An Intelligent GNSS Positioning Correction Method Based on Frequency-Domain State-Sequence Prediction

2.1 Background

(1) Adaptability of DRL: In urban environments, GNSS positioning suffers from multipath effects caused by signal reflections off buildings and signal blockage due to high-rise obstructions, leading to degraded accuracy with meter-level errors in dense areas. These challenges require adaptive solutions. DRL demonstrates strong capability in deriving optimal correction strategies by establishing mappings between distorted GNSS measurements and their corresponding adjustments, effectively capturing temporal patterns in positioning error evolution. In our framework, we formulate the GNSS positioning correction task as a sequential decision-making problem under uncertainty. Given the inherent limitations in obtaining complete environmental state information during vehicle navigation, we adopt a partially observable Markov decision process (POMDP) formulation. This approach specifically accounts for (a) partial observability of the true navigation state due to sensor limitations and environmental occlusion and (b) the Markov property, where future states depend only on the current state and action, enabling efficient policy learning despite incomplete information.

(2) Value of frequency-domain characteristics: Frequency-domain analysis has been explored in GNSS signal processing, particularly for anti-jamming and interference mitigation in transform domains (Calatrava et al., 2025). For positioning correction, the advantage of frequency-domain analysis stems from its global receptive field and energy compaction properties. Notably, Zhou et al. (2022) observed that signals dense in the time domain often admit sparse frequency representations. This sparsity enables models to effectively decouple the signal of interest, i.e., the vehicle motion, from interference. While urban multipath does not typically produce clean harmonic lines, prior studies on dynamic urban multipath channels have shown that its delay and Doppler-fading characteristics vary with platform motion and propagation geometry (X. Chen, He, & Pei, 2020; X. Chen, Morton, et al., 2020; S. Li et al., 2024). These observations suggest that frequency-domain representations may help capture time-varying error components that are difficult to characterize using only local time-domain features. By transforming the state sequence into the frequency domain, we introduce a global consistency constraint that encourages the agent to model longer-range variation and reduces the influence of transient, non-physical error spikes that might cause confusion in a purely time-domain policy.

2.2 Related Work

Recent data-driven technologies have led to significant advances in GNSS positioning. We categorize related works into three types:

(a) GNSS/INS-integrated positioning methods: GNSS/INS integration remains a mainstream solution in high-precision navigation because it can compensate for temporary GNSS outages and improve short-term continuity. Nevertheless, this hardware-centric route shows clear limitations in mass-market urban scenarios: INS drift accumulates over time, tactical-grade IMUs are prohibitively expensive, and low-cost MEMS sensors are highly sensitive to bias instability. Moreover, many GNSS/INS implementations rely on the Gaussian noise assumptions inherent to the extended KF. When multipath errors become strongly nonlinear and non-Gaussian, the estimator is more prone to inconsistency and even divergence (Siebert et al., 2023). These limitations motivate software-defined correction strategies that are more robust to complex urban noise.

(b) DNN-driven positioning methods: In the work by Kanhere et al. (2021), a DNN based on the set-transformer architecture utilized pseudorange residuals and line-of-sight (LOS) vectors of GPS L1 signals as inputs to predict an initial position-correction estimate. F. Liu et al. (2024) proposed a GNSS/INS tight integration positioning method that employs LSTM and factor graph technology to predict GNSS positions during signal interruptions, significantly reducing the error metrics of combined positioning. To capture complex error patterns, Mohanty and Grace (2022) introduced a hybrid framework integrating a graph convolutional neural network (GCNN) with a KF, which extracts topological relationships between satellite signals that are typically overlooked. However, most existing DNN-based methods depend on single GNSS observations as input, limiting their ability to accurately reconstruct the current state of a vehicle.

(c) DRL-based dynamic positioning: Zhao et al. (2023) developed a multi-input LTSM PPO (multi-LSTMPPO) module to integrate satellite data into multi-input time-series observations, significantly improving positioning accuracy. P. Li et al. (2024) enhanced the generalization of the model in data-scarce urban environments by using a self-supervised learning-based method with data augmentation. To tackle the challenge of insufficient available data, Tang et al. (2023) proposed the adaptive reward augmentation method, which leverages sufficient additional data from the source domain to boost target domain performance.

Contemporary data-driven methodologies predominantly focus on temporal-domain feature extraction and analysis, yet exhibit suboptimal capability in capturing long-term temporal dependencies, consequently failing to adequately distill the inherent structural patterns with physical interpretability embedded within complex data sets. To address this limitation, our work develops a self-supervised learning method that integrates temporal-frequency domain information to extract intrinsic patterns and local features from observed signals.

3 DRL FOR GNSS POSITIONING CORRECTION WITH SFPN

This section presents an overview of the proposed method. As illustrated in Figure 2, the overall pipeline primarily consists of three steps: step 1: the multi-input observation setup; step 2: the proposed method processing; and step 3: the correction and update stage. Details regarding the frequency-domain and time-domain branches are shown in Figure 3. During training, the self-supervised learning module provides auxiliary supervision to improve representation learning, enabling the primary network to effectively capture long-term dependencies.

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FIGURE 2

Framework of the SFPN method for positioning correction The red dashed line indicates the gradient backpropagation for updating parameters.

