Ionospheric error correction method, system and device based on beidou satellite signal

Abstract

The application provides a method, system and device for ionospheric error correction based on Beidou satellite signals, the method comprising: collecting Beidou satellite multi-frequency pseudorange and carrier phase original observation data from Beidou satellites in a Beidou observation network; taking the multi-frequency pseudorange and carrier phase original observation data as input sources of three-dimensional tomographic modeling; constructing a design matrix including ray geometric path integral information by using a three-dimensional tomographic modeling strategy combining a layered function base and a projection function; solving a linear equation set constructed by the design matrix by using a robust solver with a regularization term to suppress rank deficiency of the design matrix caused by uneven distribution of observation rays; reconstructing an ionospheric three-dimensional electron density field according to the solved layered modeling coefficients, extracting a line-of-sight delay in the direction of each observation satellite, and generating ionospheric corrections for correcting Beidou positioning and synchronization errors.

Description

Technical Field

[0001] This application relates to the field of BeiDou satellite positioning and correction technology, and in particular to an ionospheric error correction method, system and device based on BeiDou satellite signals. Background Technology

[0002] With the widespread application of global navigation satellite systems such as the BeiDou Navigation Satellite System in surveying, navigation, UAV inspection, and power grid time-frequency synchronization, the reliance on the continuity and accuracy of satellite positioning has significantly increased. However, in ionospherically active regions such as southern China, rapid or abrupt changes in TEC (Total Electron Content) often occur at noon or in the afternoon due to solar radiation, geomagnetic disturbances, and seasonal variations. This leads to significant deviations in pseudorange and phase observations, resulting in problems such as RTK / PPP fixation failure, positioning jitter of mobile inspection equipment, and unstable time synchronization based on GNSS. The traditional Single-Layer Equivalent Ionospheric Model (SLM) has been widely used in the construction of regional correction products and global ionospheric products due to its simplicity, high computational efficiency, and ability to provide an overall estimate of VTEC (Zenith TEC). However, it cannot characterize the vertical morphology of the ionosphere and is prone to introducing systematic errors when faced with significant layered or vertical structural changes, making it difficult to meet the engineering requirements of high availability and high-precision positioning for critical services such as power grids.

[0003] In related technologies, three-dimensional ionospheric tomography has been proposed to recover the three-dimensional distribution of electron density. These methods are mainly divided into two categories: pixel-based (grid-based) and function-based algorithms. Pixel-based methods are intuitive but have numerous parameters, while function-based methods have fewer parameters and better smoothness. Regardless of the method, both face the problems of uneven distribution of observed rays and ill-posed normal equations in actual calculations, manifesting as typical rank deficiency or ill-conditioned matrices, thus affecting the reconstruction accuracy and stability. Summary of the Invention

[0004] In view of this, it is necessary to provide an ionospheric error correction method, system and equipment based on BeiDou satellite signals, which can at least overcome one of the above defects.

[0005] In a first aspect, embodiments of this application provide an ionospheric error correction method based on BeiDou satellite signals, the method comprising:

[0006] Collect raw observation data of BeiDou satellite multi-frequency pseudorange and carrier phase from multiple BeiDou satellites within the BeiDou observation network;

[0007] The multi-frequency pseudorange and the original carrier phase observation data are used as the input source for three-dimensional tomographic modeling;

[0008] A three-dimensional tomographic modeling strategy combining layered function basis and projection function is used to construct a design matrix including ray geometric path integral information. The three-dimensional tomographic modeling strategy includes: dividing the ionosphere into multiple layers along the vertical height direction, applying spatial horizontal basis functions to characterize the horizontal distribution characteristics of electron density in each layer, and setting a vertical projection function to characterize the vertical distribution pattern of electron density as it changes with height.

[0009] A robust solver with regularization terms is applied to solve the linear equation system constructed from the design matrix to suppress the rank deficiency of the design matrix caused by uneven distribution of observed rays. The robust solver introduces constraint terms based on smoothing operators and adaptively adjusts the regularization parameters according to the observation environment.

[0010] The three-dimensional electron density field of the ionosphere is reconstructed based on the solved layered modeling coefficients. The line-of-sight delay in each observation satellite direction is extracted to generate ionospheric correction numbers for correcting BeiDou positioning and synchronization errors.

[0011] In one embodiment, the spatial horizontal basis function is a locally correlated basis function;

[0012] The three-dimensional tomographic modeling strategy employing a combination of hierarchical function bases and projection functions constructs a design matrix including ray geometric path integral information, comprising:

[0013] Within each of the aforementioned layers, non-uniformly distributed basis function nodes are established based on the site distribution topology of the BeiDou observation network.

[0014] By combining the coefficients of each basis function node with the corresponding basis function in a weighted linear combination, a continuous horizontal distribution field of electron density within each layer is constructed to characterize the asymmetric features of ionospheric perturbation in horizontal space.

[0015] In one embodiment, the method further includes:

[0016] According to the preset vertical profile rules of the ionosphere, the slope of the change of electron density of each layer with height is constrained by the attenuation characteristics of the vertical projection function;

[0017] The prior physical constraints provided by the vertical projection function are applied to reduce the vertical degrees of freedom in the modeling process, so as to restore the three-dimensional profile structure of the ionosphere while maintaining the horizontal resolution.

[0018] The vertical projection function is used to define the proportion of electron density energy distribution in the vertical direction for each layer.

[0019] In one embodiment, the method further includes:

[0020] The regularization term is used to perform spatial interpolation constraints on the sparse region covered by the observed rays, so as to eliminate the ill-posedness of the linear equation system caused by insufficient ray overlap, thereby improving the spatial continuity of the electron density field reconstructed during the active period of the ionosphere and in the sparse region of the station.

[0021] The robust solver evaluates the ill-conditioning of the design matrix during the solution process and adjusts the regularization parameter according to the ill-conditioning to balance the residual distribution of the observations with the spatial constraints of the smoothing operator.

[0022] In one embodiment, the method further includes:

[0023] A multi-frequency observation combination detection mechanism is used to identify and repair cycle slips in the carrier phase in real time;

[0024] A joint dynamic weighting model based on signal-to-noise ratio and satellite elevation angle is established to perform quality assessment and classification processing on the raw carrier phase observation data. Phase smoothing technology is applied to suppress the original pseudorange measurement noise, providing input observations with spatiotemporal continuity for the tomographic modeling.

[0025] In one embodiment, estimating the instrument bias between the satellite and the receiver on an intraday timescale includes:

[0026] Within a preset time window, establish the system difference analytical equation that includes the hardware delay at the satellite end and the hardware delay at the receiver end;

[0027] By establishing constraint criteria to separate the spatiotemporal evolution components of the ionosphere from the hardware measurement bias components, extracting the hardware differential code bias parameters, and applying the differential code bias parameters to perform an absolute correction on the relative total electron content after phase smoothing, in order to correct the impact of systematic measurement bias on modeling accuracy.

[0028] In one embodiment, extracting the line-of-sight delay in each observed satellite direction includes:

[0029] Based on the instantaneous spatial geometric relationship between the BeiDou satellite and each station in the BeiDou observation network, the trajectory of the signal transmission path in the three-dimensional electron density field of the ionosphere is determined;

[0030] The path integral operation is performed on the electron density of each layer reconstructed along the crossing trajectory to obtain the tilted total electron content in the line of sight direction, and then converted into the signal propagation path delay at the corresponding frequency.

[0031] In one embodiment, the method further includes:

[0032] The residual distribution characteristics of the generated ionospheric corrections are evaluated, and the change index reflecting the drastic fluctuations in ionospheric space is calculated.

[0033] When the change index exceeds a preset threshold, the update frequency of the three-dimensional tomographic modeling is adjusted, and the constraint term weights in the robust solver are optimized to improve the stability of the ionospheric correction number under strong ionospheric disturbances.

