This paper describes the application of various optimization methods in the problem of high-frequency point-to-point ray tracing in inhomogeneous media using the direct variational method. Modern implementations of the conjugate gradient method, as well as quasi-Newton methods, are considered. The analysis made it possible to determine the most optimal parameters of convergence to the desired ray path.

An implementation of the generalized force method, which is best suited for ray tracing in an anisotropic ionosphere with given positions of the transmitter and receiver, is presented. The ray search is based on the optimization of the extended functional of the phase path, taking into account both the spatial position of the ray and the direction of the wave front. Note that, regardless of the chosen mode configuration, the optimization problem, as in the isotropic case, comes down to determining the set of extrema of the objective function: as a rule, minima for the high rays and saddle points for the low rays. The generalized force method, implemented as a software model, has been tested and applied to solve oblique sounding problems in a realistic ionosphere using the NeQuick2 and IGRF13 models.

A direct variational method for solving the problem of finding ocean wave rays reflected from the coastline with given positions of the source and the point of observation is considered. It is shown that the original boundary value problem can be reduced to the direct search of stationary points of the functional equal to the time of wave propagation along the ray. Information about the objective function in the area of solutions to the ray tracing problem allows us to construct a systematic procedure for finding minima, saddle points, and maxima. A feature of the proposed approach is the optimization of the ray reflection point along a given coastline.

We apply two promising approaches, namely, the shooting method based on global optimization and the generalized force method, to the problem of point-to-point ionospheric ray tracing where the ionospheric modes tend to the minimum frequency. In most cases, both methods are capable of finding all rays connecting the given transmitter and receiver. However, the generalized force approach is more stable in locating highly divergent high rays. Application of the methods to a long-range single-hop radio channel demonstrates their potential for expanding the simulated frequency range, refining the model ionograms in the ranges of the minimum frequency of reflection from ionospheric layers, and significantly improving the overall forecast of the lowest usable frequency. The results of numerical simulations were also compared with the experimental data. Joint application of the presented methods provides valuable means for solving applied problems of radio communication forecast.

При исследовании туннельных асимптотик для нижних уровней оператора Шредингера (таких как энергетическое расщепление в симметричной двойной яме или ширина спектральной зоны в периодической задаче) естественным образом возникают либрации, т. е. периодические решения классической системы с перевернутым потенциалом, которые дважды на периоде достигают границы области возможных движений. В пределе они дают двояко асимптотические решения с двумя симметричными неустойчивыми положениями равновесия (инстантоны). Туннельные асимптотики можно записать двумя способами: либо в терминах действия на инстантоне и линеаризованной динамики в его окрестности, либо в терминах некоторой либрации, называемой туннельной. Второй способ более конструктивен, поскольку его использование для численных расчетов сводится к двум операциям: нахождение либрации с данной энергией и вычисление коэффициентов Флоке для данной либрации. Для применения этого подхода на практике предлагается находить либрации с данной энергией, используя вариационный численный метод, обобщающий идеи метода упругой нити. В качестве приложения найдена асимптотика для ширин нижних спектральных зон и лакун, выраженная через туннельную либрацию в четырехмерной системе, описывающей димер в тригонально-симметричном поле, которая была предложена М. И. Кацнельсоном.

The paper presents two approaches for the traditional two-point ray-tracing problem in anisotropic inhomogeneous ionosphere: the homing-in method based on global optimization and the direct approach based on Fermat's principle. While in most cases both methods are capable of finding all rays connecting the given transmitter and receiver, the direct approach tends to be more stable in locating highly divergent high rays, but the homing-in method usually offers a better tradeoff between the accuracy and computational cost of the low ray calculations. Joint application of the presented methods provides valuable means for solving applied problems of radio communication forecast and ionospheric models validation and adaptation.

A comparative analysis of two methods for two-point ionospheric ray tracing problem is presented. For the chosen model conditions, the accuracy and computing time parameters are compared using the shooting method and the variational approach.

Представлен сравнительный анализ возможностей двух методов решения краевой задачи о расчете лучевых траекторий коротких радиоволн. Для выбранных модельных условий проведено сравнение параметров точности и времени расчетов методом пристрелки и вариационным методом.

