A Short-Term Prediction Method for Tropospheric Delayed Products Based on Multi-Scale Sliding Window LSTM in PPP-RTK

Abstract

This invention discloses a short-term prediction method for tropospheric delay products based on a multi-scale sliding window LSTM in PPP-RTK, comprising: extracting the zenith tropospheric delay based on PPP-RTK calculation and generating the original tropospheric delay product using an interpolation model; constructing different subsets based on the generated original tropospheric delay product using a multi-scale sliding window, the number of subsets being related to the number of stations; inputting the constructed subsets into corresponding long short-term memory networks, and outputting the prediction results of each long short-term memory network; and fusing the prediction results of each long short-term memory network by combining the distance between the base station and the rover station to obtain the short-term prediction result of the tropospheric delay product. This invention achieves high-precision short-term tropospheric delay prediction by introducing a multi-scale sliding window and an LSTM network, thereby enhancing real-time positioning performance.

Description

Technical Field

[0001] This invention belongs to the field of satellite navigation and positioning, and specifically relates to a short-term prediction method for PPP-RTK tropospheric delay products based on multi-scale sliding window LSTM. Background Technology

[0002] Global Navigation Satellite Systems (GNSS) have been widely used in various fields such as atmospheric monitoring, environmental surveillance, climate change detection, disaster early warning, precision agriculture, and autonomous driving due to their automation, all-weather operation, and high precision [1-6]. Among GNSS positioning technologies, Real-Time Kinematics (RTK) is one of the most mature and widely used methods. RTK uses a double-difference positioning model, which effectively eliminates most of the errors and ambiguities in the original observation equations, enabling fast and high-precision positioning [7]. However, RTK relies on nearby reference stations, which must be located in open environments, a requirement that constitutes a significant limitation [8]. In complex urban environments, this requirement is often difficult to meet, thus limiting the widespread application of this technology. In contrast, Precise Point Positioning (PPP) is another mature GNSS positioning model that can operate without reference stations. PPP can achieve centimeter-level accuracy using only a single receiver [9]. However, the high-precision positioning capability of PPP depends on the correct resolution of ambiguities

[10] . This prerequisite usually results in a long convergence time, making PPP less suitable for applications that require real-time and fast positioning solutions.

[0003] PPP-RTK is a GNSS positioning model developed in recent years, aiming to solve the limitations of traditional RTK and PPP technologies, especially the dependence on nearby reference stations and the problem of excessively long convergence time [11, 12]. This innovative approach has attracted widespread attention from researchers worldwide. For example, Teunissen et al. extended the PPP-RTK theory by running the transmitter at different frequencies, thereby enhancing its applicability

[13] . Wang et al. proposed a tightly coupled PPP-RTK / INS / visual integrated model to improve the positioning performance of traditional methods in complex urban environments

[14] . Li et al. introduced a PPP-RTK framework based on factor graph optimization, which can achieve a more robust and accurate positioning solution with lower outlier sensitivity

[15] . In addition, Hou et al. proved that the accuracy of ambiguity floating-point combined PPP-RTK products is affected by the choice of pivot receiver, providing valuable insights for the optimization model

[16] . These studies have jointly promoted the development and improvement of PPP-RTK technology, paving the way for its widespread application in complex environments.

[0004] In PPP-RTK technology, the ability to quickly resolve ambiguities depends heavily on the availability of high-precision ionospheric and tropospheric delay products

[17] . These products are typically derived from data collected from a sparsely distributed and stable network of reference stations with known coordinates within the target area. Once high-precision and high-reliability ionospheric and tropospheric delays are estimated at the regional network level, atmospheric product generation models are used to generate the required delay products. Commonly used atmospheric product generation models include inverse distance weighting (IDW)

[18] , radial basis function (RBF)

[19] , surface fitting (SF)

[20] , and kriging interpolation

[21] . Lyu et al. provided a comprehensive comparison of the performance of IDW, RBF, and kriging interpolation at different levels of ionospheric activity, providing valuable insights into their applicability

[22] . In addition, Li et al. proposed a grid-based atmospheric product generation model designed to accommodate different distances between reference stations, enhancing its versatility

[23] . Zhang et al. explored atmospheric generation methods within the framework of optimal linear unbiased prediction, further advancing the field

[24] . These studies have collectively contributed to the development of robust and accurate atmospheric product generation technologies, which are crucial for improving PPP-RTK performance.

[0005] Despite significant progress in PPP-RTK atmospheric product generation methods, research on short-term prediction of tropospheric delay products remains relatively limited. This issue is particularly critical for designing PPP-RTK models for real-time applications, as the latency of network-side computation of tropospheric delay products and the time required for data transmission are not negligible. Fortunately, the spatiotemporal distribution of tropospheric delay exhibits relatively stable characteristics, making short-term prediction possible. For example, Hu et al. proposed a zenith tropospheric delay prediction method based on the Informer model, demonstrating the potential of advanced machine learning techniques in this field

[25] . Similarly, Bi et al. developed a deep learning model for tropospheric wet delay prediction based on convolutional neural networks (CNNs), further highlighting the applicability of deep learning methods

[26] . In addition, time series prediction methods based on deep learning have gained attention and are increasingly being applied to various aspects of GNSS data processing [27, 28]. However, most existing research on tropospheric delay prediction focuses on long-term prediction, which is inconsistent with the short-term prediction requirements of PPP-RTK.