[0034] Secondly, one embodiment of this application provides an ionospheric error correction system based on BeiDou satellite signals, the system comprising:

[0035] The data acquisition module is used to collect raw observation data of BeiDou satellite multi-frequency pseudorange and carrier phase from multiple BeiDou satellites in the BeiDou observation network;

[0036] The deviation construction module is used to take the multi-frequency pseudorange and the original carrier phase observation data as the input source for three-dimensional tomographic modeling;

[0037] The model building module is used to construct a design matrix including ray geometric path integral information using a three-dimensional tomographic modeling strategy that combines layered function basis and projection function. The three-dimensional tomographic modeling strategy includes: dividing the ionosphere into multiple layers along the vertical height direction, applying spatial horizontal basis functions to characterize the horizontal distribution characteristics of electron density in each layer, and setting a vertical projection function to characterize the vertical distribution pattern of electron density as it changes with height.

[0038] The solver module is used to solve the linear equation system constructed by the design matrix using a robust solver with regularization terms to suppress the rank deficiency of the design matrix caused by uneven distribution of observed rays. The robust solver introduces constraint terms based on smoothing operators and adaptively adjusts the regularization parameters according to the observation environment.

[0039] The correction output module is used to reconstruct the three-dimensional electron density field of the ionosphere based on the solved layered modeling coefficients, extract the line-of-sight delay in the direction of each observed satellite, and generate ionospheric corrections for correcting BeiDou positioning and synchronization errors.

[0040] Thirdly, embodiments of this application provide an electronic device, including:

[0041] Processor; and

[0042] The memory stores computer-readable instructions for controlling the processor to execute the ionospheric error correction method based on BeiDou satellite signals as described in the first aspect.

[0043] This application provides an ionospheric error correction method, system, and device based on BeiDou satellite signals. Based on multi-frequency observations and intraday instrument bias absolutization using the BeiDou satellite navigation system, combined with three-dimensional tomographic modeling using layered function bases and vertical projection functions, and an adaptive regularization robust solution strategy, it can achieve high-precision ionospheric electron density reconstruction with vertical resolution under engineering conditions. Through cycle slip repair, phase smoothing, and observation weighting based on carrier-to-noise ratio / elevation angle, the spatiotemporal continuity and reliability of the input observations are significantly improved. The introduction of projection functions and spatial smoothing constraints effectively alleviates ill-conditioned problems caused by uneven ray distribution, maintaining spatial continuity and robustness of the reconstruction even with sparse observations or strong ionospheric disturbances. Finally, the line-of-sight correction output according to the observed satellite direction can be sent in real-time to power grid terminals such as RTK / PPP, UAV inspection, and PMU, thereby reducing the positioning failure rate, improving positioning and time-frequency synchronization accuracy, and enhancing service availability. It also possesses anomaly detection and adaptive update capabilities, facilitating deployment and integration in power grid operation and maintenance and emergency scenarios. Attached Figure Description

[0044] Figure 1 This is a schematic flowchart of an ionospheric error correction method based on BeiDou satellite signals provided in an embodiment of this application.

[0045] Figure 2 This is a schematic diagram of the geometric distribution of ionospheric chromatography provided in an embodiment of this application.

[0046] Figure 3 A schematic diagram of a module for an ionospheric error correction system based on BeiDou satellite signals is provided for one embodiment of this application.

[0047] Figure 4 This is a schematic diagram of the modules of an electronic device provided in an embodiment of this application.

[0048] Explanation of main component symbols

[0049] Detailed Implementation

[0050] The technical solutions of the embodiments of this application will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only a part of the embodiments of this application, and not all of them.

[0051] It should be noted that, in the embodiments of this application, "at least one" refers to one or more, and "more than one" refers to two or more. Unless otherwise defined, all technical and scientific terms used herein have the same meaning as commonly understood by one of ordinary skill in the art to which this application belongs. The terminology used in the specification of this application is for the purpose of describing particular embodiments only and is not intended to be limiting of this application.

[0052] It should be noted that in the embodiments of this application, the terms "first," "second," etc., are used only for descriptive purposes and should not be construed as indicating or implying relative importance, nor as indicating or implying order. Features specified as "first" or "second" may explicitly or implicitly include one or more of the stated features. In the description of the embodiments of this application, words such as "exemplary" or "for example" are used to indicate examples, illustrations, or explanations. Any embodiment or design scheme described as "exemplary" or "for example" in the embodiments of this application should not be construed as being more preferred or advantageous than other embodiments or design schemes. Specifically, the use of words such as "exemplary" or "for example" is intended to present the relevant concepts in a concrete manner.

[0053] Based on the embodiments described in this application, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of this application.

[0054] With the widespread application of global navigation satellite systems such as the BeiDou Navigation Satellite System in surveying, navigation, UAV inspection, and power grid time-frequency synchronization, the reliance on the continuity and accuracy of satellite positioning has significantly increased. However, in ionospherically active regions such as southern China, rapid or abrupt changes in TEC (Total Electron Content) often occur at noon or in the afternoon due to solar radiation, geomagnetic disturbances, and seasonal variations. This leads to significant deviations in pseudorange and phase observations, resulting in problems such as RTK / PPP fixation failure, positioning jitter of mobile inspection equipment, and unstable time synchronization based on GNSS. The traditional Single-Layer Equivalent Ionospheric Model (SLM) has been widely used in the construction of regional correction products and global ionospheric products due to its simplicity, high computational efficiency, and ability to provide an overall estimate of VTEC (Zenith TEC). However, it cannot characterize the vertical morphology of the ionosphere and is prone to introducing systematic errors when faced with significant layered or vertical structural changes, making it difficult to meet the engineering requirements of high availability and high-precision positioning for critical services such as power grids.

[0055] In related technologies, three-dimensional ionospheric tomography has been proposed to recover the three-dimensional distribution of electron density, mainly divided into two categories: pixel-based (grid-based) and function-based algorithms. Pixel-based methods are intuitive but have many parameters, while function-based methods have fewer parameters and better smoothness. Regardless of the method, in actual calculations, both face the problems of uneven distribution of observed rays and ill-posed normal equations, manifesting as typical rank deficiency or ill-conditioned matrices, thus affecting reconstruction accuracy and stability. For engineering real-time applications, it is also necessary to address the influence of sources such as observation noise, multipath effects, instrument bias, and phase ambiguity, and to provide robust degradation strategies when observations are sparse or perturbations are severe. Therefore, how to recover meaningful vertical profiles while ensuring horizontal fitting accuracy and meeting the requirements of real-time correction latency and availability is a core challenge that urgently needs to be overcome in current technology.

[0056] In view of this, the ionospheric error correction method, system, and equipment based on BeiDou satellite signals provided in this application, based on multi-frequency observations and intraday instrument bias absolutization of the BeiDou satellite navigation system, three-dimensional tomographic modeling combining layered function basis and vertical projection function, and a robust solution strategy with adaptive regularization, can achieve high-precision ionospheric electron density reconstruction with vertical resolution under engineering conditions. Through cycle slip repair, phase smoothing, and observation weighting based on carrier-to-noise ratio / elevation angle, the spatiotemporal continuity and reliability of input observations are significantly improved. The introduction of projection function and spatial smoothing constraints effectively alleviates the ill-conditioned problem caused by uneven ray distribution, and can still maintain the spatial continuity and robustness of reconstruction even when observations are sparse or there are strong ionospheric disturbances. Finally, the line-of-sight correction amount output according to the direction of the observed satellite can be sent in real time to power grid terminals such as RTK / PPP, UAV inspection, and PMU, thereby reducing the positioning failure rate, improving the positioning and time-frequency synchronization accuracy, and enhancing service availability. At the same time, it has anomaly detection and adaptive update capabilities, which facilitates deployment and integration in power grid operation and maintenance and emergency scenarios.

[0057] Figure 1 This is a schematic flowchart of an ionospheric error correction method based on BeiDou satellite signals provided in an embodiment of this application. Figure 1The ionospheric error correction method based on BeiDou satellite signals shown includes at least the following steps: S100: Collect raw observation data of BeiDou satellite multi-frequency pseudorange and carrier phase from multiple BeiDou satellites in the BeiDou observation network; S200: Use the raw observation data of multi-frequency pseudorange and carrier phase as the input source for three-dimensional tomographic modeling; S300: Construct a design matrix including ray geometric path integral information using a three-dimensional tomographic modeling strategy that combines layered function basis and projection function; S400: Apply a robust solver with regularization terms to solve the linear equation system constructed by the design matrix to suppress the rank deficiency of the design matrix caused by uneven distribution of observed rays; S500: Reconstruct the three-dimensional electron density field of the ionosphere based on the solved layered modeling coefficients, extract the line-of-sight delay in each observation satellite direction, and generate ionospheric correction numbers for correcting BeiDou positioning and synchronization errors.

