Decomposition work of Loran-C resultant waves propagating in the Earth-ionosphere waveguide is concentrated on in this paper. Resultant waves of different Earth-ionosphere models are predicted by the FDTD method first. The inverse fast Fourier Transform (IFFT) spectral-division technique combined with a filtering process is employed to estimate the delays of different wave-propagation modes in the resultant waves, whereas the amplitudes are solved with the least-square method. Finally, accurate field strengths and delays of different modes (ground waves and skywaves of diverse hops) are obtained, which are important for Loran-C receivers to optimize their performances.

To expand the space applications of low-frequency sky-wave signals, Loran-C electromagnetic fields at low and high altitudes in the earth–ionosphere waveguide are studied in this letter. In order to consider the influence of the earth's surface on one-hop sky-wave fields at different altitudes, the wave-hop theory that used for field predictions on the ground is improved, and the wave-hop equation based on the frequency domain is further transformed into the time domain. Then, the proposed approach is applied to calculate the sky-wave fields at the altitudes 0–50 km under given earth–ionosphere models. The accuracy and efficiency of the method are verified by comparison to the finite-difference time-domain method.

One-hop sky-wave signals in the Earth–ionosphere waveguide are studied based on the traditional wave-hop theory and the finite-difference time-domain (FDTD) method. Field-strength variations of 100 kHz one-hop sky waves are given under three different waveguide models. The great-circle distance between the transmitter and the receiver is within 200 km. Together with a sky- and ground-wave separation technique in time domain, the narrowband Loran-C signals are employed in the FDTD method. Compared to the results of wave-hop theory, field-strength variations of low-frequency one-hop sky waves are better characterized by the FDTD method as the waves propagate in an irregular Earth–ionosphere waveguide.

The analysis of propagation characteristics of low-frequency(LF) radio waves in the Earth-ionosphere waveguide is of great significance for timing, navigation and many other fields. Because the ionosphere changes with different time and geographical location, there are random changes besides regular time-varying effects, which limits the application of traditional finite-difference time-domain (FDTD) model. In this letter, the polynomial chaos expansion FDTD (PCE-FDTD) method based on cylindrical coordinates is used to predict the statistical characteristics of LF radio-wave propagation in isotropic stochastic ionosphere environment with dielectric constant and conductivity as random variables. Compared with Monte Carlo FDTD (MC-FDTD) method and wave-hop theory, the accuracy and efficiency of PCE-FDTD method are verified. And the influence of different distributions and amplitude fluctuations of random electromagnetic parameters in the ionosphere and irregular terrain on the variance field of LF sky-wave signals is further analyzed. It provides some reference for ionospheric detection and research on propagation characteristics of LF radio waves.

The study of low-frequency (LF) radio-wave propagation phase in the Earth-ionosphere waveguide is of great significance for the accuracy enhancement of related systems such as ionospheric sounding, time service, and lightning location. In this article, the stochastic finite-difference time-domain (S-FDTD) algorithm based on the Taylor series expansion idea is used to propose a new method for measuring the uncertainty of LF radio-wave propagation phase in the random media. We study how to obtain the mean value and standard deviation of the propagation phase of the Loran-C one-hop sky waves when the ionosphere is considered to be the random medium. At the same time, the effects of different ionospheric parameters with random characteristics on the propagation phases of sky waves are analyzed. The results of the wave-hop theory are used for validation. The numerical experimental results show that the method can obtain the statistical characteristics of LF radio-wave propagation phase only by a single FDTD simulation. The method can be further extended to obtain the stochastic characteristics of LF radio-wave propagation phases in the complex Earth-random ionosphere waveguide.

Studying the impact of terrain undulation on low frequency (LF) sky-wave propagation delay is of great signifi-cance for exploring the potential applications in low-ionospheric detection, remote navigation, and timing service. This letter combines gradient ionospheric parameters and uses the finite-difference time-domain (FDTD) method to calculate the reflected signal of LF radio waves from the ionosphere. Using the sky- and ground-wave separation technology in time domain, the variation law of LF sky-wave propagation delay with terrain changes is given in both daytime and nighttime models. At the same time, this letter also analyze the influence of different ionospheres (in the presence or absence of natural magnetic field) on LF one-hop sky-wave propagation delay, providing some reference significance for studying the propagation characteristics of LF sky waves.