Aim. There are many methods for modeling a regional gravitational field in which the Legendre spherical functions of integer degree of the real order are used. They relate, however, mainly to the region which form represents a segment of the sphere. In addition, for their use, the input data must be transformed into a sphere segment with its center at the north pole. The aim of this work is to find a system of functions that would have orthogonal properties on an arbitrary spherical trapezium, as well as researching the properties of such a system. Method. Based on the Legendre spherical functions on the spherical segment, an orthogonal system of functions to an arbitrary spherical trapezoid was developed. Such functions can not be explicitly stated, nor do they have recurring relationships. Results. This article examines the associated Legandre spherical functions on the spherical trapezium where the functions are orthogonal and provide the formulas for defining the norms of these functions. The obtained functions can be used to build regional models of the gravitational fields on the arbitrary spherical trapezium. The orthogonality of the functions ensures a sustainable solution when determining the unknown model coefficients. To model the regional gravitational field with high accuracy, it is necessary to grid the input data (define the transformants of the geopotential), and then use the partial discrete orthogonality of these functions in longitudial direction or full discrete orthogonality similar to the second Neumann's method. This allows significant reduction of computing time without any loss of accuracy, as the functions under study do not have any recursive relations and it is required to use the decomposition into the hypergeometric series to define these functions. The scientific novelty and practical significance. In this paper we first obtained a system of functions that were orthogonally consistent to an arbitrary spherical trapezium. It can be used to construct a regional gravitational field, a regional magnetic field, and also for high-precision interpolation or approximation tasks, for example the construction of a regional ionosphere model.

Запропоновано метод для апроксимації параметра іоносфери VTEC. Для цього як базова використовується система сферичних функцій, ортогональних на сферичній трапеції. Вхідні дані, а саме значення параметрa іоносфери VTEC, отримано на ряді ГНСС-станцій. Виконано оцінку точності апроксимації при вибраних параметрах системи сферичних функцій на ряді незалежних даних.Ключові слова: іоносфера, сферичні функції, сферична трапеція, параметр VTEC. Космічна й атмосферна фізика

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Aim. The widespread use of global navigation satellite systems (GNSS) has led to the development of new methods designed to determine and accumulate the index of ionosphere ionization (VTEC). Using these data it is possible to significantly improve the accuracy and reliability of determining the coordinates of the observation point. Therefore, the task of constructing a model of ionization index is relevant. Method. To construct a spatial model, we used the spherical Legendre functions of the first kind of real order with integer degree as a basic system of functions. We found the magnitude of order using the Sturm-Liouville theory since it depended on the size of the investigated region. Such system of functions form two orthogonal systems of functions in the region under study (the sphere segment), but does not have recurrence relations between functions, therefore, it is necessary to use function expansion in a hypergeometric series to find them. Also in order to find unknown coefficients of the model it is necessary to use the Tikhonov regularization parameter, since the matrix of normal equations will not be stable. For calculating the time model of the ionosphere the coefficients of different spatial models were expanded in series of power polynomials. Results. Based on the data of the ionization parameter values obtained of 19 permanent stations of the ZAKPOS network using the Trimble Pivot Platform software, the spatial-temporal model of this parameter was constructed using the Legendre spherical functions up to the 3rd order as well as with power polynomials up to 3rd order. The standard deviation between the measured and model values of the VTEC parameter does not exceed 1TECU. The scientific novelty and practical significance. We developed algorithm for construction of the spacetime model of the ionosphere parameter. A ionosphere model of high resolution is obtained, which can be used to solve geodetic tasks in order to provide the necessary accuracy in determining the coordinates of the point, as well as to study and forecast the space weather.

In this paper we investigate the possibility of combining least squares collocation (LSC) and spherical functions (spherical harmonic analysis SHA) with real eigenvalues methods for calculating the geoid model. A comparison of local analytical covariance functions calculated on the basis of Legendre polynomials and polynomials with real eigenvalues is carried out. We computed geoid model for the region of Central Ukraine using the proposed method and assessed its accuracy. The input data for the computation of model are gravity anomalies, obtained after filtering the results of digitization of gravimetric maps by the 3σ criterion, functionals of the Earth's gravity field from the global gravity model EGM2008 and GNSS leveling data. The standard deviation was found between the model values of the geoid height and the values, obtained from GNSS leveling at the points of the state geodetic network. The accuracy of the geoid model obtained using the proposed method is approximately 2 cm. Such accuracy is primarily conditioned by the quality of initial data. To increase the accuracy of the model it is necessary to carry out a complex of gravimetric works in the studied region.