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FIGURE 3

Dual-domain architecture of SFPN: the frequency-domain self-supervised learning module and the time-domain actor-critic network

3.1 Multi-Input Observation Setup

Given that the global environmental information is not fully accessible, the reinforcement learning (RL) problem for positioning correction is modeled as a POMDP, which can be defined as a tuple (O, A, T, R, γ), where O={oi,i=1,…,t} is the observation space, representing the state space of the vehicle agent, A={ai,i=1,…,t} is the set of action space, T:O×A×O→[0,1] is the transition probability, R is the reward function used to guide the agent in optimizing its policy, and γ∈[0,1] is a discount rate, which measures the importance of current versus future rewards. Then, trajectories with T steps are encoded as trajt={(ot,at,rt,ot+1)t=1T}.

Conventional GNSS positioning methods estimate a receiver position from raw satellite measurements and then treat that position as the primary observation in subsequent correction or filtering steps. In urban scenarios, this position can be biased by multipath effects and blockage and thus may not reflect the true vehicle state. To obtain enhanced observations, we fuse multi-constellation satellite measurements, denoted as follows:

Mt={(Pt⟨i⟩,ρt⟨i⟩)}i=1Nt, ∀t∈{1,…,T}1

where N_t is the number of visible satellites at time t,Pt⟨i⟩ denotes the estimated visible satellite position, and ρt⟨i⟩ represents the corresponding pseudoranges. Conventional model-based methods Ψ can use a series of measurement data Mt, such as KF or RTK data, to obtain an initial position in either the Earth-centered, Earth-fixed (ECEF) frame or the geodetic (GEO) frame:

loctinit=Ψ(Mt)={(xt,yt,zt)(ECEF), ∀t∈{1,…,T}(latt,lont)(GEO), ∀t∈{1,…,T}2

where the variables x_t, y_t, and z_t represent the Cartesian coordinates in the ECEF frame. Meanwhile, lat_t and lon_t denote the angular coordinates in the GEO frame. To fully represent the state of the agent, we designed a composite observation framework integrating the temporal characteristics otloc∈Oloc and the characteristics of the GNSS observation features otgnss∈Ognss, denoted as {Oloc,Ognss}. To model the motion state of the vehicle agent at step t, the historical positioning sequence otloc, containing l − 1 steps of RL-corrected positions {loct−q}q=1l−1, serves as the primary observation feature, which can be defined as follows:

otloc=[loct−l+1,…,loct−1,loctinit]3

To quantify environmental impacts on positioning accuracy, we adopt two key metrics: (1) LOS vectors L_t that characterize the spatial distribution of satellites and (2) pseudorange residuals Rs_t, which reflect signal error characteristics induced by multipath effects and atmospheric delays. These metrics are combined to construct the second observation otgnss=[Lt,Rst] at time t. Owing to the dynamic changes in satellite visibility, the number of visible satellites in each vehicle trajectory is usually inconsistent. We set a maximum number of visible satellites, denoted as N_max, and fill the positions of invisible satellites with zeros.

3.2 Self-Supervised Fourier Prediction Network

For frequency-domain analysis, we design an SFPN to capture the inherent characteristics in observational data, thereby effectively extracting long-term trend information. For the time-domain analysis, we employ a PPO algorithm integrated with LSTM modules to capture local features and transient variations in the state sequence. With the fusion of dual-domain features, the positioning accuracy is significantly improved. This section elaborates on the design and implementation of these two core modules.

3.2.1 Learning via the Frequency Domain of the State Sequence

To theoretically justify the integration of frequency-domain analysis, we leverage the insight that state sequences contain structural information reflecting policy performance. While time-domain differences between policies can be subtle and difficult to measure directly, their frequency-domain representations often reveal distinct spectral signatures. Ye et al. (2023) found that by leveraging Fourier transforms, policy differences can be quantified through frequency-domain analysis, where performance discrepancies between two policies are bounded by the distribution difference of the state sequence. Mathematically, policy differences in the time domain can be represented as follows:

|L(π1)−L(π2)|≤Rmax1−γ‖P(h0,h1,…∣π1,G)−P(h0,h1,…∣π2,G)‖D4

where L(π) denotes the expected cumulative reward under policy π. The term P(⋅∣π,G) is the distribution of state sequences induced by executing policy π in environment G. The scalar Rmax is the maximum attainable single-step reward, and D denotes the state-space dimensionality. Equation (4) states that if two policies generate similar state-sequence distributions, their long-term returns must be close. To make this comparison more interpretable, we map the state sequences into the frequency domain using the DTFT and restrict the analysis to [0, 2π], which yields a tighter spectrum-based bound:

|L(π1)−L(π2)|⩽D1−γ⋅∑k=1∞‖Qk(0)‖Dk!⋅max1≤i≤Dsupωi∈[0,2π]|Fπ1(k)(ωi)−Fπ2(k)(ωi)|5

where Qk(0) weights the k-th frequency-domain basis at the initial state and Fπ(k)(ωi) represents the DTFT of the state sequence for any integer k ∈ [1, n]. The bounds in Equations (4)–(5) are intended as theoretical motivation for introducing frequency-domain auxiliary supervision; they are not used as direct experimental metrics in this work. This theoretical bound implies that minimizing the difference in policy performance is equivalent to minimizing the discrepancy in the frequency-domain representation of state sequences. In the context of GNSS positioning, this relationship suggests that if we can train the network to accurately predict the frequency components of the state sequence, e.g., by distinguishing low-frequency systematic errors from high-frequency noise, we can effectively guide the policy to converge toward a better solution. Low-frequency components typically correspond to slowly varying biases, such as orbital errors or multipath noise in static scenes, whereas high-frequency components correspond to transient noise. Explicitly modeling these components helps the agent learn more robust correction strategies.

