Dandan Wang

This paper investigates the effects of frequency, polarization, and incidence angle on very low frequency (VLF) reflection and transmission in an anisotropic ionosphere. By deriving the refractive indices and altitude evolution of the propagation direction of the ordinary (O) and extraordinary (E) modes, we show that, unlike the O mode whose wave normal bends toward the horizontal and approaches an effective turning condition with increasing altitude, the E-mode wave-normal direction progressively turns toward the vertical and is governed by the coupling among the pseudo-Brewster effect (PBE), magnetic guidance, and ionospheric parameter gradients. Accordingly, a dynamic pseudo-Brewster angle (PBA) matching window is proposed to quantify the matching relationship between the E-mode propagation angle and the local PBA condition. Simulations show that this window has a strong frequency dependence: as the frequency increases, the window shifts upward and broadens; under oblique incidence, the local overlap between the window and the upward E-mode flux profile becomes more evident. Further energy-flux analysis indicates that the overall upward E-mode flux transmission in the lower-ionospheric entrance region and the PBA-window flux-overlap ratio do not reach their peaks at the same incidence angle. This suggests that the upward energy transfer of VLF waves into the lower ionosphere is influenced by the combined effects of propagation path length, collisional absorption, and PBE-related local coupling. These results provide a theoretical reference for understanding the local incidence mechanism of VLF waves entering the lower ionosphere, and for developing future ground-to-satellite VLF propagation models that incorporate source angular spectra, lower-ionospheric entrance coupling, and in-orbit receiving geometry.

Receiver system delay is embedded in additional secondary factor (ASF) observations and coupled with the propagation term at the observation level, which is a key factor limiting high-accuracy ASF measurement in enhanced Loran (eLoran) systems. Existing system delay calibration methods either can only calibrate the internal delay of the receiver and therefore cannot cover the complete delay chain involved in field measurements, or rely on single-point theoretical compensation, making the estimated system delay susceptible to errors in theoretical ASF values. To address this issue, this paper proposes a system delay calibration method based on multipoint ASF measurements for eLoran receivers. By exploiting the structural characteristic that the system delay acts as a shared constant bias in multipoint ASF observations, whereas the ASF spatial component varies with the propagation path, a joint inversion model of system delay and equivalent ground conductivity is constructed to separate the two in the absence of true ASF values. On this basis, a rapid calibration procedure is further established, including calibration area selection, reference receiver calibration, determination of the ASF spatial reference value at a fixed calibration site, and rapid calibration of other receivers. Experimental results in the Xi’an and Zhoukou test areas show that the optimal estimates of the reference receiver system delay are 79.046 μs and 79.048 μs, respectively, with a difference of only 2 ns. After correction for system delay and conductivity, the mean absolute error (MAE) and root mean square error (RMSE) between the ASF spatial component at the validation points and the theoretical ASF are 0.039 μs and 0.045 μs, respectively, in the Xi’an test area, and 0.076 μs and 0.091 μs, respectively, in the Zhoukou test area. Furthermore, the rapid calibration method is validated in the Zhoukou area. For two ASF measurement systems, one using the reference receiver calibrated through joint inversion and the other using a receiver rapidly calibrated at a fixed calibration site, the mean bias, MAE, and RMSE at the same measurement points are reduced by 98.9%, 85.9%, and 84.4%, respectively, compared with the corresponding values before calibration. The results demonstrate that the proposed method can provide relatively consistent system delay estimates under different propagation conditions and improve the consistency of ASF measurements obtained by different receivers.

In Global Navigation Satellite System (GNSS)-denied environments, positioning accuracy is critical for unmanned aerial vehicle (UAV) swarms. A particle filter-based cooperative localization algorithm is proposed to improve UAV positioning accuracy. To address the issues of particle impoverishment and loss of diversity inherent in traditional particle filters, a Selective Correction Adaptive Particle Filter (SCA-PF) with an enhanced resampling strategy is introduced. First, the algorithm incorporates a threshold-based resampling improvement to mitigate particle impoverishment common in conventional particle filters. Second, an adaptive threshold feedback strategy, based on the residual variation rate, dynamically adjusts the weight threshold to more effectively select high-weight particles. Additionally, a firefly-inspired attraction mechanism is applied to fine-tune neighboring particles, thereby improving convergence and robustness under complex noise conditions. Simulation results demonstrate that the adaptive threshold strategy increases localization accuracy by an additional 10% over fixed-threshold methods. The proposed SCA-PF enhances cooperative navigation accuracy by approximately 15% compared to other resampling-based algorithms.

