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

Abstract

Tropospheric delay products play a critical role in achieving high-precision positioning in Precise Point Positioning Real-Time Kinematic (PPP-RTK) applications. The short-term prediction of these products remains a significant challenge that warrants further exploration. This study proposes a novel short-term prediction method for tropospheric delay products in PPP-RTK applications, leveraging a multi-scale sliding window and Long Short-Term Memory (LSTM) network. The multi-scale sliding window approach effectively captures data features across different temporal scales, while LSTM, a well-established and robust time series forecasting technique, ensures the accurate modeling of temporal dependencies. The integration of these two methods significantly enhances the precision of short-term tropospheric delay predictions. Experimental analysis utilizing one week of data from the Hong Kong Continuously Operating Reference Stations (CORS) network demonstrates that the proposed method achieves a maximum prediction error of less than 1.5 cm. Furthermore, compared to the standard LSTM approach, the Root Mean Square Error (RMSE) values are improved by 18.9% and 36.6% for different reference values, respectively. PPP-RTK positioning experiments reveal that the predicted products generated by this method exhibit notable improvements in Root Mean Square (RMS) values for the east, north, and up directions, with enhancements of 10.7%, 19.1%, and 4.1%, respectively, over those obtained using the conventional LSTM method. These results comprehensively validate the effectiveness and superiority of the proposed approach.

1. Introduction

The Global Navigation Satellite System (GNSS) has gained widespread adoption across various fields, including atmospheric monitoring, environmental surveillance, climate change detection, disaster early warning, precision agriculture, and autonomous driving, owing to its automation, all-weather capability, and high precision [1,2,3,4,5,6,7,8]. Among GNSS positioning technologies, Real-Time Kinematic (RTK) is one of the most mature and widely used methods. RTK employs a double-differencing positioning model, which effectively eliminates most errors in the original observation equations, enabling rapid and high-accuracy positioning [9]. However, RTK’s reliance on nearby reference stations, which must be situated in open environments, poses a significant limitation [10]. This requirement is often challenging to meet, particularly in complex urban settings, thereby restricting the broader application of this technology. In contrast, Precise Point Positioning (PPP) represents another well-established GNSS positioning model that operates without the need for reference stations. PPP can utilize precise products pre-calculated by International GNSS Service (IGS) analysis centers, enabling users to achieve precise positioning with just a single receiver [11]. Nevertheless, the high-precision positioning capability of PPP is contingent upon the correct resolution of ambiguities [12]. This prerequisite often results in extended convergence times, making PPP less suitable for applications requiring real-time and fast positioning solutions.

PPP-RTK, a recently developed GNSS positioning model, has emerged as a promising solution to address the limitations of traditional RTK and PPP techniques, particularly the dependency on nearby reference stations and the issue of prolonged convergence times [13,14]. This approach has attracted significant attention from researchers worldwide. For instance, Teunissen et al. extended PPP-RTK theory by enabling transmitters to operate on different frequencies, thereby enhancing its applicability [15]. Wang et al. proposed a tightly coupled PPP-RTK/INS/Vision integration model to improve positioning performance in complex urban environments, where traditional methods often struggle [16]. Li et al. introduced a factor graph optimization-based PPP-RTK framework which achieves more robust and accurate positioning solutions with reduced outlier susceptibility [17]. Additionally, Hou et al. demonstrated that the precision of ambiguity-float combined PPP-RTK products is influenced by the selection of the pivot receiver, providing valuable insights into optimizing the model [18]. Collectively, these studies have significantly contributed to the advancement and refinement of PPP-RTK technology, paving the way for its broader application in challenging environments.

Achieving rapid ambiguity resolution in PPP-RTK technology relies heavily on the availability of high-precision ionospheric and tropospheric delay products [19]. These products are typically derived from data collected by a network of sparsely distributed and stable reference stations with known coordinates within the target region. Once high-precision and high-reliability ionospheric and tropospheric delays are estimated at the regional network level, atmospheric product generation models are employed to produce the required delay products. Commonly used models for atmospheric product generation include Inverse Distance Weighting (IDW) [20], Radial Basis Functions (RBFs) [21], surface fitting (SF) [22], and kriging interpolation [23]. Lyu et al. conducted a comprehensive comparison of the performance of IDW, RBF, and kriging interpolation under varying levels of ionospheric activity, providing valuable insights into their applicability [24]. Furthermore, Li et al. proposed a grid-based atmospheric product generation model designed to accommodate varying distances between reference stations, enhancing its versatility [25]. Psychas et al. utilized the best linear unbiased predictor to compute undifferenced user ionospheric corrections on the network side, demonstrating and analyzing the performance of real-time PPP-RTK under moderate ionospheric activity conditions [26]. These studies collectively contribute to the development of robust and accurate atmospheric product generation techniques, which are essential for improving PPP-RTK performance.