3.2.2 Frequency-Domain Representation Prediction

To realize the frequency-domain guidance above, we design a self-supervised auxiliary task that predicts the expected future spectral content of the vehicle’s state sequence. Intuitively, the model is encouraged to capture not only the current state but also the long-term motion pattern encoded by its spectrum.

We begin by defining a discounted future state sequence. Given the latent feature h_t from the LSTM and the current action a_t from the actor, we form a sequence that weights future states by an exponential discount. Because the infinite sum is intractable, we use a recursive formulation to define the expectation of the future state sequence h∼t as follows:

[h∼t]n=[h∼(ht,at)]n={γnEπ,p[ht+n+1∣ht,at],n=0,1,2,…0,n=−1,−2,−3,…6

where Eπ,p represents the expected value under policy π and the probability of environment transition p. For n ≥ 0, the sequence records discounted future states, and for n < 0, the sequence is set to zero to enforce causality.

The SFPN architecture comprises four components, as detailed below:

(a) Encoder: In our proposed model, the encoder H is designed to transform h_t into a more informative latent representation z_t. Concretely, we first form h∼t=H(ht) and then combine it with the current action a_t to obtain zt=H(h∼t,at), which facilitates subsequent prediction and RL tasks. The encoder comprises two parts: the online encoder and the target encoder. The primary distinction between these components lies in their input sources. The target encoder processes the next-step pair h∼t+1 and at+1 to produce zt+1=H(h∼t+1,at+1), which stabilizes training. Each layer of the encoder utilizes a multilayer perceptron (MLP)-DenseNet, a slightly modified version of DenseNet.

(b) Predictor: The predictor operates on the DTFT of the discounted sequence, Fh∼t(ω)=∑n=0+∞[h∼t]ne−jωn, where ω is the frequency variable. Because the DTFT of a discrete-time signal is 2π -periodic, i.e., F(ω+2π)=F(ω), all information is contained within the interval [0, 2π]. Thus, we approximate the continuous spectrum by sampling L equally spaced frequencies, ωl=2πlL,∀l∈{l=0,1,…,L−1}. This approach produces a frequency-domain prediction target represented as a matrix of size L×D, where L corresponds to the frequency bin and D represents the feature dimension. The predictor F can be denoted as follows:

F=[F(0),F(2πL),…,F(2π−2πL)]T∈CL×D7

Although F has finite dimensions (L×D), it constitutes a recursively defined frequency-domain representation that implicitly encodes infinite-horizon temporal dependencies. This representation is achieved through a Bellman-like fixed-point relation, where each prediction step propagates information from all future states via discounted recursion. Importantly, the term "infinite-horizon" in this case does not imply processing an infinitely long data sequence; rather, it refers to the mathematical property that our recursive formulation captures the cumulative influence of arbitrarily distant future states through the convergent discounted sum. The recursive computational form is given by the following:

Fπ,at(ht,at)=T∼t+ΓEπ,p[F(h∼t+1,a∼t+1)]8

This relation shows that each spectral component F(ωl) is coupled through a Bellman-like recursion in the frequency domain. Hence, the L × D matrix F serves as a compact spectral signature that implicitly encodes the infinite-horizon temporal structure of the state sequence. The key insight is that infinite-horizon dependencies are not stored explicitly but emerge naturally from the recursive propagation in Equation (8), analogous to how value functions in RL encode infinite-horizon cumulative rewards through Bellman equations. Here, Tt∼=∑n=0+∞γnrt+n denotes the discounted reward encoding at time t, with rt+n representing the immediate reward at future step t + n. The frequency-domain decay matrix Γ=diag(γ0,γ1,…,γL−1) applies component-wise discounting. Each diagonal element γ1 attenuates the corresponding frequency bin: high-frequency components (ωl→π,γ1→0) decay rapidly and capture short-term fluctuations, whereas low-frequency components (ωl→0,γ1→1) decay slowly and encode long-term trends. This separation allows the policy to reason differently about transient noise and persistent bias. Thus, each prediction step requires only the DTFT estimate of the next state-action pair. The expected next-step spectrum under policy π and dynamics p is as follows:

Eπ,p[F(h∼t+1,at+1)]=∫h∼t+1∫at+1F(h∼t+1,at+1)⋅p(h∼t+1∣h∼t,at)⋅π(at+1∣h∼t+1)dat+1dh∼t+19

The predictor F takes the encoder output z_t and produces the DTFT estimate of the discounted sequence. Because the prediction target lies on the complex plane, the prediction network employs two separate output modules, FRe and FIm, to represent the real and imaginary parts, respectively. Therefore, the output of the predictor is represented as follows:

FRe(h∼t,at)=T∼t+ΓReERe[FRe(h∼t+1,at+1)],FIm(h∼t,at)=T∼t+ΓImEIm[FIm(h∼t+1,at+1)]10

where ΓRe and ΓIm denote the real and imaginary parts of the constant Γ=γdiag(1,e−j2πL,e−j4πL,…,e−j(L−1)πL).