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[0031]

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[0033]

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[0034] Given the lack of research on short-term prediction of tropospheric delay products, this invention provides a PPP-RTK short-term prediction method for tropospheric delay products based on a multi-scale sliding window LSTM. By introducing a multi-scale sliding window and an LSTM network, high-precision short-term tropospheric delay prediction is achieved, enhancing real-time positioning performance.

[0035] According to one aspect of the present invention, a short-term prediction method for PPP-RTK tropospheric delay products based on multi-scale sliding window LSTM is provided, comprising:

[0036] Zenith tropospheric delay is extracted based on PPP-RTK calculation, and the original tropospheric delay product is generated using an interpolation model;

[0037] Based on the generated raw tropospheric delay product, different subsets are constructed using a multi-scale sliding window, and the number of subsets is related to the number of stations.

[0038] The different constructed subsets of data are input into the corresponding long short-term memory networks, and the prediction results of each long short-term memory network are output.

[0039] By combining the distance between the base station and the rover, the prediction results of each long short-term memory network are fused to obtain the short-term prediction results of the tropospheric delay product.

[0040] As a further technical solution, the construction of the multi-scale sliding window includes: designing the front part of the sliding window to have a variable length, while keeping the rear part at a fixed length.

[0041] As a further technical solution, the different subsets of datasets constructed using a multi-scale sliding window are as follows:

[0042] ,

[0043] in, This represents the first part of the first set of data at the first scale, with a length of... l The superscript indicates the scale number, and the subscript indicates the data group number; This represents the latter part of the first set of data at the first scale, with a length of... l .

[0044] As a further technical solution, the zenith tropospheric delay is extracted based on PPP-RTK calculation, and the original tropospheric delay product is generated using an interpolation model, including:

[0045] Construct a reference station network and extract the zenith tropospheric delay from each reference station;

[0046] Select an interpolation model to generate the original tropospheric delay product based on the extracted zenith tropospheric delay.

[0047] As a further technical solution, the method also includes: selecting the required interpolation model based on the geographical features and inter-station distances in the reference station network.

[0048] As a further technical solution, the method also includes:

[0049] The number of sub-data sets to be constructed is determined based on the number of test stations, and a corresponding long short-term memory network is established for each sub-data set.

[0050] According to one aspect of the present invention, a short-term prediction device for PPP-RTK tropospheric delay products based on multi-scale sliding window LSTM is provided to implement the aforementioned short-term prediction method for PPP-RTK tropospheric delay products based on multi-scale sliding window LSTM.

[0051] As a further technical solution, the device includes:

[0052] The first main module is used to extract the zenith tropospheric delay based on PPP-RTK calculation and to generate the original tropospheric delay product using an interpolation model.

[0053] The second main module is used to construct different subsets of data based on the generated original tropospheric delay products using a multi-scale sliding window. The number of subsets is related to the number of stations.

[0054] The third main module is used to input the different constructed subsets of data into the corresponding long short-term memory networks and output the prediction results of each long short-term memory network.

[0055] The fourth main module is used to combine the prediction results of various long short-term memory networks by combining the distance between the base station and the rover station to obtain the short-term prediction results of the tropospheric delay product.

[0056] As a further technical solution, a computing device includes a memory and a processor. The memory stores program instructions that are executed by the processor. The processor calls the program instructions to execute the PPP-RTK short-term prediction method for tropospheric delay products based on multi-scale sliding window LSTM.

[0057] As a further technical solution, a non-transitory computer-readable storage medium is provided, wherein the non-transitory computer-readable storage medium stores computer instructions, the computer instructions causing the computer to execute the PPP-RTK short-term prediction method for tropospheric delay products based on multi-scale sliding window LSTM.

[0058] Compared with the prior art, the beneficial effects of the present invention are as follows:

[0059] 1. This invention utilizes a combination of multi-scale sliding windows and Long Short-Term Memory (LSTM) networks for short-term prediction of PPP-RTK tropospheric delay products. The multi-scale sliding window method effectively captures data features at different time scales, while LSTM ensures accurate modeling of time dependencies. The combination of these two methods significantly improves the accuracy of short-term tropospheric delay prediction.

[0060] 2. Experimental analysis using one week's worth of data from the Hong Kong Continuously Operating Reference Station (CORS) demonstrates that the maximum prediction error of the proposed method is less than 1.5 cm. Furthermore, compared to the standard LSTM method, the root mean square error (RMSE) is improved by 18.9% and 36.6% for different reference values, respectively. PPP-RTK positioning experiments show that the prediction products generated by this method significantly improve the RMS values ​​in the east, north, and up directions, increasing them by 10.7%, 19.1%, and 4.1% respectively compared to the RMS values ​​obtained using the traditional LSTM method. These results comprehensively validate the effectiveness and superiority of the proposed method. Attached Figure Description

[0061] 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. Obviously, the accompanying drawings described below are some embodiments of the present invention. For those skilled in the art, other drawings can be obtained based on these drawings without creative effort.

[0062] Figure 1 This is a schematic diagram of the short-term prediction method based on multi-scale sliding window LSTM provided in an embodiment of the present invention.

[0063] Figure 2 The flowchart of the PPP-RTK algorithm provided in the embodiment of the present invention is shown.

[0064] Figure 3 This is a schematic diagram of the LSTM network model algorithm framework provided in an embodiment of the present invention.

[0065] Figure 4 This is a schematic diagram of a multi-scale sliding window provided in an embodiment of the present invention.

[0066] Figure 5 This is a schematic diagram of the distribution of monitoring stations provided in an embodiment of the present invention.