Research on the short-term prediction of tropospheric delay products is particularly critical for PPP-RTK models designed for real-time applications, as the latency in computing tropospheric delay products at the network end and the time required for data transmission cannot be overlooked. However, relevant research remains limited. Fortunately, the temporal and spatial distribution of tropospheric delays exhibit relatively stable characteristics, making short-term prediction feasible. For instance, Hu et al. proposed a zenith tropospheric delay forecasting method based on the Informer model, demonstrating the potential of advanced machine learning techniques in this domain [27]. Similarly, Bi et al. developed a deep learning-based model for tropospheric wet delay prediction using convolutional neural networks (CNNs), further highlighting the applicability of deep learning approaches [28]. Additionally, deep learning-based time series prediction methods have gained traction and are increasingly being applied to various aspects of GNSS data processing [29,30]. However, most existing studies on tropospheric delay prediction focus on long-term forecasting, which does not align with the short-term prediction requirements of PPP-RTK. In contrast, the short-term prediction of ionospheric delay products has received more attention in the literature [31,32]. Given this research gap, this paper specifically addresses the development of short-term prediction methods for tropospheric delay products in PPP-RTK, aiming to bridge the current knowledge gap and enhance real-time positioning performance.

Long Short-Term Memory (LSTM), a deep learning model that optimizes memory cells through the use of gates, has been extensively applied to time series prediction tasks due to its ability to capture long-term dependencies [33,34]. For instance, Tang et al. proposed a novel approach combining the Local Attention Mechanism with LSTM for short-term ionospheric prediction, demonstrating its effectiveness in handling complex temporal patterns [35]. Similarly, Wang et al. developed a new ionospheric mapping function based on LSTM, further showcasing its versatility in atmospheric modeling [36]. In the context of GNSS/INS integration, Chen et al. introduced an LSTM-based measurement system that significantly enhances positioning accuracy, particularly in the horizontal direction [37]. Additionally, Li et al. improved the GPT-3 model by integrating LSTM with Radial Basis Functions (RBFs), enabling the accurate forecasting of zenith tropospheric delay over Antarctica [38]. These studies collectively highlight the potential of LSTM as a robust and well-established deep learning model for short-term tropospheric delay product forecasting in PPP-RTK applications. Given its proven capabilities in related domains, LSTM is well suited to addressing the challenges associated with real-time tropospheric delay prediction.

This study proposes a novel short-term prediction method for PPP-RTK tropospheric delay products based on a multi-scale sliding window and Long Short-Term Memory (MSSW-LSTM) model. The proposed method leverages multiple sliding windows of varying scales to preprocess tropospheric delay products, enabling the extraction of comprehensive information from the data across different temporal resolutions. When integrated with the LSTM model, this approach generates high-accuracy short-term forecasts of PPP-RTK tropospheric delay products.

The specific chapter arrangement is as follows. Section 2 and Section 3 introduce the fundamental principles of the MSSW-LSTM method and outline the data processing workflow for short-term tropospheric delay prediction in PPP-RTK. Section 4 presents the experimental validation using data from the Hong Kong Continuously Operating Reference Stations (CORS) network, along with a detailed analysis of the results. Finally, Section 5 summarizes the key findings and conclusions of the study.

2. Methods

2.1. PPP-RTK

PPP-RTK, a cutting-edge GNSS positioning technology that has gained prominence in recent years, is fundamentally structured into two integral components: the network segment and the user segment. The network segment is tasked with the generation of a suite of critical satellite-related products, including, but not limited to, satellite orbit determinations, satellite clock corrections, satellite attitude parameters, and satellite pseudorange, as well as phase observable-specific signal bias (OSB) corrections. Additionally, it produces regional ionospheric and tropospheric delay corrections. With the exception of the regional atmospheric delay corrections, the majority of these products are readily accessible through International GNSS Service (IGS) analysis centers. The generation of regional ionospheric and tropospheric delay corrections, however, necessitates the utilization of a dense regional reference station network. The GNSS observation equations, which are pivotal for the derivation of these regional atmospheric corrections, can be mathematically represented as follows:

$$\left\{\begin{array}{l}{P}_{r,i}^s={\rho }_r^s+c(t_r-t^s)+{\mu }_i\cdot I_r^s+M\cdot T_r+d_{r,i}-d_i^s+\sigma \hfill \\ L_{r,i}^s={\rho }_r^s+c(t_r-t^s)-{\mu }_i\cdot I_r^s+M\cdot T_r+{\lambda }_i(N_{r,i}^s+b_{r,i}-b_i^s)+\epsilon \hfill \end{array}\right.,$$

(1)

where $P_{r,i}^s$ and $L_{r,i}^s$ are the pseudorange and carrier phase at frequency $i$ of satellite $s$ and receiver $r$. ${\rho }_r^s=‖X_r-X^s‖$ is the geometric distance between the satellite $s$ and the receiver $r$. The coordinates $X_r$ at the receiver end are usually known. The satellite position $X^s$ can be calculated using precise satellite orbit products. $c$ represents the speed of light. $t_r$ and $t^s$ denote the receiver clock bias and satellite clock bias, respectively. The satellite clock bias can be obtained using precise satellite clock products. ${\mu }_i$ is the coefficient related to the frequency. $I_r^s$ represents the slant ionospheric delay. $M$ and $T_r$ denote the mapping function and zenith tropospheric delay, respectively. $d_{r,i}$ and $d_i^s$ are the pseudorange hardware delays at the receiver and satellite ends, respectively. The satellite-end hardware delay $d_i^s$ can be obtained using OSB products. ${\lambda }_i$ and $N_{r,i}^s$ represent the carrier wavelength and ambiguity, respectively. $b_{r,i}$ and $b_i^s$ are the carrier-phase hardware delays. Similarly, $b_i^s$ can be obtained from OSB products. $d_{r,i}$ and $b_{r,i}$ will be absorbed by the receiver clock bias, ionospheric delay, and ambiguity. $\sigma$ and $\epsilon$ are the pseudorange and phase noise, respectively. Furthermore, the observation equations are subject to various systematic errors that must be accurately modeled and subsequently eliminated to ensure precise positioning. These errors encompass relativistic effects, antenna phase center offsets (PCOs) and variations (PCVs), Earth rotation parameters, and solid Earth tides, among others. After accounting for these error sources through appropriate modeling and correction strategies, the remaining parameters requiring estimation can be mathematically formulated as follows:

