Stochastic models for the completeness of stratigraphic sections

Abstract

Properties of stratigraphic completeness are determined here from a Brownian motion model of sediment accumulation. This avoids flaws inherent in application of a discrete-time random walk to the time span, rather than thickness, of sediment layers. Both discrete and continuous models show that the concept of stratigraphic completeness is meaningful only when the time scale is specified. From the discrete model, not surprisingly, completeness improves with increasing relative frequency and average thickness of depositional increments and the error of completeness estimation should decrease for longer sections. The continuous model shows that two dimensionless products determine the probability that a given time interval will be recorded by some preserved sediment. The first is the ratio of the age of the interval to its time span; the second is the product of the square root of the time span and ratio of the mean to the standard deviation of accumulation rate. Expected completeness is the average of these probabilities for all successive intervals of the given time span. For long sections, completeness may be estimated from the second dimensionless product alone. The two dimensionless products are sufficient to predict the relationship of accumulation rate to time span, the distribution of bed thickness, and the weak association of completeness and section thickness.