A General Framework for Constraint Minimization for the Inversion of Electromagnetic Measurements

In this paper, we developed a general framework for the inversion of electromagnetic measurements in cases where parametrization of the unknown configuration is possible. Due to the ill-posed nature of this nonlinear inverse scattering problem, this parametrization approach is needed when the available measurement data are limited and measurements are only carried out from limited transmitter-receiver positions (limited data diversity). By carrying out this parametrization, the number of unknown model parameters that need to be inverted is manageable. Hence the Newton based approach can advantageously be used over gradient-based approaches. In order to guarantee an error reduction of the optimization process, the iterative step is adjusted using a line search algorithm. Further unlike the most available Newton-based approaches available in the literature, we enhanced the Newton based approaches presented in this paper by constraining the inverted model parameters with nonlinear transformation. This constrain forces the reconstruction of the unknown model parameters to lie within their physical bounds. In order to deal with cases where the measurements are redundant or lacking sensitivity to certain model parameters causing non-uniqueness of solution, the cost function to be minimized is regularized by adding a penalty term. One of the crucial aspects of this approach is how to determine the regularization parameter determining the relative importance of the misfit between the measured and predicted data and the penalty term. We reviewed different approaches to determine this parameter and proposed a robust and simple way of choosing this regularization parameter with aid of recently developed multiplicative regularization analysis. By combining all the techniques mentioned above we arrive at an effective and robust parametric algorithm. As numerical examples we present results of electromagnetic inversion at induction frequency in the deviated borehole configuration. 266 Habashy and Abubakar