Современные методы аналитических решений обратных задач гравиметрии и магнитометрии

Abstract

Предложен конкретный алгоритм реализации аналитического метода интерпретации одиночных локальных аномалий высокой интенсивности, по идее профессора А.А. Юнькова, на множестве точек, принадлежащих изолиниям эквипотенциальных поверхностей гравитационного потенциала вокруг трехосного эллипсоида. Для этого тела решения прямой задачи, выражаемого элементарными функциями, не существует. На основе интеграла Пуассона получены формулы для пересчета измеренного магнитного поля и поля силы тяжести в карты потенциалов на вышележащих плоскостях.

Запропоновано конкретний алгоритм реалізації аналітичного методу інтерпретації одиноких локальних аномалій високої інтенсивності, за ідеєю професора А.А. Юнькова, на множині точок, що належать до ізоліній еквіпотенціальних поверхонь гравітаційного потенціалу навколо тривісного еліпсоїда. Для цього тіла розв’язку прямої задачі, що виражається елементарними функціями, не існує. За допомогою інтеграла Пуассона отримано формули для перерахунку вимірюваного магнітного поля і поля сили тяжіння в карти потенціалів на площинах у верхньому півпросторі.

The purpose of these article is to develop modern analytical methods to solve inverse problems of gravimetry and magnetometry and to implement them using linear optimization algorithms in the presence of large measurement errors and incorrect filling of the model by abnormal masses. Design/metodolody/approach. Practical application of analytical methods in solving inverse problems has encountered difficulties since the very beginning of the development of gravimetry and magnetometry. They are connected with the field measurement error and the absence of naturally occurring anomalous bodies of regular geometric shape, as well as the lack of constancy of the physical parameters in abnormal bodies. Moreover, the lack of computer equipment up to the 90s of the last century made it practically impossible to solve the inverse problem, even for bodies of a simple form. In actual measurements of a field aggravated at all points by errors of varying intensity, the obtained solutions were often incorrect. Since depth to the lower boundary of the anomalous body and its abnormal density (magnetization) are always interrelated based on almost exact inverse proportion, why choose the right solution was not possible. The problem is further complicated by the presence of a constant or linear anomaly background. For the same reasons, the grid method, to solve inverse problems was poorly developed, particularly for ore geophysics. The presence in the geological section of the bodies with very high or low density or magnetization leads to large errors in solving the inverse problem with the help of mesh models over the entire map of the measured field. Findings. To determine the actual structure based on solutions of inverse problems modern methods for an optimized grid, it is necessary to create reliable methods, in particular, analytical methods to interpret individual local anomalies with high intensity. We offer a concrete realization of the analytical method algorithm for a set of points belonging to the contour lines of the equipotential surfaces of the triaxial ellipsoid gravitational potential. The inverse problem is solved without the algorithm of the direct problem as it is not described by elementary functions. On the basis of the Poisson integral, we obtain the formulas for converting the measured magnetic field and gravitational field to the maps of potentials on the overlying levels. We can now calculate the gravitational potential at different height levels. Then, in any vertical plane it is necessary to construct a map of potentials contours; and after that to take , on each loop, coordinates of any set of points (no less than 10). Further, for each set of points, using an optimization criterion, three parameters can be calculated, which have a triaxial ellipsoid. If we know the depth of the upper boundary of the body we can calculate the half-axes length. Application problems of analytical methods for the solution of inverse problems have been studied. Their shortcomings and relevance to ore geophysics have been defined. Practical value/implications. Methods are used as an auxiliary tool to solve inverse problems for large models optimized the mesh methods. We regard as promising the method using the coordinates of points at different elevations of one isoline. These points are located on a closed contour of the equipotential surface of the gravitational potential, or in a closed contour of the equipotential surface of the similar function calculated for a magnetic field. Further investigation is recommended to study the characteristics of the analytical method to determine the best ways to interpret anomalies. This method provides a stable solving of the inverse problem and a good agreement of results on the size and physical properties of real geological bodies.