Interval maximum likelihood inversion for well logging data and statistical properties of estimated parameters

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.