MODEL DISTRIBUTION OF POTENTIAL FOSSILS

A uniform distribution of fossils is a good initial assumption, but there are several reasons not to expect the distribution of fossils representing the temporal duration of a group of organisms to be uniform through geological time: (1) the number of organisms in the group and their geographic distribution may have changed through time, changing the probability of preservation as fossils; (2) the outcrop area accessible for sampling today is different for rocks of different ages; and (3) the proportions of marine and non-marine environments on the surface of the earth have changed through time. All of these may affect distributions of fossils and make them nonuniform. Here we consider diversity to have increased at a constant rate during the interval between the time of origin of archaeocetes and their first appearance in the fossil record. We consider that the outcrop area of fossil-bearing sedimentary strata decreases exponentially for rocks of increasing age through the course of Phanerozoic time, as shown in Figure 4 (based on data in Blatt and Jones 1975). We use an exponential model fit to data for the Phanerozoic because we lack detailed knowledge of changing exposure of sedimentary strata during epochs of Cenozoic time—when detailed information about Cenozoic strata is available it can be substituted to permit a more refined calculation.

Our nonuniform model distribution of potential fossils is shown graphically in Figure 4. It has a geological age or time dimension, a diversity dimension, and a fossiliferous strata dimension. The fossiliferous strata and time dimensions define an area of potential fossils under the exponential curve (stippled), and this area plus the diversity dimension define a volume of potential fossils. Regions of interest within the whole volume of potential fossils have been lettered A, B, C, D, and E: A is the partially shaded volume at the left representing the distribution of sedimentary rocks lacking the taxon of interest because it had not yet originated and diversified; B is the narrow, hatched, wedge-shaped volume representing diversification after origination but before any appearance in the known fossil record; C is the cross-hatched volume of observed temporal density OTD representing the known fossil record of archaeocetes; D is not used here (because the temporal extension is left-tailed); and E is the partially shaded volume at the right representing the distribution of sedimentary rocks lacking the taxon of interest because it had by now given rise to something else or become extinct. Volumes B and C together correspond to the expected temporal density ETD. Total volumes and normalized proportions of A, B, C, D, and E are given in Figure 4 for the 95% confidence limit calculation shown graphically.

It is important to emphasize that a model distribution of potential fossils is a model assuming 'all else' not represented here to be equal. Fossils may or may not be evenly distributed though time, and we have explicitly built some of the ways that they are not evenly distributed into our model. When there is knowledge of additional structure shaping the fossil record such factors can and should be built into a better model. All inference about stratigraphic ranges depends on random sampling in the context of some model, explicit or not, and explicit models are always better than vaguely-conceived implicit models. The actual ages of most fossil samples and the temporal differences between most samples are often poorly known or not known at all, but only two ages are important in our calculations: the age of the oldest known sample (t₂), the age of the youngest known sample (t₃). We require, in addition, some estimate of the number of independently-sampled fossiliferous horizons (n). Finally, the purpose of this exercise is less calculation of precise cut-off ages than comparison of the relative likelihoods of different possibilities.