(c) Projector: Because directly using the output obtained from the predictor for loss calculation may lead to overfitting, the projector is designed to map the DTFT values from a high-dimensional space to a low-dimensional space, thereby improving generalization and stabilizing training. The recursive relationship can be reformulated as contraction mapping S, with output:

S(F(h∼t,at))=MLP(F(h∼t,at))=F∼t+ΓEπ,P[F(h∼t+1,at+1)]11

where the MLP term denotes a two-layer MLP architecture specifically designed for feature dimensionality reduction.

(d) Auxiliary Network Loss: In Fourier analysis, low-frequency DTFT components capture long-term trends, whereas high-frequency components capture short-term variations. To preserve both types of information, we compute the cosine similarity directly on the primary components and use projected features for the remaining residual components. The final auxiliary loss is the sum of the two cosine similarities:

Laux(H,F)=d(h∼t+[ΓRe⋅S{FRe(h∼t+1,π(h∼t+1))}−ΓIm⋅S{FRe(h∼t+1,π(h∼t+1))}],FRe(h∼t,at))+d([ΓIm⋅S{FRe(h∼t+1,π(h∼t+1))}+ΓRe⋅S{FIm(h∼t+1,π(h∼t+1))}],FIm(h∼t,at))12

where d denotes the cosine similarity. The hidden state h_t extracted from the LSTM network layer and the corrected action a_t output by the actor network are utilized as inputs for training the SFPN auxiliary network, which is trained end-to-end through the encoder, predictor, and projector. The LSTM parameters θLSTM are dynamically adjusted via gradient ascent on an auxiliary loss Laux with learning rate lr:

θLSTM←θLSTM+lr∇θLaux(θ)13

3.2.3 Temporal Actor-Critic Module

(a) Action Space Setting: The action space is designed to apply continuous positional corrections to the initial output loc^init, avoiding the inefficiencies of discrete action spaces. The position-correction process is detailed as follows:

The actor outputs Gaussian distribution parameters (μ,σ) for each direction, i.e., atECEF=(μxt,σxt,μyt,σyt,μzt,σzt), in ECEF coordinates. Meanwhile, in GEO coordinates, the parameters can be represented as atGEO=(μlatt,σlatt,μlont,σlont), atECEF∈R6,atGEO∈R4. The continuous actions representing the position correction for each coordinate axis, Δloct=(Δxt,Δyt,Δzt) and Δloct=(Δlont,Δlatt), are sampled from the Gaussian distributions and clipped to a maximum magnitude m, i.e., Δxt∼N(μxt,σxt2),Δyt∼N(μyt,σyt2),Δzt∼N(μzt,σzt2) or Δlatt∼N(μlat,σlat2),Δlont∼N(μlont,σlont2). Then, by using the scaling factors μECEF and μGEO, the corrected position output is obtained as follows:

loc^t={(xt+μECEFΔxt,yt+μECEFΔyt,zt+μECEFΔzt)(ECEF)(latt+μGEOΔlatt,lont+μGEOΔlont) (GEO)14

(b) Reward Space Setting: To quantify the effectiveness of the positioning correction strategy learned by the RL agent, we employ the correction advantage error as the reward function, defined as follows:

rt:={rtECEF=α⋅(‖loctinit−loctref‖2−‖loc^tECEF−loctref‖2)rtGEO=α⋅(Vincenty(loctinit−loctref)2−Vincenty(loc^tGEO−loctref)2)15

where α scales the reward intensity, rtECEF and rtGEO are the reward functions in ECEF coordinates and GEO coordinates, respectively, and the reference position loctref serves as the ground truth for reward computation. The reference position can be obtained by using map-matching algorithms. The Vincenty(⋅) operation is based on Vincenty’s formula (Vincenty, 1975). By integrating multi-modal observations and this error-driven reward mechanism, our RL framework continuously optimizes GNSS positioning in complex urban environments, achieving enhanced accuracy through iterative correction.

(c) Actor-Critic Network: A temporal actor-critic framework is constructed to learn an optimal policy by leveraging the latent representation h_t, generated from two independent LSTM modules, and updates the actor-critic parameters via PPO. The actor network πθactor, parameterized by θactor, generates continuous correction a_t from Gaussian distributions by interacting with the RL environment at time t, which can be defined as follows:

at=πθactor(ht actor)16

where htactor denotes the actor-specific hidden state at time step t. The critic network Vθcritic, parameterized by θcritic, estimates the value function Vcriticπ(ot) for observations, formalized as follows:

Vcriticπ=Vcriticπ(ot)≈Vθcritic(htcritic)17

where htcritic denotes the critic-specific hidden state at time step t. Hence, the parameter set of the model can be represented as θ={θLSTMa,θLSTMc,θactor,θcritic,w}, among which θLSTMa and θLSTMc parameterize the LSTM modules for the actor and critic branches, respectively, and w denotes the weight. To stabilize training and evaluate action efficacy, we employ the generalized advantage estimation (Schulman et al., 2015) to compute advantage values:

A^t=∑k=tT−1γT−kμt+118

where μt=rt+γ⋅(Vθcritic(ht+1critic)−Vθcritic(htcritic)) represents the temporal difference errors. The actor loss combines clipped policy ratios and advantages:

Lθactor=−Et[min(ratiot(θactor)⋅A^t,clip(ratiot(θactor),1±ϵ)⋅A^t)]19

where ϵ is a hyper-parameter based on the clipping range and ratiot(θactor) is used to measure the difference between the new and old policies.