[0058] S100: Collects raw observation data of BeiDou satellite multi-frequency pseudorange and carrier phase from multiple BeiDou satellites within the BeiDou observation network.

[0059] In this embodiment of the application, the ionospheric error correction method based on BeiDou satellite signals further includes, in step S100, acquiring raw observation data of BeiDou satellite multi-frequency pseudorange and carrier phase from multiple BeiDou satellites in the BeiDou observation network.

[0060] Specifically, the data acquisition items should include at least: multi-frequency pseudorange With multi-frequency carrier phase (subscript) The data includes frequency band (e.g., L1, L2), satellite and receiver time stamps (UTC / UTC-TAI), receiver / antenna model and antenna phase center (APC) calibration constant, signal-to-noise ratio (SNR) per observation epoch, receiver observation elevation / azimuth angle, observation sampling rate, and time stamp accuracy. Sampling and aggregation strategy: The receiver samples locally at 1Hz (or higher), and data is packaged and uploaded to the processing center in 1s or 30s packets. To support a 5-minute product cycle, the data link and buffering mechanism must be able to tolerate short-term packet loss and perform retransmission / retransmission. Observation preprocessing requirements include recording: cycle slip / link break detection flags, pseudorange / phase smoothing status (e.g., phase-smoothed pseudorange parameters), and multipath indicators (MP1 / MP2 or residual-based multipath exponents).

[0061] Understandably, multi-frequency (≥2-frequency) observations and complete metadata (SNR, elevation angle, antenna constants) are fundamental for subsequent pseudorange combination absolutization, phase smoothing, and construction of the observation weight matrix; without this information, the recovery accuracy of absolute TEC and the stability of subsequent tomography will be significantly reduced. In engineering implementation, all time / length units must be consistent (e.g., TEC uses electrons / Or TECU, 1 TECU = electrons / (The delay is in meters), and the unit and dimension of the field are clearly specified in the data interface.

[0062] S200: Uses raw observation data of multi-frequency pseudorange and carrier phase as the input source for three-dimensional tomographic modeling.

[0063] In this embodiment of the application, the ionospheric error correction method based on BeiDou satellite signals further includes in step S200 constructing a geometrically non-combined observation based on the original observation data of multi-frequency pseudorange and carrier phase, estimating the instrument deviation between the satellite and the receiver on an intra-day time scale, and converting the relative total electron content into the absolute total electron content based on the instrument deviation as the input source for three-dimensional tomographic modeling.

[0064] Specifically, a geometry-free combination is constructed: for dual-frequency (1, 2) observations, the pseudorange difference and phase difference are defined:

[0065]

[0066]

[0067] in (rice), Frequency (Hz); The pseudorange group delay bias (satellite / receiver, the sum of both in meters); Phase deviation (meters); Wavelength (meters); For integer phase ambiguity; For noise / multipath.

[0068] Determining the absolute TEC from pseudo-distance (if the deviation is known): Define the coefficient.

[0069]

[0070] Then the absolute TEC of the pseudo-distance method is

[0071]

[0072] If we use the BeiDou L1 frequency of approximately 1575.42MHz, then the corresponding delay for 1 TECU is approximately... m (calculated using precise frequency in engineering implementation).

[0073] High-precision relative TEC is obtained from the phase difference (set as follows). The difference between the two is:

[0074]

[0075] Assuming that the instrument bias is approximately constant over short periods of the day, then within a given time window (e.g., several hours or a single-day overhead arc), the bias... A robust average (such as the median or weighted least squares) can estimate the constant shift. This leads to a high-precision TEC with absolute accuracy.

[0076]

[0077] Instrument bias extraction (engineering steps): Constructing a system of linear equations within an intraday time window to simultaneously fit the trajectory of multiple satellites / stations. and The satellite-end and receiver-end biases are separated by constraints (such as smoothing between adjacent time points and continuous satellite-end bias); robust estimation (such as Huber or LAD) is used to suppress the influence of anomalous observations on bias estimation.

[0078] Understandably, pseudorange provides an absolute scale but has significant noise; phase provides fine-grained temporal variations but contains constant offsets. Combining both through intraday averaging or least squares decomposition methods can achieve a balance between absoluteness and high precision, thus providing downstream tomography with a TEC input that is both absolute and highly accurate. In implementation, cycle slips and sudden anomalies (SNR drops, low elevation noise) should be removed before bias estimation; the bias unit should be uniformly set to meters in the calculation link or directly converted to TECU (using coefficients). (Do the conversion).

[0079] S300: A three-dimensional tomographic modeling strategy combining layered function basis and projection function is used to construct a design matrix that includes ray geometry path integral information.

[0080] In this embodiment of the application, the ionospheric error correction method based on Beidou satellite signals further includes in step S300 the construction of a design matrix including ray geometric path integral information by adopting a three-dimensional tomographic modeling strategy that combines layered function basis and projection function; wherein, the three-dimensional tomographic modeling strategy includes: dividing the ionosphere into multiple layers along the vertical height direction, applying spatial horizontal basis functions to characterize the horizontal distribution characteristics of electron density in each layer, and setting a vertical projection function to characterize the vertical distribution pattern of electron density as height changes.

[0081] Please refer to the following: Figure 2 . Figure 2 This is a schematic diagram of the geometric distribution of ionospheric chromatography provided in an embodiment of this application.

[0082] Specifically, the layering and projection function: [details about height]. Direction is divided into Layers (e.g.) ), for the first Layer definition center height With thickness parameters And select the vertical projection function. The most common choice is the normalized Gaussian:

[0083]

[0084] Alternatively, normalize rectangular / triangular windows to represent layer thickness distribution. The unit is km (or m, which is consistent in engineering).

[0085] Horizontal basis functions and non-uniform nodes: Define a set of locally relevant basis functions at each layer. (e.g., B-spline, radial basis function (RBF), or local polynomial), nodes The observation stations can be arranged non-uniformly according to their density: dense observation areas have dense nodes to improve resolution, while sparse observation areas have sparse nodes to avoid over-parameterization.

[0086] The formation of the design matrix element: For the first... ray (path) ), No. Layer The matrix elements corresponding to the basis functions are path integrals:

[0087]

[0088] In the numerical implementation, the ray path is divided into small segments. The approximate calculation is a summation:

[0089]

[0090] in For the first Midpoint coordinates of the segment The length of this segment (in meters or kilometers, consistent with the scale of the basis functions).

[0091] Constructing the complete observation vector and matrix: [This involves] all rays... Expanding by combining all layers / basis functions yields a linear system. ,in It is a concatenated vector of all basis function coefficients in all layers (the time index can be solved independently by sliding window or time step by time).

[0092] Understandably, employing a "layered + projection function" approach reduces the vertical degrees of freedom (from a continuous function space to a low-dimensional parameter space of layered projections), thus ensuring physical interpretation even with insufficient observations (e.g., each layer is constrained by the physically expected height distribution); simultaneously, local basis functions allow the model to represent horizontal asymmetric perturbations (such as coastal / inland differences or topographical effects). In numerical implementation... Accurate calculations depend on satellite / receiver geometry, Earth curvature correction, and analytical expressions of basis functions. In engineering, it is necessary to ensure that the path segmentation resolution matches the basis function scale to avoid numerical integration errors affecting the solution.

[0093] S400: A robust solver with regularization terms is applied to solve the linear equations constructed from the design matrix to suppress the rank deficiency of the design matrix caused by uneven distribution of observed rays.

[0094] In this embodiment of the application, the ionospheric error correction method based on Beidou satellite signals further includes step S400 of applying a robust solver with regularization terms to solve the linear equation system constructed by the design matrix, so as to suppress the rank deficiency of the design matrix caused by the uneven distribution of observed rays. The robust solver introduces constraint terms based on smoothing operators and adaptively adjusts the regularization parameters according to the observation environment.

