A critical problem in inversion of geophysical data is developing a stable inverse problem solution that can si multaneously resolve complicated geological structures. The traditional way to obtain a stable solution is based on maximum smoothness criteria. This approach, however, provides smoothed unfocused images of real geoelectri cal structures. Recently, a new approach to reconstruc tion of images has been developed based on a total varia tional stabilizing functional. However, in geophysical ap plications it still produces distorted images. In this paper we develop a new technique to solve this problem which we call focusing inversion images. It is based on specially selected stabilizing functionals, called minimum gradi ent support (MGS) functionals, which minimize the area where strong model parameter variations and disconti nuity occur. We demonstrate that the MGS functional, in combination with the penalization function, helps to generate clearer and more focused images for geologi cal structures than conventional maximum smoothness or total variation functionals. The method has been suc cessfully tested on synthetic models and applied to real gravity data. TIKHONOV REGULARIZATION AND STABILIZING FUNCTIONALS Consider the inverse problem d= Am, (1) where A is the forward modeling operator; m = m(r), a scalar function describing geological model parameter distribution in some volume V in the earth (m EM, where M is a Hilbert space of models with L z norm); and d = d(r), a geophysical data set (d E D, where D is a Hilbert space of data).

We develop a method of 3-D magnetic anomaly inversion based on traditional Tikhonov regularization theory. We use a minimum support stabilizing functional to generate a sharp, focused inverse image. An iterative inversion process is constructed in the space of weighted model parameters that accelerates the convergence and robustness of the method. The weighting functions are selected based on sensitivity analysis. To speed up the computations and to decrease the size of memory required, we use a compression technique based on cubic interpolation.Our method is designed for inversion of total magnetic anomalies, assuming the anomalous field is caused by induced magnetization only. The method is applied to synthetic data for typical models of magnetic anomalies and is tested on real airborne data provided by Exxon-Mobil Upstream Research Company.

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[1] The integral equation method has been proven to be an efficient tool to model threedimensional electromagnetic problems. Owing to the full linear system to be solved, the method has been considered effective only in the case of models consisting of a strongly limited number of cells. However, recent advances in matrix storage and multiplication issues facilitate the modeling of horizontally large structures. Iterative methods are the most feasible techniques for obtaining accurate solutions for such problems. In this paper we demonstrate that the convergence of iterative methods can be improved significantly, if the original integral equation is replaced by an equation based on the modified Green's operator with the norm less or equal to one. That is why we call this technique the Contraction Integral Equation (CIE) method. We demonstrate that application of the modified Green's operator can be treated as a preconditioning of the original problem. We have performed a comparative study of the convergence of different iterative solvers applied to the original and contraction integral equations. The results show that the most effective solvers are the BIGGSTAB, QMRCGSTAB, and CGMRES algorithms, equipped with preconditioning based on the CIE method.

The Born approximation in electromagnetic (EM) numerical modeling has limited application for solving 3-D electromagnetic induction problems, because in structures with high conductivity contrasts and at high frequencies, this approximation is inaccurate. In this paper, we develop a new and relatively simple approxi mation for the EM field called a quasi-linearapproxima tion, which is based on the evaluation of the anomalous field E a by a linear transformation of the normal (primary) field: E a = ~En, where ~ is called the electrical reflectivity tensor. The reflectivity tensor inside inhomogeneities can be approximated by a slowly vary ing function that can be determined numerically by a simple optimization technique. The new approximation gives an accurate estimate of the EM response for conductivity contrasts of more than one hundred to one, and for a wide range of frequencies. It also opens the possibility for fast 3-D electromagnetic inversion. * reflectivity tensor ~ rn determination for every subdomain D rn Oip(A, OA) =-2 Re OAI3't L.J ~E ClI3'Yd rj) j=l of the structure D with anomalous conductivity.

A rigorous physical-mathematical model of heterogeneous conductive media is based on the effective-medium approach. A generalization of the classical effective-medium theory (EMT) consists of two major parts: (1) introduction of effective-conductivity models of heterogeneous, multiphase rock formations with inclusions of arbitrary shape and conductivity using the principles of the quasi-linear (QL) approximation within the framework of the EMT formalism and (2) development of the generalized effective-medium theory of induced polarization (GEMTIP), which takes into account electromagnetic-induction (EMI) and induced polarization (IP) effects related to the relaxation of polarized charges in rock formations. The new generalized EMT provides a unified mathematical model of heterogeneity, multiphase structure, and the polarizability of rocks. The geoelectric parameters of this model are determined by the intrinsic petrophysical and geometric characteristics of composite media: the mineralization and/or fluid content of rocks and the matrix composition, porosity, anisotropy, and polarizability of formations. The GEMTIP model allows one to find the effective conductivity of a medium with inclusions that have arbitrary shape and electrical properties. One fundamental IP model of an isotropic, multiphase, heterogeneous medium is filled with spherical inclusions. This model, because of its relative simplicity, makes it possible to explain the close relationships between the new GEMTIP conductivity-relaxation model and an empirical Cole-Cole model or classical Wait’s model of the IP effect.

Abstract. It is often argued that 3D inversion of entire airborne electromagnetic (AEM) surveys is impractical, and that 1D methods provide the only viable option for quantitative interpretation. However, real geological formations are 3D by nature and 3D inversion is required to produce accurate images of the subsurface. To that end, we show that it is practical to invert entire AEM surveys to 3D conductivity models with hundreds of thousands if not millions of elements. The key to solving a 3D AEM inversion problem is the application of a moving footprint approach. We have exploited the fact that the area of the footprint of an AEM system is significantly smaller than the area of an AEM survey, and developed a robust 3D inversion method that uses a moving footprint. Our implementation is based on the 3D integral equation method for computing data and sensitivities, and uses the re-weighted regularised conjugate gradient method for minimising the objective functional. We demonstrate our methodology with the 3D inversion of AEM data acquired for salinity mapping over the Bookpurnong Irrigation District in South Australia. We have inverted 146 line km of RESOLVE data for a 3D conductivity model with 310 000 elements in 45 min using just five processors of a multi-processor workstation.

We develop a new method for interpretation of tensor gravity field component data, based on regularized focusing inversion. The focusing inversion makes its possible to reconstruct a sharper image of the geological target than conventional maximum smoothness inversion. This new technique can be efficiently applied for the interpretation of gravity gradiometer data, which are sensitive to local density anomalies. The numerical modeling and inversion results show that the resolution of the gravity method can be improved significantly if we use tensor gravity data for interpretation. We also apply our method for inversion of the gradient gravity data collected by BHP Billiton over the Cannington Ag‐Pb‐Zn orebody in Queensland, Australia. The comparison with the drilling results demonstrates a remarkable correlation between the density anomaly reconstructed by the gravity gradient data and the true structure of the orebody. This result indicates that the emerging new geophysical technology of the airborne gravity gradient observations can improve significantly the practical effectiveness of the gravity method in mineral exploration.