Because employing the single-step temporal difference error in the value-estimation method may hinder effective policy learning of the intermediate steps, the mean squared return error (Le et al., 2017) is utilized as the loss formulation of the critic network, which encompasses the cumulative discount reward over an episode with T steps:

Lθcritic=Et[12‖∑k=tTγk−trk−Vcritic(htcritic)‖22]20

The general architectural framework of the SFPN algorithm for positioning correction is presented in Figure 2, and Algorithm 1 provides a concise summary of the SFPN algorithm.

4 EXPERIMENTS

In this section, the proposed SFPN algorithm is evaluated using two real-world multi-constellation data sets. One data set is derived from the Android Raw GNSS Measurements data set, which is utilized in the GSDC. The second data set is the GZGNSS measurement data set, collected by multiple N307-5D receivers at various locations in Guangzhou, China. These data sets encompass a wide range of scenarios, enabling a comprehensive evaluation of our algorithm. The experiments utilized an NVIDIA GeForce RTX 3090 graphics processing unit for model training, with a multi-core central processing unit to manage data loading, preprocessing, and serial computing tasks. The experimental results are presented as the mean of more than five repeated trials, with error data reported as both the mean and standard deviation.

Our evaluation focuses on two key objectives: (1) assessing whether frequency-domain auxiliary supervision provides consistent robustness gains that are compatible with environment-included GNSS error patterns and (2) quantifying the positioning improvement of SFPN over model-based and state-of-the-art learning-based methods.

4.1 Data Set for Experiments

(1) The GSDC data set is constructed from GNSS measurements of GPS signals in the L1 band collected by various Android smartphones, comprising driving trajectories recorded near San Francisco International Airport and Los Angeles International Airport. The data set includes high-precision positioning information from the NovAtel SPAN system, which served as the ground-truth reference position loctref during the validation process, and each trajectory contains approximately 2 × 10³ positioning time steps. Considering the uncertainty in the number of visible satellites across different trajectories and the variance in environmental characteristics, we eliminate the trajectories with zero visible satellites at certain time steps and categorize the remaining 79 trajectories with complete reference positions into three scenarios, i.e., open environment, semi-urban environment, and urban environment. These categories are primarily differentiated by their structural complexity, quantified through buildings and tree density metrics. Based on these three major types of data sets, we further construct 9 specialized RL environments, each containing 3–7 representative trajectories. For positioning initialization, we implement a KF method to generate coarse ECEF coordinate estimates, which serve as the correction baseline in our positioning framework.

(2) The GZGNSS data set was collected by multiple N307-5D receivers at various locations within the urban area of Guangzhou, China, covering multiple road trajectories. To achieve high-precision positioning estimates, we use data obtained from the NovAtel SPAN+LCI system as the reference for ground-truth positions locLLtref and employ the model-based RTK method as the baseline for our proposed model, SFPN, to determine the initial positions locLLtinit. Considering that errors in the elevation direction are typically greater than those in the horizontal direction and that most vehicular navigation applications primarily require horizontal positioning accuracy, our correction framework focuses exclusively on the horizontal components. Additionally, the data set is systematically classified into four distinct scenarios based on environmental complexity: open road, urban canyon, urban forest, and overpass. In the experiments, we use 70% of the data set from a given environment for training and the remaining 30% for testing, whereas data from other scenarios are used entirely for testing.

4.2 Methods for Comparison

To validate the performance of the proposed SFPN method, we conducted a comparative study against prevalent model-based algorithms and learning-based algorithms.

a) Model-Based Methods: For the GSDC data set, which features smartphonegrade GNSS observations with significant multipath error and stochastic noise, we required a baseline capable of dynamically fusing imprecise measurements with a motion model. Therefore, as the baseline model, we adopted the KF algorithm, a recursive state-space estimator optimizing dynamic system predictions via prediction correction steps and multi-temporal feature fusion. For the GZGNSS data set, which uses professional-grade N307-5D receivers capable of centimeter-level positioning, we adopted the RTK method (Z. Li et al., 2022) as a high-precision industry standard. This method utilizes a Doppler smoothing code filter to reduce noise obtained from the reference station and combines a constant acceleration model to predict the motion state to obtain the initial position. The use of two fundamentally different baselines (KF for consumer-grade data, RTK for geodetic-grade data) underscores the flexibility and generalizability of our approach across device types and foundational algorithms.

b) DNN-Based Methods: A set-transformer (Kanhere et al., 2021) and a GCNN (Mohanty & Grace, 2022), which utilize distinct neural network architectures to enhance the accuracy of model-based positioning solutions, were chosen for validation. The set-transformer leverages the attention mechanism inherent to the transformer architecture, focusing its learning process on the GPS L1 constellation, with an LOS vector and pseudorange residuals as inputs. In contrast, the GCNN utilizes a graph isomorphism network module to process data from different constellation systems, including GPS, GLONASS, Galileo, and BeiDou.

c) DRL-Based Methods: In our comparative study of positioning correction task performance, two state-of-the-art DRL algorithms are applied, i.e., asynchronous advantage actor-critic (A3C) (Zhang & Masoud, 2020) and multi-LSTMPPO (Zhao et al., 2023). A3C only uses vehicle trajectories as observations and trains in a discrete action space, estimating the values of 441 actions. Multi-LSTMPPO employs multiple input observations to construct the RL environment and combines PPO with two LSTM modules to implement an effective positioning correction strategy.