$$\left\{\begin{array}{l}X=\left[\begin{array}{lllll}{\widehat{t}}_r\hfill & T_r\hfill & {\widehat{I}}_r^s\hfill & {\widehat{N}}_{r,1}^s\hfill & {\widehat{N}}_{r,2}^s\hfill \end{array}\right]\hfill \\ {\widehat{t}}_r=t_r+({\displaystyle \frac{{\mu }_2}{{\mu }_2-1}}{d}_{r,1}-{\displaystyle \frac{1}{{\mu }_2-1}}{d}_{r,2})/c\hfill \\ {\widehat{I}}_r^s=I_r^s-{\displaystyle \frac{1}{{\mu }_2-1}}(d_{r,1}-d_{r,2})\hfill \\ {\widehat{N}}_{r,1}^s=N_{r,1}^s+b_{r,1}-({\displaystyle \frac{{\mu }_2+1}{{\mu }_2-1}}{d}_{r,1}-{\displaystyle \frac{2}{{\mu }_2-1}}{d}_{r,2})/{\lambda }_1\hfill \\ {\widehat{N}}_{r,2}^s=N_{r,2}^s+b_{r,2}-({\displaystyle \frac{2{\mu }_2}{{\mu }_2-1}}{d}_{r,1}-{\displaystyle \frac{{\mu }_2+1}{{\mu }_2-1}}{d}_{r,2})/{\lambda }_2\hfill \end{array}\right.,$$

(2)

in which $X$ denotes the parameters to be estimated, and ${\widehat{t}}_r$, ${\widehat{I}}_r^s$, ${\widehat{N}}_{r,1}^s$, and ${\widehat{N}}_{r,2}^s$ denote the receiver clock offset, ionospheric delay, and ambiguities for L1/L2 frequencies, respectively, all incorporating hardware delays. Then, by using appropriate ionospheric and tropospheric interpolation models, the regional atmospheric products can be obtained. After the user side receives these products, the function model can be expressed as follows:

$$\left\{\begin{array}{l}{\tilde{P}}_{r,i}^s=v_r^s\cdot x_r+c\cdot {\overline{t}}_r+{\gamma }_i\cdot {\overline{I}}_r^s+M\cdot T_r+\sigma \hfill \\ {\tilde{L}}_{r,i}^s=v_r^s\cdot x_r+c\cdot {\overline{t}}_r-{\gamma }_i\cdot {\overline{I}}_r^s+M\cdot T_r+{\lambda }_i\cdot {\overline{N}}_{r,i}^s+\epsilon \hfill \\ \delta T_r-T_{r,0}=T_r+\tau \hfill \\ (\delta {\widehat{I}}_r^k-\delta {\widehat{I}}_r^s)-({\overline{I}}_{r,0}^k-{\overline{I}}_{r,0}^s)={\overline{I}}_r^k-{\overline{I}}_r^s+\upsilon \hfill \end{array}\right.,$$

(3)

in which ${\tilde{P}}_{r,i}^s$ and ${\tilde{L}}_{r,i}^s$ are the observed values of pseudorange and phase minus the computed values. $v_r^s$ is the unit vector between the receiver $r$ and the satellite $s$. $\delta T_r$ and $T_{r,0}$ are the tropospheric delay product and the tropospheric delay initial value, respectively. $\delta {\widehat{I}}_r^k$ and ${\overline{I}}_{r,0}^k$ are the ionospheric delay product and the ionospheric delay initial value, respectively. Given that the ionospheric delay products inherently incorporate a component of the network-side receiver hardware delay, it is imperative to implement a differencing constraint through the selection of a reference satellite at the user end. In this study, we adopted a strategic approach by designating the satellite with the maximum elevation angle as the reference satellite, thereby optimizing the observation geometry. The comprehensive algorithmic workflow of the PPP-RTK methodology is systematically illustrated in Figure 1.

2.2. LSTM

LSTM represents a sophisticated deep learning architecture specifically designed for processing and analyzing sequential data. As a specialized variant of Recurrent Neural Networks (RNNs), LSTM demonstrates superior capability in mitigating the challenges of vanishing and exploding gradients that commonly occur during the training of extended sequences. In contrast to conventional RNN architectures, LSTM exhibits enhanced performance in long-term temporal dependency modeling and sequence prediction tasks. The architecture of an LSTM network model is shown in Figure 2. The fundamental innovation of the LSTM architecture resides in its unique cell-state mechanism and the incorporation of three specialized gating structures—the input gate, forget gate, and output gate—which collectively facilitate adaptive information retention and propagation.

