Inverse Problem and Solution of the Kolmogorov Model for Bed Thickness Distribution
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Inverse Problem and Solution of the Kolmogorov Model for Bed Thickness Distribution Noritaka Endo
Received: 30 March 2009 / Accepted: 25 July 2010 / Published online: 25 August 2010 © International Association for Mathematical Geosciences 2010
Abstract 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. Keywords Kolmogorov model · Stochastic sedimentary model · Stratigraphy 1 Introduction When studying geologic records to determine the past environment, bed thickness is an important factor that is related to the sediment transport and debris production rates. When analyzing strata, it is difficult to avoid encountering factors that are N. Endo () School of Natural System, Kanazawa University, Kakuma-machi, Kanazawa City 920-1192, Japan e-mail: [email protected]
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Math Geosci (2010) 42: 955–968
not explained by the expected deterministic processes. Whether such unexplainable factors are caused by unknown deterministic processes or are essentially stochastic phenomena, a probabilistic model is useful to deal with them. The simplest and probably oldest probabilistic model targeting bed thickness distribution was proposed by Kolmogorov (1951). The Kolmogorov model is a simple mathematical model and stochastically deals with only the sedimentation and erosion magnitudes without considering the physical reasons for the sedimentation and erosion, that is, this model is independent of mechanical processes. The Kolmogorov model is therefore considered to be widely applicable and is still important as a basic model. This model is regarded as a kind of random walk model that considers discrete-time and continuousdisplacement variables (increment or decrement of the surface of layers) where
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