[0067] Figure 6 This is a schematic diagram comparing the IDW interpolation result with the true value provided in an embodiment of the present invention.

[0068] Figure 7 This is a schematic diagram illustrating the short-term forecast results for four stations on December 1st, provided as an embodiment of the present invention.

[0069] Figure 8 A schematic diagram of the cumulative distribution function (CDF) of a one-week short-term forecast provided in an embodiment of the present invention.

[0070] Figure 9 This is a schematic diagram of the displacement of four stations using a short-term prediction product, provided as an embodiment of the present invention. Detailed Implementation

[0071] Long Short-Term Memory (LSTM) networks are deep learning models that optimize memory cells using gates and are widely used in time series forecasting tasks because they can capture long-term dependencies [31, 32]. For example, Tang et al. proposed a novel approach combining local attention mechanisms with LSTM for short-term ionospheric forecasting, demonstrating its effectiveness in handling complex temporal patterns

[33] . Similarly, Wang et al. developed a new ionospheric mapping function based on LSTM, further demonstrating its versatility in atmospheric modeling

[34] . In GNSS / INS integration, Chen et al. introduced an LSTM-based measurement system that significantly improved positioning accuracy, especially in the horizontal direction

[35] . Furthermore, Li et al. improved the GPT-3 model by integrating LSTM with radial basis functions (RBF), achieving accurate prediction of zenith tropospheric delay in Antarctica

[36] . These studies collectively highlight the potential of LSTM as a robust and mature deep learning model for predicting short-term tropospheric delay products in PPP-RTK applications. Given its proven capabilities in related fields, LSTM is well-suited to address the challenges of real-time tropospheric delay prediction.

[0072]

[31] S. Hochreiter and J. Schmidhuber, "Long short-term memory," Neural Comput, vol. 9, no. 8, pp. 1735-80, Nov 15 1997, doi: 10.1162 / neco.1997.9.8.1735.

[0073]

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[0074]

[33] J. Tang, L. Xu, X. Wu, and K. Chen, "A Short-Term ForecastingMethod for Ionospheric TEC Combining Local Attention Mechanism and LSTMModel," IEEE Geoscience and Remote Sensing Letters, vol. 21, pp. 1-5, 2024,doi: 10.1109 / lgrs.2024.3373457。

[0075]

[34] Y. Wang, Y. Qin, J. Kong, Y. Yao, and X. Gao, "IonosphericRefined Mapping Function Construction Based on LSTM," IEEE Transactions on Geoscience and Remote Sensing, vol. 62, pp. 1-15, 2024, doi: 10.1109 / tgrs.2023.3338777。

[0076]

[35] Y. Chen et al. , "An LSTM-Assisted GNSS / INS Integration SystemUsing IMU Recomputed Error Information for Train Localization," IEEE Transactions. Aerosp. Electron. Syst., vol. 60, no. 3, pp. 2658-2671, 2024, doi: 10.1109 / taes.2023.3328318。

[0077]

[36] S. Li, T. Xu, Y. Xu, N. Jiang, and L. Bastos, "Forecasting GNSSZenith Troposphere Delay by Improving GPT3 Model with Machine Learning inAntarctica," Atmosphere,vol. 13, no. 1, 2022, doi: 10.3390 / atmos13010078.

[0078] This invention proposes a novel method for short-term prediction of tropospheric delay products in PPP-RTK based on a multi-scale sliding window and a long short-term memory (MSSW-LSTM) model. This method utilizes multiple sliding windows at different scales to preprocess tropospheric delay products, thereby extracting comprehensive information from data at different temporal resolutions. When combined with an LSTM model, this method can generate high-precision short-term predictions of PPP-RTK tropospheric delay products.

[0079] To make the objectives, technical solutions, and advantages of the embodiments of the present invention clearer, the technical solutions of the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some embodiments of the present invention, not all embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of the present invention. In addition, the technical features of the various embodiments or individual embodiments provided by the present invention can be arbitrarily combined to form new technical solutions. Such combinations are not bound by the order of steps and / or structural composition patterns, but must be based on the ability of those skilled in the art to implement them. When the combination of technical solutions is contradictory or cannot be implemented, it should be considered that such a combination of technical solutions does not exist and is not within the scope of protection claimed by the present invention.

[0080] This invention provides a short-term prediction method for tropospheric delayed products based on multi-scale sliding window LSTM using PPP-RTK, such as... Figure 1 As shown, firstly, the zenith tropospheric delay is extracted based on PPP-RTK calculation, and the original tropospheric delay product is generated using an interpolation model. Next, based on the generated original tropospheric delay product, different subsets are constructed using a multi-scale sliding window, the number of which is related to the number of stations. Then, the constructed subsets are input into their respective long short-term memory (LSTM) networks, and the prediction results of each LTM network are output. Finally, the prediction results of each LTM network are fused together by combining the distance between the base station and the rover station to obtain the short-term prediction result of the tropospheric delay product.

[0081] This invention, by combining the number of monitoring stations, utilizes a multi-scale sliding window strategy to generate diverse time-series subsets, effectively extracting multi-scale features and facilitating the discovery of complex patterns in the data. Furthermore, this invention introduces an LSTM architecture to effectively alleviate the gradient vanishing problem during parameter optimization, overcoming the inherent limitations of traditional methods and thus enhancing the model's learning ability and prediction stability. In addition, this invention sets the fusion weights based on the distance between the base station and the rover station, using an inverse distance weighting method, to obtain more accurate short-term prediction results for tropospheric delay products.