The LSTM cell structure comprises two principal components, a linear computational unit and a nonlinear transformation unit, working in tandem to regulate information storage and transmission. The linear unit operates as a cumulative adder, integrating historical cell-state information with current input data. The nonlinear unit employs a sigmoid activation function to modulate information flow through precise gating mechanisms. Particularly, the forget gate plays a crucial role in determining the extent of information preservation or discarding within the memory cell, thereby enabling selective memory functionality. The mathematical formulation governing the forget gate operation is expressed as follows:

$$f_t=\sigma (W_f\cdot [h_{t-1},x_t]+b_f),$$

(4)

where $f_t$ is the output of the forget gate; $\sigma$ is the sigmoid activation function, which limits the value between 0 and 1; $W_f$ is the weight matrix of the forget gate; $h_{t-1}$ represents the hidden state of the previous time step; $x_t$ denotes the input of the current time step; and $b_f$ represents the bias vector of the forget gate.

The input gate serves as a critical regulatory mechanism that governs the degree to which incoming input information influences the memory cell’s state. This gate performs a dual function: it evaluates both the current input vector and the hidden state from the preceding time step to determine the specific information that should be integrated into the memory cell. The computational procedure of the input gate involves two distinct yet interconnected operations. Initially, a sigmoid activation function is employed to generate a gating vector that determines the proportion of information to be written to the cell state.

$$i_t=\sigma (W_i\cdot [h_{t-1},x_t]+b_i),$$

(5)

where $W_i$ is weight matrix and $b_i$ represents the bias vector of the input gate. Then, a new candidate value ${\tilde{C}}_t$ is generated and activated using the tanh function:

$${\tilde{C}}_t=\tanh (W_C\cdot [h_{t-1},x_t]+b_C),$$

(6)

in which $\tanh$ is the activation function and $W_C$ and $b_C$ are the weight matrix and the bias vector of the input gate. Finally, the input gate and the candidate value are multiplied to obtain the updated input value.

The results of the forget gate and the input gate are combined to update the memory cell state $C_t$, as follows:

$$C_t=f_t\cdot C_{t-1}+i_t\cdot {\tilde{C}}_t.$$

(7)

The output gate determines the output information $o_t$ of the current time step, as follows:

$$o_t=\sigma (W_o\cdot [h_{t-1},x_t]+b_o).$$

(8)

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

$$h_t=o_t\cdot \tanh (C_t).$$

(9)

2.3. Multi-Scale Sliding Window LSTM

The fundamental concept underlying the sliding window technique revolves around the incremental computation of target results through the maintenance of a dynamic window over the data structure. This window mechanism can be implemented with either fixed or adaptive dimensions. Typically, the sliding window paradigm employs dual pointers to demarcate the left and right boundaries of the window, which traverse the data structure in synchronization with the algorithm’s progression to facilitate result computation.

This study introduces a multi-scale sliding window strategy for dataset construction, which represents a significant advancement over conventional single-scale approaches. The proposed multi-scale framework demonstrates enhanced feature extraction capabilities and superior adaptability, enabling more precise characterization and learning of signal variation patterns. Within this multi-scale architecture, each sliding window is designed with variable lengths in its initial segment while maintaining consistent lengths in its latter portion. This distinctive configuration empowers the network to concurrently capture and analyze features across multiple temporal scales, thereby enriching the learning process. The structural representation of the constructed dataset is illustrated as follows:

$$\left[\begin{array}{l}{X}_{1,l1}^1,Y_{1,m}^1\hfill \\ X_{2,l1}^1,Y_{2,m}^1\hfill \\ \dots \hfill \\ X_{n1,l1}^1,Y_{n1,m}^1\hfill \end{array}\right],\left[\begin{array}{l}{X}_{1,l2}^2,Y_{1,m}^2\hfill \\ X_{2,l2}^2,Y_{2,m}^2\hfill \\ \dots \hfill \\ X_{n2,l2}^2,Y_{n2,m}^2\hfill \end{array}\right],\dots ,\left[\begin{array}{l}{X}_{1,lk}^k,Y_{1,m}^k\hfill \\ X_{2,lk}^k,Y_{2,m}^k\hfill \\ \dots \hfill \\ X_{nk,lk}^k,Y_{nk,m}^k\hfill \end{array}\right],$$

(10)

where $X_{1,l1}^1=[x_1,x_2,x_3,\dots ,x_{l1}]$ represents the front part of the first set of data at the first scale, with a length of $l1$. The superscript numbers indicate the scale number, and the data group number are indicated in subscript. $Y_{1,m}^1=[x_{l1+1},x_{l1+2},x_{l1+3},\dots ,x_{l1+m}]$ represents the back part of the first set of data at the first scale, with a length of $m$. An illustration of a multi-scale sliding window is shown in Figure 3.

