Automatic Estimation of Regularization Parameter by Unbiased Predictive Risk Estimator (UPRE) Method in 3-D Constrained Inversion of Magnetic Data

Summary Inversion of magnetic data is one of the important steps in the interpretation of practical magnetic data. The inversion result can be obtained by minimization of Tikhonov objective function. The determination of an optimal regularization parameter is highly important in magnetic data inversion. In this paper, an attempt has been made to use unbiased predictive risk estimator (UPRE) method in selecting the best regularization parameter for 3D constrained inversion of magnetic data using gradient projection reduced Newton (GPRN) algorithm. To achieve this goal, an algorithm has been developed to estimate this parameter. The validity of the proposed algorithm has been evaluated by magnetic data acquired from a synthetic model. The results have been compared with the results of generalized cross validation (GCV) method. The GCV method failed to estimate the regularization parameter, but the UPRE method could find the best regularization parameter. Then, the algorithm was used for inversion of real magnetic data obtained from Allah Abad iron deposit. The results of three-dimensional (3-D) inversion of magnetic data from this iron deposit show that the GPRN algorithm can provide an adequate estimate of magnetic susceptibility and geometry of subsurface structures of mineral deposits. A comparison of the inversion results with drilling data clearly indicate that the proposed algorithm can be used for 3-D inversion of magnetic data to estimate precisely the magnetic susceptibility and geometry of magnetized ore bodies.
Introduction Inversion of magnetic data is one of the most important steps in the interpretation of practical magnetic data. The goal of 3-D inversion is to estimate magnetic susceptibility distribution of an unknown subsurface model from a set of known magnetic observations measured on the surface. Inversion of magnetic data is an underdetermined and ill-posed problem. In addition, the non-uniqueness of the solution is the main issue of the inversion. One way to achieve a suitable model result in the inversion is to carry out the inversion with smoothness and smallness constraint. The solution can then be obtained by minimization of an objective function that consists of a misfit function and one of Tikhonov regularization functions. Regularization parameter makes a trade-off between misfit and regularization function. The determination of an optimal regularization parameter is highly important in magnetic data inversion. There are different methods for automatic estimation of regularization parameter in 3-D inversion. The GCV method is one of the most popular methods for choosing optimal regularization parameter in inversion of magnetic data. This method sometimes fails to find optimal regularization parameter. Therefore, it is suggested to use other methods. In this paper, we have applied the UPRE method to choose the best regularization parameter for 3-D constrained inversion of magnetic data using the GPRN algorithm.
Methodology and Approaches The UPRE method has been adapted for the solution of inverse problems. The UPRE method is based on a statistical estimator of the mean squared norm of predictive value. In this method, the optimal regularization parameter minimizes the UPRE function. We have developed an algorithm for 3-D inversion of magnetic data that uses the UPRE method for choosing optimal regularization parameter, and then, the inverse problem is solved by the GPRN algorithm under flatness and positivity constraints. To evaluate the reliability of the introduced method, the magnetic data of a synthetic model contaminated by 3 percent random noise have been inverted using the developed method. The GCV method is also applied for comparison of its results with the UPRE results. The obtained results indicate that the GCV method fails to choose regularization parameter but the UPRE method finds a unique optimal regularization parameter. Finally, The introduced algorithm has been used for 3-D inversion of magnetic data from Allah Abad iron deposit. The results are consistent with borehole information.
Results and Conclusions In this paper, the UPRE method has been developed for choosing optimal regularization parameter in 3-D constrained inversion of magnetic data using the GPRN algorithm. Data from synthetic model have been inverted using the introduced algorithm and acceptable results have been obtained. Geometrical parameters of synthetic model have been obtained from the constrained inversion process with acceptable accuracy. After validation of the algorithm performance on synthetic model, it has been applied for 3-D inversion of magnetic data from Allah Abad iron deposit. The results of drilling boreholes in the area confirm the results of the 3-D inversion.