Abstract
During the conventional petrophysical interpretation fixed (predefined) zone parameters are applied for every interpretation zone (depth intervals). Their inclusion in the inversion process requires the extension of a likelihood function for the whole zone. This allows to define the extremum problem for fitting the parameter set to the full interval petrophysical model of the layers crossed by the well, both for the parameters associated with the depth points (e.g. porosity, saturations, rock matrix composition etc.) and for the zone parameters (e.g. formation water resistivity, cementation factor etc.). In this picture the parameters form a complex coupled and correlated system. Even the local parameters associated with different depths are coupled through the zone parameter change. In this paper, the statistical properties of the coupled parameter system are studied which fitted by the Interval Maximum Likelihood (ILM) method. The estimated values of the parameters are coupled through the likelihood function and this determines the correlation between them.
Highlights
In the process of well logging interpretation and inversion, the data are usually analyzed in predefined, disjoint depth intervals
If the likelihood function extended over the entire data set of a depth interval, the zone parameters can be included in the inversion
In this study the covariance matrix has been generated in a form that separates the covariance structure associated with conventional inversion and the correction from parameter coupling
Summary
In the process of well logging interpretation and inversion, the data are usually analyzed in predefined, disjoint depth intervals (interpretation intervals). In the process of inversion (which means the minimizing of the functional Q), the current element of My (the measurement data vector for the interval) is projected onto Mf and, due to the homeomorphic relationship between Mf and Mp, into the parameter space. This projection determines the parameter estimation (point estimate), and the confidence interval of the parameters through the projection of the confidence interval associated with the measurement error distribution. The inverse function between manifolds Mf and Mp is just approximated using Taylor series (mostly only up to the linear term)
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