A novel finite-difference time-domain (FDTD) method based on Newton’s second law of motion and bilinear transform (BT) technique is presented for modeling electromagnetic (EM) wave propagation in time-varying magnetized plasma. The procedure precisely expresses the time-varying electron density through its temporal derivative, enabling a more accurate capture of the rich dynamic characteristics of time-varying magnetized plasma. The numerical stability of the scheme is further examined through a von Neumann analysis. Numerical simulations indicate that the proposed BT approach has higher accuracy and computational efficiency for plasmas with strong electron-density variations compared with both the Newton-ADE-FDTD and the traditional ADE-FDTD methods. In addition, the proposed Newton-BT-FDTD scheme provides the improved accuracy relative to the conventional BT-FDTD method, especially in the case of time-varying magnetized plasma with electron-density drastic changes. Finally, the propagation characteristics of EM waves in a time-varying magnetized plasma are analyzed.

Traditional two‐dimensional (2D) modelling of Loran‐C additional secondary factor (ASF) fails to capture transverse terrain effects. In this paper, a parameter‐optimised three‐dimensional finite‐difference time‐domain (PO‐3D‐FDTD) method accelerated by multiple graphics processing units (multi‐GPU) is applied to solve Loran‐C ASF over irregular terrain. By introducing anisotropic parameters into the 3D‐FDTD formulas, the numerical dispersion error is significantly reduced. A multi‐GPU implementation enables accelerated computation and large‐scale ASF prediction. For the tested large‐scale model, the four‐GPU PO‐3D‐FDTD simulation required 0.79 h, whereas the conventional CPU‐based 3D‐FDTD was estimated to require several thousand hours. The transverse terrain effects on Loran‐C ASF are investigated through Gaussian‐shaped mountain paths. Pronounced differences between the 2D‐ and 3D‐FDTD predictions are observed in the shadow and flank regions, particularly for higher mountains. These differences are associated with lateral propagation and transverse field redistribution that are not represented by the 2D formulation, reaching several hundred nanoseconds in the tested single‐mountain cases. Furthermore, more complex ASF interference patterns are observed between two separated mountains, with the maximum 2D‐3D difference reaching approximately 1.55 μs.

The flexibility of the integral equation (IE) method is limited by the available numerical solutions with uniform‐grid schemes. In this paper, a general numerical solution of the IE is presented to predict long‐range low‐frequency (LF) groundwave propagation over irregular terrain. The general numerical solution is re‐derived with the aid of Legendre–Gauss quadrature and interpolation quadrature. The great advantage of the resulting algorithm is that it can freely use non‐uniform grid sizes in the computation area without loss of accuracy. Numerical results in two surface types, including triangular mountain and actual terrain are used to demonstrate the performance of the presented algorithm. Furthermore, it is ascertained that the non‐physical perturbations at long propagation ranges in the IE predictions are caused by the inherent uniform‐grid schemes of the numerical solution rather than the several approximations in the derivation of the IE form.

The capability of the narrow-angle and wide-angle parabolic equations (PEs) in the physical coordinate system are well known. For use on the irregular surface, a shift-map (SM) PE (SMPE) form was formulated in terms of the transformed coordinates. In sharp contrast, little is known of the accuracy of the narrow-angle and wide-angle SMPEs in the transformed coordinates. It was thought that the wide-angle SMPE is entirely superior to the narrow-angle SMPE according to the behavior in the narrow-angle and wide-angle propagators. In this paper, we analyze the accuracy of the wide-angle and narrow-angle SMPEs in the transformed coordinates through studying their artificial dispersion properties. It is found that the artificial dispersion of the narrow-angle SMPE is independent of the terrain parameters, whereas for the wide-angle SMPE, the dependence of the dispersion characteristics on the terrain slope is very strong. Also, as the terrain slope increases, its wide-angle capability deteriorates dramatically. Particularly, when the shady terrain slope is larger than about 5° , the accuracy of the wide-angle SMPE is even worse than the narrow-angle SMPE. To substantiate the accuracy analysis, numerical experiments involving wave propagation over irregular terrain are also presented.

The parabolic equation (PE) method and finite‐difference time‐domain (FDTD) method are combined to predict long‐range Loran‐C additional secondary factors (ASFs) in situations with near‐source complex topography. The non‐uniform FDTD method is employed in the complex source region, while the PE method is used for propagation over smooth terrain to a long distance. By using the advantages of each algorithm, the hybrid FDTD–PE approach is capable of modelling the near‐source topographic details, while saving much computational resources as compared to a full FDTD solution. Numerical results along several paths are used to validate the hybrid algorithm. Furthermore, the hybrid method is applied to analyse the effect of near‐source topographic complexities, including non‐flat land, land–sea transition, and island (cliff)–sea transition, on the navigational coverage for the planning of the Loran‐C systems.