4.3 Parameter Selection

In this section, we evaluate the effectiveness of the proposed SFPN by varying the learning rate of the RL training lr and the learning rate of the auxiliary network lraux to identify the optimal parameters for the model on the GSDC data set and the GZGNSS data set. The evaluation criteria are determined by calculating the mean squared error between the reference and the predicted positions, i.e., ‖loc^−locref‖2. Figure 4 presents the mean distance errors for the model trained in semi-urban and urban conditions. It is observed that a suitable reduction of lr^aux, which is controlled to be approximately one-tenth of lr, can achieve smaller horizontal errors. Consequently, we selected lr = 1e⁻⁴, aux_lr = 5e⁻⁵ for semi-urban and lr = 5e⁻⁴, aux_lr = 1e⁻⁵ for urban conditions. Because the different data set distributions and baseline methods may require distinct parameter settings, we applied aux_lr = 3e⁻⁵ for open-road and aux_lr = 5e⁻⁵ for urban-canyon conditions. A learning rate of lr = 1e⁻⁴ can achieve optimal model learning performance.

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FIGURE 4

Performance of the proposed SFPN method over ranges of the learning rate of the RL training lr and learning rate of the auxiliary network lr^aux, where lower values indicate better positioning accuracy: (top) models trained on the GSDC2022 data set; (bottom) models trained on the GZGNSS data set. (a) Semi-urban, (b) Urban, (c) Open road, (d) Canyon.

In addition, the encoder Ha and Hc share the same architecture, comprising 6 fully connected layers with 256 neurons in the hidden layer. Both the predictor F and the projection S apply a two-layer MLP architecture. Given the complex-valued prediction targets inherent in the auxiliary task, the last layer of the predictor is split into two heads as the real part FRe and the imaginary part FIm. To improve the frequency solution, we use the rectified linear unit function (Krizhevsky et al., 2012) for nonlinear activation, whereas network parameter optimization is performed via the Adam method with a learning rate of lr=3e−4.

4.4 Performance Evaluation and Comparison on the GSDC Data Set

4.4.1 Training Performance

Our initial performance evaluation focuses on the discounted returns during the training phase. Figure 5 displays convergence curves of discounted returns during training in semi-urban and urban scenarios for multi-LSTMPPO, A3C and SFPN, each configured with corresponding optimal parameters. The curves exhibit inherent oscillations due to two primary factors: (1) The distribution differences between trajectories in different areas increase the randomness of state transitions, leading to fluctuations in discounted rewards. (2) Variations in the distribution of positioning errors caused by environmental interference can also result in the curves not being smooth. Despite these fluctuations, SFPN demonstrates superior convergence characteristics. In the semi-urban environment, both SFPN and multi-LSTMPPO achieve comparable cumulative rewards at approximately 150. However, in the more challenging urban environment, SFPN achieves significantly higher converged rewards. This superiority stems from its frequency-domain analysis capability, which enables the agent to identify stable periodic error patterns amidst drastic environmental changes, thereby guiding the policy toward more effective correction strategies. In contrast, A3C, which relies on a simple fully connected network and lacks temporal memory, exhibits significantly lower cumulative rewards, highlighting the necessity of temporal and structural modeling for this task. Figure 6 compares the positioning accuracy of the SFPN algorithm and the KF algorithm in different scenarios. The results clearly show that the SFPN algorithm achieves significantly lower positioning errors than the traditional KF method in all three typical environments (open, semi-urban, and urban), demonstrating its ability to consistently maintain higher positioning accuracy, particularly in complex urban settings.

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FIGURE 5

Cumulative rewards of different DRL-based methods with corresponding optimal parameters trained in semi-urban and urban environments of the GSDC2022 data set. (a) Semi-urban, (b) Urban.

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FIGURE 6

Distance error for KF and SFPN methods applied to the GSDC data set, with each plot presenting a single trajectory example and corresponding average errors detailed in the legends. (a) Open, (b) Semi-urban, (c) Urban.

4.4.2 Testing Performance

We assessed the positioning accuracy of various methods by comparing the positioning errors of trajectories after the training phase. The evaluation performances of different methods in semi-urban and urban environments in terms of the distance error in the ECEF coordinate system are presented in Table 1. The experimental results demonstrate the superior performance of the SFPN compared with other methods across all test environments, obtaining an average improvement in positioning accuracy of 17.7% and 16.1% for semi-urban and urban environments, respectively. These results indicate that the proposed method can effectively adjust the initial positions obtained from the baseline KF across the open, semi-urban, and urban settings. A depiction of positioning trajectories within the GSDC data set is shown in Figure 7. SFPN performs best in open environments when trained in semi-urban settings, as open roads have fewer obstructions and less complex signal reflections. When trained in urban settings, the best results are generated in urban environments, indicating that positioning errors follow certain patterns in complex settings, with SFPN effectively capturing these inherent characteristics.

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FIGURE 7

Depiction of positioning trajectories within the GSDC data sets, where the corrected trajectories are closer to the reference paths. (a) US-LAX-5-GooglePixel5, (b) US-MTV-1-GooglePixel5.

DNN-based methods can reduce positioning errors, but rely on static data sets, which cannot adapt to dynamic changing environments. Thus, the average position error correction rate decreases in the urban training setting. Among DRL-based approaches, the A3C algorithm exhibited the lowest mean performance across all test settings, which might be due to its lack of consideration of GNSS measurement features in modeling complex environmental errors and its inability to effectively capture temporal dependencies. Although multi-LSTMPPO delivers remarkable performance, its performance is inferior to that of the SFPN framework. This limitation most likely stems from the reliance of the LSTM module on historical trajectory continuity in the time domain, which makes it prone to converge on local optima.