[0082] It should be noted that PPP-RTK is an advanced GNSS positioning technology that has attracted much attention in recent years. Its basic structure consists of two core components: the network segment and the user segment. The main task of the network segment is to generate a series of key satellite-related products, including but not limited to satellite orbit determination, satellite clock correction, satellite attitude parameters, and satellite pseudorange and observation-specific signal bias (OSB) correction. In addition, it generates regional ionospheric and tropospheric delay corrections. Except for regional atmospheric delay corrections, most of these products can be easily obtained from the International GNSS Service (IGS) Analysis Centre. However, generating regional ionospheric and tropospheric delay corrections requires the use of a dense network of regional reference stations. GNSS observation equations, crucial for deriving these regional atmospheric corrections, can be mathematically expressed as follows:

[0083]

[0084] in and These are pseudorange and carrier phase, respectively. The superscript 's' represents the satellite, and the subscripts 'r' and 'i' represent the receiver and frequency, respectively. This represents the geometric distance between the satellite and the receiver. C represents the speed of light in a vacuum. and These represent the clock biases at the receiver and satellite ends, respectively. Represents the ionospheric delay factor. M represents the ionospheric delay, and M represents the projection function. Indicates tropospheric delay, and These represent the receiver and satellite pseudorange hardware delays, respectively. Indicates wavelength. Indicates the ambiguity of the integer period. and These represent the phase hardware delay of the receiver and the satellite, respectively. and Let represent pseudorange and phase noise, respectively. Furthermore, the observation equations are susceptible to various systematic errors, which must be accurately modeled and subsequently eliminated to ensure precise positioning. These errors include relativistic effects, antenna phase center offsets (PCOs) and variations (PCVs), Earth rotation parameters, and solid Earth tides. After considering these error sources through appropriate modeling and correction strategies, the remaining parameters to be estimated can be mathematically expressed as follows:

[0085]

[0086] Then, by using appropriate ionospheric and tropospheric interpolation models, regional atmospheric products can be obtained.

[0087] After the user receives these products, the function model can be represented as follows:

[0088]

[0089] Since the ionospheric delay product itself includes a portion of the network-side receiver hardware delay, differential constraints must be implemented at the user end by selecting a reference satellite. This invention selects the satellite with the largest elevation angle as the reference satellite, thereby optimizing the observation geometry. The full algorithmic workflow of the PPP-RTK method is described in [link to full article]. Figure 2 The text provides a systematic explanation.

[0090] LSTM represents a complex deep learning architecture specifically designed for processing and analyzing sequential data. As a special variant of Recurrent Neural Networks (RNNs), LSTM excels at training long sequences, effectively mitigating the vanishing and exploding gradient problems. Compared to traditional RNN architectures, LSTM demonstrates superior performance in long-term temporal dependency modeling and sequence prediction tasks. The fundamental innovation of the LSTM architecture lies in its unique cell state mechanism and three specialized gating structures: the input gate, the forget gate, and the output gate. These structures work together to facilitate the adaptive retention and propagation of information.

[0091] LSTM network model algorithm framework, such as Figure 3 As shown, the LSTM cell structure comprises two main components: a linear computation unit and a nonlinear transformation unit, which work together to regulate the storage and transmission of information. The linear unit acts as an accumulator, combining historical cell state information with the current input data. The nonlinear unit uses the Sigmoid activation function to regulate the information flow through a precise gating mechanism. In particular, the forget gate plays a crucial role in determining the degree to which information is retained or discarded in the memory unit, thus enabling selective memory. The mathematical formula for the forget gate operation is shown below:

[0092]

[0093] f is the output of the forget gate. It is the Sigmoid activation function. It is the weight matrix of the forget gate. This represents the hidden state of the previous epoch. The input represents the current time step. This represents the bias vector of the forget gate.

[0094] The input gate, as a key regulation mechanism, controls the degree to which incoming information affects the state of the memory cell. This gate performs a dual function: it evaluates the current input vector and the hidden state from the previous time step to determine the specific information that should be integrated into the memory cell. The computation of the input gate involves two interrelated operations. First, a gating vector is generated using the sigmoid activation function, which determines the proportion of information written into the cell state.

[0095]

[0096] It is a weight matrix. This is the input vector for the paranoia gate. Then, a new candidate value is generated and activated using the tanh function:

[0097]

[0098] Finally, the gating vector generated by the input gate is multiplied by the candidate value to obtain the updated input value. The results of the forget gate and the input gate are combined to update the state of the memory cell as follows:

[0099]

[0100] The output gate determines the output information for the current input step size, as follows:

[0101]

[0102] Then, the state of the memory cell is processed by the tanh activation function and multiplied by the value of the output gate to obtain the final hidden state output:

[0103]

[0104] The basic concept of the sliding window technique revolves around incremental computation of the target result by maintaining a dynamic window on the data structure. This window mechanism can use a fixed or adaptive size. Typically, the sliding window paradigm uses two pointers to mark the left and right boundaries of the window, which synchronously traverse the data structure as the algorithm progresses to facilitate the computation of the result.

