Computational aspects of gravity anomalies modeling by a point masses system

Abstract

Introduction: Approximation of the Bouguer gravity anomalies by a system of equivalent sources in regional geophysical surveys was considered. The field values are set at the nodes of a regular network of points on the surface of a spherical Earth - the Kavraisky sphere. Each node corresponds to 4 parameters: latitude, longitude, geodetic height and the first radial derivative of the gravitational potential. Material points are located under all nodes of the network. The mass values are determined by the approximate solution of a system of linear algebraic equations. Materials and Methods: The initial materials were two global models of the gravity field in the Bouguer reduction WGM2012 and the earth's relief ETOPO1 in the system of geodetic parameters WGS84 for the Kuril island arc and adjacent water areas. The resolution of the models was 1° and 20 . The study area was limited by coordinates 40°–54° N, 142°–162° E; its area is about 2.4 million km2. Estimations of conditionality of matrices of coefficients of systems of equations at different depths of placement of equivalent sources were carried out. For the condition numbers, the secondorder Schultz method and a new method developed by the authors which does not require explicit calculation of the inverse matrix were used. The latter is based on Hager's approach for estimations of the norms of the inverse matrix based on the available coefficients of the system. It is designed for the operation with high-dimensional data. The proximity of the condition numbers calculated by two different methods for matrices of size 314×314 and 2623×2623, respectively, was demonstrated. The comparison of the calculating speed was carried out at different depths of placement of material points. Also an experimental assessment of the effect of rounding errors and noise in the original data on the system solution vector was performed. Results and Discussion: Quantitative estimates of the norms of coefficient matrices and condition numbers of various approximation structures are presented. There was a sharp increase in the condition numbers when the sources were immersed to a depth exceeding the grid step of setting the field in latitude. This was accompanied by a decrease in the speed of the iterative Seidel method when solving systems of equations. The high stability of the numerical solution of systems of equations, probably due to selfregularization, was revealed. Therefore, for the modelling of regional gravity anomalies, additional regularization methods may not be used. Under the conditions of low and middle latitudes when forming an approximation structure it is recommended to observe an approximate equality between the network step along the meridian and the depths of point masses. Conclusions: The specific features of solving systems of equations arising from the source wise approximation of gravity anomalies on the Kavraisky sphere were studied. The results obtained will improve the accuracy and the calculating speed of the transformants of the anomalous gravitational field of large areas.