The dual representation and its application to seismic reservoir analysis

Brian H. Russell

Geophysical and machine learning algorithms are based on finding the best weights that predict a set of observations from a set of measured features or attributes. A good example of this is the prediction of seismic reservoir parameters, such as density or porosity, from a set of seismic attributes. Many problems can be written as a linear regression model in which the weights can be determined using a generalized inverse. Although this involves a linearly weighted combination of the input attributes, we can apply a nonlinear function to the attributes themselves, which allows us to get a much better fit on the output. In this study I will consider two different forms of the generalized inverse, the primal and dual, and show the dual form leads to several very powerful analysis techniques, called kernel regression methods, that give much better fits than the multi-linear regression techniques used in many applications. I will illustrate these techniques using two simple datasets, one with only three points and the other with ten points, and then apply them to seismic reservoir analysis. The figure below shows that application of several machine learning algorithms to a dataset from Blackfoot, Alberta.