An Algorithm for Improving the Condition Number of Matrices and Its Application for Solving the Inverse Problems of Gravimetry and Magnetometry

The paper considers one of the possible statements of inverse problems in gravimetric and magnetometric remote sensing, proposes a new approach to its solution and formulates algorithms that implement this approach. The problem under consideration consists of finding hypothetical sources of the corresponding potential fields at a given depth based on these fields measured on the Earth’s surface. The problem is reduced to solving systems of linear algebraic equations (SLAE) with ill-conditioned matrices. The proposed approach to the numerical solution is based on improving the condition number of the SLAE’s matrix. A numerical algorithm implementing the proposed approach that is applicable to the stable solution of degenerate and ill-conditioned SLAEs with an approximately given right-hand side is formulated in general form. The algorithm uses the SVD decomposition of the SLAE’s matrix and constructs a new matrix close to the original one with a better (smaller) condition number. An approximate solution to the original SLAE is calculated using the pseudoinverse of the new matrix. The results of a theoretical study of the algorithm are presented and the main properties of the new matrix are given. In particular, the reduction of its condition number is estimated. Several implementations of this algorithm are considered, in particular, the MPMI method, which is based on the use of so-called minimal pseudoinverse matrices. For the model problem, the advantage of the MPMI method over a number of other common methods is shown. The MPMI method is applied to solve the considered problems of gravity exploration and magnetic exploration both in the separate solution of these inverse problems and in their joint solution when processing geophysical data for the Kathu region, in the Northern Cape area of South Africa.