Mathematical analysis to discriminate two types of sandstone-shale alternations



Sedimentation processes forming a series of bipartite layers have been studied mathematically. Two types of sedimentation processes are recognized, i.e. concurrent deposition of sandstone and shale by a turbidity current (Type I) and alternate deposition of each of them (Type II). A time series of events in a sedimentation process is reasonably considered a first-order Markov chain, and the process is described with a Markov matrix including four state-variables such as deposition of sandstone, erosion of sandstone, deposition of shale, and erosion of shale. Analysis of Markov matrix yields a fixed probability vector, which for Type I process is different from that for Type II process. The vector bears a close relation to Kolmogorov's coefficient, which is the ratio of the number of beds deposited and the number of beds preserved in a given sedimentary section. This coefficient can be computed on the basis of field observations. Substitution of the computed data determined the values of the fixed probability vector for two sedimentary sections in Japan. The results permitted a theoretical conclusion as to the genesis of observed sandstone-shale alternations. This conclusion is in good agreement with the deductions from more conventional sedimentological methods.