Study on 3-D resistivity inversion using conjugate gradient method

We present a 3-D minimum structure inversion algorithm for d.c. resistivity data, using conjugate gradient (CG) relaxation techniques. Firstly, the minimum structure inverse equations are solved. In doing so, the CG relaxation technique only requires the results of derivative matrix G and its transpose G(T) multipled by a vectors, i.e. Gx and G(T)y, respectively. Then the Rodi method is introduced to compute, and the Jcobian matrix G, Gx and G(T)y can be got after one forward calculation in each inverse iteration. Therefore, only one forward calculation is required in each inversion iteration, the inversion is computed much quickly. Additionally, we are able to bypass the actual computation of the derivative matrix G and the inversion of G(T)G, also to avoid the huge storage for G and G(T)G. On the other hand, since there are so many inversion parameters, we impose a smoothness constraint on the model in minimum structure inversion to solve the smoothest solution which effectively erases unnecessary structures, obtains trustworthy results close to true model. We firstly merge these methods and techniques into 3-D resistivity inversion and obtain very promising results, which have been tested on 3-D synthetic data. A new iterative technique is also, developed to deal with the problem of very slow convergence rate in traditional minimum structure inversion, it shows that only about ten iterations or less are required to reach the convergence.