[0105] This invention proposes an innovative multi-scale sliding window strategy for dataset construction, which significantly improves upon traditional single-scale methods. The proposed multi-scale framework exhibits enhanced feature extraction capabilities and better adaptability, enabling more accurate characterization and learning of signal variation patterns. In this multi-scale architecture, the initial portion of each sliding window is designed to be of variable length, while the subsequent portion maintains a fixed length. This unique configuration allows the network to simultaneously capture and analyze features across multiple time scales, thereby enriching the learning process. The structural representation of the constructed dataset is shown below:

[0106]

[0107] This represents the first part of the first set of data at the first scale, with a length of... l Superscripts indicate scale numbers, and subscripts indicate data group numbers. This represents the latter part of the first set of data at the first scale, with a length of... l A schematic diagram of a multi-scale sliding window is shown below. Figure 4 As shown.

[0108] The proposed method first constructs different subsets of data by applying a multi-scale sliding window technique. The number of subsets is determined by the number of monitoring stations, i.e., the number of subsets is based on the number of stations. This process involves segmenting and systematically processing the tropospheric delay products generated by the network, thereby effectively capturing the temporal correlations and evolutionary trends in the data. Subsequently, these differentiated subsets are independently input into corresponding LSTM networks. The inherent long-term memory capability of the LSTM architecture facilitates the comprehensive learning of multi-scale feature representations. The process finally enters the integration stage, where the prediction outputs of each network are intelligently weighted and aggregated through an optimized fusion strategy, ultimately generating short-term predictions of tropospheric delay products with higher accuracy and reliability.

[0109]

[0110] This represents the output of different prediction results. This represents the weights of different subsets.

[0111] In this embodiment of the invention, the weights of the subsets are set based on the distance between the base station and the rover, combined with the inverse distance weighting method, so that the weighted fusion results have higher accuracy.

[0112] To verify the proposed algorithm, this embodiment of the invention used GNSS observation data collected by the Hong Kong Continuously Operating Reference Stations (CORS) network (https: / / www.geodetic.gov.hk) between December 1 and 7, 2024. The geographical distribution of the monitoring stations is as follows: Figure 5 As shown. Through a random selection process, four stations (hkss, hkqt, hksc, and hkmw) located in the central area were designated as rover stations. Figure 5 (marked in red), while the remaining nine sites serve as reference sites ( Figure 5 (The blue markers in the diagram) are used to generate regional ionospheric and tropospheric delay corrections. The average inter-station distance of the reference station network is approximately 25 km, and the average elevation difference is 137 m. The experimental setup used GPS observation data with a sampling interval of 30 seconds.

[0113] Accurate products, including satellite orbit, clock correction, attitude parameters, and pseudorange / phase OSB correction, were obtained from the Wuhan University IGS Data Center (http: / / www.igs.gnsswhu.cn). A comprehensive PPP-RTK processing strategy is detailed in Table 1. Efficient integer ambiguity resolution is a key component of PPP-RTK implementation, crucial for both network-side processing and user-side positioning. In this study, wide-lane ambiguities were resolved through direct rounding, while narrow-lane ambiguities were handled using the least squares ambiguity decorrelation adjustment (LAMBDA) algorithm, ensuring optimal integer ambiguity estimation.

[0114] Table 1 PPP-RTK Calculation Strategy

[0115]

[0116] Figure 1 This paper demonstrates the computational workflow for short-term prediction of tropospheric delay products using a multi-scale sliding window LSTM method in PPP-RTK. The process begins by extracting the zenith tropospheric delay (ZTD) from each reference station, and then generating the raw tropospheric delay product using an appropriate interpolation model. In this embodiment, the inverse distance weighted (IDW) method is used to generate the initial tropospheric product. Subsequently, the raw tropospheric delay data is segmented into multiple independent subsets using a multi-scale approach, and independent short-term predictions are performed using an LSTM architecture.

[0117] The experimental setup specifically divided the tropospheric delay data into three subsets with different time scales: the first subsets had time steps of 10, 15, and 20, while the latter subsets maintained a uniform length of 1. The prediction model used an 18-hour observation window to predict the tropospheric delay for the next 3 hours. To optimize computational efficiency and reduce resource consumption, a single-layer LSTM network architecture was implemented for both training and prediction phases. The optimization process employed the Adam (Adaptive Moment Estimation) optimizer as a stochastic optimization algorithm, with a maximum of 1000 iterations and a learning rate of 0.01. The final product integration was achieved through an equal-weighted fusion strategy, which combined the prediction outputs from multiple subsets to generate a comprehensive tropospheric delay prediction.

[0118] Acquiring and processing raw tropospheric delay data is crucial before making short-term predictions of tropospheric delay products. This process involves calculating tropospheric delays at each reference station and then transmitting these results to a central data processing center to generate a comprehensive tropospheric delay product. Selecting an appropriate interpolation model for tropospheric delay estimation is a key step in this process, as it fundamentally determines the accuracy and reliability of the final product. It is important to note that the applicability of the interpolation model varies depending on the geographical features and inter-station distances within the reference network.

[0119] In this embodiment, the inter-station distances in the Hong Kong CORS network are relatively small, resulting in minimal differences between different interpolation models. To optimize computational efficiency, the inverse distance weighted (IDW) method was selected as the preferred interpolation model. Figure 6 A comparative analysis of the IDW interpolation results and measured values ​​from four rover stations on December 1st is presented. The analysis shows that the original tropospheric delay products generated by the IDW interpolation model are very close to the true values, with a maximum deviation of less than 2 cm. This level of accuracy confirms that the model can generate reliable products for subsequent processing.

[0120] Furthermore, empirical evidence is provided regarding the short-term stability of tropospheric delay products, supporting the feasibility of using deep learning techniques for short-term prediction applications. The stable performance of the IDW model, combined with the inherent stability of tropospheric delay over short timescales, lays a solid foundation for implementing advanced prediction methods.