[0095] Specifically, the weighted regularized least squares form: defines the observation vector. (Depend on Composition), Design Matrix coefficient vector Let the observation weight matrix be... weight The weighting is determined by the combination of the ray's elevation angle, SNR, and multipath index (higher elevation angle, higher SNR, lower multipath → larger weight). Define the smoothing operator. (e.g., first-order spatial difference or Laplace operator), the objective is to solve for:

[0096]

[0097] The analytical normal solution (if the matrix is ​​invertible) is:

[0098]

[0099] Adaptive Regularization Parameters: Online Evaluation Matrix The condition number or singular value distribution (SVD) automatically increases according to rules when the ill-conditioned degree increases (the condition number is large or there are too many small singular values). Rules can be triggered using L-curve, residual variance ratio, or threshold. Alternatively, truncated SVD (retaining only singular values) is supported. ) or sparse regularization (L1) can be used to obtain sparse solutions under a specific prior.

[0100] Robustness and outlier handling: Before solving, the observation residuals are progressively eliminated (e.g., 3σ elimination) or a weighted robust objective (Huber loss) is used to suppress the influence of outliers on coefficient estimation.

[0101] Understandably, the regularization term is achieved through... The forced coefficient vector is spatially smooth (or sparse on the prior scale), and its relationship with the observation fit term is determined by... This balance prioritizes fitting data details when observations are abundant, while ensuring the continuity and physical interpretability of the solution when observations are sparse or noise is high. The adaptive adjustment mechanism enables the system to provide higher spatial resolution reconstructions during periods of ionospheric tranquility, while automatically degenerating into smoother, more robust solutions during periods of geomagnetic storms or sparse observations, thus meeting the engineering requirements for stability in real-time operations.

[0102] S500: Reconstruct the three-dimensional electron density field of the ionosphere based on the solved layered modeling coefficients, extract the line-of-sight delay in the direction of each observed satellite, and generate ionospheric correction numbers for correcting BeiDou positioning and synchronization errors.

[0103] In this embodiment of the application, the ionospheric error correction method based on BeiDou satellite signals further includes in step S500 reconstructing the three-dimensional electron density field of the ionosphere according to the solved layered modeling coefficients, extracting the line-of-sight delay in each observation satellite direction, and generating ionospheric correction numbers for correcting BeiDou positioning and synchronization errors.

[0104] Specifically, 3D reconstruction: using the obtained coefficients To recover the electron density at any spatial point (x,y,z):

[0105]

[0106] Line-of-sight delay calculation: For any satellite-receiver pair to be corrected, its propagation path Calculate the tilted TEC (STEC) by passing through the reconstruction field:

[0107]

[0108] in This is the integral projection of the path onto the basis functions (consistent with the definition in S300). Convert the STEC to the corresponding frequency. The path delay (meters) below:

[0109]

[0110] Typically, two types of corrections are output: direct path delay in meters. and TECUs (For scientific research / monitoring and analysis purposes).

[0111] Distribution format and interface: Package the corrections and quality indicators (residual estimates, confidence intervals, number of valid observations, and publication timestamps) for each satellite-receiver pair and distribute them in real time to the RTK / PPP server, UAV navigation controller, and PMU time synchronization unit via NTRIP, industry API (JSON / binary), or private network protocol.

[0112] Quality control: Simultaneously calculate the distribution of the corrected observation residuals (e.g. The statistics are obtained and the change index (such as the maximum relative gradient or the regional TEC change rate) is output to trigger the fault tolerance strategy on the terminal side (such as increasing the local filtering window, switching to redundant time sources, etc.).

[0113] Understandably, the satellite orientation correction obtained through the above process can be directly incorporated into the observation model of the RTK algorithm or used as the ionospheric correction input for PPP, thereby significantly reducing the impact of pseudorange / phase errors on positioning and synchronization during ionospheric disturbances. The output confidence and residual information are crucial for downstream systems (such as UAV navigation control and dispatch centers): when the confidence is low or the exponential change is high, the terminal can switch between fault-tolerant and alarm modes; when the confidence is high and the latency meets real-time requirements (e.g., end-to-end latency <10s), the correction can seamlessly improve the fixation success rate and time-frequency synchronization accuracy.

[0114] In this embodiment, the spatial horizontal basis functions are locally correlated basis functions. A three-dimensional tomographic modeling strategy combining layered basis functions and projection functions is used to construct a design matrix including ray geometric path integral information. This includes: establishing non-uniformly distributed basis function nodes within each layer based on the site distribution topology of the BeiDou observation network; and constructing a continuous horizontal distribution field of electron density within each layer by weighted linear combination of the coefficients of each basis function node and the corresponding basis function to characterize the asymmetric characteristics of ionospheric perturbation in horizontal space.

[0115] Specifically, in the first In-layer selection Locally related basis function nodes Each node corresponds to a local basis function. Common configurations include compact-supported radial bases (RBFs, such as Wendland functions), local B-splines, or local polynomial bases. A typical form of RBF is:

[0116]

[0117] in For the first The radius of influence of a layer (in meters or kilometers). To standardize compact support cores (e.g.) Such as Wendland polynomials, or Gaussian kernels The compact support property ensures that each basis function is non-zero only within its influence domain, thereby controlling the sparsity of the design matrix and the solution efficiency.

[0118] The horizontal field is represented as a linear combination of coefficients:

[0119]

[0120] in The estimated coefficients (electrons / ) are updated over time. Multiplying the dimensions of the quantity by the normalized scale of the basis functions will give the correct physical quantity.

[0121] Basis function nodes are non-uniformly distributed according to station location: in densely observed areas (small station spacing), smaller... With denser nodes to improve resolution; magnification in sparse observation regions. Alternatively, sparse nodes can be used to avoid parameter overfitting. Numerically, node arrangement can employ Delaunay triangulation with local refinement or empirical density function mapping.

[0122] The matrix element is designed to be obtained by numerical integration along the ray path: for the ... ray and the first basis functions

[0123]

[0124] in For the first Midpoint of each integral segment The segment length. The resolution of numerical integration should be consistent with... It matches the scaling of the basis function.

[0125] Understandably, locally relevant basis functions possess both expressive power and computational controllability: compactly supported kernels enable... It is sparse, has high solution efficiency, and its localized representation is beneficial for characterizing asymmetric perturbations near coastal / inland areas, terrain, or sites; basis function nodes and influence radii The choice of [value] directly determines the horizontal resolution and numerical stability, and should be set adaptively based on the site topology.

[0126] The physical dimensions of the basis function coefficients should be consistent (e.g., the coefficients multiplied by the basis functions and then integrated to obtain electrons / (STEC); In engineering implementation, it is necessary to unify the basis function normalization (or retain the normalization factor) to ensure and Dimensional matching is achieved, thus yielding coefficient solutions with clear physical meaning. Non-uniform node arrangement can save parameters and computational overhead while ensuring high resolution in critical areas, which is especially important for real-time systems.

[0127] In this embodiment, the method further includes: constraining the slope of electron density variation with height for each layer by using the attenuation characteristics of the vertical projection function, according to a preset vertical profile rule for the ionosphere. The prior physical constraints provided by the vertical projection function reduce the vertical degrees of freedom in the modeling process, thereby restoring the three-dimensional profile structure of the ionosphere while maintaining horizontal resolution. The vertical projection function is used to limit the proportion of electron density energy distribution in the vertical direction for each layer.

[0128] Specifically, for the first Select vertical projection function for the layer The most common choice is the normalized Gaussian:

[0129]

[0130] in (m or km) represents altitude. The height of the center layer. For layer thickness (standard deviation), the normalization factor guarantees (Or, as required by the project, normalized to the total energy ratio of the layer). Alternative solutions include normalized rectangular or triangular windows to represent approximately uniform or gradually varying layer thickness distributions.

[0131] The prior constraints introduced by the projection function can be represented as weights or hard constraints in the inversion: for example, restricting the vertical gradient norm, which can be formalized as adding a vertical derivative constraint to the regularization term.