4.5 Result Verification on the GZGNSS Data Set

Our algorithm was further applied to the collected GZGNSS measurement data set for positioning correction. We evaluated different GNSS positioning methods by comparing their average horizontal errors, as shown in Table 2, where the error is calculated by Vincenty (loc^tGEO,loctref). In general, the proposed SFPN can effectively reduce the positioning errors of initial positions obtained by RTK in all environments, achieving a better performance than the other comparison algorithms. While the set-transformer and GCNN methods demonstrate comparatively low horizontal errors in the urban-forest setting, the positioning errors are large in urban-canyon scenarios, indicating that DNN-based methods struggle to adapt to dynamic environments. Multi-LSTMPPO shows slightly better positioning accuracy than SFPN in the open-road environment when trained on the open-road data set. The reason for this result might be that the open-road environment has continuous LOS signals and fewer dynamic obstructions, resulting in a relatively simple positioning error pattern, which makes it difficult to fully leverage frequency-domain analysis for extracting internal structural features from time-domain data.

4.6 Complexity and Runtime Analysis

Table 3 summarizes the computational overhead and inference latency of the proposed SFPN method and baseline models. Under the same configuration, SFPN achieves an inference time of 1.63 ms, which is slightly higher than that of A3C and multi-LSTMPPO, but significantly lower than that of the computationally intensive set-transformer method. In terms of computational complexity, SFPN requires only 0.42 M floating point operations (FLOPs), comparable to that of multi-LSTMPPO and much lower than that of set-transformer, indicating its lightweight computational characteristics. However, the peak memory consumption of SFPN is 28.84 MB, higher than that of the baseline models, which primarily stems from its multi-scale feature aggregation and auxiliary network design.

Despite the increased memory usage, the inference time of SFPN is still far below the 1000-ms budget corresponding to the 1-Hz sampling rate of GNSS data, fully meeting the requirement for real-time processing. In summary, SFPN achieves efficient real-time positioning with limited computational resources, providing a viable solution for edge deployment.

4.7 Confidence Interval Analysis

The 95% confidence intervals (CIs) in Table 4 are used to assess the reliability of the observed error reductions, not only their magnitude. The calculation is performed on the test set, i.e., the untrained data, as the point-wise difference between the model output and the baseline result. Specifically, for each evaluation point i and each trial r, we first compute a point-wise improvement relative to the corresponding baseline as Δei,r=ei,rbaseline−ei,rmethod. We then pool all tested point-wise improvements in each subset, compute their sample mean Δ¯ and standard deviation s, and report the 95% CI of the mean improvement as Δ¯±1.96⋅s/N, where N is the number of pooled point-wise samples. Under this criterion, a CI entirely above zero indicates a statistically reliable improvement over the baseline at the 95% confidence level, whereas a CI crossing or below zero indicates that the improvement is not reliable. The results show that SFPN is the most reliable method, with all of its CIs being strictly positive. The LSTMPPO, set-transformer, and GCNN methods also show positive CIs in all subsets, but their intervals are generally lower than that of SFPN, indicating weaker yet still reliable gains. In contrast, A3C yields negative CIs for semi, urban-canyon, and open-road training scenarios, indicating statistically reliable degradation relative to the corresponding baselines in those subsets. In addition, the CI widths are relatively narrow for SFPN, suggesting low estimation variance and stable behavior across evaluation points. Therefore, these results indicate not only that SFPN achieves higher gains, but also that these gains are statistically credible and consistently reproducible under the defined protocol.

4.8 Spectral-Error Attribution Analysis

To further verify whether the learned frequency-domain representations capture information related to positioning error, we conduct a post-hoc spectral–error attribution analysis that examines the statistical relationship between spectral features of the corrected trajectory and the residual positioning error. Specifically, for each test trajectory corrected by the trained model, we compute the velocity profile from the predicted position sequence, apply a fast Fourier transform to obtain the spectral amplitude vector, and record the mean positioning error of that trajectory. By collecting (spectrum, error) pairs from all test trajectories, we perform two complementary statistical tests for each frequency bin: (1) Pearson correlation between the spectral amplitude in each bin and the trajectory-level positioning error and (2) a two-sample t-test that compares the spectral amplitudes of high-error trajectories (above the 75th percentile) with those of low-error trajectories (below the 25th percentile).

Figure 8(a) presents analysis results for the model trained in the semi-urban environment. Here, the association is notably stronger: 32 frequency bins are significantly correlated with positioning error, and 30 bins show a significant amplitude difference between high- and low-error trajectories. The strongest individual correlations (|r|>0.5) appear in medium-to-high frequency bins (e.g., bin 69, bin 85, and bin 148), which likely correspond to transient disturbances or rapid signal-quality fluctuations that the model has learned to encode. Figure 8(b) shows results for the model trained in an urban environment. The left panel displays the Pearson correlation coefficient between each frequency bin and the positioning error. Specifically, 14 bins reach statistical significance (p<0.05), indicating that the spectral amplitude at these frequencies has a measurable linear association with trajectory-level error. The right panel compares the mean spectra of high-error and low-error trajectories: 28 bins exhibit a statistically significant amplitude difference between the two groups, suggesting that the spectral profile can discriminate between low-error trajectories and trajectories that retain larger residual errors.