[0121] To fully verify the effectiveness of the proposed multi-scale sliding window LSTM algorithm in short-term prediction of tropospheric delay products in PPP-RTK, this embodiment of the invention conducted extensive experimental evaluations using raw tropospheric delay data from four strategically selected CORS stations in the Hong Kong network. The experimental framework employed an 18-hour observation window to predict the tropospheric delay for the next 3 hours; detailed processing methods are described above.

[0122] Figure 7This paper presents a comparison of short-term forecast results from four monitoring stations on December 1st. Experimental results show that the multi-scale sliding window LSTM method achieves significant forecast accuracy. Specifically, the predicted time series exhibits extremely high temporal consistency with the original observation data, maintaining the same variation patterns, and outperforms traditional LSTM implementations in capturing subtle temporal changes and maintaining forecast stability. This comparative advantage is particularly evident in the algorithm's ability to capture subtle temporal changes and maintain forecast stability.

[0123] Empirical results confirm the effectiveness of the proposed method, highlighting its enhanced ability in short-term tropospheric delay prediction. This superior performance is attributed to the algorithm's unique multi-scale feature extraction mechanism, which enables a more comprehensive learning of temporal patterns across different time scales, thereby improving prediction accuracy and reliability in PPP-RTK applications.

[0124] To quantitatively evaluate the algorithm's performance, root mean square error (RMSE) was used as the primary metric for comprehensive evaluation. Statistical analysis utilized two different reference datasets: ground-based measured tropospheric delay and raw tropospheric delay products. Table 2 presents a comparison of the RMSE results for the MSSW-LSTM and traditional LSTM models on December 1st.

[0125] Table 2. RMSE results of MSSW-LSTM and LSTM on December 1st.

[0126]

[0127] Analysis shows that the RMSE values ​​based on the original tropospheric delay product exhibit superior performance compared to ground-based measurements. This can be attributed to the use of the original tropospheric delay product as input during model training, thereby establishing a stronger statistical relationship between the predicted values ​​and the reference dataset. Statistical evaluation indicates that the maximum RMSE values ​​of all prediction methods are below 0.7 cm, while the MSSW-LSTM method consistently outperforms the traditional LSTM implementation among all prediction methods.

[0128] Notably, when using the original tropospheric delay product as a reference, the MSSW-LSTM method improved the average RMSE by 36.4% compared to the traditional LSTM method across the four monitoring stations. This significant improvement in prediction accuracy highlights the effectiveness of the multi-scale sliding window method in capturing and modeling the complex temporal characteristics of tropospheric delay.

[0129] To fully validate the generalization ability and robustness of the MSSW-LSTM method in short-term tropospheric delay prediction, an extended evaluation was conducted using a comprehensive dataset spanning seven consecutive days from December 1st to December 7th. The experimental configuration remained consistent with previous analyses, employing an 18-hour observation window to predict the tropospheric delay for the next 3 hours. Three different evaluation scenarios were implemented: (1) using ground-based measurements as a reference; (2) using the raw tropospheric delay product as a reference; and (3) employing a continuous prediction method, where the last available raw tropospheric delay product was used as the prediction for the next 3 hours.

[0130] Figure 8 The cumulative distribution function (CDF) of prediction errors for all four monitoring stations during the entire week-long evaluation period is presented. The results show that the maximum error for all prediction methods is less than 1.5 cm, meeting the stringent accuracy requirements of the PPP-RTK tropospheric delay product. Notably, the MSSW-LSTM method consistently outperforms the traditional LSTM method and the continuous prediction method at most stations, with all methods exhibiting similar performance only at the HKQT station. This superior performance highlights the MSSW-LSTM method's superior ability to capture temporal variations and maintain prediction stability, especially during long-term operations.

[0131] A comprehensive week-long evaluation not only confirmed the robustness of the method under different time conditions but also established its practical feasibility in operational PPP-RTK applications. The consistent performance of the MSSW-LSTM method, maintaining sub-centimeter accuracy across multiple sites and over multiple days, highlights its potential as a reliable solution for real-time tropospheric delay prediction.

[0132] Table 3 presents a comparative analysis of the mean root mean square error (RMSE) values ​​of the MSSW-LSTM and traditional LSTM methods over a continuous seven-day evaluation period. Statistical results show that the MSSW-LSTM method achieves a significant performance improvement. Specifically, compared to ground-based measurements, the MSSW-LSTM method improves the mean RMSE value at the four monitoring stations by 18.9% compared to the traditional LSTM method. This performance advantage is even more pronounced when using the original tropospheric delay product as a reference, with the MSSW-LSTM method reducing the mean RMSE value by 36.6%.

[0133] These significant improvements in prediction accuracy remained consistent throughout the week-long evaluation period, providing strong evidence for the effectiveness and versatility of the MSSW-LSTM method in short-term tropospheric delay prediction in PPP-RTK applications. The superior performance of this method under different reference datasets and time conditions highlights its robustness and practical applicability to operational precision positioning systems. Furthermore, the improved consistency observed over the extended evaluation period further enhances the reliability and stability of the method in real-world operational scenarios.

[0134] Table 3. Average RMSE results for MSSW-LSTM and LSTM over one week.