The proposed methodology initiates the construction of distinct sub-datasets through the application of the multi-scale sliding window technique. This process involves the segmentation and systematic processing of tropospheric delay products generated by the network, enabling the effective capture of both temporal correlations and evolutionary trends within the data. Subsequently, these differentiated sub-datasets are independently fed into their corresponding LSTM networks, where the inherent long-term memory capability of the LSTM architecture facilitates the comprehensive learning of multi-scale feature representations. The process culminates in an ensemble phase, where predictive outputs from individual networks are intelligently weighted and aggregated through an optimized fusion strategy, ultimately yielding the final short-term forecast of tropospheric delay products with enhanced accuracy and reliability.

$$T=w_1{h}_t^1+w_2{h}_t^2+\dots +w_k{h}_t^k,$$

(11)

where $h_t^1$, $h_t^2$, and $h_t^k$ represent the predicted outputs of different subsets.

$$w_1+w_2+\dots +w_k=1,$$

(12)

$w_1$, $w_2$, and $w_k$ are the different subsets’ weights. The subset weights can be set based on the actual situation. If the differences between the subsets are small, the weights can be set to be equal, as follows:

$$w_1=w_2=w_3=\dots =w_k.$$

(13)

3. Data and Calculation Strategy

For the experimental validation of the proposed algorithm, this study utilized GNSS observation data acquired from the Hong Kong Continuously Operating Reference Stations (CORS) network (https://www.geodetic.gov.hk (accessed on 1 December 2024)) during a seven-day period from 1 to 7 December 2024. The geographical distribution of the monitoring stations is illustrated in Figure 4. Through a randomized selection process, four stations (hkss, hkqt, hksc, and hkmw) located in the central region were designated as rover stations (represented by red markers in Figure 4), while the remaining nine stations served as reference stations (indicated by blue markers in Figure 4) for generating regional ionospheric and tropospheric delay corrections. The reference station network demonstrates an average inter-station distance of approximately 25 km, with a mean elevation difference of 137 m. The experimental configuration employed GPS observations with a 30 s sampling interval.

Precise products, including satellite orbits, clock corrections, attitude parameters, and pseudorange/phase OSB corrections were obtained from the IGS data center at Wuhan University (http://www.igs.gnsswhu.cn (accessed on 1 December 2024)). The comprehensive PPP-RTK processing strategy is detailed in

Table 1. Specific calculation strategy of PPP-RTK.

Item	Strategy
Observation	GPS L1, L2, P1, P2
Sampling interval	30 s
Orbit and clock product	WUM rapid product
Cutoff angle	7°
Observation prior variance	Pseudorange: 1 m; carrier phase: 0.01 cycle
Parameter estimation method	Sequential least squares
Receiver clock	Estimated as white noise
Ionospheric delay	Estimated with random walk
Tropospheric delay	Estimated with random walk
User-side constraint	Ionospheric delay: 5 cm; tropospheric delay: 2 cm
Antenna correction	Igs20.atx
Ambiguity resolution strategy	Partial resolution with LAMBDA method

. Efficient and precise ambiguity resolution constitutes a critical component of PPP-RTK implementation, essential 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 processed using the Least-squares AMBiguity Decorrelation Adjustment (LAMBDA) algorithm, ensuring optimal integer ambiguity estimation.

Figure 5 illustrates the computational workflow for the short-term prediction of PPP-RTK tropospheric delay products utilizing the multi-scale sliding window LSTM approach. The process initiates with the extraction of zenith tropospheric delay (ZTD) at each reference station, followed by the generation of raw tropospheric delay products through an appropriate interpolation model. In this investigation, the Inverse Distance Weighting (IDW) method was employed for the generation of preliminary tropospheric products. Subsequently, the raw tropospheric delay data undergoes multi-scale segmentation, where it is partitioned into distinct subsets for independent short-term prediction using the LSTM architecture.

The experimental configuration specifically divided the tropospheric delay data into three subsets, characterized by varying temporal scales: the antecedent segments having lengths of 10, 15, and 20 time steps, respectively, while maintaining a uniform length of 1 for the subsequent segments. The accuracy of short-term predictions decreases as forecast duration increases. A 3 h short-term forecast duration adequately supported real-time user applications and balanced the accuracy of the product. A large volume of data can enhance the accuracy of time series forecasting models, but it also increases the time and computational power required for calculations. Therefore, to strike a balance between these factors, this paper, after numerous experiments, utilized 18 h of historical data to predict outcomes for the next 3 hours. To optimize computational efficiency and minimize resource expenditure, a streamlined single-layer LSTM network architecture was implemented for both the training and prediction phases. The Adam optimizer is widely utilized due to its adaptive learning rate characteristics, particularly excelling in addressing non-stationary objective functions and sparse gradient issues. In the current optimization task, Adam demonstrated rapid convergence and exhibited superior stability and efficiency compared to other optimizers. Therefore, the optimization process employed the Adam (Adaptive Moment Estimation) optimizer as its stochastic optimization algorithm, configured with a maximum iteration count of 1000 and a learning rate of 0.01. Given the minimal differences between each subnet dataset, the final product integration was achieved through an equal-weight fusion strategy, which combined the predictive outputs from multiple subsets to generate a comprehensive tropospheric delay forecast.

4. Results and Discussion

4.1. Tropospheric Delay Raw Product

Prior to conducting the short-term forecasting of tropospheric delay products, it is essential to acquire and process the raw tropospheric delay data. This process involves the computation of tropospheric delays at individual reference stations, followed by the transmission of these calculations to a centralized data processing center for the generation of comprehensive tropospheric delay products. The selection of an appropriate interpolation model for tropospheric delay estimation represents a critical step in this process, as it fundamentally determines the precision and reliability of the final products. It should be noted that the suitability of interpolation models varies according to geographical characteristics and inter-station distances within the reference network.

