] Figure 3 In the diagram, the blue line represents the reference benchmark for GIM interpolation of VTEC, the red line represents the result calculated using the traditional least squares method, and the green line represents the result calculated using the perception-regularized VTEC estimation method based on a single-station, single-epoch model at sea. Figure 3 As shown in (a) (first-order planar linear model), even in the first-order model with the fewest parameters, the red line still shows severe underestimation and oscillation. The green line, on the other hand, not only transitions smoothly at night, but also closely matches the peak shape of the blue GIM baseline during the day, achieving a reconstruction of the physical trend. Figure 3 As shown in (b) (spherical harmonic function model), the red line is more oscillating and chaotic, while the green line runs closer to the blue line than the red line. Figure 3 As shown in (c) (asymmetric quadratic polynomial model), as the parameters increase, the normal equations become ill-conditioned, the red line oscillates randomly with large amplitude, and the green line reconstructs a relatively smooth curve that closely matches the blue line. Figure 3 As shown in (d) (full quadratic polynomial model), as the parameters increase further, the red line oscillation amplitude increases again and moves away from the baseline curve, while the green line remains closer to the blue line curve, and the oscillation also conforms to the daily variation pattern.

[0167] Through seven consecutive days of experimental verification and analysis at a single sea station, it was found that the method successfully suppressed the numerical divergence of traditional methods, effectively recovered the true physical fluctuation trend of VTEC, and maintained a good fit with the reference benchmark, fully demonstrating the high robustness of the method under extreme observation environments.

[0168] The above description of the disclosed embodiments enables those skilled in the art to make or use the invention. Various modifications to these embodiments will be readily apparent to those skilled in the art, and the general principles defined herein may be implemented in other embodiments without departing from the spirit or scope of the invention. Therefore, the invention is not to be limited to the embodiments shown herein, but is to be accorded the widest scope consistent with the principles and novel features disclosed herein.

Claims

1. A method for offshore single-station single-epoch model-aware regularized VTEC estimation, characterized in that, Includes the following steps: Step 1: Using GNSS observation data, navigation ephemeris, and PPP-B2b or HAS correction information, extract and correct the total oblique ionospheric delay; Step 2: Based on the projection function, the corrected total oblique ionospheric delay is mapped to VTEC, and a VTEC mathematical model is constructed; Step 3: Establish the observation equation based on the VTEC mathematical model, extract the least squares design matrix, construct the normal equation matrix and calculate its condition number to evaluate the ill-conditioned nature of the matrix; Step 4: Construct a model-aware regularization method consisting of a selectively structured matrix, a regularization factor, and a numerically stable term, and adaptively adjust the regularization strength based on the condition number of the normal equation matrix. Step 5: Superimpose the selective structuring matrix, regularization factor, and numerical stability term into the least squares expression, solve for the VTEC model coefficients, and then calculate the VTEC value for the current epoch. Step 2 specifically includes the following sub-steps: Step 2.1: Calculate the zenith distance z at the receiver using the satellite elevation angle and the standard geometric transformation formula; Step 2.2: Calculate the projection function value based on the improved ionospheric monolayer projection function model: ； ； In the formula, Represents the projection function value. Let be the zenith distance at the IPP in the ionosphere, then , The average radius of the Earth The height of a single ionosphere. This is a correction factor for the single-layer projection function; Step 2.3: Using spherical trigonometry formulas and coordinate transformation, calculate the geographic coordinates of the ionospheric puncture point based on the satellite azimuth, elevation angle, and receiver position. Then, transform these coordinates sequentially to the geomagnetic coordinate system and the solar-fixed geomagnetic coordinate system to obtain the solar-fixed geomagnetic latitude of the puncture point. And the geomagnetic longitude of the sun ; Step 2.4: Substitute the sun-fixed geomagnetic coordinates into the selected VTEC mathematical model to construct a VTEC mathematical model suitable for a single station; The VTEC mathematical model is one of the following: a first-order planar linear model, an asymmetric quadratic polynomial model, a full quadratic polynomial model, or a spherical harmonic function model; the forms after substituting the coordinates are as follows: First-order planar linear model: ； Asymmetric quadratic polynomial model: ； All quadratic polynomial model: ； Spherical harmonic function model: ； In the formula, Indicates the IPP location value, and These refer to the sun-fixed geomagnetic latitude and sun-fixed geomagnetic longitude of the ionospheric puncture point, respectively. and These are the maximum order and degree of the spherical harmonic function, respectively; This is the regularized Legendre function; and is the spherical harmonic coefficient.

