Three-dimensional Focusing Inversion of Magnetic Gradient Tensor Data Based on the χ² Principle

The method of magnetic gradient tensor three-dimensional pose inversion is researched. Three-dimensional focusing inversion of magnetic gradient tensor data based on the χ² principle was proposed for regularization parameter adaptively selected issues of small target ferromagnetic materials in the inversion process. Depth weighting matrix and minmal support matrix were added in the inversion model of Tikhonov regularization theoretical framework to obtain the objective function to avoid the multiple solutions of inversion problem caused by the number of inverse parameter far than observational points and compensate kernel function avoid diminshing rapidly with depth. This function was calculated the optimal physical parameters by singular value decomposition. and according the χ² principle to adaptively determine the regularization parameter in the iterative process, thereby this principle increased the iterative speed and the solver accuracy. Three-dimensional focusing inversion of magnetic gradient tensor data produces models with non-smooth properties, for which typical implementations in this field use the iterative minmum support stabilizer and both regularizer and regularizing parameter are updated each iteration. The χ² principle generalizes the Morozov discrepancy principle to the augmented residual of the Tikhonov regularized least squares problem. For weighting of the data fidelity by a known Gaussian noise distribution on the measured data and, when the stabilizing, or regularization, term is considered to be weighted by unknown inverse covariance information on the model parameters, the minmum of the Tikhonov functional becomes a random variable that follows a χ²-distribution. Simulation and experimental results show that:this inversion method can reflect the outline shade of the magnetic anomaly and has good model resolution, which provide the theoretical foundation and practical experience for the application of the inversion method of magnetic gradient tensor.