In the context of this study, the Hong Kong CORS network exhibits relatively small inter-station distances, resulting in minimal variations among different interpolation models. To optimize computational efficiency, the Inverse Distance Weighting (IDW) method was selected as the preferred interpolation model. Figure 6 presents a comparative analysis between the IDW interpolation results and ground truth measurements for the four rover stations on December 1st. The analysis reveals that the raw tropospheric delay products generated through the IDW interpolation model demonstrate close alignment with true values, with the maximum deviations remaining below 2 cm. This level of accuracy confirms the model’s capability to produce reliable raw products for subsequent processing.

Furthermore, the observed short-term stability of the tropospheric delay products provides empirical evidence supporting the feasibility of employing deep learning techniques for short-term forecasting applications. However, as shown in Figure 6, the tropospheric delay can vary by more than 2 cm within a 3 h period. If a standard method is used instead of deep learning forecasting, this could introduce significant errors, thereby degrading positioning accuracy and convergence speed. The consistent performance of the IDW model, coupled with the inherent stability of tropospheric delays over short timescales, establishes a solid foundation for the implementation of advanced forecasting methodologies.

4.2. Multi-Scale Sliding Window LSTM Prediction

To comprehensively validate the efficacy of the proposed multi-scale sliding window LSTM algorithm for the short-term prediction of PPP-RTK tropospheric delay products, an extensive experimental evaluation was conducted using raw tropospheric delay data from four strategically selected CORSs within the Hong Kong network. The experimental framework employed an 18 h observation window to forecast tropospheric delays for the subsequent 3 h period, with the detailed data processing methodologies being outlined in the preceding section.

Figure 7 presents the comparative short-term forecasting results for the four monitoring stations on December 1st. These experimental results demonstrate that the multi-scale sliding window LSTM approach achieves remarkable prediction accuracy. Specifically, the predicted time series exhibits exceptional temporal coherence with the original observations, maintaining consistent variation patterns while demonstrating superior performance compared to conventional LSTM implementations. This comparative advantage is particularly evident in the algorithm’s ability to capture subtle temporal variations and maintain prediction stability throughout the forecast horizon.

The empirical results substantiate the effectiveness of the proposed methodology, highlighting its enhanced capability in short-term tropospheric delay prediction. The superior performance can be attributed to the algorithm’s unique multi-scale feature extraction mechanism, which enables the more comprehensive learning of temporal patterns across different timescales, thereby improving prediction accuracy and reliability in PPP-RTK applications.

To quantitatively assess the algorithm’s performance, a comprehensive evaluation was conducted using Root Mean Square Error (RMSE) as the primary metric. The statistical analysis employed two distinct reference datasets: ground truth tropospheric delay measurements and raw tropospheric delay products.

Table 2. RMSE results of MSSW-LSTM and LSTM on 1 December.

Model	MSSW-LSTM		LSTM		Standard Methods
Reference Value	Truth (cm)	Raw (cm)	Truth (cm)	Raw (cm)	Truth (cm)	Raw (cm)
hkss	0.69	0.27	0.30	0.52	0.49	0.42
hksc	0.25	0.20	0.41	0.37	0.45	0.40
hkqt	0.35	0.47	0.35	0.66	0.36	0.45
hkmw	0.62	0.30	0.53	0.40	0.58	0.41

presents the comparative RMSE results between the MSSW-LSTM and traditional LSTM models for December 1st. The term ‘standard methods’ refers to using the data from the last epoch as the predicted result for the next 3 h.

The analysis reveals that the RMSE values calculated against the raw tropospheric delay products demonstrate a superior performance compared to those using the ground truth measurements. This phenomenon can be attributed to the model training process, which utilized raw tropospheric delay products as its input, thereby establishing a closer statistical relationship between the predicted values and the reference dataset. The statistical evaluation demonstrates that all the prediction methods maintain maximum RMSE values below 0.7 cm, with the MSSW-LSTM approach consistently outperforming conventional LSTM implementations.

Notably, when using raw tropospheric delay products as reference values, the MSSW-LSTM method achieves a substantial 36.4% improvement in average RMSE across the four monitoring stations compared to the traditional LSTM method. This significant enhancement in prediction accuracy underscores the effectiveness of the multi-scale sliding window approach in capturing and modeling the complex temporal characteristics of tropospheric delays.

To thoroughly validate the generalizability and robustness of the MSSW-LSTM method for short-term tropospheric delay prediction, an extended evaluation was conducted using a comprehensive dataset spanning seven consecutive days from 1 to 7 December. The experimental configuration maintained consistency with previous analyses, employing an 18 h observation window to forecast tropospheric delays for the subsequent 3 h period. Three distinct evaluation scenarios were implemented: (1) using ground truth measurements as reference values, (2) utilizing raw tropospheric delay products as reference values, (3) employing a persistence forecast approach where the last available raw tropospheric delay product was propagated as the prediction for the next three hours.

