2-D cross-hole electromagnetic inversion algorithms based on regularization algorithms

The cross-hole electromagnetic (EM) method, which is currently at the forefront of electric logging technology, fundamentally solves the problems of the lateral imaging ability of single well logging and the lack of detection of inter-well physical properties. However, due to the complexity of underground reservoir distribution and the non-uniqueness problem of geophysical inversion, there remains a lack of practical and effective cross-hole electromagnetic inversion methods. Our goal is to develop an efficient method to reduce the non-uniqueness of the physical property model recovered in the inversion. It is worth noting that the regularization algorithm (RA), as a means to approximately solve inversion problems, can obtain different solutions by changing the form of the regularization function, so as to ensure the stability of inversion results and conform to the smooth or non-smooth characteristics in known geology or geophysics. We adjust the features of the final inversion model in a defined framework by changing the values of the $\alpha $coefficient in the regularization and using the Lawson norm as a ${l}_p$-norm approximation form for $p \in [ {0,2} ]$. At the same time, the iteratively reweighted least squares method is used to solve the optimization problem, and the gradient in the Gauss-Newton solution is adjusted successively to ensure that every term in the regularization contributes to the final solution. Compared with the traditional ${l}_2$-norm inversion method, the sparse inversion method can make more effective use of information regarding known physical properties and obtain better inversion results. Then, the effectiveness of our inversion method is verified by model tests and inversion of measured data in a mining area.