Jiangfan Liu

The enhanced long-range navigation (eLoran) system serves as a reliable backup to the global navigation satellite system (GNSS) and holds strategic significance in critical applications. To address the limitations of the traditional eLoran system under cold start and incomplete chain conditions, this paper proposes a two-stage pseudorange positioning framework. First, this framework projects the Earth to a standard sphere, enabling the derivation of closed-form solutions based on measured pseudorange values. Second, a Newton-Raphson method is employed for iterative refinement, utilizing closed-form solutions as an initial position to obtain the accurate position. This framework requires less prior position information and supports flexible station combinations. Also, the framework is capable of positioning under cold start conditions and in areas beyond the coverage of the entire chain. Third, the framework is comprehensively validated through theoretical simulations and practical applications. The results demonstrate significant improvements in positioning availability and accuracy, compared to the traditional eLoran system that utilizes time difference of arrival (TDOA) and pseudorange-based local optimization methods. The proposed framework exhibits superior robustness and broader applicability, establishing a solid foundation for eLoran-based resilient navigation.

The long-range navigation (Loran) system serves as a reliable backup to the global navigation satellite system (GNSS) and holds strategic significance in critical applications. However, the Loran system faces positioning ambiguity and range limitation challenges due to its time difference positioning mechanism and the constraints of the station chain signal in the additional time difference ambiguity discrimination (ATDAD) approach. To address these issues, this paper proposes a received electric field strength ambiguity discrimination (REFSAD) approach based on the electric field strength (EFS) attenuation characteristics of the Loran signal. The research constructs a comprehensive attenuation model applicable to mixed sea-land propagation paths and designs an ambiguity resolution strategy that does not require additional station information, thus enabling reliable identification of the true position. Through comprehensive Monte Carlo simulations and field-based experimental validation, this study demonstrates that the REFSAD approach exhibits statistically significant performance advantages, with 98.5% of positioning results falling within the high success rate interval (80-100%), compared to 97.4% for the ATDAD approach. When the coverage area is expanded by a factor of 3.8, the REFSAD approach maintains robust performance, with 98.2% of results within the high success rate range. Field experiments conducted in the South China Sea further validated the efficacy of REFSAD in regions where ATDAD is inapplicable, yielding a 100% positioning success rate with a root mean square error (RMSE) consistently maintained below 20 meters, thereby satisfying maritime navigation accuracy requirements.

In this study, Loran-C signals were collected throughout the day, and the characteristics of the received signals at different propagation distances were analyzed. Because the signal amplitude is small and difficult to recognize at a long distance and there is mutual interference between stations, a linear average method is used to process the received signal. At locations closer to the receiver, clear observations of the time delay and amplitude variation in the one-hop sky wave can be made by using the ground wave as a reference, which can be applied to studying the characteristics and parameter inversion of the lower ionosphere. When the distance is further, the significant enhancement of the sky-wave signal during the night may lead to decreased accuracy in timing and positioning. When the distance is much larger than the propagation range of the ground wave, only the sky-wave signal can be received, and the signal is more stable than when the distance is closer. During the night, multiple amplitude-comparable multi-path signals may appear in the sky wave, making the identification of the one-hop sky wave more difficult.

In this study, the effect of random characteristics of the ionosphere on the enhanced Loran (eLoran) sky-wave signal was analyzed. First, the bilinear transform stochastic finite-difference time-domain (S-BT-FDTD) method in layered half-space isotropic plasma is derived, and the correctness of the calculation of the mean value and standard deviation of the field components of the eLoran sky-wave signal is verified. Second, the mean value and standard deviation of the amplitude and time delay of the eLoran sky-wave signal were analyzed by combining the International Reference Ionosphere, pseudo-1-D BT-FDTD method, and Monte Carlo method. The simulation results show that the mean value of the amplitude of the reflection coefficient amplitude (RA) in the eLoran signal bandwidth remains unchanged. The larger the standard deviation of the electron density, the larger is the standard deviation of RA. The standard deviation of RA during the day is greater than that at night. The mean value of the time delay remains unchanged, and the standard deviation of the time delay is related to the way the electron density varies randomly. When the incident angle increases, the RA increases, the standard deviation of RA decreases, the time delay decreases, the standard deviation of the time delay decreases, and the signal becomes more stable.

In this study, a real‐time high accuracy signal simulation technique for an enhanced Loran (eLoran) signal simulator is proposed, which can be used to verify the performance of the eLoran receiver. The proposed method is based on a maximumly flat design of the low‐pass filter, and high simulation accuracy can be obtained in the low‐frequency band. The simulator structure is based on the variable fractional delay Farrow structure, which can regenerate eLoran signals with high‐precision time of arrival (TOA) in real‐time based on discrete samples on a hardware platform. By comparison of existing methods and the true TOA data validation, the proposed method has high performance for low‐frequency narrow‐band eLoran signals and the proposed technology can apply to arbitrary TOA real‐time simulations.

In this letter, Wait's exponential (EXP) model and international reference ionosphere (IRI) model are combined to study the diurnal variation characteristics of very low frequency (VLF) wave propagation. The EXP model is used below the IRI model to compensate for the sub-ionosphere electron density lack of the IRI. The major EXP parameters are reconstructed from the lower IRI electron density for different times of the day. Being capable of characterizing the sub-ionosphere details, the combined IRIEXP model gets higher accuracy than the pure IRI model for VLF diurnal variation prediction. The measurement result is replicated by using the finite-difference time-domain (FDTD) method to validate the presented model.

It is important to study the electron density distribution of plasma sheath to inhibit the blackout phenomenon of reentry vehicle. In this article, we propose a new method to predict the electron density distribution based on a backpropagation neural network (BPNN). We build a dataset based on the hybrid matrix method of computational electromagnetics (EMs) and train a three-layer network to map the relationship between EM wave propagation characteristics and electron density. This method is more accurate than the traditional plasma diagnosis method and reduces the complicated measurement process. Experimental results show that the neural network method has a performance with an average error of less than 3% and can reduce twice the training time using multicore CPU than low computing capability graphics card. This study sheds new light on electron density diagnosis to find more accurate and effective methods.

This study proposes a high‐precision algorithm used in enhanced Loran sky–ground wave separation based on the Levenberg–Marquart algorithm. The simulation results show that, compared with the existing algorithms, the algorithm proposed in this study has higher accuracy, especially in the case of a low signal‐to‐noise ratio. Furthermore, when the skywave of enhanced Loran has at least a 35 μs time delay compared with the groundwave, the performance of the algorithm is significantly better than that of other algorithms. Analysis of an actual signal showed that the average correlation coefficient was greater than 99% and that the signal was successfully separated.

A new approach for using polynomial chaos‐based expansion finite‐difference time‐domain (PCE‐FDTD) is presented to calculate the uncertainty of electromagnetic wave propagation in dispersive materials. Based on the bilinear transform method, this approach performs polynomial expansion on random electromagnetic fields by PCE‐FDTD method. The proposed algorithm has a simple formulation and is very easy to extend to isotropic dispersive material. It has a general form for different types of random dispersive materials and can efficiently calculate the mean value and the SD of electromagnetic field components in a single run. Two examples of different dispersive media with two random variables are listed to show the generality of the algorithm, and a radar cross‐section of a perfectly conducting cylinder coated by a layer of random plasma is illustrated as an example to show the practicability of the approach. The results are validated by comparing it with the Monte Carlo method.