Figure 8 presents the cumulative distribution function (CDF) of the prediction errors across all four monitoring stations throughout the week-long evaluation period. The results demonstrate that all the prediction methods maintain maximum errors below 1.5 cm, satisfying the stringent accuracy requirements for PPP-RTK tropospheric delay products. Notably, the MSSW-LSTM method consistently outperforms both the traditional LSTM approach and the persistence forecast method at most stations, with the exception of the hkqt station, where comparable performance is observed across all methods. This exceptional performance profile underscores the MSSW-LSTM method’s superior capability in capturing temporal variations and maintaining prediction stability across extended operational periods.

The comprehensive week-long evaluation not only confirmed the method’s robustness across different temporal conditions but also established its practical viability for operational PPP-RTK applications. The consistent performance advantage demonstrated by the MSSW-LSTM approach, particularly in maintaining sub-centimeter-level accuracy across multiple stations and days, highlights its potential as a reliable solution for real-time tropospheric delay prediction in precise positioning applications.

Table 3. Average RMSE results of MSSW-LSTM and LSTM for one week.

Model	MSSW-LSTM		LSTM
Reference Value	Truth (cm)	Raw (cm)	Truth (cm)	Raw (cm)
hkss	0.41	0.22	0.46	0.39
hksc	0.33	0.25	0.52	0.31
hkqt	0.28	0.31	0.42	0.51
hkmw	0.48	0.26	0.45	0.43

presents the comparative analysis of the average Root Mean Square Error (RMSE) values between the MSSW-LSTM and traditional LSTM methods over a continuous seven-day evaluation period. The statistical results reveal the significant performance enhancements achieved by the MSSW-LSTM approach. Specifically, when evaluated against ground truth measurements, the MSSW-LSTM method demonstrates an 18.9% improvement in average RMSE across the four monitoring stations compared to the conventional LSTM method. This performance advantage becomes even more pronounced when using raw tropospheric delay products as reference values, with the MSSW-LSTM method achieving a remarkable 36.6% reduction in average RMSE. The weekly averaged mean absolute error (MAE) results for both approaches are presented in

Table 4. Average MAE results of MSSW-LSTM and LSTM for one week.

Model	MSSW-LSTM		LSTM
Reference Value	Truth (cm)	Raw (cm)	Truth (cm)	Raw (cm)
hkss	0.38	0.19	0.44	0.36
hksc	0.31	0.22	0.48	0.27
hkqt	0.26	0.29	0.38	0.47
hkmw	0.44	0.22	0.43	0.39

. These outcomes lead to conclusions consistent with Table 3, demonstrating that the prediction products generated by the MSSW-LSTM method achieve superior accuracy compared to conventional LSTM approaches.

These substantial improvements in prediction accuracy, consistently maintained throughout the week-long evaluation period, provide compelling evidence of the MSSW-LSTM method’s effectiveness and generalizability for short-term tropospheric delay prediction in PPP-RTK applications. The method’s superior performance across different reference datasets and temporal conditions underscores its robustness and practical applicability for operational precise positioning systems. Furthermore, the consistent performance gains observed over an extended evaluation period reinforce the method’s reliability and stability in real-world operational scenarios.

4.3. PPP-RTK Experiment with Short-Term Forecast Tropospheric Products

To comprehensively evaluate the impact of short-term predicted tropospheric delay products on PPP-RTK positioning performance, a series of controlled experiments were conducted using both the MSSW-LSTM and traditional LSTM predicted products. The experimental data utilized the 3 h predicted dataset for 1 December. Recognizing the substantial influence of ionospheric delay accuracy on PPP-RTK performance, the experimental design incorporated true ionospheric delay values to isolate the specific effects of tropospheric delay prediction quality. In the PPP-RTK data processing pipeline, a fixed constraint of 2 cm was applied to the tropospheric delay products, while maintaining the calculation strategies detailed in Table 1.

To rigorously assess the convergence characteristics, an artificial data interruption was introduced at hourly intervals, forcing the positioning solution to reinitialize and converge repeatedly. Figure 9 illustrates the displacement patterns in the east (E), north (N), and up (U) components at four monitoring stations on 1 December, utilizing the short-term predicted tropospheric delay products. For comparative purposes, the most recent tropospheric delay product was treated as the tropospheric delay for the subsequent 3 h period (termed the standard method), with PPP-RTK positioning likewise performed. For enhanced visual clarity in the comparative analysis, the displacement results for the hksc, hkqt, and hkmw stations are vertically offset by 10 cm, 20 cm, and 30 cm, respectively.

The experimental results demonstrate two key findings: First, the utilization of short-term predicted tropospheric delay products enables rapid convergence, typically achieved within a minimal number of epochs. Second, the MSSW-LSTM-derived tropospheric products consistently outperform those generated by the traditional LSTM and standard methods, albeit with marginal yet statistically significant improvements. This performance advantage is particularly evident in the stability and consistency of the convergence patterns across multiple reinitialization cycles.

These findings not only validate the practical utility of the predicted tropospheric delay products in operational PPP-RTK scenarios but also highlight the superior performance of the MSSW-LSTM approach in enhancing precise positioning solutions. The demonstrated capability to maintain positioning accuracy while accommodating frequent re-convergence requirements underscores the robustness of the proposed methodology in real-world applications.

Table 5. RMSs of four stations for two models.