2. The method according to claim 1, wherein, Step 1 specifically includes the following sub-steps: Step 1.1: Calculate the satellite position coordinates based on the navigation ephemeris data, and calculate the satellite elevation angle and azimuth angle based on the receiver position; Step 1.2: The pseudorange and carrier phase observations of the current epoch are processed using non-differential, non-combined precise point positioning technology. The total oblique ionospheric delay, including satellite differential code bias and receiver differential code bias, is extracted and expressed as follows: ； In the formula, The total oblique ionospheric delay includes both satellite DCB and receiver DCB. The total electron content is slanted. , It is the coefficient of the difference in ionospheric delay between two frequencies. and These are the frequencies of satellite observations L1 and L2, respectively. r For receiver identification, c At the speed of light, For receiver DCB, For satellite DCB; Step 1.3: Receive and decode BeiDou PPP-B2b or Galileo HAS data streams in real time to obtain satellite DCB correction numbers; Step 1.4: Use the satellite DCB correction to eliminate the satellite DCB in the total oblique ionospheric delay, and obtain the corrected total oblique ionospheric delay: ； In the formula, is the corrected slant ionospheric total delay.

3. The method according to claim 1, wherein, Step 3 specifically includes the following sub-steps: Step 3.1: Substitute the projection function value, VTEC mathematical model, and receiver DCB term into the corrected oblique ionospheric total delay formula to construct the ionospheric VTEC inversion model, as follows: ； In the formula, For the corrected total delay of the oblique ionosphere, Represents the improved projection function of a single ionospheric layer. Representing the VTEC mathematical model, For the receiver DCB; This is further transformed into an observation equation: ； wherein is a physical constant coefficient having a value of ; Step 3.2: Write the observation equations in matrix form. ,in This is a vector consisting of the total oblique ionospheric delay observations of all visible satellites at a single epoch. Design the least squares matrix. The parameter vector to be estimated includes VTEC model coefficients and receiver DCBs for each GNSS system; Step 3.3: Construct the normal equation matrix using the least squares principle. ; Step 3.4: Perform singular value decomposition on the normal equation matrix and calculate its maximum singular value. and minimum singular value This leads to the condition number. This is used to quantify the ill-conditioned nature of the design matrix; 。 4. The method according to claim 3, wherein, Step 4 specifically includes the following sub-steps: Step 4.1, construction of the selective structuring matrix The matrix is a diagonal matrix whose diagonal elements impose no constraint on the constant term and unit constraint on the higher order parameters, defined as: ； In the formula, for diagonal matrix, This represents the diagonalization operator, used to diagonalize an input vector. The mapping is a diagonal matrix, where the diagonal elements are the components of the input vector and the off-diagonal elements are 0. This represents the total number of valid observations at that epoch. For elements on the main diagonal, the piecewise function is defined as: ； In the formula, is the serial number of the parameter to be estimated; step 4.2, construct a numerical stability term : ； In the formula, For numerically stable terms, The total number of unknowns to be solved in the observation equation. It is a very small constant. The trace operator; Step 4.3, determine the regularization factor in a sub-strategy way according to the type of the selected VTEC mathematical model , establish the mapping relationship between the condition number and the regularization strength: increase the regularization strength when the condition number increases, and vice versa; Step 4.4, the selective structured matrix , a regularization factor , and a numerical stability term are combined to form an adaptive regularization term used to modify the original least squares problem.

5. The method according to claim 4, wherein, Regularization factor The calculation formula is: ； In the formula, This represents taking the median. For matrix The set of singular values, The total number of unknowns to be solved in the observation equation. For the maximum singular value, It is the second largest singular value. The minimum singular value is LP; the LP, SH, AQ, and FQ models are respectively the first-order planar linear model, the spherical harmonic function model, the asymmetric quadratic polynomial model, and the full quadratic polynomial model.

6. The method according to claim 4, wherein, Step 5 specifically includes the following sub-steps: Step 5.1, selectively structure the matrix Regularization factor Sum of numerical stability terms Superimposed onto the original least squares normal equations, the final solution expression for the perception-regularized VTEC estimation method of the single-station, single-epoch model at sea is constructed: ； In the formula, for An identity matrix of order 1; Step 5.2, solving the above expression to obtain the estimated parameter vector which contains the VTEC model coefficients and the receiver DCBs for each GNSS system; Step 5.3: Obtain the geographic coordinates of the receiver's location using precise point positioning technology, and convert them to latitude in the sun-fixed geomagnetic coordinate system. and longitude ; Step 5.4: Substitute the obtained VTEC model coefficients and the solar-fixed geomagnetic coordinates at the receiver into the selected VTEC mathematical model to calculate the VTEC value at the receiver at the current epoch.

7. The sensing-regularized VTEC estimation method for a single-station, single-epoch model at sea according to claim 2, characterized in that, The non-difference non-combination precise point positioning technology utilizes GNSS pseudo-range and carrier phase observation values, and solves receiver position, clock difference, troposphere delay and total slant ionosphere delay containing hardware delay through Kalman filtering or least square filtering per epoch.

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