[0132]

[0133] Or in the discrete version using a matrix Indicate and add to the overall regularization:

[0134]

[0135] in For horizontal smoothing operators, For vertical slope constraint operators, Balance the horizontal and vertical smoothing intensity respectively.

[0136] Preset vertical profile rules can be set based on historical average profiles or physical models (e.g., the statistical pattern of the center height of layer F2 with latitude / longitude / time of day). The prior values ​​are used to allow for fine-tuning with near real-time data during runtime.

[0137] Understandably, the vertical projection function projects continuous data... The directional dimension is reduced to a hierarchical coefficient (from infinite to finite), which reduces the uncertainty of the solution while preserving the crucial vertical response; the center height of the projection function With width The setting of the vertical constraint determines the sensitivity of the layer to electron densities at different heights and should be selected in conjunction with the regional ionospheric characteristics (such as the center height of the F layer) and the observation geometry. Vertical constraints are particularly important when observing sparse or ray-dependent structures that primarily traverse a certain height band: they prevent the appearance of vertical oscillatory solutions (non-physical solutions) and improve the physical interpretability of the reconstruction results. However, excessively strong vertical constraints can weaken the response to sudden vertical structures (such as perturbation-induced uplift / sinking). It needs to be adaptively adjusted based on the change index.

[0138] In this embodiment, the method further includes: using a regularization term to perform spatial interpolation constraints on sparse regions covered by the observed rays, in order to eliminate the ill-posedness of the linear equations due to insufficient ray overlap, thereby improving the spatial continuity of the electron density field reconstructed during periods of ionospheric activity and in sparse regions of the station. The robust solver evaluates the ill-conditioning of the design matrix during the solution process and adjusts the regularization parameter according to the ill-conditioning to balance the residual distribution of the observations with the spatial constraints of the smoothing operator.

[0139] Specifically, a weighted Tikhonov regularization framework is adopted (see S400), where the smoothing operator can be a two-dimensional Laplacian (the sparse matrix corresponding to the discretization). Alternatively, a difference operator based on grid adjacency can be used to perform spatial interpolation / smoothing on the coefficient vector:

[0140]

[0141] The degree of pathologicalness can be assessed by obtaining a matrix through singular value decomposition (SVD). Spectral information: If Singular values ​​are listed in descending order. Then the condition number (like Very small (Very high) can be used as an ill-conditioned indicator. Example of an adaptive rule:

[0142]

[0143] in and It is an empirical constant. Control the rate of increase; or use the consistency criterion / discrepancy principle to select. This ensures that the sum of squared residuals matches the variance of the observation noise. In extremely ill-conditioned cases, truncated SVD (TSVD) can be used: only retaining... Greater than the threshold The components, or the use of sparse constraints (L1) to recover solutions with locally concentrated characteristics. The spatial interpolation effect is numerically manifested as follows: in sparse observation regions, the coefficients are strongly constrained by the neighborhood, forming a smooth transition; in dense observation regions, due to the observation weights... The improvement makes it closer to the real local structure.

[0144] Understandably, regularization acts as an "interpolator for observational gaps": where ray coverage is insufficient, the regularization term fills in the information with physical / mathematical priors (smoothness or neighborhood consistency), thereby avoiding non-physical interpretations and numerical oscillations; adaptive adjustment The system can relax constraints to preserve details when observations are plentiful, and strengthen constraints to ensure continuity and robustness when observations are scarce or noise is high. The SVD / condition number-driven adaptive strategy enables the system to automatically switch solution modes under different observation qualities / densities (e.g., from ordinary Tikhonov to truncated SVD or stronger smoothing), which is crucial for ensuring the reliability of real-time product delivery and availability to terminals.

[0145] In this embodiment, the method further includes: using a multi-frequency observation combination detection mechanism to identify and repair cycle slips in the carrier phase in real time; establishing a joint dynamic weighting model based on signal-to-noise ratio and satellite elevation angle; performing quality assessment and classification processing on the raw carrier phase observation data; and applying phase smoothing technology to suppress raw pseudorange measurement noise, providing spatiotemporally continuous input observations for tomographic modeling.

[0146] Specifically, cycle slip detection and repair:

[0147] Constructing multiple frequency combinations to utilize redundant information: geometrically independent combinations (such as...) , () is used to eliminate geometric terms and amplify ionospheric / bias effects; wide-lane or multi-frequency redundant combination is used to sensitively detect integer step changes.

[0148] Time difference detection: for any combination of quantities (For example Calculate the difference .like Then it is determined that a cycle slip may occur; the threshold can be set according to the observed noise and expected physical changes:

[0149]

[0150] in The standard deviation of the observation noise of the combined quantity (estimated by SNR / receiver performance). The maximum permissible rate of physical change (given by historical TEC rate of change or model). is the confidence factor.

[0151] Multi-frequency consistency detection: Utilize the redundancy of three or more frequency bands to construct independent detection statistics (e.g., wide wavelength combinations and geometrically independent combinations do not simultaneously satisfy stationarity). When multiple statistics are triggered simultaneously, cycle slips are determined with high confidence.

[0152] Repair strategy: If cycle slips are detected, the number of integer cycle slips is estimated and the phase is compensated using the Melbourne-Wübbena / wide-lane combination (or multi-frequency difference). If the redundancy is insufficient, the phase is reset using pseudorange within a short window and the epoch is marked as suspicious for subsequent weighting and filtering.

[0153] Joint dynamic weighting model (quality assessment): Assigning dynamic weights to each observation. (For smoothing, bias estimation, and tomographic input), a normalized expression available for an engineering project is:

[0154]

[0155] in: (Linearized SNR score, truncated at [0,1]); (Elevation angle weight, el is the satellite elevation angle,) (Control elevation angle sensitivity) (Multi-path metric MP, Controlled attenuation); Used for normalization .parameter Determined by on-site verification or historical statistics.

[0156] For low-weighted observations ( If the value is less than the threshold, its influence can be reduced or it can be directly removed from the phase smoothing or tomographic design matrix.

[0157] Phase smoothing (pseudorange denoising): Using phase-smoothed pseudorange: Pseudoranges are denoised using a multi-epoch phase average at each epoch. Filtering is typically implemented using recursive filtering:

[0158]

[0159] Alternatively, weighted fusion can be used to transform high-precision phase changes (from...) This is used to correct for short-term noise in pseudoranges, resulting in input observations that are both absolute and low-noise. Weighting It can adaptively adjust based on the current SNR and cycle slip detection results (reset when a cycle slip occurs).

[0160] Understandably, multi-frequency redundancy significantly improves the detection capability of cycle slips and reduces misjudgments; the time difference threshold must simultaneously consider both observation noise and the physical rate of change of the ionosphere to avoid misjudging real, intense ionospheric disturbances as cycle slips. Joint weighting incorporates factors such as SNR, elevation angle, and multipath into a unified index, which not only increases the weight of reliable observations in tomographic solutions but also provides a basis for dynamically eliminating noisy observations, thereby ensuring the spatiotemporal continuity and quality of subsequent TEC inputs. Phase smoothing combined with cycle slip repair can significantly reduce the interference of pseudorange noise on tomographic solutions without sacrificing the absolute scale (guaranteed by the absoluteification step of S200).

[0161] In this embodiment, estimating the instrument bias between the satellite and receiver on an intraday timescale includes: establishing a systematic difference analytical equation including the hardware delay at the satellite end and the hardware delay at the receiver end within a preset time window; separating the ionospheric spatiotemporal evolution component and the hardware measurement bias component by establishing constraint criteria; extracting the hardware differential code bias parameter; and applying the differential code bias parameter to perform an absolute correction on the phase-smoothed relative total electron content to correct the impact of systematic measurement bias on modeling accuracy.

[0162] Specifically, the bias model and linear decomposition: for each observation (by satellite) With receiver (Identifier) ​​in time Establish the relationship between the pseudorange-phase combination difference:

[0163]

[0164] in Satellite instrument deviation (expressed in TEC units or converted meters). For receiver-side instrument deviation, To account for measurement errors and high-frequency residuals, a linear system is obtained by stitching together all observations within a time window: ,in .