Model	MSSW-LSTM			LSTM			Standard Method
Direction	E (cm)	N (cm)	U (cm)	E (cm)	N (cm)	U (cm)	E (cm)	N (cm)	U (cm)
hkss	2.10	2.17	2.87	1.86	2.52	3.23	2.23	2.76	3.50
hksc	3.61	2.01	4.40	5.22	2.54	4.88	5.00	2.51	5.28
hkqt	1.35	2.42	3.59	2.14	3.34	4.00	1.89	3.87	4.49
hkmw	3.32	2.11	6.48	2.40	2.36	5.98	2.40	2.36	7.03

presents a comprehensive statistical analysis of Root Mean Square (RMS) values across the four monitoring stations, comparing the performance of both models in the east (E), north (N), and up (U) directions over a continuous seven-day evaluation period. The statistical results reveal the consistent performance advantages of the MSSW-LSTM method over the traditional LSTM approach and standard method. Specifically, the MSSW-LSTM method demonstrates significant improvements in positioning accuracy compared with the LSTM approach, with average RMS reductions of 10.7% in the east component, 19.1% in the north component, and 4.1% in the vertical component.

These substantial improvements across all three coordinate components provide robust empirical evidence supporting the superior performance of the MSSW-LSTM method in PPP-RTK applications. The particularly notable enhancement in the horizontal components (E and N) suggests that the method excels in capturing the spatial–temporal characteristics of tropospheric delays that predominantly affect horizontal positioning accuracy. The consistent performance gains observed throughout the week-long evaluation period further validate the reliability and effectiveness of the proposed methodology for precise positioning applications.

4.4. Discussion of Experimental Results

The proposed method demonstrates superior performance in our Hong Kong experiments compared to conventional LSTM approaches and standard methods. The maximum error of the predicted 3 h tropospheric delay products remains below 1.5 cm, fully meeting PPP-RTK user-end positioning requirements. The experimental results conclusively validate the method’s effectiveness. However, it is important to note that tropospheric delay characteristics demonstrate significant regional variability due to their strong dependence on local atmospheric conditions. While the present investigation has specifically examined the Hong Kong region, our future research aims to achieve the following: (1) extend this methodology to other geographically distinct areas, (2) develop global-scale short-term prediction capabilities for PPP-RTK tropospheric delay products, (3) systematically evaluate the method’s generalizability across diverse climatic and topographic conditions. This expanded validation framework will provide more comprehensive insights into the technique’s operational applicability for Global Navigation Satellite systems.

The generation of precise satellite orbits, clock offsets, OSB, and atmospheric products on the PPP-RTK network side requires a certain amount of time, leading to a non-negligible latency when these products are delivered to the user side. As demonstrated by Wang et al., the user’s positioning performance degrades with increasing latency, which motivated their work on short-term satellite clock prediction [39]. This aligns with the primary objective of our study. As discussed in the paper, tropospheric delay products exhibit high short-term stability. A delay of a few minutes has a negligible impact on PPP-RTK users. It neither affects the network-side tropospheric delay prediction nor compromises the user-side positioning performance.

Additionally, it should be noted that the tropospheric delay predicted by the method proposed in this study currently lacks reliable uncertainty quantification. This limitation may impact PPP-RTK users’ positioning performance to some extent. We will conduct in-depth investigations into this issue in future work.

5. Conclusions

The accurate short-term prediction of tropospheric delay products represents a critical technical challenge in the operational implementation of PPP-RTK systems. This study introduces an effective approach, termed MSSW-LSTM (multi-scale sliding window Long Short-Term Memory), for short-term tropospheric delay prediction. The main contributions of this paper are as follows:

• To address the short-term prediction challenge of PPP-RTK tropospheric delay products, this paper proposes the MSSW-LSTM method. The proposed methodology leverages a multi-scale sliding window strategy to generate diverse temporal subsets, enabling the effective extraction of multi-scale features and facilitating the discovery of intricate patterns within the data. Furthermore, the incorporation of LSTM architecture addresses the inherent limitations of traditional methods by effectively mitigating the gradient vanishing problem during parameter optimization, thereby enhancing the model’s learning capability and prediction stability.
• To rigorously validate the algorithm’s effectiveness and generalizability, an extensive experimental evaluation was conducted using observation data from the Hong Kong CORS network spanning 1–7 December 2024. The experimental results demonstrate significant performance improvements, with the MSSW-LSTM method achieving average RMSE reductions of 18.9% and 36.6% compared to conventional LSTM when evaluated against ground truth measurements and raw tropospheric delay products, respectively. Notably, the maximum prediction error remains below 1.5 cm, well within the stringent accuracy requirements for PPP-RTK tropospheric delay products.
• Subsequent PPP-RTK positioning experiments utilizing the predicted products further substantiate the method’s superiority. Weekly statistical analyses reveal consistent performance enhancements across all coordinate components, with average RMS improvements of 10.7%, 19.1%, and 4.1% in the east, north, and up directions, respectively, compared to results obtained using traditional LSTM predictions. These comprehensive experimental findings not only validate the effectiveness and superiority of the MSSW-LSTM approach but also establish its practical viability as a reliable solution for short-term tropospheric delay prediction in PPP-RTK applications. The method’s demonstrated capability to enhance both prediction accuracy and positioning performance positions it as a valuable tool for advancing the state of the art in precise positioning technologies.