These attribution results provide direct evidence that the frequency-domain representations learned by SFPN encode information that is statistically associated with the magnitude of residual positioning error.

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FIGURE 8

Spectral-error attribution analysis

Each panel pair shows, on the left, the Pearson correlation between each frequency-bin amplitude and positioning error (red bars: p<0.05) and, on the right, a mean spectrum comparison between high-error (above 75th percentile) and low-error (below 25th percentile) trajectories. (a) The semi-urban-trained model yields 32 significant correlation bins and 30 significant difference bins. (b) The urban-trained model yields 14 significant correlation bins and 28 significant difference bins.

4. 9 Ablation Study

To verify whether the gains observed for SFPN come from meaningful spectral structure rather than merely the addition of an auxiliary loss, we conducted a targeted ablation with three variants: SFPN, SFPN w/o aux (no auxiliary supervision), and SFPN w/random aux (auxiliary branch retained but frequency-domain targets randomly corrupted). For clarity, in our experiments, SFPN w/o aux is equivalent to the multi-LSTMPPO backbone, i.e., the same dual-LSTM PPO correction network with the frequency-domain auxiliary branch removed. Therefore, multi-LSTMPPO serves as the no-auxiliary-task ablation of SFPN, whereas SFPN w/random aux is used to test whether the auxiliary branch acts merely as generic regularization. The results in Figure 9(a) show that SFPN consistently performs best. In the semi-urban training, SFPN achieves an error of 3.97 m, outperforming the model without auxiliary supervision (4.24 m) and with random auxiliary supervision (4.31 m). Moreover, SFPN w/random aux is slightly worse than SFPN w/o aux, indicating that mismatched spectral targets can introduce conflicting gradients and degrade the representation for correction. In urban training, SFPN remains best at 3.85 m, whereas SFPN w/random aux (4.38 m) is better than SFPN w/o aux (4.55 m) but still clearly inferior to the full SFPN. This finding suggests that in highly disturbed urban settings, random auxiliary signals may provide a weak regularization effect, yet the dominant improvement still comes from physically meaningful spectral-structure guidance rather than from simply the addition of an auxiliary loss.

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FIGURE 9

Ablation analysis of SFPN variants in terms of positioning error and SSA periodiccomponent correlation. (a) Ablation results of SFPN variants trained on the semi-urban and urban data sets and evaluated across all environments. Lower values indicate better performance. (b) SSA periodic-component correlation ablation. Higher values indicate better performance.

To evaluate whether SFPN learns meaningful periodic structure, we applied singular spectrum analysis (SSA). The SSA periodic-component correlation measures the alignment between the corrected and ground-truth velocity sequences (embedding window: 50 steps). A value closer to 1 indicates a better capture of the true motion’s periodic structure. The results are presented in Figure 9(b), showing that the full SFPN consistently achieves the highest correlation across all scenarios. Removing the auxiliary branch causes a clear decrease, confirming that the frequency-domain task is central to learning periodic structure—not a minor addition. Replacing the frequency-domain targets with random targets degrades performance further (0.27 in semi-urban, 0.18 in urban), showing that meaningful spectral guidance is essential and that random signals disrupt rather than regularize the learning.

These trends hold across two data sets with different receivers and baselines, suggesting that the gains observed for SFPN are not due to a specific device alone. The stronger periodic correlation with structured spectral supervision supports the hypothesis that the structured spectral supervision helps mitigate errors, especially in challenging urban environments.

4.10 Discussion

The experimental results provide both indirect performance-based and direct statistical evidence that frequency-domain state-sequence modeling contributes to robust GNSS correction. Through rigorous testing on two distinct data sets, i.e., the GSDC and GZGNSS measurement data sets, SFPN exhibits consistent performance improvements across various urban scenarios. Comparative analysis reveals three key findings:

1. Compared with model-based approaches (KF, RTK), SFPN shows superior performance across all test scenarios. Figure 6 illustrates this advantage in urban environments, while Table 1 quantitatively confirms the improvement in both open and complex urban settings. These results validate the ability of SFPN to overcome the limitations of traditional filtering methods in handling nonlinear error dynamics and temporal noise correlations.

2. When compared with DNN-based methods (GCNN, set-transformer), SFPN maintains more consistent performance across different environmental conditions. As evidenced in Table 2, SFPN outperforms these approaches, particularly in challenging urban-canyon scenarios, where conventional DNNs struggle with dynamic error patterns.

3. Compared against state-of-the-art DRL methods (A3C, multi-LSTMPPO), SFPN achieves better convergence properties and final performance. The cumulative rewards plots in Figure 5 further demonstrate that by applying the Fourier transform to the state sequence to capture its intrinsic structural information in the frequency domain, the proposed method significantly outperforms both the traditional DRL algorithm (A3C) and another advanced LSTM-based PPO algorithm (multi-LSTMPPO) in terms of final performance. Moreover, Table 1 documents the superior test time accuracy of SFPN. This improvement stems from the unique capability of SFPN to capture both temporal dynamics and frequency-domain patterns, addressing key limitations of existing DRL approaches.

A remaining limitation of the current study is that while the spectral-error attribution analysis demonstrates a statistically significant association between frequency-domain features and residual positioning error, it does not isolate the specific physical error source (e.g., multipath noise) responsible for each spectral component. Controlled simulation or ray-tracing studies would be needed for such fine-grained physical attribution.