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This paper aims to investigate the impact of the overdetermination (data-to-unknowns) ratio on the global inversion of wireline logging data. In the course of the so-called interval inversion method, geophysical data measured in a borehole over a longer depth range is jointly inverted and the depth variation of the investigated petrophysical parameters are expanded into series using Legendre polynomials as basis functions resulting in a highly overdetermined inverse problem. A metaheuristic Particle Swarm Optimization (PSO) approach is applied as a first phase of inversion for decreasing the starting model dependence of the interval inversion procedure. In the subsequent linear inversion steps, by using the measurement error of logging tools and the covariance matrix of the estimated petrophysical parameters, we can quantify the accuracy of the model parameters. The dataset used in this study consists of nuclear, resistivity and sonic logs which are inverted to compute porosity, shale volume and water saturation along the investigated interval. For increasing the data-to-unknowns ratio of the inverse problem, shale volume is estimated separately by a PSO-based factor analysis and fixed as known parameter for the interval inversion process. Since the shale volume has been described as high degree Legendre polynomial, a significant increase of the overdetermination ratio considerably decreases the uncertainty of the remaining model parameters allowing for a more reliable calculation of hydrocarbon content.

Uncertainty reduction of interval inversion estimation results using a factor analysis approach