[0165] Constraints and reference datums: To eliminate degrees of freedom (e.g., by adding the same constant to all...) Subtract the same constant to all No impact Set constraints such as Or fix a certain reference receiver Solve using weighted least squares or robust estimation. :

[0166]

[0167] in For observation weights, It can be a smoothing or damping matrix to avoid overfitting. This is the microregularization parameter. Robust losses (such as Huber) can be used to suppress the effects of anomalous observations.

[0168] Absolute correction: obtained Then, it is applied to the relative TEC obtained by phase-smoothing to make it an absolute TEC:

[0169]

[0170] Understandably, by combining and constraining multi-satellite and multi-station data within a daily window, stable hardware biases can be separated from rapidly changing ionospheric components. Appropriately selecting a time window (with a constant ranging from tens of minutes to several hours) can balance bias stability with the timeliness of ionospheric changes. The reliability of the results depends on observational coverage (number of satellites and stations) and observational quality. When observations are sparse, regularization should be added or external priors (such as real-time broadcast satellite hardware delay estimations) should be introduced to improve robustness.

[0171] In this embodiment, extracting the line-of-sight delay for each observation satellite direction includes: determining the traverse trajectory of the signal transmission path in the three-dimensional electron density field of the ionosphere based on the instantaneous spatial geometric relationship between the BeiDou satellite and each station within the BeiDou observation network; performing path integration on the reconstructed electron density of each layer along the traverse trajectory to obtain the tilted total electron content in that line-of-sight direction, and converting it into the signal propagation path delay at the corresponding frequency.

[0172] Specifically, the path integral is approximated as follows: for the ray path of the satellite-receiver pair Segment by numerical values This indicates that the tilted TEC is approximately:

[0173]

[0174] If it has been expressed using layered projection Then it can be reconstructed using the design matrix projection:

[0175]

[0176] in That is, the projection integral of the specific path in S300.

[0177] Convert STEC to path delay (meters) at frequency Down:

[0178]

[0179] In engineering projects, it is usually provided at the same time (TECU) and (meters), and can be converted to commonly used receiver frequencies (such as L1, L2). The recommended output format includes timestamp, satellite number, receiver number, and... , Observation weights / confidence levels, number of effective integration segments M, and residual indices (e.g., the residual corresponding to a single ray after reconstruction).

[0180] Understandably, the numerical accuracy of path integrals is affected by piecewise resolution, basis function interpolation errors, and the approximation of Earth's curvature; in engineering implementation, it should be ensured that... The delay is smaller than the basis function variation scale and takes into account the Earth's curvature and ionospheric georeference (Earth-fixed / celestial coordinate transformation). The output delay can be used not only directly for downstream positioning / synchronization correction, but also as quality control information—for example, if the residual of a certain path is high or the confidence level is low, the terminal can fall back to a redundant time source or expand the local filtering window.

[0181] In this embodiment of the application, the method further includes: evaluating the residual distribution characteristics of the generated ionospheric corrections and calculating a change index reflecting the drastic fluctuations in ionospheric space. When the change index exceeds a preset threshold, the update frequency of the three-dimensional tomographic modeling is adjusted, and the constraint term weights in the robust solver are optimized to improve the stability of the ionospheric corrections under strong ionospheric disturbances.

[0182] Specifically, residual statistics and the exponent of change are defined as follows: given the reconstructed observation residual vector... Calculate the following statistic: mean residuals residual standard deviation The maximum absolute value of the spatial residual The change index CI can be defined using the maximum relative rate of change over a given time period, for example:

[0183]

[0184] in Traverse the ground grid points or region of interest, in units of TECU / s (or TECU / min). A spatial gradient index can also be defined:

[0185]

[0186] Triggering logic and response: Set threshold and ,when or Time-based execution: Shorten product release cycles (e.g., from 5 minutes to 1 minute) to improve time resolution; increase regularization weights. (or vertical constraint ratio) To improve smoothness; prompt the terminal to take fault-tolerant measures (such as expanding the local filter window, switching to a redundant time source, and recording events for offline playback). Adaptive weight optimization: During the strong disturbance phase, the observation weight matrix can be adjusted. The penalty for low elevation angle observations is increased, or the weight of high SNR observations and dense sites is temporarily increased, so as to maintain a stable correction output when data availability decreases.

[0187] Understandably, the change index provides an operational means of "perturbation quantification": when the ionosphere fluctuates drastically over short spatiotemporal scales, the system should prioritize the robustness of the correction (smoothing, extending the confidence interval) rather than over-tracking noise. Parameter adjustments after triggering (update frequency, regularization strength, weight allocation) must be performed in real-time and recorded for post-event verification and model iteration; threshold settings can be based on historical event statistics to obtain the optimal trade-off between recall and precision.

[0188] The complete workflow of the ionospheric error correction method based on BeiDou satellite signals provided in this application is described below with an exemplary embodiment.

[0189] When a regional power grid operator deployed drones for inspection and substation PMUs (Phasor Measurement Units) along a critical 220kV transmission corridor, the accident rate increased due to daytime ionospheric disturbances in southern China, resulting in drone positioning jitter and decreased PMU synchronization accuracy. To address this, the ionospheric error correction system proposed in this application, based on multi-frequency observation and hierarchical tomographic modeling using the BeiDou satellite navigation system, was adopted to ensure operational continuity. The end-to-end workflow, from on-site data acquisition to real-time correction distribution and application at the power grid, is described below.

[0190] System Deployment and Assumptions

[0191] Coverage area: A 200km transmission corridor with 12 enhanced observation stations (CORS / baseline stations) along the line, with an average spacing of ≈18km; covering 3 substations (each equipped with RTK / PPP access and PMU).

[0192] Receiver / Antenna: Dual-band (L1 / L2) high-precision receiver + phase center calibration antenna. Data sampling is 1Hz, locally buffered and uploaded every 30 seconds; real-time product release cycle is normally 5 minutes, reduced to 1 minute during disturbances.

[0193] Example of tomographic model parameters: number of layers L=3 (layer center height z=90km, 200km, 350km); the basis function nodes of each layer are arranged non-uniformly according to the site topology, and the influence radius r0 is taken as 50km (near field details) to 120km (sparse region) within the layer.

[0194] Target business metrics (pilot targets): Increase RTK fixation success rate by ≥20% during periods of ionospheric activity, reduce UAV positioning RMS by ≥0.5m, and reduce PMU synchronization jitter to <1μs (depending on the existing network baseline and local hardware).

[0195] Complete workflow (step-by-step, including formulas, parameters, and decision points)

[0196] S100—Field Data Acquisition (Triggering and Synchronization)

[0197] Action (on-site): Each CORS / augmentation station, along with on-site drones and substation receivers, acquires multi-frequency pseudorange data at 1Hz. carrier phase SNR, elevation / azimuth, antenna APC metadata, and timestamps. Data is locally cached for a short period and uploaded to the regional calculation center in 30-second batches. Engineering values / thresholds: Minimum SNR reception threshold set at 25dB-Hz; sampling packet loss rate <1%. Purpose: To ensure the integrity and time synchronization of the original observations in subsequent steps.

[0198] S200—No Geometric Combination, Weekslip Correction, and Intraday Bias Estimation (Preprocessing and Absoluteization)

[0199] Specifically (preprocessing):

[0200] Cycle slip detection (multi-frequency differential and time differential detection) is performed on each satellite-station observation and repaired immediately (see Melbourne–Wübbena or wide-lane combination); example of detection threshold: .

[0201] Calculate geometry-free combinations:

[0202]

[0203] And use pseudo-distance difference Obtain a rough absolute TEC:

[0204]

[0205] Due to phase difference Obtain high-precision relative TEC Within the intraday window (e.g., 2 hours) Perform a robust average and estimate the instrument bias term (satellite end). With receiver end Solve the linear equations. (See S200 above) Weighted least squares or Huber estimation is used to obtain .

[0206] Absolutization: obtaining , as the input for tomography.

