Dmitry V. Churbanov

The paper proposes a neural network approach to the joint three-dimensional inversion of gravitational and magnetic data aimed at the recovery of a single ore body. The method is based on a convolutional U-Net architecture trained on a large synthetic dataset generated by stochastic combinations of rectangular prisms simulating typical geological structures. Various loss functions are considered and compared: mean squared error (MSE), mean absolute error (MAE), Huber loss, and a loss with a forward-problem physics residual. It is shown that, in the conducted synthetic experiments, this loss provides the lowest error among the considered variants and improves the physical consistency of the recovered density and magnetization models, while data augmentation by rotations improves the reconstruction quality. The results of neural network recovery are also compared with the classical solution of the inverse problem within the framework of discrete potential theory. In the considered test example, the neural network approach achieved an error of 4.23% for the loss with a physics residual with a processing time of 0.2 s on the fixed 21³ grid. В работе предложен нейросетевой подход к совместной трехмерной инверсии гравитационных и магнитных данных, ориентированный на восстановление единого рудного тела. Метод основан на сверточной архитектуре U-Net, обучаемой на большом синтетическом датасете, сгенерированном с помощью стохастической комбинации прямоугольных призм, имитирующих типичные геологические структуры. Рассмотрены и сравнены различные функции потерь: среднеквадратичная ошибка (MSE), средняя абсолютная ошибка (MAE), функция Хьюбера и функция потерь с физической невязкой прямой задачи. Показано, что в проведенных синтетических экспериментах применение этой функции потерь обеспечивает наименьшую ошибку среди рассмотренных вариантов и повышает физическую согласованность восстановленныхnмоделей плотности и намагниченности, а аугментация данных поворотами улучшает качество реконструкции. Также сравнены результаты восстановления с помощью нейронных сетей с классическим решением обратной задачи в рамках теории дискретных потенциалов. В рассмотренном тестовом примере нейросетевой подход позволил добиться ошибки 4.23% для функции потерь с физической невязкой при времени обработки 0.2 с на фиксированной сетке 21³.

We consider the optimization of a neural network previously developed by the authors for the joint inversion of 3D gravitational and magnetic fields in the context of mineral exploration. The distinctive feature of this neural network is that it solves ill-posed (ill-conditioned) inverse problems. The neural network implements a special two-level algorithm. The lower level of the algorithm uses two neural networks with equivalent architectures. The first of them computes the gravitational field sources in a given domain from measurements of this field on a remote surface. The second neural network processes magnetic field measured on the same surface to find magnetic sources in the same domain. The found source distributions are used at the upper level of the algorithm to calculate their structural residual, which determines the degree of difference (closeness) of their geometries. As a result, minimizing this residual, when training a neural network at the upper level, implements a computational algorithm that yields geometrically close source distributions of different fields. The article examines in detail the possibilities of optimizing some elements of the neural networks and the algorithms used (datasets, training process, specific form of loss functions, etc.) Test calculations for model problem demonstrate high quality of joint inversion by our optimized neural networks approach. Calculations were also carried out for the joint processing of real-feald data from gravity and magnetic exploration in Jussara region, Goias State, Brazil. The article also considers the issue of determining in joint field inversion not only the geometric distribution of sources, but also their physical intensities.

The paper proposes a new two-level neural network approach to joint inversion of gravitational and magnetic fields in the problem of mineral exploration. The lower level uses two neural networks with equivalent architecture. The first network calculates the distribution of gravitational field sources in a given domain based on measurements of this field on a certain remote surface. The second one finds the distribution of magnetic field sources in the same area based on magnetic field measurements. At the upper level, a structural residual between the two obtained source distributions is calculated, which determines the “degree of difference” of their geometries. The resulting neural network is trained to minimize this structural discrepancy. As a result, this two-level system implements a computational algorithm that finds geometrically close sources of fields of different nature. Test simulations demonstrate the high quality of joint inversion of model gravitational and magnetic fields within the proposed aproach. В работе предлагается новый двухуровневый нейросетевой подход к совместной инверсии гравитационных и магнитных полей в адаче поиска полезных ископаемых. Нижний уровень использует две эквивалентные по архитектуре нейронные сети. Одна вычисляет распределение источников гравитационного поля в заданной области по измерениям этого поля на некоторой удаленной поверхности. Вторая находит в той же области распределение источников магнитного поля по измерениям магнитного поля. На верхнем уровне вычисляется структурная невязка двух полученных распределений источников, которая определяет “степень различия” их геометрий. Результирующая нейронная сеть обучается таким образом, чтобы структурная невязка была минимальной. В итоге эта двухуровневая система реализует вычислительный алгоритм, который находит геометрически близкие источники полей различной природы. Тестовые расчеты демонстрируют высокое качество совместной инверсии модельных гравитационных и магнитных полей в рамках предложенного подхода.

We present the project of a 2U CubeSat format spaceborne multichannel laser heterodyne spectroradiometer (MLHS) for studies of the Earth’s atmosphere upper layers in the near-infrared (NIR) spectral range (1258, 1528, and 1640 nm). A spaceborne MLHS operating in the solar occultation mode onboard CubeSat platform, is capable of simultaneous vertical profiling of CO₂, H₂O, CH₄, and O₂, as well as Doppler wind measurements, in the tangent heights range of 5–50 km. We considered the low Earth orbit for the MLHS deployment and analyzed the expected surface coverage and spatial resolution during one year of operations. A ground-based prototype of the MLHS for CO₂ and CH₄ molecular absorption measurements with an ultra-high spectral resolution of 0.0013 cm⁻¹ is presented along with the detailed description of its analytical characteristics and capabilities. Implementation of a multichannel configuration of the heterodyne receiver (four receivers per one spectral channel) provides a significant improvement of the signal-to-noise ratio with the reasonable exposure time typical for observations in the solar occultation mode. Finally, the capability of building up a tomographic picture of sounded gas concentration distributions provided by high spectral resolution is discussed.

The problem of using an additional boundary condition to find a coefficient that depends on the spatial variable is considered. The existence and uniqueness of the solution to the direct problem is studied. The solution to the direct problem is proved to be stable with respect to the sought coefficient. Uniqueness conditions for the solution to the coefficient inverse problem are described.

For a partial differential equation simulating population dynamics, the inverse problem of determining its nonlinear right-hand side from an additional boundary condition is studied. This inverse problem is reduced to a functional equation, for which the existence and uniqueness of a solution is proven. An iterative method for solving this inverse problem is proposed. The accuracy of the method is estimated, and restrictions on the number of steps are obtained.

The inverse problem of reconstructing the right-hand sides of an elliptic equation in a circular domain is studied. The right-hand sides describe point sources. Under certain conditions, numerical experiments reveal there is a second solution. The results from experiments are illustrated by figures reflecting the juxtaposition of the two solutions.