Inverse Problem and Solution of the Kolmogorov Model for Bed Thickness Distribution

Bed thickness is an important factor when interpreting geologic records to determine the past environment; it is related to the sediment transport and debris production rates. Because of the inherent uncertainty of these phenomena, a probabilistic model is useful for dealing with the problem. Many probabilistic models are variations of the Kolmogorov model, which is a type of random-walk model. The Kolmogorov model is a simple mathematical model that has a wide range of applications. However, when estimating paleo-environments from geologic records, the inverse problem is more practical than the forward problem, but the former has not been well discussed. Previous applications have estimated the probability density function (PDF) for stochastic steady states (including virtual cumulative erosion that is not physically observable) but not for independent events, and the difference between the results for these two kinds of PDFs has not been analyzed. This study considers the inverse problem of the Kolmogorov model and the properties of its solution. This study found that (1) the inverse problem can be solved analytically in a general form; (2) the difference between the above two PDFs can exceed 10% in some cases; and (3) different PDFs for the deposition and erosion magnitudes of independent events can reproduce the same bed thickness distribution of preserved layers.