[0207] Understandably, pseudorange provides an absolute scale but has significant noise; phase provides a fine timing sequence but carries a constant bias. Intra-day bias estimation combines the two to obtain a TEC input that is both absolute and smooth, reducing the systematic impact of hardware bias on the tomographic solution.

[0208] S300—Constructing a design matrix (modeling) for hierarchical function basis and projection functions.

[0209] Specifically (model building):

[0210] Vertically divide into L=3 layers, and set the layer center. =[90,200,350]km, layer thickness =[30,50,70]km (Gaussian projection example):

[0211]

[0212] The horizontal basis functions are compactly supported RBF (Wendland) or local B-spline, with nodes distributed non-uniformly according to the observation station topology: 10–20 km between nodes in densely populated sections and 50–120 km in sparsely populated sections. The radius of influence of the basis functions is... Take a distance of 50–120 km.

[0213] Numerical integration over each satellite ray yields the design matrix element:

[0214]

[0215] It is understandable that the projection function constrains the vertical shape (reducing the degrees of freedom), while the local basis functions allow for high-resolution modeling in key areas, thus achieving both physical consistency and meeting the sparsity requirements of real-time engineering solutions.

[0216] S400—Robust Solution (Weighted Regularization and Adaptive)

[0217] Specifically (solving):

[0218] Construct the observation weight matrix The weights are calculated jointly based on SNR, elevation angle, and multipath metrics (example: Normalized SNR× × ).

[0219] This leads to the weighted regularization problem:

[0220]

[0221] Analytical form (if invertible):

[0222]

[0223] Adaptive rule: Calculate before each solution SVD / condition number ,like (Example) Then increase Or enable TSVD.

[0224] Output reconstruction coefficients and residual statistics .

[0225] It is understandable that the weighted term prioritizes high-quality observations, the smoothing term ensures the continuity of the solution under sparse / perturbed conditions, and the condition number-driven adaptive regularization can automatically ensure the stability of the solution and control noise amplification in real-time systems.

[0226] S500—3D Reconstruction, Path Integration, and Real-time Data Delivery (Application Side)

[0227] Specifically (restructuring and distribution):

[0228] use Reconstruct the electron density at any (x,y,z):

[0229]

[0230] Calculate the line-of-sight STEC for each on-network terminal (RTK base station, drone, PMU) and convert it to frequency. Path delay:

[0231]

[0232] Packaged distribution: For each terminal, the correction amount is based on the satellite output ( inm、 (inTECU), confidence / residual, timestamp, and distributed via NTRIP / private network API (target end-to-end latency <10s).

[0233] End-user application examples: UAV navigation controllers add corrections to the observation model to reduce instantaneous positioning errors; RTK servers apply corrections in baseline calculations to improve fixation success rates; PMUs use corrections to improve the stability of GNSS time sources or as a detection source to determine whether to switch to a backup time source.

[0234] It is understandable that real-time corrections not only directly improve positioning / synchronization, but also provide confidence information that allows the terminal to adopt fault-tolerant strategies (such as expanding the filtering window or issuing an alarm) based on the current quality.

[0235] Runtime monitoring, adaptation, and contingency strategies

[0236] Change index monitoring: Calculate CI and spatial gradient SG (e.g., TECU / min, TECU / 100km), threshold example CI_thr=0.5TECU / min; if the threshold is exceeded, trigger: update cycle decreases to 1min, increases... The system also issues a "high disturbance" flag on the terminal. Logs and playback: Observations, coefficients, and residuals are saved, supporting replay for parameter optimization and event analysis. Fault tolerance: If the data link is interrupted, the edge station retains the last correction locally and allows the terminal to switch to local filtering or a backup time source.

[0237] This embodiment, based on multi-frequency observations from the BeiDou Navigation Satellite System, intraday instrument bias absolutization, and three-dimensional tomographic modeling using a "layered function basis + projection function," exhibits significant engineering benefits: First, it improves the availability and accuracy of power grid positioning and time-frequency synchronization—significantly reducing the RTK / PPP fixation failure rate during periods of ionospheric activity (target improvement of approximately ≥20% in pilot tests), decreasing the RMS of UAV horizontal positioning by approximately 0.5m, and significantly reducing PMU synchronization jitter (target magnitude <1μs); Second, it enhances the observability and physical interpretability of the ionospheric vertical structure, recovering meaningful Ne vertical profiles through layered projection functions, which is beneficial for space environment monitoring and subsequent event analysis; Third, it improves the system's robustness and real-time response. Capabilities: Multi-frequency cycle slip detection, SNR / elevation angle joint weighting, phase smoothing, and adaptive regularization work together to output spatially continuous and confident correction products even when observations are sparse or subject to severe disturbances. It also automatically triggers encrypted updates and terminal fault-tolerant strategies through the change index. Fourth, it facilitates engineering deployment and operation and maintenance: The design balances the sparsity of the sparse matrix with the scalability of local basis functions, supporting real-time NTRIP / industry API distribution, log playback, and adaptive parameter adjustment, reducing the frequency of on-site manual intervention and shortening anomaly recovery time. Fifth, it has direct value for power grid emergency and operation and maintenance decisions, improving daily inspection and positioning quality, and providing reliable positioning / synchronization guarantees and early warning data for scheduling and maintenance during strong disturbances or extreme events.

[0238] Figure 2 This is a schematic diagram of an ionospheric error correction system module based on BeiDou satellite signals provided in one embodiment of this application. Figure 2The ionospheric error correction system 10 based on BeiDou satellite signals shown includes at least the following components: a data acquisition module 11, a deviation construction module 12, a model construction module 13, a solution module 14, and a correction output module 15.

[0239] In this embodiment, the data acquisition module 11 is used to acquire raw observation data of BeiDou satellite multi-frequency pseudorange and carrier phase from multiple BeiDou satellites within the BeiDou observation network. Please refer to [link / reference] for details. Figure 1 , 2 The details and their corresponding descriptions are not repeated here.

[0240] In this embodiment, the deviation construction module 12 is used to construct geometrically uncombined observations based on the original observation data of multi-frequency pseudorange and carrier phase, estimate the instrument deviation between the satellite and the receiver on an intraday timescale, and convert the relative total electron content into the absolute total electron content based on the instrument deviation, serving as the input source for three-dimensional tomographic modeling. Please refer to [link to details]. Figure 1 , 2 The details and their corresponding descriptions are not repeated here.

[0241] In this embodiment, the model building module 13 is used to construct a design matrix including ray geometric path integral information using a three-dimensional tomographic modeling strategy that combines layered function basis functions and projection functions. The three-dimensional tomographic modeling strategy includes: dividing the ionosphere into multiple layers along the vertical height direction; applying spatial horizontal basis functions to characterize the horizontal distribution characteristics of electron density within each layer; and setting a vertical projection function to characterize the vertical distribution pattern of electron density as it changes with height. Please refer to [link / reference] for details. Figure 1 , 2 The details and their corresponding descriptions are not repeated here.

[0242] In this embodiment, the solver module 14 is used to solve the linear equation system constructed from the design matrix using a robust solver with regularization terms, in order to suppress the rank deficiency of the design matrix caused by uneven distribution of observed rays. The robust solver introduces constraint terms based on smoothing operators and adaptively adjusts the regularization parameters according to the observation environment. Please refer to [link to details]. Figure 1 , 2 The details and their corresponding descriptions are not repeated here.

[0243] In this embodiment, the correction output module 15 is used to reconstruct the three-dimensional electron density field of the ionosphere based on the solved layered modeling coefficients, extract the line-of-sight delay in each observation satellite direction, and generate ionospheric corrections for correcting BeiDou positioning and synchronization errors. Please refer to [link to details] for further information. Figure 1 , 2 The details and their corresponding descriptions are not repeated here.

[0244] Figure 3 This is an electronic device 20 provided in one embodiment of this application. For example... Figure 3 As shown, the electronic device 20 includes at least the following components: a processor 21 and a memory 22.

[0245] In this embodiment, the memory 22 is used to store executable instructions of the processor 21, which, when configured to execute instructions, implement... Figure 1 The diagram shows an ionospheric error correction method based on BeiDou satellite signals.

