High-order optimization methods for large-scale 3D CSEM data inversion
Abstract
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 phaseencoded
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 finitedifference
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.
Has parts
Paper 1: Amaya Benitez, Manuel; Boman, Linus; Morten, Jan Petter. A Low-rank Approximation to the Hessian for 3D CSEM Gauss-Newton Inversion. EAGE Conferences & Exhibitions - The final published version is available at http://dx.doi.org/10.3997/2214-4609.20141099Paper 2: Amaya Benitez, Manuel; Morten, Jan Petter; Boman, Linus. Efficient computation of approximate low-rank Hessian for 3D CSEM inversion. SEG annual meeting; 2014 - The final published version is available at http://dx.doi.org/10.1190/segam2014-0493.1
Paper 3: Manuel Amaya, Jan Petter Morten, and Linus Boman., A low-rank approximation for large-scale 3D CSEM Gauss-Newton inversion
Paper 4: M. Amaya, K. R. Hansen and J.P. Morten., 3D CSEM data inversion using Newton and Halley class methods - This is an author-created, un-copyedited version of an article accepted for publication/published in [insert name of journal]. IOP Publishing Ltd is not responsible for any errors or omissions in this version of the manuscript or any version derived from it