[0135]

[0136] To comprehensively evaluate the impact of short-term predicted tropospheric delay products on PPP-RTK positioning performance, this embodiment of the invention conducted a series of controlled experiments using MSSW-LSTM and traditional LSTM prediction products. Considering the significant impact of ionospheric delay accuracy on PPP-RTK performance, actual ionospheric delay values ​​were introduced into the experimental design to isolate the specific influence of tropospheric delay prediction quality. In the PPP-RTK data processing flow, a fixed constraint of 2 cm was applied to the tropospheric delay products while maintaining the calculation strategy described in detail in Table 1.

[0137] To rigorously evaluate convergence characteristics, artificial data interruptions were introduced every hour, forcing the positioning solution to reinitialize and converge repeatedly. Figure 9 The displacement patterns of the four monitoring stations on December 1st are presented in the three components of East (E), North (N), and High (U), using short-term predicted tropospheric delay products. To enhance visual clarity in the comparative analysis, the displacement results for the hksc, hkqt, and hkmw stations were vertically offset by 10 cm, 20 cm, and 30 cm, respectively.

[0138] The experimental results reveal two key findings: First, tropospheric delayed products using short-term predictions achieve rapid convergence, typically within the fewest observation periods. Second, tropospheric products generated by MSSW-LSTM exhibit consistent performance advantages across multiple reinitialization periods, although the improvements are small but statistically significant. This performance advantage is particularly evident in the stability and consistency of the convergence patterns.

[0139] These findings not only validate the practicality of the predicted tropospheric delay product in real-world PPP-RTK scenarios but also highlight the superior performance of the MSSW-LSTM method in enhancing precise positioning solutions. The ability to maintain positioning accuracy while meeting frequent reconvergence requirements further demonstrates the robustness of the proposed method in real-world applications.

[0140] Table 4 presents a comprehensive statistical analysis of the root mean square (RMS) values ​​of the four monitoring stations over a seven-day evaluation period in the east (E), north (N), and altitude (U) directions, comparing the performance of the two models. The statistical results show that the MSSW-LSTM method outperforms the traditional LSTM method in all directions. Specifically, the MSSW-LSTM method reduces the average RMS value by 10.7% in the east component, 19.1% in the north component, and 4.1% in the vertical component.

[0141] Significant improvements across all three coordinate components provide strong empirical evidence supporting the superior performance of the MSSW-LSTM method in PPP-RTK applications. In particular, the significant improvements on the horizontal components (E and N) demonstrate the method's excellence in capturing the spatiotemporal features of tropospheric delay, which are key factors influencing horizontal positioning accuracy. The consistent performance improvements observed throughout the one-week evaluation period further validate the reliability and effectiveness of the method in precise positioning applications.

[0142] Table 4. RMS values ​​of the two models at four sites.

[0143]

[0144] The implementation of the various embodiments of the present invention is based on programmed processing by a device with processor functionality. Therefore, in practical engineering, the technical solutions and functions of the various embodiments of the present invention are encapsulated into various modules. Based on this reality, and building upon the above embodiments, the embodiments of the present invention provide a PPP-RTK tropospheric delay product short-term prediction device based on a multi-scale sliding window LSTM. This device is used to execute a PPP-RTK tropospheric delay product short-term prediction method based on a multi-scale sliding window LSTM as described in the above method embodiments.

[0145] The device includes: a first main module for extracting zenith tropospheric delay based on PPP-RTK calculation and generating the original tropospheric delay product using an interpolation model; a second main module for constructing different subsets of data based on the generated original tropospheric delay product using a multi-scale sliding window, wherein the number of subsets is related to the number of stations; a third main module for inputting the constructed subsets into corresponding long short-term memory networks and outputting the prediction results of each long short-term memory network; and a fourth main module for fusing the prediction results of each long short-term memory network by combining the distance between the base station and the rover station to obtain the short-term prediction result of the tropospheric delay product.

[0146] This invention provides a PPP-RTK short-term prediction device for tropospheric delay products based on a multi-scale sliding window LSTM. Addressing the current state of research on short-term prediction of ionospheric delay products, this device employs several modules and introduces a multi-scale sliding window and LSTM network to achieve high-precision short-term tropospheric delay prediction, thereby enhancing real-time positioning performance.

[0147] It should be noted that the device embodiments provided by the present invention are used not only to implement the methods in the above method embodiments, but also to implement the methods in other method embodiments provided by the present invention. The only difference is that corresponding functional modules are set. The principle is basically the same as that of the above device embodiments provided by the present invention. As long as those skilled in the art can improve the modules in the above device embodiments by referring to the specific technical solutions in other method embodiments and combining technical features to obtain corresponding technical means and technical solutions composed of these technical means, on the basis of the above device embodiments, and on the premise of ensuring the practicality of the technical solutions, they can obtain corresponding device-type embodiments for implementing the methods in other method-type embodiments.

[0148] Based on the same inventive concept as the foregoing embodiments, this embodiment of the invention also provides a computing device, including a memory and a processor. The memory stores program instructions that are executed by the processor. The processor calls the program instructions to execute the PPP-RTK short-term prediction method for tropospheric delay products based on multi-scale sliding window LSTM.

[0149] Based on the same inventive concept as the foregoing embodiments, this embodiment of the invention also provides a non-transitory computer-readable storage medium storing computer instructions that cause the computer to execute the PPP-RTK short-term prediction method for tropospheric delay products based on multi-scale sliding window LSTM.