[0246] It is understood that the ionospheric error correction method, system, and equipment based on BeiDou satellite signals provided in this application, based on multi-frequency observations and intraday instrument bias absolutization of the BeiDou satellite navigation system, three-dimensional tomographic modeling combining layered function basis and vertical projection function, and a robust solution strategy with adaptive regularization, can achieve high-precision ionospheric electron density reconstruction with vertical resolution under engineering conditions. Through cycle slip repair, phase smoothing, and observation weighting based on carrier-to-noise ratio / elevation angle, the spatiotemporal continuity and reliability of input observations are significantly improved. The introduction of projection function and spatial smoothing constraints effectively alleviates the ill-conditioned problem caused by uneven ray distribution, and can still maintain the spatial continuity and robustness of reconstruction when observations are sparse or there are strong ionospheric disturbances. Finally, the line-of-sight correction amount output according to the direction of the observed satellite can be sent to power grid terminals such as RTK / PPP, UAV inspection, and PMU in real time, thereby reducing the positioning failure rate, improving the positioning and time-frequency synchronization accuracy, and enhancing service availability. At the same time, it has anomaly detection and adaptive update capabilities, which facilitates deployment and integration in power grid operation and maintenance and emergency scenarios.

[0247] Those skilled in the art should recognize that the above embodiments are only used to illustrate this application and are not intended to limit this application. Any appropriate changes and variations made to the above embodiments within the essential spirit and scope of this application fall within the scope of protection claimed in this application.

Claims

1. A method for ionospheric error correction based on BeiDou satellite signals, characterized in that, The method includes: Collect raw observation data of BeiDou satellite multi-frequency pseudorange and carrier phase from multiple BeiDou satellites within the BeiDou observation network; The multi-frequency pseudorange and the original carrier phase observation data are used as the input source for three-dimensional tomographic modeling; A three-dimensional tomographic modeling strategy combining layered function basis and projection function is used to construct a design matrix including ray geometric path integral information. The three-dimensional tomographic modeling strategy includes: dividing the ionosphere into multiple layers along the vertical height direction, applying spatial horizontal basis functions to characterize the horizontal distribution characteristics of electron density in each layer, and setting a vertical projection function to characterize the vertical distribution pattern of electron density as it changes with height. A robust solver with regularization terms is applied to solve the linear equation system constructed from the design matrix to suppress the rank deficiency of the design matrix caused by uneven distribution of observed rays. The robust solver introduces constraint terms based on smoothing operators and adaptively adjusts the regularization parameters according to the observation environment. The three-dimensional electron density field of the ionosphere is reconstructed based on the solved layered modeling coefficients. The line-of-sight delay in each observation satellite direction is extracted to generate ionospheric correction numbers for correcting BeiDou positioning and synchronization errors.

2. The ionospheric error correction method based on BeiDou satellite signals according to claim 1, characterized in that, The spatial horizontal basis functions are locally correlated basis functions; The three-dimensional tomographic modeling strategy employing a combination of hierarchical function bases and projection functions constructs a design matrix including ray geometric path integral information, comprising: Within each of the aforementioned layers, non-uniformly distributed basis function nodes are established based on the site distribution topology of the BeiDou observation network. By combining the coefficients of each basis function node with the corresponding basis function in a weighted linear combination, a continuous horizontal distribution field of electron density within each layer is constructed to characterize the asymmetric features of ionospheric perturbation in horizontal space.

3. The ionospheric error correction method based on BeiDou satellite signals according to claim 2, characterized in that, The method further includes: According to the preset vertical profile rules of the ionosphere, the slope of the change of electron density of each layer with height is constrained by the attenuation characteristics of the vertical projection function; The prior physical constraints provided by the vertical projection function are applied to reduce the vertical degrees of freedom in the modeling process, so as to restore the three-dimensional profile structure of the ionosphere while maintaining the horizontal resolution. The vertical projection function is used to define the proportion of electron density energy distribution in the vertical direction for each layer.

4. The ionospheric error correction method based on BeiDou satellite signals according to claim 3, characterized in that, The method further includes: The regularization term is used to perform spatial interpolation constraints on the sparse region covered by the observed rays, so as to eliminate the ill-posedness of the linear equation system caused by insufficient ray overlap, thereby improving the spatial continuity of the electron density field reconstructed during the active period of the ionosphere and in the sparse region of the station. The robust solver evaluates the ill-conditioning of the design matrix during the solution process and adjusts the regularization parameter according to the ill-conditioning to balance the residual distribution of the observations with the spatial constraints of the smoothing operator.

5. The ionospheric error correction method based on BeiDou satellite signals according to claim 1, characterized in that, The method further includes: A multi-frequency observation combination detection mechanism is used to identify and repair cycle slips in the carrier phase in real time; A joint dynamic weighting model based on signal-to-noise ratio and satellite elevation angle is established to perform quality assessment and classification processing on the raw carrier phase observation data. Phase smoothing technology is applied to suppress the original pseudorange measurement noise, providing input observations with spatiotemporal continuity for the tomographic modeling.

6. The ionospheric error correction method based on BeiDou satellite signals according to claim 5, characterized in that, The estimation of instrument bias between the satellite and the receiver on an intraday timescale includes: Within a preset time window, establish the system difference analytical equation that includes the hardware delay at the satellite end and the hardware delay at the receiver end; By establishing constraint criteria to separate the spatiotemporal evolution components of the ionosphere from the hardware measurement bias components, extracting the hardware differential code bias parameters, and applying the differential code bias parameters to perform an absolute correction on the relative total electron content after phase smoothing, in order to correct the impact of systematic measurement bias on modeling accuracy.

7. The ionospheric error correction method based on BeiDou satellite signals according to claim 1, characterized in that, The extraction of the line-of-sight delay from each observed satellite direction includes: Based on the instantaneous spatial geometric relationship between the BeiDou satellite and each station in the BeiDou observation network, the trajectory of the signal transmission path in the three-dimensional electron density field of the ionosphere is determined; The path integral operation is performed on the electron density of each layer reconstructed along the crossing trajectory to obtain the tilted total electron content in the line of sight direction, and then converted into the signal propagation path delay at the corresponding frequency.

8. The ionospheric error correction method based on BeiDou satellite signals according to claim 7, characterized in that, The method further includes: The residual distribution characteristics of the generated ionospheric corrections are evaluated, and the change index reflecting the drastic fluctuations in ionospheric space is calculated. When the change index exceeds a preset threshold, the update frequency of the three-dimensional tomographic modeling is adjusted, and the constraint term weights in the robust solver are optimized to improve the stability of the ionospheric correction number under strong ionospheric disturbances.

9. An ionospheric error correction system based on BeiDou satellite signals, characterized in that, The system includes: The data acquisition module is used to collect raw observation data of BeiDou satellite multi-frequency pseudorange and carrier phase from multiple BeiDou satellites in the BeiDou observation network; The deviation construction module is used to take the multi-frequency pseudorange and the original carrier phase observation data as the input source for three-dimensional tomographic modeling; The model building module is used to construct a design matrix including ray geometric path integral information using a three-dimensional tomographic modeling strategy that combines layered function basis and projection function. The three-dimensional tomographic modeling strategy includes: dividing the ionosphere into multiple layers along the vertical height direction, applying spatial horizontal basis functions to characterize the horizontal distribution characteristics of electron density in each layer, and setting a vertical projection function to characterize the vertical distribution pattern of electron density as it changes with height. The solver module is used to solve the linear equation system constructed by the design matrix using a robust solver with regularization terms to suppress the rank deficiency of the design matrix caused by uneven distribution of observed rays. The robust solver introduces constraint terms based on smoothing operators and adaptively adjusts the regularization parameters according to the observation environment. The correction output module is used to reconstruct the three-dimensional electron density field of the ionosphere based on the solved layered modeling coefficients, extract the line-of-sight delay in the direction of each observed satellite, and generate ionospheric corrections for correcting BeiDou positioning and synchronization errors.

10. An electronic device, characterized in that, include: processor; as well as A memory having computer-readable instructions stored thereon for controlling the processor to execute the ionospheric error correction method based on BeiDou satellite signals as described in any one of claims 1 to 8.

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