A multiplicative regularized gauss-newton algorithm and its application to the joint inversion of induction logging and near-borehole pressure measurements

Abstract

Due to the ill-posed nature of nonlinear inverse problems of borehole geophysics, a parameterization approach is necessary when the available measurement data are limited and measurements are only carried out from sparse transmitter-receiver positions (limited data diversity). A potential remedy is the joint inversion of multi-physics measurements. A parametric inversion approach has desirable attributes for multi-physics measurements with different resolutions. It provides a flexible framework to put the sensitivities of multi-physics multi-resolution measurements on equal footing. In addition, the number of unknown model parameters to be inverted is rendered tractable with parameterization. Consequently, a Gauss-Newton based inversion algorithm taking advantage of the Hessian information can be advantageously employed over inversion approaches that rely only on gradient information. We describe a new dual-physics parametric joint-inversion algorithm to estimate near-borehole fluid permeability and porosity distributions of rock formations from fluid-flow and electromagnetic measurements. In order to accommodate the cases in which the measurements are redundant or lack sensitivity with respect to certain model parameters causing nonuniqueness of the inverted solution, the objective functional to be minimized is regularized with a penalty term. One of the central aspects of this approach is the determination of the regularization parameter. The latter must be chosen in such a way that the relative importance of the misfit between measured and predicted data and the penalty term are effectively balanced over the course of minimization. We propose a new method of adaptively choosing the regularization parameter within a Gauss-Newton method based joint-inversion algorithm using a multiplicative regularization strategy. The multiplicative regularization method is tested against additive regularization in joint-inversion problems involving wireline formation tester transient pressure and induction-frequency electromagnetic logging measurements. The multiplicative regularization method delivers improved convergence rates over additive regularization for all investigated problems. Inversions of relatively more noise-contaminated measurements benefit more from multiplicative regularization.