GEOLOGICAL PROCESS RATES ESTIMATED FROM RATES ON OBSERVABLE SCALES OF TIME

Two studies are essential for understanding rates in geography and geology: (1) Mandelbrot’s 1967 classic on the fractal coastline of Britain; and (2) Sadler’s 1981 classic on sediment accumulation rates. Both demonstrate dependence on scale, whether spatial or temporal. Rates are ratios with real-world dependence on denominators as well as numerators. To be comparable, ratios must be calculated for the same denominator, and rates must be calculated for the same scale. Rates should always be indexed by their scale, which would preclude many alarmist findings when rates are unwittingly compared on different time scales.

A process involves a series of events leading to a result. This has a characteristic time scale, typically short, but not instantaneous. A random walk is a time series generated by a random process involving a spatial or temporal step of some size, and a corresponding random deviation, left or right. The base rate of the random process is the ratio of the deviation to the step. Random walks are informative because they show how rates decline systematically when observed over intervals longer than the step size. They show too, conversely, how rates appear to accelerate when observed over shorter and shorter intervals.

Evolutionary time series are often compared to random walks, but the comparison is appropriate when and only when the base rate of the random process matches the base rate of the process being modeled. The latter is rarely known directly, but must be estimated from multiple observations on longer scales of time. It is always informative to plot rates against their denominators (e.g., log rates versus log intervals). A well-behaved LRI plot means that the base rate of an evolutionary or geological process can be estimated with reasonable precision. As geologists and paleontologists we are accustomed to long intervals and low rates that are observable but rarely representative of the time scales or the base rates of the processes we study. Earth and life processes are dynamic. Models worth exploring are those on the time scale of the process for any process of interest.