Statistical determination of geophysical well log response functions

Abstract

The vertical response function of induction logging tools is shown to be derivable from a power spectrum analysis of the measurement. The vertical response function is the one‐dimensional sequence of weights that characterizes how the tool combines the rock conductivities along the borehole to form an output called the apparent conductivity; it is the system impulse response. The value of knowing this function lies in the possible use of filter theory to aid in data processing and interpretation. Two general notions establish the framework for the analysis. The first is that logging is a linear, convolutional operation. Second, the earth’s conductivity profile forms a stochastic process. The probabilistic component is fleshed out by reasonably based assumptions about the occurrence of bed boundaries and nature of conductivity changes across them. Brought together, these tenets create a characterization of the conductivity sequence that is not a stationary process, but rather is intrinsic, as defined in the discipline of geostatistics. Such a process is described by a variogram, and it is increments of the process that are stationary. The connection between the power spectrum of the measurement and the system response function is made when the convolutional model is merged with the conductivity process. Some examples of induction log functions are shown using these ideas. The analysis is presented in general terms for possibly wider application.