[0150] In summary, accurate short-term prediction of tropospheric delay products is a key technical challenge in the practical application of PPP-RTK systems. This invention proposes an innovative method called MSSW-LSTM (Multi-Scale Sliding Window Long Short-Term Memory) for short-term tropospheric delay prediction. The proposed method utilizes a multi-scale sliding window strategy to generate diverse temporal subsets, effectively extracting multi-scale features and facilitating the discovery of complex patterns in the data. Furthermore, the introduction of the LSTM architecture effectively alleviates the gradient vanishing problem during parameter optimization, overcoming the inherent limitations of traditional methods, thereby enhancing the model's learning ability and prediction stability.

[0151] To rigorously verify the effectiveness and universality of the algorithm, extensive experimental evaluations were conducted using observational data from the Hong Kong CORS network from December 1st to 7th, 2024. Experimental results show that, compared to the traditional LSTM method, the MSSW-LSTM method reduces the average RMSE by 18.9% and 36.6% when evaluated against ground-based measurements and raw tropospheric delay products, respectively. Notably, the maximum prediction error remains within 1.5 cm, fully meeting the stringent accuracy requirements of PPP-RTK tropospheric delay products.

[0152] Subsequent PPP-RTK positioning experiments using the predicted products further validated the superiority of this method. Weekly statistical analysis showed consistent performance improvements across all coordinate components, with average RMS improvements of 10.7%, 19.1%, and 4.1% in the east, north, and altitude directions, respectively, compared to results using traditional LSTM predictions. These comprehensive experimental results not only validated the effectiveness and superiority of the MSSW-LSTM method but also established its practical feasibility for short-term tropospheric delay prediction in PPP-RTK applications. The method's performance in improving prediction accuracy and positioning performance makes it a valuable tool for advancing precise positioning technology.

[0153] Finally, it should be noted that the above embodiments are only used to illustrate the technical solutions of the present invention, and not to limit them; although the present invention has been described in detail with reference to the foregoing embodiments, those skilled in the art should understand that modifications can still be made to the technical solutions described in the foregoing embodiments, or equivalent substitutions can be made to some or all of the technical features therein; and these modifications or substitutions do not cause the essence of the corresponding technical solutions to deviate from the technical solutions of the embodiments of the present invention.

Claims

1. A short-term prediction method for PPP-RTK tropospheric delayed products based on multi-scale sliding window LSTM, characterized in that, include: Zenith tropospheric delay is extracted based on PPP-RTK calculation, and the original tropospheric delay product is generated using an interpolation model; Based on the generated raw tropospheric delay product, different subsets are constructed using a multi-scale sliding window, the number of which is related to the number of stations. The construction of the multi-scale sliding window includes designing the first part of the window to have a variable length, while the second part maintains a fixed length. The different subsets constructed using the multi-scale sliding window are as follows: ， in, This represents the first part of the first set of data at the first scale, with a length of... l The superscript indicates the scale number, and the subscript indicates the data group number; This represents the latter part of the first set of data at the first scale, with a length of... l ; The different constructed subsets of data are input into the corresponding long short-term memory networks, and the prediction results of each long short-term memory network are output. By combining the distance between the base station and the rover, the prediction results of each long short-term memory network are fused to obtain the short-term prediction results of the tropospheric delay product.

2. The short-term prediction method for PPP-RTK tropospheric delayed products based on multi-scale sliding window LSTM according to claim 1, characterized in that, Zenith tropospheric delay is extracted based on PPP-RTK calculation, and the original tropospheric delay product is generated using an interpolation model, including: Construct a reference station network and extract the zenith tropospheric delay from each reference station; Select an interpolation model to generate the original tropospheric delay product based on the extracted zenith tropospheric delay.

3. The short-term prediction method for PPP-RTK tropospheric delayed products based on multi-scale sliding window LSTM according to claim 1, characterized in that, The method further includes selecting the required interpolation model based on the geographical features and inter-station distances in the reference station network.

4. The short-term prediction method for PPP-RTK tropospheric delayed products based on multi-scale sliding window LSTM according to claim 1, characterized in that, The method further includes: The number of sub-data sets to be constructed is determined based on the number of test stations, and a corresponding long short-term memory network is established for each sub-data set.

5. A short-term prediction device for PPP-RTK tropospheric delay products based on multi-scale sliding window LSTM, characterized in that, This method is used to implement the PPP-RTK tropospheric delay product short-term prediction method based on multi-scale sliding window LSTM as described in any one of claims 1-4.

6. The PPP-RTK tropospheric delay product short-term prediction device based on multi-scale sliding window LSTM according to claim 5, characterized in that, The first main module is used to extract the zenith tropospheric delay based on PPP-RTK calculation and to generate the original tropospheric delay product using an interpolation model. The second main module is used to construct different subsets of data based on the generated original tropospheric delay products using a multi-scale sliding window. The number of subsets is related to the number of stations. The third main module is used to input the different constructed subsets of data into the corresponding long short-term memory networks and output the prediction results of each long short-term memory network. The fourth main module is used to combine the prediction results of various long short-term memory networks by combining the distance between the base station and the rover station to obtain the short-term prediction results of the tropospheric delay product.

7. A computing device, characterized in that, It includes a memory and a processor, the memory storing program instructions that are executed by the processor, and the processor calling the program instructions to execute the PPP-RTK tropospheric delay product short-term prediction method based on multi-scale sliding window LSTM as described in any one of claims 1 to 4.

8. A non-transitory computer-readable storage medium, characterized in that, The non-transitory computer-readable storage medium stores computer instructions that cause the computer to perform the PPP-RTK tropospheric delay product short-term prediction method based on multi-scale sliding window LSTM as described in any one of claims 1 to 4.

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