### Sammendrag

Marine controlled-source electromagnetic (CSEM) method is a non-invasive offshore technique used, in association with magnetotelluric and seismic data, for the study of the oceanographic lithosphere and hydrocarbon reservoir exploration. CSEM data are often used in optimization processes that produce an electrical resistivity imaging of the subsurface. CSEM research shows interest for developing high-order optimization methods, able to achieve faster convergences without investing too much manual effort building initial inversion models. As a result, 3D CSEM industry has started a transition from quasi-Newton to Gauss-Newton methods.
The large numerical complexity is a limiting factor when applying the Gauss- Newton method for the 3D inversion of CSEM data. These problems can involve O(106) inversion parameters and O(105) forward simulations, resulting in a Jacobian matrix of O(100 TB) and a Gauss-Newton Hessian matrix of O(1 TB). There are some papers that propose methods to reduce the memory complexity and others that present schemes to reduce the time complexity. However, there is not a proposal to significantly reduce the total numerical complexity of the 3D Gauss-Newton optimization method without affecting the parameterization of the problem.
The first main contribution of this thesis is a method for obtaining a low-rank approximation of the Gauss-Newton Hessian matrix that dramatically reduces the numerical complexity of the 3D CSEM Gauss-Newton optimization without altering the parameterization of the resistivity models. For large-scale surveys, it can reduce the number of forward simulations between 10-100 times, and it also reduces the memory complexity, from O(TB) to O(GB). It is based on simulating groups of distant phase-encoded sources, instead of single-source simulations. The resultant small number of simulations motivated the development of a matrix free recursive direct solver to obtain the model updates at each iteration with a reduced memory usage. A study of the associated cross-talk noise and inversion results validates this proposal.
The second main contribution of this thesis is the introduction, apparently for the first time in 3D CSEM, of the Newton and the Halley class methods. This opens the state-of-the-art frontiers to higher-order methods where the computation of a Green function per model parameter is required. Initially, the numerical complexity of these methods makes their use unapproachable. In this research it is concluded that it is possible to apply these methods with the same memory complexity as in a Gauss-Newton method, and with a contained time complexity. It is proposed the use of a finite-difference frequency-domain direct solver for on-the-fly computations of the Green functions, a reduced memory construction of the systems matrices and the modification of a trust-region solver to handle the indefiniteness of these matrices. Synthetic 3D CSEM survey inversion results demonstrate the feasibility of this method.

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