Global Penalty Functional Methods
The linearized approach discussed in Section 5.4 constructs models built up by adding perturbations to the current best estimate of σ(z). The method minimizes a penalty functional of the perturbation. For example, the "small" penalty functional is the integrated square of the perturbation values, and the "fiat" functional is the integrated square of the first derivatives of the perturbations with respect to depth; however, these do not require the final σ(z) to be smallest or flattest in any sense. Clearly, it is advantageous to reformulate the inverse problem so that the Fredholm equation of the first kind to be solved has the model in the integrand rather than a perturbation of the model. We call penalty functionals of the entire model "global" to contrast them with the "local" functionals which involve a model perturbation. In the global case, the arsenal of linear inverse theory techniques are applied to the equation to compute weighted smallest or flattest models, as well as models deviating least from a given σ(z) profile. These minimum structure solutions are valuable since they are less likely to have spurious features to mislead an interpreter. Interesting features of such solutions are hopefully essential characteristics required by the data, and not just artifacts of noise or the inversion method.
An important benefit of reformulating the equations to optimize directly a global functional of the total model is that additional constraints on the conductivity are straightforward to incorporate. For instance, the conductivity may be known to be bounded between two functions, or the average value of conductivity over a depth range may be known to within a given uncertainty. Conductivity models which optimize a global penalty functional, satisfy the observations, and satisfy additional imposed constraints are more likely to be representative of the true Earth.
The negative aspect of the optimization is the iterative solution required because the kernels are model dependent. Consequently, there is concern about convergence of the algorithm and the possibility of being trapped in a local minimum. There is no way of determining analytically or numerically whether the true global minimum has been achieved. If there is doubt, the pragmatic approach is to repeat the inversion beginning with different starting models and different partitions of the depth axis. In most cases explored by us, the algorithms converged to the same conductivity model and thus provided optimism that the global solution had been found.
Figures & Tables
In this short monograph, 1D inversion methods are examined collectively using a uniform notation. One-dimensional inversion methods are still important because there are geologic regions where lateral variation is small and 1D interpretation is directly applicable; 1D inverse solutions provide good starting models for 2D inversion; and understanding the 1D inverse problem provides a foundation for solving inverse problems in higher dimensions. The 10 chapters are “Introduction”, “Existence”, “Uniqueness,” “Asymptotic Methods,” “Linearized/Iterated Methods,” “Global Penalty Functional Methods,” “Monte Carlo Methods,” “Exact Methods,” “Appraisal Methods,” and “Conclusion.”