Optimal Transformations for Multiple Regression: Application to Permeability Estimation From Well Logs

Abstract

Summary Conventional multiple regression for permeability estimation from well logs requires a functional relationship to be presumed. Because of the inexact nature of the relationship between petrophysical variables, it is not always possible to identify the underlying functional form between dependent and independent variables in advance. When large variations in petrological properties are exhibited, parametric regression often fails or leads to unstable and erroneous results, especially for multivariate cases. In this paper, we describe a nonparametric approach for estimating optimal transformations of petrophysical data to obtain the maximum correlation between observed variables. The approach does not require a priori assumptions of a functional form, and the optimal transformations are derived solely based on the data set. Unlike neural networks, such transformations can facilitate physically based function identification. An iterative procedure involving the alternating conditional expectation (ACE) forms the basis of our approach. The power of ACE is illustrated using synthetic as well as field examples. The results clearly demonstrate improved permeability estimation by ACE compared to conventional parametric-regression methods.