Since 1964, the Kaliningrad branch of Pushkov Institute of Terrestrial Magnetism, Ionosphere and Radio Wave Propagation, Russian Academy of Sciences (IZMIRAN) has been conducting regular observations of the parameters of the lower atmosphere, ionosphere, and variations in the Earth’s magnetic field. The base of the measuring equipment, located in Ulyanovka village, Kaliningrad region (54° N, 20° E) consists of ionosonde for the vertical sounding of the ionosphere; two-frequency GPS/GLONASS receivers, which are used to determine the total electron content in the ionosphere; magnetic variation stations for the continuous recording of variations of the three components of the Earth’s magnetic field vector and calculating the local K-index; and a weather station. The simultaneous observations of variations in the magnetic field, as well as the ionospheric and meteorological parameters in the same observatory, offer a wide range of opportunities to study the atmospheric-ionospheric relations. This article provides a brief description of all the tools, provides a method for transmitting and storing the time series of measured parameters, and also presents examples of using these data for various geophysical studies.

Global optimization of phase distance functional is a promising approach to the multipath problem of the point-to-point ionospheric ray tracing. The approach is involves systematic algorithm where all relevant rays between the fixed points are found one after another in a systematic manner, without the need to provide an accurate initial estimation for each solution. In particular for each 10-cal ray searching the direct variational method is applied. The global optimization is applied to a point-to-point ionospheric ray tracing, where the modeled ionosphere is three-dimensional inhomogeneous medium.

В работе представлены результаты моделирования загоризонтной радиолокации при заданной диаграмме направленности (ДН) плоской вертикальной антенной решётки. При моделировании лучевых траекторий и поглощения коротких волн в ионосфере использовалась численная модель распространения радиоволн, построенная в приближении геометрической оптики. Для описания среды распространения использовались эмпирическая справочная модель ионосферы IRI-2012 и модель нейтральной атмосферы MSIS-86.

A variational approach for solving the boundary value problem of computing ray trajectories and fronts of ocean waves is presented. The solution method is based on Fermat’s principle (of stationary time). A distinctive feature of the proposed approach is that the Fermat functional is optimized directly without solving the Euler–Lagrange equation; moreover, the locations of the wave source and receiver are fixed. Multipath propagation in the boundary value problem is addressed by finding various types of stationary points of the Fermat functional. The technique is numerically tested by applying the method of bicharacteristics with the use of analytical seabed models. The advantages of the variational approach and the prospects of its further development as applied to ocean wave computation are described. The relations between various types of stationary points of the travel time functional, caustics, and foci are discussed.

A variant of the direct optimization method for point-to-point ionospheric ray tracing is presented. The method is well suited for applications where the launch direction of the radio wave ray is unknown, but the position of the receiver is specified instead. Iterative transformation of a candidate path to the sought-for ray is guided by a generalized force, where the definition of the force depends on the ray type. For high rays, the negative gradient of the optical path functional is used. For low rays, the transformation of the gradient is applied, converting the neighborhood of a saddle point to that of a local minimum. Knowledge about the character of the rays is used to establish a scheme for systematic identification of all relevant rays between the given points, without the need to provide an accurate initial estimate for each solution. Various applications of the method to isotropic ionosphere demonstrate its ability to resolve complex ray configurations including 3-D propagation and multi-path propagation where rays are close in the launch direction. Results of the application of the method to ray tracing between Khabarovsk and Tory show good quantitative agreement with the measured oblique ionograms.

We developed two new global empirical models of the F2 layer critical frequency that discribe the main features of high latitude ionosphere in details: (1) Main Ionospheric Trough (MIT) model based on satellite data of topside ionosphere sounding and in-situ measurements; (2) CHanged of the foF2 calculation in IRI model (Kaliningrad version) - CHIRIK model based on radio occultation measurements. Using radio occultation and ground-based ionosonde database with an archive of F10.7 values we obtained the F10.7-optimized index for better global reproduction of the foF2 dependence on solar activity for all temporal intervals. Another applied product of our team is the ray tracing technique in 3D weakly inhomogeneous ionosphere.

Direct optimization of the optical path functional is a promising approach to the point-to-point ionospheric ray tracing problem. The approach involves a systematic transformation of the ray trajectory to an optimal configuration satisfying the Fermat's principle, while the endpoints are kept fixed according to the boundary conditions. Here, a strategy is proposed for the identification of both high and low rays using a direct variational approach. High rays are obtained by minimizing the optical path of ionospheric radio rays. Low rays which correspond to saddle points of the optical path are found using the minimum mode following method, where the saddle points are essentially converted to local minima. The method is applied to a point-to-point ionospheric ray tracing, where the propagation medium is obtained with the International Reference Ionosphere model.

Point-to-point ray tracing is an important problem in many fields of science. In direct variational methods, some trajectory is transformed to an optimal trajectory. While these methods are routinely used in calculations of pathways of seismic waves, chemical reactions, diffusion processes, etc., these approaches are not widely known in ionospheric point-to-point ray tracing. A two-dimensional rep...