1D TDEM Inversion Using an L-curve Criterion
Eiichi Arai
Metal Mining Agency of Japan, JAPAN.
Contacts: earai@mmaj.go.jp (Eiichi Arai)
Abstract
A simple numerical method is described here to invert electromagnetic fields one-dimensionally in the TDEM survey induced by an infinite current line source which is laid over the horizontal layered earth, and in which the electric field is measured along the line perpendicular to a current line. The regularization technique is commonly applied to a linearized least-squares inversion to achieve a stable computation. A crucial aspect of this technique is the choice of the regularization parameter that controls the trade-off between a model objective function and a data misfit. To optimize a regularization parameter, I have applied an L-curve criterion developed by Hansen(1992) for linear inverse problems, and developed a practical approach to the nonlinear inversion of TDEM data. This is a heuristic method that utilizes relative change of a model objective function as a function of a data misfit with regularization parameter. When a model objective function as a function of a data misfit with regularization parameter is plotted on a log-log scale, this exhibits a characteristic corner. As the degree of regularization decreases towards this corner point, a model objective function changes very little while a data misfit is reduced. Further decrease in the degree of regularization beyond this point would result in rapid increase in a model objective function with little reduction in a data misfit. As a result the curve appears L-shaped. Any solution within a range around this corner provides a reasonable model for the inversion. This is because solutions by the regularization parameters near the corner of the curve approximate the unregularized solution as closely as possible without being dominated by the contribution from the perturbation, and data misfits by these regularization parameters are expected to be almost equal to the square of the L-2 norm of the perturbation. The developed 1-D inversion program was applied to the synthetic data from the five-layered resistivity structure to check validity of my algorithm and an L-curve criterion. Numerical tests concluded that a set of inverted resistivities from the synthetic data without any noise was almost coincident with the true resistivity model enough for my 1-D inversion program to be assessed to work well. Although the inverted resistivity structures from the contaminated data have some deviations from the true resistivity model, trends of change in the inverted resistivities with depth are in accordance with those of the resistivity model.
raeg2003@tansa.kumst.kyoto-u.ac.jp
Last modified: Wed Dec 04 14:24:30 2002